Determination of Vapor Pressure of Chemical Compounds: A Group

Article pubs.acs.org/IECR

Determination of Vapor Pressure of Chemical Compounds: A Group Contribution Model for an Extremely Large Database Farhad Gharagheizi,† Ali Eslamimanesh,‡ Poorandokht Ilani-Kashkouli,† Amir H. Mohammadi,*,‡,§ and Dominique Richon‡,§ †

Department of Chemical Engineering, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran MINES ParisTech, CEP/TEP-Centre Énergétique et Procédés, 35 Rue Saint Honoré, 77305 Fontainebleau, France § Thermodynamics Research Unit, School of Chemical Engineering, Howard College Campus, University of KwaZulu-Natal, King George V Avenue, Durban 4041, South Africa ‡

S Supporting Information *

ABSTRACT: In the present study, a group contribution model is developed for determination of the vapor pressure of pure chemical compounds at temperatures from 55 to 3040 K. About 42 000 vapor pressure values belonging to around 1400 chemical compounds (mostly organic ones) at different temperatures are treated to propose a reliable and predictive model. A three-layer artificial neural network is optimized using the Levenberg−Marquardt (LM) optimization algorithm to establish the final relationship between the functional groups and the vapor pressure values. The obtained results indicate the average absolute relative deviation (AARD%) of the calculations/estimations from the applied data to be about 6% and a squared correlation coefficient of 0.994. Furthermore, the outliers of the model are detected using the leverage value statistics method.

1. INTRODUCTION Of great significance in chemical process design, modeling, and simulation is the accurate estimation of vapor pressure of chemical compounds.1−3 This property not only defines the volatility of chemicals but also leads to explaining the air−water partition coefficient, transport characteristics of the species, and fate of environmental pollutants in the atmosphere.1−6 Other applications of the vapor pressure values have been already wellestablished.1−14 The Clausius−Clapeyron equation is normally applied to evaluate the vapor pressure as follows1,2 d(ln Pvp) d(1/T )

=−

ΔH v R ΔZv

state through satisfying the equality of fugacities (equilibrium criteria) of pure components throughout the liquid and vapor phases; and (3) molecular approaches such as quantitative structure− property relationship (QSPR) algorithms and group contribution (GC) methods.45−61 A detailed description of the aforementioned categories can be found in our previous paper,3 in which we have developed a QSPR model to estimate the vapor pressure values of pure compounds on the basis of an extremely large data set (around 45 000 data). The following can be generally inferred from the literature:3 (1) Some of the methods, especially those that are based on Clapeyron equations or corresponding states methods, need a large amount of experimental vapor pressure data to tune their adjustable parameters for each chemical compound at wide ranges of temperatures. Therefore, for each chemical, special adjusted parameters are required to use these equations. (2) Although applications of equations of state have been demonstrated to provide generally accurate vapor pressures, they require phase equilibrium calculations. Furthermore, the α function parameters have not been tuned for many of chemical species up to now. (3) The group contribution models have not so far been developed on the basis of very large vapor pressure databases. This is probably because of some mathematical limitations during calculation steps of these techniques.

(1)

where ΔHv and ΔZv denote the differences in the enthalpies and compressibility factors of saturated vapor and saturated liquid. Having at least one experimental Pvp-T datum and the calculated or experimental ΔHv(T) and ΔZv(T) values of one compound, it is possible to determine its vapor pressure by applying eq 1. The significance of the vapor pressure property for the chemical industry has justified lots of efforts for its experimental measurements and also for developing several estimation techniques. It can be inferred on the basis of a literature survey that the so far theoretical attempts for representation or prediction of the vapor pressure property can be grouped into three main categories as follows:3 (1) modifications or integrations of the Clapeyron equation for extension of the corresponding results to different compounds or low/high temperature regions, such as integrating the total enthalpy of vaporization or the twoor three-parameter corresponding states methods;14−44 (2) application of equations of state, e.g., using the previously tuned parameters for the α function of cubic equations of © 2012 American Chemical Society

Received: Revised: Accepted: Published: 7119

January 24, 2012 April 17, 2012 April 18, 2012 April 18, 2012 dx.doi.org/10.1021/ie3002099 | Ind. Eng. Chem. Res. 2012, 51, 7119−7125

Industrial & Engineering Chemistry Research

Article

chemical functional groups and the desired vapor pressure values.63 The simplest method for this purpose is to relate the functional groups to the value of the vapor pressure of each compound through a multilinear relationship. Our calculations show that application of this methodology for the current problem does not lead to accurate results within an acceptable range of the deviations from the treated data, mainly due to the number of datum within the data set and the nonlinearity of the problem.50−66 Thus, a nonlinear mathematical approach such as artificial neural network (ANN) should be used for pursuing this objective. Coupling this mathematical algorithm and the group contribution method generally leads to accurate representation/prediction of physicochemical properties, e.g., surface tension,64 parachor,65 critical properties, and acentric factors,66 to name a few. The numbers of the selected functional groups in each compound and the vapor pressure values62 have been normalized between −1 and +1 to decrease the computational errors because the applied vapor pressure data62 have a wide range of values over different orders of magnitude. Furthermore, the main objective of this method, which is performed in the optimization process, has been to evaluate the optimal values of the parameters of the neural networks (W1, W2, b1, b2). In the next step, the database has been divided into three subsets, including the “training” set, the “validation” (optimization) set, and the “test” (prediction) set. In the present study, the training set has been implemented to generate the ANN structure, the validation (optimization) set has been used for optimization of the model parameters in order to avoid underfitting or overfitting problems, and the test (prediction) set has been employed to investigate the prediction capability and validity of the obtained model. The division of database into three subsets has been performed randomly. For this purpose, about 80%, 10%, and 10% of the main data set are randomly selected for the training set (about 34 000 data), the optimization set (4000 data), and the prediction set (4000 data). The effect of the percent allocation of the three subdata sets from the database on the accuracy of the ANN model has been studied elsewhere.67 In the later step, we generally consider many distributions to prevent the local accumulations of the data in the feasible region of the problem.63 Hence, the acceptable distribution is the one with homogeneous distributions of the data on the whole domain of the three subdata sets.63 There are normally two weight matrices and two bias vectors in a three-layer feed-forward ANN (FFANN): W1 and W2, and b1 and b2.64−66 The optimum values of these parameters are determined through minimization of an objective function. The objective function used in this study is the sum of squares of errors between the outputs of the FFANN (represented/ predicted vapor pressures) and the target values (the applied data62). This minimization process has been performed using the Levenberg−Marquardt (LM)68,69 algorithm. The LM68,69 is, perhaps, the most widely used nonpopulation based optimization methodology.64−66 In most cases, the number of neurons in the hidden layer (n) is fixed. Therefore, the main goal is to produce an ANN relationship, which is capable of predicting the target values as accurately as possible. This step is repeated until the best ANN is obtained. However, it is more efficient that the number of neurons in the hidden layer is optimized according to the accuracy of the obtained FFANN and to avoid the underfitting or overfitting drawbacks of ANN-based structures.64−66

According to the ever-growing demands of the chemical industry to develop more general and accurate group contribution methods to represent/predict thermophysical properties, our objective herein is to propose a comprehensive and predictive group contribution model to determine the vapor pressure values of around 1400 compounds as a function of temperature.

2. THE DATABASE AND MATHEMATICAL METHODS 2.1. Database. Around 42 000 vapor pressures available in the DIPPR 801 database,62 belonging to 1405 chemical compounds, have been treated to develop and validate the model. To the best of our knowledge, this database is one of the largest databases applied for proposing a group contribution model. The schematic structure of the investigated compounds are sketched and presented as Supporting Information. It should be noted that the implemented data set in this work is not as large as the data set applied in our previous work,3 which contained around 45 000 vapor pressure data. We can refer to the following criteria to clarify this concept: (1) The recently reported model3 has been developed on the basis of the QSPR approach, which applies molecular descriptors, while the current model uses the chemical functional groups (group contributions). It should be noted that there are some chemicals present in the database62 for which the functional groups or molecular descriptors have not been defined so far. In other words, it is not currently possible to define both the molecular descriptors and functional groups in all of the compounds in the database. (2) We have detected the suspended experimental vapor pressure data in our previous study.3 We have eliminated those points from the treated database in this work. 2.2. Determination of Group Contributions. Having defined the compounds present in the database, the chemical structures of all of the studied compounds have been analyzed to identify the functional groups. These functional groups are generally selected from a series including 500 various chemical groups as follows: (1) Functional groups are divided in different categories, each one containing two pairs from all of the groups. (2) A mathematical algorithm is used to establish a linear relationship between the two groups in a pair GCi = a × GCj + b

(2)

where GC denotes the functional groups, a and b are the parameters of the linear regression, and subscripts i and j refer to ith and jth functional groups. (3) In the case where the squared correlation coefficient of eq 2 is greater than a selected value (here it is 0.95), one of the groups is omitted from the investigated pair because it has no significant effects on the final developed model and results in increasing the model parameters (final functional groups). The preceding procedure has been followed until the most efficient contributions for evaluation of the corresponding property (vapor pressure) have been determined. As a result, 202 functional groups have been defined for the problem of interest, which are reported in Supporting Information. 2.3. Developing the Model. Perhaps, the most important calculation step is to search for a relationship between the 7120

dx.doi.org/10.1021/ie3002099 | Ind. Eng. Chem. Res. 2012, 51, 7119−7125


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3. RESULTS AND DISCUSSION An optimized GC-FFANN model has been developed following the previously described procedure to represent/ predict the vapor pressure values. For this purpose, several 3FFANNs modules were generated by assuming numbers 1−50 for n (number of neurons in a hidden layer). The most accurate results were observed at n = 10, which contributes to no underfitting or overfitting of the final model. It should be noted that this value is not a global one because the optimization method used to train the ANN has non-negligible effects on the optimal numbers of neurons.64−66 Therefore, the developed three-layer GC-FFANN structure has the structure of 203−10−1 (temperature + 202 chemical groups are regarded as the inputs of the model). For each investigated molecule, only a few chemical groups are present, and consequently, the number of model parameters is reasonable. The .mat file (MATLAB file format) of the obtained model containing all the parameters and the instruction for running the program are freely available upon request to the authors. The represented/predicted vapor pressure values are shown in Figures 1 and 2 compared with the applied data62 in both

Figure 2. Comparison between the represented/predicted results of the developed model and the applied vapor pressure data62 of the investigated pure chemical compounds: VP, vapor pressure; rep, represented; pred, predicted; exp, vapor pressure values reported in DIPPR 801;62 ×, training set; +, optimization set; ○, prediction set.

Figure 1. Comparison between the represented/predicted results of the developed model (in logarithmic values) and the applied vapor pressure data62 of the investigated pure chemical compounds: VP, vapor pressure; rep, represented; pred, predicted; exp, vapor pressure values reported in DIPPR 801;62 ×, training set; +, optimization set; ○, prediction set.

Figure 3. Relative deviations between the represented/predicted vapor pressure values and the reported ones in DIPPR 80162 (in logarithmic values): VP, vapor pressure; rep, represented; pred, predicted; exp, vapor pressure values reported in DIPPR 801;62 ×, training set; +, optimization set; ○, prediction set.

logarithmic scales and the real values for better illustrations. Figures 3 and 4 show the deviations between the calculated/estimated and the vapor pressure values reported in DIPPR 801.62 The statistical results of the proposed GC model are reported in the Supporting Information. Moreover, the average absolute relative deviations (AARD) of the results from the applied vapor pressure data62 for each chemical family are shown in Supporting Information. The ranges of the temperatures, the ranges of calculated/estimated vapor pressure values for each compound, the numbers of the functional groups in each compound, and the references of the applied experimental data have also been presented in the Supporting Information. It is worth it pointing out that there are some data points for which the presented model results contribute to high

absolute relative deviations from the applied data.62 It is, thus, of significance to have a deep investigation about these points. Detailed scrutiny of the chemical families related to these results of the model would not lead to the conclusion that these points belong to special chemical groups (chemical families). Hence, it is probable that the vapor pressure values for these compounds are not accurate or may be somehow erroneous because of the difficulties and possible errors in experimental measurements, which result in high uncertainties in the obtained data (refer to the Supporting Information for observing the uncertainties of the data for each data source). In this paper, we have applied the mathematical strategy of leverage value statistics70,71 to identify the probable doubtful 7121



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Figure 5. Detection of the probable doubtful data and the outliers of the developed model using the leverage value statistics70,71 method: green ex, training set; pink circle, optimization set; blue plus, prediction set; red ex, suspended data in training set; red circle, suspended data in optimization set; red plus, suspended data in prediction set, black ex, outliers in training set.

Figure 4. Relative deviations from the applied vapor pressure data:62 VP, vapor pressure; rep, represented; pred, predicted; exp, vapor pressure values reported in DIPPR 801;62 ×, training set; +, optimization set; ○, prediction set.

To recapitulate, we may conclude the following factors: (1) The developed GC-FFANN model is statistically correct and valid. (2) There is only one point that may be designated as an outlier of the developed model including one of the data of hydrogen peroxide (see the Supporting Information files for observing this datum). It is of interest to point out that this detected outlier is also among the outliers that had been detected in our previous work.3 The effects of this point on the prediction capability of the developed GC-FFANN model is negligible because it consist only very small portion of the applied data set. (3) The number of suspended data (see the Supporting Information for observing these data) include around 2.3% of the whole data set.62 The results of the leverage value statistics70,71 algorithm interprets that elimination of these data does not have much effect on the prediction capability of the model, because they had been already assigned (through the applied mathematical algorithms) to have less effects in development of the model than the other data. It should be noted that the number of parameters of the GCFFANN module are as follows W1 (weight 1) = 203 × 10, W2 (weight 2) = 1 × 10, b1 (bias 1) = 10 × 1, and b2 (bias 2) = 1 × 1. Therefore, the ratio of (experimental data)/(parameters of the module including the weights and biases) is equal to 42 126/ 2051, which leads to around 21. Therefore, the model is also acceptable to high extent according to this mathematical concept. To summarize, the pros and cons of the proposed model with respect to available methods are as follows: Pros (1) The current model is more general and comprehensive than the available GC methods because it has been developed and tested on the basis of a very large data set (about 42 000 data points) of the vapor pressure values of about 1400 various chemical species at different temperatures, while the previous ones have been developed either for a limited number of compounds or particular chemical groups.

experimental vapor pressure data and the outliers of the developed model as follows: The leverage or hat indices are calculated based on hat matrix (H) with the following definition:3

H = X(XTX)−1XT

(3)

where X is a two-dimensional matrix comprising n chemicals (rows) and k parameters of the model (columns) and T stands for the transpose matrix. The hat values of the chemicals in the feasible region of the problem are, as a matter of fact, the diagonal elements of the H value. Having evaluated the H values by eq 3, the Williams plot is sketched as shown in Figure 5 for graphical identification of the suspended data and the outliers. This plot shows the correlation of hat indices and standardized cross-validated residuals (R), which are defined as the difference between the represented/predicted vapor pressure values and the implemented data. All of the calculated H and R values are presented in the Supporting Information for all of data points. A warning leverage (H* = 0.0181), indicated by the black vertical line, is generally fixed at the value equal to 3n/p, where n is number of training chemicals and p the number of model parameters plus one. The leverage of 3 is normally considered as a “cutoff” value to accept the points within ±3 range (two horizontal red lines) standard deviations from the mean (to cover 99% normally distributed data). The existence of the majority of training, optimization, and test data points in the ranges 0 ≤ H ≤ 0.0181 and −3 ≤ R ≤ 3 reveals that both model derivation (development) and its predictions are done in the applicability domain, which results in a statistically valid model. “Good high leverage” points are located in domain of 0.0181 ≤ H and −3 ≤ R ≤ 3. These points can be designated as the ones that are outside of the applicability domain of the developed model. In other words, the developed correlation is not able to represent/predict the following data at all. The points located in the range of R < −3 or 3 < R (whether they are larger or smaller than the H value) are designated as outliers of the model or “bad high leverage” points. These erroneous representations/predictions can be attributed to the doubtful vapor pressure data. 7122



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(2) This model is rapid in calculations because it is presented as computer software. (3) The functional groups are new and more general than the first-, second-, or third-order groups proposed in the methods in the literature. (4) The number of the outlier of the model is very small compared to the numbers of the evaluated vapor pressures. However, the number of probable doubtful data is considerable (around 2.3% of the whole data set). (5) The GC models are very familiar compared to Cons QSPR ones for engineers to apply.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: + (33) 1 64 69 49 70. Fax: + (33) 1 64 69 49 68. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.E. wishes to thank MINES ParisTech for providing him a Ph.D. scholarship.

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(1) The number of evaluated vapor pressure data62 is less than that treated in our previous model,3 as described earlier.

REFERENCES

(1) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (2) Smith, J. M.; Van Ness, H.; Abbott, M. Introduction to Chemical Engineering Thermodynamics, 7th ed.; McGraw-Hill: New York, 2005. (3) Gharagheizi, F.; Eslamimanesh, A.; Ilani-Kashkolli, P.; Mohammadi, A. H.; Richon, D. QSPR molecular approach for representation/prediction of very large vapor pressure dataset. Chem. Eng. Sci. 2012, 76, 99−107. (4) Katritzky, A. R.; Kuanar, M.; Slavov, S.; Hall, C. D.; Karelson, M.; Kahn, I.; Dobchev, D. A. Quantitative correlation of physical and chemical properties with chemical structure: Utility for prediction. Chem. Rev. 2010, 110, 5714−5789. (5) Shiu, W. Y.; Doucette, W.; Gobas, F. A. P. C; Andren, A.; Mackay, D. Physical−chemical properties of chlorinated dibenzo-pdioxins. Environ. Sci. Technol. 1988, 22, 651−658. (6) Gharagheizi, F.; Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Empirical method for estimation of Henry’s law constant of nonelectrolyte organic compounds in water. J. Chem. Thermodyn. 2012, 47, 295−299. (7) Coquelet, C.; Chapoy, A.; Richon, D. Development of a new alpha function for the Peng-Robinson equation of state: Comparative study of alpha function models for pure gases (natural gas components) and water-gas systems. Int. J. Thermophys. 2004, 25, 133−158. (8) Gasem, K. A. M.; Gao, W.; Pan, Z.; Robinson, R. L., Jr. A modified temperature dependence for the Peng−Robinson equation of state. Fluid Phase Equilib. 2001, 181, 113−125. (9) Gasem, K. A. M.; Robinson, R. L., Jr. Evaluation of the simplified perturbed hard chain theory (SPHCT) for prediction of phase behavior of n-paraffins and mixtures of n-paraffins with ethane. Fluid Phase Equilib. 1990, 58, 13−33. (10) Hartono, R.; Mansoori, G. A.; Suwono, A. Prediction of molar volumes, vapor pressures and supercritical solubilities of alkanes by equations of state. Chem. Eng. Commun. 1999, 173, 23−42. (11) Khalil, Y. F.; Anderson, T. F.; Lovland, J. Improved temperature dependence for the attractive term in the Peng−Robinson equation of state. American Chemical Society (ACS) national meeting, Washington, DC, United States, Aug 26−31, 1990, 807−812. (12) Kontogeorgis, G. M.; Smirlis, I.; Yakoumis, I. V.; Harismiadis, V.; Tassios, D. P. Method for estimating critical properties of heavy compounds suitable for cubic equations of state and its application to the prediction of vapor pressures. Ind. Eng. Chem. Res. 1997, 36, 4008− 4012. (13) Li, H.; Yang, D. Modified α function for the Peng−Robinson equation of state to improve the vapor pressure prediction of nonhydrocarbon and hydrocarbon compounds. Energy Fuels 2011, 25, 215−223. (14) Reinhard, M.; Drefahl, A. Handbook for Estimating Physicochemical Properties of Organic Compounds; Wiley & Sons: New York, 1999. (15) Antoine, C. Tensions de vapeurs; nouvelle relation entre les tensions et les températures. C. R. Seances Acad. Sci., Paris 1888, 107, 681−684 , 778−780, 836−837. (16) Cox, E. R. Pressure-temperature chart for hydrocarbon vapors. Ind. Eng. Chem. 1923, 15, 592−593.

4. CONCLUSION We proposed a group contribution model to estimate the vapor pressure of pure chemical compounds (mostly organic ones) at different temperatures (a temperature-dependent GC model). A very large data set of vapor pressure values of pure compounds (DIPPR 801)62 was used to develop and validate the model. Its required parameters are temperature and the numbers of functional groups present in each investigated compound. A three-layer feed forward artificial neural network was generated using the Levenberg−Marquardt (LM)68,69 optimization algorithm to develop the final model. The leverage value statistics70,71 method proved that there is only one point designated as an outlier of the obtained GC model and around 960 probable doubtful data existing in the main data set. It can be interpreted from the final results that a reliable and predictive model was developed to calculate/ estimate the vapor pressure of many of pure compounds (mostly organic ones), which are especially applied in chemical and petroleum industries, although there are still some limitations. The model has a wide range of applicability, but its prediction capability is restricted to the compounds that are similar to those ones applied to develop the model. Application of the model for the totally different compounds than the investigated ones is not recommended, although it may be used for rough estimation of the vapor pressure of this kind of compound. Another issue that cannot be ignored is the effect of the uncertainties of the data applied for developing the model on the predicted results. Greater accuracy in the experimental measurements of the data would contribute to a more accurate model.

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ASSOCIATED CONTENT

S Supporting Information *

The selected functional groups for the developed GC model, the 202 functional groups from all 1405 pure compounds in the main data set, the statistical parameters of the presented model, the distributions of the data in three data sets, the average absolute relative deviations of the obtained results from the applied data62 of vapor pressure for various chemical families, the whole obtained results, sources of experimental data accompanied by the related uncertainties, the statistical hat and residual values in Excel files (.xls spreadsheets), and the schematic chemical structures of the investigated compounds in a .pdf file. This material is available free of charge via the Internet at http://pubs.acs.org. 7123



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NOTE ADDED AFTER ASAP PUBLICATION After this paper was published online May 10, 2012, the spelling of author Poorandokht Ilani-Kashkouli was corrected. The corrected version was reposted May 23, 2012.

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Determination of Vapor Pressure of Chemical Compounds: A Group

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