Article pubs.acs.org/IECR
Representation and Prediction of Vapor−Liquid Equilibrium Using the Peng−Robinson Equation of State and UNIQUAC Activity Coefficient Model Younas Dadmohammadi, Solomon Gebreyohannes, Agelia M. Abudour, and Brian J. Neely School of Chemical Engineering, Oklahoma State University, Stillwater, Oklahoma 74078-0537, United States
Khaled A. M. Gasem* Department of Chemical & Petroleum Engineering, University of Wyoming, Laramie, Wyoming 82071, United States S Supporting Information *
ABSTRACT: Many important processes in the oil and gas industry (e.g., distillation, absorption, extraction) involve contact between liquid and vapor phases. The reliable design of these industrial processes requires accurate thermodynamic models to describe the vapor−liquid equilibrium (VLE) of the mixtures of interest. Two common approaches, γ−ϕ and ϕ−ϕ, are utilized to describe such VLE behavior. In this study, we present a comprehensive assessment of the representation and predictive capability of these two approaches, utilizing the UNIQUAC model to determine the activity coefficients and the Peng−Robinson (PR) equation of state to calculate the fugacity coefficients. The assessment was completed using a diverse binary VLE database consisting of 916 binary systems involving 140 compounds belonging to 31 chemical classes. Both the overall results and the results categorized for highly nonideal systems and for aqueous systems are presented within the context of internal and external consistency tests. Specifically, regressed and generalized parameters are utilized in internal and external consistency tests, respectively. Further, the phase behavior of sample systems was analyzed using Danner’s molecular classification method based on the mNRTL1 parameter and GE/RT values. For the systems considered, the regression results show that the γ−ϕ approach represents the VLE behavior more precisely compared to the ϕ−ϕ approach. The overall results using the γ−ϕ approach exhibit an absolute average deviation (% AAD) of 1.6, 0.1, 4.5, and 5.7 for the pressure, temperature, mole fraction, and equilibrium constant (K), respectively. The ϕ−ϕ approach regression results are within 3 times the error of the γ−ϕ approach. A similar trend was observed for the quantitative structure−property relationship generalized predictions. The γ−ϕ approach predicts the VLE behavior more accurately compared to the ϕ−ϕ approach. The overall results based on the γ−ϕ approach exhibit % AADs of 5.1, 0.4, 5.9, and 8.1 for the pressure, temperature, mole fraction, and K, respectively. The ϕ−ϕ approach generalized predictions are within 2 times the error obtained from the γ−ϕ approach. The results of Danner’s molecular classification of the phase behavior indicated that systems with similar components are more likely to produce nearly ideal mixture behavior and systems involving dissimilar components are more likely to produce nonideal mixture behavior. Further, the quality of the representations for the ϕ−ϕ approach are generally good for most system classifications with the exception of adequate or poor fits observed for strongly polar−strongly polar and aqueous−strongly polar systems. Cubic EOSs, such as Soave−Redlich−Kwong1 and Peng− Robinson (PR),2 and ACMs, such as nonrandom two liquid (NRTL)3 and universal quasichemical (UNIQUAC),4 are among the most widely used models in the process industry. In general, such models contain one or more binary interaction parameters, which must be provided for the particular system of interest. The theory-framed quantitative structure−property relationship (QSPR) approach is utilized for binary interaction parameter prediction. Because of recent advances in computational chemistry, the QSPR method has the potential to reduce the burden of time and resources associated with a conventional experimental approach. This methodology relies on the
1. INTRODUCTION The optimum design and simulation of commercial-phase separation units require accurate models for use in phase-equilibrium computations. Contemporary process simulators typically use fugacity coefficients (ϕ) to account for the nonideal behavior of fluids in general (vapor or liquid), and activity coefficients (γ) to describe the nonideal behavior of liquids. Accordingly, two common approaches, γ−ϕ and ϕ−ϕ, are utilized to describe the vapor−liquid equilibrium (VLE) behavior. Traditionally, the γ−ϕ approach is used for low-pressure VLE systems involving mixtures containing both polar and nonpolar components. In contrast, the ϕ−ϕ approach is usually applied for high-pressure systems following careful selection of the applicable equation of state (EOS) and mixing rules. To implement these approaches, EOSs and activity coefficient models (ACMs) are employed. The selection of a suitable EOS and ACM is the key step in applying the two approaches. © 2016 American Chemical Society
Received: Revised: Accepted: Published: 1088
September 17, 2015 December 29, 2015 January 5, 2016 January 5, 2016 DOI: 10.1021/acs.iecr.5b03475 Ind. Eng. Chem. Res. 2016, 55, 1088−1101
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Industrial & Engineering Chemistry Research
respectively. P is the total pressure for the system, and ϕ̂ vi is the component partial fugacity coefficient in the vapor phase. The activity coefficient, γi, is calculated from an ACM (also known as a molar excess Gibbs energy model). The relationship to the molar excess Gibbs energy, GE, is given by eq 4:
principle that the thermophysical properties have an explicit functionality in terms of molecular properties (or descriptors). Specifically, in this approach, theoretical frameworks are used to develop the behavior models and the QSPR aided with artificial neural networks is used to generalize the substance-specific parameters of the models. Initially, the known molecular descriptors are used to train the QSPR model. The trained QSPR model can be applied for a priori prediction over external data sets. Reliable database development, molecular structure generation and optimization, quantification of the molecule descriptors, optimization of the number of relevant descriptors, and subsequent model validation are the main steps in building a robust QSPR model. Many pure-fluid molecular physical properties may be found in the literature.5−10 Detailed studies on the development of generalized models based on the PR EOS and UNIQUAC ACM were presented in our previous publications.11−13 In the current work, we assess the representation and predictive capability of the two approaches using the PR EOS and UNIQUAC ACM. In a review of the literature, much effort has been focused on the improvement of EOSs and ACMs; however, these types of investigations using the two approaches are limited generally to a small number of binary mixtures and certain temperature and pressure conditions. In this work, we assessed the representation and predictive capability of the two approaches, γ−ϕ and ϕ−ϕ, based on UNIQUAC and PR, respectively, on a diverse VLE database comprised of 916 systems. This comparison is conducted for chemicals belonging to 31 functional-group interactions and based on the molecular polarity level (Danner classification). To date and to our knowledge, a study of this scope is not available in the literature, and as such, it should prove useful in demonstrating the merits of using the γ−ϕ and ϕ−ϕ approaches for different types of mixtures producing different phase behavior classifications.
⎛ ∂(∑ n GE) ⎞ k k ⎟⎟ RT ln γi = ⎜⎜ ∂ ni ⎝ ⎠P , T , n
j≠i
where R is the universal gas constant and T is the temperature. The number of moles for each component in the system is defined by nk. f ri can be categorized for subcritical and supercritical components in eqs 5 and 6, respectively:
i = 1, N
(1)
where, for each species i of N components, f ̂ is the partial fugacity in the vapor (v) and liquid (l) phases, respectively. When the fugacities are expressed in terms of measurable quantities and mass balance constraints are applied, the desired equilibrium properties are then determined. In the following sections, the γ−ϕ and ϕ−ϕ approaches for characterizing the VLE are presented. Specifically, the section describes the considerations and treatment of the two approaches in this study. 2.1. γ−ϕ Approach. The γ−ϕ calculation approach relies on an ACM for the liquid phase, while retaining an EOS model for the vapor phase. The activity coefficients are usually obtained from an excess Gibbs free energy model. Specifically, eq 1 is rewritten for this approach as follows: v v fi ̂ = ϕî Pyi
(5)
f ir = Hi(T , P)
(6)
⎛ u − u 22 ⎞ ⎛ a12 ⎞ ⎟ = exp⎜ − ⎟ τ12 = exp⎜ − 12 ⎝ ⎠ ⎝ RT ⎠ RT
(7)
⎛ u − u11 ⎞ ⎛ a 21 ⎞ ⎟ = exp⎜ − ⎟ τ21 = exp⎜ − 21 ⎝ ⎝ RT ⎠ RT ⎠
(8)
In this study, the parameters a12 and a21 in the above equations are regressed for each binary mixture. These parameters account for the differences in mixed (u12 and u21 and pure (u11 and u22)) component characteristic energy interactions. The Bondi groupcontribution method is used to determine the values of the van der Waals surface area and volume.15 2.1.2. UNIQUAC Interaction Parameter Regressions. The two adjustable parameters (a12 and a21) in the ACM were obtained by nonlinear regression of the experimental VLE data. Regression of the interaction parameters in the UNIQUAC model was performed previously in our group for evaluation of the representation capabilities of the models.13 For this purpose, eq 1 was recast to perform the phase-equilibrium calculations as follows:
(2)
l
fi ̂ = γi f ir xi
f ir = fil (T , P)
where Hi is Henry’s constant. The main advantage of this approach is that it references the fugacity in the liquid phase to an ideal liquid solution rather than an ideal gas. Traditionally, this approach is used for lowpressure VLE systems involving mixtures containing both polar and nonpolar components. The advantages of this approach are its simplicity and ability to handle a large variety of mixtures of interest in the chemical industries. This approach, however, has limited applicability for high-pressure systems and near the critical region. Prausnitz et al.14 detailed some of the advantages and disadvantages of this approach. 2.1.1. UNIQUAC ACM. In this study, the UNIQUAC model is selected for the present evaluation. Abrams and Prausnitz derived the UNIQUAC equation for nonrandom mixtures containing molecules of different sizes.4 This model is applicable to a wide range of liquid mixtures that contain both polar and nonpolar fluids. Details of the UNIQUAC model and its correlative capabilities have been presented widely in the literature [see, e.g., refs 4, 13, 14, and 37]. The UNIQUAC model has two binary-specific parameters, τ12 and τ21, which are defined based on characteristic energy parameters, Δu12 and Δu21:
2. REPRESENTATION METHODOLOGY Generally, phase-equilibrium relations determine the partitioning of a particular molecular species among coexisting phases. For a closed, VLE system at constant temperature (T) and pressure (P), the equal-fugacity criterion states v l fi ̂ (T , P) = fi ̂ (T , P)
(4)
(3)
where f ri is the liquid reference fugacity of component i and xi and yi are the liquid- and vapor-phase mole fractions,
v
ϕî Pyi = γiPi °ϕi v xiλi 1089
(9) DOI: 10.1021/acs.iecr.5b03475 Ind. Eng. Chem. Res. 2016, 55, 1088−1101
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Industrial & Engineering Chemistry Research where, for any component i, ϕ̂ vi is the component fugacity coefficient in the vapor phase, yi is the vapor mole fraction, γi is the component activity coefficient in the liquid phase, Pi° is the pure-component vapor pressure, ϕvi is the pure-component fugacity coefficient in the vapor phase, which is calculated at the system temperature and saturation pressure of pure component i, xi is the liquid mole fraction, and λi is the Poynting factor. Equation 9 was employed to regress the parameters of the UNIQUAC model for all of the VLE systems included in this study. Because practically all of the systems considered were at low pressure (99% of the data are at less than 10 bar), the vapor-phase fugacity coefficients were assumed to be 1. The parameter regression analyses used an objective function, OF, which is based on the sum of squares of relative errors in the pressure as follows: n
OF =
⎛ P exp − P cal ⎞ 2 ⎟ exp ⎠ ⎝ P
Table 1. Representation Capability of the γ−ϕ Approach for VLE Property Predictions data set ideal solution
highly nonideal systems
aqueous systems
∑⎜ i=1
i
(10) all data
where n is the number of data points and the superscripts exp and cal refer to experimental and calculated values, respectively. After evaluation of the VLE property predictions calculated using various forms of objective functions, this objective function was selected. The OF shown in eq 10 was found to provide a reasonable balance of the model prediction errors for the temperature (T), pressure (P), equilibrium constant (K), activity coefficient (γ), and vapor mole fraction (y). 2.2. ϕ−ϕ Approach. In this approach, the fugacities in eq 1 are expressed as follows: v
data set all data; no binary interactions are used
(11)
l l fi ̂ = ϕî Pxi
(12)
where ϕ̂ vi and ϕ̂ li are the fugacity coefficients of a specific component, i, in the vapor and liquid phases, v and l, respectively. The vapor and liquid mole fractions for each component are represented by y and x, respectively. Equation 13 is applied to calculate the fugacity coefficients using the selected EOS:14 1 ln ϕî = RT
∫P
0
⎛ RT ⎞⎟ ⎜v ̂ − dP ⎝i P ⎠
no. of systems
no. of points
% AAD
P (bar) 916 33283 0.68 T (K) 916 33283 9.29 y1 675 18200 0.10 K 675 18200 6.79 Results Using Regressed a12 and a21 P (bar) 348 14929 0.20
−0.13 4.15 −0.01 −0.82
13.5 1.5 15.3 19.2
0.00
2.5
T (K) y1 K P (bar)
348 262 262 55
14929 8203 8203 4344
1.46 0.04 7.51 0.33
0.14 0.00 −0.64 −0.01
0.2 5.0 6.2 4.2
T (K) y1 K P (bar) T (K) y1 K
55 47 47 916 916 675 675
4344 2313 2313 33283 33283 18200 18200
2.24 0.06 15.76 0.13 1.17 0.03 4.98
0.30 −0.01 −3.12 0.00 0.04 0.00 −0.18
0.4 9.4 10.5 1.6 0.1 4.5 5.7
RMSE
Table 2. Representation Capability of the ϕ−ϕ Approach for VLE Property Predictions
v
fi ̂ = ϕî Pyi
bias
property
highly nonideal systems
aqueous systems
(13)
where v̂i is the partial molar volume of component i and R is the universal gas constant. This approach is used traditionally for high-pressure systems following the careful selection of the applicable EOS and mixing rules. For example, the cubic EOSs are used to describe the phase behavior of mixtures of normal fluids involving relatively small molecules with little polarity. When an appropriate EOS is applied, this approach is highly efficient and provides for the calculation of volumetric equilibrium, caloric, and auxiliary thermodynamic properties of multicomponent, multiphase systems. Further, the approach can handle near-critical region calculations while avoiding the complexity of standard-state requirements. For greater detail in the development of this approach and the associated advantages and disadvantages, see the work of Abrams and Prausnitz.4 2.2.1. PR EOS. In this work, we focus on the PR EOS.2 While many EOSs exist, the computational efficiency of the PR EOS offers a distinct advantage in many applications. The pressureexplicit form of the equation is given as
all data
P=
property
no. of systems
no. of points
RMSE
bias
% AAD
P (bar)
916
33283
19
2.7
18
2.5 −0.08 −0.02
3 39 15
T (K) 916 33283 23 y1 675 18200 0.2 K 675 18200 0.09 Results Using Regressed C12 and D12 P (bar) 348 14929 0.22
−0.02
4.8
T (K) y1 K P (bar) T (K) y1 K P (bar) T (K) y1 K
0.43 0.00 −0.25 −0.05 1.14 0.01 −0.67 −0.01 0.22 0.00 −0.10
0.5 11.1 13.8 10.0 1.1 22.0 24.9 3.9 0.4 9.5 11.7
348 262 262 55 55 47 47 916 916 675 675
14929 8203 8203 4344 4344 2313 2313 33283 33283 18200 18200
a(T ) RT − v−b v(v + b) + b(v − b)
5.09 0.07 8.52 0.23 9.42 0.12 19.47 0.15 4.25 0.08 5.29
(14)
where a( T ) =
b=
0.457235α(T )R2Tc 2 Pc 2
0.077796RTc Pc
(15)
(16)
and P is the pressure, T is the temperature, v is the molar volume, a and b are the EOS parameters, Tc is the critical temperature, Pc is the critical pressure, and R is the universal gas 1090
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Figure 1. Pressure representation of the γ−ϕ and ϕ−ϕ approaches by the types of interactions based on the regressed UNIQUAC model and PR equation.
Table 3. Polarity Combination of Compounds Based on the Danner Classification no.
component 1
component 2
1 2 3 4 5 6 7
nonpolar weakly polar nonpolar nonpolar weakly polar strongly polar very strongly polar (aqueous)
nonpolar weakly polar weakly polar strongly polar strongly polar strongly polar strongly polar
b=
i
aij = bij =
2
(17)
where ω is the acentric factor, Tr is the reduced temperature, and A−E are correlation parameters with values of 2.0, 0.836, 0.134, 0.508, and −0.0467, respectively. 2.2.2. Mixing Rules. When an EOS is applied to mixtures, the mixing rules for the parameters of the EOS must be specified. The familiar one-fluid mixing rules were used in the current work and are given as14 a=
j
aiaj(1 − Cij) bi + bj 2
(1 + Dij)
(20)
(21)
3. QSPR METHODOLOGY Reliable database development, molecular structure generation and optimization, quantification of the molecule descriptors, optimization of the number of relevant descriptors, and subsequent model validation are the main steps in building a
∑ ∑ zizjaij i
(19)
j
where z is the mole fraction of compound i in the phase of interest. Using experimental VLE data, the empirical binary interaction parameters, Cij and Dij, can be determined through regression.17 The essential input variables for the PR EOS then include the critical temperature, critical pressure, and acentric factor for each fluid, along with two binary interaction parameters (Cij and Dij). For simplicity, while not sacrificing accuracy, classical mixing rules involving one or two interaction parameters are often employed in EOS applications, although more complex mixing rules have been proposed.18,19 A comprehensive study of the PR equation, data reduction methods, and mixing rules has been published recently by the Oklahoma State University Thermodynamics Group.17
constant. Earlier work at Oklahoma State University provided an expression for calculation of the term a(T) shown in eq 15 as follows:16 a(T ) = exp[(A + BTr)(1 − TrC + Dω + Eω )]
∑ ∑ zizjbij
(18) 1091
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ϕ−ϕ approach
γ−ϕ approach
system
nearly ideal: max | GE/RT| ≤ 0.15 and | g12| ≤ 100
nonideal: max |GE/RT| ≈ 0.15 and −180 < g12 < −100; 100 < g12 < 220
highly nonideal: max |GE/RT| > 0.15 and g12 ≤ −180; max |GE/RT| < −0.15 and g12 ≥ 220
quality of fita
Nonpolar−Nonpolar benzene (1)−toluene (2) n-heptane (1)−ethylbenzene (2) n-octane (1)−ethylbenzene (2) 1-heptene (1)−toluene (2)
√ √ √ √
3 3 3 2
Weakly Polar−Weakly Polar propionic aldehyde (1)−acetone (2) propionic aldehyde (1)−2-butanone (2) acetone (1)−vinyl acetate (2) acetone (1)−propyl acetate (2) acetaldehyde (1)−vinyl acetate (2) acetaldehyde (1)−methyl acetate (2) diethyl ether (1)−acetone (2) acetaldehyde (1)−diethyl ether (2)
√ √ √ √ √ √
1 2 3 3 3 3 3 2
√ √ Nonpolar−Weakly Polar
benzene (1)−acetone (2) n-heptane (1)−thiophene (2) n-heptane (1)−butanone (2) tetrachloromethane (1)−acetonitrile (2)
√ √ √ √
3 3 3 2
√
3 3 3 3 3 1
Nonpolar−Strongly Polar benzene (1)−2-propanol (2) toluene (1)−1,2-dichloroethane (2) benzene (1)−diethylamine (2) n-heptane (1)−ethyl iodide (2) n-octane (1)−pyridine (2) n-octane (1)−methanol (2) diethyl ether (1)−methyl iodide (2) 1,4-Dioxane (1)−2-Propanol (2) methanol (1)−2-methyl-1-propanol (2) ethanol (1)−2-methyl-1-propanol (2) butylamine (1)−1-butanol (2) bromobenzene (1)−cyclohexanol (2) 1,2-dichloroethane (1)−2-methyl-1-propanol (2) methanol (1)−1,2-dichloroethane (2)
√ √ √ √ √ Weakly Polar−Strongly Polar √ √ Strongly Polar−Strongly Polar √ √ √
2 3
√ √ √
2 2 2 1 1 1
Aqueous−Strongly Polar water water water water a
√
(1)−diethylamine (2) (1)−methanol (2) (1)−2-propanol (2) (1)−ethanol (2)
√ √ √
2 3 2 2
Quality of the fit: 3 = good; 2 = adequate; 1 = poor.
proportions, the computational algorithm monitors the data distribution and may supersede the given proportions. Specifically, priority is given to the assignment of data to the training set followed by the validation and internal test sets for those interactions with a limited amount of data. More details on the QSPR models for generalizing the UNIQUAC and PR EOS model parameters were presented in our previous work.11,12 Accordingly, only a brief description is provided herein. The main tasks for developing the QSPR models include the following: (1) database development; (2) parameter regression analyses for VLE systems using UNIQUAC/ PR EOS; (3) molecular structure generation and optimization; (4) descriptor reduction; (5) QSPR model development using neural networks.11,12
robust QSPR model. The compiled database consisted of 35000 data points for VLE measurements belonging to 916 binary systems. The detailed information regarding the data collection is presented in the next step. The ChemBioDraw Ultra,20 Open Babel,21,22 DRAGON,23 and CODESSA24 software are used to generate the 2D and 3D molecule structures, optimize the 3D structure, and quantify the descriptors, respectively. During QSPR model development, the database is divided into four different sets, which are the training, validation, internal test, and external test sets with 50%, 15%, 10%, and 25% of the data, respectively. This division ensures adequate representation of all functional-group interactions in each of the data sets. While the data is assigned randomly in the given 1092
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Figure 2. Phase-equilibrium composition predictions for the nonpolar−nonpolar system classification: (a) benzene (1) + toluene (2); (b) n-heptane (1) + ethylbenzene (2); (c) n-octane (1) + ethylbenzene (2); (d) 1-heptene (1) + toluene (2).
4. VLE LITERATURE DATABASE EMPLOYED
5. RESULTS AND DISCUSSION
To facilitate the development of a robust model and the evaluation of its subsequent performance, a reliable database is essential. An expanded database was constructed containing low-pressure (99% of data are less than 10 bar) VLE measurements for the 916 binary systems composed of 140 diverse chemical species belonging to 31 chemical classes. The diversity of the molecular species in the assembled databaseassured by variations in the molecular size, shape, asymmetry, and polarityprovide a suitable means of evaluating the efficacy of this approach. Overall, the database contains more than 35000 data points for VLE measurements. More details about the assembled database were presented in our previous work.13 For the vapor-pressure data, we relied on the DIPPR pure-fluid property database.25 In such cases, a careful check is made on the fits obtained for the pure-fluid limits. The details regarding the experimental data are presented in the Supporting Information.
5.1. Representation Assessment. We evaluated the abilities of the γ−ϕ and ϕ−ϕ approaches to represent the desired equilibrium properties (P, T, y1, and K) employing a database of 916 VLE binary mixtures, which involve 31 chemical classes. The ideal solution model and no-binaryinteraction case study are selected as a reference point for both the UNIQUAC model and PR EOS, respectively. Tables 1 and 2 provide the property prediction errors obtained for different scenarios for both the UNIQUAC ACM and PR EOS. As expected, the lack of binary interaction parameters results in large errors in phase-equilibrium property calculations. When using interaction parameters, the overall results for all systems, as well as the results for aqueous systems and highly nonideal systems, are reported separately to facilitate the comparison. As indicated by the summary results, the γ−ϕ approach provided lower predication errors (percent average-absolute deviation, % AAD) than the ϕ−ϕ approach for the systems 1093
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Figure 3. Phase-equilibrium composition predictions for the weakly polar−weakly polar system classification: (a) propionic aldehyde (1) + acetone (2); (b) propionic aldehyde (1) + 2-butanone (2); (c) acetone (1) + vinyl acetate (2); (d) acetone (1) + propyl acetate (2); (e) acetaldehyde (1) + vinyl acetate (2); (f) acetaldehyde (1) + methyl acetate (2); (g) diethyl ether (1) + acetone (2); (h) acetaldehyde (1) + diethyl ether (2).
considered. Specifically, the overall % AADs for the γ−ϕ approach are 1.6, 0.1, 4.5, and 5.7 for P, T, y1, and K, respectively. This level of precision compares to 3.9, 0.4, 9.5, and 11.7% AAD for the ϕ−ϕ approach. Thus, the deviations from the ϕ−ϕ approach are, on average, within 2.5 times those produced by the γ−ϕ approach. Further, both approaches exhibit significant deviations for the aqueous systems. Figure 1 shows the distribution of errors in pressure regression for the ϕ−ϕ and γ−ϕ approaches by functional-group interactions. For each functional-group interaction, the results are shown using gray tones based on the % AAD ranges as given in the figure key. For 26 systems (less than 3% of all systems), of which 9 systems were aqueous, the ϕ−ϕ approach resulted in an error of greater than 10% AAD. As expected, the precise representation is provided for both approaches when the components
involved have identical functional groups (diagonal elements of the triangular matrix). This is due to the fact that components with the same functional groups show nearly ideal behavior. Both approaches result in relatively large errors for most of the aqueous systems. In particular, the errors were above 20% AAD for systems containing sulfide, bromoalkanes, nitro compounds, pyridine derivatives, and chlorobenzene. The higher errors for aqueous systems can be attributed to either highly nonideal behavior exhibited when polar molecules, such as water, interact in a mixture or a trace amount of mole fraction of water in a mixture. Although these results provide insight into the model’s performance for aqueous systems, more data are needed to adequately examine each functional-group interaction and to provide conclusive comparisons. In the following section, a thorough analysis of the capabilities of these two approaches is provided where the database was examined 1094
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Figure 4. Phase-equilibrium composition predictions of the nonpolar−weakly polar system classification; (a) benzene (1) + acetone (2); (b) n-heptane (1) + thiophene (2); (c) n-heptane (1) + 2-butanone (2); (d) tetrachloromethane (1) + acetonitrile (2).
The types of molecular interactions present in a system of interest determine the degree of nonideality. When components of a system exhibit nearly ideal behavior, there usually exist similar functional groups, polarities, and molecular sizes; conversely, when components of a system exhibit highly nonideal behavior, there usually exists a high degree of polarity difference. While binary system observations of the molecules and interactions involved can provide a qualification of the mixture behavior, a precise evaluation of the degree of nonideality is not possible.26 When model parameter values are related with the behavior of the system, a qualitative approach can be utilized as an alternative. In this manner, the type of system behavior is determined, and then the most suitable thermodynamic approach or model can be utilized for determination of the system properties. Previously, Gebreyohannes and co-workers28 applied the same approach as Danner26 to determine the behavior of the systems. A model parameter, g12, for modified NRTL, mNRTL1, and the maximum |GE/RT| values was utilized to represent the behavior of the systems in our database.28 After evaluation of the systems in the current database, systems with
further using the Danner classification of mixture phase behavior.26 The details regarding the experimental data and regressed UNIQUAC model and PR EOS parameters are presented in the Supporting Information. 5.1.1. Danner Classification: Representation. The Danner classification was applied to categorize the low-pressure VLE database described earlier based on the mNRTL1 g12 parameter. On the basis of the component polarity, the database, which includes a total of 916 nonelectrolyte binary systems, has been evaluated and divided into seven groups. The compound polarity was defined based on the Heat Transfer and Fluid Flow Service (HTFS) Handbook.27 The atom size, molecular ability for the donation or acceptance of a hydrogen atom, and evidence of hydrogen bonding are considered when a polarity code, which ranges from 0 to 9, is assigned. Code “0” represents the nonpolar, and codes “1” and “2” represent the weakly polar. The strongly polar class was assigned codes “3”, “4”, “5”, “6”, “7”, and “8”, while code “9” was assigned to the very strongly polar compound. Table 3 shows the seven compound polarity combinations that may result from formation of a binary system. 1095
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Figure 5. Phase-equilibrium composition predictions for the nonpolar−strongly polar system classification: (a) benzene (1) + 2-propanol (2); (b) toluene (1) + 1,2-dichloroethane (2); (c) benzene (1) + diethylamine (2); (d) n-heptane (1) + ethyl iodide (2); (e) n-octane (1) + pyridine (2); (f) n-octane (1) + methanol (2).
Figure 6. Phase-equilibrium composition predictions for the weakly polar−strongly polar system classification: (a) diethyl ether (1) + methyl iodide (2); (b) 1,4-dioxane (1) + 2-propanol (2).
a maximum |GE/RT| value of ≤0.15 are classified as nearly ideal, while |GE/RT| values of >0.15 are classified as highly nonideal systems. In addition, the results show that the g12 range of the nearly ideal systems is approximately between −100 and +100, while for highly nonideal systems, it is >220 and
