Extension of the VTPR Group Contribution Equation of State: Group

Article pubs.acs.org/IECR

Extension of the VTPR Group Contribution Equation of State: Group Interaction Parameters for Additional 192 Group Combinations and Typical Results Bastian Schmid,† Andre Schedemann,† and Jürgen Gmehling†,*,‡ †

DDBST GmbH, Oldenburg D-26129, Germany Carl von Ossietzky Universität Oldenburg, Technische Chemie, Oldenburg D-26111, Germany

‡

S Supporting Information *

ABSTRACT: Today development, design, and optimization of the various processes is carried out with the help of process simulators. The reliability of the results mainly depends on the quality of the thermodynamic model and the model parameters used. While gE models can be applied for calculation of the phase equilibrium behavior of multicomponent systems using only binary experimental data, group contribution methods like UNIFAC or modified UNIFAC Dortmund allow prediction of the required thermophysical properties using only a limited number of group interaction parameters. For systems containing supercritical components equations of state like Soave−Redlich−Kwong or Peng−Robinson or group contribution equations of state (GCEOS) like the predictive Soave−Redlich−Kwong (PSRK) or the volume-translated Peng−Robinson group contribution equations of state (VTPR) can be applied. In different papers it was already shown that VTPR is a very powerful thermodynamic model. In this paper new group interaction parameters for 192 group combinations are presented, so that the actual matrix now contains group interaction parameters for 252 group combinations. In this paper predicted results of the VTPR group contribution equation of state are compared with the results obtained using modified UNIFAC Dortmund or the PSRK method.

1. INTRODUCTION

of the pure component i can directly be calculated from critical data. The temperature dependency of the attractive parameter aii is taken into account with the help of the Twu α-function8

Approximately 10 years after the GCEOS PSRK model was published by Holderbaum and Gmehling in 1991,1 Ahlers and Gmehling presented the VTPR group contribution equation of state.2−4 In the VTPR GCEOS, most of the weaknesses of the PSRK method, such as poor results for liquid densities, excess enthalpies, activity coefficients at infinite dilution, and asymmetric systems, were removed. The group contribution equation of state VTPR (eq 1) is a combination of the volume-translated Peng−Robinson equation of state5 with the group contribution method modified UNIFAC (Do)6 using improved mixing rules suggested by Chen et al.7 and temperature-dependent VTPR group interaction parameters. P=

αi(T ) = Tr,Nii .(Mi − l)·exp[Li · (l − Tr,Nii·Mi)]

which leads, compared to the Mathias−Copeman α-function9 used in PSRK, to improved results especially at high reduced temperatures. The parameters Li, Mi, and Ni of the α-function are usually obtained by fitting eq 4 to experimental pure component vapor pressure data. Since liquid heat capacities also deliver important information about the temperature dependency especially at low temperatures, liquid heat capacities can additionally be taken into account during the parameter-fitting procedure.10 The c parameter proposed by Peneloux et al.11 can be calculated from the mole fractions and the component-specific translation parameters ci

R·T (v + c − b) −

a(T ) (v + c) ·(v + c + b) + b·(v + c − b)

ci = vPR,i − vexp , i (at Tr, i = 0.7)

c=

aii(T ) = ac , iαi(T ) = 0.45724

R

Pc, i

. αi(T )

(6)

(2)

whereby the component-specific translation parameters ci are obtained as the difference of the volume calculated with the Peng−Robinson equation of state and the experimental one

(3)

Received: Revised: Accepted: Published:

and the covolume bii bii = 0.0778

∑ xici i

·Tc,2i

RTc, i Pc, i © 2014 American Chemical Society

(5)

using a linear mixing rule

(1)

The attractive parameter aii(T) 2

(4)

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Industrial & Engineering Chemistry Research

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Figure 1. Experimental and predicted pure component densities using VTPR () and Peng−Robinson EOS (···).

Figure 2. Experimental (⧫)16 and predicted densities for a synthetic LNG using VTPR () and Peng−Robinson (···).

Rackett equation as published by Ahlers and Gmehling2,13 can only be applied for nonpolar components with a moderate chain length as known from hydrocarbon processing applications, such as gas processing, refineries, and petrochemical processes and therefore should be no longer applied.14,15 With this translation parameter, the systematic error in the volume calculation caused by cubic equations of state can be distinctly reduced. Figure 1 shows on the left-hand side the improved density prediction for carbon dioxide, hydrogen sulfide, and methane at reduced temperatures from 0.5 to 1. While the effect of volume translation in the case of these small molecules is comparatively small, it can be seen that greater improvements are obtained for larger molecules, such as o-nitrotoluene and 1-octadecanol shown on the right-hand side of Figure 1. Automatically introduction of the concept of volume translation leads also to a significant better description of mixture densities over the whole composition and temperature range. As an example for the improvement of the liquid density prediction of multicomponent mixtures, a synthetic LNG with nitrogen, n-butane, methane, ethane, propane, and 2-methylpropane was chosen. The experimental data together with the prediction results using VTPR and Peng−Robinson are shown in Figure 2. It can be seen that with the help of VTPR nearly perfect results are obtained in contrast to PSRK. In VTPR improved mixing rules for parameters a and b suggested by Chen et al.7 were introduced. In the mixing rule for the parameter a(T)

Figure 3. P/T curves for the symmetric system propane (1)−n-butane (2).17 () Prediction result obtained using VTPR. (···) Prediction result obtained using PSRK.

Figure 4. P/T curves for the strong asymmetric system ethane− hexadecane.17 () Prediction result obtained using VTPR. (···) Prediction results obtained using PSRK.

E ⎞ ⎛ gres a (T ) ⎟ a(T ) = b·⎜⎜∑ xi· ii + bii −0.53087 ⎟⎠ ⎝ i

determined with the help of the fitted DIPPR parameters (eqs 105 and 116 in ref 12) at a reduced temperature Tr,i = 0.7. An estimation of the translation parameter using a modified 3394

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dx.doi.org/10.1021/ie404118f | Ind. Eng. Chem. Res. 2014, 53, 3393−3405


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Figure 5. Overview of the VTPR group contribution equation of state.

only the residual part of the excess Gibbs energy is used. The exponent 3/4 in the nonlinear combination rule for the crossparameter bij bij3/4 =

(bii3/4 + bjj3/4) 2

(8)

of the quadratic mixing rule for parameter b b=

∑ ∑ xixjbij i

j

(9)

was taken from modified UNIFAC (Do) and leads to a significantly better description in the case of asymmetric systems.

Figure 6. Temperature range covered by the different data types.

Figure 7. Flow sheet of the group interaction parameter-fitting procedure. 3395



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Figure 8. Actual group interaction parameter matrix for VTPR: (blue) two parameters, (green) four parameters, (orange) six parameters.

Figure 9. Experimental (⧫, (···).

▲)17

and predicted properties for different aromatic−alcohol systems using VTPR () and modified UNIFAC (Do) 3396



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Figure 10. Experimental (⧫, ▲)17 and predicted data for sub- as well as supercritical VLE’s, azeotropic, and critical line for the system ethanol(1)− water(2) using VTPR ().

Figure 11. Experimental data (●)17 and predicted critical curvature for different binary systems using VTPR ().

eutectic systems, and activity coefficients at infinite dilution. In VTPR, the temperature function for the group interaction parameters is the same as the one known from modified UNIFAC (Do) (eq 10).

In case of symmetric systems like propane−n-butane (Figure 3) nearly the same VLE results are obtained for VTPR and PSRK. Both models provide VLE, which are in good agreement with the experimental findings. From Figure 4, it can be seen that only VTPR is able to predict the VLE behavior of the strong asymmetric system ethane (1)−hexadecane (2) reliably. While for the n-alkane−n-alkane systems shown before no group interaction parameters are required, the gE information for all other group combinations has to be taken into account using temperature-dependent VTPR parameters, which in the case of VTPR are fitted simultaneously to different binary experimental data, like, for example, vapor−liquid equilibria, gas solubilities, excess enthalpies, solid−liquid equilibria of simple

⎛ a + b T + c T2 ⎞ nm nm ⎟ Ψnm = exp⎜ − am T ⎝ ⎠

(10)

Dependent on the strength of the temperature dependency, at least two (anm, amn) but also up to six parameters (anm, amn, bnm, bmn, cnm, cmn) are fitted to a comprehensive database covering a wide temperature range. For a better understanding, the different parts of the VTPR model and their connections are summed up in Figure 5. 3397



Article E ⎛ ∂ ln γ ⎞ hi̅ i⎟ ⎜ = ⎜ ∂1 ⎟ R ⎝ T ⎠P , x

This means they contain valuable information about the temperature dependence. Heats of mixing data above 363.15 K are furthermore important, supporting data at high temperatures, and solid−liquid equilibrium data of simple eutectic systems (SLE) can be used as supporting data at low temperatures. Finally, liquid−liquid equilibrium data (LLE) often deliver the only information available for systems with a strong deviation from Raoult’s law. The first step of the parameter-fitting procedure is the selection of appropriate experimental data for the chosen group combination using the VTPR fragmentation rules.15 This raw database is then evaluated optically and in the case of vapor− liquid equilibrium data additionally with the help of consistency tests, before the fitting procedure takes place. A uniform distribution of the data with respect to compounds, temperature, pressure, and composition is a precondition for fitting reliable group interaction parameters. During the simultaneous parameter regression using the simplex Nelder−Mead method,18 the sum of the deviations Δ between experimental and calculated values for all phase equilibrium data and excess properties taken into account is minimized. The contribution of the different data types to the objective function F can be adjusted with the help of weighting factors wi.

Figure 12. Experimental and predicted azeotropic data (⧫)17 using VTPR () and PSRK (···) .

A prerequisite for successful extension of the group interaction parameter matrix is a comprehensive database. In our case, the Dortmund Data Bank (DDB)17 is used in the parameter-fitting procedure. Figure 6 shows the temperature range covered by the different data types taken into account during the parameterfitting procedure. Since a lot of activity coefficients at infinite dilution (γ∞) were measured using gas−liquid chromatography, these data provide information about asymmetric systems (stationary phase, solute). They are related to the fugacity coefficients in the following way ln γi = ln φi mix − ln φi pure

(12)

(11)

F = WVLE ∑ ΔVLE+WAZD ∑ ΔAZD

Gas solubilities (GLE) cover the dilute region, and vapor− liquid equilibrium data of normal- (VLE) and low-boiling components with a boiling point below 273.15 K as well as azeotropic data (AZD) provide information about the real behavior as a function of composition. Excess enthalpies (hE) describe the temperature dependency of the activity coefficients following the Gibbs−Helmholtz equation.

+WhE ∑ ΔhE + WcPE ∑ ΔcPE + WGLE ∑ ΔGLE+Wγ ∞ ∑ Δγ ∞ +WSLE ∑ ΔSLE + WLLE ∑ ΔLLE =minimum

(13)

Figure 13. Experimental VLE data (⧫)17 and predicted influence of n-butane (C4) as entrainer for the binary systems H2S−propane, CO2−ethane, and H2S−ethane using VTPR () (solvent-free basis). 3398



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Figure 14. Experimental data (⧫)17 and predicted solid−liquid equilibrium behavior of the systems ethane (propane) with CO2 and H2S using VTPR ().

After each run, the resulting parameters are checked against the database. Then the weighting factors of the different data types can be adjusted or data removed until a reliable prediction of all thermophysical data is obtained. Sometimes also systematic measurements are performed. Figure 7 shows the flow sheet of the parameter-fitting procedure in detail. The first step in the parameter-fitting procedure is the selection of the main groups, e.g., for n-alkanes−alcohols, for which group interaction parameters should be fitted. With the help of the structural information stored in the pure component data files of the Dortmund Data Bank, suitable components (alkanes, alcohols) are selected and binary experimental data are retrieved from the comprehensive mixture data bank. Pure component properties like the Twu α-function parameter Li, Ni, and Mi, the critical temperature, critical pressure and critical volume, the volume translation parameter ci, the acentric factor ωi, the melting temperature, the heat of fusion,..., are also directly taken from the Dortmund Data Bank. Then the raw database is carefully checked using consistency and plausibility tests. Before the regression is performed, the number of data is reduced and graphically evaluated whereby an equal distribution of temperature, pressure, composition, and compounds is realized. The evaluation, reduction, and fitting procedure is then repeated until the results are satisfying. In some cases, e.g., in particular, for strong polar systems, the parameters obtained from the fitting procedure did not allow a reliable description of the different thermophysical properties. Here, the van der Waals group surface areas had to be simultaneously adjusted with the group interaction parameters keeping in mind their physical meaning. This fact was already observed during development of the UNIFAC group contribution method.5

Figure 15. Experimental data (⧫)17 and predicted solid−liquid equilibrium behavior of the systems benzene (1)−methane (2) using VTPR ().

Figure 16. Experimental data (⧫, ●)17 and predicted values for the global phase diagram (type IIa) of the binary system CO2−n-octane using VTPR.

2. NEW PARAMETERS Using the sophisticated software package and the comprehensive database (Dortmund Data Bank) a large number of the required group interaction parameters have been fitted. In Figure 8 the current status of the group interaction parameter matrix for VTPR is shown. The numerical values of the new and revised parameters are given in the Supporting Information together with the required information about the main and subgroups and the surface area parameters. Carefully fitted group interaction parameters are the precondition for a successful application19 of VTPR for tasks like reliable prediction of binary phase equilibria and excess properties, identification of separation problems, e.g., azeotropic

points in multicomponent systems, construction of residual curves and borderlines, design of separation columns, selection of suitable solvents for separation processes,20 chemical processes, etc., selection of working fluids for thermodynamic cycles, consideration of the real behavior on the chemical equilibrium conversion (Kγ, Kφ), prediction of flash points of flammable liquid mixtures, prediction of the fate of a chemical in the environment, bioaccumulation effects, etc., reliable description of the diffusional mass transfer (using Δai, Δf i, Δμi instead of Δci), prediction of various thermodynamic properties (h, Δhv, s, etc.), e.g., for thermodynamic cycles, etc. 3399



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Figure 17. (a) Residual curves and azeotropic behavior for the ternary system CO2 (1)−ethane (2)−n-butane (3) at atmospheric pressure calculated using VTPR. (b) Contour lines for α12 and azeotropic point for the system CO2 (1)−ethane (2)−n-butane (3) at 243.15 K calculated using VTPR.

The results for VLE, SLE, and γ∞ of both models are in good agreement with the experimental data. For azeotropic data and excess enthalpies slightly better results are obtained for VTPR. Also, the case of systems containing polar and sub- or supercritical components VTPR reliably predicts the phase equilibrium behavior. For example, the binary system ethanol (1)−water (2) was chosen. From Figure 10 it can be seen that all investigated properties are very close to the experimental data. Even the curvature of the azeotropic line and the

Below typical results for prediction of different thermophysical properties using the new fitted group interaction parameters are given.

3. TYPICAL RESULTS Different experimental and predicted thermophysical properties for various aromatic−alcohol systems using VTPR and modified UNIFAC (Do) are shown in Figure 9. 3400



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disappearance of the azeotropic behavior at low temperatures (∼ 303 K) is reliably predicted using the volume-translated Peng−Robinson group contribution equation of state. In Figure 11, predicted critical lines of different binary systems together with experimental data are presented. Again, it can be seen that the experimental critical lines are well described using VTPR. VTPR can directly be used for selection of suitable solvents for separation processes (extraction, absorption, azeotropic and extractive distillation, solution crystallization, etc.). Below results are shown for selection of an entrainer for separation of azeotropic systems like ethane−CO2, ..., in the gas processing industry. As can be seen from Figure 12 VTPR as well as PSRK can be used to predict the appearance of the azeotropic behavior of compounds present in natural gas. Although the experimental data are scattering,21 it seems that the predicted azeotropic data are in good agreement with the experimental findings in a wide temperature range. One possibility for separation of these azeotropic mixtures is addition of a selective entrainer. With VTPR and a sophisticated software package it can be found that, e.g., n-butane (C4) which is present in the natural gas stream directly can be applied as selective entrainer as realized in the Ryan−Holmes distillation process.22 From Figure 13 one can see that adding 20−40 mol % C4 to the liquid phase results in an increase of the relative volatility of H2S and CO2 and a disappearance of the azeotropes. The Ryan−Holmes process is especially important at high CO2 concentrations in the natural gas stream, as, e.g., in the case of enhanced oil recovery. Another important property for the design and development of cryogenic processes, e.g., the controlled freeze zone (CFZ) process by Exxon,23 is a reliable knowledge of the solid−liquid equilibrium behavior. In Figure 14 experimental SLE data together with the prediction results for the four simple eutectic systems propane− CO2, ethane−CO2, propane−H2S, and ethane−H2S are shown. As can be seen, the predicted solid−liquid equilibria for all four systems using VTPR are in nearly perfect agreement with the experimental findings. The phase equilibrium behavior of the system benzene− methane at low temperatures is of a special interest with a view to the design of cryogenic heat exchangers for LNG processing. While the straight line in Figure 15 liquid−solid equilibrium occurs, the section with the linear slope is a three-phase line. Here, the vapor, liquid, and solid phases are in equilibrium with each other.24 In this area the liquid phase is unstable and pure benzene precipitates. VTPR is able to reliably predict the experimental (V)SLE behavior. Full characterization of a binary mixture requires knowledge of the global phase diagram as shown in Figure 16 for the system CO2−n-octane. The pure component vapor pressure curves of CO2 and n-octane are connected by the critical line. These three lines surround the vapor−liquid phase region. At low temperatures the system shows VLLE behavior, whereby the pressure is similar to the vapor pressure of pure CO2. The vapor phase disappears at the upper critical end point (UCEP), and a LLE is formed. The resulting global phase diagram is a type IIa diagram following the nomenclature by van Konynenburg et al.25 As can be seen the global phase diagram is reliably predicted using VTPR. Residual curves for the ternary system CO2−ethane−n-butane as shown in Figure 17 are of a great importance for development

Figure 18. Experimental (⧫)17 and predicted excess enthalpies for the system ethane (1)−propane (2) using VTPR ().

Figure 19. Pressure dependency of the excess volumes for ethane (1)−CO2 (2) at 291.6 K.

Figure 20. Experimental (⧫)17 and predicted molar entropies for toluene using VTPR ().

and design of distillation processes. The so-called topology map as a result of simultaneous presentation of the residual curves and the singular points (pure compounds, azeotropic points) and, if present, a borderline in one diagram is a helpful tool to better understand the investigated separation process.26 The contour lines for the ternary system CO2−ethane− n-butane at 243.15 K shown in Figure 17 indicate the course of the separation factors α12 for the components CO2−ethane predicted using VTPR. Starting from the azeotropic point of the binary system CO2−ethane (x1,az = 0.6385), the corresponding contour line α12 = 1 is colored red. With the help of the 3401



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Figure 21. Experimental (⧫)17 and predicted Joule−Thomson coefficients and inversion curves using VTPR ().

Figure 22. Relative deviations between calculated and experimental data for some selected group combinations for VTPR (red), PSRK (green) and modified UNIFAC Dortmund (blue).

solvent-free y−x diagrams on the right-hand side of Figure 17 the influence of different amounts of n-butane on the relative volatility becomes clear. While addition of 20 mol % n-butane

to the binary mixture shifts the azeotropic composition from x1,az = 0.6385 to x1,az ≈ 0.9. The azeotrope is broken in the presence of 40 mol % n-butane. 3402



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were basically only valid for nonpolar components, e.g., hydrocarbons. With the development of gE mixing rules by Huron, Vidal,29 and Michelsen,30 suddenly the advantages of gE models and equations of state could be combined and the solution of groups concept became available for equations of state.31 From Figure 22, which shows the relative deviation between calculated and experimental data for a few selected group combinations for VTPR, PSRK, and modified UNIFAC Dortmund, it becomes clear that today the VTPR group contribution equation of state as a result of systematic improvements of cubic equations of state and mixing rules shows an improved prediction quality compared to PSRK. The quality of the predicted results is comparable to that of modified UNIFAC (Do) using the consortium matrix,32 but compared to modified UNIFAC (Do), VTPR shows a lot of advantages: the real behavior of the vapor and the liquid phase is automatically taken into account, handling of systems containing supercritical components is possible, and at the same time additional pure component and mixture properties like densities, enthalpies, entropies, or heat capacities can be predicted. In this paper, the VTPR parameter matrix was extended by 192 group combinations. This means that the actual parameter matrix now contains group interaction parameters for 252 group combinations. From several results for different pure component and mixture properties it becomes clear that the volume-translated Peng−Robinson group contribution equation of state is indeed a tool which can cope with most of the requirements of modern process design. In the meantime, the first commercial vendors of process simulation software have implemented VTPR in their simulator. It would be desirable that also other simulator companies will do this.

With VTPR, also the pressure dependency of excess enthalpies and excess volumes can be successfully predicted. As an example the change of the excess enthalpies with temperature and pressure together with the experimental data for the binary system ethane (1)−propane (2) are presented in Figure 18. For every temperature, the heats of mixing were determined at two different pressures. All predicted results at the given conditions using VTPR are in agreement with the experimental data taken from ref 17. The same conclusion can be drawn for the excess volumes shown in Figure 19. Although experimental excess volumes which include information about the pressure dependency of the activity coefficients following eq 14 ⎛ ∂ ln γi ⎞ v iE ⎜ ⎟ = ̅ RT ⎝ ∂P ⎠T , x

(14)

are not taken into account during the parameter-fitting procedure the predicted and experimental values at different pressures and states (LL, both components liquid; VL, one component vapor; VV, both components vapor) are in a good agreement (Figure 19). For prediction of the behavior of working fluids for thermodynamic cycles knowledge about the entropies is required. The entropy can be calculated using the following equation v

S=

⎡⎛

⎞

∫∞ ⎢⎣⎝ ∂∂TP ⎠ ⎜

⎟

− v

R⎤ ⎥dv + R ln(z) + Sid v⎦

(15)

The results for pure toluene over a wide pressure range are shown in Figure 20. The bubble and the dew point curves separate the three areas liquid, vapor−liquid, and vapor from each other and meet in the critical point. Information about the Joule−Thomson coefficients (JTC, eq 16) is μJT =

⎛ ∂T ⎞ ⎜ ⎟ ⎝ ∂P ⎠h

■

ASSOCIATED CONTENT

S Supporting Information *

Information about the main and subgroups and van der Waals surface parameters Qk and the new and revised group interaction parameters for the VTPR group contribution equation of state as well as an example VLE calculation in the form of a MathCad33 sheet. This material is available free of charge via the Internet at http://pubs.acs.org.

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essential for development and optimization of refrigeration processes like, for example, the Lindeor the RESS process (rapid expansion of supercritical solutions). 27 Prediction of the Joule−Thomson inversion curve (JTIC) is of a great importance since experimental determination at extreme conditions (approximately 5 times the critical temperature and 12 times the critical pressure) is very difficult. The JTIC is defined as curvature where the Joule−Thomson coefficient μJT is equal to zero. This means it is the borderline between negative and positive JTC. Figure 21 shows the JTC and JTIC for nitrogen, CO2, and methane. It can be seen that not only the JTC but also the JTIC can be reliably predicted using the VTPR model.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +49-441-798-3831. Fax: +49-441-798-3330 E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ a

4. CONCLUSION With the presentation of the first cubic equation of state by Johannes Diderik van der Waals in 1873 it was for the first time possible to describe the PVT behavior including condensation, evaporation, and critical phenomena including supercritical phenomena using only two parameters a and b.28 Since then a lot of work was spent on the improvement of cubic equations of state especially for vapor pressures and densities. However, the applicability of equations of state using classical mixing rules was very limited: The results obtained from equations of state

anm b bnm c cnm F 3403

6. LIST OF SYMBOLS cohesive energy parameter of the Peng−Robinson equation of state, dm6 bar mol−2 temperature-independent group interaction parameter, K covolume parameter of the Peng−Robinson equation of state group interaction parameter translation parameter of the volume-translated Peng−Robinson equation of state, dm3 mol−1 group interaction parameter, K−1 objective function dx.doi.org/10.1021/ie404118f | Ind. Eng. Chem. Res. 2014, 53, 3393−3405


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cP molar heat capacity, J mol−1 K−1 g molar Gibbs energy, J mol−1 h molar enthalpy, J mol−1 h̅ partial molar enthalpy, J mol−1 Li, Mi, Ni parameters of the Twu−α-function N number of data P pressure bar Q group surface area R universal gas constant, J mol−1 K−1 S entropy, J mol−1 K−1 T absolute temperature, K v molar volume, dm3 mol−1 v̅ partial molar volume, dm3 mol−1 w weighting factor x liquid-phase mole fraction X group mole fraction z compressibility

LLE MG NRTL PSRK RESS SG SLE UCEP UNIFAC

liquid−liquid equilibrium main group nonrandom two liquid predictive Soave−Redlich−Kwong rapid expansion of a supercritical solution subgroup solid−liquid equilibrium upper critical end point universal quasichemical theory functional group activity coefficients UNIQUAC universal quasichemical theory VL vapor−liquid VLE vapor−liquid equilibrium VTPR-GCEOS volume-translated Peng−Robinson group contribution equation of state VV vapor−vapor

■

Greek Letters

αii γ ω ψ Δ ν Γ Θ ρ μJT

temperature-dependent function of a(T) activity coefficient acentric factor temperature function difference between two values stoichiometric factor group activity coefficient surface area liquid-saturated density kg m−3 Joule−Thomson coefficient, K atm−1

(1) Holderbaum, T.; Gmehling, J. PSRK: A Group-Contribution Equation of State based on UNIFAC. Fluid Phase Equilib. 1991, 70, 251. (2) Ahlers, J.; Gmehling, J. Development of an universal group contribution equation of state. I. Prediction of liquid densities for pure compounds with a volume translated Peng-Robinson equation of state. Fluid Phase Equilib. 2001, 191, 177. (3) Ahlers, J.; Gmehling, J. Development of a Universal Group Contribution Equation of State. III. Prediction of Vapor-Liquid Equilibria, Excess Enthalpies, and Activity Coefficients at Infinite Dilution with the VTPR Model. Ind. Eng. Chem. Res. 2002, 41, 5890. (4) Ahlers, J.; Yamaguchi, T.; Gmehling, J. Development of a Universal Group Contribution Equation of State. 5. Prediction of the Solubility of High-Boiling Compounds in Supercritical Gases with the Group Contribution Equation of State Volume-Translated PengRobinson. Ind. Eng. Chem. Res. 2004, 43, 6569. (5) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (6) Weidlich, U.; Gmehling, J. A Modified UNIFAC Model.1. Prediction of VLE, hE, and gamma Infinite. Ind. Eng. Chem. Res. 1987, 26, 1372. (7) Chen, J.; Fischer, K.; Gmehling, J. Modification of PSRK mixing rules and results for vapor-liquid equilibria, enthalpy of mixing and activity coefficients at infinite dilution. Fluid Phase Equilib. 2002, 200, 411. (8) Twu, C. H.; Coon, J. E.; Cunningham, J. R. A New Generalized Alpha Function for a Cubic Equation of State. Part 1. Peng-Robinson Equation. Fluid Phase Equilib. 1995, 105, 49. (9) Mathias, P. M.; Copeman, T. W. Extension of the Peng-Robinson Equation of State to Complex Mixtures: Evaluation of the Various Forms of the Local Composition Concept. Fluid Phase Equilib. 1983, 13, 91. (10) Diedrichs, A.; Rarey, J.; Gmehling, J. Prediction of Liquid Heat Capacities by the Group Contribution Equation of State VTPR. Fluid Phase Equilib. 2006, 248, 56. (11) Peneloux, A.; Rauzy, E.; Freze, R. A Consistent Correction For Redlich - Kwong - Soave Volumes. Fluid Phase Equilib. 1982, 8, 7. (12) Design Institute for Physical Properties, Sponsored by AIChE (2005; 2008; 2009; 2010; 2011; 2012). DIPPR Project 801 - Full Version. Design Institute for Physical Property Research/AIChE. (13) Rackett, H. G. Equation of State for Saturated Liquids. J. Chem. Eng. Data 1970, 15, 514. (14) Schmid, B. Application of a modern group contribution equation of state for the synthesis of thermal separation processes. Ph.D. Thesis, Carl von Ossietzky University Oldenburg, Germany, Nov 2011. (15) Schmid, B.; Gmehling, J. Revised parameters and typical results of the VTPR group contribution equation of state. Fluid Phase Equilib. 2012, 317, 110.

Subscripts

calcd c exp h i,j,k max m n P PR r res T v v x

calculated critical experimental constant enthalpy component i,j,k maximum melting number of atoms constant pressure Peng−Robinson reduced residual constant temperature vaporization constant volume constant composition

Superscripts

E (i) m, n ∞  mix

excess pure component main groups m, n infinite partial mixture

Abbreviations

AZD CFZ DIPPR GCEOS GLE HE JTC JTIC LL

REFERENCES

azeotropic data controlled freeze zone Design Institute for Physical Properties group contribution equation of state gas−liquid equilibrium excess enthalpy Joule−Thomson coefficient Joule−Thomson inversion curve liquid−liquid 3404



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(16) Haynes, W. M. Measurements of Orthobaric - Liquid Densities of Multicomponent Mixtures of LNG Components (N2, CH4, C2H6, C3H8, CH3CH(CH3)CH3, C4H10, CH3CH(CH3)C2H5, and C5H12) between 110 and 130K. J. Chem. Thermodyn. 1982, 14, 603. (17) Dortmund Data Bank, 2013, www.ddbst.com. (18) Nelder, J. A.; Mead, R. A Simplex Method for Function Minimization. Comp. J. 1965, 7, 308. (19) Schmid, B.; Gmehling, J. The universal group contribution equation of state VTPR: present status and potential for process development. Fluid Phase Equilib. 2011, 302, 213. (20) Schedemann, A. Weiterentwicklung von thermodynamischen Vorhersagemethoden als Grundlage für die Lösungsmittelauswahl bei der Flüssig-Flüssig-Extraktion und der Absorption. Ph.D. Thesis, Carl von Ossietzky University Oldenburg, Germany, March 2013. (21) Gmehling, J.; Menke, J.; Krafczyk, J.; Fischer, K. Azeotropic Data; VCH: Weinheim, 2004. (22) Holmes, A. S.; Ryan, J. S. U.S. Patent 4,318,723, Mar 9, 1982. (23) Valencia, J. A.; Denton, R. D. U.S. Patent 4,533,372, Aug 6, 1985. (24) de Hemptinne, J. C. Benzene Crystallization Risks in the LIQUEFIN Liquefied Natural Gas Process. Proc. Safety Prog. 2005, 24, 203. (25) van Konynenburg, P. H.; Scott, R. L. Critical lines and phase equilibria in binary van der Waals mixtures. Philos. Trans. R. Soc. London, Ser. A 1980, 298, 495. (26) Seader, J. D.; Henley, E. J. Separation Process Principles; Wiley & Sons: New York, 1997. (27) Hannay, J. B.; Hogarth, J. On the solubility of solids in gases. Proc. R. Soc. London 1880, 30, 178. (28) van der Waals, J. D. Over de continuiteit van den gas-en vloeistoftoestand. Ph.D. Thesis, University Leiden, Netherlands 1873. (29) Huron, M.-J.; Vidal, J. New Mixing Rules in Simple Equations of State for Representing Vapor-Liquid Equilibria of Strongly Non-Ideal Mixtures. Fluid Phase Equilib. 1979, 3, 255. (30) Michelsen, M. L. A Modified Huron-Vidal Mixing Rule for Cubic Equations of State. Fluid Phase Equilib. 1990, 60, 213. (31) Gmehling, J..; Kolbe, B.; Kleiber, M.; Rarey, J. Chemical Thermodynamics for Process Simulation; VCH: Weinheim, 2012. (32) The UNIFAC − Consortium, www.unifac.org. (33) MathSoft Mathcad, www.mathcad.com.

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Extension of the VTPR Group Contribution Equation of State: Group

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