Application of the Simplified Perturbed-Chain SAFT to Hydrocarbon

May 12, 2009 - A new group contribution (GC) method has been developed to estimate parameters of the simplified perturbed-chain statistical associatin...
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Ind. Eng. Chem. Res. 2009, 48, 5867–5873

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Application of the Simplified Perturbed-Chain SAFT to Hydrocarbon Systems with New Group-Contribution Parameters Zhi-Yong Zeng,†,‡ Yuan-Yuan Xu,† Xu Hao,† and Yong-Wang Li*,† State Key Laboratory of Coal ConVersion, Institute of Coal Chemistry, Chinese Academy of Sciences, Taiyuan 030001, P.R. China, and Graduate UniVersity of Chinese Academy of Sciences, Beijing 100049, P.R. China

A new group contribution (GC) method has been developed to estimate parameters of the simplified perturbedchain statistical associating fluid theory (sPC-SAFT) for hydrocarbon. A key advantage of this method is that the binary interaction parameter between groups has been adopted in estimating the molecular parameter of pure component. Besides, parameters of all single groups are set to be the same for simplifying the model. Using genetic algorithm, twelve group interaction parameters are estimated on the basis of 71 sets of pure components of sPC-SAFT parameters. Tests and comparisons with other methods were performed in calculating the PVT and phase equilibrium data of heavy and branched hydrocarbon systems. The results show that the present GC method is better than the other predictive approaches in calculating parameters of sPC-SAFT. 1. Introduction Thermodynamic properties are necessary for the design of chemical and petrochemical industrial processes. Although the properties of hydrocarbons are studied continually, the experimental PVT and phase equilibrium data of heavy and branched alkanes are still scarce. Even the data of linear alkanes are available mainly for small molecules (up to and including decane) and for branched alkanes, the limited data are mainly accessible to isomers of nonane.1,2 However, the experimental data of heavy and branched alkanes are important substances in chemical and petrochemical industries.3 For instance, many kinds of heavy and branched alkanes exist in the products of Fischer-Tropsch synthesis (FTs),4 the computation of PVT and phase equilibrium data are required in various unit operations of chemical engineering. Therefore, the precise knowledge of phase equilibrium property is vital for designing and optimizing the separation and processing technologies. Since experimental measurements are costly and timeconsuming, reliable and efficient models play an important role in the description of the thermodynamic properties. Generally, the properties of each substance depend directly on the nature of its molecule. Therefore, a complete understanding of molecular behavior is necessary for predicting the physical properties of fluids. The molecule-based equations of state (EoSs) have advanced the molecular models greatly in recent years. Among them, a family of equations of state known as statistical association fluid theory (SAFT)5-7 is promising. These models not only provide a useful thermodynamic basis for deriving chemical potentials or fugacities that are needed for phase equilibrium simulations but also allow for recognizing and quantifying the effect of molecular structure and interactions on bulk properties and phase behavior. One of the most successful modifications is the perturbed-chain SAFT (PCSAFT) EoS proposed by Gross and Sadowski8 in 2001, with its simplified version proposed later by von Solms et al.9 The PC-SAFT adopts a more realistic dispersion term accounting * To whom correspondence should be addressed. E-mail: [email protected]. † State Key Laboratory of Coal Conversion, Institute of Coal Chemistry. Chinese Academy of Sciences. ‡ Graduate University of Chinese Academy of Sciences.

for dispersive interaction between chains. Both the original and simplified PC-SAFT (sPC-SAFT) have been applied in many complex systems,10-16 such as vapor-liquid equilibria (VLE) and liquid-liquid equilibria (LLE) of polymers,10-12 alkanols,13,15 and water16 systems. In the sPC-SAFT EoS, a set of three purecomponent parameters is required for nonassociative compounds, namely, the segment number, m, the segment diameter, σ, and the segment energy, ε/k. Although the sPC-SAFT has been studied plentifully, the lack of reliable pure-component parameters is still an obstacle to extend the model in more practical applications. The typical method8 is by fitting vapor pressure and liquid-density data, simultaneously. For these substances that are difficult or even impossible to measure the data of vapor pressure, some researchers11,17 obtained the parameters by fitting both the experimental PVT and binary phase equilibrium data. Unfortunately, experimental data are scarce and the substance number is too large, and the parameters obtained in this way are often not unique for different sources of experimental data.18 Thus, it is imperative to develop a more efficient, convenient, and predictive method for obtaining the pure-component parameters. One of the possible solutions9,13,19 is the extrapolation method, which extrapolates the parameters from lower-molecular-weight compounds to the similar unknown or high-molecular-weight compounds. This approach is often used widely for homogeneous series, the parameters of m, mσ3, and m(ε/k) increase linearly with molar mass increasing. The parameters of several homologous series have been studied in the literature, such as n-alkanes9 and 1-alcohols,13 and promising results had been obtained for the studied systems. However, it is difficult to obtain parameters for the unusual compounds, such as branched and heavy hydrocarbons. The other suggested solution20-22 is the group contribution (GC) method. This method has been adopted for computing many thermodynamic properties and satisfactory results have been yield achieved for many systems.1,23-27 In the SAFT-type models, GC method has also been suggested in these years. Tamouza et al.28 have obtained parameters of six groups for the original SAFT and the SAFT variable range (SAFT-VR), and they were applied for calculating vapor-liquid equilibrium (VLE) of five hydrocarbon families. Lymperiadis et al.22 have developed a predictive group-contribution SAFT-γ by extending SAFT-VR EoS, in which each functional group is modeled as

10.1021/ie8019246 CCC: $40.75  2009 American Chemical Society Published on Web 05/12/2009

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a united-atom spherical square-well segment; thus, the different groups are characterized by size diameter, energy well depth and range parameters representing the dispersive interaction, and by shape factor parameters. Tihic et al.20,21 fitted parameters of 200 new compounds for sPC-SAFT EoS and then estimated parameters of 45 kinds of first-order groups (FOG) and 26 kinds of second-order groups (SOG) by combining the ConstantinouGani group contribution method29 with the sPC-SAFT EOS. The results are satisfactory for the studied polymers. The aim of this work is to apply sPC-SAFT in calculating PVT and phase equilibrium data of heavy and branched hydrocarbon systems with a reliable, convenient, and efficient parameter-estimating method. For this purpose, a new GC method was developed to obtain the characteristic parameters of sPC-SAFT EoS. Then, the PVT and binary phase equilibrium data of heavy and branched alkane systems were calculated by sPC-SAFT with the new GC parameters. Simultaneously, comparisons were carried out between the proposed GC model and other models in the literature. 2. Model and Assumptions The sPC-SAFT EoS has a clear physical molecular model, which assumes that the molecule is composed of chains of freely jointed spherical segments. Several intermolecular forces are considered, and they are divided into different contributions, correspondingly.8 For nonassociating compounds, it consists of an ideal gas contribution, a hard-chain contribution, and a perturbation contribution, which accounts for the attractive interactions. The equation of state in the expression of compressibility factor is given as Z ) Z id + Z hc + Z disp

(1)

where Z denotes the compressibility factor and Z ) 1. More details of sPC-SAFT EoS are described clearly in the original literature works.8,9 For the sPC-SAFT EoS, combining rules are required by segment diameter and energy parameter in calculating mixture systems. The Lorentz-Berthelot rules are employed in this study, and its expressions are shown in eqs 2 and 3. id

εij ) √εiiεij(1 - kij) σii + σjj σij ) 2

(2)

tively, which are the approximately average values of 71 pure-compounds.8,9,20,30,31It should be noticed that deviations from the factual values would be compensated by the group interaction parameters. For example, parameter of CH3 - CH2 ) kCH3-CH2 × (parameter of CH3 + parameter of CH2) where, kCH3-CH2 is an adjustable coefficient and could be regressed from the experimental data. Thus, no matter how to determine the parameters of CH3 and CH2, it would not affect the parameter of CH3-CH2. Second, this method considers the contributions of the nearest neighbors, only. Third, the interaction parameters of the binary groups, no matter where they are in the molecule, possess the same parameters. These measurements and assumptions could not only ensure that the model is reasonable and the result is accurate but also decrease the number of parameters. Assuming that a given molecule is assembled by nt single groups, which contains N kinds of binary groups, and there are ni for each kind of binary group. Then, it could be expressed as. N

nt )

∑n

i

+1

(4)

i)1

The parameter of a single group has been normalized in the first assumption. Then, the parameters of binary groups should be multiplied by a factor two, namely 2, 3.9, and 250 for segment number, diameter, and energy, respectively. In the new group contribution method, an interaction coefficient is used for each parameter of a binary group. Thus, parameters of each g g g , 3.9km,i , 250km,i for the sPC-SAFT binary group turn out as 2km,i EoS, and the molecular parameter is obtained by adding parameters of all the binary groups one by one. For the segment g . It should be noticed number, the accumulated result is 2∑i nikm,i that nt - 2 units (the segment number of each single group has been set to 1 unit) have been counted twice in the binary groups; thus, this part should be reduced. Then, it could be expressed as N

(3)

where kij denotes the binary molecular interaction parameter. Considering that molecule is treated as a chain of segments in the sPC-SAFT model, the group contribution method is easy to be adapted. The GC method, adopted by Tihic et al.21 is based on the Constantinou-Gani group contribution method,29 which uses the simple first-order groups (FOG) as the basic level and second-order groups (SOG) as the higher level, the contribution of SOG is expected to reduce the deviations caused by ignoring the effect of the neighboring groups. In this work, a new GC model is proposed to estimate the pure-component parameters of sPC-SAFT, which attaches importance to the neighbor effect directly. The single group is defined as one of the elements C, N, O, and S with its bonded atoms, such as CH3. The binary group is defined as two single groups directly linked with one bond, such as CH3-CH2, and the special group interaction parameter is used for each property of a binary group. Moreover, three simple assumptions were made for simplifying the model: First, every single group has the same segment parameters, and its values are set to 1, 1.95 (Å), and 125 (K) for segment number, diameter, and interaction energy parameters, respec-

mmolecule ) 2

∑nk

g i m,i

- (nt - 2)

(5)

i)1

For the segment diameter and energy parameters, they could be expressed as N

σmolecule ) 3.9

∑nk

g i σ,i /nt

(6)

i)1

N

(ε/k)molecule ) 250

∑nk

g i ε/k,i /nt

(7)

i)1

g g g where km,i , kσ,i , and kε/k,i are the temperature-independent group interaction parameters for m, σ, and ε/k, respectively. In this work, parameters fitted from experimental data are taken as reliable ones. Thus, they were taken as reference values in the regression process. The parameters of pure compounds8,9,20,30,31 were used to regress all group interaction parameters with genetic algorithm (GA)32 in this study. Table 1 shows the results of group interaction parameters for different binary groups, and Table 2 presents the standard deviations (SD), the average deviations (AAD), and the average absolute percent deviations (AAPD). Figure 1 shows the molecule weight (MW)

Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009 Table 1. Group Interaction Parameters of Different Binary Groups for sPC-SAFT

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Table 3. AAPD between the Calculated and Experimental Vapor Pressure of Different Hydrocarbons

interaction parameters

AAPD%

data model 1 model 2a model 321 model 49 source 8

binary group

g km,i

g kσ,i

g kε/k,i

total no. occurrences

compounds

CH3sCH2 CH2sCH2 CH3sCH CH2sCH CHsCH CH3sC CH2sC CHsC CH2dCH CH2dC CHtC CtC

0.751265 0.67777 0.709178 0.620360 0.534979 0.651035 0.666652 0.516002 0.773943 0.524163 0.922072 0.629156

1.367125 1.037853 1.296729 0.955237 0.869011 1.274564 0.824969 0.92353 1.406445 1.632903 1.115119 1.325473

1.241908 1.064812 1.16711 1.056809 1.164682 1.261236 0.791241 1.019675 1.246933 1.829789 1.200265 2.088711

82 349 45 24 10 50 20 7 12 6 7 4

n-hexadecane n-eicosane n-docosane n-tetracosane n-octacosane n-triacontane n-hexatriacontane n-octatriacontane 2-methylheptane 2,3-dimethylhexane 2,2,3-trimethylpentane average

1.88 1.98 0.320 0.920 2.420 19.430 14.120

5.8

2.0 3.3 5.2 6.7 14.0 7.5 17.8 20.5 2.4 6.7 6.8 8.5

3.0 11.0 17.9 23.9 36.4 47.1 50.9 55.1 15.2 32.3 18.3 28.3

3.3 12.8 18.2 23.0 34.3 40.2 47.3 51.4

33 33 33 33 33 34 35 35 36 36 36

28.8

a

Table 2. Statistical Results with the New GC Method parameter

Npt

SD%a

AAD%a

AAPD%a

m σ ε/k

71 71 71

0.16 0.07 5.64

0.09 0.03 3.26

2.41 0.85 1.31

SD ) [∑(Xest - Xexp)2/(Npt - 1)]1/2; AAD ) (1/Npt)∑|Xest - Xexp|; AAPD ) (1/Npt)∑(|Xest - Xexp|/Xexp). a

The parameters are estimated by the proposed GC method in this work. Table 4. AAPD between Experimental37 and Calculated Liquid Density by Different Models AAPD/Fliq compounds

T range (K) model 2

2-methyloctane 2-methylnonane 2-methylundecane 2-methyltridecane 2-methylpentadecane 2,2-dimethylheptane 2,2-dimethyloctane 2,2-dimethyldecane 2,2-dimethyldodecane 2,2-dimethylbutadecane average

442-512 400-530 490-580 480-600 580-670 450-525 475-525 490-580 530-610 550-640

a

model 321 model 538

0.4 1.7 2.3 2.3 0.5 1.6 1.5 3.2 4.2 0.2 1.8

0.4 1.2 2.0 2.3 0.4 10.4 10.5 7.8 5.6 2.2 4.3

3.1 3.6 7.7 11.4 20.4 6.1 5.6 9.6 13.2 6.7 8.7

a The parameters are estimated by the proposed GC method in this work.

Figure 1. Molecule weight versus the ratio of PGC/Plit.8,9,20,30,31 The 9, •, and 2 symbols represent the parameter of segment number, segment diameter, and segment energy, respectively.

versus the ratio of the parameters computed by group contribution method (PGC) to ones provided by literatures (Plit).8,9,20,30,31 The result indicates that most of ratios are very close to 1, which means the regressed parameters are coincident well with ones provided by literatures. Thus, the regressed parameters are reliable. Moreover, it is noticed that the group interaction parameters are different obviously for the different bonds between the same single groups. Taking CH2 and CH as examples, when they are linked with the single bond and double bond, the group interaction parameters of segment number result in 0.620360 and 0.773943, respectively. It is revealed that the bond effect is very important. On the basis of eqs 4-7 and Table 1, the sPC-SAFT parameters of pure-components are able to be calculated. The detailed example of applying the new GC method is given in the Appendix. 3. Results and Discussion In this section, the above proposed GC method was tested in calculating the PVT and phase equilibria of heavy and branched

alkanes. In the following parts, sPC-SAFT with the parameter, fitted from experimental data of vapor pressure and liquid density, was called as the fitted-parameter model (model 1) and detailed values could be found in the literature.8,9,20,30,31 The sPC-SAFT with the GC parameter, provided in this work, was noted as the proposed GC model (model 2), sPC-SAFT with the GC parameter, provided by Tihic et al.,21 was named as the original GC model (model 3), and detailed information is shown clearly in the original paper.19 sPC-SAFT with the correlated parameter, which was provided by Solms et al.9 and shown in eqs 8-10, was labeled as the correlated-parameter model (model 4), which holds true only for normal alkanes. m ) 0.02537MW + 0.9081

(8)

mε ) 6.918MW + 127.3 k

(9)

mσ3 ) 1.732MW + 18.44 (10) 3.1. PVT of Heavy and Branched Alkanes. First, the sPCSAFT model was used to calculate vapor pressure and liquid density of different heavy and branched alkanes with different sources of parameters. The results are shown in Tables 3 and 4. It is obvious that the proposed GC model is able to reproduce the experimental data of vapor pressure and liquid density over a wide range of temperature. The AAPD are 5.8%, 8.5%, 28.3%, and 28.8% for the fitted-parameter model, the proposed GC model, the original GC model, and the correlated-parameter model, respectively. Thus, the proposed GC model is better than the original GC model and the correlated-parameter model in calculating the vapor pressure of the considered heavy and

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Figure 2. Experimental excess volumes of n-pentane + 2,6-dimethylheptane39 versus the calculated ones with the proposed GC model: (9) experimental data. The solid line represents the calculated data.

branched n-alkanes. Deservedly, the fitted-parameter method got the best result. Moreover, it could be observed that the deviations increase when the carbon number (Cn) of alkanes increases. There are two possible reasons: One is that this effect is caused by the computed parameters deviating from the real values, because the contributions from the nonlinked groups effects the group interaction parameter for the long-chain hydrocarbons. The other is that the vapor pressure of heavy alkane is very small, so it is hard to measure the precise value, which is also confirmed by the fact that the fitted-parameter model has similar AAD increments with an increasing Cn, too. The liquid densities of several branched alkanes were calculated with different models, and the total AAPD are 2.4%, 4.2%, and 9.2% for the proposed GC model, original GC model, and Peng-Robinson EoS (PR EoS, model 5), respectively. The detailed information is shown in Table 4. This result indicates that the proposed GC model is better in calculating liquid densities of considered branched hydrocarbons. Moreover, it shows that the AAPD of the proposed GC model are 1.4% and 3.4% for the one-branch and two-branch alkanes, respectively. While for the original GC model, they are 1.2% and 7.3%. Combining with the results of branched alkanes in Table 3, it is suggested that the branch number of alkanes would affect the group interaction parameters and consequently cause larger deviations from the experimental data.37 Next, the liquid densities and excess volumes of mixtures, including heavy and branched alkanes, were calculated with the proposed GC model. Figure 2 shows the calculated and experimental excess volume of systems, which contain normal alkanes and branched alkanes.39 The AAPD are 5.6% for pentane + 2,6-dimethylheptane with trivial interaction parameters. Figure 3 shows the calculated and experimental densities of n-hexadecane + 2,2,4-trimethylpentane,40 the AAPD is 0.5% with kij ) 0.0028. It should be noticed that all the nonzero kij values are regressed from the experimental data in this work. Therefore, the proposed GC model could satisfactorily reproduce densities of systems including heavy and branched alkane systems. 3.2. Phase Equilibria of Heavy and Branched Alkanes. In this part, vapor-liquid equilibrium calculations are performed with sPC-SAFT EoS for several binary systems. Figures 4-6 show the experimental and calculated vapor-liquid equilibria of three binary systems at different temperature conditions. It is clear that the proposed GC model is able to calculate the

Figure 3. Experimental densities of n-hexadecane + 2,2,4-trimethylpentane40 versus the calculated ones with the proposed GC model: (9) experimental data. The solid line represents the calculated data.

Figure 4. VLE for the system n-hexane + squalane. Experimental data are from the work of Joyce et al.41 The symbols 9 and • represent the experimental molar fraction of suqalane at T ) 574.6 and 623.3 K, respectively. The solid lines denote calculated molar fraction of suqalane with the proposed GC model at T ) 574.6 and 623.3 K, respectively.

experimental data accurately, even if the interaction parameter is set to zero. Then, the effect of the interaction parameter was further studied and the result of methane + n-hexatriacontane is shown in Figure 7. It could be concluded that the interaction parameter is important for the system with high degree of asymmetry. Moreover, the temperature effect on the molecular interaction parameter is studied, and the result is shown in Figure 8; the molecular interaction parameters increase with the increasing temperature for n-hexane + n-hexadecane and methane + n-hexatriacontane systems. 4. Conclusions In this work, a new GC method is developed for estimating parameters of sPC-SAFT for hydrocarbon, which requires the information of molecular structure, only. In this method, the parameters of each single group are assumed to be the same, and efforts were made on optimizing the group interaction parameters. A group interaction parameter table has been derived with small average deviations from 71 sets of pure-component parameters. The data of PVT and phase equilibria were calculated with sPC-SAFT EoS for heavy and branched alkane systems, and

Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009

Figure 5. VLE for the system ethane + n-hexatriacontane. Experimental data are from the work of Chao et al.42 The symbols 9, •, and 2 represent the experimental molar fraction of ethane at T ) 373.15, 473.05, and 573.05 K, respectively. The solid lines denote the calculated molar fraction of ethane with the proposed GC model.

Figure 6. VLE for the system n-hexane + n-hexadecane. Experimental data are from the work of Joyce et al.43 The symbols 9, •, and 2 represent the experimental molar fraction of n-hexadecane at T ) 572.5, 524.3, and 472.1 K, respectively. The solid lines denote calculated molar fraction of n-hexadecane with the proposed GC model.

the promising results confirmed the proposed GC method is reliable and accurate. Moreover, the proposed GC model is more accurate than the original GC model and the correlatedparameter model in calculating the data of vapor pressure and liquid density for the considered systems. Therefore, the proposed GC method is reliable in estimating parameters of sPC-SAFT for hydrocarbon when the experimental data are insufficient, which would make for the wider applications of sPC-SAFT in calculating various thermodynamic properties. Acknowledgment The authors gratefully acknowledge the financial support from the Key Program of the National Outstanding Young Scientists Foundation of China under Grant No. 20625620, National Natural Science Foundation of China under Grant No. 20590361, and the Knowledge Innovation Program of the Chinese Academy of Sciences Grant No. 2007YQNRC18. This work is also supported by Synfuels CHINA. Co., Ltd.

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Figure 7. Effect of the molecular interaction parameter to the VLE calculation with the proposed GC model. The symbol 9 represents the experimental data.42 The solid and dashed lines represent the calculated data at kij ) 0.083 and 0.00, respectively.

Figure 8. Temperature effect to the molecular interaction parameters. The symbol 9 represents the methane + n-hexatriacontane system,42 and • represents the n-hexane + n-hexadecane system.43 Table 5 binary groups CH3-C CH2-CH2 CH3-CH2 CH2-C total value

ni

g km,i

g kσ,i

g kε/k,i

3 0.651035 1.274564 1.261236 3 0.67777 1.037853 1.064812 1 0.751265 1.367125 1.241908 1 0.666652 0.824969 0.791241 g g 8 ∑nikm,i ) 5.404332 ∑nikσ,i ) 9.129345 ∑nikgε/k,i ) 9.011293

Appendix Here is presented the typical example of the application of the proposed group-contribution (GC) method for calculating the PC-SAFT parameters for the alkanes. Taking 2,2-dimethylheptane as an example, the molecular structure is (CH3)3-C-CH2-CH2-CH2-CH2-CH3 Then, on the basis of the proposed GC method, this molecule could be divided into eight binary groups, and the detailed information is shown in Table 5. The PC-SAFT parameters could be calculated using eqs 1-4, and the expressions are shown here: nt ) 8 + 1 ) 9

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m ) 2 × 5.404332 - 9 + 2 ) 3.808664 ´

σ ) 3.9 × 9.129345/9 ) 3.956050 Å ε/k ) 250 × 9.011293/9 ) 250.31700 K With this parameter set, the PC-SAFT EOS calculated liquid densities deviate from the experimental data with an average absolute deviation 1.59% in the temperature range 450525 K. List of Symbols kij ) molecular interaction parameter g km,i ) group interaction parameter of segment number g kσ,i ) group interaction parameter of segment diameter g kε/k,i ) group interaction parameter of segment energy Cn ) linear or branched alkane containing n carbon atoms MW ) molecular weight nt ) total number of groups T ) temperature P ) pressure Z ) compressibility factor m ) the segment number parameter of the PC-SAFT EOS k ) Boltzmann constant ≈ 6.023 × 1023 Npt ) number of data points x ) molar frction Greek Letters ε/k ) the segment energy parameter of the PC-SAFT EOS σ ) the segment diameter parameter of the PC-SAFT EOS ω ) acentric factor F ) density Superscripts g ) group hc ) hard chain disp ) dispersion id ) ideal sat ) saturated Subscripts c ) critical i,j ) group or molecule i,j est ) estimated exp ) experimental

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ReceiVed for reView December 15, 2008 ReVised manuscript receiVed April 5, 2009 Accepted April 13, 2009 IE8019246