Prediction of the Phase Behavior and Properties of Hydrocarbons with

Article pubs.acs.org/IECR

Prediction of the Phase Behavior and Properties of Hydrocarbons with a One-Parameter PC-SAFT Approach Assisted by a Group Contribution Method Renato F. Evangelista and Francisco M. Vargas* Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: Modeling thermodynamic properties can be challenging when the data availability for parameters identification is limited. Fully predictive group contribution (GC) methods have been developed as an alternative to overcome data scarcity. However, in order to provide a higher degree of accuracy, most recent GC approaches require detailed information on the molecule’s structure, which is not acquirable for systems with unspecified components. This work intends to establish the foundation to overcome this limitation. The proposed PC-SAFT approach assisted by a homosegmented group contribution scheme permits parameters calculation with accuracy due to one adjustable parameter without requiring meticulous information regarding the molecular structure, for instance, the relative position between carbon-centered groups. This semipredictive approach is especially suitable for cases in which some data are available but are not taken into consideration by current GC models. A genetic algorithm-based routine was developed to determine the group contribution parameters by simultaneously optimizing vapor pressure and saturated liquid density calculations of sixty-nine pure hydrocarbons. Further analysis indicated good predictive capabilities of the model in the extreme case where a single vapor pressure data point was provided to adjust the model parameter, with an average percentage absolute deviation of 3.79% in saturation pressure and 2.40% in saturated liquid density over 18 additional compounds that were not included in the training database. Overall, the results have shown that satisfactory predictions are possible provided a simple quantification of different carbon groups based on the type of bonds they form and a single saturation pressure point. Therefore, given that the proposed approach does not require the relative position between groups in a molecule, the method may extend the applicability of the group contribution concept to some specific industry applications.

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INTRODUCTION Modeling thermodynamic phase equilibrium is necessary for the design and development of mixtures in various industrial sectors, ranging from oil and gas exploration, chemical synthesis, material processing, to even transportation. Hydrocarbons are the focus of extensive thermodynamic modeling mostly due to the large economic impact that their derivatives have, from fuels to specialty polymers. This significant need to understand hydrocarbon behavior has driven important advances in thermodynamic modeling. In the past, due to computational limitations and their conceptual simplicity, cubic equations of state (EoS) such as Soave−Redlich−Kwong1 (SRK) and the Peng− Robinson2 (PR) equations were found to provide acceptable results to model a large variety of processes and, therefore, remained for many years as the most widely used EoS in industry. However, robust statistical mechanics-based EoS that offered improved modeling results were later formulated. Particularly the statistical associating fluid theory (SAFT), originally developed by Chapman et al. 3,4 based on Wertheim’s first-order perturbation theory5−8 has emerged as a promising approach to handle more complex systems that conventional cubic equations of state fail to accurately represent. Among several © 2017 American Chemical Society

variations of SAFT, the perturbed-chain SAFT (PC-SAFT) proposed by Gross and Sadowski9 has shown enhanced predictive potential over cubic EoS,10 especially for modeling high-molecular-weight fluids.11 The PC-SAFT has been extensively used for modeling polymers systems and petroleum fluids, with versions readily available in commercial packages such as KBC’s Multiflash,12 Calsep’s PVTsim,13 and VLXE Blend.14 For these reasons, the PC-SAFT EoS was selected as the foundation for this work. The reader is referred to the original work for detailed information and for the complete formulation of the model.9 In general, SAFT models represent a real molecule by a chain of hard spherical segments. For nonassociating compounds, the PC-SAFT EoS requires three model parameters: the number of segments in the chain (m), the spherical segment diameter (σ), and the dispersion interaction energy between segments (ε/k). These parameters can be tuned to match available experimental Received: Revised: Accepted: Published: 9227

April 12, 2017 June 18, 2017 July 20, 2017 July 20, 2017 DOI: 10.1021/acs.iecr.7b01541 Ind. Eng. Chem. Res. 2017, 56, 9227−9236

Article

Industrial & Engineering Chemistry Research

parameters of nonassociating compounds could be determined based on first-order groups (FOG) and, occasionally, to an additional second order group (SOG) term. Burgess et al.35 later extended their GC method for calculating PC-SAFT parameters of pure compounds to accurately predict phase density and derivative properties of hydrocarbons at a broader pressure range. Their hybrid approach could predict properties of normal and branched alkanes, aromatics, and cycloalkanes at temperatures to 533 K and pressures to 276 MPa. Peters et al.36,37 also proposed a homosegmented method focused on polymers, in which the PC-SAFT parameters were obtained using simple mixing rules over only five specific groups to predict not only thermodynamic properties of pure polymers but also their liquid−liquid phase equilibria with solvents. The main downside of homosegmented approaches is their inherent inability to account for fine structural differences, such as proximity effects and isomerism, without any sort of secondorder correction. These approaches have the group contribution concept employed as a mixing rule for estimating the model parameters instead of intrinsically implementing the method into the statistical associating fluid theory. More recently, heterosegmented SAFT versions11,38,39 have been developed as well as group contribution formulations founded on these models. In these versions, the model parameters are assigned to building blocks representing a number of recurring segments instead of to the molecule. Therefore, the chains consist of distinct groups, while the homosegmented approach assumes all the segments have equal segment diameter and interaction energy for a given compound. Peng et al.40 proposed a group contribution scheme with molecules described as chains formed from tangent segments to the heterosegmented SAFT-VR. As expected, the GC-SAFT-VR carries a more sophisticated formulation than the homonuclear approaches, requiring cross-interaction parameters such as the segment−segment energy range and well depth parameters, although the model is capable of accurately predicting the phase behavior of mixtures without requiring binary interaction parameters. An analogous group contribution scheme was adopted by Paduszyński and Domańska on the development of the hs-PC-SAFT.41 Alternatively, Lymperiadis et al.42 developed a group contribution scheme founded on the fused-heteronuclear united-atom representation and implemented into a SAFT-VR-based model, referred to as SAFT-γ. Besides two energy and range parameters that describe association between sites, an additional shape parameter was introduced along with unlike energy parameters between some groups in a compound. Later, Papaioannou43 et al. incorporated the Mie interaction potential between segments44 into the SAFT-γ group contribution formulation instead of the squarewell model of variable range. Unfortunately, the structure of complex molecules or hypothetical pseudocomponents is not completely well understood, and consequently, the knowledge regarding specific molecular arrangements required for current group contribution approaches is not easily achievable. Moreover, the implementation of the most sophisticated group contribution methods may require exhausting structural analysis of the analyte or partial modification of the original equation of state. Therefore, the objective of this work it to develop a simple method that permits straightforward computational implementation to existing PCSAFT algorithms without the requirement of more than conventional carbon characterization as input for the group contribution scheme. This conceptual work moves against the evolution of group contributions methods, which have become

data, most commonly the saturated liquid density and vapor pressure of the pure substances. Several researchers have obtained the PC-SAFT EoS parameters for a number of pure compounds, therefore a large database of simulation parameters is currently available.9,15 However, the estimation of parameters for compounds with limited experimental data available is still a challenge. In some cases, the required experimental data is either nearly nonexistent or it is not economically feasible to be extensively generated, thus the determination of model parameters for individual species present in a mixture is often inviable. For instance, there are thousands of hydrocarbon compounds in a given petroleum fluid. The sampling of reservoir fluids is expensive and even more so the required experimental evaluation of their thermodynamic properties. Fortunately, the minimum sets of experiments required for the parameter determination can be lowered if the thermodynamic characterization model is simplified. In addition, empirical correlations have been proposed to estimate the PC-SAFT parameters of pseudocomponents with a reduced number of tunable parameters applied to petroleum systems.16−18 Alternatively, group contribution (GC) methods have been successfully applied to SAFT parameters estimation, where the parameters can be calculated once structural information on the analyte molecule is available. Succinctly, these methods segment molecules into a number of functional groups defined based on their hypothetical contribution to a particular molecular property of interest. Though this is not a new concept, the integration of group contribution to equations of state, and more specifically to SAFT models, is fairly recent. In 2004, Vijande et al.19 pioneered this line of research on fully predictive SAFT approaches by applying the group contribution formalism to the PC-SAFT EoS in order to describe liquid density and saturation pressure of hydrofluoroethers. The approach was founded on mixing rules for parameters that represent the contribution of different homonuclear (or homosegmented) functional groups for each of the three molecular PC-SAFT parameters. These parameters were simply trained based on previously fitted PC-SAFT parameters and still provided reliable estimation of the PVT behavior for the investigated pure compounds. Later that same year, Tamouza et al.20 selected both the original SAFT4 and SAFT variable-range (SAFT-VR)21 models to implement a homosegmented group contribution method for vapor−liquid equilibrium calculations of n-alkanes, alkylbenzenes, alkyl-cyclohexanes, α-olefins, and 1-alkanols. The obtained parameters were able to yield satisfactory prediction of the vapor−liquid equilibria of binary mixtures, bearing in mind that the predictions were performed without considering binary interaction parameters22 (kij). Their work initiated a series of research efforts aimed to expand and enhance the proposed approach, such as by applying the concept to other SAFT versions,23−25 introducing more specific groups26 and including terms to account for dipole−dipole,23 quadrupole−quadrupole,26 and multipolar24,25,27−29 interactions. Other homosegmented GC methods accounting for different structural aspects or focused on particular categories of molecules were also developed over the past decade. The former concept proposed by Vijande et al.19 was recently refined to account for proximity effect without indefinitely increasing the number of functional groups.30,31 Their approach included an additional perturbation term to their original mixing rules to reflect the influence of neighboring groups given their relative positions. Tihic et al.32 combined the simplified PC-SAFT33 to the Constantinou and Gani group contribution method34 so that the PC-SAFT 9228

DOI: 10.1021/acs.iecr.7b01541 Ind. Eng. Chem. Res. 2017, 56, 9227−9236

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Figure 1. Representation of the PC-SAFT parameters calculation procedure based on group contribution theory.

more specific and input demanding over the past decade, to revisit the possibility of achieving reasonable accuracy in predicting phase equilibria with a single tunable parameter.

Table 1. Optimized Group Contribution Constants

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C CH CH2 CH3 Cunsat CHunsat CH2unsat Caro CHaro

METHODOLOGY From Molecules to Group Contribution. Group contribution methods are founded on segregating molecules into recurrent groups pursuing a generalization of similar structures’ influence on the intended molecular property. In this work, the molecular properties of interest correspond to the PC-SAFT EoS chain volume and segment dispersive energy, while the group contributions to the overall properties are quantified as constants. These constant values, along with mixing rules, can then be used to calculate group contribution parameters associated with pure fluid molecules, as will be described in the current section. In the following section, a procedure to calculate the PC-SAFT EoS parameters from the GC parameters and a single tunable parameter is explained. A representation of the concept is pictured in Figure 1. The selection of total number of groups and their nature were defined aiming for a balance between accuracy, when predicting liquid density and vapor pressure of pure compounds, and simplicity, regarding potential experimental analysis. Good balance was achieved with nine groups centered in a single carbon (united-atom) and with similar groups discriminated depending on their molecular geometry. The groups are defined as follows: two groups with aromatic structure (Caro and CHaro) and seven groups with aliphatic configurations; four with saturated configuration (C, CH, CH2, and CH3) and three comprising unsaturated carbons (Cunsat, CHunsat, and CH2unsat). Three constants for each of the nine groups defined above are required in this procedure: the group volume (VGCi), the group energy per effective superficial area (ε/kAsGCi) and the group area correction (A*GCi). The area correction was implemented to account for intrinsic steric effects because of the presence of adjacent groups, which may cause a reduction in the extent of intermolecular forces due to distortion of the electron cloud as molecules grow in size. Finally, the GC parameters for a pure compound can be calculated using the GC constants, whose values are presented in Table 1. Eqs 1−3 show the additive rules to calculate the molecule volume, segment energy, and area correction.

( ) ε kA s

VGCi (Å3)

group

3.04 7.37 12.55 17.68 3.24 10.23 17.55 6.00 10.21

GCi

(K/Å2)

40.40 27.42 14.82 7.94 34.55 20.58 8.04 23.27 15.32

A*GCi (Å2) 6.13 8.10 11.77 17.34 7.62 11.84 15.68 3.58 9.50

ng

VGC =

∑ niVGC

i

(1)

i=1

⎛ ε ⎞ ⎜ ⎟ = ⎝ kA s ⎠GC

n

∑i =g 1 ni

( ) ε kA s

GCi

n

∑i =g 1 ni

(2)

ng

A*GC =

* ∑ niAGC

i

i=1

(3)

where VGC is the GC molecule volume, (ε/kAs)GC is the GC segment energy per area, AGC * is the GC overall area correction, ni is the number of groups i present in the molecule, and ng is the total number of groups. From Group Contribution to PC-SAFT EoS. The PCSAFT EoS parameters σ and ε/k are expressed as a function of m and the GC parameters as defined in eqs 4 and 5. ⎛ 6V ⎞1/3 σ = ⎜ GC ⎟ ⎝ πm ⎠

(4)

⎛ ε ⎞ ⎛ AS − A GC * ⎞ ⎟ ε/k = ⎜ ⎟ ⎜ m ⎝ kA s ⎠GC⎝ ⎠

(5)

Where As is the cumulative surface area of m segments and can be expressed as a function of m and VGC. Therefore, eq 5 can be alternatively expressed as by eq 6. 9229


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Figure 2. Schematic of the genetic algorithm-based optimization routine, where Gen, i, and k are the generation, component, and temperature count, respectively.

⎛ ε ⎞ ⎛ 1/3⎛ VGC ⎞2/3 A* ⎞ ⎟ − GC ⎟⎟ ε/k = ⎜ ⎟ ⎜⎜π ⎜6 ⎝ m ⎠ m ⎠ ⎝ kA s ⎠GC⎝

method submits a set of solutions, represented as individuals, to a series of modifications through simulated generations (iterations). Concisely, these individuals may have their chromosomes (encoded strings) altered either by mutation or via crossover until the best individual (global optimum) satisfies the termination criteria. Overall, global optimization methods are suitable for nonsmooth and potentially discontinuous fitness (objective) functions.45 In addition, multiobjective genetic algorithms46,47 generally allow efficient implementation of parallel computing and output a set of Pareto solutions, therefore eliminating the necessity for specifying appropriate weighting

(6)

Estimation and Optimization of the GC Constants. Sixty-nine compounds, including n-alkanes, n-alkenes, cyclic compounds, branched alkanes/alkenes, polynuclear aromatics, and benzene derivatives were considered for estimating and optimizing the group contribution constants. Multiobjective genetic-algorithm optimization was selected as the base for the optimization routine due to the high nonlinearity of the problem addressed in this work. This population-based evolutionary 9230


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Table 2. PC-SAFT Parameters of Representative Pure Compounds Obtained Using the Optimized Group Contribution Constants and the Corresponding Deviations between DIPPR Data and Predicted Properties fitted aromatics

cyclic

alkanes and alkenes

calculated

AAD%

compound

CAS number

m

σ (Å)

ε/k (K)

vapor pressure

liquid density

toluene tert-butylbenzene 1-methylnaphthalene 1-phenylnaphthalene acenaphthene pyrene cyclohexane cyclooctane 1,1-dimethylcyclohexane decylcyclohexane cyclooctene 1,3-cyclohexadiene n-heptane n-hexadecane 2,2,3-trimethylbutane 2,2,4,4-tetramethylpentane 2,2-dimethyloctane 1-octene 2,3-dimethyl-1,3-butadiene

108−88−3 98−06−6 90−12−0 605−02−7 208−96−8 129−00−0 110−82−7 292−64−8 590−66−9 1795−16−0 931−88−4 592−57−4 142−82−5 544−76−3 464−06−2 1070−87−7 15869−87−1 111−66−0 513−81−5

2.75 3.30 3.45 4.03 3.10 4.07 2.61 3.58 2.87 5.61 3.34 2.72 3.87 6.43 2.98 3.45 4.00 3.62 2.79

3.73 4.03 3.90 4.11 4.08 4.02 3.81 3.77 4.07 4.09 3.80 3.59 3.65 3.97 3.99 4.10 4.02 3.85 3.75

290.1 294.4 342.7 367.6 388.9 404.0 273.2 271.9 281.3 282.5 277.5 270.1 225.8 259.1 246.4 250.7 253.0 248.1 252.7

0.13 1.57 1.24 8.87 5.59 4.17 0.99 6.71 2.00 3.28 1.08 2.49 4.16 3.10 2.42 2.62 1.95 0.73 1.17

1.04 0.94 1.68 2.27 1.86 2.73 1.29 6.25 0.62 2.98 4.66 1.03 0.22 2.45 1.79 1.99 1.38 0.84 0.84

factors. The procedure was performed in two steps: first a preliminary population was selected based on the projected σ and ε/k values for the nc species and later seeded into a second stage for optimizing the actual vapor pressure and liquid density predictions. The complete procedure, including the optimization key parameters, is schematized in Figure 2. In the first step, the GC constants were adjusted for each compound to achieve the best match between the calculated and the reference PC-SAFT EoS parameters σ and ε/k. In this work, the reference parameters correspond to the set of PC-SAFT parameters directly fitted to experimental data, as they are expected to yield the best achievable results for a model. Although the PC-SAFT parameters for the 69 compounds were readily available elsewhere9,15 even for a wider range of temperatures, they were retuned to DIPPR database48 values for vapor pressure and saturated liquid density, first, to ensure uniformity over the same source of data and, second, aiming for enhanced balance between the prediction accuracy of these two properties. DIPPR data were also used for parameter training and are referred as reference values throughout this work. The selected set of data for each component was contained between reduced temperature of 0.5 and 0.9, similar to methodologies adopted elsewhere.41,42 The list of PC-SAFT parameters obtained and used as a reference in this work is provided in the Supporting Information that accompanies this manuscript. The initial population for the genetic algorithm at this stage was created through uniform distribution within widely spaced boundaries, which shrinks along with the population size over each iteration via two types of selection: (1) natural selection, limited by the range enclosing any resulting variable value of the remaining individuals at the end of each generation (2) or by a post selection that eliminates undesired anchor points of the Pareto solutions domain (i.e., set of parameters that would yield at least one anomalous fitness/objective score and, therefore, unrealistic PC-SAFT parameters). The probability rate that an individual entry undergoes mutation into a random value selected uniformly over the intervals enclosed by the moving boundaries was 0.1. The crossover was performed by taking a

weighted average of the parents, where the weighting factors were randomly assigned at each occurrence, and the termination criteria was defined based on the average variation in the spread of Pareto solutions. Eqs 7 and 8 correspond to the average absolute percentage deviations (AAD%) for σ and ε/k averaged over all the 69 compounds and, along with their respective largest AAD% at each generation, were independently considered during the multiobjective optimization procedure. These four fitness functions were tailored to minimize the deviation between the calculated and reference parameters as well as to even out the discrepancies over the 69 compounds. y0 (1) =

y0 (2) =

nc

1 nc

∑

1 nc

∑

σcalcn − σrefn σrefn

n=1

nc

100% (7)

(ε /k)calcn − (ε /k)refn (ε /k)refn

n=1

100% (8)

The second optimization stage was performed with the same configurations but with two exceptions: the fitness functions were built around the vapor pressure (P calc * ) and the corresponding saturated liquid density (ρcalc) for each compound instead of the PC-SAFT parameters (eqs 9 and 10) and adaptive mutation was applied rather than uniform. Note that the properties were evaluated at nT points within a specific temperature range for each compound. 1 y(1) = nc

⎛ 1 ∑ ⎜⎜ n n=1 ⎝ T

1 nc

⎛ 1 ∑ ⎜⎜ n n=1 ⎝ T

y(2) =

nc

nc

Tf

∑ T = T0

Tf

∑ T = T0

⎞ ⎟100% ⎟ ⎠

(9)

⎞ * − Pref * Pcalc n,T n,T ⎟ 100% ⎟ * Pref ⎠ n,T

(10)

ρcalc

n,T

− ρref

ρref

n,T

n,T

Prioritizing robustness, the m parameters were tuned for every compound and for every individual in the population, either by 9231


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Table 3. Calculated PC-SAFT EoS Parameters of Pure Compounds Not Included in the GC Constants Training Database and the Corresponding Deviations between DIPPR Data and Predicted Properties fitted aromatics

cyclic

alkanes and alkenes

calculated

AAD%

compound

CAS number

m

σ (Å)

ε/k (K)

vapor pressure

liquid density

m-xylene o-xylene indane p-tert-butylstyrene 1-butylnaphthalene 1-hexylnaphthalene 1-decylnaphthalene cycloheptane 1,1-dimethylcyclopentane isopropylcyclopentane butylcyclopentane methylcyclopentadiene n-pentadecane n-octadecane n-eicosane 3,4-dimethylhexane 2,2-dimethylheptane 3-methyl-1-pentene

108−38−3 95−47−6 496−11−7 1746−23−2 1634−09−9 2876−53−1 26438−27−7 291−64−5 1638−26−2 3875−51−2 2040−95−1 26519−91−5 629−62−9 593−45−3 112−95−8 583−48−2 1071−26−7 760−20−3

2.82 2.92 2.65 3.80 4.11 4.71 5.76 3.13 2.51 2.91 3.35 2.77 6.14 7.01 7.59 3.49 3.67 2.62

3.91 3.86 4.03 4.10 4.07 4.10 4.18 3.77 4.07 4.04 4.01 3.54 3.95 4.01 4.04 3.93 4.01 3.93

304.6 302.5 347.7 305.6 334.0 327.6 319.3 271.9 281.0 285.3 282.3 262.2 257.3 261.9 264.2 249.2 250.2 247.4

3.72 2.28 4.32 1.03 6.32 8.23 0.71 4.26 2.73 1.07 2.50 0.92 2.58 4.77 5.77 0.64 0.45 0.21

1.30 1.12 2.91 1.55 1.17 1.91 3.55 3.90 2.63 2.18 2.74 5.47 2.46 2.75 3.44 1.61 1.53 1.06

golden section search or parabolic interpolation,49,50 and the vapor pressures were calculated using the bisection method. To minimize the computation time, a PC-SAFT version simplified for pure compounds was implemented along with parallel computing. This methodology allowed defining all nine GC parameters by simultaneously optimizing vapor pressure and liquid density calculations for the 69 compounds. Therefore, this approach can prevent possible error accumulation that may occur when the parameters are progressively identified, from simpler molecules that consist of a limited number of groups to more comprehensive structural configurations.

adjacent carbon than the corresponding group in aromatic configuration. Several optimized m values along with their corresponding σ and ε/k parameters and the resulting AAD% of both vapor pressure and liquid density for each compound are provided in Table 2. The selection of 19 compounds displayed in this table attempts to provide good representation of all considered species and, therefore, were chosen bearing in mind the diversity in structural arrangement. The values for all 69 compounds are provided in the Supporting Information that accompanies this manuscript. The overall AAD% was of 2.14% in vapor pressure and 1.60% in saturated liquid density. Maximum deviations of 8.87% in vapor pressure for 1-phenylnaphthalene and 6.25% in saturated liquid density for cyclooctane were observed. The highest deviations are most probably due to the fact that less recurrent structures in the training database but present in these compounds were less prioritized, given the averaging nature of the group contribution concept and that all groups’ constants were simultaneously optimized. Besides achieving satisfactory accuracy, considering the simplicity of the homosegmented group contribution scheme, the approach was also able to capture isomeric differences and, to some extent, compensate for proximity effects without requiring second-order corrections due to the partial adjustment of the model parameters to the training data, even though it cannot explicitly account for fine structural differences.

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RESULTS The optimal GC constants were selected by inspection of the Pareto solutions at the last generation based on the overall AAD % as well as considering the largest deviation among compounds and are reported in Table 1. Consistent and physically meaningful trends can be observed over the resulting optimized group contribution parameters. As the number of hydrogens in united-atom groups increases the normalized interaction energy monotonically decreases, whereas the volume and area correction increase. Higher values of volume and area corrections are observed for isomeric groups as they go from aliphatic to unsaturated or aromatic conformation, while the interaction energy becomes less accentuated. Clearly, the volume among groups of similar nature is incremented by the presence of additional hydrogen atoms. However, the volume of groups comprising the same atoms with distinct arrangements decreases in the presence of a double bond or aromatic assembly. This behavior agrees with the volume increment trends estimated by Slonimskii et al.51 In their work, the volume increments of repeating polymer units or atomic groups were calculated as the volume of the atom, represented as a sphere, minus the volume of segment cuts by overlapping adjacent valence-bonded atoms. Therefore, even though the length of an aromatic bond is slightly shorter than the aliphatic, groups centered in aliphatic carbons will always have an additional

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PREDICTABILITY AND SENSITIVITY ANALYSIS

In order to evaluate its predictive ability, the method was applied to compounds that were not part of the parameter training database. The additional compounds were selected comprising all the structural categories considered before and a variety of molecular sizes. The PC-SAFT EoS parameters obtained for the additional 18 compounds along with their corresponding AAD% of both vapor pressure and liquid density are provided in Table 3. The overall AAD% among the extra compounds was 2.92% in vapor pressure and 2.41% in saturated liquid density, with maximum deviations of 8.23% and 5.47%, respectively. 9232


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considered. In these three cases, the accuracy in density prediction was not significantly affected. The vapor pressure predictions obtained using a single density value in the fitting procedure were not satisfactory, even though significant improvements in density calculation were observed. Once again, the one-point case was not capable of capturing the trends for all compounds as well as when one additional data point is taken into account. Although AAD% values as high as 10.03% on 3-methyl-1pentene vapor pressure prediction and 5.47% on methylcyclopentadiene density calculation in the one-point case using only vapor pressure data were observed, the method has shown the capability of predicting the overall trend of physical properties for all pure fluids with average deviations of 3.79% and 2.40% for vapor pressure and liquid density, respectively. It is worth mentioning that all the results presented in this work are the direct output of a systematic optimization procedure, therefore no compound was addressed individually nor had their parameters exclusively fine-tuned in a case-by-case basis. The predicted values for representative compounds in this case are plotted in Figures 3, 4, and 5 along with the reference data. The 1653 vapor pressure and 1740 liquid density predicted values for the 87 compounds were plotted against their respective reference data and are shown in Figure 6. The 87 data points used for the m parameter identification were also included. Inputting only density data for the parameter training may be considered in situations that this is the most relevant property. The flexibility of the methods permitted further optimization of density predictions when saturated liquid density was solely considered in the m parameter identification process, even though the ability of predicting other properties was compromised. The method presented in this manuscript is intended to extend the applicability of the group contribution concept to cases where the information that is conventionally required is not available. For instance, a complete understanding of the molecular configuration of some complex petroleum components may not be achievable, but the quantification of different carbon groups based on the type of bonds they form may be feasible. The proposed group contribution method does not require the relative position between groups in a molecule and therefore can fulfill this specific gap of application while yet providing satisfactory predictions. Moreover, there is always a compromise between the prediction accuracy of a group contribution method and its cost in terms of input requirement, thus the most suitable method varies from case to case. Therefore, this approach is not intended to be more accurate

With the purpose of comprehending the sensitivity of the method regarding the amount of training data provided for the m parameter identification, a systematic procedure was adopted for all 18 additional compounds considered in this work and at nine different scenarios. The group contribution approach was applied inputting the data points corresponding to all considered temperatures, to the upper and lower extremities and to a single temperature at the center of the range for each compound. Besides considering both vapor pressure and saturated liquid density, each property was also independently tested in each case. The results for all scenarios are shown in Table 4. Table 4. Sensitivity Analysis of the One-Parameter PC-SAFT Approach Applied over Species Not Included in the Group Contribution Parameters Training Databasea AAD% vapor pressure fitted to both properties

vapor pressure

liquid density

all points 2 points 1 point all points 2 points 1 point all points 2 points 1 point

liquid density

mean

max

mean

max

2.918 4.257 3.243 2.914 4.164 3.786 46.803 40.158 67.036

8.225 11.157 9.484 8.227 11.237 10.026 114.340 114.340 198.450

2.405 2.484 2.369 2.409 2.486 2.398 1.189 1.351 3.710

5.474 5.407 5.470 5.466 5.408 5.470 2.921 3.405 12.423

In “both properties” case, 1 and 2 points are considered for each property.

a

Satisfactory results were observed in all cases where both properties were addressed. Naturally, the best results were achieved when all the points were considered. However, the predictions obtained using a single point for each property were more accurate than on the two-points case. The observed behavior is most likely due to the fact that the middle points are more reliable for parameter identification, as higher uncertainty might be experienced in extreme regions of the saturation curves. Remarkably, the values obtained using only vapor pressure are comparable to the results attained when including the saturated liquid density. As expected, slight improvement in vapor pressure calculation was observed when the liquid density was not considered in the m parameter identification except for the 1 point case, where the overall trend for some compounds may not have been captured as well as when an additional point is

Figure 3. Vapor pressure and saturated liquid density of aromatics. The red lines correspond to predicted values and the open circles to reference data. The points used for the m parameter fitting are highlighted (solid black circles). m-Xylene (m-X), Indane (ind.), p-tert-butylstyrene (ptb-sty), and 1butylnaphthalene (1b-naph) were selected as representative molecules. 9233


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Figure 4. Vapor pressure and saturated liquid density of alkanes and alkenes. The red lines correspond to predicted values and the open circles to reference data. The points used for the m parameter fitting are highlighted (solid black circles). n-Pentadecane (n-c15), n-eicosane (n-c20), 2-2dimethylheptane (2dm-c7), and 3-methyl-1-pentene (3m-1c5) were selected as representative molecules.

Figure 5. Vapor pressure and saturated liquid density for cyclic compounds. The red lines correspond to predicted values and the open circles to reference data. The points used for the m parameter fitting are highlighted (solid black circles). 1-1-Dimethylcyclopentane (1dmc-c5), isopropylcyclopentane (ip-c5), and butylcyclopentane (bc-c5) were selected as representative molecules.

Figure 6. Predicted vapor pressure and saturated liquid density plotted against the respective reference data for all 87 compounds. The open red circles are the predicted values and the open black circles are the points used for the m parameter identification.

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than other currently available methods. Given the simple formulation of the proposed method, the predictions may be even less accurate than group contribution methods that individually address a larger number of molecular structures and functional groups. For these reasons, any comparison performed here would be either merely speculative or an attempt to trace a comparison among distinct scenarios. Besides, a fair comparison to group contribution methods would be hardly achieved due to several possible reasons: (1) the other method is implemented to a different variation of SAFT; (2) it was developed for specific hydrocarbon family series; (3) it is focused on polymer systems applications; (4) it either requires very refined molecular structure information or it was built over a much larger number of groups; (5) methods differ in optimum temperature and pressure ranges of application; and (6) the method has additional limitations such as the inability to distinguish isomers.

CONCLUSIONS

The proposed method has shown promising predictive potential by satisfactorily calculating vapor pressures and saturated liquid densities at various temperatures for pure hydrocarbons comprising a diverse collection of molecular structures. A single vapor pressure data point is needed as input to the model. The predictions can still be significantly improved when a larger number of reliable vapor pressure data points are provided for the model parameter tuning or when liquid density data is also taken into consideration in the fitting process. Therefore, the proposed method may be a preferred alternative over models that involve a large number of adjustable parameters and do not provide the same degree of accuracy or that are restricted to specific families of molecules. This method may also be advantageous over fully predictive group contribution approaches when some experimental data are available or in scenarios where these approaches are not applicable, as the proposed concept does not require thorough molecular structure identification and still holds 9234


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Industrial & Engineering Chemistry Research satisfactory robustness. Due to a flexible adjustable parameter, the method is, for instance, capable of distinguishing between two isomers without requiring information regarding the relative position of carbon-centered groups. Besides, the method does not alter the original PC-SAFT formulation and can be easily implemented with available commercial thermodynamic modeling packages. With a flexible yet predictive model, this work aims to provide the foundation to predict the phase behavior of fluids with very limited data.

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Abbreviations

AAD% DIPPR EoS GC PC-SAFT m-X ind ptb-sty 1b-naph n-c15 n-c20 2dm-c7 3m-1c5 1dmc-c5 ip-c5 bc-c5

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b01541. Tables S1−S8: PC-SAFT parameters fitted via standard procedure and obtained using the optimized group contribution constants (PDF)

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AUTHOR INFORMATION

*Address: 6100 Main MS-362, Houston, Texas 77005-1827, United States. Tel.: +1 (713) 348-2384. E-mail: fvargas@rice. edu. ORCID

Renato F. Evangelista: 0000-0002-2942-016X Francisco M. Vargas: 0000-0001-5686-5140 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Caleb Sisco for providing the thermodynamic modeling code and to Miguel Garcia for helpful discussions. R.E. thanks CAPES Foundation for the financial support (Grant 18644/12-0).

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NOMENCLATURE m number of segments (PC-SAFT EoS) VGC GC molecule volume, Å3 A*GC GC area correction, Å2 ni number of groups i present in the molecule ng total number of groups i component count (optimization context) k temperature count (optimization context) Gen generation count (optimization context) nc total number of compounds nT total number of temperatures T temperature, K P*, Psat vapor pressure, bar Greek Symbols

spherical segment diameter (PC-SAFT EoS), Å dispersion interaction energy (PC-SAFT EoS), K GC segment energy per area, K/Å2 saturated liquid density, kg/m3

Subscripts & Superscripts

aro unsat i n T calc ref

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Corresponding Author

σ ε/k (ε/kAs)GC ρ

percentage average absolute deviation Design Institute for Physical Properties (database) equation of state group contribution perturbed-chain statistical associating fluid theory m-xylene indane p-tert-butylstyrene 1-butylnaphthalene n-pentadecane n-eicosane 2-2-dimethylheptane 3-methyl-1-pentene 1-1-dimethylcyclopentane isopropylcyclopentane butylcyclopentane

aromatic unsaturated group compound temperature calculated reference 9235


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Prediction of the Phase Behavior and Properties of Hydrocarbons with

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