Correlation to predict solubility of hydrogen and carbon monoxide in heavy paraffins

Fluid Phase Equilibria 320 (2012) 11–25

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Correlation to predict solubility of hydrogen and carbon monoxide in heavy paraffins Seethamraju Srinivas a , Randall P. Field a,∗ , Suphat Watanasiri b , Howard J. Herzog a a b

MIT Energy Initiative, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA Aspen Technology Inc., 200 Wheeler Rd., Burlington, MA 01803, USA

a r t i c l e

i n f o

Article history: Received 26 October 2011 Received in revised form 9 February 2012 Accepted 10 February 2012 Available online 18 February 2012 Keywords: Modeling Solubility Hydrocracker Fischer–Tropsch synthesis Peng–Robinson equation of state

a b s t r a c t The Fischer–Tropsch (FT) reactor and hydrocracker, which are important elements of a synthetic liquid fuel process, are multi-phase reactors having gas–solid–liquid reactions involving H2 and/or CO. To predict the reactor performance, it is therefore important for simulation models used in the technoeconomic or feasibility studies of such processes to accurately capture the solubility of H2 and CO in the reactant–product mixture. This poses challenges in terms of the asymmetric nature of the mixture and accurate characterization of the components involved. Using solubility experimental data and validated component characterization equations from literature, correlations to determine the Peng–Robinson binary interaction parameters and hence, predict the solubility of H2 and CO are presented. These binary interaction parameter correlations are expressed as a function of the solvent carbon number and temperature. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Processes for conversion of low-value feed stocks like coal, biomass, residual oil, etc., into liquid fuels are gaining renewed interest in the current energy situation. The indirect liquefaction process involving Fischer–Tropsch (FT) synthesis converts these raw materials into synthetic liquid fuels via gasification. FT synthesis is a heterogeneous catalytic process that converts syngas (a mixture of CO and H2 ) predominantly to n-paraffins as represented in (1). nCO + (2n + 1)H2 → Cn H2n+2 + nH2 O

(1)

This polymerization reaction can generate a product distribution with the carbon number (n) varying from 1 to 70, or more. The FT raw product mixture is quite asymmetric in nature with respect to the size and the nature of the components present with components ranging from light gas (H2 , CO, etc.) to heavy paraffins. Such a mixture provides challenges in obtaining phase equilibrium experimental data and in predicting phase behavior from existing thermodynamic models. This motivates us to derive a correlation to predict the solubility of H2 and CO in the heavy paraffin solvents with an extrapolative capability. After a short review on the importance of solubility, a discussion on the property methods considered with some comments on water-handling is presented. It

∗ Corresponding author. Tel.: +1 617 324 2391. E-mail address: rpfi[email protected] (R.P. Field). 0378-3812/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2012.02.008

then addresses the issue of estimation of critical parameters and their role in solubility prediction. The methodology used in arriving at the binary interaction parameters which in turn are used to develop the solubility correlation is described later. The final sections present the results and their validation. Investigations by different researchers on Fischer–Tropsch synthesis in slurry reactors led to the conclusion that solubility of syngas components (H2 and CO) is needed in understanding the reaction rate and selectivity since solubility plays a role in the synthesis chemistry (e.g., Satterfield and Stenger [1]) as well as the hydrodynamics (e.g., Quicker and Deckwer [2]). The solubility information is also valuable in the design and operation of the reactors. Caldwell and van Vuuren [3] noted the importance of vapor–liquid equilibria (VLE) in FT process, and derived a criterion for prediction of maximum operating temperature for slurry systems, based on Anderson–Schulz–Flory product distribution. It is important to note that the solubility of H2 and CO increases with increase in temperature, an unusual behavior as compared to other gases. This temperature dependence must be captured well by the chosen thermodynamic model. The solubility of H2 and CO also increases with solvent molecular weight or carbon number. And, as expected, solubility increases with pressure. Further, there is no correlation between the solubility of these two gases. While CO is more soluble than H2 , the sensitivity towards temperature is higher for H2 [4]. In establishing a solubility correlation for H2 and CO in the heavy paraffins, Wang et al. [5] point out the two major challenges. The first challenge is the inherent asymmetric nature of the components involved in these mixtures which makes

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S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

• Both PR and SRK EoS were developed for lighter components in their standard forms. Despite improvements to account for the size and shape of the molecules (ω), they do not reproduce the vapor pressure of heavy n-paraffins due to the absence of accurate critical properties. • The SRK EoS over-predicts the vapor pressure of n-octacosane by ∼30% in the range of 475–500 K when extrapolated values for the critical properties are used.

Nomenclature Avg. Devn. average deviation Max. Devn. maximum deviation RMSE root mean square error average absolute deviation AAD Y physical property correlated ˇ,  correlating parameters carbon number n Subscripts 0 at effective carbon number of zero ∞ as carbon number approaches infinity

the cubic equation of state (EoS) limited in its scope of application, and is further discussed in Sections 2 and 3. The second concerns the characterization of the Fischer–Tropsch wax which is a complex mixture of paraffins that is normally modeled as a lumped component based on carbon number. In short, the liquid phase properties need to be defined. Thus, correlations to predict critical and physical properties based on carbon number are needed as discussed in Section 4.

2. Choice of property method We restrict ourselves here to the use of cubic equations of state (EoS) like Soave–Redlich–Kwong (SRK) [6] and Peng–Robinson (PR) [7] because they are widely used in engineering calculations for high pressure VLE calculations. Predictions from cubic EoS depend on two major factors—critical constants and mixing rules. Tsonopoulos and Heidman [8] compare SRK, PR and CCOR (cubic chain-of-rotators) EoS concluding that the vapor pressure predictions are similar for all of them. The liquid density predictions improve as we go from SRK to PR to CCOR due to the increasing complexity of the volume dependence term. The binary interaction parameter (BIP) used in these EoS is close to zero for most hydrocarbon–hydrocarbon binaries. However, the BIPs must be non-zero when considering highly asymmetric mixtures like H2 or CO and hydrocarbons [8]. Marano and Holder [9] cite references which show the following:

There have been few attempts in the past to regress the BIPs for hydrogen with some of the heavy paraffin solvents to correlate the solubility. In one of the earliest works, Gray et al. [10] analyzed VLE data for binaries involving H2 using mixing rules and true critical constants to correlate BIPs with the critical temperature of the second component. Park et al. [11] and Srivatsan et al. [12] use PR and SRK EoS to regress BIPs for both H2 and CO in n-C10 , n-C20 , n-C28 and nC36 paraffins. They found the BIPs for these heavy paraffins to be significantly different from the values obtained for lighter components. The study was repeated for the solvents n-C16 and n-C18 by Graaf et al. [13]. A summary of the estimated BIP values is given in Table 1 which does not depict any clear trends as the solvent carbon number varies. Graaf et al. [13] and Breman et al. [14] show that the accuracy of the calculated vapor-phase fugacity coefficients is affected by the BIP values for the PR and SRK EoS. Using similar binaries [10,11] in their work, Chao and Lin [15] combine the Huron–Vidal mixing rule with the SRK EoS to develop BIPs as smooth functions of solvent carbon number. Marano and Holder [9] predict solubilities of light gases in terms of mole fractions using correlations for Henry’s constants. After comparing these values with those obtained using PR EoS, they conclude that BIPs should be regressed from experimental solubility data to improve the prediction accuracy. It is interesting to note that a few investigators tried to use models different from those based on EoS. The regular solution theory and a group-contribution based EoS were used to develop optimized parameters by Graaf et al. [13]. Riazi and Roomi [16] also work with regular solution theory to predict hydrogen solubility in hydrocarbons. Using a hydrogen solubility parameter calculated from solvent type or molecular weight, they claim that the same degree of accuracy can be obtained in the predicted values as compared to an EoS. The advantage of the Riazi and Roomi method [16] is its simplicity and the elimination of a need for solvent critical

Table 1 Summary of fitted BIPs [11–13]. Authors

Binary mixture

Temperature range (K)

EoS and No. of parameters used

Values

RMSE*

Park et al. [11]

H2 –n-C10 H2 –n-C20 H2 –n-C28 H2 –n-C36

344–423 323–423 348–423 373–423

PR, 1 PR, 1 PR, 1 PR, 1

0.4082 0.3706 0.285 0.1509

0.0023 0.0009 0.0043 0.0059

Srivatsan et al. [12]

CO–n-C10

311–378

CO–n-C20

323–423

CO–n-C28

348–423

CO–n-C36

373–423

PR, 2 PR, 1 PR, 2 PR, 1 PR, 2 PR, 1 PR, 2 PR, 1

0.0676, 0.0118 0.1184 −0.1557, 0.0323 0.1271 −0.0142, 0.0051 0.0448 0.2199, −0.0162 −0.0021

0.0014 0.0014 0.0053 0.0052 0.0028 0.003 0.0075 0.0085

CO–n-C16 CO–n-C18 H2 –n-C16 H2 –n-C18

298–353 303–566 293–353 303–566

PR, 1 PR, 1 PR, 1 PR, 1

0.115 0.129 0.342 0.535

n.a.$ n.a. n.a. n.a.

Graaf et al. [13]

* $

Indicates error in mole fraction. Errors reported in a form different from RMSE.

BIP values

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

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Table 2 Representative methods used to model solubility. Authors

Application

Gases used

Model/correlation used

Modification done

Yermakova and Anikeev [17]

Fischer–Tropsch reactor (Slurry) Fischer–Tropsch reactor (Fixed-bed)

H2 and CO H2 and CO

SRK EoS with modified interaction parameters Modified SRK EoS

Ahon et al. [19]

Fischer–Tropsch reactor (Slurry)

H2 and CO

SRK EoS

Iliuta et al. [20]

Fischer–Tropsch reactor (Slurry)

H2 and CO

PR EoS

Pellegrini et al. [21]

Hydrocracker

H2

SRK EoS

Kim et al. [22]

GTL Process having FT reactor

H2 and CO

SRK EoS

Express BIP in terms of Pr and Tr of the solvent Express ˛ as a function of the solvent molecular weight Input critical properties calculated using Marano’s ABCs Use Marano’s ABCs to find the critical and physical properties Calculate critical constants using an equation of the form suggested by [29]; BIP values from relations given in [8] Model implemented in Aspen HYSYS; standard form of EoS has zero BIP values between H2 or CO and hydrocarbons

Wang et al. [18]

properties which as discussed previously are difficult to estimate accurately. The correlation proposed holds well for carbon numbers in the range of 6–46. Representative works using an EoS for modeling the solubility in FT and hydrocracker reactors are listed in Table 2. Except for the work of Iliuta et al. [20], the rest of the papers use SRK EoS. The modified SRK EoS used by Wang et al. [18] aims to express the ˛ function in terms of the molecular weight of the solvent instead of the usual dependency on acentric factor, ω, and assumes BIP value of zero. However, they advocate the use of non-zero BIPs to improve the prediction accuracy. Most of the references cited in Table 2 use models developed in high-level languages to carry out simulations. For techno-economic or feasibility studies, the use of process simulation software is more convenient. Aspen Plus V7.1 [23] is the simulator chosen in this work. Another important consideration in choosing a property method for the FT reactor is its ability to handle water, a major by-product of the FT reaction. In their comparison study using the EoS methods, Tsonopoulos and Heidman [8] point out that PR provided a better fit for water solubility (K-value of water in the hydrocarbon liquid) compared to CCOR or SRK when using rigorous three phase flash calculations. It is worth mentioning that most of the other references cited in our work do not address the water-handling issue, which is an integral part of the phase equilibrium modeling for FT systems. For the sake of completeness, we mention that the SRK EoS in combination with the Kabadi–Danner [24] correlation is recommended to handle water–hydrocarbon mixtures.

3. Conventional vs. predictive EoS Based on the discussion in Section 2, PR EoS appears to be a suitable candidate to be used in this work. However, in the recent past, predictive EoS like the predictive SRK (PSRK) [25] and the PR EoS with modified Huron–Vidal mixing rule (PRMHV) [26] have been proposed for modeling phase equilibria of the systems under discussion. These predictive methods possess the capability to predict phase behavior of polar components like water, small molecules like H2 and CO, light gases like CH4 , etc., and heavy paraffins by the use of specially defined UNIFAC functional groups. In fact, Marano and Holder recommend the use of PRMHV model [9]. The option chosen with the PR EoS is the Boston–Mathias alpha function [27] represented by PR-BM in Aspen Plus. It is also to be noted that most of the experimental data has been fitted by other

investigators using PR with either one or two BIPs (see Table 1). The reported fit in Table 1 will be used to ascertain whether or not the BIP values obtained in this work follow a similar trend. In screening these three potential candidates, the primary selection is based on their ability to predict solubility of H2 and CO with reasonable accuracy, and secondly, to handle water solubility. Trials are carried out with the default BIPs value of zero for all binaries involving H2 or CO and heavy paraffins. Trials with PRMHV showed solubility trends opposite to what was expected with respect to temperature, viz., H2 solubility decreases with increasing temperature. With PSRK, the trends are as expected but are not as accurate as PR-BM. Thus, PSRK and PRMHV are ruled out. The accuracy of the predictive EoS depends on the UNIFAC groups defined and on the accuracy of the low pressure activity coefficient models. Since such data is limited for H2 and CO with the heavy paraffins, the authors believe that the predictive EoS does not perform well. The rest of this work uses PR-BM method. As regards to handling water, the calculations when using PR-BM can be simplified using the free water solubility method with an appropriate unsaturation correction method [23]. This is a reasonable simplification for engineering approximations instead of using the rigorous three-phase flash calculations during flowsheet modeling.

4. Asymptotic behavior correlations (ABCs) The cubic equations of state (EoS) like Soave–Redlich–Kwong (SRK) [6] and Peng–Robinson (PR) [7] require the pure component critical parameters like critical pressure, critical temperature, etc., to perform the calculations. For compounds whose critical and thermophysical properties are unavailable due to lack of experimental data, one technique is to apply group contribution methods to predict these parameters for homologous series. However, Gasem et al. [28] discuss the failure of these methods when used for extrapolation to higher carbon numbers resulting in unreasonable behavior. Riazi and Roomi [16] emphasize the importance of choice of method used in critical property prediction, especially for heavier paraffin solvents (C20+ ), and its impact on estimating the solubility. As an alternative, equations called asymptotic behavior correlations (ABCs) were proposed by Kreglewski and Zwolinski [29] to estimate the properties of higher carbon-number components through extrapolation, using known property values of the lowercarbon number homologs. Despite the expectation of being more accurate and consistent than the group contribution methods, these

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initial ABCs are shown to perform poorly at low carbon numbers [30]. This led Riazi and Al-Sahhaf [30] to develop equations that can predict behavior over a wide carbon number range of n-alkanes for the estimation of their critical constants and physical properties like boiling point, etc., in terms of their molecular weight. The equations were shown to have internal consistency and were proposed for estimation of properties of heavy petroleum mixtures. However, these equations do not provide vapor pressure calculations whose importance is discussed in Gray et al. [10]. They conclude that EoS methods lose their accuracy when the critical property and acentric factor values used do not match the vapor pressure in the region of interest. They recommend that the critical properties be fit to be more consistent with the vapor pressure data. It is also to be noted that vapor pressure is temperaturedependent; whereas none of the property parameters calculated from the equations proposed by Riazi and Al-Sahhaf [30] have a temperature dependency. Marano and Holder [31–33] revisited the ABCs and proposed a more generalized form based on carbon number, instead of molecular weight for n-paraffins as shown in Eqs. (2) and (3). They also included temperature-dependent terms to calculate properties like vapor pressure, heat capacity, etc. Y = Y∞ − Y0 exp(−ˇ(n ± n0 ) )

(2)

Y∞ = Y∞,0 + Y∞ (n − n0 )

(3)

n ≥ n0 , ˇ > 0,  > 0 Readers are referred to their papers [31–33] for a detailed description and the basis used in developing these ABCs. The extrapolative ability of their correlations was tested by predicting properties of components outside the range of carbon numbers and temperature used in developing the relationships and comparing them with results from molecular simulations. Reasonable agreement is reported. It suffices to say that their work is comprehensive and well-validated, and provides correlations for estimating critical constants, physical properties and transport properties based on carbon number and temperature. We have hence chosen

Table 3 Estimated critical properties of heavy hydrocarbons. Carbon number

Tc (K)

Pc (bar)

ω

20 28 36 43 46

769.88 837.27 882.06 910.10 919.87

11.2 7.7 5.65 4.47 4.08

0.88 1.15 1.38 1.57 1.64

correlations proposed by Marano and Holder in this work to calculate the relevant inputs (critical temperature, Tc ; critical pressure, Pc ; acentric factor, ω) as shown in Table 3. The vapor pressure is calculated by the PR EoS using these critical parameters in this work. To confirm the consistency of the calculations, the predicted vapor pressures for three components (C20 , C28 and C36 ) are compared against TRC (thermodynamic research centre) data which has an extensive database for long-chain hydrocarbons. These components are chosen since they are used for regression of the interaction parameters in later sections. The comparison in each case is depicted in Fig. 1 which shows good agreement between the experimental data and the model predictions. Though the match in Fig. 1 is not good in the lower temperature range, it is not of much concern since the proposed correlations in this work are intended for use at high temperature (473–653 K) applications where the match is good. 5. Methodology Following the previous discussion, it can be concluded that it is necessary to regress BIPs using experimental data available in literature. The sources used in finding experimental data for H2 and CO solubility in hydrocarbons are given in Table 4, which provides a summary of the experimental methods used, the range of parameters covered (pressure, temperature and solubility) and the error measurements for each of them. The data covers more than 700 experimental points for 12 different binary pairs. To start

Fig. 1. Comparison of vapor pressure of paraffins. (Symbols represent experimental data from TRC database. Line represents prediction using PR EoS with Marano’s parameters.)

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

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Table 4a Solubility experimental data references. Authors

Apparatus and experiment type

Gases used

Solvents used

Theoretical work

Huang et al. [34]

Semi-flow; stationary liquid with flowing gas stream

H2 and CO

n-Eicosane (C20 )

None

n-Octacosane (C28 ) n-Hexatriacontane (C36 ) Huang et al. [35]

H2 and CO

Semi-flow; stationary liquid with flowing gas stream

n-Eicosane (C20 )

Correlation developed for use with SRK EoS

n-Octacosane (C28 ) n-Hexatriacontane (C36 ) Mobil FT wax Park et al. [11]

H2

Variable-volume, static-type blind equilibrium cell

n-Decane (C10 )

Single binary interaction parameter fitted to PR and SRK EoS

n-Eicosane (C20 ) n-Octacosane (C28 ) n-Hexatriacontane (C36 ) Srivatsan et al. [12]

CO

Variable-volume, static-type blind equilibrium cell

n-Decane (C10 )

Two binary interaction parameters fitted to PR and SRK EoS

n-Eicosane (C20 ) n-Octacosane (C28 ) n-Hexatriacontane (C36 ) Florusse et al. [36]

Cailletet apparatus

H2

n-Decane (C10 )

SAFT approach used to model the binaries and compare with experimental data

n-Hexadecane (C16 ) n-Octacosane (C28 ) n-Hexatriacontane (C36 ) n-Hexatetracontane (C46 ) Gao et al. [37]

Variable-volume, static-type blind equilibrium cell

H2 and CO

n-Dodecane (C12 )

Single binary interaction parameter fitted to PR and SRK EoS

Sokolov and Polyakov [38]

Constant volume autoclave

H2

n-Decane (C10 )

Correlation used to find H2 solubility as a function of the reduced parameters of the state of the solvent

Chou and Chao [39]

Semi-flow; stationary liquid with flowing gas stream

H2 and CO

n-Tetradecane (C14 )

with, the solubility of H2 in C16 was predicted using the PR-BM model in Aspen Plus. Using a default value of zero for the BIP in the predictions, the predicted values are compared with the experimental data from Florusse et al. [36] at three different temperatures. Though the predictions meet the expected trends of solubility increasing with temperature and pressure, they are significantly different from experimental values. This adds strength to the conclusion that non-zero BIPs are needed to model such asymmetric systems when using EoS like PR. The references listed in Table 4 cover pressure and temperature data in the range of operation of the FT and hydrocracker reactors. Also, these papers provide solubility data in terms of the mole fraction of the solute (H2 or CO) making it easier for use in

SASOL FT wax

Verified Huang’s correlation [35]

regression analysis. For the sake of completeness, we mention a few sources which provide solubility data but have not been used in our work. Deimling et al. [4] use a transient absorption technique to calculate solubilities and mass transfer coefficients in three FT liquid fractions as functions of temperature and pressure. The data is, however, presented in the form of graphs and Henry parameters. Ostwald coefficients are used by Makranczy et al. [40] in presenting the solubility data for different gases in n-alkanes (n-C6 to n-C16 ). They also propose a linear relationship between gas solubility and reciprocal parachor values of the solvent. The references from Beenacker’s group [13,14] discuss in detail the experimental set-up and calculations used in measuring solubility data, which are the most rigorous to date. However, they are not presented in a

Table 4b Solubility experimental data references (range of data with errors). Authors

Range of experimental data Pressure

Huang et al. [34] Huang et al. [35] Park et al. [11] Srivatsan et al. [12] Florusse et al. [36] Gao et al. [37] Sokolov and Polyakov [38] Chou and Chao [39] * † ¢

10–50 atm 10–50 atm Up to 17.4 MPa Up to 10.2 MPa Up to 16 MPa Up to 13.2 MPa 40–300 atm 10–50 atm

Denotes reproducibility of the samples. Not specified. Mole fraction of H2 or CO in liquid phase.

Temperature 100–300 ◦ C 100–300 ◦ C 323.2–423.2 K 311–423 K 280–450 K 344–410 K 55–210 ◦ C 200–300 ◦ C

Errors on experimental data ¢

Solubility

Pressure

Temperature

Solubility

0.01–0.175 0.0113–0.173 0.0273–0.2271 0.0385–0.2099 Up to 0.3 0.0113–0.1493 0.033–0.345

±0.05 atm ±0.05 atm ±0.05 MPa ±0.05 MPa n.a.† ±0.06 MPa n.a.† ±0.05 atm

±0.1 ◦ C ±0.1 ◦ C ±0.1 K ±0.1 K n.a.† ±0.1 K ±1 ◦ C ±0.1 ◦ C

1.5 mol%* 1.5 mol%* ±0.001 ±0.001 n.a.† ±0.001 n.a.† 1.5 mol%*

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Table 5 Summary of regressed BIPs.† No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. ¢ @ * †

T (K) CO and C10 CO and C12 CO and C20 CO and C28 CO and C36 H2 and C10 H2 and C12 H2 and C14 H2 and C16 H2 and C20 H2 and C28 H2 and C36 H2 and C46

P (MPa)

310.9–377.6 344.3–410.9 323.2–573.15 348.2–573.15 373.2–573.15 283.17–448 344–411 328–473 298.13–448 323–573 343–573 358–573 373–447

2.22–10.2 0.69–8.75 0.989–8.38 0.994–8.41 1–8.95 1.238–17.39 1.42–13.24 4–30 1.151–15.134 0.981–12.91 1–13.1 1–16.75 2.293–11.84

Kij (1)¢

SD(1)

Kij (2)¢

SD(2)

No. of points*

0.3932 0.4792 0.6704 0.7242 0.7827 0.2099 0.2273 0.3829 0.6945 0.6466 0.6929 0.7593 0.7370

0.0282 0.0514 0.0072 0.0109 0.0148 0.0245 0.1045 0.2684 0.0044 0.0132 0.0107 0.0246 0.0468

−0.00103 −0.001204 −0.00184 −0.00184 −0.00184 0.00034 0.000366 −7.92E−5 −0.00104 −0.00104 −0.00104 −0.00104 −0.00104

8.65E−5 0.000139 n.a.@ n.a.@ n.a.@ 6.62E−5 0.000279 0.000622 n.a.@ n.a.@ n.a.@ n.a.@ n.a.@

18 (13) 26 (19) 50 (37) 46 (35) 42 (28) 123 (75) 24 (10) 12 (12) 114 (58) 52 (51) 106 (91) 83 (43) 36 (18) 706 (471)

Kij = Kij (1) + Kij (2) × T. n.a.: not applicable since Kij (2) is fixed. Number of available data points (number of data points used in regression). Actual data points used in the regression are shown in Table A1 in Appendix.

format that can be used readily for regression of the BIPs. It is interesting to note that their work was performed within the context of deriving VLE data for methanol and higher alcohols synthesis from syngas. Comparison with PR and SRK EoS is also shown. The data from all the references in Table 4 is screened for anomalies, if any, and then grouped into isotherms for each binary as shown in Table A1 in the Appendix. The data regression system (DRS) tool in Aspen Plus [23] is used to perform the regression analysis and obtain the BIP values. The data input type used is TPxy. The experimental data is normally reported in TPx form, with x being the mole fraction of H2 or CO in the solvent. Since most of the heavy n-paraffins have very low volatility under the range of temperature and pressure considered, the mole fraction of H2 or CO in the vapor phase is fixed at 0.999 which is a reasonable assumption. Further, the standard deviation (SD) in each of these variables needs to be specified. Since the vapor phase mole fractions of the solute are not measured, we associate a higher value of SD (15%) with it. The SD values on temperature (T) and liquid phase mole fraction (x) are set to zero. A SD of 0.1% is used on the pressure measurements. This conforms to the standard practice in which the BIP values are determined by fitting the experimental data to minimize an objective function based on the sum of squared errors in predicted bubble point pressures. The last input needed by the DRS is the choice of algorithm to be used in the regression, which is the Britt–Luecke algorithm in this case. The other option is the Deming algorithm. A brief comparison of these algorithms can be found in Luecke et al. [41]. After completing all the required inputs, the regression is

performed and results analyzed ensuring that no errors are present for the converged cases. 6. Regression results In the first step of regression, a single BIP was used in each of the isotherms for the PR-BM model based on Peng–Robinson EoS. It is to be noted that the property parameters (Tc , Pc and ω) for the heavier n-paraffins are calculated from Marano’s correlations [31,32] and are provided as inputs (see Table 3). The fitted BIPs and their accuracy are presented in Tables A1 and A2 in the Appendix which show that the fit is quite accurate in all the cases. The regressed BIP values are significantly different in each case. This leads us to the conclusion that the interaction parameters have temperature dependence and that a single parameter cannot adequately describe the solubility behavior. Hence, as the next step, two BIP values are regressed by combining data from all the isotherms. The parameters and their accuracy are shown in Tables 5 and 6, respectively. The numbers in brackets in the last column of Table 5 represent the actual number of points used for regression out of the total available experimental points. The remaining data points are used to validate the model as described in Section 7. Gray et al. [10] too concluded that the use of two interaction parameters improves the VLE predictions for asymmetric mixtures of light gases with heavy paraffins. The obtained interaction parameters are then used in Pxy analysis (i.e., flash calculations) to compare the solubility under the same conditions used in the regression. The comparison as seen in representative plots in

Table 6 Accuracy of the regressed parameters. No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. $

CO and C10 CO and C12 CO and C20 CO and C28 CO and C36 H2 and C10 H2 and C12 H2 and C14 H2 and C16 H2 and C20 H2 and C28 H2 and C36 H2 and C46

The deviation is for pressure (MPa).

Avg. Devn.$

Max. Devn.$

RMSE$

RMSE%$

AAD$

AAD%$

0.00 0.03 0.00 0.01 0.00 0.01 0.01 0.25 −0.01 0.05 0.01 0.01 −0.02

−0.05 −0.21 0.32 −0.30 −0.22 0.21 0.13 1.07 −0.19 0.68 0.32 −0.39 −0.35

0.03 0.09 0.10 0.11 0.07 0.06 0.06 0.61 0.06 0.20 0.14 0.16 0.12

0.49 2.41 2.56 3.92 2.29 1.16 1.56 4.07 1.08 5.21 3.04 4.59 2.48

0.02 0.07 0.07 0.09 0.05 0.04 0.05 0.51 0.04 0.15 0.10 0.13 0.09

0.45 1.93 1.85 2.75 1.69 0.95 1.36 3.49 0.89 3.71 2.35 3.83 2.01

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

17

Fig. 2. Quality of regression fit and validation test for CO BIPs (CO in C36 ). (Symbols are experimental data from Srivatsan et al. [12], Huang et al. [34,35]. Lines are model calculations using PR-BM with Kij values from Table 5. Solid lines represent regression fit and dotted lines represent validation test.)

Figs. 2 and 3 shows a reasonably good fit. The quality of the fits can be judged by the graphs provided for all the binaries (as a pdf file) in the supporting information to this article. Thus, the accuracy of the fitted interaction parameters is checked both qualitatively and quantitatively. The BIPs temperature dependency used in this work is provided in the footer of Table 5. One may note that Kij (2) is fixed in some of the cases in Table 5, especially for carbon numbers beyond C16 . Initial regressions were

performed with both the BIPs left free. Analysis of the obtained BIP values led to the following observations: • Values of the temperature-associated parameter, Kij (2) for CO were almost the same for C20 , C28 and C36 . • For H2 , the trend was also dominated by C20 , C28 and C36 paraffins (see symbols representing Kij values in Fig. 6). However, regressed values of Kij (2) were different for these components unlike the

Fig. 3. Quality of regression fit and validation test for H2 BIPs (H2 in C16 ). (Symbols are experimental data from Florusse et al. [36]. Lines are model calculations using PR-BM with Kij values from Table 5. Solid lines represent regression fit and dotted lines represent validation test.)

18

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

Fig. 4. Parity plot for solubility of H2 and CO in C28 . (Experimental values are from Huang et al. [35] and model predictions are using PR-BM with Kij values from Table 5. Gas Feed1, Feed2 and Feed3 have 40.01, 50.01, and 66.64 mol% H2 , respectively, with the rest being mol% CO. gas feed is devoid of C28 .)

case with CO. In addition, more data points are available and used in the regression of these H2 systems. Of the three components (C20 , C28 and C36 ) in case of H2 , C28 has the most number of data points available (106) and used in regression (91). Therefore, more confidence is associated with the value of Kij (2) obtained for the H2 –C28 binary. This value (−0.00104) is therefore used as the “fixed” value for binaries including H2 with C16 and above while regressing for Kij (1) only. To maintain

consistency, the Kij (2) value from the CO–C28 binary (−0.00184) is also used as the “fixed” value in regressing Kij (1) only for the CO binaries for C20 , C28 and C36 . 7. Validation of BIP correlations To further validate the BIPs, independent tests were performed. The first involved the comparison of experimental data not used in fitting the parameters with the predictions. These are mostly at

Fig. 5. Kij for binaries with CO as a function of carbon number and temperature. (Symbols represent regressed Kij obtained from Kij (1) and Kij (2) in Table 5. Line represents prediction using the fitted correlation in Table 7.)

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

different temperatures than those used in the regression (represented by the dotted lines in Figs. 2 and 3) and show that the fitted interaction parameters indeed capture the temperature dependence reasonably well across different solvents, both for CO and H2 . In a second test, the solubility predictions are compared for a syngas mixture instead of H2 or CO alone. Huang et al. [35] report the solubility of synthesis gas mixtures (H2 and CO) in C28 determined at three feed gas compositions: 40.01, 50.01, and 66.64 mol% H2 . The phase equilibrium data presented by them (liquid phase mole fractions of H2 and CO) are at 200 and 300 ◦ C covering a pressure range of 20–50 atm. The parity plot in Fig. 4 compares the predicted values using the fitted interaction parameters with the experimental data. The dotted lines in the plot represent the ±5% deviation from the diagonal within which most of the comparisons lie. This shows that the fitted parameters work well with mixtures too—the real application they are intended for.

7.1. Correlation of BIPs FT liquid mixtures are predominantly n-paraffins, and often n-olefins when an iron catalyst is used in the reaction. This mixture has a wide range of carbon numbers extending beyond C70 . However, the experimental solubility data is available only for a handful of them (see Table 4). Therefore, it is essential to have a correlation that can predict the solubility of H2 and CO in the paraffins as a function of carbon number and temperature. In the past, researchers have tried to establish this correlation using EoS like PR and SRK with different sets of mixing rules based on carbon number, molecular weight, parachor values, etc. Readers are referred to Tables 2 and 4a for details on some such studies. In the present work, the fitted BIPs are used to develop an extrapolation rule based on carbon number and temperature (see Table 7). The correlations are linear or quadratic polynomials or power-law functions fitted using MS-Excel® . One can observe that there may be under- or over-predictions because of the nature of the fit. Two

19

Table 7 Summary of correlations developed for Kij .¢ Component

Carbon number (n)

Correlation#

CO CO CO CO H2 H2 H2 H2

C10 C10 C20 C20 C10 C10 C16 C16

Kij (1) = 0.026n + 0.1482 Kij (2) = −0.000079n − 0.000252 Kij (1) = 0.3052n0.2616 † Kij (2) = −0.00184 † Kij (1) = 0.01242n2 − 0.25675n + 1.53176 Kij (2) = −0.0000462n2 + 0.001009n − 0.005117 Kij (1) = 0.4953n0.1069 D Kij (2) = −0.00104 D

¢ # † D

to C20 to C20 to C36 to C36 to C16 to C16 to C46 to C46

Kij = Kij (1) + Kij (2) × T. n = carbon number. Used to extrapolate beyond C36 . Used to extrapolate beyond C46 .

piece-wise expressions are used to describe the behavior over the entire carbon number range for both CO and H2 , with a break at carbon number 20 and 16, respectively. Each of the parameters in Table 7 is represented as a function of the carbon number. Since the two interaction parameters when used together capture the temperature effect, it essentially results in a correlation in terms of both the solvent carbon number and the temperature. This behavior is depicted graphically in Figs. 5 and 6 for CO and H2 , respectively, with predictions extending to component C100 . The behavior of CO is more uniform (Fig. 5) as compared to H2 (Fig. 6) since it exhibits a single inflexion point in the carbon number range considered. As pointed out earlier (Table 5), the interaction parameter, Kij , is a linear combination of two parameters: Kij (1) and Kij (2) with the latter associated with the temperature dependence. The functional form is shown in Eq. (4). Kij = Kij (1) + Kij (2) × T

(4)

Kij for CO decreases with increasing temperature over the entire carbon number range. For any given temperature, it decreases until C20 and then increases again which led to breaking the piece-wise functions at this carbon number. Hence, a linear function is used

Fig. 6. Kij for binaries with H2 as a function of carbon number and temperature. (Symbols represent regressed Kij obtained from Kij (1) and Kij (2) in Table 5. Line represents prediction using the fitted correlation in Table 7.)

20

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

Fig. 7. Validation of the model for CO (extrapolation capability using C43 ). (Symbols are experimental data from Chou and Chao [39]. Lines are model calculations using PR-BM with Kij values extrapolated as shown in Fig. 5.)

to interpolate between C10 and C20 and a power-law function is used between C20 and C36 . Note that the power-law form is also used for extrapolating beyond C36 for binaries with CO. Kij for H2 appears to change behavior at carbon number 14 as shown in Fig. 6. Below carbon number 14, it increases with temperature. At carbon number 14, the Kij value appears to be almost independent of temperature. Beyond this, the trend reverses with an increase in temperature leading to decrease of Kij values. A minimum is observed at carbon number 20, and two maxima at carbon numbers 12 and possibly 36. To have better continuity, C16 is chosen as the limit in the piece-wise functions. A quadratic polynomial

describes the behavior between C10 and C16 and a power-law form is used to fit the trend between C16 and C46 . Though the powerlaw fit does not show a good R2 value, this is the simplest and best choice possible. Higher order polynomials tend to predict unrealistic Kij values at higher carbon numbers and are therefore avoided. Further, with differences in the experimental data points (range of pressure, temperature and composition) and lack of high temperature data for C46 (beyond 447 K), it may not be justified to have a function that passes through all of the regressed Kij (1) values. The power-law form is used for extrapolating Kij (1) beyond C46 for binaries with H2 . The use of these correlations for Kij (1) and the

Fig. 8. Validation of the model for H2 (interpolation capability using C43 ). (Symbols are experimental data from Chou and Chao [39]. Lines are model calculations using PR-BM with Kij values interpolated as shown in Fig. 6.)

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

21

Fig. 9. Solubility prediction at 573 K for H2 and paraffins. (Kij (1) and Kij (2) are calculated from correlations in Table 7.)

constant-value for the temperature-associated term (Kij (2)) result in more conservative predicted Kij values at high carbon numbers as shown in Figs. 5 and 6. The correlations developed in this work are summarized in Table 7. Chou and Chao [39] provide solubility data for H2 and CO in SASOL wax which is characterized by an average carbon number of 43. This is justified owing to the highly paraffinic nature of the FT waxes. Using the correlations developed (Table 7), the solubility of H2 and CO is predicted in C43 solvent. The comparisons in Figs. 7 and 8 show that the predictions are reasonably accurate and serve as a validation case for the developed correlations. As a word of caution, it is to be noted that this is the best possible trend built from available data, and the correlation can be improved when solubility data for intermediate carbon numbers (e.g., C24 , C32 , C40 and C44 ) is available. Also, the reason for the observed behavior in Figs. 5 and 6 cannot be explained and needs further investigation, possibly from molecular simulations and more basic studies at a theoretical level. Such trends have also been reported in literature (references in Table 4a). As a test of the extrapolative ability of the correlations to higher carbon number predictions, the values of Kij till C100 at 373, 473 and 573 K are shown in Figs. 5 and 6, for CO and H2 , respectively. The overall values, trends, and temperature dependency seem reasonable. To further test the magnitude of the Kij values obtained, the solubility of CO and H2 in the following paraffins – C36 , C48 , C56 , C64 and C72 – is predicted using Pxy analysis in Aspen Plus. The results for H2 at 573 K are shown in Fig. 9 which seem reasonable and follow the expected trends—increase in solubility with temperature, pressure and carbon number. Similar behavior is observed in case of CO too.

(PR-BM variation). The critical parameters needed in the EoS for the heavy paraffins are estimated using equations from Marano and Holder [31]. Using these parameters, vapor pressure predictions for heavy solvents (C20 , C28 and C36 ) compare reasonably well with data from TRC. BIPs for individual binary pairs are regressed which are further used to develop a solubility correlation as a function of temperature and carbon number. The BIPs for the individual binary pairs are validated using data points not used in the regression. Further, they are also tested for a mixture of H2 and CO in C28 , for which data was available. Finally, the predictions from the correlations are validated for a syngas mixture in SASOL wax solvent characterized based on an average carbon number (C43 ). These validations, which are reasonably accurate, prove the predictive capability of the developed correlation. The correlation captures the increase in solubility with temperature, pressure and molecular weight. Future work should be aimed at acquiring solubility experimental data for the intermediates (e.g., C24 , C32 , C40 and C44 ) and higher carbon numbers (C > 36 for CO binaries; C > 46 for H2 binaries) to improve the functional form of the correlation.

Acknowledgements The authors thank Dr. Susan Little from BP for her help in this work. The funding provided by BP as a part of the Conversion Research Program at MIT is also gratefully acknowledged.

Appendix A. Appendix See Tables A1 and A2.

8. Summary Appendix B. Supplementary data Based on experimental data, a correlation to predict the solubility of H2 and CO in n-paraffin solvents as a function of temperature and carbon number is determined for the Peng–Robinson EoS

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.fluid.2012.02.008.

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S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

Table A1 Summary of regressed BIPs. No.

T (K)

1.1 1.2 1.3 1

CO and C10

2.1 2.2 2.3 2

P (MPa)

Kij (1)¢

SD(1)

Kij (2)¢

SD(2)

310.9 344.3 377.6 310.9–377.6

2.84–7.57 2.36–7.69 2.22–7.37 2.22–10.2

0.0719 0.0374 −0.0006 0.3932

0.003 0.001 0.005 0.0282

CO and C12

344.3 377.6 410.9 344.3–410.9

1.52–7.87 3.34–8.54 2.47–7.75 0.69–8.75

0.4758 0.4886 0.4722 0.4792

0.005 0.007 0.004 0.0514

3.1 3.2 3.3 3.4 3.5 3

CO and C20

323.2 423.2 373.2 473.15 573.15 323.2–573.15

2.8–6.05 2.83–5.94 1.977–5.93 1.978–5 1.984–5 0.989–8.38

0.0741 −0.0978 0.0064 −0.2081 −0.3781 0.6704

0.016 0.014 0.010 0.018 0.023 0.0072

4.1 4.2 4.3 4.4 4.5 4

CO and C28

348.2 373.2 423.2 473.15 573.15 348.2–573.15

2.33–4.62 1.973–5.02 2.19–5.59 1.965–5 1.974–5.006 0.994–8.41

0.0536 0.0546 −0.0572 −0.1617 −0.3308 0.7242

0.032 0.020 0.030 0.019 0.018 0.0109

5.1 5.2 5.3 5

CO and C36

373.2 473.15 573.15 373.2–573.15

1.8–5 2–5.014 2–5.004 1–8.95

0.0968 −0.0884 −0.2807 0.7827

0.025 0.023 0.009 0.0148

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6

H2 and C10

313 328 343 358 373 388 403 418 433 448 283.17–448

2.158–7.893 1.993–7.273 1.843–8.235 1.718–7.635 1.608–8.215 1.508–7.665 1.418–8.276 1.343–7.736 1.283–7.246 1.238–6.806 1.238–17.39

0.3279 0.3243 0.3229 0.3261 0.3326 0.3298 0.3374 0.3493 0.3669 0.3890 0.2099

0.006 0.005 0.003 0.003 0.007 0.007 0.009 0.011 0.014 0.017 0.0245

7.1 7.2 7.3 7

H2 and C12

344 378 411 344–411

3.37–5.81 1.42–5.59 1.77–5.62 1.42–13.24

0.3448 0.3901 0.3656 0.2273

0.009 0.008 0.012 0.1045

8.1 8.2 8.3 8

H2 and C14

328 403 473 328–473

4–30 4–30 4–30 4–30

0.2775 0.3510 0.3454 0.3829

0.025 0.028 0.033 0.2684

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9

H2 and C16

313 328 358 388 403 418 433 448 298.13–448

2.071–6.893 1.921–6.343 1.656–7.856 1.451–7.98 1.361–7.783 1.286–7.333 1.216–8.077 1.151–8.074 1.151–15.134

0.3583 0.3429 0.3129 0.2841 0.2733 0.266 0.2580 0.2540 0.6945

0.033 0.030 0.011 0.009 0.007 0.005 0.005 0.004 0.0044

−0.00104

n.a.@

10.1

H2 and C20

323 373 423 373 473 573 373 473 573 323–573

3.26–12.91 2.23–11.82 2.81–9.3 1–5 1–5 1–5 0.996–5.015 0.989–4.969 0.981–4.999 0.981–12.91

0.5242

0.123

−0.000078

0.0003

1.1224

0.224

−0.002009

0.0005

0.7655

0.134

−0.001248

0.0003

0.6466

0.0132

−0.00104

n.a.@

373.2 423.2 343–447 373 473 573

4.02–12.43 2.86–11.24 1.461–12.581 1–5 1–5 1–5

1.0412

0.394

−0.00189

0.001

0.9317 0.7521

0.163 0.195

−0.00171 −0.00105

0.0004 0.0004

10.2

10.3

10 11.1 11.2 11.3

H2 and C28

−0.00103

−0.001204

−0.00184

−0.00184

−0.00184

0.00034

0.000366

−7.92E−5

No. of points*

8.65E−05

4 5 4 18 (13)

0.000139

7 5 7 26 (19)

n.a.@

5 5 11 8 8 50 (37)

n.a.@

4 12 3 8 8 46 (35)

n.a.@

12 8 8 42 (28)

6.62E−05

6 5 6 10 9 7 8 8 8 8 123 (75)

0.000279

3 3 4 24 (10)

0.000622

4 4 4 12 (12) 3 3 10 7 8 8 9 10 114 (58) 7 9 6 5 5 5 5 5 4 52 (51) 5 9 47 5 5 5

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

23

Table A1 (Continued) No.

T (K) 373 473 573 343–573

11.4

11 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12

H2 and C36

13.1 13.2 13.3 13.4 13.5 13.6 13

H2 and C46

Kij (1)¢

P (MPa) 1–5 1–5 1–5 1–13.1

SD(1)

Kij (2)¢

SD(2)

No. of points*

0.6107

0.149

−0.00078

0.0003

−0.00104

n.a.@

5 5 5 106 (91)

n.a.@

13 3 3 4 4 8 8 83 (43)

n.a.@

3 3 3 3 3 3 36 (18)

0.6929

0.0107

373 388 403 432 447 473 573 358–573

1.845–5.778 1.735–5.418 1.625–5.098 1.455–5.791 1.375–5.511 2–5 2–5 1–16.75

0.4247 0.2951 0.2574 0.2055 0.1776 0.2799 0.2149 0.7593

0.049 0.099 0.094 0.068 0.063 0.047 0.031 0.0246

373 388 403 418 433 447 373–447

3.063–6.741 2.853–6.311 2.683–5.941 2.533–5.601 2.403–7.421 2.293–5.041 2.293–11.84

0.3979 0.3532 0.3145 0.2754 0.2591 0.2109 0.7370

0.146 0.141 0.136 0.130 0.120 0.118 0.0468

−0.00104

−0.00104

706 (471) ¢ @ *

Kij = Kij (1) + Kij (2) × T. n.a.: not applicable since Kij (2) is fixed. Number of available data points (number of data points used in regression).

Table A2 Accuracy of the regressed BIPs. No.

Avg. Devn.$

Max. Devn.$

RMSE$

RMSE%$

AAD$

AAD%$

1.1 1.2 1.3 1

CO and C10

−0.01 0.00 −0.01 0.00

−0.05 0.03 −0.04 −0.05

0.03 0.02 0.03 0.03

0.54 0.45 0.62 0.49

0.03 0.02 0.03 0.02

0.53 0.39 0.58 0.45

2.1 2.2 2.3 2

CO and C12

0.01 0.07 0.01 0.03

−0.26 0.26 0.07 −0.21

0.12 0.12 0.03 0.09

2.39 2.87 0.97 2.41

0.09 0.09 0.02 0.07

2.09 2.66 0.69 1.93

3.1 3.2 3.3 3.4 3.5 3

CO and C20

−0.01 0.00 0.01 0.03 0.00 0.00

−0.11 −0.16 −0.26 0.29 0.15 0.32

0.05 0.09 0.12 0.13 0.07 0.10

1.01 2.45 2.99 3.54 1.77 2.56

0.03 0.07 0.09 0.10 0.06 0.07

0.75 1.82 2.50 2.89 1.41 1.85

4.1 4.2 4.3 4.4 4.5 4

CO and C28

−0.02 0.03 0.00 0.01 0.01 0.01

−0.09 −0.27 0.05 −0.13 0.10 −0.30

0.06 0.15 0.04 0.07 0.06 0.11

1.75 5.05 0.83 2.78 1.86 3.92

0.06 0.13 0.03 0.06 0.04 0.09

1.75 3.83 0.77 2.13 1.34 2.75

5.1 5.2 5.3 5

CO and C36

0.00 0.01 0.00 0.00

−0.22 0.07 0.05 −0.22

0.10 0.04 0.02 0.07

3.28 1.36 0.62 2.29

0.09 0.04 0.02 0.05

2.83 1.11 0.53 1.69

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6

H2 and C10

0.02 0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

0.08 0.05 −0.04 0.05 0.18 0.10 0.08 0.08 0.07 0.08 0.21

0.05 0.04 0.03 0.03 0.09 0.04 0.04 0.03 0.03 0.03 0.06

0.91 0.76 0.67 0.67 1.54 0.76 0.74 0.69 0.69 0.73 1.16

0.04 0.03 0.03 0.03 0.07 0.03 0.03 0.03 0.03 0.02 0.04

0.89 0.72 0.64 0.61 1.29 0.68 0.65 0.60 0.60 0.61 0.95

7.1 7.2 7.3 7

H2 and C12

0.00 0.01 0.01 0.01

0.00 0.06 0.08 0.13

0.00 0.04 0.04 0.06

0.03 0.86 1.13 1.56

0.00 0.03 0.04 0.05

0.02 0.74 1.01 1.36

24

S. Srinivas et al. / Fluid Phase Equilibria 320 (2012) 11–25

Table A2 (Continued) Avg. Devn.$

No.

Max. Devn.$

RMSE$

RMSE%$

AAD$

AAD%$

8.1 8.2 8.3 8

H2 and C14

0.30 0.30 0.19 0.25

1.52 1.07 0.91 1.07

0.77 0.61 0.61 0.61

4.25 4.21 3.92 4.07

0.49 0.48 0.53 0.51

3.26 3.46 3.51 3.49

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9

H2 and C16

−0.01 0.00 −0.01 0.00 0.00 0.00 0.00 0.00 −0.01

−0.04 −0.02 −0.03 0.08 −0.08 −0.08 0.09 0.09 −0.19

0.02 0.02 0.02 0.04 0.04 0.04 0.04 0.04 0.06

0.71 0.47 0.51 0.69 0.75 0.67 0.71 0.63 1.08

0.02 0.01 0.02 0.03 0.04 0.03 0.03 0.03 0.04

0.60 0.39 0.41 0.63 0.69 0.56 0.59 0.53 0.89

10.1 10.2 10.3 10

H2 and C20

0.01 0.03 0.01 0.05

0.12 0.43 −0.23 0.68

0.05 0.18 0.12 0.20

0.67 6.88 3.51 5.21

0.03 0.15 0.09 0.15

0.53 5.85 3.10 3.71

11.1 11.2 11.3 11.4 11

H2 and C28

0.00 −0.01 0.02 0.01 0.01

−0.07 −0.22 0.29 0.11 0.32

0.03 0.11 0.14 0.05 0.14

0.49 1.86 4.24 1.79 3.04

0.03 0.09 0.11 0.04 0.10

0.40 1.59 3.66 1.51 2.35

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12

H2 and C36

0.01 0.00 0.00 −0.01 −0.02 0.03 0.02 0.01

0.40 −0.01 −0.01 −0.10 −0.11 0.19 0.25 −0.39

0.17 0.01 0.00 0.05 0.06 0.12 0.10 0.16

4.80 0.47 0.19 0.93 1.14 3.46 2.39 4.59

0.13 0.01 0.00 0.04 0.04 0.10 0.07 0.13

3.88 0.39 0.17 0.77 0.96 2.84 1.86 3.83

13.1 13.2 13.3 13.4 13.5 13.6 13

H2 and C46

−0.03 −0.03 −0.03 −0.02 −0.03 −0.02 −0.02

−0.14 −0.15 −0.14 −0.14 −0.09 −0.12 −0.35

0.09 0.09 0.09 0.08 0.07 0.07 0.12

1.36 1.57 1.67 1.69 1.74 1.65 2.48

0.07 0.07 0.07 0.07 0.07 0.06 0.09

1.20 1.41 1.50 1.53 1.63 1.54 2.01

$

The deviation is for pressure (MPa).

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