Thermodynamic Properties of Light Synthetic Natural Gas Mixtures

liquid equilibrium (VLE) data covering wide temperature and pressure ranges. ..... (experimental data from ref 26); (;) values from RK-PR and (--) PC-...
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Ind. Eng. Chem. Res. 2006, 45, 3684-3692

Thermodynamic Properties of Light Synthetic Natural Gas Mixtures Using the RK-PR Cubic Equation of State Santiago Aparicio Martinez* and Kenneth R. Hall Artie McFerrin Department of Chemical Engineering, Texas A&M UniVersity, College Station, Texas 77843

In this work, we test the recently developed generalized Redlich-Kwong/Peng-Robinson (RK-PR) threeparameter cubic equation of state for predicting thermodynamic properties of light, synthetic natural gas mixtures using the simple one-fluid mixing rule. The required binary interaction parameters (considered temperature-independent) come from correlation of literature vapor-liquid equilibrium data for the involved binary systems covering wide temperature and pressure ranges. Densities and phase envelopes of natural gas mixtures calculated with the equation of state are compared to highly accurate literature data to check the validity of the model. We also report comparisons with the predictions obtained from the Patel-Teja (PT) and the molecular-based, perturbed-chain SAFT (PC-SAFT) equations of state. RK-PR has good predictive ability for these systems considering its simplicity. Introduction Natural gas is an important energy1 and feedstock source, and it appears poised to be the fastest growing source of energy soon. From an environmental viewpoint, it is the least offensive fossil fuel because it burns clean and because it produces less greenhouse gases than coal or oil.2 The importance of natural gases requires an accurate knowledge of their thermodynamic properties over wide ranges of pressure and temperature to optimize their production, transportation, and utilization. These properties can be measured with high accuracy using state of the art techniques, and this is the most reliable and accurate approach.3 Nevertheless, the composition, and thus properties, of natural gases can vary widely depending upon the reservoir from which the fluid comes. Hence, it is almost impossible, because of economical and time constraints, to measure all the properties for all the possible mixtures over the wide temperature and pressure ranges required. The measurement of the required volumetric properties and vapor-liquid equilibrium in multicomponent mixtures, especially for high-pressure conditions such as those for deep reservoir gases, is expensive and difficult. Measurements require considerable experimental skill and complex equipment, with the attendant financial and time constraints.3,4 Thus, the demand for properties requires reliable models that can both correlate and predict the properties with the required accuracy. Natural gases are multicomponent mixtures, and their thermodynamic behavior is complex. Some of the principal components of the mixtures, such as methane and nitrogen, are always supercritical under reservoir conditions. Also, new technologies allow exploration and extraction of nonconventional, hyperbaric reservoirs in which temperatures and pressures are extremely high.5 Thus, the development of reliable and accurate models is not simple. From an historical viewpoint, equations of state (EOS) have played a pivotal role in modeling the thermodynamic properties of complex mixtures, and they have been able to predict with some success phase equilibrium for hydrocarbon reservoir * To whom correspondence should be addressed. Permanent address: Department of Chemistry, University of Burgos, 09001 Burgos, Spain. Tel.: +34 947 258 062. Fax: +34 947 258 831. E-mail: [email protected].

fluids.6,7 Recently, accurate semiempirical EOS have appeared that predict volumetric properties of natural gases. Within that group is AGA8-DC92,8 an international standard for density predictions. Nevertheless, this multiparameter EOS is not applicable for phase equilibrium calculations or for liquid properties.9 Another promising approach to calculate properties for multicomponent mixtures is the BACKONE EOS;10 although this model has predicted accurately thermodynamic properties, it is complex compared to simple cubic EOS commonly used in the natural gas industry. In addition, a comparison of BACKONE to simple cubic EOS is unnecessary because BACKONE has significantly better structure, and it should have superior performance. Cubic EOS are common choices in the chemical and gas industries to model complex phase behavior and to design chemical processes because of their computational simplicity and the relatively good results obtained.11-13 Although cubic EOS are used widely for phase equilibrium calculations in hydrocarbon fluids, they have drawbacks and limitations, such as low accuracy for volumetric properties, that reduce their applicability for complex mixtures such as natural gas.14 In a recent paper, Cismondi and Mollerup15 report a new, threeparameter, cubic EOS that combines the Redlich-Kwong and Peng-Robinson EOS density dependences, termed the RKPR EOS. One of the main goals of this EOS is improved volumetric correlations and predictions. We test this ability in this work by analyzing both volumetric and phase envelope correlations and predictions. The objective of this preliminary work is to check the capabilities of RK-PR EOS for predicting the properties of natural gas mixtures in comparison to some selected EOS whose reliabilities are well-known, and which the chemical industry uses. The selected models are the cubic threeparametric Patel-Teja EOS,16 which is a simple but frequently applied equation for industrial purposes, and the perturbed-chain SAFT (PC-SAFT)17 model, which is among the most recent modifications to the successful SAFT theory. Our purpose is to compare the quality of predictions of a simple EOS to a complex molecular-based EOS. In this work, we use the one-fluid monoparametric mixing rule. The binary interaction parameters, considered temperatureindependent, come from correlation of experimental vaporliquid equilibrium (VLE) data covering wide temperature and pressure ranges. We predict thermodynamic properties of light synthetic natural gases (SNG) with n-octane as the heaviest

10.1021/ie051241o CCC: $33.50 © 2006 American Chemical Society Published on Web 04/11/2006

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Figure 1. (a) Fugacity coefficient, φ, and (b) density, F, of methane. Symbols for literature values:20 (s) RK-PR EOS and (--) PC-SAFT EOS. (b) 200, (9) 240, ([) 280, (2) 320, (f) 360, (O) 400, and (0) 440 K.

component, and we report the binary interaction parameters. Highly accurate experimental densities and phase envelopes available in the open literature are compared to the predicted values. RK-PR Equation of State The RK-PR EOS purports to reduce the limitations of twoparameter, cubic EOS. The authors of RK-PR consider that any two-parameter cubic EOS must have the same drawbacks that can be corrected by introducing a third parameter. Cismondi and Mollerup combine a van der Waals-type attractive term with a three-parameter density dependence to connect the RedlichKwong and Peng-Robinson EOS. The expressions for the EOS and residual Helmholtz energy, Ares, from which all the derived properties could be calculated are

P)

a RT V - b (V + δ1b)(V + δ2b)

( )

(

(1)

)

V + δ1b b a Ares ) -ln 1 - ln RT V RTb(δ1 - δ2) V + δ2b

(2)

where

a ) ac

( )

(3)

1 - δ1 1 + δ1

(4)

δ2 )

3 2 + Tr

k

More detailed information about the calculation of pure fluid parameters appears in the original paper.15 The model parameters are a, b, and δ1; thus the extension to mixtures requires adequate mixing rules. Natural gases have minimal deviations from ideality; hence, the one-fluid model mixing rule should be adequate to describe their properties:7

am )

∑i ∑j

zizjaij )

∑i ∑j

zizj(aiaj)0.5(1 - kij)

(5)

bm )

∑i zibi

(6)

δ1,m )

∑i ziδ1,i

(7)

where kij are the binary interaction parameters and zi are the mole fractions of the liquid or vapor phases.

We do not consider more complex or multiparametric mixing rules, because we want to check if RK-PR constitutes an improved cubic EOS for natural gas systems. Many EOS produce better results without binary interaction parameters or more complex mixing rules, and this is one measure of the quality of these EOS.7,14,18 Results and Discussion Pure Fluids. Although the thermodynamic behavior of multicomponent mixtures is more difficult to describe theoretically than that of their constituent pure compounds, it is necessary that any model applied to a complex mixture describe properly the properties of the pure fluids. In particular, two of the main components of natural gas mixtures, methane and nitrogen, are supercritical under reservoir conditions; hence, their behavior is more difficult to describe with available models. Prediction of the properties of these compounds is important because methane comprises the major fraction of natural gases and nitrogen interacts significantly with hydrocarbons. Thus, we have calculated the densities and fugacities of methane and nitrogen as an initial test of suitability of the EOS. We calculate the properties over wide temperature and pressure ranges with the RK-PR EOS and compared them to those obtained from the three-parameter Patel-Teja16 EOS and with the molecularbased PC-SAFT17 EOS. Critical properties and acentric factors required in the calculations are from Poling et al.,19 whereas the PC-SAFT-specific parameters come from Gross et al.17 Calculated results for methane and comparison to the recommended IUPAC values20 appear in Figure 1. Figures 2 and 3 present the percentage deviations (eq 8) for methane and nitrogen, while the percentage average absolute (eq 9) and maximum deviations are in Table 1.

% Deviation ) 100

AAD )

100

N

∑ N r)1

propexp - propcalc propexp

(8)

|propexp,r - propcalc,r| propexp,r

(9)

Figure 1 reveals that RK-PR gives good predictions for methane fugacity and density over the pressure-temperature range considered, although PC-SAFT and PT (Figure 2) produce lower deviations. The quality of the predictions for all of the EOS decreases for P > 40 MPa. The results for nitrogen are similar with the predictions from RK-PR being better with those from PT and PC-SAFT being worse, Figure 3. RK-PR describes the supercritical behavior of these fluids well, and

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Figure 2. Percentage deviation between literature20 and RK-PR, PT, and PC-SAFT values for fugacity coefficient, φ, and density, F, for methane: (0) 200, (4) 240, (]) 280, (×) 320, (+) 360, (/) 400, and (-) 440 K.

the quality of the RK-PR predictions for these pure fluids is very close to that of the PC-SAFT EOS, but the PT EOS produces predictions superior to those from RK-PR. Predictive Ability of RK-PR EOS. As an initial test of RK-PR, we have calculated the vapor-liquid equilibrium of some selected binary systems involved in natural gas mixtures setting the corresponding kij ) 0. We compare the RK-PR predictions to those obtained from the PC-SAFT. Figure 4 presents experimental and calculated vapor-liquid equilibrium for methane + propane, methane + n-pentane, and methane + nitrogen. From Figure 4, we deduce that the purely predictive ability of PC-SAFT is almost equal to that of RK-PR for these systems. We note that when the asymmetry of the mixture increases, as for methane + n-pentane, the quality of RK-PR and PC-SAFT predictions remains almost the same. Figure 5 presents the VLE predictions for both EOS as a function of temperature. We see that the predictions from RKPR are slightly worse when the temperature increases, whereas PC-SAFT gives good predictions for low and high temperatures

and pressures. Although it is not apparent in Figures 4 and 5, the quality of the PT predictions is similar to those from PCSAFT. Considering the simplicity of RK-PR, this EOS maintains satisfactory predictive ability over wide temperature ranges. We conclude that RK-PR compares well to PC-SAFT for these natural gas constituent binary systems. While binary interaction parameters would improve the predictions from RKPR, it performs reasonably well without them. Binary Interaction Parameters. Using binary interaction parameters with the EOS provides more flexibility and reliability for multicomponent mixture calculations.7 Usually, these parameters result from least-squares minimization of experimental binary VLE data. In this work, we compare densities27 and phase envelopes28,29 for multicomponent mixtures to the EOS predictions. We have considered literature SNG mixtures with hydrocarbons no higher than n-octane (light SNG mixtures). Tables 2 and 3 contain the compositions of 10 light SNG mixtures considered in this work.

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Figure 3. Percentage deviation between literature21 and RK-PR, PT, and PC-SAFT values for fugacity coefficient, φ, and density, F, for nitrogen; (0) 200, (4) 250, (]) 300, (×) 350, (+) 400, and (/) 450 K. Table 1. Percentage Average Absolute Deviations for Fugacity Coefficient, AAD(O), and Density, AAD(G), and Maximum Deviation between Literature Values20,21 and Those Calculated with Different EOS AAD(φ) RK-PR PT PC-SAFT

AAD(F)

methanea

nitrogenb

methanea

nitrogenb

3.33 1.16 2.01

1.78 1.39 3.97

1.26 1.59 1.08

1.19 2.28 2.41

with the temperature ranges and the number of data points considered in the correlation of each binary system. In Table S1, we note that a wide temperature range is available for the correlation of binary interaction parameters. In this work, we consider these parameters temperature-independent. The objective function for fits to obtain binary interaction parameters is eq 10, the sum of the squared relative deviations of the K ()y/x) values for N data points for the two components.17,31

maximum deviation RK-PR PT PC-SAFT

N

methanea

nitrogenb

methanea

nitrogenb

-6.20 2.33 3.95

-6.00 7.31 9.97

9.04 6.88 2.50

3.75 -5.86 -4.57

Number of points ) 84, T range ) 200-440 K, P range ) 0.1-100 MPa. b Number of points ) 72, T range ) 200-450 K, P range ) 0.1100 MPa. a

These mixtures require 66 binary interaction parameters for the RK-PR. The PT and PC-SAFT parameters come from the literature.16,17,30 In Table S1 (Supporting Information), we report the literature sources of experimental binary VLE data together

Objfunc )

∑ i)1

[

]

Ki exp - Kicalc Ki exp

2

(10)

Parameter optimization results from a least-squares procedure combined with the Marquardt algorithm.32 Phase equilibrium compositions for binary systems and multicomponent mixtures use a Gibbs minimization procedure previously described.33 Table S2 (Supporting Information) contains calculated parameters together with AADs obtained from eq 9. From the analysis of Table S2 (Supporting Information), we infer that the correlative ability of RK-PR is high for these binary systems. The average values for AAD are 2.44 for K1 and 3.38 for K2;

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Figure 5. Vapor-liquid equilibrium for the binary system methane + ethane at different temperatures: (b) 158.15 (experimental data from ref 25), (9) 230.00 (experimental data from ref 26), and (2) 270.00 K (experimental data from ref 26); (;) values from RK-PR and (--) PCSAFT EOS with kij ) 0. Table 2. Composition of Light Synthetic Natural Gas Mixtures Considered in This Work for Density Predictionsa composition (mole fraction) component

SNG 1

SNG 2

SNG 3

SNG 4

SNG 5

methane ethane propane isobutane n-butane isopentane n-pentane n-hexane nitrogen carbon dioxide

0.812 99 0.032 94 0.006 37 0.001 01 0.001 00

0.812 03 0.043 06 0.008 94 0.001 48 0.001 55

0.858 98 0.084 99 0.022 96 0.003 51 0.003 47 0.000 51 0.000 53

0.135 75 0.009 94

0.057 03 0.075 92

0.010 07 0.014 98

0.965 80 0.018 15 0.004 05 0.000 99 0.001 02 0.000 47 0.000 32 0.000 63 0.002 69 0.005 89

0.906 44 0.045 53 0.008 33 0.001 00 0.001 56 0.000 30 0.000 45 0.000 40 0.031 34 0.004 66

a

Data obtained from ref 27.

Table 3. Composition of Light Synthetic Natural Gas Mixtures Considered in This Work for Phase Envelope Predictionsa composition (mole fraction)

Figure 4. Vapor-liquid equilibrium for the binary systems (a) methane + propane (T ) 213.71 K, experimental values from ref 22), (b) methane + n-pentane (T ) 273.16 K, experimental values from ref 23), and (c) methane + nitrogen (T ) 127.59 K, experimental values from ref 24). (b) Experimental values, (s) values from RK-PR and (-- PC-SAFT EOS with kij ) 0.

these deviations are much lower than those obtained with other cubic EOS and of the same order as those obtained from PCSAFT.17,30 RK-PR can correlate successfully very asymmetric binary mixtures from low to high temperatures and pressures (Figure 6) with a constant binary interaction parameter. RKPR and PC-SAFT correlate the complex phase behavior of mixtures with nonlinear alkanes (Figure 7) accurately. Additionally, RK-PR correlates the critical behavior of these binary mixtures well. So, it is possible to correlate with very good accuracy the critical locus of binary systems with this EOS and the simple one-parameter mixing rule. Natural Gas Density Predictions. Density is one of the most important properties required in the natural gas industry. The gas density frequently comes from three-parameter EOS such as RK-PR,18 although the quality of the predictions decreases with increasing temperature and/or pressure. In this work, we have tested density predictions obtained with RK-PR for some key natural gas mixtures whose PVT behavior is available in

component

SNG 6

SNG 7

SNG 8

SNG 9

SNG 10

methane ethane propane isobutane n-butane isopentane n-pentane n-hexane n-heptane n-octane nitrogen carbon dioxide

0.691 14 0.026 2 0.004 23 0.001 05 0.001 04 0.000 34 0.000 23 0.001 10

0.904 183 0.080 38 0.008 01 0.000 81 0.001 23 0.000 1 0.000 079 0.000 047 0.000 011

0.84446 0.08683 0.03297 0.00293 0.00589 0.00084 0.00086 0.0005

0.015 59 0.259 08

0.003 13 0.002 02

0.00772 0.017

0.881 882 0.027 2 0.008 5 0.001 7 0.003 2 0.000 85 0.000 94 0.001 19 0.000 258 0.000 18 0.069 0.005 1

0.833 482 0.075 26 0.020 09 0.003 05 0.005 2 0.001 2 0.001 44 0.000 68 0.000 138 0.000 11 0.056 51 0.002 84

a

Data obtained from refs 28 and 29.

the literature with state-of-the-art accuracy.27 Table 2 contains the compositions of the selected multicomponent mixtures; these are light synthetic natural gas mixtures whose heaviest component is n-hexane. Figure 8 compares the experimental densities for SNG1 and SNG2 mixtures to those obtained from the RK-PR. We infer that RK-PR describes properly the PVT behavior of these complex mixtures in the wide pressure-temperature range considered. Table 4 reports the percentage average absolute and maximum deviations obtained with RK-PR, PT, and PCSAFT, we have also included in this table the predictions obtained with the AGA8-DC92 model for comparative purposes. AGA8-DC92 gives rise to lower deviations among the tested models, as we might expect for such a complex

Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3689 Table 4. Percentage Average Absolute (AAD) and Maximum (MAX) Deviations between Experimental and Calculated Densities for SNG1-SNG5 Mixturesa SNG1

SNG2

SNG3

SNG4

SNG5

AAD MAX AAD MAX AAD MAX AAD MAX AAD MAX RK-PR PT PC-SAFT AGA8 a

Figure 6. Vapor-liquid equilibrium for the binary system methane + n-heptane at different temperatures: (b) 277.59 (experimental data from ref 34) and (9) 410.93 (experimental data from ref 34); (s) values from RK-PR (kij ) 0.00200) and (--) PC-SAFT EOS (lik ) 0.001 60 17).

Figure 7. Vapor-liquid equilibrium for the binary system propane + isobutane at different temperatures: (b) 266.54, (9) 299.82, (2) 338.71, and ([) 366.48 (experimental data from ref 35); (s) values from RK-PR (kij ) 0.0040) and (--) PC-SAFT EOS (kij ) 0.0026, this work).

multiparametric model, within the three other models considered PC-SAFT has the lowest deviations, but predictions from RKPR are only slightly worse and substantially better than those from PT. Figure 9 shows the percentage deviations for SNG1 using RK-PR and PC-SAFT (SNG2-SNG5 possess similar trends); RK-PR has higher deviations for the low-temperaturehigh-pressure regions, while PC-SAFT has greater deviations in the high-temperature ranges. RK-PR appears to be a notable improvement for cubic EOS. It is well-known that other cubic EOS are not good choices for predicting densities of complex systems. However, RK-PR appears to predict densities in these complex multicomponent mixtures with an accuracy close to that of PC-SAFT.36

1.60 2.58 1.03 0.04

4.81 8.20 2.80 0.24

1.17 -3.68 1.98 5.25 2.37 2.37 7.69 3.05 9.05 3.61 1.05 -2.35 1.31 3.26 1.11 0.08 0.34 0.07 0.22 0.03

5.91 9.51 3.51 0.11

2.16 5.33 3.44 9.43 1.24 -3.82 0.05 -0.39

Experimental values obtained from ref 27.

Natural Gas-Phase Envelope Predictions. Prediction of phase envelopes in multicomponent mixtures such as natural gas is a stringent test for any theoretical model. It is a common practice in the chemical and gas industries to calculate phase envelopes using simple two-parameter EOS such as PengRobinson. Most software for process design also uses these simple models. An analysis of the scarce, highly accurate experimental phase envelopes available in the literature shows that these EOS do not produce accurate predictions, with errors up to 5 K at the cricondentherm and up to 3 MPa at the cricondenbar.14 These errors are too high for most practical purposes, and many industrial designs are overcompensated. In our opinion, it is not legitimate to apply these EOS to such systems. Here, we test the RK-PR phase envelope predictions, comparing them to those obtained from the PT and PC-SAFT. Figure 10 shows the phase envelopes calculated from the different EOS for SNG6-SNG10. We have calculated the full phase envelope, from low to high temperatures, although experimental literature data only cover the dew points, to illustrate the different behavior of the three EOS. The phase envelope of SNG6, Figure 10a, reveals huge differences from the three EOS. All the EOS predict dew points with almost the same good quality, but the differences in the cricondenbar (Table 5) and in the bubble point curve are remarkable. Because no literature data exist for the bubble point curve, we cannot deduce which of the EOS is more suitable for this sample, although it seems that RK-PR and PC-SAFT produce better results, the shape of the phase envelope predicted by PT shows a strange curvature and an anomalous behavior near the predicted cricodenbar. Figure 10b contains predictions for SNG7, and the results for the dew point curve are essentially equal from the three EOS, but none of the EOS can predict the cricondenbar for this system. RK-PR has lower deviations (Table 5), and PC-SAFT provides the worst cricondenbar prediction for this system. All EOS predict the low-temperature bubble point curve equally. Results for SNG8 plotted in Figure 10c show that all the EOS predict the phase envelope well. Once again, however, higher

Figure 8. PVT behavior of (a) SNG1 and (b) SNG5 samples. (b) Experimental data from ref 27. Surface is calculated with RK-PR EOS.

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Figure 9. Percentage deviations for SNG1 sample with (a) RK-PR and (b) PC-SAFT EOS. Experimental data from ref 27. Surface is a smoothing of the deviations.

Figure 10. Phase envelopes for (a) SNG6, (b) SNG7, (c) SNG8, (d) SNG9, and (e) SNG10: (9) experimental data form refs 28 and 29; (s) values from RK-PR, (--) values from PT; and (s -s) values from PC-SAFT.

deviations appear at the cricodenbar, but for this sample PT and RK-PR are not too far removed from the experimental results. For the low-temperature bubble points, the predictions from all three EOS are almost identical.

The predictions for SNG9, Figure 10d, are not very good. The three EOS only produce good results for the dew points at low pressures. When the pressure increases and approaches the cricondenbar, huge differences appear with all the EOS, although

Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3691 Table 5. Calculated Cricodentherm and Cricodenbar for SNG6-SNG10 Samples with Different EOSa SNG6 P/MPa

SNG7 T/K

P/MPa

SNG8 T/K

P/MPa

SNG9 T/K

P/MPa

T/K

P/MPa

T/K

3.9619 3.7805(4.58) 3.3734(14.85) 3.4764(12.25)

277.36 277.23(0.04) 274.45(1.00) 276.81(-0.86)

4.5573 4.2332(7.11) 3.6186(20.60) 3.7302(18.15)

273.59 274.11(-0.19) 270.05(1.48) 272.92(-1.06)

243.07 10.5968 246.68(-1.49) 9.8391(7.15) 239.71(2.83) 8.9883(15.18) 242.04(-0.97) 8.8455(16.53)

244.58 239.42(2.11) 237.13(0.96) 239.01(-0.79)

9.2361 9.4804(-2.65) 8.5799(7.11) 8.2040(11.17)

238.63 243.49(-2.04) 239.38(1.69) 243.08(-1.55)

Cricodentherm 3.5616 252.23 3.1128 228.93 5.0106 261.31 exptlb RK-PR 3.8061(-6.87) 253.15(-0.35) 3.8800(-24.65) 229.65(-0.31) 5.0427(-0.64) 262.79(-0.57) PT 3.2162(9.70) 250.66(0.97) 3.4719(-11.54) 225.57(1.78) 4.5760(8.67) 258.64(1.58) PC-SAFT 3.3839(4.99) 251.81(-0.45) 3.4790(-11.76) 228.18(-1.16) 4.5196(9.80) 260.61(-0.76) exptlb RK-PR 9.3494 PT 7.3832 PC-SAFT 8.1452 a

219.72 233.94 219.64

6.9753 6.3687(8.70) 6.1518(11.81) 5.8247(16.50)

219.60 219.97(-0.17) 215.51(2.03) 217.54(-0.95)

Cricodenbar 8.1670 8.5141(-4.25) 7.9668(2.45) 7.4932(8.25)

SNG10

Parenthesized values are % deviations defined according to eq 8. b Calculated from data in refs 28 and 29.

the predictions obtained with RK-PR are better than those obtained with the other EOS (Table 5). As with the other samples, all the EOS predict the low-temperature bubble point curve behavior. The phase envelopes for SNG10 in Figure 10e indicate that RK-PR gives the best results among the three EOS. While PT and PC-SAFT produce low deviations for the dew point curve, when the pressure increases to the cricodenbar, they have large deviations. From the analysis of phase envelope predictions for SNG6SNG10 using the EOS, we arrive at several conclusions: (i) RK-PR appears to be superior compared to PT and PCSAFT. (ii) The three EOS predict for the low-pressure dew point curves and cricodentherms with almost the same quality, although RK-PR produces values closer to the experimental ones (Table 5). (iii) Cricodenbars are usually underestimated by all the EOS, but RK-PR predicts cricodenbars with the lowest errors. (iv) Predictions for the cricodentherm for heavier samples, such as SNG7 (with n-heptane) and SNG9-SNG10 (with n-heptane and n-octane), are worse than for the lighter samples with the three EOS especially for the cricodenbar. (v) Predictions for the low-temperature bubble points are almost the same with the three EOS except for sample SNG6 for which PT show huge differences from the other two EOS. (vi) The presence of higher alkanes, e.g. in samples SNG7, -9, and -10, has little effect upon the densities, but they have a strong effect upon the phase envelope of the mixtures and decrease the accuracy of predictions with the studied models. We believe that RK-PR EOS exhibits clear superiority over the other two EOS studied, although its cricodenbar highpressure predictions are not stellar, principally for systems with heavy hydrocarbons. Concluding Remarks RK-PR is a simple, three-parameter cubic EOS that may be as an alternative to other engineering cubic EOS. It produces reliable predictions for pure component properties over wide temperature and pressure ranges; its correlative ability for VLE in binary systems using the single parameter, one-fluid mixing rule is high compared to complex molecular models such as PC-SAFT. Density predictions obtained with RK-PR are better than those obtained with other common EOS, such as the threeparameter PT, and of the same quality as those obtained with PC-SAFT. This result is very important because usually molecular-based EOS are better choices for density predictions in multicomponent mixtures than cubic EOS such as RK-PR.

Phase envelope predictions with RK-PR are better than those with the other EOS considered, but predictions in the highpressure ranges close to the cricodenbar are less good. This effect occurs for all the EOS considered. We consider this as the main drawback of the RK-PR EOS, as well as the other EOS, but this EOS exhibits a clear superiority over the others even in the cricodenbar region. Thus, RK-PR is potentially a more accurate alternative to traditional cubics and to complex molecular-based EOS. It is a simple EOS that requires little computational effort and which produces better correlations and predictions for pure fluids and binary and multicomponent mixtures than the EOS usually considered for chemical engineering purposes. Acknowledgment We gratefully acknowledge financial support from the following: the USA-Spain Fulbright Commission and Ministerio de Educacio´n y Ciencia (Spain) (S.A.M.), Texas A&M University, and the Texas Engineering Experiment Station. Supporting Information Available: Literature sources for binary vapor-liquid equilibrium optimization (Table S1) and binary interaction parameters for RK-PR EOS and one-fluid mixing rule (Table S2). This material is available free of charge via the Internet at http://pubs.acs.org. List of Symbols a ) attractive force constant in RK-PR EOS, co-pressure b ) repulsive force constant in RK-PR EOS, co-volume k ) pure component parameter of RK-PR EOS kij ) binary interaction parameter N ) number of data points P ) pressure propexp,r ) experimental property propcalc,r ) EOS calculated property R ) gas constant T ) temperature Tr ) reduced temperature V ) molar volume xi ) mole fraction in the liquid phase of i compound yi ) mole fraction in the vapor phase of i compound zi ) mole fraction in the liquid or vapor phase of i compound Z ) compressibility factor Greek Letters δ1 ) third parameter in RK-PR EOS δ2 ) function of δ1 φ ) fugacity coefficient F ) density

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ReceiVed for reView November 8, 2005 ReVised manuscript receiVed February 15, 2006 Accepted March 9, 2006 IE051241O