Thermodynamic Consistency Test of Experimental Data - American

Mar 18, 2011 - Thermodynamics Research Unit, School of Chemical Engineering, University of ... Campus, King. George V Avenue, Durban 4041, South Afric...
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Wax Solubility in Gaseous System: Thermodynamic Consistency Test of Experimental Data Amir H. Mohammadi,*,†,‡ Ali Eslamimanesh,† and Dominique Richon† † ‡

nergetique et Procedes (CEP/TEP), 35 Rue Saint Honore, 77305 Fontainebleau, France MINES ParisTech, Centre E Thermodynamics Research Unit, School of Chemical Engineering, University of KwaZulu-Natal, Howard College Campus, King George V Avenue, Durban 4041, South Africa ABSTRACT: Wax deposition from natural gas can cause severe problems in production, transmission and processing operations. Accurate knowledge of wax solubility in natural gas system is required to avoid operating problems. Unfortunately, experimental measurements of solubilities of these compounds in gaseous systems are quite challenging. This is partly because concentrations of these solid substances in the gas phase are extremely low; this generally may result in generation of highly uncertain experimental data. In this paper, we present a thermodynamic consistency test based on the Gibbs-Duhem equation to determine the reliability of experimental solubility data of paraffin waxes (n-C24H50 to n-C33H68) available in the open literature. This test uses the PengRobinson equation of state and two-fluid van der Waals (vdW2) mixing rules to represent the solubilities of solid waxy compounds in supercritical CO2 and ethane. The results show that all the investigated experimental data that are well represented by the applied thermodynamic model seem to be thermodynamically consistent.

’ INTRODUCTION An ever-growing demand for natural gas has made it one of the significant products of the petroleum industry. Natural gas is considered to be an environmentally friendly clean fuel, offering important environmental benefits when compared to other fossil fuels.1 It is also a relatively safe source of energy when transported, stored, and used.1 Certain pressure and temperature variations in some natural gas production fields may result in precipitation/deposition of some heavy hydrocarbons, which are normally extracted by supercritical components of natural gases in the reservoirs. One important fraction, which has great deposition potential during production, transmission, and processing, is wax. Paraffin waxes are mixtures of alkanes normally in a homologous series of chain lengths, typically in the range n-C15-n-C50.2-4 Wax deposition in oil systems has been the subject of many experimental and theoretical studies.4,5 Wax precipitation from gas condensates has been investigated by a few researchers.3,6-8 Consequently, there is indeed very limited information on wax deposition/solubility in natural gas systems.9 The deposition of waxy compounds in natural gases is of interest in the development of production from some gas reserves such as deep wells or acid gas fields.9 Failure to account for the presence of these compounds can lead to significant increases in operating costs, lost production, or failure of piping systems.9 Overdesign of facilities to account for waxy compounds can bring about higher capital and operating costs, and may prevent the development of otherwise economic natural gas resources.9 Accurate experimental data and reliable thermodynamic models on solubility of wax in natural gas system are required to avoid the stated problems. In this work, we focus on modeling the solubilities of paraffin waxes in supercritical CO2 and ethane and investigating the reliability of the iterature data applying a thermodynamic consistency test. It is expected that the current study provides a better understanding of wax solubility in natural gas systems. r 2011 American Chemical Society

’ THERMODYNAMIC CONSISTENCY TEST The thermodynamic relationship that is frequently used to analyze thermodynamic consistency of experimental phase equilibrium data is the fundamental Gibbs-Duhem equation.10-12,14,15 This equation, as presented in the literature, interrelates the activity/fugacity coefficients of all components in a given mixture. If this equation is not obeyed within the defined criteria, then the data are declared to be thermodynamically inconsistent. This means that this relation imposes a constraint on the activity/fugacity coefficients that is not satisfied by the experimental data.10-12 This is due to various errors occurring during experimental works, especially those dealing with high pressure and very low solubilities.13 The ways in which the Gibbs-Duhem equation10-12,14,15 is arranged and applied to the experimental data have resulted in the origination of several “consistency test methods”, most of them designed for low-pressure data. Among these are the “slope test”, the “integral test”, the “differential test”, and the “tangent-intercept test”.10-12,14-17 Good reviews of these methods are given elsewhere.14,15 In the past decade, Valderrama and co-workers17-21 have investigated the applications of numerical thermodynamic consistency methods to various systems including incomplete phase equilibrium data of high-pressure gas-liquid mixtures,17 highpressure ternary mixtures of compressed gas and solid solutes,18 high-pressure gas-solid solubility data of binary mixtures,19 vaporliquid equilibrium data for mixtures containing ionic liquids,20 and high-pressure gas-liquid equilibrium data including both liquid and vapor phases.21 Recently, Eslamimanesh et al.12,22,23 Received: November 1, 2010 Accepted: February 14, 2011 Revised: February 5, 2011 Published: March 18, 2011 4731

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Industrial & Engineering Chemistry Research have applied almost the same approach for performing the thermodynamic consistency test on significant systems encountered in oil and gas industries, e.g., the water content of methane in equilibrium with gas hydrate, liquid water, or ice,12 the sulfur content of hydrogen sulfide,22 and the solubility data of carbon dioxide and methane with water inside and outside the gas hydrate formation region.23 A method based on rewriting the Gibbs-Duhem equation10-12,14,15 in terms of fugacity coefficients,24 has been used in this work. The consistency method employed here can be considered as a modeling procedure. This is because a thermodynamic model that can accurately represent the experimental data (i.e., the average deviations of the model results from experimental values are within the acceptable range according to the studied system and for a desired purpose) must be used to apply the consistency test.12,22,23 The fitting of the experimental data requires the calculation of some model parameters using a defined objective function that must be minimized.12 As stated by Valderrama and Alvarez17 and Eslamimanesh et al.,12,22,23 a good consistency test method to analyze highpressure data must fulfill 10 basic requirements:12,17,22,23 (i) It must use the Gibbs-Duhem equation.10-12,14,15 (ii) It must use the fundamental equation of phase equilibrium. (iii) It must use for testing all the experimental P-T-y (pressure-temperature-gas phase composition) data available. (iv) It does not necessarily require experimental data for the whole concentration range and be applicable for data in any range of concentration. (v) It must be able to correlate the data within acceptable limits of deviations, deviations that must be evenly distributed. (vi) It requires few additional calculated properties. (vii) It must be able to detect erroneous experimental points. (viii) It makes appropriate use of necessary statistical parameters. (ix) It must be simple to be applied, with respect to the complexity of the problem to be solved. (x) It must be able to conclude about consistency with regard to defined criteria. Equations. The Gibbs-Duhem10-12,14,15 equation for a binary homogeneous mixture at constant temperature can be written as12,17-25 " # vE ð1Þ dP ¼ y1 dðln γ1 Þ þ y2 dðln γ2 Þ RT where vE is the excess molar volume, T represents temperature, R stands for the universal gas constant, γ is the activity coefficient, y represents the solute mole fraction, P stands for pressure, and “d” is the derivative symbol. In this equation, subscripts “1” and “2” refer to components 1 and 2 in the present phases, respectively. Equation 1 is rewritten in terms of fugacity coefficients as follows:12,17-23   Z-1 ð2Þ dP ¼ y1 dðln j1 Þ þ y2 dðln j2 Þ P where Z is the compressibility factor and j stands for the fugacity coefficient in the related phase.

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This equation can be written in terms of the composition of paraffin wax in the gas phase. If the latter compound is considered as species 2 in the binary mixture of paraffin wax þ supercritical CO2/ethane, the latter equation becomes12,17-23 1 dP y2 dðln j2 Þ 1 - y2 dðln j1 Þ ¼ þ P dy2 Z - 1 dy2 Z - 1 dy2

ð3Þ

or in integral form: Z Z Z 1 1 1 - y2 dP ¼ dj2 þ dj ð4Þ Py2 ðZ - 1Þj2 y2 ðZ - 1Þj1 1 j1, j2, and Z can be calculated using an equation of state and suitable mixing rules (thermodynamic model). In eq 4, the left-hand side is designated by Ap and the righthand side is designated by Aj, as follows:12,17-23 Z 1 Ap ¼ dP ð5Þ Py2 Aj ¼ Aj1 þ Aj2 Z Aj1 ¼

ð6Þ

1 - y2 dj y2 ðZ - 1Þj1 1

ð7Þ

1 dj ðZ - 1Þj2 2

ð8Þ

Z A j2 ¼

Thus, if a set of data is considered to be consistent, Ap should be equal to Aj within acceptable defined deviations. To set the margins of errors, a percent area deviation (ΔAi%) between experimental and calculated values is defined as12,17 " # Aji - Api ΔAi % ¼ 100 ð9Þ Api where i refers to given experimental data. The maximum values accepted for these deviations regarding the proposed systems are calculated using appropriate mathematical procedure. Thermodynamic Model. For phase equilibrium calculations, the equality of the fugacity of pure solute to its fugacity in supercritical fluid has been assumed, i.e.26 pure solid

fi

supercritical

¼ fi

ð10Þ

where f refers to the fugacity and i stands for the ith component in the mixture. In this study, the derivation of the required equations is based on the following assumptions:26 1. The supercritical fluid is assumed to be insoluble in the solid (solute-containing) phase. 2. The fugacity of pure solid i stands for the fugacity of the solute i in the mixture. 3. The molar volume of the solute is pressure-independent. 4. The solid phase is incompressible. 5. The fugacity coefficient of the solute at saturation is unity. Therefore, eq 10 is rewritten as s  vi ðP - Pisat Þ sat ð11Þ ¼ yi j i P Pi exp RT where vs is the solid molar volume, superscript “sat” stands for saturation conditions, and yi and ji are the mole fraction and fugacity of the solute i in supercritical phase, respectively. 4732

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Table 1. Experimental Data Ranges Used for the Consistency Test in This Work range of data Na

system

T (K)

P (MPa)

y2  106 (mole fraction)c

refb

mixtures with supercritical CO2 n-C24H50

9

310

8.85-26.06

544-1090

9

n-C25H52

8

308-313

10.36-20.75

215-1000

9

n-C28H58

13

308-318.15

7.52-22

10.1-435

9, 31

n-C29H60

6

308

7.57-21.51

11.4-66.2

9

n-C28H58

6

308.1

6.57-20.20

1890-15200

9

n-C29H60 n-C30H62

5 10

308.1 308.1-313.1

6.47-16.67 6.57-20.20

2320-14200 594-3200

9 9

n-C32H66

16

308.1-319.1

6.57-20.20

149-2180

9

n-C33H68

16

308.1-318.1

6.47-20.20

183-2970

9

mixtures with supercritical ethane

a

Number of data points. b Reference of experimental data. c Wax solubility

To accurately evaluate the fugacity coefficient of the solute, reliable equations of states are needed. The cubic equations of state (CEoSs) have been widely used in phase equilibrium calculations due to their application simplicity. It was previously shown9 that the Peng-Robinson (PR)27 equation of state (EoS) with two-fluid van der Waals (vdW2) mixing rules28 leads to reliable results for calculation of the solubility of paraffin wax in supercritical CO2 and ethane. Methodology. The following algorithm is applied for the thermodynamic consistency test:12,17,22,23 1. Determine Ap from eq 5 using the experimental P-T-y data. Use a numerical integration for this purpose. In this work, Simpson’s 3/8 rule29 was used. Valderrama and Alvarez17 have demonstrated that the deviations between the calculated values of the integrals by the simple trapezoidal integration rule and a fitted polynomial function are below 2%. Therefore, a simple numerical integration method, e.g., the trapezoidal rule, can be applied for the cases when there are only two available experimental data points.12,22,23 2. Evaluate Aj by eqs 6-8 using the obtained values for j2 and Z from the thermodynamic model for the proposed system and y2 from experimental data. 3. For every set of the experimental data, determine an absolute percent area deviation (ΔAi%) between experimental and calculated values using eq 9. Consistency Criteria. First and perhaps most important is the fact that the thermodynamic model should lead to the average absolute deviations of the results from experimental values to be within the acceptable range.12 In this work, the accepted deviations in gas-phase mole fraction calculations (defined by the following equation), are considered to lie between 0 and 25%:18 exp

AD% ¼ 100

jycal i - yi j exp yi

ð12Þ

where superscripts “cal” and “exp” refer to calculated and experimental values, respectively. For determination of the acceptable percentages of the two evaluated areas deviations from each other, the error propagation was performed on the existing experimental data. This was done using the general equation of error propagation,31 considering the temperature and mole fraction of paraffin wax in supercritical

CO2 and ethane phase as the independent measured variables.17 The calculated individual area (Aj) is the dependent variable of interest. The error in the calculated areas, EA and the percent error EA% are calculated as follows:12,19,22,23 " # " # DAjj DAjj EA ¼ ΔT þ Δy ð13Þ DT Dy 2  3 E   A EA % ¼ 1004 5 Ajj 

ð14Þ

where subscript j refers to the jth calculated area. We assume maximum uncertainties of 0.3 K for the experimental temperature and 5% for the experimental solubility data.9 However, these uncertainties depend on the method of experimental measurements; e.g., the method used by Teja and co-workers9 is based on dynamic method. The maximum acceptable errors are much dependent on the uncertainty of solubility measurements, and one can also neglect the first right-hand-side term of eq 13. However, the uncertainty for the measurement of the solubility of paraffin wax is high, and that is why it is justified to perform our thermodynamic consistency test on such data. The partial derivatives of the two preceding equations have been evaluated using the central finite difference29 method. It results in the relative average absolute deviations range between 0 and 22% for the data of solubilities of investigated compounds in supercritical CO2 and ethane. Therefore, the range [0, 22]% is established as the maximum acceptable error for the areas ([Ai]) of the left- and right-hand sides of eq 6. Regarding these facts, the thermodynamic consistency test criteria are applied through the following instructions:12,17-23 1. Check the percentage Δy2 to see that it is not outside the margins of errors [0, 25]%. If it is, change the thermodynamic model or eliminate the weak predictions until the absolute deviations of the results from experimental values are within the acceptable range. In this work, we have not changed the thermodynamic model as it is demonstrated to bring about accurate calculation results for most of the investigated data. 4733

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Table 2. Sublimation Pressures of the Investigated Paraffin Waxes at Different Temperatures9 T (K)

substance

Table 4. Acentric Factor (w) and Critical Properties (Tc and Pc) of the Investigated Pure Compounds9

sublimation press. (MPa) -11

compound

Tc (K)

Pc (MPa)

ω

n-C24H50

310

4.90  10

C2H6

305.32

4.872

0.099

n-C25H52

308 313

1.58  10-11 5.39  10-11

CO2 n-C24H50

304.19 809.96

7.382 1.0496

0.2276 1.032

n-C28H58

308

1.71  10-13

n-C25H52

818.56

1.0256

1.066

318

2.67  10-12

n-C28H58

842.11

0.9694

1.163

325

1.68  10-11

n-C29H60

849.29

0.9549

1.195

n-C29H60

308

1.14  10-13

n-C30H60

856.17

0.9421

1.226

n-C30H66

308

1.50  10-14

n-C32H66

869.12

0.9208

1.287

313

6.46  10-14

n-C33H68

875.22

0.9119

1.317

308 313

1.02  10-15 4.74  10-15

n-C32H66

n-C33H68

-14

318

2.12  10

319

2.86  10-14

308

1.15  10-15

313

5.43  10-15

318

2.44  10-14

Table 3. Solid Molar Volumes (vs) of the Paraffin Waxes9 solute

vs  103 (m3/mol)

n-C24H50

0.4238

n-C25H52

0.4513

n-C28H58

0.4894

n-C29H60

0.5058

n-C30H66

0.5222

n-C32H66

0.5550

n-C33H68

0.5714

2. If the model correlates the data within the acceptable error ranges of the calculations and the area test is fulfilled for all points in the data set, the proposed model is reliable and the data are thermodynamically consistent. 3. In the case that the model acceptably correlates the data, which are not proved yet to be thermodynamically consistent, and the area test is not accomplished for most of the data set (more than 75% of the areas), the applied model is conclusive but the experimental data are considered to be thermodynamically inconsistent. 4. In the case that the model acceptably correlates the data and some of the area deviations (e25% of the areas) are outside the error range [0,22]%, the applied method declares the experimental values as being not fully consistent. 5. The determined data in the previous step could be further analyzed to check, if after eliminating some points, the remaining data fulfill the criteria described before and these remaining data are consistent or inconsistent. Experimental Data. We have reviewed the existing data related to solubilities of paraffin waxes in supercritical natural gas fluids. Most of the data (alkanes from n-C24 to n-C33) have been reported in GPA Research Report 1719 by Prof. Teja’s group of the Georgia Institute of Technology. It should be noted that there are some other data sets such as solubilities of triacontane and n-C36H68 in supercritical carbon dioxide,9,31 which are not used in this study due to lack of the required experimental properties for modeling including solid molar volumes and sublimation pressures. Eighteen (isothermal)

experimental data sets have been investigated for the consistency test. Table 1 summarizes the ranges of the data along with the references. The sublimation pressures and solid molar volumes of the paraffin waxes are reported in Tables 2 and 3, respectively. Table 4 shows the acentric factor and critical properties of the compounds investigated in this study.

’ RESULTS AND DISCUSSION The results of calculations of solubilities of investigated paraffin waxes in supercritical CO2 and ethane are shown in Table 5. The tuned binary interaction parameters using the proposed thermodynamic model are also reported in Table 5. It is inferred that the applied model is able to represent many of the experimental solubility values within the acceptable absolute deviation range of approximately [0, 25]% requested for a successful consistency test. Table 6 reports the results of the thermodynamic consistency test for solubilities of investigated paraffin waxes in supercritical CO2 and ethane. It should be noted that the data sets for which the proposed thermodynamic model does not lead to the deviations within the acceptable range are ignored for the consistency test. The results show that all of the studied experimental data that are well represented by the applied thermodynamic model seem to be thermodynamically consistent. This fact demonstrates the capability of the experimental procedure to measure these solubilities using the dynamic method in spite of several difficulties in such measurements. Another element inferred from the test results is that these measurements have been done with careful calibration of the measuring devices such as pressure transducers and temperature probes by the group of Prof. Teja. Furthermore, the results of such a test introduce a procedure to select the experimental data by which a thermodynamic model is supposed to be tuned and optimal values of the model parameters are supposed to be obtained. Thermodynamically inconsistent data (sometimes not fully consistent data) used for tuning of the models will bring about inaccurate predictions of the model in further applications and the cause of such deviations may not be easily determined. The final point to consider is that the data on which the proposed thermodynamic consistency test was applied should be reported as isotherms because the main assumption in the development of eqs 1-8 is similar to that assumed in developing the original Gibbs-Duhem equation10-12,14,15 at constant temperature. This fact assigns some limitations to choosing the experimental data sets for the consistency test by the applied procedure (constant temperature) especially for scarce data of the solubilities of paraffin waxes in natural gas systems. One way of solving this problem of few data is to generate more data in a 4734

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Table 5. Results of Tuning the Thermodynamic Model system

T (K)

P (MPa)

6 yexp 2  10

6 ycalc 2  10

(mole fraction)

(mole fraction)

kijc

lijc

0.118

0.267

ARDa (%)

refb

14.1

9

mixtures with supercritical CO2 n-C24H50

310

8.85

544.0

620.7

11.03

572.0

651.5

13.9

14.31

806.0

918.4

13.9

15.74

863.0

985.5

14.2

17.53

926.0

1053.4

13.8

19.18 22.62

999.0 1008.0

1139.0 1149.0

14.0 14.0

24.41

996.0

1132.3

13.7

26.06

1090.0

1242.6

14.0

10.36

215.0

236.2

12.68

383.0

422.1

15.20

574.0

629.4

9.7

20.23 10.34

602.0 321.0

663.3 324.3

10.2 1.0

13.79

659.0

715.3

8.5

16.82

1000.0

1086.5

8.7

20.75

952.0

1043.8

7.52

10.1

11.1

10.00

39.0

42.7

9.5

14.98

52.3

57.0

9.0

17.51 19.98

64.7 68.9

70.9 75.1

9.6 8.9

14.0 n-C25H52

308

313

n-C28H58

308

318

n-C28H58

308.15

318.15

n-C29H60

308

0.184

0.438

9.9

9

10.2

0.200

0.400

9.6 0.118

0.308

9.7

21.51

72.4

79.9

11.96

172.0

189.8

12.67

207.0

227.9

10.1

15.65

240.0

263.9

10.0

16.58

307.0

336.5

9.6

17.69

390.0

428.2

9.8

20.24 8.00

435.0 12.3

439.5 1.9

9.00

34.5

46.4

34.6

10.00

51.7

73.2

41.5

11.00

54.6

82.3

50.8

12.00

59.1

81.2

37.4

15.00

68.3

56.2

17.7

18.00

81.3

31.8

60.9

20.00 22.00

93.6 99.0

20.7 13.2

77.9 86.7

10.50

39.6

42.7

11.00

76.0

69.3

12.50

127.8

129.6

1.4

13.50

179.0

146.6

18.1

9

10.3 0.112

0.035

0.036

0.284

-0.031

-0.033

10.4

1.0 84.2

7.7

9

31

31

8.8

7.57

11.4

15.0

10.07

19.8

25.1

0.118

0.308

27.0

32.0

12.65 17.51

31.4 38.1

40.8 50.7

30.0 33.0

20.03

45.5

60.1

32.0

21.51

66.2

87.1

31.6

9

mixtures with supercritical ethane n-C28H58

308.1

6.57

1890.0

2234.0

10.10

3380.0

3985.7 4735

-0.006

0.190

18.2

9

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Table 5. Continued system

n-C29H60

n-C30H62

T (K)

308.1

308.1

313.1

n-C32H66

308.1

313.1

319.1

n-C33H68

308.1

313.1

318.1

a

6 yexp 2  10

6 ycalc 2  10

P (MPa)

(mole fraction)

(mole fraction)

12.02

6430.0

7716.0

20.0

13.64

7530.0

8985.2

19.3

16.67

10080.0

12092.0

20.0

20.20

15200.0

18237.0

6.47

2320.0

2767.1

10.20

4320.0

5178.4

19.9

12.12

8290.0

9840.2

18.7

13.64 16.67

9910.0 14200.0

11786.0 16969.0

18.9 19.5

kijc

lijc

ARDa (%)

refb

20.0 0.016

-0.048

0.123

0.024

19.3

6.57

549.0

667.0

10.10

1240.0

1483.7

19.7

21.5

12.02

1450.0

1744.1

20.3

13.64

1710.0

2066.4

20.8

16.67

2240.0

2706.8

20.8

20.20

3200.0

3852.5

6.57 10.10

486.0 1450.0

610.8 1801.0

12.02

1450.0

1832.8

13.64

1710.0

2171.4

6.57

216.0

258.4

10.10

713.0

855.6

20.0

12.02

801.0

962.9

20.2

13.64

959.0

1140.5

18.9

16.67 20.20

1260.0 1810.0

1527.1 2184.7

21.2 20.7

9

9

20.4 -0.079

-0.111

25.7 24.2

9

26.4 27.0 -0.083

0.111

-0.067

-0.218

19.7

6.57

177.0

208.9

10.10

933.0

1110.1

18.0 19.0

12.02

1150.0

1370.8

19.2

13.64

1440.0

1709.7

18.7

16.67

1730.0

2062.2

19.2

20.20

2180.0

2581.1

6.57 10.10

149.0 1280.0

196.8 1675.1

12.02

1440.0

1889.3

13.64

2140.0

2818.0

6.47

371.0

441.4

10.20

963.0

1155.6

20.0

12.12

1140.0

1359.7

19.3

13.64

1360.0

1627.1

19.6

16.67 20.20

1640.0 2370.0

1966.9 2846.4

19.9 20.1

9

9

18.4 -0.154

-0.455

32.1 30.9

9

31.2 31.7 -0.026

-0.032

-0.029

-0.080

19.0

6.47

288.0

344.4

10.20

1540.0

1844.9

19.6 19.8

12.12

1470.0

1754.9

19.4

13.64

1720.0

2062.1

19.9

16.67

2240.0

2658.2

18.7

20.20

2930.0

3508.4

6.47 10.20

183.0 1540.0

239.2 2005.1

12.12

1960.0

2565.6

30.9

13.64

2970.0

3890.7

31.0

9

9

19.7 -0.048

-0.159

30.7 30.2

9

c exp exp b ARD = 100[(|ycalc 2 - y2 |)/y2 ]. References of experimental data. Binary interaction parameter.

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ARTICLE

Table 6. Detailed Results of Thermodynamic Consistency Test on the Experimental Data of Solubilities of Paraffin Waxes in Supercritical CO2 and Ethane system

T (K)

P (MPa)

Z

jG 1

jG 2

Ap

Aj

ΔA (%)

test resulta

8.85 11.03

0.276 0.278

0.598 1 0.510 0

2.66  10-7 1.65  10-7

1229.556

1337.910

8.8

TC

14.31

0.319

0.42

2.21  10-7

15.74

0.340

0.398 7

2.75  10-7

17.53

0.366

0.371 9

3.73  10-7

19.18

0.390

0.351

5.04  10-7

22.62

0.440

0.319 2

9.93  10-7

24.41

0.46

0.306 2

1.44  10-6

26.06 10.36

0.489 0.264

0.295 0.521 0

2.00  10-6 1.21  10-6

1844.934

1559.377

15.5

TC

12.68

0.292

0.450 3

1.80  10-6

15.2

0.327

0.397 4

3.06  10-6

20.23

0.401

0.331

1.01  10-5

10.34

0.292

0.552 7

3.65  10-6

1117.351

1028.382

8.0

TC

13.79

0.32

0.452 5

5.77  10-6

16.82

0.361

0.397 1

1.08  10-5

20.75 7.52

0.417 0.453

0.349 3 0.653 3

2.81  10-5 2.68 10-6

24187.923

23265.347

3.8

TC

10

0.262

0.534

1.08  10-8

14.98

0.32

0.401

1.78  10-8

14.98

0.325

0.401

1.78  10-8

17.51

0.362

0.362

2.93  10-8

19.98

0.397

0.333 7

5.00  10-8

21.51

0.420

0.319

7.07  10-8

11.96 12.67

0.329 0.331

0.527 4 0.507

4.19  10-8 3.80  10-8

2132.006

1925.925

9.7

TC

15.65

0.361

0.441 9

4.30  10-8

16.58

0.372

0.426 1

4.79  10-8

17.69

0.386

0.409 3

5.58  10-

20.24

0.420

0.377 7

8.69  10-8

10.5

0.343

0.576

5.96  10-9

3125.123

2876.067

8.0

TC

11

0.334

0.558 6

3.50  10-9

12.5 13.5

0.333 0.339

0.512 0.487 3

1.65  10-9 1.35  10-9

0.243

0.528 7

6.78  10-11

256.507

221.806

13.5

TC

10.1

0.319

0.389 1

2.19  10-11

12.02

0.362

0.348 1

1.63  10-11

13.64

0.400

0.322 3

1.67  10-11

16.67

0.474

0.288 3

2.01  10-11

20.2 6.47

0.565 0.241

0.263 3 0.534 8

2.88  10-11 7.77  10-11

143.561

138.305

3.7

TC

10.2

0.323

0.386 7

2.50  10-11

12.12

0.369

0.346 6

1.81  10-11

13.64

0.408

0.322 7

1.82  10-11

16.67

0.488

0.289 1

2.09  10-11

6.57

0.251

0.528 2

4.54  10-12

884.162

841.806

4.8

TC

10.1

0.325

0.388 7

8.77  10-13

12.02 13.64

0.368 0.405

0.34 0.321 1

7.02  10-13 6.62  10-13

16.67

0.475

0.286

7.25  10-13

20.2

0.555

0.261 0

9.91  10-13

mixtures with supercritical CO2 n-C24H50

n-C25H52

n-C25H52

n-C28H58

310

308

313

308

318

318.15

mixtures with supercritical ethane n-C28H58

n-C29H60

n-C30H62

308.1

308.1

308.1

6.57

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ARTICLE

Table 6. Continued system n-C32H66

n-C33H68

T (K)

P (MPa)

Z

jG 1

jG 2

Ap

Aj

ΔA (%)

test resulta

308.1

6.57 10.1

0.254 0.326

0.528 2 0.388

2.61  10-13 3.34  10-14

1757.077

1558.892

11.3

TC

12.02

0.369

0.346

2.41  10-14

13.64

0.406

0.321

2.14  10-14

16.67

0.474

0.286

2.15  10-14

20.2

0.554

0.261 0

2.77  10-14

0.251

0.534 3

1.56  10-12

1196.326

1135.358

5.1

TC

10.2

0.328

0.386 1

2.44  10-13

12.12 13.64

0.372 0.407

0.345 0.321 1

1.96  10-13 1.86  10-13

16.67

0.475

0.286

2.06  10-13

20.2

0.555

0.261 0

2.83  10-13

0.279

0.563

1.27  10-13

1067.608

929.362

12.9

TC

10.2

0.337

0.410 1

6.50  10-13

12.12

0.379

0.367 1

4.85  10-13

13.64

0.413

0.341 8

4.35  10-13

16.67 20.2

0.482 0.561

0.305 7 0.278 4

4.46  10-13 5.85  10-13

308.1

313.1

a

6.47

6.47

TC, thermodynamically consistent data.

statistical form using statistical software. The generated data are treated as pseudoexperimental. However, it is not recommended to generate data based on doubtful data which are not yet thermodynamically tested.12,22,23 Therefore, one has to perform such a test by the used procedure with the existing experimental data, even if only two isothermal data points are available.12,22,23

In eq A1, Z represents compressibility factor and, a and b are attractive and repulsive parameters of the equation of state, respectively, and subscript i refers to ith components in mixture.26   Dðnam Þ ^ai ¼ ðA2Þ Dðni Þ T , P, nj6¼i

’ CONCLUSIONS In this work, a thermodynamic consistency test was applied on the related 12 isothermal experimental data sets. The PR EoS27 with vdW2 mixing rules28 was applied to calculate the solubilities of the investigated compounds in supercritical CO2 and ethane. The consistency test was based on the area test approach derived from the original Gibbs-Duhem equation10-12,14,15 at constant temperature.12,17-24 The results show that 100% of the investigated experimental data of solubilities, which are well represented by the applied model, seem to be thermodynamically consistent. In addition, the results indicated that the measurements of such data must be done accurately and deliberately to be able to use them in tuning of future models for prediction/representation of such solubilities in natural gas systems. The presented test also leads to better understanding of the importance of solubilities of solid compounds in natural gas fluids produced in the petroleum industry.

ðA3Þ

’ APPENDIX The following equation has been used to evaluate the fugacity coefficients of species in the mixture using the PR EoS27 and vdW2 mixing rules:28 lnðji Þ ¼

^bi am =ðbm RTÞ ðZ - 1Þ - lnðZ - βÞ σ-ε bm " #   ^bi ^ai Z þ σβ 1þ ðA1Þ ln Z þ εβ am bm

  ^bi ¼ Dðnbm Þ Dðni Þ T , P, nj6¼i pffiffiffi 2 pffiffiffi ε ¼ 1þ 2

σ ¼ 1-

β¼

ðA4Þ ðA5Þ

bm P RT

ðA6Þ

am ¼

∑i ∑j yi yj aij

ðA7Þ

bm ¼

∑i ∑j yi yj bij

ðA8Þ

aij ¼

pffiffiffiffiffiffiffiffi aii ajj ð1 - kij Þ

ðA9Þ

bij ¼

bi þ bj ð1 - lij Þ 2

ðA10Þ

where kij and lij are interaction parameters between the ith and jth compounds in the mixture, and n is the mole number of compounds. Applying eqs A2 and A3 yields the following relations:26   N Dðnam Þ ^ai ¼ ¼2 yj aij ðA11Þ Dðni Þ T , P, nj6¼i j¼1



  N ^bi ¼ Dðnbm Þ ¼2 yj bij Dðni Þ T , P, nj6¼i j¼1



4738

ðA12Þ

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Industrial & Engineering Chemistry Research

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] Tel.: þ (33) 1 64 69 49 70. Fax: þ (33) 1 64 69 49 68.

’ ACKNOWLEDGMENT The financial support of the ANR (Agence Nationale de la Recherche) and OSEM (Orientation Strategique des Ecoles des Mines) is gratefully acknowledged. A.E. wishes to thank MINES ParisTech for providing a Ph.D. scholarship. The authors are grateful to Prof. Jose O. Valderrama for the fruitful discussions done on the issue. ’ NOMENCLATURE A = area (m2) AD = absolute deviation ARD = absolute relative deviation a = attractive parameter of the equation of state (MPa 3 m6/mol2) ^a = parameter of equation of state defined by eq A11 b = repulsive parameter of the equation of state (m3/mol) ^b = parameter of equation of state defined by eq A12 CEoS = cubic equation of state d = derivative operator E = error EoS = equation of state k = binary interaction parameter f = fugacity l = binary interaction parameter N = number of experimental data points and number of components in the mixture n = number of moles P = pressure (MPa) PR = Peng-Robinson R = universal gas constant (MPa 3 m3/mol 3 K) T = temperature (K) TC = thermodynamically consistent data v = molar volume (m3/mol) vdW2 = van der Waals 2 fluid mixing rule y = mole fraction in gas phase Z = compressibility factor Greek Symbols

β = parameter of the equation of state defined by eq A.6 γ = activity coefficient j = fugacity coefficient Δ = difference value σ = parameter of the equation of state defined by eq A.4 ε = parameter of the equation of state defined by eq A.5 ω = acentric factor Subscripts

A = area c = critical property i = ith component in a mixture or ith experimental data set j = jth component in a mixture or jth individual calculated area m = refers to total value of the EoS attractive and repulsive parameters p = refers to experimental P-T-y data j = refers to calculated parameters of the model for evaluations of the integrals in eqs 6-8

ARTICLE

1 = refers to supercritical CO2 or ethane 2 = refers to paraffin wax Superscripts

calc = calculated E = excess property exp = experimental s = solid G = gas sat = saturation (sublimation) pressure (MPa)

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