ARTICLE pubs.acs.org/IECR
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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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
4737
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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)
’ REFERENCES (1) Mokhatab, S.; Poe, W. A.; Speight, J. G. Handbook of Natural Gas Transmission and Processing; Elsevier: Burlington, MA, USA, 2006. (2) Firoozabadi, A. Thermodynamics of Hydrocarbon Reservoirs; McGraw-Hill: New York, 1999. (3) Leontaritis, K. J. The wax deposition envelope of gas condensates. Presented at the Offshore Technology Conference, OTC8776, Houston, TX, USA, 1996. (4) Merino-Garcia, D.; Correra, S. Cold flow: A review of a technology to avoid wax deposition. Pet. Sci. Technol. 2008, 26, 446–459. (5) Azevedo, F. A.; Teixeira, A. M. A critical review of the modeling of wax deposition mechanisms. Pet. Sci. Technol. 2003, 21, 393–408. (6) Thou, S.; Ruthammer, G.; Potsch, K. Detection of asphaltenes flocculation onset in a gas condensate system. SPE 78321; 2002. (7) Nichita, D. V.; Goual, L.; Firoozabadi, A. Wax precipitation in gas condensate mixtures. SPE Prod. Facil. 2001, No. Nov, 250–259. (8) Jeirani, Z.; Lashanizadegan, A.; Ayatollahi, S.; Javanmardi, J. The possibility of wax formation in gas fields: a case study. J. Nat. Gas Chem. 2007, 16, 293–300. (9) Teja, A. S.; Smith, V. S.; Sun, T. S.; Mendez-Santiago J. Solids Deposition in Natural Gas Systems; Research Report 171, GPA Project 905-93; 2000. (10) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid Phase Equilibria; Prentice Hall: New York, 1999. (11) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 6th ed.; McGraw-Hill: New York, 2003. (12) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic consistency test for experimental data of water content of methane. AIChE J. 2011, DOI: 10.1002/aic.12462. In press. (13) De Stefani, V.; Baba-Ahmed, A.; Richon, D. A review of experimental methods for solid solubility determination in cryogenic systems. Cryogenics 2004, 44 (9), 631–641. (14) Van Ness, H. C.; Abbott, M. M. Classical Thermodynamics of Non-electrolyte Solutions; McGraw-Hill: New York, 1982. (15) Raal, J. D.; M€uhlbauer, A. L. Phase Equilibria: Measurement and Computation; Taylor & Francis: Abingdon, U.K., 1998. (16) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (17) Valderrama, J. O.; Alvarez, V. H. A versatile thermodynamic consistency test for incomplete phase equilibrium data of high-pressure gas-liquid mixtures. Fluid Phase Equilib. 2004, 226, 149–159. (18) Valderrama, J. O.; Robles, P. A. Thermodynamic consistency of high pressure ternary mixtures containing a compressed gas and solid solutes of different complexity. Fluid Phase Equilib. 2006, 242, 93–102. (19) Valderrama, J. O.; Zavaleta, J. Thermodynamic consistency test for high pressure gas-solid solubility data of binary mixtures using genetic algorithms. J. Supercrit. Fluids 2006, 39, 20–29. (20) Valderrama, J. O.; Reategui, A.; Sanga, W. E. Thermodynamic consistency test of vapor-liquid equilibrium data for mixtures containing ionic liquids. Ind. Eng. Chem. Res. 2008, 47, 8416–8422. (21) Valderrama, J. O.; Faundez, C. A. Thermodynamic consistency test of high pressure gas-liquid equilibrium data including both phases. Thermochim. Acta 2010, 499, 85–90. 4739
dx.doi.org/10.1021/ie1022145 |Ind. Eng. Chem. Res. 2011, 50, 4731–4740
Industrial & Engineering Chemistry Research
ARTICLE
(22) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic consistency test for experimental data of sulfur content of hydrogen sulfide. Ind. Eng. Chem. Res. [Online early access]. DOI: 10.1021/ie1017332. Published online: February 18, 2010. (23) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic consistency test for experimental solubility data of carbon dioxide and methane with water inside and outside gas hydrate formation region. J. Chem. Eng. Data, accepted manuscript. 2011. (24) Bertucco, A.; Barolo, M.; Elvassore, N. Thermodynamic consistency of vapor-liquid equilibrium data at high pressure. AIChE J. 1997, 43, 547–554. (25) Tester, J. W.; Modell, M. Thermodynamics and its Applications. 3rd ed. Prentice Hall PTR, Upper Saddle River, New Jersey. (26) Yazdizadeh, M.; Eslamimanesh, A.; Esmaeilzadeh, F. Thermodynamic modeling of solubilities of various solid compounds in supercritical carbon dioxide: Effects of equations of state and mixing rules J. Supercrit. Fluids 2011, 55, 861–875, DOI: 10.1016/j.supflu.2010.10.019. (27) Peng, D. Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59–64. (28) Mukhopadhyay, M. Natural Extracts Using Supercritical Carbon Dioxide; CRC Press: Boca Raton, FL, USA, 2003. (29) Constantinides, A.; Moustofi, N. Numerical Methods for Chemical Engineers with MATLAB Applications; Prentice Hall PTR: New York, 1999. (30) Mickley, H. S.; Sherwood, T. K.; Reed, C. E. Applied Mathematics in Chemical Engineering; McGraw-Hill: New York, 1957. (31) Reverchon, E.; Russo, P.; Stassi, A. Solubilities of solid octacosane and triacontane in supercritical carbon dioxide. J. Chem. Eng. Data 1993, 38, 458–460.
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