RESEARCH NOTE pubs.acs.org/IECR
Determination of Sulfur Content of Various Gases Using Chrastil-Type Equations Ali Eslamimanesh,† Amir H. Mohammadi,*,†,‡ and Dominique Richon† † ‡
nergetique et Procedes, 35 Rue Saint Honore, 77305 Fontainebleau, France MINES ParisTech, CEP/TEP—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: Accurate knowledge of the sulfur content of gases is a significant factor for prediction of the sulfur precipitation conditions in production of sour natural gases. However, because the concentration of sulfur in the gas phase is very low, the measurement difficulties consequently result in scarce experimental solubility values available in the literature. In this work, application of four commonly used Chrastil-type equations including the original Chrastil, Adachi and Lu, del Valle and Aguilera, and Mendez-Santiago and Teja, which consider the effects of temperature, density of gas (solvent), and pressures on the sulfur solubility, are investigated to calculate the sulfur content of different gases at various temperature, pressure, and sulfur content ranges. In this study, sulfur is treated as single molecule S8. The Valderrama-Patel-Teja equation of state with non-density dependent mixing rules is used to calculate the densities of gases. It is found out that the absolute average deviations (AAD %) of the evaluated sulfur contents by the aforementioned equations for 260 investigated experimental values are 21%, 12%, 19%, and 20%, respectively.
1. INTRODUCTION One of the imperative problems in production of sour natural gases is precipitation of the sulfur in the formation, well bores, and production facilities especially at high temperatures, pressures, and at high hydrogen sulfide concentrations.14 For instance, sulfur deposition can cause a substantial and drastic reduction in the permeability of the formation near the well bore.2 The occurrence of sulfur precipitation has also been observed when transporting natural gases.17 Sulfur may exist as a number of polymeric species ranging typically up to S8 in the gas and combines with other gases to produce polysulfides or sulfanes such as H2S9.1,2,818 The amount of each molecule depends on both pressure and temperature.2,4,1215 Under atmospheric pressure and at temperatures greater than the fusion temperature, liquid sulfur is composed of about 99% (mole basis) of S8 and traces of lighter molecules.1,2,1115,1820 Nevertheless, if temperature increases, the ring molecules of sulfur polymerize. Liquid sulfur begins to polymerize at about 430 K.1,2,1215,21 Several experimental studies on the sulfur content of various gas samples have been reported in the literature. Detailed investigations of the results of these measurements are well reviewed elsewhere.1,2,17 Modeling sulfur þ sour/acid gas phase equilibrium by conventional thermodynamic models requires the use of unknown parameters considering the complex chemical behavior of sulfur, as mentioned earlier.1,2,17 The various thermodynamic models reported in the literature typically use the PengRobinson equation of state (PR-EoS)22 and focus on ranges of pressure, temperature, and hydrogen sulfide content at natural gas production conditions, typically higher than transportation conditions.17 Recently, artificial neural network (ANN) method has been used to calculate/predict the sulfur content of hydrogen sulfide r 2011 American Chemical Society
vapor and other gas samples.1,17 However, development of such a method is not generally so easy. Therefore, there is still a need for applying new relations, which can be simply applied in the petroleum industry. In this study, the capability of four Chrastil-type equations, which are generally applied in supercritical fluid extraction processes, are examined and compared for determination of the solubility of sulfur in various gases and gas mixtures at the conditions generally faced in production and transportation of sour natural gases. It is shown that Chrastil-type equations can be considered as alternative tools to represent sulfur content of gases.
2. EQUATIONS 2.1. Chrastil. As Chrastil23 assumed in 1982, the solute
molecules associated with the gas molecules are in chemical equilibrium with the resulting complex (solvate complex) as follows:23,24 A0 þ kB0 T A0 B0 k
ð1Þ 0
where k represents the association term, A denotes the solute molecule, and B0 is the gas (solvent) molecule. Therefore, the equilibrium concentration of the solute can be calculated from the mass action law as follows:23,24 a þb ð2Þ c ¼ Fk exp T In the preceding equation, c is the concentration of a solute in a Received: January 26, 2011 Accepted: April 20, 2011 Revised: March 25, 2011 Published: April 20, 2011 7682
dx.doi.org/10.1021/ie200187v | Ind. Eng. Chem. Res. 2011, 50, 7682–7687
Industrial & Engineering Chemistry Research
RESEARCH NOTE
Table 1. Experimental Data Ranges of Sulfur Content of Various Gases Used in This Work gas compositions/mole fraction set no.
temperature ranges (K)
1
303.2363.2
pressure
sulfur content ranges
ranges (MPa)
(mole fraction 10 )
3045
0.0492.48
H2S
N2
CO2
C1
C2
C3
i-C4
n-C4
refb
0.0495
0
0.074
0.8765
0
0
0
0
28
6
8
2
303.2363.2
3045
0.0783.11
8
0.0993
0
0.0716
0.8291
0
0
0
0
3
303.2363.2
3045
1.035.820
8
0.1498
0
0.0731
0.7771
0
0
0
0
4
303.2363.2
2045
0.0117.11
9
0.1771
0
0.0681
0.7584
0
0
0
0
5
303.2363.2
3045
1.6912.70
8
0.2662
0
0.07
0.6638
0
0
0
0
6
303.2363.2
3045
0.07122.77
8
0.1
0
0.0086
0.8914
0
0
0
0
7
303.2363.2
3045
0.0793.20
8
0.1003
0
0.1039
0.7958
0
0
0
0
8
316.3383.2
7.0331.14
16701020
21
1
0
0
0
0
0
0
0
29
9
363.2
11.8336.21
483010500
6
1
0
0
0
0
0
0
0
30
0
10
363.2383.2
12.0740.52
5.34201
12
0
1
0
0
0
0
0
11
363.2
11.4732.78
23408150
6
0.9509
0.0001
0.0372
0.0103
0.0009
0.0004
0.0001
0.0001
12
363.2
18.3834.71
10.20225
6
0.4411
0.0068
0.0072
0.4797
0.0409
0.0191
0.0028
0.0024
13
398
12.8548.8
23.78270
10
0.84
0.09
0.07
0
0
0
0
0
14
473
8155
1178680
9
0.42
0
0.02
0.56
0
0
0
0
15
485
7.576
95.201050
9
0.09
0
0.05
0.86
0
0
0
0
16
398458
6.7134.2
62.205110
11
0.35
0
0.08
0.57
0
0
0
0
17
398
7.0043.00
1.84687
8
0.35
0
0.08
0.45
0.12
0
0
0
18
398
7.0043.50
4.62952
8
0.35
0
0.08
0.45
0
0
0.12
0
19 20
373.15433.15 373.15433.15
10.060.0 10.060.0
142026700 9.39301
28 18
1 0.07
0 0.08
0 0.2
0 0.65
0 0
0 0
0 0
0 0
21
373.15433.15
10.060.0
3.67281
20
0.06
0.04
0.09
0.81
0
0
0
0
22
373.15433.15
10.060.0
8.36594
15
0.01
0.04
0.14
0.81
0
0
0
0
23
373.15433.15
10.060.0
101163872
16
0.2
0.04
0.1
0.66
0
0
0
0
total a
N
a
31
32
260
Number of experimental data. b References of experimental data.
gas (g 3 L1), F denotes the density of the solvent (g 3 L1), T is the temperature in K, and a and b are the adjustable parameters of the equation, respectively. It has been demonstrated that because the solvate complex is not stoichiometric in most cases, the association term is a characteristic of the proposed system and can be treated as an adjustable parameter.24 2.2. Adachi and Lu. A modification to the Chrastil’s equation23 was proposed by Adachi and Lu.25 They considered the association term (k) of eq 2 to be a function of gas (solvent) density as follows:25 k ¼ e 1 þ e 2 F þ e 3 F2
ð3Þ
where e13 are treated as adjustable parameters to reduce the relative deviations of the calculated results from experimental values for different solutes. Although these researchers were met with some divergences in their calculation algorithm,25 they reported overall improvement in the calculated solubilities of different solids in supercritical carbon dioxide. 2.3. del Valle and Aguilera. Fitting the experimental values of solubilities of vegetable oils in supercritical carbon dioxide has contributed to obtaining a correlation by del Valle and Aguilera26 that considers the effects of temperature and density of gas
(solvent) as the following form:26 b d c ¼ Fk exp a þ þ 2 T T
ð4Þ
where a, b, and d are the adjustable parameters determined for a system of interest. ndez-Santiago and Teja. The effects of pressure on 2.4. Me the solubilities of different solids in gases have been considered by Mendez-Santiago and Teja, who have presented the following equation:27 1 a bF þ þd ð5Þ y2 ¼ exp P T T where y denotes the solubility (mole fraction), subscript 2 refers to the solute, P is the pressure in MPa, and a, b, and d are the adjustable parameters of the equation.
3. EXPERIMENTAL DATA For performing a study on the capabilities of the aforementioned equations in evaluation of the sulfur content of gases, we have tried to gather all reliable experimental data in open literature based on our previous studies.1,2,17 Table 1 shows the 7683
dx.doi.org/10.1021/ie200187v |Ind. Eng. Chem. Res. 2011, 50, 7682–7687
Industrial & Engineering Chemistry Research
RESEARCH NOTE
Table 2. Optimal Values of the Adjustable Parameters of the Equation of Chrastil23
Table 4. Optimal Values of the Adjustable Parameters of the Equation of del Valle and Aguilera26
optimal values of the parameters
optimal values of the parameters
a
set no.
k
1 2 3 4 5 6 7 8, 9, 19 10 11 12 13 14 15 16 17 18 20 21 22 23 Total
3.75 3.8 4.63 4.98 4.98 3.18 3.64 5.05 3.12 10.14 10.59 4.11 3.42 3.08 3.75 4.96 4.23 3.61 3.68 3.46 5.26
AAD = (100/N)
a 2692.6 2820.7 4180.7 4376.0 4376.0 2633.2 2793.0 3206.3 5660.3 0.2 3.2 0.3 0.3 0.5 3582.0 0.3 0.2 4733.0 4822.0 4947.0 6138.0
((|ycalc 2
a
b
AAD %
set no.
18.2 18.1 18.7 20.0 20.0 15.0 17.5 20.5 5.8 64.4 64.7 22.9 17.5 15.9 12.3 28.3 24.2 9.9 9.5 8.2 10.2
5 9 14 24 24 10 10 20 40 8 1.4 102 2 102 16 24 27 20 20 23 33 53 29 21
a
b
d
k
1
17.51
2540
13
3.55
2
17.05
2637
9
3.51
8
3 4
18.04 19.59
3998 4244
1 2
4.42 4.84
14 26
5
15.63
4786
19
4.47
7
6
14.18
2456
8
2.95
9 9
7
16.56
2618
6
3.39
47.23
1.89 104
4.06 106
4.55
17
5.81
5658
30
3.12
40
11
64.39
11
7
10.14
7
12 13
73.22 22.92
99 2
140 8
12.03 4.11
1.2 102 2.0 102
14
17.51
4
6
3.42
15
16.00
35
35
3.08
25
16
13.39
3555
21
3.93
29
17
28.33
5
4
4.96
20
18
24.25
30
37
4.23
20
20
9.90
4732
160
3.61
22
21 22
9.47 7.25
4820 5744
349 9321
3.68 1.03
33 27
23
20.50
2759
1.78 106
5.11
27
total
Table 3. Optimal Values of the Adjustable Parameters of the Equation of Adachi and Lu25
a
b
e1
e2
16
19
Table 5. Optimal Values of the Adjustable Parameters of the Equation of Mendez-Santiago and Teja27
optimal values of the parameters set no.
6
8, 9, 19 10
expt yexpt 2 |)/(y2 )).
AAD %
optimal values of the parameters e3
set no.
AAD %
a
b
d
AAD %
1
2862.1 5 105
0.25 1.0 106 5 106
3
1
4901.9
97346.2
1
4
2 3
2049.2 4.0 2768.4 14.2
0.71 1.0 106 2 106 3.09 1.0 106 2 1012
6 7
2 3
3908.1 5696.1
73432.0 105586.0
1 3
6 10
4
4842.2 31.1
7.15 1.0 106 1 1012
15
4
8193.9
164821.4
8
13
5
4758.1 8.1
3.94 5.7 103 1 105
4
5
7194.5
114843.6
8
8
6
2096.4 4 1010
0.42 3.5 103 1 105
7
6
3979.9
76392.6
1
7
7
2108.4 11.2
3.02 5.9 103 1 105
7
7
4180.1
77537.4
0
7
8, 9, 19
3584.1 15.5
4.48 7.6 105 1 1012
14
8, 9, 19
7534.8
149734.5
11
8
10
8224.5 3 1013
3.16 1.1 104
1 1010
14
10
12021
133976.9
21
14
11 12
30769 51601
108599.7 639928.4
68 106
23 60
11 12
10000.0 100.0 10000.0 100.0
20.92 1.7 103 1 1012 25.39 1.2 102 1 1012
6 54
13
9374.5 46.6
13.02 2.6 103 1 107
17
13
43257
209114.0
98
42
14
3476.8 4.8
7
14
33098
170954.6
63
29
15
45458
16 17
2.33 4.5 104
1 1012
4124.0 0.1
1.44 1.0 103
1 107
5
15
3459.8 2.4
1.66 9.7 104
1 1012
14
16
1098.9 19.4
3.68 6.0 104
1 107
6130.9 40955
184562.2
86
16
170493.5
5
27
5
17
258847.3
91
39
18
7406.5 0.5
2.78 1.2 10
1 1012
11
18
5429.2
248704.4
2
27
20 21
5132.1 0.9 6307.5 1.0
1.55 9.8 104 2.20 9.3 104
1 107 1 107
8 20
20 21
7773.2 9069.4
149533.3 164573.2
9 14
16 19
22
6636.0 1.0
2.28 1.0 103
1 107
23
22
8979.2
164873.9
12
21
5049.5 0.9
4
1 107
15
23
9137.5
184886.6
17
12
total
23 total
3
2.55 8.1 10
7684
16 20
dx.doi.org/10.1021/ie200187v |Ind. Eng. Chem. Res. 2011, 50, 7682–7687
Industrial & Engineering Chemistry Research
RESEARCH NOTE
Table 6. Results of the Investigated Equations for Data Sets 8, 9, and 19 after Eliminating the Thermodynamically Inconsistent Data optimal values of the parameters set no. 8, 9, 19
k
correlation
a
b
d
e1
e2
e3
AAD %
Chrastil23
4.09
4247
11.4
-
Adachi and Lu25
-
3584.1
15.5
-
4.48
8 105
1 1012
1
del Valle and Aguilera26 Mendez-Santiago and Teja27
4.09 -
11 12698.2
4246 196860.6
29.98 22
-
-
-
5 6
temperature, pressure, and sulfur content ranges, the compositions of the gas samples, and the references of the experimental data.
4. METHODOLOGY The Valderrama-Patel-Teja equation of state (VPT-EoS)33 with nondensity-dependent (NDD) mixing rules34 is used to calculate the density of the gases at the desired conditions. These values are freely available upon request from the authors. Detailed description of the applied EoS can be found elsewhere.35,36 In this work, sulfur is treated as single molecule S8. The optimal values of adjustable parameters of the investigated correlations are evaluated through minimization of the following objective function: expt 100 N ucalc 2i u2i ð6Þ OF ¼ expt N i∑ u2i ¼1 where N stands for the number of experimental data, i denotes the ith experimental data, u is either equal to c or y2 and superscripts calc and expt refer to calculated and experimental sulfur content values, respectively. In this work, the robust differential evolution37,38 (DE) optimization algorithm is applied for obtaining the optimum values of the correlation parameters since it has been already proven to converge to the global optimum without sensivity to the calculation starting point.3945
5. RESULTS AND DISCUSSION Table 2 reports the optimal values of the Chrastil23 equation along with the absolute average deviations (AAD %) of the calculated sulfur content values from experimental ones. Similarly, Tables 35 indicate the obtained results using the equations proposed by Adachi and Lu,25 del Valle and Aguilera,26 and Mendez-Santiago and Teja,27 respectively. Detailed investigations of the AAD % of the aforementioned equations results lead to several important points. First and perhaps most importantly is the fact that we have used almost all of the available data sets for sulfur content of gases for evaluation of the correlation parameters. However, several reasons including low concentrations of sulfur in gas phase, high temperatures, experimental procedures, design of experimental apparatuses, errors in calibrations of temperature probes and pressure transducers, possible errors during the measurements, failing instruments, etc. lessen our confidence in experimental data that have been obtained without significant repeatability and reproducibility tests and not through various experimental equipment and methods for comparisons. For instance, the authors have previously performed thermodynamic consistency tests on the experimental data of sulfur content of hydrogen sulfide vapor.2 Eliminating the inconsistent data from the applied data of
11
solubility of sulfur in hydrogen sulfide vapor for obtaining the adjustable parameters of the equations reduce the AAD % of the calculations results as shown in Table 6. Another factor to consider is that considering more adjustable parameters for representation of the equilibrium constant (k) has a great effect on the evaluated sulfur content of the investigated gas samples, although the values of the (e3) parameter are negligible in comparison with the other two (e1 and e2). As a matter of fact, the correlation presented by Adachi and Lu25 results in a better description of the physical behavior of these kinds of systems because it interrelates that the equilibrium constant varies with the density of the investigated gas. However, taking into account the effects of pressure on the solubility of sulfur in the gas phase regarding the equation proposed by Mendez-Santiago and Teja27 has weak effects on the average absolute deviations of the results from experimental values. Neglecting the effects of gas density predictions by the VPTEoS33 with non-NDD mixing rules34 and the common errors in optimization procedure on the obtained results would be an oversight although using the DE optimization strategy37,38 has contributed to overcome the drawbacks of other statistical procedures used in original articles e.g. in that of Adachi and Lu.25 Regarding the preceding discussions, the investigated equations, which are generally used for correlating the solubilities of different compounds in supercritical carbon dioxide lead us to have just a rough estimation of the sulfur content of gases (the lowest calculated AAD % of the results is contributed from the Adachi and Lu25 equation that is 12%). However, using more reliable and consistent experimental data would yield more accurate adjustable parameters and consequently more reliable sulfur content results.
6. CONCLUSIONS Four widely used equations namely as original Chrastil,23 Adachi and Lu,25 del Valle and Aguilera,26 and Mendez-Santiago and Teja,27 were used in this article to determine the solubilities of sulfur in gaseous systems at the temperature range of 303.2485 K and pressure range of 6.7155 MPa; conditions of interest in the petroleum industry and especially for sour gas production. The VPT-EoS33 with nondensity-dependent NDD mixing rules34 was used to calculate the density of the gases (solvents) at the desired conditions and the DE optimization method37,38 was applied to obtain the optimum values of the adjustable parameters. In this work, sulfur is treated as single molecule S8. The absolute average deviations (AAD %) of the results from experimental values were found to be 21%, 12%, 19%, and 20%, regarding the results of the original Chrastil,23 Adachi and Lu,25 del Valle and Aguilera,26 and MendezSantiago and Teja,27 equations, respectively. It was argued that the effects of unreliable experimental data on the calculated results are not negligible. Furthermore, it was inferred that considering the pressure 7685
dx.doi.org/10.1021/ie200187v |Ind. Eng. Chem. Res. 2011, 50, 7682–7687
Industrial & Engineering Chemistry Research dependency of the sulfur solubilities in investigated gases would lead to a slight reduction of the deviations of the evaluated results from experimental values. To recapitulate, it should be noted that the aforementioned equations can be used to obtain just rough values of the sulfur content of gas samples. More reliable experimental data are needed to present more accurate and comprehensive correlations for this purpose.
’ AUTHOR INFORMATION Corresponding Author
*Email:
[email protected] Tel.: þ (33) 1 64 69 49 70. Fax: þ (33) 1 64 69 49 68.
’ ACKNOWLEDGMENT The financial support of the Orientation Strategique des Ecoles des Mines (OSEM) is gratefully acknowledged. Mr. Ali Eslamimanesh wishes to thank MINES ParisTech for providing a Ph.D. scholarship. ’ NOMLENCLATURE a = adjustable parameter of the correlations A0 = solute molecule AAD = absolute average deviation, % ANN = artificial neural network b = adjustable parameter of the correlations B0 = solvent molecule c = solubility of sulfur (g/L) d = adjustable parameter of the correlations DE = differential evolution e1e3 = adjustable parameter of the Adachi and Lu’s25correlation k = association term N = number of experimental data NDD = non-density dependent mixing rules OF = objective function P = pressure, (MPa) PR-EoS = the PengRobinson equation of state T = temperature, (K) u = parameter in eq. 6 VPT-EoS = the Valderrama-Patel-Teja equation of state y = mole fraction of solute in gas phase Greek Letters
F = density of gas (solvent) (g/L) Subscripts
i = stands for ith experimental data point k = association term 2 = refers to sulfur Superscripts
calc = calculated expt = experimental
’ REFERENCES (1) Mohammadi, A. H.; Richon, D. Estimating sulfur content of hydrogen sulfide at elevated temperatures and pressures using an Artificial Neural Network algorithm. Ind. Eng. Chem. Res. 2008, 47, 8499–8504. (2) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic consistency test for experimental data of sulfur content of hydrogen sulfide. Ind. Eng. Chem. Res. 2011, 50, 3555–3563.
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