A Chemical Equilibrium Equation of State Model for Elemental Sulfur

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Ind. Eng. Chem. Res. 2001, 40, 2160-2167

A Chemical Equilibrium Equation of State Model for Elemental Sulfur and Sulfur-Containing Fluids Robert A. Heidemann,* Aaron V. Phoenix,† Kunal Karan,‡ and Leo A. Behie Department of Chemical and Petroleum Engineering, The University of Calgary, 2500 University Drive, N.W., Calgary, Alberta, Canada T2N 1N4

The phase behavior of natural gases containing elemental sulfur has been modeled using a chemical equilibrium model and the Peng-Robinson equation of state. Sulfur in the fluid phases is modeled as a mixture of eight species, S1 through S8, and reactions of each with hydrogen sulfide to form eight different sulfane species are anticipated. A proposal is made for constructing the Peng-Robinson parameters for all species from the parameters for source substances including S8 and H2S and interaction parameters only between the source substances. A separate model is used for sulfur as a solid. Comparisons are made with data for pure sulfur and for sulfur in contact with a number of gas mixtures containing hydrogen sulfide, carbon dioxide, methane, and nitrogen. The equilibrium sulfur content of the gases is well predicted over a wide range of gas composition, temperature, and pressure. Introduction Natural gases containing hydrogen sulfide may also contain elemental sulfur in solution. Production of these natural gases may result in precipitation of solid sulfur in well bores and in above-ground equipment.1 At higher temperatures, liquid solutions with sulfur as the principal component can occur. This paper reports on an equation of state model that correlates the equilibrium behavior in such systems. Sulfur is known to exist as a number of species ranging up to S8 in the vapor, and it combines with other gases to produce polysulfides or sulfanes such as H2S2.2 In the liquid, polymerization is responsible for a dramatic increase in viscosity with increasing temperature above 433 K.3-6 Neutron scattering experiments7 confirm that sulfur is a molecular liquid between 386 and 433 K, made up almost wholly of S8 rings, but above 433 K, the S8 rings open to polymerize into very long S8 chains. Swift et al.8 developed a reaction equilibrium model to interpret and extrapolate sulfur solubility data. In an earlier paper,9 a Peng-Robinson equation of state model was proposed for sulfur and sulfur-containing gases. In that model, the sulfur was treated as the compound S8 because ideal gas reaction equilibrium calculations at moderate temperatures and pressures showed that S8 was by far the most abundant equilibrium species. In this paper, eight sulfur species ranging from S1 to S8 are fully accounted for. When hydrogen sulfide is a mixture component, equilibrium formation of the sulfanes from H2S2 to H2S9 is included, as is the dissociation of hydrogen sulfide to produce molecular hydrogen. The Peng-Robinson equation is used to describe the fluid phases, as was done in Karan et al.9 However, * To whom correspondence should be addressed. E-mail: [email protected]. Telephone: (403) 220-8755. Fax: (403) 284-4852. † Current address: Department of Chemical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada. ‡ Current address: D. B. Robinson Research Ltd., Edmonton, Alberta, Canada.

chemical reaction equilibria determine the distribution of the sulfur between the various possible species in all of the fluids. The vapor- and liquid-phase nonidealities are accounted for through the equation of state. A separate model is employed for solid sulfur, which is treated as S8. Equation of State Parameters The Peng-Robinson parameters for the mixture as a whole are calculated in the usual way: Nc

bm )

yibi ∑ i)1

(1)

and Nc N c

am )

∑ ∑yiyjaij i)1 j)1

(2)

For the usual components of natural gas mixtures, the pure component aii and bi parameters can be calculated from the critical temperature, critical pressure, and acentric factor.10 However, this is not possible for the individual sulfur species or the sulfanes. The S8 species is the most abundant in the temperature range considered, and for the purpose of calculating the parameters, we regard all of the other sulfur species as being formed from the S8 molecule. The sulfanes are formed from S8 and H2S. The pure species parameters aii and bi and the cross-parameters aij are calculated from the parameters for S8 and H2S. The Peng-Robinson parameters for hydrogen sulfide, hydrogen, and any inert species are evaluated from the critical temperature, critical pressure, and acentric factor. This leaves only the S8 parameters to fit pure sulfur vapor pressure and liquid density data and one interaction parameter to fit hydrogen sulfide-sulfur phase behavior. The number (and identity) of fitting parameters is the same as that in the earlier paper.9 The reactions to form the sulfur species and the sulfanes from S8 and H2S are

10.1021/ie000828u CCC: $20.00 © 2001 American Chemical Society Published on Web 03/30/2001

Ind. Eng. Chem. Res., Vol. 40, No. 9, 2001 2161

(k/8)S8 ) Sk; k ) 1, ..., 7 H2S +

k-1 S8 ) H2Sk; k ) 2, ..., 9 8

(3) (4)

The approach taken to calculating the Peng-Robinson parameters can be generalized in the following manner. It is assumed that there are a limited number of source species from which all others are formed. The species actually present in the mixture are denoted by Ai, i ) 1, 2, ..., Nc, where Nc is the total number. The first Ns are the “source species”. The others can be thought of as being formed from the source species in chemical reactions such as (3) and (4): Ns

νjiAj ) Ai; ∑ j)1

i e Nc

(5)

Obviously, the νji stoichiometric coefficients need not be integers (and will not be in the case of a sulfur species Sk being formed from S8), so this model is more general than “clustering” of monomers. The source species are assumed to have equation of state parameters aii and bi and cross-parameters aij ) aji calculated using interaction parameters in the usual way:

aij ) (1 - kij)xaiiajj; i, j e Ns

(6)

For the other species present in the mixture, the equation of state parameters are calculated from the parameters for the source species in a generalization of the method used by Heidemann and Prausnitz.11,12 Specifically,

νjibj ∑ j)1

(7)

N s Ns

aij )

Reaction and Phase Equilibria To solve the combined chemical reaction and phase equilibrium problem, we used a method reported by Phoenix15 and Phoenix and Heidemann.16 A nested calculation method is used. In the inner loop, the phases are treated like ideal solutions and a so-called nonstoichiometric method17 proposed by Michelsen18 is used to find phase amounts and compositions. The phase nonidealities are updated, and convergence is checked in an outer loop. A brief description is given in the appendix. Thermochemical data required to evaluate the standard state chemical potentials of reacting species were obtained from the JANAF tables19 and from the Ph.D. thesis of Raymont20 (for the sulfanes). Coefficients were found in expressions of the form

c1 µ0 ) + c2 ln T + c3 + c4T + c5T2 + c6T3 (12) RT T The coefficients were fitted to values at six temperatures ranging from 300 to 1300 K for the various substances. Numerical values for the coefficients are given in Table 1. Correlation for Pure Sulfur Vapor and Liquid

Ns

bi )

Hendriks et al.13 suggested that the mixing rule for the equation of state am parameter could have the form of (11). Here we have shown how that this form follows from reasonable assumptions for the cross-terms between the actual species. Rizvi and Heidemann14 used this same kind of mixing rule for the specific system of ammonia and water combining to form two different hydrated ammonia species.

∑ ∑νkiνljakl

(8)

k)1l)1

We used the coefficients found from (7) and (8) to evaluate the mixture parameters in (1) and (2) and the fugacities of the individual components. Compact expressions can be obtained for the mixture constants through writing the mass balances. Let n0T be the total numbers of moles, and let zj (j ) 1, Ns) be the mole fractions of source species in a phase before any association takes place. After association, the total number of moles is nT and the mole fractions are xj (j ) 1, Nc). Conservation of mass for each of the source species requires that Nc

νjixi ) n0Tzj ∑ i)1

nT

(9)

These mass balances result in

The Peng-Robinson a and b parameters for S8 can be fitted to sulfur vapor pressure and liquid density. The reference density and vapor pressure data are derived from various original sources21-23 and are tabulated in The Sulfur Data Book.24 The b parameter was fixed at the value used in the previous paper9 where sulfur was presumed to be S8 only, i.e., b ) 0.131 22 m3/kmol. This assured that the calculated liquid density was within 1% of the data over a wide temperature range, as is shown in Table 3. Vapor densities depend strongly on the details of the equilibrium species present and not so strongly on the value used for b. The coexisting vapor and liquid phases are different complex mixtures of eight sulfur species with all chemical reactions in equilibrium in both phases. Even so, there is only 1 degree of freedom in this two-phase equilibrium. In our fitting process, the temperature was fixed and the a parameter for S8 was varied until an equilibrium was reached with two phases present. A different value of a was required to fit the data at each temperature. The a values correlate well with temperature from 393 to 913 K with the function

Ns

nTbm )

n0T

zj bj ∑ j)1

(10)

Ns N s

nT2am ) (n0T)2

∑ ∑zizjaij i)1 j)1

(11)

a ) 10.1835 - 508.1/T + 589.6 × 103/T2

(13)

Vapor pressures, heats of vaporization, and coexisting volumes are shown in Tables 2 and 3. Except for one low-pressure datum, vapor pressures varying over 5 orders of magnitude are reproduced within 1% up to 800

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Ind. Eng. Chem. Res., Vol. 40, No. 9, 2001

Table 1. Standard State Chemical Potential Functions µ0/RT ) c1/T + c2 ln T + c3 + c4T + c5T2 + c6T3 H2S H2 S8 S1 S2 S3 S4 S5 S6 S7 H2S2 H2S3 H2S4 H2S5 H2S6 H2S7 H2S8 H2S9

c1 × 10-3

c2

c3

c4 × 103

c5 × 106

c6 × 109

-3.360 61 -0.914 359 6.663 84 32.426 9 14.403 5 15.345 5 15.219 7 10.026 9 8.363 92 9.001 15 1.541 61 1.150 66 0.506 284 -0.291 066 -1.134 71 -2.433 26 -4.033 75 -5.686 56

-1.408 15 -2.542 76 -16.196 9 -3.101 54 -3.100 28 -4.996 09 -6.623 62 -9.157 89 -11.495 1 -13.961 4 4.511 25 -6.368 82 -17.248 9 -28.129 0 -39.009 0 -49.889 1 -60.769 2 -71.649 2

-11.780 3 2.473 94 60.675 3 0.321 071 -5.695 28 2.239 31 9.296 09 26.883 0 37.556 0 47.899 9 -48.290 7 12.515 1 73.320 8 134.127 194.932 255.738 316.544 377.350

-7.811 73 -2.524 25 -7.376 00 0.483 562 -1.912 17 -2.182 73 -4.317 79 -4.970 53 -5.898 27 -6.379 54 -28.169 4 -9.950 62 8.268 14 26.486 9 44.705 7 62.924 4 81.1432 99.3619

4.855 97 1.423 02 2.287 41 -0.082 605 3 0.649 654 0.607 482 1.376 43 1.571 84 1.892 53 2.002 82 14.745 0 5.607 72 -3.529 51 -12.666 7 -21.804 0 -30.941 2 -40.0784 -49.2157

-1.689 73 -0.398 144 -0.368 491 0.004 067 -0.120 749 -0.099 469 -0.237 240 -0.260 135 -0.315 273 -0.332 698 -3.788 54 -1.637 65 0.513 239 2.664 13 4.815 02 6.965 91 9.11680 11.2677

Table 2. Comparison of Predicted Vapor Pressure and the Heat of Vaporization of Sulfur with Data (The Sulfur Data Book24); Peng-Robinson b ) 0.13122 m3/kmol heat of vaporization, kJ/kg T, K 393.15 413.15 433.15 473.15 513.15 553.15 593.15 633.15 673.15 713.15 753.15 793.15 833.15a 873.15a 913.15a a

calcd 364 348 335 314 299 288 280 275 272 275 277 285 298 314 334

vapor pressure, bar

data 355 343 332 315 302 293 285 281 282 286 295 307 322 339 358

calcd

data

10-5

4.01 × 1.46 × 10-4 4.49 × 10-4 2.85 × 10-3 0.01228 0.0402 0.1074 0.246 0.504 0.943 1.646 2.729 4.35 6.73 10.17

4.05 × 10-5 1.42 × 10-4 4.46 × 10-4 2.83 × 10-3 0.01216 0.0402 0.1074 0.247 0.507 0.949 1.652 2.716 4.30 6.53 9.54

Extrapolations.

Table 3. Saturated Liquid and Vapor Specific Volumes of Sulfur Calculations and Data (The Sulfur Data Book24) liquid vol, (m3/kg) × 103

vapor vol, m3/kg temp, K 393.15 413.15 433.15 473.15 513.15 553.15 593.15 633.15 673.15 713.15 753.15 793.15 833.15 873.15 913.15

model

data

3312 3252 958 983 328 331 57.0 57.5 14.5 14.4 4.81 5.06 1.95 2.04 0.916 0.943 0.481 0.506 0.276 0.293 0.169 0.1810 0.1095 0.1217 0.0737 0.0843 0.0511 0.0624 0.0363 0.0442

diff %

model

data

diff %

-1.8 2.6 0.8 0.9 -0.8 4.8 4.5 2.9 4.8 6.0 6.5 10.1 12.6 18.1 17.9

0.552 0.555 0.559 0.567 0.575 0.584 0.593 0.603 0.614 0.625 0.638 0.652 0.668 0.686 0.705

0.556 0.559 0.564 0.571 0.579 0.587 0.596 0.604 0.614 0.622 0.637 0.649 0.681 0.712 0.755

0.7 0.7 1.0 0.7 0.8 0.6 0.5 0.2 0 -0.6 -0.2 -0.5 1.8 3.7 6.6

K. The liquid volume is slightly too small at lower temperatures and slightly too large between 673 and 800 K. The volume of the saturated vapor is much improved over results with the simpler model that ignored chemical transformations between the sulfur species.9 The calculated volumes are within 10% of the data up to 800 K. The heats of vaporization were calculated at each temperature as the difference between the enthalpies

Figure 1. Mole fractions of sulfur species in the vapor at the vapor pressure.

of the equilibrium vapor and liquid, which were in turn calculated from the mixture equation of state and the ideal gas contribution. The ideal gas enthalpies included enthalpies of formation and were obtained from temperature derivatives of the chemical potential expression in (12). The results, which are in Table 2, show the same behavior as the data taken from The Sulfur Data Book. In an intermediate temperature range, the heat of vaporization passes through a minimum. This behavior is a consequence of the various chemical reactions between the sulfur species. The correspondence between the calculated and tabulated values is very good at lower temperatures. The minima are reached at slightly different points, and errors at the higher temperatures are around 7.5%. The tabulated values with which the calculations are compared were calculated from vapor pressure and volumetric data using the Clausius-Clapeyron equation. Because the model provides good vapor pressures and phase volumes, the heats of vaporization would also be expected to show no larger errors. Figures 1 and 2 show the mole fractions of the sulfur species in the calculated equilibrium vapor and liquid, respectively. The dominant species is S8 in both phases, especially at low temperatures. The S1 species has a mole fraction below 1 × 10-4 and does not appear on the figures. Solid Sulfur. As in our previous paper,9 we have not attempted to account for the two different crystalline forms that sulfur takes in the solid. Rather, we have


Figure 2. Mole fractions of sulfur species in the liquid at the vapor pressure.

assumed that sulfur in the solid is S8 and have developed a simple model for this phase from solid-fluid equilibrium data in mixtures with hydrogen sulfide. The fugacity of the S8 solid is fitted by

Figure 3. Temperature-pressure phase diagram for sulfurhydrogen sulfide.

ln(fS8, MPa) ) 24.4814 - 14499.48/T + 0.127454P/RT (14) Coefficients in this expression were obtained from two points in the three-phase solid-liquid-liquid and solidliquid-vapor temperature-pressure measurements of Woll25 and one point in the solid-fluid equilibrium data for sulfur in hydrogen sulfide of Roof.26 Hydrogen Sulfide-Sulfur. The interaction parameter between S8 and hydrogen sulfide (kij in (6)) was set at kij ) 0.0812. This value was determined from the data of Brunner and Woll,27 by minimizing the sum of squares of the error in the calculated sulfur mass fraction in hydrogen sulfide. The model used takes into account the decomposition of hydrogen sulfide to produce hydrogen; hence, interaction parameters between hydrogen and the other species might be included in the equation of state. However, the calculated hydrogen fraction is very small under all circumstances, and all of the interaction parameters with hydrogen were set to zero. The hydrogen sulfide-sulfur temperature-pressure phase diagram is shown in Figure 3. Included on the figure are four calculated three-phase lines and the fourphase quadruple point where they intersect. Also shown on the diagram are the data of Woll25 for two of the three-phase lines involving solid sulfur. The data of Woll are reproduced very well. It can be seen that the model calls for liquid-liquid separations between sulfur and hydrogen sulfide, behavior that is consistent with the data. The liquid-liquid-vapor three-phase behavior terminates at a critical end point not far from the hydrogen sulfide critical point. All of the multiphase behaviors shown in Figure 3 were calculated using the procedure described in the appendix. A condition where more than two phases are present was found by adjusting the temperature or pressure (or both in the case of the quadruple point). The somewhat tedious trial and error calculations required locating conditions where intermediate mole fraction values summed to unity in three or more Nc Xij ) 1 in (A.4). phases; ∑i)1 Figures 4 and 5 compare model calculations with the

Figure 4. Comparison of model calculations with data of Roof26 for hydrogen sulfide-sulfur, including solid-fluid equilibria.

data on the solubility of elemental sulfur in hydrogen sulfide of Roof26 and Brunner and Woll,27 respectively. The two lower temperature isotherms in Figure 4 represent equilibria with solid sulfur. The isotherm at 366.5 K is below the melting temperature of sulfur but above the quadruple point temperature. This 366.5 K isotherm in Figure 4 and the three isotherms in Figure 5 describe equilibria with sulfur as a liquid. The data vary from 1 to 17 mass % sulfur in the hydrogen sulfide rich phase at pressures up to 60 MPa and over a temperature range from 316.3 to 433.1 K. Sour Gases-Sulfur. In addition to hydrogen sulfide, the components of natural gases that have been considered are methane, carbon dioxide, and nitrogen. The Peng-Robinson interaction parameter for nitrogensulfur was set to zero in the absence of any specific data. Kennedy and Wieland28 reported sulfur solubility in several pure and mixed gases, including pure carbon dioxide. The accuracy of their data has been questioned,26 but theirs is the only comprehensive data for sulfur in CO2. An interaction parameter of kij ) 0.135 for CO2-S8 matches the Kennedy and Wieland28 reported solubility at 34.47 MPa and 121.11 °C (5000 psi and 250 °F). The methane-S8 interaction parameter was fitted to data of Brunner et al.29 for a gas that was

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Figure 5. Comparison of model calculations with data of Brunner and Woll27 for liquid sulfur-hydrogen sulfide equilibria.

Figure 7. Comparison of model calculations with data. Methanerich gas 1.

Figure 6. Calculated sulfur content of sour natural gases. Comparison with data of Brunner et al.29 Gas 1: 84% H2S, 9% CO2, 7% CH4. Gas 2: 42% H2S, 2% CO2, 56% CH4. Gas 3: 9% H2S, 5% CO2, 86% CH4. Gas 4: 35% H2S, 8% CO2, 57% CH4.


Table 4. Peng-Robinson Interaction Parameters, kij H2S H2 S8 CH4 CO2 N2

H2S

H2

S8

CH4

CO2

N2

0 0 0.0812 0.08 0.097 0.176

0 0 0 0 0 0

0.0812 0 0 0.155 0.135 0

0.08 0 0.155 0 0.092 0.031

0.097 0 0.135 0.092 0 -0.017

0.176 0 0 0.031 -0.017 0

86 mol % methane (their gas number 3). A least-squares fit of the data for this gas at 485 K resulted in kij ) 0.155. The interaction parameters used are summarized in Table 4. Figure 6 compares model calculations with five sets of data reported by Brunner et al.29 Temperatures range from 398 to 485 K, and experimental pressures vary from 6.7 to 155 MPa. The H2S content of the four gases

ranges from 9 to 84 mol %. The other constituents are methane and carbon dioxide, with the carbon dioxide amount lower than 10 mol %. One isotherm was reported for three of the gases, and two isotherms were given for the fourth. Model calculations show good agreement with all of the data. Brunner and Woll27 presented data showing the effect of temperature on the equilibrium sulfur content of four gases containing 65%, 66%, and 81% methane, with the remainder being hydrogen sulfide, carbon dioxide, and nitrogen in varying ratios. Model comparisons with these data are shown in Figures 7-10 . Figure 7 also shows a more detailed plot of the Brunner et al.29 gas number 3 data that were used to fit the methane-S8 interaction parameter. Note that equilibrium sulfur is computed to be a solid at 100 °C, which is the lowest isotherm on the four figures. This temperature falls




between the melting temperature of sulfur and the quadruple point in the hydrogen sulfide-sulfur phase diagram. Measured sulfur contents are below 0.45 mass % in the four Brunner and Woll27 methane-rich gases and are as low as 0.005 mass %. The model represents the data over the whole range of temperature and pressure in this dilute sulfur content range. Discussion The liquid-vapor phase behavior of pure sulfur is correlated well by the chemical equilibrium model. Vapor pressures and liquid densities have been correlated effectively. The vapor density and the heat of vaporization of liquid sulfur calculations can be regarded as predictive, because the data were not used in the correlating effort. The model shows the correct qualitative behavior of the heat of vaporization, includ-

ing a peculiar minimum that can only be accounted for by a redistribution of sulfur between different molecular species as the temperature is increased along the vapor pressure curve. This kind of behavior could not be seen in the earlier model9 that treated sulfur as S8 only. The vapor density is considerably improved over the earlier model, also because chemical reactions between the sulfur species have been accounted for. As is shown in Figures 1 and 2, the equilibrium phases are complex mixtures with significant contributions from many of the sulfur species. The model represents a good correlation of binary data for sulfur and natural gas components. Two of the three-phase lines involving the solid on the sulfurhydrogen sulfide phase diagram are well represented. Sulfur solubility in hydrogen sulfide, even up to 20 mass % sulfur, is correlated with good accuracy, as is shown in Figures 4 and 5. The predictive (or interpolating) powers of the model are demonstrated in Figures 6-10. Sulfur is much less soluble in methane than in hydrogen sulfide. The model represents data well under conditions where the sulfur solubility is well below 0.1 mass %. The match of data with hydrogen sulfide alone and in the mixed natural gases is clearly improved over the previous model9 in which sulfur treated as S8 only. The model employed has taken into account eight sulfur species and eight sulfane species, all in chemical reaction equilibria with hydrogen and hydrogen sulfide. The thermochemical data for all of the sulfur species in their ideal gas states were obtained from the JANAF tables.19 These data are based primarily on the work of Rau et al.30 Lenain et al.31 measured Raman spectra data of sulfur vapor and suggest that the Rau model is not completely accurate. Also, Migdisov et al.32 report measurements on sulfur-hydrogen sulfide behavior that they interpret in terms of sulfane formation reactions. Such new information might have affected the Raymont20 model for sulfanes that was used in the present calculations. If modified thermochemical data were available for any of the 16 sulfur and sulfane species, there might be changes required in the equation of state correlations reported in this paper. The model is complex as it is but does not consider the possibility of more than one isomer of any of the Sn species. The well-known existence of two different crystalline forms of sulfur was also not accounted for. Although additional refinement is possible, the results obtained from the model do represent a variety of data for pure sulfur and for sulfur with acid gases. There are available data of interest with which we have not compared our model, notably the measurements of Davis et al.33 For the data considered, there is clear improvement over the results reported by Karan et al.,9 particularly in the heat of vaporization of pure sulfur. This earlier work involved a much simpler physical picture in which the sulfur was treated as a single species. We have also not made comparisons with the performance of the model by Tomcej et al.34 because the parameters in their model have not been reported. Acknowledgment This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

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Appendix: Reaction and Phase Equilibrium Method There are π phases with phase amounts βj g 0 (j ) 1, ..., π). The mole fraction of substance i in phase j is Xij. Mass balances (and possibly other constraints) require that M linear equations be satisfied: Nc

π

βjXij ) ck; ∑ ∑ i)1 j)1 Rki

k ) 1, ..., M

(A.1)

where ck is a constant (possibly the number of atoms of type k) in the linear constraint equations and the Rki coefficients are also constants (possibly the number of atoms of type k per mole of species i). Minimization of the Gibbs free energy at constant temperature, pressure, and overall composition requires M

µij/RT g

∑ Rkiλk k)1

(A.2)

where the equality sign must be satisfied if phase j is actually present with βj > 0. The new variables, λk, are Lagrange multipliers that are introduced in the constrained minimization. The solution procedure makes use of Michelsen’s proposal for the case when the phases are either pure or ideal solutions. (A.2) is solved for the mole fraction in terms of the ideal gas standard chemical potential, the fugacity coefficient, and the reference pressure for the standard pressure: M

ln Xij )

∑ Rkiλk - µ0i /RT - ln(φijP/P0) k)1

(A.3)

A nested solution procedure is employed. In the inner loop, the fugacity coefficients are regarded as independent of the phase compositions. The procedure starts with initial values for the fugacity coefficients. (A.3) is used to define the dependence of the mole fractions on the Lagrange multipliers. The independent variables are the phase amounts (the βj) and the Lagrangian multipliers (the λk). The equations that must be satisfied are

{

Nc

[

Xij - 1] ∑ i)1

) 0; < 0;

βj > 0 βj ) 0

(A.4)

and the M linear mass balance constraints, written here as Nc

π

Rki(∑βjXij) - ck ) 0 ∑ i)1 j)1

(A.5)

These equations are used to solve for M Lagrangian multipliers and π phase amounts by the NewtonRaphson iteration. In the outer loop calculations, the fugacity coefficients are updated and convergence is checked. New phase amounts and Lagrangian multipliers are then found in the inner loop. Literature Cited (1) Hyne, J. B. Controlling Sulfur Deposition in Sour Gas Wells. World Oil 1983, 35.

(2) Hyne, J. B.; Muller, E.; Wiewiorowski, T. K. Nuclear Magnetic Resonance of Hydrogen Polysulfide in Molten Sulfur. J. Phys. Chem. 1969, 11, 3733. (3) Poulis, J. A.; Massen, C. H.; v. d. Leeden, P. Magnetic Susceptibility of Liquid Sulphur. Trans. Faraday Soc. 1962, 58, 474. (4) Wiewiorski, T. K.; Touro, F. J. The Sulfur -Hydrogen Sulfide System. J. Phys. Chem. 1966, 70, 234. (5) Touro, F. J.; Wiewiorski, T. K. Viscosity-Chain Length Relationship in Molten Sulfur Systems. J. Phys. Chem. 1966, 70, 239. (6) Meyer, B.; Oomen, T. V.; Jensen, D. The Color of Liquid Sulfur. J. Phys. Chem. 1971, 75, 912. (7) Descoˆtes, L.; Bellisent, R.; Pfeuty, P.; Dianoux, A. J. Dynamics of Liquid Sulphur and the Equilibrium Polymerization Transition. Physica A 1993, 201, 381. (8) Swift, S. C.; Manning, F. C.; Thompson, R. E. Sulfur Bearing Capacity of Hydrogen Sulfide Gas. Soc. Pet. Eng. J. 1976, 16, 57. (9) Karan, K.; Heidemann, R. A.; Behie, L. A. Sulfur Solubility in Sour Gas: Predictions with an Equation of State Model. Ind. Eng. Chem. Res. 1998, 37, 1679. (10) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1985. (11) Heidemann, R. A.; Prausnitz, J. M. A van der Waals-type equation of state for fluids with associating molecules. Proc. Natl. Acad. Sci. U.S.A. 1976, 73, 1773. (12) Heidemann, R. A.; Prausnitz, J. M. Phase Equilibria from an Equation of State for Fluids with Associating Molecules. AIChE Annual Meeting, Chicago, Nov 1976. (13) Hendriks, E. M.; Walsh, J.; van Bergen, A. R. D. A General Approach to Association Using Cluster Partition Functions. J. Stat. Phys. 1997, 87, 1287. (14) Rizvi, S. S. H.; Heidemann, R. A. Correlation of AmmoniaWater Equilibrium Data with Various Modified Peng-Robinson Equations of State. Fluid Phase Equilib. 1986, 29, 439. (15) Phoenix, A. V. Multiphase Chemical and Polymer Phase Equilibria Calculations. Ph.D. Dissertation, The University of Calgary, Calgary, Alberta, Canada, 1998. (16) Phoenix, A. V.; Heidemann, R. A. A nonideal multiphase chemical equilibrium algorithm. Fluid Phase Equilib. 1998, 150151, 255. (17) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms; John Wiley & Sons: New York, 1982. (18) Michelsen, M. L. Calculation of Multiphase Ideal Solution Chemical Equilibrium. Fluid Phase Equilib. 1989, 53, 73. (19) JANAF Thermochemical Tables. J. Phys. Chem. Ref. Data 1985, Supplement 1. (20) Raymont, M. E. D. The Thermal Decomposition of Hydrogen Sulfide. Ph.D. Dissertation, The University of Calgary, Calgary, Alberta, Canada, 1974. (21) West, J. R. Thermodynamic Properties of Sulfur. Ind. Eng. Chem. 1950, 42, 713. (22) West, W. A.; Menzies, A. W. C. The Vapor Pressures of Sulfur between 100° and 550° with Related Thermal Data. J. Phys. Chem. 1929, 33, 1880. (23) Kellas, A. M. Determination of Molecular Complexity of Liquid Sulphur. J. Chem. Soc. 1918, 113, 903. (24) Tuller, W. N. The Sulphur Data Book; McGraw-Hill: New York, 1954. (25) Woll, W. The Influence of Sour Gases Upon the Melting Curve of Sulfur. Erdo¨ l-Erdgas-Z. 1983, 99, 297. (26) Roof, J. G. Solubility of Sulfur in Hydrogen Sulfide and in Carbon Disulfide at Elevated Temperature and Pressure. Soc. Pet. Eng. J. 1971, 11, 271. (27) Brunner, E.; Woll, W. Solubility of Sulfur in Hydrogen Sulfide and Sour Gases. Soc. Pet. Eng. J. 1980, 20, 377. (28) Kennedy, H. T.; Wieland, D. R. Equilibrium in the Methane-Carbon Dioxide-Hydrogen Sulfide-Sulfur System. Pet. Trans. AIME 1960, 219, 166. (29) Brunner, E.; Place, M. C., Jr.; Woll, W. H. Sulfur Solubility in Sour Gas. J. Pet. Technol. 1988, 40, 1587. (30) Rau, H.; Kutty, T. R. N.; Guedes De Carvalho, J. R. F. Thermodynamics of Sulfur Vapor. J. Chem. Thermodyn. 1973, 5, 833. (31) Lenain, P.; Picquenard, E.; Corset, J.; Jensen, D.; Steudel, R. Raman Spectrometric Study of the Composition of Sulfur Vapor at Temperatures up to 700 °C. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 859.

Ind. Eng. Chem. Res., Vol. 40, No. 9, 2001 2167 (32) Migdisov, A. A.; Suleimenov, O. M.; Alekhin, Y. V. Experimental study of polysulfane stability in gaseous hydrogen sulfide. Geochim. Cosmichim. Acta 1998, 62, 2627. (33) Davis, P. M.; Lau, C. S. C.; Hyne, J. B. Data on the Solubility of Sulfur in Sour Gases. Alberta Sulphur Res. Ltd. Q. Bull. 1992-93, XXIX, 1. (34) Tomcej, R. A.; Kalra, H.; Hunter, B. E. Prediction of Sulphur Solubility in Sour Gas Mixtures. Presented at the 39th

Annual Technical Meeting of the Petroleum Society of CIM, Calgary, Alberta, Canada, June 12, 1988; Paper 88-39-14.

Received for review September 20, 2000 Revised manuscript received February 22, 2001 Accepted February 22, 2001 IE000828U

A Chemical Equilibrium Equation of State Model for Elemental Sulfur

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