Ind. Eng. Chem. Res. 2003, 42, 3857-3864
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CORRELATIONS Gas-Solid Equilibrium in Mixtures Containing Supercritical CO2 Using a Modified Regular Solution Model Jose´ O. Valderrama,*,†,§ Nelson A. Gonza´ lez,‡ and Vı´ctor H. Alvarez§ Department of Mechanical Engineering, Faculty of Engineering, and Department of Chemistry, Faculty of Sciences, University of La Serena, Casilla 554, La Serena, Chile, and Center for Technological Information, Casilla 724, La Serena, Chile
A model that includes an equation of state and an excess Gibbs free energy model is proposed and applied to the correlation of phase equilibria in gas + solid systems containing supercritical carbon dioxide. A modified “regular solution” model for the excess Gibbs free energy that considers nonpolar, polar, and hydrogen-bonding contributions and a generalized three constant equation of state is proposed. The mixing rule derived for the model includes a concentration-dependent interaction parameter and an interaction parameter into one of the volume constants of the equation of state. Literature data for nine binary gas + solid systems containing supercritical carbon dioxide are used for testing the proposed model. The systems studied were binary mixtures containing carbon dioxide as solvent with 2,3-dimethylnaphthalene, 2,6-dimethylnaphthalene, caffeine, cholesterol, anthracene, naphthalene, phenanthrene, pyrene, and benzoic acid. The proposed model correlates the solute concentration in the gas phase better than several other models presented in the literature that use the same number of interaction parameters. Introduction The most common method used for the correlation and prediction of phase equilibria in mixtures containing a supercritical fluid, such as carbon dioxide, is the use of equations of state (EoS). Common and industrially important EoS are the cubic equations derived from van der Waals equation of state (VdW). Among the many generalized cubic EoS of VdW type available nowadays, the Soave-Redlich-Kwong,1 the PengRobinson,2 and the Patel-Teja-Valderrama,3 among others, have proven to combine the simplicity and accuracy required for the prediction and correlation of volumetric and thermodynamic properties of fluids.4-7 These cubic equations of state can be written in a general form as follows:
P)
a RT V - b V(V + b) + c(V - b)
a ) acR(Tr) (1)
For the SRK equation c ) 0, for the PR equation c ) b, and for the PTV equation c * b * 0. The function R(Tr) adopts the popular model of Soave:1 R(Tr) ) [1 + F(1 xTr)]2, F being a function of the acentric factor for the SRK and PR equations and a function of the product [acentric factor × critical compressibility factor] for the PTV equation. The parameters Ωa, Ωb, and Ωc assume constant values for the SRK and PR equation, while for the PTV equation these Ωi’s are a function of the critical compressibility factor. * To whom correspondence should be addressed. † Faculty of Engineering. ‡ Faculty of Sciences. § Center for Technological Information.
For mixtures, the equation of state parameters a, b, and c are expressed as functions of the concentration of the different components in the mixture, through the so-called mixing rules. Until recent years, most of the applications of EoS to mixtures considered the use of the classical mixing rules for the mixture force parameter a and volume parameters b and c: N N
a)
∑i ∑j
N N
xixjaij
b)
∑i ∑j
N N
xixjbij
c)
∑i ∑j xixjcij
(2)
An interaction parameter has been introduced into the force parameter a in VdW equations to improve predictions of mixture properties.8 It has been recognized, however, that even with the use of the interaction parameter the classical VdW mixing rules do not give accurate results for complex systems.9-11 During the last 25 years, efforts have been put into extending the applicability of cubic equations of state to obtain accurate representations of phase equilibria in highly polar mixtures, associated mixtures, mixtures containing supercritical components, and other very complex systems. The different approaches presented in the literature include the use of multiple interaction parameters in the mixing rules,12,13 the introduction of the local-composition concept,14 the connection between excess Gibbs free energy models and equations of state,15 the use of nonquadratic mixing rules,16-18 and the introduction of interaction parameters into the volume constants of the equation of state.19
10.1021/ie020797y CCC: $25.00 © 2003 American Chemical Society Published on Web 07/03/2003
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Table 1. Solubility of Several Solid Solutes in Supercritical CO2 Reported in the Literature, Calculated Using the Peng-Robinson Equation of Statea system CO2 +
range of T (K)
range of P (MPa)
range of y2 × 104
|∆y2| (%)
mixing rule
reference
2,3-dimethylnaphthalene 2,6-dimethylnaphthalene caffeine cholesterol naphthalene anthracene phenantrene benzoic acid pyrene
308-328 308-328 313-353 313-333 308-318 303-343 318-338 318-338 308-343
10-28 10-28 20-35 10-25 11-35 9-41 12-28 12-28 8-48
3-90 19-45 3-11 0.1-1.5 74-299 0.1-3.5 3-38 3-98 0.1-9.4
7-20 9-16 16-24 53-63 16-20 7-19 41-48 10-26 1-84
VdW VdW KTK KTK P&R KTK VdW VdW KTK
Kurnik et al.21 Kurnik et al.21 Meyer et al.34 Meyer et al.34 Mendes et al.35 Meyer et al.34 Kurnik et al.21 Kurnik et al.21 Meyer et al.34
a
In the table, VdW is the classical quadratic mixing, KTK is the model of Kurihara et al.,26 and P&R is Panagiotopoulos and Reed.16
Modeling Mixtures Containing Supercritical CO2
ized equation of state. Kurihara et al. proposed a mixing rule based on a convenient separation of the excess Gibbs free energy, as follows:
The first efforts to model the phase behavior in systems containing a supercritical fluid were done using the virial EoS,20 but these attempts were not successful. Best results have been obtained using cubic EoS such as SRK and PR.21-23 Caballero et al.23 have applied the classical EoS (RK, SRK, and PR), including quadraticand cubic-type mixing rules. These authors found better results with the use of more than one interaction parameter. Table 1 shows selected results from the literature for several binary systems containing supercritical carbon dioxide. The numbers presented in that table indicate that the concentration of the solute in the gas phase has not been well predicted and this should be improved. We believe that the concentration of the solute in the gas phase is the most important variable for the modeling and the design of supercritical fluid extraction processes since it determines the maximum amount of solute that can be extracted by the supercritical solvent. As seen in the table, errors of over 80% are found in some cases.
gE ) gERS + gERES
Here, gERS is the excess Gibbs free energy for a E is the “residual” (RES) “regular solution” (RS) and gRES excess Gibbs free energy. Kurihara et al. derived an expression for the regular solution contribution using a general cubic equation of state and for the residual contribution they used a Redlich-Kister expansion. In this paper an expression for the PTV equation of state for the pure regular solution model has been derived. E is determined by a The residual contribution gRES regular solution model that considers the polar and hydrogen-bonding contributions, as explained in the following section. For the nonpolar part the concepts of regular solution (the excess volume and the excess entropy are zero) are used to derive a mixing rule for the force parameter “a” in the equation of state. The PTV equation of state can be written as
P)
Solubility Calculations The theory of solid solubility in a gas is found in standard books,24,25 so a summary only is given in what follows. For the solid (component 2), at a given temperature, the fugacity at the sublimation condition must be equal to the fugacity of the pure gas at the sublimation pressure Ps2. Some approximations are usually introduced to obtain an expression for the concentration of the solute in the gas phase, also called solubility. If the solid phase is considered to be pure and the volume of the solid is considered to be pressure-independent, the following equation is obtained for the solubility of the solid in the gas phase:24 sol s V 2 (P-P 2)/RT
y2 ) (Ps2/φˆ v2P)e
(3)
In this expression, the fugacity coefficient φˆ V2 is also a function of y2 and is calculated using an EoS and the chosen mixing rules. Therefore, an iterative procedure is required to accurately evaluate the solubility of the solid in the gas phase, y2. The Proposed Model The model proposed in this work follows a treatment similar to that of Kurihara et al.,26 but uses only the regular solution model combined with a cubic general-
(4)
a RT V - b [V(V + b) + c(V - b)]
(5)
in which
a ) acR(Tr) ac ) Ωa(R2 Tc2/Pc2) R(Tr) ) [1 + F(1 - xTr)]2
(6)
b ) Ωb(RTc/Pc) c ) Ωc(RTc/Pc) The parameters Ωi and F are
Ωa ) 0.66121 - 0.761057Zc Ωb ) 0.02207 + 0.20868Zc
(7)
Ωc ) 0.57765 - 1.87080Zc F ) 0.46283 + 3.58230ωZc + 8.19417(ωZc )2 Following Kurihara et al.26 and using the PTV equation of state, with the following mixing rules for the volume
Ind. Eng. Chem. Res., Vol. 42, No. 16, 2003 3859
constants,
Table 2. Lydersen Group Contributions Used in the Regular Solution Method of Martin and Hoy28 N N
b)
∑i ∑j xixjbij
bij ) (b/i + b/j )/2 b/i ) bi(1 - βi) b/j ) bj(1 - βj)
(8)
N N
c)
∑i ∑j xixjcij
for a solvent “i”, βi ) 0 an expression is found for “a” in a mixture: N N
a)
∑i ∑j xixjaij -
{ ( )} b-c
ln
2b
gERES
aij ) xaiaj(1 - kij) (9)
b+c
The parameters kij and βj in the proposed mixing rules for the equation of state are calculated by regression analysis of experimental phase equilibrium data, as detailed later here. It can be easily shown that, for solubility calculations, the proposed mixing rules defined by eqs 8 and 9 do not suffer from the so-called Michelsen-Kistenmacher syndrome.27 Modified Regular Solution Model Expressions for the Gibbs free energy for a regular solution model for multicomponent mixtures25 reduce to the following equation for the binary mixtures of interest in this work:
gE ) (x1V1 + x2V2)ξ1ξ2(δ1 - δ2)2
(10)
Vi being the pure component volume, δi the solubility parameter, and ξi the volume fraction. The volume fractions of components 1 and 2 are defined as
ξ1 )
x1V1 x1V1 + x2V2
x2V2 ξ2 ) x1V1 + x2V2
group
aliphatic ∆T
-CH3 -CH2>CH>C< dCH2 -CHd >Cd -CHd aromatic >Cd aromatic -O>O epoxide -COO>CdO -CHO >(CO2O) -COOH -OH f -OH primary -OH sec. -OH tert, -OH phenolic -NH2 -NH>N-CdN -NCO HCON< -CONH-CON< -CONH2 -OCONH-S-SH-Cl 1 -Cl 2 Cl aromatic Cl2 twin -Br -Br aromatic -F -I
0.020 0.020 0.012 0.0 0.018 0.018 0.0
cyclic ∆T
0.021 0.047 0.040 0.048 0.086 0.039 0.082 0.082 0.082 0.082 0.035 0.031 0.031 0.014 0.060 0.054 0.062 0.071 0.054 0.071 0.078 0.015 0.015 0.017 0.017 0.017 0.034 0.010 0.010 0.018 0.012
0.013 0.012 -0.007
0.011 0.011 0.011 0.014 0.027 0.033
0.024 0.007
0.008
found for a nonpolar contribution δNP, a polar contribution δP, and a hydrogen-bonding contribution δH. This residual contribution δRi to the solubility parameter is calculated considering a polar contribution δP and a hydrogen-bonding contribution δH. These contributions are determined following Martin and Hoy,28 as follows:
(δT)i2 ) (δNP)i2 + (δP)i2 + (δH)i2 (11)
gE
In the proposed model the excess Gibbs free energy is expressed as the sum of two terms: a nonpolar conE , which includes tribution gEo and a residual part gRES polar and hydrogen-bonding contributions. This is done as follows:
gE ) gEo + gERES
(12)
gERES ) (x1V1 + x2V2)ξ1ξ2(δR1 - δR2 )2
(13)
The residual contribution is introduced into eq 9, to complete the definition of the mixing rule for the force constant “a”. The total solubility parameter is
(δR)i2 ) (δP)i2 + (δH)i2
(14)
An aggregation number η was defined by Martin and Hoy28 as a function of the liquid density (FL), the molecular weight (M), and the relation (Tb/Tc), as follows:
log η ) 3.39066
Tb M - 0.15848 - log Tc FL
(15)
The relation Tb/Tc is estimated according to the method given by Martin and Hoy28 from group constants given in Table 2 using the equation
Tb/Tc ) 0.567 +
∑ ∆T - (∑ ∆T)2
(16)
Using this defined aggregation number η, the polar and
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Table 3. Molar Cohesion and Volume Constants Used in the Regular Solution Method of Martin and Hoy28 group -CH3 -CH2>CH>C< dCH2 -CHd >Cd -CHd >Cd -O-O>O-COO>CdO -CHO >(CO)2O -COOH -OHf -OH -OH -OH -OH NH2 -NH>N-CdN -NCO
bond type saturated saturated saturated saturated olefin olefin aromatic aromatic ether acetal epoxide ester ketone aldehyde anhydride acid H-bond OH primary (not H-bonded) second tert phenolic amino 1° amino 2° amino 3° nitrile isocyanate
FT
FP
group
148.30 131.50 86.00 32.0 126.5 121.50 84.50 117.30 98.10 115.00 115.50 176.20 326.60 263.00 292.64 567.30 276.10 237.50 329.40 289.20 390.40 171.00 226.60 180.00 61.10 354.60 358.70
0 0 0 0 32.7 29.1 30.8 30.4 31.7 105.6 5.0 76.2 256.2 257.0 259.9 567.0 203.2 237.5 329.4 289.2 390.4 171.0 226.6 180.0 61.1 354.2 4.0
bond type
FT
FP
HCON< -CONH-CONH2 OCONH
formamide amide amide urethane
497.20 554.70 589.90 616.90
354.0 437.0 483.6 436.2
-SCl Cl Cl Cl2 Br Br F conjugation cis trans 4 mem. ring 5 mem. ring 6 mem. ring 7 mem. ring bicycloheptane ring tricyclodecane ring base value ortho sub meta sub para sub
thioether primary secondary aromatic twinned primary aromatic primary
209.40 205.10 208.30 161.00 342.70 257.90 205.60 41.30 23.26 -7.13 -13.50 77.76 20.99 -23.44 45.10 22.56 62.47 135.10 9.70 6.60 40.30
209.4 150.0 154.0 39.8 275.0 60.0 49.0 35.8 -9.7 -7.1 -13.5 98.0 41.5 29.8 0
-6.5 -11.9 -16.5
Table 4. Summary of the EoS and the Mixing Rules Proposed in This Work for Binary Mixtures of Interest in This Work; Component 2 Is the Solutea P)
a RT (V - b) [V(V - b) + c(V - b)]
b ) x1b1 + x2b2(1 - β2)
a ) x12 a1 + x22 a2 + 2x1x2(a1a2)0.5(1 - k12) -
{(
(x1b1 + x2b2(1 - β2)) - (x1c1 + x2c2)
ln
E g RES ) (x1V1 + x2V2)ξ1ξ2(δR1 - δR2 )2
a
ξ1 )
c ) x1c1 + x2c2
2(x1b1 + x2b2(1 - β2))
x1V1 x1V1 + x2V2
ξ2 )
x2V2 x1V1 + x2V2
}
)
(x1b1 + x2b2(1 - β2)) + (x1c1 + x2c2)
E gRES
Note: The residual solubility parameter is calculated according to Martin and Hoy,28 defined by eqs 14-18.
hydrogen-bonding solubility parameters are defined as
xη -η 1
δH ) δT
δP ) δT
(17)
∑ FP η∑ F T
x
Here, FP and FT are the cohesion and volume constants, respectively, determined by group contribution. Table 3 gives the values of the FT and FP contributions for several chemical groups as reported by Martin and Hoy.28 The total solubility parameter is calculated at 298.15 K as
[ x
RTFL δT ) M
{
}]
PTc3 2.303BT 1-1 PcT 3 (T + C)2
designated as MRS110, where the number “110” stands for one parameter for constant “a”, one parameter for constant “b”, and no parameter for constant “c” in the EoS. Other options such as a MRS100 (one parameter for constant “a” and none for “b” and “c”) or MRS111 (one parameter for constant “a”, one for “b”, and one for “c”) could be studied. For the calculation of the volumetric fraction ξi, the molar volume for the solute was taken as a constant at T using acceptable values given in the literature. For the volume of the solvent (supercritical CO2), the volume was calculated from the PTV equation of state. Table 5 gives the values for the molar volume for the CO2 as calculated from the PTV equation for temperatures and pressures within the ranges of the experimental data used in this study.
1/2
(18)
Here, T is 298.15 K, Tc the critical temperature, P the vapor pressure at 298.15 K, Pc the critical pressure, R the ideal gas constant, M the molecular weight, and FL the liquid density at 298.15 K and B and C are Antoine’s constants in the vapor pressure expression: Log P(mmHg) ) A - B/(T(K) + C). Table 4 gives details on the mixing rules used for the binary mixtures studied in this work. The model is
Systems Studied Literature data for the nine binary gas-solid systems have been used for analysis and application of the proposed model. Tables 6 and 7 show the pure component data employed in the calculations while Table 8 presents some details on the experimental binary mixture data used in the applications of the models. In Table 6, M is the molecular weight, Vsol 2 is the solid molar volume, Tc is the critical temperature, Pc is the critical pressure, ω is the acentric factor, Zc is the critical
c 0.016198
Ps(Pa)
In the table, AS and BS are the constants in the sublimation pressure expression Log ) AS - BS/T(K), and, A, B, and C are the constants in the vapor pressure expression Log P(mmHg) ) A - B/(T(K) + C). b Caballero et al.23 c Extrapolated from Daubert et al.36 d Xu et al.37 e Weast and Astle.38 f Calculated by Singh et al.39 g Calculated from the correlation of vapor pressure from Daubert et al.36 h Extrapolated correlation of vapor pressure from Riedel equations.40 i Calculated of extended Clausius-Clapeyron equation.25 j Calculated from Rackett equation.25 k Calculated from acentric factor. l From Z ) P V /RT , with V ) 0.51 m3/kmol given by Daubert et al.36 c c c c c
h -20.88 9.450 c 0.00290 f 5633.4 14.418 c c 0.274 b 0.225 304.2 44.0 CO2 carbon dioxide
7.38
c 0.212 c 0.9477 959.0 386.7 C27H46O cholesterol
1.25
l 0.297 d 0.5550 4.15 194.2 C8H10N4O2 caffeine
855.6
c 0.246 c 0.6039 4.47 122.1 C7H6O2 benzoic acid
751.0
c 0.211 c 0.5090 2.66 202.3 C16H10 pyrene
936.0
c 0.221 c 0.4892 178.2
873 C14H10 anthracene
2.9
c 0.222 c 0.4949 2.90 869.2 178.2 C14H10 phenanthrene
a
i -0.01 6.861 j 0.00718 d 5781.0 15.031 c
4679.90
g -30.03 9.098 c 0.00964 b 4618.1 14.408 c
2026.60
g -49.29 8.427 c 0.00626 e 4904.0 13.395 e
3034.37
g -33.66 8.339 c 0.00611 b 4397.6 12.147 c
3247.17
g -46.10 8.219 c 0.00624 b 4873.4 14.631 c
3043.46
g -35.76 7.716 c 0.00796 b 3733.9 13.583 748.4 128.2 C10H8
4.05
0.3020
c
0.265
c
117.7 (313) 165.2 (323) 148.8 (323) 159.1 (296) 92.8 (297) 157.9 (291) 362.4 (293)
c
2893.41
g -27.40 8.082 c 0.00639 b 4419.5 14.429 b 154.7 c 0.255 b 0.42.01 156.2 C12H12
3.22
785.0 156.2 C12H12
777.0
2173.55
h
2600.27
footnote C
2612.37 8.174 j 0.00635 b 4302.5 14.065 b 154.7 k 0.255
B footnote
b 0.4240
vapor pressure
A footnote BS
footnote
(19)
2,3-dimethyl naphthalene 2,6-dimethyl naphthalene naphthalene
cal exp |(yexp ∑ 2 - y2 )/y2 |i i)1
Table 6. Properties of the Solids Involved in This Study
N
W)
Zc
Calculation of Interaction Parameters Parameters such as kij and βj in the proposed mixing rules are usually calculated by regression analysis of experimental phase equilibrium data, although some authors have attempted to obtain binary interaction parameters from pure component property data.29-31 None of these proposals, however, have proven to be of general applicability and at present there is no accurate predictive way for evaluating the interaction parameters. To obtain good correlation of the variables of interest in a complex application such as mixtures containing a supercritical component, these parameters must be calculated by regression analysis of experimental phase equilibrium data. The basic idea in the regression analysis usually employed to determine the interaction parameters is to apply the EoS to the calculation of a particular property and then minimize the differences between predicted and experimental values of that property, according to a specified objective function. For the description of the phase equilibrium in gas-solid systems the objective function includes directly the solute concentration in the gas phase. The objective function employed in the calculations is
AS footnote
Vsol 2 (cm3/mol) (T in K)
footnote
compressibility factor, and FL is the liquid density at 25 °C. The symbols A, B, and C are Antoine’s constants in the vapor pressure expression Log P(mmHg) ) A - B/(T(K) + C) and AS and BS are the constants in the sublimation pressure expression Log Ps(Pa) ) AS BS/T(K). The tables also show the references from which the data were taken. In Table 7, δT is the total solubility parameter, δH is the hydrogen-bonding parameter, δP is the polar parameter, and δNP is the nonpolar parameter. In Table 8 the interval of temperature, pressure, and the solute mole fraction in the vapor phase for the data used are listed.
ω
333
150.6 78.8 64.1 57.6 53.7 51.0 48.9 45.9
353
196.2 107.5 78.7 67.0 60.6 56.5 53.5 49.2
formula
10 15 20 25 30 35 40 50
substance
318
95.2 63.3 56.0 52.1 49.5 47.5 46.0 43.6
343
175.6 92.7 70.9 62.0 57.0 53.6 51.3 47.5
FL (298 K) (mol/cm3)
10 15 20 25 30 35 40 50
V (cm3/mol)
sublimation vapor
10 15 20 25 30 35 40 50
298
57.3 51.6 48.5 46.5 44.9 43.7 42.7 41.0
T (K)
Pc (MPa)
10 15 20 25 30 35 40 50
P (MPa)
Tc (K)
10 15 20 25 30 35 40 50
V (cm3/mol)
T (K)
M (g/mol)
P (MPa)
3.22
Table 5. Molar Volume for Carbon Dioxide Calculated from the PTV Equation
-36.20
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Table 7. Solubility Parameters for the Substances Involved in This Study; Solubility Parameters Were Calculated Using the Method of Martin and Hoy28 substance
δT
2,3-dimethylnaphthalene 2,6-dimethylnaphthalene caffeine cholesterol naphthalene anthracene phenanthrene benzoic acid pyrene carbon dioxide
9.721 9.407 7.895 8.378 9.874 10.220 10.582 12.634 11.396 6.011a
a From Prausnitz et al.25 of nonpolar fluids.
δP
δNP
(δP2 + δH2)0.5
0
4.580
8.574
4.580
0
4.432
8.297
4.432
3.757 1.551 0 0 0 4.816 3.831 0
6.533 2.679 5.133 5.361 5.551 7.414 5.685 4.877
4.433 7.785 8.435 8.701 9.009 9.025 9.104 3.524b
6.533 3.096 5.133 5.361 5.551 8.841 6.855 3.524
δH
b
optimum value of the interaction parameters depend on the searching interval and of the initial value of the interaction parameters used to start the iterative procedure. The method consumes more computer time but guarantees the finding of an optimum parameter, which gives the lowest deviation in a given property. The programs developed for this study explores for multiple acceptable solutions to choose as a final solution that which gives the lowest deviation of the established objective function, defined by eq 19. Results
Estimated by regression of δNP data
The nonlinear regression programs developed for this study represent an important improvement to what is normally described in the literature as optimization procedures. The program considers the use of a modified Marquardt method,32 as a basic numerical algorithm. It has been demonstrated33 that for cases such as those containing supercritical carbon dioxide, multiple solutions (local optimum values) are found in a range which seem to be “acceptable” for correlation purposes. The
For the mixtures studied the concentration of the solute in the gas phase has been correlated and results presented in Table 9. The deviation in the solvent concentration is in all cases below 0.2%, so they are not included. The table also give the deviations for the solute mole fraction in the gas phase, |∆y2| (%), defined for a set of N experimental points, as N
|∆y2| (%) ) (100/N)
exp exp | [(ycal ∑ 2 - y2 )/y2 ]i| i)1
(20)
A percentage average error (Averr %), defined by Caballero et al.,23 has been also included in the results. This Averr % is defined for a set of N experimental
Table 8. Details on the Literature Data Used mixture CO2 +
N
range of T (K)
range of P (MPa)
range of y2 × 104
reference
2,3-dimethylnaphthalene 2,6-dimethylnaphthalene caffeine cholesterol naphthalene anthracene phenanthrene benzoic acid pyrene
28 30 43 26 43 49 23 23 36
308-328 308-328 313-353 313-333 308-318 303-343 318-338 318-338 308-343
10-28 10-28 20-30 10-25 10-35 10-41 12-28 12-28 8-48
22.0-90.1 19.0-92.1 2.8-7.2 0.02-1.5 2.8-11.3 0.03-3.5 3.3-31.9 11.4-98.3 0.03-9.4
Kurnik et al.21 Kurnik et al.21 Johannsen and Brunner41 Yun et al.42 Mendes et al.35 Johnston et al.43 Kurnik et al.21 Kurnik et al.21 Johnston et al.43
a
In the table, N is the number of data points.
Table 9. Optimum Interaction Parameters Obtained with the MRS110 Mixing Rule Model; Component 2 Is the Solute system CO2 +
T (K)
range y2 × 104
k12
β2
|∆y2| (%)
Averr %
2,3-dimethylnaphthalene
308 318 328
22.0-64.3 12.8-71.9 3.4-90.1
-0.59785 -0.01246 0.00349
0.14489 -0.18868 -0.15085
4.8 2.8 5.2
3.1 1.8 1.4
2,6-dimethylnaphthalene
308 318 328
19.0-44.7 7.57-67.7 3.05-92.1
0.02674 0.04052 0.02331
-0.10980 -0.06027 0.10943
4.2 2.8 5.0
5.6 2.9 6.4
caffeine
313 333 353
2.95-5.4 3.04-7.22 2.83-11.3
-0.48724 -0.55529 -0.58889
-0.42796 -0.58046 -0.60874
2.4 2.3 7.7
3.3 3.2 7.8
cholesterol
313 323 333
0.13-0.94 0.023-1.24 0.106-1.45
0.36794 0.39103 0.4109
0.46779 0.52214 0.56881
4.9 10.8 12.9
4.4 2.9 8.5
naphthalene
308 313 318
110-184.0 98.0-226.0 74.0-299.0
-0.02901 -0.0357 -0.0289
-0.09327 -0.10209 -0.07061
1.2 2.1 2.6
1.3 2.5 2.8
anthracene
303 323 343
0.29-0.79 0.03-1.72 0.14-3.49
0.08243 0.12340 0.03131
0.11835 0.26135 0.03788
6.4 26.1 6.9
6.6 5.8 6.6
phenanthrene
318 328 338
8.49-22.8 4.65-31.9 3.28-38.4
-0.03916 -0.04109 -0.02918
-0.06806 -0.06086 -0.02001
5.7 2.4 6.4
5.8 3.3 3.0
benzoic acid
318 328 338
11.4-43.8 4.9-73.4 3.2-98.3
-0.50589 -0.45662 -0.42306
-0.19575 0.08916 0.30555
3.2 4.1 3.9
4.3 7.2 1.6
pyrene
308 323 343
0.30-3.47 0.15-5.4 0.03-9.38
-0.12130 -0.12446 -0.14329
0.12655 0.14692 0.13333
8.8 4.1 28.0
9.8 3.1 1.5
Ind. Eng. Chem. Res., Vol. 42, No. 16, 2003 3863
Figure 1. Percent relative deviation in the gas-phase solute concentration for benzoic acid and naphthalene at 318 K.
points as
100 Averr % )
x
1
N
exp 2 ∑ [(ycal 2 - y2 ) ]i
N i)1 1
(21)
N
∑
N i)1
[yexp 2 ]i
As seen in Table 9, the deviations in the solute concentration in the gas phase are lower than the results found in the literature for similar systems, as presented in Table 1. The overall deviation is 6.6 in |∆y2| % and 4.3 in Averr %. The highest deviations remain for the systems CO2 + pyrene, while the lowest deviations are below 5% for the systems CO2 + benzoic acid and CO2 + naphthalene. Also, Table 9 shows that the definitions of Averr % and |∆y2| % (eqs 20 and 21) lead to values so different that opposite conclusions about the accuracy of the model used should be drawn. Figure 1 show the percent relative deviation for the solute gas-phase concentration at 323 K for the systems benzoic acid + CO2 and cholesterol + CO2.
ac ) parameter in the general EoS (eq 1) ai, aj, bi, bj, ci, cj ) EoS constants for pure components aij, bij, cij ) cross parameters in an EoS A, B, C ) Antoine constants for vapor pressure As, Bs ) Antoine constants for sublimation pressure Averr % ) percentage average error F ) parameter in the R(T) function FP, FT ) molar cohesion constants gE ) excess Gibbs free energy gE0 ) excess Gibbs free energy for nonpolar contribution E gRES ) residual excess Gibbs free energy E gRS ) excess Gibbs free energy for a regular solution kij ) binary interaction parameter for “a” in an EoS M ) molecular weight N ) number of the points in a data set P ) pressure Pc ) critical pressure Ps2 ) sublimation pressure R ) ideal gas constant (1.987 cal/mol-K) T ) temperature Tc ) critical temperature Tb ) normal boiling temperature Tr ) reduced temperature, Tr ) T/Tc ∆T ) Lydersen group constant V1, V2 ) molar volume for pure components 1 and 2 Vsol 2 ) solid molar volume xi, xj ) mole fraction of components i and j x, y ) liquid and vapor phase mole fractions y2 ) solute mole fraction in the gas phase Zc ) critical compressibility factor W ) objective function Abbreviations EoS ) equation of state KTK ) Kurihara et al., mixing rule MRS ) Modified Regular Solution PR ) Peng-Robinson EoS P&R ) Panagiotopoulos and Reed mixing rule PTV ) Patel-Teja-Valderrama EoS SRK ) Soave-Redlich-Kwong EoS VdW ) van der Waals EoS
Conclusions Important improvements in the correlation of the gasphase solute concentration in gas-solid systems have been done by proposing a consistent separation of the regular solution model for the excess Gibbs free energy and a generalized cubic equation of state. The inclusion of an interaction parameter into the volume constant of the EoS improves correlation, giving lower deviations in the solute concentration in the gas phase. Consistent with other results found in the literature, one cannot avoid, however, including more than one empirical adjustable parameter if accurate correlation of the solute concentration in the gas phase is needed. On the basis of the results obtained in this study, the following conclusions are obtained: (i) the concentration of the solute in the gas phase can be reasonably correlated using the proposed model; (ii) the method developed for the calculation of the optimum parameters in this work proved to be efficient in determining the absolute optimum value for the interaction parameters; and (iii) to unequivocally decide about the accuracy of an EoS model, the deviation |∆y2| % must be used.
Greek Letters
Nomenclature
The authors thank the support of the National Commission for Scientific and Technological Research (CONICYT-Chile), through the research grant FONDECYT 1000031, the Direction of Research of the University of
Symbols a, b, c ) constants in an EoS
R ) temperature function in an EoS βi, βj ) empirical parameters for “b” in an EoS δi ) solubility parameter δH ) hydrogen bonding contribution to δT δNP ) nonpolar contribution to δT δP ) polar contribution to δT δT ) total solubility parameter δRi ) residual solubility parameter (δR)i2 ) (δP)i2 + (δH)i2 η ) aggregation number ∆ ) deviation ξi ) volume fraction of component i FL ) liquid density φˆ V2 ) fugacity coefficient for the solute in the vapor phase ω ) acentric factor Ωa, Ωb, Ωc ) parameters in the PTV EoS Superscripts/Subscripts cal ) calculated exp ) experimental L, V ) liquid, vapor
Acknowledgment
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Ind. Eng. Chem. Res., Vol. 42, No. 16, 2003
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Received for review October 7, 2002 Revised manuscript received April 4, 2003 Accepted May 6, 2003 IE020797Y
