Estimation of solubilities in supercritical carbon dioxide: A correlation for the peng-robinson interaction parameters

220

The Journal

of Supercritical

Fluids,

1992, 5, 220-225

Estimation of Solubilities in Supercritical Carbon Dioxide: A Correlation for the Peng-Robinson Interaction Parameters? Keith

D. Bartle,

Anthony

A. Clifford,*

and Gavin

F. Shilstone

School of Chemistry, University of Leeds, Leeds, LS2 9JT, UK Received August 26, 199 1; accepted in revised form July 27, 1992

The isotherms of published solubilities of 19 compounds of varied chemical type in pure CO2 have been fitted to the Peng-Robinson equation-of-state using the binary interaction parameter as an adjustable variable. The parameters thus obtained at temperatures close to 308 K are found to be approximately correlated to a particular function of some of the physical properties of the solute and solvent. This allows rough estimates to be made of solubilities of compounds in supercritical CO2, where there are no published experimental measurements. Keywords:

solubility, supercritical, equation-of-state, carbon dioxide

INTRODUCTION Prediction of solubilities in supercritical fluids is difficult, even when experimental data are available to refine the equations used.’ One approach is to use an equation-of-state, and of these the Peng-Robinson (P-R) equation* has been one of the most widely used, although even this equation will not accurately cover a wide range of conditions. This equation has one interaction parameter, &t2, for a binary mixture, to which the predicted solubility is sensitive. For this reason, although at2 can be obtained in principle from many types of physical property data for the mixture, it usually needs to be obtained by fitting experimental solubilities to the equation. Further difficulties are (a) that the interaction parameters are often found to be temperature dependent,3*4 (b) the equation does not fit the data equally well at all temperatures and pressures, and (c) reliable values of the other physical parameters needed for the equation-of-state are not always available. In spite of these problems, it is worthwhile to search for a method of making approximate predictions of solubilities in a supercritical fluid, when no experimental measurements have been made. In this regard, a cubic technique has been previously proposed for obtaining P-R binary interaction parameters from the ratio of the critical volumes of both components for mixtures of CO2 and some hydrocarbons5 In this study, a correlation is sought for wider range of compounds, which could be “Paper presented at the 2nd International Symposium on Supercritical Fluids, May 20-22, 199 1, Boston, MA, USA.

used to estimate P-R binary interaction parameters and hence, solubilities in supercritical CO*. CALCULATION OF P-R INTERACTION PARAMETERS The first stage is to fit all available good solubility data in supercritical CO2 to the P-R equation-of-state. A simplified set of equations is used, arising from the assumptions that (a) the solute in equilibrium with the saturated solution remains as a pure solid in the presence of CO2 under pressure, (b) the saturated solution is dilute, allowing some terms to be dropped from the equation-ofstate, and (c) that the molar volume of the solid is essentially unchanged at the pressures considered. The equations used are given in the first section of Table I, and although they are well known, it is worthwhile giving the precise simplified form to which the parameters, that are given and discussed below, refer. The critical parameters and acentric factor used for COz are also given in the bottom section of Table I. Of the more than 60 compounds for which tabulated data for solubilities in CO2 have been published by the end of 1989,6 there are 19 for which reliable values of the physical data necessary for the calculations are available. Table II lists these compounds with the critical parameters, molar volumes, and acentric factors used in the calculations below. Table III gives the equations used for the vapor pressures of the compounds. Vapor pressure data for these involatile compounds may be unreliable and will contribute significantly to the error in the values of the interaction parameters obtained. The simplified P-R

0896-8446/92/0503-0220$4.00/O

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The Journal of Supercritical Fluids, Vol. 5, No. 3, 1992

Simplified Solute in Section)

P-R Equations a Supercritical

for Fluid

In x=ln(p,lp)-ln

Dilute (Top

Solubilities in Supercritical Carbon Dioxide

TABLE I Solutions, Giving Section) With the

the Mole Parameters

221

Fraction, x, at Saturation of a for Solutions in CO2 (Bottom

q++pV,/RT

ln & =(6,/b,)(Z-l)-ln(Z-pbi/RT)-b,/b,)

(a,,12~2blRT)(2al,Ia,,

ln[(Z+(1+~2)b,p/RT)I{Z+(l-42)b,pIRT)]

Z = pVl RT; b, = 0.0778RT,,, I pC,,; b, = O.O778RT,,,p,,

a12

= [0.45724(1-~,2)R2T,,,T,,2K,K2]l~~

a 11= (0.45724R2Tc,,2~,2)/p,,, K, =1+(1-d-)(

0.37464 + 154226w, - 0.26992q’)

K2 = 1+ (1 - dT)(

0.37464 + 1.542260, - 0.26992w,‘)

V is the molar volume of the pure solvent, pVis the vapor pressure,and V,,, the molar volume of the pure solute pc,i, T,,i, and 61,are critical pressures,temperatures, and acentric factors Subscripts: 1 = solvent; 2 = solute

For CO, : pC,, = 7.383 MPa;

42

=0.51B;

B=

4~2

T,,, = 304.2 K;

-01)(v,.,Wz,,)(~c.2

w, = 0.225

lpc,~)'

A = l/2 for compounds containing -OH groups and capable of hydrogen bonding A = 1 for other compounds. V,.; = (0.2918 - 0.928wi)RT,,i / pC,i

equations and parameters of Tables II and III were used to fit published isotherms of experimental solubilities by adjusting the binary interaction parameters, using the principle of least squares. Sixty-seven temperature-dependent P-R interaction parameters were obtained, which varied in value from close to zero to 0.2. Probable errors in the parameters varied with the quality of the experimental solubilities and the ability of the model to fit the data, but were typically 0.01 to 0.02. TEMPERATURE DEPENDENCE OF P-R INTERACTION PARAMETERS For some compounds, the P-R interaction parameter, found by best fit of the data at one temperature, was

found to be strongly dependent on temperature. An example is naphthalene, where the values, shown in Table IV, fall from 0.085 to 0.013 as the temperature rises from 308 to 338 K. Similar behavior is shown by phenol, and also, to a smaller extent over the same temperature range, by octadecanol, biphenyl and benzoic, and stearic and oleic acids. Other compounds give interaction parameters which are constant within the probable errors over this same temperature range. An example is phenanthrene, for which the data are also given in Table IV, and are seen to be all within 0.01 of a value of 0.12. This approximately constant behavior is also exhibited by 2,3- and 2,6dimethylnaphthalene, fluorene, pyrene, anthracene, and acridine. The other compounds investigated have pub-

222 Bade et al.

Physical

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TABLE II Used in the Calculation of Peng-Robinson

Interaction

Parameters

Refs.

VII3 (cm3 mol-I) n-docosane n-tetracosane 1-octadecanol steak acid palmitic acid oleic acid naphthalene 2,3-dimethylnapthalene 2,6-dimethylnapthalene phenanthrene anthracene fluorene pyrene biphenyl carbazole acridine phenol p-chlorophenol benzoic acid

787.1 806.1 790.0 810.0 791.0 797.0 749.8 769.2 779.3 882.6 869.3 826.4 894.8 769.1 899.1 890.1 692.2 730.0 752.0

9.93 9.20 14.4 16.5 19.0 17.0 41.14 29.06 29.06 31.72 21.9 29.91 26.04 33.91 32.65 29.72 61.30 47.5 45.6

TABLE III Parameters for Solute Vapor Pressures, pv, Given by loglO(pJbar) = -0.218%/T + d, Used in the Calculation of Peng-Robinson Interaction Parameters c 6)

d

28641.4 32762.6 53040.0 43518.3 38335.2 29735.7 17065.2 19679.6 20228.8 20595.0 242 10.5 9588.0 22654.0 19816.9 22134.1 22084.5 11891.5 16346.2 15253.3

13.79 15.40 27.45 20.85 18.41 12.39 8.57 9.02 9.42 8.42 9.72 8.32 8.52 9.62 8.25 9.30 5.63 8.25 6.15

0.976 1.066 0.892 1.085 1.047 1.120 0.302 0.417 0.417 0.437 0.370 0.406 0.465 0.416 0.496 0.428 0.450 0.490 0.620

390.9 441.8 332.9 302.4 300.7 316.1 125.0 155.7 155.0 181.9 138.9 138.2 159.1 178.0 151.5 178.3 89.0 101.6 96.5

IO,1 1 10,ll 10.11 10.11 10,ll 10,ll 8,10 8,12 8.12 8,10,12 10,ll 8,10 8,10,12 IO,1 1 8,12 8.10 8,lO 10,ll 10.11

TABLE IV Calculated Values of the P-R Interaction Parameter for Naphthalene and Phenanthrene at Various Temperatures T W)

6 12

Sol. refs.

308 318 323 328 332 333 338

0.085 0.075 0.072 0.052 0.055 0.014 0.013

7,17 17,18 19 7,17,18,19 19 7 7

308 313 318 323 328 338 343

0.126 0.123 0.117 0.122 0.111 0.107 0.127

20 21 22 23 22 22 23

Ref. Naphthalene

n-docosane n-tetracosane 1-octadecanol steak acid palmitic acid oleic acid naphthalene 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene phenanthrene anthracene fluorene pyrene biphenyl carbazole acridine phenol p-chlorophenol benzoic acid

10 10 11 11 11 11 10 13 13 14 14 14 14 14 15 16 10 11 10

lished solubilities at only one temperature or two temperatures within 10 K. Contributory, and possibly the most important, factors in obtaining these results is that the simplified equa-

Phenanthrene

tions used assume that (a) the solute is a solid, which does not absorb C02, and (b) the solutions are dilute. The compounds with the most temperature-dependent calculated parameters are those with the low melting points and/or solubilities at the highest temperatures and pressures above x = 0.01, well above the concentrations for which solutions in supercritical fluids may be considered

The Journal

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Fluids,

Calculated Values of the P-R Interaction Factor, A, and the Group of Parameters, B

TABLE V Parameter,

812 n-docosane II-tetracosane 1-octadecanol steak acid palmitic acid oleic acid naphthalene 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene phenanthrene anthracene fluorene pyrene biphenyl carbazole acridine phenol p-chlorophenol benzoic acid

Solubilities in Supercritical Carbon Dioxide

Vol. 5, NO. 3, 1992

0.128 0.140 0.056 0.071 0.083 0.103 0.085 0.084 0.091 0.125 0.090 0.090 0.092 0.095 0.193 0.126 0.104 0.098 0.127

to be dilute. For example, naphthalene has a melting point of 80 “C and can form a liquid phase, in the presence of CO2 under pressure, above 65 “C (the temperature of some of the data used).7 Phenol, octadecanol, biphenyl, and oleic and stearic acids have melting points below that of naphthalene and phenol, biphenyl, and benzoic acid have solubilities above x = 0.01 amongst the data used for the fitting. On the other hand, phenanthrene, 2,3- and 2,6-dimethylnaphthalene, fluorene, pyrene, anthracene, and acridine all have melting points above 100 “C. The fall in the best-fit interactions parameters therefore appears to be largely due to an attempt to fit the simplified equations to rising solubilities at higher temperatures and pressures. Conversely, as a working hypothesis for approximate CO;? solubility calculations, the P-R binary interaction parameters may be regarded as constant for compounds with higher melting points and critical points, that are not expected to show rising solubilities at the highest temperatures and pressures considered. CORRELATION OF THE P-R INTERACTION PARAMETERS Because of their temperature dependence for some compounds, interaction parameters at a low temperature, close to 308 K, were investigated to find correlations with other properties. In fact, the data used for all compounds were for 308 K, except for stearic, oleic and palmitic acids, and carbazole, where the closest data are at 310 K, and octadecanol, n-docosane, and n-tetracosane, where the closest data are at 313 K. Table V gives the values of 6 at these temperatures, obtained by fitting experimental solubilities, for which references are also given.

s12, at Temperatures

A

Close

the

Sol. refs.

B

1 1 l/2 l/2 l/2 l/2 1 1 1 1 1 1 1 1 1 1 l/2 l/2 l/2

to 308 K,

223

0.194 0.198 0.130 0.180 0.198 0.187 0.103 0.181 0.179 0.245 0.117 0.187 0.202 0.208 0.322 0.223 0.223 0.186 0.261

24 24 25 25 26 25 7.17 22 22 20 20,27,28 23 23 7 21 20.24 29 29 24

A number of correlations were investrgated, the most successful of these (for which there is no theoretical justification) being a plot against a group, B, of physical parameters for CO2 (subscript 1) and the solute (subscript 2), where B = 4%

- @I ,( F.2 / %I)( PC,2 ’ PC.1)2 ’

(1)

and A is a parameter which is l/2 for compounds containing -OH groups and 1 for other compounds, this distinction being discussed further below. The critical volumes used, for all compounds including C02, were calculated from the equations V,,i = (0.2918-0.9280;)RT,,i

/ pc,i .

(2)

Table V gives the values of A chosen for each compound and the calculated values of B, and Figure 1 shows a plot of at2 vs. B. There appears to be an approximate linear correlation, the slope of the best fit straight line being 0.51, that is, 4, = OSlB .

(3)

It seems appropriate to force this empirical correlation through the origin as in the limit where the solute and solvent become identical, both B and 42 would become zero. The root-mean-squared deviation from the line vertically, that is, in terms of &, is 0.015, which is a typical probable error in the calculated values of at2. However, some individual points deviate from the line well outside

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Bartle et al. I

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0.20

Fluids, Vol. 5, No. 3, 1992 1

I

.

I

l

0.20 . 0.15 . l

. a

0.15 s,,

0.10

I . /

.

l

.

l a

.

l

.

. .

.

6 .

‘* 0.10

0.05 ///

,1/l 0

0.05 0.2

0.1

0.3

B

Figure 1. Plot of best-fit P-R interaction parameters, a,?, vs. the group of parameters, B, defined by eq 1. 0, compounds with A = 1; W compounds with A = l/2.

their calculated probable errors, the worst case being naphthalene, with a deviation of 0.03 1 compared with a probable error of 0.006. Solubilities for naphthalene at 308 K, calculated using a value of al2 obtained from eq 3, the simplified P-R equations, and the parameters of Tables II and III, are higher than experimental solubilities by around 25%. In comparison, by using the best-fit value of s12, the solubilities predicted by the P-R equation are within around 10% of the experimental values at pressures between 100 and 280 bar. There is only one deviation from the line which is greater than that for naphthalene, that for n-tetracosane at 0.038. The factor A was introduced because previously the points were found to exhibit two correlations: one for the carboxylic acids, phenols, and octadecanol; and another for the remaining compounds. Introduction of a factor of l/2 moved the two correlations together with approximately the same scatter exhibited by the two separate correlations. The use of a factor of somewhat less than l/2 decreases the scatter of the combined correlation somewhat, but fine-tuning of the factor does not seem to be justified by the data. The compounds for which the factor l/2 needs to be applied all contain -OH groups. However, the factor A is put forward only as an empirical quantity. Equation 3 can only be put forward tentatively as a method of obtaining values of 8,* and hence predicting solubilities in this difficult area. This and other correlation equations require further study. The correlation is scattered, but it does cover a wide range of types of compound. Because of the temperature dependence of S,, for

0

Figure 2. Plot of best-fit P-R interaction parameters, a,?, obtained from experimental solubility data for naphthalene at 318 K using various values for the acentric factor difference (co?- %J.

some compounds, the correlation is not valid above 308 K for compounds with lower melting and critical temperatures. In spite of these problems, the authors have used eq 3 to make estimates of solubilities in CO2 of very complex molecules, such as agrochemicals.9 In making these calculations it was also necessary to use prediction methods8 for many of the other data needed for the solutes. The predicted solubilities are therefore very approximate, but these values have been found useful in understanding the extraction behavior of these solutes. The complete equations needed to make these solubility estimates are given in Table I, again with the preliminary and tentative nature of these equations being stressed. One satisfactory feature of the equations is that the solubilities calculated are not very sensitive to the value chosen for the parameter 02. This is because an erroneously high value chosen for w2 will predict a higher value for &12 and there is a partial cancellation of these two errors in the predicted solubilities. This effect is illustrated in Figure 2, which shows the best-fit values of 812 obtained from naphthalene solubility in CO2 at 3 18 K as the y value is varied. This shows that similar solu-

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bilities will be obtained if both O, and &I2 are increased together. CONCLUSIONS The correlation, discussed above, can therefore be used, in conjunction with the Peng-Robinson equationof-state, to obtain rough estimates of the solubility of involatile compounds in supercritical carbon dioxide. The appropriate equations are given in Table I.

Solubilities in Supercritical Carbon Dioxide

(10) CRC Handbook of Chemistry and Physics; 62 ed.; CRC Press: Florida, 197 1. Properties Data Service; Institution of (11) Physical Chemical Engineers: Rugby, UK, 1990. (12) The Coal Tar Data Book; The Coal Tar Research Association: Leeds, UK, 1965. (13) Osborn, A. G.; Douslin, P. R. J. Chem. Eng. Data 1975,

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