Experimental solubilities of two lipid derivatives in supercritical carbon dioxide and new correlations based on activity coefficient models

Neha Lamba, Ram C. Narayan, Joy Raval, Jayant Modak and Giridhar Madras*
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India. E-mail: giridhar@chemeng.iisc.ernet.in; Fax: +91 80 23600683; Tel: +91 80 22932321

Received 9th October 2015 , Accepted 4th February 2016

First published on 5th February 2016


Abstract

The solubilities of two lipid derivatives, geranyl butyrate and 10-undecen-1-ol, in SCCO2 (supercritical carbon dioxide) were measured at different operating conditions of temperature (308.15 to 333.15 K) and pressure (10 to 18 MPa). The solubilities (in mole fraction) ranged from 2.1 × 10−3 to 23.2 × 10−3 for geranyl butyrate and 2.2 × 10−3 to 25.0 × 10−3 for 10-undecen-1-ol, respectively. The solubility data showed a retrograde behavior in the pressure and temperature range investigated. Various combinations of association and solution theory along with different activity coefficient models were developed. The experimental data for the solubilities of 21 liquid solutes along with geranyl butyrate and 10-undecen-1-ol were correlated using both the newly derived models and the existing models. The average deviation of the correlation of the new models was below 15%.


1. Introduction

A fluid, whose temperature and pressure has been raised above the critical point is called a supercritical fluid (SCF). These fluids have properties between those of gases and liquids and find a number of applications in chemical engineering and biotechnology.1–7 In particular, carbon dioxide as a supercritical fluid has been investigated extensively as it is non-flammable compared to other fluids such as hydrocarbon gases and fluorine based compounds. As the critical temperature of supercritical carbon dioxide is 304.25 K, it has been extensively utilized in processing biological materials, many of which are heat-labile.8 Numerous studies pertaining to extraction and reaction of non-polar lipids (like vegetable oils, algal oil, animal fat and fatty acids) indicate the applicability of supercritical carbon dioxide.9–12

With advances in oleochemistry, chemical derivatives of naturally occurring fatty acids and triglycerides are gaining immense importance in the areas of surfactants, perfumes, flavors, lubricants and reaction precursors. Supercritical carbon dioxide could be potentially used in many of these reactions and separations.13–15 In this context, the importance of solubility data in designing such unit operations/processes cannot be undermined. The solubilities of many solids are reported in the literature but the solubilities of liquids in SCFs are relatively scarce. The solubilities of fatty acids and triglycerides in supercritical fluids have been reported in literature.16–19 The solubilities of geraniol and 10-undecenoic acid has been reported.20 The chemical derivatives of these compounds will find applicability in food, flavors and pharmaceutical industries but the solubilities of these compounds have not been reported.

In this work, the solubilities of two lipid derivatives, geranyl butyrate and 10-undecen-1-ol, have been reported. Geranyl butyrate (synthesized by esterification of naturally occurring geraniol and butyric acid21,22) is used as a plasticizer and as a flavoring substance. The synthesis of geranyl butyrate in supercritical carbon dioxide has been reported earlier.23 The fatty alcohol, 10-undecen-1-ol is derived by reduction of 10-undecen-1-oic acid (obtained from the alkaline pyrolysis of castor oil24) and it has been used in synthesizing several biodegradable polymers.25 Further, it could also be used as a surfactant and emulsifier like many other fatty alcohols. To the best of our knowledge, this is the first report on the solubilities of these compounds in supercritical carbon dioxide.

The semi-empirical models are known for their simplicity in correlating solubility data.19,26–30 Models such as MT model,31 Chrastil model32 and association model28 have been used to correlate the experimental data. However, these correlations are empirical and may not provide insights into the nature of interactions between the solute and the SCF phase. Recently, new models based on solution and association theories29,33 have been developed. However, a comprehensive survey and comparison of various models is lacking.

The objectives of this study are two-fold. Firstly: determining the solubilities of 10-undecen-1-ol and geranyl butyrate in a wide range of temperatures and pressures. Secondly: deriving several new models based on combinations of solution/association theory and activity coefficients. Thus, this work makes an earnest attempt to compare, evaluate and draw parallels among various combinations of models in their ability to correlate the experimental data of geranyl butyrate, 10-undecen-1-ol and several other liquid solutes reported in literature.

2. Materials

Geranyl butyrate (CAS no. 106-29-6) with purity >95%, having 4% geraniol and 1% nerol as impurities and 10-undecen-1-ol (CAS no. 112-43-6) with purity >98% were supplied by TCI chemicals, Japan. Carbon dioxide with 99% purity was purchased from Noble Gases, (Bangalore, India). Chemical structures of the compounds used in the study are shown in Fig. 1.
image file: c5ra20959e-f1.tif
Fig. 1 Solubility of (a) 10-undecen-1-ol & (b) geranyl butyrate at different temperatures, ■, 308.15 K; ●, 313.15 K; ▲, 323.15 K; ▼, 333.15 K. Solid line represents the MT model fit.

3. Experiment and procedure

A detailed description and procedure along with the flow diagram of the experimental setup has already been reported in our earlier study.15 Flow saturation technique was used in this study to determine the solubilities of the solutes in supercritical carbon dioxide (SCCO2). CO2 from the cylinder was passed through the silica gel bed to increase its purity to 0.999 mass fraction. A chiller maintained at 263 K followed by a HPLC pump (PU-2080-CO2, Jasco International Company, Japan) was used to deliver CO2 at a flow rate of 0.1 ml min−1. This flow rate was optimized after conducting a number of experiments at different flow rates to ensure saturation.15 A liquid column of volume 50 ml (EV-3-50-2, Jasco) was placed inside the thermostat, pre-filled with 30 ml of the liquid solute. After passing through the chiller, CO2 was pumped to the 50 ml liquid column. The operating conditions of temperature and pressure were maintained by the thermostat and the back pressure regulator (BP-1580-81, Jasco), respectively. A six way Rheodyne valve was used to switch the flow between the online and offline modes. After the complete saturation of the solvent (CO2) stream, the liquid solute depressurizes into a collection trap at regular intervals. The equilibrium solubility was then calculated using the amount collected. All the experiments were repeated in triplicate and the uncertainty was within ±5%.

4. Models & correlations

4.1. Existing models

4.1.1. Mendez-Santiago and Teja (MT) model. This model is an empirical model which relates the solubility of the solute to the operating temperature, pressure and the solvent density. This was derived on the basis of dilute solution theory that relates the enhancement factor and the density of the solvent to the solubility of solute.31 The simplified form for the liquid solutes is,
 
T[thin space (1/6-em)]ln(Py2) = A1 + B1ρ + C1T (1)

In eqn (1), T is the temperature in K, P is pressure in MPa, y2 is solubility and ρ is density of solvent in kg m−3. A1 (K), B1 (K m3 kg−1) and C1 are the temperature independent constants. This model can be used to check the consistency of the solubility data by plotting the variation of T[thin space (1/6-em)]ln(Py2) − C1T with ρ. All points at different temperatures and pressure fall on to a straight line.

4.1.2. Solution theory plus Wilson activity coefficient model. This model was derived for correlating the solubility of liquid solutes using the solution theory by comparing the fugacity of solute in both the liquid and the supercritical fluid (SCF) phase at equilibrium.15 Wilson activity coefficient model at infinite dilution was used to correlate activity coefficient for the solute in SCF phase. The simplified expression for the solubility is
 
image file: c5ra20959e-t1.tif(2)

In eqn (2), Φ is the pure component molar volume ratio of solvent to solute, y2 is solubility, T is temperature in K and A2, B2 and C2 (K) are temperature independent constants.

4.1.3. Association model. This model was derived using the association theory (similar to solids28), where a molecule of solute associates with ‘k’ molecules of solvent to form a solvato complex at equilibrium with the SCF phase. This is a simple four parameter model relating the solubility of solute in SCF phase to the system pressure, temperature, solvent density and the association number,20,28
 
image file: c5ra20959e-t2.tif(3)
4.1.4. Association theory plus van Laar activity coefficient model20. This is based on the association theory i.e., formation of a solvato complex that is in equilibrium with the SCF phase. Further, van Laar activity coefficient model was used to correlate the activity coefficient of liquid solute in the liquid phase. A five parameter model for solubility was obtained,
 
image file: c5ra20959e-t3.tif(4)

In eqn (3) and (4), P is pressure in MPa, P* is the reference pressure, T is temperature in K, image file: c5ra20959e-t4.tif is the ratio of pure component molar density of the solvent (B) to solute (A) in kg m−3, k3 and k4 are the association numbers. A3 (K), B3 and C3 and A4 (K), C4 (K), E4 and F4 are temperature independent constants in eqn (3) and (4), respectively.

4.2. New models

4.2.1. Association theory plus Wilson activity coefficient model. In accordance with the association theory, if a solute molecule ‘A’ associates with ‘k’ molecules of solvent ‘B’ and forms ‘ABk’ solvato complex then at equilibrium,
 
A + kB ⇄ ABk (5)

The equilibrium constant (K) for eqn (5) can be written as

 
image file: c5ra20959e-t5.tif(6)

In eqn (5), [f with combining circumflex]ABk is the fugacity of the solvato complex in SCF phase ([f with combining circumflex]ABk = yAγAfABk), [f with combining circumflex]A is the fugacity of solute in liquid phase ([f with combining circumflex]A = xAγAfA) and [f with combining circumflex]B is the fugacity of solvent in SCF phase ([f with combining circumflex]B = [small phi, Greek, circumflex]BPyB). f* represents the reference fugacities. The subscripts, ‘SCF’ and ‘S’ represent the supercritical and the solute phase, respectively.

Eqn (6) can be rewritten by expanding the individual terms of the equation as,20

 
image file: c5ra20959e-t6.tif(7)

In eqn (7), fABk is the pure component fugacity of the solvato complex in SCF phase and fA is the pure component fugacity of solute in liquid phase. P* and ϕ* are the reference pressure and reference fugacity. [small phi, Greek, circumflex]B is the fugacity of solvent in SCF phase. yA and xA are the solubilities of the liquid solute in SCF and liquid phase, respectively. In SCF phase (γA = γA as yA is order of 10−3), γA is the infinite dilution activity coefficient of solute in SCF phase and γA is the activity coefficient of solute in liquid phase.

 
image file: c5ra20959e-t7.tif(7a)
 
fABk = ϕABkP (7b)
 
image file: c5ra20959e-t8.tif(7c)
 
image file: c5ra20959e-t9.tif(7d)
 
image file: c5ra20959e-t10.tif(7e)

In above equations, MA and MB are the molecular weights of the solute and solvent, respectively. PvapA is the vapor pressure of liquid solute. ρA and ρB are the densities of solute and solvent respectively. VA is the molar volume of solute, K is the equilibrium constant, ΔHs is heat of solvation and Av, Bv (K) and qs are constants. ϕABk is the fugacity coefficient of the solvato complex.

The fugacity coefficient of solvent in SCF phase can be written as,34

 
ln[thin space (1/6-em)][small phi, Greek, circumflex]B = 2ρB(yBBBB + yABkBABkB + B) (7f)
Bij is the second virial coefficient of the pair ij, B is the second virial coefficient of the mixture in the SCF phase and k5 is the association number. The activity coefficient for the solute in SCF phase is given by Wilson activity coefficient model as,
 
image file: c5ra20959e-t11.tif(7g)
aAB and aBA represent the interaction potential between solute (A) and solvent (B). Combining all the equations from eqn (7a) to (7g) and assuming that the contribution of higher powers in density is negligible,15,20 eqn (7) can be rewritten as a four parameter model,
 
image file: c5ra20959e-t12.tif(8)

In eqn (8), ρ′ is the ratio of pure component molar density of the solvent (B) to solute (A) in kg m−3, y is solubility, P is pressure in MPa, P* is the reference pressure, T is temperature in K, k5 is the association number and A5 (K), B5 and C5 are temperature independent constants.

In eqn (8),

 
image file: c5ra20959e-t13.tif(8a)
 
image file: c5ra20959e-t14.tif(8b)
 
image file: c5ra20959e-t15.tif(8c)
 
image file: c5ra20959e-t16.tif(8d)

4.2.2. Association theory plus van Laar plus Wilson activity coefficient model. For eqn (5), the equilibrium constant can be written in terms of ratio of fugacities (eqn (6)) and further expanded in terms of activity coefficients, as discussed in the previous section,
 
image file: c5ra20959e-t17.tif(9)

[f with combining circumflex]AB can be expanded in terms of activity coefficient of solute in SCF phase using the Wilson activity coefficient model at infinite dilution as given in eqn (7g). [f with combining circumflex]A is also written in terms of activity coefficient for the solute in liquid phase expanded using van Laar activity coefficient model as,20

 
image file: c5ra20959e-t18.tif(10)

In eqn (10), aBB and aAA are the interaction potentials between solvent–solvent and solute–solute molecules, respectively. xA and xB are the solubilities of solute (A) and solvent (B), respectively in liquid phase. Using all the equations from eqn (7a) to (7g) and (10), a simplified expression for solubility is obtained,

 
image file: c5ra20959e-t19.tif(11)

In eqn (11),

 
image file: c5ra20959e-t20.tif(11a)
 
image file: c5ra20959e-t21.tif(11b)
 
image file: c5ra20959e-t22.tif(11c)
 
image file: c5ra20959e-t23.tif(11d)
 
image file: c5ra20959e-t24.tif(11e)
 
image file: c5ra20959e-t25.tif(11f)
 
image file: c5ra20959e-t26.tif(11g)

Eqn (11) simplifies to a 5 parameter model,

 
image file: c5ra20959e-t27.tif(12)

In eqn (12), k6 is the association number and A6 (K), C6 (K), E6 and F6 are constants.

4.2.3. Solution theory plus Wilson plus van Laar activity coefficient model. At equilibrium, the fugacity of solute in liquid and SCF phase can be equated as,15
 
[f with combining circumflex]lpA = [f with combining circumflex]SCFA (13)

In eqn (13), ‘A’ is the solute and ‘lp’ and ‘SCF’ represent the liquid and the supercritical phase respectively. Eqn (13) can be rewritten in terms of the activity coefficients as,

 
[f with combining circumflex]lpA = γlpAxAflpA (13a)
 
[f with combining circumflex]SCFA = γSCFAyAflA (13b)

γlpA and γSCFA are the activity coefficients of solute in liquid and SCCO2 phase. xA and yA are the mole fractions of solute in the liquid and the SCF phase, respectively. flpA and flA are the pure component fugacity of solute in the respective phases. For the supercritical fluid–liquid equilibria, the logarithm of fugacity ratio can be related as,15

 
image file: c5ra20959e-t28.tif(14)

Using eqn (13a) and (13b) in (14),

 
image file: c5ra20959e-t29.tif(14a)

In eqn (14a), xA and yA are the mole fractions of solute in the liquid and the SCF phase, respectively. A7 and B7 are constants and T is temperature in K. The solute forms a dilute solution in the SCF phase. Therefore, the activity coefficient of solute in SCF phase (γSCFA) can be expanded using Wilson activity coefficient model at infinite dilution as,

 
image file: c5ra20959e-t30.tif(14b)

The activity coefficient of solute in liquid phase (γlpA) can be expressed using the van Laar activity coefficient model as,20

 
image file: c5ra20959e-t31.tif(14c)

From eqn (14a), (14b) and (14c),

 
image file: c5ra20959e-t32.tif(15)

In eqn (15),

 
image file: c5ra20959e-t33.tif(15a)
 
image file: c5ra20959e-t34.tif(15b)
 
image file: c5ra20959e-t35.tif(15c)
 
image file: c5ra20959e-t36.tif(15d)
 
image file: c5ra20959e-t37.tif(15e)
 
image file: c5ra20959e-t38.tif(15f)
 
F8 = A7 − 1 − ln[thin space (1/6-em)]xA (15g)

As solubility cannot depend on fractional powers of density, eqn (15) reduces to a four parameter model,

 
image file: c5ra20959e-t39.tif(16)

In eqn (16), A8 (K), C8 (K), E8 and F8 are constants.

5. Results and discussion

5.1. Experimental data

The solubilities of geranyl butyrate and 10-undecen-1-ol in SCCO2 were measured from 308.15 to 333.15 K and at pressures varying from 10 to 18 MPa (Table 1). At temperatures above 333.15 K, solubilities are normally very low. Similarly, there is no significant difference in solubilities at higher pressures (above 18 MPa). Therefore, the solubilities of the compounds studied are determined in the range of Tc to 1.1Tc and Pc to 3Pc as there is significant change of densities (and solubilities) only in these regions. In order to verify the internal consistency of the experimental data, MT model (eqn (1)) was used. The term T[thin space (1/6-em)]ln(Py2) − C1T was plotted against ρ, the density of SCCO2 as shown in Fig. 1. It can be observed from Fig. 1 that different isotherms fall onto a single straight line indicating self consistency of the experimental data obtained in this study.
Table 1 Experimental solubilities (y2/mol mol−1) for the liquid solutes, 10-undecen-1-ol and geranyl butyrate, at temperature T (K) and pressure P (MPa)a
T/K P/MPa Geranyl butyrate 10-Undecen-1-ol
(y2 × 103) (y2 × 103)
a Standard uncertainties (u) are u(T) = 0.1 K, u(P) = 0.2 MPa and relative uncertainty (ur), ur(y2) = 0.05.
308.15 10 17.3 4.8
14 20.6 12.5
18 23.2 25.0
313.15 10 16.5 4.3
12 16.7 6.4
14 19.7 9.4
16 19.9 12.4
18 22.7 21.0
323.15 10 3.3 3.7
12 14.6 5.2
14 16.2 8.3
16 17.8 10.5
18 18.4 14.1
333.15 10 2.1 2.2
12 4.2 4.7
14 12.5 6.0
16 15.8 10.2
18 16.7 14.4


Fig. 2(a) and (b) show the variation of solubilities of 10-undecen-1-ol and geranyl butyrate as a function of pressure. It was observed that solubility increases with pressure along an isotherm. This is due to increase in density of the supercritical carbon dioxide with pressure, when the temperature and thereby the vapor pressure is constant.35 This has been observed for all solid and liquid solutes analyzed in literature. However, the variation of solubility with temperature is not straightforward. This is due to the interplay of two variables: density and vapor pressure, both of which vary with temperature. In case of experimental data reported in this work, at a constant pressure, increase in solubility was observed with decrease in temperature (in the given pressure and temperature range). This phenomenon is called retrograde behavior, where the density plays a dominant role as compared to the vapor pressure.36 In other words, the operating pressures seem to be lower than the cross over pressure.37 This can be attributed to the low vapor pressures of the compounds in the given range of temperatures.


image file: c5ra20959e-f2.tif
Fig. 2 Variation of solubilities of (a) 10-undecen-1-ol & (b) geranyl butyrate with pressure at different temperatures, ■, 308.15 K; ●, 313.15 K; ▲, 323.15 K; ▼, 333.15 K. The solid line represents the correlation based on association + van Laar + Wilson activity coefficient model eqn (12).

The solubility of geranyl butyrate was higher than 10-undecen-1-ol, although the former has a higher molecular weight. The solubility data of geranyl butyrate was compared with geraniol and myristic acid and the solubilities of 10-undecen-1-ol can be compared to 10-undecenoic acid. The solubility of geraniol, 10-undecenoic acid and myristic acid has been reported earlier.20,38 The solubility of geranyl butyrate was found to be comparable to geraniol.20 The solubility of 10-undecen-1-ol (C11 alcohol with single double bond) was lower than the solubility of geraniol (C10 alcohol with 2 double bonds). This indicates, solubility decreases with increase in molecular weight, and it increases with an increase in unsaturation.16,17,39 Further, the solubility of 10-undecen-1-ol was higher than the solubility of 10-undecenoic acid.20 The solubility of geranyl butyrate was higher than the solubility of myristic acid (a fatty acid with the 14 carbon atoms as geranyl butyrate) and geraniol. As carbon dioxide is non-polar, it is expected that non-polar compounds will have higher solubilities than polar solutes in SCCO2.

5.2. Models

Many empirical and semi empirical models have been reported in the literature.40 These models relate the solubility of solute in SCF (solvent) with temperature, pressure or density of solvent (eqn (1)–(4)). The SCF phase or liquid phase non-ideality is expressed using either of the expressions for the activity coefficient. Different possible combinations of solution or association theory with activity coefficient models were tried, as shown in Table 2.
Table 2 All possible combinations of the association and solution theory with different activity coefficient models
Theory Activity coefficient model for solute in SCF phase Activity coefficient model for solute in liquid phase Correlation
Association Constant Constant Eqn (3)
Wilson Constant Eqn (8)
Wilson van Laar Eqn (12)
Constant van Laar Eqn (4)
van Laar Constant Eqn (3)
van Laar Wilson Case A
Constant Wilson Case A
Solution van Laar Wilson Case A
van Laar Constant Case B
Constant van Laar Case C
Constant Wilson Case C
Constant Constant Case C
Wilson van Laar Eqn (16)
Wilson Constant Eqn (2)


Among the various combinations, some of them are either redundant or lead to a highly complex non-linear expression. Three such cases are possible,

Case A: when Wilson activity coefficient model was used for the liquid phase was coupled with association theory or solution theory, the final expression becomes non-linear and complex.

Case B: when van Laar model was used to express the activity coefficient in the SCF phase or when solution model is used as a stand-alone model, expressions of the form image file: c5ra20959e-t40.tif were obtained. This expression is trivial as it does not account the effect of pressure or density.

Case C: these combinations of solution model are not possible as they do not account for non-ideality in the SCF phase.

In another case when van Laar is used in the SCF phase along with association model, a similar expression like eqn (3) was obtained. Hence, all the possible combinations of association or solution theory with different activity coefficients (eqn (2)–(4), (8), (12) and (16)) were considered. Thus, in this study, three new models (eqn (8), (12) and (16)) have been derived for correlating the solubilities of various compounds at different temperatures and pressures. These models have been derived by considering liquid–SCF phase equilibria by using association theory and solution theory along with the van Laar and Wilson activity coefficient models for the solute.

In the first case, association theory was used along with the Wilson activity coefficient model for solute in SCF, resulting in a four parameter model (eqn (8)). The experimental data were fitted to the model using multiple linear regression using Origin® 8.5. In eqn (8b), A5 can either be positive and negative depending on the interaction potential between solute and solvent and the Clausius Clapeyron equation's constant. A5 can be positive or negative depending on whether ΔHs/(aAB + BvR) is greater than or less than unity, respectively. B5 in eqn (8c) accounting for the virial coefficients, interaction potential and association number was always found to be positive. Similarly, C5 in eqn (8d) having activity coefficient of the solvate complex and the reference activity coefficients along with the Clausius Clapeyron equation's constant can be either positive or negative depending on whether Av is greater than or less than the quantity, image file: c5ra20959e-t41.tif, respectively. Thus, two parameters (k5, B5) need to be always positive and two parameters (A5, C5) can be either positive or negative.

In the second case, association theory was used along with the van Laar (for solute in liquid phase) and Wilson (for solute in SCF phase) activity coefficient models and a model with five parameters was obtained (eqn (12)). The constants were obtained by the multiple linear regression. A6 in eqn (11b) is similar to A5 and can be positive or negative depending upon the value of interaction potentials, heat of solvation and the Clausius Clapeyron equation's constant. Positive value of A6 implies that ΔHs > (aAAaABBvR). In eqn (11d), C6 accounts for the interaction potential between the solute–solute and the solvent–solvent molecules, respectively. This can have positive or negative values depending upon which type of interactions are dominant. C6 will be positive if aBB > (2xA/xB)aAA and negative when aBB < (2xA/xB)aAA. E6 in eqn (11f) is similar to B5 (eqn (8c)) and also accounts for the virial coefficients and the intermolecular interaction between solute and the solvent particles. F6 (eqn (11g)) similar to C5 (eqn (8d)) can be positive or negative depending on whether Av is greater or less than the quantity, image file: c5ra20959e-t42.tif. Thus, in this model, among the five parameters, only k6 needs to be positive.

In the last case, the model was derived using solution theory (instead of association theory) along with the van Laar and Wilson activity coefficient models and a four parameter model (eqn (16)) was obtained. Eqn (16) differs from eqn (12) due to the absence of the term (k6 − 1)ln(P/P*), thereby reducing a parameter in eqn (16). In eqn (15b), A8 accounts for the interactions between solute–solvent and solute–solute interactions. The value of E8 in eqn (15f) will depend upon the interaction potential of solute–solvent pair. F8 in eqn (15g) will have either positive or negative value depending upon the constant A7. There are no constraints on the parameters in this model.

The deviation of experimental data from the model was quantified using AARD% (average absolute relative deviation) and given by,

 
image file: c5ra20959e-t43.tif(17)

In eqn (17), ycal2 is the solubility value calculated by the model and yexp2 denotes the solubility obtained from experimental data and N is the number of data points. To determine the best model among all the models, solubility data of geranyl butyrate and 10-undecen-1-ol were regressed using the various models ((eqn (2)–(4), (8), (12) and (16))). In addition, the solubilities of 21 compounds were chosen based on their relevance in lipid processing and biodiesel/biolubricant production and correlated using the above models. Among the combinations discussed, all solution theory based models and the association + Wilson activity coefficient model resulted in one parameter less than the other association theory based models. The regression was performed using multiple linear regression routine in Origin® 8.5 and the AARD% values for all the compounds were calculated. The AARD (%) of all the six models are reported in Table 3.

Table 3 Solubility data and AARD% values from all the models of all the compounds reported
Compound Temperature (K) Pressure (MPa) Solubility (y2 × 103) AARD%
Association Association + van Laar Association + Wilson Association + van Laar + Wilson Solution + Wilson Solution + van Laar + Wilson References
10-Undecenoic acid 308–333 10–18 0.38–17.4 9.8 10 10 9.1 11.5 11.7 20
Geraniol 308–333 10–18 2.74–25.15 4.8 4.8 4.2 4.4 4.4 4.5 20
Butyl-stearate 308–323 10–16 0.31–5.54 9.8 8.3 9.4 8.1 9.9 8.7 41
Butyl laurate 313–328 10–16 1.3–23 12.1 9.3 13.1 9.9 22.0 12.3 41
Dimethyl sebacate 308–328 10–18 4.5–11.8 5.8 5.8 5.9 5.9 7.7 6.1 15
Bis-2-ethylhexyl-sebacate 308–328 10–18 0.08–5.5 11.8 10.8 11.7 11.1 14.4 11.9 15
Ethyl stearate 313–333 8.94–18.26 1–15 12.4 12.2 12.3 12.2 17.1 19.4 42
Ethyl oleate 313–333 9.54–18.62 1–23 12.3 11.9 12.0 11.8 12.7 12.7 42
Ethyl linoleate 313–333 9.30–16.97 1–19 13.4 13.4 13.3 13.2 14.5 13.7 42
Methyl stearate 313–343 9.08–20.42 0.2–25 20.3 22.6 20.4 22.3 15.1 23.3 43
Methyl oleate 313–333 9.55–18.12 3–33 11.4 10.6 11.4 10.5 26.8 14.7 44
Methyl palmitate 313–343 6.72–18.29 0.1–27 13.0 10.7 13.3 9.4 16.5 16.4 45
Nonanoic acid 313–333 10–30 0.13–7.82 19.0 19.1 20.5 20.7 35.9 26.1 19
Ethyl ester-EPA 313–333 9.03–21.07 0.2–19.4 12.9 12.4 13.0 12.5 13.0 12.9 42
Ethyl ester-DHA 313–333 9.03–21.07 0.6–12.9 8.3 7.0 8.4 7.0 10.2 9.1 42
1-Octanol 313–348 7–19 0.39–80.8 13.1 10.9 16.7 13.5 18.2 14.2 46
Oleic acid 308–318 9.6–18.62 0.48–2.57 4.9 3.6 5.0 3.8 5.0 4.4 47
Styrene 323–333 8–24 4.95–170.24 14.3 14.2 17.3 17.3 22.9 22.8 34
Methyl laurate 313–333 8–12 0.2–5.2 19.9 19.9 19.7 19.7 24.3 24.0 48
Methyl linoleate 313–333 10.44–18.03 4.1–19.4 2.0 2.0 2.0 2.0 9.6 7.6 49
Methyl myristate 313–343 7.87–15.97 0.3–13 14.2 11.5 14.8 11.3 14.9 14.8 50
Geranyl butyrate 308–333 10–18 2.1–23.19 11.0 10.1 12.4 9.0 19.0 12.9 Present work
10-Undecen-1-ol 308–333 10–18 2.2–24.9 8.7 8.5 9.7 9.3 19.0 16.3 Present work
Average AARD%       11.5 10.9 12.0 11.0 15.9 13.9  


Table 3 shows that for some systems such as geraniol, oleic acid and methyl linoleate, all the models showed AARDs less than 5%. Further systems such as nonanoic acid, styrene and methyl laurate have shown AARDs greater than or around 25% for the solution theory based models. The models that are derived using a particular theory have similar AARDs. Thus models based on association theory have similar AARDs (10–12%) while the solution theory based models (14–16%) have similar AARDs. However, there is a significant difference between these models and, generally, association theory based models (eqn (3), (4) and (8), (12)) are superior to solution theory based models (eqn (2) and (16)) as can be observed from the Table 3. Further, among the association theory based models, AARD (%) obtained for the association theory based model using van Laar and Wilson activity coefficient model (eqn (12)) gave the least AARD for many compounds.

In the new models derived in the present work, the non ideality in both the phases was considered. Further, the non ideality of the liquid phase can be neglected for the higher molecular weight lipids and lipid derivatives (liquid solutes) discussed in the present study, as it does not make any significant difference in the AARD values. Thus, the AARD values of the models discussed in this study are not significantly different. However, the AARD values for these models can be significantly different for lower molecular weight compounds, where the non ideality in the liquid phase cannot be neglected.

The experimental solubility data was fitted using eqn (12) as shown in Fig. 2(a) and (b) for the compounds investigated in this study (10-undecen-1-ol and geranyl butyrate). The parameters obtained from the model using eqn (12) are reported in Table 4, and the physical interpretation of the parameters and an insight into the model has been discussed below.

Table 4 Model parameters of eqn (12) obtained using multiple linear regression
Compound Parameters
k6 A6 C6 E6 F6
10-Undecenoic Acid 2.42 7656 −7140 27.78 −37.36
Geraniol 0.91 −690 −863 8.62 −6.79
Butyl-stearate 0.51 −11896 11[thin space (1/6-em)]228 −26.11 25.12
Butyl laurate 0.20 9624 −14900 52.41 −36.95
Dimethyl sebacate 0.58 6093 −10457 37.51 −26.34
Bis-2-ethylhexyl-sebacate 1.92 −4723 6641 −12.74 −0.80
Ethyl stearate 5.23 5255 1058 0.43 −35.53
Ethyl oleate 2.35 −1354 2140 3.69 −12.24
Ethyl linoleate 2.81 2626 −2607 17.44 −24.92
Methyl stearate 2.38 −4869 −0.06 30.26 −1.50
Methyl oleate 10.21 31[thin space (1/6-em)]786 −17314 56.79 −129.71
Methyl palmitate 7.52 4525 10[thin space (1/6-em)]730 −28.23 −38.93
Nonanoic acid Case 1 −2.6 −5099 −3288 24.70 9.04
Case 2 0.69 −5099 330[thin space (1/6-em)]948 24.70 −0.92
Ethyl ester-EPA 1.87 −850 3481 0.52 −13.83
Ethyl ester-DHA 2.53 8024 −9100 37.00 −41.52
1-Octanol 0.29 −2679 −4804 25.05 −1.31
Oleic acid 0.23 −25993 22[thin space (1/6-em)]068 −58.85 68.20
Styrene 3.62 275 −3998 14.72 −13.39
Methyl laurate 10.01 5302 −1397 5.01 −44.78
Methyl linoleate 11.52 22[thin space (1/6-em)]563 751 −9.72 −96.55
Methyl myristate 13.03 2664 30[thin space (1/6-em)]633 −92.24 −44.38
Geranyl butyrate 0.18 30 −3544 17.29 −7.10
10-Undecen-1-ol 3.73 5517 −3897 13.51 −30.47


It was found that A6 and E6 were positive and C6 and F6 were negative for both the compounds, as reported in Table 4. As discussed earlier the model has constraints that require k6 to be positive while the other parameters (A6, C6, E6 and F6) can be either positive or negative. Based on multiple linear regression, it was observed that F6 was negative for most of the compounds. For nonanoic acid, it was observed that k6 was negative when the solubility data was regressed using multiple linear regression (Case 1, Table 4). Therefore, the data was regressed using non linear regression with a constraint of k6 > 0. This resulted in similar values of A6 and E6 but a different value for F6 (Case 2, Table 4) and also resulted in an increase of AARD% from 20.7 to 37.7%. For all other compounds, k6 was positive. Based on the phenomenon of association, k6 can take values in two ranges, 0 < k6 < 1 or k6 > 1. Association number in this range has also been reported in case of solubilities of solids in supercritical carbon dioxide.28 In the first condition, when 0 < k6 < 1, the solvent molecule i.e., CO2 is surrounded by the solute molecules. When k6 > 1, the solute molecule is surrounded by the solvent molecules. Thus, the parameters in Table 4 provide physical insights into the solubilities of these compounds.

All the existing models based on association and solution theory for liquids have been selected for a comparative study. However, all these theories assume that the activity coefficient of the solute in liquid phase is constant. Therefore, in the newly developed correlations, the variation of the activity coefficient of the solute in both the phases was accounted. Different possible correlations were derived using van Laar and Wilson activity coefficient models to understand the non idealities present in both the phases. Further, one can understand the interaction potential between the like pairs of molecules from the interaction parameters present in the van Laar model (aAA and aBB) and similarly between the unlike pair of molecules from the Wilson model (aAB). Different parameters of the models give the physical interpretations for these interactions. The new models have been fitted to lipids and lipid derivatives to indicate their applicability, as shown in Table 3 and various parameters have been discussed for the three new models with their physical insights. However, the models derived in this study are applicable to all the liquid solutes reported in the literature. Therefore, these new models are capable of correlating a wide variety of systems as well as providing insights into the molecular interactions of these compounds with SCCO2.

6. Conclusions

The solubilities of two lipid derivatives, 10-undecen-1-ol and geranyl butyrate in SCCO2 for a temperature range of 308.15 to 333.15 K and pressures from 10 to 18 MPa was experimentally determined and were in the order of 10−3 to 10−2. The solubility of geranyl butyrate was similar to 10-undecen-1-ol. Three new models based on association and solution theories were derived. The models based on association theory were slightly more capable than the solution model in correlating experimental data. The association model plus van Laar activity plus Wilson activity coefficient model resulted in lower AARD (%) for many compounds compared to all the other models.

Acknowledgements

The authors thank Council of Scientific and Industrial Research (CSIR), India for financial support. Corresponding author thanks the Department of Science and Technology (DST), India for J. C. Bose fellowship.

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