32 D Stent FaR- (2.19) P F For mixtures, a and b can be evaluated from the pure components values (ai, bi)using mixing rules(equations(2. 21)-(2.23)) a=∑xx叫 21) a计;=(a;a 2(1-k1) b=∑xb where ki an adjustable interaction parameter. One important feature of the van der Waals equation is that it can be expressed in an alternative form that is cubic in volume and, given all other parameters, can be solved analytically for Vi and fi. In the quest to provide improved predictive models for NCF systems the van der Waals equation has provided the starting point for many related cubic quations of state. Two commonly used examples are the Redlich-Kwong(Redlich and Kwong, 1948)and Peng-Robinson(Peng and Robinson, 1976)equations(see Table 2.2) which both introduce a more sophisticated expression for the attraction term but retain the original form of the repulsion term. Both equations can easily be solved analytically or numerically for volume and fugacity coefficient and have been used in numerous correlations of solubility data in NCFs(McHugh and Krukonis, 1986; Prausnitz, 1965: Rizvi et al., 1986). Many variations on the mixing rules(equations(.2 1)-(2.23))have so been proposed(Kwak and Mansoor, 1986; Deiters, 1982; Mathias and Copeman, 1983). A comparative study of the effectiveness of a variety of commonly used equations of state in representing phase equilibrium data in NCF mixtures has been made by Haselow et al. (1986) The simple EOS approach represents the solubilities of volatile, low molecular weight solutes in NCFs reasonably well, as shown by the fit obtained using the Peng-Robinson EOS to the data for naphthalene in SCF CO2 shown in Fig. 2.8. Inspection of this data reveals how the components of equation(2. 15)used in the EOS method contribute to define the solubility curve. In the low-pressure region the behaviour of the Nc is pproximated to that of a perfect gas, for which the lowest pressure obtainable is that of the sublimation pressure of the pure solid, The low-pressure limit for the mole fraction of naphthalene in the mixture is simply given by the osition of the mixture at this limiting pressure. As the pressure is increased, y2 initially declines in accord with the first'perfect gas'term in equation(2. 15). As the pressure is further increased y2 begins to increase as the SCF progressively solvates the solute and 2 decreases. Transfer of solute to the SCf phase is also enhanced by the effect of hydrostatic pressure through the Poynting correction term
32 D. Steytler a=- F,R~T,~ (2.19) PC PC b=- Fb RTc (2.20) For mixtures, a and b can be evaluated from the pure components values (ai, bi) using mixing rules (equations (2.2 1)-(2.23)): a = cxi xj ajj a,. = (ai U,)*/~(I - k..) (2.21) (2.22) (2.23) ij I/ V 0 = c X; b; ij where kij = an adjustable interaction parameter. One important feature of the van der Waals equation is that it can be expressed in an alternative form that is cubic in volume and, given all other parameters, can be solved analytically for Vi andfi. In the quest to provide improved predictive models for NCF systems the van der Waals equation has provided the starting point for many related cubic equations of state. Two commonly used examples are the Redlich-Kwong (Redlich and Kwong, 1948) and Peng-Robinson (Peng and Robinson, 1976) equations (see Table 2.2) which both introduce a more sophisticated expression for the attraction term but retain the original form of the repulsion term. Both equations can easily be solved analytically or numerically for volume and fugacity coefficient and have been used in numerous correlations of solubility data in NCFs (McHugh and Krukonis, 1986; Prausnitz, 1965; Rizvi er ul., 1986). Many variations on the mixing rules (equations (2.21)-(2.23)) have also been proposed (Kwak and Mansoori, 1986; Deiters, 1982; Mathias and Copeman, 1983). A comparative study of the effectiveness of a variety of commonly used equations of state in representing phase equilibrium data in NCF mixtures has been made by Haselow et al. (1986). The simple EOS approach represents the solubilities of volatile, low molecular weight solutes in NCFs reasonably well, as shown by the fit obtained using the Peng-Robinson EOS to the data for naphthalene in SCF CO, shown in Fig. 2.8. Inspection of this data reveals how the components of equation (2.15) used in the EOS method contribute to define the solubility curve. In the low-pressure region the behaviour of the NCF is approximated to that of a perfect gas, for which the lowest pressure obtainable is that of the sublimation pressure of the pure solid. The low-pressure limit for the mole fraction of naphthalene in the mixture is simply given by the composition of the mixture at this limiting pressure. As the pressure is increased, yz initially declines in accord with the first ‘perfect gas’ term in equation (2.15). As the pressure is further increased y;’ begins to increase as the SCF progressively solvates the solute and $; decreases. Transfer of solute to the SCF phase is also enhanced by the effect of hydrostatic pressure through the Poynting correction term
Supercritical fluid extraction 33 10 065-10-120250 CO? at 25"C by the Peng-Robinson equation of state(reproduced from Paulaitis et aL., 1983) The application of the EOS approach to liquid solutes is more involved since the NCF can dissolve in the liquid phase and the fugacities of both components in two coexisting phases must then be considered(Prausnitz and Benson, 1959). Relatively simple compul er methods are however available(Mchugh and Krakonis, 1986 )for an iterative solution lation(2.7))of each phase Phase equilibrium data for liquids is usually represented at constant temperature as function of pressure(or vice versa)as shown in Fig. 2.9 for the butanol/CO2 system. The region enclosed by the loop represents a two-phase region in which a CO2-rich gas phase coexists with a butanol-rich ' liquid phase. At constant pressure the compositions of the coexisting liquidand 'gas phases are given by the points of intersection of a horizontal"tie line'within the loop. The relative proportions of the coexisting phases can be obtained in the usual way using the 'lever rule. In this example, increasing the pressure increases the solubility of butanol in CO2 until at a critical pressure of approx mately 160 bar complete miscibility occurs. This represents the particularly simple behaviour of a liquid with low molecular weight, but as molecular weight and/ or polarity increased the phase behaviour becomes more complex. A rich variety of phase dia- grams have been observed and systematically classified for binary NCF mixtures (Schneider, 1970) One limiting factor which restricts application of EOS models for food-related pplications is the lack of available data for the fundamental properties of pure components required for input. Another problem concerns the ambiguity of mixing rules and associated 'adjustable parameter(s)(e.g. ky in equation(2. 22))which are ofter floated' in fitting experimental data. Although attempts have been made to define such parameters in terms of pure component properties, EOS models have been more
Supercritical fluid extraction 33 lo-’ 10-2 ._ s .!l 10-3 ? i% 10-~ 2 P 10-5 c u - 0 E z 1o-6 0 50 100 150 200 250 Pressure (bar) Fig. 2.8. Representation of experimentally determined (0) solubility data for naphthalene in SCF C02 at 25OC by the Peng-Robinson equation of state (reproduced from Paulaitis et nl., 1983). The application of the EOS approach to liquid solutes is more involved since the NCF can dissolve in the liquid phase and the fugacities of both components in two coexisting phases must then be considered (Prausnitz and Benson, 1959). Relatively simple computer methods are however available (McHugh and Krakonis, 1986) for an iterative solution of the condition of equal fugacities (equation (2.7)) of each component in both phases. Phase equilibrium data for liquids is usually represented at constant temperature as a function of pressure (or vice versa) as shown in Fig. 2.9 for the butanol/C02 system. The region enclosed by the loop represents a two-phase region in which a C02-rich ‘gas’ phase coexists with a butanol-rich ‘liquid phase’. At constant pressure the compositions of the coexisting ‘liquid’ and ‘gas’ phases are given by the points of intersection of a horizontal ‘tie line’ within the loop. The relative proportions of the coexisting phases can be obtained in the usual way using the ‘lever rule’. In this example, increasing the pressure increases the solubility of butanol in CO2 until at a critical pressure of approximately 160 bar complete miscibility occurs. This represents the particularly simple behaviour of a liquid with low molecular weight, but as molecular weight and/or polarity is increased the phase behaviour becomes more complex. A rich variety of phase diagrams have been observed and systematically classified for binary NCF mixtures (Schneider, 1970). One limiting factor which restricts application of EOS models for food-related applications is the lack of available data for the fundamental properties of pure components required for input. Another problem concerns the ambiguity of mixing rules and associated ‘adjustable parameter(s)’ (e.g. kc in equation (2.22)) which are often ‘floated’ in fitting experimental data. Although attempts have been made to define such parameters in terms of pure component properties, EOS models have been more