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Introduction to Solid-Fluid Equilibrium Modeling [Chemical Engineering Progress](Chemical Engineering Progress Via Acquire Media NewsEdge) This article introduces the thermodynamic property models that are commonly used in process simulators to perform solid-liquid equilibrium and solid-supercritical fluid equilibrium calculations. (ProQuest: ... denotes formulae omitted.) Most chemical products are produced and consumed in solid forms, from basic chemicals and petrochemicals to pharmaceuticals. Solid-liquid equilibrium (SLE) and solid-supercritical fluid equilibrium (SFE) represent the thermodynamic limits of chemical processes involving solids. Knowledge of these critical phase-equilibrium phenomena combined with proper modeling allow for the development of processes that include the formation of polymorphs, crystallization, selective dissolution, and freezing point depression. SLE phase diagrams Many chemical process operations involve solids - e.g., mixing, reaction, crystallization, extraction, filtration, washing, drying, and granulation, to name a few. Among them, crystallization is the most important separation and purification operation, and understanding the solid-liquid phase diagram is the first step toward the successful development of a crystallization process (I). SLE phase diagrams facilitate the interpretation of crystal solubility data and the control of crystal growth, as well as enable the predictions of desired crystal polymorphs, maximum crystal recovery, and kinetics of crystal growth (2). SLE knowledge is also critically important for flow assurance in pipelines in the oil and gas industry. Many process issues - such as wax deposition, pipeline clogging, asphaltene precipitation, and mineral scale formation in oil reservoirs during water injection (3) - can be prevented if sufficient SLE knowledge, including data and models, are available. Another important application of SLE phase diagrams is freezing point depression of fluid inside pipelines. In a piping network such as a chilled-water piping system, water cannot be circulated below its freezing point or ice will form inside the pipe, which may cause damage or necessitate shutdown. By mixing another liquid (e.g., ethylene glycol) into the system, the freezing point of water can be reduced to a temperature below 0°C. SLE phase behavior is only mildly affected by pressure, so solubility plots are often presented as temperaturecomposition, or T-x, diagrams. Figure 1 shows a solubility plot of caffeine in three different solvents - hexane, water, and 1,4-dioxane. The region under the solubility curve is unsaturated liquid for a specific solvent. Because the activity coefficient of the solute - caffeine - is a weak function of temperature, this T-x diagram is generated from the solubility data at 25°C (4) with the assumption that the caffeine activity coefficient remains approximately unchanged at the saturation points. Solid-supercritical fluid equilibrium Supercritical fluid exists at temperatures and pressures above its critical point. It has a density comparable to that of a liquid and a viscosity comparable to that of a gas, and it has a high diffusion coefficient. These properties make supercritical fluids efficient extraction solvents. When a supercritical fluid is at conditions far from its critical point, its density increases rapidly with increasing pressure. When solvent density increases, its solvation power increases and thus solute dissolution increases rapidly (5). Solid-supercritical fluid extraction has attracted much attention in the food processing and pharmaceutical industries in recent years. For example, supercritical C02 (SCD) is used for large-scale decaffeination of green coffee beans, for extraction of hops in beer production (6), and as an effective solvent for extracting hydrocarbon from crude oil (7). SCD can also replace conventional organic dry cleaning solvents, as well as increase the solvation power of detergents. SCD is nontoxic and nonflammable, has low critical temperature and pressure, is relatively inexpensive, and is environmentally friendly. All of these properties make SCD an excellent extraction solvent. In the pharmaceutical industry, supercritical fluid has enabled single-step particle formation, which is difficult or even impossible by conventional methods such as sonication and Soxhlet extraction (8). Using SFE, microor even nanoparticles with a narrow size distribution can be formed. Simulating SLE and SFE Today's process simulators are capable of rigorous and accurate modeling of chemical systems and processes involving solids. In addition to vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) calculations, process simulators can readily perform SLE and SFE calculations and predictions. This article briefly introduces the thermodynamic framework and thermophysical property models that are commonly used in process simulators to perform SLE and SFE calculations. To illustrate the work process for thermodynamic modeling of SLE and SFE with process simulators, caffeine solubility is modeled in pure solvents, mixed solvents, and supercritical fluids at different operating conditions. Activity coefficient models are used to correlate and predict SLE of caffeine, and equation-of-state (EoS) models are used to perform SFE calculations for caffeine. Thermodynamic framework To understand SLE and SFE, we need to start with some basic thermodynamics. Fugacity. Chemical potential is conveniently expressed in terms of fugacity. In an isothermal, ideal or nonideal system of pure or mixed solid, liquid, or gas, fugacity (/) is defined in terms of chemical potential (pf.) as (9): ...(1) where R is the gas constant, T is the system temperature, p(. is the chemical potential of component /', and fi is the fugacity of component /'; p;0 and Jf are the chemical potential and fugacity at the reference state, respectively. Fugacity is commonly expressed in terms of pressure. For a particular gas, fugacity is a measure of corrected partial pressure of that gas in equilibrium. When substance / is in equilibrium in two phases a and ß, the fugacities of the two phases are equal: ...(2) Fugacity coefficient. The normalized term fugacity coefficient ((p;.) is used to characterize the nonideal behavior of a substance in an equilibrium system: ...(3) where Pi is the partial pressure of component i at T, P is the system pressure, and yi is the mole fraction of component / in the gas phase. Activity. Activity (a) is a measure of liquid-phase nonideality. It is the ratio of the fugacity of a liquid component, i, at a given temperature to the component's fugacity at a reference state: ...(4) Activity coefficient. Activity coefficient (y) is a measure of deviation from Raoult's law. The activity coefficient of component / is related to activity and the composition of component / in solution by: ...(5) where x(. is the mole fraction of component i in solution. The fugacity coefficient is related to the activity coefficient by: ...(6) where pffi' is the saturation pressure of component i at T. Melting point. Melting point (Tm) is the temperature at which a specific solid polymorph melts (fuses) to form liquid at atmospheric pressure. At this temperature, the solid and liquid phases exist in equilibrium. Heat of fusion. Heat of fusion (AH0^) is the heat required to melt (fuse) a specific solid polymorph to form a liquid. Solid solubility. At a specific temperature and pressure, a solid polymorph (a solute) is in equilibrium with a solvent, forming a saturated solution. The concentration of solute in that solution is known as the solubility of the solid polymorph. Solid solubility in solution depends on the composition of the solvent and the operating conditions. Building on Eq. 2, the solubility of a solute can be expressed in terms of fugacity equality in both phases.The phases can be liquid, supercritical fluid, or pure solid polymorph. For solid-liquid equilibrium: ...(7) where f* is the fugacity of solute / as a solid polymorph. fi is the fugacity of solute i in the saturated liquid solution, and y, is the activity coefficient of solute / in the saturated liquid solution. For solid-supercritical fluid equilibrium: ...(8) where ff is the fugacity of solute / in supercritical fluid. The ratio offP to f. is related to the Gibbs energy change in fusion, AG°, ...(9) The full thermodynamic expression for AG0^ is: ...(10 where A//1^ is the enthalpy change (i.e., heat of fusion), AC0 , is the heat capacity change for the dissolution of a specific solid polymorph, and Tnt} is the reference temperature, normally taken as Tm. At Tm. AG°/((v is zero and AC°^v is often approximated as zero. Equation 10 is then simplified to Eq. 11 or expressed in terms of solubility product constant, Ksp, in Eq. 12: ...(11) ...(12) ...(12a) ...(12b) To cover wider temperature ranges and pressure effects, Eq. 12 is further expanded to: ...(13) where C, D, and E are additional polynomial coefficients MPa. Solubility index. Solubility index (ST) is a measure of the saturation of a solute in a solution. It is expressed in terms of the ratio of activity of solute / in solution to the activity of solute / in saturated solutions: ...(14) where ai is the activity of solute i in solution and a(TM)1 is the activity of solute / in saturated solution. SI > 0 indicates supersaturated solution, SI = 0 indicates saturated solution, and SI <0 indicates unsaturated solution. Process simulation and thermodynamic models Process simulators are essentially heat and mass balance calculators (10). They perform calculations based on user input, process constraints, and required output. A process simulator integrates the mass and energy balance equations with the process flowsheet's constraints and the chosen thermodynamic models. Thermodynamic models are the scientific foundation of process simulators (11). Process simulators integrate databanks, correlative models, reference states, estimation techniques, and flash algorithms. They provide well-tested and consistent thermodynamic property calculations for process modeling and simulation. Engineers can choose which model to apply to their simulation. Whenever a reliable thermodynamic model is established, experts identify model parameters from regression of available experimental data. The parameterized thermodynamic model is then used to provide robust interpolation within - and extrapolation beyond - available experimental data ranges with confidence (11). Modeling a particular mixture requires an understanding of the mixture's thermodynamic limits - i.e., temperature and pressure ranges (12). Additionally, one needs to choose the thermodynamic model that best describes the thermodynamic properties of the mixture and ensure that it is valid for the temperature and pressure ranges of interest. Once a thermodynamic model has been selected and validated for a specific process, it becomes a corporate asset for that company. Because significant investment is involved in validating the thermodynamic model and in modeling and simulating the chemical process, companies do not modify or switch thermodynamic models unless significant advantages can be substantiated. In short, selecting an appropriate thermodynamic model plays a critical role in the success of modeling and simulation for any process in industry (11). Modeling SLE and SFE As discussed earlier, solid-liquid equilibrium occurs when the fugacity of the solid solute and the fugacity of the solute in the saturated liquid solution are equal. Similarly, solid-supercritical fluid equilibrium occurs when the fugacity of the solute as a solid is equal to the fugacity of the solute in the supercritical fluid. In SLE, liquid-phase fugacity is more conveniently calculated with activity coefficient models (ACMs), while in SFE, supercritical-fluid-phase fugacity is calculated with equations of state (EoS). Table 1 lists some of the activity coefficient models and equations of state most commonly used in process simulation. The activity coefficient models include: nonrandom two-liquid (NRTL), UNIQUAC functional-group activity coefficients (UNIFAC), conductor-like screening models - realistic solvation (COSMO-RS), conductor-like screening models - segment activity coefficient (COSMO-SAC), and nonrandom two-liquid - segment activity coefficient (NRTL-SAC). The equations of state include: Peng Robinson, Soave-Redlich-Kwong (SRK), and perturbed-chain statistical association fluid theory (PC-SAFT). The activity coefficient corrects for the ideal solution assumption of Raoult's Law. In ACMs, which are designed to describe liquid-phase nonideality, activity coefficients are calculated as a function of temperature and composition of the mixture. The effect of pressure is assumed to be negligible, which is quite reasonable for modeling SLE. To model SFE systems, real gas behavior needs to be taken into account, so EoS models are typically used. Derived from thermodynamic relationships, EoS models are usually expressed in terms of pressure, P = function ( T. v, x), where P, T, v, and x represent pressure, temperature, molar volume, and composition, respectively. Additionally, EoS models are able to calculate enthalpy of departure, which is the difference between the enthalpy of the real state and that of the ideal gas state. This strength makes EoS the right choice for modeling compressible fluids (12). On the other hand, EoS models are less reliable in accounting for liquid nonideality, especially when the mixture is highly nonideal. Nonrandom two-liquid (NRTL) activity coefficient model. The NRTL ACM is based on the hypothesis that local composition around a molecule is different from the bulk composition in liquid mixtures (13). This model has perhaps been the most widely used activity coefficient model to correlate VLE and LLE data. The model parameters include a nonrandomness factor (cl.) and two asymmetric binary-interaction parameters (x^.and x/(.) for each binary pair of molecules. With a y fixed at a constant value of 0.2 or 0.3 (cc. = 0.2 for binary systems that form two liquids and a. = 0.3 for binary systems that do not form two liquids), two binary-interaction parameters are often sufficient to correlate experimental data within engineering accuracy for binary systems. The model mixing rule allows extrapolation from the binary-interaction parameters for multicomponent systems. However, SLE data alone are not sufficient to identify NRTL binary-interaction parameters because they do not provide enough information on the composition dependency of the liquid-phase nonideality. VLE and/or LLE data must be used in conjunction with SLE data to properly identify NRTL binary-interaction parameters and solubility product constants. Therefore, the suitability of NRTL for SLE is very limited, except for those well-studied solutes for which extensive phase equilibrium data are available. UNIQUAC functional-group activity' coefficients (UNIFAC) model (i4). This activity coefficient model treats a nonelectrolyte liquid mixture as a solution of the structural functional groups (such as >C=C< and -CH,-) from which the molecules are formed, rather than a solution of the molecules themselves. The model predicts activity coefficients for solvents and solutes as long as these molecules are structured as a set of UNIFAC functional groups for which the group-group binary-interaction parameters have been determined. Otherwise, the model should not be used. For small linear organic molecules, UNIFAC yields acceptable quantitative or semi-quantitative predictions. For complex molecules with rigid molecular configurations, UNIFAC predictions are often, at best, qualitative. UNIFAC functional group information is readily available for many molecules in public databanks and commercial simulators. The molecular structure of caffeine is depicted in Figure 2. Table 2 lists the UNIFAC functional groups used to represent eight common solvents as well as the solute caffeine. While the solvents can be readily structured from existing UNIFAC functional groups, the caffeine molecule cannot. Such a molecule must instead be represented as the combination of functional groups that the modeler considers most closely resembles it. However, different choices of UNIFAC functional groups will yield different predictions. For this example, the caffeine molecule is represented as one >C=C< group, two -CH^-COin ring groups, three CH3-N< groups, and one -Cfaromaticj-NH, group. This illustrates a major uncertainty associated with the use of UNIFAC. Conductor-like screening models - realistic solvation (COSMO-RS) and conductor-like screening models - segment activity coefficient (COSMO-SAC). COSMO-RS (15) and COSMO-SAC (i6) are two variants of the COSMO model that originated from the use of solvation thermodynamics and computational quantum mechanics. These predictive models rely on molecule-specific "sigma profiles" - probability distributions of molecular surface charge density - as the only input to predict activity coefficients. The generation of sigma profiles from quantum mechanical calculations can be time-consuming. Commercial process simulators offer sigma profile databanks for many molecules. including caffeine and the four solvents in Figure 3. Popular web-based sigma profile databases include VT-2005 Sigma Profile Database and VT-2006 Solute Sigma Profile Database (17, 18). In general, the quality of COSMO-RS/SAC predictions are qualitative or semi-quantitative; these models correctly predict activity coefficient trends and values that are of the same order of magnitude as the experimental data. Nonrandom two-liquid segment activity coefficient (NRTL-SAC) model. The NRTL-SAC model (i9) suggests that, in lieu of structural functional groups, four conceptual segments should suffice to account for all major distinct molecular surface interaction characteristics: hydrophobic (A), polar attractive (T-), polar repulsive (T+), and hydrophilic (Z). This model uses inherent segment-segment binary-interaction parameters, as well as conceptual segment numbers, to determine the activity coefficients for all molecules in solution. Inherent segment-segment binaryinteraction parameters among the four conceptual segments are first established from phase equilibrium data of representative reference molecules that exhibit such molecular surface interaction characteristics. These parameters are incorporated in the model simulators and are constant. Conceptual segment numbers are pure-component model parameters that quantify molecular surface interactions in terms of the conceptual segments. They are determined from regression of experimental phase equilibrium or solubility data. Table 3 presents the conceptual segment numbers for the eight solvents and caffeine. The values for the solvents are already published and can be taken from simulator databanks, while those for caffeine are regressed from available caffeine solubility data in the solvents. NRTL-SAC provides semi-quantitative predictions for phase behavior of solvents and solutes in solution. Peng-Robinson and Soave-Redlich-Kwong (SRK) EoS. The Peng-Robinson (20) and Soave-Redlich-Kwong (21) equations of state, which express energy and volume as functions of critical temperature (T), critical pressure (Pc), and acentric factor (to), are widely used in industry for lowand medium-pressure systems because of their simplicity and accuracy. Using symmetric binary-interaction parameters and the quadratic mixing rule, these two equations of state are able to perform correlation and extrapolation of fluid phase equilibrium, including SFE. Table 4 gives the critical properties of caffeine and supercritical CO., for both Peng-Robinson and SRK modeling. Perturbed-chain statistical association fluid theory (PC-SAFT) EoS. The PC-SAFT equation of state was developed by Gross and Sadowski (22) from advanced molecular theory. For non-associating systems, the model requires only three component-specific parameters - segment number (m), segment diameter (a), and segment energy parameter (e) - to model bulk properties and phase equilibria even for high-pressure systems. As shown in Table 5, PC-SAFT component-specific parameters (m, a, and e) are available in simulators for C02, but not for caffeine. Pure-component parameters for caffeine were identified by regression of liquid density and vapor pressure data. Since the model has its foundation in statistical mechanics, it accounts for molecule size and shape. It has excellent predictive capabilities and good precision for correlating mixture properties. Correlative vs. predictive models The ACM and EoS models in Table 1 can be categorized into two classes: correlative models and predictive models. NRTL ACM, Peng-Robinson EoS, SRK EoS, and PCSAFT EoS are correlative models. These models incorporate adjustable binary-interaction parameters that are designed to correlate experimental data so that the models can then be used to both interpolate within data ranges and extrapolate beyond data ranges. UNIFAC, COSMO-SAC, and COSMO-RS are predictive models. They make use of purecomponent structural parameters that have been identified from quantum chemistry calculations or experimental data during the development of the models. The NRTL-SAC ACM is a hybrid model, i.e., it is both correlative and predictive. It has no adjustable binaryinteraction parameters. Rather, the model makes use of purecomponent parameters that are identified from regression of experimental phase equilibrium data. Guidelines for models and model parameters Activity coefficient models are appropriate for modeling nonideal liquid mixtures, especially when polar and hydrophilic molecules are involved. Experimental data should always be the first priority and the basis for thermodynamic modeling. If sufficient phase equilibrium data are available, they should be regressed with correlative models such as NRTL to obtain accurate interpolation and extrapolation, and binary-interaction parameters should be determined by regression of the experimental data. When experimental data are missing for minor binary systems (i.e., binary systems with trace components), NRTL binary-interaction parameters may be estimated from supplemental binary system "data" generated from predictive models such as UNIFAC. Predictive models such as UNIFAC and COSMO-SAC can be very useful when no experimental data are available. Preliminary SLE studies can help design experiments or guide screening of candidate solvents. However, very rarely can these predictive models be directly applied in process or product design. Note that UNIFAC and COSMO-SAC activity coefficient models require basic molecular structure information, which is not always available (especially for natural products and compounds). Models developed with actual experimental data, however limited those data are, are always preferred over predictive models. For SLE studies in the pharmaceutical industry or specialty chemicals industry, limited solute solubility data are available for pure solvents and binary solvents. While such solubility data are insufficient to determine NRTL binary-interaction parameters, they can be very useful for determining conceptual segment numbers for NRTL-SAC. These limited solubility data are first regressed to identify NRTL-SAC conceptual segment numbers for the solute, and then used to predict solubility in other pure solvents or mixed solvents. The ability to make use of limited solubility data is the unique advantage of NRTL-SAC among the activity coefficient models. In some commercial simulators, NRTL-SAC conceptual segment numbers are stored in the databank for common solvents, and solute conceptual segment parameters can be regressed from solubility data for at least four solvents with different molecular characteristics. Interestingly, researchers are developing methodologies to generate sigma profiles for COSMO-SAC/COSMO-RS based on the conceptual segment concept and available solubility data. In lieu of time-consuming quantum mechanical calculations, this approach generates effective sigma profiles from actual experimental data and therefore should better represent the data. When the effect of pressure is important, such as in solid-supercritical fluid equilibrium, equations of state should be used. Because it is an advanced equation of state (derived by applying perturbation theory to chain molecules), PC-SAFT usually provides more accurate results than a cubic equation of state. The three pure-component parameters required for PC-SAFT (m, a, and c) are available in process simulators for many light gas molecules. For solute molecules and special solvents, these parameters can be regressed from pure-component physical property data such as liquid density and vapor pressure. Peng-Robinson and SRK are two widely used cubic equations of state. These models require critical properties such as T., Pc, and to. While critical properties are often available in process simulators for light gas and hydrocarbon molecules, such critical properties need to be estimated by relatively rudimentary semi-empirical group contribution methods for solutes because their critical properties cannot be measured due to thermal instability. Both the physical meaning of critical properties for solutes and the reliability of the group contribution estimation methods must be questioned. Binary-interaction parameters are required for the correlative models (NRTL, PC-SAFT, Peng-Robinson, and SRK). Pure-component conceptual segment numbers are required for the NRTL-SAC hybrid model. These parameters should be regressed from, and checked against, experimental data. While models with specific binary-interaction parameters may interpolate well within certain ranges of temperature or composition, they may not extrapolate well. For example, if experimental data are available only for a limited temperature range, the parameter for temperature dependency may be unreliable if the temperature of interest is far outside the range in which the experimental data were obtained. Process simulators often provide databanks of such binary-interaction parameters. Users of these databanks should be aware that these parameters have been obtained by regression of experimental data over specific temperature, pressure, and composition ranges. These parameters should be properly validated for the specific application, as they may not be appropriate for the process conditions of interest. The solubility product constant, Ksp, is unique for each solid polymorph. Ksp can either be calculated from the heat of fusion and melting temperature through thermodynamic relationships (Eq. 11 ), or be regressed from experimental data. Temperature dependency and/or pressure dependency of the solubility product constant should be considered for both SLE and SFE. Because solid crystals can form multiple polymorphs, modelers should be aware of the specific polymorphs in the system and the possibility that multiple polymorphs might be present. Most process simulators incorporate the commonly used ACM and EoS models and model parameter databanks, and can be very useful in performing various thermodynamic calculations for SLE and SFE. Aspen Plus V8.4 was used to perform regression of pertinent parameters and thermodynamic calculations for the following examples. Example 1. Solid-liquid equilibrium This example demonstrates solid-liquid equilibrium modeling of caffeine solubility in pure solvents and solvent mixtures. Caffeine has a melting temperature (Tm) of 512.15 K and heat of fusion ( AH°^v) of 216,000 kJ/kmol (23). The available data on caffeine solubility in the eight pure solvents discussed earlier are summarized in the second and third columns of Table 6 (23). The multiple values for caffeine solubility in water at 298.15 K are a reflection of the data's uncertainty. While ample caffeine solubility data are available, such SLE data alone are not sufficient to determine NRTL binary-interaction parameters. Therefore, the predictive activity coefficient models - UNIFAC, COSMO-SAC, and NRTL-SAC - must be used to model caffeine's solid-liquid equilibrium. The modeling employed the UNIFAC functional groups shown in Table 2 and the NRTL-SAC parameters in Table 3, as well as the COSMOSAC sigma profiles in the simulator databank. Table 7 presents the solubility product constant for caffeine obtained by regression of solubility data with UNIFAC, COSMO-SAC, and NRTL-SAC modeling. The variables A, B, C, D, and E are determined via modeling, then plugged into Eq. 13 to calculate ln K . (The anhydrate polymorph is assumed for all of the solubility data.) For reference, Table 7 also includes the K value calculated from the melting temperature and heat of fusion of caffeine by Eq. 12. Table 6 reports the results from UNIFAC, COSMOSAC, and NRTL-SAC modeling. A parity plot (Figure 4) compares the model results (y-axis) with the experimental data (x-axis). Figure 4 shows that the hybrid model NRTLSAC provides the most accurate results, COSMO-SAC gives relatively good predictions, and the UNIFAC predictions have much larger variations. The models were also used to predict caffeine solubility in binary solvents. Figure 5 shows the predictions of caffeine solubility in an ethyl-acetate/ethanol binary solvent at 25°C. The NRTL-SAC model predictions for the system are of the same order of magnitude as the experimental data, but the COSMO-SAC and UNIFAC predictions should be considered only qualitative. Example 2. Solid-supercritical fluid equilibrium Caffeine extraction by supercritical C02 is used as an example to illustrate how to model SFE. The Peng-Robinson, SRK, and PC-SAFT equations of state are used to correlate the experimental data. Experimental data on caffeine solubility in supercritical CO., at 15.2-35.2 MPa and temperatures of 50°C, 60°C, and 70°C are reported by Azevedo, et al. (24) and others. The critical properties of C02 (obtained from simulator databanks) and of caffeine (estimated by the Joback (25) and Lee-Kesler (26) group contribution methods) required by the Peng-Robinson and SRK EoS are shown in Table 4, and the pure-component parameters (m, a, and e) for PC-SAFT modeling are shown in Table 5. PC-SAFT purecomponent parameters for caffeine are regressed from liquid density and vapor pressure data (27). Table 8 presents the symmetric binary-interaction parameters determined by regression of the experimental data, and Table 9 reports the regressed solubility product constant (Ksp) for caffeine dissolving in supercritical C02. The values of obtained using different EoS models reflect the assumptions associated with each model and the varying estimation methods that are related to model parameters such as critical properties. For example, the PC-SAFT purecomponent parameters are regressed from experimental data, whereas the critical properties for the Peng-Robinson and SRK models are estimated by group contribution methods. Therefore, Km values from the PC-SAFT regression are considered more reliable. Figure 6 compares the model correlations with experimental data. It shows that SRK results should be consid- ered qualitative, while Peng-Robinson and PC-SAFT EoS provide quantitative representations of SFE for caffeine. However, PC-SAFT modeling is preferred over both PengRobinson and SRK because it uses experimental data rather than group contribution estimates. Nomenclature at = activity of component / At = total cavity surface area, A2 a sai = activity of solute / in saturated solution AC® = heat capacity change for the dissolution of a solid polymorph e = elementary charge (1.6022 x 10 10 coulomb) fi = fugacity of component / f0i= fugacity of component i at the reference state f* = fugacity of component i in phase a ff = fugacity of component i in phase ß fi = fugacity of solute / in saturated liquid solution f* = fugacity of solute / as a solid polymorph ff = fugacity of solute / in supercritical fluid AG®, = Gibbs energy change in fusion AH". = heat of fusion (enthalphy change) Ksp = solubility product constant P = system pressure Pi = partial pressure of component / p(o) = sigma profile pf* = saturation pressure of component / at T R = gas constant SI = solubility index T = system temperature Tm = melting point T. = reference temperature, normally assumed as Tm x. = mole fraction of component i in solution y, = mole fraction of component / in the gas phase Greek Letters Y, = activity coefficient of component /' e = segment energy parameter p( = chemical potential of component i H® = chemical potential at the reference state o = surface-segment charge-density distribution, e/A2 <p( = fugacity coefficient of component / cu = acentric factor LITERATURE CITED 1. 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