Diffusion coefficients of solids in supercritical carbon dioxide: Modelling of near critical behaviour

Most of the models proposed in literature for binary diffusion coefficients of solids in supercritical fluids are restricted to infinite dilution; this can be explained by the fact that most of experimental data are performed in the dilute range. However some industrial processes, such as supercritical fluid separation, operate at finite concentration for complex mixtures. In this case, the concentration dependence of diffusion coefficients must be considered, especially near the upper critical endpoint (UCEP) where a strong decrease of diffusion coefficients was experimentally observed. In order to represent this slowing down, a modified version of the Darken equation was proposed in literature for naphthalene in supercritical carbon dioxide. In this paper, the conditions of application of such a modelling are investigated. In particular, we focus on the order of magnitude of the solubility of the solid and on the vicinity of the critical endpoint. Various equations proposed in literature for the modelling of the infinite dilution diffusion coefficients of the solutes are also compared. Ten binary mixtures of solids with supercritical carbon dioxide were considered for this purpose.     

Contactless Weighing Up to High Pressures For Analysing Adsorption And Density of Fluids

Adsorption of supercritical fluids methane, nitrogen and argon by active carbons was studied up to a pressure of 500 bar. A three-parameter isothermal equation was used to represent the adsorption equilibrium. This isothermal equation is based on a physical model conception which had already been used for the modelling of adsorption processes with a pressure up to 150 bar. Beside the exact knowledge of the measurable parameters pressure, temperature and fluid composition, the density of the adsorbate are essential for the evaluation of the adsorption analysis. The fluid density can be determined either via equations of state, which is normally the most practicable and fastest way, or via lift measurements of a lowering body in the fluid based on the principle of Archimedes. This work represents and discusses the question of to what extent the fluid density determined under real conditions via equations of state, using, for example, equation of Bender, corresponds to the fluid density measured under high-pressure. This revised version was published online in August 2006 with corrections to the Cover Date.     

A generalized correlation for Henry’s Law constants of nonpolar solutes in four polymers

In this work, Henry’s Law constants of nonpolar solutes in four polymers were obtained by extrapolation of finite concentration vapor–liquid equilibrium (VLE) data using the UNIQUAC equation to infinite dilution condition. The consistency of the results was confirmed by comparing them with infinite dilution data through the linear relationship between logarithm of Henry’s Law constants and inverse of temperatures. This verification provided good method for crosschecking a reliability of finite concentration VLE data with infinite dilution data. Henry’s Law constants were correlated based on both types of data as a function of temperature using the classical van’t Hoff equation. Generalized correlations of the Henry’s Law constants of solutes were proposed for polyisoprene (PI), polyisobutylene (PIB) and poly(n-butyl methacrylate) (PBMA).     

Endothermic character of toluene adsorption from supercritical carbon dioxide on activated carbon at low coverage

Recasens, P., Velo, E., Larrayoz, M.A. and Puiggené, J., 1993. Endothermic character of toluene adsorption from supercritical carbon dioxide on activated carbon at low coverage. Fluid Phase Equilibria, 90: 265-287.

Heat effects and volumetric properties are analyzed for the adsorption of toluene from supercritical carbon dioxide onto activated carbon at the limit of zero coverage, based on existing data for the system. Using values of the adsorption equilibrium constant at different temperatures as a function of fluid density, large, negative partial molar volumes for toluene in the fluid were obtained, which were previously unavailable.

Numerical integration of the differential equation that expresses the isobaric temperature dependence of the equilibrium constant, coupled with parameter optimization, enabled us to estimate the differential enthalpy of toluene adsorption onto the surface from the ideal gas at the same pressure and temperature, in addition to the enthalpy of transfer from the fluid to the surface. This is found to be large and positive near the critical conditions. Using the thermodynamic analysis of Kelley and Chimowitz, our results show that in terms of the enthalpy of transfer, the isothermal adsorption from a supercritical fluid is an endothermic process, thus explaining the retrograde behavior experimentally observed for the regeneration of carbon with supercritical CO₂ at conditions not far from the solvent's critical point.     

Analysis of retrograde behavior and the cross-over effect in supercritical fluids

In this paper, results based upon thermodynamic stability theory are developed which lead to a set of both necessary and sufficient conditions for the existence of retrograde behavior in a multicomponent solute system dissolved in a pure supercritical fluid. While experimental evidence of retrograde behavior in single solute systems has been known for some time, recently data have been obtained showing the retrograde effect in binary solute systems dissolved in a pure supercritical fluid.In such systems, cross-over regions may be defined. These are pressure—temperature regions where the solubility of one solute increases while that of the other decreases with a change in temperature at constant pressure. The existence of cross-over regions in multicomponent mixtures can have implications for a new separation technique using pure supercritical fluids. In conjunction with an equation of state, the results derived here allow cross-over regions to be predicted, thus enabling one to identify candidate systems and thermodynamic conditions for the cross-over process. For this work a variation of a perturbed hard sphere model equation of state was used for the calculations.     

Partial molar volume of n-alcohols at infinite dilution in water calculated by means of scaled particle theory

The partial molar volume of n-alcohols at infinite dilution in water is smaller than the molar volume in the neat liquid phase. It is shown that the formula for the partial molar volume at infinite dilution obtained from the scaled particle theory equation of state for binary hard sphere mixtures is able to reproduce in a satisfactory manner the experimental data over a large temperature range. This finding implies that the packing effects play the fundamental role in determining the partial molar volume at infinite dilution in water also for solutes, such as n-alcohols, forming H bonds with water molecules. Since the packing effects in water are largely related to the small size of its molecules, the latter feature is the ultimate cause of the decrease in partial molar volume associated with the hydrophobic effect.     

Vapor‐Liquid Phase Equilibria for Carbon Dioxide+Isopentanol Binary System at Elevated Pressure

High‐pressure vapor‐liquid phase equilibrium data for carbon dioxide+isopentanol were measured at temperatures of 313.2, 323.1, 333.5 and 343.4 K in the pressure range of 4.64 to 12.71 MPa in a variable‐volume high‐pressure visual cell. The experimental data were well correlated with Peng‐Robinson equation of state (PR‐EOS) together with van der Waals‐2 two‐parameter mixing rule, and the binary interaction parameters were obtained. Henry coefficients and partial molar volumes of CO₂ at infinite dilution were estimated based on Krichevsky‐Kasarnovsky equation, and Henry coefficients increase with increasing temperature, however, partial molar volumes of CO₂ at infinite dilution are negative and the magnitudes decrease with temperature.     

Volumes of aqueous alcohols, ethers, and ketones to T = 523 K and p = 28 MPa.

Densities of dilute aqueous solutions of isopropanol, 1,5-pentanediol, cyclohexanol, benzyl alcohol, diethyl ether, 1,2-dimethoxyethane, acetone, and 2,5-hexanedione were measured by means of a vibrating-tube flow densimeter at temperatures near T = (302, 373, 423, 473, and 521) K at a pressure of p = 28 MPa. At the lowest and highest temperatures, measurements were also made close to the saturation vapour pressure of water to investigate the effect of pressure on the volumes of solutes. Apparent molar volumes were calculated for each solute and extrapolated to give partial molar volumes at infinite dilution. The variation of the volume with temperature, pressure, and structure of solute is discussed qualitatively, and group contributions are determined at the temperatures of measurements and p = 28 MPa. Several equations proposed in the literature for correlating the partial molar volumes at infinite dilution as a function of state parameters are tested. Parameters of one selected equation are tabulated allowing calculation of the partial molar volumes at infinite dilution at temperatures and pressures up to T = 573 K and p = 40 MPa. respectively.     

Using supercritical fluid chromatography to determine diffusion coefficients of 1,2-diethylbenzene, 1,4-diethylbenzene, 5-tert-butyl-m-xylene and phenylacetylene in supercritical carbon dioxide

The binary diffusion of 1,2-diethylbenzene, 1,4-diethylbenzene, 5-tert-butyl-m-xylene and phenylacetylene at infinite dilution in supercritical carbon dioxide were measured between 15.0 and 35.0 MPa and in the temperature range of 313.16 to 333.16K by the Taylor-Aris chromatographic method. The effect of temperature, pressure, viscosity and density was discussed. In the case of temperature dependence, additional measurements were done for 5-tert-butyl-m-xylene from 308.16 to 398.16K at 35.0 MPa. The measured diffusivities of the four solutes were compared with the calculated ones by several predictive formulas.     

Modeling of solid–liquid equilibria in naphthalene,normal-alkane and polyethylene solutions

A solid–liquid equilibrium (SLE) model is developed on the basis of an equation of state referred to as copolymer SAFT. This SLE model is demonstrated for hydrocarbon solutions containing totally and partially crystallizable solutes. Initially regressed and tested on the solubility data for naphthalene, normal-alkane, and polyethylene, this model is used in a sensitivity study to understand the effects of crystallizability, melting temperature, molecular weight, and pressure on solid–liquid and liquid–liquid transitions of polyethylene in subcritical and supercritical propane.     

Application of the perturbed Lennard–Jones chain equation of state to solute solubility in supercritical carbon dioxide

The perturbed Lennard–Jones chain (PLJC) equation of state is a thermodynamic model based on the perturbation theory of liquid state. This equation has been shown in the past to be a successful model for phase equilibria calculations of binary and ternary fluid mixtures and polymer solutions. In this work, we employed for the first time the PLJC equation to model the solubility of 39 solids in supercritical carbon dioxide. It was shown that the model achieves good correlation with three temperature independent parameters. A comparison of the PLJC with the commonly used Peng–Robinson equation reveals the PLJC equation gives better correlation to the solubility data than the Peng–Robinson model that utilizes temperature dependent parameters.     

Calculations of phase equilibria for mixtures of triglycerides,fatty acids,and their esters in lower alcohols

The objects of study were mixtures containing triglycerides and lower alcohols and also the products of the transesterification of triglycerides, glycerol and fatty acid esters. The Redlich-Kwong-Soave equation of state was used as a thermodynamic model for the phase state of the selected mixtures over wide temperature, pressure, and composition ranges. Group methods were applied to determine the critical parameters of pure substances and their acentric factors. The parameters obtained were used to calculate the phase diagrams and critical parameters of mixtures containing triglycerides and lower alcohols and the products of the transesterification of triglycerides, glycerol and fatty acid esters, at various alcohol/oil ratios. The conditions of triglyceride transesterification in various lower alcohols providing the supercritical state of reaction mixtures were selected.     

Correlation and prediction of liquid-liquid distribution coefficients in aqueous systems using a modified Wilson model

A modified Wilson model is tested for its ability to correlate and predict distribution coefficients in two representative systems: 1-butanol-water and cyclohexanewater. The model is fitted to ternary equilibrium data for various solutes in these systems using a procedure involving minimization of the least-squares distance between calculated and experimental logarithmic distribution ratios. In addition, benzene-water, hexane-water, and cyclohexane-water distribution coefficients for infinitely diluted liquid solutes are predicted using only binary system information. All computations involve using both van der Waals and molar volumes as structural parameters to account for the geometry of the molecules studied. Satisfactory representations of experimental distribution ratios and fairly accurate distribution coefficients at infinite dilution are obtained for both systems. However, in a number of cyclohexane-water systems, miscibilities of constituent binary mixtures are poorly predicted from ternary system information when van der Waals volumes are used. Replacement of van der Waals volumes by molar volumes has little influence on the fit, but significant improvement is observed for the prediction of both binary miscibility properties and for distribution coefficients at infinite dilution in all the solvent-water systems considered.Presentation to First International Symposium on Solubility Phenomena, University of Western Ontario, London, Ontario, August 21–23, 1984.     

Development of equations for differential and integral enthalpy change of adsorption for simulation studies

We present equations to calculate the differential and integral enthalpy changes of adsorption for their use in Monte Carlo simulation. Adsorption of a system of N molecules, subject to an external potential energy, is viewed as one of transferring these molecules from a reference gas phase (state 1) to the adsorption system (state 2) at the same temperature and equilibrium pressure (same chemical potential). The excess amount adsorbed is the difference between N and the hypothetical amount of gas occupying the accessible volume of the system at the same density as the reference gas. The enthalpy change is a state function, which is defined as the difference between the enthalpies of state 2 and state 1, and the isosteric heat is defined as the negative of the derivative of this enthalpy change with respect to the excess amount of adsorption. It is suitable to determine how the system behaves for a differential increment in the excess phase adsorbed under subcritical conditions. For supercritical conditions, use of the integral enthalpy of adsorption per particle is recommended since the isosteric heat becomes infinite at the maximum excess concentration. With these unambiguous definitions we derive equations which are applicable for a general case of adsorption and demonstrate how they can be used in a Monte Carlo simulation. We apply the new equations to argon adsorption at various temperatures on a graphite surface to illustrate the need to use the correct equation to describe isosteric heat of adsorption.     

The Full Pressure–Temperature Phase Envelope of a Mixture in 1000 Microfluidic Chambers

Knowing the thermodynamic state of complex mixtures—liquid, gas, supercritical or two‐phase—is essential to industrial chemical processes. Traditionally, phase diagrams are compiled piecemeal from individual measurements in a pressure–volume–temperature cell performed in series, where each point is subject to a long fluid equilibrium time. Herein, 1000 microfluidic chambers, each isolated by a liquid piston and set to a different pressure and temperature combination, provide the complete pressure–temperature phase diagram of a hydrocarbon mixture at once, including the thermodynamic phase envelope. Measurements closely match modeled values, with a standard deviation of 0.13 MPa between measurement and model for the dew and bubble point lines, and a difference of 0.04 MPa and 0.25 °C between measurement and model for the critical point.     

Effective interaction between two ion-adsorbing plates: Hofmeister series and salting-in/salting-out phase diagrams from a global mean-field analysis

Interactions between ions and solutes are key to ion-specificity. A generic model in which ions interact via square well potentials of finite range with charged plates is solved analytically on the Poisson-Boltzmann level and analyzed globally for varying surface charge, salt concentration, and ion-surface affinity. Ion adsorption as well as depletion can lead to stably bound plates at finite separation, relevant for the equilibrium salting-out of small solutes such as proteins. The interplate pressure at large plate separation, relevant for aggregation kinetics of large solutes, exhibits direct as well as indirect Hofmeister ordering, depending on surface charge and salt concentration. A simple method for mapping explicit ion-surface potentials of mean force as obtained from solvent-explicit molecular dynamics simulations onto square-well potential parameters is demonstrated.     

Modeling adsorption in binary associating solvents using the extended MPTA model

Application of the MPTA model has been extended to associative liquid adsorption. The MPTA model describes fluid–fluid interactions using an equation of state (EoS) term, and fluid–solid interactions using a potential equation. In order to extend the application to associative liquid adsorption, an association term has been considered for fluid–fluid interactions. Sixteen binary mixtures containing associating and non-associating components in equilibrium with various adsorbents have been studied; fluid–fluid interactions have been modeled using the Peng–Robinson, Soave–Redlich–Kwong, volume-translated SRK and CPA equations of state, while the effects of fluid–solid interactions have been taken into account using Dubinin–Radushkevich–Astakhov (DRA) and Steele potential functions. The model parameters have been obtained by fitting the model to experimental data on surface excess. For the studied systems, the accuracy of fitted isotherms has been found to be more dependent on the fluid–solid potential equation rather than the applied EoS. Calculations show that the SRK equation is a suitable choice for non-associating systems, while the CPA equation is found to be more appropriate for associating systems, as would be expected. The results also show that the Steele potential function is in better agreement with experimental data than the DRA potential function.     

PSRK: A Group Contribution Equation of State Based on UNIFAC

A group contribution equation of state called PSRK (Predictive Soave-Redlich-Kwong) which is based on the Soave-Redlich-Kwong equation (Soave, 1972) has been developed. It uses the UNIFAC method to calculate the mixture parameter a and includes all already existing UNIFAC parameters. This concept makes use of recent developments by Michelsen (1990b) and has the main advantage, that vapor-uquid-equilibria (VLB) can be predicted for a large number of systems without introducing new model parameters that must be fitted to experimental VLB-data. The PSRK equation of state can be used for VLB-predictions over a much larger temperature and pressure range than the UNIFAC γ--approach and is easily extended to mixtures containing supercritical compounds. Additional PSRK parameters, which allow the calculation of gas/gas and gas/alkane phase equilibria, are given in this paper. In addition to those mixtures covered by UNIFAC, phase equilibrium calculations may also include gases like CH₄ C₂H₆, C₃H₆, c₄H₁₀, CO₂, N₂, H₂ and CO.