Pure-component parameters determination

The evaluation of the sublimation pressure is a problem since most of the compounds to be extracted with the supercritical fluids exhibit sublimation pressures of the order of 10 14 bar, and as a consequence these data cannot be determined experimentally. The sublimation pressure is thus usually estimated by empirical correlations, which are often developed only for hydrocarbon compounds. In the correlation of solubility data this problem can be solved empirically by considering the pure component parameters as fitting-parameters. Better results are obviously obtained [61], but the physical significance of the numerical values of the parameters obtained is doubtful. For example, different pure component properties can be obtained for the same solute using solubility data for different binary mixtures. 

Determination of pure component parameters. In order to use the EOS to model real substances one needs to obtain pure component below its critical point, a technique suggested by Joffe et al. (18) was used. This involves the matching of chemical potentials of each component in the liquid and the vapour phases at the vapour pressure of the substance. Also, the actual and predicted saturated liquid densities were matched. The set of equations so obtained was solved by the use of a standard Newton s method to yield the pure component parameters. Values of exl and v for ethanol and water at several temperatures are shown in Table 1. In this calculation vH and z were set to 9.75 x 10"6 m3 mole"1 and 10, respectively (1 ). The capability of the lattice EOS to fit pure component VLE was found to be quite insensitive to variations in z (6[Pg.90]

The thermodynamic state of a pure component is determined by three variables the pressure P, the volume V, and the temperature T. The relationship between these three variables is known as the state equation and is represented by a surface in the three-dimensional plotting of P, V, and T. Any pure component, following the value of these three parameters, will be either a solid (S), a liquid (L), or a gas (G). A plot of P against T, or P against V is generally preferred because of its easier application. 

Simulations of ternary systems were performed using the pure component parameters in Table I and the cross parameters for the systems acetone/ CO2 and water/C02 determined previously (fi j - 1 and 0.81 respectively). Because of expected difficulties similar to the ones mentioned for the water/C02 system, no attempt was made to simulate the system acetone/water near room temperature. Thus, we set the acetone/water interaction parameters to the values from the Lorenz-Berthelot rules with fi j-l. Direct simulations of ternary phase equilibria have not been previously reported to the best of our knowledge. 

For biomaterials that are thermally unstable and decompose before reaching the critical temperature, several estimation techniques are available. We have used the Lydersen group contributions method ( ). Other techniques available for predicting critical properties have been reviewed and evaluated by Spencer and Daubert ( ) and Brunner and Hederer Qfi). It is also possible to determine the EOS parameters from readily measurable data such as vapor pressure, and liquid molar volume instead of critical properties (11). We used the Lydersen method to get pure component parameters because the vapor compositions we obtained were in closer agreement with experiment than those we got from pure component parameters derived by Brunner s method. The critical properties we used for the systems we studied are summarized in Table II. 

These interaction parameters are used in place of the corresponding pure-component parameters to determine the B,j values. 

Equations (4.345) and (4.346) are the van Laar activity-coefQcient equations that are used for fitting data by adjusting the parameters A21 and A 2- Although these parameters are derived from pure-component parameters, as shown in Equations (4.347) and (4.348), they are, nevertheless, considered mixture-specific binary interaction parameters and are thus indicated with subscripts because they are determined by fitting binary mixture data. Equations (4.347) and (4.348), while showing the source of derivation of the parameters, are not used for their determination. 

Water Solute in Hydrocarbon-Rich Vapor and Liquid. The pure component parameters of water solute AP(TC) and a were determined by using Equations 5 and 16 to fit the gas-phase volumetric properties of steam (5) and the second virial cross coefficients of steam and light gases such as methane, ethane, and nitrogen (6). The least-squares minimization technique was used to find the parameters that gave the minimum deviations between calculated and experimental pressures and second virial cross coefficients. (Table I lists the parameters for pure steam and of other compounds used in this study.)... 

Methanol in Hydrocarbon-Rich Vapor and Liquid. The volumetric properties of methanol gas (12) and the second virial cross coefficients of methanol and light gases (13) were used to determine the pure-component parameters AP(TC) and a for methanol. Table II shows the enthalpy departure of gaseous methanol from ideal gas at three temperatures and several pressures. For comparison, the experimental values (14) and the values calculated by the Soave equation (1) are also shown. Table II indicates that the Won modified equation of state predicts the enthalpy departure of methanol very well at low temperatures and fairly well at high temperatures, but that the original Soave equation considerably underestimates the enthalpy departure at all temperatures and pressures. Since the original Soave equation was meant to be applied only to hydrocarbons, we are not surprised at this result. Comparison of calculated and experimental second virial cross coefficients between methanol and methane (and also C02) is presented elsewhere (15).)... 

When volumetric data are available for a polar gas or for hydrocarbon gas mixtures containing polar species, the Won modification of the Soave equation of state can be used to determine the two pure-component parameters Ap( Tc) and a for the polar compound. With these parameters, the Won modified equation of state provides good estimates of VLE K-ratios of trace polar compounds in hydrocarbon-rich mixtures. 

Numerical values for the pure-component parameters 9 and b ate determined from Eqs. (1.3-26)-(1.3-28) ... 

The masses of these have to be approximately determined from pure adsorption isotherms of both components (1, 2) and calculations of coadsorption equilibria including only pure components parameters as for example the lAST-formalism [4.10,4.4,4.15,4.17]. 

This equation contains two parameters a and b that are related to the interaction energy of the molecules and to the size of the molecules, respectively. Therefore, they are called pure-component parameters and are usually determined by fitting to experimental liquid-density and vapor-pressure data. Applying equations of states to mixtures is, in most cases, done by applying a one-fluid theory. This means that the parameters of a virtual mixture molecule are obtained by so-called mixing rules from the model parameters of the pure compraients, e.g., by ... 

Among the three pure-component parameters for nonassociating components there are two parameters that are related to the size of the molecule the segment diameter (7 and the segment number m. The third parameter, the energy parameter a, decribes the attractive interactions between two molecules. For volatile components, these parameters are determined by simultaneously fitting to physical properties, which can on the one hand be calulated by an equation of state and are on the other hand related to the size and the interactions of the molecules. Such properties are, e.g., liquid-density data (related to molecule size) and vapor pressures (related to the intermolecular interactions). These parameters have already been determined for a huge number of relevant solvents and can be found in extensive parameter tables, e. g in [15] (SAFT) and [24, 39] (PC-SAFT). 

Although the polymer parameters were also fitted to binary data, they still have the character of the pure-component parameters. This is confirmed by calculations of other polymer/solvent systems, which are illustrated in Fig. 5b. Using the same polycarbonate parameters as determined for the chlorobenzene system, the... 

Figure 12 shows the results for the modeling of the solubility of the copolymer poly(ethylene-co-l-butene) in propane. The pure-component parameters for poly (ethylene) (HOPE), poly(l-butene), and propane as well as the binary parameters for HDPE/propane and poly(l-butene)/propane were used as determined for the homopolymer systems. 

The BWRS equation is Starling s modification of the Benedlct-Webb-Rubin equation of state. It contains eleven adjustable pure component parameters plus a binary interaction parameter for each component pair. Thus, a typical 20 component mixture would be characterized by 220 pure component parameters and 180 different binary interaction parameters—a total of AOO constants. Exxon s set of constants were determined by multi-property regression, a procedure in which parameters are adjusted until available data on density, enthalpy, vapor pressure, K-values, sonic velocity, and specific heats are all matched simultaneously. The large number of constants to be determined requires that these data be accurate and that they cover a wide range of conditions. Nearly 20,000 data points were used to determine our set of constants. The large amount of data required limits the components that can be handled to the relatively few for which such data exist and may also place a practical limit on the number of parameters desirable in an equation of state. 

Equatioa-of-state theories employ characteristic volume, temperature, and pressure parameters that must be derived from volumetric data for the pure components. Owiag to the availabiHty of commercial iastmments for such measurements, there is a growing data source for use ia these theories (9,11,20). Like the simpler Flory-Huggias theory, these theories coataia an iateraction parameter that is the principal factor ia determining phase behavior ia bleads of high molecular weight polymers. 

For each binary pair, there are two adjustable parameters that must be determined from experimental data, that is, (uy - ujj), which are temperature dependent. Pure component properties rl and ql measure molecular van der Waals volumes and surface areas and have been tabulated6. 

The two adjustable parameters, (u 2 - u22) and ( 2i - n), must be determined experimentally6. Pure component properties rx,r2, qi and q2 have been tabulated6. 

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