Interaction parameters rules

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

The mixing rule is given by Eq. (2-100) with the interaction parameter Q for each pair of components defined by Eq. (2-101). 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

The Trebb/e-Bishnoi EoS is a cubic equation that may utilize up to four binary interaction parameters, k=[ka, kb, kc, kcombining rules is presented next (Trebble and Bishnoi, 1987 1988). 

In this section we consider typical examples. They cover all possible cases that could be encountered during the regression of binary VLE data. Illustration of the methods is done with the Trebble-Bishnoi (Trebble and Bishnoi, 1988) EoS with quadratic mixing rules and temperature-independent interaction parameters. It is noted, however, that the methods are not restricted to any particular EoS/mixing rule. 

The calculated critical points of the binary pairs, particularly the critical pressures, are quite sensitive to the values used for the interaction parameters in the mixing rules for a and b in the equation of state. One problem in undertaking this study is that no data are available on the critical lines of any of the binary pairs except for CO2 - H2O. Even for C02 - H2O, two sets of critical data available (18, 19) are in poor quantitative agreement, though they present the same qualitative picture of the critical phenomena. 

The interaction parameters for binary systems containing water with methane, ethane, propane, n-butane, n-pentane, n-hexane, n-octane, and benzene have been determined using data from the literature. The phase behavior of the paraffin - water systems can be represented very well using the modified procedure. However, the aromatic - water system can not be correlated satisfactorily. Possibly a differetn type of mixing rule will be required for the aromatic - water systems, although this has not as yet been explored. 

It has been shown that the selected water model in conjunction with the equation of state provides a uniform method to calculate VLE and LLE of aqueous systems over a wide temperature and pressure range. The remaining discrepancies could possibly be eliminated with modified rules for the interaction parameters. 

For dilute solutions, Equations 4 and 5 reduce to the Bronsted-Guggenheim equations, and the parameters a23 and cu2 can be expressed in terms of the interaction parameters of tne Bronsted-Guggenheim theory. For concentrated solutions, Harned s rule is a simple empirical extension of the Brb nsted-Guggenheim theory. Thus, 1t 1s surprising how well the rule describes activity coefficients 1n highly concentrated solutions. 

Figure 2. Interaction parameters for Harned s Rule as a function of temperature...

Rule 6. The interaction parameter e/Zcg may be estimated from the boiling point Tb (K) at one atmosphere... 

The previous section discussed techniques for obtaining the molecular potential interaction parameters <7 and e based on pure species physical properties of molecule i. Interactions between unlike molecules (i.e., all i-j pairs) must also be considered in the calculation of transport properties (notably, binary diffusion coefficients). The following is a set of combining rules to estimate the i- j interaction parameters, assuming that the pure species values are known. 

The solute-solvent and the solvent-solvent interaction potentials are assumed to be given by Lennard-Jones potential. The Lennard-Jones interaction parameters for dissimilar solvent (i) and solute (/ ) spheres are estimated from those of the interaction of similar spheres through the combining rule ey = ( ,/ ,y) 2 and oi = (tr,-,- + ojj)/2 [121, 124]. 

Fig. 4. The temperature dependence of the rate of non-radiative transitions y for some (given) values of the interaction parameter w parameters co0 and cr0 are the same as in Fig. 1(a). The sharp peaks result from the divergence of the resolvent in equation (31) for w > 10 at some positive co > o>m- F°r comparison, the golden rule result is also presented (the thick line below).

Equations 2 and 3 are called the van der Waals mixing rules. In these equations, a j t and b j j (i j) are parameters corresponding to pure component ti) while ay and by (i j) are called the uni ike-interaction parameters. It is customary to relate the uni ike-interaction parameters to the pure-component parameters by the following expressions ... 

Figure 4. Three-phase equilibrium LtL2V in the system carbon dioxide-water-1-propanol at 333 K and 13.1 MPa exp., this work — Calculated with Peng-Robinson EOS using Panagiotopoulos and Reid mixing rule, left side prediction from pure component and binary data alone, right side interaction parameters fitted to ternary three-phase equilibria at temperatures between 303 and 333 K...

The good agreement between experimental and calculated values is pointed out by the average absolute deviation, AAD. The results demonstrate the possibility to compute the solubility as well as the partition coefficients with the Peng-Robinson-equation of state and a mixing rule with two interaction parameters. 

Momentary mass of a sample may be derived from momentary weight only if the density of the gas, and thus the buoyancy of the sample, are known. Volumetric data of pure gases are calculated from precise equations of state, as they exist for carbon dioxide and other gases, or taken from tables. Cubic equations of state are used to calculate densities of gas mixtures. We have always employed van-der-Waals mixing rules and fitted the interaction parameter to vapor-liquid equilibria determined by ourselves or taken from literature. 

Here, k is the Boltzmann constant. The Lorentz-Bcrthelot rule has been adopted for the parameters working between different species of the mixture and the values are thus yilk= 9.% K and a j2=3.405 x 10 1 m, which correspond to the interaction parameters of neat argon. The critical temperature of the Lennard-Jones monoatomic fluids ) evaluated by the integral equation theory is 1.321 multiplied by ( f/k ), which is equal to 111.9 K and 223.8 K for the component 1 and 2, respectively. The temperature set in our present calculation is higher than any of these critical temperature. Thus the fluids discussed in this work are always at supercritical state. 

Another common approach to reduce the number of parameters is to assume some form of combining rules, so that the hetero-interaction parameters are given by homo-parameters ... 

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