Phase equilibria approximate methods

The object of Chapter 4 was to provide an overview of phase equilibria concepts, which are more easily obtained through phase diagrams and the approximate, historical methods. With Chapter 4 as background, the subject of the present chapter is the phase equilibrium calculation method that is both most accurate and most comprehensive. 

For most SPR appHcations the dissociation constant is calculated from the Kd = koff/kon proportion. It can also be evaluated from the steady-state phase (equilibrium). This method, however, is apphcable only for kinetically fast interactions. The time required for reaching the equihbrium at a protein concentration equal to JCj can be expressed as l/ko . For an interaction with off =1x10 s it will take approximately 1 h to reach equilibrium. The steady state is reached faster with increased protein concentrations. Using a high protein concentration results, however, in the collection of only a narrow range of Req points, close to saturation (Rmax)- In this situation the Kj is... 

The equations are transcendental equations, which must be solved by approximation methods. This presents no problem with the use of modem computers. However, it is still appropriate to discuss graphical aids to the solution of the equations. We discuss only one example. Let us consider the equilibrium between a solid solution and a liquid solution at a constant pressure. For the present we choose the pure solid phases and the pure liquid phases as the standard states of the two components for each phase. The equation for equilibrium for the first component is... 

This formulation, while of absolutely general validity, is so complicated that approximate methods of solution of phase equilibrium problems have to be developed. A few essential aspects of these approximate methods are discussed in the next subsection. 

Cotterman and Prausnitz (1991) have reviewed the approximate methods that have been used in the solution of phase equilibrium problems within the framework of a continuous description (or, often, a semicontinuous one, where a few components are dealt with as discrete ones). These methods are based on relatively trivial extensions of classical methods (which make use of Gibbs free energy equations, equations of state, and the like) to a continuous description. 

Relationships governing the distribution of a substance between gas and liquid phases are the subject matter of phase-equilibrium thermodynamics and, for the most part, fall beyond the scope of this text. However, we will cover several simple approximate relationships that provide reasonably accurate results over a wide range of conditions. Such relationships form the bases of more precise methods that must be used when system conditions require them. 

The calculations reported in this paper and a related series of publications indicate that it is now quite feasible to obtain reasonably accurate results for phase equilibria in simple fluid mixtures directly from molecular simulation. What is the possible value of such results Clearly, because of the lack of accurate intermolecular potentials optimized for phase equilibrium calculations for most systems of practical interest, the immediate application of molecular simulation techniques as a replacement of the established modelling methods is not possible (or even desirable). For obtaining accurate results, the intermolecular potential parameters must be fitted to experimental results, in much the same way as parameters for equation-of-state or activity coefficient models. This conclusion is supported by other molecular-simulation based predictions of phase equilibria in similar systems (6). However, there is an important difference between the potential parameters in molecular simulation methods and fitted parameters of thermodynamic models. Molecular simulation calculations, such as the ones reported here, involve no approximations beyond those inherent in the potential models. The calculated behavior of a system with assumed intermolecular potentials is exact for any conditions of pressure, temperature or composition. Thus, if a good potential model for a component can be developed, it can be reliably used for predictions in the absence of experimental information. 

Special consideration is given to the effect of completely miscible cosolvents such as methanol and ethanol. A new approximate method of predicting cosolvent effects is presented. The results should be useful in supplying necessary phase equilibrium data to complex computer programs for modeling transport and fate of sparingly soluble organics in the environment. 

Using one of these activity coefficient equations it is possible to calculate liquid-liquid equihbrium (LLE) behavior of multicomponent hquid systems. Consider, for example, the ternary system of Figure 1. A system of overall composition A splits into two liquid phases B and C. The calculation of compositions of B and C is analogous to the flash ciculation of vapor-liquid equilibrium problems. By using the UNIQUAC equations to obtain the partition coefficients, Kj, this problem can be solved for any composition A of the overall system. The calculations are lengthy but computer programs for this purpose (2) have been published. In this paper simpler approximate methods for phase equilibrium problems of environmental interest is sought. For the moment it is sufficient to note that the activity coefficients provide the means of complete liquid-liquid equihbrium computations. 

The conditions for phase equilibrium were presented equalities of temperature, pressure and chemical potential of each species in all phases. The evaluation of chemical potentials of mixtures was discussed, and the following methods and approximations were presented ideal mixture, Henry s law, and simple correlations for activity coefficients. 

The McCabe-Thiele constructions described in Chapter 8 embody rather restrictive tenets. The assumptions of constant molal overflow in distillation and of interphase transfer of solute only in extraction seriously curtail the general utility of the method. Continued use of McCabe-Thiele procedures can be ascribed to the fact that (a) they often represent a fairly good engineering approximation and (b) sufficient thermodynamic data to justify a more accurate approach is often lacking. In the case of distillation, enthalpy-concentration data needed for making stage-to-stage enthalpy balances are often unavailable, while, in the Case of absorption or extraction, complete phase equilibrium data may not be at hand. 

Yet another topic involves the extension of the synthesis methods to processes with multiple solutes. Here, the impact of concentration on the slope of phase equilibrium Eq. (11.4) may become a factor for highly nonideal solutions at high concentrations. The analysis techniques presented herein can be extended when the slopes of the equilibrium curves can be approximated as constant, independent of mixture composition. Also, the analyses are simplified when the minimum external MSA for the principal solute is capable of removing the other solutes as well (El-Halwagi and Manousiouthakis, 1989). 

In the mentioned works, it is suggested that tray by tray method should be used only for the part of the column located between zones of constant concentrations. The special equations, taking into account phase equilibrium between the meeting vapor and liquid flows, are applied to such zones. Approximations to the mode of minimum reflux are estimated by means of gradual increase of theoretical plates number in that part of the column for which tray by tray method is used. 

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