Activity coefficient-models thermodynamic model

The principle of distillation is the use of differences in volatiHties of the components to be separated. Distillation processes are usually carried out in countercurrent mode in multistage units. The differences that can be obtained in concentrations of the components in the vapor and liquid phases are determined by the vapor-liquid equihbrium (VLE). Until the 1970s reliable data for vapor-liquid equilibria could only be obtained by measurement, which, for a mixture containing more than two components, required a large number of time-consuming measurements. Advances in chemical thermodynamics have resulted in methods activity coefficient models (g models or equations of state) for the calculation of the phase-equihbrium behavior of multicomponent mixtures on the basis of binary subsystems. In the case that no information about the binary subsystems is available, predictive methods (group contribution methods) are available to allow estimation of the required phase equilibria. 

Obtaining Activity Coefficients from Thermodynamic Models... 

Traditional activity coefficient based thermodynamic models have been successfully used to describe several LLE systems. The nonrandom two-liquid (NRTL) model of Renon and Prausnitz (1968) and the universal quasi-chemical (UNIQUAC) method of Abrams and Prausnitz (1975) models have been used to correlate LLE data for the many multi-component mixtures (Ghanadzadeh et al., 2009 Se and Aznar, 2002), while a group contribution method (UNIFAC) (Fredenslund et. al., 1977) has been widely used to predict the LLE systems. 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

Several activity coefficient models are available for industrial use. They are presented extensively in the thermodynamics literature (Prausnitz et al., 1986). Here we will give the equations for the activity coefficients of each component in a binary mixture. These equations can be used to regress binary parameters from binary experimental vapor-liquid equilibrium data. 

This expression provides the basis for vapor-liquid equilibrium calculations on the basis of liquid-phase activity coefficient models. In Equation 4.27, thermodynamic models are required for cf>y (from an equation of state) and y, from a liquid-phase activity coefficient model. Some examples will be given later. At moderate pressures, the vapor phase becomes ideal, as discussed previously, and fj = 1. For... 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

Since the degree of coupling is directly proportional to the product Q (D/k)in, the error level of the predictions of q is mainly related to the reported error levels of Q values. The polynomial fits to the thermal conductivity, mass diifusivity, and heat of transport for the alkanes in chloroform and in carbon tetrachloride are given in Tables C1-C6 in Appendix C. The thermal conductivity for the hexane-carbon tetrachloride mixture has been predicted by the local composition model NRTL. The various activity coefficient models with the data given in DECHEMA series may be used to estimate the thermodynamic factors. However, it should be noted that the thermodynamic factors obtained from various molecular models as well as from two sets of parameters of the same model might be different. 

Existing activity coefficient models such as NRTL and UNIQUAC have been very successful correlative tools that provide versatile, flexible thermodynamic frameworks to correlate available experimental data. Once properly parameterized, the models are then used to perform both interpolation and extrapolation. 

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. 

The parameters A, B, and C in the above models are, in general, functions of temperature and pressure, but are independent of composition. These parameters are typically obtained by fitting experimental data. Given the parameters of these activity coefficient models, we can predict all the thermodynamic properties of the system. 

Figure 4.3. (a) Thermodynamic factor for the system ethanol-water at 40°C obtained from different activity coefficient models. Parameters from Gmehling and Onken (1977ff Vol. I/la p. 133). (h) Thermodynamic factor for the system ethanol-water at 50°C obtained using the NRTL equation using parameters fitted to isothermal vapor-liquid equilibrium data. Parameters from Gmehling and Onken... 

Taylor, R. and Kooijman, H. A., Composition Derivatives of Activity Coefficient Models (For the Estimation of Thermodynamic Factors in Diffusion), Chem. Eng. Commun., 102, 87-106 (1991). 

Many mixtures of interest in the chemical indu.stry exhibit strong nonidealities that can not be described by the EOS with any form of the van der Waals mixing rules. Mixing rules that combine equations of state with liquid excess Gibbs free-energy (or, equivalently, activity coefficient) models are more suitable for the thermodynamic... 

The use of the so-called EoS/G mixing rules, which were suggested in the early 1990s. This is a very powerful tool of applied thermodynamics, which permits a predictive use of cubic equations of state, when a GC (or other predictive) activity coefficient model is used for estimating the energy term of the equation of state. These methodologies are briefly reviewed in the next sections. 

Isothermal or isobaric activity coefficient relationships are modeled from experimental data by a variety of equations, all with a thermodynamic basis. The more useful of these equations are Van... 

Liquid mixtures at low pressure are generally described using the activity coefficient models as described in Table 9.11-1, and the behavior of a liquid mixture is generally not much affected by pressure, unless the pressure is very high. However, as we will see in Sec. 10.3, for phase equilibrium calculations at high pressures, especially as the critical point of a mixture is approached, there are important advantages to using the same thermodynamic model for both phases. In such cases the same equation-of-state model should be used for the vapor and liquid phases. 

These results suggest that, although not quite as good as P-T-x-y data, P-T-x data can be useful for estimating parameters in an activity coefficient model that can then be used to estimate the missing vapor compositions. An important disadvantage of P-T-x data, however, is that we cannot test its thermodynamic consistency since the activity coefficients are obtained from a model, not directly from experimental data. 

While using an activity coefficient model will provide a quantitative relationship between the mutual solubilities, we can get a qualitative understanding of how the presence of one dissolved species affects others by examining the interrelation between mixed second derivatives. In particular, the Maxwell equations in Chapter 8 and some of the pure fluid equations in Chapter 6 were derived by examining mixed second derivatives of thermodynamic functions. Another example of this is to start with the Gibbs energy and note that at constant temperature, pressure, and all other species mole numbers,... 

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