Thermodynamic factor coefficient models

Both L coefficients and / factors can, in principle, be calculated from microscopic models. For the evaluation of L,j, the random-alloy model [J. R. Manning (1968) A. R. Allnatt, A. B. Lidiard (1987)] is sometimes used. For the evaluation of thermodynamic factors, one takes advantage of the empirical rule that in extended solid solutions AO-BO, the cation vacancy concentration and the oxygen potential are related to each other as... 

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. 

In fact, X is a correction parameter for the Pick diffusion coefficient. This correction has a similar effect on the apparent diffusivity as the correction given in Eq. (14). When X is less then 1, the diffusivity increases with occupancy. This correction can also be applied to the Maxwell-Stefan diffusivity, which results in an even larger effect of concentration on the flux. The concentration dependence of the flux in the Maxwell-Stefan equations depends largely on the adsorption isotherm chosen, since this isotherm determines the thermodynamic factor. For Langmuir adsorption the concentration dependence of the flux increases in the following order using different models ... 

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... 

It should be noted that the Maxwell-Stefan D calculated from Eq. 4.1.5 can be quite sensitive to the model used to compute T, an observation first made by Dullien (1971). One of the reasons for this sensitivity is that E involves the first derivative of the activity coefficient with respect to composition. Activity coefficient model parameters are fitted to vapor-liquid equilibrium (VLE) data (see, e.g., Prausnitz et al., 1980 Gmehling and Onken, 1977). Several models may provide estimates of In % that give equally good fits of the vapor-liquid equilibrium data but that does not mean that the first derivatives of In % (and, hence, E) will be all that close. To illustrate this fact we have calculated the thermodynamic factor, E, for the system ethanol-water with several different models of In %. The results are shown in Figure 4.3 a). The interaction parameters used in these calculations were fitted to one set of VLE data as identified in the figure caption. Similar illustrations for other systems are provided by Taylor and Kooijman (1991). 

Sanni and Hutchison (1973) presented data on the binary Fick dilfusivity for the systems benzene-chloroform, cyclohexane-carbon tetrachloride, cyclohexane-toluene, benzene-cyclohexane, benzene-toluene, and diethyl ether-chloroform. Calculate the thermodynamic factor F for these systems using parameters from Gmehling and Onken (1977ff). Hence, estimate the Maxwell-Stefan diffusion coefficients and test the applicability of the Vignes model. 

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). 

The thermodynamic factor is evaluated for liquid mixtures from activity coefficient models. For a regular solution, for example,... 

Tables 7.2 and 7.3 display the heats of transports and thermal diffusion ratio (Kj) of chloroform in binary mixtures with selected alkanes and of toluene (1), chlorobenzene (2), and bromobenzene at 30 °C and 1 atm. Concentration-dependent thermal conductivity, mutual diffusion coefficients, and heats of transport of alkanes in chloroform and in carbon tetrachloride are given by Rowley et al. (1988). The polynomial fits to these coefficients for the alkanes in chloroform and in carbon tetrachloride are used to estimate the degree of coupling and the thermal diffusion ratio Kn from Eqns (7.46) and (7.47), and shown in Figures 7.1 and 7.2 (Demirel and Sandler, 2002). The thermal conductivity and the thermodynamic factors for the hexane-carbon tetrachloride mixture have been predicted by the local composition model of NRTL.

Activity-coefficient models, however, can only be used to calculate liquid-state fugacities and enthalpies of mixing. These models provide algebraic equations for the activity coefficient (y,) as a function of composition and temperature. Because the activity coefficient is merely a correction factor for the ideal-solution model (essentially Raoult s Law), it cannot be used for supercritical or noncondensable components. (Modifications of these models for these types of systems have been developed, but they are not recommended for the process simulator user without consultation with a thermodynamics expert.)... 

For the PC model, adsorption can be seen as purely partitioning behavior. This model ascribes a thermodynamic partitioning coefficient ( adsorption separation factor ) to each solute to describe adsorption quantitatively. No explicit thermodynamic contribution of the surface site to the adsorption process nor competitive adsorption effects are considered. 

Ba, Sr, and B. Consistency between programs was evaluated by comparing the log of the molal concentrations of free ions and complexes for two test solutions a hypothetical seawater analysis and a hypothetical river water analysis. Comparison of the free major ion concentrations in the river water test case shows excellent agreement for the major species. In the seawater test case there is less agreement and for both test cases the minor species commonly show orders of magnitude differences in concentrations. These differences primarily reflect differences in the thermodynamic data base of each chemical model although other factors such as activity coefficient calculations, redox assumptions, temperature corrections, alkalinity corrections and the number of complexes used all have an affect on the output. 

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