Van der Waals one-fluid mixing rule

The greatest use of cubic equations of state is for phase equilibrium calculations involving mixtures. The assumption inherent in such calculations is that the same equation of state as is used for the pure fluids can be used for mixtures if we have a satisfactory way to obtain the mixtures parameters. This is most commonly done using the van der Waals one-fluid mixing rules,... 

Mixture, size, and energy parameters can be evaluated using van der Waals one-fluid mixing rules and usual combining rules with binary parameters, as in... 

Until recently, the most common way of choosing mixture parameters was to satisfy only eqn. (3,3,2) with the van der Waals one-fluid mixing rules as follows ... 

In Figure 3.4,1 the results for the methane and n-pentane (Knapp et al. 1982) binary system are presented. This is a typical mixture for which the van der Waals one-fluid mixing rules with a single constant binary interaction parameter performs very well... 

Why the van der Waals one-fluid mixing rules cannot describe highly nonideal mixtures can be understood by starting with the relation between the molar excess... 

A desirable characteristic of an excess free-energy-based mixing rule is that it goes smoothly to the conventional van der Waals one-fluid mixing rule for some values of its parameters. This is useful because in multicomponent mixtures only some of the binary pairs may form highly nonideal mixtures requiring mixing rules such as... 

D.3. Program VDW Binary VLE vyith the van der Waals One-Fluid Mixing Rules (IPVDW and 2PVDW)... 

VDW BINARY VI.E WITH VAN DER WAALS ONE-FLUID MIXING RULES 1 CONVENTIONAL (IPVDW)... 

Example D,3,B Fitting Binary VLE Data with the Two-parameter van der Waals One-fluid Mixing Rule... 

The program VDWMIX is used to calculate multicomponent VLE using the PRSV EOS and the van der Waals one-fluid mixing rules (either IPVDW or 2PVDW see Sections 3.3 to 3.5 and Appendix D.3). The program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is the number of components (up to a maximum of ten), the liquid mole fractions, the temperatures at which the calculations are to be done (for as many sets of calculations as the user wishes, up to a maximum of fifty), critical temperatures, pressures (bar), acentric factors, the /f constants of the PRSV equation for each compound in the mixture, and, if available, the experimental bubble point pressure and the vapor phase compositions (these last entries are optional and are used for a comparison between the experimental and calculated results). In addition, the user is requested to supply binary interaction parameteifs) for each pair of components in the multicomponent mixture. These interaction parameters can be... 

Other researchers " have used other mixing rules similar to the van der Waals one-fluid rules. Some have been shown to yield improved results over the van der Waals one-fluid mixing rules. 

By using standard thermodynamics, the activity coefficient expression that corresponds to the van der Waals equation of state can be derived. When the van der Waals one-fluid mixing rules for the energy and co-volume parameters are employed, the van der Waals equation can be expressed as a sum of a combinatorial-FV and a regular solution term ... 

The repulsive, chain, and hydrogen bonding terms are extended to mixtures rigorously, and so mixing rules are needed only for the dispersion term of the equation. The van der Waals one-fluid mixing rules are used, which involve only one temperature-independent interaction parameter... 

The mixing rules described above for vdW eos s, known as van der Waals one-fluid mixing rules, or simply van der Waals mixing rules, apply as well to the vdW-type cubic equations described next. 

Here a new parameter jiry, known as the binary interaction parameter, has been introduced to result in more accurate mixture equation-of-state calculations. This parameter is found by fitting the equation of state to mixture data (usually vapor-liquid equilibrium data, as discussed in Chapter 10). Values of the binary interaction parameter k - that have been reported for a number of binary mixtures appear in Table 9.4-1. Equations 9.4-8 and 9.4-9 are referred to as the van der Waals one-fluid mixing rules. The term one-fluid derives from the fact that the mixture is being described by the same equation of state as the pure fluids, but with concentration-dependent parameters. 

As has already been mentioned, simple cubic equations of state with the van der Waals one-fluid mixing rules of Eqs. 9.4-8 and 9.4-9 are applicable at all densities and temperatures, but only to mixtures of hydrocarbons or hydrocarbons with inorganic gases. That is, this model is applicable to relatively simple mixtures. On the other hand, excess... 

Repeat the calculations of the previous problem with the regular solution model. Compare the two results. Develop an expression for the activity coefficient of a species in a mixture from the Peng-Robinson equation of state with the van der Waals one-fluid mixing rules, a. Show that the minimum amount of work, W , necessary to separate 1 mole of a binary mixture into its pure components at constant temperature and pressure is... 

Derive the expression for the partial molar volume of a species in a mixture that obeys the Peng-Robin.son equation of state and the van der Waals one-fluid mixing rules. 

It should be pointed out that the results in Figs. 10.3-8 to 10.3-11 are examples of a successful application of the mixing rule of Sec. 9.9 to highly nonideal systems. For comparison, we show in Fig. 10.3-13 the results that would be obtained for the acetone-water system using the van der Waals one-fluid mixing rule, EqS 9.4-8 and... 

Figure 10.3-13 Vapor-liquid equilibria of the acetone-water binary mixture described by the Peng-Robinson equation of state and the van der Waals one-fluid mixing rules. The solid lines result from the value of the binary parameter xy being fit to the data at 298 K. The dotted lines are the highly nonideal (and unrealistic) behavior predicted by setting k j = 0.

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