Equation multicomponent mixture

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

The creatmenc of the boundary conditions given here ts a generali2a-tion to multicomponent mixtures of a result originally obtained for a binary mixture by Kramers and Kistecnaker (25].These authors also obtained results equivalent to the binary special case of our equations (4.21) and (4.25), and integrated their equations to calculate the p.ressure drop which accompanies equimolar counterdiffusion in a capillary. Their results, and the important accompanying experimental measurements, will be discussed in Chapter 6 ... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

Equation (6.60) extends equation (6.59) to calculate [Pg.267]

Extending the method to a multicomponent mixture, the total mass balance remains the same, but separate component balance equations must now be written for each individual component i, i.e.. 

A significant advantage of the Wilson equation is that it can be used to calculate the equilibrium compositions for multicomponent systems using only the Wilson coefficients obtained for the binary pairs that comprise the multicomponent mixture. The Wilson coefficients for several hundred binary systems are given in the DECHEMA vapour-liquid data collection, DECHEMA (1977), and by Hirata (1975). Hirata gives methods for calculating the Wilson coefficients from vapour liquid equilibrium experimental data. 

The UNIQUAC equation developed by Abrams and Prausnitz is usually preferred to the NRTL equation in the computer aided design of separation processes. It is suitable for miscible and immiscible systems, and so can be used for vapour-liquid and liquid-liquid systems. As with the Wilson and NRTL equations, the equilibrium compositions for a multicomponent mixture can be predicted from experimental data for the binary pairs that comprise the mixture. Also, in the absence of experimental data for the binary pairs, the coefficients for use in the UNIQUAC equation can be predicted by a group contribution method UNIFAC, described below. 

For multicomponent mixtures the temperature that satisfies these equations, at a given system pressure, must be found by trial and error. 

Example 10.5 illustrates the trade-offs for gas permeation. The feed was assumed to be a binary mixture, which simplifies the calculations. For multicomponent mixtures, the same basic equations can be written for each component and solved simultaneously12. The approach is basically the same, but numerically more complex. 

In classical thermodynamics, there are many ways to express the criteria of a critical phase. (Reid and Beegle (11) have discussed the relationships between many of the various equations which can be used.) There have been three recent studies in which the critical points of multicomponent mixtures described by pressure-explicit equations of state have been calculated. (Peng and Robinson (1 2), Baker and Luks (13), Heidemann and Khalil (14)) In each study, a different statement of the critical criteria and... 

If A and B are the light and heavy key components of a multicomponent mixture, then applying the method given earlier for binary mixtures, equation 11.53, the minimum reflux... 

The derivation of the mixture-balance laws has been given by Chapman and Cowling for a binary mixture. Its generalization to multicomponent mixtures, as in Equation 5-1, uses a determination of the invariance of the Boltzmann equation. This development has been detailed by Hirschfelderet These derivations were summarized in the notes of Theodore von Karmin s Sorbonne lectures given in 1951-1952, and the results of his summaries were stated in Pinner s monograph. For turbulent flow, the species-balance equation can be represented in the Boussinesq approximation as ... 

First a derivative is given of the equations of change for a pure fluid. Then the equations of change for a multicomponent fluid mixture are given (without proof), and a discussion is given of the range of applicability of these equations. Next the basic equations for a multicomponent mixture are specialized for binary mixtures, which are then discussed in considerably more detail. Finally diffusion processes in multicomponent systems, turbulent systems, multiphase systems, and systems with convection are discussed briefly. 

Generally speaking, a fluid can be a liquid or a gas, where an important difference is in the equation of state that provides a relationship among the pressure, temperature, and mass density. Gases, of course, are compressible in the simplest case an ideal gas law provides the equation of state for a multicomponent mixture as... 

The perfect-gas equation of state for multicomponent mixtures depends on the species composition. Representing the composition as either mass fraction Yk or mole fraction Xk leads to... 

In later chapters we discuss the evaluation of transport properties for multicomponent mixtures of gases. At this point, however, the intent is to provide only a brief introduction on viscosity, since it is the principal fluid property that appears in the Navier-Stokes equations. The discussion here is limited to single-component fluids. 

Working with multicomponent mixtures requires quantifying the amounts of various chemical constituents that comprise the mixture. In the conservation equations a mass fraction will be the most appropriate measure, since mass is a conserved quantity. By definition, the mass fraction is... 

When considering the mass continuity of an individual species in a multicomponent mixture, there can be, and typically is, diffusive transport across the control surfaces and the production or destruction of an individual species by volumetric chemical reaction. Despite the fact that individual species may be transported diffusively across a surface, there can be no net mass that is transported across a surface by diffusion alone. Moreover homogeneous chemical reaction cannot alter the net mass in a control volume. For these reasons the overall mass continuity need not consider the individual species. At the conclusion of this section it is shown that that the overall mass continuity equation can be derived by a summation of all the individual species continuity equations. 

The shapes of the curves in these figures are characteristic of most multicomponent mixtures. At low pressures, the slope of each curve is approximately — 1.0. A slope of — 1.0 for ideal K-factors is predicted by Equation 14-1. Each curve passes through a value of unity at a pressure very close to the vapor pressure of the component at the temperature of the graph. This also is predicted for ideal-solution behavior. 

Binary systems of course can be handled by the computer programs devised for multicomponent mixtures that are mentioned later. Constant molal overflow cases are handled by binary computer programs such as the one used in Example 13.4 for the enriching section which employ repeated alternate application of material balance and equilibrium stage-by-stage. Methods also are available that employ closed form equations that can give desired results quickly for the special case of constant or suitable average relative volatility. 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

The differential heat of adsorption for each component in the mixture is estimated using the Clapeyron equation, extended to multicomponent mixtures and assuming ideal behavior of the gas phase (fugacity of i-th components Pi.)> that is,... 

With the advent of the personal computer, the numerical solution to these equations is straightforward even for multicomponent mixtures. Figure 2.10 shows an example calculation for the separation of a 20 wt% solution of n-decane in MEK. In these calculations, the ratio of the permeabilities of MEK... 

The generic balance relations and the derived relations presented in the preceding section contain various diffusion flux tensors. Although the equation of continuity as presented does not contain a diffusion flux vector, were it to have been written for a multicomponent mixture, there would have been such a diffusion flux vector. Before any of these equations can be solved for the various field quantities, the diffusion fluxes must be related to gradients in the field potentials . 

In order to predict correctly the fluxes of multicomponent mixtures in porous membranes, simplified models based solely on Fields law should be used with care [28]. Often, combinations of several mechanisms control the fluxes, and more sophisticated models are required. A well-known example is the Dusty Gas Model which takes into account contributions of molecular diffusion, Knudsen diffusion, and permeation [29]. This model describes the coupled fluxes of N gaseous components, Ji, as a function of the pressure and total pressure gradients with the following equation ... 

Several authors have recently reported on the use of Newton-Raphson procedures as applied to the solution of the equation describing a complex fractionator for the separation of multicomponent mixtures. However, it appears that the flexibility to handle various types of electrolyte problems, or to handle... 

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