Equations of change, for a multicomponent

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. 

Table 8.4-1 The Differential Form of the Equations of Change for a Multicomponent System...

An important observation in this chapter is that the equations of change for a multicomponent mixture are identical, in form, to those for a pure fluid. The difference between the two is that the pure fluid equations contain thermodynamic properties (U, H, S, etc.) that can be computed from pure fluid equations of state and heat capacity data, whereas in the multicomponent case these thermodynamic properties can be computed only if the appropriate mixture equation of state and heat capacity data or enthalpy-concentration and entropy-concentration data are given, or if we otherwise have enough information to evaluate the necessary concentration-dependent partial molar quantities. at all temperatures, pressures, and compositions of interest Although this represents an important computational difference between the pure fluid and mixture equations. 

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

B Equation of Change for Temperature for a Multicomponent System Subtraction then gives ioi a. N ... 

Thus, the interrelationships provided by Eqs. 8.2-8 through 8.2-15 are really restrictions on the mixture equation of state. As such, these equations are important in minimizing the amount of experimental data necessary in evaluating the thermodynamic, properties of mixtures, in simplifying the description of multicomponent systems, and in testing the consistency of certain types of experimental data (see Chapter 10). Later in this chapter we show how the equations of change for mixtures and the Gibbs-Duhem equations provide a basis for the experimental determination of partial molar properties. 

The thermodynamics of irreversible processes begins with three basic microscopic transport equations for overall mass (i.e., the equation of continuity), species mass, and linear momentum, and develops a microscopic equation of change for specific entropy. The most important aspects of this development are the terms that represent the rate of generation of entropy and the linear transport laws that result from the fact that entropy generation conforms to a positive-definite quadratic form. The multicomponent mixture contains N components that participate in R independent chemical reactions. Without invoking any approximations, the three basic transport equations are summarized below. 

Section 4.1 via Section 4.1.2 formally illustrates vapor-Uquid equilibria vis-a-vis distillation in a closed vessel along with bubble-point and dew-point calculations for multicomponent systems. How vapor-liquid equilibrium is influenced by chemical reactions in the liquid phase is treated in Section 5.2.1.2, where two subsections, 5.2.1.2.1 and 5.2.1.2.2, deal with reactions influencing vapor-Uquid equilibria in isotopic systems. We next encounter open systems in Chapter 6. The equations of change for any two-phase system (e.g. a vapor-Uquid system) are provided in Section 6.2.1.1 based on the pseudo-continuum approach for the dependences of species concentrations... 

The conservation equations (mass, momentum, energy) are primarily considered for fluids of uniform and homogeneous composition. Here, we examine how these conservation equations change when two or more species are present and when chemical reactions may also take place. In a multicomponent mixture, transfer of mass takes place whenever there is a spatial gradient in the mixture properties, even in the absence of body forces that act differently upon different species. In fluid flows, the mass transfer will generally be accompanied by a transport of momentum and may further be combined with a transport of heat. For a multicomponent fluid, conservation relations can be written for individual species. Let m, be the species velocities and Pi the species density where the index i, is used to represent the fth species. 

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

Consider a fluid of molecules Interacting with pair additive, centrally symmetric forces In the presence of an external field and assume that the colllslonal contribution to the equation of motion for the singlet distribution function Is given by Enskog s theory. In a multicomponent fluid, the distribution function fi(r,Vj,t) of a particle of type 1 at position r, with velocity Vj at time t obeys the equation of change (Z)... 

To this point the equations of change have been set down for pure fluids under both isothermal and nonisothermal conditions, and for multicomponent fluids and charged species. The boundary and initial conditions have, however, been considered only to a limited extent. They will be discussed in the context of the specific subject areas for example, diffusion, chemical reaction, surface tension, and heat transfer. Here, the form of the equations of change will be analyzed so that some of the more important characteristic similarity parameters can be brought out and the stage set for subsequent analyses over restricted ranges of these parameters. 

In the following developments, we rely on the results of Section 6.2.1.1 and identify the equations of change of concentration of a species i in a countercurrent two-region/two-phase system we focus on two-phase systems. Next we consider the equations for operating lines in such devices. The multicomponent separation capability of sucb systems is treated next in tbe context of a two-pbase system. 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

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