Multicomponent system, equations

For multicomponent systems, Equation 10.4 can be written for the limiting component, that is, the component with the highest Kt. Having determined the number of stages, the concentrations of the other components can be determined from Equation 10.6. 

Equations 10.8 to 10.10 can be written in terms of a stripping factor (S), where S = 1/A. For multicomponent systems, Equations 10.8 and 10.9 can be used for the limiting component (i.e. the component with the lowest K,) and then Equation 10.10 can be used to determine the distribution of the other components. 

The equations are derived from the differential of the Gibbs energy for a one-phase, multicomponent system Equation (2.33). The differential is exact, and therefore the condition of exactness must be satisfied. Two equations... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The preceding material of this section has focused on the most important phenomenological equation that thermodynamics gives us for multicomponent systems—the Gibbs equation. Many other, formal thermodynamic relationships have been developed, of course. Many of these are summarized in Ref. 107. The topic is treated further in Section XVII-13, but is worthwhile to give here a few additional relationships especially applicable to solutions. 

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... 

The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Adsorption Chromatography. The principle of gas-sohd or Hquid-sohd chromatography may be easily understood from equation 35. In a linear multicomponent system (several sorbates at low concentration in an inert carrier) the wave velocity for each component depends on its adsorption equihbrium constant. Thus, if a pulse of the mixed sorbate is injected at the column inlet, the different species separate into bands which travel through the column at their characteristic velocities, and at the oudet of the column a sequence of peaks corresponding to the different species is detected. 

Examples of ideal binary systems ate ben2ene—toluene and ethylben2ene—styrene the molecules ate similar and within the same chemical families. Thermodynamics texts should be consulted before making the assumption that a chosen binary or multicomponent system is ideal. When pressures ate low and temperatures ate at ambient or above, but the solutions ate not ideal, ie, there ate dissimilat molecules, corrections to equations 4 and 5 may be made ... 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

The local-composition models have hmited flexibility in the fitting of data, but they are adequate for most engineering purposes. Moreover, they are implicitly generalizable to multicomponent systems without the introduction of any parameters beyond those required to describe the constituent binaiy systems. For example, the Wilson equation for multicomponent systems is written ... 

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

Only the Wilson, NRTL, and UNIQUAC equations are suited to the treatment of multicomponent systems. For such systems, the parameters are determined for pairs of species exactly as for binary systems. 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

Cullinan This is an extension of Vignes equation to multicomponent systems ... 

These equations apply also to multicomponent systems, where the enthalpies are found for each phase from the component enthalpies. 

The OZ equation was solved using the closure of Rogers and Young [50], Eq. (42), which for the multicomponent system takes the form... 

Wilson s [77] equation has been found to be quite accurate in predicting the vapor-liquid relationships and activity coefficients for miscible liquid systems. The results can be expanded to as many components in a multicomponent system as may be needed without any additional data other than for a binary system. This makes Wilson s and... 

Martinez-Ortiz, J. A., and D. B. Manley, Direct Solution of the Isothermal Gibbs-Duhem Equation for Multicomponent Systems, Ind. Eng. Chem. Process Des. Dev., 17, 3, (1978) p. 346. 

The equation of Krichevsky and Ilinskaya can readily be extended to high-pressure solutions of a gas in a mixed solvent, as shown by O Connell (01) and discussed briefly by Orentlicher (03). This extension makes it possible to predict the behavior of simple multicomponent systems using binary data only. 

Equivalent expressions for equations (3) and (6) exist for the polymer (7). The Flory-Huggins expressions can also be extended to multicomponent systems (2) ... 

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 Fenske equation (Fenske, 1932) can be used to estimate the minimum stages required at total reflux. The derivation of this equation for a binary system is given in Volume 2, Chapter 11. The equation applies equally to multicomponent systems and can be written as ... 

These equations can be solved simultaneously with the material balance equations to obtain x[, x, xf and x1,1. For a multicomponent system, the liquid-liquid equilibrium is illustrated in Figure 4.7. The mass balance is basically the same as that for vapor-liquid equilibrium, but is written for two-liquid phases. Liquid I in the liquid-liquid equilibrium corresponds with the vapor in vapor-liquid equilibrium and Liquid II corresponds with the liquid in vapor-liquid equilibrium. The corresponding mass balance is given by the equivalent to Equation 4.55 ... 

In fact, the development of the equation does not include any assumptions limiting the number of components in the system. It can also be written for a multicomponent system for any two reference Components i and j1 r> 8 ... 

Equation (10.8) can be readily generalized to multicomponent systems. The only difference is that the number of particles of species j in each of the two regions, Mj and Nu,j replace Nj and Nu, respectively. In simulations of multicomponent systems dilute in one component, it is possible that the number of particles of a species in one of the two regions becomes zero after a successful transfer out of that region. Equation (10.8) in this case is taken to imply that the probability of transfer out of an empty region is zero. 

Brinkley (1947) published the first algorithm to solve numerically for the equilibrium state of a multicomponent system. His method, intended for a desk calculator, was soon applied on digital computers. The method was based on evaluating equations for equilibrium constants, which, of course, are the mathematical expression of the minimum point in Gibbs free energy for a reaction. 

Fortunately, few of these variables are truly independent. Geochemists have developed a variety of numerical schemes to solve for equilibrium in multicomponent systems, each of which features a reduction in the number of independent variables carried through the calculation. The schemes are alike in that each solves sets of mass action and mass balance equations. They vary, however, in their choices of thermodynamic components and independent variables, and how effectively the number of independent variables has been reduced. 

This choice of basis follows naturally from the steps normally taken to study a geochemical reaction by hand. An aqueous geochemist balances a reaction between two species or minerals in terms of water, the minerals that would be formed or consumed during the reaction, any gases such as O2 or CO2 that remain at known fugacity as the reaction proceeds, and, as necessary, the predominant aqueous species in solution. We will show later that formalizing our basis choice in this way provides for a simple mathematical description of equilibrium in multicomponent systems and yields equations that can be evaluated rapidly. 

Having derived a set of equations describing the equilibrium state of a multicomponent system and devised a scheme for solving them, we can begin to model the chemistries of natural waters. In this chapter we construct four models, each posing special challenges, and look in detail at the meaning of the calculation results. 

Because it is based on chemical reactions, the double layer model can be integrated into the equations describing the equilibrium state of a multicomponent system, as developed in Chapter 3. The basis appears as before (Table 3.1),... 

The Underwood and Fenske equations may be used to find the minimum number of plates and the minimum reflux ratio for a binary system. For a multicomponent system nm may be found by using the two key components in place of the binary system and the relative volatility between those components in equation 11.56 enables the minimum reflux ratio Rm to be found. Using the feed and top compositions of component A ... 

Assuming the liquid phases remain immiscible, the modelling approach for multicomponent systems remains the same, except that it is now necessary to write additional component balance equations for each of the solutes present, as for the multistage extraction cascade with backmixing in Section 3.2.2. Thus for component j, the component balance equations become... 

---