Mass Transfer in Multicomponent System

In practice, most of the mass transfer processes involve multicomponent, and the mass transfer rate should be calculated individually by each component. In some cases for simplifying the calculation, two influential components, called key components, are taken as if a two-component system. However, such simplification may lead to serious error, and the rigorous method is preferable. The calculation of multicomponent mass transfer is by the aid of Maxwell-Stefan equation which is introduced briefly in the section below. 

Most of the multicomponent systems are non-ideal. From thermodynamic viewpoint, the transfer of mass species i at constant temperature and pressure from one phase to the other in a two-phase system is due to existing the difference of chemical potential 7t, x p between phases, in which /t,- p =p + T Fln where y, is the activity coefficient of component i is p at standard state. In other words, for a gas (vapor)-liquid system, the driving force of component i transferred from gas phase to the adjacent liquid phase along direction z is the 

The transfer (diffusion) of component i from gas phase to the liquid-phase should overcome the resisting force from the adjacent component xj in the gas phase, such resistance is represented by the frictional force between two fluid molecules, which is proportional to the velocity difference (m, — uj) and the activity of component j (denoted by aj, aj = yjXj). If the system under consideration contains moles of species i per m, then the balance between driving force and resisting force on m (one cubic meter) basis is as follows [37]  

Multiplying foregoing equation by X( and noting that molar mass flux of component i is equal to Ni = = Ji + XiNu we obtain  

The Maxwell-Stefan dififusivity D, , obeys the Onsager reciprocal relation of irreversible thermodynamics, i.e., 

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Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

In fact, through use of matrix models of mass transfer in multicomponent systems (as opposed to effective diffusivity methods) it is possible to develop methods for estimating point and tray efficiencies in multicomponent systems that, when combined with an equilibrium stage model, overcome some of the limitations of conventional design methods. The purpose of this chapter is to develop these methods. We look briefly at ways of solving the set of equations that model an entire distillation column and close with a review of experimental and simulation studies that have been carried out with a view to testing multicomponent efficiency models. 

In any event, we hope it is now well understood that mass transfer in multicomponent systems is described better by the full set of Maxwell-Stefan or generalized Fick s law equations than by a pseudobinary method. A pseudobinary method cannot be capable of superior predictions of efficiency. For a simpler method to provide consistently better predictions of efficiency than a more rigorous method could mean that an inappropriate model of point or tray efficiency is being employed. In addition, uncertainties in the estimation of the necessary transport and thermodynamic properties could easily mask more subtle diffusional interaction effects in the estimation of multicomponent tray efficiencies. It should also be borne in mind that a pseudobinary approach to the prediction of efficiency requires the a priori selection of the pair of components that are representative of the... 

Diffusion and mass transfer in multicomponent systems are described by systems of differential equations. These equations are more easily manipulated using matrix notation and concepts from linear algebra. We have chosen to include three appendices that provide the necessary background in matrix theory in order to provide the reader a convenient source of reference material. Appendix A covers linear algebra and matrix computations. Appendix B describes methods for solving systems of differential equations and Appendix C briefly reviews numerical methods for solving systems of linear and nonlinear equations. Other books cover these fields in far more depth than what follows. We have found the book by Amundson (1966) to be particularly useful as it is written with chemical engineering applications in mind. Other books we have consulted are cited at various points in the text. 

Stewart WE, Prober R. Matrix calculation of multicomponent mass transfer in isothermal systems. Ind Eng Chem Fundam 1964 3 224-235. 

There is a large body of literature that deals with the proper definition of the diffusivity used in the intraparticle diffusion-reaction model, especially in multicomponent mixtures found in many practical reaction systems. The reader should consult references, e.g.. Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley Sons, New York, 2002 Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993 and Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, 1997. 

In this section we review the experimental studies that have been carried out with a view to testing models of multicomponent condensation. There is a great shortage of experimental data on mass transfer in multicomponent vapor (plus inert gas)-liquid systems. Most published works deal with absorption (or condensation or evaporation) of a single species in the presence of a nontransferring component. Thus, this review is necessarily brief. 

M.4 Frey (1986) has analyzed mass transfer within the liquid phase in an ion exchanger. He presents a detailed comparison of the approaches of Krishna and Standart (1976a), Toor (1964a) and Stewart and Prober (1964), Van Brocklin and David (1972), and Tunison and Chapman (1976). Try to reproduce the results of Frey and satisfy yourself of the need for rigorous modeling of mass transfer in multicomponent ionic systems. 

Krishna, R. and Standart, G.L., Mass and Energy Transfer in Multicomponent Systems, Chem. Eng. Commun., 3, 201-275 (1979). 

Chemical engineers frequently have to deal with multicomponent mixtures that is, systems containing three or more species. Conventional approaches to mass transfer in multicomponent mixtures are based on an assumption that the transfer flux of each component is proportional to its own driving force. Such approaches are valid for certain special cases. 

W. E. Stewart, R. Prober, Matrix Calculation of Multicomponent Mass Transfer in Isothermal Systems, Ind. Eng. Chem. Fundam., 1964, 3, 224-235. 

The mass transfer coefficient of two-component system is the basic information necessary for the prediction of mass transfer rate in the process. The calculation of mass transfer for multicomponent system is also based on the mass transfer coefficients of the correspondent binary pairs (see Sect. 4.1.3). 

Curtiss CF (1968) Symmetric gaseous diffusion coefiBcients. J Chem Phys 49(7) 2917-2919 Curtiss CF, Bird RB (1999) Multicomponent diffusion. Ind Eng Chem Res 38 2515-2522 Curtiss CF, Bird RB (2001) Additions and corrections. Ind Eng Chem Res 40 1791 Cussler EL (1997) Diffusion mass transfer in fluid systems, 2nd edn. Cambridge University Press, Cambridge... 

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. 

MULTICOMPONENT FILM MODEL FOR MASS TRANSFER IN NONIDEAL FLUID SYSTEMS... 

Using an entirely different approach to the modeling of multicomponent mass transfer in distillation (an approach that we describe in Chapter 14), Krishnamurthy and Taylor (1985c) found significant differences in design calculations involving nonideal systems. For an almost ideal system (a hydrocarbon mixture), pseudobinary methods were found to be essentially equivalent to a more rigorous model that accounted for diffusional interaction effects. 

In this textbook we have eoneentrated our attention on mass transfer in mixtures with three or more species. The rationale for doing this should be apparent to the reader by now multicomponent mixtures have characteristics fundamentally dijferent from those of two component mixtures. In fact, a binary system is peculiar in that it has none of the features of a general multicomponent mixture. We strongly believe that treatments of even binary mass transfer are best developed using the Maxwell-Stefan equations. We hope that this text will have the effect of persuading instructors to use the Maxwell-Stefan approach to mass transfer even at the undergraduate level. 

Discuss how the fundamental models of mass transfer in Sections 12.1.7 (binary systems) and 12.2.4 (multicomponent systems) may be used to estimate mass transfer rates for use in a nonequilibrium simulation of an existing distillation column. Your essay should address the important question of how the model parameters are to be estimated. 

The 15 chapters fall into three parts. Part I (Chapters 1-6) deals with the basic equations of diffusion in multicomponent systems. Chapters 7-11 (Part II) describe various models of mass and energy transfer. Part III (Chapters 12-15) covers applications of multicomponent mass transfer models to process design. 

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation of rates of mass and energy transport in multicomponent systems. Multicomponent mass transfer coefficients are defined in Chapter 1, Chapter 8 develops the multicomponent film model, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considers models based on turbulent eddy diffusion. Chapter 11 shows how the additional complication of simultaneous mass and energy transfer may be handled. 

We also feel that portions of the material in this book ought to be taught at the undergraduate level. We are thinking, in particular, of the materials in Section 2.1 (the Maxwell-Stefan relations for ideal gases). Section 2.2 (the Maxwell-Stefan equations for nonideal systems). Section 3.2 (the generalized Fick s law). Section 4.2 (estimation of multicomponent diffusion coefficients). Section 5.2 (multicomponent interaction effects), and Section 7.1 (definition of mass transfer coefficients) in addition to the theory of mass transfer in binary mixtures that is normally included in undergraduate courses. 

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