Multicomponent system, mathematical

Distillation Columns. Distillation is by far the most common separation technique in the chemical process industries. Tray and packed columns are employed as strippers, absorbers, and their combinations in a wide range of diverse appHcations. Although the components to be separated and distillation equipment may be different, the mathematical model of the material and energy balances and of the vapor—Hquid equiUbria are similar and equally appHcable to all distillation operations. Computation of multicomponent systems are extremely complex. Computers, right from their eadiest avadabihties, have been used for making plate-to-plate calculations. 

In contrast to quantitative analyses, the results of qualitative tests and of identifications cannot be evaluated by means of mathematical statistics. Instead, information theory is a helpful tool to characterize qualitative analyses, in particular in case of multicomponent systems. 

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. 

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. 

Separation systems include in their mathematical models various vapor-liquid equilibrium (VLE) correlations that are specific to the binary or multicomponent system of interest. Such correlations are usually obtained by fitting VLE data by least squares. The nature of the data can depend on the level of sophistication of the experimental work. In some cases it is only feasible to measure the total pressure of a system as a function of the liquid phase mole fraction (no vapor phase mole fraction data are available). 

Empirical models are often mathematically simpler than mechanistic models, and are suitable for characterizing sets of experimental data with a few adjustable parameters, or for interpolating between data points. On the other hand, mechanistic models contribute to an understanding of the chemistry at the interface, and are very often useful for describing data from complex multicomponent systems, for which the mathematical formulation (i.e., functional relationships) for an empirical model might not be obvious. Mechanistic models can also be used for interpolation and characterization of data sets in terms of a few adjustable parameters however, mechanistic models are often mathematically more complicated than empirical relationships. 

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

The mathematical evaluation of the stability problem in multicomponent systems is most complicated, even if we assume that local equilibrium prevails at the boundaries. The result is a relation for the concentrations at the boundary of the following form... 

Langmuir isotherm or model Simple mathematical representation of a favorable (type I) isotherm defined by Eq. (2) for a single component and Eq. (4) for a binary mixture. The separation factor for a Langmuir system is independent of concentration. This makes the expression particularly useful for modeling adsorption column dynamics in multicomponent systems. 

Through their parallel and independent efforts on both sides of the Atlantic, which began in the 1950s with mathematically modeling known phase diagrams for unary and binary systems, Kaufman and Hillert are considered founding fathers of the CALPHAD method, the field of computational thermodynamics concerned with the extrapolation of phase diagrams for multicomponent systems. (Source L. P. Kaufman, personal communication, February 08, 2004.)... 

Calculations Assuming Ideal Solution Behavior for Multicomponent Systems. The oaloulation of the bubble-point pressure and the composition of the vapor at the bubble point for an ideal solution consisting of more than two components involves no new principles or procedures. If Raoult s Law is applicable the partial pressure of each component in the vapor can be calculated and their sum is equal to the bubble-point pressure. Stated mathematically... 

Rhee et al. developed a theory of displacement chromatography based on the mathematical theory of systems of quasi-linear partial differential equations and on the use of the characteristic method to solve these equations [10]. The h- transform is basically an eqmvalent theory, developed from a different point of view and more by definitions [9]. It is derived for the stoichiometric exchemge of ad-sorbable species e.g., ion exchange), but as we have discussed, it can be applied as well to multicomponent systems with competitive Langmuir isotherms by introducing a fictitious species. Since the theory of Rhee et al. [10] is based on the use of the characteristics and the shock theories, its results are comprehensive e.g., the characteristics of the components that are missing locally are supplied directly by this theory, while in the /i-transform they are obtained as trivial roots, given by rules and definitions. 

A general mathematical treatment of system peaks and of the closely related method of vacancy chromatography was given by Helfferich and Klein [8]. This work includes a detailed analysis of the phenomena that take place upon injection of a sample into a chromatographic column. It is based on the use of the solution of the ideal model of chromatography for multicomponent systems, with competitive Langmuir isotherms (see Chapters 8 and 9), and of the ft-transform. 

The starting point for studying the diffusion with chemical reactions for multicomponent systems in porous catalyst pellets is to derive the mathematical models that describe the system under study. 

Should it prove impossible to effect a satisfactory separation by changing the distillation pressure, the next step will be to find a suitable additive with which one of the constituents forms a.heteroazeotrope, or a homoazeotrope that is easily split up [34]. An approximate method for separating heteroazeotropic mixtures based on the mathematical model for the liquid-liquid-vapour equilibrium of two multicomponent systems was elaborated by BrU et al. [49a]. 

The terminal model for copolymerization can be naturally extended to multicomponent systems involving three or more monomers. Multicomponent copolymerizations And practical application in many commercial processes that involve three to five monomers to impart different properties to the final polymer (e.g., chemical resistance or a certain degree of crosslinking) [134]. There is a classical mathematical development for the terpolymerization or three-monomer case, the Alfrey-Goldfinger equation (Eq. 6.43)... 

On the basis of the results obtained, one can say that the SEFS results are very useful in analyzing the atomic structure of superthin surface layers of matter. But whenever studies of the local atomic structure can be performed by other methods, for example by EELFS, the complexity of the SEFS technique becomes an important disadvantage. However, in the experimental study of atomic PCF s of surface layers of multicomponent atomic systems within the formalism of the inverse problem solution, a complete set of integral equations is necessary to provide mathematical correctness. This set of equations can be solved by the methods of direct solution only. In this case the use of the SEFS method may be a necessary condition for obtaining a reliable result. Besides, the calculations made can be used as a test when studying multicomponent systems. 

In the first part of this book we developed the principles Qaws) of the thermodynamics and applied them to pure fluids. We now we want to extend their application to multicomponent systems. The thermodynamics laws themselves are general and apply to all systems, whether pure or multicomponent. Their mathematical expressions were developed in Chapter 6 and are summarized below ... 

In principle the enthalpy of vaporization may obtained by integration of the peak area in the curve of heat flow versus temperature. This mathematical calculation will not be precise, since the baseline drifts a certain amount due to the change of the specific heat during the heating. In addition the baseline does not return to the same level after the reaction due to mass loss during the reaction. Exact determination of the integration limits is therefore diffieult, especially with the multicomponent systems (petroleum and its products) described below. 

The mathematical model, which makes it possible to consider the influence of the hydrodynamic conditions of flow on the processes of mixing and chemical transformations of reacting substances in a liquid phase, assumes that the average flow characteristics of a multicomponent system can be described by the equations of continuum mechanics and will satisfy conservation laws. 

Adsorption kinetics of a single particle (activated carbon type) is dealt with in Chapter 9, where we show a number of adsorption / desorption problems for a single particle. Mathematical models are presented, and their parameters are carefully identified and explained. We first start with simple examples such as adsorption of one component in a single particle under isothermal conditions. This simple example will bring out many important features that an adsorption engineer will need to know, such as the dependence of adsorption kinetics behaviour on many important parameters such as particle size, bulk concentration, temperature, pressure, pore size and adsorption affinity. We then discuss the complexity in the dealing with multicomponent systems whereby governing equations are usually coupled nonlinear differential equations. The only tool to solve these equations is... 

We will now address the formulation of a kinetic model for a heterogeneous particle with single component systems first to illustrate the concept of energy distribution, and then logically extend the mathematical formulation to multicomponent systems. 

Longuet-Higgins, H. C. 1951. The statistical thermodynamics of multicomponent systems. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 205, 247. 

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