Stage calculations multicomponent systems

Stage-by-stage calculations for systems with constant relative volatilities are relatively easy, and the resulting profiles illustrate most of the behaviors observed with multicomponent systems. Fortunately, extending the calculation to nonconstant relative volatility systems is not difficult and is discussed in Section 5.4. We return to these calculations in Chapter 8 for total reflux systems (see Example 8-3). 

One of the more widely used methods for calculating the number of theoretical stages in multicomponent systems was developed by Lewis and Matheson [6]. Again, the molar flow rates of vapor and liquid in each section are assumed to be constant. On each equilibrium stage, the summation of the concentrations of all vapor components must equal unity. The same is true for the summation of the concentrations of all liquid components. Further, the vapor and liquid compositions of each component are related by the K value for that component. 

The calculation of single-stage equilibrium separations in multicomponent systems is implemented by a series of FORTRAN IV subroutines described in Chapter 7. These treat bubble and dewpoint calculations, isothermal and adiabatic equilibrium flash vaporizations, and liquid-liquid equilibrium "flash" separations. The treatment of multistage separation operations, which involves many additional considerations, is not considered in this monograph. 

In order to determine the packed height it is necessary to obtain a value of the overall number of transfer units methods for doing this are available for binary systems in any standard text covering distillation (73) and, in a more complex way, for multicomponent systems (81). However, it is simpler to calculate the number of required theoretical stages and make the conversion ... 

In this chapter consideration is given to the theory of the process, methods of distillation and calculation of the number of stages required for both binary and multicomponent systems, and discussion on design methods is included for plate and packed columns incorporating a variety of column internals. 

The general method of stage-by-stage calculation for multicomponent systems was first shown by Lewis and Matheson (LI) and by Underwood (Ul) in 1932. The method of Lewis and Matheson was further improved by Robinson and Gilliland (Rl), but substantially unchanged. In its most basic form, the concept of the method is simple. Consider the example cited above. If the amounts of each component in both of the products could be exactly calculated, it would only be necessary to start at one end and calculate until a stage was reached at which the composition matched that of the other product. 

The lacking special description of the Gibbs phase rule in MEIS that should be met automatically in case of its validity is very important for solution of many problems on the analysis of multiphase, multicomponent systems. Indeed, without information (at least complete enough) on the process mechanism (for coal combustion, for example, it may consist of thousands of stages), it is impossible to specify the number of independent reactions and the number of phases. Prior to calculations it is difficult to evaluate, concentrations of what substances will turn out to be negligibly low, i.e., the dimensionality of the studied system. Besides, note that the MEIS application leads to departure from the Gibbs classical definition of the notion of a system component and its interpretation not as an individual substance, but only as part of this substance that is contained in any one phase. For example, if water in the reactive mixture is in gas and liquid phases, its corresponding phase contents represent different parameters of the considered system. Such an expansion of the space of variables in the problem solved facilitates its reduction to the CP problems. 

The classic papers by Lewis and Matheson [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes [Ind. Eng. Chem., 25, 290 (1933)] represent the first attempts at solving the MESH equations for multicomponent systems numerically (the graphical methods for binary systems discussed earlier had already been developed by Pon-chon, by Savarit, and by McCabe and Thiele). At that time the computer had yet to be invented, and since modeling a column could require hundreds, possibly thousands, of equations, it was necessary to divide the MESH equations into smaller subsets if hand calculations were to be feasible. Despite their essential simplicity and appeal, stage-to-stage calculation procedures are not used now as often as they used to be. 

Can we do the internal stage-by-stage calculations first and then solve the overall balances To begin the stage-by-stage calculation procedure in a distillation column, we need to know all the compositions at one end of the column. For ternary systems with the variables specified as in Table 6.3, these compositions are unknown. To begin the analysis we would have to assume that one of them is known. Therefore, internal calculations for multicomponent distillation problems are necessarily by trial and error. In a ternary system, once an additional composition is assumed, both the overall and internal calculations are easily done. The results can then be compared and the assumed composition modified as needed until convergence is achieved. 

If the assunption of negligible holdup is not valid, then the holdup on each stage and in the accumulator acts like a flywheel and retards changes. A different calculational procedure is required for this case and for multicomponent systems fBarton and Roche. 1997 Diwekar. 1995 Mujtabe, 2004). 

The analysis for cross-flow can easily be extended to multiconponent systems. Now each stage is a perfectly mixed multicomponent system with a feed from the previous stage. The results for each stage can be calculated using the procedure in Section 17.5.5. Thus, a trial-and-error is needed on each stage, but the entire cascade is only calculated once. Coker et al. (1998) illustrate a different procedure for multiconponent cross-flow. 

We have considered only binary distillation to this point because multicomponent systems increase complexity by orders of magnitude. Furthermore, we continue to fight the battle of too little data. In order to properly design a multicomponent column, we would need equilibrium data for the multicomponent system and enthalpy data. The latter is usually not even available for binaries. Even with all of the required data available, the column design would be an extraordinarily complicated calculation requiring a stage-by-stage determination. 

In concentrated systems the change in gas and liquid flow rates within the tower and the heat effects accompanying the absorption of all the components must be considered. A trial-and-error calculation from one theoretical stage to the next usually is required if accurate results are to be obtained, and in such cases calculation procedures similar to those described in Sec. 13 normally are employed. A computer procedure for multicomponent adiabatic absorber design has been described by Feintuch and Treybal [Ind. Eng. Chem. Process Des. Dev., 17, 505 (1978)]. Also see Holland, Fundamentals and Modeling of Separation Processes, Prentice Hall, Englewood Cliffs, N.J., 1975. 

As it follows from the present review, a rather complete and experimentally well-grounded quantitative theory of radical copolymerization of an arbitrary number of monomers has been developed. This theory allows one to calculate various statistical copolymers characteristics using the known values of reactivity ratios. The modern stage of the development of this theory is characterized by new approaches applying, for example, the apparatus of graph theory and theory of the dynamic systems which permit to widen the area of theoretical consideration involving the multicomponent copolymerization at high conversions. 

This chapter introduces how continuous distillation columns work and serves as the lead to a series of nine chapters on distillation. The basic calculation procedures for binary distillation are developed in Chapter 4. Multicomponent distillation is introduced in Chapter 5. detailed conputer calculation procedures for these systems are developed in Chapter 6. and sinplified shortcut methods are covered in Chapter 7. More complex distillation operations such as extractive and azeotropic distillation are the subject of Chapter 8. Chapter 9 switches to batch distillation, which is commonly used for smaller systems. Detailed design procedures for both staged and packed columns are discussed in Chapter 10. Finally, Chapter 11 looks at the economics of distillation and methods to save energy (and money) in distillation systems. 

The previous chapters served as an introduction to multicomponent distillation. Matrix methods are efficient, but they still require a fair amount of time even on a fast conputer. In addition, they are simulation methods and require a known number of stages and a specified feed plate location. Fairly rapid approximate methods are required for preliminary economic estimates, for recycle calculations where the distillation is only a small portion of the entire system, for calculations for control systems, and as a first estimate for more detailed simulation calculations. 

A column comprises individual separation stages in which the purification of the product is carried out by means of the effect that vapor and liquid have different compositions at equilibrium. Accordingly, the column design calls for knowledge of the phase equilibria of the systems [5]. Normally, phase equilibrium calculations are based on binary parameters describing the interactions of two different molecules. If multicomponent mixtures are considered, some of these interactions might be unknown. To obtain better simulation results, they should at least be estimated. This was the main reason for the development of the UNIFAC group contribution method twenty years ago. 

Qualitative leap to the second stage (i.e., to the distillation theory of ideal multicomponent mixtures) was realized by Underwood (1945,1946a, 1946b, 1948). Underwood succeded in obtaining the analytical solution of the system of distillation equations for infinite columns at two important simplifying assumptions -at constant relative volatilities of the components (i.e., which depend neither on the temperature nor on mixture composition at distillation column plates) and at constant internal molar flow rates (i.e., at constant vapor and liquid flow rates at all plates of a column section). The solution of Underwood is remarkable due to the fact that it is absolutely rigorous and does not require any plate calculations within the limits of accepted assumptions. 

Improper feed point location may require more theoretical stages than calculated. For a binary system, the feed point is located where the feed liquid composition matches the downflowing liquid composition in the column. With a multicomponent feed, the theoretical stage that matches the light-key concentration in the feed probably will not match the heavy-key concentration. In such a case, the feed stage should be selected so that the ratio of light key to heavy key in the liquid feed is the same as the ratio in the downflowing liquid within the column. 

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