Equilibrium stage

Staged contact of two phases (a) equilibrium stage (b) non-equilibrium or actual stage. 

This chapter is divided into two parts In the first, we take up the topic of equilibrium stages in their various configurations and apply them to a number of different mass transfer operations. The second part, which is less extensive, considers the effect of mass transfer resistance expressed through an appropriate stage efficiency. 

Separation calculations for distillation are based on the concept of a process that involves contacting of vapor and liquid in one or more equilibrium stages. In this process one or more feed streams enter a stage, and one or more streams leave the stage. Energy may be added to or withdrawn from the stage. Importandy, thermodynamic equilibrium is required to exist on the stage, and this concept has been discussed in the previous section (e.g.. Fig. S.2-1). 

the summation of mole fractions must equal unity  

Finally, an enthalpy balance must prevail on the stage  

FIGURE 53-1 Component mass flows on a contacting stage. 

FIGURE 5.3-1 Comporeem mass flows on a cor I acting stage. 

These four relationships, the M-E-S-H eqneiions. are the critical ones fur equilibrium stage calculations in distillation, and they are sufficient if the pressure effect in Eqs. (5.3-3), (5.3-6), and (5.3-7) can be determined deductively. Otherwise, momentum balance eqnetions must be added. 

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Smith, B. D., "Design of Equilibrium Stage Processes," McGraw-Hill, New York (1963). 

At best only one equilibrium stage achievable for each WFE or SPE unit separation efficiency decreases as pressure decreases. 

Fig. 21. Single absorption equilibrium-stage diagram where the equiUbrium curve is for 8% SO2, 12.9% the diagonal lines represent the adiabatic temperature rise of the process gas within each converter pass the horizontal lines represent gas cooling between passes, where no appreciable conversion...

Fig. 7. Distillation column with stacked multiple equilibrium stages. Terms are defined in text.

E. J. Henley and. D. Seader, Equilibrium-Stage Separation Operations in ChemicalEngineering ]ohn. Wiley Sons, Inc., New York, 1981. 

N Number of equilibrium stages X Mole fraction in liquid ... 

A simple equilibrium stage (no feed or sidestreams) is depicted in Fig. 13-22. Four mass streams and a heat-leak (or heat-addition) stream provide the following number of variables ... 

The results of the analyses for all the various elements commonly encountered in distillation processes are summarized in Table 13-5. Details of the analyses are given by Smith (Design of Equilibrium Stage Processes, McGraw-Hul, New York, 1967) and in a somewhat different form by Henley and Seader (op. cit.). 

The simple absorber column shown in Fig. 13-23 will be analyzed here to illustrate the procedure. This unit consists of a series of simple equilibrium stages orthe type in Fig. 13-22. Specification of the number of stages N utilizes the single repetition variable and... 

FIG. 13-30 Material-balance envelope around the bottom end of the column. The partial rehoiler is equilibrium stage 1,... 

The application of a 50 percent Murphree vapor-phase efficiency on a y-x magram is illustrated in Fig. 13-40. A pseudo-equilibrium cui ve is drawn halfway (on a vertical line) between the operating hnes and the true-equilibrium cui ve. The true-equilibrium cui ve is used for the first stage (the partial reboiler is assumed to be an equilibrium stage), but for 1 other stages the vapor leaving each stage is assumed to approach the equilibrium value only 50 percent of me way Consequently, the steps in Fig. 13-40 represent actual trays. 

Apphcation of a constant efficiency to each stage as in Fig. 13-40 will not give, in general, the same answer as obtained when the number of equilibrium stages (obtained by using the true-equilibrium cui ve) is divided by the same efficiency factor. 

However, the total number of equilibrium stages N, N/N,n, or the external-reflux ratio can be substituted for one of these three specifications. It should be noted that the feed location is automatically specified as the optimum one this is assumed in the Underwood equations. The assumption of saturated reflux is also inherent in the Fenske and Underwood equations. An important limitation on the Underwood equations is the assumption of constant molar overflow. As discussed by Henley and Seader (op. cit.), this assumption can lead to a prediction of the minimum reflux that is considerably lower than the actual value. No such assumption is inherent in the Fenske equation. An exact calculational technique for minimum reflux is given by Tavana and Hansen [Jnd. E/ig. Chem. Process Des. Dev., 18, 154 (1979)]. A computer program for the FUG method is given by Chang [Hydrocarbon Process., 60(8), 79 (1980)]. The method is best applied to mixtures that form ideal or nearly ideal solutions. 

Equations (13-40) and (13-41) are plotted in Fig. 13-43. Components having large values of or absorb or strip respectively to a large extent. Cooresponding values of and approach a value of 1 and are almost independent of the number of equilibrium stages. 

An estimate of the minimum absorbent flow rate for a specified amount of absorption from the entering gas of some key component K for a cascade with an infinite number of equilibrium stages is obtained from Eq. (13-40) as... 

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

Naplitali-Sandholm SC Method This method employs the equilibrium-stage model of Figs. 13-48 and 13-49 but reduces the number of vari les by 2N so that only N(2C + 1) equations in a hke number of unknowns must be solved. In place of Vj, Lj, Xij, and iji j, component flow rates are used according to their definitions ... 

When it is desired to compute, with rigorous methods, actual rather than equilibrium stages, Eqs. (13-69) and (13-94) can be modified to include the Murphree vapor-phase efficiency T ij, defined by Eq. (13-29). This is particularly desirable for multistage operations involving feeds containing components of a wide range ol volatility and/or concentration, in which only a rectification (absorption) or stripping action is provided and all components are not sharply separated. In those cases, the use of a different Murphree efficiency for each component and each tray may be necessary to compute recovery accurately. 

The iC values (vapor-liquid equihbrium ratios) in Equation (13-123) are estimated from the same equation-of-state or aclivity-coefficient models that are used with equilibrium-stage models. Tray or packed-section pressure drops are estimated from suitable correlations of the type discussed by Kister (op. cit.). 

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