McCabe-Thiele method determination

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starling from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. 

Hengstebeck [137] presents a simplified procedure for reducing a multicomponent system to an equivalent binary using the key components. From this the number of stages or theoretical plates and reflux can be determined using conventional binary procedures and involving the McCabe-Thiele method. 

For these conditions there are two basic methods for determining the number of plates required. The first is due to Sorel(25) and later modified by Lewis126 , and the second is due to McCabe and Thiele(27). The Lewis method is used here for binary systems, and also in Section 11.7.4 for calculations involving multicomponent mixtures. This method is also the basis of modem computerised methods. The McCabe-Thiele method is particularly... 

Figure 11.15. Determination of number of plates by the McCabe-Thiele method (Example 11.8)...

For a binary mixture under constant pressure conditions the vapour-liquid equilibrium curve for either component is unique so that, if the concentration of either component is known in the liquid phase, the compositions of the liquid and of the vapour are fixed. It is on the basis of this single equilibrium curve that the McCabe-Thiele method was developed for the rapid determination of the number of theoretical plates required for a given separation. With a ternary system the conditions of equilibrium are more complex, for at constant pressure the mole fraction of two of the components in the liquid phase must be given before the composition of the vapour in equilibrium can be determined, even for an ideal system. Thus, the mole fraction yA in the vapour depends not only on X/ in the liquid, but also on the relative proportions of the other two components. 

Equilibrium diagram for benzene-toluene mixtures at total pressure of 760 mm Hg (McCabe-Thiele method for determining number of theoretical plates). 

The feed, at a flow rate of 100 kmol/h, is sent as saturated vapor to the distillation column. The column is equipped with a partial condenser with a vapor product, and a reboiler. For a solvent rate of 500 kmol/h, it is required to determine the required number of equilibrium stages and the optimum feed location for a reflux ratio of 1.5 times the minimum. The McCabe-Thiele method may be used on a solvent-free basis. 

According to the McCabe-Thiele method, the system for separation is considered on a quasi-binary basis. In this approach, it must be possible to neglect the influence of the solvent, which is acceptable if the phase boundary lines (solubilities) do not change much with concentration during separation. In this case, the number of theoretical stages, the minimal reflux (ratio), the minimum number of theoretical stages, and their mutual dependence can be determined. 

Determination of separating stages by the McCabe-Thiele method for equilibrium curves with an inflection point or an azeotropic point... 

For the separation of the liquid feed M (composition x ) into two streams of composition Xg (product, distillate) and Xf, (bottom product), the required number of theoretical plates can be determined graphically with the aid of the McCabe-Thiele method, which was used in the past because no computers were available to solve the extensive systems of equations for mass balances and equilibrium relationships. This method is no longer of practical importance, but it is an excellent didactic aid for understanding the basic principles of rectification. 

Fig. 2.3.4-2 Mass balance of an extraction (a) and application of the McCabe-Thiele method (b) for determining the theoretical number of stages N and the minimum amount of solvent...

Both the Lewis and McCabe-Thiele methods are based on the CMO assunption. When the CMO assumption is valid, the energy balances are automatically satisfied, which greatly sinplifies the stage-by-stage calculations. The reboiler and condenser duties, and Q, are determined from the balances around the entire column. If CMO is not valid, and Q, will be unaffected if the same external reflux... 

Determine the number of stages required, using the Lewis method and the McCabe-Thiele method... 

If the number of theoretical stages N, is determined, (for example by the graphical McCabe-Thiele method or by computation), and n, or HETS for the chosen packing are known, the height Z for mass and heat transfer of the packed column can now be calculated with Eqs. (1-194) or (1-195), respectively. 

For a given rectification problem the minimum reflux ratio v ,jn and the minimum number of separation stages may be determined by the McCabe-Thiele method or calculated by Eqs. (2-74) and (2-77). With these, the required number of theoretical stages for any chosen reflux ratio V is found from the Gilliland diagram. The best economic operating range of rectification columns is... 

S.3.4 McCabe-Thiele Method to Determine the Number of Theoretical Separation Stages... 

With the known course of the equilibrium curve for fixed extraction conditions, the required number of theoretical stages N, is determined by drawing steps between the equilibrium curve and the balance line, analogous to the McCabe-Thiele method (see Chapter 2.S.2.4 and Fig. 6-18). 

Of significance is that multistage membrane separations for binary systems can be treated graphically, in a fashion similar to the classical McCabe-Thiele method for binary distillation. This is developed and illustrated in Chapter 4 and affords a convenient means for evaluating separation possibilities, in determining the effect of permeability, reflux... 

Based on the same equations for the distillation columns design presented in Section 4.3.2, we formulate the problem to determine the number of trays as function of the L/D ratio. We use McCabe-Thiele method to compute the number of theoretical trays, where the equilibrium composition is given by the following equation ... 

To design such a process, the McCabe-Thiele method may be used to determine the number of theoretical separation stages, as examined in Sections 3.3.2-3.3.4 for distillation, absorption (gas scrubbing), and liquid-liquid-extraction. Thus, we obtain the number of theoretical extraction stages of a countercurrent extraction column based on the equilibrium curve (solubility of extract in the solvent for a given content in the solid) and the operating line. The latter depends on the extract content of the solid feed and residue, and on the in- and outlet extract concentration in the solvent The extract content of the feed is fixed, and the value of the residue is specified by the required degree of extraction. The inlet content of the extract in the solvent is also fixed, as either pure solvent is used or the value is specified by separation of the extract from the used solvent after the extraction. Therefore, the only parameter that is left is the outlet concentration of the extract in the solvent, which depends on the ratio of the solvent flow to the feed rate of the solid feedstock (mass balance). 

The number of theoretical stages required can be determined from Figure 11-8 by means of a stepwise procedure similar to the McCabe-Thiele method used in distillation calculations (see Chapter 7). An operating line, QT, first must be constructed below the equilibrium curve, NO, developed from equilibrium data. Line QT can be drawn by... 

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