McCabe-Thiele method equilibrium stage

For the first time through a liqmd-liquid extrac tion problem, the right-triangular graphical method may be preferred because it is completely rigorous for a ternary system and reasonably easy to understand. However, the shortcut methods with the Bancroft coordinates and the Kremser equations become valuable time-savers for repetitive calculations and for data reduction from experimental runs. The calculation of pseudo inlet compositions and the use of the McCabe-Thiele type of stage calculations lend themselves readily to programmable calculator or computer routines with a simple correlation of equilibrium data. 

One way of calculating the number of equilibrium stages (or number of theoretical plates, NTP) for a mass exchanger is the graphical McCabe-Thiele method. To illustrate this procedure, let us assume that over the operating range of compositions, the equilibrium relation governing the transfer of the pollutant from the... 

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. 

The McCabe-Thiele method uses a graphical approach, where the equilibrium data, the operating lines, and the stepping of stages are done graphically. In this example, where the relative volatility is assumed constant, a numerical approach may be implemented. 

To apply the McCabe-Thiele method to extraction, the equilibrium data are shown on a rectangular graph, where the mass fraction of solute in the extract or V phase is plotted as the ordinate and the mass fraction of solute in the raffinate phase as the abscissa. For a type I system, the equilibrium line ends with equal compositions at the plait point. The use of only one concentration to characterize a ternary mixture may seem strange, but if the phases leaving a given stage are in equilibrium, only one concentration is needed to fix the compositions of both phases. 

Extraction, liquid-liquid, 8,10,13 equilibrium diagrams, 103 equipment, 76-84 extract reflux, 415 group method, 475-480 infinite stages, 415 McCabe-Thiele method, 315-318... 

Calculate by the Fenske equation the number of equilibrium stages required. Compare the result to that calculated from the McCabe-Thiele method. 

The equilibrium curve reveals the relationship between the concentration of the liquid, Xb, and the corresponding vapour concentration, y, in equilibrium. Thus the equilibrium curve provides the basis for the calculation of separating stages by the McCabe-Thiele method [77] which is no doubt most commonly used due to its... 

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

In this case it is not possible to derive the required number of separating stages by the McCabe-Thiele method because, firstly, the equilibrium curve is too flat and, secondly, extreme concentrations are involved, as shown by the following example ... 

Both absorption and stripping can be operated as equilibrium stage operations with contact of liquid and vapor. Since distillation is also an equilibrium stage operation with contact of liquid and vapor, we would expect the equipment to be quite similar. This is indeed the case both absorption and stripping are operated in packed and plate towers. Plate towers can be designed by following an adaptation of the McCabe-Thiele method. Packed towers can be designed by use of HETP or preferably by mass transfer considerations (see Chapter 16T... 

Equilibrium data must be converted to ratio units, Y vs. X. These values can be greater than 1.0, since Y = y/(l - y) and X = x/(l - x). The Y = X line has no significance in absorption. As usual the stages are counted at the equilibrium curve. A minimum S/G ratio can be defined as shown in Figure 12-9. If the system is not isothermal, the operating line will not be affected, but the equilibrium line will be. Then the McCabe-Thiele method must be modified to include changing equilibrium curves. 

The McCabe-Thiele method is based on the idea of theoretical stages (theoretical separation unit, theoretical tray). From Chapter 1.1, a theoretical stage is that part of a separation apparatus in which heat and/or mass transfer occurs between two phases in contact. Both phases leaving the theoretical separation unit are at phase equilibrium. 

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). 

Minimum reflux, by definition, is that condition whereby a zone of an infinite number of stages exists immediately on each side of the feed location. In the McCabe-Thiele method of graphical representation and calculation, both operating lines intersect with the locus of the feed partitioning, all at a point on the fC-value plot (or equilibrium curve). It is therefore impossible to calculate away from this point of intersection, in either direction. 

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... 

Figure 8.1.20. Vapor-liquid equilibrium curve, operating lines for enriching section and stripping section and construction of ideal stages and q Une for a feed consisting of vapor and liquid in a multistage distillation column having ideal equilibrium stages the McCabe-Thiele method.

FIGURE 8.4 Data from Table 8.1 plotted as an y-x plot for the more volatile species (which is in this case is acetone, which boils at 56.15°C, compared to 100.0°C for water). The data points from the table are shown, and a simple interpolation curve is added. The 45° line represents the condition ya Xa- This line is customarily shown because the equilibrium curve and this line interact in the McCabe-Thiele method of calculation of equilibrium stages in distillation. The plot is aU for 1 atm pressure (as are most such plots). The temperatures from Table 8.1 are not shown. 

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