McCabe-Thiele model

Another variation on the manipulated distillate scheme is to use a setpoint for the steam/feed ratio to establish the separation power base. Generally, the feed flow rate signal should be lagged with an 8 to 20 min capacitance lag (filter), so the steam flow is proportional to a trailing average of the feed rate. Sometimes, this falls into the category of model-based predictive control because the McCabe-Thiele model, or a computer simulation model shows that the separation power base can be established by the steam/feed ratio as shown in Table 3.3. 

Rather than using a continuous model and representing the system in terms of the dispersion model, one may consider each bed as equivalent to a number of theoretical equilibrium stages. A McCabe-Thiele diagram then provides a convenient representation of the system and shows clearly the effects of the process variables [10,30]. 

One approach for selecting the operation conditions consists in applying a McCabe-Thiele-like analysis to an ideal stage-by-stage model of the unit [13]. This approach can be applied to systems described by any kinds of isotherm but it is limited to binary separations. 

The McCabe-Thiele analysis can be used to model physical absorption and stripping processes that use equilibrium stages. 

Figure 1-13 Model 1. Assumed in the McCabe-Thiele method.

Fig. 4.6 Binary data fit for the six binary subsystems of the system methanol + acetic acid + methyl acetate + water at 1 bar with two variants of the UNIQUAC model. (McCabe Thiele Diagrams for p = 1 bar in mole fractions of the component appearing first in the system title)...

This problem was also run on the Aspen Plus process simulator (see Problem 4.G1 and chapter appendix). Aspen Plus does not assume CMO and with an appropriate vapor-liquid equilibrium (VLE) correlation (the nonrandom two-liquid model was used) should be more accurate than the McCabe-Thiele diagram, which assumes CMO. With 5 equilibrium stages and feed on stage 4 (the optimum location), = 0.9335 and Xg = 0.08365, which doesn t meet the specifications. With 6 equilibrium stages and feed on stage 5 (the optimum), Xq = 0.9646 and Xg = 0.0768, which is slightly better than the specifications. The differences in the McCabe-Thiele and process simulation results are due to the error involved in assuming CMO and, to a lesser extent, differences in equilibrium. 

The separation power base in the classic McCabe-Thiele graphical model of a binary distillation column is established by the reflux ratio, R/D, which is the ratio of the reflux flow rate divided by the distillate flow rate. For example, with a distillation column that is fed 1,000 kg/h of feed that produces 85 kg/h of distillate with 425 kg/h of reflux, the reflux ratio is 425/85 = 5. A minimum reflux ratio is required to achieve the desired separation with an infinite number of theoretical stages. The maximum reflux ratio, called total reflux, with zero distillate flow rate can be used in design calculations to determine the minimum number of theoretical stages required to achieve a desired separation. 

The simulation results presented below are based on the 26th-order model in Table G.l that includes variable liquid holdups in the reactor, the distillate receiver, and the reboiler, but not on the individual equihbrium stages. Thus, the dynamic column model reduces to the steady-state model used in the McCabe-Thiele analysis (Seader and Henley, 1997) if the column accumulation terms are all zero. With a nominal internal reflux ratio (yiD) of 1.455,J he column yields a separation concentration ratio, S = xjylxs, of 90.5. 

There are two types of extraction processes, namely, leaching where solids are extracted and liquid-liquid extraction. The commonly used leaching equipment is the extraction battery where counter current flow is established resulting in a number of extracting stages. This can be modeled as equilibrium process. The McCabe-Thiele procedure can be used to model this process. In liquid-liquid extraction, ternary diagrams and tie lines are used to determine the extent of extraction. Batch extractive distillation is a commonly used extractive method for liquid-liquid extraction. 

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. 

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