McCabe-Thiele analysis

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]. 

And in the limiting case when the mass transfer resistances are small and St Pe, 

It is important to know that the HETP in SMB H ) and in elution chromatography (H) are not identical, although both are related to the combined effects of axial dispersion and the mass transfer resistances. According to the above equation 17.46, H depends on the flow rate ratio. 

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Figure 31. McCabe-Thiele analysis of equilibrium stages for the ethanol-water system.

Like distillation, the McCabe-Thiele analysis is strictly valid only for a binary system. However, only two components are usually present at significant concentrations within each individual section of the coliunn (and, besides, in practice, the SMB process is essentially used to separate binary mixtrues). A preliminary analysis in which each section is considered as a pseudo binary McCabe-Thiele system can therefore provide useful guidance in the design of a multicomponent adsorption system. 

The equilibrium relationship is given. The slope of the operating line is also given. To use the McCabe-Thiele analysis, one point on the operating line is needed. The exit gas composition is 1% of H2S (y = 0.01) and the entering water does not contain any H2S (x = 0). Since the points represent passing streams on one end of the cascade, they represent one point on the operating line. 

Total reflux is similar to minimum reflux in that it is not usually a real condition. In total reflux, all of the overhead vapor is returned to the column as reflux, and all of the liquid is returned as boilup, so that there are no distillate and bottom flows out of the column. At steady-state, this means that the feed stream flowrate is also zero. Total reflux is used in actual columns during start up and also to test their efficiency. Total reflux is useful in a McCabe-Thiele analysis in order to find the minimum number of stages required for a given separation. 

Figure 4.10 McCabe-Thiele analysis for total reflux and minimum number of plates.

The material balances give a slightly different answer than that obtained from the McCabe-Thiele analysis shown in Figure 4.15. The McCabe-Thiele analysis is only an approximate solution, and it may be that the discrepancy comes from the need for more equilibrium data points to obtain a more accurate equilibrium curve. 

Figure 4.19 McCabe-Thiele analysis for a system with a side stream.

Some extraction systems are such that the solvent and diluent phases are almost completely immiscible in each other. Hence, separation yields an extract phase essentially free of diluent and a raffinate phase that is almost pure diluent. This greatly simplifies the characterization of the system. When partial miscibility for an extraction process is very low, the system may be considered immiscible and application of McCabe-Thiele analysis is appropriate. It is important to note that McCabe-Thiele analysis for immiscible extraction applies to a countercurrent cascade. The McCabe-Thiele analysis for immiscible extraction is analogous to the analysis for absorption and stripping processes. Consider the flow scheme shown in Figure 5.23,... 

All extraction systems are partially miscible to some extent, but when partial miscibility is very low, the system may be treated as completely immiscible and McCabe-Thiele analysis is appropriate. 

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

This linear relationship between the total pressure, P, and the mole fraction, x, of the most volatile species is a characteristic of Raoult s law, as shown in Figure 7.18a for the benzene-toluene mixture at 90°C. Note that the bubble-point curve (P-x) is linear between the vapor pressures of the pure species (at x, = 0, 1), and the dew-point curve (P-yJ lies below it. When the (x, yi) points are graphed at different pressures, the familiar vapor-liquid equilibrium curve is obtained, as shown in Figure 7.18b. Using McCabe-Thiele analysis, it is shown readily that for any feed composition, there are no limitations to the values of the mole fractions of the distillate and bottoms products from a distillation tower. 

Example 4-4. McCabe-Thiele analysis of open steam heatii ... 

Second, we would like to assume CMO so that we can use the McCabe-Thiele analysis procedure. An easy way to check this assunption is to conpare the latent heats of vaporization per mole rHimmelblau. 1974k... 

C. Plan. We will use a McCabe-Thiele analysis. Plot the equilibrium data on a y-x graph. 

Exanq)le 12-1. McCabe-Thiele analysis for dilute absorber... 

McCabe-Thiele Analysis for More Concentrated Systems... 

If absorption or stripping can be assumed 1) to be isothermal and 2) to have negligible heat of absorption, then the energy balances will be satisfied. In this case the McCabe-Thiele analysis procedure can be adapted to more concentrated systems where the total flow rates L and V are not constant. We will have the desired straight operating line if we define... 

The McCabe-Thiele analysis for dilute immiscible extraction is very similar to the analysis for dilute absorption and stripping discussed in Chapter 12. It was first developed by Evans (1934) and is reviewed by Robbins (1997). In order to use a McCabe-Thiele type of analysis we must be able to plot a single equilibrium curve, have the energy balances automatically satisfied, and have one operating line for each section. 

When there is some partial miscibility of diluent and solvent, the McCabe-Thiele analysis can still be used if the following alternative assunption is valid. 

All extraction systems are partially miscible to some extent. When partial miscibility is very low, as for toluene and water, we can treat the system as if it were conpletely immiscible and use McCabe-Thiele analysis or the Kremser equation. When partial miscibility becomes appreciable, it can no longer be ignored, and a calculation procedure that allows for variable flow rates must be used. In this case a different type of stage-by-stage analysis, which is very convenient for ternary systems, can be used. For multiconponent systems, corrputer calculations are required. 

The McCabe-Thiele procedure has been applied to flash distillation, continuous countercurrent distillation, batch distillation, absorption, stripping, and extraction. What are the common factors for the McCabe-Thiele analysis in all these cases ... 

Of course, the assumptions required to use a McCabe-Thiele analysis or the Kremser equation may not be valid for a given separation. If the assumptions are not valid, the results of the analysis could be garbage. To determine the validity of the assumptions, the engineer has to examine each specific case in detail. The more dilute the solute, the more likely it is that the assumptions will be valid. 

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