Cascade of equilibrium stages

A countercurrent multistage extraction system is shown below, which is to be modelled as a cascade of equilibrium stages. 

Figure 9.1 A cascade of equilibrium stages with refluxing and reboiling. (From Smith R and Jobson M, 2000, Distillation, Encyclopedia of Separation Science, Academic Press reproduced by permission).

Suppose one wanted to remove more spices by this same method. The oil and spices phase from the first equilibrium stage could be mixed with more vinegar. This is the idea behind a cross-flow cascade of equilibrium stages. 

This concerns essentially steps 2 and 4 of the design procedure from Table 2 and can be restricted to the basic fractionating configuration in Fig. 5. In the following, we will approximate each of the counter-current sections by a cascade of equilibrium stages, as is shown in Fig. A-1. 

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... 

This will also be the mixing point for the two product streams L and Vj, although these streams are not coming from the same stage (i.e. they contain the same material as in the two feed streams). Assuming all the stages in the cascade are equilibrium stages, we also know that the points L and Vj lie somewhere on the equilibrium curve, but not necessarily on opposite ends of the... 

Extraction involves the transfer of components between two liquid phases, much as absorption or stripping involves the transfer of components from liquid to vapor phase or vice versa. As in vapor-liquid multistage separation processes, the device employed to carry out liquid-liquid extraction is usually a counterflow column that performs the function of a number of equilibrium stages interconnected in counterflow configuration. In each stage, two inlet liquid streams mix, reach equilibrium, and separate into two outlet liquid streams. As in vapor-liquid columns, the lack of complete equilibrium in liquid-liquid extractors is accounted for by some form of tray efficiency. Liquid-liquid extraction may also be carried out in a cascade of mixing vessels connected in series in counterflow. 

The normal method of cascading the equilibrium stages is using countercurrent flow. With two streams, they would enter the cascade at opposite ends of the cascade. 

Another important concept is the calculation of the minimum number of equilibrium stages to perform a given separation. This is a theoretical concept since it occurs when one or both flowrates in a countercurrent cascade are infinite. It is useful to calculate since the actual number of stages will be larger (i.e., this is a limit). An analytical equation can be... 

The case of minimum number of equilibrium stages corresponds to infinite flow from and back into the cascade, such that no product is removed. In this case... 

Equilibrium processes are limited by the degree of separation in a single contact, so cascades of contact stages are used to achieve a higher degree of separation. 

The minimum number of equilibrium stages is a theoretical concept based upon infinite flows within a cascade and no feed or product streams. 

The calculation of the concentration of extractable components in a countercurrent cascade of equilibrium solvent extraction stages is first developed for the simple countercurrent extraction section of Fig. 4.3. The theory is then extended to the extracting-scrubbing system of Fig. 4.4 for fractional extraction and is illustrated by a numerical calculation for the separation of zirconium from hafnium, using TBP in kerosene as solvent. 

Assume that the cascade is to reduce the concentration of the extractable component from x to X] by extraction with organic of relative volume /F. The point Xi, yo is thereby specified. Beginning at Xi, jvo and projecting upward in vertical and horizontal steps, the compositions for all of the other equilibrium stages are determined. The number of vertical projections between the operating line and equilibrium line necessary to step from Xi to x gives the required number of equilibrium stages. 

Thus, by specifying 0 for the cascade and the ratio yolDx and recovery p for any one of the extractable components, the required number of equilibrium stages N can be calculated from (4.48). 

Figure 4.11 Number of equilibrium stages in an extraction cascade. (Adapted from Sherwood et aL [S3. )...

Apply the McCabe-Thiele graphical method to determine the number of equilibrium stages required for a given separation and the optimal location along the cascade for introduction of the feed. 

The presentation of equilibrium-stage calculation techniques is preceded by Chapter 6, which covers an analysis of the variables and the equations that relate the variables so that a correct problem specification can be made. Single-stage calculation techniques are then developed in Chapter 7, with emphasis on so-called isothermal and adiabatic flashes and their natural extension to multistage cascades. 

This relationship describes the operating line of multistage distillation. It formulates a relation between hquid and gas concentration within the cascade. In Fig. 5.2-3, the operating line and the equilibrium line are plotted in an y/x diagram. The distance between these two lines represents the driving force of the interfacial mass transfer. Distance and length of these lines indicate the difficulty of the separation performed in the cascade. It can be expressed either by the number of equilibrium stages or by the number of transfer units. 

Acetone can be removed from acetone-air mixtures using simple countercurrent cascades, by adsorption onto charcoal (Foust et al. 1980). We wish to find the required number of equilibrium stages to reduce a gas stream carrying 0.222 kg acetone per kg air to a value 0.0202 kg acetone per kg air. Qean charcoal = 0) enters the system at 2.5 kg/sec, and the air rate is constant at 3.5 kg/sec. Equilibrium between the solid and gas can be taken to obey the Langmuir-type relationship... 

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