Key and Nonkey Components

This topic is best illustrated using an example. The example used here is the depropanizer described by King (7), which has previously been analyzed by Jenny (20), Hengstebeck (21), and Edmister (22). 

Example 2.4 A material balance for the column is shown in Table 2.7. Tbe column operates at a pressure of 315 peia. The feed is 66 percent vapor at the column inlet- Tlie relative volatilities of the components at 206°F (feed plate temperature) are shown in Table 2.3. The column is equipped with a partial condenser, and the reflux ratio is 1.5. It is required to determine the number of theoretical stages. 

Key components are the two components in the feed mixture whose separation is specified. The more volatile of these components is the light key. and the less volatile is the heavy key. In Example 2.4, propane is the light key and n-butane the heavy key. Other components are termed turnkeys. The nonkeys which are more volatile than the keys are termed light nonkeys (methane and ethane in Example 2.4), and those less volatile are heavy nonkeys (pentane and hexane in Example 2.4). 

The key components appear to a significant extent in both overhead and bottom products. Light nonkeys end up almost exclusively in the overhead product, and heavy nonkeys end up almost exclusively in the bottom product, 

In many separations, components are present whose relative volatilities are intermediate betwsen the light key and the heavy key. These components are termed intermediate keys or distributed keys. Intermediate keys are split between the top and bottom products. 

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The mechanism by which nonkey components affect a given separation is more complex in practice than the broad arguments presented here. There are complex interrelationships between the volatility of the key and nonkey components, etc. Although the argument presented is thus not rigorous, it is broadly correct. 

Figure 21.11 The overall flowrate decomposed into key and nonkey components. (From Smith R and Linnhoff B, 1988, Trans IChemE ChERD, 66 195 reproduced by permission of the Institution of Chemical Engineers.)...

The mechanism by which nonkey components affect a given separation is more complex in practice than the broad arguments presented here. There are complex interrelationships between the volatility of the key and nonkey components, and so on. Also, it is often the case that the distillations system has constraints to prevent certain heat integration opportunities. Such constraints will often present themselves as constraints over which the pressure of the distillation columns will operate. For example, it is often the case that the maximum pressure of a distillation column is restricted to avoid decomposition of material in the reboiler. This is especially the case when reboiling high molar mass material. Distillation of high molar mass material is often constrained to operate under vacuum conditions. Clearly, if the pressure of the distillation column is constrained, then this restricts the heat integration opportunities. Another factor that can create... 

Consider a mixture consisting of the isomers C-C4H8, t-C4H8, and the dimer C8Hi6- In order to determine which are the key elements, and the key and nonkey components, we first construct the molecular matrix as... 

Contrary to the key components, which occur in pivot columns, the reactions in pivot columns are the nonkey ones. In the present example, the second and third reactions are key (nonpivot) and the first is nonkey (pivot). As before, the ordering of the reactions will influence the choice of key and nonkey reactions, and the earlier a reaction occurs in the ordering, the more likely it is not to be a key reaction. Again, we can count the number of pivot columns in two ways on the one hand, because the matrix was augmented with a unit matrix, it equals the number of reactions. On the other hand, each pivot column either corresponds to a key component or to a nonkey reaction. Consequently,... 

In general, the flow of key components is constant and independent of the sequence, while the flow of nonkey components varies according to the choice of sequence, as illustrated in Fig. 5.8. 

The (x, i )), values in Eq. (13-37) are minimum-reflux values, i.e., the overhead concentration that would be produced by the column operating at the minimum reflux with an infinite number of stages. When the light key and the heavy key are adjacent in relative volatihty and the specified spht between them is sharp or the relative volatilities of the other components are not close to those of the two keys, only the two keys will distribute at minimum reflux and the Xi D),n values are easily determined. This is often the case and is the only one considered here. Other cases in which some or all of the nonkey components distribute between distillate and bottom products are discussed in detail by Henley and Seader (op. cit.). 

To solve Equation 9.51, it is necessary to know the values of not only a ,-j and 9 but also x, d. The values of xitD for each component in the distillate in Equation 9.51 are the values at the minimum reflux and are unknown. Rigorous solution of the Underwood Equations, without assumptions of component distribution, thus requires Equation 9.50 to be solved for (NC — 1) values of 9 lying between the values of atj of the different components. Equation 9.51 is then written (NC -1) times to give a set of equations in which the unknowns are Rmin and (NC -2) values of xi D for the nonkey components. These equations can then be solved simultaneously. In this way, in addition to the calculation of Rmi , the Underwood Equations can also be used to estimate the distribution of nonkey components at minimum reflux conditions from a specification of the key component separation. This is analogous to the use of the Fenske Equation to determine the distribution at total reflux. Although there is often not too much difference between the estimates at total and minimum reflux, the true distribution is more likely to be between the two estimates. 

Find the key components If any components have similar volatilities to one of the keys, and end up in the same product, lump them with the keys. Convert all mole fractions to the equivalent binary [Eqs, (2.46) and (2.47)], An alternative, simpler procedure is to lump all light keys and light nonkeys into a single light pseudocomponent, and all heavy keys and heavy nonkeys into a single heavy pseudocomponent, This procedure (used in Fig, 2.22) is preferred by the author and others (28), Whichever method is preferred, it must be consistently applied. 

The Underwood equation requires a trial-and-error solution and a subsequent material balance to estimate the minimum reflux ratio. First, the unknown, q>, is determined by trial and error, such that both sides of the following equation are equal. The unknown value of q> should lie between the relative volatilities of the light and heavy key components. The key components are those that have their fractional recoveries specified. The most volatile component of the keys is the light key and the least volatile is the heavy key. All other components are referred to as nonkey components. If a nonkey component is lighter than the light key component, it is a light nonkey if it is heavier than the heavy key component it is a heavy nonkey component. 

For multicomponent systems, an approximate value of (he minimum number of stages (at total reflux) may be obtained from the Fanske relationship [Eq. (5.3-28)]. In the use of this relationship for multicomponent mixtures, die mole fractions and die relative volatility refer to die light and heavy keys only. However, values for the nonkey components may be inserted in the equation to determine (heir distribution after the numbet of minimum stages has been determined through the use of the key components. For a more rigorous approach to the determination of minimum stages, see die paper by Chien/... 

DISTRIBUTED AND UNDISTRIBUTED COMPONENTS. A distributed component is found in both the distillate and bottoms products, whereas an undistributed component is found in only one product. The light key and heavy key are always distributed, as are any components having volatilities between those two keys. Components more volatile than the light key are almost completely recovered in the distillate, and those less volatile than the heavy key are found almost completely in the bottoms. Whether such components are called distributed or undistributed depends on the interpretation of the definition. For a real column with a finite number of plates, all components are theoretically present in both products, though perhaps some are at concentrations below the detectable limit. If the mole fraction of a heavy nonkey component in the distillate is 10 or less, the component may be considered undistributed from a practical standpoint. However, in order to start a plate-by-plate calculation to get the number of plates for the column, this small but finite value needs to be estimated. 

Between the lower and upper invariant zones, the mole fraction of both keys in the vapor phase decreases, and the ratio of light key to heavy key decreases. This region of the column serves to remove the light nonkey components from the liquid flowing down and the heavy nonkey component from the material that will flow up and form the distillate. The small amount of reverse fractionation shown for the key components is an interesting phenomenon that is often found in real columns operating at close to the minimum reflux ratio. 

This appears to be a straightforward application of overall material balances, except that there are two variables too many. Thus, we will have to assume the recoveries or concentrations of two of the components. The normal boiling point of propane is 231.1 K, that of n-butane is 272.7 K, 309.2 for n-pentane, and 341.9 for n-hexane. Thus, the order of volatilities is propane > n-butane > n-pentane > n-hexane. This makes propane the light key, n-butane the heavy key, and n-pentane and n-hexane the heavy nonkeys (HNKs). Since the recoveries of the keys are quite high, it is reasonable to assume that all of the HNKs appear only in the bottoms. Based on this assumption, the overall material balances yield D = 907.9 kmol/h, W = 1092.1 kmol/h. The estimated compositions are as follows. 

In this case, propane and pentane are the light key and heavy key, respectively. Methane and ethane are LNK, hexane is a HNK, while butane is a sandwich component, meaning that it has a volatility intermediate between the keys. This problem fits into Case C, where the distribution of the nonkeys at minimum reflux must be determined, simultaneously with the minimum reflux ratio. Shiras et al. (1950) developed the following equation to determine whether or not a component is distributed at minimum reflux... 

For multicomponent feeds, specification of two key components and their distribution between distillate and bottoms is accomplished in a variety of ways. Preliminary estimation of the distribution of nonkey components can be sufficiently difficult as to require the iterative procedure indicated in Fig. 12.1. However, generally only two and seldom more than three iterations are necessary. 

In practice, the deisobutanizer is usually placed first in the sequence. In Table 12.1, the bottoms for Case 1 then becomes the feed to the debutanizer, for which, if nC4 and iCs are selected as the key components, component separation specifications for the debutanizer are as indicated in Fig. 12.3 with preliminary estimates of the separation of nonkey components shown in parentheses. This separation has been treated by Bachelor. Because nC4 and Cg comprise 82.2 mole% of the feed and differ widely in volatility, the temperature difference between distillate and bottoms is likely to be large. Furthermore, the light-key split is rather sharp but the heavy-key split is not. As will be shown later, this case provides a relatively severe test of the empirical design procedure discussed in this section. 

Solution. The two key components are n-butane and isopentane. Distillate and bottoms conditions based on the estimated product distributions for nonkey components in Fig. 12.3 are... 

If any nonkey components are suspected of distributing, estimated values of Xf,D cannot be used directly in (12-35). This is particularly true when nonkey components are intermediate in volatility between the two key components. In this case, (12-34) is solved for m roots of 0 where m is one less than the number of distributing components. Furthermore, each root of 0 lies between an adjacent pair of relative volatilities of distributing components. For instance, in Example 12.4, it was found the nCs distributes at minimum reflux, but nQ and heavier do not and /C4 does not. Therefore, two roots of 0 are necessary where... 

The Lewis-Matheson method is also an equation-tearing procedure. It was formulated according to the Case I variable specification in. Table 6.2 to determine stage requirements for specifications of the separation of two key components, a reflux ratio and a feed-stage location criterion. Both outer and inner iterations are required. The outer loop tear variables are the mole fractions or flow rates of nonkey components in the products. The inner loop tear variables are the interstage vapor (or liquid) flow rates. The Lewis-Matheson method was widely used for hand calculations, but it also proved often to be numerically unstable when implemented on a digital computer. 

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