Nonkey component distribution

Application of Underwood s equation to systems containing distributed nonkey components is as follows ... 

Treat the mole fraction of each distributed nonkey component in the distillate as an unknown. Write Eq. (3.11) fbr each value of B calculated above. (L/Z ) is also unknown. Solve the equations simultaneously to get the mole fraction of each distributed component in the distillate and L/D)min. In the above example, there are five values of 0 and therefore five equations. There are also five unknowns—the mole fractions of DKl, DK2, DKS, and DK4 in the distillate, and (L/D)mjn. 

Even when (12-27) is invalid, it is useful because, as shown by Gilliland, the minimum reflux ratio computed by assuming a Class 1 sepmation is equal to or greater than the true minimum. This is because the presence of distributing nonkey components in the pinch-point zones increases the difficulty of the separation, thus increasing the reflux requirement. 

Solution. From the results of Example 12.4, assume the only distributing nonkey component is n-pentane. Assuming the feed temperature of 180°F is reasonable for computing relative volatilities in the pinch zone, the following quantities are obtained from Figs. 12.3 and 12.6. 

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. 

The lower bounds on the recoveries were set to 0.85 on the grounds that a number of simulations performed showed that to avoid the distribution of nonkey components it was necessary to keep the recoveries of the key components greater than or equal to 0.85. 

Figure 2,23 Application of dlb plots to examine nonkey component distribution in products, Depropanizer example. 20 theoretical stages, = 1.40.

A component is said to be distributed (or distributing) at minimum reflux if it appears both in the distillate and the bottoms at minimum reflux. Usually, nonkeys are nondistributed (or nondistributing), that is, at minimum reflux the heavy nonkeys are totally contained in the bottoms and the light nonkeys in the distillate. A nonkey component may be distributed if... 

The nonkey components that appear in both top and bottom products are known as distributed components and those that are not present, to any significant extent, in one or another product, are known as nondistributed components. 

Kister says that d/b plots are primarily used when there is a tight spec, on a nonkey component or a concern about the distribution of an intermediate key component. His book shows d/b curves for various feed stage locations on a plot of the mole ratio of a reference component in the distillate to the bottom product, versus the relative volatility of each component to this reference component. This plot is made on log-log paper. The optimum feed produced a curve closest to linear. The d/b plot is suggested as a troubleshooting tool in the subsection of the Troubleshooting section, Fractionation Operating Problems. ... 

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/... 

Determine the distribution of nonkey components and the ratio of rectifying to stripping stages, both by the Fanske relationship. 

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. 

For the case of minimum reflux, the distinction between distributed and undistributed components is clearer, since heavy nonkey components are generally absent from the distillate, and light nonkey components are not present in the bottoms. The concentrations of these species can go to zero because of an infinite number of plates in the column and conditions that lead to a progressive reduction in concentration for each plate beyond the feed plate. 

We will determine which of the nonkey components distribute according to the Shims criterion. From the statement of the problem, FRLKD = 0.98 and FRHKD = 0.01. For... 

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. 

For example, suppose we specify that 13 lbmole/hr of iCj in the feed is allowed to appear in the distillate. Because the split of iCs is not sharp and nCs is close in volatility to iCj, it is probable that the quantity of nCs in the distillate will not be negligible. A preliminary estimate of the distributions of the nonkey components... 

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... 

Equations (12-17) and (12-18) give the distribution of nonkey components at total reflux as predicted by the Fenske equation. 

Example 12.3. Estimate the product distributions for nonkey components by the Fenske equation for the problem of Example 12.2. 

Distribution of nonkey components in the feed is determined by (12-28). The most likely nonkey component to distribute is nCs because its volatility is close to that of iCs(HK), which does not undergo a sharp separation. For nCs, using data for X-values from Fig. 12.6, we have... 

Therefore, Dx cs.d = 0.1963(13.4) = 2.63 Ibmole/hr of nCs in the distillate. This is less than the quantity of nCs in the total feed. Therefore, nCj distributes between distillate and bottoms. However, similar calculations for the other nonkey components give negative distillate flow rates for the other heavy components and, in the case of iCt, a distillate... 

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... 

It might be expected that a product distribution curve for actual reflux conditions would lie between the two limiting curves. However, as shown by Stupin and Lockhart, product distributions in distillation are complex. A typical result is shown in Fig. 12.15. For a reflux ratio near minimum, the product distribution (Curve 3) lies between the two limits (Curves 1 and 4). However, for a high reflux ratio, the product distribution for a nonkey component (Curve 2) may actually lie outside of the limits and an inferior separation results. 

For the distillation operation shown below, establish the type condenser and an operating pressure, calculate the minimum number of equilibrium stages, and estimate the distribution of the nonkey components. Obtain K-values from Fig. 7.5... 

For the conditions of Problem 12.7, compute the minimum external reflux rate and the distribution of the nonkey components at minimum reflux by the Underwood equation if the feed is a bubble-point liquid at column pressure. 

The Fenske and Winn equations are not restricted to the two key components. Once JVmin is known, they can be used to calculate mole fractions and Xi for all nonkey components. These values provide a first approximation to the actual product distribution when more than the minimum stages are employed. A knowledge of the distribution of nonkey components is also necessary when applying the method of Winn because (12-15) cannot be converted to a mole ratio form like (12-12). 

This equation is attributed to Underwood and can be applied to subcooled liquid or superheated vapor feeds by using fictitious values of Lp and x p computed by making a flash calculation outside of the two-phase region. As with the Fenske equation, (12-27) applies to components other than the key components. Therefore, for a specified split of two key components, the distribution of nonkey components is obtained by combining (12-27) with the analogous equation for... 

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