Component efficiencies. In binary mixtures, the efficiencies of each of the two components are identical. In multicomponent separations, component efficiencies are all different because

1. Each component has a different diffusivity, both in the vapor and in the liquid.

2. In a multicomponent mixture, the diffusion rate of a component depends not only on its own concentration in the mixture, but also on the concentration of other components. This may lead to coupling and interaction of the mass transfer among various components. Some examples are (192)

а. Reverse diffusion—Mass transfer opposite to the concentration driving force.

б. Diffusion barrier—No net mass transfer even though a concentration driving force exists.

c. Osmotic diffusion—Mass transfer in the absence of a concentration driving force.

3. The effective slope of the equilibrium curve, m, and therefore \ [Eq. (7.5)1 differs for each component. Therefore, each component has a different ratio of gas-phase resistance to liquid-phase resistance [Eq. (7.13)] and a different ratio of overall column efficiency to Murphree tray efficiency [Eq. (7.4)].

Design practice. A computational case study by Toor and Burchard

(192) demonstrates that accounting for the above factors can alter the stage requirement for a multicomponent separation by 30 to 40 percent. Several other authors (12,145,193,194) also warn against assuming equal component efficiencies in multicomponent distillation design. Nonetheless, individual component efficiencies are seldom used in design practice, due to the following reasons:

■ Multicomponent efficiency prediction methods are based on theoretical binary efficiency methods. As previously stated (Sec. 7.2.1), the reliability of those methods leaves a lot to be desired. This difficulty can be bypassed when reliable efficiency data are available for the binary pairs making up the multicomponent mixture. As demonstrated by Vogelpohl (193), binary efficiency data can be extended to multicomponent systems using a multicomponent computation method.

■ Few commercial simulations are geared to handle rigorous multi-component efficiency computations,

■ Rigorous methods for computing multicomponent efficiencies are complex, difficult to use for design, and often of unknown reliability. The ideal method, which is simple enough, yet reliable, is still being sought. The main bottleneck here is the availability of adequate commercial-scale data that will permit proper testing of the various methods.

Pseudo binary method. The most common and generally the simplest procedure used for multicomponent efficiencies, it proceeds in the following steps (12):

1. From the column simulation in terms of theoretical stages, locate representative stages in each section of the column.

2. For each representative stage, select light key and heavy key components, and calculate the composition of the pseudo binary mixture as

Some judgment is required in selecting the pseudo keys, and the two components selected are often not the same for different parts of the column. The light- and heavy-key approach can be extended to allow for multi-pair efficiencies that may be different. The choice of binary pairs depends on feed and product compositions, volatility differences, components of major interest for design, and the components which are the majority fraction of the mixture.

3. Predict the binary diffusion coefficients of the keys in each phase at the mixture temperature and pressure.

4. Calculate YLK and XLK on adjacent theoretical trays n and n + 1, and determine the slope of equilibrium curve m from m = ~ ^ (7.39)

5. Use the binary correlations (Sec. 7.2.1) to predict £0G and E^v, possibly also E0. The section efflcien<y E0 is then used to determine the number of trays in each section of column when used in conjunction with a theoretical-stage simulation.

6. There are three options for applying the efficiency:

a. Use the section efficiency Ea in conjunction with a theoretical-stage simulation to determine the number of trays in each section. This is least accurate, but can be used with commercial theoretical-stage simulations.

b. Assume the Murphree tray efficiency calculated in item 5 above is the same for all components, then apply the Murphree tray efficiency in a column simulation. This option is simple, more accurate than the previous, but requires a simulation that can use Murphree efficiencies. This option reliably predicted composition profiles both for similar (195) and dissimilar (145) components, but it is unknown whether it always works so well. Chan and Fair (145) expect it to generally work well, especially if the key components dominate the feed mixture to the column.

c. Repeat the above steps, calculating Murphree efficiencies for many binary pairs. This requires the solution to a linear set of equations in order to obtain the component mole fractions in the mixture. Ognisty and Sakata (195) show that for systems of either similar or dissimilar components, this option predicts composition profiles practically as accurately as the rigorous diffusional interaction method below. This option, however, increases the complexity of the pseudo binary method, and also makes it difficult to use with most commercial simulations. With a large number of components, this option becomes as complex as a rigorous diffusion method (195).

Individual component efficiency method. Another simple procedure, it was recommended for the AIChE tray efficiency correlation (125). It yields individual component efficiency but takes no account of diffusional interaction. It proceeds as follows (12):

1. Predict gas phase diffusion coefficient DG i for each component i in the presence of all other components, using Wilke's equation ( 196).

j = 1 * i where Z)Gy is the gas-phase diffusion coefficient for each binary pair.

2. Predict liquid-phase diffusion coefficient DL i of each component as solute through the other components as solvent.

3. Determine the slope of the equilibrium curve, m£ for each component, from the values of vapor and liquid compositions given by a theoretical stage simulation and from

4. Calculate SOGi, and ¿5mv,i f°r each component using the equations for binary mixtures (Sec. 7.2.1).

5. is used in a column simulation program that has provision to accept individual component efficiencies.

Biddulph (197-199) applied a simplified version of this method for an oxygen-nitrogen-argon column and a benzene-toluene-xylene column. In each case, the components were similar to one another, and there was experimental evidence to show that the component individual point efficiencies EOG;1 were the same. Making the assumption of constant point efficiency in each column section permitted Biddulph to eliminate the first two steps. In a later paper, Biddulph (194) shows that this assumption, and therefore his simplification, do not always hold where components are dissimilar.

Diffusional interaction methods. These calculate component efficiencies, but account for diffusional interactions. The calculation procedure is based on the Maxwell-Stefan diffusion equations, as developed by Krishna et al. (200,201). The equations are complex and are presented in the original reference. Lockett (12) has am excellent summary. For a ternary system, the steps below are followed (12):

1. Estimate diffusion coefficients for each binary pair in each phase.

2. Calculate the number of transfer units for each of the binary pairs.

3. Using a special weighted averaging procedure, calculate the number of ternary transfer units for the gas phase. Repeat for the liquid phase.

4. Determine the slopes of the equilibrium curve.

5. Combine the number of ternary transfer units in each phase to obtain the number of ternary overall gas-phase transfer units.

6. Evaluate the elements of the ternary overall gas-phase transfer unit matrix,

7. Calculate point efficiencies based on this matrix.

8. Using the slopes of the equilibrium curve obtained in (4) above, and the appropriate mixing model, convert point to Murphree tray efficiencies.

Burghardt et al. (202), Ognisty and Sakata (195), and Chan and Fair (145) show that the diffusional interaction method gives good and reliable prediction of experimental multicomponent tray efficiency data. While Ognisty and Sakata recommend this method for precise calculations, Chan and Fair express preference for a simpler procedure if good accuracy can be assured. Burghardt et al. (202) closely examined simplifying assumptions inherent in diffusional interaction models and show that these lead to negligible errors for several cases studied.

Vogelpohl (193) and Medina et al. (203) applied the diffusional interaction method for predicting ternary distillation composition profiles using binary data. They achieved this by eliminating the first two steps and assuming that all the mass transfer resistance is in the vapor. This procedure was shown to give excellent agreement with experimental data for dissimilar components. Biddulph and Kalbassi (194), however, report some discrepancies between prediction and experiment due to this assumption.

Diffusional interaction methods have also been applied successfully to packed columns. Gorak (204) found that the Krishna-Standart model (205) is relatively simple and sufficiently accurate to predict multicomponent composition profiles. Gorak's own variation of the diffusional interaction method was also reported to predict experimental data well, while use of HETP was reported to give poor data predictions.

Multicomponent efficiency profiles. Figure 7.11 (198,199) shows typical variations of Murphree plate efficiency for a ternary system made up of a light key (LK) component, a heavy key (HK) component, and a heavy nonkey (HNK) component (benzene, toluene, and m-xylene, respectively). On the basis of experimental data, point efficiency was assumed the same for all components throughout the column.

Figure 7.11 shows the same Murphree tray efficiency for the binary pair of key components separated. In the upper part of the column, separation is between the LK and the HK. These behave like a "hi-

Tray number

Figure 7.11 Variation of Murphree tray efficiency in multicomponent distillation. (From M. W. Biddulph, Hydrocarb. Proc., October, p. 145, 1977, reprinted courtesy of Hydrocarbon Processing.}

Figure 7.11 Variation of Murphree tray efficiency in multicomponent distillation. (From M. W. Biddulph, Hydrocarb. Proc., October, p. 145, 1977, reprinted courtesy of Hydrocarbon Processing.}

nary pair" and show the same efficiency. In the bottom few trays, separation takes places between the HK and HNK, and these components have the same efficiency. The efficiency of the LK component is higher in the region where the HK is separated from the HNK, while the efficiency of the HNK is lower in the region where the HK is separated from the LK.

The efficiency of the HK component undergoes a sharp change in the lower part of the column. This occurs when the HK component reaches its peak composition (Sec. 2.3.2). This sharp change in efficiency has no physical significance (193,203), but is a consequence of applying the Murphree efficiency concept to multicomponent systems [i.e., applying Eq. (7.3) across a concentration peak]. Sometimes, this may even lead to negative efficiencies. In this region, predictions from all multicomponent efficiency prediction methods can be in gross error (194).

The previous observations were made for the benzene-toluene-xylene system (198,199). Similar observations were made for the nitrogen-argon-oxygen system (197). In both cases, point efficiencies were assumed constant for all components throughout the column.

7.3.6 Efficiency scaleup: process factors

This section focuses only on process scaleup—i.e., scaling up from one chemical system to another, and also scaling up from one set of process conditions to another using the same system. Equipment factors such as column diameter, operating regime, and tray geometry are considered separately in the following section.

Different type of system. Sections 7.3.1 and 7.3.4 suggest that if liquid viscosity, relative volatility, surface tension, and surface tension gradients are similar, one can expect similar efficiencies. Since it is difficult to attain similarity in all these, it is best to scale up from the same chemical system. Further, in aqueous-organic systems where surface tension gradients can be steep, concentration has a marked and unpredictable effect on efficiency (116,184,187). Data collected in one concentration range may scale up poorly to another.

One type of system scaleup which can be done with confidence is extending efficiency data from one pressure to another. The small increase of efficiency with pressure (Sec. 7.3.4) can be allowed for using the O'Connell correlation (Sec, 7.2.2). Caution is required when going to high pressures (> 150 psi), as vapor recycle may reduce efficiency, and this effect is difficult to predict.

Different process conditions. Here we discuss staying with the same system, but extrapolating test data to different process conditions (e.g., feed composition, reflux ratios, feed temperature, etc.). Concentration and pressure effects were discussed separately above; flow regimes are discussed separately in the next section.

In plant or pilot tests, separation data are analyzed by computer simulation that reproduces test conditions. The number of theoretical stages in the simulation is varied until the simulated product compositions and temperature profile match those measured. Tray efficiency is determined from the number of stages that give a good match to test data.

In the above procedure, errors in VLE are compensated by equivalent errors in tray efficiency. If the relative volatility calculated by the simulation is too high, fewer stages will be needed to match the measured test compositions, i.e., efficiency will be lower. Scaleup will be good as long as the VLE and efficiency errors continue to equally offset each other. This requires that process conditions (feed composition, feed temperature, reflux ratio, etc.) remain unchanged during scale-up.

When process conditions are changed, the VLE and efficiency errors no longer offset each other equally. If the true relative volatility is higher than the simulated relative volatility, then the scaleup will be conservative. If the true volatility is lower than the simulated relative volatility, scaleup will be optimistic, and often dangerous. This is illustrated using the following example.

Example 7.2 A stripping section of a column separating propane from isobutylene needs to handle two future feeds (a) 72 percent propane, (6) 88 percent propane. Field-test data show that when column feed contained 80 percent propane, its bottoms contained 2 percent propane. Based on the test data, what will be the purity of isobutylene with the future feeds?

solution The example is solved using a McCabe-Thiele diagram (Fig. 7,12). For simplicity, the following arbitrary assumptions are made:

■ The column operates at an arbitrarily fixed stripping ratio (i.e., all the operating lines on Fig. 7.12a to /"have the same slopes).

■ A straight component balance line (i.e., constant molar heat of vaporization, Sec. 2.2.2).

» The feed to the column is 50 percent vaporized (i.e., slopes of all the ^-lines on Fig. 7.12a to/"are -1).

The test separation data are analyzed using a McCabe-Thiele diagram (Fig. 7.12a), with an equilibrium curve based on experimental data (average relative volatility of about 1.7). The analysis shows that the stripping section requires 12 theoretical stages.

Consider an analysis of the same test data, but with an equilibrium curve based on a VLE prediction which gives higher relative volatilities (average of about 2.5) than the experimental data. With the calculated VLE, the McCabe-Thiele diagram (Fig. 7.126) requires only eight theoretical stages.

Both Figs. 7.12a and b correctly predict the test conditions. Either Fig. 7.12a (using experimentally derived VLE and 12 stages) or Fig. 7.126 (using predicted VLE and 8 stages) will correctly predict product compositions under process conditions similar to those experienced in the test.

Consider now a propane concentration of 72 percent in the feed. The McCabe-Thiele diagram, based on the Fig. 7.12a interpretation of the test data, predicts that the propane concentration in the column bottoms will decline to 1 percent (Fig. 7.12c). On the other hand, if the McCabe-Thiele diagram were based on the Fig. 7.126 interpretation of the test data, it would predict 0.5 percent propane ill the column bottoms (Fig. 7.12d). This prediction is optimistic.

Consider a propane concentration of 88 percent in the feed. The McCabe-Thiele diagram, based on the Fig. 7.12a interpretation of the test data, predicts a pinch just below the feed (Fig. 7.12e). Due to the pinch, the concentration of propane in the tower bottom will be 17 percent, i.e., much higher than the 2 percent propane in the test data. In practice, this pinch will probably be eliminated by increasing the boilup ratio (i.e., reducing the slope of the operating line). However, increasing the boilup ratio means more liquid and vapor traffic, a greater heat load on the reboiler, and possibly, a premature capacity bottleneck.

If the test data were interpreted using Fig. 7.126, the pinch will not be seen. Based on the Fig. 7.126 interpretation, the McCabe-Thiele diagram (Fig. 7.12f)

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