Examples Of Mislocated Feeds

distillation do not occur at the feed points, as they do in binary distillation, but above and below the feed points. The stages between the lower pinch zone and the feed point fractionate the light nonkeys from the bottom section liquid, and those between the upper pinch zone and the feed fractionate the heavy nonkeys from the rectifying section liquid.

2.3.7 Best leed stage location

Due to the presence of nonkeys near the feed, determination of the best feed point in multicomponent distillation is difficult. King (7) surveyed published criteria and their shortcomings. The following rules of thumb are often used;

1. The key ratio in the feed stage liquid should be as close as possible to the key ratio in the liquid portion of the feed (flashed to tower pressure).

2. The binary equivalent composition of the light key in the feed should correspond to the intersection of the component balance lines on a Hengstebeck diagram (e.g., Fig. 2.186).

3. The feed point should give the most equal slopes on both sides of the feed stage in a key ratio plot (Fig. 2.20). Too high a feed causes excessive retrograde distillation below the feed, and too low a feed causes excessive retrograde distillation above it. In order to reach the optimum feed point, the feed stage should be moved from the "sharp" maximum (or minimum) toward the flat one (Fig. 2.20).

4. Applying an analytical technique, shortcut (Sec. 3.2.6) or rigorous.

Hanson and Newman (25) demonstrated the following shortcomings of the first two rules of thumb:

1. Although rules 1 and 2 often work well, there are many instances where they give poor feed locations. The rules become less reliable as minimum reflux is approached.

2. Light nonkeys raise the optimum key ratio at the feed stage, while heavy nonkeys lower it. Rules 1 and 2 therefore become less reliable when there are a lot more light than heavy nonkeys or vice versa, or when the amount of nonkeys exceeds the amount of keys.

3. Nonkeys whose volatility is either close to the keys or far removed from the keys tend to shift the optimum key ratio (or light key concentration) at the feed stage to a lesser degree than nonkeys whose volatility is moderately removed from the keys.

4. The first rule of thumb tends to work better when there are more light than heavy nonkeys, while the second tends to work better when there are more heavy than light nonkeys.

Shortcut analytical techniques tend to be just as unreliable as the rules above. The third rule of thumb accounts for the nonkeys and is a favored shortcut method by the author and others (7). While a rigorous analytical technique (Sec. 3.1.7) is generally the most reliable pro-

Figure 2.20 Effect of feed stage on key ratio plot. De-propanizer example, feed composition same as Example 2.4, 20 theoretical stages, R/Rmio - 1.40.

cedure for predicting the best feed point, the third rule of thumb provides an invaluable check.

2.3.8 Distribution of nonkeys (d/b plots)

The d/b plot provides a useful estimate of the distribution of nonkeys and intermediate keys in the products. Hengstebeck (14) and Geddes

(26) noted that at total reflux, application of Fenske's equation (Sec. 3.2.1) for each component gives

Thus at total reflux, a log-log plot of xd!xb (or d/b) for each component against relative volatility gives a straight line with a slope equal to the minimum number of stages (Fig. 2.21). Stupin and Lockhart (27) noted that this relationship is nonlinear at minimum reflux and at finite reflux ratios (Fig. 2.21). At minimum reflux, the curve becomes asymptotic to the dashed lines at the relative volatilities of the light and heavy keys, corresponding to a total recovery of light nonkeys and heavy nonkeys in the distillate and bottoms, respectively. On the other hand, intermediate keys are separated to a better degree at total reflux than at minimum reflux.

Figure 2.21 Distribution of components at various reflux ratios. (Reproduced from W. J. Stupin and F. J. Lockhart, paper presented at the annual meeting of the AIChE, Los Angeles, 1968. Reprinted courtesy of Dr. W. J. Stupin.)

Figure 2.21 Distribution of components at various reflux ratios. (Reproduced from W. J. Stupin and F. J. Lockhart, paper presented at the annual meeting of the AIChE, Los Angeles, 1968. Reprinted courtesy of Dr. W. J. Stupin.)

Stupin and Lockhart (27) also noted that as reflux is lowered from total to minimum, the separation of nonkeys first worsens (curve 2, Fig. 2.21), then improves (curve 3, Fig. 2.21). The intermediate keys follow the converse pattern. At a reflux ratio of about 1.2 to 1.5 times the minimum, component distribution resembles that of the total reflux component distribution. Detailed discussion is elsewhere (7,27). Figure 2.16 demonstrates that light nonkeys are fractionated out in the stripping section and heavy nonkeys in the rectifying section. The dlb plot depicts this behavior (Sec. 2.4.2).

2.4 Analyzing Computer Simulation Results by Graphical Techniques

A prime application of graphical methods in modern distillation technology is for analyzing the results of computer simulations. Several of the graphical construction rules can be bent in order to benefit from computer accuracy and to reduce effort. Johnson and Morgan (28) described several key considerations; their work is expanded here using the author's experience.

2.4.1 Use of x-y diagrams (McCabe-Thiele and Hengstebeck)

It has been the author's experience that an x-y diagram is the most useful graphical technique for analyzing computer simulation results. The x-y diagram is capable of

1. Detecting pinched regions (Sec. 2.2.5): Pinching and its cause (minimum reflux, mislocated feed, tangent pinch, etc.) are readily visible on an x-y diagram. Figure 2.22 compares a well-located feed point in the depropanizer example with a mislocated feed point. The pinch is clearly seen in Fig. 2.226, while no pinch exists in Fig. 2.22a.

The x-y diagram should be examined for pinching both for design conditions and for some deviations (variation in feed temperature, feed composition, errors in relative volatility, etc.). For deviations, a simple construction (e.g., a g-line with a slightly different slope) on the design conditions x-y diagram is usually sufficient. When the threat of a pinched region is detected, a more detailed analysis (e.g., a computer run for the deviated conditions) is warranted.

2. Identifying mislocated feed points: The feed point should be where the <?-line intersects the equilibrium curve. This rule works well for binary distillation (Sec. 2.2.5), but not so well for multicomponent distillation (Sec. 2.3.7). In Fig. 2.22 it works well; the x-y diagram

Mccabe ThieleHengstebeck Diagrams

Figure 2.22 Application of x-y diagrams to analyze a computer simulation. Diagrams were prepared using computer composition printouts. Depropanizer example, 20 theoretical stages, R/Rmili = 1.4. (a) Correct feed point. (ft) Feed point too high.

Pseudo light mole fraction in liquid (b)

Figure 2.22 Application of x-y diagrams to analyze a computer simulation. Diagrams were prepared using computer composition printouts. Depropanizer example, 20 theoretical stages, R/Rmili = 1.4. (a) Correct feed point. (ft) Feed point too high.

suggests stage 8 is the best feed point. Simulation showed that stage 9 is slightly better, but stage 8 is adequate. However, the author has encountered many multicomponent distillations where the simulated optimum feed point did not match the intersection of the g-line with the equilibrium curve.

3. Identifying excessive reflux and reboil: This can be recognized by too wide a gap between the component balance line and the equilibrium curve throughout the column. Errors in estimating minimum reflux and/or convergence difficulties in the computer simulation are often the culprit. The author has experienced a case where the design reflux rate was lowered by 30 percent after an x-y diagram identified excess reflux. Figure 2.22 suggests that there is some room for reducing reflux and reboil in this example. Simulation confirmed that adding five stages can reduce reflux and reboil by 15 percent while still maintaining adequate margin from pinching.

4. Identifying cases where feed or intermediate heat exchangers are attractive: A wide gap between the component balance line and the equilibrium curve in the bottom section indicates a potential for a preheater or interreboiler; a wide gap in the top section suggests a potential for a precooler or intercondenser. In Fig. 2.22, the gaps are wide in both sections, suggesting excessive reflux and reboil rather than a potential for adding an intermediate heat exchanger.

5. Guiding column optimization, and showing the effects of changing feed or product composition, thermal state of the feed, use of side draws, multifeed arrangements, etc.

Construction. Once a converged computer simulation is available, construction of an x-y diagram is far easier than constructing one from scratch. Many computer simulations provide the option of plotting an x-y diagram. If this option is unavailable, the following sequence of steps can be followed:

1. 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 pseudo-component, 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.

2. Plot the equilibrium curve, using compositions printed out by the computer

In multicomponent systems, these compositions must first be converted to the appropriate binary equivalent or pseudo-light-component as per item 1 above. Since vapor and liquid leaving each stage are in equilibrium, plotting the composition of the vapor leaving a stage against the composition of liquid leaving the same stage will give a point on the equilibrium curve. Repeating for several stages will define the equilibrium curve.

3. Plot the component balance lines, using compositions printed out by the computer (expressed as the appropriate binary equivalents or pseudo-light- component compositions as per item 1 above)

Equation (2.9) states that the component balance line can be obtained by plotting the vapor composition entering a stage against liquid composition leaving the same stage. The plotted component balance "line" may turn out to be a curve. The computer accounts for enthalpy variations and does not assume constant molar overflow (Sec. 2.2.2), The resulting component balance curves overcome the greatest accuracy problem (Sec. 2.2.2) of conventional x-y diagrams. Figure 2.22 shows some curvature of the component balance lines.

4. Draw the 450 diagonal line

Plot the feed composition on the diagonal (expressed as the appropriate binary equivalent or pseudo-light-component composition as per item 1 above).

5. Calculate q

If using the equivalent binaiy method, use Eq. (2.58). For *LNK,]im obtain an average value from near the middle of the rectifying section; for *HNK4im obtain an average value from near the middle of the stripping section. If using the pseudo-light key component method, take q as the liquid leaving the feed stage minus liquid entering the feed stage from the stage above, and the difference divided by the feed rate. Note that this is slightly different from and more appropriate than the normal definition of q. Construct the g-line using Eq. (2.27).

6. Step the stages off in the normal manner. 2.4.2 Use of key ratio and d/b plots

Key ratio plots are primarily for identifying mislocated feed stages in multicomponent distillation. For this purpose, they are superior to x-y diagrams. Key ratio plots are easy to construct; all it takes is calculating the key ratio in the liquid for a few stages in the feed region and plotting these against stage number on semilog paper. Figures 2.19 and 2.20 are key ratio plots prepared from compositions calculated by a computer simulation.

d/b plots (Sec. 2.3.8) are primarily used when there is a tight spec on a nonkey component or a concern about the distribution of an intermediate key component, d/b plots are easy to construct. One component (say the heavy key) is selected as the reference component, and the relative volatility of each of the other components is calculated in relation to it. Then the mole fraction of each component in the top product is divided by its mole fraction in the bottom product, and the quotient is plotted against its relative volatility on log-log scales. Often, the ratio of number of moles in the top to number of moles in the bottom is used instead of the mole fraction ratio. In other cases (28), recovery is plotted on an inverse log scale [(100%/overhead recovery) - 1] instead of the mole fraction ratio.

The d/b ratio plot is frequently non-linear, but should be smooth ( Sec. 2.3.8). The prime cause of bumps is a poor estimate of relative volatility. If a refined estimate (see Sec. 3.2.1 for estimating guidelines) does not improve things, the bump may reflect anomalies or a need to relocate a feed. The d/b plot gives a measure of how relocating the feed point affects the nonkey component split.

Figure 2.23 shows dfb plots for the depropanizer example, based on compositions calculated by the computer. The diagram shows that for a

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