Murphree Plate Efficiency Graph
4% greater
2.2.6 Minimum reflux ratio
Using Eqs. (2.11) and (2.16), Eq. (2.9) for the rectifying section component balance line can be expressed in terms of the reflux ratio
As the reflux ratio decreases, so does the slope of the upper component balance line. The effect of reflux ratio on the component balance lines is illustrated in Fig. 2.11, using the benzenetoluene system in Example 2.1.
Any practical separation requires that the component balance lines intersect below the equilibrium curve, as for a reflux ratio of 3.0 in Fig. 2.11c. The McCabeThiele construction corresponding to this ratio is shown in Fig. 2.9c. If insufficient reflux is provided, the component balance lines intersect above the equilibrium curve, as for a reflux ratio of 1.0 in Fig. 2.11a. The McCabeThiele construction (Fig. 2.116) for these conditions shows that even with an infinite number of stages, the separation cannot be achieved.
The separation is theoretically possible if the component balance lines intersect at a point just below the equilibrium curve. The corresponding reflux ratio is termed minimum reflux. The separation at minimum reflux requires an infinite number of stages. In Fig. 2.11, the minimum reflux ratio is 2.0. The McCabeThiele construction for this ratio is shown in Fig. 2.11c.
At minimum reflux, the pinch occurs at the intersection of the component balance line and the 5line when the equilibrium curve has no inflection points (Fig. 2.11c). This would be expected because the component balance lines intersect on the gline. When the equilibrium curve has a point of inflection (Fig. 2.12), the pinch between the equilibrium curve and the component balance line may occur at the point of tangency instead of the intersection of the qline and the component balance line. This condition is termed tangent pinch.
To determine minimum reflux, construct the gline and identify its point of intersection with the equilibrium curve. Then draw a line from the product composition point on the 45" diagonal line to this intersection point. From Eq. (2.29), the slope of the line is i?min/ (J?min + 1), and the intercept of this line on the y axis is *o/(Rmjn + 1). Minimum reflux can be determined from either of these. If minimum reflux occurs at a tangent pinch, the minimum reflux is independent of the 7line and the feed composition. It can then be determined from the equilibrium curve alone (13).
Neither minimum reflux nor a tangent pinch is an operable condition. Either will require an infinite number of stages in the column, and this is physically impossible. Nevertheless, operation can some
x. mole fraction benzene in liquid (b)
Figure 2.11 Effect of reflux ratio on component balance lines, (a) Overall; (6) ii < Rmiri, impossible operation; (c) R = Rmi„, minimum reflux; <d) R = total reflux.
x, mole fraction benzene in liquid (d)
Figure 2.11 (Continued)
x, mole fraction benzene in liquid (d)
Figure 2.11 (Continued)
times approach minimum reflux when a column contains a large excess of stages.
2.2.7 Minimum stripping
Using Eqs. (2.12) and (2.17), the stripping section component balance line [Eq. (2.10)] can be expressed in terms of the stripping ratio
x, mole fraction in liquid
Figure 2.12 Tangent pinch. (From C. J. King, Separation Processes, 2d ed., Copyright © by McGrawHill, Inc. Reprinted by permission.)
x, mole fraction in liquid
Figure 2.12 Tangent pinch. (From C. J. King, Separation Processes, 2d ed., Copyright © by McGrawHill, Inc. Reprinted by permission.)
Minimum stripping is analogous to minimum reflux. The discussion in Sec. 2.2.6 is also valid to minimum stripping. To determine minimum stripping, draw a line from the bottom composition point on the 45° diagonal line to the intersection of the gline with the equilibrium curve. From Eq. (2.30), the slope of this line is (1 + S^VS^ and its intercept on the y axis is ~xB/Smin.
2.2.8 Total reflux and minimum stages
As reflux ratio increases, so does the slope of the upper component balance line. The effect of increasing reflux ratio on the component balance lines is illustrated in Fig. 2.11, using the benzenetoluene system of Example 2.1. At the limit, where reflux ratio approaches infinity, the rectifying section component balance line [Eq. (2.29)] reduces to
and the component balance line becomes the 45° diagonal line. The reflux ratio [Eq. (2.16)] can approach infinity only if the product rate
D approaches zero. Similarly, when the stripping ratio approaches infinity, the stripping section component balance line [Eq. (2.30)] reduces to
and the component balance line becomes the 45° diagonal. The stripping ratio [Eq. (2.17)] can approach infinity only when the bottoms rate B approaches zero.
The condition where the reflux and stripping ratios approach infinity is termed total reflux. No feed enters the column and no product leaves. Both component balance lines coincide with the 45° diagonal line and are therefore furthest away from the equilibrium curve. Total reflux sets the minimum number of stages required for the separation. For Example 2.1, Fig. 3.lid shows that the minimum number of stages required for the separation is 6.
2.2.9 Allowance for stage efficiencies
Distillation stage calculations are usually performed with ideal stages. The number of ideal stages required for the separation is divided by the overall column efficiency (Sec. 7.1.1) to obtain the required number of trays. In packed towers, the number of stages in the column is multiplied by the HETP (Height Equivalent of a Theoretical Plate, see Sec. 9.1.2) to obtain the packed height.
For tray columns, an alternative approach uses Murphree tray efficiencies (Sec. 7.1.1). This efficiency is easy to incorporate into an xy diagram, and the diagram construction can be performed using actual rather than ideal stages. The Murphree tray efficiency is defined as yn y«1
On an xy diagram, the denominator equals the spacing between the equilibrium curve and the component balance line (Fig. 2.13). yn is given by
„ rspacing between equilibrium curve*] ,0 QA,
Jn = "Mand component balance line J + y"1 U d4J
2.2.10 Extension to complex columns
Extension of the xy diagram to columns containing a second feed or a side product has been discussed by numerous authors (7, 8, 1417). Yaws et al. (18) extended the McCabeThiele diagram to threefeed columns. Kister (19) extended the McCabeThiele diagram to columns
Figure 2.13 Use of Murphree tray efficiency in xy diagram construction, te) An enlarged section of the diagram; (6) an xy diagram incorporating Murphree efficiencies. (Part b from C. J. King, Separation Processes, 2d ed., Copyright © by McGrawHill, Inc. Reprinted by permission.)
Figure 2.13 Use of Murphree tray efficiency in xy diagram construction, te) An enlarged section of the diagram; (6) an xy diagram incorporating Murphree efficiencies. (Part b from C. J. King, Separation Processes, 2d ed., Copyright © by McGrawHill, Inc. Reprinted by permission.)
having multiple feeds, multiple side products, multiple points of heat removal or addition, and any combination of these.
A complex fractionator is divided into N + 1 sections (Fig. 2.14). The partition between each two adjacent sections occurs either at a feed point, or a sidedraw point, or a heat removal point, or a heat addition point (Fig. 2.14). Table 2.5 shows the equations applying to each section of a complex fractionator (19).
Two other useful relationships derived from applying Eq. (2.35) to Lj and Lj _ 1 and Eq. (2.36) to Vj and Vj _ x are (19)
Figure 2.14 Dividing a complex fractionator into N + 1 sections at points of feed entry, sidedraw removal, and heat removal and addition. (From Henry Z. Kister, Chemical Engineering, January 21, 1985, pp. 97104. Reprinted courtesy of Chemical Engineering.)
Figure 2.14 Dividing a complex fractionator into N + 1 sections at points of feed entry, sidedraw removal, and heat removal and addition. (From Henry Z. Kister, Chemical Engineering, January 21, 1985, pp. 97104. Reprinted courtesy of Chemical Engineering.)
table 2.5 Equations Applying to Each Section "J" of a Complex Fractlonator
The following equations have been derived for a complex tractionator. For each Section J, these equations apply:
Liquid flowrate:
Vapor flowrate:
Component balance line:
Point of intersection of the component balance line with the 45 deg diagonal line:
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