Infinite stages

Figure 8 illustrates the performance of a system with three equilibrium stages in the absorber and six in the stripper. The actual steam requirement is 147 moles/mole SO2 (41.3 kg/kg). The use of a finite number of stages increases the steam requirement a factor of 2.5 from the case of infinite stages with a nonlinear equilibrium. 

Table IV gives minimum steam requirement (infinite stages) at several different solution capacities. The factor attribu-able to equilibrium nonlinearity increases as more SO2 is absorbed, because the buffer capacity is consumed to a greater extent. Any capacity for SO2 absorption can be achieved by varying Na concentration (pH) in the solution. At low pH ([Na] = 1.5 M) the solution capacity for SO2 absorption is small, but the nonlinearity factor is also small (1.05). Solution capacity can be increased by operating at higher pH ([Na] = 2.5 M), but nonlinearity is more severe (1.32).

Na](M) pS02/pH20 0.02, infinite stages Steam Requirement Moles H20/Mole SO2 90% SO2 removal,, 1.0 M Citrate Nonlinearity Factor Capacity Moles S02/liter... 

Figure 14.7. Locations of operating points P and Q for feasible, total, and minimum extract reflux on triangular diagrams, and stage requirements determined on rectangular distribution diagrams, (a) Stages required with feasible extract reflux, (b) Operation at total reflux and minimum number of stages, (c) Operation at minimum reflux and infinite stages.

Minimum reflux corresponds to the overlap of an operating line and a tie-line (infinite stages at a pinch point). This concept is similar to minimum solvent flowrate for an extraction process without reflux. Total reflux corresponds to the minimum number of stages. Remember that total reflux means that no streams are going into or out of the column, so that F, B, and D are zero, and A = A. ... 

The next step is to find (L/G)min- This is the case where the column would have infinite stages, and corresponds to a pinch point. To draw the operating line with minimum slope, connect (Xq, Y ) and the point where the equilibrium line crosses the value Tat+i. This second point is easily found from the equilibrium data and the two points are used to find the slope ... 

The other limiting parameter, minimum reflux ratio at infinite stages, may be calculated by means of the Underwood " equations ... 

Minimum reflux conditions correspond to a situation of minimum L/V, maximum product, and infinite stages. In the McCabe-Thiele constructions minimum reflux was determined either by feed conditions or an equilibrium line pinch condition as in Fig. 8.15. 

The entering flow rate of the stripping agent Vo is not specified. The minimum rate at infinite stages can be computed from (12-78), provided that a key component is selected. Suppose we choose B2, which is the heaviest component to be stripped to a specified extent. At 70°C, the vapor pressure of B2 is computed from (4-83) to be 90.4 psia. From (4-75) at 30 psia total pressure... 

Extraction, liquid-liquid, 8,10,13 equilibrium diagrams, 103 equipment, 76-84 extract reflux, 415 group method, 475-480 infinite stages, 415 McCabe-Thiele method, 315-318... 

The distribution coefficient of ethanol (E) between water (IV) and ester (5) is roughly 2 = (mole% E in S)/(mole% E in W) = x lx" at 20°C. A 10mole% solution of E in IV is to be extracted with S to recover the ethanol. Compare the separations to be obtained in countercurrent, cocurrent, and crosscurrent (with equal amounts of solvent to each stage) contacting arrangements with feed ratios of S to W of 0.5, 5, and 50 for one, two, three, and infinite stages. Assume the water and ester are immiscible. 

Consider the case of minimum reflux ratio (infinite stages). As the amount of solvent is reduced, point M (equal to S + F) in Fig. 11.9 moves towards F, and P (equal to D + So) moves towards D. Point P" (equal to B - S) moves away from the equilibrium curve. The maximum distance that points M, P, and P" can be moved is determined by the slope of the tie lines. The minimum solvent ratio, which corresponds to the minimum reflux ratio, is reached when a tie line and an operating line coincide. A pinch point can occur either in the enriching or in the stripping section of the column, so it is necessary to seek the highest value of the minimum reflux ratio by trial and error. In this example, it occurs at the feed stage. The minimum reflux ratio is 0.58 and the corresponding minimum solvent ratio is 0.74. 

Minimum reflux is based on the specifications for the degree of separation between two key components. The minimum reflux is finite and feed product withdrawals are permitted. However, a column can not operate under this condition because of the accompanying requirement of infinite stages. Nevertheless, minimum reflux is a useful limiting condition. 

This juxtaposition, however, necessarily excludes the point where the X locus intersects the X-value representation (or equilibrium curve), since this would lead to a condition of minimum reflux and infinite stages, as described in the next subsection. 

In the limit, on the one hand, there would be a condition of minimum reflux or recycle, giving infinite stages, and on the other, total reflux or recycle, whereby no finite product streams are obtained. 

It is possible to attain minimum reflux if the shape of the equilibrium curve causes the occurrence of infinite stages. This case is shown in Figure 12-13. 

Figure 12-12. Infinite stages and minimum reflux with intersection of equilibrium curve feed line and rectifying mass balance. (Reproduced with permission from reference 18. Copyright 1997, American Chemical Society.)...

The two limiting reflux conditions in a distillation column are minimum and total reflux. Suppose we have an infinite staged column with a constant binary flow of constant composition. 

Infinite stages is also a limiting condition for absorption column operation. This condition coincides with minimum (L ) liquid rate as shown in Figure 13-3. 

Note that the tangency of the mass balance line with the equilihrium hne causes a pinch point, which gives infinite stages. The significance of the minimum liquid rate is similar to that of minimum reflux in distillation that is, 1.2 to 1.5 times this value is a reasonable operating level for the absorption column. 

As with distillation, the cases of minimum reflux (infinite stages) and minimum stages (infinite reflux ratio) occur in extraction. The minimum reflux takes place in the extract section when an extended tie line meets a 2 as close to Se as possible (in Figure 13-14). Also, for the raffinate section an extended tie line... 

Figure 11.21 Operating line and minimum adsorbent/solvent ratio for infinite stages.

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