Rl xri

Figure 8.2. Quasi-steady State Off-cut Recycle Strategy in Binary Batch Distillation

Mayur et al. measured the benefits of recycling in terms of a potential reduction in batch time. However the benefits may be measured (Mujtaba, 1989) as follows:

1. Reduction in batch time: For a given fresh feed and a given separation, the column performance is measured in terms of minimum batch time required to achieve a desired separation (specified top product purity (x D/) and bottom product purity (x B2) for binary mixture). Then an optimal amount and composition of recycle, subject to physical bounds (maximum reboiler capacity, maximum allowable purity of the off-cut) are obtained in an overall minimum time to produce the same separation (identical top and bottom products as in the case without recycle). The difference in the minimum times obtained with and without recycle shows the benefit of recycling. 2. Increase in productivity: For a fixed reboiler charge it is wished to obtain the optimal amount of fresh feed and the composition of the recycle (note: optimal amount of feed will automatically determine the optimal amount of recycle) to produce a given separation (specified top and bottom product purity for binary mixture) in a minimum batch time. This will give the productivity in terms of fresh feed processed per unit time. For the same given separation and no recycle, the productivity is obtained by processing a charge of fresh feed equal to the reboiler capacity. The difference in these productivities with and without recycle on a fixed reboiler charge basis will show the benefit of recycling.

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition {Rl, xRj) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rrft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x D1) and a residue (87, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR]) and a bottom product (B2, x 82)- Both rj(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xR/ can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR] which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut.

Christensen and Jorgensen (1987) investigated the possible economic impact of off-cut recycling on some difficult binary separations. For tray column they defined a measure of the degree of difficulty (q) of a given separation in a given column (described in detail in Chapter 3). With several binary examples they showed that this measure could qualitatively predict the profitability of using an off-cut recycle. They also measured the benefits of recycling in terms of reduction in batch time. They found that the greater the measure of the difficulty of separation, the larger the benefits of using off-cut recycle.

It is quite clear from the previous works that recycling of off-cut material is particularly interesting if a given separation is to be performed in an existing column. The number of trays in the column may not be quite appropriate for the distillation task at hand, and running the column in a conventional manner (without recycling) may need a very long time. In such a case recycling offers a possibility of reducing the distillation time (Mujtaba and Macchietto, 1993). In addition, recycling an off-cut may be used advantageously to reduce capital investment for a given batch separation by allowing a smaller column to be used than would otherwise be necessary with ordinary operation.

The problem of choosing whether and when to recycle each off-cut and the size of the cut is a difficult one. Liles (1966) considered dynamic programming approach and Luyben (1988) considered repetitive simulation approach to tackle this problem. Mayur et al. (1970) and Christensen and Jorgensen (1987) tackled it as a dynamic optimisation problem using Pontryagin's Maximum Principle applied to very simplified column models as mentioned in Chapters 4 and 5.

Mujtaba (1989) used the measure of the degree of difficulty of separation proposed by Christensen and Jorgensen (1987) to decide whether or not an off-cut is needed. The optimal control algorithm of Morison (1984) was then used to develop operational policies for reflux ratio profiles and amount and timing of off-cuts which minimise the total batch time. A more realistic dynamic column model (type IV as presented in Chapter 4) was used in the optimisation framework.

8.2. Classical Two-Level Optimisation Problem Formulation for Binary

Before going to a detailed formulation of the problem the following discussion is worthwhile:

Consider a problem with no recycle as shown in Figure 8.3 where an initial charge to the column is BO with composition xB0. The charge is separated into two fractions, the overhead as (D7, xD]) and the bottom as (B2, xB2)-The overall mass balance is therefore:

Mixtures

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