stages are added, the incremental decrease in reboiler heat input gets smaller and smaller. The cost of the shell continues to increase (to the 0.802 power as shown in Table 4.1). Figure 4.1 shows how the variables change with the number of stages.
The total annual cost reaches a minimum of $4,823,000 per year for a column with 44 stages. Thus in this numerical case, the optimum ratio of the actual number of trays to the minimum is 42/15 = 2.8 instead of the heuristic 2. The reflux ratio is 3 at the optimum 44-stage design, which gives a ratio of actual to minimum of 3/2.9 = 1.04 instead of the heuristic 1.2.
These differences may seem quite large and indicate that the heuristics are not very good. However, good engineers always build in some safety factors in their designs.
Building a column that is larger in diameter and has more heat exchanger area than the real economic optimum is good conservative engineering. The number of trays in a column can sometimes be increased by going to smaller tray spacing or installing more efficient contacting devices. But changing the diameter requires a completely new vessel. Therefore, the heuristics give a pretty good design.
It should also be noted that the optimum is quite flat. The TAC decreases only from 5.09 to 4.82 x 106 $/year as the number of stages is increased from the heuristic 34 stages to the optimum 44 stages. This is only 5%.
If the cost of energy is reduced, the optimum number of stages becomes smaller. Using an energy cost of half that assumed above, the optimum number of stages is 42 instead of 44 and the TAC drops from $4,823,000 to $2,980,000 per year. It is clear that energy costs dominate the design of distillation columns.
Stainless steel is used in the cost estimates given in Table 4.1. If the materials of construction were more exotic, the optimum number of stages would decrease.
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