This paper considers the selection of controlled variables for the unconstrained degrees of freedom, such that near-optimal operation is achieved with constant setpoints ("self-optimizing control"). From a second-order Taylor approximation around the optimal point, we derive an exact local method. This may be used to find the optimal linear combination of measurements to use as controlled variables. We also derive the approximate singular value rule, which is very useful for quick screening and elimination of poor candidate variables.

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We show that the minimum energy requirement for separation of a multicomponent mixture in a three-product Petlyuk arrangement is equal to the minimum energy for the most difficult of the two separations (top/middle or middle/bottom product) in a conventional single column. In the V min diagram (part 1 of this series), this is simply the highest peak. These results are based on an analytical solution for columns with an infinite number of stages, assuming constant relative volatilities and constant molar flows. The previous analytical results for the Petlyuk column are extended to include nonsharp separations, multicomponent feeds, and any feed quality.

The V min diagram is introduced to effectively visualize how the minimum energy consumption is related to the feed-component distribution for all possible operating points in a two-product distillation column with a multicomponent feed. The classical Underwood equations are used to derive analytical expressions for the ideal case with constant relative volatility and constant molar flows. However, the diagram can also be used for nonideal mixtures. The V min diagram is very insightful for assessing multicomponent separation in a single column and is even more powerful for complex column arrangements, such as Petlyuk columns (parts 2 and 3 of this series of papers).

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