Minimum Energy Consumption in Multicomponent Distillation. 2. Three-Product Petlyuk Arrangements Ivar J. Halvorsen † and Sigurd Skogestad* Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 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. 1. Introduction In this paper, the minimum-energy expressions for the three-product Petlyuk arrangement 1 shown in Fig- ure 1 are generalized to handle any feed quality and nonsharp product splits. We also illustrate by examples that we can easily handle more than three feed compo- nents. We use the simplifying assumptions of constant pressure, constant relative volatility (R), and constant molar flow and consider the limiting case with an infinite number of stages. The ternary feed (F) with components A (light), B (in- termediate), and C (heavy) is supplied to the prefrac- tionator (column C1), which performs the “easy” A/C split. The minimum vapor flow in the prefractionator column is obtained for a particular distribution of the intermediate B component, denoted as the preferred split. 2 This split also results in a minimum overall ener- gy requirement in the Petlyuk column. Interestingly, this solution is not unique, and several authors, e.g., Fidkowski and Krolikowski 3 and Christiansen and Sko- gestad, 4 have shown that the optimum can be obtain- ed by operating the prefractionator in the whole region between the preferred split and the so-called “balanced” split where the vapor flow requirements in the bottom of column C21 and in the top of column C22 are equal. This implies that there is a “flat” optimality region and that the minimum vapor flow can be obtained not only at a single operating point but also along a line segment in the space spanned by the 2 degrees of freedom. An analytical expression for the minimum vapor flow in a Petlyuk arrangement with a ternary feed and liquid side stream was obtained independently by Fidkowski and Krolikowski 3 and Glinos et al. 5 for the case of a saturated liquid feed (q ) 1) and sharp product splits: Here, θ A and θ B are the two common Underwood roots, obtained from (3) for the prefractionator feed. Fidkowski and Krolikowski 3 derived (1) by a quite detailed algebraic procedure, via expressions for pinch zone compositions at the connection points as functions of the operating point of the prefractionator. Here we will use another approach, more directly based on the Underwood equations. Such an approach was first presented by Carlberg and Westerberg, 6,7 who also extended the solution to more than one intermediate component. An important finding in our work is that the minimum energy requirement (V min Petlyuk ) and the detailed vapor flow requirements may be obtained by just a glance at the V min diagram for a single two-product column. This was presented in part 1 of the series 8,9 and is computed based on Underwood’s equations 10-13 for multicompo- nent distillation in conventional columns. The most im- portant results from part 1 are reviewed in section 2. In the directly coupled sections of the Petlyuk ar- rangement, we have recycle flows from the main column into the top and bottom of the prefractionator. This is a new situation compared to the conventional arrange- ments, and we must really check if Underwood’s meth- * To whom correspondence should be addressed. Phone: +47 73594030. Fax: +47 73594080. E-mail: Sigurd.Skogestad@ chemeng.ntnu.no. † Current address: SINTEF Electronics and Cybernetics, N-7465 Trondheim, Norway. E-mail: Ivar.J.Halvorsen@ sintef.no. V min Petlyuk ) max ( R A z A R A - θ A , R A z A R A - θ B + R B z B R B - θ B ) F (1) Figure 1. Integrated Petlyuk arrangement for separation of ternary mixtures. 605 Ind. Eng. Chem. Res. 2003, 42, 605-615 10.1021/ie0108649 CCC: $25.00 © 2003 American Chemical Society Published on Web 01/01/2003