* Corresponding author. Tel.: #1-617-253-6526; fax: #1-617-253- 9695. E-mail address: pib@mit.edu (P.I. Barton) Chemical Engineering Science 55 (2000) 3835 } 3853 Computation of heteroazeotropes. Part II: e$cient calculation of changes in phase equilibrium structure John E. Tolsma, Paul I. Barton* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 15 March 1999; accepted 13 December 1999 Abstract This paper, the second in a two-part series, discusses how a new approach for the computation of the heteroazeotropes present in a multicomponent mixture can be extended to compute e$ciently the changes in phase equilibrium structure with parameters such as pressure. One particular advantage of the approach is the capability to detect incipient azeotropes and heteroazeotrope (i.e., azeotropes and heteroazeotropes that do not appear under current conditions but may appear under di!erent conditions). The algorithm for computing the azeotropes and heteroazeotropes is analyzed in the "rst paper in this series: Computation of Heteroazeotropes. Part I: Theory. This paper includes a brief discussion of the implementation of this approach and some numerical examples as well as examples illustrating the e!ectiveness in determining the bifurcation pressures at which azeotropes and heteroazeotropes appear and disappear or switch between each other.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Heteroazeotropy; Continuation methods; Bifurcation analysis; Interval arithmetic 1. Introduction Computation of the azeotropes and heteroazeotropes present in a multicomponent mixture is a necessary task when analyzing and designing separation systems. In addition to computing the azeotropes and hetero- azeotropes at a given pressure (or temperature), it is often valuable to know how the azeotropic composi- tion and temperature vary with pressure (or, equiva- lently, how the azeotropic composition and pressure vary with temperature). A systematic approach for ana- lyzing such changes can be incorporated directly into design algorithms, dramatically increasing the space of alternative designs. Given the composition, temperature, and pressure of an azeotrope or heteroazeotrope, standard continuation methods can be applied to the necessary conditions for azeotropy to determine how the state variables change with a given parameter (e.g., temperature or pressure). Applying a phase stability test during the continuation, it is possible to determine at what conditions an azeotrope becomes a heteroazeotrope or vice versa. However, there often arises situations where at the speci"ed temperature or pressure, the azeotrope or heteroazeotrope does not even exist. Furthermore, performing a phase stability test at each point during the continuation is extremely costly. This paper describes a systematic approach for identifying incipient azeotropes and heteroazeotropes and the computation of the bifurcation pressures at which they appear and disappear or switch between each other. Section 2 brie#y describes the algorithm analyzed in Tolsma and Barton (2000a) and includes a brief descrip- tion of the implementation. Section 3 contains a descrip- tion of how the approach can be extended to analyze the phase equilibrium structure under parameter variation. Section 4 includes numerical examples for both the computation of the azeotropes and heteroazeotropes present in a multicomponent mixture as well as examples illustrating the ability to analyze the phase equilibrium structure. 0009-2509/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 3 3 - 6