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    of the selected parameter on the total number of stages in
    the column. This graphical method is highly intuitive, fast
    and efficient. However, if there is more than one degree of
    freedom, the more efficient NLP or MINLP optimization
    method should be applied to find the best solution.
    In this paper, we show how the examination of the
    overall separation space can help in the creation of feasibleseparation schemes. We focus on the design of complex het-
    erogeneous azeotropic distillation columns with decanters,
    multiple feeds, and side stream draws, where an extra degree
    of freedom requires a one-parameter optimization in order
    to find the optimal solution among several feasible ones.
    2. Getting feasible designs by examining separation
    space
    Let us consider separation spaces for two examples of
    ternary azeotropic mixtures by labeling the azeotropes,
    distillation boundaries, two-liquid phase regions and the
    corresponding vapor lines. In Fig. 1, the water–acetic
    acid–n-butyl acetate system shows one azeotrope and one
    distillation region (the azeotrope is an unstable node, n-butyl
    acetate is a stable node and the other components are sad-
    dles). In the water–butanol–n-butyl acrylate system, shown
    in Fig. 2, there are three binary azeotropes and one ternary
    azeotrope. These azeotropes give rise to three distillation
    regions (the ternary azeotrope is an unstable node, all bi-
    nary azeotropes are saddles, and all pure components are
    stable nodes).
    In order to characterize a system, one requires knowledge
    of the nature of all the azeotropes and pure components in
    the system. This information is also necessary for distilla-
    tion boundary calculation [3] since, for ternary systems, the
    distillation boundaries are special residue curves that con-
    nect azeotropes. The boundaries between distillation regions
    restrict the products that can be obtained from a simple
    distillation column (one feed, two products). As a rule, both
    products should lie in the same distillation region [4,5].
    Distillation boundaries can be crossed by mixing streams
    or by decanting if the mixture forms multiple liquid phases
    (e.g. the water–butanol–n-butyl acrylate system in Fig. 2).Fig. 2. Separation space for water–butanol–n-butyl acrylate as predicted by
    a NRTL-Dimer model at 26.34 kPa. The system’s azeotropes, distillation
    boundaries, liquid–liquid equilibrium region and vapor line has been
    labeled.
    The information about all azeotropes among a consid-
    ered component set is of critical importance to the design
    of azeotropic distillation systems. We compute tempera-
    tures, compositions and stabilities (stable node, unstable
    node or saddle) of azeotropes predicted by thermodynamic
    models for multicomponent mixtures at a specific pressure
    by the most reliable homotopy method combined with an
    arc-length continuation [6]. In this method, we exploit an
    efficient scheme for finding all stationary points of the boil-
    ing surface, starting from pure component solutions for a
    hypothetical ideal mixture described by the Raoult Law at
    the beginning of the homotopy path (homotopy parameter
    hD0). Then, we gradually ‘add non-idealities’ to our model
    by increasing the homotopy parameter and, eventually, end-
    ing with the rigorous model for hD1 [7]. This calculation
    approach is very robust, since all solution branches are
    connected. The eigenvalues of the Jacobian calculated at
    stationary points give the stabilities of azeotropes and pure
    components. Thereafter, we check the topological consis-
    tency of the residue curve map via the Zharov and Serafi-
    mov topological constraint [8]. The homotopy continua-
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