Although the SC and inside-out methods are reasonably robust, they are not guaranteed to converge and sometimes fail, particularly for very nonideal liquid solutions and when initial guesses are poor. A much more robust, but more time-consuming, method is differential arclength homotopy continuation, the basic principles and applications of which are discussed by Wayburn and Seader [Comp. Chem. Engng., 11, 7-25 (1987); Proceedings Second Intern. Conf. Foundations of Computer-Aided Process Design, CACHE, Austin, TX, 765-
862 (1984); AIChE monograph Series, AIChE, New York, 81, No. 15 (1985)]. Homotopy methods begin from a known solution of a companion set of equations and follow a path to the desired solution of the set of equations to be solved. In most cases, the path exists and can be followed. In one implementation, the set of equations to be solved, call it f(x), and the companion set of equations, call it g(x), are connected together by a set of mathematical homotopy equations:
where t is a homotopy parameter. An appropriate function is selected for g(x) such asf(x) - f (x0), where x0 are the initial guesses, which can be selected arbitrarily. At the beginning of the path, t = 0 and Eq. (13-110) becomes h(x,t) = f(x) -f(x0) = 0 orf(x) = f(x0). The homotopy parameter is then gradually moved from 0 to 1. At a value of t = 1, Eq. (13-110) becomes h(x,t) = f(x) = 0, which corresponds to the desired solution.
The movement along the path is accomplished by a predictor-corrector continuation procedure, where the corrector is often a numerical Euler integration step of the differential-arclength form of Eq. (13-110) along the arclength of the path (rather than a step in t), as proposed by Klopfenstein [J. Assoc. Comput. Mach., 8, 366 (1961)]. The arclength of the path is preferred, over the homotopy parameter, as a continuation parameter because the path may make one or more turns in the homotopy parameter, making it difficult to take an integration step. The predictor step is accompanied by a truncation error that is reduced in the corrector step, which employs Newton's method with Eq. (13-110) to return to the path. If the predictor steps along the path are not too large, the corrector steps always converge.
Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickery and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to actual values so as to solve the MESH equations when very nonideal liquid solutions were involved. Taylor, Wayburn, and Vickery [I. Chem. E. Symp. Ser. No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where little separation occurs, to a higher and more reasonable efficiency.
Continuation methods, also called imbedding and path-following methods, were first applied to the solution of separation models involving large numbers of nonlinear equations by Salgovic, Hlavacek, and Ilavsky [Chem. Eng. Sci., 36, 1599 (1981)] and by Byrne and Baird [Comp. Chem. Engng., 9, 593 (1985)]. Since then, they have been applied successfully to problems involving interlinked distillation (Wayburn and Seader, op. cit.), azeotropic and three-phase distillation [Kovach, III and Seider, Comp. Chem. Engng., 11, 593 (1987)], and reactive distillation [Chang and Seader, Comp. Chem. Engng., 12, 1243 (1988)], when SC and inside-out methods have failed. Today, many computer-aided distillation-design and simulation packages include continuation techniques to make the codes more robust.
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