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108 Erika R. Frits, Ali Baharev, Zoltán Lelkes, Mihály Markót, Zsolt Fonyó, Endre Rév, and Tibor Csendes5. SummaryAs it has earlier been shown by researchers of Budapest Univ., Hungary, feasibility of theseprocesses, and the feasible domain of the process parameters, can be well estimated by studyinga system of an explicite autonomous first order ordinary differential equation system coupledwith a system of algebraic equations. Existence and location of the singular points playa crutial role in feasibility of these processes.From mathematical point of view, the main problem consists of reliably finding the existenceand loci of all the singular points in a given domain, and determining the bifurcations ofthe phase map. This problem is solved by applying an interval-arithmetic based branch andbound optimization algorithm developed at Univ. Szeged, Hungary.Feasibility study of extractive distillation variants is a successful application area of intervalarithmetics based reliable computation of zeroes and global extrema of real functions.Singular points in the neighborhood of bifurcation is rather difficult to find because thecomputation time increases enormously. Determination of the bifurcation points in this way,i.e. by repeated computation of singular points and plotting bifurcation diagrams, is unreliable.Instead, criterion of bifurcation is applied, based on linearization of the differentialequation in the neighborhood of the singular points. Implicite function theorem was alsoapplied to compute elements of the Jacobian.The mathematical model is detailed, and the capability of the method is demonstrated onexample results related to the separation of acetone – methanol mixture (forming minimumboiling azeotrope) with water as a heavy boiling entrainer, in both batch and continuous extractivedistillation processes. All the singular points inside (and even partially outside) thephysically valid composition domain are found. Bifurcation points are also found, and domainsof radically different maps are determined, including the feasible domain.AcknowledgmentsThis research was partially supported by OTKA F046282, T048377, T046822, and T037191.References[1] Lelkes, Z., P. Lang, B. Benadda and P. Moszkowicz (1998) ”Feasibility of Extractive Distillation in a BatchRectifier”, AIChE Journal 44 810.[2] Lelkes, Z., E. Rév, C. Stéger and Z. Fonyó (2002) ”Batch Extractive Distillation of Maximal Azeotrope withMiddle Boiling Entrainer”, AIChE Journal 48 2524.[3] Rév, E., Z. Lelkes, V. Varga, C. Stéger and Z. Fonyó (2003) ”Separation of minimum boiling binary azeotropein batch extractive rectifier with intermediate boiling entrainer”, Ind. Eng. Chem. Research 42 162.[4] Csendes, T. and D. Ratz. (1997) ”Subdivision direction selection in interval methods for global optimization”,SIAM J. Num. Anal. 34, 922.[5] Csendes, T. (2001) ”Interval Analysis: Subdivision Directions in Interval B&B Methods” in: Floudas, C.A.and Pardalos, P.M. (eds.): Encyclopedia of Optimization, Kluwer, Dordrecht[6] Knüppel, O. (1993) PROFIL - Programmer’s Runtime Optimized Fast Interval Library. Bericht 93.4, TechnischeUniversität Hamburg-Harburg[7] Hammer, R., M. Hocks, U. Kulisch and D. Ratz (1993) Numerical Toolbox for Verified Computing I. Springer-Verlag, Berlin