A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

A new homotopy-based strategy is presented that can be used in the robust determination of multiple steady-state solutions for continuous stirred tank reactor (CSTR) systems. The strategy relies on the features of homotopy parameter and variables bounding, and requires only that one feasible solution of the system is either known beforehand or can be solved with an existing solving algorithm. The strategy systematically results in all the multiple solutions, or alternatively confirms that the problem does not have multiple solutions, within the predefined problem domain. The strategy was successfully demonstrated with CSTR cases gathered from the literature. Finding all the feasible solutions was verified in simple CSTR systems by applying tools available in the literature. Variables bounding constrained the homotopy path to travel only within the pre-defined variable domain. The strategy is applicable for determining multiple steady states for a variety of chemical engineering systems.

[1]  R. Seydel.  From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis , 1988 .

[2]  B. Wayne Bequette,et al.  Impact of process design on the multiplicity behavior of a jacketed exothermic CSTR , 1995 .

[3]  Willy Govaerts,et al.  Cl_matcont: a continuation toolbox in Matlab , 2003, SAC '03.

[4]  J. W. Kovach.  Heterogenous azeotropic distillaion-homotopy-continuation methods , 1987 .

[5]  Angelo Lucia,et al.  Multivariable terrain methods , 2003 .

[6]  M. Laborde,et al.  Calculation of multiplicity of steady states in a catalyst pellet by homotopic continuation method , 2011 .

[7]  Steven A. Johnson,et al.  Mapped continuation methods for computing all solutions to general systems of nonlinear equations , 1990 .

[8]  Achieving Stability under Inaccessible Conversions Using CSTR Cascades , 2008 .

[9]  R. Aris,et al.  “Yet Who Would Have Thought the Old Man to have had so Much Blood in Him?”—Reflections on the Multiplicity of Steady States of the Stirred Tank Reactor , 1986 .

[10]  Dan Luss,et al.  Multiplicity features of reacting systems. Dependence of the steady-states of a CSTR on the residence time , 1983 .

[11]  Juha Tanskanen,et al.  Modified bounded homotopies to enable a narrow bounding zone , 2008 .

[12]  Ralph E. White,et al.  A Robust Homotopy Continuation Method for Seeking All Real Roots of Unconstrained Systems of Nonlinear Algebraic and Transcendental Equations , 2011 .

[13]  J. Markoš,et al.  Steady States Analysis and Dynamic Simulation as a Complement in the Hazop Study of Chemical Reactors , 2005 .

[14]  Aubrey B. Poore,et al.  The classification of the dynamic behavior of continuous stirred tank reactors—influence of reactor residence time , 1976 .

[15]  A. Yermakova,et al.  Analysis of the CSTR with two-phase flow under phase equilibrium conditions , 2005 .

[16]  L. Razon,et al.  Multiplicities and instabilities in chemically reacting systems — a review , 1987 .

[17]  M. Kuno,et al.  Computing all real solutions to systems of nonlinear equations with a global fixed-point homotopy , 1988 .

[18]  R. Jackson,et al.  General mass action kinetics , 1972 .

[19]  Rüdiger Seydel,et al.  Role of continuation in engineering analysis , 1987 .

[20]  Angelo Lucia,et al.  Global Terrain Methods , 2002, Encyclopedia of Optimization.

[21]  T. L. Wayburn,et al.  Homotopy continuation methods for computer-aided process design☆ , 1987 .

[22]  Jorge R. Paloschi,et al.  Bounded homotopies to solve systems of sparse algebraic nonlinear equations , 1997 .

[23]  J. D. Seader,et al.  Computing multiple solutions to systems of interlinked separation columns , 1987 .

[24]  Juha Tanskanen,et al.  A rigorous minimum energy calculation method for a fully thermally coupled distillation system , 2007 .

[25]  Milan Kubíček,et al.  Book-Review - Computational Methods in Bifurcation Theory and Dissipative Structures , 1983 .

[26]  Jani Kangas,et al.  A new Newton homotopy based method for the robust determination of all the stationary points of the tangent plane distance function , 2012 .

[27]  M. Kubicek,et al.  Modeling of chemical reactors—XVIII stability and oscillatory behaviour of the CSTR , 1970 .

[28]  J. Seader,et al.  Global homotopy continuation procedures for seeking all roots of a nonlinear equation , 2001 .

[29]  Martin Golubitsky,et al.  A QUALITATIVE STUDY OF THE STEADY-STATE SOLUTIONS FOR A CONTINUOUS FLOW STIRRED TANK CHEMICAL REACTOR* , 1980 .

[30]  Rutherford Aris,et al.  An analysis of chemical reactor stability and control—I: The possibility of local control, with perfect or imperfect control mechanisms , 1958 .

[31]  A. B. Poore,et al.  On the dynamic behavior of continuous stirred tank reactors , 1974 .

[32]  Jorge R. Paloschi,et al.  Using sparse bounded homotopies in the SPEEDUP simulation package , 1998 .

[33]  Bruce A. Finlayson.  Introduction to Chemical Engineering Computing , 2006 .

[34]  V. Balakotaiah,et al.  Global mapping of parameter regions with a specific number of solutions , 1988 .

[35]  Dan Luss,et al.  Structure of the steady-state solutions of lumped-parameter chemically reacting systems , 1982 .

[36]  Juha Tanskanen,et al.  Homotopy parameter bounding in increasing the robustness of homotopy continuation methods in multiplicity studies , 2010, Comput. Chem. Eng..

[37]  A. C. Dimian,et al.  State multiplicity in CSTR-separator-recycle polymerisation systems , 2002 .

[38]  Dan Luss,et al.  ANALYSIS OF THE MULTIPLICITY PATTERNS OF A CSTR , 1981 .

[39]  M Shacham,et al.  Exothermic CSTRs. Just How Stable Are The Multiple Steady States , 1994 .

[40]  Achim Kienle,et al.  Multiple steady states in two-phase reactors under boiling conditions , 2003 .

[41]  Manuel Pérez-Molina,et al.  Increasing the reactant conversion through induced oscillations in a continuous stirred tank reactor by using PI control , 2013 .

[42]  Dan Luss,et al.  Global analysis of the multiplicity features of multi-reaction lumped-parameter systems , 1984 .

[43]  Mohd Ali Hashim,et al.  Input Multiplicity Analysis in a Non‐Isothermal CSTR for Acid‐Catalyzed Hydrolysis of Acetic Anhydride , 2010 .

[44]  R. E. White,et al.  A new homotopy for seeking all real roots of a nonlinear equation , 2011, Comput. Chem. Eng..

[45]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[46]  Chen Bingzhen,et al.  Modeling, simulation and analysis of the liquid-phase catalytic oxidation of toluene , 2010 .

[47]  Ilkka Malinen.  Improving the robustness with modified bounded homotopies and problem-tailored solving procedures , 2010 .

[48]  L. Watson.  Numerical linear algebra aspects of globally convergent homotopy methods , 1986 .

[49]  Neima Brauner,et al.  A web-based library for testing performance of numerical software for solving nonlinear algebraic equations , 2001 .

[50]  J. D. Seader,et al.  A generalized framework for computing bifurcation diagrams using process simulation programs , 2001 .

[51]  Dan Luss,et al.  Exact steady-state multiplicity criteria for two consecutive or parallel reactions in lumped-parameter systems , 1982 .