NOTE TO USERS This reproduction is the best copy available. TIME OPTIMAL CONTROL OF A HIGH-DIMENSIONAL, NONLINEAR BINARY DISTILLATION COLUMN USING THE LUUS-JAAKOLA OP'MMIZATION PROCEDURE Manoj Rajagopalan A thesis submitted in confomiity with the requirements for the degree of Master of Applied Science. Graduate Department of Chernical Engineering and Applied Chemistry, ia the University of Toronto Copyright by Manoj Rajagopalan 200 1 National Library Bibliothèque nationale I*I ofCanada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 35WeDingon Slreet 395. rue WeUington Ottawa ON K1A ON4 OaawaON KIA ON4 Curado CeMda The author has granted a non- L'auteur a accordé une licence non exclusive licence ailowing the exclusive permettant à la National Library of Canada to Bibiiothèque nationale du Canada de reproduce, loan, disüi'bute or se1 reproduire, prêter, distribuer ou copies of this thesis in microfom, vendre des copies de cette thèse sous paper or electronic formats. La forme de microfichelfilm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts ftom it Ni la thèse ni des extraits substantiels may be printed or otherwise de cde-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Title: Time Optimal Control of a High-Dimensional, Nonlinear Binary Distillation Column using the Luus-Jaakola Optimization Procedure Degree and year of convocation: Master of Applied Science (MASc), 2001 Full Name: Manoj Rajagopalan Graduate ûepartment: Department of Chernical Engineering and Applied Chemistry University: University of Toronto ABSTRACT The dynarnic model of a methanol-isopropanol binary distillation column is set up from first principles with the aim of perfonning a time optimal control study. With 21 stages, the state of each king described by 2 variables, the time optimal control problem is 42 dimensional in its state space, with two control variables: the reflux flowrate and the reboiler heat duty. The model poses many computational challenges in its complex fomulation and the nature and number of constraints on the control and the state variables. It is desired to take the system from an initial steady state to a desired steady state in minimum possible time. The Luus-Jaakola (U)optimization procedure is used to solve this tirne optimal control problem using piecewise constant control and flexible tirne-stage lengths. The complex mode1 is directly used without any simplifications or transformations. In light of the speciai computational challenges involved, the LJ optimization procedure is found to perform very well: the best result obtained takes the system to within 4.319~10~(1- nom) of the desired state in 52.494 minutes employing a &stage control policy. The large computation time requirements limit experimentation with this model and possible avenues for improvements are identifiai. ACKNOWLEDGEMENTS This work was performed under the guidance of Prof. Rein Luus whose advice and suggestions are highly valued, and encouragement, gratefully acknowledged. 1 also express my gratitude to my colleague Mr. Ramasubramanian Sundaraiingarn for his technical advice. corrections and support. Profuse th& go out to my fiends Mr.Harpreet Singh Dhariwal and Mr. Parmjit Singh Kanth for their immensely kind and timely help, and advice on cornputational issues. Financial support from the Naturai Science and Engineering Research Council of Canada and from the University of Toronto in the form of the Differential Fee Waiver. Mary H. Beatty Fellowship and University of Toronto Open Fellowship are gratefuily acknowledged. 1 thank Prof.Mark Kortschot for his invaluable assistance at the time of need. TABLE OF CONTENTS ABSTRACT ............................................................................... ACKNOWLEDGEMENTS ....................................... ...................... TABLE OF CONTENTS ................................................................ LIST OF TABLES ....................................................................... LIST OF FIGURES ...................................................................... MTRODUCTION .................................................................. Part 1: Distillation Column Model DISTILLATION COLUMN DYNAMICS ....................................... 2.1 Ordinary stages .............................................................. 2.2 Extraordinary stages ............................................................. 1.2.1 Condenser ............................................. ......... .... 2.2.2 Feed stage ..................... ............................................. 2.2.3 Reboiler ..................................................................... 3 PROPERTY MODELS ............................................................. 3.1 Vapor-liquid equilibrium. ..................................................... 3.2 Enthdpy ........................................................................... 3.2.1 Saturateci Iiquid phase entfialpy ........................................... 3 -2.3 Mixing eEects ............................................................... 3.2.3 Composition derivative of saturated liquid phase enthalpy ........... 3.2.4 Samted vapor phase enthaipy ........................................... 3.3 Liquid molar overflow .......................................................... Part 11: Time Optimal Contml 4 . THEORETICAL DEVELOPMENT .............................................. 25 4.1 The tirne optimal contrd pmblem ............................................. 25 4.2 The Luus-Jaakola optimization procedure .................................... 27 5 . THE DISTILLATION TIME OPTIMAL CONTROL PROBLEM .......... 30 5.1 Solving for the steady states ................................................... 31 5.2 Constraint handling .............................................................. 35 6. NUMERICAL RESULTS .......................................................... 6.1 Prelirninary computations ...................................................... 6.1.1 Effect of number of random points ..................................... 6.1.2 Effectofthepenaltyfunction factor .................................... 6.1.3 Effectofnumberoftimestages ......................................... 6.2 intense computations .......*..................... ..............*.............* 6.2.1 Effeçt of number of random points ..................................... 6.2.2 Effect of penalty function factor ........................................ 6.2.3 Effect of number of tirne stages ......................................... 6.2.4 The best result obtained .................................................. 7. DISCUSSION ........................................................................ 63 8. CONCLUSIONS ................................................................... 65 APPENDDC A: Degree of freedom analysis of the culumn at steady state 66 APPENDiX B: FORTRAN program ............................................. 68 NOTATION ....................... .... ......................................... 81 REFERENCES ....................................................................... 85 LIST OF TABLES Table 1.1: Summary of assurnptions ........................................................ Table 3.1. Antoine constants for the components .......................................... Table 3.2. Enthalpy seference data .................................... .... .................. Table 33: Coefficients in the specific heat polynomial in temperature ................ Table 3.5. Heat of mixing at 298.15 K [3] ................................................... Table 3.5: Heat of mixing [34] ................................................................ Table 3.6. Coefficients in the enthalpy of vaporization polynomial in temperature ... Table 5.1. Conditions within the column during the initial steady state ................. Table 5.2. Conditions within the column during the final steady state .................. Table 6.1. Effect of the number of random points on result obtained ................... Table 6.2. Effect of penalty Function factor on preliminary results obtained ........... Table 63: Effect of the number of time stages on the resuit obtained ................... Tabk 6.4. Effect of the number of random points on result .............................. Table 6.5. Effect of the number of initially chosen tirne stages .......................... Table 6.6. Control policies hmruns with 400 random points .differing in P ......... LIST OF FIGURFS Fig .2.1: Schematic diagram of an ordinary stage Fig. 6.1. Effect of nurnber of randorn points on convergence for 8= 106 ............... Fig .6.2. Effect of number of random points on convergence for 8= Io8 ............... Fig. 63: Effect of 8on convergence for R = 150 ........................................... Fig. 6.4. EFfèct of 80n convergence for R = 300 ......................................... Fig. 6.5. Final time and convergence profile for the case 8= 10' and R = 300 ......... Fig. 6.6: Cornparison of state trajectories (mole fraction) for different cases of P and R ....................................................................................... Fig .6.7. Cornparison of state tmjectories (liquid holdup) for different cases of P ....