batch process design: an overview from controlbdigital.unal.edu.co/51256/1/1039454473.2015.pdf ·...

Batch process design: an overview from control

Christian Camilo Zuluaga BedoyaChemical Engineer

Universidad Nacional de ColombiaFacultad de Minas

Departamento de Procesos y EnergıaMedellın

2015


Christian Camilo Zuluaga BedoyaChemical Engineer

Thesis work presented as partial requirement for the degree ofMaster of Engineering – Chemical Engineering

AdvisorLina M. Gomez Echavarrıa

Doctor of Engineering

CoadvisorHernan D. Alvarez Zapata

Doctor of Engineering

Universidad Nacional de ColombiaFacultad de Minas

Departamento de Procesos y EnergıaMedellın

2015

To my mother ...A mi madre ...

Acknowledgment

I am greatly grateful with my director Lina Marıa Gomez, for her permanent supporting andmotivation to work in this project, and her teaching about science and research, to orientatemy ideas to accomplish my objectives.I would like to thank my coadvisor Hernan Alvarez, who always has been there for an adviceand his positive ideas to improve my academic and scientific life.I extend my acknowledgments to Dynamic Processes Research Group - KALMAN and itsteamwork, for encouraging the active discussion and leaning of research within an open andreceptive environment.I would like to express my sincerely gratitude to Marce, Gloria, Alex, Andres; who throughtheir friendship and fellowship, have contributed with creative ideas and excellent advises.To my friends Juanda, Eli, Fabian, Cindy, Robert, Nina, Yeison, Florez and specially Diana,for their friendship and permanent motivation to achieve this goal.Also, I would also like to express my gratitude to the Universidad Nacional de Colombiafor granting me Academic Fees Exemption Scholarship (“Beca de Excencion de DerechosAcademicos”), and to COLCIENCIAS - Young Researchers Program Scholarship (“Beca Pro-grama Jovenes Investigadores”) without which I would not had the chance of being exclusivelydedicated to this project.Finally, express my gratitude to my mother Luz Mery who taught me to be persistent and myfather Jairo who have motivated me to pursuit my goals without doubt of my abilities and tomy sister Eli and my brother Mauro for staying next to me in all situations.

Abstract


In this work, the topic of batch process design is addressed, through an analysis of thephenomenological-based model and using set-theoretic methods to deduce process constraintsand parametric effects in state controllability. A review of literature is presented about char-acterization of batch process from point of view of design problem. Furthermore simultaneousprocess and control design is also reviewed, considering the main contributions for batch pro-cesses case. The drawbacks of these works are found when batch process characteristics arenot considered, specially the irreversibility. Even design parameters, initial state and batch timeare seldom related with this dynamic characteristic. Within the proposed methodology a statecontrollability index is presented (N ) which takes advantage of control sets characterization,particularly the controllable trajectories set (Tt(Ω0,Ωtf )). This set can give a measurement ofcontrollability and irreversibility of the process. In that sense, the design parameters and initialstate are selected according to controllability index maximization, and designing the processwith these dynamic criteria. Finally, the methodology is applied to a benchmark of a batchreactor, determining the best value for global heat transfer coefficient and initial state. Thisprocedure is also applied to a more realistic case, which is a convective coffee dryer, improvingcontrollability changing size parameters of the equipment.

Keywords: Irreversibility, batch process, state controllability, set-theoretic methods, designparameters, initial state, phenomenological-based model.

Resumen

Diseno de procesos por lotes: una mirada desde el control

En este trabajo se busca abordar el tema de diseno de procesos por lotes, atraves del analisisdel modelo de base fenomenologica y usando los metodos de la teorıa de conjuntos paraencontrar las restricciones del proceso y los efectos parametricos en la controlabilidad deestado. Una revision de la literatura es peresentada acerca de la caracterizacion de los procesospor lotes desde el punto de vista del problema de diseno. Tambien se aborda el tema dediseno simultaneo de proceso y control, considerando las principales contribuciones para elcaso de los procesos por lotes. Las desventajas de dichos metodos son debidas a que lascaracterısticas de los procesos por lotes no son consideradas, especialmente la irreversibilidad.Incluso parametros de diseno, condiciones iniciales y tiempo de lote muy pocas veces sonrelacionados con dicha caracterıstica dinamica. Dentro de la metodologıa propuesta se presentaun ındice de controlabilidad de estado (N ), el cual se apropia de la caracterizacion de losconjuntos en control, particularmente el conjunto de trayectorias controlables (Tt(Ω0,Ωtf )).Este conjunto puede dar una medida de la controlabilidad e irreversibilidad del proceso. En estesentido, se seleccionan los parametros de diseno y condiciones iniciales que otorguen un ındicede controlabilidad maximo, disenando el proceso con dichos criterios dinamicos. Finalmente,la metodologıa es aplicada a un benckmark de un reactor por lotes, determinando los mejoresvalores para el coeficiente global de transferencia de calor y el estado inicial. Esta metodologıatambien fue aplicada a caso mA¡s realAstico, que es un secador convectivo de cafe, mejorandola controlabilidad por medio de cambios en los parametros del equipo.

Palabras clave: Irreversibilidad, proceso por lotes, controlabilidad de estado, metodos dela teor ıa de conjuntos, parametros de diseno , condiciones iniciales, modelo de basefenomenologica

Contents

1. Introduction 1

1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Research problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4. Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5. Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Batch process design 5

2.1. Batch processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2. The design problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1. Unique trajectory problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2. Constrained positioning problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3. Heuristic design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4. Model-based design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1. Batch process modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.2. Model and design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.3. Other uses of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3. Simultaneous process and control design 12

3.1. Simultaneous process and control design . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2. Methods that consider controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3. Methods with simultaneous optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4. Simultaneous process and control design in batch processes . . . . . . . . . . . . . . 143.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4. Irreversibility and controllability in batch processes 16

4.1. Reversibility and controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.1. Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.2. Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2. Dynamics of a batch process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Contents iv

4.3. About controllability of batch processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3.1. Batch-output controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3.2. Nonlinear state controllability: Lie algebra . . . . . . . . . . . . . . . . . . . . 194.3.3. Controllability analysis via set-theoretic methods . . . . . . . . . . . . . . . . 20

4.4. Quantifying controllabillity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4.1. Controllability index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4.2. Key parameters that modify controllability in a batch process . . . . . . . 22


5. Design parameters with optimal state controllability 25

5.1. Design Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2. Model selection and classification of variables . . . . . . . . . . . . . . . . . . . . . . . 265.3. Computation of controllable trajectories set . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3.1. Reachable and Controllable Sets: . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3.2. Controllable trajectories set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4. Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4.1. Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4.2. Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.5. Formulation of optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5.1. Constraint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5.2. Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5.3. Numerical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


6. Applications of proposed batch process design methodology 34

6.1. Batch reactor process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.1.1. Batch reactor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.1.2. Constraint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.1.3. Nominal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.1.4. Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.1.5. Optimal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.1.6. Irreversibility discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2. Convective coffee dryer process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2.1. Coffee dryer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2.2. Constraint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Contents v

6.2.3. Nominal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2.4. Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2.5. Optimal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2.6. Irreversibility discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56


7. Conclusions and future work 58

7.1. General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.2. Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Bibliography 62

A. Convective Coffee Dryer Model 66

List of figures

2.1. Differences among types of operation a) batch process, b) discontinuous pro-cess with material adding (semi-batch), c) discontinuous process with productremoval (semibatch), d) continuous process. . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1. Classification of simultaneous design methodologies [1] . . . . . . . . . . . . . . . . . 13

4.1. Effect of design parameter for Lee et al. [2] batch reactor . . . . . . . . . . . . . . . 224.2. Effect of initial state for Lee et al. [2] batch reactor . . . . . . . . . . . . . . . . . . . 234.3. Effect of batch time for Lee et al. [2] batch reactor . . . . . . . . . . . . . . . . . . . 23

5.1. SPCD algorithm for the proposed methodology. . . . . . . . . . . . . . . . . . . . . . . 26

6.1. Process flow diagram for Lee et al. [2] batch reactor . . . . . . . . . . . . . . . . . . . 346.2. Dynamic response of batch reactor model . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3. Reachable set of batch reactor with nominal parameters . . . . . . . . . . . . . . . . 386.4. Controllable set of batch reactor with nominal parameters . . . . . . . . . . . . . . . 396.5. Controllable trajectories set of batch reactor with nominal parameters . . . . . . . 396.6. Convex controllable trajectories set of batch reactor with nominal parameters . 406.7. Optimization results of batch reactor optimization . . . . . . . . . . . . . . . . . . . . 416.8. Reachable set of batch reactor with optimal parameters . . . . . . . . . . . . . . . . 426.9. Controllable set of batch reactor with optimal parameters . . . . . . . . . . . . . . . 436.10. Controllable trajectories set of batch reactor with optimal parameters . . . . . . . 436.11. Convex controllable trajectories set of batch reactor with optimal parameters . . 446.12. Nominal control sets for irreversibility analysis . . . . . . . . . . . . . . . . . . . . . . . 446.13. Optimal control sets for irreversibility analysis . . . . . . . . . . . . . . . . . . . . . . . 456.14. Process flow diagram for the convective coffee dryer. . . . . . . . . . . . . . . . . . . 466.15. Dynamic response of coffee dryer - Coffee layer (x1 and x3) . . . . . . . . . . . . . . 476.16. Dynamic response of coffee dryer - Drying air (x2 and x4) . . . . . . . . . . . . . . . 486.17. Reachable set for coffee dryer with nominal parameters . . . . . . . . . . . . . . . . . 506.18. Controllable set for coffee dryer with nominal parameters . . . . . . . . . . . . . . . 516.19. Controllable trajectories set for coffee dryer with nominal parameters . . . . . . . 516.20. Convex controllable trajectories set for coffee dryer with nominal parameters . . 52

LIST OF FIGURES vii

6.21. Optimization results of coffee dryer optimization . . . . . . . . . . . . . . . . . . . . . 536.22. Reachable set for coffee dryer with optimal parameters . . . . . . . . . . . . . . . . . 546.23. Controllable set for coffee dryer with optimal parameters . . . . . . . . . . . . . . . . 546.24. Controllable trajectories set for coffee dryer with optimal parameters . . . . . . . . 556.25. Convex controllable trajectories set for coffee dryer with optimal parameters . . 556.26. Nominal control sets (red: Reachable Set, black: Controlable Trajectories Set)

for irreversibility analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.27. Optimal control sets (red: Reachable Set, black: Controlable Trajectories Set)

for irreversibility analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.1. Process flow diagram for for convective coffee drying. . . . . . . . . . . . . . . . . . . 66A.2. Process system diagram for convective coffee drying. . . . . . . . . . . . . . . . . . . 67

List of tables

6.1. Model parameters for batch reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2. Model parameters for coffee dryer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.1. Model parameters of coffee drying process . . . . . . . . . . . . . . . . . . . . . . . . . 68

List of symbols and abbreviations

Latin letters

Symbol DescriptionX Admissible states setU Admissible control action setN Controllability indexR Reachable setC Controllable setT Controllable trajectories setJ Objective function for optimization problemf Nonlinear function modelS Number of samples according to Chernoff boundd Euclidean distance between two pointst Timeu Manipulated variables vectorul l-th manipulated variablex State variables vectorxk k-th state variabley Output variables vectoryk k-th output variable

Greek letters

Symbol Descriptionθ Design parameterΘ Design parameter setη Hyper-volume of controllable trajectories setδ Probability associated to Chernoff boundε Error associated to Chernoff boundλ Weighing trade-off coefficientφ Transition mapping of a systemΩ State variables set

Subscripts

Symbol Descriptiontf Final condition

List of symbols and abbreviations x

Symbol Description0 Initial condition

max Maximum valuemin Minimum value

nominal Nominal valueτ Arbitrary time instant

Superscripts

Symbol Description− Backward in time∗ Optimal value

Abbreviations

Abbreviation DescriptionARX Autoregresive ModelFIS Fuzzy Inference System

ANN Artificial Neural NetworkNPW New Present WorthLTI Linear-Time-InvariantEPP Equipment-Process-Plant

SPCD Simultaneous Process And Control DesignSQP Sequential Quadratic ProgrammingSA Simulated Annealing

ODE Ordinary Differential EquationsMIDO Mixed-Integer Dynamic OptimizationMINLP Mixed-Integer Non-Linear Programming

CHAPTER 1Introduction

1.1. Motivation

Batch processes are a considerable part of the chemical process industry. In these processesmaterials with high economical value are processed, such as pharmaceutics and polymers.Also, they are used when raw materials are toxic while transformation occurs. Batch processeshave characteristics that permit high flexibility, making different products in the same set ofequipment (multi-product) or multiple tasks in same units (multi-purpose).The main motivation for this investigation is that batch processes have dynamic propertiesthat nowadays are only mentioned, without going deeper into control system theory. A firstchallenge is to complement the irreversibility understanding in batch processes [3], lookingfor tools from control system theory and set-theoretic methods with the aim to characterizethe design problem of a batch process. That means, finding the optimal values of designparameters, initial states and batch time which can extend controllability, flexibility and repet-itivity. Finally, this proper operation of the equipment will lead to raw material savings, fewerdiscarded runs and energy savings.Regarding simultaneous process and control design methodologies in batch process, the re-viewed literature are only focused in simultaneous optimization approach, without taking intoaccount dynamic properties as controllability or reversibility [4; 5; 6; 7]. Only the work ofGomez-Perez et al. [8] gives a methodology for trajectory design considering control aspects.Another motivation for this work is that nonlinear controllability establishes that weak re-versibility is a requirement for a system to be controllable [9]. However, batch processes areinherently irreversible, because there is a time from which transformation can not return to itsinitial value. Despite this behavior, batch processes react to an input excitation, which meansthat a certain input (control action) can affect the states of the system [10].The research of Garcıa [11] is a valuable advance in the area of batch processes with the studyof the optimal initial states for controllability purposes. Analogous to continuous processes,where Alzate [12] developed a methodology for simultaneous design of process and control forcontinuous plants, this thesis will develop an approximation for batch process design consideringstate controllability.

1.2. Research problem

Batch processes play a fundamental role inside industry, because they are the direct way toscale-up a process from laboratory to pilot or industrial scale [13]. This gives an advantageto produce different products in the same equipment (multi-product) and saving operativecosts compared with continuous processes. Particularly, batch processes have a time-varying

1

CHAPTER 1. Introduction 2

operation point, nonlinear behaviors and they are inherently irreversible [10]. This featuresgenerate questions from control theory framework and which pose as starting point for newresearches.When chemical (continuous or batch) processes are designed, dynamic components are notaccounted for and sometimes the implementation of a control system will require a considerableeffort, for achieving an acceptable performance [14]. In batch process design there are somesequential steps: (i) design of the necessary equipment for fulfilling the production demands,(ii) planning ofcapacity for assigning the different units to produce the required products, (iii)units time-sequence scheduling, and (iv) unit operation to meet safety, quality and productivityrequirements. [13]. This implies that when batch processes are designed, their irreversiblebehavior is not taken into account. This fact has many significant consequences in reachingthe desired product quality, and unfortunately rejecting high amounts of inadequate product,affecting environment and the economy of the company.Moreover, current design methodologies do not give information about how and to what extentthe design or operational parameters affect controllability of the process. Considerations aboutirreversibility and initial state effects are absent in design algorithms. With this background,there are some difficulties in batch process design area:

1. There are only a few works where dynamic properties and features of batch processes arestudied and analyzed, with the objective to consider them for process design [10; 15].

2. The current approach of simultaneous process and control design is pretty general andno distinction is made between its applicability to continuous or batch processes [16; 17].

3. Recent works in simultaneous process and control design in batch processes address thedesign problem as a trajectory tracking, so they focus in solving an optimization problem[4; 5].

4. The existing controllability index based on control set theory implies the manipulationof large data sets, thus demanding huge computational efforts [11].

These facts evidence the need for a precise methodology for batch process design, that includestheir dynamic properties such as controllability and irreversibility. Therefore, the next researchquestion guides the present work:Is it possible to design a batch process, using a phenomenological-based model andset-theoretic methods to obtain the design parameters and initial conditions thatmaximize state controllability, achieving a controllable process?

1.3. Objectives

The objectives that conduct this thesis are:

General objective:

Propose a methodology to design a batch process, through the study and determination of therelevant design parameters, maximizing the state controllability from set-theoretic methodsframework, and allow obtaining a controllable process design.


Specific objectives:

1. Describe the main characteristics of batch processes, that complicate their design andcontrol, compared with continuous processes.

2. Identify recent developments in the area of batch process design and its integration withcontrol system design.

3. Select a reference problem (benchmark) decribed by a phenomenological-based modelof a batch processes, classifying the design parameters.

4. Adapt concepts of the control set theory for evaluating state controllability for batchprocesses using an adequate index.

5. Propose a methodology of simultaneous process and control design for batch processes,determining the effect of the design constraints in the state controllability of the process.

6. Evaluate the proposed methodology with a simultaneous design of the process and controlin the benchmark problem.

1.4. Main contributions

The main outcome of this work is a new methodology for batch process design, through theaccounting of the effects of design parameters and initial state to state controllability. With thedefinition of controllability index (N ), a relative percent of improvement is calculated basedon nominal design parameters, contributing to obtain a controllable design. Additionally, themain publication related to this work is:

Conference papers:

1. Zuluaga-Bedoya, C. C. and Gomez, L. M. (2014). Dynamic considerations in process andcontrol design integration in batch processes (in Spanish). In 24 Congreso Argentinode Control Automatico AADECA. Buenos Aires, Argentina.

1.5. Thesis outline

This work uses principles of modeling in chemical engineering and elements of set theory todevelop a batch process design methodology. The thesis is organized as follows:

• In Chapter 2 the principles of batch process design are exposed, with these intentions:

. Presenting two design problem approaches: one with unique trajectory design anda positioning problem. The last one is the used in this thesis coupled with set-theoretic methods.

. Reviewing the main approaches of batch design, one based on heuristic criteria andother in the process model. Also, process model can give information to use inother areas, like is explained.


• Chapter 3 a review of literature for simultaneous process and control design is presented.These are the main purposes:

. Describing the main definitions of simultaneous process and control design. Thisis when process design procedure is combined with dynamic analysis and controlsystem design, considering economic and controllability criteria.

. Exposing two approaches of integrated design and control: methods based oncontrollability or performance indexes and methods derived from simultaneous op-timization.

• Chapter 4 presents the main frameworks of controllability and a definition of irreversibilityin batch processes. In that sense, the next points are treated:

. Reviewing the concept of irreversibility in batch process, and how controllable tra-jectories set can account some of this topic.

. Introducing set-theoretic methods permits a controllability quantification. Withthis approach, analysis over design parameters can be developed, in order to selectthe design parameter with the highest impact in controllability.

. Remarking that not only design parameters change the controllable trajectoriesset, also the initial states and batch time are fundamental because they affect thereversibility of the process.

• Afterwards, Chapter 5 propose the methodology for batch process design consideringcontrollability of the process. These items are highlighted:

. Showing how reachable and controllable sets are calculated using Monte Carlosampling and then how controllable trajectories set is computed.

. Presenting a controllability index based on controllable trajectories set, that isbasic for optimization purpose, due to it accounts the variability respect to nominalparameter.

. Remarking that the objective function presented can include controllability andeconomic criteria, using a weighing coefficient.

• In Chapter 6 the proposed methodology is applied to a batch reactor and a coffee dryer,in order to analyze their controllability, standing out the next points:

. Describing both process model and their variable classification.

. Defining the constraint sets to compute reachable and controllable sets.

. Calculating controllable trajectories set and measuring controllability from the pro-posed index.

. Formulating the optimization problem in order to find the optimal design parametersand initial states for both processes.

• In Chapter 7 the general conclusions of this work are presented with the main contribu-tions and advantages. Besides, some items are presented as future work.

CHAPTER 2Batch process design

This chapter remarks main developments in the area of batch process design, from theoreticaldesign to heuristic thumb rules in order to get a proper design. In first place, in Section 2.1 abrief description of batch processing is given, with common arrangements in batch equipment.Also, a distinction between batch and continuous processing is presented. Then, in Section 2.2two points of view respect to batch design objectives are shown. In Section 2.3 presents anoverview related to criteria for achieving a preliminary design. Then, Section 2.4 presentscommon uses of a process model when a batch process is designed. Finally, Section 2.5presents some concluding remarks.

2.1. Batch processing

Batch processes are characterized by raw material is loaded in predefined amounts and trans-formed through a specific sequence of process activities (recipe) for a given period, either usinga single unit or multiple unit arrangement [10; 11]. The complete terminology and standardsrelated to batch processes are found in ANSI/ISA-88.01-1995.Batch processes also include process where feed or outlet flow is intermittent. This kind ofprocesses is known as semi-batch processes. Generally, batch operation requires consideringload time of raw material, starts, runs, and discharge time of the product, and often cleaningand maintenance times.In the other hand, continuous processes are those where raw materials are transported in acontinuous flow through the equipment while products leave the equipment without intermit-tence. Once a continuous process reaches an operating point, the process is time-independent,until a new operating point is forced or a disturbance is applied [15].Batch processes are commonly used in pharmaceutic, foods and polymer industries, where ahigh product quality is required. Sometimes is better to select batch operation over continuousone. In [18] some advices and recommendations are presented in order to select a batch process:

• Low volume of production. Under 500 ton/year , a batch process should be selected.

• When product is seasonal or raw material is only available during certain periods.

• Products with short lifetime, due to inventory is difficult to implement.

• When reactions taking place have slow kinetics, and a batch process is the only feasibleoption.

• Process with high fouling rate, due to batch processes must have cleaning and mainte-nance times among runs.

5

CHAPTER 2. Batch process design 6

• When process design is hard to achieve for low raw material transformation.

As the main difference between batch and continuous processes is the presence of inlet oroutlet reactant/product flows, the Figure 2.1 shows types of batch processes and a continuousprocess in order to give a better explanation.

Heating/cooling fluid outlet

Heating/cooling fluid inlet

Materialinlet



Material outlet





Material outlet

Materialinlet

Figure 2.1. Differences among types of operation a) batch process, b) discontinuous process with ma-terial adding (semi-batch), c) discontinuous process with product removal (semibatch), d)continuous process.

In the same way, there are three main features that highlight batch processes in process designstudies:

1. Flexibility: For batch processes, temperature profile and run batch time can be modifiedin order to get different products and specifications. Similarly, semi-batch processes areincluded on batch processes class, but they have an additional degree of freedom, whichis the input raw material flow, controlling the advance rate of the transformation [19].This flexibility can be used to reject fluctuations or quick changes in demand [20]. Inaddition, batch processes have a high robustness to the lack of knowledge of the process[21].

2. Multi-product equipment: Batch processes can be adjusted to process different productsin the same set of equipment. Multi-product characteristic is the best advantage ofbatch processes over continuous processes [18]. This can be achieved by sharing thesame units for different products, and even if the market demand ask for it, it is possibleto operate a batch plant in a parallel multi-product arrangement. This must considerthe proper plant scheduling for each unit.

3. Multi-purpose equipment: Unlike as happens in continuous processes, where every pro-cess unit has only a defined task, batch plants allow using process units when different


task or objectives are settled. In the work of Gorsek and Glavic [18] a batch plant designanalysis with multi-purpose equipment was proposed, concluding that depending on pro-duction volume, a batch plant with this feature can be more efficient than a plant withcontinuous operation. However, implementation of multi-purpose equipment requires anoptimal arrangement and location for the units, in order to reduce connection cost.

2.2. The design problem

Batch process design has four steps, where some research gaps can be identified: the first stepconsists in sizing of the equipment depending on market demand, the second step is aboutplanning of required capacity according to the number of available units and types of productsto process, the third step that is related to scheduling or getting the operational sequence foreach set of equipment, and the final operation step that guarantees the proper performanceof the process, maintaining the quality specifications of the product [13].In the first step, equipment is commonly designed using standards design parameters [22]but without considering dynamic features of batch processes. Then, when a control system isimplemented after process design, the process could have a low controllability. For that reason,it is important for designing a batch process to considerate dynamic behavior and its analysisthrough a dynamic property such as controllability and stability. In this way, after designingthe process will have high controllability and a variety of control system could be implemented.From dynamic point of view, a designed batch process must ride along a trajectory that reachesthe desired final state, namely, the state when all desired specifications are met. However, whenthe batch process has not been designed there is no defined trajectory. Then, this analysis hasto be focused from two points of views, as it is explained below.

2.2.1. Unique trajectory problem

Most authors treat batch process design as an optimization problem, where manipulated vari-able is set in a specific trajectory, in order to push the states variables until desired properties.This final properties are related to product quality or a required conversion. Most of works ofmodel-based batch process design, only find this optimal trajectory that meets all constraintsand it is explicit, so it can be achieved using a control system. Optimal control problem arethe main approach in this type of design problem.In [22] a brief discussion is presented about trajectories and optimal control sequences. In thatwork, two design problem are presented, maximum conversion and minimum time problems.Both return an optimal trajectory as a result. For a given volume, the optimization findsthe values of batch time (or conversion) that maximize (or minimize) the objective function.These problems are subject to path and end point constrains, and they are solved using dynamicprogramming.

2.2.2. Constrained positioning problem

Another framework to approach design problem is to analyze processes from set-theoreticmethods, that means, considering all constraints of the batch process (physical, economic,safety, etc.) and define a set of feasible trajectories. After that, the process can be drivenalong this set, and if it goes out, then final state will not be reached. So, if we look the problem


as a positioning objective at the final state, then the process could follow any trajectory insidethe set (not always the same), and it would reach the final desired state and meet the requiredspecifications.Analogous to design, Gomez [15] proposes a discussion to change the control problem in a batchprocess, from tracking control to positioning control, where do not matter states trajectoriesguaranteeing that always it will have an admissible control action, to take the process to thefinal desired state. This is done using a model predictive control with knowledge of set ofcontrollable trajectories.The work developed in this thesis exposes a proposal for design problem, with the objective tomake wider this set, improving controllability. Then a control system can be designed to meetcontrol objectives.

2.3. Heuristic design

In process engineering, batch process design has been studied mainly from heuristic criteria, re-lated with the steps of sizing procedure of the process equipment and even control-manipulatedvariable pairing. These rules do not involve neither phenomenological-based models usage noranalysis from dynamic properties such as controllability or stability. Particularly, in batch plantdesign other topics could be included such plant capacity, scheduling, multi-purpose equipment.The last one is an advantage of batch over continuous processes.In the hierarchical methodology exposed by [23], it is advised to start the batch process designas it would be a continuous process. Consequently, a second step is to replace a continuousunit by a batch unit, considering inventory calculations and optimal cycle times for each unit,in order to reduce the net annual cost of the whole process. After that, is relevant to check fornearly equal cycle times and size factors, to merge units in a multi-task equipment. This allowthe designer to select the best process flow diagram and checking operability in a computationalsimulator.Other works, such [24], highlight that understanding of process leads to a recipe optimization,including the required tasks for process synthesis, depending on the type of plant that is selected(single product or multi-product). Simultaneously, scheduling analysis is planned using Ganttcharts [21], combining cycle times and process stages. Later, parallel units are analyzed andthen studies of inventory are performed. Finally size factors are determined [20] and a flowsheetis selected according to net present worth (NPW) estimates. Also, scheduling problem hasbeen formulated mathematically as an optimization problem.

2.4. Model-based design

This section explains how modeling is developed from chemical process engineering and howto use these models to get many designs and other types of applications as control design andoptimization.

2.4.1. Batch process modeling

Chemical process modeling is fundamental for analysis and design tasks. Some objectives ofmodeling are prediction, control, signal processing, design, optimization and fault detection


[25]. Specifically in batch process, a proper design must include information given by the model,such as kinetic phenomena (reaction, adsorption, etc.), transport phenomena, thermodynamicequilibrium, and mainly material and energy balances [20]. However, it is not always possibleto get those detailed information about the process, so information based on experimentaldata is also used. The development of batch process models is still an open research topic,because available models are not able to represent a variety of operational conditions [19].

Types of models

In [26], two types of models are used in batch process design: physic-chemical models whichare used to specify the task in each equipment, and dynamic models that are used for studyingplant capacity scheduling. These models can have a high level of information, and shouldbe evaluated at simulation environment, finding important design parameters [27]. In [22]the importance of using simple models to preliminary design step is remarked, where somevariables can be kept explicit, in order to adjust the optimal operation of the system. Whena optimization framework is set, the use of detailed material and energy balances model isneeded.According to Ljung [28], dynamic models are classified into three groups: white-box models,black-box models and gray-box models.

• White-box models: They also are called phenomenological-based models or first prin-ciples model. This kind of models only depends on phenomenological abstraction ofthe process. That means, a full knowledge of the process is needed and it obeys toconservation laws and transport or transference principles.

• Black-box models: They are based on pure empiric knowledge from experimental data.Both structure and parameters are empiric and they do not have any direct mathematicalrelation with phenomena that happen in the process. However, they can give informationabout operational regimes, with the objective to generate a mathematical structure thatcontains parameters adjusted with experimental data [25]. In this group, auto-regressivemodels (ARX), which are linear and only depend on analyzed. Also, NARMAX modelsbelong to this group, which are nonlinear and include analyzed variable and exogenousinput to the model. Additionally, there are some models developed in artificial intelligenceschemes, such artificial neural networks (ANN) or fuzzy logic inference systems (FIS)[15].

• Gray-box models: This type of models combines the process understanding from mathe-matical abstraction of phenomena and the usage of experimental data. Generally, thesemodels have a structure based on phenomenology of the process and their parametersare adjusted with the available experimental data. They have an average interpretationdegree, so they are suitable for batch process analysis and design.

Specifically chemical process models are addressed in [25; 29].

2.4.2. Model and design

The leading works in design of batch processes based on dynamic model are focused on findingpathways and trajectories, operation times, that leads the transformation to a high productivityof the desired product. This formulation is abstracted to a optimization problem that solvesthe optimal operation policies [22].


In [6; 30] an optimization problem is presented using operational envelopes of the batch process,this means, finding the regions of variables that guarantee a feasible and profitable operationin the process. This strategy permits that operational variables are kept inside permissiblelimits, achieving desired objectives.Despite there are some works relating batch process model and its design, there is no a certaincriterion to select design parameters of the process. Another important fact, is that few worksrelate dynamic characteristics of the batch process with the design step and which are onlyaddressed when control system is designed. This is in part due to the concept of designproblem explained before, namely, the main design strategies have the objective to define aunique trajectory over the run time.

2.4.3. Other uses of the model

Besides process design, a dynamic model can be used to develop control, optimization or scale-up calculations. In the last case, Monsalve-Bravo [25] proposed a methodology for scale-up ofbatch processes, through the usage of Hankel matrix of the discrete process model.Control problems are one application for process model. Regardless what type of controller(feedback, feedforward or combined), process model can give information for controller pre-diction. In the batch process case, model predictive control is the most used strategy due tosome variables are not available for measurement or only can be known at the end of the run.This controller commonly uses gray-box models coupled with a moving time-horizon to makepredictions about process behavior.Another usage for process model is the state estimation, due to measurement of all variables isneither possible nor feasible for high costs of sensors and budget limitations. Furthermore, thecontrolled variables affect final product quality, so this measurement only can be obtained whenrun has finished [19]. There are some state estimation schemes whose model can be trans-formed in order to get an unique dependency of measured variables. For a further descriptionof state estimation in batch process refer to [31].

2.5. Concluding remarks

This chapter has focused on the theoretical aspects of the design of chemical processes, espe-cially, batch processes. Here, it is worth highlighting:

• Batch processes have time-varying features that difference them with continuous pro-cesses, and when production is low, batch processing could be more efficient than con-tinuous.

• Two design problem approaches were considered: a unique trajectory design and a po-sitioning problem. The last one can view all the feasible trajectories, and then definingthe desired final product specifications.

• The heuristic approaches do not consider dynamic aspects of the process and they areonly focused on previous designs, experimental data or some scales relations like sizefactors. They are centered in scheduling and capacity designs.


• Model-based design are the most suitable for batch process design, because they accountconstraints and time-dependence of the process. Depending on the type of the selectedmodel, it will give a good estimation of process behavior.

CHAPTER 3Simultaneous process and control design

Simultaneous process and control design (SPCD) is still an open research topic around processcontrol, and its main advantage is that an integrated design will favor the operation andefficiency of the process. In Section 3.1 the basic definition of simultaneous process andcontrol design is exposed. Then, two approaches of integration are presented in Section 3.2and Section 3.3. Afterwards, batch process works in the area are addressed in Section 3.4.Finally, in Section 3.5 some conclusions are presented.

3.1. Simultaneous process and control design

When chemical processes are designed, there is a trend to not account dynamic characteristicsof the process, and sometimes it turns difficult the implementation of control systems in thoseplants, investing lots of money trying to control them [14]. The objective of simultaneousprocess and control design is to join the design of control and process design at the same time,favoring that the equipment operates inside a controllable region, trying that disturbances andset point changes do not throw the system out this region.Specifically, there are some definitions of simultaneous (integrated) design of process andcontrol and the following are highlighted:

• Alvarez [32] defines simultaneous design of process and control as the task developedsimultaneously the equipment, process or plant (EPP) design and the control systemdesign for this EPP.

• Simultaneous design of process and control is a methodology which initial stages ofdesign are integrated, not only economic criteria, also controllability indexes and othercriteria [14].

• The research area that combines steady state design and considerations of dynamiccontrol in an optimization is called integrated design and control [16].

• The use of economic relations of steady state, and dynamic controllability in all stagesof process development is called simultaneous design [33].

Those definitions given above explains that simultaneous design, integrated design of processand control or integration of design and control are all the same concepts, and they are referredto the coupling of process design procedures with control system design.The simultaneous design classifications have been covered by several authors [16; 17; 32; 34;35]. In a general way, there are two types, a first approach where controllability indexes areused in the process design, taking advantage of empiric procedures that only consider a set of

12

CHAPTER 3. Simultaneous process and control design 13

scenarios or arrangements. In the other hand, it is the simultaneous optimization approach,where both designs are generalized in a dynamic optimization problem. Some authors, classifythis last branch in a wider group.The Figure 3.1 shows other classification for simultaneous design of process and control:

Simultaneous design of process and control

Controllability indexes

Optimization-basedmethods

Mixed-integer dynamic optimization (MIDO)

Embedded control approach

Robust theory approach

Black boxOptimization

Optimization based on controllability index

Selection of one of different arrangements

Figure 3.1. Classification of simultaneous design methodologies [1]

3.2. Methods that consider controllability

Most of the advances in that kind of methodologies are focused in developing new indexes forcontrollability, robustness and stability, improving the convergence of optimization problem.Also, another parallel focus is the use of sequential methodologies to face the problem, thatmeans, a set of successive solutions are found in the same simultaneous design problem. In [36]a cascade approach is shown, with dynamic nature due to decisions taken in previous stagesinfluence design decisions of subsequent stages design decisions. These stages are processsynthesis, instrumentation of the whole process and finally process control [36].In [16] a simultaneous design method is proposed, and it approximates dynamic nonlinearprocess model using a model with uncertainties. Then robust stability is evaluated, in order tocalculate the process variability. They also highlight that the biggest problem of integration ofdesign and control is to find the worst case, for conditioning of the optimization problem.Other authors have taken other way to generate new concepts that favor the simultaneousdesign. In [37] practical local controllability is defined, which is based on state controllabilityof Kalman [38] and compared with input-output controllablility of classical control theory, andthen using this information for integration of design and control.Lately, new researches point to generate new indexes for state controllability, that quantifyhow controllable is a process [12] and not only return a binary answer.


3.3. Methods with simultaneous optimization

The research of [39] remarks the importance to use dynamic optimization approaches whentraditional designs can get unstable solutions, which with a MIDO problem can suggest aredesign of the unit. In this optimization the complete nonlinear model is discretized withthe objective to get a form of mixed-integer nonlinear programming (MINLP) optimizationproblem, where convexitiy must be guaranteed to find global solutions. If the system couldnot get that form, this methodologies only will give local solutions.In [35] two types of design are considered: sequential design and simultaneous design, findingthat for the last one better results in economical and controllability matters. Besides, the ideaof optimal control is applied instead of fixing a control scheme like PI controller. There aretwo types of simultaneous optimization problems: (i) Optimal control, where the MinimumPontryaing Principle is addressed, and (ii) where optimal control is merged to design problemand then all is solved using sequential quadratic programming (SQP).Finally, in [34] a simultaneous design is presented focused in MIDO approach (see Figure3.1). Also an advanced control system is proposed using a parametric programming technique.Another important fact is to consider parametric uncertainty [40], because batch processeshave this characteristic, and sometimes is complex due to parameters can change as time goeson.Many papers have been developed in classification and reviews of simultaneous optimizationin the area of integration of design and control, some those are [1; 17].

3.4. Simultaneous process and control design in batch processes

Regarding batch processes, the research material about simultaneous design is very limited.Most of the works are only focused on the optimization methods of the Figure 3.1, takingaway the concepts of controllability, without going deeper on control structures, only findingthe optimal profiles of the manipulated variables.In [4] a simultaneous design for a batch distillation column is proposed. In the first step,feasible operation regions are found, using a short-cut method. Then a dynamic optimizationis executed, formulated as nonlinear programming problem (NLP). However, Low and Sorensen[5] argues with this approach, because the optimization regions are non-convex and this canreturn a local minimum, so they propose resolve the same problem but using genetic algorithms.Furthermore, [6] present an interesting approach of multi-objective optimization, fixing a per-missible ranges for operation parameters, which can guarantee the proper operation of theprocess. This also permits a low variability with control disturbances.Finally, in [7] a semi-batch styrene polymerization is designed, using an algorithn of mixed-integer dynamic optimization (MIDO) that join control actions and design parameters, guar-anteeing constrained operation and the feasibility in final state.In a different approach, Gomez-Perez et al. [8] gives a way to select an optimal trajectory fora batch process taking into account state controllability criteria. This publish uses set theoryin control to calculate Available Control Action set, in order to get an optimal input profile.However, it is only related to find the best profile without taking design parameters of theprocess.



In this chapter an introduction to simultaneous process design and control has been presented.In the case of batch processes all related researches are pretty specific and they do not generalizebatch process as well as continuous ones. Here are some remarks:

• Simultaneous process design and control is when process design procedure is combinedwith dynamic analysis and control system design, considering economic and controlla-bility criteria.

• Two approaches of integrated design and control are taken: some methods based oncontrollability or performance indexes and methods derived of simultaneous optimization.

• The advances in indexes-based methods are only new proposals to account some proper-ties like stability, controllability and robustness. However, most of them work orientatedto input-output framework, namely, under classic control theory.

• Simultaneous optimization approaches are almost formulated as nonlinear dynamic op-timization, where answer is an optimal profile of manipulated variable.

• The advances in simultaneous design in batch processes are too specific and there is nogeneralization about integrated design considering dynamic characteristics.

CHAPTER 4Irreversibility and controllability in batch processes

As it was presented in the previous chapter simultaneous design and control strategies are onlyfocused in continuous processes, and they neither formalize controllability in batch processesnor establish differences between batch applications considering batch process features.In this chapter, irreversibility and controllability notions are exposed in the Section 4.1. After-wards, Section 4.2 presents a generalization of batch processes dynamics and features. Laterin Section 4.3, a review of the methods for nonlinear controllability is presented, highlight-ing set-theoretic methods due to their capacity to controllability quantification (Section 4.4).Finally, some conclusion of this chapter are presented in Section 4.5.

4.1. Reversibility and controllability

From a dynamic point of view reversibility and controllability are concepts which definitions areapplied regardless to what kind of processes are formulated. So, it is necessary to give basicdefinitions and understand the reasons why they are not directly applicable to batch processes.Let us consider the nonlinear system (4.1):

.x(t) = f(x(t), u(t)) (4.1)

where t ∈ R+, x(t) as state vector and u(t) is the control action. With x(t) ∈ X ⊆ Rm,f (·, ·) : X × U → X and an arbitrary time horizon tf ≥ 0. Let U be the admissible controlaction set and X the admissible state set, which are compact and simply-connected. Also, itis assumed that f is bounded and Lipschitz. Then, for each t ∈ [0, tf ], x ∈ X , u ∈ U existsan unique solution x(·) : [0, tf ] → X . For each τ ∈ [0, tf ], the solution path is given byφ(0, τ , x, u).Besides, consider the following backward-time system:

.x(t) = −f(x(t), u(t)) (4.2)

where for each τ ∈ [tf , 0], x ∈ X , u ∈ U exists a solution in backward-time, given byφ−(τ , tf , x, v) = x and v = u(tf − τ).

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CHAPTER 4. Irreversibility and controllability in batch processes 17

4.1.1. Reversibility

Before going into batch processes controllability, it is necessary to define reversible systems.According to [9] a strongly reversible system means that if a state x0 can drive to state z0, thesame path can be traveled backward.A better explained concept is formulated according to [9; 10].Weakly reversibility: A weakly reversible system exists iff the following statements areaccomplished:

• x(t) can be reached from x(t).

• If x(t + N) can be reached from x(t), x(t) can be reached from x(t + N).

• If x(t + p) can be reached from x(t) and x(t + N) can be reached from x(t + p), sox(t + N) can be reached from x(t).

Then, a system is irreversible if is not reversible, namely, the system can not return to an initialstate, due to nonexistence of an admissible control action sequence.Another important fact about reversibility is that a system can contains reversible or irreversiblestate variables. With a presence of at least an irreversible state variable the full system will beirreversible.The irreversibility in a batch process can be acquired by existence of an irreversible statevariable in the whole state vector, namely, once a specific time has reached, this state variablecan not return to its initial value. In [10] a definition of irreversibility in batch process is given.Irreversibility of batch processes: Let .x(t) = f (x(t), u(t)) be a nonlinear system, with anyi = 1, ..., n and for all t ≥ 0, it holds .xi(t) ≥ 0 or .xi(t) ≤ 0; that means, .xi is a increasing ordecreasing function for all x ∈ X and all u ∈ U .

4.1.2. Controllability

Controllability is a dynamic property that describes capacity to control a variable using a certainmanipulated variable. A first definition of state controllability was presented in [38]:State controllability: An state is said to be controllable if there exists a control signalu1(t) defined in time interval 0 ≤ t ≤ t1, such the transition mapping φ(0, t1, x, u1) = 0. Ingeneral, time t1 depends on x. If each state is controllable, it is said that plant is “completelycontrollable”.

Particularly, linear state controllability of batch processes can not be evaluated as continuousprocesses, because linear analysis requires linearization around equilibrium point, and in batchprocesses there is no such equilibrium points.In nonlinear case, many authors refers to reachability as the capacity to reach a desired statewith certain control action sequences. Reachability is equivalent to controllability in lineartime-invariant (LTI) systems, but in nonlinear systems, controllability is the ability to reach astate, being an equilibrium point.Moreover, Sontag [9] established that a controllable system must meet two requirements: fullaccessibility rank and weakly reversibility. Reachability in batch processes can be evaluated


using Lie algebra method, but due to their irreversible behavior, controllability are not possibleto be evaluated.In next sections both methods are presented, finding that they are not able to give informationabout the irreversibility of batch processes and consequently of controllability. Even though,if a batch process is excited with an admissible control action sequence, it reacts to thisinput, which means that there is a capacity of control. So, it is needed a different notion ofreversibility, than can validate controllability analysis.

4.2. Dynamics of a batch process

From dynamic context, a batch process has many significant differences respect to a continuousprocess, especially due to its time-dependency nature. These differences are hardly ever takeninto account when control schemes for batch processes are proposed. The main features ofbatch processes are summarized as follows [13]:

1. Time-varying characteristics: Unlike continuous processes, batch processes have tran-sient behavior along all run time. Also, they can perform sequential task that involvedifferent dynamic responses.

2. Nonlinear behavior: Batch processes have high variable interaction that leads to a cou-pling effect and non-minimum phase dynamics.

3. Irreversible behavior: This feature is caused by run history of a batch process, namely,once a state is reached, it is not possible to execute corrective actions in order to returnto its initial state.

4. Limited corrective actions over time: This effect reduces the effectiveness of the con-troller on batch process. When a batch is started there is a high range of correctiveactions, however this capacity is decreased as time goes on.

5. Repetitive nature: Batch processes are run many times and between them model cor-rections are performed. This nature allows other uses of previous run data.

4.3. About controllability of batch processes

The classic notion of the controllability concept was made by [38], who developed state spacerepresentation for linear time-invariant systems. That methodology gives a simple binaryanswer about controllability (controllable or not controllable). However, due to time-varyingnature of batch process, this classic definition can not be directly used. For that reason, someauthors have researched a compatible method to evaluate controllability of batch processes.

4.3.1. Batch-output controllability

In [41] it is said that controllability represent the corrective actions existence problem thatimproves desired process performance, independently whether those control action are appliedat the batch run or between runs. When controllability of run-time variable is calculated, itimplies if a specific trajectory can be followed by state variables, at least in a given vicinity.


That is called a realizability problem. In the other side, controllability of run-end variablesis related to reach a certain final condition, which it probably would be a variant of statecontrollability.Thus, with a batch process representation of the following manner, permits define batch-outputcontrollability [41]:

.xk(t) =F (xk(t), uk(t), vk), xk(0) = xick (vk) (4.3)

yk(t) =H(xk(t), uk(t), vk) (4.4)zk =H(xk [0, Tk ], uk [0, Tk .], vk) (4.5)

Definition: Taking y ik , i = 1, ..., p as i-th output of the system (4.3) to (4.5) and r i

as the constant relative grade ∂∂uk

d j y ik

dt j = 0 ∀j < r i , ∂∂uk

d r i y ik

dtr i 6= 0∀t. The system (4.3) to(4.5) is locally batch-output controllable from t0, around the independent trajectories y[t0, T ]and the final output z, it there is a δy , δz > 0 such that for all ‖yk [t0, Tk ] − y[t0, T ]‖L <

δy , yk [t0, Tk ] ∈ C r i−1 for i = 1, ..., p, ‖zk − z‖ < δz , exists a uk [t0, Tk ] ∈ U and vk ∈ Vthat drives the outputs yk [t0, Tk ] and zk .This kind of controllability can be analyzed by linear tools (linearization).

4.3.2. Nonlinear state controllability: Lie algebra

Despite research about nonlinear state controllability is open, it is possible to verify it locallyusing differential geometry approach. A limitation of differential geometry is that only canbe applied to nonlinear control-affine systems, and nowadays mathematical models are morecomplex and coupled. Another disadvantage of differential geometry is that requires evaluationof Lie brackets, that have a high computational complexity.The following proposition is presented in [42]:Proposition: Let ∆ be a non-singular involutive distribution with dimension d and it isassumed that ∆ is invariant under the vectorial fields f , g1, ..., gm of the system .x = f (x) +g(x)u = f (x)+∑m

i=1 gi(x)ui . In the other hand, suppose that the distribution spang1, ..., gmis in ∆. Then, for each point x0 it is possible to find a vicinity U0 of x0 and a local coordinatestransformation z = Φ(x) defined in U0 such that the system is represented in the new set ofcoordinates:

.ζ1 =f1(ζ1, ζ2) +

m∑i=1

g1i(ζ1, ζ2)ui (4.6).ζ2 =f2(ζ2) (4.7)yi =hi(ζ1, ζ2) (4.8)

After that, reachability analysis can be evaluated according to [42; 43]:


∆0 =spang∆1 =∆0 + [f ,∆0]

=spang , [f , g ]∆2 =∆1 + [f ,∆1] + [g ,∆1]

=spang , [f , g ], [f , [f , g ]], [g , [f , g ]]

∆k =∆k−1 +m∑

i=0[gi ,∆k−1]

where [f , g ] = ∂g∂x f (x)− ∂f

∂x g(x).When ∆k has the same dimension of the system, then system is reachable. If there is anysingularity point, it is necessary to find the points of state space representation that makes ∆knon-singular. This need a coordinate transformation and solving the partial derivative problemwith the method of characteristics.The main issue is that according to [9] a system has to be reachable and weakly reversible tobe controllable. Batch processes are inherently irreversible so differential geometry approachis not feasible to apply due to reversibility is not guaranteed.

4.3.3. Controllability analysis via set-theoretic methods

In order to analyze state controllability of batch processes via set-theoretic methods, is nec-essary to define the reachable set, the controllable set and the controllable trajectories set[10; 11].Reachable setReachable set shows where is the reachability of the system according to admissible controlaction set U . It depends on initial state (Ω0) and it tends to become wider as time goes on.

Definition (Reachable set in t time from Ωτ set, Rt(Ωτ )). Given Ωτ , the reachable setRt(Ωτ ) from Ωτ in a time t > τ is the set of all state vectors x, such that there existsx(τ) ∈ Ωτ and u(·) ∈ U such that x(t) = x, where U is the admissible control actions set andX the admissible state set:

Rt(Ωτ ) =z ∈ X | ∃x ∈ Ωτ ∧ u ∈ U : z = φ (τ , t, x , u) (4.9)

Controllable setControllable set is computed in order to analyze which of the admissible state set (X ) drivethe system to the desired final state set (Ωtf ). For different final state set, controllable set isgoing to change.Definition (Controllable set in t time from Ωτ , Ct(Ωτ )). Given Ωτ , the controllable set Ct(Ωτ )toward Ωτ in a time t < τ is the set of all state vectors x, for whose exists u(·) ∈ U such thatif x(t) = x then x(τ) ∈ Ωτ .

Ct(Ωτ ) =z ∈ X | ∃u ∈ U : φ (τ , t, z , u) ∈ Ωτ (4.10)


Controllable trajectories setControllable trajectories set can show what trajectories of reachable and controllable set drivethe system from initial state (Ω0) to the desired final state (Ωtf ). This set also gives a notionof reversibility of the process, because these trajectories indicate that a state can travel eitherforward or backward while it is inside this set.Definition (Controllable trajectories set from Ω0 to Ωtf , Tt(Ω0,Ωtf )). Sequence of sets Ωtof reachable states at a t time from Ω0, the initial states set, and controllable ones at atf − t time, from Ωtf , the final states set; such that there exists an admissible control actionsequence, such that for all initial state x(0) ∈ Ω0 it is possible to guarantee that system canreach a final state x(tf ) ∈ Ωtf through a state trajectory x(·) ∈ X .

Tt(Ω0,Ωtf ) =x ∈ X | x ∈Rt(Ωτ ) ∧ x ∈ Ct(Ωtf )∀t ∈ [0, tf ] (4.11)

In that sense, controllable trajectories set represents a way to account controllability, becauseit has reachable states and reversible states, so controllability is guaranteed inside it. However,from physical point of view once an irreversible transformation, lie an irreversible reaction hasbeen happened, it is not possible to return it.

4.4. Quantifying controllabillity

As it was presented set-theoretic methods can account for reversibility and controllability of abatch processes, so this method is going to be used for controllability quantification. This givesan advantage because controllability evaluation are not only a binary answer, it will return aspecific value for controllability, which can be used later for optimization in the design problem.

4.4.1. Controllability index

Some authors have addressed controllability using a long variety of indicators or indexes. How-ever this indexes are only related with input-output controllability and fundamentally for con-tinuous processes [16; 17]. Moreover, this indicators are calculated from linear models, thatcan not give information about irrevesibility.Other indexes, related with control sets, are defined in [12; 44]. This indexes are for continuousprocess, which are reversible. Also, in [44] different kind of indexes are proposed. One relatedwith size and other related with shape of the reversible set boundary. In [11] an size-index forcontrollable trajectories set is proposed for irreversible systems, which is defined as follows:Definition (Controllability hyper-volume from Ω0 to Ωtf , ηTt(Ω0,Ωtf )). Hyper-volume in Rn+1

occupied by all state vectors that are reachable from Ω0, the initial state set, and that arecontrollable toward Ωtf , the final state set; such that there exists an admissible control actionsequence u(·) ∈ U , such that for all initial state x(0) ∈ Ω0 it is possible to guarantee thatsystem can reach a final state x(tf ) ∈ Ωtf through a state trajectory x(·) ∈ X .

ηTt(Ω0,Ωtf ) =∫ tf

0

∫Tt(Ω0,Ωtf )

1dxdt (4.12)


The controllability hyper-volume ηTt(Ω0,Ωtf ) is the volume that bounds all controllable trajec-tories for a system, namely, any state vector x ∈ ηTt(Ω0,Ωtf ) will be reachable from Ω0 andtoward Ωtf . In that way, controllability acquires a quantifiable value, that allows it integratingin analysis problems and optimization.

4.4.2. Key parameters that modify controllability in a batch process

There are some key parameters that make batch processes different of continuous ones. Thefirst type of key parameter is the design parameters, that despite on be shared with continuousprocesses, they acquire a different connotation when there is an irreversible behavior [3]. Thesecond one is the initial state conditions, which are only important in batch processes, dueto they act as disturbances for the process. Finally, the last key parameter is the batch time,which depends on desired product specifications.

Design parameters:

The effect of design parameters are critical for process dynamics. It is very common to applyonly steady state calculations when a process is designed. As a illustrative example, the batchreactor model from [2]. This model is detailed in Section 6.1. With these models, severalsimulations were performed in order to analyze variations of reversible (x2 : T ) and irreversible(x1 : Ca) variables change with key parameters. In the Figure 4.1, a short trial with differentvalues of design parameter is presented. Depending on how large is heat transfer parameter,the dynamics of the batch process change respect to nominal values. Sometimes, this is anadvantage because a less effort of control action is required.

Figure 4.1. Effect of design parameter for Lee et al. [2] batch reactor

Initial state:

Similarly, initial state affects irreversibility of the process. In the Figure 4.2, this effect isrepresented. If a reversible state variable is initialized in a different value, it would reachdifferent final condition without changing control action, but possibly a correct sequence ofcontrol action can return it to its nominal value. If the initialized variable is irreversible, it onlycan be affected by another reversible state variable.


Figure 4.2. Effect of initial state for Lee et al. [2] batch reactor

Batch time:

Another key parameter is batch time, because when the run progress, the effects of correctiveactions are significantly decreased. If the physical process is analyzed, there is a point wherea longer run time results impractical (complete conversion or reactant depletion). Also, if thecontrol action and initial state are the same, the only alteration due to batch time is in thefinal states. In the Figure 4.3, the variation between batch time and state variables is shown.

Figure 4.3. Effect of batch time for Lee et al. [2] batch reactor


This chapter has focused on the study of irreversibility and controllability of batch processesand how a special notion of controllability (via set-theoretic methods) can give a quantifiedvalue for design purposes. Finally, there are some aspects to stand out:


• Batch processes have a high dependency of time, initial conditions and internal distur-bances. These effects is caused by a highly irreversible behavior and it is difficult to getsimple control strategies with desired performance.

• Controllability in nonlinear systems is an open branch of research and new procedures areneeded for its evaluation. From the available methods, only set-theoretic methods arerobust to work with the majority of nonlinear process model, without making coordinatetransformations.

• As set-theoretic methods permits a controllability quantification, a recent index is pre-sented accounting the hyper-volume of the controllable trajectories set. With this index,analysis over design parameters can be developed, in order to select the most impactabledesign parameter.

• Not only design parameter change the controllable trajectories set, also the initial statesare fundamental because they affect the reversibility of the process. Also, batch time orrun time is important when design is proposed.

CHAPTER 5Design parameters with optimal state controllability

This chapter describes the methodology for batch process design, considering irreversibilityand controllabilty. This is achieved by analyzing design parameters in order to get a control-lable design of the process. In first place, Section 5.1 presents the algorithm of the proposedmethodology. In Section 5.2 the basic concepts for model selection and possible design param-eters are presented. Afterwards, Section 5.3 shows how to compute reachable set, controllableset and controllable trajectories set. Then, in Section 5.4, two criteria for simultaneous pro-cess and control design are presented before the optimization problem formulation. Next stepis to formulate the optimization problem, subject to constraint sets and defining the desiredobjective function (Section 5.5). Finally in Section 5.6 the main conclusions of this chapterare presented.

5.1. Design Algorithm

In the Figure 5.1 the design algorithm of this methodology for simultaneous process designand control for batch processes is presented.In first place, model selection is performed. This step should be coherent with variable ofinterest for the designer. Secondly, a model classification is done, selecting parameters andinput variables. Next, constraint sets are settled according to process limitations, quality ofraw materials and economic restrictions over the process operation. Consequently an optimiza-tion loop is performed, where iteration design parameters are fixed, reachable and controllablesets are computed, controllable trajectories set is found through crossed-intersection and thenhyper-volume approximation is calculated with convex controllable trajectories set. With thisvalue objective function is computed. This loop is repeated until convergence criteria is full-filled.

25

CHAPTER 5. Design parameters with optimal state controllability 26

Is J minimum?

Yes

END

BEGIN

Choose a phenomenological-based model for the batch

process

Classify parameters and variables of the model

Evaluate J

Calculate Controllable

Trajectories Set

Calculate Reachable Set

Calculate Controllable Set

No

Define admissible set X and U

Fix θ parameter inside ϴ set

Figure 5.1. SPCD algorithm for the proposed methodology.

5.2. Model selection and classification of variables

For a proper process design and operation, it is necessary to have a phenomenological-basedmodel which gives information about main variables for the designer. This process can bedecisive, because the selection of a large-scale model with high interactions among variables,will lead to a deficient solution or will difficult the set computations.Once the model has been selected, a classification is suggested, in manipulated variables, statevariables and all parameters of the model. These parameters can be divided as design pa-


rameters and operational parameters. The design parameters include equipment sizing (areas,volumes, diameters) and settings or equipment arrangements, etc. The operational parame-ters include kinetic equations, constitutive equations and other parameters that are not ableto change in the design step.

5.3. Computation of controllable trajectories set

With design parameters selected, the limit conditions for those parameters are needed (Θ),depending on physical, operational or budget limitations. Another important step is to definethe admissible control action set (U) and the admissible state set (X ). The admissible stateset depends on quality, safety and economic criteria. Firstly, reachable and controllable setneed to be computed and intersected, in order to get the controllable trajectories set.

5.3.1. Reachable and Controllable Sets:

Reachable set (Rt(Ω0)) is able to find all possible states where the system can be driven ina specific time tf , following the control action sequence, from the initial state x0. The formaldefinition was presented in Section 4.3.3. Reachable set can size the effect of control actionover state variables. For doing the calculation, a randomized sampling (Monte Carlo) is appliedin admissible control action set. Then, the process model is solved for those sequences, usingan ODE solver. The Algorithm 1 is applied to find the reachable set for a nonlinear model, ina time t and an initial condition set Ω0

Algorithm 1 Computation of reachable set from Ω0 in a time t

Given Ω0, f , X , U , ε and δ :

1. Begin.

2. Find S, the sample size given by the error ε and a failure probability δ, through the boundof Chernoff [10; 45]:

S ≥ 12ε2 log 2

δ(5.1)

3. Generate S random samples uj ∈ U con j = 1, 2, ..., S.

4. Calculate rj = φ(0, t, x, uj).

5. End: the point cloud formed by rj is the representation of Rt(Ω0) with error ε andfailure probability δ.

6. Return: Rt(Ω0)

Controllable set (Ct(Ωtf )) is found when all points of the admissible state set make, in aspecific time, through a sequence of admissible control actions, take the system to the desiredfinal state set. The formal definition was presented in Section 4.3.3. This set can analyzethe capacities of admissible control actions set when the process must be in desired conditionin the final time (tf ). The calculation of controllable set the system model has to be solved


backward in time, through a ODE solver. As reachable set, controllable set also need arandomized control actions input. In the Algorithm 2 is used for the calculation of controllableset.

Algorithm 2 Computation of controllable set to Ωtf in a time t

Given Ωtf , f , X , U , ε and δ :

1. Begin.

2. Find S, the sample size given by the error ε and a failure probability δ, through the boundof Chernoff with Equation (5.1).

3. Generate S random samples vj ∈ U con j = 1, 2, ..., S.

4. Calculate cj = φ−(t, tf , x, vj).

5. End: the point cloud formed by cj is the representation of Ct(Ωtf ) with error ε andfailure probability δ.

6. Return: Ct(Ωtf ).

When a large-scale system is used or there is a high coupling effect of the variables, analternative algorithm could be used to calculate controllable set to (Ωtf ). Sometimes, thenumeric algorithm for solution of ODE system does not robust for backward integration, sothis alternative approach should be used. Therefore, a MonteCarlo sampling is made overadmissible state set, and taking as initial state (xt0). Then the system is solved forward intime and then a selection of the controllable set point is made. In the Algorithm 3 thisalternative computing is presented.

Algorithm 3 Alternative computation of controllable set to Ωtf in a time t

Given Ωtf , f , X , U , ε and δ :

1. Begin.

2. Find S, the sample size given by the error ε and a failure probability δ, through the boundof Chernoff with Equation (5.1).

3. Generate S random samples vj ∈ U con j = 1, 2, ..., S.

4. Sample the admissible state set with s random samples xk ∈ X con k = 1, 2, ..., S.

5. Calculate c∗j = φ(t, tf , xk , vj).

6. Check which trajectories cj of c∗j arrive to desired final state.

7. End: the point cloud formed by cj is the representation of Ct(Ωtf ) with error ε andfailure probability δ.

8. Return: Ct(Ωtf ).


5.3.2. Controllable trajectories set

Controllable trajectories set is defined the intersection of reachable and controllable sets. Theformal definition was presented in Section 4.3.3. For its computation, a crossed-intersectionis required, taking both point cloud of the reachable and controllable set for each time step.Then each point of the controllable set is used to calculate the distance to each point of thereachable set. After, those distances are compared against a maximum distance criterion, todecide what points belong to the controllable trajectories set. The Algorithm 4 presents howcontrollable trajectories set is computed.

Algorithm 4 Computing of controllable trajectories set from Ω0 to Ωtf in a time tf :

Given Ω0, Ωtf , f , X , U , ε and δ :

1. Begin.

2. Compute Rt(Ω0).

3. Compute Ct(Ωtf ).

4. Compare each point of Rt(Ω0) with the whole vector Ct(Ωtf ) using the Euclideandistance:

dj =[ n∑

k=1(xk,R − xk,C)2

]1/2

(5.2)

5. Check dj smaller than dmin, and append xk,C to T (Ω0,Ωtf ).

6. End: This approximation for Rt(Ω0)⋂Ct(Ωtf ) with an error ε, probability δ and distancedmin.

7. Return: Tt(Ω0,Ωtf ).

5.4. Design criteria

The design of batch processes should include both controllability and reversibility criteria.Despite batch processes are irreversible, controllable trajectories set gives an idea of this ir-reversibility, it means, instead of reaching initial point after evolution, process should reachdesired final state.

5.4.1. Controllability

A set-theoretic framework is used to quantify controllability. An indicative of changes incontrollable trajectories set is the basic criterion for determine changes in controllability.Definition (controllability index): The controllability index is a relation between the differencebetween the current hyper-volume of controllable trajectories set with the iteration design


parameter and the hyper-volume of the set with nominal parameters, all over the hyper-volumeof the set with nominal parameters.

In the Equation (5.3) the controllability index is exposed.

N =ηTt(Ω0,Ωtf ,θ) − ηTt(Ω0,Ωtf ,θ0)

ηTt(Ω0,Ωtf ,θ0)× 100 =

∫ tf0∫Tt(Ω0,Ωtf ) 1dxdt − ηnominal

ηnominal× 100% (5.3)

This index give a estimate of the change respect to nominal or initial dynamic conditions. Thatmeans, when the index is below 0, a decrease in the hyper-volumes is observed. If the index isabove 0 an increase in controllability is achieved. If the index is equal to 0, the controllabilitydid not change. This index is only valid when the hyper-volume of the system with nominalparameters is not equal to zero.

N =

> 0 increase0 equal< 0 decrease

Here it is worth clarifying that all hyper-volumes must be calculated according to Equation(4.12), using normalized states and normalized time. As the numerical ODE solver gives atrajectory by discrete time, the hyper-volume can be approximated as a polytope volume,assuming that the obtained controllable trajectories set is convex. This is due to the definitionof the set boundary will not give much information for the purposes of this methodology andwould require area calculation. Another reason is that it cloud take a high computationalburden for each iteration step of the optimization problem.

5.4.2. Irreversibility

As it was explained before, controllability explains possibility of a system to travel from Ω0 toΩtf and return to initial condition. In the case irreversible is not possible to return to initialcondition.An irreversible system is a batch process whose characteristics were explained in the Section 4.2and in this processes the aim is to drive system to a desired final state, and not return to initialstate. Nevertheless, it is desired to drive the system through a safe trajectory that guaranteesthe desired final condition.The controllable trajectories set is computed by intersection of reachable set from the initialstate with controllable set to final state. In reversible systems like continuous processes orrobotic systems, initial and final states are the same, so controllability is guaranteed whenreachable set is included in controllable set. In irreversible systems (batch processes) thatresult is impossible to achieve but from design parameters the size and shape of the controllabletrajectories set can be changed.


5.5. Formulation of optimization problem

During the simultaneous design, it is necessary formulation of an optimization problem, maxi-mizing the controllability index defined above (see Equation 5.3). For a defined constraint setand an objective function, numerical optimization can be solved.

5.5.1. Constraint sets

The problem requires a proper definition of the constraints and limits values of the designparameters. In these regions the optimal parameter value is found. Also, this step is neededfor three calculation of control sets (admissible states set). All trajectories that goes out ofadmissible states set are rejected. This could be quite subjective, but if we consider that inall iterations this constraint sets will not change. That permits a comparison among iterationswith same process model.Design parameter set: The design parameter bounds are defined a priori to design dependingon physical and economic constraints. Sometimes, a design parameter must be coupled withothers. For example, if a batch reactor has diameter as design parameter in this methodology,then heat transfer area, height and agitator diameter must be coupled. Other way to studycoupling is to repeat the methodology for different design parameters and then find the bestvalues for each one. This also can be used to determine which design parameter has a highereffect in the controllability.

Θ = φ ∈ Rn | θmin ≤ φ ≤ θmax (5.4)

Moreover, bounds of initial state need a specification taking quality of raw material and con-ditions for a safe operation of the equipment.Admissible states set: the set X , in a first glance, is limited by non-negativity constraint ofstate variables, however, a subset of Rn

+ was chosen, based in engineering criteria that give apermissible operation. This set must be compact and simply connected.

X = z ∈ Rn+ | xmin ≤ z ≤ xmax (5.5)

Admissible control actions set: this set must have limits where the transformation canbe speeded up or slowed down. This interval is determined by engineer experience and inaccordance with final control element limitations. This set must be a compact set.

U = u ∈ R|u ≥ umin ∧ u ≤ umax (5.6)

5.5.2. Objective function

The definition of the objective function is determined by controllability, reversibility and eco-nomic criteria. However, a better characterization of reversibility changes in controllable tra-jectories set is needed. An alternative is calculating the size of the portion of controllable


trajectories set inside reachable set. The work of Alzate [12] made a similar deduction forcontinuous processes. Regarding economic part it depends of which kind of costs are included,and how are they affected by the design parameter or initial condition.The iteration variables are the design parameter, and the initial state. These are coupled inthe variable θ and it can move along Θ set.In the other hand, most of the optimization algorithms minimize, but in that case a maximiza-tion is needed. So the sign of the objective function is changed. With that stated two kind ofobjective function could be used:

• Controllability objective function: this function only considers the controllability indexas optimization criteria, taking away economic part. This is useful, when preliminarystudies are done, or when analyzed costs do not depend of design parameters.

J = −N (5.7)

• Combined objective function: this type of function has compensation coefficient λthat relates magnitudes of controllability and economic terms.

J = −N + λc(θ) (5.8)

where c(θ) is the cost in function of design parameter. In this last formulation is thatλ can be used as weighing coefficient that gives a equality magnitude degree betweencosts and controllability.

Then the optimization problem formulation is represented by the Equation (5.9). In thisexpression, restriction dim(Ωtf ) > 0 refers to the numerical limitation about the final statecan not be a point. That is due to the probability of intersect a single point is zero.

minθ∈Θ

J (5.9)

s. t..x = f(x, u, θ), x(0) = θ

dim(Ωtf ) > 0x ∈ Xu ∈ U

5.5.3. Numerical optimization

Numerical optimization is performed using stochastic methods, because the sets generatedby objective function are non-convex, in order to guarantee a global optimal value. So theuse of global optimization methods is required, because this methods find global instead oflocal minims, without violating system constraints. Another reason is found, if it is considerthat hyper-volumes are calculated from a point cloud, there is no mathematical expression


to get gradient or Hessian matrix, which are necessary elements of an optimization throughgradient-based methods.Among global optimization methods, there are aspects to consider: (i) the evaluation of theobjective function has a considerable level of computational burden, so global algorithms thatdiscards some trial points (population) are not suitable to use, (ii) It is necessary to define aconvergence criterion for the algorithm used, some methods are based on objective functiondecreases and other are based on a maximum iteration number and (iii) the method shouldhave an effective local minima skip strategy, because this formulations could have many localsolutions.


This chapter has proposed the methodology for the simultaneous design of process and controlin batch processes. There are some items to stand out:

• The proper model selection involves a phenomenological-based model that includes themain variables of interest and design parameters.

• Reachable and controllable sets are calculated using Monte Carlo sampling and then atreatment to calculate controllable trajectories set is done.

• Controllability index is basic for optimization purpose, due to it accounts the variabilityrespect to initial parameter.

• The constraint limits are defined based on physical, economical, quality and safety mat-ters. The admissible control action set is also influenced by limitation of the final controlelements.

• Global optimization algorithms are required when control sets are treated, because thereis no an explicit expression that permits get a gradient of the objective function.

CHAPTER 6Applications of proposed batch process design methodology

This chapter exposes the implementation of the proposed methodology on two examples. InSection 6.1 a benchmark of batch reactor is considered. The phenomenological-based modeldeveloped by Lee et al. [2] is presented and a parameter uncoupling is shown in order toease the design parameter selection. Constraint sets and simulation settings are fixed to solveoptimization problem. The second application, presented in Section 6.2, is a convective coffeebeans dryer. The model is developed using material and energy balances. Some assumptionsare applied to simplify the model. After that, two design parameters are selected, constraintsets are specified and optimization is performed.

6.1. Batch reactor process

This first application is intended to give a general and easy understanding of methodology. Dueto its simpleness and wide applications in literature, this process can favor the visualization ofcontrol sets and their changes when design variables are adjusted.

6.1.1. Batch reactor model

This reference problem consists in a batch reactor where reaction A→ B is taking place, It isexothermic and has a cooling jacket with negligible dynamics [2]. In the Figure 6.1, a processflow diagram is shown.

x1=CA

x2=Tu=Tj

Figure 6.1. Process flow diagram for Lee et al. [2] batch reactor

• Process modelThe Equations (6.2) and (6.1) that govern the model are specified by [2]. It is assumedthat jacket has an instant dynamic, so the model is represented by reactant concentration

34

CHAPTER 6. Applications of proposed batch process design methodology 35

x1 and reactor temperature x2. The input u is the cooling fluid temperature that passesthrough the jacket.

dx1dt =− k0e−E/Rx2x2

1 (6.1)dx2dt =− UA

MCp(x2 − u) + (−∆H)V

MCpk0e−E/Rx2x2

1 (6.2)

The parameters of the model are shown in Table 6.1

Table 6.1. Model parameters for batch reactor

Symb. Value UnitUA/MCp 0.09 min−1

(−∆H)V /MCp 1.64 KL/molk0 2.53× 1019 L/mol/minE/R 13550 Kx1(0) 0.9 mol/Lx2(0) 299 Ktf 4 min

where UA/MCp is the heat transfer term, (−∆H)V /MCp is the reaction heat term, k0 isthe reaction kinetic coefficient, E/R is the activation energy term, x1(0) is the nominalinitial state for concentration of reactive A, x2(0) is the nominal initial state for reactortemperature, and tf is the batch run time.

• Model simulationThe model presented was simulated in MATLAB ® using the built-in function ode15sand for a batch time of 4 min. This time was selected considering previous works [15]in order to not affect productivity. In Figure 6.2 the simulation result is shown, for twostate variables and a constant input of Tj = 313 K .


0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

Tem

pera

ture

[di

men

sion

less

]

0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

0.3

Time [min]

Con

cent

ration

[di

men

sion

less

]

x1

x2

Figure 6.2. Dynamic response of batch reactor model

Another important fact is that all simulations were performed under normalized variables(in X ), with the purpose to speedup the numerical algorithms. The reason is thatnormalized variables allow to get an equal magnitudes in each variables.

• Uncoupling of design parameterThe process model shown before in Equations (6.2) and (6.1) contains coupled parame-ters that do not have physical interpretation as a whole term. Therefore, a uncoupling ofthis terms is needed, with the aim to ease the design parameter selection and definitionof their bounds.Suppose a reactor volume of V = 0.1 m3 and that the liquid inside reactor has a densityof ρr = 866 kg/m3. Also assume that reactor heat capacity is Cpr = 2 kJ/kgK and theheat of reaction is ∆H = −2840.48 kJ/mol . The diameter-height relation is 1. Withthese data the following relations could be determined:

M = ρr V (6.3)

D = 3√

4V /π (6.4)Ai = 5/4πD2 (6.5)

P1 = UAiMCpr

(6.6)

P2 = (−∆H)VMCpr

(6.7)

where M is reactive mass in [kg ], D is the reactor diameter in [m], Ai is the heat transferarea in [m2], U is the global heat transfer coefficient in [kJ/minm2K ], P1 is the valueof UA/MCp and P2 is (−∆H)V /MCp in the Equation (6.2). The parameter of V wasnot changed due to it is a capacity parameter and its variation should be linked to batchtime, without affecting production capacity of the plant. In this work the batch time isconstant as well as the volume.


After that, heat transfer coefficient was selected as a design parameter, due to physically,it can be incremented or decreased by flow rate in the jacket. Other reasons are: (i)this parameter is directly related with control input and it will favor the effect of controlaction over states variables, (ii) the selection of other parameter like reactor volume Vwill change the productivity of the plant, as it was mentioned, due to its relation withbatch time. However numerical treatment of batch time exceed the scope of this work.


In this section, all the constraints for simulation and optimization of the reference problem arepresented. In first place admissible sets, then conditions and limits of design variables (designparameter and initial state). Finally the bounds for desired final state are shown.

• Admissible states setThe admissible states set X , in a first glance, is limited by non-negativity constraint ofstate variables, however, a subset of Rn

+ was chosen, based on designer criteria that givea proper reaction progress. This set must be compact and simply connected.

X = z ∈ R2 | 0 ≤ z1 ≤ 4 mol/L ∧ 293 ≤ z2 ≤ 393 K (6.8)

• Admissible control actions setThe admissible control actions set must have limits where reactions can be speededup or slowed down. This interval is determined by experience of the designer and inaccordance with final control element limitations. For this reactor considering reactionis exothermic, the bounds presented in Equation (6.9) permits a shutdown and speedingup of the reaction. This set must be a compact set.

U = u ∈ R|293 ≤ u ≤ 369 K (6.9)

• Design parameter setThis set is defined by main values of the literature for heat transfer coefficient (U). Theoptimization algorithm finds the optimal value inside this interval.

Θ = θ ∈ R|12.1987 ≤ θ ≤ 52.28 kJ/min/m2K (6.10)

This values of the global heat transfer coefficient are equivalent from 203.31 W /m2Kup to 871.33 W /m2K which are reasonable values for reactor jackets [46].

• Initial state setThe initial state set is also defined for optimization purposes. This set contains all thefeasible initial state values for the process. This definition depends on quality specifica-tions of the raw material.


X′ = z ∈ X | 0.5 ≤ z1 ≤ 3.5 mol/L ∧ 294 ≤ z2 ≤ 325 K (6.11)

• Final state setThis set is used to define the main specifications of the desired product. It is used tocalculate controllable set and it is the set of final states when t = tf

Ωtf = z ∈ X | z1 = 0.1 mol/L ∧ 293 ≤ z2 ≤ 330 K (6.12)

6.1.3. Nominal Case

In first place, control sets for the batch reactor with nominal parameter are shown.

• Nominal control setsThe nominal control sets are computed using the nominal value of the selected designparameter θ = 15.684 kJ/min/m2K which is the value reported in Table 6.1.The model was simulated in MATLAB ® using the built-in function ode15s and a iterativeprocess with Monte Carlo sampling.The reachable set (R(Ω0)) is represented in the Figure 6.3. For that set 35000 trajec-tories were simulated (according to Chernoff bound ε = 0.01, δ = 0.0085) and with azero-dimension initial set Ω0 = [0.9 298], given by initial conditions of the process.

Figure 6.3. Reachable set of batch reactor with nominal parameters

The controllable set (C(Ωtf )) is shown in the Figure 6.4. A number of 70000 trajectorieswere used for the calculation of this set, that is considered due to high number ofdiscarded points, namely, points out of the admissible states set X . Also, a final set


was taken with one-dimension size (Ωtf ), defined by required final conditions in order toachieve the conversion.

Figure 6.4. Controllable set of batch reactor with nominal parameters

The controllable trajectories set is defined as the intersection between reachable andcontrollable set, R(Ω0) ∩ C(Ωtf ) (see definition in Section 4.3.3). With the purposeof intersect both sets, an euclidean length criteria was used (dmin = 0.002), making acomparison between points clouds of reachable set and controllable set for each time.The controllable trajectories set for a reference problem is shown at Figure 6.5.

Figure 6.5. Controllable trajectories set of batch reactor with nominal parameters


• Controllability IndexIn order to calculate the presented controllability index is necessary to use a method forarea approximation. First, a Monte Carlo integration method was taken into account,but due to nonexistence of a boundary explicit expression, it was rejected. So, the useof convex tools was considered, because despite the reachable or controllable set arenon-convex set, it is assumed that their intersection, the controllable trajectories, setis convex. MATLAB ® convex hull function was used. This tool find the set that arealways convex, and if the set is non-convex, it performs a convex bounding of this setthrough the most minimum convex set which enclose the non-convex set. The excessportions accumulate a deviation of the real volume of the tube, but as this tool is usedin all iterations and controllability index accounts a relative measure of controllability,so this deviations can be neglected.

• Convex controllable trajectories setIn the Figure 6.6, there are both graphics, controllable trajectories set and its approxi-mation by convex sets.

Figure 6.6. Convex controllable trajectories set of batch reactor with nominal parameters

This figure shows that approximation of the controllable trajectories set by convex toolsare good enough. The green points represent the bound that encloses the set and itsvolume is computed by MATLAB ® convex hull function.

6.1.4. Design criteria

In this step, the iteration variables are the design parameter (θ) and initial state (Ω0). Thesevariables are moving along Θ and X0 sets in the optimization problem. The objective functionwas defined as Equation (5.7), because benchmark problem does not provide information aboutcosts.


The optimization problem shown before is solved using Simulated Annealing Algorithm (SA).This algorithm does not requires numerous function evaluations, without the problem of otheralgorithms that when best individuals are selected, many function evaluations are discarded.Another fact of Simulated Annealing Algorithm is that is based on temperature decreasing ofa liquid until it reaches a minimum energy state and liquid becomes a crystal. In [47] thisalgorithm is explained and some characteristic are remarked:

• It is a global optimization algorithm.

• Has random movements of the particle.

• It skips local minima by “upward” movements.

• Has an acceptance probability function.

• It has a temperature decreasing function.

The optimization algorithm presented in Section 5.5.3 is implemented in MATLAB ® consid-ering bound constraints of the design variable.After 30 iterations the algorithm was stopped, In the Figure 6.7, convergence and algorithmtemperature is presented. It it shows a quick convergence around 2200%, that means con-trollability index increased by 22 times. That is also due to initial value is closer to lowerbound.

0 5 10 15 20 25 3013

28

43

58

73

88

Alg

orithm

Tem

pera

ture

[K

]

0 5 10 15 20 25 30-2300

-1900

-1500

-1100

-700

-300

Iterations [-]

Bes

t O

bjec

tive

Fun

ctio

n [-]

Best Objective FunctionAlgorithm Temperature

Figure 6.7. Optimization results of batch reactor optimization

The initial algorithm temperature was kept in its default value 100 K . The initial guesswas system at the nominal parameter (θ = 15.684 kJ/minm2K ) and at nominal initial state(Ω0 = [0.9 298]). The initial objective function was 0% and a hyper-volume of 0.0004678.The final value of the objective function is −2243.47% and 0.010979266 as the hyper-volumeof the controllable trajectories set. The optimal parameter is θ∗ = 50.19618 kJ/minm2K andan optimal initial state Ω∗0 = [3.46929 294.025], which give an improvement of 2243.47%equivalent to an increment of 22 times of nominal volume.


6.1.5. Optimal Case

After optimization procedure, the control sets were calculated for batch reactor model withoptimal parameters. After controllability index is treated.

• Optimal control setsOptimal reachable set (R∗(Ω0)) is affected in shape by initial condition. In that sense,reachable set could include more points that in nominal case. The Figure 6.8 shows thereachable set for the batch reactor with optimal parameters. It has the same simulationsettings as nominal reachable set, with (Ω∗0) instead of (Ω0).

Figure 6.8. Reachable set of batch reactor with optimal parameters

If Figure 6.8 is compared with Figure 6.3, some differences appear in shape and size ofthe reachable set. It is clear that with optimal parameter the system can reach morefinal states.The case of controllable set (C∗(Ωtf )) does not have an appreciable change in shape, onlyits hyper-volume is increased. The Figure 6.9 shows the controllable set with optimalparameter (optimal design parameter and initial state).


Figure 6.9. Controllable set of batch reactor with optimal parameters

As it was mentioned before the same final state (Ωtf ) makes that controllable set preserveits shape, contrary as happens with reachable set.For controllable trajectories set, intersection was performed as it was explained before.The Figure 6.10 presents the controllable trajectories set where it is observed the optimalinitial state.

Figure 6.10. Controllable trajectories set of batch reactor with optimal parameters

When Figure 6.10 is compared with Figure 6.5 appreciable differences are found. A firstdifference is initial state, which in optimal case it generates a shape change in controllabletrajectories set. Another difference is noticed in the second slice of the set, indicating asmall controllability.


• Convex controllable trajectories setThe same convex approximation for computing controllable trajectories set was appliedin optimal case. A slim convex boundary was found in the second slice of the set.

Figure 6.11. Convex controllable trajectories set of batch reactor with optimal parameters

6.1.6. Irreversibility discussion

As it was mentioned before an irreversible system contains reversible and irreversible statevariables. In the case of batch reactor x1 = Ca is the irreversible variable due to always it isdecreased. In x2 = T is the reversible variable, so through its manipulation can be used toregulate the process along desired trajectory.In the Figures 6.12 and 6.13 nominal and optimal cases are compared. Reachable and con-trollable trajectories sets are plotted together with the aim to analyze how are they includedby each other.

Figure 6.12. Nominal control sets for irreversibility analysis


Figure 6.13. Optimal control sets for irreversibility analysis

With optimal parameters a bigger piece of the controllable trajectories set is contained inreachable set, getting a better controllability. This proves that changes in size and shape ofthe controllable trajectories set can improve process regulation and decrease irreversibility.

6.2. Convective coffee dryer process

This example will show that the methodology can be applied to more complex systems, witha higher number of state variables and much nonlinearities. This is considered as a real caseapplication which was developed for studying of coffee drying dynamics and the literatureabout this process is continuously growing.

6.2.1. Coffee dryer model

This process has a drying chamber with three meshes, fed by a hot air current, which comesfrom heat exchanger or a burner. This air is put in contact with coffee beans layer, favoringmass transfer phenomena and eliminating partially water inside coffee beans. The process flowdiagram is shown in Figure 6.14.

• Process modelThe detailed model development is presented in Appendix A. It is assumed that there isno appreciable pressure changes in the ascendant air. Also, convective mass transfer isneglected and the diffusion is taken as an effective diffusivity provided by [48]. The statevariables of the model are: x1 as moisture content of the coffee beans layer, x2 as the airhumidity, x3 as the temperature of coffee layer and x4 as air temperature. Control action(u) is defined as air inlet humidity, which comes from heat exchanger. The followingEquations (6.13) to (6.21) constitute process model:


Air

Burner

Drying Chamber

Fan

Wet coffee layer

Dryinglayer

Heat Exchanger

F1 , T1

F2 , T2

Combustion gases

Plenum

Pre-drying layer

Figure 6.14. Process flow diagram for the convective coffee dryer.

dx1dt = − .m3

1 + hc,0MT

[kg/s] (6.13)

dx2dt =

[ρ1

u1 + u + .m3 − ρ2

x21 + x2

]1

VIIρa[kg/s] (6.14)

dx3dt =

[−λw

.m3 +.Q] 1

MT Cpmez,1[C/s] (6.15)

dx4dt =

[ρ1F 1

1 + u H1 + .m3λ3 − ρ2F 11 + x2

H2 −.Q]

1VIIρaCpmez,2

[C/s] (6.16)

U = 0.2755CpaGa

(dpGa

0.06175 + 0.000165x4

)−0.34[m/s] (6.17)

Kya = 2 ∗Da−g/dp [m/s] (6.18)

Da−g = 4.1582× 10−8 exp[(0.1346x3 + 2.2055) x1 −

(1184

x3 + 273.16

)][m2/s]

(6.19).Q = UasVI(x4 − x3) [W ] (6.20)

.m3 = KyaasVI

[ρc

(x1

1 + x1− heq,c

1 + heq,c

)ρa (ha,sat − x2)

]0.5[kg/s] (6.21)

where hc,0 is the initial moisture content of the coffee layer. This value must be the samehc(0). In other side, λw is the heat of vaporization of water, F is the volumetric flow ofdrying air, dp is the particle diameter of coffee beans, ρc is the coffee bean density, heq,cis the equilibrium moisture content of the coffee beans, ha,sat saturation humidity of thedrying air, MT is the solid material accumulated in the volume, s is the bed height, Dtis diameter of drying chamber, as is the heat and mass transfer area in the coffee beanslayer and ε is the bed porosity.In Table 6.2 the model parameters and initial conditions for the process are shown:


Table 6.2. Model parameters for coffee dryer.

Symb. Value Unithc,0 0.45 kgH2O/kgDCλw 2510598.308 J/kgF 0.0531 m3/sdp 0.0112 mheq,c 0.04982 kgH2O/kgDAha,sat 0.08631 kgH2O/kgDAs 0.1 mDt 0.6 mas 0.2827 m2

ε 0.45 −x1(0) 0.45 kgH2O/kgDCx2(0) 0.03187 kgH2O/kgDAx3(0) 27 Cx4(0) 50 Ctf 6 h

• Model simulationThe model presented was simulated in MATLAB ® using the built-in function ode15sand for a batch time of 6 hours. This simulation time is because this is the time in eachmesh of the drying chamber and also one mesh is analyzed. In Figure 6.15 the simulationresult is shown for state variables x1 and x3 which belong to coffee bean layer. In Figure6.16 shows results for state variables x2 and x4. Both figures correspond to a constantinput of h1 = 0.01312 kgH2O/kgAS.

Figure 6.15. Dynamic response of coffee dryer - Coffee layer (x1 and x3)


Figure 6.16. Dynamic response of coffee dryer - Drying air (x2 and x4)

• Design parameter selectionIn this model, there are many parameters that could affect controllability. In a first sight,feed air velocity could act as a design parameter, but a model inspection suggest thatmass transfer equation only depends on diffusion phenomenon, so feed air velocity willnot affect the moisture content. For that reason, size parameters are selected: s as thebed height which affects volume of the system and Dt as the drying chamber diameterwhich also affects volume and cross-sectional area.


With the complete model, it is possible advance in second step of methodology, which con-sists in variable and parameter classification. The variables were classified as state variables,manipulated variables as was mentioned before. The model parameter are this variables whichare not state or manipulated variables, namely, this which are part of constitutive equations.The constraints for design parameters (Θ) are defined in this section. Also, admissible stateset and admissible control action set are defined.

• Admissible states setThe admissible states set X , is limited by non-negativity constraint of state variablesand by physical restrictions as h2,sat and hc,eq. As same as previous example, this setmust be compact and simply connected.

X = x ∈ R4|Lx ≤ x ≤ Ux, (6.22)

where Lx = [0.0498239, 0, 10, 10], is the lower bound of states variables and the upperbound is given by Ux = [0.9, 0.08631, 69, 69]. The respective units of these bounds are[kgH2O/kgDC , kgH2O/kgDA, K , K ]

• Admissible control actions set


The admissible control actions set must be coherent with air humidity constraints. Alsothis set is limited by heat exchanger design because it has to work as a heat pump, inorder to be able of extract humidity of the environment air. This set is defined by aninterval of 20% to 100% in relative humidity at 50C . It must be a compact set.

U = u ∈ R|0.01554 ≤ u ≤ 0.08631, (6.23)

• Design parameter setThis design parameter set is defined as a reasonable limits in drying chamber sizing andit can be modified by designer. The optimization algorithm finds the optimal value insidethis interval.

Θ = θ ∈ R+ | 0.05 ≤ θ1 ≤ 0.19 ∧ 0.1 ≤ θ2 ≤ 0.9, (6.24)

• Final state setThis set is used to define the main specifications of the desired product. It is used tocalculate controllable set and it is the set of final states when t = tf

Ωtf = z ∈ X | zmin ≤ z ≤ zmax (6.25)

where zmin = [0.15, 0.06, 49, 49] and zmax = [0.18, 0.078, 50, 50] are the lower and upperbounds of final state set given by product quality.

6.2.3. Nominal case

In first place, control sets for the coffee dryer with nominal parameter are shown.

• Nominal control setsThe nominal control sets are calculated using the nominal design parameters θ =[0.1, 0.6] m which are values reported in Table 6.2. The initial state is always thesame, since it is not taken as a design parameter.For reachable set (R(Ω0)) 25000 trajectories were simulated (according to Chernoffbound δ = 0.02, ε = 0.01) and with a zero-dimension initial states set Ω0 =[0.45, 0.03187, 27, 50], given by initial conditions of the process. The Figure 6.17 showsthe reachable set of coffee dryer with nominal parameters.


Figure 6.17. Reachable set for coffee dryer with nominal parameters

For controllable set (C(Ωtf )) 50000 trajectories were simulated, taking into account highnumber of rejected trajectories (A trajectory is rejected when it goes out of the admissiblestates set X ). Another important fact around controllable set is that when the systemdimension is increased, more difficult and computationally expensive is the computingof this set. As was explained before (Section 5.3), an alternative method can be usedtaking a Monte Carlo sample of the admissible state set (X ) and shooting trajectorieswith admissible control sequences checking which ones reach the final state condition(Ωtf ).


Figure 6.18. Controllable set for coffee dryer with nominal parameters

The Figure 6.19 shows the controllable trajectories set for coffee dryer set. This set has5th order (4 state variables and the time), so its visualization is only possible takingviews with pairs of state variables.

Figure 6.19. Controllable trajectories set for coffee dryer with nominal parameters

The convex approximation of controllable trajectories set is presented in Figure 6.20.


Figure 6.20. Convex controllable trajectories set for coffee dryer with nominal parameters

In Figure 6.20, it is important to highlight that state variables of temperatures (x3 andx4) remain coupled. That suggests a fast temperature dynamics, probably caused byhigh heat transfer rate, which is faster than mass transfer dynamic. Apparently thisbehavior will reveal a less controllable behavior, but if it is considered that temperatureof the coffee beans layer (x3) is almost equal to drying air temperature (x4) which meansthat heat transfer dynamics are coupled and the coffee beans layer will get the necessaryenergy to evaporation of the water in coffee beans. In that sense, it could represent adesirable behavior for conrollability increasing.

6.2.4. Design criteria

In the optimization problem, the iteration variable is the design parameter (θ). This variableis moving along Θ set. The objective function was defined as Equation 5.7, because costsof coffee dryer chambers are quite difficult to find and they could depend on geometry andcapacities of the equipment.As same that previous example, the selected optimization algorithm was Simulated Annealingand it was implemented in MATLAB ® considering bound constraints of the design variablefor the coffee dryer process.After 30 iterations the algorithm was stopped, In Figure 6.21, convergence and algorithmtemperature is presented. It it shows a quick convergence around −32.1417%, that meanscontrollability index increased by 0.3214 times.


Figure 6.21. Optimization results of coffee dryer optimization

The initial algorithm temperature was kept in its default value 100 K . The initial guess wassystem at the nominal parameter (θ = [0.1, 0.6] m). The initial objective function was 0% anda hyper-volume of 1.7480×10−6 . The final value of the objective function is −32.1417% and2.53129×10−6 as the hyper-volume of the controllable trajectories set. The optimal parameteris θ∗ = [0.055, 0.692] m, which gives an improvement of 32.1417% equivalent to an incrementof 0.32 times of nominal volume. This result also shows that the design parameter θ1 affectsmuch than the other one θ2, as is observed with the change respect to their nominal values.

6.2.5. Optimal case

After optimization procedure, the control sets were calculated for batch reactor model withoptimal parameters. After controllability index is treated.

• Optimal control setsOpposed to previous example, the optimal control sets of coffee beans dryer are onlyaffected in width because initial state remains equal. So shape is changed only due tocontrollability increment. The Figure 6.22 shows the reachable set for the system withoptimal parameters.


Figure 6.22. Reachable set for coffee dryer with optimal parameters

For the controllable set, it can be noticed a higher density of points in the initial contourof the set. That implies that a higher number of feasible trajectories compound this set,contributing to a higher number of controllable trajectories. Figure 6.23 presents thecontrollable set for system with optima parameters.

Figure 6.23. Controllable set for coffee dryer with optimal parameters


The same intersection approximation for computing controllable trajectories set wasapplied in optimal case. Figure 6.24 shows controllable trajectories set of the coffeedryer process with optimal parameters.

Figure 6.24. Controllable trajectories set for coffee dryer with optimal parameters

The convex approximation of controllable trajectories set is presented in Figure 6.25.

Figure 6.25. Convex controllable trajectories set for coffee dryer with optimal parameters

When Figure 6.20 is compared with Figure 6.25, differences are slightly visible, probablybecause increment in controllability is not as much as in batch reactor example. Also sets


visualization makes difficult to appreciate changes in the set. In a detail, the differencesare around second slice of the set when controllability is smaller and the process hasto travel along few trajectories. In Figure 6.20, this sectional view of the controllabletrajectories set has feer points when is compared with Figure 6.25 when is noticed agreater points density.

6.2.6. Irreversibility discussion

When reachable and controllable trajectories sets are compared in both cases (nominal andoptimal), it is evidenced that controllable trajectories set occupy a high portion of the reachableset. That means whether reachable set increases, also controllable trajectories set does. Inthe Figure 6.26 these sets are shown for nominal parameters. However after half batch time,controllable trajectories set decreases in size due to controllable set reduction.

Figure 6.26. Nominal control sets (red: Reachable Set, black: Controlable Trajectories Set) for irre-versibility analysis

With optimal parameters the reachable set acquire a higher point density, which makes thatcontrollable trajectories set expands almost covering all reachable set before half batch time,which is the critical step of the batch run.


Figure 6.27. Optimal control sets (red: Reachable Set, black: Controlable Trajectories Set) for irre-versibility analysis


This chapter has shown the application of the proposed methodology for the simultaneousdesign of process and control in two examples. There are some items to highlight:

• Model selection affects how design parameters are varied. In the case of batch reactoran uncoupling was needed in order to get independent parameters and associate themto reasonable values.

• The computation of reachable and controllable sets gets numerically difficult when di-mension of the system is high. This numerical computation should be improved.

• Controllability index is based on hyper-volumes, and that means, it will get a lower valueas dimension of the system increases.

• Coffee dryer process showed that some design parameters have more impact in control-lability than others, and this methodology could be used as design parameter selection.

• Simulated Annealing algorithm proved that is a suitable global optimization algorithmsfor purposes of this methodology, because it has a local skip minima strategy and it doesnot discard objective function evaluations.

CHAPTER 7Conclusions and future work

7.1. General conclusions

This thesis developed a methodology for batch process design considering dynamic criteria.Within concepts and definitions, it worth to highlight controllable trajectories set interpretation,either from controllabiliy and reversibility.Another important definition for controllability is presented. The set-theoretic methods areable to represents reachability and reversibility aspects, so controllability is deduced.The proposed methodology: (i) considers dynamic features of the batch process like time-varying behavior and irreversible trajectories taking place in the transformation, (ii) it usesphenomenological-based model of the process to retrieve information about reachability, con-trollability, and irreversibility of the process for a fixed set of parameters, (iii) determining theeffect of each design parameter in the controllability index of the process and (iv) choosingthe best values in the constraint set.

The main contributions of this work are:

• The study of controllability index in a batch process, with a lot of advantages overdesign analysis. One of these is that designer can study independent effects of differentmanipulated variables u to the process with their respective admissible control actionsets.

• Considering that phenomenological-based models are increasingly used as a tool fordesign chemical processes, and no formal procedure about how to use them has been in-troduced until today, this thesis presents a clear and simple way to use phenomenologicalmodels as a tool for finding design parameters when a batch process is designed.

• An extension of key parameters that affects controllability in batch processes is presented.A first key is the design parameters which are dependent on process constraints andbudget limits. The second key is the initial state, that could change the shape of thecontrollable trajectories set and with that affect controllability. Final key is the batchtime which can modify the final state, affecting reachable and controllable sets.

While the proposed methodology allows the determination of optimal design parameters ofbatch processes, its main limitations are related to:

• For applying of the methodology, a dynamic lumped-parameter model is required. De-pending of the dimension of the model, computing of the control sets becomes numer-ically more difficult. Besides, some models have coupled values of the parameters andfor its individualization some assumptions are required.

58

CHAPTER 7. Conclusions and future work 59

• Despite control sets are calculated from nonlinear model and with and efficient numericalintegration algorithms, dealing of control sets as point clouds are not suitable for large-scale systems because computer memory is limited. In that sense, a different approachfor control sets computing is needed.

Regarding the objectives completion, it can be concluded:

General objective:Propose a methodology to design a batch process, through the study and determination of therelevant design parameters, that maximize the state controllability from set-theoretic methodsframework, and allow obtaining a controllable process design.

• The methodology for batch process design was proposed, considering controllability cri-teria. At this point, is important to remark the wide use of controllability index, not onlyfor design purposes, also for dynamic analysis of uncertainty in initial state.

• Controllability index can be also mixed with an economic criterion. This is explained whenobjective function is presented, including a trade-off coefficient in order to equilibrateeconomic and controllability criteria.

Specific objectives:

1. Describe the main characteristics of batch processes, that complicate their design andcontrol, compared with continuous processes.

• The main characteristics of batch processes are treated in Chapter 4, highlight-ing time-varying features and irreversible behavior as whose which difference fromcontinuous processes.

2. Identify recent developments in the area of batch process design and its integration withcontrol system design.

• As it was explained in Chapter 2 the developments in the batch process designarea are based on heuristic approach. Two design problems are presented: uniquetrajectory design and positioning problem.

• In Chapter 3 the main approximations to simultaneous design of process and con-trol were shown. Two approaches of this integrated design and control are taken:some methods based on controllability or performance indexes and methods derivedof simultaneous optimization. Another important fact is that there is no distinc-tion bewteen methodologies that are applicable to continuous and which to batchprocesses.

• Also the advances in simultaneous design for batch processes are widely specificand all are related to simultaneous optimization approach.

3. Select a reference problem (benchmark) decribed by a phenomenological-based modelof a batch processes, classifying the design parameters.


• In Chapter 6 the batch reactor benchmark was presented. After some treatments,heat transfer coefficient was uncoupled, with the purpose to ease the design pa-rameter selection.

• Also, a convective coffee dryer model was presented, with the intention to applythe methodology to a real case problem. The full model is deduced in Appendix A

4. Adapt concepts of the control set theory for evaluating state controllability for batchprocesses using an adequate index.

• Controllability and reversibility definitions were presented in Chapter 4, remark-ing the main differences between reversible (continuous) processes and irreversible(batch) processes.

• Set-theoretic methods permit account the controllability of irreversible systems,through calculation of hyper-volume of controllable trajectories set. This index canbe used for design procedure or initial state specification.

5. Propose a methodology of simultaneous process and control design for batch processes,determining the effect of the design constraints in the state controllability of the process.

• In the optimization problem that return optimal parameters, controllability andeconomic criteria can be used, considering a trade-off coefficient.

• In Chapter 5 a global optimization is presented due to control sets do not have anexplicit expression that allows the use of gradient-based methods.

6. Evaluate the proposed methodology with a simultaneous design of the process and controlin the benchmark problem.

• The proposed methodology was evaluated in two examples: a widely studied bench-mark and a proposed convective coffee drying model for a real case application.It was evidenced that both design parameters and initial state have incidence overstate controllability.

• The analysis over in Chapter 6 showed that some design parameters affect morethan others, so this information could be used as a criteria for the selection ofdesign parameter. A prior analysis can be included as a part of the second step ofthe presented methodology.

7.2. Future work

This thesis leaves some topics as a further work which does not meet the aims planned.However these achieved outcomes open the following topics:

• A controllability index for high-dimension system is required. The index proposed inthis thesis works properly, but when dimension increases, computation of reachable andcontrollable set becomes much difficult.

• The majority of design algorithms of batch processes do not have a step for variablepairing, where manipulated variable is selected. That is probably due to irreversibility andnature of batch processes, that in many cases shows it as obvious selection. However, ananalysis from set-theoretic methods will be interesting considering controllability index.


• Extend this methodology to more than single control action case. This will help to realizewhich one gives a higher controllability.

• Incorporate batch time to this optimization problem and show how controllability indexis affected by batch time.

• Apply this methodology to different processes, with the aim to prove its generality.

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APPENDIX AConvective Coffee Dryer Model

Inside coffee post-harvest treatment, drying occupies the most important role when energyefforts are analyzed. Coffee beans enter to dryer from washing process and the moisture contentis decreased from 53%w .b. to 12%w .b.. If this process is badly performed, microorganism cangrow over coffee bean surface, turning it exposed to biochemical reactions that could affectsugar content [49], and modifying organoleptic properties of cup coffee such as taste, aroma,color, body [50]. Also, coffee dealing requires a proper dried product.With that perspective, a coffee dryer modeling is presented:The silo dryer consists in a drying enclosure, a heat exchanger and a blow. Drying enclosure isformed by three meshes, named as: drying layer, pre-drying layer and wet coffee layer. Coffeegrains have an initial moisture content (hc), and they are placed into wet coffee layer. After aperiod of time tf , they are moved to pre-drying layer and stay there by a same period of time.Finally, they are carried to drying layer with the aim of achieve the desired moisture content.Air flow (F1) is continually by blower, injecting it into heat exchanger, reaching a necessarytemperature (T1) and absolute moisture (h1). Air flow ingresses into drying enclosure, passingby plenum, guarantying an uniform flow regime. Air flows up, keeping contact with the firstmesh (drying layer). Here, dry air capture moisture from coffee grains. Then, the mixture flowof air and water (F2), at a temperature T2 and absolute moisture h2 pass by pre-drying layer,retiring more moisture form this layer. Finally, the gaseous flow passes by the wet coffee layerbefore to go out form the process. Figure A.1 shows the process flow diagram for this process.

Air

Burner

Drying Chamber

Fan

Wet coffee layer

Dryinglayer

Heat Exchanger

F1 , T1

F2 , T2

Combustion gases

Plenum

Pre-drying layer

Figure A.1. Process flow diagram for for convective coffee drying.

Following the modeling procedure presented in Alvarez et al. [29]. One of the most importantstep of this methodology is to define process systems. With the aim to find the number of pro-cess systems, it might find boundaries like physical separation, interphase layers or sites wherephysic-chemical transformations take place. Only the first layer will be examined, because

66

APPENDIX A. Convective Coffee Dryer Model 67

modeling this layer, the equations will be extended to other layers, by modifying operationalconditions. First process system will be associated with wet coffee layer, and second processsystem will be the air.

PS: SI

PS: SII

1

2

3

Figure A.2. Process system diagram for convective coffee drying.

In the Figure A.2 it is shown mass and energetic interactions between both process systems.Mass interaction is represented by flow 3, integrated by steam flow from coffee grains to air,describing moisture loss of coffee grains. Big arrow represents energetic interaction betweensystems. It describes the transference of latent heat form air to moisture to evaporate it.The full mathematical model is described by Equations (A.1) to (A.12). This model is writtenin process variables.

dhcdt = − .m3

1 + hc,0MT

[kg/s] (A.1)

dT3dt =

[−λw

.m3 +.Q] 1

MT Cpmez,1[C/s] (A.2)

dh2dt =

[ρ1

h11 + h1

+ .m3 − ρ2h2

1 + h2

] 1VIIρa

[kg/s] (A.3)

dT2dt =

[ρ1F 1

1 + h1H1 + .m3λ3 − ρ2F 1

1 + h2H2 −

.Q]

1VIIρaCpmez,2

[C/s] (A.4)

U = 0.2755CpaGa

(dpGa

0.06175 + 0.000165T2

)−0.34

[m/s] (A.5)

Kya = 2 ∗Da−g/dp [m/s] (A.6)

Da−g = 4.1582× 10−8 exp[

(0.1346T3 + 2.2055) hc −( 1184

T3 + 273.16

)][m2/s] (A.7)

.Q = UasVI(T2 − T3) [W ] (A.8)

.m3 = KyaasVI

[ρc

(hc

1 + hc− heq,c

1 + heq,c

)ρa (ha,sat − h2)

]0.5

[kg/s] (A.9)

VI = As(1− ε) [m3] (A.10)VII = As(ε) [m3] (A.11)

MT = VI ∗ ρc [kg ] (A.12)

APPENDIX A. Convective Coffee Dryer Model 68

The model parameters are shown in the Table A.1

Table A.1. Model parameters of coffee drying process

Symb. Value Unit.hc,0 0.45 kgH2O/kgDCλw 2510598.308 J/kgF 0.0531 m3/sdp 0.0112 mheq,c 0.04982 kgH2O/kgDAha,sat 0.08631 kgH2O/kgDAs 0.1 mA 0.2827 m2

ε 0.45 −hc(0) 0.45 kgH2O/kgDCT2(0) 50 Ch2(0) 0.03187 kgH2O/kgDAT3(0) 27 Ctf 6 h

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