Table of Contents

Cover

Title Page

Preface

Acknowledgments

Prior Knowledge

How This Book Is Structured

Icons

Software and Companion Website

Nomenclature

Section 1: Basic Theory

Chapter 1: Introduction
1. 1.1 Introduction
2. 1.2 Motivation
3. 1.3 Reactor Network Synthesis
4. 1.4 Solving the Reactor Network Synthesis Problem
5. 1.5 Chapter Review
6. References
Chapter 2: Concentration and Mixing
1. 2.1 Introduction
2. 2.2 Concentration Vectors and Dimension
3. 2.3 Mixing
4. 2.4 Chapter Review
5. References
Chapter 3: The Attainable Region
1. 3.1 Introduction
2. 3.2 A Mixing and Reaction Game
3. 3.3 The AR
4. 3.4 Elementary Properties of the AR
5. 3.5 Chapter Review
6. References
Chapter 4: Reaction
1. 4.1 Introduction
2. 4.2 Reaction Rates and Stoichiometry
3. 4.3 Reaction from a Geometric Viewpoint
4. 4.4 Three Fundamental Continuous Reactor Types
5. 4.5 Summary
6. 4.6 Mixing Temperatures
7. 4.7 Additional Properties of the AR
8. 4.8 Chapter Review
9. References
Chapter 5: Two-Dimensional Constructions
1. 5.1 Introduction
2. 5.2 A Framework for Tackling AR Problems
3. 5.3 Two-Dimensional Van De Vusse Kinetics
4. 5.4 Multiple CSTR Steady States and ISOLAS
5. 5.5 Constructions in Residence Time Space
6. 5.6 Chapter Review
7. References

Section 2: Extended Topics

Chapter 6: Higher Dimensional AR Theory
1. 6.1 Introduction
2. 6.2 Dimension and Stoichiometry
3. 6.3 The Three Fundamental Reactor Types Used in AR Theory
4. 6.4 Critical DSRs and CSTRs
5. 6.5 Chapter Review
6. References
Chapter 7: Applications of AR Theory
1. 7.1 Introduction
2. 7.2 Higher Dimensional Constructions
3. 7.3 Nonisothermal Constructions and Reactor Type Constraints
4. 7.4 AR THEORY for Batch Reactors
5. 7.5 Chapter Review
6. References
Chapter 8: AR Construction Algorithms
1. 8.1 Introduction
2. 8.2 Preliminaries
3. 8.3 Overview of AR Construction Methods
4. 8.4 Inside-out Construction Methods
5. 8.5 Outside-in Construction Methods
6. 8.6 Superstructure Methods
7. 8.7 Chapter Review
8. References
Chapter 9: Attainable Regions for Variable Density Systems
1. 9.1 Introduction
2. 9.2 Common Conversions to Mass Fraction Space
3. 9.3 Examples
4. 9.4 Chapter Review
5. References
Chapter 10: Final Remarks, Further Reading, and Future Directions
1. 10.1 Introduction
2. 10.2 Chapter Summaries and Final Remarks
3. 10.3 Further Reading
4. 10.4 Future Directions
5. References

Appendix A: Fundamental Reactor Types

A.1 The Plug Flow Reactor
A.2 The Continuous-Flow Stirred Tank Reactor
A.3 The Differential Sidestream Reactor

Appendix B: Mathematical Topics

B.1 Set Notation
B.2 Aspects of Linear Algebra
B.3 The Complement Principle
References

Appendix C: Companion Software and Website

C.1 Introduction
C.2 Obtaining Python and Jupyter
1. Index

End User License Agreement

List of Illustrations

How This Book Is Structured

Figure P.1 Organization of chapters.

Chapter 1: Introduction

Figure 1.1 Overview of reactor network.
Figure 1.2 Donald's investigation, summarized in a matrix for the maximum toluene concentration achieved.
Figure 1.3 Coffee and fed-batch reactors.
Figure 1.4 Toluene concentration data obtained in a batch experiment and a continuous-flow stirred tank reactor (CSTR).
Figure 1.5 Options (a)–(f) represent a number of different reactor configurations. Only PFRs and CSTRs are used with a maximum of three reactors per configuration.
Figure 1.6 Overview of different approaches. Each approach will terminate in an optimal design, although the particular design achieved may differ depending on the order of steps taken.
Figure 1.7 Metabolic pathway showing how glucose and glycerol are converted in a series of complex biological reactions to produce a number of different by-products.
Figure 1.8 (a) A CSTR configuration that approximates a plug flow reactor (PFR).
Figure 1.9 Order of tasks using the AR approach.
Figure 1.10 A maze with one entry point and multiple exit points.
Figure 1.11 Hierarchy of design for reactors.

Chapter 2: Concentration and Mixing

Figure 2.4 Converting concentration–time data into concentration–concentration data. (a) Concentration profiles for benzene and toluene. Note that ethylene, xylene, diphenyl, and hydrogen are not shown. (b) Benzene and toluene plotted in the phase plane.
Figure 2.1 Concentration data plotted in the phase plane.
Figure 2.2 Concentration data plotted in three-dimensional space.
Figure 2.3 (a) Two-dimensional plot of the five component data in c_A–c_B space. (b) Three-dimensional plot of the five component data in c_B–c_D–c_E space.
Figure 2.3 Graphical interpretation of the NaCl–H₂O system for different beaker concentrations.
Figure 2.6 NaCl–H₂O–sugar mass fractions.
Figure 2.7 Graphical interpretation of molar concentration for the NaCl–KOH–H₂0 system.
Figure 2.8 Geometric representation of concentration using concentration vectors from the origin.
Figure 2.9 Some examples of what concentration vectors might represent in reality. (a) Reaction in a beaker, (b) points along a reactor, (c) mixtures, and (d) points in a distillation column.
Figure 2.10 Physical interpretation of mixing for batch and continuous processes. (See color plate section for the color representation of this figure.)
Figure 2.11 (a) Concentrations present in the NaCl–KOH system plotted as concentration vectors from the origin. (b) The graphical meaning of mixing. Mixtures lie on a straight line joining the two concentrations.
Figure 2.12 Mixture concentrations are formed by a linear combination of concentration vectors.
Figure 2.13 Position of the mixture concentration C^* in relation to C₁ and C₂.
Figure 2.14 The possible set of mixtures achieved when three beaker concentrations are used.
Figure 2.15 Mixtures for experiment 1 (beakers 1 and 3) and experiment 2 (beakers 2 and 3) when equal portions are used. (a) Achieving C_A, (b) achieving C_B, and (c) achieving C_C.
Figure 2.16 Forming a mixture concentration C_C as a series of three mixing steps. (See color plate section for the color representation of this figure.)
Figure 2.17 Achieving C_C in only two mixing steps.
Figure 2.18 Physical steps needed to achieve C_C as well as the corresponding geometric interpretation in NaCl–KOH space. (See color plate section for the color representation of this figure.)
Figure 2.19 There are infinitely many combinations that might be employed to achieve concentrations in the interior of the three mixing lines defined by C₁, C₂, and C₃.
Figure 2.20 Possible drinks recipe formulations (x's) and available operating concentrations (b's) plotted in “bang” and “zing” concentration space.
Figure 2.21 (a) The region of new concentrations achieved if x₂ is made achievable and (b) the region of new concentrations achieved if x₄ is made available.
Figure 2.22 (a) A set of many points in a two-dimensional concentration space and (b) the region of achievable points achieved by traversing on the outermost boundary points. The perimeter points generate all points in the interior region.
Figure 2.23 Multiple ways of achieving an interior point.
Figure 2.24 (a) A collection of concentrations in NaCl–KOH–Sugar space and (b) the corresponding convex hull for the set of points given in (a).
Figure 2.25 Five sample points in a two-dimensional space. The points do not necessarily represent concentrations.
Figure 2.26 Identifying extreme points.
Figure 2.27 Top-down view of a garden. (a) Loose rope enclosing the garden and (b) tightened rope representing a convex hull that encloses all eight lights. (See color plate section for the color representation of this figure.)
Figure 2.28 (a) A three-dimensional CAD drawing of a rubber duck and (b) the associated convex hull, given in a wireframe representation. (See color plate section for the color representation of this figure.)
Figure 2.29 Mixing fills in concavities and joins regions together. (a) Two separate (nonconvex) regions A₁ and A₂, (b) mixing between regions A₁ and A₂, and (c) a single (convex) region containing both regions and the space separating A₁ and A₂.

Chapter 3: The Attainable Region

Figure 3.1 (a) Batch concentration profile for the BTX beaker experiment and (b) the convex hull of the set of points for the BTX beaker experiment.
Figure 3.2 Batch profiles for subsequent batches run from points generated in region A from Figure 3.1b. None of the profiles achieve a larger maximum toluene concentration compared to the original experiment.
Figure 3.3 (a) Example of a typical mixing line used to achieve point x₁. Point x₁ is located in the concave section of the profile (region B₁). (b) The batch profile obtained when a new experiment is run using a starting concentration equal to point x₁.
Figure 3.4 (a) The concave region obtained (shown as the filled region B₂) belonging to the updated batch profile initiated at point x₁. (b) Batch profile obtained from a third batch experiment from x₂. Concentration x₂ is formed by mixing concentration y₂ with the feed F.
Figure 3.5 The physical steps required to achieve a mixture concentration equal to point x₂.
Figure 3.6 (a) Fourth batch experiment and (b) fifth batch experiment. All five previous batch profiles are displayed on a single plot.
Figure 3.7 The result of many successive batches.
Figure 3.8 A comparison between the maximum toluene concentration achieved via successive batches (point H) to that produced by the optimal CSTR recommendation from Chapter 1 (point I).
Figure 3.9 Property 1. The AR is convex. (a) A nonconvex region and (b) a convex region generated via mixing operations.
Figure 3.10 Property 2. (a) The AR is generally composed of straight (mixing) and curved (reaction) lines. (b) Sometimes, when the profile is already convex, mixing is not required. Hence, there is no requirement that the ARs contain both reaction and mixing sections.
Figure 3.11 Property 3. The AR must contain the feed. (a) An AR containing a single feed point and (b) an AR associated with multiple feeds containing all feed points.
Figure 3.12 Property 4. The AR is unique, simply connected, and a single region. (a) Three separate ARs AR₁, AR₂, and AR₃, (b) mixing between the three regions, and (c) the final (single) region obtained contains all regions and is simply connected.
Figure 3.13 Property 5. The AR is a constrained subset of ℝⁿ. Mass balance restricts systems from achieving all concentrations in the range [0, ∞), such as that shown (a) for a system in ℝ², or similarly for (b) representing a constrained region in ℝ³.

Chapter 4: Reaction

Figure 4.1 (a) Rate vectors in c_B–c_T space and (b) rate vectors in c_B–c_E space.
Figure 4.2 Vector field for the BTX system in c_B–c_T space.
Figure 4.3 (a) Rate field in x–y space and (b) rate field in x–z space.
Figure 4.4 Concentration profiles approaching equilibrium.
Figure 4.5 Differential segments of material inside a PFR. Each segment is assumed to have constant concentration.
Figure 4.6 (a) Three large-scale PFRs and (b) the symbol used commonly to represent a PFR in process flow diagrams.
Figure 4.7 Beakers (batch reactors) on a conveyor belt, which approximate PFR behavior. (See color plate section for the color representation of this figure.)
Figure 4.8 PFR trajectories for two different initial conditions (feed points) and integration ranges (residence times).
Figure 4.9 (a) Concentration profiles for the BTX system in a PFR and (b) PFR trajectory for the BTX system from the feed point to equilibrium.
Figure 4.10 Approximating the gradient of a PFR trajectory.
Figure 4.11 (a) Geometric interpretation of the PFR trajectory and (b) rate vectors are tangent to the PFR trajectory. (See color plate section for the color representation of this figure.)
Figure 4.12 (a) PFR trajectories can never intersect each other, and (b) this phenomenon would imply that multiple rate vectors exist at the intersection point.
Figure 4.13 Matryoshka dolls.
Figure 4.14 PFR trajectories on the boundary of a mixing line. Trajectories are not allowed to cross for the same kinetics.
Figure 4.15 (a) A laboratory-scale CSTR and (b) symbol used to represent a CSTR.
Figure 4.16 A CSTR locus and PFR trajectory from the feed point.
Figure 4.17 A plot of the cubic expression for f(c_B) versus c_B for a residence time of τ = 3.0 s and feed point C_f = [5.0, 0.25]^T.
Figure 4.18 (a) Geometric interpretation of the CSTR. (b) CSTR solutions are collinear with the rate vector evaluated at that point and the feed. (See color plate section for the color representation of this figure.)
Figure 4.19 (a) Computing CSTR solutions in the autocatalytic system by searching for colinear points, (b) full CSTR locus for the autocatalytic system, and (c) concentration profiles for components A and B achieved in the CSTR for different residence times. (See color plate section for the color representation of this figure.)
Figure 4.20 (a) A CSTR locus forming the majority of a candidate AR boundary and (b) PFR trajectories from CSTR points on the boundary. The region is often expanded by PFRs when sections of the CSTR locus are found on the region boundary.
Figure 4.21 Reactor structures for a hypothetical system. (See color plate section for the color representation of this figure.)
Figure 4.22 (a) Reactor structure required to achieve point A, (b) reactor structure required to achieve point C and (c) reactor structure required to achieve point F.
Figure 4.23 Geometric difference between (a) bypass and (b) recycle reactors.
Figure 4.24 (a) A laboratory approximation of a DSR and (b) symbols used to represent a DSR.
Figure 4.25 Five PFRs in series with sidestream addition that approximate a DSR.
Figure 4.26 A DSR may be viewed as a beaker and retort stand combination on a conveyor belt. (See color plate section for the color representation of this figure.)
Figure 4.27 Mass balance over a differential segment of a DSR.
Figure 4.28 DSR trajectories obtained for different parameters for the same kinetics.
Figure 4.29 DSR trajectories for constant values of α over the integration range. (See color plate section for the color representation of this figure.)
Figure 4.30 A graphical representation of the possible directions a DSR could pursue. The DSR direction is defined by the span of the vectors C⁰ − C and r(C).
Figure 4.31 (a) DSR trajectories obtained with different α policies and (b) a simple illustration showing how different DSR α policies result in different solution trajectories. Some of the policies serve to grow the region, whereas others do not.
Figure 4.32 (a) Physical interpretation of mixing two beakers of different temperature and (b) geometric interpretation of mixing temperatures.
Figure 4.33 (a) Hypothetical AR boundary showing three rate vectors evaluated at specific points on the boundary and (b) the resulting PFR trajectories that are obtained from the three points of interest.
Figure 4.34 (a) Three rate vectors residing in the complement region; (b) rays extended backward from the three points of interest; and (c) only rate vectors extended backward, which intersect the candidate AR boundary, serve to expand the region (via CSTR).

Chapter 5: Two-Dimensional Constructions

Figure 5.1 Species concentrations as a function of PFR residence time for Van de Vusse kinetics, with a₁ = 1 and a₂ = 1.
Figure 5.2 AR for the two-dimensional Van de Vusse system when a₁ = a₂. (a) The entire AR boundary is generated from a PFR from the feed point F. (b) When the rate field for the system is plotted over the region, we can visually inspect the AR boundary to show that no rate vectors on the AR boundary point out of the region, indicating that the AR has been found.
Figure 5.3 Maximizing for c_B using the AR when a₁ = 1 and a₂ = 1.
Figure 5.4 Optimizing for a new objective function. (a) Points intersecting the AR that satisfy the objective function. (b) The required optimal reactor structures that produce concentrations x₁ (i), x₂ (ii), and x₃ (iii).
Figure 5.5 (a) Species concentrations as a function of PFR residence time and (b) PFR trajectory in c_A − c_B space for with a₁ = 1 and a₂ = 10.
Figure 5.6 (a) Concentration profiles of all species as a function of time and (b) PFR trajectory in c_A − c_B space when a₁ = 20 and a₂ = 2.
Figure 5.7 Species concentrations profiles achieved in a CSTR for Van de Vusse kinetics, with a₁ = 20 and a₂ = 2.
Figure 5.8 (a) PFR trajectory and CSTR locus from the feed point with a₁ = 20 and a₂ = 2. (b) CSTR with bypass of the feed.
Figure 5.9 (a) Optimal reactor AR boundary structures (See color plate section for the color representation of this figure.) and (b) optimal reactor network required to generate the AR.
Figure 5.10 Overall reactor network.
Figure 5.11 (a) Graphical interpretation of yield in c_A − c_B space and (b) yield lines overlaid on the AR for the Van de Vusse system.
Figure 5.12 (a) Y_B contours for the Van de Vusse system, (b) Σ_BD contours, and (c) Σ_BC contours.
Figure 5.13 Determining the volume of a PFR using optimal AR boundary concentrations.
Figure 5.14 (a) The set of achievable concentrations for a PFR from the feed and (b) the AR for a CSTR–PFR structure without mixing. (See color plate section for the color representation of this figure.)
Figure 5.15 (a) Concentration profiles of components A and B in a PFR from the feed. After τ ∼ 80 s, little change in the species concentrations is observed. (b) PFR trajectory, corresponding to a PFR from the feed point, plotted in c_A–c_B space.
Figure 5.16 CSTR locus and PFR trajectory from the feed operating from the feed concentration.
Figure 5.17 Convex hull representing the set of achievable concentrations for the CSTR–PFR structure.
Figure 5.18 (a) The CSTR locus from the feed, produced with various initial guesses. (b) AR obtained using the full set of CSTR points. The optimal reactor structure is still a CSTR from the feed followed by a PFR; however, this region is significantly larger than that obtained initially. (c) Rate field for the isola system.
Figure 5.19 (a) AR for complex isola kinetics. The objective function is simply the concentration of component B. (b) A quadratic objective function in c_A representing operating profit from the sale of component B.
Figure 5.20 Physical interpretation of mixing two different reactors of unique residence times (reactor volumes).
Figure 5.21 Geometric interpretation of mixing in residence time space. Mixtures also lie on a straight line when residence time is used since residence time also obeys a linear mixing law.
Figure 5.22 Concentration profiles in a beaker after the reaction has been halted.
Figure 5.23 Two identical ARs. The region given by Figure (b) contains a larger upper bound than that given by Figure (a). Both regions contain the exact same concentrations and thus both regions represent the same set of achievable states.
Figure 5.24 PFR trajectory from the feed, plotted now in c_A–τ space.
Figure 5.25 Convex hull for the PFR from the feed.
Figure 5.26 (a) CSTR locus from the feed, plotted together with the PFR trajectory obtained previously. The CSTR appears to achieve the majority of concentrations in a much smaller residence time when compared to the PFR. (b) The candidate region produced for a CSTR and PFR from the feed point.
Figure 5.27 (a) The area of the AR varies depending on where a PFR is initiated on the CSTR locus. (b) The optimal point is found by initiating PFRs at each point on the CSTR locus and computing the corresponding AR area.
Figure 5.28 Full AR for the residence time problem.
Figure 5.29 Different optimal reactor structures obtained depending on the desired exit concentration. (a) A CSTR with bypass of the feed, (b) a CSTR operated at point X in Figure 5.28 (C_A = 10 mol/L), and (c) a CSTR followed by a PFR.
Figure 5.30 Different payback period objective functions overlaid onto the AR to determine the optimal product concentration and reactor volume. The line corresponding to 1.3 years, and marked with an asterisk (*), represents the minimum payback period for the process.

Chapter 6: Higher Dimensional AR Theory

Figure 6.1 Concentrations that are stoichiometrically compatible with the ammonia reaction. The set of concentrations lies on a line in $c06-math-0014$ – $c06-math-0015$ − $c06-math-0016$ space. The gradient of the line is defined by the vector a, whereas its position is determined by the feed vector C_f.
Figure 6.2 (a) The set of concentrations that are stoichiometrically compatible with the feed point C_f = [1, 1, 0]^T and Equation 6.2. (b) Any AR generated with kinetics using this feed results in a set of compositions that lie on a plane in c_A–c_B–c_C space.
Figure 6.3 PFR trajectory associated with different kinetics, but the same feed point and reaction stoichiometry. (See color plate section for the color representation of this figure.)
Figure 6.4 A hypothetical set of five points and associated convex hull. The extreme points either belong to the feed set, or else they must be derived from a part of the plant where reaction has taken place.
Figure 6.5 Protrusions and nonprotrusions, (a) extreme points and (b) exposed points.
Figure 6.6 The AR boundary for the Van de Vusse system.
Figure 6.7 Different proposed optimal reactor structures. (a) PFRs in parallel, (b) CSTR with feed bypass, (c) CSTR-PFR, (d) DSR-PFR with feed bypass (e) PFR-CSTR, and (f) DSR-PFR.
Figure 6.8 (a) The subspace spanned by r(C) and (C⁰ − C) in a DSR. (b) An inconsistent AR boundary shape.
Figure 6.9 The role of connectors on the AR boundary. Mixing lines meet PFR trajectories at a connector transversely. The connector is an initiating point for PFR trajectory segments generated on the AR boundary.
Figure 6.10 Connectors on the AR boundary manifest themselves either as critical CSTR effluent compositions, or critical DSR solution trajectories. (See color plate section for the color representation of this figure.)
Figure 6.11 Controlling a rocket's movement through space via four input parameters: pitch, yaw, roll, and thrust.
Figure 6.12 Trajectories in the presence of external interferences. The lack of intervening controls in the presence of a strong wind results in an undesired trajectory that does not reach the intended target.
Figure 6.13 DSR trajectories on the AR boundary. All nonoptimal changes in the α policy serve to move the resulting trajectories into the region. The DSR is thus not locally controllable on the AR boundary. (See color plate section for the color representation of this figure.)
Figure 6.14 Critical DSR trajectory for the Van de Vusse system.
Figure 6.15 A critical CSTR connected to a DSR.
Figure 6.16 Determinant function for a critical CSTR. Only certain points along a residence time range satisfy the condition for a CSTR to be critical.
Figure 6.17 Critical CSTR surface (Λ(C) = 0) and CSTR locus. The CSTR locus intersects Λ(C) = 0 at two points, indicating a CSTR from the feed is critical at two points. (See color plate section for the color representation of this figure.)

Chapter 7: Applications of AR Theory

Figure 7.1 Convex hull of PFR trajectory from the feed.
Figure 7.2 (a) Convex hull of a PFR and CSTR from the feed and (b) convex hull of the CSTR–PFR reactor structure from the feed. Note that this is not the full AR for the three-dimensional Van de Vusse system. (See color plate section for the color representation of this figure.)
Figure 7.3 A CSTR in series with a PFR from the feed point.
Figure 7.4 The determinant of the controllability matrix E as a function of CSTR residence time from the feed point. Two roots exist: one at τ ∼ 36.7 s and the other at the CSTR equilibrium point.
Figure 7.5 (a) Unfilled candidate region for the three-dimensional Van de Vusse kinetics including a critical DSR trajectory from the feed point and (b) the full AR for the three-dimensional Van de Vusse system in c_A–c_B–c_D space. Mixing lines have been removed from the plot to make interpretation of the AR boundary structures easier to identify. (See color plate section for the color representation of this figure.)
Figure 7.6 Optimal reactor structures associated with the three-dimensional Van de Vusse system. Note that both structures terminate with a PFR.
Figure 7.7 The three-dimensional Van de Vusse system for different terminating objective functions: (a) c_D = 0.3 mol/L and (b) c_D = 0.4 mol/L. (See color plate section for the color representation of this figure.)
Figure 7.8 PFR trajectory from the feed for the BTX system.
Figure 7.9 (a) CSTR from the feed for the BTX system and (b) candidate AR for a CSTR followed PFR from the feed.
Figure 7.10 CSTR determinant function for the BTX system. A root of the determinant function exists at τ ∼ 1.33 h. For τ > 10 h, the CSTR effluent concentrations appear to produce determinant values close to zero, indicating that the CSTR lies close to the AR boundary.
Figure 7.11 The AR for the BTX system (a) computed by an automated AR construction algorithm. The region obtained is in agreement with the theoretical prediction given by (b), although there is still a moderate difference between the two regions.
Figure 7.12 (a) Critical DSR trajectories from the feed (curve AB) and from the CSTR equilibrium point (curve CB). (See color plate section for the color representation of this figure.) (b) Full AR for the BTX system.
Figure 7.13 Optimal reactor structures for the BTX system.
Figure 7.14 Maximum toluene surface for the BTX system.
Figure 7.15 (a) Optimal reactor structures in c_B–c_T–c_H space and (b) filled AR for the BTX system in c_B–c_T–c_H space. (See color plate section for the color representation of this figure.)
Figure 7.16 (a) PFR trajectory and CSTR locus from the feed for the adiabatic example and (b) AR for the adiabatic example.
Figure 7.17 Optimal reactor structure for the adiabatic system.
Figure 7.18 AR obtained for isothermal operation (T = 560 K).
Figure 7.19 Sliding a PFR trajectory downward.
Figure 7.20 Rate field for the current candidate region.
Figure 7.21 Generating a candidate AR using only PFRs and a base trajectory. Extension via (a) two PFRs (b) three PFRs, and (c) four PFRs in series, respectively.
Figure 7.22 (a) AR generated using only PFRs and (b) the rate field showing that there are no rate vectors pointing out of the boundary.
Figure 7.23 Interstage cooling without bypass.
Figure 7.24 Adiabatic PFR trajectories for three different inlet temperatures.
Figure 7.25 Recommended arrangement for three adiabatic PFRs.
Figure 7.26 A family of PFR trajectories for different starting inlet temperatures T⁰.
Figure 7.27 (a) The roots of function f when T = 450 K and (b) optimal temperature profiles for a differentially cooled reactor for traversal along boundary ABCDE.
Figure 7.28 Cold-shot cooling for an adiabatic exothermic reaction.
Figure 7.29 (a) Cold-shot cooling for different mixing fractions using two reactors and (b) candidate region obtained for all mixing fractions for the two reactor arrangement.
Figure 7.30 (a) Points on the DCR AR boundary, (b) mixing lines representing cold-shot cooling between DCR effluent with the feed, and (c) a DCR in series in a PFR corresponding to the PFR trajectories in (b).
Figure 7.31 A segment of the AR boundary in ℝ³ when temperature is a control parameter.
Figure 7.32 Graphical interpretation for various fed-batch reactor trajectories. Each trajectory corresponds to a different α policy. The sidestream composition of all trajectories is given by point O.
Figure 7.33 A fed-batch trajectory map for complex kinetics containing multiple nodes (multiple CSTR solutions). This map is generated by plotting many constant α trajectories for different feed point and the same sidestream composition C⁰.
Figure 7.34 Comparison between continuous and batch reactive equipment. All three continuous reactors required to form the AR share equivalent batch structure.
Figure 7.35 Feeding rate and volume profiles for an equivalent fed-batch mimicking a DSR.
Figure 7.36 (a) Concentration profiles for components A, B, and C, achieved in a constant α fed-batch initiated at the seed concentration. (b) Trajectories and locus achieved in the equivalent fed-batch. (See color plate section for the color representation of this figure.)
Figure 7.37 Optimal batch reactor structures for the Van de Vusse system.
Figure 7.38 Potential paths traversed on the AR boundary that satisfy the objectives functions for a batch Van de Vusse system.
Figure 7.39 α (F(t)/V(t)) policies required to achieve the desired objective function values. (a) Feeding policy for when c_D = 0.3 mol/L is the objective function. Structure 2 is required. (b) Feeding policy for when c_D = 0.4 mol/L is the objective function. Structure 1 is required.

Chapter 8: AR Construction Algorithms

Figure 8.1 Geometric interpretation of the hyperplane equation: (a) Hyperplanes in ℝ² are simply straight-line segments (a one-dimensional subspace) and (b) hyperplanes in ℝ³ are planes (a two-dimensional subspace).
Figure 8.2 Hyperplanes in x–y spac
Figure 8.3 Geometric interpretation of vertex and facet enumeration.
Figure 8.4 (a) Stoichiometric subspace for the steam reforming system projected onto different component spaces: CH₄–H₂O (top left), CH₄–CO (top right), and H₂–CO₂ (bottom left). The stoichiometric subspace, projected onto ϵ₁–ϵ₂ space, is also given (bottom right). (b) Stoichiometric subspace for methane steam reforming now given for a different feed point.
Figure 8.5 Stoichiometric subspace for multiple feeds. (See color plate section for the color representation of this figure
Figure 8.6 Stoichiometric subspace with residence time.
Figure 8.7 Summary of AR construction algorithms.
Figure 8.8 The AR may be enclosed by a larger region, containing both attainable and unattainable points.
Figure 8.9 (a) The inside-out construction process and (b) the outside-in construction process.
Figure 8.10 DSR trajectories for constant α values using the feed point as the DSR sidestream composition. The CSTR locus from the feed point is also shown for comparison. It is clear that the equilibrium points for the DSR trajectories coincide with the CSTR locus points. Van de Vusse kinetics is used here. (See color plate section for the color representation of this figure.)
Figure 8.11 The RCC method applied to the three-dimensional Van de Vusse. The method may be summarized into four broad construction phases: (a) Initialization (PFR from the feed point), (b) growth (initial DSR trajectories from the feed), (c) iteration (constant α DSR trajectories from the extreme points), and (d) polish (PFR trajectories from the convex hull points).
Figure 8.12 The RCC method for the isola example from Chapter 5: (a) Constant α DSR trajectories from the feed point, (b) the candidate region obtained by the RCC method, and (c) the true AR for the isola example. Note that the RCC method fails to identify the isola in this instance.
Figure 8.13 (a) Three-dimensional Van de Vusse AR, projected onto different component planes: (b) c_A–c_B space, (c) c_A–c_D space, and (d) c_B–c_D space.
Figure 8.14 (a, b) Two shapes, projected onto different planes. The shape of the resulting projection differs between objects and the plane that it is projected on.
Figure 8.15 Two-dimensional slices in ℝ³ taken in the (a) x–z plane, (b) y–z plane, and (c) x–y plane.
Figure 8.16 Reaction in a two-dimensional plane in ℝ³. Reactors that produce effluent concentrations that lie within a plane of a fixed composition are termed iso-compositional reactors. These reactors may be viewed as DSR with specialized sidestream policies.
Figure 8.17 (a) Iso-compositional trajectories for planes in c_B–c_D space at fixed values of c_A. The trajectories all lie within a two-dimensional plane. (See color plate section for the color representation of this figure.) (b) Final candidate region produced for the Van de Vusse system using the iso-state method.
Figure 8.18 The AR construction process for one iteration of the complement method. Rate vectors that intersect the region are included in the list of achievable points and a newer, larger region is then computed.
Figure 8.19 Talking CSTRs.
Figure 8.20 Results for three-dimensional Van de Vusse kinetics, taken at snapshots during construction.
Figure 8.21 Graphical representation of the method of bounding hyperplanes. Hyperplanes are introduced at the corners of the polytope.
Figure 8.22 Rate vectors evaluated relative to a bounding hyperplane.
Figure 8.23 Summary of construction results for the three-dimensional Van de Vusse kinetics, using the method of bounding hyperplanes.
Figure 8.24 Computing candidate ARs using bounding hyperplanes, rotated about an edge.
Figure 8.25 Comparison of constructions between (a) bounding hyperplanes and (b) rotated hyperplanes.
Figure 8.26 Alternate construction scenarios using rotated hyperplanes: (a) Unbounded constructions in τ space and (b) nonisothermal constructions with temperature-dependent kinetics.
Figure 8.27 The shrink-wrap method. Invalid extreme points are eliminated via boundary intersections.
Figure 8.28 Candidate AR constructed using the shrink-wrap method. Results are generated for (a) 25 × 25, (b) 50 × 50, and (c) 100 × 100 grid sizes.
Figure 8.29 The total connectivity model. Approximation of PFRs (a) and DSRs (b) using a number of CSTRs.
Figure 8.30 The total connectivity model: (a) Streams around the ith CSTR in the total connectivity model and (b) overview of the total connectivity model.
Figure 8.31 Three CSTRs in series.
Figure 8.32 ROP and DN blocks for use in the IDEAS framework.
Figure 8.33 Two-dimensional AR for Van de Vusse kinetics, constructed using the IDEAS framework, as a function of grid size.

Chapter 9: Attainable Regions for Variable Density Systems

Figure 9.1 Reactor network involving a variable density system.
Figure 9.2 Gases compressed into a single chamber.
Figure 9.3 Parallel reactors in a variable density system.
Figure 9.4 Stoichiometric subspace for methane steam reforming for different component pairs in mass fraction space: (i) CH₄–H₂O, (ii) CH₄–CO₂, (iii) CO–H₂, and (iv) ϵ₁–ϵ₂.
Figure 9.5 Comparison of stoichiometric subspaces in concentration space obtained for the methane steam reforming system for constant density (hatched regions) to the region obtained via mass fractions (clear region). (See color plate section for the color representation of this figure.)
Figure 9.6 Stoichiometric subspace for the Van de Vusse system, generated in mass fraction space.
Figure 9.7 PFR trajectory and CSTR locus from the feed point.
Figure 9.8 (a) Full AR for the Van de Vusse system in mass fraction space. (b) AR for the Van de Vusse kinetics, converted to concentration space. The transparent region is the AR obtained for an identical feed point when constant density is assumed. (See color plate section for the color representation of this figure.)
Figure 9.9 (a) AR for the methane steam reforming reaction in mass fraction space. (b) Comparison of the stoichiometric subspace and two-dimensional projection of the AR onto z_CO– $c09-math-0132$ space.

Chapter 10: Final Remarks, Further Reading, and Future Directions

Figure 10.1 A hybrid AR construction method. This method of construction may assist in identifying the true AR boundary when both construction methods agree (a), or signal when complex structures form the boundary if there is disagreement in the two methods (b).

List of Tables

Chapter 1: Introduction

Table 1.1 Sam's First Few Experiments
Table 1.2 All Discoveries and Recommendations Made by Sam, Alex, and Donald
Table 1.3 Experiments for Minimum Hydrogen Production
Table 1.4 Rate Expressions for the BTX System

Chapter 2: Concentration and Mixing

Table 2.1 Selected Concentration Data for the BTX Data Obtained in a Batch System
Table 2.2 Concentration Profile Data for a Two-Component System
Table 2.3 Concentration Profile Data for a Three-Component System
Table 2.4 Concentration Profile Data for a Five-Component System
Table 2.5 Mixture Compositions for Beakers 1 and 2
Table 2.6 List of Possible Concentrations Given by the R&D Department
Table 2.7 Achievable and Unachievable Points for the R&D Investigation
Table 2.8 Six-Dimensional Concentration Data

Chapter 3: The Attainable Region

Table 3.1 Summary of Batch Experiments and the Associated Maximum Toluene Concentration Achieved

Chapter 4: Reaction

Table 4.1 Four Arbitrary (Possibly Feasible) Concentrations and Associated Rate Vectors
Table 4.2 A List of Concentration Vectors. Can You Identify Which Vector Is an Equilibrium Point?
Table 4.3 Summary of Corresponding Rate Vectors
Table 4.4 Parameters for Different DSRs
Table 4.5 Summary of the Three Fundamental Reactor Types Used in AR Theory, Based on Physical, Mathematical, and Geometric Properties

Chapter 5: Two-Dimensional Constructions

Table 5.1 Rate Constants for Van de Vusse Kinetics in the Equal Rate Case
Table 5.2 Rate Constants for the a₁ < a₂ Case for Van de Vusse Example
Table 5.3 Rate Constants for a₁ > a₂ Case for Van de Vusse Kinetics
Table 5.4 Conversion, Selectivity, and Reactor Yield for the Points in Figure 5.9
Table 5.5 Information Necessary to Numerically Evaluate the Integral in Equation 5.6b
Table 5.6 Summary of Required Operating Conditions for Payback Period

Chapter 7: Applications of AR Theory

Table 7.1 Mixing Results for Different Values of λ
Table 7.2 Summary of Fed-Batch Operating Parameters

Chapter 9: Attainable Regions for Variable Density Systems

Table 9.1 Rate Constant for Methane Steam Reforming and Water–Gas Shift Reactions
Table 9.2 Pure Component Properties for the Steam Reforming and Water–Gas Shift Reaction
Table 9.3 Binary Interaction Parameters for Use in the Peng–Robinson Equation of State

Appendix B: Mathematical Topics

Table B.1 Set-Builder Notation

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Names: Ming, David, 1985- author. | Glasser, David, 1936- author. | Hildebrandt, Diane, 1961- author. | Glasser, Benjamin John, 1968- author. | Metzger, Matthew, 1983- author.

Title: Attainable region theory : an introduction to choosing an optimal reactor / by David Ming, David Glasser, Diane Hildebrandt, Benjamin Glasser, Matthew Metzger.

Identifiers: LCCN 2016022700 (print) | LCCN 2016025203 (ebook) | ISBN 9781119157885 (cloth) | ISBN 9781119244714 (pdf) | ISBN 9781119244707 (epub)

Subjects: LCSH: Chemical reactors-Design and construction. | Statistical tolerance regions.

Classification: LCC TP157 .M528 2016 (print) | LCC TP157 (ebook) | DDC 660/.2832–dc23

Preface

The problem that this book sets out to solve can be formulated very simply. Given a set of chemical reactions with known kinetics, what is the best reactor that can be used to carry out the set of reactions. As easy as the question is to ask, the answer is not obvious, as on the face of it there are an infinite number of possibilities. What this preface will do is to outline the 50-year journey to solve this problem. This history also embodies a cautionary tale for those doing (and wanting to do) research.

I did my PhD looking at a reaction and its kinetics. As a result of this, I became interested in chemical reactor theory. I remember, in particular, the excellent books of Denbigh and Levenspiel. Both of them talked about the aforementioned problem and documented some heuristics to help industrialists to come up with some solutions. However, it was recognized that these heuristics were sometimes contradictory.

At that stage, I became interested in optimization in general. Also, on my first sabbatical leave, I was able to work with some of the greats of chemical engineering (Stanley Katz, Reuel Shinnar, and Fritz Horn) and became involved solving some more limited optimization problems, such as using contact times for catalytic reactions, and minimizing holding times for a series of continuous-flow stirred tank reactors (CSTRs). Also it was at this time that I became interested in using Pontryagin's maximum principle (developed for space exploration problems) on chemical reactor problems. This was the stage that I, at first, thought we could solve the main problem by extending residence time distribution theory to nonlinear kinetics. There was the well-known result for segregated systems that one could work out conversions for all kinetics using the residence time distribution. I thought we should be able to extend this theory for all possible reactors and then use the maximum principle to solve the general problem.

On the next two sabbaticals, I worked with Roy Jackson and Cam Crowe to take this idea further, and in the end, much to our joy, Roy Jackson and I solved the problem (after 15 years!). The joy soon evaporated as we realized that even though we had a complete description for all possible reactors with nonlinear kinetics, in order to solve the main problem we had to find a function that could have a non-countable infinity of changes with possibly some of the values going to infinity. This was clearly an impossible problem to solve.

This meant going back to the drawing board! I started to fiddle with some simple problems from the literature by drawing simple two-dimensional graphs. What I soon realized is that if there were concavities on the graphs, they could be filled in with straight lines to make what are called convex hulls. What was really exciting was that these lines were nothing more than mixing between two points on the graph. Out of this came the idea that a reactor was a system that was made up of two processes: reaction and mixing. Each of these processes can be represented on a graph as vectors. Suddenly, the problem I was looking at changed to a geometric one, as I was now looking at making the region in the space as large as possible using the two vector processes. When visiting Martin Feinberg at Rochester, he drew my attention to a paper by Fritz Horn that talked about what he termed attainable regions (ARs). That is, in this context, the largest region in some component space that one could obtain using any processes. He showed that if one had this AR, the optimization problem was relatively simple. Without at first realizing what I was doing, I had found a geometric way of finding the AR for reacting systems. At this stage, I was only doing it graphically on problems that could be represented in two dimensions.

At this stage, it became clear why one could not solve this problem using standard optimization methods. This was because mixing is not a differential process in the ordinary sense. That is, one could not use methods that only looked in the immediate neighborhood as one could mix from any point that was itself attainable, and this could be far from the neighborhood of the point we were examining. An interesting point is that the AR method can solve these non-continuous problems, and so is essentially a new method of optimization, but to the best of my knowledge this idea has not been taken up in other fields of study.

When I arrived back from sabbatical, I looked for a student to work on the problem to extend the results. At this stage, Diane Hildebrandt, who I had taught as an undergraduate, and had done her MSc with me , was looking for a PhD project. I warned her that the project was a little open-ended. However, she was sufficiently interested in the topic (was this brave or foolhardy?) to want to work on it. In the end, out of her work, came the foundations of our work on AR theory. At last, we had a method that could solve the problem I had started out to look at more than 20 years earlier.

What is really interesting (and important) is that we were able to solve a problem that was generally regarded as impossible by thinking about it in a very different way. Instead of trying to answer the problem of finding the optimal reactor in one step, we had first asked what processes are occurring in a chemical reactor, how can we represent them, and how can we then use these processes to ask what are all possible outcomes. Only at this stage do we look at how to perform the optimization. By breaking the problem up into smaller stages, in the end, we were able to solve it.

1. Passion
2. Patience
3. Perseverance
4. Persuasion

I believe the history of this book typifies all of these aspects.

Passion

If I (and others) had not believed that this project was worthy of spending time on it, it would not have been done.

Patience

I worked patiently on it for 15 years to get an answer that was of no real use.

Perseverance

Perseverance is the hard work you do after you get tired of doing the hard work you already did. —Newt Gingrich. I believe this speaks for itself in the context of this work.

Persuasion

It is not sufficient to do good work on its own; one needs to persuade the rest of the world. We really struggled with this! Thomas Grey said it all.

Full many a gem of purest ray serene,
the dark unfathomed caves of ocean bear,
full many a flower is born to blush unseen
to waste its sweetness on the desert air
David Glasser

Attainable Region Theory

An Introduction to Choosing an Optimal Reactor

Preface

Passion

Patience

Perseverance

Persuasion