Identification of the Attainable Region for Batch Reactor Networks

Benjamin J. Davis, Larry A. Taylor, and Vasilios I. Manousiouthakis* Department of Chemical and Biomolecular Engineering, UniVersity of California, Los Angeles, California 90095

In this work, we describe a method for automatically identifying the set of all points in concentration space that represent outlet compositions of some network of discretely fed batch reactors for a given reaction set with known kinetics. This so-called batch attainable region (BAR) is dependent on the batch network’s feed and total operating time, and it is shown to be quantifiable using the Infinite DimEnsionAl State-space (IDEAS) framework. We first establish that a simple batch reactor model possesses the properties that allow application of the IDEAS framework. We then formulate the resulting IDEAS Infinite Linear Program (ILP) whose solution is guaranteed to identify the globally optimal network of batch reactors. We subsequently use a simple transformation of this IDEAS ILP that leads us to propose two algorithms that are related to the construction of the true BAR. The first is a “Shrink-Wrap”-like algorithm that is similar to that previously reported [Manousiouthakis et al. The Shrink-Wrap Algorithm for the Construction of the Attainable Region: Application of the IDEAS Framework. Comput. Chem. Eng. 2004, 28, 1563] and creates increasingly accurate approximations of a set guaranteed to contain the true BAR for all network operating times. The second is a breadth-first algorithm that creates increasingly accurate inner approximations to the BAR for a given network operating time. These two algorithms are applied to an example from the literature and are shown analytically to converge in the limit to the true BAR.

Introduction to an example from the literature and are shown to converge in the limit to the true BAR. Evaluation of limits on the performance of reactors and reactor networks is crucial to the economic success of any Background chemical process network. Consequently, the analysis and design of reactors and reactor networks have been the primary foci of Automatic (computer-based) reactor network synthesis (RNS) process systems engineering research. Previous works on non- evolved as a field of its own starting in the 1980s. Chitra and steady-state reactor networks have addressed techniques for Govind2 performed work in 1981 on the identification of optimal analyzing, modeling, or optimizing single-batch units, but they reactor types and configurations using a super-structure-based make little mention of how units can be used in conjunction or approach. They later expanded on their earlier work, applying what theoretical limits exist on the performance of these types a superstructure-based method for optimal RNS for both of non-steady-state systems. Because non-steady-state networks isothermal3 and non-isothermal4 reaction systems. Ong5 con- are fundamentally different from their steady-state counterparts, sidered the optimization of continuously stirred tank reactors identification of these performance limits requires careful (CSTRs) in series using Bellman’s6 dynamic programming. consideration of the effect of time (both reaction and holding Pibouleau et al.7 proposed a mixed-integer nonlinear program- time) and causal relationships between reactors. ming (MINLP) formulation for the automatic synthesis of The goal of this work is to apply the Infinite DimEnsionAl networks featuring CSTRs and single-stage separation units. In State-space (IDEAS) framework to construction of the batch 1994, Omtveit et al. presented a RNS method that also included attainable region (BAR) for non-steady-state networks of batch a separation network as a separate sub-problem;8 that paper also reactors; this is the first application of IDEAS to a network of gives an extensive literature review of work on RNS to that dynamic process units. The remainder of the work is structured point. Smith and Pantelides9 gave another reactor-separator as follows: first, we give background information on reactor network superstructure-based formulation in their 1995 work. network synthesis (RNS), the IDEAS conceptual framework, Bikic and Glavic produced a series of papers on a superstructure- and attainable region (AR) construction. Next, the applicability based nonlinear programming (NLP) method for RNS for of IDEAS to batch RNS is established and the relevant IDEAS networks with multiple multicomponent feeds,10 non-isothermal infinite linear program (ILP) is formulated. We then outline a complex reaction schemes,11 and reactor/separator networks.12 variable transformation which leads to two algorithms that are Their 1996 work stated that “the proposed design procedure related to the construction of the true BAR: the first is similar can also be used to support the design of batch processes”, but to the “Shrink-Wrap” algorithm that was developed by Man- that work did not specifically address the claim. Esparta et al.13 ousiouthakis et al.1 for the construction of the steady-state AR proposed a superstructure-based method for RNS using iso- and creates increasingly accurate approximations of a set thermal two-phase CSTRs in 1998. Hua et al. proposed a NLP guaranteed to contain the true BAR for all network operating model for RNS that included “differential recycling DSR” times. The second is a breadth-first algorithm that creates reactor units.14 Mehta and Kokossis15 proposed a stochastic increasingly accurate inner approximations to the BAR for a optimization approach to non-isothermal and multiphase RNS. given network operating time. These two algorithms are applied Pahor et al. proposed a superstructure/MINLP approach for optimization16 and then, in later work, applied the method to * To whom correspondence should be addressed. Tel.: 310 206 0300. the non-isothermal production of allyl chloride.17 Moreover, in Fax: 310 206 4107. E-mail address: [email protected]. 1999, Grossman et al. gave a review of advances in mathemati-

10.1021/ie071664l CCC: $40.75 © xxxx American Chemical Society Published on Web 04/23/2008 PAGE EST: 13 B cal programming for process systems synthesis;18 this review mum mixed reactor networks,40 exothermic reversible reaction has sections devoted to both RNS and AR construction theory, kinetics,41 and reaction systems with external heating and respectively. cooling.42 The RNS problem was also approached using the methods In 1997, Feinberg and Hildebrandt43 reported on the deter- of optimal control; the work by Aris19 was significant in this mination of the optimal reactor network configuration using field, addressing the problem of optimal control of a batch the geometric properties of the AR extreme points in the first reactor. In 1970, Paynter and Haskins20 formulated an optimal part of a three-part series. This work was later expanded by control problem for determining the optimal reactor type for a Feinberg in 1999 to a more extensive set of mathematical 44 single reactor using an axial dispersion model. Waghmare and properties of the AR boundary and in 2000 to properties of 45 46 Lim applied optimal control theory to single isothermal reactor critical DSRs and CSTRs whose outlets are on the AR 47 systems21 and later applied the same techniques to complex boundary. In 1999, McGregor et al. examined the relationship reaction schemes.22 Achenie and Biegler23-25 proposed an NLP between the geometric AR identification method and Pontrya- 48 method for RNS that used superstructures in a “target-based” gin’s maximum principle. Rooney et al. extended the geo- approach. Godorr et al.26 outlined optimal control policies for metric AR identification method to higher dimensions by extending the 2D subspace. In 1997, Nisoli et al.49 outlined a reactor structures on the AR boundary, using temperature as method for identifying the AR for a two-phase reaction- the control variable. Hillestad formulated the RNS problem as separation system; they applied the method to the production an optimal control problem and then examined its solution for and separation of dimethyl ether (DME) from methanol and the isothermal27 and non-isothermal28 cases. methyl tert-butyl ether (MTBE) from isobutene and methanol. The IDEAS conceptual framework was proposed by Man- That same year, Smith and Malone50 outlined an application of ousiouthakis et al.1 in an effort to overcome two limitations of AR identification in the free-radical polymerization of poly- superstructure-based optimal process network synthesis meth- (methyl methacrylate) (PMMA). Later, in 2002, Gadewar et al.51 ods: (i) the considered superstructure may impose unforeseen analyzed networks of two-phase CSTRs that are surrogates for limitations on the eventually obtained optimal network, and (ii) reactive distillation units to find an AR for such networks. the nonconvex nature of the resulting superstructure-based Kauchali et al.52 used the earlier methods of Nisoli et al.67 to optimization formulations (NLP, MINLP, etc.) only guar- identify candidate ARs for the water-gas shift (WGS) reaction, antees local optimality of the obtained optimal network. which is a problem that also was previously studied by Omtveit IDEAS overcomes these limitations by considering all possible et al.53 process network configurations and establishing that most More recently, the IDEAS conceptual framework has also commonly applied process models can be used to yield been applied in the construction of the AR for reactor networks. optimization formulations with an infinite number of variables Burri et al.54 first presented several IDEAS-based infinite linear and an infinite number of linear constraints. The ability of programming (ILP) formulations of the AR construction prob- IDEAS to address several long-standing process network lem in 2002. That same year, Kauchali et al.55 independently synthesis problems has been demonstrated on the minimum developed an IDEAS-like linear programming model for utility cost (MUC) problem for mass exchange networks29 extending candidate ARs. Manousiouthakis et al.1 then presented (MENs), the minimum plate area30 and MUC31 problems for properties of one of the aforementioned IDEAS ILPs, which heat-integrated complex distillation networks, the minimum total allowed construction of the true AR without explicit solution annualized cost (MTAC) problem for separation networks32 and of the ILP using a so-called “Shrink-Wrap” algorithm. Concur- power cycle networks,33 the MUC problem for heat and power rent and independent work by Abraham and Feinberg56 proposed integrated complex distillation networks,34 and the minimum a method of bounding hyperplanes to identify subsets of a total liquid hold-up problem for complex reactive distillation superset containing the AR that are guaranteed not to contain networks.35 the AR. More recently, Zhou and Manousiouthakis demonstrated that variants of the Shrink-Wrap algorithm are applicable to The attainable region (AR) for a given set of reactions and the AR construction for nonideal axial dispersion reactor reactor technologies is defined as the set of all points in models57 and variable-density fluid reactor models,58 respec- concentration space that are attainable through reaction and tively. mixing from a given feed point; this definition has been widely There is extensive work in the literature that addresses single- credited to Horn in 1964.36 Quantification of the AR for reactor batch reactor optimization and batch process scheduling. Rip- networks is an important problem in chemical process optimiza- pin59 gave a review of studies of individual batch units and their tion, because knowledge of the AR quantifies, for process optimization in 1983. That same year, he also wrote an overview designers, the fundamental limitations on the performance of of general structures for batch process systems.60 He followed chemical process flowsheets. Identification of the particular up these reviews 10 years later with the current progress in the s s reactor or, more generally, the reactor network whose output field of engineering and design of batch processes.61 Reklaitis62 concentration vector is an extreme point of the AR is often the gave a review of progress and issues in computer-aided batch 37 objective of reactor engineering studies. Work by Gavalas in process design in 1990. Levien63 wrote on the optimal design 1968 on nonlinear differential equations for chemical reactors of batch, discrete semi-batch, and continuous semi-batch reactor introduced the concept of the invariant manifold and gives units in his 1992 work. Terwiesch et al.64 surveyed industry example two-dimensional (2D) plots of the conversion of needs for batch processes and suggested both optimal control different species. Many of the proposed AR construction methods for improvements and also further research problems methods in the literature are based on a geometric approach to to be addressed. Yi and Reklaitis65 have performed work on AR identification that was outlined in the 1987 work by Glasser the optimal synthesis of batch storage networks for chemical et al.38 This geometric approach has spawned numerous other processes. Maravelias66 examined the problem of optimal works whose objective was to identify the AR for adiabatic, scheduling of single-stage and multistage batch processes using variable density reactor networks,39 segregated and maxi- a mixed-integer sequencing algorithm. Later work by Sung and C Maravelias67 defines a “process attainable region” for production outlet time, λ the reactor technology flag, Fh the reactor planning and scheduling problems with limited equipment volumetric “feed” (reactor volume), F the reactor volumetric capacity. “effluent” (constant density, F ) Fh), and N the number of species considered. Application of IDEAS to Batch Reactor Network The “reactor technology flag” (λ) is a convention that is used Synthesis to allow holding tanks to be part of the formulation. If λ ) 1, the unit is a batch reactor with nonzero Ri defined by model The fundamental difference between batch and steady-state eqs 2-5. If λ ) 0, there is no reaction and the unit is a holding reactor network synthesis problems is the time dependence of tank that is defined by eq 6 with holding time tout - tin. For the underlying batch reactor process. In a network context, this IDEAS to be applied to this model, we must the identify vectors time dependence immediately raises the issue of “causality”; u1, u2, y1, and y2 and information maps M1 and M2 (which, in a batch reactor network, a reactor cannot “feed” another together, comprise M) that satisfy the following properties: reactor unless its contents are unloaded before the other reactor’s loading time. To examine causality in a straightforward manner, u y M (u )‚u we consider that batch reactor loading, unloading, and mixing M: u ) [ 1 ] f y ) [ 1 ] ) [ 1 2 1 ] (7) u2 y2 M2(u2) operations can only occur at prespecified discrete times, {t0, t1, t2, ..., tn} and are instantaneous. For the remainder of this work we will refer to the reactor inlet and outlet volumes (which are We will define these vectors as follows: equal, because of our assumption of constant density) as flow u ) [Fh] ) [F] ) y (8) rates; this is a slight misnomer, because each reactor’s inlet and 1 1 outlet are discrete (i.e., there is no temporal nature to the flow u ) [Ch(1) Ch(2) ... Ch(N) tin tout λ ] (9) into or out of a reactor), but the convention of calling them 2 ) flow rates makes the analogy between batch reactor networks y2 [C(1) C(2) ... C(N) ] (10) and other steady-state reactor networks clearer. The inlet and outlet volumes of a batch reactor unit are extensive properties Because y1 ) u1, the map M1 is just the identity map. We that do not affect the units’ intensive properties, such as species uniquely define the map M2 by assuming in this work that the concentrations, temperature, pressure, etc. set of differential equations in eq 2 with initial conditions in eq The standard model for a batch reactor is outlined in almost 3 admits a solution and that the solution is unique. Sufficient any reaction engineering textbook (see the work of Levenspiel,68 conditions on the properties of the rate vector for this to be Froment and Bischoff,69 Schmidt,70 Fogler,71 Nauman,72 Raw- true can be found in theorem 2.4 of Khalil.75 Each of the outlet 73 lings, etc.). More recently, there have been more-complicated concentrations can be found using the information in u2; batch reactor models proposed in the literature that account for therefore, we can define the map M2 as follows: imperfect mixing and continuously fed batch operation;74 however, we will not specifically address these complications eqs 2-6 M : u 98 y in this work. To apply IDEAS to the batch reactor network 2 2 2 synthesis problem, we must first prove that the batch reactor model is flow-invariant with respect to its intensive properties. Therefore, this batch reactor model satisfies the first necessary Second, we must prove that the operations of mixing and property for the applicability of the IDEAS framework. The second propertysthat the operations of mixing, splitting, and splitting in the distribution network are linear. A more formal s mathematical method for showing the applicability of IDEAS recycle are linear is obvious, given that the intensive properties to a unit model has been previously outlined in Zhou et al.57 of the units are considered to be known. We define our IDEAS inlet-outlet information map (M) for the batch reactor model as follows: Graphical Representation of Batch Reactor Networks As an example, consider a simple batch reactor network that M u ) [Ch(1) Ch(2) ... Ch(N) tin tout λ Fh ] 98 y ) consists of three reactors and two holding tanks. Reactor 1 is [C(1) C(2) ... C(N) F ] (1) fed at some initial time (we will call it t0) and runs until its outlet time t1. At that time, the reactor 1 outlet is split into three dCˆ (t) ) i ) ˆ ˆ ˆ ˆ parts, which are fed into reactor 2, reactor 3, and holding tank if λ 1: Ri(C1(t), C2(t), ..., Ci(t) ..., CN(t)) dt 1, respectively. Reactor 2 operates from t1 (which is the output ∀i ) 1, ..., N (2) time of reactor 1) until time t2 and then its output enters holding tank 2. Reactor 3 operates from t until t , at which time its ) h ≡ ˆ in ∀ ) 1 3 if λ 1: C(i) Ci(t ) i 1, ..., N (3) outlet is mixed with the contents of holding tanks 1 and 2 to form the network outlet. This simple batch reactor network can if ) 1: C(i) ≡ Cˆ (tout) ∀i ) 1, ..., N (4) λ i be visualized in a time-axis form, as shown in Figure 1. if λ ) 1: Cˆ (t) g 0 ∀i ) 1, ..., N; ∀t ) [tin, tout] (5) We can also represent this same process flowsheet in an i OP/DN form, by rearranging the process network diagram such if λ ) 0: C(i) ) Ch(i) ∀i ) 1, ..., N (6) that all the mixing and splitting operations are contained in the distribution network (DN) and all unit operations (the three batch where u is the reactor inlet information vector, y the reactor reactors and two holding tanks) are contained in the operator outlet information vector, M the nonlinear reactor information block (OP). This method of representing a flowsheet can be input-output map, Ch i(t) the concentration of species i in the used to best visualize the IDEAS formulation of the batch reactor reactor at time t, Ri the rate of generation of species i (in units network problem. In the IDEAS formulation, all possible of moles per volume per unit time), Ch(i) the concentration of reactors are considered in the formulation of the optimization species i in the reactor inlet, C(i) the concentration of species problem; therefore, instead of a diagram with three reactors, i in the reactor outlet, tin the reactor inlet time, tout the reactor there is an infinite number of batch reactors and holding tanks D

outlet time is tn. Thus, n designates the number of times that the reactor/holding tank contents are allowed to mix (n ) 3in the example above). Also note that holding tanks are implicitly included in this formulation as units in the OP block (no holding occurs in the DN).

IDEAS ILP Formulation for Batch Reactor Network Synthesis The IDEAS optimization problem outlined below considers every possible reactor and holding tank unit and all causally feasible interconnections in the DN, thereby guaranteeing that all possible batch reactor networks are included in the math- ematical problem formulation. Without any loss of generality, each unit’s input information vector u2 is considered to be known, and thus, using the information map M, its output information vector y2 is also known. Knowledge of each unit’s Figure 1. Batch reactor network flowsheet in time axis form. input and output information vectors y2 and u2 will be shown below to lead to an IDEAS formulation of the optimal batch reactor network synthesis (RNS) problem that is an infinite linear program (ILP). Before we can outline this IDEAS ILP, we first define (infinite) sets of reactors that are organized by their inlet and outlet times:

Sij ≡ the set of all reactors and holding tanks that operate from time ti to time tj (11)

Using these sets as defined, the formulation for the batch reactor network synthesis problem can be set up as follows:

V) 0k in ijk ij h ij minimize f(F*m* , F *l, F ml, F m, F m)

subject to: (1) Distribution Network Inlet Mass Balance (Splitting)

n n n ) 0k ) h 0k ) 0k F * ∑ ∑ F*m* ∑ ∑ F m ∑ ∑ F m k)1 m∈S0k k)1 m∈S0k k)1 m∈S0k

Figure 2. IDEAS representation for batch reactor network problem. (2) OP Outlet Mass Balances (Splitting) Table 1. Definition of Symbols Used in Figure 2 n variable definition ij ) ijk F m ∑ ∑ F lm ) jk Fh* flow out of the DN/overall network k 2 l∈S k>j>i F* flow into the DN/overall network hij ∀ ∈ ij ∀ ) - ∀ ) - < Fm flow into the mth reactor/holding tank, m S ; i 0, ..., n 2; j 1, ..., n 1(i j) operating from time ti until time tj ij flow out of the lth reactor/holding tank, Fl (3) Distribution Network Outlet Mass Balance (Mixing) operating from time ti until time tj ijk flow from the lth reactor/holding tank, which Fml n-1 n-1 operates from time ti until time tj to the h ) in ) in mth reactor/holding tank which operates F* ∑ ∑ F *l* ∑ ∑ F l ) in ) in from time tj until time tk i 0 l∈S i 0 l∈S 0k Fm** flow from DN inlet to the mth reactor/ holding tank, operating from time t0 until time tk (4) Distribution Network Outlet Component Balances (Mix- in flow to DN outlet from the lth reactor/ F*l* ing) holding tank, operating from time ti until time tn h ij hij ) h ij h ij h ij T Cm property vector for the flow Fm [Cm(1) Cm(2) ... Cm(N)] n-1 n-1 ij ij ) ij ij ij T in in in in Cl property vector for the flow Fl [Cl (1) Cl (2) ... Cl (N)] h h ) ) T C*F* ∑ ∑ Cl F *l* ∑ ∑ Cl F l Ch * property vector for the DN outlet ) [Ch *(1) Ch *(2) ... Ch *(N)] T i)0 l∈Sin i)0 l∈Sin C* property vector for the DN inlet ) [C*(1) C*(2) ... C*(N)] (5) OP Inlet Mass Balances (Mixing) included in OP as in Figure 2. The variables shown in Figure n-2 2 are defined in Table 1. h jk ) ijk F m ∑ ∑ F ml Note that (i) the network outlet and all reactors in the network i)0 l∈Sij may only be fed at their designated inlet time, (ii) the network i

n n ) 0k 0k ) 0k 1. F* ∑ ∑ Fm am** ∑ ∑ Fm k)1 m∈S0k k)1 m∈S0k

n ij ) jk ijk 2. Fm ∑ ∑ Fl alm k)2 l∈Sjk k>j>i ∀m ∈ Sij; ∀i ) 0, ..., n - 2; ∀j ) 1, ..., n - 1(i < j)

n-1 ) in 3. 1 ∑ ∑ a*l* Figure 3. Matrix A from eq 24. i)0 l∈Sin The previously mentioned constraint sets 1 and 2 then lead to - n 1 the matrix equation: where A is a matrix of mixing ratios (shown h ) in in 4. C* ∑ ∑ Cl a*l* i)0 l∈Sin AF ) F (24)

n-2 in Figure 3) and F is given as: ) ijk 5. 1 ∑ ∑ aml i)0 l∈Sij F ) i

To approximately identify the solution to the aforementioned n-1 infinite problem, we discretize the concentration space with a ) in 3. 1 ∑ ∑ a*l* finite number of grid points. We then consider all units with an i)0 l∈Sin inlet composition vector at one of the grid points, some at n-1 starting time ti, and some at ending time tj. Let U be the finite h ) in in number of units that operate from each starting time ti to each 4. C* ∑ ∑ Cl a*l* ij i)0 ∈ in ending time tj; we will define new finite sets G with car- l S dinality U, which approximate the infinite sets Sij. The vector ij of all flows through each of these units in set G can be defined n-2 as follows: ) ijk 5. 1 ∑ ∑ aml i)0 l∈Sij ij ) ij ij ij ‚‚‚ ij T < < F [F1 F2 F3 FU ] i j k jk ∀i ) 0, ..., n - 1; ∀j ) 1, ..., n (i < j) (23) ∀m ∈ S ; ∀j ) 1, ..., n - 1; ∀k ) 2, ..., n (j < k) G n-2 eliminates points in concentration space (identifying them as h jk ) ij ijk 6. Cm ∑ ∑ Cl aml “unattainable”) by explicitly calculating from where each point i)0 l∈Sij must have come. An explicit description of this algorithm is < < i j k given as follows: ∀ ∈ jk ∀ ) - ∀ ) < m S ; j 1, ..., n 1; k 2, ..., n (j k) (1) Identify a suitable superset in concentration space for the system in question, based on knowledge of its physical 7. 0 e aijk e 1 lm constraints. ij jk ∀m ∈ S ; ∀l ∈S ; ∀i ) 0, ..., n - 2; ∀j ) 1, ..., n - (2) Discretize the superset in all directions; the level of 1; ∀k ) 2, ..., n (i < j < k) accuracy of the method is improved as this discretization becomes finer. 0 e ain* e 1 *l (3) Start from an extreme point of the current set and travel ∀i ) 0, ..., n - 1; ∀k ) 1, ..., n; ∀m ∈ S0k; ∀l ∈ Sin backward along the path of the batch reactor in concentration space from t1 to t0. If this new point is in the current set, keep Constraint 4 can be thought of as a specification on the outlet it. If it is not, remove it from the set. concentration for the feasible solution. The set of mixing ratios (4) Repeat step 3 until no more extreme points can be that satisfy the aforementioned constraints defines the AR for removed; this region is guaranteed to contain the true BAR for the batch reactor network for a given n; this feasible set is similar all network operating times n and for a given reaction/holding 1 to that obtained in Manousiouthakis et al. for steady-state RNS. time, t1 - t0. Because of the fact that there is only one network inlet, there can only be one possible inlet concentration set for each reactor/ Breadth-First BAR Construction Algorithm holding tank that is fed at time t0 (the feed concentration). Similarly, all reactors fed at time t1 can only be fed by the The aforementioned Shrink-Wrap-like algorithm is only effluent of the reactor with inlet time t0 and outlet time t1 or guaranteed to converge to a superset of the BAR. An alternative the holding tank with inlet time t0 and outlet time t1. Reactors algorithm that is guaranteed to converge to BARn (for any given that feed at time t2 can be fed by some linear combination of n) is given next. BARn is “grown” from time t0 out to time tn, the effluents of reactors/holding tanks that outlet at t2 and so to respect the time hierarchy of the unit operations in a forward on. Generally, to ensure causality, a reactor or holding tank that dynamic programming-like manner.6 The algorithm is an feeds at ti can be fed only by the linear combination (mixing) instance of a breadth-first search (see the work of Cormen et 78 of effluents of reactors that outlet at ti. The fact that the lower al. ) with the network feed as the first point in concentration triangle (excluding the first column) of matrix A (Figure 3) is space in the candidate BAR. At each stage of generation, the zero is a manifestation of the network’s causality, namely that points or nodes form a directed acyclic graph (DAG). Because unit feeds at time ti can only be comprised of unit effluents at of this, it is easy to construct an actual network of operations time ti. from the nodes of the graph. Each node of the DAG, other than the first, can either have a single parent or multiple parents, Shrink-Wrap BAR Construction Algorithm because of mixing. If the node is generated by a batch reactor simulation, it has only one parent and that batch reactor An algorithm that is computationally similar to the “Shrink- 1 operation will be added to the resulting network of reactors to Wrap” algorithm (outlined in Manousiouthakis et al. ) can be create that node. If the node is generated by a linear combina- applied to the construction of a superset of the true BAR. tion of other nodes (mixing), it will have two or more parents. Because of the causal nature of the batch RNS problem, this At least one of the parent nodes is a newly generated vertex method can only be guaranteed to find a superset to the true from the previous time step, because we have assumed that BAR. One of the steady-state RNS problem characteristics that mixing is instantaneous. An outline of the algorithm is given leads to the Shrink-Wrap steady-state AR construction method as follows: is that any unit’s feed can be constructed, through mixing, as a (1) Start from the feed pointsthis is the BAR at time t or linear combination of the feed and reactor unit exits that belong 0 BAR . to the candidate steady-state AR. This was due to the fact that 0 all points in the candidate AR could be constructed indepen- (2) Travel along the path of the batch reactor in concentration dently of each other (through mixing) without consideration of space from t0 to t1. This will be the exit concentration vector of “when” each point was generated. Because unit outlets in the the first batch reactor. batch reactor network case cannot be used arbitrarily to (3) Find the convex hull of all the points from the previous reconstruct other unit inlets (because of causality), the avail- time step and the points from the new time step to find the ability of any arbitrary unit exit in the candidate BAR to mix BAR at that particular time. For the first time step, this will be and convexify the candidate BAR cannot be guaranteed unless a line. This is BAR1. one knows when that unit operates. However, a Shrink-Wrap- (4) Discretize this line and travel from time t1 to time t2 along like algorithm can be used to identify increasingly accurate the reactor trajectories starting at all discretized points on the approximations of a convex set that is guaranteed to contain line BAR1 that are not part of BAR0. the BAR for all network operating times n, although it is not (5) Form the convex hull of this new set and BAR1; this is guaranteed to necessarily converge to the true BAR. This an approximation of BAR2. algorithm removes extreme points (vertices) from an initial (6) For every BARi, (a) generate the reactor trajectories from superset by following the batch reactor trajectory backward (for time ti to time ti+1 for all points of BARi on an appropriately a given time, t1 - t0) and evaluating from where it started. If defined grid that are not contained in BARi-1, and (b) form the the starting point of the vertex’s trajectory is not a point in the convex hull of this new set of points with the approximation of interior of the superset, then no batch reactor can exist that BARi (this is an approximation of BARi+1). creates that point and the vertex is removed. The algorithm then (7) Repeat step 6 until time tn; this final convex hull is an updates the set as points are removed. The method essentially approximation of BARn, the true BAR at tn. H

Table 2. List of Points Used in Forming BAR0, BAR1, BAR2, BAR3, and BAR point first generated in C(A) C(C) formed by

A BAR2 0.2486 0.3204 mixing B BAR2 0.1667 0.3493 map of point C C BAR1 0.375 0.2466 map of point F D BAR3 0.0625 0.3931 map of point B E BAR3 0.2158 0.3495 mixing F BAR0 1.0 0.0 network feed point G BAR3 0.1064 0.3882 map of point A H BAR 0.0 0.1851 see Appendix C W BAR 0.05752 0.4568 map of point Z Y BAR 0.0 0.4666 see Appendix D Z BAR 0.1579 0.4156 see Appendix D Case Study for Trambouze Kinetics Figure 4. Shrink-Wrap-like algorithm superset for tin - tout ) 2 min and We will apply the two aforementioned algorithms to the varying grid size. construction of the BAR for a case study that exhibits Tram- bouze79 reaction kinetics (example 4 from his 1959 work), which is a reaction scheme that is often studied in the literature:1,38,55

k 918 ) -1 -1 A B (zeroth order, k1 0.025 mol L min ) (26)

k 928 ) -1 A C (first order, k2 0.2 min ) (27)

k 938 ) -1 -1 A D (second order, k3 0.4 L mol min ) (28)

These are the same reaction kinetics and parameters as those given in the work of Manousiouthakis et al.1 The rates of generation for each species for this reaction pathway are Figure 5. Attainable region (AR) for steady-state networks. )- - - 2 RA k1 k2CA k3CA (29) described by Appendices C and D. A table of points that define R ) k (30) B 1 these regions is given as Table 2. Table 3 lists the points and ) curves that define BAR , BAR , BAR , BAR , and BAR, for RC k2CA (31) 0 1 2 3 easy reference. ) 2 RD k3CA (32) Shrink-Wrap BAR Construction Algorithm Results for The batch reactor equations can be solved analytically, as the Trambouze Example described in Appendix A. This makes this example ideal for testing our candidate BARn identification algorithms, because Figure 4 shows increasingly accurate approximations of a they can be compared analytically to the true BARn at each superset to the BAR for species A (x-axis) and species C (y- time. For the purpose of creating BARns for this example, we axis), ranging from a 100 × 100 grid size to a 10 000 × 10 000 consider that the batch reactor time is 2 min, i.e., that τ ) tn - grid size. The identified BAR superset is only slightly smaller tn-1 ) tn-1 - tn-2 ) ... ) t1 - t0 ) 2 min. than the steady-state AR found in the work of Manousiouthakis 1 For this batch reactor time, BAR0, BAR1, BAR2, and BAR3 et al. (shown in Figure 5) at the high end of the y-axis are created analytically for this example as described by intersection of the AR (0.4666 for BAR, versus 0.4705 for Appendix B and BAR (BARn as n f ∞) is identified as steady-state AR), but significantly smaller at the lower end of

Table 3. List of Points and Relevant Equations for Curves That Define BAR0, BAR1, BAR2, BAR3, and BAR region defined by relevant equations

BAR0 point F F ) (1.0,0.0) BAR1 points C to F (line f1) f1(C1(A)) ) C1(C) ) 0.8(0.2 - ln 2)(C1(A) - 1) - BAR 8C2(A) 3 5 2 points A to B (g1), B to C (line), g (C (A)) ) C (C) ) (0.2 - ln 2) + ln - 0.1 1 2 2 - (x - ) C to F (line), and F to A (line) 4 4C2(A) 4 4C2(A)

BAR3 points E to G (h1),GtoD(h2), 15 - 40C (A) ) ) 3 + 5 - D to C (line), C to F (line), h1(C3(A)) C3(C) 0.4264( ) ln( ) 0.1 16 - 16C (A) x4 - 4C (A) F to E (line) 3 3 - 12C3(A) 2 5 h (C (A)) ) C (C) ) (0.2 - ln 2) + ln - 0.2 2 3 3 - (x - ) 3 8C3(A) 3 8C3(A)

BAR points Z to W (j1),WtoY(j2), 40C(A) - 15 CZ(C) 5 j (C(A)) ) C(C) ) ( ) + 0.5 ln( ) - 0.1 Y to H (line), H to F (line), 1 16 - 16C(A) C (A) - 1 4 - 4C(A) F to Z (line) Z 30C(A) - 5 CZ(C) 5 j (C(A)) ) C(C) ) + 0.5 ln - 0.2 2 ( - ) - ( - ) 6 16C(A) CZ(A) 1 3 8C(A) I the y-axis intersection of the AR (0.1851 for BAR, versus 0.0 for steady-state AR). Fundamentally, the reason for this dis- crepancy is the causality restriction that has been imposed on the batch network; the point (0.0, 0.0) cannot be part of the BAR because no batch reactor trajectory with a feed in BAR can have this point as an exit. Similarly, the point (0.0, 0.1) cannot be part of BAR for the same reason. Repeated application of this argument within the aforementioned Shrink-Wrap-like algorithm leads to the lower boundary of the BAR superset being the line that goes through the network feed point (1.0, 0.0), a batch reactor exit that lies on the y-axis, and its corresponding feed. Solving the batch reactor equations in Appendix A for a reaction time of 2 min and an outlet concentration of A of 0.0 mol/L gives an inlet concentration of A of 0.0625 or 1/ mol/ 16 - ) L. Also using Appendix A, a batch reactor that feeds at (0.0625, Figure 6. Breadth-first algorithm results for BAR3 with tin tout 2 min and varying grid size. Ch(C)) will exit at (0, Ch(C) + 0.01157). The value of Ch(C) for which the network feed point, aforementioned reactor inlet, and reactor outlet points are all on the same line is: h h + C(C) ) C(C) 0.01157 w 0.0625 - 1 -1 Ch(C) ) 15(0.01157) ) 0.1736

Thus, the batch reactor exit is:

C(C) ) 0.01157 ) 16(0.01157) ) 0.1851

This is consistent with the Shrink-Wrap-like BAR code, which identifies the lowest concentration attainable on the C(C) axis as point H (0.0, 0.1851). The BAR superset identified in Figure 4 is, in fact, the BAR itself. As discussed in detail in Appendix C, if the point of BAR3 with the lowest value of C(A) is considered to be a batch reactor feed point, then this batch reactor’s exit is on the y-axis. Mixing this exit with the network feed results in a point with C(A) ) Figure 7. Enlargement of Figure 6. 0.0625 that is below the point in BAR3 with the same value of C(A). Repeated application of this batch reaction/mixing process isothermal kinetic model. We have shown that a batch reactor is shown in Appendix C to reach the lower boundary of the model can be incorporated into the IDEAS framework, which previously identified BAR superset. A similar procedure is is proven to identify the globally optimal network of batch outlined in Appendix D for the upper boundary of the superset, reactors. thus establishing that the superset given in Figure 4 is the actual We have demonstrated that a Shrink-Wrap-like algorithm can BAR. be used to identify increasingly accurate approximations of a These results suggest that if species C were an unwanted superset of the batch attainable region (BAR). We have also byproduct of this reaction set, choosing any network of batch outlined a novel breadth-first algorithm that can be used to reactors with a reaction/holding time of 2.0 min would give identify the batch reactor network BARn. We have applied both unacceptable results; a network of steady-state reactors (e.g., a algorithms to a case study with Trambouze kinetics, and we CSTR) could yield practically no generation of C and, thus, have demonstrated that the breadth-first algorithm accurately would be superior to a network of batch reactors (except when identifies BARn (for n ) 3), whereas the Shrink-Wrap-like the reaction/holding time becomes very, very small.) algorithm accurately identifies not only a superset of BAR but the actual BAR itself. Iterative procedures that demonstrate how Breadth-First BAR Construction Algorithm Results for the BAR boundary, as identified by the Shrink-Wrap-like the Trambouze Example algorithm, can be reached with causal (forward moving in time) batch reaction and mixing operations are discussed in detail. A “breadth-first” algorithm can be used to identify the BARn for any given n; we show the case for n ) 3 as depicted in Appendix A: Solution of Batch Reactor Model for Figure 6 and an enlargement of the top of the same region as Trambouze Kinetics Figure 7 to illustrate the convergence of the algorithm with an increasing number of grid points. The considered Trambouze kinetics (k1 ) 0.025, k2 ) 0.2, 2 1/2 This result matches very well with the analytical calculation k3 ) 0.4) satisfy the relation (k2 - 4k1k3) ) 0. Therefore, of BAR3 that is given in Appendix B. any batch reactor trajectory satisfies the following relations:

Conclusions - h -R (2 k2τ)C(A) k2τ C(A) ) We have outlined a method for automatically identifying the 2k τCh(A) + 2 + k τ attainable region for batch reactor networks for a given overall 3 2 ) h +R +R + h - network time, for any number of components, and for any C(C) C(C) [2 ln(1 k3τ k3τC(A)) k2τ] J where τ is the reaction time of unit, R is defined as:

k R ≡ 2 ) 1 2k3 4

Ch(A) and C(A) respectively represent the inlet and outlet concentrations of species A, and Ch(C) and C(C) respectively represent the inlet and outlet concentrations of species C. For τ ) 2.0:

Ch(A) - 0.0625 1 + 24C(A) C(A) ) w Ch(A) ) Ch(A) + 1.5 16 - 16C(A) and: Figure B1. BAR2 generated analytically.

C(C) ) 0.5 ln(0.8Ch(A) + 1.2) - 0.1 where: 3 Appendix B: Analytical Generation of BAR0, BAR1, g ) ln(x2) - 0.1 ) 0.2466 1( ) BAR2, and BAR3 8 ) { } BAR0 consists solely of the feed point: BAR0 (1.0,0) 1 ) x - ) g1 ln( 3) 0.2 0.3493 ) {(Ch 0(A),Ch 0(C))}. (6) BAR1 is the convex hull (which, in this case, is a line) between two points: the feed and the exit of a batch reactor Thus: with τ ) 2.0 and with the feed point as its inlet. Based on the ∈ formulas given previously, the latter point is (C0(A), C0(C)) ) C2(C) [0.2466, 0.3493] (0.375, ln(x2) - 0.1). We will call this point C. BAR1 can be defined as We must determine if there is a point in the domain of g1 whose tangent intersects the feed point: BAR ) {(C (A), C (C)) 1 1 1 g (C (A)) - 0 dg (C (A)) 1 2 ) 1 2 w 5 ) - ln( - ) where: C2(A) 1 dC2(A) 4 4C2(A) 1 - (ln 2) - 4(ln 2)C (A) C (C) ) f (C (A)) 2 w C (A) ) 0.2486 1 1 1 - 2 C2(A) 1 C (A) ∈ [0.375, 1.0]} 1 1.6(0.2486) + 5(ln 2) - 2.6 g (0.2486) ) ( ) - 1 and: 4 - 4(0.2486) 0.1 ) 0.3204 ≡ - - f1(C1(A)) 0.8(0.2 ln 2)(C1(A) 1) This result shows that such a point exists; we will call this point (0.2486, 0.3204) point A. The function g is concave on the BAR2 is the convex hull of all batch reactor trajectories 1 domain; therefore, BAR is the area formed by the line from starting from points in BAR1 and BAR1. It can be defined as 2 follows: the feed point to point A, the part of the batch reactor trajectory 1 from point A to point B ( /6, 0.5(ln 3) - 0.2), the line from 3 - C (A) - 0.0625 point B to point C ( /8, 0.5(ln 2) 0.1), and the line from point C (A) ) 1 ) f (C (A)) w C to the feed point (see Figure B1). 2 C (A) + 1.5 2 1 1 BAR3 for this problem is comprised of all points attainable ∈ C2(A) [0.1667, 0.375] from batch reactor trajectories that start on the boundary of BAR2. We only need to consider extreme points, because the ) + + - ) C2(C) C1(C) 0.5 ln(0.8C1(A) 1.2) 0.1 Trambouze reaction vector is dependent only on the concentra- - - + + - tion of species A; any reactor inlet point on a given vertical 0.8(0.2 ln 2)(C1(A) 1) 0.5 ln(0.8C1(A) 1.2) ) line that satisfies the relation: 0.1 f3(C1(A)) Ch l(C) e Ch(C) e Ch u(C) The convexification of this region will be a line from one endpoint of the BAR1 line to either another endpoint of the new will be mapped to a point (C(A), C(C)) that lies between the set, or a point intermediate on that new set, depending on points (C(A), Cl(C)) and (C(A), Cu(C)). Therefore one must whether the new set has an inflection point in the domain: only consider the reactor trajectory that extends the region the furthest for each point on the C(A) axis. Three separate segments 8C (A) - 3 C (C) ) (0.2 - ln 2) 2 + need to be mapped (i.e., travel along the batch reactor trajectories 2 ( - ) 4 4C2(A) for 2 min) to create BAR3: line AF, curve AB, and line BC. Line CF does not need to be mapped again, because its map 5 0.5 ln - 0.1 ≡ g (C (A)) was calculated previously and has already been considered in ( - ) 1 2 4 4C2(A) BAR2. The convex hull of the region that is defined by these K G to point D, the line from point D to point C, and line CF (previously defined). The final graph of BAR3 appears almost identical to the result from the breadth-first algorithm (Figure 6) and will not be reproduced.

Appendix C: Analytical Generation of the Bottom of BAR

The point in BAR3 with the lowest value of C(A) is (0.0625, 0.3931). This point maps to the point (0, 0.4047). The equation for the line from this point to the feed point is C(C) ) 0.4047- (1 - C(A)). The point on this new line at C(A) ) 0.0625 is C(C) ) 0.3794. Mapping this new point back to the axis gives the point (0, 0.3910). This procedure can be repeated iteratively

Figure B2. New mappings needed to make BAR3. as follows; define Ci(C) as the ith C(C) value in the sequence and ∆C(C) as the increase gained by mapping, where C0(C) ) three mappings and BAR2 will give BAR3. Mapping line AF 0.3931 and ∆C(C) ) 0.01157. The first three points in the gives: sequence are then: 15 - 40C (A) ) 3 + 5 - ) 15 + C3(C) 0.4264( ) 0.5 ln( ) C (C) [C (C) ∆C(C)] - - 1 0 16 16CC(A) 4 4C3(A) (16) ≡ ∈ 0.1 h1(C3(A)), C3(A) [0.1064, 0.375] ) 15 2 + 15 + 15 C2(C) C0(C) 1 ∆C(C) Mapping curve AB gives (16) [(16) ](16) - - 15 3 15 2 15 15 (12C3(A) 2)(0.2 ln 2) 5 ) + + + C (C) ) + ln - C3(C) C0(C) 1 ∆C(C) 3 - x - (16) [(16) (16) ](16) 3 8C3(A) 3 8C3(A) ≡ ∈ 0.2 h2(C3(A)), C3(A) [0.0625, 0.1064] and Cn(C), in the limit as n f ∞, can be expressed as:

Mapping line BC gives 15 n 15 n 15 i lim C (C) ) C (C) lim ( ) + ( )∆C(C) lim ∑ ( ) ) n 0 + nf∞ nf∞ nf∞ 2 1 24C3(A) 16 16 i)0 16 C (C) ) 2.4 ln + 0.2 + 3 ( ( ) )( - ) 15∆C(C) ) 0.1736 3 16 16C3(A) 5 This point maps forward along a reactor trajectory for 2.0 min 0.5 ln + 0.9(ln 3) - 0.4(ln 2) - ( - ) to point H (0, 0.1851), which is also the lowest point on the 4 4C3(A) ≡ ∈ C(C) axis obtained through the Shrink-Wrap-like algorithm. 0.38 h3(C3(A)), C3(A) [0.0625, 0.1667]

A plot of these three mappings is given as Figure B2. Appendix D: Analytical Generation of the Top of BAR To form BAR3, we must make the convex hull of these new As shown in the construction of BAR2 and BAR3, at high points with the points of BAR2. We first determine if h1 has a values of C(A) (C(A) > 0.375), the upper BARn boundary is a tangent that intersects the feed point: straight line that is tangent to the curve generated when each point of the upper BAR - boundary (which, again, is a straight h (C (A)) - 0 dh (C (A)) n 1 1 3 ) 1 3 line for large values of C(A)) is considered a feed to a batch - , C3(A) 1 dC3(A) reactor with a reaction time of τ ) 2.0 min. BAR2 has, as its ) ∈ w upper boundary, the line AF, which is tangent to the curve CAB C3(A) 0.2158 [0.1064, 0.375] (at point A), which is generated when the points of the upper - 15 - + + ) boundary of BAR1 (line CF) are considered as feeds to batch 0.4264( 5C3(A)) 0.8C3(A) 0.5325 8 reactors with τ ) 2.0. Similarly, BAR3 has, as its upper 5 boundary, the line EF, which is tangent (at point E) to the curve [1 - C (A)] ln 3 (4 - 4C (A)) h1(C3(A)). This curve consists of points that are outlets of batch 3 reactors that have, as feeds, the points on the upper boundary We see that h does, in fact, intersect the feed point and is of BAR2 (line AF). This process will stop expanding BARi 1 f ∞ concave, so there will be a convexification line from the feed upward (as i ) only when the line ZF (that is, the upper boundary of BARi) is tangent at point Z to the curve that results point to point E (0.2158, 0.3495). Because h2 is also concave and its derivative is equal to the derivative of h at their inter- from mapping the line ZF through the batch reactor map with 1 ) ) section, there will be no further convexifications at the top of τ 2.0. To identify this point Z (Z (CZ(A), CZ(C)), the following conditions must hold: the region. The function h3 is concave and the value of C3(C) (1) All points on the line ZF satisfy: for h3 at point B is greater than that of line CD at the C3(A) of point B; therefore, there is a convexification line from point C Ch(C) C (C) to point D (0.0625, 0.393147) to finish BAR3. This completes ) Z - the analytical generation of BAR3; it is the region formed by Ch(A) - 1 CZ(A) 1 the line from the feed to point E on h1, the function h1 from point E to point G (0.1064, 0.3882), the function h2 from point (2) The line ZF maps to the following curve: L

- C (C) ) 40C(A) 15 Z + 5 - C(C) ( - ) - 0.5 ln( - ) 16 16C(A) CZ(A) 1 4 4C(A) ≡ 0.1 j1(C(A))

(3) At C(A) ) CZ(A), C(C) ) CZ(C) and dC(C)/dC(A) is given as:

C (C) dC(C) ) Z - dC(A) CZ(A) 1 where:

2 8(1 - C (A)) Figure D1. Point Z, line ZF, curve ZW, point W, curve WY, and point Y C (C) ) Z Z - - 2 (complete top of BAR). 25 16(1 CZ(A)) from point W to point Y. This turns out to be approximated Taking the derivative of the function j1 from condition 2 above extremely well by the Shrink-Wrap-like algorithm, which, in and evaluating the derivative at CZ(A) yields: this case, has successfully identified the BAR itself. dj (C(A)) 1 | ) Literature Cited dC(A) ) C(A) CZ(A) (1) Manousiouthakis, V.; Justanieah, A.; Taylor, L. The Shrink-Wrap 25 Algorithm for the Construction of the Attainable Region: Application of 0.5(C (A))2 - C (A) - C (C) + 0.5 Z Z (16) Z the IDEAS Framework. Comput. Chem. Eng. 2004, 28, 1563. (2) Chitra, S.; Govind, R. Yield Optimization for Complex Reaction - 3 (1 CZ(A)) Systems. Chem. Eng. Sci. 1981, 36, 1219. (3) Chitra, S.; Govind, R. Synthesis of Optimal Serial Reactor Structures for Homogeneous Reactions. Part I: Isothermal Reactors. AIChE J. 1985, This point must be on the map of line ZF (condition 2), giving 31, 177. the coordinates of point Z, (CZ(A),CZ(C)) ) (0.1579, 0.4156). (4) Chitra, S.; Govind, R. Synthesis of Optimal Serial Reactor Structures Because, by definition, line ZF and its map have the same slope for Homogeneous Reactions. Part II: Non-Isothermal Reactors. AIChE J. at point Z, there is no need to convexify the top of the BAR 1985, 31, 185. (5) Ong, S. Optimization of CSTRs in Series by Dynamic Programming. near point Z. Next, we define point W (0.05752, 0.4568) as the Biotechnol. Bioeng. 1986, 28, 818. point to which point Z maps after a batch reaction time of 2.0 (6) Bellman, R. Dynamic Programming; Princeton University Press: min; the portion of the mapping of line ZF between point Z Princeton, NJ, 1957. and point W will now be called curve ZW, as defined by the (7) Pibouleau, L.; Floquet, P.; Domenech, S. Optimal Synthesis of - function j . The second derivative of j , with respect to C(A), Reactor Separator Systems by Non-Linear Programming Method. AIChE 1 1 J. 1988, 34, 163. is negative between point Z and point W, so the curve ZW is (8) Omtveit, T.; Wah, P.; Lien, K. Decomposed Algorithmic Synthesis concave between point Z and point W. We can now map curve of Reactor-Separation-Recycle Systems. Comput. Chem. Eng. 1994, 18, ZW, through the batch reactor map with τ ) 2.0, until it reaches 1115. the C(A) axis to complete the top of BAR: (9) Smith, E.; Pantelides, C. Design of Reaction/Separation Networks Using Detailed Models. Comput. Chem. Eng. 1995, 19, S83. (10) Bikic, D.; Glavic, P. Synthesis of Reactor Networks with Multiple - C (C) ) 30C(A) 5 Z + Multi-Component Feeds. Comput. Chem. Eng. 1995, 19, S161. C(C) ( - ) - (11) Bikic, D.; Glavic, P. Innovative Designs of Reactor Networks from 6 16C(A) CZ(A) 1 Reaction and Mixing Principles. Comput. Chem. Eng. 1996, 20, S455. 5 (12) Bikic, D.; Glavic, P. Towards Automatic Generation of Novel 0.5 ln - 0.2 ≡ j (C(A)) - ( - ) 2 Reactor Separator Networks with Multiple Multi-Component Feeds. Com- 3 8C(A) put. Chem. Eng. 1997, 21, S41. (13) Esparta, A.; Obertopp, T.; Gilles, E. Synthesis of Isothermal Fluid- The map hits the C(A) axis at C(C) ) 0.4666. We will call Fluid Two-Phase CSTR Networks. Comput. Chem. Eng. 1998, 22, S671. this point Y (Y ) (0.0, 0.4666)) and refer to the portion of the (14) Hua, K.; Li, Y.; Hu, S.; Shen, J. Three-Distribution-Parameter map of ZW for τ ) 2.0 min that remains in the physical region General Model for Reactor Network Synthesis. Comput. Chem. Eng. 2000, g 24, 217. (i.e., C(A) 0) as curve WY. A plot of line ZF, curve ZW, (15) Mehta, V.; Kokossis, A. Non-Isothermal Synthesis of Homogeneous and curve WY is given as Figure D1. The derivatives of curve and Multiphase Reactor Networks. AIChE J. 2000, 46, 2256. ZW at point W and curve WY at point W are equal: (16) Pahor, B.; Irsic, N.; Kravanja, Z. MINLP Synthesis and Modified Attainable Region Analysis of Reactor Networks in Overall Process Schemes dj (C(A)) 4(6C (A) - 1) Using More Compact Reactor Superstructure. Comput. Chem. Eng. 2000, 1 | ) W ) 24, 1403. (17) Pahor, B.; Kravanja, Z.; Bedenik, N. Synthesis of Reactor Networks dC(A) C(A))C (A) (1 + 4C (A))(7 - 12C (A)) W W W in Overall Process Flowsheets Within the Multilevel MINLP Approach. dj2(C(A)) Comput. Chem. Eng. 2001, 25, 765. | (18) Grossmann, I. E.; Caballero, J. A.; Yeomans, H. Advances in dC(A) ) C(A) CW(A) Mathematical Programming for Automated Design, Integration and Opera- tion of Chemical Processes. In Proceedings of the International Conference In addition, curve WY is concave between point Y and point on Process Integration (PI ‘99), Copenhagen, Denmark, 1999 (19) Aris, R. The Optimal Design of Chemical Reactors; Academic W, so no convexification line is necessary to fill in the region; Press: New York, 1961. the top of the BAR is the line from the feed point F to point Z, (20) Paynter, J.; Haskins, D. Determination of Optimal Reactor Type. the function j1 from point Z to point W, and the function j2 Chem. Eng. Sci. 1970, 25, 1415. PAGE EST: 13 M

(21) Waghmare, R. S.; Lim, H. C. Optimal Operation of Isothermal (51) Gadewar, S.; Malone, M.; Doherty, M. Feasible Region for a Reactors. Ind. Eng. Chem. Fundam. 1981, 20, 361. Counter-Current Cascade of Vapor-Liquid CSTRs. AIChE J. 2002, 48, (22) Parulekar, S. J.; Waghmare, R. S.; Lim, H. C. Yield Optimization 800. for Multiple Reactions. Chem. Eng. Sci. 1988, 43, 3077. (52) Kauchali, S.; Hausberger, B.; Hildebrandt, D.; Glasser, D.; Biegler, (23) Achenie, L.; Biegler, L. Algorithmic Synthesis of Chemical Reactor L. Automating Reactor Network Synthesis: Finding a Candidate Attainable Networks Using Mathematical Programming. Ind. Eng. Chem. Res. 1986, Region for the Water-Gas Shift (WGS) Reaction. Comput. Chem. Eng. 2004, 25, 621. 28, 149. (24) Achenie, L.; Biegler, L. Developing Targets for the Performance (53) Omtveit, T.; Tanskanen, J.; Lien, K. Graphical Targeting Procedures Index of a Chemical Reactor Network: Isothermal Systems. Ind. Eng. Chem. for Reactor Systems. Comput. Chem. Eng. 1994, 18, S113. Res. 1988, 27, 1811. (54) Burri, J.; Wilson, S.; Manousiouthakis, V. Infinite Dimensional (25) Achenie, L.; Biegler, L. A Superstructure Based Approach to State-Space Approach to Reactor Network Synthesis: Application to Chemical Reactor Network Synthesis. Comput. Chem. Eng. 1990, 14, 23. Attainable Region Construction. Comput. Chem. Eng. 2002, 26, 849. (26) Godorr, S.; Hildebrandt, D.; Glasser, D.; McGregor, C. Choosing (55) Kauchali, S.; Rooney, W.; Biegler, L.; Glasser, D.; Hildebrandt, Optimal Control Policies Using the Attainable Region Approach. Ind. Eng. D. Linear Programming Formulations for Attainable Region Analysis. Chem. Chem. Res. 1999, 38, 639. Eng. Sci. 2002, 57, 2015. (27) Hillestad, M. A Systematic Generation of Reactor Designs I: (56) Abraham, T.; Feinberg, M. Kinetic Bounds on Attainability in the Isothermal Conditions. Comput. Chem. Eng. 2004, 28, 2717. Reactor Synthesis Problem. Ind. Eng. Chem. Res. 2004, 43, 449. (28) Hillestad, M. A Systematic Generation of Reactor Designs. II: Non- (57) Zhou, W.; Manousiouthakis, V. Non-Ideal Reactor Network Isothermal Conditions. Comput. Chem. Eng. 2005, 29, 1101. Synthesis Through IDEAS: Attainable Region Construction. Chem. Eng. (29) Wilson, S.; Manousiouthakis, V. IDEAS Approach to Process Sci. 2006, 61, 6936. Network Synthesis: Application to Multi-Component MEN. AIChE J. 2000, (58) Zhou, W.; Manousiouthakis, V. Variable Density Fluid Reactor 46, 2408. Network Synthesis: Construction of the Attainable Region Through the (30) Drake, J.; Manousiouthakis, V. IDEAS Approach to Process IDEAS Approach. Chem. Eng. J. 2007, 129, 91. Network Synthesis: Minimum Plate Area for Complex Distillation Networks (59) Rippin, D. Simulation of Single- and Multi-Product Batch Chemical with Fixed Utility Cost. Ind. Eng. Chem. Res. 2002, 41, 4984. Plants for Optimal Design and Operation. Comput. Chem. Eng. 1983, 7, (31) Drake, J.; Manousiouthakis, V. IDEAS Approach to Process 137. Network Synthesis: Minimum Utility Cost for Complex Distillation (60) Rippin, D. W. T. Design and Operation of Multi-product and Multi- Networks. Chem. Eng. Sci. 2002, 57, 3095. purpose Batch Chemical Plants. Comput. Chem. Eng. 1983, 7, 463. (32) Justanieah, A.; Manousiouthakis, V. IDEAS Approach to Separation (61) Rippin, D. Batch Process Systems Engineering: A Retrospective Network Synthesis: Application to Chromium Recovery Form Wastewater. and Prospective Review. Comput. Chem. Eng. 1993, 17, S1. V V Ad .En iron. Res. 2003, 7, 549. (62) Reklaitis, G. V. Progress and Issues in Computer-Aided Batch (33) Martin, L.; Manousiouthakis, V. Globally Optimal Power Cycle Process Design: from Foundations of Computer-Aided Process Design; Synthesis via the Infinite-Dimensional State-Space (IDEAS) Approach Elsevier: New York, 1990. Featuring Minimum Area with Fixed Utility. Chem. Eng. Sci. 2003, 58, (63) Levien, K. Maximizing the Product Distribution in Batch Reac- 4291. tors: Reactions in Parallel. Chem. Eng. Sci. 1992, 47, 1751. (34) Holiastos, K.; Manousiouthakis, V. Infinite-Dimensional State-Space (64) Terwiesch, P.; Agarwal, M.; Rippin, D. Batch Unit Optimization (IDEAS) Approach to Globally Optimal Design of Distillation Networks with Imperfect Modeling: A Survey. J. Process Control 1994, 4, 238. Featuring Heat and Power Integration. Ind. Eng. Chem. Res. 2004, 43, 7826. (65) Yi, G.; Reklaitis, G. Optimal Design of Batch-Storage Network (35) Burri, J.; Manousiouthakis, V. Global Optimization of Reactive with Recycle Streams. AIChE J. 2003, 49, 3084. Distillation Networks Using IDEAS. Comput. Chem. Eng. 2004, 28, 2509. (66) Maravelias, C. A Decomposition Framework for the Scheduling (36) Horn, F. Attainable and Non-Attainable Regions in Chemical of Single- and Multi-Stage Processes. Comput. Chem. Eng. 2006, 30, 407. Reaction Technique. In Third European Symposium on Chemical Reaction (67) Sung, C.; Maravelias, C. An Attainable Region Approach for Engineering; Pergamon Press: London, 1964. Production Planning of Multi-Product Processes. AIChE J. 2007, 53, 1298. (37) Gavalas, G. Nonlinear Differential Equations of Chemically React- (68) Levenspiel, O. Chemical Reaction Engineering; Wiley: New York, ing Systems: Springer Tracts in Natural Philosophy; Springer: New York, 1972. 1968. (38) Glasser, D.; Hildebrandt, D.; Crowe, C. A Geometric Approach to (69) Froment, G.; Bischoff, K. Chemical Reactor Analysis and Design; Steady Flow Reactors: The Attainable Region and Optimization in Wiley: New York, 1979. Concentration Space. Ind. Eng. Chem. Res. 1987, 26, 1803. (70) Schmidt, L. Engineering of Chemical Reactions; Oxford University (39) Hildebrandt, D.; Glasser, D.; Crowe, C. Geometry of the Attainable Press: New York, 1998. Region Generated by Reaction and Mixing: With and Without Constraints. (71) Fogler, H. S. Elements of Chemical Reaction Engineering; Prentice Ind. Eng. Chem. Res. 1990, 29, 49. Hall: Upper Saddle River, NJ, 1999. (40) Glasser, D.; Hildebrandt, D.; Godorr, S. The Attainable Region (72) Nauman, E. B. Chemical Reactor Design, Optimization, and for Segregated Maximum Mixed and Other Reactor Models. Ind. Eng. Scaleup; McGraw-Hill: New York, 2002. Chem. Res. 1994, 33, 1136. (73) Rawlings, J.; Ekerdt, J. Chemical Reactor Analysis and Design (41) Hopley, F.; Glasser, D.; Hildebrandt, D. Optimal Reactor Structures Fundamentals; Nob Hill Publishing: Madison, WI, 2002. for Exothermic Reversible Reactions with Complex Kinetics. Chem. Eng. (74) Zhang, J.; Smith, R. Design and Optimization of Batch and Semi- Sci. 1996, 51, 2399. Batch Reactors. Chem. Eng. Sci. 2004, 59, 459. (42) Nicol, W.; Hernier, M.; Hildebrandt, D.; Glasser, D. The Attainable (75) Khalil, H. K. Nonlinear Systems; Prentice Hall: Upper Saddle River, Region and Process Synthesis: Reaction Systems with External Cooling NJ, 1996. and Heating. Chem. Eng. Sci. 2001, 56, 173. (76) Justanieah, A. On the Convergence of a Finite-Dimensional (43) Feinberg, M.; Hildebrandt, D. Optimal Reactor Design from a Approximation Algorithm for a Class of Infinite-Dimensional Linear Geometric Viewpoint Part I: Universal Properties of the Attainable Region. Programs: An Application to the IDEAS Framework, Ph.D. Thesis, Chem. Eng. Sci. 1997, 52, 1637. University of California at Los Angeles (UCLA), Los Angeles, CA, 2004. (44) Feinberg, M. Recent Results in Optimal Reactor Synthesis via (77) Bapat, R. B.; Raghavan T. E. S. Non-NegatiVe Matrices and Attainable Region Theory. Chem. Eng. Sci. 1999, 54, 2535. Applications: Encyclopedia of Mathematics and its Applications; Cambridge (45) Feinberg, M. Optimal Reactor Design from a Geometric Viewpoint. University Press: New York, 1997. Part II: Critical Sidestream Reactors. Chem. Eng. Sci. 2000, 55, 2455. (78) Cormen, T. H.; Leiserson, C. E.; Rivest, R. L. Introduction to (46) Feinberg, M. Optimal Reactor Design from a Geometric Viewpoint. Algorithms; McGraw-Hill: New York, 1990. Part III: Critical CFSTRs. Chem. Eng. Sci. 2000, 55, 3553. (79) Trambouze, P.; Piret, E. Continuous Stirred Tank Reactors: Designs (47) McGregor, C.; Glasser, D.; Hildebrandt, D. The Attainable Region for Maximum Conversions of Raw Material to Desired Product: Homo- and Pontryagin’s Maximum Principle. Ind. Eng. Chem. Res. 1999, 38, 652. geneous Reactions. AIChE J. 1959, 5, 384. (48) Rooney, W.; Hausberger, B.; Biegler, L.; Glasser, D. Convex Attainable Region Projections for Reactor Network Synthesis. Comput. ReceiVed for reView December 5, 2007 Chem. Eng. 2000, 24, 225. ReVised manuscript receiVed March 10, 2008 (49) Nisoli, A.; Malone, M.; Doherty, M. Attainable Regions for Accepted March 11, 2008 Reaction with Separation. AIChE J. 1997, 43, 374. (50) Smith, R.; Malone, M. Attainable Regions for Polymerization Reaction Systems. Ind. Eng. Chem. Res. 1997, 36, 1076. IE071664L