Computers chem. Engng, Vol. 21, Suppl., pp. $41-$46, 1997

P e r g a m o n ¢ 1997 Elsevier Science Ltd All rights reserved

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Towards Automatic Generation of Novel Reactor-Separator Networks with Multiple

Multicomponent Feeds

D. Biki6 and P. Glavk3 Faculty of Chemistry and Chemical Engineering, PO, Box 224, Smetanova 17, SI-2000 Maribor, Slovenia

E-mail: DUSKO.BIKIC@UNI-MB.Sl, Fax: ++386-62-227-774

A b s t r a c t - In the present paper we address the problem of synthesis and integration of chemical reactor networks into the overall process when separation and heat effects take place simultaneously with a complex set of reactions in a reactor system with multiple multicomponent feeds. To solve this problem an algorithmic procedure consisting of two stages is proposed. In the first stage a reactor network presentation which involves series-parallel combinations of differential sidestream reactors with intermediate separations has been used to generate a comprehensive set of process alternatives. Optimal process configuration and parameters have been obtained in the second stage with the use of conventional optimisation techniques. The present methodology has been derived from basic principles of re- action, mixing and separation. The use of the proposed methodology has been illustrated in detail with a simple ex- ample problem.

INTRODUCTION

It is well known that operation of reactor system domi- nates economy of a plant. Even slight losses of valuable reactant and reversible side products can claim devastat- ing impact on the economy of a plant because competi- tion forces management of chemical and petrochemical companies to work at relatively low profit margins. Re- actor system is usually operated at intermediate degree of conversion to reduce losses. Complex separation systems are then employed to recover and reuse valuable components from the reactor exit. Recycling is the most popular and powerful way to improve the economy of a plant. However, the existence of new streams that con- nect reactors with the separation sub-system and addi- tional process units increases complexity of a flowsheet. The complexity of a plant further increases when reac- tion and separation can take place simultaneously in the reactor network. A task of process engineer is to under- stand the behaviour of the reactor-separator system and its interactions with the down-stream separation sub- system in order to extract the maximum performance from chemical reactions to be utilised in a commercial scale production.

This is a non-trivial and non-routine task because com- plex relationships among different phenomena taking place simultaneously are not yet clearly understood while there is practically no reliable qualitative expert knowledge to support the design task. It is surprising that an adequate solution to this problem has not yet been published. This problem is of a paramount interest of process synthesis because: (i) this is the problem we usually meet in the industrial practice while (ii) proper operation of reactor system often provides means of at-

taining massive savings of valuable reactants. In this pa- per we present a methodology which can be effectively used to provide solution to this class of design problems.

P R E V I O U S W O R K

Several generalised approaches for the synthesis of reac- tor networks have been suggested over the last decade. Notable among those are: targeting strategies (Balakri- shna and Biegler, 1991; Lakshmanan and Biegler, 1996), superstructured approach supported by the MINLP formulation (Kokossis and Floudas, 1990) and geometric approach, well known as the Attainable Re- gion (Glasser et al. 1987). Omtveit et al. (1994) have addressed possible use of attainable region to solve de- sign problems when complex reactions take place in a process with a recycle stream originating from the downstream separation subsystem. Kravanja and Grossmann (1994) have presented new developments in the fields of automated topology and parameter process synthesis. All these earlier studies are extremely impor- tant for development of synthesis and process integra- tion of chemical reactor networks.

In our previous work (Biki6 and Glavi~, 1995) we have presented a design procedure for integrating reactor networks when several multicomponent feeds exist. The use of the design procedure has been later extended to non-isothermal systems (Biki6 and Glavi~, 1996). In the present paper we address a more general problem of placing separators into the reactor network with view to selecting and optimising reactor networks with interme- diate separations in the context of the overall process. By using the methodology presented in this paper it is

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possible to approach the design problem without the need for decomposing a process into reactor and sepa- ration subsystems.

PROBLEM STATEMENT

The problem to be addressed in this paper can be stated as follows:

Given:

• rate and heat balance equations of complex set of chemical reactions,

• feed flow rates, • compositions of feed streams and • physical properties of components

find:

• separation strategy, • the number and types of reactors, • connections among reactors in the network, • locations of feeds and • feeding scenarios

that provide means of attaining the most comprehensive product distribution!

Our aim is not to find optimal solution to some particu- lar problem in the first place. Instead, we investigate performance of chemical reactions in order to select a model that comprises the richest presentation of phe- nomena taking place in the process. In this way we are able to generate a set of design variables and process presentation which is sufficient to provide the most comprehensive picture of the process. This model is in turn optimised to extract the optimal process.

It is also possible to approach this problem by treating a system where separation takes place truly simultane-

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ously with chemical reactions. Such situation could be realised, for example, with reactive distillation or with a batch mode of operation. However, such operation is often difficult or even impossible to realise in a continu- ous operation. We do not address this class of problems in the present paper. Instead, we investigate more realis- tic problem of placing separators into the reactor net- work to enhance its performance.

BASIC REACTOR MODEL WITH INTERMEDIATE SEPARATION

To solve the above problem we use a reactor network presentation which involves series-parallel combinations of the basic reactor models with the intermediate sepa- ration step (Fig. la). The basic reactor model consists of differential reactor compartments with sidestreams and intermediate separator. To make our ideas clear, we have presented the basic reactor model behaviour in the concentration space (Fig. lb). Consider two arbitrary but distinct feed streams A and B originating from the concentration space. Fluids from those feed streams could be mixed to attain point C which is located at the line connecting feed streams, A and B. Fluid with com- position C can be processed in a reactor to attain the point in the concentration space denoted with D. Feed to the separation unit (D) is achieved by reaction and mix- ing processes along the path CD which has been drawn with a dotted curve. Operation of the reaction system can be manipulated in various ways: e.g. with different feeding, with heat addition or removal, with simultane- ous reaction and separation, utilising different type and/or amount of catalyst etc. Separation unit is located at the reactor exit to split stream D, which also belongs to the attainable region, into streams ~. and G which may be shifted out of the attainable region. In this way we provide means of extending solution space into regions

Recycle

CB

°. E

r 1'~ % %

"'.% " :ib~

H'q..... ""'"'.. ",,,

CA A

Figure la. The basic reactor-separator model consists of reactor compartments and an intermediate separation.

Feed stream is attained by mixing streams in the attain- able region.

Figure lb. Graphical presentation of the basic reactor model behaviour in the concentration space. Note that mixing and separation processes can be presented with

straight lines.

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which could not have been attained by using exclusively mixing and reaction. Separation can be drawn in the concentration space as a straight line connecting feed point and composition of exit streams E and G which may be further processed in separate reactors to attain points F and I-I. Each of these new streams form their own attainable regions which could be in turn connected by a mixing process to form the attainable region of the entire process. Hence, we deal with reactor network and intermediate separations in contrast to the traditional approach where separation subsystem is treated as a set of down-stream process units which serve just to recover unreacted valuable reactants and reversible side prod- ucts. Instead, we investigate benefits of utilising separa- tors to shift concentration profiles inside the reactor system in order to improve the performance of the overall process. In this way we provide means of dis- continuous changes of concentration profiles in the reac- tor network which could have not been attained by using exclusively reaction and mixing processes. It is conceiv- able that such approach results in more flexible and novel design ideas.

Benefit of such problem presentation is not only that separation could improve performance of the reactor system. Essential advantage of putting separators into the reactor system is that we can investigate and identify possible directions of extending the solution space and its bounds. Once we find configurations and a set of de- sign variables that bind the solution space, a process could be optimised on the basis of the most comprehen- sive problem formulation. In this way we investigate also process alternatives which could have not been in- vented by using only sound human reasoning.

To provide means of choice between different modes of mixing and feeding strategies we use a differential side- stream reactor model. Such presentation of the reactor enables choice between different modes of macromixing and various feeding strategies. Previous studies (Hilde- brandt and Biegler, 1995) have indicated that such model is rich enough to present all possible concentra- tions which could be attained with using only reaction and mixing.

Recycle stream originating from the bottom of the sepa- rator is drawn with a dashed line to indicate that other possible connections exist.

METHOD OUTLINE

The problem of reactor-separator network synthesis has been solved in two main steps. In the first step we gen- erate a set of physically feasible design variables and process structures that bind the solution space. The de- rived set of process alternatives consists of a subset that binds (i) concentration space and a subset that binds (ii) an overall yield space. It is crucial to find those two sets because plants are usually operated between the maxi- mum selectivity and the maximum overall yield to trade- off investment vs. operating costs. In the second stage we construct a superstructure of feasible process devices and extract the optimal solution by using conventional optimisation techniques. By using this approach we are

able to compare process alternatives on the basis of the most comprehensive problem presentation. By using simple words, we are investigating the performance of the basic driving forces to extend the solution space to its limits and then we contract it with constraints of ac- tual devices to find a feasible set of process units that could be used to find the optimal performance of a plant.

Initially, the methodology presented in this paper has been developed from geometric interpretations of opti- mality conditions of the optimal reactor-separator sys- tems. Despite of obvious strengths of geometric presen- tations, restriction to only 2 and 3-dimensional problems is too stringent for a vast majority of real industrial problems. Therefore, we have instead developed an al- gorithmic methodology capable handling problems with more than 3-dimensional kinetics models. Even though algorithmic methods may converge to undesirable local extremes, we have adopted a simple methodology which starts from a single basic model and then sequentially adds new refinements into the process structure until the increase of the objective function ceases. In each step the refined structure has been optimised to guarantee that currently investigated structure has been driven to its limits. In this way the solution space can be extended to its limits in a finite number of steps.

Limitations of the proposed methodology are: (i) only reaction, mixing, separation and heat effects can take place simultaneously in the reactor system, (ii) possible separation methods are known in advance, (iii) structure of the downstream separation system must be known in advance and (iv) rate equations must be twice continu- ously differentiable.

Our approach is related to the lstPRINCE (Aelion et al. 1991) methodology by the way it uses formal optimisa- tion theory while it is in complement with the attainable region technique by extending its use to simultaneous reaction-separation systems. The use and the purpose of the presented methodology is to generate the most com- prehensive set of feasible process alternatives from which we can in turn derive the optimal solution by us- ing conventional optimisation techniques.

The derived methodology eliminates the requirement for anticipated superstructure and the use of qualitative knowledge for finding the optimal solution.

Methodologies for treating problem of synthesis and in- tegration of non-isothermal reactor networks with mul- tiple feeds have been presented in our previous studies (Biki6 and Glavi~; 1995, 1996) and will be omitted here. In this paper we focus our attention only to the problem of placing separators into the reactor system.

ALGORITHM

Expansions of the solution space have been performed by sequential enrichments of the process structure. The currently derived structure has been enriched in each step by adding new basic reactor-separator units in- series and in-parallel. The number of optimisation vari- ables increases in consecutive steps. Values of the adopted set of variables have been preserved to ensure

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better starting point and faster convergence. Execution of the program stops when the enriched structure fails to provide higher value of the objective function.

Separation has been presented with simple mass-balance equations. Variable side stream flow rates have been presented by dividing each reactor into several com- partments with constant side stream flow rate. Hence, the differential sidestream reactor has been presented by a sequence of systems of ordinary differential equations (ODEs):

dc - f ( s , c , fl ) s e [s F ..... SE] (1)

ds

where c is concentration vector, s is a scalar related with retention time and fl is a scalar related with the side stream feed flow rate. Different levels of mixing and feeding scenarios have been achieved by adjusting val- ues of s and fl in each sub-reactor. The resulting se- quence of initial value problems (IVPs) has been solved with the use of the Bulirsch-Stoer method (Stoer and Bulirsch, 1980). The use of orthogonal collocation may have significant advantages when constraints on state variables are to be considered (Cuthrell and Biegler, 1987). The use of Gear's method becomes indispensable to solve problems involving stiff kinetic models.

The resulting differential-algebraic optimisation prob- lem (DAOP) has been solved with the use of the suc- cessive quadratic programming (SQP) (Han, 1976). We have solved this problem in a conventional way: sensi- tivity equations have been calculated simultaneously along with the mass balance equations in the inner loop. Optimisation has been performed in the outer loop (Cuthrell and Biegler, 1987). Complexity of the optimi- sation problem and the number of IVPs increase in con- secutive steps.

In our future work we intend to solve this problem much more efficiently with the use of mixed integer non-linear programming (MINLP).

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5.8

c B

PFR exit

0 cA 5.8

Figure 2. The attainable region for the Van de Vusse re- action scheme in the example problem.

ous yield, ~ , can be written as a function of concentra- tion of the reactant, CA, and the desired product, cB, as follows:

~ B(CA,CB ) _ klCA-k2CB klC A +k3c 2 (4)

Feed flow rate is 100 L s l , feed concentrations of com- ponents A and B are 5.8 mol/L and 0 mol/L, respec- tively. Reaction constants are: kl=10 s l , k2=l s l and k3=l L s l mol l (Chitra and Gowind, 1981).

Relative volatilities of components are in the following order: (ZB>aA>(ZC>(XD. The desired product B can be separated as practically pure substance with more than 99.9 % recovery.

For simplicity we assume that only one distillation col- umn can be placed into the reactor system.

E X A M P L E P R O B L E M

The proposed methodology will be illustrated in detail with a simple but non-trivial example problem. We have compared our results with those from the previously published study (Chitra and Gowind, 1981). Although our results provide substantially higher value of the ob- jective function, this does not point to any deficiencies of the previously published study whatsoever since our results were obtained with a different set of assumptions.

Consider the Van de Vusse reaction scheme (Van de Vusse, 1964):

A kl > B k2 > C (2)

2A k3 ) D (3)

taking place in an isothermal, constant density continu- ous-flow reactor system. Reactions in series (2) are of the first order while reaction in parallel (3) is of the sec- ond order. The desired component is B. The instantane-

a. Operation without separation

The attainable region of this Van de Vusse's scheme is shown in Fig. 2.

A continuous stirred tank reactor (CSTR), followed by a plug-flow reactor (PFR) enclose the attainable region. The maximum product yield (63.48 %) in a reactor sys- tem involving only reaction and mixing is achieved in the PFR exit.

b. Operation without the recycle stream

It is easy to visualise (Fig. 3) the role of separation by drawing a line connecting output streams from the sepa- ration unit. Instantaneous yield functions are drawn with dotted lines. The attainable region for this problem is enclosed with a path ABCOA.

Top composition of the distillation column is situated above the attainable region at the upper left corner of the concentration space (CA = 0, CB = 5.8 mol LI). Bottom composition of the column will always remain in the in-

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5.8 S ) ".

~ ,"" . •.. • . . . . . . . . , . . "..

~ , , . ' " ' .

' C . . . . "

B ' " . '.

0 c A 5.8

Figure 3. Graphical presentation of the role of the inter- mediate separation in the example problem.

terior of the attainable region since the feed point must originate at least from the boundary of the attainable region. It is obvious that the use of separation will im- prove operation of the reactor system because the de- sired product withdrawn from the column will not suffer any further decomposition in the consecutive reaction step. Meanwhile, the bottom composition falls deeper into the attainable region where the instantaneous yield increases. It is apparent that we would have to process the bottom stream in another reactor to achieve the maximum overall yield. Furthermore, the exit composi- tion of that reactor must correspond to the zero value of the instantaneous yield.

Process structure has been derived in a single step be- cause we have assumed, for simplicity, that just one

Table 1. Stream data for the derived process structure with intermediate separation

S t r e a m cA (mol L 1) cB (mol L l ) qv (L s 1)

A 5.800 0.000 100.00 B 2.408 2.455 100.00 C 1.218 3.292 100.00 D 2.817 0.000 43.23 E 1.679 0.921 43.23 F 0.194 1.942 43.23 G 0.000 5.800 56.77

distillation column could be placed into the reactor net- work. Optimisation results are summarised in table 1; the derived process structure is drawn in Fig. 4. Per- formance of a plant could be further increased by plac- ing additional reactors with intermediate separations into the process. Additional pieces of equipment would also increase investment costs. Therefore, proper topology of a plant can be determined by optimising annualised costs. Selection among process alternatives has well de- veloped over past decades and we focus our attention in this paper just to the problem of generating sound proc- ess alternatives.

With the use of the derived structure it is possible to achieve 71.25 % overall yield. This is 12.24 % higher than that obtained without intermediate separation (63.48 %). Therefore, placement of the distillation col- umn into the reactor system can be justified. Note also that we have been able to increase an overall product yield significantly without recycle stream which would require larger process units and more expensive process- ing. This simple example problem illustrates advantages of the integrated approach to the design; we have been able to generate a novel design idea which may signifi- cantly improve operation of the entire process even in

Product

To the separation subsystem

Recycle from the separation subsystem L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4. The derived process structure in the example problem is a cascade of 4 idealised reactors with the intermediate separation step.

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cases when proper design is not obvious.

c. Operation with recycle stream

Should we attempt to further increase an overall yield, the use of the recycle stream would have been indispen- sable. With the use of the recycle stream we would be able to shift the operation of the reactor system in the neighbourhood of the origin of the concentration plane where the instantaneous yield function approaches the value of 1. Behaviour of the instantaneous yield function at the point (0,0) is, from the mathematical point of view interesting. In this point, it has a pole. It is possible to approach the value of one only at its right-hand limit. This means that the zero losses of the desired product could be achieved with an infinite recycle of reactant A originating from the down-stream recycle system. At this extreme the reactor system would be operated at nearly zero conversion. This extension of the solution space is drawn with a dashed line in figure 4. The derived struc- ture of the reactor-separator network is rich enough to present the performance of chemical reactions with and without recycle streams.

It should be understood that visualisation of the example problem has been used just to make our ideas more clear. Results have been obtained by solving a differen- tial-algebraic optimisation problem (DAOP), hence we can solve other problems without departing from ideas presented in this paper.

CONCLUSIONS AND REMARKS

A similar approach has been suggested by Omtveit et al. (1994). In contrast to the earlier work, however, we place separation into the reactor network. This strategy may have significant advantages compared to a simple recycle stream from the separation sub-system because in this way we can, as it has been shown in the example problem by comparing process alternatives, reduce load on downstream units by reducing the need for large re- cycle ratios.

There is practically an infinite number of ways to pro- duce compositions that belong to the interior of the at- tainable region. Hence, we reduce the solution space to the minimum number of configurations required to de- scribe potentials of a set of reactions taking place in a process.

A, B, C, D A, B, C ...

C

CA, CB

qv S

NOMENCLATURE

reaction components s t r e a m s

concentration vector concentrations of components A and B, respectively, mol L t reaction constant of the i th reaction, in re- spective units volume flow rate, L s t scalar related with retention time

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Greek letters

t~

eB

Subscripts

F,E

relative volatility scalar related with the side stream feed flow rate instantaneous product yield of the desired product

values at the reactor feed and exit, respec- tively

REFERENCES

Aelion, V., J. Cagan and G. Powers, 1991, Inducing Optimally Directed Innovative Designs from Chemical Engineering First Principles, Comp. chem. Eng., 15(9), 619-627. Balakrishna, S. and L. T. Biegler, 1991, Constructive Targeting Approaches for the Synthesis of Chemical Reactor Networks, Ind. Eng. Chem. Res., 31, 2152- 216z9. Biki6, D. and P. Glavi~, 1995, Synthesis of Reactor Networks with Multiple Multicomponent Feeds, Comp. chem. Eng., 19, S161-S166. Biki6, D. and P. Glavi~, 1996, Innovative Designs of Reactor Networks from Reaction and Mixing Principles, Comp. chem. Eng., 20, $455-$460. Cuthrell, J. E. and L. T. Biegler, 1987, Simultaneous Solution and Optimization of Process Flowsheets with Differential Equation Models, AIChE J., 33, 282. Chitra, S. P. and R. Gowind, 1981, Yield Optimization for Complex Reaction Systems, Chem. Eng. Sci., 36, 1219-1225. Glasser, D., C. Crowe and D. Hildebrandt, 1987, A Geometric Approach to Steady Flow Reactors: The At- tainable Region and Optimization in Concentration Space, AIChE J. 33, 282. Han, S. P., 1976, Superlinearly Convergent Variable Metric Algorithms for General Programming Problems, Math. Prog., 11, 263. Hildebrandt, D. and L. T. Biegler, 1995, Synthesis of Chemical Reactor Networks, AIChE Syrup. Ser., 91, 52- 67. Kokossis, A. C. and C. A. Floudas, 1990, Optimization of Complex Reactor Networks-I Isothermal Operation, Chem. Eng. Sci., 45, 595-614. Kravanja, Z. and I. E. Grossmann, 1994, New Devel- opments and Capabilities in PROSYN-an Automated Topology and Parameter Process Synthesizer, Comp. chem. Eng., 18, 1097-1114. Lakshmanan, A. and Biegler, L. T., 1996, Synthesis of Optimal Chemical Reactor Networks, Ind. Engng. Chem. Res., 35, 1344-1353. Omtveit, T., J. Tanskanen and K. M. Lien, 1994, Graphical Targeting Procedures for Reactor Systems, Comp. chem. Eng., 18, Sl13-Sl18. Stoer, J. and R. Bulirsch, 1980, Introduction to Numeri- cal Analysis, Springer-Verlag, New York. Van de Vusse, J. G., 1964, Plug-flow Type Reactor ver- sus Tank Reactor, Chem. Eng. Sci., 19, 994-997.