# Process Synthesis for a Reactor-Separator-Recycle System using the Attainable Region Approach.

код для вставкиСкачатьDev. Chem. Eng. Mineral Process., 6(1/2), pp.21-39, 1998. Process Synthesis for a Reactor-SeparatorRecycle System using the Attainable Region Approach C. McGregor, D. Hildebrandt* and D. Glasser School of Process and Materials Engneering, University of the Witwatersrad Private Bag 3, Wits 2050, SOUTH AFRICA The optimisation of a system of reactors and separators with recycle is an important problem in chemical engineering. This work examines the synthesis and optimisation of a reactor-separator-recycle system using the geometric ideas of the attainable region technique. The approach is applied to a simple idealised system where the available process operations are limited. Because the structure being considered has been constrained, the reaction and separation sections have effectively been decoupled. It is shown that because of this decoupling the reaction and separation sections can be individually synthesised using whatever methods are available. However, the overall system must still be optimised as a whole. Three examples are considered, each with different types of objective firnctions. Introduction In most chemical processes, a reactor system is followed by a separation system which separates the desired components from the reactor product stream and recycles the unconverted reactants back to the feed of the reactor system The optimisationof such a system of reactors, separators and recycle is thus an important problem. The capital cost of the reaction and separationunits, energy requirements, product yield and product selechty are all important criteria for the process engineer when examining the * Author for correspondence (e-mail: dihil@chemeng.chmt.wits.ac.za). 21 C.McGregor, D. Hildebrandt and D. Glasser economics of Merent process options. Despite the fact that this type of process is common, very few techniques exist to handle reactor-separator-recycle systems in an integrated manner. Most systematic procedures rely on heuristics [l] based on studymg the reactor and separator systems separately. The potential benefits resulting from integrating the reaction and separation processes may then not be realised. The process synthesis technique using mixed-integer non-linear programming (MINLP) has been applied to systems of these types [2-4]. In much of the work, the reaction and separation systems are treated separately and then combined into an overall process. As with the heuristic rules, the potential benefits of the integrated system may not be realised. A general process synthesis strategy has generally been considered difficult because of the large number of structural alternatives and non-linear design equations for the reaction and separation units. The Attainable Region (AR) approach differs from traditionalmethods of process synthesis. The concept was introduced by Horn [5] as a method of representing all the possible states that can be reached by a given system of reactors. Further work developed the AR approach using geometric principles [6-91 to generate a boundary which defines the AR. The attainable region concept has also been applied to reactor-separator-recycle systems which include a sharp split separator [lo]. This paper will examine the attainable region for reactor-separator-recycle systems where sharp slit separators are not permitted, but where simple separators based on vapour-liquid equilibrium are used. Principles of the Attainable Region The Characteristic Vector The conditions of a process stream are typically characterjsed by a number of variables that, for instance,might include the concentration of the various oomponents, the stream temperature and reactor residence time or space time. If the stream cau be uniquely characterised by n variables, these variables can be listed together in the f m of an n-dimensionalvector c, termed the characteristic vector. The elements of c must be sufl6cient to descriie the fundamental processes that are allowed to occur, and must 22 Process synthesisfor a reactor-separator-recycle system include all the variables in the objective function that is to be optimised. In particular, the elements of c should be chosen such that they obey linear mixing laws. The m o n for this will become evident. Fundamental Processes An attainable region is constructedby starting with the feed point and making use of the allowable processes, operating within the system, to move to other points witbin the space. The processes operating within the system are represented geometrically as vector fields acting throughout the space. These fields move the system from one point or state to another point or state according to the representation of the process ocmrring. Common to all systems is the process of mixing. Jf two streams with states CIand cz are mixed, then the resultant mixture c' is p e n by: c* = UCl + (1 - a)cz O l U l l Equation (1) is often referred to as the level arm rule. Note that in order for Equation (1) to be saWed, all the elements of the characteristic vector must obey linear mixing rules. Most previous AR work has considered reaction as one of the funQmental processes. Reaction is designated by a vector field, r(c). The reaction vector r, gives theinstantaneous direction of change in the n-dimensional space if material of state c is allowed to react differemally. Once the reaction vector has been dehed, it is then possiile to characterise the geometric properties of simple ideal reactors [6]. A new aspect in this paper is that the fundamentalprocess of phase change which is incorporated into the AR. Vapour-liquid equiliirium (VLE) is assumed to exist between the vapour and liquid phases resulting fkom partial phase change. Since the vapour and liquid phases have different compositions, partial phase change can be used to achieve separation. Partial phase change is designated by a vector field,s(c) . The separation vector s, gives the insta~~taueous direction of change if material of state c undergoes differential boiltng. A Werent, but equally consistent separation vector, could also Ik defined based on Werentid condensation. Once the separation vector has been 23 C. McGregor, D. Hildebrandt and D. Glasser defined, it is then possible to characterise the geometric properties of ideal partial boilen and condensers. Definition of the Attainable Region The attainable region is now defined as the region in the space of the characteristic vector that can be achieved from the feed by any system, operating at steady-state, that satisfies the all the constraints placed on the system. The attainable region is constructed by starting from some achievable point, typically the feed. The fundamental processes are then used to extend the region. Bemuse of the linear mirdng properties of the space, the AR is a closed, connected, convex region Also, any interior point in the region can be found by mixing two points on the boundary of the region. Thus, in order to descn'be the attainable region, all that is needed is to define the boundary of the attainable region, &R. Because the processes have here been h t e d to mction, phase change and mixing, the AR is the set of all achievable products from any possible system of reactors and VLE separators sa-g the constraints placed on the system Problem Definition The Attainable Region Space In this paper, the objective is to find the attainable region for a reactor-separator-recycle system. Flow diagrams for the possible system configurations are shown in Figure 1 with the following constraints placed on these system: the system has a single liquid feed stream and must have a single liquid product stream; the reaction occurs in the liquid phase. The reactor network may consist of any combination of steady-state reactor units. The network may include a plug-flow reactor (PFR), a continuously-stinred tank reactor (CSTR) or any other reactof that can be defined in terms of the fundamental processes of reaction and mixing. However, the separation is constrained to occur within an equiliirium stage boiler. Total condensers are used to ensure that the product and recycle streams are in the liquid phase. The variables requiredto describe the characteristicvector must now be specified. It is well known that mole fiadom x, obey the lever arm rule and are so suitable for inclusion in the charactenistcvector. In a system with n componentsn-1 mole hctions 24 Process synthesisfor a reactor-separator-recyclesystem CQ Network F i e 1. System configurations. Y L are required to charaderise the composition of any stream. The space time, r, is typically used in attainable region analysis to characterise the reactor system. If the space time is defined by T = VpolFmowhere Pp is some reference density (for instauce, the density of the liquid feed), and m0 is the corresponding reference molecular mass, then r obeys linear mixing rules [S]. Finally, a variable is requixed which will characterise the amount of phase change that has occurred in the system. The phase change extent, w ,is now introduced, and is defined as the fraction of the feed stream that has undergone phase change, that is: y = PIF (2) where P is the molar rate of phase change and F the total feed flowme - in other words, the Merence in vapour product and vapour feed flowrates in a boiler or the difference between liquid product and liquid feed flowrates in a condenser. In a partd boiler with liquid feed only, w is the fraction of the feed which is vaporised, and in a partial condenser with vapour feed only, v/ is the h a i o n of the feed which is condensed. For a total boiler or a total condenser, the phase change extent is one. The phase change extent is similar to the space time of a reactor - like r, w i s a capacityvariable. The magnitude of y/ is also related to the energy required for phase change. 25 C. McGregor, D. Hildebrandt and D. Glasser Consider mixing between any of the attainable streams in a system where all streams contain one of two possiile phases. A restriction is placed on mixing: only streams of the same phase can be mixed. If these restrictions are adhered to, the phase change extent y can be shown to obey linear mixing rules and is thus also included in the characteristicvector. Now, if the objective function can be written in terms of the mole fkactions, the space time and the phase change extent, then there are sufficient variables to charactexise the system. The general form of the characteristic vector for an n component system can then be written as: Process Vectors Now, assume that in the system a set of reactions occur and that the kinetics for each of the reactions is known. The kinetics are such that a species i has a rate of formation ri. These rates of formation can then be mitten together as a vector functios r(c), such that: where r, is the molar rate of formation of component i . Unlike for reaction processes, phase change processes will have two product streams, each of a different phase. Each ofthe products of a equilibrium stage process can be represented by a characteristic vector. If these products of different phases could actually be mixed, they would give the feed to the separation process. The mass balance for an equiliirium stage boiler can then be written for all components as the vector equatiox 26 Process synthesis for a reactor-separator-recycle system where y and @ are the phase change extents of the product and feed streams respectively ( y / = y o+ wS where fi is the phase change extent of the equilirium stage boiler) and s, the separation vector, is given by: where E, is the mole fraction of component I in equiliirium with a liquid of composition LZ, that is: Except for the fact that fi and thus y-g is in the range of zero to one, Equation (7) has the same form as the mass balance for a CSTR Thus, an equilibrium stage boiler can be seen as analogous to the CSTR An equilibrium stage boiler also has the same geometric properties as a CSTR. These properties will determine what size of ecphbrium stage sepmtor is required to reach any point in the attainable region. Therefore, just as the space time is a useful variable for characterising systems with reaction, so too is the phase change extent a usefbl variable for characterking systems with phase separators. Having determined the liquid product of the ecphirium stage, the vapour product can be determined through the equilibrium relatiodup. It should also be noted that the phase change extents of the liquid and vapour product streams are equal, as both strams have passed through the same separator unit Thus, having found the liquid product CL by Equation (6),the correspondingvapour product CG is defined Examples Consider the two component system in which the exothermic reversiile reaction A + B occu~s.Now let the mole -on of B in the liquid phase be x. Assuming that the reaction occurs only in the liquid phase, the normalised rate of reaction is defined in terms of ra, the rate of formation of A, by: 27 C.McGregor, D.Hildebrandt and D.Glasser where T is the reaction temperature. If the heat of mixing is .small compared to the enthalpy of reaction, and the reaction occurs adiabatically, then the reaction temperature can be related to x by: where T f is the tempemure of the feed to the reactor and ;x;f is the mole fraction of B in the feed to the reactor, Tad is the adiabatic temperature rise if pure A reacts complete@to form B. This example has been previously examined [S] where the values of the constantsused are the same as are used now and are shown in Table 1. Table 1. Parameters usedfor examples. k2 EZ 5 x los 5 x lo8 4000 8000 P 0 300 a 200 ‘Is The example is now extended so that the product of the reactor passes into a phase separation process which separates the more volatile A fiom the less volatile B. One of the products of the separator m then be recycled, mixed with the feed and passed to the reactor structure. Ifthe vapour product is recycled, it mu‘st Grst be totally c o n d m before it can be mixed with the liquid f e d If the liquid phase is recycled, the vapur product must be condensed so as to form the desired liquid product. These flowsheet options are shown in Figure 1. Now,the reactor feed -on xf and reactor feed temperam T,, will vary as the recycle ratio and separation products are changed. To fixthe reactor feed temperature, a heat exchanger is placed before the reactor. This heat exchanger heats the reactorfeed to the temperature that would be obtained if the process feed was adiabatically reacted to form a stream of mole fraction xd that is: where P is the basis temperature, the temperame of the feed to the process. Therefore, the reactor operates on the adiabat which originates at the process feed. 28 Process synthesisfor a reactor-separator-recyclesystem Because a phase separation process is being used, the vapour-liquid equiliirium relationshrp must be speatied. To simpllfy the example the components were assumed to have a constant relative volatility, a,that is: U E(x) = (a- l)x+ 1 where E(x) is the fraction of B in the vapour phase in equiliirium with x, the fraction of B in the liquid phase. Neglecting Phase Change Capacity The attainable region can now be constructed. Because the sum of the mole fractions of A and B must be unity, only one of the mole fractions needs to be included in the characteristicvector. Also, the product is only in the liquid phase, so the mole fraction of B in the product x, is the only composition variable which must be included in the charactexisticvector. As the feed to the process consists of pure A, the mole fraction x, is the same as the overall conversion. In this example the cost of phase separators and heat exchangers are assumed to be small in comparison to the cost of reacton. Thus y is not part of the objeciive -on and need not be included in the characteristicvector. The vectors then needed to describe this system are c = s = [x - E(x), 03. The processes allowed to o [+TI, r = [r,, 11 and m are limited to reaction and phase change. Reaction occurs in a CSTR and s e p t i o n in an equili’brium stage boiler. It can be shown that: where x, is the mole hction of B withm the CSTR and x, is the mole hction of B in the system product (that is the system conversion). Thus the separation should be chosen so as to maximise the reaction rate as this will minimise the space time, even if this requires an infinite recycle ratio. The loci representing the various systems can be -oc and are shown in Figure 2. In Figure 2, the feed to the process is represented by point 0. Curve 0 29 C.McGregor, D. Hildebrandt and D. Glasser 8 A 0,04 0.02 0 0.0 0.2 0.4 0.6 Conversion (xp) 1cutves I Process Flow D i a m OP ICurves Process Flow D i a m I OR Figure 2. AR neglecting y. represents the liquid-recycle system, while curve Q represents the vapour-recycle sysThe reaction rate has a maximum r,, at a composition x-. The operating conditions should be chosen so that the reaction rate is as high as possiile. On section PQ of m e 0, a composition of x,, can be achieved in the CSTR using the liquid-recycle system with a finite recycle. On section OP of curve 0, x, 30 cannot be Process synthesisfor a reactor-separator-recycle system achieved but if the product rich stream is recycled then the highest rate possible is obtained. This however requires an infinite recycle rate. On section OQ of m e 8, the recycle of the reactant rich phase results in a lower reaction rate in the CSTR and thus the space times for the vapour-recycle systems are larger. This result is surprising as recycling the product rich phase is counterintui~e. On section QR of curve 8 ,a compositionof x,, can be achieved in the CSTR using the vapour-recycle system with a finite recycle. On section RS of m e 8,x,, cannot be achieved but if the reactant rich stream is recycled then the highest rate possible is obtained. Thts however requires an inlinite recycle rate. On section QT of curve 0 , the recycle of the product rich phase results in a lower reaction rate in the CSTR and thus the space times for the liquid-recycle systems are larger. This is the standard result in terms of choosing which phase to recycle. The reactant rich phase is recycled when hgh conversion is required The segment PQR is a straight line, which if extended back would passes through 0. This segment represents the range over which the separator can achieve the maximum reaction rate, r,,,,, in the CSTR. The fact that PQR is a straight line passing through 0 can be shown by substituting r-, a constant, for r, in Equation (12), that is r = xplrmax.. Figure 3 shows optmum fkaction of the separator feed which must be vaporiSed. The fraction vaporised isjust v - the separator phase change extent required to give the maximum reaction rate in the CSTR. From the values of y/. it can be seen that point Q can be achieved without using a separator. This point represents the composition at which the CSTR naturally operates at maximum rate. Now, the necessary conditions for the attainable region require that the region is convex. As can be seen in Figure 2, both system loci include concavities. These concavities can be removed by introducing a bypass which takes some of the feed and mixes it with the system product The system bypass is represented by the straight line 6 or OPQR, which gives a boundary for the attainable region of OPQRS. The bypass acts to ensure that the CSTR always operates at the maximum rate. Note that a bypass JW$ around the CSTRwill achieve the same result, is indicatedby line 6. It can 31 C. McGregor, D. Hildebrandt and D. Glasser Conversion (+ Fiwe 3. Optimum vaporisationj-actions. also be noted that while section OP of the boundary can be achieved using a liquid-recycle system with bypass, it can also be achieved by using a vapour-reqcle system with bypass. Thus the whole boundary of the attainable region can be achieved using a system with vapour-phase recycle. The section OP can also be achieved by using a CSTR with bypass only, no separator is required. It s h d d be noted that component A, the reactauc is more volatile than component B, the reaction product. Thus the phase which is more conaxhated in reactant is recycled in the system, which is as expected. On curve QT, the separator is operating at the bubble point of the reactor product; while on curve RS, the separator is o p e a m at the dew point of the reactor product Both these sections represent in6nite recycle conditions where the reactor uperates at the fastest rate possible. If a PFR was consided it would only be a candidate for the boundary in the sections where the separator cannot achieve the maximum rate in a CSTR As these sections have infinite recycle, the reactor feed and recycle streams have the same composition as the reactor product. At infinite recycle ratios, the PFR behaves IX as the behaviour of a recycle reactor approaches that of a CSTR at large like a CSTR, JE recycle ratios. The phase cbange extent y, is 1 for the sections QT and RS. However, the recycle streams have dinite flowrate, and so the system 32 iy is infmite. Thus, by Process synthesis for a reactor-separator-recycle system including y in the characteristicvector, an attainable region will be obtained with does not include infinite recycles. Neglecting reactor capacity In this example, unlike the first example where the cost of phase change was excluded, the cost of the reactor will now be assumed small and the cost of phase change included. Thus, ly is now part of the objective function and 7 is no longer included in objective function. The vectors needed to describe this system are c = [ x p . w ] , ;r = [rx,01 and s = [ x - E ( x ) , 11. The attainable region can now be constructed. First, it should be noted that a product with conversion x, less than the eqmhbrium conversion x., be obtained without using a separator. Thus, for x, < x, y = 0. To obtain a product with x, > x, for the reaction can the product is obtained with requires some separation. As componentA is more volatile than component B, a product with higher B mole hction is obtained in a system with vapour recycle. A system with liquid recycle will produce a product with a lower B mole W o n , x,. The phase change extent y required, is then given by rewriting the definingvector equation, Equation ( 5 ) as: The system phase change extent y and the recycle ratio for either a vapour or liquid recycle system can be shown to be the following fimctions of v/.: I 2 . Vapour recycle : Liquidrecycle 1 33 C. McGregor, D. Hildebrandt and D. Glasser A Line I ProcessFlowDiagram I Area I _ I I I - I I Process Flow Diagram * L I Figure 4. AR neglecting r. The system phase change extent w required to produce any product can be calculated, thus giving the attainableregion shown in Figure 4. As the region is already convex, the region cannot be extended by mixing. In the figure, point 0 is the process feed, while point P represents the product &om an equilibrium reactor. Including costs of reaction and separation Now the phase change extent as well as the reactor space time will be included in the objective function required to optimise the system. The vectors needed to describe the s = [x-E(x), 1,Ol and r = [rx,0,ll. The reaction in this region are c = [xp,y,~], example is allowed to occur in any network of reactors which minimises the space time. The separation process is still restricted to occur in an equilibrium stage boiler. 34 Process synthesis for a reactor-separator-recyclesystem The attainable region can now be constructed. First consider the vapom-recycle system. Select a product composition x, and system phase change extent Equat~ons(14) and (15) the boiler phase change extent determined. From y/, y/. From and recycle ratio R can be the composition of the liquid and vapw products of the separator can be determined The reactor feed composition x8 and reactor product composition x, can then be determined fkom the recycle ratio and the separator product compositions. Next, the space time zof the system must be determined. This gives a point in three dimension space. To find the boundary of the attainable region, the minimum space time must be found This requires the AR constructed for an adiabatic exothermicreactor network. The attainable region for the adiabatic exothermic reaction studied in these examples has been previously constructed [S]. There it is shown that when the reactor product composition x, is less than the maximum rate conversion x-, then minimum space time is obtained in a CSTR with bypass, where the CSTR operates at the maximum rate. ~f x, is h t e r than the maximum rate composition, then the minimum space time is obtained in a mctor network CoIlSisting of a CSTR operating at the maximum rate, followedby a PFR. Thus, if the value determined for x, as explained above is less than x- then a bypass around the CSTR must be used so that the CSTR operates at r,, and the space time is given z = xp/rmax.Consequently, the space time is independent of values of y/ for which x, < x-. y/ for all A flat inclined plane then forms part of the boundary of the attainable region as shown in Figure 5 . If the determined value of x, is greater than x, then the space time can be determinedby: where R is the recycle ratio given by Equation (15) and r: is the normalised rate of reaction experienced at a given extent - constant for those concentrations occurring in the CSTR and equal to r, for those in the PFR. 35 C.McGregor, B.Hildebrandt and L). Gasser The specral case for the liquid-recyclesystem must be considered separatelybecause y =2 by Equation (14). Because of this the phase change extent of the liquid-recycle system is independent of the recycle ratio. Thus,for the licpid recycle system the phase change extent y/s should be chosen so as to maximise the rate, just as was done for the first example. However, as in the first example, the values for the proctuct composition x, for which the maximm rate can be achieved is larger for the vapour-recycle system than for the liquid-recycle system. Thus a vapour-recycle system with y = 2 can be used to achieve the same section of the boundary as that which can be achieved using the liquid-recycle qktem. The vapour-recycle system also forms part of the boundary that cannot be achieved using the liquid-recycle system. Thus, the entire boundary of the attainable region can be obtained using a vapour-recycle system. The system phase change extent given by E m o n (14) is infinite only when ys= 1, when the recycle ratio is infinite. Therefore, by including y/ in the cbaraaeristic vector, infinite recycle ratios do not form part of the boundary of the attainable region. The attainable region for the system is shown in Figure 5 as a three-dimensional plot with COntDuT lines on the 3D boundary of the region. Figure 6 is a projection of the AR Figure 5. AR including y and 7: 36 Process synthesisfor a reactor-separator-recycle system Fiure 6. Projection of AR including y and r. boundary onto the [xp, y]-plane. The figure shows the various surfaces which make up the boundary. In Figure 6 , m e 0 represents the vapour-nxycle systems in which a CSTR operates naturally at the maximum rate. SurfaceA canbe generated by filling in the mixing fan hull between the feed point 0 and curve 0 . This represents a bypass around the vapour-recycle system such that the CSTR operates at the maximum rate. Alternatively, surface A can be generated by introducing a bypass around the CSTR in 37 C. McGregor, D.Hilakbrandt and D.Glasser the reactor network, ensuring that the CSTR operates at the maximum rate. Both of these systems can represent the boundary of the AR because surface. I is a flat inched plane. Point P on m e 0 represents a CSTR operating at the maximum rate; there is no separation and recycle in this system. Curve 8 represents those products which can be obtained using an equilibrium reactor and thus requires an infinite space time. Curve 8 marks the limit of the conversions which can be achieved in any system with separation, reaction and recycle. Point Q on curve 8 represents the product from an equilibrium reactor, without separation and recycle. Surface I! shows those products that can be achieved optmdly in the vapour-recycle system where the reactor network consists of a CSTR (operating at maximum rate) followed by a PFR Discussion Of particular interest is the Merent boundary structures obtained by considering different objective functions. Neglecting the capacity of separation results in s t ~ ~ c t u r e s most of which require infinite recycle, while neglecting the capacity of reaction results in all structures having equilibrium reactors, that is reactors of infinite volume. Consideringboth the capacity of reaction and separation results in stmctum which only have infinite reactor volumes or infinite recycies if the extremes of collvefson axe desired. Almost all structures have finite reactor volume and recycle. These systems have a reactor network consists of a CSTR operating at maximurnrate followed by PFR, if necessary, which is the same as that obtained by f i n e the AR for the reaction only. Conclusions Because the structure being considered has been constrained, the d o n and separation sections have effectively been decoupled. The reaction and separation sections can then be individually synthesised using whatever methods are available. Note that by decoupling the system, the option of simultaneous reaction and separation, such as by c&alyl~distillation, is excluded. Nonetheless, by decoupling the system, the process synthesis procedure is simplified. This is not necessarily a limitation since not all systems are amenable to simultaneous reaction and separation. In order for a system to be optunal, not only must the system structure be optunal, but the control parameters 38 Process synthesis for a reacror-separator-recyclesystem must ensure optunal performance of the system Although the structure of the system could be synthesised by decoupling, the system itself has to be optimised as a whole. No dmupling here is possiile. This work has thus shown how the geometric ideas of the Attainable Region approach can be incorporated into synthesising an optimal reactor-separator-recycle system. Acknowledgements We are grateful for research funding provided by SASOL and the FRD. References 1. Mizsey, P. and Fonyo, Z.1990. Towards a more Realistic Overall Process Synthesis -the Combined Approach. Comput. Chem. Eng., 14(11),pp.1213-1236. 2. Kokossis, A and Floudas C. 199 1. Synthesis of Isothermal Reactor-Separator-Recycle Systems. Chem. Eng. Sci., 46,pp.1361-1383. 3. Pibouleau, L., Floquet, P. and Domenech, S. 1988. Optimal Synthesis of Reactor-SeparatorSystems by Non-linear ProgrammingMethods. NChFJ., 34 (I), pp.163-166. 4. Balakrishna,S. and Biegler, LT. 1993. A Unified Approach for the Simultaneous Synthesis of Reaction, Energy and S e p d o n Systems. fnd. Eng. Chem. Res., 32, pp.1372-1382. 5. Horn, F. 1964. Proc.3rd Eur. Symp. on Chemical ReactionEnghe&. Pergarnos New York, p.1. 6. Hildebrandt, D., Glasser, D. and Crowe, C. 1990. Geometry ofthe Attainable Region Generated by Readion and Mimng: With and Without Constraints. Chem. Eng. Sci., 29 (l), pp.49-58. 7. Giasser, D., Hildebrancb, D. and Crowe, C. 1987. A Geomdric Approachto Steady Flow Reactors: The Attainable Region and Optimisation in ConcentrationSpace. Ind. Eng. Chem. Res., 26 (9), pp. 1803-18 10. 8. GI-, B., Hildehdt, D. and Glasser, D.1992. OptimalMixing for Exothermic Reversible Reactions. Ind. Eng. Chem. Res.,31(6),pp.1541-1549. 9. Feinberg. M and Hildebmdt, D. 1997. Optimal Reactor Design &oma Geometric Viewpoint: I. Universal Properties ofthe Aaainable Region Chem. Eng. Sci., 52 (lo), pp.1637-1665. 10. M e i t , T., Transkanen, J. and Lien, K. 1994. Graphical Targeting Procedum for Reactor Systems. Comput. Cbem.Eng., 18,Supplement,pp. 113-118. Received 12 F&ruary, 1997; Accepted after revision: 10 March, 1997. 39

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