- Email: [email protected]
Innovative designs of reactor networks from reaction and mixing principles
Computers chem. Engng Vol. 20. Suppl.. pp. $455-$460, 1996 Copyright © 1996 Elsevier Science Ltd S0098-1354(96)00086-5 Printed in Great Britain. All rights reserved 0098-1354/96 $15.00+0.iX)
Pergamon
INNOVATIVE DESIGNS OF REACTOR NETWORKS FROM REACTION AND MIXING PRINCIPLES DU~KO BIKI~ and PETER GLAVI(~ Faculty of Chemistry and Chemical Engineering PO. Box 224, Smetanova 17, SI-62001 Maribur, Slovenia
Abstract - In the present paper an algoriflnnic procedure for generating innovative reactor network designs for complex reaction schemes is presented. Computational methodology is derived from the basic principles of reaction and mixing. Candidate reactor networks are generated by expanding the solution space into regions where different levels of mixing are required. Directions of extending the design space are identified by using formal optimization theory. A set of candidate reactor networks is generated in a finite number of the solution space extensions. The derived methodology eliminates the requirement for an postulated superstructure and the use of qualitative knowledge for finding the optimal solution. With certain modifications the proposed design procedure can also be used to support the design of batch processes. The generated reactor structures are truly optimal. The proposed methodology is illustrated by an example of integrating chemical reactors into the overall process.
INTRODUCTION Design of reactor networks has been addressed in many papers over past decades. Published research can be classified into two main categories: perfornmnce targeting and integration of chemical reactors into the overall process. A vast majority of papers deal with the problem of generating and selecting among candidate solutions. It is surprising that, except for simple reaction schen~es, there is almost no qualitative knowledge that could effectively support the design task. Several generalized approaches for solving this problem have been suggested over past decades. Notable among those are: targeting strategies (Balakrishna and Biegler; 1991, 1992, 1993), superstructured approach supported with the MINLP formulation (Kokossis and Floudas, 1991), and geometric approach, well known as the Attainable Region (Glaser et at., 1987; Hildebrandt et al., 1990). Recent research 0-Iildebrandt and Biegler, 1994) of the performance targeting supported by geometric interpretations of reaction and mixing promise substantial progress in this important field of process synthesis. PROBLEM STATEMENT In the present paper we focus attention to the problem of design and integration of chemical reactor networks for complex reaction schemes into the overall process when multiple multicomponent feeds with fixed flow rates exist. This is one of the central questions of process synthesis. It is also a realistic design problem because it is conceivable that several multicomponent feeds to the reactor subsystem will exist in practically every flow-sheet. The problem to be addressed in this paper can be stated as follows: given (i) rate equations of a set of chemical reactions, (ii) feed flow rates and (iii) compositions of feedsfind (i) number and types of reactors, (ii) connections among reactors in the network and (iii) locations of feeds along ,~ith feeding scenarios that provide means of attaining the most comprehensive product distribution. It is assumed that: (i) structure of the separation system is known in advance and (ii) only reactions, mixing and heat transfer take place in reactors. Our prima~ task is to find ways for establishing proper topology of the reactor network, contacting fluids of different age and feeding strategies for fresh feed and recycle streams of different compositions. It is important to realize that once the structure of reactor network and locations of feeds along with feeding strategies have been found, a solid connection among the reactor and separation subsystems has been established. At this point the invention of the process structure in the conceptual design phase is completed. The foundation of a good design is set in the conceptual phase of the process synthesis. Should one introduce too many assumptions in this critical design ~'lage, the design space would have been curtailed too soon. Hence, a great deal of sound design ideas would have been
$455
$456
European Symposiumon Computer Aided Process Engineering--6. Part A
lost. To overcome this complication we propose a methodology that efficiently creates novel designs by establishing a set of variables and process structures that dominate the design problem. Bounds on concentration and yield space are not always identical when more than one feed to the reactor subsystem with limited flow rates exist. The reason is that constraints on flow rates and compositions are unconditionally active. Hence, they influence bounds on the overall yield space while they do not affect bounds on the concentration space whatsoever. Furthermore, as it is well known, reactors are usually operated between the maximum selectivity and the maximum desired product yield to achieve the minimum total annualized costs. Therefore, it is crucial to identify bounds on both concentration and yield space to provide means for sufficiently generalized presentation of the problem. In our problem formulation we focus attention to the most comprehensive product distribution space instead of postdating the objective function that should be either minimized or maximized. In this way our approach becomes independent on the objective function. In other words, we focus our interest to extensions of the solution space to identify variables that dominate the design problem in first place and then extract the optimal solution for specific objective function on the basis of the most comprehensive problem formulation. This is non-routine and non-trivial task because: (i) the problem has combinatorially many candidate solutions and (ii) the available qualitative knowledge for simple reaction schemes does not apply for complex reaction schemes. To reveal this problem we have used formal optimization theory to derive general features of candidates for optimal reaction-mixing systems. The derived properties have been used to extend the design space by increasing the degree-of-freedom. Constraints on feed flow rates are unconditionally active, hence they always bound the optimization problem. Expansion of the solution space is used to identify critical variables and then the optimal process layout is selected with the use of conventional optimization techniques. METHOD OUTLINE The basic idea of the proposed method is to identify a set of candidate reactor net~vrks along with design variables that bind the solution space and then to extract the optimal network by optimizing a model that comprises all candidate configurations. In our previous paper (Biki6 and Glavi~, 1995) we have presented a methodology for generating candidate reactor networks for isothermal reactors. In the present paper we extend its use to more complicated cases when reactions take place in nonisothermal systems with external heat sources and sinks. A set of variables that dominate the problem can be created by manipulating the mathematical model (Aelion et al, 1991). The use of this approach renders prohibitive in practice because it would have required extremely complicated mathematical manipulations for every new set of kinetic equations. To avoid this difficulty we have derived general features of candidate variables that extend the solution space to its limits. Geometrical interpretatations of the derived features have been used to support the design task with the simple-to-use step-bystep procedure in which general features of design variables have been visualized for 2- and 3-dimensional kinetic models. By using the proposed method a set of variables that bind the solution space can be derived in a finite number of steps as follows: • •
maximize the instantaneous yield function with respect to temperature, draw lines connecting points of constant instantaneous yield function assuming that temperature could be adjusted to any chosen value, • identify bounds on the optimal temperature profile and conditions in the reaction system that drive temperature into the direction of extremes, • find regions where temperature exceeds constraints on temperature; assume tltat reactor(s) in these regions will operate isothermally; draw these profiles, • use graphical procedure (Biki6 and Glavi~, 1995) to find optimal mixing and feeding strategy,
The reactor network that can be generated with the use of the proposed teclmique provides means for attaining the maximum product yield from several feeds with fixed compositions and flow rates. To ensure that the reactor network can produce also all concentrations, structures generated with the use of the attainable region must be added in parallel. The resulting reactor network is multiple-input-multiple-output system that consists of series-parallel combination of PFRs (plug-flow reactor), CSTRs (continuous stirred-tank reactor) and DSRs (differential sidestream reactor). The derived structures are often unrealistic since they could involve an infinite number of side streams gradually distributed over the reactor length or such temperature profiles that could not be attained in available equipment. These unrealistic process alternatives can be effectively eliminated with constraints on reactor sizes and heat transfer. Once the structure of candidate reactor network Ires been set, optimal solution can be extracted with the use of conventional optimization techniques. We have used the procedure where optimization is performed in an outer loop with subsequent integration of an ODE model in the inner loop (Cuthrell and Biegler, 1985). Despite of certain disadvantages we found this technique reliable when kinetics model is stiff or unstable.
European Symposium on Computer Aided Process Engineering--6. Part A
$457
EXAMPLE
Consider file reactor synthesis problem for tile Van de Vusse reaction scheme (Clfitra and Gowind, 1985; Van de Vusse, 1964): kl k2 A >B )C (1) ks 2A )D (2) in which the desired product is denoted with B and A is the limiting reactant. Reactions in series (1) are of the first order while the parallel reaction (2) is of the second order. Kinetics data are shown in Tab. 1. The reactant is available from two feeds, F~ and F2 with compositions and flow rates as shown in Tab. 2. Table 1. Kinetics data for the Van de Vusse reactions in the example problem . .. E (J mol -I) ao fl 66,300 33,150
1.500 .106(s -~) 1.000 .102 (sl )
0.28 0.36
99,450
4.444 .108 (L tool"l s"l)
0.20
Table 2. Feed compositions and volume flow rates for the example problem c A (tool L -I ) c B (tool L "l) q (L s"l) Ft F2
0.65 1.00
0.35 0.00
100 100
The objective is to maximize an overall product yield in the reactor system consisting of 5 reactors. No assumptions have been made on reactor types and the reactor network topology. Temperature of the reactor fluid must be on the interval Te [550 K, 900 K ]. Uniform heat flow to reactors, g/, is assumed and is limited to interval 2" ~ [-27.78.103,+27.78 •l0 3] where 2" p%ro The instantaneous yield function of the desired product B can be written as follows:
k,(r),'-k2(V)p
es(r,p,r) =
k, (T)r + k3 (r)Aor 2
(3)
Let us now assume that temperature of the reactor fluid can be adjusted to any value in the interval T~ [0, Qo]. Although this is completely unrealistic assumption, our analysis will show very useful remits. Maximization of the instantaneous yield function with respect to the reactor fluid temperature has indicated that temperatut~ concentration profile (see Fig. 1) must be adjusted according to the following equation: T* -
-AE2 R e ln(z/r)
(4)
where:
z = glp+~](gtp) 2 +glP Ig2 gl =
ao,2
g2 =
Aoao.3
(6) ao,l ao.i Substituting Eqns. 4, 5 and 6 into F_,qn. 3 and rearranging gives the instantaneous yield function along the optimal temperature path: -
,
(5)
dp = q~'a(P) = - z + g , p 2 dr z+g2z
(7)
Note that ~ is function o f p only. Solution of the differential Eqn. 7 with the initial value conditions from the region r -- ro, ro~ [0,1]; p -- Po, poe [0,1 ]; 1> p +r represents a PFR trajectory that could have been attained by adjusting temperature to values defined in the Eqn. 4. Note also that the solution to Eqn. 7 is a convex, monotonically increasing function on the interval re[0,1]. This indicates that PFR trajectories hind the solution space when the temperature profile could have been adjusted according to Eqn. 4. By inspecting Eqns. 4 and 7 the following conclusions can be approached: I. ff p --~ 0 then T'--~ 0, II. f i r --~ 0 then T'--~ oo, III. if r --~ 0 then p --~ p ~ , Conclusion I indicates that the lower bound on the temperature of feed stream F2 is unconditionally active. Conclusions II and III indicate that the upper bound on the reactor fluid temperature is also unconditionally active. Hence, beth constraints will affect the design. Lower bound on the temperature of the feed F~ is also active. 20: I.I(A)-P
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European Symposiumon Computer Aided Process Engineering--6. Pan A
Let us now investigate the behaviour of the reaction kinetics at the active constraints on temperature. The instantaneous yield function over the entire concentration space is shown in Fig. 1. The instantaneous yield function in the interior of the feasible region has been calculated from Eqn. 7 while values at bounds have been calculated from Eqn. 3. We have found that there exists a region (curve Mm) where straight lines originating from the feed point Fa are tangential to the instantaneous yield function. It consists of two parts; a curve at the lower bound on temperature and a straight line in the interior of the allowable temperature interval. The reason is that the instantaneous yield functions in the interior of the feasible region are straight lines parallel to the r-axis (F.qn. 7). This implies that mixing of fluids along these lines always satisfies necessary condition for the maximum extension of the yield space since a line is tangent to itself. Feed point F2 has a single, continuous region M2 located completely at the lower bound on temperature. 1
oo
o
~
.....
I
,°0.767
:''k
I?i
"
:: - i ' . -
.:
08-"
""
-0.820
""
..-
~
'
-.o,..
\
\ :; ::
\
:0.91_5~_
./.'i/./... 2.'.- ...
0
\
"
~
0
1 F
Figure 1. Graphical presentation of basic properties of the Van de Vusse reaction in the concentration space. Let us investigate two possible reaction paths (FI~ and FI2) that could be drawn from F2. In both cases reaction paths intercept curve M2 indicating that a CSTR must be used to extend the yield space. Reaction path rim intercepts the region Mj in the interval where it is a straight line indicating that the fluid from feed point F~ must injected into the reactor as an idealized instantaneous pulse to provide optimal feeding strategy. The injection of feed F~ will discontinuously shift the concentration profile in the direction parallel to the r-axis. This will require discontinuous temperature change to provide the optimal processing. Reaction path FI2 intercepts fl~e curved part of the region Mj indicating that a DSR must be utilized to attain the maximum extension of the yield space.
F,
CSTR(r = 550 K)
PFR/DSR(/'= 550 K)
PFR (Noaisothermal)
PFR (T= 950 K)
Figure 2. Idealized reactor network that binds the solution space Therefore, there exist two possible scenarios for mixing of F~. If the composition in the reactor approach to the curved part of M~ then a DSR must be employed; otherwise a fluid from feed point FI must injected into the reactor
European Symposiumon Computer Aided Process Engineering--6. Part A
$459
as an idealized instantaneous pulse to provide optimal feeding strategy. Movements in the concentration space between regions where CSTR and/or DSR are required must be performed in PFRs. From the above analysis one can derive an idealized reactor network that binds the solution space (Fig. 2). Candidate reactors that have been found in the region at the lower bound on temperature must operate isothermally. As soon as the fluid in the PFR/DSR penetrate into the interior of the feasible region (line corresponding to 1"--550 K) temperature could be increased to follow the optimal reaction path. This path could be followed until the temperature approach the upper bound on temperature (T=900 K). From this point on an isothermal PFR operating at the upper bound on temperature must be used. The shape of the instantaneous yield function at the upper bound on temperature indicates that a PFR is optimal solution. The maximum product yield is achieved at the point where the composition in the last reactor satisfy the following condition: ~8 = 0 (8) Until this point we have treated the problem in a qualitative way to get insights into the features of the kinetic model that extends the solution space. Based on the previous analysis and the reactor structure shown in Fig. 2 we have been able to anticipate the initial guess of the reactor network consisting of 5 non-adiahatieally operated reactors shown in Fig. 3. Gradual distribution of feed along a DSR/PFR reactor would have been impossible to attain in practice and it will be omitted in our further analysis. It is reasonable to delete this detail from the network because it would have been operated at low degree of conversion. Hence, its contribution to the performance of the reactor network would have been negligible. Material and heat balance equations for PFR sectiotts can be written as:
dp
k,(T)r-k2(T)p
dr
k,(T)r+k3(T)Ao r2
....
dt dr
--
=
(9)
fl,k,(T)r + fl2k2(T)P+ fl3k3(T)Ao r2 +2" k(T)l r + k s (T)Ao r2
(1o)
The resulting differential-algebraic optimization problem has been solved with the procedure where optimization is performed in an outer loop with subsequent integration of an ODE model along with sensitivity equations in the inner loop (Cuthrell and Biegler, 1987). Differential equations have been integrated with the variable-step Bulrisch-Stoer method to ensure sufficient accuracy. Optimization has been performed with the use of the SQP method. F2
CSTR
PFR- I
PFR-2
PFR-3
PFR-4
Figure 3. Optimal reactor network structure Results of the optimization are shown in Tab. 3. The optimal CSTR temperature and the feed temperature of PFR-I have taken the value of the lower bound on temperature; exit temperature of PFR-4 approached the upper bound on temperature. It is interesting to note that heat fluxes to PFR-2 and PFR-3 have taken the upper bound value while PFR-4 was slightly cooled to follow the optimal temperature profile. Derivation of these results would have been virtually impossible without the analysis described in the present paper. Table 3. Optimization results for the reactor network in the example problem (superscripts LIB and LB denote that values are at the upper and lower bound, respectively) Feed Exit Reactor r p T (K) r p T (K) tu(W L q) CSTR 1.0000 0.0000 550.0 La 0.6160 0.3232 550.0 La PFR-I 0.6330 0.3366 550.0 La 0.2855 0.6216 585.9 100.0 tra PFR-2 0.2855 0.6216 638.5 0.1471 0.7311 651.8 100.0 t~a PFR-3 0.1471 0.7311 735.6 0.0659 0.7943 743.1 7.174 PFR-4 0.0659 0.7943 893.8 0.0047 0.8376 900.0 t~ -3.963 In the example problem rite problem of yield optimization has been studied. Hence, we have used only a subnetwork that extends the yield space. Optimization of the complete superstructure must be used for the purposes of the reactor network integration into the overall process since operation vs. investment costs relationships often require designs where reactors are operated between the maximum selectivity and the maximum yield. Although the proposed methodology has been described in terms of specified example which is set forth in considerable detail, it should be understood that this is by way of illustration only. Accordingly, any 2- and 3- dimen-
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European Symposium on Computer Aided Process Engineering---6. Part A
sional problems with non-isothermal kinetics models could be solved without departing from the spirit of the methodology described. CONCLUDING REMARKS An algorithmic methodology for innovative design and integration of chemical reactor networks into the overall process, supported by general features of candidates for the optimal reactor network derived from principles of chemical reaction engineering and mixing is presented. The methodology presented can be applied to design problems of integrating chemical reactor networks into the overall process. Arbitrary objective functions can be used since the derived reactor network structures can provide the most comprehensive problem formulation. The proposed methodology is illustrated by an example of the integration of complex chemical reactor network into the overall process. It is well known that proper selection of mixing of fluids provides means of achieving the desired product distribution. General guidelines for mixing of fluids for simple reactions unfortunately do not apply to complex reaction schemes. The reason is that suitable mixing pattern for complex reactions is generally not obvious. Complex reactor networks must often be employed to find a compromise between reaction steps that require different levels of mixing. Selection of proper mixing pattern for complex reaction schemes has traditionally been supported by postulated superstructures and by expert systems using qualitative knowledge. In spite of their strengths these techniques do not create novel design ideas. The principal advantages of the presented methodology are that: • the requirement for an anticipated superstructure and the use of qualitative knowledge for finding the optimal solution are eliminated and • it encompasses the design process by creating novel designs. The use of the proposed method is limited to 2- and 3-dimeusional reaction schemes only. NOTATION A, B, C, D Ao ao.~ Cp Ei A,//i Fi p, r Q q Rg T To t ki Ao /3, p 2" ¥
reaction components of Van de Vusse reaction scheme concentration of the reactor fluid, mol L" Arrhenius constants of the i-th reaction, see units in Tab. 1 average specific heat of the reactor fluid, J g-I K-I activation energy of the i-th reaction, J mol" heat of the i-th reaction, J tool" i-th feed stream dimensionless concentrations of components A and B ( p = cBIAo,r - c^/Ao ) heat flux, W volume flow rate, L s" ideal gass constant (= 8.314 J mol"K "l) temperature of the reactor fluid, K base temperature (= 300 K) dimensionless temperature (=T/To) rate constants for the i-th reaction (= ao.iexp(Ei/RgT), in respective units concentration of the reactor fluid, mol L" dimensionless heat for the respective reactions (-AH~4o/pceTo) density of the reaction fluid, g L" heat flow rate (=~/pcpTo), s"t volume heat flow (=Q/V), W L"1
REFERENCES Aelion V., J. Cagan, G. Powers (1991). Computers Chem. Engng, 15(9), 619-627. Balakrishna S. and L. T. Biegler (1991). I&EC Research, 31, 300-312. Balakrislma S. and L. T. Biegler (1992), lnd. Eng. Chem. Res. 31, 2152-2164. Balakrishna S. and L. T. Biegler (1993). lnd. Eng. Chem. Res., 32, 1372-1382. Biki~ D. and P. Glavi~ (1995). Computers Chem. Engng., 19, Suppl., S161-S166. Cuthrell, J. E. and L. T. Biegler (1985). lnd. Chem. Eng. b]~np. Ser. No 92, 13. Cuthrell, J. E. and L. T. Biegler (1987). AIChEJ. 33, 282. Chitra, S. P., R. Gowind (1985).AIChEJ., 31,2, 185-193. Glaser D., C. Crowe and D. Hildebrandt (1987). l&ECResearch, 26(9), pp. 1803-1810. Hildebrandt D. and Biegler L. T. (1995). AIChE Syrup. Ser., 91, 304, 52-67. Hiidebrandt, D., and D. Glasser (1990a). Chem. Engng. Sci., 45, 2161-2168. Hildebrandt, D., D. Glaser and C. Crowe (1990b). l&ECResearch. 29(1), 49. Kokossis A. C. and C. A. Fioudas (1991). Chem. Engng. Sci., 46, 1361-1383. Van de Vusse, J. G. (1964). Chem. Engng. Sci., 19, 994-997.
Pergamon
INNOVATIVE DESIGNS OF REACTOR NETWORKS FROM REACTION AND MIXING PRINCIPLES DU~KO BIKI~ and PETER GLAVI(~ Faculty of Chemistry and Chemical Engineering PO. Box 224, Smetanova 17, SI-62001 Maribur, Slovenia
Abstract - In the present paper an algoriflnnic procedure for generating innovative reactor network designs for complex reaction schemes is presented. Computational methodology is derived from the basic principles of reaction and mixing. Candidate reactor networks are generated by expanding the solution space into regions where different levels of mixing are required. Directions of extending the design space are identified by using formal optimization theory. A set of candidate reactor networks is generated in a finite number of the solution space extensions. The derived methodology eliminates the requirement for an postulated superstructure and the use of qualitative knowledge for finding the optimal solution. With certain modifications the proposed design procedure can also be used to support the design of batch processes. The generated reactor structures are truly optimal. The proposed methodology is illustrated by an example of integrating chemical reactors into the overall process.
INTRODUCTION Design of reactor networks has been addressed in many papers over past decades. Published research can be classified into two main categories: perfornmnce targeting and integration of chemical reactors into the overall process. A vast majority of papers deal with the problem of generating and selecting among candidate solutions. It is surprising that, except for simple reaction schen~es, there is almost no qualitative knowledge that could effectively support the design task. Several generalized approaches for solving this problem have been suggested over past decades. Notable among those are: targeting strategies (Balakrishna and Biegler; 1991, 1992, 1993), superstructured approach supported with the MINLP formulation (Kokossis and Floudas, 1991), and geometric approach, well known as the Attainable Region (Glaser et at., 1987; Hildebrandt et al., 1990). Recent research 0-Iildebrandt and Biegler, 1994) of the performance targeting supported by geometric interpretations of reaction and mixing promise substantial progress in this important field of process synthesis. PROBLEM STATEMENT In the present paper we focus attention to the problem of design and integration of chemical reactor networks for complex reaction schemes into the overall process when multiple multicomponent feeds with fixed flow rates exist. This is one of the central questions of process synthesis. It is also a realistic design problem because it is conceivable that several multicomponent feeds to the reactor subsystem will exist in practically every flow-sheet. The problem to be addressed in this paper can be stated as follows: given (i) rate equations of a set of chemical reactions, (ii) feed flow rates and (iii) compositions of feedsfind (i) number and types of reactors, (ii) connections among reactors in the network and (iii) locations of feeds along ,~ith feeding scenarios that provide means of attaining the most comprehensive product distribution. It is assumed that: (i) structure of the separation system is known in advance and (ii) only reactions, mixing and heat transfer take place in reactors. Our prima~ task is to find ways for establishing proper topology of the reactor network, contacting fluids of different age and feeding strategies for fresh feed and recycle streams of different compositions. It is important to realize that once the structure of reactor network and locations of feeds along with feeding strategies have been found, a solid connection among the reactor and separation subsystems has been established. At this point the invention of the process structure in the conceptual design phase is completed. The foundation of a good design is set in the conceptual phase of the process synthesis. Should one introduce too many assumptions in this critical design ~'lage, the design space would have been curtailed too soon. Hence, a great deal of sound design ideas would have been
$455
$456
European Symposiumon Computer Aided Process Engineering--6. Part A
lost. To overcome this complication we propose a methodology that efficiently creates novel designs by establishing a set of variables and process structures that dominate the design problem. Bounds on concentration and yield space are not always identical when more than one feed to the reactor subsystem with limited flow rates exist. The reason is that constraints on flow rates and compositions are unconditionally active. Hence, they influence bounds on the overall yield space while they do not affect bounds on the concentration space whatsoever. Furthermore, as it is well known, reactors are usually operated between the maximum selectivity and the maximum desired product yield to achieve the minimum total annualized costs. Therefore, it is crucial to identify bounds on both concentration and yield space to provide means for sufficiently generalized presentation of the problem. In our problem formulation we focus attention to the most comprehensive product distribution space instead of postdating the objective function that should be either minimized or maximized. In this way our approach becomes independent on the objective function. In other words, we focus our interest to extensions of the solution space to identify variables that dominate the design problem in first place and then extract the optimal solution for specific objective function on the basis of the most comprehensive problem formulation. This is non-routine and non-trivial task because: (i) the problem has combinatorially many candidate solutions and (ii) the available qualitative knowledge for simple reaction schemes does not apply for complex reaction schemes. To reveal this problem we have used formal optimization theory to derive general features of candidates for optimal reaction-mixing systems. The derived properties have been used to extend the design space by increasing the degree-of-freedom. Constraints on feed flow rates are unconditionally active, hence they always bound the optimization problem. Expansion of the solution space is used to identify critical variables and then the optimal process layout is selected with the use of conventional optimization techniques. METHOD OUTLINE The basic idea of the proposed method is to identify a set of candidate reactor net~vrks along with design variables that bind the solution space and then to extract the optimal network by optimizing a model that comprises all candidate configurations. In our previous paper (Biki6 and Glavi~, 1995) we have presented a methodology for generating candidate reactor networks for isothermal reactors. In the present paper we extend its use to more complicated cases when reactions take place in nonisothermal systems with external heat sources and sinks. A set of variables that dominate the problem can be created by manipulating the mathematical model (Aelion et al, 1991). The use of this approach renders prohibitive in practice because it would have required extremely complicated mathematical manipulations for every new set of kinetic equations. To avoid this difficulty we have derived general features of candidate variables that extend the solution space to its limits. Geometrical interpretatations of the derived features have been used to support the design task with the simple-to-use step-bystep procedure in which general features of design variables have been visualized for 2- and 3-dimensional kinetic models. By using the proposed method a set of variables that bind the solution space can be derived in a finite number of steps as follows: • •
maximize the instantaneous yield function with respect to temperature, draw lines connecting points of constant instantaneous yield function assuming that temperature could be adjusted to any chosen value, • identify bounds on the optimal temperature profile and conditions in the reaction system that drive temperature into the direction of extremes, • find regions where temperature exceeds constraints on temperature; assume tltat reactor(s) in these regions will operate isothermally; draw these profiles, • use graphical procedure (Biki6 and Glavi~, 1995) to find optimal mixing and feeding strategy,
The reactor network that can be generated with the use of the proposed teclmique provides means for attaining the maximum product yield from several feeds with fixed compositions and flow rates. To ensure that the reactor network can produce also all concentrations, structures generated with the use of the attainable region must be added in parallel. The resulting reactor network is multiple-input-multiple-output system that consists of series-parallel combination of PFRs (plug-flow reactor), CSTRs (continuous stirred-tank reactor) and DSRs (differential sidestream reactor). The derived structures are often unrealistic since they could involve an infinite number of side streams gradually distributed over the reactor length or such temperature profiles that could not be attained in available equipment. These unrealistic process alternatives can be effectively eliminated with constraints on reactor sizes and heat transfer. Once the structure of candidate reactor network Ires been set, optimal solution can be extracted with the use of conventional optimization techniques. We have used the procedure where optimization is performed in an outer loop with subsequent integration of an ODE model in the inner loop (Cuthrell and Biegler, 1985). Despite of certain disadvantages we found this technique reliable when kinetics model is stiff or unstable.
European Symposium on Computer Aided Process Engineering--6. Part A
$457
EXAMPLE
Consider file reactor synthesis problem for tile Van de Vusse reaction scheme (Clfitra and Gowind, 1985; Van de Vusse, 1964): kl k2 A >B )C (1) ks 2A )D (2) in which the desired product is denoted with B and A is the limiting reactant. Reactions in series (1) are of the first order while the parallel reaction (2) is of the second order. Kinetics data are shown in Tab. 1. The reactant is available from two feeds, F~ and F2 with compositions and flow rates as shown in Tab. 2. Table 1. Kinetics data for the Van de Vusse reactions in the example problem . .. E (J mol -I) ao fl 66,300 33,150
1.500 .106(s -~) 1.000 .102 (sl )
0.28 0.36
99,450
4.444 .108 (L tool"l s"l)
0.20
Table 2. Feed compositions and volume flow rates for the example problem c A (tool L -I ) c B (tool L "l) q (L s"l) Ft F2
0.65 1.00
0.35 0.00
100 100
The objective is to maximize an overall product yield in the reactor system consisting of 5 reactors. No assumptions have been made on reactor types and the reactor network topology. Temperature of the reactor fluid must be on the interval Te [550 K, 900 K ]. Uniform heat flow to reactors, g/, is assumed and is limited to interval 2" ~ [-27.78.103,+27.78 •l0 3] where 2" p%ro The instantaneous yield function of the desired product B can be written as follows:
k,(r),'-k2(V)p
es(r,p,r) =
k, (T)r + k3 (r)Aor 2
(3)
Let us now assume that temperature of the reactor fluid can be adjusted to any value in the interval T~ [0, Qo]. Although this is completely unrealistic assumption, our analysis will show very useful remits. Maximization of the instantaneous yield function with respect to the reactor fluid temperature has indicated that temperatut~ concentration profile (see Fig. 1) must be adjusted according to the following equation: T* -
-AE2 R e ln(z/r)
(4)
where:
z = glp+~](gtp) 2 +glP Ig2 gl =
ao,2
g2 =
Aoao.3
(6) ao,l ao.i Substituting Eqns. 4, 5 and 6 into F_,qn. 3 and rearranging gives the instantaneous yield function along the optimal temperature path: -
,
(5)
dp = q~'a(P) = - z + g , p 2 dr z+g2z
(7)
Note that ~ is function o f p only. Solution of the differential Eqn. 7 with the initial value conditions from the region r -- ro, ro~ [0,1]; p -- Po, poe [0,1 ]; 1> p +r represents a PFR trajectory that could have been attained by adjusting temperature to values defined in the Eqn. 4. Note also that the solution to Eqn. 7 is a convex, monotonically increasing function on the interval re[0,1]. This indicates that PFR trajectories hind the solution space when the temperature profile could have been adjusted according to Eqn. 4. By inspecting Eqns. 4 and 7 the following conclusions can be approached: I. ff p --~ 0 then T'--~ 0, II. f i r --~ 0 then T'--~ oo, III. if r --~ 0 then p --~ p ~ , Conclusion I indicates that the lower bound on the temperature of feed stream F2 is unconditionally active. Conclusions II and III indicate that the upper bound on the reactor fluid temperature is also unconditionally active. Hence, beth constraints will affect the design. Lower bound on the temperature of the feed F~ is also active. 20: I.I(A)-P
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Let us now investigate the behaviour of the reaction kinetics at the active constraints on temperature. The instantaneous yield function over the entire concentration space is shown in Fig. 1. The instantaneous yield function in the interior of the feasible region has been calculated from Eqn. 7 while values at bounds have been calculated from Eqn. 3. We have found that there exists a region (curve Mm) where straight lines originating from the feed point Fa are tangential to the instantaneous yield function. It consists of two parts; a curve at the lower bound on temperature and a straight line in the interior of the allowable temperature interval. The reason is that the instantaneous yield functions in the interior of the feasible region are straight lines parallel to the r-axis (F.qn. 7). This implies that mixing of fluids along these lines always satisfies necessary condition for the maximum extension of the yield space since a line is tangent to itself. Feed point F2 has a single, continuous region M2 located completely at the lower bound on temperature. 1
oo
o
~
.....
I
,°0.767
:''k
I?i
"
:: - i ' . -
.:
08-"
""
-0.820
""
..-
~
'
-.o,..
\
\ :; ::
\
:0.91_5~_
./.'i/./... 2.'.- ...
0
\
"
~
0
1 F
Figure 1. Graphical presentation of basic properties of the Van de Vusse reaction in the concentration space. Let us investigate two possible reaction paths (FI~ and FI2) that could be drawn from F2. In both cases reaction paths intercept curve M2 indicating that a CSTR must be used to extend the yield space. Reaction path rim intercepts the region Mj in the interval where it is a straight line indicating that the fluid from feed point F~ must injected into the reactor as an idealized instantaneous pulse to provide optimal feeding strategy. The injection of feed F~ will discontinuously shift the concentration profile in the direction parallel to the r-axis. This will require discontinuous temperature change to provide the optimal processing. Reaction path FI2 intercepts fl~e curved part of the region Mj indicating that a DSR must be utilized to attain the maximum extension of the yield space.
F,
CSTR(r = 550 K)
PFR/DSR(/'= 550 K)
PFR (Noaisothermal)
PFR (T= 950 K)
Figure 2. Idealized reactor network that binds the solution space Therefore, there exist two possible scenarios for mixing of F~. If the composition in the reactor approach to the curved part of M~ then a DSR must be employed; otherwise a fluid from feed point FI must injected into the reactor
European Symposiumon Computer Aided Process Engineering--6. Part A
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as an idealized instantaneous pulse to provide optimal feeding strategy. Movements in the concentration space between regions where CSTR and/or DSR are required must be performed in PFRs. From the above analysis one can derive an idealized reactor network that binds the solution space (Fig. 2). Candidate reactors that have been found in the region at the lower bound on temperature must operate isothermally. As soon as the fluid in the PFR/DSR penetrate into the interior of the feasible region (line corresponding to 1"--550 K) temperature could be increased to follow the optimal reaction path. This path could be followed until the temperature approach the upper bound on temperature (T=900 K). From this point on an isothermal PFR operating at the upper bound on temperature must be used. The shape of the instantaneous yield function at the upper bound on temperature indicates that a PFR is optimal solution. The maximum product yield is achieved at the point where the composition in the last reactor satisfy the following condition: ~8 = 0 (8) Until this point we have treated the problem in a qualitative way to get insights into the features of the kinetic model that extends the solution space. Based on the previous analysis and the reactor structure shown in Fig. 2 we have been able to anticipate the initial guess of the reactor network consisting of 5 non-adiahatieally operated reactors shown in Fig. 3. Gradual distribution of feed along a DSR/PFR reactor would have been impossible to attain in practice and it will be omitted in our further analysis. It is reasonable to delete this detail from the network because it would have been operated at low degree of conversion. Hence, its contribution to the performance of the reactor network would have been negligible. Material and heat balance equations for PFR sectiotts can be written as:
dp
k,(T)r-k2(T)p
dr
k,(T)r+k3(T)Ao r2
....
dt dr
--
=
(9)
fl,k,(T)r + fl2k2(T)P+ fl3k3(T)Ao r2 +2" k(T)l r + k s (T)Ao r2
(1o)
The resulting differential-algebraic optimization problem has been solved with the procedure where optimization is performed in an outer loop with subsequent integration of an ODE model along with sensitivity equations in the inner loop (Cuthrell and Biegler, 1987). Differential equations have been integrated with the variable-step Bulrisch-Stoer method to ensure sufficient accuracy. Optimization has been performed with the use of the SQP method. F2
CSTR
PFR- I
PFR-2
PFR-3
PFR-4
Figure 3. Optimal reactor network structure Results of the optimization are shown in Tab. 3. The optimal CSTR temperature and the feed temperature of PFR-I have taken the value of the lower bound on temperature; exit temperature of PFR-4 approached the upper bound on temperature. It is interesting to note that heat fluxes to PFR-2 and PFR-3 have taken the upper bound value while PFR-4 was slightly cooled to follow the optimal temperature profile. Derivation of these results would have been virtually impossible without the analysis described in the present paper. Table 3. Optimization results for the reactor network in the example problem (superscripts LIB and LB denote that values are at the upper and lower bound, respectively) Feed Exit Reactor r p T (K) r p T (K) tu(W L q) CSTR 1.0000 0.0000 550.0 La 0.6160 0.3232 550.0 La PFR-I 0.6330 0.3366 550.0 La 0.2855 0.6216 585.9 100.0 tra PFR-2 0.2855 0.6216 638.5 0.1471 0.7311 651.8 100.0 t~a PFR-3 0.1471 0.7311 735.6 0.0659 0.7943 743.1 7.174 PFR-4 0.0659 0.7943 893.8 0.0047 0.8376 900.0 t~ -3.963 In the example problem rite problem of yield optimization has been studied. Hence, we have used only a subnetwork that extends the yield space. Optimization of the complete superstructure must be used for the purposes of the reactor network integration into the overall process since operation vs. investment costs relationships often require designs where reactors are operated between the maximum selectivity and the maximum yield. Although the proposed methodology has been described in terms of specified example which is set forth in considerable detail, it should be understood that this is by way of illustration only. Accordingly, any 2- and 3- dimen-
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sional problems with non-isothermal kinetics models could be solved without departing from the spirit of the methodology described. CONCLUDING REMARKS An algorithmic methodology for innovative design and integration of chemical reactor networks into the overall process, supported by general features of candidates for the optimal reactor network derived from principles of chemical reaction engineering and mixing is presented. The methodology presented can be applied to design problems of integrating chemical reactor networks into the overall process. Arbitrary objective functions can be used since the derived reactor network structures can provide the most comprehensive problem formulation. The proposed methodology is illustrated by an example of the integration of complex chemical reactor network into the overall process. It is well known that proper selection of mixing of fluids provides means of achieving the desired product distribution. General guidelines for mixing of fluids for simple reactions unfortunately do not apply to complex reaction schemes. The reason is that suitable mixing pattern for complex reactions is generally not obvious. Complex reactor networks must often be employed to find a compromise between reaction steps that require different levels of mixing. Selection of proper mixing pattern for complex reaction schemes has traditionally been supported by postulated superstructures and by expert systems using qualitative knowledge. In spite of their strengths these techniques do not create novel design ideas. The principal advantages of the presented methodology are that: • the requirement for an anticipated superstructure and the use of qualitative knowledge for finding the optimal solution are eliminated and • it encompasses the design process by creating novel designs. The use of the proposed method is limited to 2- and 3-dimeusional reaction schemes only. NOTATION A, B, C, D Ao ao.~ Cp Ei A,//i Fi p, r Q q Rg T To t ki Ao /3, p 2" ¥
reaction components of Van de Vusse reaction scheme concentration of the reactor fluid, mol L" Arrhenius constants of the i-th reaction, see units in Tab. 1 average specific heat of the reactor fluid, J g-I K-I activation energy of the i-th reaction, J mol" heat of the i-th reaction, J tool" i-th feed stream dimensionless concentrations of components A and B ( p = cBIAo,r - c^/Ao ) heat flux, W volume flow rate, L s" ideal gass constant (= 8.314 J mol"K "l) temperature of the reactor fluid, K base temperature (= 300 K) dimensionless temperature (=T/To) rate constants for the i-th reaction (= ao.iexp(Ei/RgT), in respective units concentration of the reactor fluid, mol L" dimensionless heat for the respective reactions (-AH~4o/pceTo) density of the reaction fluid, g L" heat flow rate (=~/pcpTo), s"t volume heat flow (=Q/V), W L"1
REFERENCES Aelion V., J. Cagan, G. Powers (1991). Computers Chem. Engng, 15(9), 619-627. Balakrishna S. and L. T. Biegler (1991). I&EC Research, 31, 300-312. Balakrislma S. and L. T. Biegler (1992), lnd. Eng. Chem. Res. 31, 2152-2164. Balakrishna S. and L. T. Biegler (1993). lnd. Eng. Chem. Res., 32, 1372-1382. Biki~ D. and P. Glavi~ (1995). Computers Chem. Engng., 19, Suppl., S161-S166. Cuthrell, J. E. and L. T. Biegler (1985). lnd. Chem. Eng. b]~np. Ser. No 92, 13. Cuthrell, J. E. and L. T. Biegler (1987). AIChEJ. 33, 282. Chitra, S. P., R. Gowind (1985).AIChEJ., 31,2, 185-193. Glaser D., C. Crowe and D. Hildebrandt (1987). l&ECResearch, 26(9), pp. 1803-1810. Hildebrandt D. and Biegler L. T. (1995). AIChE Syrup. Ser., 91, 304, 52-67. Hiidebrandt, D., and D. Glasser (1990a). Chem. Engng. Sci., 45, 2161-2168. Hildebrandt, D., D. Glaser and C. Crowe (1990b). l&ECResearch. 29(1), 49. Kokossis A. C. and C. A. Fioudas (1991). Chem. Engng. Sci., 46, 1361-1383. Van de Vusse, J. G. (1964). Chem. Engng. Sci., 19, 994-997.