1136
Ind. Eng. Chem. Res. 1994,33, 1136-1144
The Attainable Region for Segregated, Maximum Mixed, and Other Reactor Models David Glasser,’ Diane Hildebrandt, and Sven Godorr Department of Chemical Engineering, University of the Witwatersrand, Private Bag X3, Wits 2050, Republic of South Africa
It is common practice in process synthesis to choose a reactor structure and use the free parameters associated with it to optimize the system. It is of some importance to assess how effective such an approach is. A geometric (attainable region) approach is used to find the bounds on the performance of reactor systems, and these are compared with the complete attainable region. It is shown for instance that the conversions obtained in segregated and maximimum mixed reactors are not general bounds on conversion. Even two or more environment reactor models need not be sufficiently general t o provide bounds on conversion. Given a proposed reactor system, it is thus possible to compare the achievable conversions with those of the complete attainable region. This provides a useful test t o determine whether significant improvements in reactor performance can be obtained by using alternative reactor structures. Introduction The concept of residence time distribution, or macromixing, was first clearly laid out and analyzed by Dankwerts (1953). Dankwerts showed that the residence time distribution or density function (denoted RTD) was sufficient information to calculate the conversion for firstorder kinetics. This was not true for other kinetics where the conversion would depend on the point-to-point variation in concentration which would not be described by the RTD alone. Dankwerts (1958)later addressed this problem by analyzing the concept of mixing on a molecular scale, or micromixing as we now call it. In this paper he developed the idea of degree of segregation and defined it. The upper limit of this was a completely segregated system while the lower limit depended on the RTD of the system. Zwietering (1959)examined this lower limit and defined a condition of maximum mixedness compatible with the RTD. The two bounds on mixing could be related to the time at which the mixing occurred: the earlier the mixing occurred the less the degree of segregation and the higher the degree of mixedness and, conversely, the later the mixing occurred the higher the degree of segregation and the lower the degree of mixedness. There was at this stage a mistaken belief that the limit of conversion for any arbitrary reactor system lay between that defined by the segregated reactor model (SR) and the maximum mixed reactor model (MMR). Zwietering also determined the flow configurations for the reactor systems which were compatible with the SRand the MMR, and these are shown in Figure 1. Chauhan et al. (1972)considered a single homogeneous, isothermal, constant-density reaction. They showed that, for any arbitrary RTD, the optimum micromixing depended on the convexity property of the rate vector. If the rate expression ( r ) plotted versus reactant concentration ( c ) was concave down (i.e., d2r/dc2> 0), the MMR maximized conversion. Conversely, if the rate of reaction was concave up (i.e., dWdc2 < 0), the conversion was maximized by permitting no micromixing or equivalently using a SR. They had found a very important result in that the convexity of the reaction vector determines the type of mixing which defines the bound on conversion.
* Author to whom all correspondence should be addressed. E-mail address:
[email protected].
Maximum Mixed Reactor
Segregated Reactor
Figure 1. Schematic representation of idealized reactors.
The importance of this result appears not to have been appreciated, possibly because the result was isolated in that it could not be generalized to more than one reaction or to either non-isothermal or non-constant-volume systems. Unfortunately, the part of their results stating that in certain situations the bounds on conversion are defined by the SR and the MMR was noticed and has perhaps been mistakenly generalized by some workers in the field. While some authors have not made this mistake (Shinnar, 1986;Nauman and Buffham, 1983),certainly some of the more recent textbooks on reactor engineering,such as those by Froment and Bischoff (1990)and Fogler (19921,still suggest that the bounds on mixing also represent bounds on conversion. In a slightly different vein, Denbigh (1961)introduced the concepts of instantaneous and overall reaction yields. Simple rules, based on these concepts, were developed in order to predict the flow configurations that would enhance or suppress consecutive or side reactions. These rules are not sufficient to predict optimal flow configurations when one has complex reaction schemes such as the Van de Vusse (1964)kinetics. Because of the earlier belief that the bounds on mixing represented bounds on conversion, it was appealing to use models which are combinations of MMR’s and SRs. These models have been called environment models where the number of environments refers to the number of different reactor structures that are used. Ng and Rippin (1965) for instance used a two-environment model which was a combination of a SR and NMR. Another approach that has been commonly used to try to obtain optimal results from chemical reactors is to
0888-5885/94/ 2633-1l36$04.50/0 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1137 propose some chemical reactor structure and look at the best results that one can achieve using this presupposed structure. In the early days these structures were quite simple (Ng and Rippin, 1965). As computing power has increased, it has become more popular to take some substructure and use repeated units of these to try to obtain more general results (Balakrishna and Biegler, 1991; Kokossis and Floudas, 1990). It is of importance, in the light of this work, to know how successful these approaches have been. Are they "rich" enough in content to give results which are generally applicable, or are there situations in which they do not get answers which are even close to the optimum? More recently in a series of papers Hildebrandt and Glasser (H&G) (Glasser et al., 1987; Hildebrandt et al., 1990; Hildebrandt and Glasser, 1990) have incorporated a geometric interpretation of mixing and reaction with the concept of the attainable region and have been able to use these concepts to determine the bounds on conversion, as well as the optimal flow configuration, for various systems. These ideas will be used to try to put into perspective the previous results and methods. The discussion will be restricted to steady-flow, constantpressure systems. This attainable region is limited by the kinetics of the system and, for kinetics that are thermodynamically consistent, must lie within the "accessible composition space" which is thermodynamically constrained, as defined by Shinnar and Feng (1985). This new geometric approach and some of the results will be briefly outlined below in order to provide background for the concepts that will be developed in this paper. The reader is referred to the previous papers for exact definitions and for a full derivation of these results.
Geometric Interpretation of Reaction and Mixing Consider steady flow systems with one or more input or output streams, where the only processes allowed are reaction and mixing. We will also limit ourselves to reactions which do not exhibit chemical oscillations or unstable equilibria as has been discussed by Feinberg (1980). Consider a space Rn;a point in the space will be denoted C which will represent a possible state of the material a t some point in our reactor system. The elements or variables that comprise C , which we will refer to as the characteristic vector, have the following two properties: (i) At each point C in the space we can assign a reaction uectorR(C),such that if we allow material with properties defined by C to react, the instantaneous change of C , dC, will be dC = R ( C ) d a where a is a positive scalar with units of time. (ii) C obeys a linear mixing law; that is, if we mix two materials with properties defined by C1 and Cp respectively, the resulting mixture, represented by C*, will lie on the straight line between C 1 and Cp. Furthermore, if we mix material described by Cp into material C1, the change in the characteristic vector a t C1 will be in the direction of the mixing vector Cp - C I . We can now examine which variables have the required properties specified above so that they can be elements of C and hence define the space in Rn. Certainly, the mass fractions di, the space time T, and the enthalpy B of a mixture obey these two properties and could thus be used as variables (i.e., axes) of our space. In particular if we
consider a constant-density system, we could use the concentration of species i, ci, rather than the mass fraction as well as residence time t rather than space time T. Using concentrations would simplify the functional form or dependence of R on C . If we furthermore considered a system where the enthalpy of mixing was zero and the ratio of the enthalpy of reaction to the specific heat at constant pressure of the mixture was constant, we would find that the temperature varied linearly with concentration with either adiabaticreaction or mixing. We could then use temperature Trather than enthalpy as a variable which would further simplify the dependence of R on C . The number of variables that we choose to consider in C depends on the problem that we are considering. We can minimize the dimension of the problem so as to include only those variables on which R directly depends plus any other variable(@ that we might wish to optimize with respect to, i.e., those appearing in the objective function. The local change in C due to the two processes that we allow in our reactor, namely reaction and mixing, has the property that it (i) lies in the plane defined by the reaction and mixing vectors and (ii) points in a direction in the acute angle between the mixing and reaction vectors.
Geometry of Idealized Reactors The following geometric properties have been derived in the previous papers. The plug flow reactor (PFR) is defined by dC/da = R ( C )
where at a = ao,C = Co
(1)
The PFR is thus a trajectory in the space such that the reaction vector is tangent to the trajectory at every point. The CSTR is defined by C - C" = R ( C ) (a- a"),
CY
= a",C = C o
(2)
All points on the CSTR locus have the property that the mixing vector, which is defined in this case as the difference between the feed and the exit characteristic vector (C C " ) , is collinear with the reaction vector at the exit conditions. In both of these situations, a 1 a" and monotonically increases along the curve traced out by the reactor.
Definition of the Attainable Region and the Necessary Conditions The attainable region A is defined as the set of all C that can be reached by any possible (physically realizable) steady-state reactor systems from a given feed@)Cf". It is necessary that the attainable region A is such that (i) it is convex, (ii) no reaction vectors on the boundary of A point outward, and (iii) no reaction vectors in the complement of A extend backward into A. The region that we construct that fulfills the necessary conditions cannot be extended by any series-parallel arrangement of PFR's and CSTR's or any reactor with any arrangement of differential reaction and forward mixing. We also can show that any continuous reactor curve that starts inside the region cannot cross the boundary and thus extend the region; consequently the main branch of any reactor that starts inside the region cannot extend the region. As yet we only have this limited sufficiencycondition in that we cannot show the necessary condition prevents a reactor other than a CSTR with feed in A, and which exhibits multiplicity, from having a branch that starts in the complement of A.
1138 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
We note that we can define attainable regions for more restricted situations than that described above. In particular we can define the set of all possible outlet conditions for a specified class of reactors as the attainable region for that class of reactors. This attainable region for the restricted class of reactors should be contained within or at most correpond to the attainable region for all possible reactor systems.
Geometric Interpretation of the Segregated Reactor (SR) The SR is shown schematically in Figure 1. The SR has only been defined for constant-density systems, and we will thus confine our variables to concentration and residence time in this discussion. The exit concentration C* from a SR with a RTD E ( t ) and feed Cfo is given by (3)
where
f=
t E ( t )dt = V/Q
(4)
E @ )dt = 1
(5)
and JOm
CPFR(~) is the value of C from a PFR of residence time t and feed ClotVis the total volume of the reactor, Q is the volumetricflow rate of the feed, and 1 is the mean residence time of the system. Equations 3 and 4, together with the restriction imposed by eq 5, define the convex combination of all possible PFR operating points. The limits of what is achievable by any SR, which we shall call the attainable region of SR, ASR,is thus the bound on C* for all E ( t )that fulfill eq 5. Geometrically we can see that the attainable region of a SR is given by the convex hull of the PFR from the feed point.
Geometric Interpretation of the Maximum Mixed Reactor (MMR) In the MMR we have reaction and addition of (i.e., mixing with) fresh feed occurring simultaneously. (The MMR has been defined only for the situation where there is a single feed which we will denote Cf”.) It is shown schematically in Figure 1. The two processes can be geometrically interpreted as follows: Locally reaction changes the characteristic vector of the mixture in the direction of the reaction vector R, while mixing moves the composition in the direction of the mixing vector Y with the original feed; thus Y = (C Cf”). dC/da = R(C) + q(c~)(C- Cp) = R(C) + ~ ( ( U ) V
(6a)
where q is a positive scalar which depends on a. In the case of a given RTD, q would need to be chosen so as to give rise to this distribution, namely q(4
=
E(a)
jamE(t’)dt’
the intensity function q 2 0
(6b)
If we were considering the attainable region for all possible MMRs, we could consider all possible variations in q. Thus we can see that locally the direction of the
change in the concentration vector must lie in the acute angle between the mixing and reaction vectors. This type of reactor has more degrees of freedom for movement in concentration space than either the CSTR or the PFR and in the limit can represent either of these two. It may be noticed that the boundary condition for the above equation has not been stated. Glasser et al. (1986) showed that a MMR which has a RTD which extends over an infinite time can exhibit multiplicity. The RTD of the MMR can tend to infinity if at the initial point in the MMR both the volume and the volumetric flow rate tend to zero; the concentration could do one of three things as this limit is approached. Firstly, the concentration could tend to the feed concentration, in which case the MMR will not exhibit any multiplicity. Secondly, in the limit the initial point could behave as a CSTR (whichcan exhibit multiplicity) and thus the structure of the MMR in this case would be equivalentto a CSTR followed by differential reaction and mixing. Lastly, the concentration may not tend to a limit but vary periodically as V tends to zero which could also induce multiplicity. This type of behavior, as we have already indicated, may not be included in the construction of the attainable region if the concentrations achieved by these systems with periodic variations lie outside the region that satisfies the necessary conditions. Notice that it is only the initial point in the MMR (Le., as the volume tends to zero) that can behave as a CSTR; thus a CSTR followed by differential reaction and mixing is an example of a MMR while a differential combination of reaction and mixing followed by a CSTR is not. Equation 6 would only apply once we have specified the “initial” condition-such as a CSTR operating point. The behavior and characteristics of systems with the periodically varying boundary conditions are not as yet understood and thus any multiplicity induced by these solutions may not be covered by the subsequent discussion. Although the SR and MMR have only been defined for constant-density systems, the geometric interpretation of the segregated and maximum mixed reactors allows us to generalize the concepts to any general C space in which we can define both a reaction vector and linear mixing laws. We could thus in general define the operation of these reactors geometrically in terms of mass fractions together with space time and enthalpy variables. The implications of the geometric interpretation and the limits on the operation and the achievable outputs for the SR and MMR are best illustrated by means of examples. We will begin with simple examples in order to illustrate the various aspects that have been discussed and will later generalize the results.
Example 1 In this example we consider a constant-density system where the reaction vector depends on two concentrations. The reaction scheme is
x-Y-z rz = -0.5x
+
1 300x2
ry = -rx -
(7)
0.l y 1 + 40xy
We would like to compare the set of all possible outlet concentrations achievable by (i) all possible SRs, (ii) all possible MMRs, and (iii) all possible reactors. The
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1139 0.30--
-PFR
-.- - -
C
PFR f r o m F e e d p o i n t C S T R f r o m Feedpoint Final PFR Mixing L i n e r
f r o m Feedpoint - . Mixing Line
O.2Ot
\.
’..-.*... . . *.
>-
(0
0.00 0.00
-% :\
Q
0.00
0.20
0.40
0.60
0.80
1.00
0.00
0.20
0.40
0.60
X Figure 2. Attainable region of segregated reactor ASRfor example 1.
assumptions made in this example are that the concentrations obey linear mixing laws and the reaction vector depends only on the concentrations of X and Y. We thus define our space as C = (X, Y) where the reaction vector is R = (rz,ry), Let us begin by drawing the PFR trajectory from the feed point as shown in Figure 2. This curve has a concavity in it, and we can fill this in with the straight line BC which is tangent to the PFR trajectory at both B and C. The convex region enclosed by ABCOA is thus the attainable region for the SR. Any point in the region can be obtained by mixing points on the PFR trajectory, and thus the region satisfies eqs 3-5 for all possible residence time density functions E(t). We will call the region the attainable region for the SR, ASR.It is interesting to note that points on the boundary of the region can be achieved by either the PFR trajectory itself or at most by mixing between two unique points (namely B and C or 0 and A) on the PFR trajectory. Thus the RTD’s of SR’s that make up the boundary of the region will correspond to either a single or at most two PFR trajectories in parallel. On the other hand there are infinitely many RTD’s that can produce a point inside the region, although any point inside the region can also be achieved by mixing at most three points along the PFR trajectory. We now examine the behavior of the MMR in two dimensions. We are in particular interested in finding the boundary of the attainable region for MMR’s, AMMR, as this will define the bounds on conversion for any possible MMR. We note that because a line acts as a separatrix between areas there is no possibility that the boundary of the attainable region for MMRs will contain points which were achieved by intermediate amounts of mixing. Thus either the reaction vector or the mixing vector will extend the region the most but never a combination of them since the resultant of any combination will lie in the acute angle between the reaction and mixing vectors. (The CSTR is the special case where the mixing and reaction vectors are collinear.) In order to satisfy the boundary condition for a MMR, it is necessary that a CSTR can only occur as the first reactor (Glasset et al., 1986). Thus the only possibilities we need consider in making up the boundary of the attainable region for MMRs are a combination of a CSTR (first), bypassing of feed, and a PFR where any one or two of these three processescan be absent. The PFR trajectory from the feedpoint is again shown in Figure 3. Bypassing feed material (A) allows compositions along the line EF to be achieved, where the line AEF is tangent to the PFR trajectory at F. The attainable region for a PFR with
0.80
1.00
X Figure 3. Attainable region of maximum mixed reactor A m for example 1. 0.30
-
Optimal C S T R Final PFR
0.20
> 0.10
0.00 0.00
0.20
0.60
0.40
0.80
1.00
X Figure 4. Attainable region A for example 1.
bypass is thus bounded by the PFR trajectory along AE, by the mixing line EF (which fills only part of the concavity in the PFR trajectory), and by the PFR trajectory along FO. We now compare this to what is achievable by a CSTR as the first reactor by drawing the CSTR locus from the feed point as shown in Figure 3. The CSTR locus crosses the PFR trajectory at point J; between A and J the CSTR lies inside the PFR curve but at point J it moves outaide the PFR curve. If we allow bypass around the CSTR from the feed point, we can also achieve line HGA, where the line from the feed point (A) is tangent to the CSTR locus at H. After point H a PFR starting from H will extend the region more than the CSTR, and this PFR trajectory is shown as curve HO. Our attainable region for the MMR, AMMR,is given by the region enclosed by AGHOA. We note that we may obtain any points within the region by mixing fresh feed with the exit concentration of a MMR, i.e., bypassing, and so the whole region is attainable. We also note that this region is not convex, but in order to fill in the concavity around G we would need to mix the output from two MMRs in parallel, which would itself not be a MMR. If we now use the method of H&G we can draw the full attainable region (corresponding to the set of all concentrations achievable by all possible reactors) which we will call A. Without going through the details, this gives us the region enclosed by AKLOA in Figure 4. The boundary is made up of a PFR (corresponding to AK) followed by a CSTR with feed a t K to point L (where the straight line KL is tangent to the CSTR locus (and PFR trajectory) starting at K and is also tangent to the PFR trajectory at L). We may achieve all points on the straight line KL by bypassing feed from K and mixing it with material a t L.
1140 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
ATad is a system constant = U/tP
cp
>0.10--
0.00
0.20
0.40
0.60
0.80
1.00
X Figum 5. Comparison of attainable regions ASR,A m , and A for example 1.
Finally, to complete the boundary of A, there is a PFR with feed L, giving curve LO. Thus for instance to make the most y, we have a PFRCSTR-PFR combinationwhich is neither a SR or a MMR. It can in general be seen that A contains both ASR and AM^ with a common boundary only along the x-axis and a small section of the feed PFR which is shown in Figure 5.
Interestingly there are regions where the SR lies beyond the MMR and vice versa. This example shows clearly that the bounds on mixing do not necessarily provide bounds on the conversion, or in general give optima for any objective functions which are algebraic functions of x and y only. Another point of interest is that by using the attainable region approach we have been able to find all possible outlet concentrations for all possible SRs and MMRs corresponding to all possible RTDs (i.e., E ( t ) or q(cr), respectively) without explicitly looking at the RTD's. We note that the attainable region A for this example is sufficiently simple that it could be achieved by a twoenvironment model such as that used by Ng and Rippin (1965).
Example 2 Let us now suppose that we have a single fluid-phase reaction with no volume change on reaction where the rate of formation of the single reactant is given by
C is the concentration of reactant and the kj are reaction rate constants of the Arrhenius form: k j = koi exp(-EilR7')
(10)
where koi are the preexponential factors, Ei are the activation energies, Tis the absolute temperature, and R is the gas constant. We will further assume that the reaction is adiabatic and the heat capacity and heats of reaction are constant so that we can write
T = T"+ATad(l-y)
(11)
where To is the feed temperature to the reactor, y is the normalized concentration = C/Cfo
(12)
(13)
AH is the enthalpy of reaction, is the heat capacity of the mixture, and Cfois the feed concentration. Thevalues of the constants used are given in Table 1. With the assumptions that are made in this example, concentration and residence time follow linear mixing laws and the reaction vector depends only on concentration. Consequently we can reduce the dimension of the space to R2 where C = (y,t) and R = (r(y,T(y)),l) as the temperature is uniquely related toy via the energy balance, eq 11. We can use the necessary conditions to draw the attainable region in the y-t space, where t is defined as the volume of the reactor ( V, divided by the volumetric flow rate (Q)of the feed. This is shown in Figure 6. The boundary of the attainable region, given by ABCDCT represents the minimum residence time versus rona tration relationship for all possible reactors, that L,for any possible combination of reaction and mixing and t h w in general. It can be seen that the reactor sequence which makes up the boundary of the attainable regior is gi ren by PFR (AB)-CSTR (BC)-PFR (CD)-CSTR (DE)-PFR (EF). Also on the same diagram are drawn the PFR and CSTR from the feed point. Note that the concavities between B and C and D and E respectively are filled in by straight lines and are achieved by bypassing material from B and mixing it with C and the same with D and E. Let us now examine what can be achieved by a SR and a MMR. We will as before denote the attainable region for all possible reactors as A, the attainable region for all possible SR's as ASR,and that for all possible MMR's as A m . The convex hull of the PFR from the feed point A defines ASR. The boundaries of A and ASR coincide along AB of Figure 6; after point B,ASRlies inside A. Thus the SR only defines the bounds on conversion from the feed concentration (Le., y = 1 at point A) up to the concentration at point B. Concentrations past point B can be achieved in a smaller volume of reactor (Le. smaller residence time) than that of the best SR by using the optimal reactor structure defining the boundary of A. If we refer to our example, we can see that the boundary of the attainable region for MMRs, shown on a blown-up curve in Figure 7 is (i) a plug flow reactor along ABG, (ii) a CSTR with feed A operating at H with bypass to give line GH, and (iii) a plug flow reactor with feed H operating along HI. Note that curve ABC on Figure 7 corresponds to curve ABC on Figure 6. Point H is defined as the point on the CSTR locus where the line from the feed point (A) is tangent to the locus and G is the point where line AH intersects the feed PFR trajectory. Again we see that the boundaries of AMm and A (and thus ASR) coincide along AB. Thereafter the boundary of AM.MR lies inside A and thus the volume of the best MMR required to achieve concentrations past B is larger than that of the optimal reactor structure which defines the boundary of A. We have compared the attainable regions for firstly all possible reactors A, secondly for all possible SRs ASR, and lastly for all possible MMR's AM^ and have shown that the ASRand A m are strictly subsets of A. We have thus also shown that, for some RTD, the bounds on micromixing do not in general define the bounds on conversion. We have furthermore shown that we can determine the optimal reactor structure and thus the optimal RTD (macromixing) as well as micromixing for the system. We can also from this geometric approach determine the RTD of the optimal SR and MMR as we can determine both the reactor structure and the operating points for these reactors.
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1141 Table 1. Constants for Example 2 Rate Data koi 6 EiIR koz 15.1 EdR koa 0.55 EsIR Energy Balance Constraints
To
290
Td
. .,
I 1
0
I
I
t
1
I (
1
8000 18 000 0
2000
It is again clear from this examplethat the SR and MMR do not represent bounds on the conversion. One might again ask about the two (or more) environment models such as those of Ng and Rippin (1965). This twoenvironment model allows one for instance to use a SR followed by a MMR. This allows us for example to achieve the union of ASRand A m , and in particular allows us to use a PFR-CSTR-PFR combination. We see that we can extend the region by using material at point B (achieved by SR) as feed to a CSTR operatingat C followed by a PFR (which is a MMR). Thus the two-environment model has extended the attainable region in that we can achieve the section of boundary ABCD; but the model has not found the complete boundary of the attainable region. In order to find the complete attainable region in this example, one would need to have a three-environment model that is a SR followed by two MMRs. For this particular example we could in fact have obtained the same result using the method described in Levenspiel (1972) where one plots l l r vs y as shown in Figure 8. This method is however not general as the assumption inherent in the approach of Levenspiel is that only series combinations of P F R s and CSTRs are considered and the method does not appear to beadaptable to more complex reactors or mixing schemes. Furthermore, it has not been extended to situations where the rate is a function of two concentrations. It is the strength of the attainable region approach that it can handle all of these situations. It should also be made clear that in obtaining the attainable region no assumptions have been made about any particular optimization problem. In fact once attainable region A is known (or ASRand AM^), one can in principle solve any optimization problem with an objective function that is an algebraic function of the variables of the attainable region. The Levenspiel approach is limited to finding, for a single reaction, the series arrangement of CSTR's and P F R s that minimize the total volume of the reactor system for a given exit concentration or equivalently the maximum conversion for a given volume of reactor.
1.00
-A 0.50
0.00
1.00
Concentration Figure 6. Attainable region for example 2.
CSTR f r o m Fecdpoint Optimal CSTR
1.00-al
u
al 0 . 5 0 - -
a
0.25--
0.80
0.60
0.40
1.00
Concentration
Figure 7. Enlargement of amall section of attainable region from Figure 6.
1
PFR 3
Generalizations from the Geometry of the Segregated Reactor (SR) With the understanding that we can gain from the geometrical interpretation of the SR, we can say the following in two-dimensional situations: (i) When the PFR trajectory is convex, the reactor itself is the boundary of the convex hull and thus is the bound on what is achievable by a SR. This will also in general be the bound on what is achievable, unless the system exhibits multiple steady states. We can see this using the following argument. The boundary of the convex hull of the PFR is convex and has no reaction vectors pointing outward. Thus unless the CSTR or any other reactor can exhibit multiple steady states and thus "jump" to a point outside the convex hull of the PFR, the region cannot be extended. This jump would occur if reaction vectors in the complement of the convex hull of the PFR could be extrapolated backward into the region. These points could then be achieved by
0.00
0.20
0.40
0.60
0.so
1.00
Concentration Figure 8. Levenspiel plot for example 2.
a CSTR starting from inside the convex region and the attainable region would thus include such points. (ii) When the PFR has concavities, the boundary of the convex hull of the PFR trajectory will consist of convex sections of the trajectory and straight lines between the convex sections. Thus the SR that in this case defines the bound on conversion of what is achievable by all possible
1142 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
SRs will be either a single PFR (alongthe convex sections) or two P F R s in parallel (on the straight-line sections of the boundary). Thus in two dimensions the structure of the optimal SR is always very simple, in that at most one side stream is required, and not the multitude usually shown in Figure 1. Furthermore we can see that when the PFR has concavities the SR will not in general define the bounds on conversion. Consider the end points C1 (coresponding 1 71 and C2 (corresponding to 7 2 > 71) of a line that lies in the boundary of the convex hull of the PFR, Le., tangent to the PFR curve at (CI, 71) and (C2, 72). Let us also consider the case when 7 2 is finite or in other words C2 is not an equilibrium point. We know then that a CSTR locus starting from C1 must pass through point C2, as the reaction vector at point 2 is collinear with the line through points 1and 2, and thus in general must be able to achieve points that the SR could not. We can furthermore say the following about the operating points of the two PFRs. In general a line that fills in a concavity on a curve must be tangent to a curve if the curve is continuous where the line touches it. We have thus from geometrical considerations found some very powerful results for the optimal segregated reactor structure in two-dimensional space. These geometric ideas can be extended to higher dimensional spaces. The PFR is still a curve in the space, and thus in higher dimensional spaces does not act as a separatrix between regions as it does in two dimensions. The convex hull of the trajectory will contain convex sections of the curve, lines, and planes (or hyperplanes in higher dimensional spaces). The reaction vectors at vertices (other than those correspondingto the end points of the curve) of the planes must all be coplanar. We can thus generalize the optimal SR structure: in n-dimensional space the SR that defines the bound of conversion for all possible SRs requires at most ( n - 1) side streams corresponding to the vertices of the hyperplane defining the convex hull of the PFR; equivalently, the optimal SR requires at most n parallel PFRs. The reaction vectors at the exit of the PFR’s of finite, but nonzero volume, must be coplanar to the plane defined by the operating points of the PFRs. If the rather unusual situation arises that more than n points of the PFR (and of course the associated reaction vectors) lie in the plane, the reactor structure could incorporate these points as extra PFR’s in parallel but the same result could be achieved with only n P F R s in parallel. This is the only situation in which using more than n P F R s in parallel will not actually to detrimental to the performance of the SR.
Generalizations of the Geometry of the Maximum Mixed Reactor (MMR) In two dimensions we can see that moving in the direction between the reaction and mixing vectors would do worse than either moving in the direction of the reaction vector alone or moving to points where the reaction and mixing vector are collinear. Thus the most general structure for the optimal MMR would be a CSTR-PFR with bypass. In higher dimensions this is no longer true as the reaction and mixing vectors at a point do not act as separatrixes in the space. We for example know that in three dimensions the most general structure for the optimal MMR would be a CSTR-DSR (with addition of fresh feed)-PFR with bypass, where DSR refers to a reactor where differential reaction and mixing (in this case with feed material) is occurring. The mixing (in other words
the addition of the side stream) would be done such that the reactor curve for the differential part of the process would lie in the surface defined by Y, R, and vVRcoplanar (Hildebrandt and Glasser, 1990; Glasser et al., 1992. We do not understand the underlying geometry of attainable region enough, however, to generalize the behavior of the optimal MMR in higher dimensional spaces.
Discussion There has been a large amount of literature associated with finding the best results that one can get for a given system of chemical reaction with given kinetics. The work in the past has generally suffered from the problem that there was no general model for a chemical reactor. In order to overcome this problem, many workers have proposed structures which they hoped were general enough to obtain results which were not significantly different from the true optima. Such a general chemical reactor model (Glasser and Jackson, 1984; Jackson and Glasser, 1986)has in fact been derived, but it is of such a complex nature that its use for optimization remains an unsolved problem. An entirely new approach using geometric concepts applied to the processes of mixing and reaction has been used to generate the boundary of the attainable region and in particular togenerate the optimal reactor structures. In the light of these results it is important to reevaluate the earlier work and to see how effective the previous results have been. In order to do this, we have used the geometric approach to generate the boundaries of the attainable regions for MMRs and SR’s and compared the results with those of the full attainable region. It is of interest to note that these two former regions were obtained with no reference to the residencetime distributions of each of these reactors. It is also of importanceto note that we only need to generate the boundaries of these regions, which means that we are able to find the regions using geometric methods without the need for large computers and intensive computing. It is shown from these relatively simple counterexamples that there are indeed cases of interest where bounds on mixing do not provide bounds on conversion. What is also evident from these examples is that even such models as two or higher environment models can have difficulty in matching the optimal reactor structures for quite simple reaction systems, let alone those that arise for multiple reaction schemes, and hence could give poor approximations to the optimal results. While in two-dimensional examples we have shown that one will only find PFR’s and CSTR’s with bypass making up the attainable region boundary, this is not so in higher dimensions. For instance, an example of a single exothermic reversible reaction in three dimensions has generated an attainable region whose boundary is given by a CSTRPFR-DSR (with mixing of fresh feed)-PFR (Glasser et al., 1992). This makes it clear that the optimal reactor structures are not necessarily simple structures one can predict in advance. This suggests that there might be situations where the current methods could have structures which are not “rich” enough to give results which are close to the true optima. While we are not able to predict the optimal structures in general, we are in a position to make some recommendations on the basis of these and other results which we have obtained from looking at the geometry of the attainable region. Because of the convexity of the attainable region, one tends to be dealing with plug flow curves, straight lines
Ind. Eng. Chem. Res., Vol. 33, No. 5,1994 1143 from boundary points, and hyperplanes filling concavities. Without going through a complicated discussion, one can infer from this that the optimal structure is likely to be made up of n parallel reactor branches (where n is the dimensionality of the problem) with interchange between these branches. Each of the subunits in this system is likely to be a PFR, CSTR, or DSR. Now, as the DSR in the limit can in principle go to either the CSTR or the PFR, one need in practice only consider a series parallel arrangement of DSRs. There might be some technical problems in the implementation of a calculation scheme based on this, as the CSTR limit of the DSR may not be straightforward to program. In particular, where appropriate, one would need to make allowance for the multiple solutions of the CSTR at the beginning of each DSR. Also it might be convenient to break up each DSR into a finite number of side feed points rather than a continuous feed. More work on the attainable region approach may well lead us to be able to generate more accurately the class of optimal structures such that one could then with some confidence use the older approaches to solve particular problems. This might in the long run be the best approach, as generating attainable regions in high dimensions might also become very computer intensive. Conclusions It has been shown that, for a given set of kinetics, it is possible to construct the attainable regions for specified reactor networks and compare these to the full attainable region. The performance of a proposed structure can therefore be compared to that of the globally optimal reactor system. This will assist the reaction engineer in assessing whether modifications to a design will significantly improve conversions. In particular the idealized reactors that represent the upper and lower bounds on micromixing (the MMR and SR, respectively) are shown to be entirely contained within the full attainable region and do not necessarily correspond to the bounds on achievableconversions. Constructing the attainable region does not rely on an assumed reactor structure but, by considering the processes of reaction and mixing, effectively includes all possible steady-state reactor configurations. The optimal structure may be determined from the reactors that make up the boundary of the full attainable region, and this structure will in general not consist of simple combinations of SRs and MMR's. Nomenclature CSTR = continuously stirred tank reactor DSR = differential side stream reactor, Le., one in which differential reaction and mixing are occurring simultaneously MMR = maximum mixed reactor PFR = plug flow reactor RTD = residence time distribution or density function SR = segregated reactor A = the attainable region for all physically realizable reactors A m = the attainable region for all MMRs ASR= the attainable region for all SRs AVB = change in B in the direction of A,ith component of AVB = (Aj(aBi/aCj)} C = characteristic vector Cf"= characteristic vector of the feed for the attainable region C" = characteristic vector of the feed to a reactor
c, = specific heat capacity of mixture c = concentration d = mass fraction E = residence time density function Ej = activation energy of species i ki = reaction rate constant of species i ko = preexponential factor in reaction rate expression Q = volumetric flow rate q = rate of addition of side stream = dQIda R = reaction vector r = rate expression t = residence time T = temperature AT,, = adiabatic temperature rise, defined by eq 11and 13 V = volume of reactor x , y = normalized concentrations a = positive scalar 8 = specific enthalpy Y
T
= mixing vector = (C- Cf") = space time
Subscripts PFR = a value for a plug flow reactor MMR = a value for a maximum mixed reactor SR = a value for a segregated reactor 1, 2 = specific points or variables i = species i Superscripts
- = mean value (sometimes not used if meaning is Clem)
* = a point generated by mixing " = an initial point
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Received for review June 1, 1993 Revised manuscript received December 8, 1993 Accepted February 17, 1994. ~~
* Abstract published in Advance ACS Abstracts, April 1,1994.
