Ind. Eng. Chem. Res. 1990,29, 49-58
49
PROCESS ENGINEERING AND DESIGN Geometry of the Attainable Region Generated by Reaction and Mixing: With and without Constraints Diane Hildebrandt and David Glasser* Department of Chemical Engineering, University of the Witwatersrand, Johannesburg WITS 2050, South Africa
Cameron M. Crowe Department of Chemical Engineering, McMaster University, Hamilton, Canada L8S 4L7 Given a system of reactions with known reaction kinetics and a known feed, it is of interest t o find all the possible outlet conditions (the attainable region) for an arbitrary system of steady-flow reactors with any total volume. A geometric approach was used to derive a set of necessary conditions as well as a limited, but powerful, sufficiency condition for the attainable region. These results extend those previously obtained for isothermal, constant-density systems in concentration space t o include adiabatic and variable-density systems in composition-volume space. It is also shown how the method can be used t o find the attainable region for systems with constraints. In a previous paper (Glasser et al., 1987) (called GCH), the problem of finding the best steady-flow (one that will not support sustained oscillations) system of chemical reactors, given a set of reactions with its associated kinetics, was investigated. The system was constrained to be isothermal with no density changes on reaction or mixing, and the region examined was confined to the concentration space. The problem was solved by the use of geometric techniques to find the attainable region, that is, the region in the stoichiometric subspace that can be reached by any possible reactor system. Horn (1964) showed that if one could find the attainable region for a system then, provided the objective function was a simple function of only the basis variables, the problem of the optimization was relatively straightforward. There has been much previous work in the field of optimization of such systems. The method to minimize the total residence time for a single reaction occurring in a reactor system comprised of CSTR’s and plug flow and recycle reactors in series is, for instance, well-known. The minimum residence time is found by minimizing the area, representing the residence time of the reactors, on a reciprocal of the rate versus conversion graph. This usually involves a graphical search in order to determine the best configuration, and there is no simple way of determining if the configuration is globally optimal (Levenspiel, 1972). Chitra and Govind (1985a,b)give an extensive summary of previous work done in optimizing isothermal and nonisothermal reactor systems and look a t the optimization of serial systems of reactors. Achenie and Biegler (1986) consider a general structure consisting of constant dispersion model reactors with sinks and sources and splitting points. If their calculation procedure succeeds, the optimum can be found for the given structure, but they give no way of determining the optimum structure or indeed of determining if the optimal reactor system can be described by networks of constant dispersion model reactors. Konoki (1957) and Horn (1961) developed algebraic expressions for interstage cooling. Caha et al. (1973) de0888-5885/90/2629-0049$02.50/0
veloped a numerical technique to solve the equations to determine the optimal configuration of a three-stage reactor with interstage cooling. Konoki (1960) derived the equations to minimize the reactor volume of a cold shot cooling reactor. Malengi! and Vincent (1972) developed the general equations to maximize the profit of the reactor, where the profit is considered to be a function of the heat supplied and the amount of catalyst in the system. If the cost of heat is assumed negligible, the objective function simplifies to minimizing the amount of catalyst or equivalently the volume of the reactor system. The authors show how to use the criteria to numerically find the optimal configuration of a threestage cold shot cooling reactor. The technique cannot be used in situations where the specific heat or enthalpies of reaction are functions of temperature.
Problem Statement In this paper, we examine the problem of finding all possible exit conditions (the attainable region) for a system of reactions with known reaction kinetics taking place in a steady-flow system of reactors. The only processes permitted are chemical reaction and mixing. The importance of this approach is that no assumptions concerning the structure of the network of reactors are made, but this comes directly out of the construction of the attainable region. In a previous paper (GCH), this same problem was investigated, but it was limited to isothermal systems with no density changes on reaction or mixing, and only concentrations were considered as variables. The space time (volume) of the system was not considered. In this paper, we will allow for both density changes and constantpressure adiabatic systems and include space time (volume) as a variable. Furthermore, we will look at how the approach can be used on systems on which other constraints are also placed. The relaxation of the previous limitations and the ability to handle constraints will significantly increase the range 0 1990 American Chemical Society
50 Ind. Eng. Chem. Res., Vol. 29, No. 1. 1990
of problems of interest that can be solved. As before, the approach will be geometric, and we assume that the steady-flow svstems we examine do not allow sustained oscillations. Suppose we have a system of n species, A , , ...,A,,, with a specified set of r reactions
where the rate of formation of species A, is given by rr, which depends on the concentrations of the species c, and the temperature T. In order to simplify the notation, we will write these arrays of quantities as vectors; thus,
r = ( r l , r,, ..., r n )
Let us now join all of these quantities into a single array, which we call the characteristic vector C, which is a vector with n + 2 elements; i.e.,
C=
(4,4,..., d,,
T , 0)
(5)
These elements of the vector all obey the following mixing rule; that is, if we have two streams with mass flow rates M , and M, and characteristic vectors C1 and Cz, respectively, and mix them, then after mixing we have C* where 05aI 1 cy' = ac, + (1-a)C, (64
c*= a(C, - C,) + c,
(6b)
where
M1 MI + M2 In particular, this is true since CY=
c = (e1, epr.... t,,) In particular, r = r ( c, T 1. Now let us suppose we are given a specified mass flow M of matejial with reference concentration c O, enthalpy per mass H o , density po, and temperature T o (for instance, the feed conditions). It is from this initial material that we wish to determine what final conditions we can achieve in arbitrary systems of reactors of total volume V using only the processes of reaction and mixing. In order to relate the results to the previous ones (GCH), we will find it convenient to define new variables with units corresponding to our previous ones of concentration, time, and temperature, respectively. The new quantities are defined, unlike the previous quantities, so that they all obey linear mixing laws. This point will be clarified later. Let the mass fraction of species i in a mixture be w,and its molecular mass be m,; then,
(7a)
and
To finalize our description, we note that the rate of formation vector can also be expanded to give us what we can call a reaction vector, r', such that rYc,T) = (rl(c,T),rz(c,T),..., r,,(c,T),1, 0) (84 or
VPO M
(3)
e=H / q J
(44
T =
and
where C : is the heat capacity at constant pressure per unit mass a t the reference conditions. If the density of the mixture does not change with reaction and mixing, then p = p O , di and T become the true concentrations and mean residence time of the system, respectively, and the results reduce to those of GCH. The enthalpy of a mixture at constant pressure is a function of the temperature ( T )and composition and can be written as
where Pfiis the specific enthalpy (heat) per unit masspf formation of species i at the reference conditions and AH, is the specific enthalpy (heat) of mixing. If the heat capacity of the mixture does-not change with reaction, mixing, and temperature, then Cpo= Cp and the above equation becomes
In this case, the quantity 8 is thus the change in temperature of a system of constant composition.
(8b) R(C) = (Ri(C),R2(C), ..., R,tC), 1, 0) where rr = R, for a given state of the system. Note, however, that r, and R, are different functions, as they have different independent variables. The relationships between the variables c , t , and T and the new variable Care P
c, = 4,
i
= 1, ..., n
(8c)
Po P
t = -r PO
where and
We have now set up the basis for the system we wish to discuss. We note that if we have an isothermal system we merely decrease the size of our characteristic vector by leaving out the final element (e),as the system will then be a function of composition ( d i ) and space time (7)only. As discussed, we limit ourselves to two processes which can alter our characteristic vector (C);these are mixing and reaction. Mixing C1 with Cz is characterized by a vector given by eq 6a and is in the direction of C, - C1, while reaction of C, is characterized by the vector R ( C ! ) . Provided there are no discontinuities, our characteristic
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 51 vector ( C , )must change locally in a direction given by the resultant vector: 0I p I1 (9) g = PR(C1)+ (1 - C,)
(3) The recycle reactor is a plug flow reactor, but some of the feed is recycled to the inlet. It is defined by the following equations:
Definition of the Attainable Region We define the attainable region as the region in the space of the characteristic vector ( C )that can be reached by any possible reactor system from a given feed. Now the equilibrium point(s) is never strictly attained, as it is only reached as a limit as 7 tends to infinity. In order to overcome this difficulty, we will expand the above definition to include all the limit points (that is boundaries) as well. This makes the attainable region a closed set and does away with some of the difficulties in handling the existence of open sets. Thus, we include the limit of 7 as 7 in the set as well as the equilibrium concentrations and enthalpy. Furthermore, if we mix any finite point in the space with a sequence of points which tend to the equilibrium point(s), we obtain a line that tends to a “vertical” (i.e., parallel to the time axis) line. This “vertical” line is strictly not attainable, but again we allow it as a limiting process. If these “vertical” lines form part of the boundary, then we have points on the boundary that represent the same concentrations with different times, which is clearly not possible, but we will still accept these “vertical” lines as the boundary of the region, thus closing the set.
-d = C-
p)(c,
-
Existence and Uniqueness of the Attainable Region We now define the base of the attainable region as the closure of the convex hull of the feed point(s) and the equilibrium point(s). We note that as a result of our closure of the set these points are all attainable. (We do not allow sustained oscillations.) The result is that the base is attainable, and since it is nonempty and attainable, the existence of the attainable region is assured. Once the difficulty of the limit point(s) has been cleared up, the proof of the uniqueness follows as derived in GCH. This is because infinities exist only in the time domain while the concentration and enthalpy space are bounded. We have, however, included the limits as 7 m, so we conclude that the attainable region is a unique closed region that is also compact and simply connected. An adiabatic system with a single feed has a fixed enthalpy. The attainable region for this adiabatic system will consequently be in a subspace of the total space, even though the temperature of the system may vary. -+
Geometry of Some Idealized Reactors Assume that the reference conditions are the feed conditions. (1) The plug flow reactor has as its defining equation dC/d.r = R ( C )
C(r=O) = C o
(10)
The plug flow curve is thus a trajectory in the space such that the reaction vector is tangent to the curve a t each point. These trajectories are uniquely determined by their initial points on the boundary of the region and cannot cross each other. The trajectories, starting from different initial times but with the other initial values the same, are just the same curves but shifted up or down the time axis. (2) The CSTR has the defining equation C‘ - co = R(Ce)(7- 7 0 ) C(7O)= co (11) The CSTR has the property that the vector defined as the difference between the feed and exit concentrations (Ce) is collinear with the reaction vector at the exit conditions.
d$
R(C) R + l
C(+=O) = C*; C(+=7-7O) = C‘ (12a)
c* = RC‘ + Co
R + l where R is the recycle ratio, that is, the mass flow rate ratio of the recycle flow rate to the feed rate; Ceis the exit concentration of the recycle reactor and therefore the plug flow reactor; and 7 - 7 O corresponds to the space time of the recycle reactor. The recycle reactor is a plug flow reactor with a feed at a weighted average between the exit concentration (Ce)and the given feed (CO). The variable $ is an independent, scalar quantity related to the space time of the fluid in the plug flow section of the reactor. Necessary Conditions for Attainable Region The same necessary conditions for the attainable region apply in concentration-volume space as in concentration space. The conditions will be listed here and a full discussion is given in GCH. It is necessary that the attainable region A on its base with feed concentration Cois such that (a) it is convex; (b) no reaction vector in the boundary of A (dA) points outward from A, that is, the reaction vectors in dA point inward, are tangent to dA, or are zero; (c) there is no plug flow trajectory within the stoichiometric subspace in the complement of A , which has two points such that the line joining the later point to the earlier point can be extended to intersect dA (or A ) ; and (d) no negative of a reaction vector in the complement of A , when extended, can intersect a point of dA (or A ) . Each of these necessary conditions corresponds to operations associated with mixing or to one of the reactors we have examined in the previous section. At present, there is only a necessary condition for the attainable region. In order to complete the analysis, one would need a full sufficiency condition or show that a region satisfying the necessary condition is unique. We will again construct the attainable region using points that we know are attainable. In this way, even though we will not have ensured that we have the full atainable region itself, the region we will have obtained that satisfies the necessary condition will be attainable. The attainable region that we have constructed in this way cannot be extended by the processes of mixing, a plug flow reactor, a recycle reactor, or a CSTR, and if a recycle reactor exit condition is on the boundary of the region, we can always attain this condition from a plug flow reactor starting in the region. The above sufficiency result leaves open the question of whether we could expand the attainable region by any other processes. It is easy to see that any differential process involving reaction and mixing cannot be used, as the results of eq 9 preclude this. This, however, does not exclude other more complex reactors where a concentration jump, caused by the existence of multiple steady states, can occur. Such cases could be the axial mixing reactor, a series of simple reactors with complex recycles. In all of these reactors, there may be jumps in the concentration-time space such as occur in the recycle reactor and the CSTR, and we are not able to be sure that we could not use such a reactor to extend the region we have constructed by using conditions a-d. It would seem that the only deficiency in the above sufficiency condition relates to the number of multiplicities
52 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
of solutions for reactors that have discontinuities. That is, are there reactors that have more solutions for the given kinetics than the CSTR or the recycle reactor? If this is not possible, then the above sufficiency condition would appear to be complete in that the solutions for the other reactors should map into those of the CSTR and recycle reactors via differential processes, and this is already covered via the sufficiency result.
Construction of the Attainable Region In order to find the attainable region, we must find a region that both satisfies the necessary condition and that is indeed attainable. We therefore use a construction method, in which we start with the feed point(s) and use the processes of reaction and mixing to find an attainable region that satisfies the necessary condition. As discussed above, we may be able to extend the region by using complex reactors that exhibit multiple steady states, but the region that we find by the construction method is at least attainable. The construction method will vary depending on the exact nature of the problem and the imposed constraints, but it will always more or less follow the following sequence: Construct the plug flow trajectory(s) from the feed point(s). Find the convex hull of the trajectory(s), which is equivalent to finding all the points that are achievable by mixing all the possible outlet materials of the plug flow reactor(s) in all combinations. Check whether any reaction vectors point outward on the boundary of the hull. If reaction vectors point outward, then find the feed point(s) and, if unconstrained, the family of plug flow reactors or CSTRs that will extend the region the most. Be sure to include all possible branches of the reactor curve if applicable. Usually one alternates between plug flow reactors and CSTRs. Find the new convex hull and repeat step 2 . If no reaction vectors point outward, check whether necessary conditions c and d are met. This is to check whether all the branches of the CSTR's and recycle reactors have been included. If the conditions are not met, extend the region by using the appropriate reactor which is exhibiting the multiple steady states, find the new convex hull, and repeat from step 2. If the necessary condition is met, we have a region that satisfies all the conditions and that is achievable. This region is then a candidate for the attainable region. Thus, by following this method, we can construct an attainable region for any set of reactions. The construction method is particularly easy to apply in two dimensions, but the ideas can be used to find the attainable region in any number of dimensions. This aspect is presently being examined for three-dimensional examples and will require the multidimensional convex hull theory to obtain results in higher dimensions. Examples In this paper, we will limit ourselves to examples that we can draw in two dimensions. We will be able to demonstrate many of the facets associated with the geometric approach, but it must be realized that there are certain results that are valid for two dimensions that are not true in three or higher dimensions. This is mainly because a line acts as a separatrix in two dimensions but not in higher dimensions. Example 1. In our previous paper, we looked at the Trambouze example with constant density. The kinetics are now modified to allow for the effect of density changes on the construction of the attainable region. The reaction
.
C 1
j .I
'-
'
Y
05
0 0
0.2
0.4
0.6
X
Figure 1. Trambouze example modified for nonconstant density system. (--) Plug flow from feed A. (---) CSTR from feed A. (- -- -) Plug flow from feed B. (--) Tangent lines.
is a parallel, irreversible decomposition of A to form B (zero-order reaction with rate constant 0.025 mol/ (L min)), C (first-order reaction with rate constant O.B/min based on the rate of the reaction of A), and D (second-order reaction with rate constant 0.4 L/(mol min)). The reaction rates are based on the following scheme: A-B, A-2C, A-D (13) We have modified the reaction forming C so that the number of moles is no longer conserved. We will assume the reaction occurs at constant temperature and pressure in the gas phase and that the reaction mixture behaves ideally. We wish to optimize the selectivity of A to C. Using these kinetics and objective function, we need only to work in two dimensions, namely, dA and dc. We will normalize our variables such that x = dA/dAo and y = dc/(2dAo)where dAo refers to pure A and is taken as 1 mol/L. The construction of the attainable region is essentially the same as that of the constant-density example in GCH and is shown in Figure 1. The region bounded by the straight line AB, the plug flow trajectory from B (BC),the y axis between the plug flow equilibrium (C) and (O,O), and the x axis satisfies the conditions for the attainable region. If we define selectivity as dc/(dAo- d A ) ,we find that we maximize the selectivity along line AB; i.e., an infinite number of reactors could give rise to this maximum. All points on line AB could be achieved by using a CSTR from the feed with an appropriate amount of bypass. The maximum selectivity for all these cases is 2.05. Example 2. In our previous paper, we considered a fairly artificial example where the CSTR locus had multiple branches. This type of situation arises fairly often in nonisothermal reactor systems, even with simple kinetics. We now consider an adiabatic reactor system with the following kinetics: kl
A-B
kl
A-C
where ri is the rate of formation of species i, Ci is the concentration of species i, and k , and k2 are rate constants. For this example, we will construct the attainable region in concentration space, which will allow us to determine all feasible concentrations of A and B. We can write that
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 53
I t
I
1
A
0.2
0.4
0.6
X
0.2
0.8
0.4
0.8
0.6
C
I
~
Figure 2. Example with multiple solutions for the CSTR. (--) CSTR from feed point A. Curve AF,plug flow from feed A. Curve BC,plug flow from feed B. Line AB,tangent line to CSTR a t B. Line AD,tangent line to CSTR a t D. Curve DE,plug flow from feed
Figure 3. Attainable region for adiabatic system with no constraints. (-- -) CSTR from feed point A. Curve AGBC,plug flow from feed A. Curve EHC,plug flow from feed E. Line AB,tangent to plug flow AGBC. Line AE,tangent to CSTR.
D.
We will refer to C A / C A o as x and CB/CAoas y. The plug flow equation can be written, with the temperature dependence included, as follows:
attainable region when using only plug flow reactors. We will define the rate of formation as -r = 5 x IO5 exp(-4000/T)C 5 X lo8 exp(-8000/T)(1 - C ) (18) where C is the concentration. The temperature for an adiabatic constant-pressure process with constant specific heat and heat of reaction is given by
where u = (k,CAo/k& = exp(ll.5 - 5000/T)x
T = Ti, + (540 - 1 0 0 ) ~+ lOO(1
- X)
T = T o + T,d(l (17b) (17~)
and Ti, is the inlet temperature, i.e., a t x = 1 and y = 0. This latter equation, eq 17c, is the constant-pressure energy balance where C,, and the enthalpy of the reaction are constants. The inlet concentration corresponds to x = 1 and y = 0, and we will use an inlet temperature of 290. If the attainable region is constructed in concentration-space in the manner discussed in GCH, we find that the plug flow trajectory AF from the feed concentration A lies close to the x axis, as can be seen in the bottom part of Figure 2. The CSTR locus from the feed point has three branches. If only one branch is considered, one can find a region ABC, which lies close to the x axis, that satisfies parts a and b of the necessary conditions. One, however, finds that conditions c and d are not met. The other two branches of the CSTR locus enlarge the attainable region quite considerably and form a region given by the x axis, straight line AD, the plug flow trajectory from D (DE),and the y axis from the origin to point E. It is found that the reaction vector points inward along line AD and the x axis and is zero along the y axis. Necessary conditions c and d are also satisfied by this enlarged region. It can be seen that the multiple branches of the CSTR arise even with simple kinetics for nonisothermal systems. These branches must be included in the attainable region, and they may enlarge the attainable region quite considerably when compared with the region constructed by using only one branch of the CSTR locus. This has important implications for optimization, such as if one were trying to maximize the production of B. Example 3. In this example, for a single constantdensity reaction system and an adiabatic system of reactors, we examine the following two cases: (a) construct the attainable region in concentration-time space; (b) find the
-
C)
(19)
We will use a feed concentration of 1, T oof 300, and Tad of 200. We define T oas the basis temperature, i.e., the temperature of the stream if it was pure feed material or, in other words, C = 1. (a) Construction of the Attainable Region. Starting at a feed concentration of 1 and a time of 0, we can draw the plug flow trajectory AGBC on the concentration-time axes. This trajectory is a concave curve for part of its trajectory, as shown in Figure 3. We are not able to draw the plug flow reactor all the way to the equilibrium point but have stopped at some arbitrarily large time. We fill in the concavity of the plug flow trajectory by mixing feed (given as point A on Figure 3) and material at concentration B where the straight line AB is tangent to the curve AGBC at point B. The reaction vector is in this case (r,l) and is a function of concentration only. The slope of the plug flow trajectory is equal to that of the reaction vector and, as drawn in Figure 3, is simply the inverse of the rate of formation of C or, in other words, l / r . We can therefore see directly from the plug flow trajectory AGBC by a simple downward translation of the curve how the reaction vector varies along line AB. It can be seen that there is a section of the line over which the reaction vector points outward. We, therefore, have not yet found the lower boundary of the attainable region. At points B and D , the slope of the reaction vector is equal to that of line AB, and these points are therefore solutions to the equation of the CSTR with feed A. If we draw in the CSTR locus ADEF from point A , it passes through points B and D and lies beneath line AB between the two points. The CSTR locus is itself not convex. We can draw a straight line between points A and E , which fills in the concavity and represents mixing between these two points. From the slope of the plug flow trajectory, we can see that the reaction vector points inward along line AE and is collinear at point E. Along the boundary EF formed by
54
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
c
03 t
G2
01
0 0
02
04 _
________.
_
OD
06 _
_
~
1
_-
Figure 4. Attainable region for adiabatic system using only plug flow reactors. Curve AXBC, plug flow from feed A . Curve DJKC, plug flow from feed D. Curve LMNC, plug flow from feed L. Curve EHC, plug flow from feed E. Line AB, tangent to plug flow AXBC. Line AJ, tangent to plug flow DJKC. Line AE, limiting position of tangent to plug flow reactor. (--I Tangents to plug flow trajectories.
the CSTR locus, the reaction vectors point outward. We, therefore, draw in the plug flow trajectory EHC from point E . As the reaction vector is a function of concentration only, the slope of the plug flow trajectory is the same for all times at the same concentration, and thus, the trajectory of the second plug flow is the same as that of the first plug flow but shifted downward so that point G corresponds to point E , The lower boundary of the region is now given by AEHC, and in the limit, the concentration tends to zero and 7 m. The other boundary line is the vertical from A , formed by mixing feed with the equilibrium material. No reaction vectors point outward along this boundary, and the boundary is convex. No reaction vectors in the region below this boundary, when extended backward. intersect the boundary. The plug flow trajectory in the region below this boundary can also not be intersected twice by a line from the boundary. We thus have satisfied our necessary conditions for the attainable region. If we wish, for instance, to find the minimum residence time for a given outlet concentration, the answer would lie on curve AEHC. Any point above this boundary can be reached as well. For example, if we wished to operate our system at point I , we could use a plug flow reactor operating at point B with bypass so as to give us an outlet condition corresponding to point I. It is interesting to see how in this case the residence time of the system with this outlet concentration is less than that of the plug flow reactor alone. (b) Attainable Region Using Only Plug Flow Reactors. (Bypassing is a plug flow reactor of zero space time in parallel with another reactor.) We will now use only plug flow reactors, and we will look at constructing the attainable region. We can use this attainable region to find the reactor configuration to minimize the total residence time for a given output concentration. The plug flow trajectory from the feed point is given in Figure 4 by curve AXBC and corresponds to curve AGBC in Figure 3. Notice that the time refers to the residence time of the system, which in this case is a plug flow reactor. By allowing bypassing and mixing from any point along the plug flow trajectory with any other point along the trajectory, we obtain the attainable region for a single plug flow trajectory with bypass, the boundary of which is given by ADBC and a vertical line from A. Line AB of the boundary fills in the concavity of the plug flow trajectory, and BC is part of the plug flow trajectory. The time still
-
refers to the residence time of the system, which can now consist of a plug flow reactor with some bypass. For example, we can mix fluid from the plug flow reactor with residence time B with feed material given by point A in such a ratio so as to produce material of concentration and residence time represented by point D. We now wish to choose the second plug flow reactor such that the boundary of the attainable region is lowered as much as possible. There are two points to consider when choosing the second reactor. Firstly, the plug flow trajectory for an adiabatic reactor is of fixed shape, as the initial conditions are unique. Thus, the plug flow trajectory from feed point D , for example, is just the original trajectory AXBC shifted down until point X on curve AXBC touches point D. We will refer to the trajectory AXBC as the base trajectory. Furthermore, as temperature depends only on concentration in this adiabatic system, the temperature at point D must be the same as point X. Thus, when shifting the base trajectory as described, we always automatically fulfill the adiabatic relationship. Consequently, as the plug flow trajectories cannot cross each other and there is only one base trajectory, the required second trajectory must be the one that when moved downward is the lowest for all possible feeds to the second reactor or, in other words, extends the region the most. If we do this, we find the best feed point is point D on line AB where the slope of AB equals that of the plug flow trajectory (and, therefore, the slope of the reaction vector or simply l / r ) . This result occurs because the plug flow trajectory cannot move out of the one-reactor attainable region between A and D , as the reaction vectors point inward along AD. Point D is the first point at which the plug flow trajectory may move outside the attainable region, as the gradient of the reaction vector is equal to the slope of line AB. This trajectory must also be the lowest, as plug flow trajectories cannot cross and consequently trajectories with feeds between DB must lie higher than the one from D. The trajectory of the second plug flow reactor is given by curve DJKC. We can again mix every point that can be reached by the system of reactors with every other point and so obtain the two-reactor attainable region, the boundary of which is given by ALJKC and a vertical line from A. The boundary ALJKC represents the relationship between the minimum residence time for all possible outlet concentrations for a two-stage adiabatic plug flow reactor system. The same considerations that held for the second plug flow reactor would again hold when choosing the third stage. We would thereby find the feed to the third stage to be point L , where the slope of line AJ would again be equal to that of the (base) plug flow trajectory. The third plug flow reactor would operate along curve LMNC, and the boundary of the attainable region for a three-stage reactor system would be given by line AM and the plug flow trajectory MNK. It is interesting to see that the change in the boundary of the attainable region between two and three stages is much smaller than that between one and two stages and how the tangency point moves toward the left. Each additional stage would give a smaller and smaller change in the boundary of this limited attainable region until, in the limit, we would reach the lower boundary of the whole attainable region-AEHC. A t point E , the slope of the base plug flow trajectory, and consequently line AE, is a minimum. Curve EHC is the trajectory of the plug flow reactor with a feed point at E. It would take an infinite
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 55 0.3
280
0.1
-
0
0
0.5
C
275
1
1
Figure 6. Family of adiabatic plug flow reactors. (-) Plug flow trajectories for different basis temperatures. Curve X Y Z (--), boundary of the attainable region for a one-stage adiabatic plug flow reactor system. Basis temperatures are written on the plug flow curves.
flow trajectory with a basis temperature of 330 K has a lower residence time than the plug flow trajectory with a basis temperature of 325 K. Thus, in the limit, the portion of a plug flow trajectory (of given feed temperature) that lies in the boundary is the point where the neighboring curve, that is, the curve with a infinitely small increase in the basis temperature but the same feed concentration, intersects it; i.e.,
1 dr dT
This equation is general and gives the outlet concentration (Ce) that a reactor of specified basis temperature should operate at in order to lie in the boundary of the attainable region or, in other words, describes the minimum residence time versus concentration relationship. When the specific heats are constant and the temperature-concentration relationship simplifies to eq 19, dT/dTo is unity. Furthermore, if the density of the system is constant as well, eq 20 agrees with the results obtained by conventional optimization techniques (Horn, 1961; Konoki, 1957). For single-reactor systems, the attainable region boundary is given by the envelope of the extrema of the plug flow curves with different initial temperatures, that is, the dotted line X Y Z of Figure 6. Let us now consider the best basis temperature for the next stage. We know that the outlet concentration of the first stage, which is also the inlet concentration to the second stage, must lie on the envelope shown in Figure 6. We must find the inlet temperature (or equivalently find the basis temperature) to the second stage reactor such that the trajectory of the reactor will extend the boundary of the attainable region as much as possible. The condition to determine the boundary is more easily seen if we view the problem in a different way. We will rather specify the final basis temperature and, therefore, the related plug flow reactor trajectory that fulfills the
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03
C
t
t
L
Y
01
01
0
0.5
1
C
--
0
I
05
1
C
Figure 7. Example of interstage cooling reactor. Curve ABC, second-stage plug flow trajectory. Curve AD, best first-stage reactor for second-stage reactor ABC. Curve X Y Z (--), boundary of the attainable region of a one-stage adiabatic reactor system. Curve JKL (--), boundary of the attainable region of a two-stage adiabatic reactor system.
Figure 8. Example of cold shot cooling reactor. Curve X Y Z (-4, locus of optimal operating points for a one-stage adiabatic reactor system; (--) trajectories of the first-stage plug flow reactors. Curve ABC (-), second-stage plug flow trajectory. Curves DEF, GHI,JKL (- - -) isoenthalpic feed lines; (-) lowest second-stage plug flow trajectories from the isoenthalpic feed lines.
adiabatic condition, for the second stage. As an example, we specify the second plug flow trajectory given by curve ABC in Figure 7. The envelope of the attainable region from the first stage is shown as envelope X Y Z . We must find the feed concentration to the plug flow trajectory ABC that will extend the attainable region the most. This is equivalent to finding the feed point that will move the trajectory of the second stage reactor as low as possible. We do this by moving curve ABC up and down along envelope X Y Z and in this manner find that this is the point on the envelope where the slope of the envelope (and therefore the slope of the plug flow trajectory that makes up that part of the envelope) equals that of the specified plug flow trajectory. Hence, this point occurs where the reaction vector at the outlet of the first plug flow stage equals that at the inlet to the second plug flow stage, as at point D for trajectory ABC. This result must again hold for all reactor stages and agrees with the conventional mathematical optimization results (Horn, 1961; Konoki, 1957). We can repeat the process for all possible basis temperatures and again find the envelope formed by the intersection of neighboring curves shown by envelope JKL. Envelope JKL represents the boundary of the attainable region for a two-stage intercooled reactor system. It also represents the minimum residence time vesus outlet concentration for all two-stage interstage cooled reactors of the type shown in Figure sa. We have again used geometric arguments for determining the attainable region and used this to perform the optimization. Again, the reasons behind the optimization relationships can be clearly visualized. This construction can be repeated for three and greater number of stages and the optimal performance curves found for that number of stages. Example 5. We will now look at finding the attainable region by using cold shot cooling alone and hence the optimal reactor configuration to minimize the residence time for a given exit concentration by using cold shot cooling. In this system, we may heat up the feed to the first plug flow reactor stage and add feed at its basis tem-
perature between subsequent reactor stages as shown in Figure 5b. We can thus vary the feed temperature to the first reactor and the residence time of the various reactor stages as well as the amount of bypass at each stage. We will use the same rate expression as in example 3 and the same adiabatic relation, although the method we use will hold for cases where the relation is more complex. The plug flow reactor trajectories for different enthalpies (i.e., different basis temperatures) are again those given in Figure 6. The first stage of the construction will be identical with that of the previous example, and as explained above, the dotted line X Y Z of Figure 6 is the boundary of the attainable region. In order to find the boundary of the attainable region for the second reactor stage, we will again consider the problem of which is the best feed point for a second plug flow reactor with specified basis temperature. The problem and construction are now different from the previous two examples, as the whole process is not truly adiabatic. This is so because we have heated up the feed to the first reactor, and thus, different amounts of bypass give us different energy balance equations and hence different basis temperatures. We will consider, for example, a second reactor stage with constant enthalpy such that the outlet stream has a basis temperature of 300 K. For each first reactor stage shown in Figure 6, we mix all possible outlet concentrations with the bypass (unheated feed, which is assumed to be at 250 K in this example) in just the correct proportions to give the isoenthalpic lines corresponding to a basis temperature of 300 K. These isoenthalpic lines represent possible feed points for the second reactor stage and are shown in Figure 8 as curves DEF, GHI,and JKL. For the simple isoenthalpic relation used in this example, eq 19, the feed lines are simply a constant ratio of the length of the line between the outlet point and feed point A . For each feed line, we can move the plug flow trajectory ABC, which has a basis temperature of 300 K, up and down until we find the feed point that gives the lowest possible second
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 57 trajectory or, in other words, the trajectory that extends the attainable region the most. This feed point is always the point where the plug flow trajectory ABC is tangent to the feed locus, and it can be shown that it must therefore also be tangent to the first plug flow trajectory. This condition implies that the reaction vector a t the outlet of the first plug flow reactor must be equal to that a t the inlet of the second plug flow reactor, which agrees with the mathematically obtained result for constant-density systems (Konoki, 1960, Maleng6 and Vincent, 1972). We still, however, have not completed the construction. We must still choose the best first stage for the given second reactor trajectory. We must, therefore, look a t which first reactor stage gives the lowest second trajectory (as these translated trajectories ABC cannot cross each other), and this will be the best feed reactor for the given trajectory. In this case, the feed locus GHI gave the lowest second stage plug flow reactor, and thus, the related plug flow trajectory must be the optimal first-stage reactor. We can do this for all possible second reactor basis temperatures, and we will again find the envelope of the attainable region and hence the optimal operating points for a two-stage reactor system. We can repeat this construction for as many stages as required. The construction is fairly simple and only requires that one integrate each plug flow trajectory once. The calculation for the isoenthalpic feed lines is especially easy in this example, but the construction method will hold for more complex energy balance expressions as well.
Extensions to Nonadiabatic Systems In this paper, we have essentially limited ourselves to adiabatic cases, although we have allowed for interstage cooling in one of our examples. The extension to nonadiabatic cases need not be too difficult, as enthalpy obeys a linear mixing rule. Thus, the adiabatic case is just an isoenthalpic cross section of the attainable region. In principle, by including an energy loss or gain term in the last element of the reaction vector for R in eq 8b in place of the zero, one can include nonadiabatic cases. If one requires making use of a coolant with certain properties, then one can augment the reaction vector R with another component, which will be the enthalpy of the coolant stream. If one, however, wishes to do this, care must be taken with the space time variable which will need to be made the same by a suitable transformation of the new component augmented with a further variable. In this expanded space, the necessary conditions need no longer hold, and in fact, the enthalpy space can in principle go to infinity unless we place constraints, for example, temperature, on the system. However, the projection into time-concentration space or concentration space may still provide us with a useful attainable region. Let us examine this point with a simple example. Suppose we have a single, exothermic, reversible reaction. We know that there is an optimum temperature profile which will maximize our conversion for a given size reactor. A reactor following this profile will form part of the boundary of the attainable region in concentration-time space. This reactor will, of course, not be adiabatic. If we now look at the enthalpy-concentration-time space, allow our value of zero in the last element of R in eq 8b to be a variable Q that could take on any value, solve for the geometry in the space for all the possible values of Q, and then take the projection into the concentration-time space, we obtain the same answer. We need not allow Q to be arbitrary but could impose, for instance, the form of UA(T - T J ,where T, is the fixed temperature of a condensing coolant and UA an overall heat-transfer coefficient, and hence find the at-
tainable region in the subspace, allowing for only this form of heat transfer. It is clear from this discussion that with a little care and forethought a very wide variety of nonadiabatic problems can also be handled by the geometric approach that has been outlined in this paper, even though it may involve construction of a large number of curves or moving into higher dimension.
Conclusion We have constructed the attainable region in concentration-time space as well as found the optimal configurations of several reactor systems. We have also used the geometric ideas to find the attainable regions for several systems where we limit the number of possibilities, that is, constrain the system. It is easy to visualize the relationships between the different types of reactors and the geometric relationships behind the optimal reactor configurations. Furthermore, by use of the techniques developed in this paper, the cold shot cooling problem can be solved relatively easily for even complex temperature-concentration relations, which cannot be handled by using the standard mathematical results (Maleng6 and Vincent, 1972). All the examples presented in this paper required only reactors in series and bypass to achieve all the points on the boundary of the attainable region. We can see that the tangency conditions in the geometry of the systems give rise to the well-known analytical results for some of these simpler systems. However, these tangency conditions continue to apply even when the simplifications are not possible, so the geometric methods remain quite easy to apply even when some of the simple analytic results are not available any more. It is clear that the geometric ideas that have been developed provide a very powerful tool for finding a region that obeys the necessary conditions for the attainable region and hence to "solve" a wide variety of optimization problems. As there is at present not a complete sufficiency condition, we have not yet proved that any optimum is a global one. The limitation, however, relates only to multiplicity conditions and might be a limitation in the proof rather than a real one. It would, of course, be a global optimum if there was only one such region that obeyed the necessary conditions. The real power of the approach is that it actually generates the structure of the optimal reactor system directly for all possible objective functions that are functions of our basis variables, as well as the size of the individual elements in the system. Work is in progress to extend the sufficiency condition and to construct examples in three and higher dimensions. Acknowledgment D.H. thanks the University of Potchefstroom for allowing time to pursue this work. Nomenclature Roman Letters A , B, C, D = chemical species A i= chemical species i = ratio of k l C A / k 2 ,as defined in eq 17b C, = heat capacity at constant pressure per unit mass C = characteristic vector (dl, d2, ..., d,, 7,6 ) ci = concentration of species i c = concentration vector di = composition variable of species i, defined by eq 2 g = resultant vector H = specific enthalpy per unit mass
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
58
= specific enthalpy (heat) of formation per unit mass at reference conditions Afi,,, = specific enthalpy (heat) of mixing k , = rate constant for reaction i M = total mass flow of material m i= molecular mass of species i n = number of concentration variables; Le., dimension of concentration space R ( C ) = reaction vector in n + 2 space, as a function of C R, = rate of formation of species i as a function of C r’(c.7’)= reaction vector in n + 2 space, as a function of c and ?’ r = rate of formation vector in n space, with components ri r, = rate of formation of species i as a function of c and T T = temperature Tad= adiabatic temperature rise, defined by eq 19 T o = basis temperature, defined by eq 19 t = residence time, defined by eq 8d V = vn!ume of reactor w,= mass fraction of species I x , I=I mrmalized concentration of species 1.’
Greek 1,ertw-s = mixing ratios H = temperature-like variable, defined by eq 4a r = space time defined by eq 3 p = density v = sthichiometric coefficient 4 = scalar quantitv used in eq 12a 01,
Superscripts 0 = initial or reference condition of a material
*
= value of variable after mixing e = value of variable at the exit of the reactor
Si1 bscr ipt s
i = species
i
1, 2 = stream 1 or 2 in = inlet conditions
Literature Cited Achenie, L. E. K.; Biegler, L. T. Algorithmic Synthesis of Chemical Reactor Networks Using Mathematical Programming. Znd. Eng. ChPm. Fundam. 1986,25, 621-627. Caha, J.; Hlavacek, V.; Kubicek, M.; Marek, M. Study of the Optimization of Chemical Engineering Equipment. Numerical Solution of the Optimization of an Adiabatic Reactor. Znt. Chem. Eng. 1973, 13 (3),-466-473. Chitra. S. P.: Govind. R. Svnthesis of Ootimal Serial Reactor Structures for HomogeneoGs Reactions. Part I: Isothermal Reactors. AZChE J . 1985a, 31, (21, 177-184. Chitra, S. P.; Govind, R. Synthesis of Optimal Serial Reactor Structures for Homogeneous Reactions. Part 11: Nonisothermal Reactors. AZChE J. 1985b, 31 (21, 185-194. Classer, D.; Hildebrandt, D.; Crowe, C. A Geometric Approach to Steady Flow Reactors: The Attainable Region and Optimization in Concentration Space. Ind. Eng. Chem. Res. 1987, 26, 1803-1810. Horn, F. Zur Berechnung von adiabatischen Abschittsreaktoren. I. 2. Elektrochem. 1961, 65, 295-303. Horn, F. Attainable and Non-Attainable Regions in Chemical Reaction Technique. Third European Symposium on “Chemical Reaction Engineering”; Pergamon Press: New York, 1964; pp 1-10. Konoki, K. Theory on the Operation and Design of the most Effective Multistage Reactor. Chem. Eng. Jpn. 1957, 21 (7), 408-412. Konoki, K. Control of Reactor Temperature by Means of Cold Bypass. Chem. Eng. Jpn. 1960, 24 (8), 569-571. Levenspiel. 0. Chemical Reaction Engineering, 2nd ed.; Wiles: New Yorc, 19’72; p 143. MalengB, .J. P.; Vincent, L. M. Optimal Design of a Sequence of Adiabatic Reactors with Cold-Shot Cooling. - Ind. Enn. Chem. Process Des. Deu. 1972, 11 (4), 465-468.
Receiued for review February 22, 1989 Accepted September 25, 1989
