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PROCESS DESIGN AND CONTROL Strategy of an Attainable Region Partition for Reactor Network Synthesis Zhao Wen,* Zhou Chuanguang, Zhang Zhishan, and Han Fangyu Qingdao Institute of Chemical Technology, Qingdao 266042, People’s Republic of China
Li Chengyue Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China
This paper presents a strategy of an attainable region partition for the synthesis of reactor networks. According to the definitions of objective function selectivity, instantaneous selectivity, and single yield, two concentration trajectory equations are developed under two extreme cases of continuous stirred-tank reactor (CSTR) and differential side-stream reactor (DSR) by means of their geometrical characters and those of reaction and mixing in concentration space for the Van de Vusse reaction scheme. The two equations may be regarded as the optimization criterion of maximum selectivity for the reactor system. The concept of the maximum selectivity curve is presented by extending the feed of the reactor system from pure component to arbitrary concentration. The two-dimensional concentration space is divided into three regions (CSTR region, plug-flow reactor region, and non-operation region) by the curves of maximum selectivity and maximum single yield. The property of the maximum selectivity curve is dSR/dx ) 0, and that of the maximum single yield curve is SR ) 0. Three different regions have respectively the following properties: CSTR region, dSR/dx < 0, SR > 0; PFR region, dSR/dx > 0, SR > 0; nonoperation region, dSR/dx > 0, SR < 0. The feasible flow configuration of reactor networks can be conveniently determined by taking advantage of the properties of different regions and their boundaries. 1. Introduction The synthesis of reactor networks, as a key subtask of chemical process synthesis, is to find optimal reactor types, flow configuration, and key design parameters for given feed conditions and kinetics, which will optimize a specific objective function. Because of complications and variability of the problem for the synthesis of reactor networks, the subject was studied later compared with other subtasks such as the synthesis of heat exchanger networks and separation sequences and for many years with little progress. In the mid-1980s, the literature in relation to the synthesis of reactor networks increasingly emerged. Some efficient strategies and methods such as the heuristic method,1 superstructure method,2 attainable region method,3,4 and targeting strategies method5,6 were proposed. The concept of an attainable region was first suggested by Horn.7 It was defined as the set of possible output state variables that could be achieved in all of the available designs for given input and constraint conditions of the process system. According to the range of the problem, the so-called output was either the final output or any state point between the input and final output for the reactor system. * To whom correspondence should be addressed. Phone: 0086-532-4022845. Fax: 0086-532-4863434. E-mail: zhoucgce@ public.qd.sd.cn.
Recently, Hildebrandt and Glasser,8 Glasser et al.3,9 and Hildebrandt et al.10 described the basic ideas of the attainable region more clearly by using a geometric method for the reactor system; i.e., all of the output state variables that are achieved by any reactor type and any flow configuration were determined only by reaction and mixing principles in a steady-state reactor system for given kinetics and feed conditions. Through insights into the important properties of the attainable region, the synthesis problems of some reactor networks can be rigorously and elegantly solved without setting up a superstructure. For example, Hopley et al.11 and Nicol et al.12 introduced a temperature variable into the attainable region and studied intensively the synthesis problems of reactor networks for adiabatic operation, cold operation, and out-cooling operation of the exothermic reversible reaction system. Bickic and Glavic13,14 combined differential side-stream reactor (DSR) trajectory that describes improving the product distribution with the attainable region and explored the problem of extending the solution space of thermal and isothermal processes in the case of multifeeds and multicomponents. Smith and Malone15 applied the concept of the attainable region to the macromolecule polymerization process and attempted to improve molecular weight distribution. Sund and Lien16 studied the simultaneous optimization problem of the mixing pattern and the feed distribution in a lowdensity polyethylene reactor on the basis of the concept
10.1021/ie010251w CCC: $22.00 © 2002 American Chemical Society Published on Web 12/18/2001
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Figure 1. Convex region formed by reaction and mixing.
of the attainable region. Omtveit et al.17 overcame the dimensional limitation to some extent by using the principle of atom conservation. Wen et al.18,19 combined the optimal technology equation with the attainable region and studied the influence of the separation recycle stream on the synthesis of reactor networks. Recently, Feinberg and Hildebrandt20-22 have been exploring the properties of a multidimensional attainable region on the basis of the common ones of the attainable region. The authors believe that the attainable region method demonstrates profoundly the essence of the reactor system character and is a very effective tool for the reactor network synthesis. This paper presents a partition strategy for the synthesis of reactor networks on the basis of reaction and mixing principles of this method. According to the definitions of objective function selectivity, instantaneous selectivity, and single yield, the attainable region can be studied in partition by means of their geometric concept and that of reaction and mixing. In fact, the feasible configuration of reactor networks can be conveniently determined by analyzing the properties of different regions and their boundary in concentration space without constructing the attainable region. The reactor system is assumed to be a steady, isothermal, homogeneous-phase, and constant-volume process in this paper. 2. Bases of the Attainable Region Partition Reaction and mixing are two basic processes of reactor operation. Selectivity, instantaneous selectivity, single yield, etc., are important parameters in studying the reactor performance and functions on concentration state variables. They are respectively assumed to be the objective function of the reactor system. The characters of product distribution for the reactor system can be found by means of the geometric expression of these functions and that of reaction and mixing. These are the bases of the attainable region partition method. 2.1. Convex Region of Reaction and Mixing. For a reactor system that is composed of any reactor type and any flow configuration, such as a continuous stirredtank reactor (CSTR), plug-flow reactor, recycle reactor, and the composite of them, the path of its state change may be expressed by means of the corresponding trajectory in concentration space. The concave region of the trajectory can be filled with the straight line that represents the mixing of two points. The convex region that is composed of the output state variables of the reactor system is always constructed by reaction and mixing in concentration space.3 For example, Figure 1 shows a convex region constructed by reaction and
Figure 2. Contour lines of selectivity: (a-c) numerical value reduces in turn.
mixing in a two-dimensional concentration space. In the figure the curve ADO represents the locus of concentration change, the straight line AD represents the mixing of point A and point D, and the convex region is the region surrounded by the straight line AD, the curve ADO, and x axis. 2.2. Geometric Expression of the Objective Function, Reaction and Mixing. From the definition of φR, the selectivity of objective product R, we get
y ) φR(1 - x)
(1)
where in the x-y space the contour lines of selectivity are a set of lines with parameter φR, as shown by lines a-c in Figure 2. The cross points can be obtained when Figures 1 and 2 are cross-referenced. Apparently, the negative slope of the line from any state point in the convex region ADOA (including points on the boundary) to point A represents the selectivity of this point. According to the property of the convex region, the negative slope of the contour line of selectivity that is tangential to the convex region boundary is the maximum of the selectivity that is achieved by all reaction and mixing for the convex region. Therefore, the following condition is true:
dφR ∂φR ∂φR dy ) + )0 dx ∂x ∂y dx
(2)
Substituting formula (1) for formula (2) gives
dy -y ) dx 1 - x
(3)
where it shows that the law of concentration change must be followed in two-dimensional concentration space. Now, we illustrate the problem with the Van de Vusse reaction: k1
k2
A 98 R 98 S,
k3
2A 98 D
The reaction rates of involved components are as follows:
For reactant A -rA ) -
dCA ) k1CA + k3CA2 dt
(4a)
For objective product R rR )
dCR ) k1CA - k2CR dt
(4b)
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of the two points (it accords with the law of level). So, the contour line of selectivity circling around point A is tangential to many contour lines of instantaneous selectivity. The reaction and mixing going along these tangential points make the output state variables of the reactor system move along the definite locus to ensure that the instantaneous selectivity of the system achieves the local maximum. So, the following condition is true at these tangential points:
dSR ∂SR ∂SR dy ) + )0 dx ∂x ∂y dx
(7)
Figure 3. CSTR locus and DSR trajectory.
Substituting formulas (6) and (3) for formula (7), we get
y)
Figure 4. Contour of instantaneous selectivity and selectivity: (a-d) numerical value reduces in turn; (1-7) numerical value increases in turn.
Let
R1 ) k3CA0/k1, R2 ) k2/k1 x ) CA/CA0, y ) CR/CA0 Supposing the composition of the feed is x ) 1 and y ) 0 and substituting formula (4a,b) for formula (3), we get
y)
x(1 - x) 2
R1x + (1 - R2)x + R2
(5)
Formula (5) is the CSTR equation for the feed of pure A. Its locus is shown in Figure 3 for given R1 ) 20 and R2 ) 2. Its physical concept is that the local maximum points of possible selectivity by any reaction and mixing with the feed of point A lie on the CSTR locus for the stated reaction system. Thus, the CSTR equation describes the law followed by the system composition in the reaction and mixing process when the selectivity is selected for an optimal target. Substituting formula (4a,b) for the definition formula of instantaneous selectivity, we get
SR )
x - R2y x + R1x2
(6)
Obviously, contour lines of instantaneous selectivity are a set of curves. We draw contour lines of selectivity and instantaneous selectivity in one coordinate system, as shown in Figure 4. The curve ABCO is a CSTR locus with the feed of pure A in the figure. When one of contour lines of selectivity a-d is tangential to one of the contour lines of instantaneous selectivity 1-7, the instantaneous selectivity at a tangential point is the local maximum on the contour line of the selectivity. The contour line of selectivity from point A (1, 0) to tangential point B (x, y) also represents a mixing process
R1x2(1 - x) R2(1 + 2R1x - R1x2)
(8)
When viewed from a geometric expression, every previous tangential point is a simultaneous process of reaction and mixing. So, formula (8) is a DSR equation for a differential side-stream reactor with the feed of pure A; its trajectory is shown in Figure 3. Its physical concept is that the local maximum points of instantaneous selectivity achieved by any reaction and differential mixing simultaneously with the feed of pure A lie on the DSR trajectory. Obviously, the DSR equation describes the law followed by the system composition in the simultaneous reaction and mixing process when instantaneous selectivity is selected for the optimal target. 2.3. Optimal Criteria of Maximum Selectivity of the Reactor System. From the preceding analysis, we may find an important law. That is, in the convex region (including its boundary) formed by any reaction and mixing with the feed of pure A in (x, y) space, the local maximum points of selectivity lie on the CSTR locus. However, these points are not always the maximum points of selectivity achieved by reaction and mixing. For the differential side-stream feed of pure A, all of the local maximum points of instantaneous selectivity lie on the DSR trajectory, but the reaction and mixing processes corresponding to these points cannot always form a convex region. Accordingly, these points are not always the local maximum of selectivity. Nevertheless, when the CSTR locus and DSR trajectory intersect, the selectivity achieves a maximum and the reaction and mixing process forms the maximum convex region (attainable region) that cannot be extended only by reaction and mixing. Obviously, it is the criterion of the maximum selectivity for the reactor system that the equations of CSTR and DSR are simultaneously met. Given a reaction scheme, for example, the forenamed Van de Vusse reaction (R1 ) 20 and R2 ) 2), it is convenient to determine that the maximum point of selectivity for the reaction system is point B (0.3162, 0.0587), as shown in Figure 3. According to the geometric concept of point B that is not only the switching point of the CSTR locus and PFR trajectory but also the end-point mixing with the feed of point A, then the maximum convex region achieved by any reaction and mixing is constructed by the PFR and mixing process from point B, as shown in Figure 5. In Figure 5, the curves ADO and ABO are respectively the PFR trajectory and CSTR locus with the feed of point A; the straight line AB represents mixing of points A and B; the curve BCO is the PFR trajectory with the
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Figure 5. Attainable region of the Van de Vusse reaction (a1 ) 20 and a2 ) 2).
Figure 7. Properties of the maximum selectivity point in concentration space: (A, B, and D) initial feed points; (C) maximum selectivity point; (AC) mixed line; (1-3) respectively are the CSTR locus; (4-6) respectively are the DSR trajectory.
Figure 6. Attainable region of the Van de Vusse reaction (a1 ) 2 and a2 ) 20).
feed of point B; the region ABCOA is an attainable region (AB is a straight line). Figure 5 also shows that the operation of either one CSTR process getting to the outlet state point B or the bypass CSTR process getting to any point on the straight line from point A to B can ensure the maximum selectivity for the reactor system. On the line the single yield of point B is the maximum. Other operations, for example, one PFR process with the feed of point A, one PFR process with the feed of point B, two or more CSTR in a series of processes with the feed of point A, or one CSTR process not operating at point B with the feed of point A, reduce the selectivity to some extent. According to the optimum criteria of maximum selectivity, the state point of maximum selectivity can be determined by combining the CSTR and DSR equations for given feed point A (1, 0) and kinetics. Obviously, in (x, y) space, the location of the state point of maximum selectivity on the boundary of the attainable region depends on kinetic conditions. For example, in Figure 5 R1 . R2, wihich shows that a parallel reaction is more important than a series reaction in increasing the selectivity of R. For the CSTR process, the feed concentration drops instantly to the outlet one to induce a low concentration in favor of increasing the selectivity of R. This accords with the conclusion drawn from the usual qualitative analysis. Correspondingly, when R1 , R2, the contrary result can be obtained, as shown in Figure 6. If R1 ) 2 and R2 ) 20, the curves ACO and ADO are respectively the PFR trajectory and CSTR locus, the state point of maximum selectivity for the reactor system coincides with point A, and the upper boundary of the attainable region is completely composed of the trajectory of PFR. Then the PFR process is more favorable for increasing the selectivity than the CSTR process.
Figure 8. Construction method of the maximum selectivity curve: (- -) DSR trajectory; (BME) maximum selectivity curve; (ABO, FNO) respectively are the CSTR locus.
3. Two Important Curves of an Attainable Region Partition 3.1. Concept of the Maximum Selectivity Curve. With the kinetics of chemical reaction and the feed of pure component known, the curves of CSTR and DSR equations can be plotted in (x, y) space, as shown in Figure 7. In Figure 7, point A is the feed point and point C is the maximum point of selectivity. We select any two points B and D on the straight line connecting point A to point C and plot respectively the curves of CSTR and DSR with the feed of points A, B, and D; then the maximum points of selectivity for the feed of points A, B, and D are point C. However, different feed points (x0, 0) (x0 ∈ [0, 1]) correspond to different maximum points of selectivity. The curve connecting these maximum points of the selectivity is the maximum selectivity curve in (x, y) space. 3.2. Construction of the Maximum Selectivity Curve. According to the concept of maximum selectivity, the authors present two construction methods as follows and illustrate them with the Van de Vusse reaction (R1 ) 20 and R2 ) 2). (1) Geometric Plotting Method. The procedures of the construction method are as follows: first, we select several points (x0, 0) as the virtual feed points for the reactor system in the concentration space (x, y); second, for each feed point, we plot points of intersection for their respective CSTR locus and DSR trajectory by means of the criterion of the maximum selectivity; finally, the maximum selectivity curve can be obtained through connection of these points, as EMB shown in Figure 8. In Figure 8, the region ABCOA surrounded by straight line AB, curve BCO, and straight line OA is the convex region formed by reaction and mixing with the feed of point A, and the region ABCOA surrounded
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Figure 9. Construction method of the maximum one-way yield curve.
by straight line FM, curve MGO, and straight line OF is the convex region FMGOF formed by reaction and mixing with the feed of point F. (2) PFR Equation Analytical Method. Known conditions are as follows:
PFR equation: dy x - 2y )dx x + 20x2
(9)
instantaneous selectivity equation: SR )
x - 2y x + 20x2
(10)
Substituting formulas (9) and (10) for formula (7), we get
y)
x(10x - 1) 40x - 1
(11)
For PFR, different trajectories with different feed points are not intersectional when each state point has only one vector. So, the trajectory of formula (11) in (x, y) space is named the maximum selectivity curve. When the two methods are compared, their results are accordant. The advantage of the first method consists of an explicit physical meaning, but because of the complication of the DSR equation for a complex reaction, it is usually very difficult to solve it by combining it with the CSTR equation. However, it is very convenient to determine the curve equation of maximum selectivity by the second method. Both methods may be applied to the complex reaction whose instantaneous selectivity is the function on products and key reactant concentration. 3.3. Maximum Single Yield Curve. As shown in Figure 9, we select several points on the maximum selectivity curve EMB and straight line OE as the feed point for PFR and plot their PFR trajectory. Because different PFR curves are not intersectional, every curve has a maximum point of single yield. The maximum single yield curve ONC can be obtained by connecting these points in the attainable region. We may use other methods. For instance, the line whose instantaneous selectivity is null (SR ) 0) coincides with the maximum single yield curve ONC. This shows that the maximum single yield curve means a set of maximum points of single yield that are achieved for the reactor system with different feed points. Along the curve the instantaneous selectivity is null for a complex reaction system, and for product R, the productive rate equals the reactive rate; that is, the net productive rate is null. In
Figure 10. Three zones in attainable region: I, CSTR region; II, PFR region; III, non-operation region.
general, because of the byproduct reaction, the selectivity will decrease when the operation point lies on the curve. 4. Strategy of an Attainable Region Partition When viewed from Figure 10, the maximum selectivity curve EMB and the maximum single yield curve ONC divide the attainable region ABCOA into three regions. Region I is named the CSTR region because we should use the CSTR process to get to the maximum selectivity curve to achieve maximum selectivity for any feed point in the region. Other operations such as single PFR, recycle reactor, or several CSTRs in series will induce the loss of selectivity. Region II is named the PFR region. Because the maximum selectivity curve is a slick joint23 of reaction and mixing, we may use the PFR process with the feed of any point on the curve. Along the PFR trajectory, the feed reacts to reach the maximum single yield curve ONC, and the single yield will increase and the selectivity will decrease. The reason is that at every point on the nonconcave part of the PFR trajectory the single yield and the selectivity are the maximum in all operation patterns for the same feed point and conversion fraction. Accordingly, the operation point on the curve of single yield usually has a large loss of the selectivity. So, it makes the selectivity decrease to go on with the operation along the PFR trajectory from any point on this curve. Correspondingly, the region that is surrounded by the maximum single yield curve and the attainable region boundary will not be expected in designs and operations and is named non-operation region. Obviously, the property of the maximum selectivity curve is dSR/dx ) 0 and that of the maximum single yield curve is SR ) 0. Three different regions have respectively the following properties: CSTR region, dSR/ dx < 0, SR > 0; PFR region, dSR/dx > 0, SR > 0; nonoperation region, dSR/dx > 0, SR < 0. The feasible flow configuration of reactor networks can be conveniently determined by taking advantage of the properties of different regions and their boundary. According to the idea of the attainable region partition as described above, the authors present a new strategy for the synthesis of the reactor networks in twodimensional concentration space. The procedures are as follows: (1) Given feed and kinetic conditions, we may respectively plot the curves of maximum selectivity and maximum single yield in two-dimensional concentration space.
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(2) For the feed point in the CSTR region, we should use the CSTR process starting with the feed point to reach the maximum selectivity curve and then the PFR process to reach the maximum single yield curve. So, the feasible flow configuration is CSTR + PFR. When the maximum selectivity curve tends to close to the feed point A, the CSTR region disappears, and then the feasible flow configuration is reduced to the PFR process. (3) For the feed point in the PFR region, PFR is determined as the feasible flow configuration. (4) Non-operation region is not required in the design and operation and, therefore, will not be considered. 5. Conclusions Given kinetics and feed conditions for the reactor system, according to the definitions of such different objective functions as selectivity, instantaneous selectivity, and single yield, the two-dimensional concentration space can be divided into three regions (the CSTR region, the PFR region, and the non-operation region) by means of their geometric characters and those of reaction and mixing in concentration space. The numbered feasible flow configurations for the reactor system can be conveniently determined by means of the properties of different regions and their boundary without constructing an attainable region. Another paper reports other examples for the partition strategy proposed in this paper. Acknowledgment This work was supported by the National Natural Science Foundation of China (Nos. 29776028 and 29836140). Notation C ) state variable or mole concentration k ) rate constant r ) reaction rate x ) nonconversion of component A SR ) instantaneous selectivity of component R φR ) selectivity of component R y ) single yield of component R a1, a2 ) kinetic parameters Subscripts A ) key reactant D, S ) byproducts R ) objective product
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Received for review March 19, 2001 Revised manuscript received September 28, 2001 Accepted October 16, 2001 IE010251W
