Ind. Eng. Chem, Process Des. D e v . 1980, 19, 195-197
Greek Letters shear rate, SS’ = average shear rate, eG = gas holdup, fraction of clear liquid volume, dimensionless G : = gas holdup, fraction of aerated liquid volume, dimensionless X = characteristic material time, s p = liquid viscosity, g c m - l s ~ ~ p a = apparent viscosity, g cm-’ s-l vL = kinematic viscosity of liquid = p / p L , cm2 s-l pL = liquid density, g u = surface tension, g T = characteristic process time = d v s / U B ,s $ =
Subscripts IE = inelastic
195
VE = viscoelastic
Literature Cited Akita, K., Yoshida, F., Ind. Eng. Chem. Process Des. Dev, 12, 76 (1973). Akita, K., Yoshida. F., Ind. Eng. Chem. Process Des. Dev, 13, 84 (1974). Akita, K., private communication, Tokushima University, 1978. Baykara, 2. S.,Ulbrecht, J., Biotech. Bioeng., 20, 287 (1978). Elbirli. B., Shaw, M. T., J . Rheo/ogy, 2 2 , 561 (1978). Nakanoh, M., M.Eng. Thesis, Kyoto University, 1976. Nishikawa, M., Kato, H., Hashimoto, K., Ind. Eng. Chem. Process Des. Dev., 16, 133 (1977). Perez, J. F., Sandall, 0. C..A I C h E J . . 20, 770 (1974). Prest, W. M., Porter, R. S.,O’Reilly, J. M., J . Appl. Po/ym. Sci., 14, 2697 (1970). Ranade, V. R., Ulbrecht, J. J., A I C h E J . , 24, 796 (1978). Yagi, H., Yoshida, F., Ind. Eng. Chem. Process Des. D e v . , 14, 488 (1975).
Recezued f o r reuiew August 25, 1978 Accepted October 10, 1979
COMMUNICATIONS Optimization of a Nonlinear System Consisting of Weakly Connected Subsystems Optimization of a very large complex system described by nonlinear equations is often beyond the capability of presently available procedures if the system is treated in its entirety. Many such systems are found to contain “weakly connected” subsystems which may be considered independently. A revised approach which makes use of this special structure to substantially reduce the size of overall system optimization is proposed. Under some circumstances, this method reduces a complex nonlinear integer or mixed integer problem into an equivalent small dimension continuous variable optimization problem.
Introduction Optimization of nonlinear systems in which some or all of the decision variables are integers is generally difficult. When the number of integer variables considered is relatively small there are several methods which can provide satisfactory results. For example, the authors have found computer programs based on the adaptive random search procedure of Heuckroth et al. (1976) useful in moderate size problems. However, there does not appear to be any fully satisfactory procedure for optimization of very complex nonlinear systems with many continuous and integer variables. There are a number of practical mixed integer and integer optimization problems encountered in the process industries which are too large to be successfully treated when the system is considered as a whole. One area in which difficulties have been encountered has been in the optimization of process systems subject to stochastic reliability or availability constraints. Burdick et al. (1977) have pointed out that in many such problems the system may be decomposed into stochastically independent subsystems. By using a Monte Carlo simulation technique they were able to determine bounds for the subsystem behavior and these bounds could then be used in a simplified overall system optimization. It is the purpose of this communication to present an improved procedure for the establishment of subsystem characteristics where the subsystems are “weakly connected” and to show that the general approach is applicable to a wider class of problems than originally considered. Problem Formulation In their paper, Burdick et al. (1977) optimized the design 0019-7882/80/1119-0195$01.00/0
of the shutdown heat removal system (SHRS) of a liquid metal fast breeder reactor (LMFBR) minimizing cost subject to SHRS unavailability. They used eight integer variables in their nonlinear optimization problem. They further showed that a cost-unavailability relationship for the SHRS could be obtained by generating a Monte Carlo plot of SHRS cost vs. SHRS unavailability for randomly selected designs. A lower bounding curve was obtained for the plot. The curve provided an approximate functional SHRS cost for a given SHRS unavailability. The advantage of such an approach is that it reduces a complex nonlinear integer or mixed integer system optimization problem into a simple continuous problem. Determination of the bounding curves via Monte Carlo simulation can be very time consuming since a large number of configurations must be examined if reasonably accurate bounding curves are to be obtained. However, a simpler and more exact approach can be used to obtain the functional relationship between unavailability and SHRS cost. Solution of the problem minimize C( Q) (1) subject to Q 5 h, (2) locates one point on the bounding curve. Resolving the problem for a number of different k, delineates the entire bounding curve. The revised approach has been found to be a significant improvement over that initially proposed. The cloud of points in Figure 1 was obtained by computing costs and unavailability for all possible configurations of the SHRS. The solid curve shown in Figure 1 has been drawn through the results obtained from a series of @ 1979 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980 I
I
1
I
1
i I
1 1 OE-07
1 OE-08
1 OE-06
I 1 OE-05
SHRS unavailability
Figure 1. Comparison of minimum SHRS cost curve with costs of all possible configurations.
optimizations defined by eq 1 and 2 . The optimizations were carried out readily using adaptive random search (Heuckroth et al., 1976). It is apparent that the curve of Figure 1 does correctly provide the minimum cost for a given availability. When several safety systems are involved, the nonlinear integer optimization problem may be very difficult for presently available methods. Such a problem can be reduced to a continuous problem involving only the cost and unavailability of each safety system involved by the following. (1)Obtain a minimal cost-unavailability function, C,(Qi)for each safety system from a series of optimizations defined by eq 1 and 2 . ( 2 ) Minimize
“ (3) subject to the risk constraints fj(Q1,
*.., 8,) 5 kj
0‘ = 1, *..,m )
(4)
The above formulation implies that the behavior of the individual subsystems is independent. Generalization of Suboptimization Problem The system characteristics which allowed the subsystem optimization above to be effective were the following: (a) The tasks to be accomplished lead to clearly identifiable subsystems. (b) The decision variables affecting the performance cost or state of any one subsystem do not affect the performance cost or state of any other subsystem. (c) The performance and contribution of the subsystems to system cost can be described in terms of the state variables. (d) Only a small number of state variables are required to describe the subsystem behavior (in the previous case just one-unavailability). A set of linked subsystems with the foregoing characteristics can be described as “weakly connected”. Such systems are linked only through the state variables. The definition of a “weakly connected” system implies that the decision variables for one subsystem will not appear in the constraints of any other subsystem. Further, constraints on the overall system which are related to subsystem behavior may be written in terms of the state variables. Himmelblau (1967) has considered the decomposition of large-scale process systems into so-called “irreducible systems” which can be examined independently. Himmelblau’s definition of an irreducible system corresponds to the second of the characteristics required by a weakly linked system (decision variables affect only one subsys-
tem). Himmelblau (1967) showed that an irreducible subsystem was one in which a recycle stream feeds back information or material, from some point downstream of an element, effectively forming a cyclical loop on the graph of the process. A set of irreducible subsystems may be considered to be weakly linked if their performance may be described in terms of a small number of state variables. The term “weakly connected” is consistent with the terminology of graph theory. There elements are considered to be either part of a “strongly connected” subset or outside such subsets (Sage, 1977). “Weakly connected” elements may be considered a subclassification of those elements not in strongly connected sets. The number of state variables needed to describe system performance determines the practicality of the suboptimization process. Determination of the required functional relationship in the SHRS cost-unavailability problem was relatively simple since only a single state variable was involved. The procedure is still practical if two or three state variables are involved but it is of doubtful utility for a large number of state variables. Thus, in a problem similar to that previously considered, the size of the system might not be fixed. It would then be necessary to determine the relationship between cost and unavailability for several different plant sizes. By appropriately interpolating between the curves of cost vs. unavailability for various plant sizes, the overall optimization program could obtain the data it needed for evaluation of any particular point. Once the overall optimization procedure has been completed, the most desirable cost and state variables for the individual subsystem are known. However, the detailed configurations of the subsystems are not known. The detailed configuration is readily obtained by appropriately resolving the original subsystem optimization problems. In the sample problem, SHRS cost would be minimized subject to the requirement that the unavailability be equal to or less than that obtained from the total system study. If both size and unavailability were decision variables, the foregoing problem would be solved with the additional constraint that the subsystem size be equal to or greater than the size determined in the overall analysis. Applications Although the application of the suggested approach is restricted to situations where the number of state variables is limited, there are a number of problems of practical importance where the solution is facilitated by use of the proposed suboptimization technique. For example, consider the design of a large system such as a nuclear power plant which is constrained by a requirement that the overall system unavailability shall be less than some set level, It. In that case our overall objective function is n
minimize C(Q,,
..., QN)
=
1 Ci(Qi) i=l
and the total system would be constrained by Qsystem = g(Qi, Q2, .-.,QA 5 It
(5)
(6)
Optimization of each of the individual subsystems (using eq 1 and 2 ) a t a series of unavailability levels would again provide the needed functional relationships between costs and unavailability. The corporate capital budgeting problem under conditions of risk provides another application area. Weisman and Holzman (1972a,b) considered the design of an engineering system of fixed size under risk. When the enterprise is adverse to risk and has a utility function u(r) which can be approximated by
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Ind. Eng. Chem. Process Des. Dev. 1980, 79, 197-198
u ( r ) = 1 - e-ar (7) and the total costs (C,) can be considered to be normally distributed, then the appropriate objective function for design of the fixed size individual system is a
minimize E(C,) + -(a2(C,)) (8) 2 where a2(Ct),the cost variance, includes the effects of both price uncertainties and possible failure to meet required specifications. Subsequently, Weisman et al. (1975) extended the approach to consideration of the design of an individual chemical plant where both plant size and time of construction were to be determined. These questions required a consideration of the projected sales, product selling price, feed costs, as well as the projected variation in these quantities. The more general, and probably more frequent, capital budgeting problem arises when an enterprise is required to simultaneously consider a number of new plants. Here there must be a determination of which of these plants should be built, what size they should be, and when they should be constructed. If the problem is considered as a whole, with the overall optimization program required to select the decision variables governing the configuration of the various plants, the problem is generally too large for any effective solution. The present approach may be readily applied here. The decision variables affecting the design of one plant do not affect the design or costs of another. Hence, the cost vs. size relationship can be determined for each plant independently. This can be done by selecting eq 8 as the objective function and determining the cost (C,) and cost variance a2(C,) as a function of plant size. One is justified in doing this since once plant size is assumed to be fixed one is operating under the same conditions which give rise to eq 8 as the appropriate objective function. The relationships between plant size and cost, and plant size and cost variances, serve as input to the overall capital budgeting problem. This problem is now freed of the necessity for considering the decision variables affecting the individual design and configuration of the various plants. The only variables are plant sizes, times of construction, and those designating whether the various plants should be built a t all. These latter variables may be treated as zero-one integers while sizes and times may be taken as continuous. Alternatively, only plants of fixed
throughputs may be considered and the throughputs obtained from
S I = (X,)"k
(9)
where X , is an integer which takes on the values 0, 1, 2, 3 etc. and k and m are constants set to provide the throughputs of interest. The objective function would be the maximization of the expected utility for the several investments and would follow the form previously derived by Weisman et al. (1975). For the usual case, where consideration is being given to perhaps three to five plants, the mixed integer optimization problem is of a size which can be handled readily by existing programs such as ARSTEC (Rasmuson and Marshall, 1976).
Nomenclature C( ) = cost function for the system C,( ) = cost function for subsystem i Q, = unavailability for subsystem i u = utility k, = constant f,( ) = risk constraint equations g( ) = unavailability constraint equations n = number of subsystems subscript t = total subscript i = subsystem i Literature Cited Burdick, G. R., Rasmuson, D. M., Weisman. J., "Probabilistic Approaches to Advanced Reactor Design Optimization", p 395 of "Nuclear Systems Reliability Engineering and Assessment", J. B. Fussell and G. Burdick. Ed., Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1977. Heuckroth, M. W., Gaddy, J. L., Gaines, L. D., AIChE J . , 22, 744 (1976). Himmelblau, D. M., Chem. Eng. Sci., 22, 883 (1967). Computer Program for Solving Rasmuson, D., Marshall, N., "ARSTEC-A Nonlinear Mixed-Integer, Optimization Problems Using the Adaptive Random Search Technique", RE-S-76-180, Idaho Nuclear Eng. Lab. (1976) (Program available from Argonne Code Center, 9700 S. Cass Ave., Argonne, Ill.). Sage, A. P., "Methodology of Large Scale Systems", p 124, McGraw-Hill, New York, N.Y., 1977. Weisman, J., Holzman. A., Ind. Eng. Chem. Process Des. Dev., 11, 386 (1972a). Weisman, J., Hoizman, A,, Manage. Sci., 19, 235 (1972b). Weisman, J., Pulido, H.. Khanna, A., Ind. Eng. Chem. Process Des. Dev.. 14, 51 (1975).
Joel Weisman'
University o f Cincinnati Cincinnati, Ohio 45222
EG & G Idaho Inc. Idaho Falls, Idaho 83401
Garry R. Burdick Dale M. Rasmuson Received f o r review October 2, 1978 Accepted July 12, 1979
Differentiating between Two First-Order Reactions and a Single Second-Order Reaction
Oleck and Sherry (1977) speculated that there may be several species in the hydrodemetallationof petroleum residue, each obeying first-order kinetics, so that the aggregate appears to obey second-order kinetics. The resolution between these two kinetics requires experimental data of sufficiently high conversion, or sufficient variations in initial concentrations. I t is shown here that two first-order kinetics can be used to approximate a second-order kinetic only at less than 92% conversion, and only when the ratios of the parameters of the two first-order reactions fall within very narrow ranges.
Several authors (Chang and Silvestri, 1974, 1976; Inoguchi, 1971; Oleck and Sherry, 1977) found that the hydrodemetallation reaction of petroleum residue can be described as either second-order kinetics or first-order 0019-7882/80/1119-0197$01.00/0
kinetics. Oleck and Sherry also commented that there may be several reactive species forming "lumps", each obeying first-order kinetics, so that the aggregate appears to obey second-order kinetics. 0 1979 American Chemical Society
