Ind. Eng. Chem. Process Des. Dev. and Fluid Propertles in the Chemical Process Industries, Berlin (West), 17-21 Mar 1980. Brm, H. I.; Luecke, R. H. TechmmeMcs 1973, 15, 233. Chan, W. K.; Boston, J. F. "A New Alsorfthm for the Construction of P-T Emrdopes", Accepted for publicetion in Ind. Eng. Chem. Process Des. Dev. 1981. Chao, K. C.; Un. H. M.; Nagsshwar, 0. D.; Kim, H. Y.; Oiiphant. J. L.; Sebastian, H. M.; Slmnick, J. J. "Phase Equilibrium in Coal Liquefaction Processes", Final Report, Research 367-2.Electric Power Research Institute, Palo Aito, CA. 1980. Chueh, P. L.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1967, 6 , 492. Conndiy. J. F. J. Chem. f h y s . 1962, 36,2897. Culberson. 0.L.; McKetta. H. J., Jr. Pet. Trans. AIM€ 1950, 189, 321. Culberson, 0. L.; McKetta, H. J., Jr. Pet. Trans. A I M 1051, 192, 297. Dechema Chemistry Data Series "Vapor-Uquld Equilibrium Data Collection"; Bahrens, D.; Eckermann, R., Ed.; Dechema. Deutsche Geseiischaft fur Chemisches Apparatewesen: Frankfwt/Main, West Germany, 1977;Voi. 1, Part 1. Deming, E.; Shupe, L. E. fhys. Rev. 1932, 4 0 , 848. Dymond, J. H.; Smith, E. B. "The Virlai Coefficients of Gases": Oxford-Ciarendon Press: Oxford, 1969. Gmhling, J. D.; Liu, D.; Prausnitz. J. M. Chem. Eng. Sci. 1979, 34, 951. Graboski, M. S.;Daubert, T. F. Ind. Eng. Chem. Process Des. Dev. 1976, 17, 443. Heidemann, R. A.; Khall. A. M. "The Calculation of Critical Points"; Presented at the 86th National AIChE Meeting. Houston, Apr 1979. Henderson, D. A&. Chem. Ser. 1970, No. 182, 1. Huron, M-J.; Vidal, J. HUM Phase Equiii6. 1979, 3 , 255. Jaffe, J. Ind. Eng. Chem. Process D e s . Dev. 1981, 20, 168. Kkk, B. S.; Zlegler, W. T. A&. Cryog. €ng. 1965, 10, 160. Kiink, A. E.; Cheh, H. Y.; Amick. E. H., Jr. AIChE J. 1975, 2 1 , 1142.
1983,22, 391-396
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Krichevskii, I.R.; Zhavorokov, N. M.; Tsikiis, D. S. Zh. Fiz. Khim. 1937, 9 ,
317. Martin, J. J. Ind. Eng. Chem. Fundam. 1979, 16, 81. Mathias, P. M.; Boston, J. F.; Watansiri, S. "Effective Utilization of Equations of State for Thermodynamic Properties in Process Simulation", AIChE J. in press. Meyer, C. A.; McCiintock, R. B.; Sihrestri, G. J.; Spencer, R. C., Jr. "1967 Steam Tables-Thermodynamic and Transport Properties of Steam C o n prising Tables and Charts for Steam and Water"; ASME, United Engineering Center, New York, Dec 1967. Micheison, M. L. FiuM P h a Equiib. ~ 1080, 4 , 1. Nichols, W. B.; Reamer, H. H.; Sage, B. H. AIChE J. 1057, 3 , 262. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. Reamer, H. H.; OMS, R. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1996, 2 8 , 1936. Rediich, D.; Kwong, H. N. S. Chem. Rev. 1049, 4 4 , 233. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. "The Properties of Gases and Liquids", 3rd ed., McGraw-HIII Book Company: New York, 1977;p 630. Saddington, A. W.; Krase, N. W. J. Am. Chem. Soc. 1934, 56, 353. Soave, G. Chem. Eng. Scl. 1072, 2 7 , 1197. Tokunags, J. J. Chem. Eng. Data 1975, 2 0 , 41. Whiting, W. B.; Prausnitz, J. M. "Equatlons of State for Strongly Nonideai Fluid Mixtures: Appllcation of the Local Composition Concept"; Presented at the AIChE Spring National Meeting, Houston, 1981. Wiebe, R.; Gaddy, V. L. J. Am. Chem. Soc.1934, 56, 76. Wiebe. R.; Tremearne, T. H. J. Am. Chem. Soc. 1034, 56, 2357. Williams, R. B.;Katz, D. L. Ind. Eng. Chem. 1954, 46, 2512.
Received for review June 29, 1981 Accepted October 13, 1982
Close Approximations of Global Optima of Process Design Problems C. Y. Lut and J. Welsman" Department of Chemiai & Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 4522 1
Procedures are devised for transforming process optimization problems Into signomlal programming problems. An investigation of the various algorithms proposed for finding the global optimum of such a signomiai programming problem showed the Falk algorithm to be most suitable. When the Falk procedure is coupled to an efficient optimization program, global optima to small size problems are readily obtained. In a moderate size problem, a close approximation of the global optimum can be obtained with reasonable computing effort.
Introduction The use of formal optimization or mathematical programming procedures has become quite common in process engineering design. A number of large size, nonlinear, constrained optimization computer programs are generally available. Such programs are capable of efficiently solving problems of the form max ( m i d f(x) (1) subject to g i ( m = bi (i = 1, 2, ..., 1) (24 (i = 1 + 1, 1 2, ..., m) (2b) gi(X){S,ZJbi Nonlinear problems with more than 75 variables and a similar number of constraints can now be readily handled. A wide variety of optimal design problems have been successfully solved by this approach. Essentially all of the available optimization programs are hill climbing (descending)techniques. They locate the constrained mRnimum (or minimum) closest to the starting point, but they will not necessarily determine the global maximum (or minimum) if there are multiple extrema.
+
f
Case Western Reserve University, Cleveland, OH 44106. 0196-4305/83/1122-0391$01.50/0
One can be certain that the global optimum has been obtained only if a maximization problem is a concave programming problem or a minimization problem is convex. We have a convex programming problem if (a) the function being minimized, f ( X ) is convex and (b) the contrainta are only inequalities of the form gi(X)Ibi with each g i ( X ) also being convex. A concave programming problem is the maximization of concave objective function subject to g i ( X )I bi where the g i ( X )are concave. Almost all practical design problems are neither convex nor concave and one cannot be certain that the global optimum has been found. Most designers have been willing to accept the fact that they may not have located the global optimum providing the minimum or maximum they have located is considerably better than they would have achieved without the use of the formal optimization procedure. However, in many cases the system being designed is very costly and an expediture of a considerable design effort can be justified to locate a better optimum. A common procedure in this case is to repeat the optimization process for a series of different starting points in the hope that one of the chosen starting points will be in the vicinity of the global optimum. This technique is often successful. However, Wilde (1979) points out that in some real optimization @ 1983 American Chemlcal Society
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problems the global optima are so far from what is considered a reasonable starting point that no starting point is likely to be chosen in their vicinity. Such optima are likely to remain hidden when any locally convergent optimization technique is used. Attempts have been made to develop generalized procedures which can find the global extremum of a multimodal function (see Dixon and Szego, 19751, but no real success has been attained. In view of this, attention has been given to problems of specific structure where global optima can be theoretically achieved by solving a sequence of optimization problems. Signomial Programming In the 1960's, Duffin, Peterson, and Zener (1967) originated "geometric programming". Geometric programming problems have the form K
min f ( x ) = C Cjpj(x) j=l
(3)
subject to T
where
pt,(X) = fixharib k=l
(5b)
and all the C!, Cti, and bi are positive numbers (a's may be zero, positive, or negative). The functions represented by the objective function and constraints are designated "posynomials". Duffii et al. (1967) developed specific algorithms which would find the global optimum of a geometric programming problem. They also noted that if each x k is replaced by e'k the resulting problem is convex with respect to the zk. Gottfried and Weisman (1973) and others have pointed out that any locally convergent optimization procedure may be used with the transformed program to achieve the global optimum. Geometric programming per se is of limited interest to design engineers since almost all real problems contain both positive and negative terms. We then have a "generalized polynomial" or "signomial" programming problem. That is, a programming problem defined by eq 3,4, and 5 but with the Cj and the Cti allowed to be either positive or negative. A signomial programming problem is not convex and many have a number of local optima. Falk (19731, Passy (1978), and Gochet and Smeers (1979) among others have noted that is is possible to approximate a signomial programming problem by a series of convex programming problems. If the negative terms in the transformed problem (problem in which the x k have been replaced by ezk) are replaced by convex approximations, the resulting problem is convex and has a global optimum. By solving a series of such convex problems, in which successively closer approximations of the negative terms are used, a close approximation of the global optimum of the original problem may be obtained. Objectives and Approach of Present Investigation The objectives of the present investigation were twofold, namely: (1)to determine whether the usual process design problems can be expressed in a manner such that the signomial programming techniques capable of locating a global optimum can be applied, and (2) to determine if the available signomial programming techniques are capable
of locating the global optima of typical process design problems with a reasonable computation effort. To accomplish the foregoing, a number of process design problems in the literature were examined and the means by which they could be appropriately transformed were determined. The applicability of the available algorithms was then determined by applying all of them to a series of small test problems and a small size design problem. The best of the available algorithms was then applied to a moderate size process design problem and the efficiency of the technique was evaluated. Transformation of Process Design Problems to Acceptable Form A number of optimal process design problems which have appeared in the literature [e.g. (Perry and Singer (1968), Robertson and O'Grady (1966), Weisman and Holzman (19721, Weisman et al. (1975), Rijkaert and Martens (1974)]were examined to determine the ways in which typical process design problems differ from a signomial programming problem. It was observed that if such problems were to be transformed appropriately it would only be necessary to provide procedures for dealing with (a) reversed inequality constraints (greater than constrains), (b) equality constraints, (c) reciprocals of signomials, (d) exponential functions of the form [exp(CP)] where C is a positive or negative constant and P is a posynomial, (e) logarithms of signomials, and (f) trigonometric functions. Provision for reversed inequality constraints is trivial. Multiplication of both sides of the equation by (-1) transforms the inequality appropriately. If the algorithm used requires the bi to be positive, a suitable positive constant may be added to each side of the inequality. Each equality constraint may be replaced by two inequality constraints. That is, each g,(X) = b, is replaced by gi(x) I b, + (64 (6b) -gi(x) -< -b, where t is a specified tolerance. Constraints which contain reciprocals in the form (7)
where both hi(X) and gi(X) are signomials with positive values and C is a positive or negative constant, are replaced by C - + gi(x) I bi (8) Xn+1
The additional constraint
MX) 5 0
(9) must also be imposed. If the reciprocal appears in the objective function, the same substitution and additional constraint are used. To deal with exponential and logarithmic terms, we make use of the fact that all of the signomial programming algorithms of interest replace the original x k by elk. This yields convex terms for the positive terms and concave terms for each of negative terms. The concave terms are then replaced by convex approximations. To use these algorithms, it is not necessary that the transformed program be a signomial but simply that, after replacement of the x b by e", the positive terms be convex and the negative terms concave. Thus if one of the constraints is of the form (10) Ft explhi(X)l + gi(x) 5 bi x,+1-
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 393
where hi(x)may be any signomial, the problems will still be convex after each xk is replaced by ezh providing Fi is a positive constant. If Fi is a negative constant, the negative of the exponential term will be concave and this is also satisfactory. Hence, no revision of such a constraint is required. If Fi represents a signomial rather than a constant, we rewrite the original constraint as Fixn+I
+ gi(x)
5 bi
(11)
and add the constraint exp[hi(X)] - xn+1 5 0 (12) An exponential term appearing in the objective function is handled in the same manner. The presence of logarithmic terms is treated similarly. If a constraint is of the form Fi[ln (hi(x))l+ gi(x) 5 bi (13) where Fi may represent a constant or a signomial, we replace the constraint by Fi(xn+k)
+ gi(x) 5 bi
(14)
and
hi(X)- exn+k5 0
(15)
The same treatment is used for logarithmic terms in the objective function. We may term problems with transformed logarithmic or exponential terms as modified signomial programs. “higonomet+ functions must be replaced by their series expansions. In most engineering problems, only the first few terms are required to obtain the needed accuracy. However, since the series for sine and cosine contain both positive and negative terms, this replacement may significantly increase the number of negative terms in the problem. From the foregoing, it is clear that under nearly all circumstances optimal process design problems can be converted to a form suitable for solution by signomial programming techniques. On the basis of a more limited examination, Beightler and Philips (1976) previously came to a similar conclusion. However, in making the conversion, the number of variables, constraints, and negative terms may be very appreciably increased. These additions significantly increase the time required to obtain a solution. Algorithms for Solution of Signomial Programming Problems The three algorithms [Falk (19731, Passy (19781, and Gochet and Smeers (1979)] which, at the outset of this study, appeared to have the capability of locating the global extremum of a modified signomial programming problem, have a number of major features in common. Each of the algorithms transforms the modified signomial problem into a series of convex programming problems. All the convex problems are obtained by replacing the x k by e Z k in each positive term and approximating each negative term by a convex function. In addition, each algorithm includes the logical structure required to generate a sequence of approximating problems. All of the algorithms obtain an approximation of the global optimum from the sequences of approximating problems through use of the “branch and bound” procedure [Lawler and Wood (1966)l. The “branch and bound” procedure is an intelligently structured search of the space of all feasible solutions. The space of all feasible solutions is repeatedly partitioned into smaller and smaller subsets and a lower bound (in case of minimization) is calculated for the objective function of the solutions within each
I
1
r
f m2 ml Figure 1. Linear approximations of negative terms used by Falk algorithm. subset. After partitioning, those subsets with a bound that exceeds the objective function of a known feasible solution are excluded. The partitioning continues until a feasible solution is found such that its objective function is no greater than the bound for any subset. When carried to completion, the procedure provides a close approximation of the global extremum. However, when there are a large number of alternatives which must be explored, as there will be when there are a large number of negative terms, branch and bound procedures can become very time consuming. In the Falk (1973) algorithm, which is an adaptation of the work of Falk and Soland (1969) on separable programming, each negative terms is replaced by a linear approximation. For example, if the term (ceara)is being approximated between the limits ml and m2,we have the linear approximation L1 (see Figure 1). If the solution of this problem yields zi*, and we select this term to branch upon, we use zi* as one of the new limits and create two new linear approximations, L2and Ls.The procedure is continued, each time we branch upon this term, by using the value of zi obtained at the last step to create two new problems to branch upon. Since each linear approximation overestimates the negativelterms, the constraints based on the linear approximation are more easily satisfied than the original constraints. This, plus the overestimate of any negative terms in the objective function, means that any solution obtained will be a lower bound. A feasible solution, which provides an upper bound, is obtained at each step by solving the original signomial problem using the last approximate solution as a starting point. The three algorithms differ primarily in the way in which they approximate negative terms. While the Falk (1973) method replaces each negative term by a linear approximation, the Passy (1978) method replaces the nonconvex constraint set by a union of quasiconcave functions. The resulting set (functions) is convex. The Gochet and Smeers (1979) approach replaces all of the negative terms in a given constraint by a single term posynomial. The transformed problem is then a posynomial programming problem. The several approximation methods lead to different branching (partitioning) paths and this leads to different efficiencies for the algorithms. At the onset of the study, it was expected that the Gochet and Smeers (1979) approach would be most efficient since it introduced fewer terms into the transformed program.
Evaluation of Signomial Programming Algorithms Three computer programs embodying each of the three algorithms of interest were written [see Lu (1980)l. Each of these programs was then joined to an efficient optimization program for the solution of the convex programming problems produced by the main program. Note that the transformed problems are no longer in the geometric
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programming format. Hence, we are not restricted to the selection of optimization procedures which are designed for use with geometric programming problems. We may therefore choose an optimization technique from a wide range of algorithms. The optimization technique chosen was the direct search procedure of Hooke and Jeeves (1962), as modified by Weisman et al. (1965), for unconstrained minimization. The unconstrainted minimization problems were in turn produced by using a standard exterior point penalty function approach for treating the constraints. The relative efficiency of the three signomial programming algorithms was then evaluated by solving a series of design optimization programs selected from the literature. Four very simple design problems were used for an initial evaluation. This was followed by evaluation of all of methods with a small design problem, and finally the best of the approaches was used to solve a medium size design problem. Table I illustrates the results obtained. The solution times are CPU times required on the University of Cincinnati's Amdahl470 main frame computer. The fmt four problems represent the very simple design problems used initially. From these four resulta, it is clear that the Passy method is least effective, although it computes problem 1 more rapidly, it fails to complete 2 of the problems, and in 3 out of the 4 cases it has the poorest accuracy. The Falk algorithm is clearly the best of the three. In the two longest problems, it required much less computer time than the Gochet-Smeers algorithm and it had the highest accuracy in all cases. The superiority of the Falk algorithm was confirmed in the examination of a small size design problem. The problem chosen for this purpose was the optimal design of the Williams-Otto process (1960) as formulated by Rijkaert and Martens (1974). The problem (problem 5 in Table I) which had 13 decision Variables and 11 constraints was solved in a little over 2 min by the Falk algorithm, but the other two algorithms failed to converge to a solution in 50 min. As a final test, the optimal nuclear reactor design of Weisman and Holzman (1972) was used. Although the original problem was formulated with 12 decision variables and eight inequality constraints, the complex design relationships imposed led to a signomial problem in 20 decision variables and 16 constraints. This contrasts with only 13 variables and 11 constraints for the signomial form of the Williams-Otto process optimization. Since both the Passy and Gochet-Smeers procedures failed to converge to a solution in problem 5, only the Falk algorithm was considered. Initial performance of the Falk algorithm with the Weisman and Holzman (1972) design problem was somewhat disappointing. With a very wide variation allowed for the bounds of some of the decision variables, no final solution was obtained in 40 min of computing time. The difference between upper and lower bounds on the solution were within 5.7% of each other (% based on lower bound) at the end of 40 min (see Table 11). However, the performance of the algorithm was found to depend significantly on the range over which the decision variables appearing in negative terms could vary. When more reasonable bounds were set for these variables (see Table 11), the upper and lower bounds were within 3.8% of each other in 40 min of computing time; 30.4 min was required to bring the two bounds within 5% of each other. In relatively complicated design problems, it would appear that setting a desired range for the deviation between
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Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 395
upper and lower bounds is an appropriate way to terminate the problem. Complete fathoming of every branch apparently can result in excessive computation times. Since most engineering cost estimates have significant uncertainties associated with them, a 3 to 5% range on the objective function a t the optimum will be generally acceptable. Nevertheless, the 30 to 40 min of CPU time required to bring the solution of the Weisman-Holzman problem to this level of precision is somewhat longer than some designers would consider acceptable.
Effect of Revised Optimization Procedures The computing time requirements are determined by the efficiency of the optimization procedure used to solve each convex programming problem as well as the constraint approximation procedure. If the time for solving each convex problem were reduced by a given factor, the total computing time would be reduced by nearly the same factor. It is therefore desirable to determine whether recent developments in nonlinear programming techniques can significantly reduce the computing time required. One of the major recent developments in nonlinear programming is Powell’s (1978) variable metric method for constrained optimization. The method is based on Davidon’s (1959) variable metric method for unconstrained minimization and makes use of the ideas of Han (1976) and others. The method uses estimates of the Lagrange multipliers to impose the required constraints. At each step in the program, a quadratic programming problem is solved to obtain a revised estimate of the Lagrange multipliers to be used in the next step. Application of this method to a variety of problems has shown that it generally requires the lowest number of functional evaluations of any presently available constrained optimization procedure. For this reason, it has been found very useful in large scale system optimization where each functional evaluation takes many seconds [see Geyer (198l)l. The efficiency of Powell’s (1978) optimization procedure was evaluated by using it to solve the signomial programming problem obtained from Weisman and Holzman’s (1972) nuclear reactor design. The times required for several different starting points were compared to the times needed by the original (direct search plus penalty function) optimization procedure. It was found that approximately the same time was required for both procedures indicating that the Powell method would not provide any improvement. Although the Powell method no doubt reduces the number of functional evaluations required, the considerable additional work it performs after each functional evaluation balances the reduced number of iterations required. In large-scale system optimization, where time required to evaluate the system for a given set of decision variables will be much longer, the Powell method would be expected to be superior. Another recent improvement in nonlinear programming is the use of transformed penalty functions. For the classical penalty function proceudre to work, the constraints violations must be heavily weighted with respect to the objective function. This gives rise to a very illconditioned unconstrained minimization problem which is difficult to solve. Powell (1969)suggested that this difficulty might be overcome by perturbing the constraints slightly, and recently Charamblous (1980) developed an efficient algorithm for carrying this out. By application of suitable penalty functions, a constrained minimization problem is converted to an unconstrained minimization problem whose extremum approaches that of the original problem. For problems defined by eq l and 2 but with all gi(X)I 0, the conventional
exterior penalty function approach yields an objective function of the general form min[41 = f ( ~ +) E s i ~ , [ g i ( ~ ) ~ q i=l
(16)
where Gi = 0 if g i ( X ) I0, Gi = 1 if g , ( X ) 1 0, and s, and q are positive numbers. At each iteration, the value of s, is increased for those constraints not yet satisfied. As the si approach infinity, the extremum of (16) approaches the extremum of the original problem. In the algorithm of Charamblous (1980) each constraint is perturbed inward by an amount ti 1 0. We then obtain as an objective function m
42
= f ( x ) + Cst6i[gi(X)- tilq Pl
(17)
By appropriate selection of a series of si and ti for given q, the extremum of (17) can be made equal to the extremum of the original problem with finite values for the components of si. The ill-conditioning caused by the penalty function is thus avoided. Charamblous (1980) provides a procedure for obtaining ti through the Lagrange multipliers. His numerical experience with the algorithm showed it to be substantially faster than conventional penalty function approaches. Similar results were obtained in the present study. The signomial programming problem derived from Weisman and Holzman’s (1972)nuclear reactor plant design problem was solved, for several random starting points, by using both the original penalty function approach and Charamblous (1980) revision. It was found that the Charamblous algorithm was about 2.5 to 3 times faster. Hence, with the Charamblous exterior point penalty function algorithm the appoximate global optimum (within 5%) would be obtained ill 10 to 12 min of CPU time on the University of Cincinnati’s main frame computer. Conclusion It is now feasible to obtain close approximations of the global optima to moderate size process design problems. The procedure involves expression of such problems as modified signomial programming problems and use of the Falk (1973) algorithm to obtain the global optimum via branch and bound. To achieve this result with a reasonable computing effort, it is necessary to use an efficient means for solving the nonlinear programming problems generated at each step of the branch and bound procedure. Direct search coupled to the transformed penalty function technique of Charamblous (1980) was satisfactory for solution of these problems. Although the suggested procedure’s computing time requirements are moderate, the conversion of a design problem into acceptable signomial form can require a significant amount of engineering time. Some designers may prefer resolving the original design problem (without placing it in signomial form) for a large number of random starting points and choosing the best solution. With the Weisman and Holzman (1972) design problem, results very slightly better than the lowest upper bound from the Falk algorithm were obtained in only 6 min of CPU time. However, use of random starting points does not provide the upper and lower bounds on the solution obtainable through the branch and bound features of the proposed approach. Nomenclature a, b = constants C constant f ( X )= objective function
Ind. Eng. Chem. Process Des. Dev. 1983, 22, 396-401
398
F = constant or signomial g ( X ) = constraint function h ( X ) = signomial function P ( X ) = posynomial function q = constant si = penalty factor for ith constraint ti = perturbation constant for ith constraint x k = decision variable (X)= vector of decision variables z k = transformed decision variable (zk = In x k ) 4 = modified objective function
Literature Cited Beightler, C. S.; Philllps, D. T. "Applied Geometric Programming"; Wlley: New York, 1976; p 458. Charamblous, C. Oper. Res. 1960, 28, 650. Cohrille, A. R. "A Comparative Study of Nonlinear Programming Codes"; New York Scientlflc Center, IBM Corp.: New York, 1968; Report 3202949. Davtdon, W. C. Argonne National Lab Report, ANL 5990, Argonne, IL, 1959. Dlxon. L. C.; Szego, G. P., Ed. "Towards Global Optlmlzation", N. Hdiand-Elsevler: Amsterdam, 1975. Duffin, R. J.; Peterson, E.; Zener, C. "Geometric Programmlng"; Wlley: New York, 1967. Falk, J. E. "Global Solutions of Slgnomial Programs"; George Washington Unlversky, Washington, D C Research Report T-274, 1973. Falk, J. E.; Soland, R. Manage. Sci. 1969, 75, 550. Geyer, H. "GPSAPIV2 with Applications to Open-Cycle MHD Systems"; Argonne National Lab Report ANL/MHDdO-15, argonne, IL, 1981.
W h e t , W.; Smears, Y. Oper. Res. 1979, 27, 962. Gottfrled, 8.; Weisman, J. "Introduction to Optlmlzation Theory"; Prentice Hall: Engelwood Cliffs. NJ, 1973; p 284. Han, S. P. Msth. Program. 1876, 1 7 , 263. Hooke, R.; Jeeves, T. A. J. Assoc. Comput. Mach. 1962, 8 . 212. Lawler, E. L.; Wood, D. E. Oper. Res. 1966, 14, 699. Lu, C. Y. M.S. Thesis In Chem. Eng., Unlv. of Cinclnnatl, Clnclnnatl, OH, 1980. Martens, X. M. Ph.D. Thesis, Kathoheke Unlversky Leuven, 1971. Passy, U. J. Optim. TheoryAppl. 1978, 26, 97. Perry, R. H.; Slnger. E. Chem. Eng. 1966, 75, 163. Powell, M. J. D. In Proceedhgs of 1977 Dundee Conference on Numerical Analysis, published In Lecture Notes In Mathematics, Sprlnger-Verlag: 1978; Vol. 630, p 144. Powell, M. J. D. "Optimization"; Fletcher, R., Ed.; Academic Press: London, 1969. Rljkaert, M. J. Martens, X. M. AIChEJ. 1974, 20, 742. Robertson, H. H.; O'Qrady, W. P. "Steady-State Optimization of the Oxysynthesis Process"; Proceedings of International Federation of Automatlc Control: London, 1966. Weisman, J.; Holzman, A. G. Ind. Eng. Chem. Process D e s . D e v . 1972, 7 7 , 386. Weisman, J.; Pulido, H.; Khanna, A. Ind. Eng. Chem. Process D e s . D e v . 1975, 14, 51. Weisman, J. C.; Wood, C.; Rlvlln, L. A I M Symp. Ser. 1885, 67(55), 50. Wllde, D. J. "HLdden Optlma In Engineering Deslgn", in "Constructive Approaches to Mathetmatlcal Modellng"; Coffman, C. V.; Fix, G. J., Ed.; Academlc Press: New York, 1979. Wllllams, T.; Otto, R. AI€€ Trans. Part I 1960, 79, 458.
Received for review September 2, 1981 Accepted December 9, 1982
Reactor Model for the Gasification of Black Shale in the Ftuidized Bed. Comparison with Pilot Plant Data Hans Eklund"+and Owe Svensson' Department of Chemical Engineering, Lund Institute of Technology, 5.220 07 Lund 7, Sweden
A reactor model for an oxygen Mown bubbli fluMiied bed gasifler has been developed to simulate the performance of a pilot plant gasifier. Input data to the model are kinetic parameters for the fixed carbon gasification, pyrolysis yields, steam/oxygen feed rates, bed temperature, and expanded bed height. The algorithm calculates mass and energy balances, e.g., carbon and steam conversions, together with parHal pressure profiles for six gas components throughout the bed. The solid phase is assumed to be perfectly mixed, and the gas phase flow is assumed to be nonideal, accounted for by an eddy diffuslvity parameter. Due to the particle properties of the shale, e.g., large density and coarse particles, this quite simple approach was shown to give accurate resuits. The model predictions are compared with experimental data from a 6 ton/day pilot gasifier.
Introduction A development program for exploitation of the Swedish shale deposits has been going on for a number of years. At first, the interest was primarily assinged to recovery of u "from the shales, but after the oil embargo in 1974, utilization of the fossil energy in kerogen became potentially interesting. The work was directed toward the gasification route, and the fluidized bed reactor was considered to be the best alternative to cope with the requirements on a thermal process integrated with a hydro-metallurgical treatment of the shale for recovery of uranium. The work was initiated by building small bench-scale units and laboratory equipment to bring up basic kinetic Norsk Hydro as. Research
Center, N-3901 Porsg~nn,Norway.
* Ranstad Skifferaktiebolag,S-520 50 Stenstorp, Sweden.
0196-4305/83/1122-0396$01.50/0
data and certain engineering knowledge to be used when planning a pilot plant and also as a basis for feasibility studies. This work has been reported by Bjerle et al. (1980), Berggren et al. (1980), and Bjerle et al. (1982). Parallel to these basic studies, a 6 ton/day pilot plant was erected at the site of the uranium shale deposit at Ranstad in the central part of Sweden. A simplified flow sheet of the plant is shown in Figure 1. The gasifier is operated as a bubbling fluidized bed, having a diameter of 0.4 m. The nominal bed height of the fluidized bed can be varied in three steps between 2 and 4 m, by changing the overflow level as shown in the figure. The effective bed level, however, ranges between 1.5 and 3.2 m due to the slugging properties of the bed. To avoid agglomeration from segregation of coarse materal, char is also discharged by a screw conveyer in the bottom of the bed. After the reactor, fines are separated from the gas in a cyclone. Tar and steam are condensed in a spray scrubber. A sample of the 0 1983 American Chemical Society
