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CAP= concentration of the model metal etioporphyrin in the inlet solution to the reactor, ppm C, = concentration of model metal etioporphyrin inside the catalyst, ppm C B = concentration of the reaction intermediate in the solution, ppm Cb = concentration of the reaction intermediate inside the catalyst, ppm CM = total metal concentration in the solution, (CA + CB) DA = effective diffusivity of A in the catalyst extrudate, c m z / s DB = effective diffusivity of B in the catalyst extrudate, cmz/s fA = flux of A per unit mass of the catalyst f~ = flux of B per unit mass of the catalyst kl = first-order rate constant for hydrogenation step, (mL of solution)/ (g of cat. h) k , = first-order rate constant for dehydrogenation step, (mL of solution)/(g of cat. h) k3 = first-order rate constant for the demetallation step, (mL of solution)/(g of cat. h) kl+, k2+,k3+,k4+ = diffusion disguised rate parameters md = weight % metal deposition on the catalyst m d = net weight averaged metal deposition on the catalyst, %
P = hydrogen pressure of the reactor, MPa Q = solution flow rate to the reactor, mL/h R = radius of the catalyst extrudate r = radial length coordinate from the center of the catalyst extrudate T = temperature of the reaction, "C TR = time on stream, h W = weight of the catalyst, g p = solution density, g/mL pc = bulk density of the catalyst, g/mL Registry No. COO, 1307-96-6;MOO3, 1313-27-5;Ni, 7440-02-0; V, 7440-62-2;nickel etiochlorin, 89922-68-9; vanadyl etiochlorin I, 89922-69-0.
Literature Cited
Anderson, R. E. "Experimental Methods in Catalytic Research", Academic Press; New York, 1968; Chapter 1. Arey, W. F., Jr., Blackwell, N. E., 111; Reichie. A. D. Seventh World Petroleum Congress, Mexlco, 1967; p 167. Audibert, F.; Duhaut, P. Paper at the 35th Mklyear Meeting of the American Petroleum Instltute Dhrlsion of Reflnlng, Houston, TX. May 13-15, 1970. Beuther, H.; Schmld, E. K. Proceedings. 6th World Petroleum Congress, Sec. III, FrankfurtlMain, 1963, Paper 20, p 1-11. Bridge. A. G.; Green, D. C. Paper., Dlv. Pet. Chem., Am. Chem. Soc. 1979, 24, 791. Chao, K. C.; Seader. J. D. A I M J. 1861. 7 , 598. Copeland, T. M. Ph.D. Thesis, MIT, Cambridge, MA, 1978. Cukor, P. M.; Prausnk, J. M. J. Phys. Chem. 1972, 76, 598. Dautzenberg, F. M.; Van Kilnken, J.; Pronk, K. M. A.; Sie. S. T.; Wijffels, J. 8. ACS Symp. Ser. 1978, 65, 254. Docalswamy, L. K.; Tajbl, D. 0.Catal. Rev. Sci. Eng. 1974, IO, 177. Hardin, A. H.; Packwood, R. H.; Ternan, M. Prepr., Div. Pet. Chem., Am. Chem. SOC.1978, 23, 1450. Inoguchi. M.; Kagaya, H.; D a m , K.; Sakurade, S.; Satomi, Y.; Inaba, K.; Tate, K.; Nishiyama, R.; Onishi, S.; Nagai, T. Bull. Jpn. Pet. Inst. 1971, 13, 153. Komiyama, H.; Smb, J. M. A I C M J. 1974. 20. 728. Kwan, T.; %to, M. Nlppon Kagaku ZessM 1970, 9 1 , 1103. Loev, E.; Goodman. M. M. "Progress In Separation and Purification"; Perry, E. S.; Van Oss, C. J., Ed.; Wliey: New Ywk, 1970; Vol. 3, p 73. Mears, D. E. Chem. Eng. Scl. 1971a. 26, 1361. Mears, D. E. Id. Eng. Chem. Process Des. Dev. W7lb, IO, 341. Oxenrelter, M. F.; Frye, C. G.; Hoekstra, G. E.; Sroka, J. M. Paper presented at the Japanese Pet. Inst., Nov 30, 1972. PNcher, W. H., Jr. Sc.D. Thesis, MIT, Cambridge, MA, 1972. Prathef, J. W.; Ahangar, A. M.; Pltts, W. S.; Heniey, J. P.; Tarrer, A. R.; Guin. J. A. I d . Eng. Chem. ProcessDes. Dev. 1977, 16, 287. Sato, M.; Takayoma, N.; Kurka, S.; Kwan, T. Nippon Kagaku Zasshi 1971, 9 2 , 834. Satterfield, C. N.; Coiton, C. K.; Pitcher, W. H.. Jr. A I C M J. 1973, 79, 628. Shah. Y. T.; Paraskos, J. A. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 388. Spry. J. C., Jr.; Sawyer, W. H. Paper presented at 68th annual AIChE Meeting, Lo8 Angeles. CA, Nov 16-20, 1975. Tamm, P. W.; Harnskger, H. F.; Bridge, A. G. Ind. Eng. Chem. Process Des. Dev. 1981. 20. 262. Todo, N.; Kabe, T.;.Ogawa, K.; Kurka, M.; Sato, T.; Slhmada, K.; Kurikl, Y.; Oshima, T.; Takematsu, T.; Kotera, Y. Kogyo Kagaku Zesshi 1971, 74, 563. Wel, J. J . Catal. 1982a, 1 , 526. Wei, J. J . Catal. 1962b, 1 , 538. W e k , P. E. Doctoral Dissertation, Swiss Federal Institute of Technology, Zurlch, 1966.
Agrawai. R. Sc.D. Thesls, MIT, Cambrldge, MA, 1980. Aqawai, R.; Wei, J. Ind. Eng. Chem. Process Des. Dev. 1984, Preceding article In this issue.
Received for review March 28, 1983 Accepted August 4, 1983
Design of Regenerative Noncontinuous Processes via Simulation F. Carl Knopf,'t 0. V. Reklaltlr,' and M. R. O k o d L b w f i m t of Chemlml E m , Lordslena Stet8 Unhrslty, Baton Rouge, Loulsiana 70803, Lbpartment of Chemlcal Englnwing, and Depafimnt of Agicuttwal Enghwing, pvro'ue Untverslty, West Lafayene, Indiana 47907
A generalized noncontinuous system co118/8tkls of M parallel semicontinuous trains and their input queues, followed first by N parallel intermediate storage units and then by another stage of P parallel semicontinuous trains, is c-onsbred. A SLAM si" model is devekped which incorporates both multiproduct scheduting via dispatching rules as we# as optimal selection of processing rates and Hermediate storage volumes for regenerative processes, i.e., those that operate with cyclically reoccurring shutdown periods. A response surface strategy is used for executing the design optimization. An application with a fluid milk plant is described.
Introduction to Noncontinuous Processes Noncontinuous processes are representative of a considerable segment of the processing industry and are used to produce some of the highest unit price products. Department of Chemical Engineering, Louisiana State University. * Department of Chemical Engineering, Purdue University. 9 Department of Agricultural Engineering, Purdue University. 0196-4305/84/1123-0522$01.50/0
Noncontinuous operations are particularly prevalent in the food, pharmaceutical, polymer, and fine chemicals processing industries in which virtually all operations are of the batch semicontinuous type. Such operations are used in these industries because of the flexibility that they provide in accommodating the variable nature of the feed stock materials, the large number of products produced using similar recipes, the nature of the processing steps (e.g., bacterial culture), the inherent variability and seasonality of the market demand, as well as the short al0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
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processing some products. (8) Set-up and change over times may be sequence dependent. At the present time, no general solution procedure exists for the scheduling of complex batch processes. However, literature does exist for the two related problems of flowshops and jobshops. The classical flowshop consists of operation k of any job always being performed on machine k. This relation eliminates the need to distinguish between operation number and machine number. The jobshop allows the flow of product to be multi-directional, It is possible to allow any number of operations for a given job. Thus, it is appropriate to consider a jobshop in terms of operation j of job i performed on machine k. The books by Baker (1974), Coffman (1976), and Elmaghraby (1973) list hundreds of references on solution approaches to these problems. The majority of this literature deals with flowshop problems having infinite intermediate storage, with almost no attention being given to flowshop problems with finite intermediate storage. In this paper the design problem with finite intermediate storage will be examined and product scheduling will be treated via dispatching rules. Several papers (Karp, 1975; Lenstra and Rinnooy Kan, 1976; Carey et al., 1976; Lenstra et al., 1977) have been published which show that most flowshop and jobshop problems belong to the class of NP (nonpolynomial) decision problems. NP represents a class of decision problems which cannot be solved by algorithms whose solution time is bounded by a polynomial in the number of variables considered in the problem. This does not mean that all NP problems cannot be solved with reasonable efficiency. Often by taking special advantage of the problem structure, near-optimal solutions to NP problems can be found within reasonable computation times. Linear programming problems represent an interesting anomaly as far as NP problems are concerned in that they are routinely solved with very reasonable computational effort.
! SERVER P
QUEUE L
SERVER M
M SEMI-CONTINUOUS UNITS
N INTERMEDIATE STORAGE TANKS
P SEMI-CONTINUOUS UNITS
Figure 1. Typical noncontinuous processing train.
lowable inventory life of the products. Because of these immutable factors even the increasingly larger processing plants now being built, especially in the food industry, are noncontinuous by design and will be so for the foreseeable future. In a previous paper (Knopf et al., 1982) we divided noncontinuous processes into two general classes, those that could be designed independently of scheduling considerations and those in which the scheduling of products must be considered in the design. In that work a general formulation of the optimization problem arising in design of batch semicontinuous operations without scheduling considerations was presented. This paper will consider the more complex problem of combined scheduling and design in noncontinuous processes. A typical noncontinuous process of the type considered in this work is shown in Figure 1. It consists of two stages of processing with multiple intermediate storage vessels. A key point is that, in general, the parallel processing lines are not identical. Applications with this general structure can be found throughout the process industry, including fluid milk processing, polymer processing plants with multiviscosity or multicolor products, and even offshore oil storage facilities with multiquality crude inputs and demands. The technique developed in this work is intended for systems that show extended periods of nonoperation a t regular intervals, often termed regenerative systems. These periods of nonoperation are useful in that they decouple production campaigns and eliminate the need for long term production horizons. State of the Art for Noncontinuous Processes with Scheduling Considerations The addition of scheduling considerations to batch semicontinuousdesign introduces an entire set of problems that are generally avoided in the literature. However, an optimal production schedule can reduce the time and cost of production campaigns in batch plants so its consideration in the design phase is warranted. The general problem in planning and scheduling of batch semicontinuous plants can be summarized as follows: Find the slate of products, processing order, and route of produds through the system for each batch campaign which will optimize the cost or profit. The circumstances which are significant to multiproduct batch semicontinuous scheduling include the following: (1) Product demands must be considered over several time periods to account for inventory costs. (2) Parallel equipment may be available at any stage; thus batch splitting may occur. (3) Produds may be allowed to bypass each other at intermediate storage areas. (4) Producta may wait in intermediate storage while high priority downstream operations are being performed. (5) Products may have different production priorities due to fouling, deterioration, and other considerations. (6) Processing times may vary with product recipe. (7) Parallel units may not be identical and thus some units may not be suitable for
The Scheduling Problem in Batch Chemical Processes Little attention has been given to scheduling problems within batch plants, especially those plants with finite interstage storage (FIS). Two of the more comprehensive analyses which highlight the computational difficulties encountered when trying to schedule FIS batch chemical processes are due to Swanson (1972) and Prabhakar (1974). These authors tried to take advantage of the structure of batch scheduling problems to find optimal scheduling solutions. Swanson (1972) considered two related scheduling problems which are unique to batch processing. The first is the problem of scheduling operations if more than one batch can be processed in the same unit operation. The other problem is that of determining what scheduling sequence should be used and how much intermediate inventory should be kept. His solution technique involved using the dual of a linear programming scheduling model. The dual was found to have only a small number of variables that took on nonzero values for any feasible solution. The solution method also allowed one scheduling decision to imply another to improve solution efficiency. The main shortcoming of Swanson’s work was that the storage level for intermediate produds was allowed to become negative. When additional constraints to maintain inventories positive were added, solution times became excessive P1.5 h) . A alternate approach to the scheduling of batch operations was presented by Prahbakar (1974), who considered a two-reactor system with intermediate and final storage.
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The production scheduling problem with sequencing considerations was formulated as a mixed integer programming problem and solved by using a branch and bound search algorithm. Two unique features of Prabhakar's work were allowing a product to be split up among reactors (job splitting) and simultaneously determining the sequence on each given piece of equipment. The major difficulty with Prabhakar's solution technique was excessive computer time. A simplified process with 20 products and 4 reactors required 939 s of CPU time on an IBM 370/165 to get within 10% of the optimum solution. A more recent effort by Mauderli and Rippin (1979) reports a product scheduler for multipurpose batch plants. The user is required to specify the available equipment modules, the processing steps in specific order for each product, and which equipment modules may be used with each product. The optimal schedule is found as the solution of a linear program or a mixed integer program if penalties are included to account for product change over times or costs. A major limitation of this work is that no intermediate storage is included in the optimization. It is important to note that these authors were concerned with the scheduling of products in batch plants. The combined problem of scheduling and design was not examined. For a more detailed discussion of scheduling in noncontinuous processes the reader is referred to the review paper by Reklaitis (1982).
The Combined Scheduling and Design Problem In the absence of suitable analytical techniques, simulation is the appropriate way to analyze the complex combined scheduling and design problem that typically occurs in batch processes. A now perhaps dated review of simulation of transient processes was given by Hlavacek (1977). In a section devoted to the simulation of batch plants, he points out that batch chemical plants often operate in modes where scheduling and transient conditions are important. To handle these processing situations, he points out that combined simulation languages such as GASP IV may be employed. It is also noted that these problems had been almost completely overlooked in chemical engineering computer aided design. GASP IV (Pritsker, 1974) represents the most extensively developed general simulation language that is able to handle the variable types found in batch processing. When one models any system, two variable types may occur. One is the type associated with discrete events (batch completions) and the other is those which can be modeled as rate equations (continuous operations). GASP IV provides the framework for writing discrete events, continuous and combined discrete events, and continuous simulation models. Basically, in an event simulation written within the GASP IV framework, time is advanced to the next event by external specification, calculations within a program, or sampling from a distribution. At the next event, any system parameter, such as equipment utilization, can be changed or any attribute associated with an entity moving through the system can be changed. Changes in the system can, however, only occur at event times. In a continuous simulation written within the GASP IV framework, rate or state equations are numerically integrated from initial conditions until a decision point is reached. These rate equations can be difference equations or an ODE model. The decision points termed state events occur when state variables have reached a certain threshold level. A t the threshold level the rate equations representing the system may be changed. GASP IV also d o s for the combination of discrete and continuous variable types within the simulation. Pritsker and Pegden
(1979) have updated GASP IV into SLAM (Simulation Language for Alternative Modeling), which is the version used in this work. The literature contains very few studies in which simulations were used to solve noncontinuous design and scheduling programs. Reklaitis and co-workers (Fruit et al., 1974; Embury et al., 1976,1977,Overturf et al., 1978a,b) incorporated scheduling models within GASP IV simulations to evaluate various operating practices in polymer processing plants. Overturf et al. (1978a) tested the operating policy set by Davis and Kermode (1975)for a single train polymer system. It was shown that long simulation runs were necessary to identify optimal operating policies. However, it should be noted that long simulation times will not always be necessary if the system is regenerative (Fishman, 1973). In a companion paper, Overturf et al. (1978b) consider the problem of a polymer plant processing 38 products with four nonidentical processing lines. Both a heuristic and a mixed integer linear program were evaluated as scheduling mechanisms. The scheduling mechanisms, subject to various resource constraints, were tested within a GASP IV simulation of the processing lines. Finally, Embury et al. (1977) considered a multi-stage processing plant involving both parallel equipment and extensive in-process storage. A GASP IV simulation was used to study the effects of scheduling in the plant operations. The GASP IV simulation allowed consideration of fluctuations in product quality and the recycling of off-grade material. These papers [Overturf et al. (1978a,b) and Embury et al. (1976,1977)] showed that mixed integer programming was effective for scheduling noncontinuous processes. However, they also indicated that heuristic scheduling approaches often termed "rules of thumb" can give good suboptimal solutions. All of the simulations discussed could be used to model stochastic variations in operating times and product quality. Smith and Rudd (1964) considered the variations in batch time on process performance and solved several simple problems using queuing theory. However, for realistic variations such as occur in their example of a rubber reclaiming product, process simulation was found to be necessary. In this brief review we have outlined the difficulties that occur in obtaining a solution to the combined scheduling and design problem that is encountered in batch semicontinuous processes. Although each of the articles was confined to specific process examples, the case studies performed highlight several important considerations in batch processing. These include the role of inventory costs in scheduling as well as the difficulties that arise due to fluctuations in product characteristics and product processing time. The effects of intermediate product storage in the processing trains were avoided because of the increased complications they added. It should also be noted that all efforts in improving the design of the simulated processes were through case studies. A more efficient approach to determining the optimal design of batch semicontinuous plants with scheduling considerations and intermediate storage is by direct optimization of equipment size within a simulation and heuristic scheduling framework. The considerations involved in implementing such an approach will be discussed in the following sections.
Development of the Simulation Model The process to be optimized involves a configuration of operations common to the polymer, food, and other noncontinuous processing industry. As shown in Figure 1, it consists of M parallel processing devices (extruders, pasteurizing units, etc.), each being fed from their respective
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 525 Table I. Simulation State Events event index
1. 2.
3.
4.
5.
6.
7. 8. 9.
event definition and consequent action A storage tank has become full. Shut-off valve inputing material from M subtrain. Positive threshold value in a tank has been crossed. This allows the P subtrain to begin removing product. A storage tank has become empty. Shut off valve removing material, Initiate search for tank with same material A P subtrain has completed its current product Initiate search of storage tanks for its next product, An M subtrain has completed its current product. Remove next product from queue and initiate search for empty tank or tank with same product. Check for an empty tank. Check if there is an idle P subtrain for acceptance of product (called by event index 2). Error detection event. Negative threshold value in a storage tank has been crossed, This allows the M subtrains t o begin refilling the tank.
storage queues. The processing conditions of the M units may vary making their product slates different. The storage queues may be individual storage tanks or volume allocations within one common storage tank. From the semicontinuous processors the product will travel to N intermediate storage tanks of equal size. From the N storage tanks, product will be sent to P packaging or final processing semicontinuous units. The P semicontinuous units may be nonidentical. For example, each machine may produce products with different characteristics. The system was simulated using SLAM (Pritsker and Pegden, 1979) as a combined discrete-continuous process. The discrete continuous representation was found necessary to account for the complex situations that can occur in systems of this type. The continuous portions of the system were represented in terms of state variables. Typically the state variables represented the quantity of current product remaining to be processed on each semicontinuous subtrain or the current level in each storage tank. Changes in the state variables were all described by first-order difference equations (state equations). Important decision points in the time history of the system were identified in terms of specific values of the associated state variables. These decision points or events and their consequences are summarized in Table I. It is interesting to note that state events 2 and 9 are necessary to prevent rapid starting and stopping of the semicontinuous subtrains. The logical consequences of each event are encoded in separate event subroutines. These 8 subroutines interact with each other to keep track of product completions and to make decisions as to appropriate valve positions. The valves essentially represent the status of the pathway to and from each storage tank (zero if closed, one if open). Thus the valve positions can be used within the firsborder state equations to control tank levels and the operating status of each semicontinuous subtrain. User Inputs Although the model is written in general form to allow any number of processing entities, it only requires a min-
imal number of user inputs. The user inputs to the model consist of: (1)the number of intermediate storage tanks; (2) an initial rate of each of the semicontinuous subtrains and an initial size of each storage tank; (3) the product processing order from each initial storage queue, defined in terms of user selected dispatching rules; (4) the quantity of each product to go through each of the final (P)semicontinuous subtrains. (5) the stopping and starting levels for the interstage storage tanks. For example, a tank may have to be 75% full before downstream units can begin withdrawing product. In addition, once a tank has become full, a level of 75% empty may be arbitrarily used before the same product can again be put in the tank. These controls are necessary to prevent rapid on-off cycling of upstream and downstream units.
Simulation Assumptions The four key assumptions incorporated within the simulation model are: (1)Products are to be removed from their initial storage queue based on the order determined by any of four heuristic dispatching rules which will be discussed in the next section. (2) Once a product enters the system it cannot be passed by a product from the same initial queue. (3) From the interstage storage units a heuristic sequencing decision based on maximizing downstream equipment utilization is used. Thus, job splitting at the interstage storage tanks can occur. Also different products running parallel at the final semicontinuous units can occur. However, once a product begins on a semicontinuous unit, the unit remains dedicated to that product until the product production quota is completed. (4) No product mixing in interstage storage tanks is allowed. Simulation Output The output from the model allows the user to evaluate the efficiency of each unit operation. This evaluation is through statistics collected on the time of operation for each unit. The output also shows the status of the system at each process change (i.e., a tank becoming full or empty, etc.). This helps the user to evaluate system performance. Scheduling and Optimization The purpose of this section is to illustrate an approach to solve noncontinuous design problems in which the scheduling of products is considered. Essentially, the proposed approach incorporates a heuristic scheduler within the simulation model and the combination is used within an overall optimization strategy. By employing heuristic scheduling rules good suboptimal solutions can be obtained much more efficiently than is the case with exact techniques (Overturf et al., 1978b; Fruit et al., 1974). The use of direct optimization methods allows an “optimal” design rather than a best case study to be found. The incorporation of a scheduler within a simulation framework combined with an overall design optimization is an approach which is not specifically treated in the literature. However, the literature does discuss the problems related with the optimization of simulation models. Basically, two approaches have been suggested for optimizing the independent variables of a simulation model. The first method involves a direct evaluation of the independent variables and the second method involves response surface techniques. Farrell (1977), in his review of simulation optimization, notes that direct evaluation approaches are of two types. The first type consists of naive techniques such as heuristic search, complete enumeration, and random search. The second type uses the more efficient nonlinear programming techniques that are nonderivative based. Successful efforts
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have included coordinate searches (Lefkowita and Schriber, 1971) and pattern searches (Nelson and Krisbergh, 1964). Pegden and Gately (1977) have used a Hooke Jeeves pattern search to optimize both a discrete and a combined discrete continuous simulation model, each written in GASP IV. Both examples considered only three decision variables. A t-test was incorporated to determine if the difference in the optimal means of simulation runs represented an improved search direction. The second method used in simulation optimization involves response surface methodology (Wilde, 1964; Meyer, 1971; Smith, 1975; Bard, 1974). A surface is fit to a factorial design in the vicinity of an initial search point. Optimization techniques such as gradient searches are then used to predict new points for the next search area. With the large number of variables typically found in batch plant design, response surface techniques currently provide the most economical method of obtaining the optimal batch design. Also, because of the nonlinearities that typically occur in batch designs, second or higher order response surfaces should be used. Box and co-authors (Box and Hunter, 1957; Box and Draper, 1959; Box et al., 1973) have reported experimental design techniques for use with response surfaces. Various response surface techniques have been used in optimizing simulation models (Hunter and Naylor, 1970; Biles, 1974). However, the majority of these simulation/optimization studies only examined a few simulation variables. For batch semicontinuous design optimization, a comparatively large number of variables will have to be optimized. For models written within a SLAM or GASP IV structure, the user can select one of four heuristic scheduling rules already incorporated into the executive. These scheduling rules can be applied to any storage queue and allow sequencing products by: (1) first product into queue first out; (2) last product into queue first out; (3) low value first based on an attribute awociated with the produd; this attribute could be the amount of product to be processed; (4) high value first based on an attribute associated with the product. As will be demonstrated, for systems that are regenerative, heuristic scheduling rules can produce good suboptimal schedules. Of course, as the time between regenerations becomes longer or if the system is nonregenerative, then inventory considerations and costs will become increasingly important. Under such conditions scheduling is best accomplished with models which employ longer planning horizons. The simulation model can be used with any semicontinuous rates and tank size (system variables) to evaluate how long the processing of products will take. This fact can be used to generate a response surface, which will yield the system production time for any value of system variables. To accomplish this, the simulation program includes a section of code which can generate a factorial design of the process variables. The fadorial design allows a second order quadratic response surface to be determined Y = bo + blTX + XrbllX (1) where Y = the total production time, bo = constant coefficient, bl = a vector of linear coefficients, bll = a s y m metric matrix of quadratic and interaction coefficients,and x = a column vector of the system variables. The program keeps track of the total processing time Y for each design. After the necessary number of designs are evaluated, the quadratic coefficients are generated and a reduced gradient optimization program is called. Within the gradient optimization model the capital cost of the sysem is minimized subject to the production time con-
Table 11. Fluid Dairy Plants Nominal Equipment Rates and Production Slate equipment
rate or size
pasteurization unit no. 1 pasteurization unit no. 2 pint packaging machine no. 1 quart packaging mqchine no. 2 $/,-gal packaging machine no. 3 gallon packaging machine no. 4
6000 gal/h 6000 gal/h 1000 gal/h 2500 gal/h 3500 gal/h 1500 gal/h 3500 gal
storage tanksa
Nominal Daily Production Requirements (All Units in Total Gallons) product type 3.2%milk 3% milk 2% milk skimmilk orangejuice fruit drink
total
pints
quarts
'/,-gal
gal
20000
2000 500 1000 1000 3000 100
5000 1000 2000 2000 5000 400
10000
3000
3000 5000 10000 10000 500
500 2000
5000 10000 15000
20000 5000
2000 2000 4000
a Generally products are removed from the storage tanks when they are a/,, full, and the tanks are refilled with product when they are I/,, full.
straint supplied by eq 1. The capital costs in our work were represented by power law expressions (Knopf et al., 1982). It is of course possible to include system energy costs, as will be shown in the example section. The gradient optimization returns an optimal set of conditions for the system variables (the semicontinuous rates and the tank size). These values are input to the simulation model and the proceas is continued until the convergence criterion on capital costs is met.
Application within a Fluid Milk Processing Plant In order to show the potential of the combined design and scheduling approach, we applied it to the design of an actual fluid milk processing plant. The plaot in question employs two HTST pasteurizer-separator-homogenizer units. One unit is devoted to fluid milk products and the other unit is used to process fruit drink products. Two different initial processors are frequently used as products containing high sugar content (fruit drinks, ice cream, etc.) must be held at pasteurization temperatures longer than nonsugar products such as milk. Thus in this example job splitting will not occur on the first processors as each product must be restricted to a specific initial processor. There are eight interstage storage tanks at the dairy, followed by four packaging machines. Packaging of each product container size is dedicated to a specific machine. Table I1 gives the actual rates and size of the dairy plant equipment. The typical daily production requirements are also given. As shown, there are seven process variables, the six rates of the semicontinuous units and the size of the storage tanks, once the production slate is fixed. The number of HTST pasteurizer-separator-homogenizers, storage tanks and packaging machines (M,N , and P ) must be a priori fixed as these are discrete variables. A fluid milk plant is an example of a regenerative system. Fluid milk production is typically 16 h per day, with the remaining 8 h of downtime devoted to sanitation procedures. The slate of products produced during an operational day is generally fixed by customer demands for the following day. Thus, fluid milk processing is a regenerative noncontinuous process in which inventory costa are minimal. Scheduling of products from the initial queues was accomplished by the dispatching rule of
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 3, 1984 527
Table IV. Results from the Combined Simulation Optimization of the Fluid Milk Process
Table 111. Results from the Simulation of the Fluid Milk Plant
4
Semicontinuous Units units pasteurization unit no. 1 pasteurization unit no. 2 packaging unit no. 1 packaging unit no. 2 packaging unit no. 3 packaging unit no. 4
utilization 0.702 0.351 0.641 0.519 0.926 0.758
Interstage Storage Tanks mean volume, unit gal tank no. 1 1320.6 tank no. 2 1953.0 tank no. 3 1396.4 tank no. 4 2256.4 tank no. 5 1708.0 tank no. 6 1088.3 tank no. 7 1324.3 tank no. 8 919.3
rate, gal/h 6000 6000
1000 2500 3500 1500
size, gal 3500 3500 3500 3500 3500 3500 3500 3500
smallest volume first, based on the total amount to be processed. Other dispatching rules including largest volume first were examined and found to be not as effective as smallest volume first. For the dairy plant the typical production schedule and user variables shown in Table I1 were set in the simulation program. The results given in Table I11 show that pasteurizer unit 1 was utilized 70% of the simulated time and pasteurizer unit 2 was used 35% of the simulated time. The utilization of the packaging machines varied from 52 to 93%. Altogether, 11.9 h was required to process the typical production day. The wide variation in equipment utilization indicates some equipment is sized far from optimum. The general noncontinuous processing model was validated by comparing the simulated dairy plant to the actual plant. The total simulation time of 11.9 h is comparable to the 16 h processing day normally found in the dairy. The difference in time can be attributed to the stopping and starting rules for the interstage storage tanks. The general rules of 75% full tank for packaging to begin and 25% full tank for refilling to begin are not accurately maintained in the dairy. The use of a monitoring subroutine allowed the system variables to be recorded at each event time. Careful examination of these events with plant personnel and other people knowledgeable in the dairy industry indicated that the simulated performance is in good qualitative agreement with actual plant operation. Once the simulation model was validated, the optimization of the fluid milk plant was undertaken. The optimization was performed first with respect t~ the capital equipment cost of the system and then the combined effect of energy and capital equipment costs were examined. In order to determine an optimal design for the plant, seven process variables were identified for the dairy process: X(1) and X(2), the rates of the first semicontinuous servers; X ( 3 ) ,the size of the interstage storage tanks; X(4) to X(7) the rates of the final semicontinuous servers. These seven variables require a factorial design of 36 levels to evaluate the quadratic, interaction, and linear terms of the response surface. This quadratic response surface is used to represent the constraint surface of total production time for the seven variables. For the fluid milk process examined, the production constraint is limited to 16 h per day. The remaining 8 h are devoted to sanitation procedures.
pasteurizing storage units, gal/h tanks, gal run 1 2 - 1 1 2 3 4 5
6000 4000 5000 5500 5400
6000 3000 3763 4000 4094
3000 1500 1000 800 800
packaging machines, gal/h 1000 500 500 500 499
2
3
4
2500 1900 1300 1116 1033
3500 2900 2600 2450 2400
1500 1200 1000 900 900
run
simulation time, h
system cost, $
1 2 3 4 5
12.12 17.04 18.05 16.26 16.74
897 490 743 314 67 2 045 644 172 639 855
Table V. Energy Equation Coefficients for F!uid Milk Production ck, BLk, K, energy unit operation Btu/ft3 Btu/ft3 $/Btu type 10.58 24.68 8.793-06 electrical separation homogenization 15.26 61.04 8.793-06 electrical 730.58 81.17 7.003-06 steam pasteurization 4.74 42.70 8.793-06 electrical 6.54 58.86 8.7893-06 electrical cooling 1079.14 119.90 2.933-06 refrigeration 18.59 167.35 8.793-06 electrical packaging
The actual dairy plant (Table 11) semicontinuous unit rates and tank sizes were input to the simulation model. A modification of reducing the number of storage tanks from 8 to 4 was made because it was clear that excess tankage was available. The results are given as run no. 1 in Table IV. Table IV shows that this slightly modified actual system requires 12.12 h to process the six products. The totalsystem capital cost is $897 490. The optimization program was then used to optimize the system subject to the production constraint of 16 h. The optimum system levels became run no. 2. These vales are input to the simulation, a factorial design is created, and a response surface representing the system time is again generated. This process is continued until a convergence criteria of 1% change in capital costs is reached. Run no. 5 shows the optimal system cost of $639 855, which represents a 29% savings over the original design of run no. 1. An even greater savings (47%)is realized when the optimal system with four tanks ($639 855) is compared to the actual plant with eight tanks ($1206 248). The computer time necessary to generate the response surface for each run was about 600 s on a CDC 6500. The gradient optimization procedure generally required about 20 s on the CDC 6500. It is possible to extend the capital equipment cost optimization procedure to include energy costs. In our predecessor paper (Knopf et al., 1982) we discussed energy data collection and correlation techniques for batch plants. Table V shows the correlated energy requirements for the fluid milk plant example. A continuous base line energy load (BL,) was assumed and the product energy requirementa (c,) were considered linear with rate. Use energy cost term K was obtained from monthly utility rates for each type of energy. The present value of the combined energy and capital equipment costs were used as the objective function. The optimum system based only on capital equipment cost (run no. 5, Table IV) was used as starting conditions.
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However, no improvement in this design could be found by considering the combined energy and capital costs objective function. This result can be explained by the fact that equipment is already sized as small as possible (minimum capital cost) within the production constraint. The addition of linear energy costs is not sufficient to shift the system optimum. It can be expected that if the cost parameters are selected to reflect an energy economy of scale, a shift in the design will occur in a manner similar to that observed in our previous work (Knopf et al., 1982). Conclusion As shown in this work, a combined continuous discrete simulator is an effective tool for studying the design and operation of noncontinuous multi-product processes. The simulation readily allows simple scheduling rules to be tested and selected, and alternate unit configurations and rates to be evaluated. Since production scheduling does have a significant impact on equipment utilization, both scheduling and equipment selection decisions must, in general, be jointly considered. This work demonstrates that the use of a response surface strategy allows process design optimization involving a moderate number of variables to be performed using the simulation model. The key limitation of the response surface approach lies in the time required to execute a simulation run. With a regenerative process, simulation of a single production cycle suffices for the determination of mean system performance. With nonregenerative operation simulation times can mount considerably because runs must be executed over multiple production cycles to compensate for the effects of initial conditions on calculated mean system performance measures. Although this work was confined to an M X N 2-stage parallel unit process with intermediate storage, the basic approach clearly is suitable for more general configurations providing that an appropriate simulation model is developed. However, since the completion of this work, a modular alternative to the SLAM simulation executive has been reported which considerably facilitates this task. The BOSS simulator (Joglekar and Reklaitis, 1982; Joglekar et al., 1983) allows unit by unit assembly of noncontinuous multiproduct process models in a manner analogous to existing steady-state flowsheeting systems and reduces the need for user coding of event logic to exceptional situations. Literature Cited Baker, K. ”Introductlon to Sequencing and Scheduling”; Wlley: New York. 1974. Bard, Y. “Nonllnear Parameter Estimation”; Academlc Press: New York, 1974. Biles, W. E. “A Gradlent-Regresslon Search Procedure for Simulation Experimentation”, Proceedings of the 1974 Winter Slmulatlon Conference,
Society for Computer Slmulation, La Jolla, CA, 1974. Box, G. E. P.; Hunter, J. S. Ann. Math. Staf. 1957, 2 8 , 195. Box, G. E. P.; Draper, N. R. J. Am. Stat. Assoc. 1959, 54. 287. Box, G. E. P.; Hunter, W. G.; MacGregor, J. F.; Erjavec, J. Technomehlcs 1973, 15, 33. Carey, M. R.; Johnson, D. S.; Sethl, R. Math. Opns. Res. 1078, 1 , 117. Coffman, E. G. “Computer and Job Shop Scheduling Theory”; Wlley: New York, 1976. Davis, J. J.; Kermode, R. I. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 459. Elmaghraby, S., Ed. “Symposium on the Theory of Scheduling”; SpringerVerlag: Berlin. 1973. Embury, M. C.; Reklaitls, G. V.; Woods, J. M. “Simulation of the Operation of a Staged Multl-Product Process with Llmited Interstage Storage Buffers”, Proc. 9th Hawali Conf. Sys. Sci.. 348, 1976. Embury, M. C.; Reklaltls. G. V.; Woods, M. J. “Schedullng and Slmuletiin of a Staged Seml-Continuous Muiti-Roduct Process”, Proceedhgs, 2nd Pacific Area Chemical Englneering Congress, Denver, CO. Aug 1977. Farrell, W. “Literature Review and Blbliography of Simulation Optimization”, Proceedings, 1977 Wlnter Simulation Conference, Society for Computer Simulation, La Jolla, CA, 1977. Fishman, G. S. “Concepts and Methods in Discrete Event Digital Simulation”, Wlley: New York, 1973. Fruit, W. M.; Reklaltis, G. V.; Woods. J. M. The Chem. fng. J. 1974, 8 , 199. Hlavacek. V. Comput. Chem. €nu. 1977, 1 , 75. Hunter, J. S.; Naylor, T. H. Management Scl. 1070, 76, 7. Joglekar, G. S.; Reklaitls, G. V. “A Slmulator for Batch and Semlcontinuous Processes”, paper 20f, AIChE Annual Meeting, Los Angeles, CA. Nov 1982; Comput. Chem. Eng. in press. Jogbkar, G. S., Clark, S. M.; Reklaltls, G. V. “BOSS: A Process Simulator for Noncontlnuous Operatlons”. Proceedings 3rd International Congress on Computers and Chemical Engineering, Paris, France, April 1983; p C161-1. Karp, R. M. Networks 1975, 5 , 45. Knopf, F. C.; Okos. M. R.; Reklaitis, G. V. Ind. Eng. Chem. Process Des. Dev. 1982, 2 1 , 79. Lefkowits, R. M.; Schrlber, T. J. “Use of an External Optimization Algorlthm with a QPSS Model”, Proceedings, 1971 Winter Simulatlon Conference, Society for Computer Slmulatlon. La Jolla, CA, 1971. Lenstra, J. K.; Rinnooy Kan, A. H. G. Opns. Res. 1078, 26. 22. Lenstra, J. K.; Rinnooy Kan, A. H. G.; Bruckner, P. Ann. Dlscrete Math. 1977, 1 , 343. Mauderli. A.; Rlppln, D.W. T. ”Production Planning and Scheduling for MultiPurpose Batch Chemical Plants”, Proceedlngs, 12th Symposium on “Computer Applications In Chemlcal Englneering”, Vol. 1, Monreaux, Swltzerland, Aprll 1979. Meyer, R. H. “Response Surface Methodology”; AHyn and Bacon, 1971. Nelson, C. W.; Krlsbergh. H. M. Manege. Scl. 1984, 2 0 , 1164. Overturf, 8. W.; Reklaitls, G. V.; Woods, J. M. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 161. Overturf, B. W.; Reklaitis, G. V.; Woods, J. M. Ind. Eng. Chem. Process Des. D e v . 1978. 17, 166. Pegden, C. D.; Gately, M. P. “Decision Optimization for GASP I V Simulation Models”, Proceedlngs, 1977 Wlnter Slmulation Conference, Soclety for Computer Simuiatlon, La Jolla, CA, 1977. Prabhakar, T. Manage. Scl. 1974, 2 1 , 34. Pritsker, A. A. B. “The GASP I V Simulation Language”; Wlley: New York, 1974. Pritsker, A. A. B.; Pegden. C. P. “Introduction to Simulation and SLAM”; Systems Publishing Corporatlon and Halsted Press (John Wlley): New York, 1979. Reklaitls, G. V. AIChESymp. Ser. 1982, 78(214) 119. Smith, D.E. “Automated Response Surface Methodology in Digital Computer Simulation, Volume I: Program Description and User’s GuMe“; Office of Naval Research, Arlington, VA, Sept 1975. Smith, N. H.; Rudd, D. F. Chem. Eng. Scl. 1984, 19, 403. Swanson, H. X. W.D. Thesls, University of Florida, Gainsvllle, FL, 1972. Wllde, D. J. “Optimum Seeklng Methods”; Prentice-Hall: Englewood Cliffs, NJ, 1964.
Received for review April 28, 1982 Accepted October 14, 1983
