ISSN 10642307, Journal of Computer and Systems Sciences International, 2014, Vol. 53, No. 3, pp. 410–419. © Pleiades Publishing, Ltd., 2014. Original Russian Text © A.B. Borisenko, S.V. Karpushkin, 2014, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2014, No. 3, pp. 113–123.
CONTROL SYSTEMS FOR TECHNOLOGICAL PROCESSES
Hierarchy of Processing Equipment Configuration Design Problems for Multiproduct Chemical Plants A. B. Borisenko and S. V. Karpushkin Tambov State Technical University, Tambov, Russia email:
[email protected] Received February 19, 2013; in final form, January 22, 2014
Abstract—A procedure for performing one stage of designing multiproduct chemical plant—defini tion of the processing equipment configuration is proposed. At this stage, a threelevel hierarchical structure of problems is formed—at the upper level, the parameters of the operational mode of the chemical engineering system that ensures the desired output of each product are found; at the middle level, the size and number of equipment units and methods of product processing for all processing stages are chosen; at the lower level, technological and mechanical design of individual pieces of equipment (machines) are performed. The decomposition of the processing equipment design prob lem into three problems described above reduces the original mixed integer nonlinear programming problem to one nonlinear programming problem and a set of integer programming problems, where the number of integer programming problems is equal to the number of processing stages. The reduced problems can be solved using available optimization methods, which considerably reduces the solution time and improves the quality of design solutions. Basic information links between the problems of dif ferent hierarchical levels are determined, general problem statements and an algorithm for the simul taneous solution of the upper and middle level problems is proposed. By way of example, the problem of selection and design of a mechanical agitator for a vertical bulkcapacity storage is considered. DOI: 10.1134/S1064230714030046
INTRODUCTION The equipment configuration design, that is, the selection of types, number, geometrical dimensions, and operational parameters of the basic and auxiliary equipment units of chemical engineering systems (CESs) is a key problem in designing new chemical plants and updating the existing ones so as to modify the range of manufactured products and overall production. Most difficulties arise when designing multi product chemical plants (MPCs) that manufacture chemical dyes, intermediate products, additives to polymer materials, pharmaceuticals, chemical agents, etc. MCPs have the following features: a wide range of manufactured products, small output, short manufacturing times of most types of prod ucts and, as a consequence, the necessity to design multiproduct plants that can produce several products with similar synthesis technology; frequent changes in the range and amount of products, and the need to adapt the available equipment to manufacturing new products; primarily periodic mode of operation of MCPs (products are manufactured in batches, which sequen tially pass through all the processing stages), while some processing stages can be performed by continuous machines operating in semicontinuous mode; the parameters of the MCPs operational mode (size of batches, duration of processing operations, duration of production cycles, etc.) drastically affect the configuration of equipment used in different pro cessing stages. Chemical engineering systems of MCPs consist of a set of isolated equipment subsets designed for the execution of physical–chemical processing stages that are provided in process regulations (preparation of primary materials, chemical transformations, and extraction of components). These equipment subsets have a wide range of equipment and technological process types, while the information about the kinetic laws and physical and chemical characteristics of the materials are far from being complete, and the equip ment load is quite different in different periods of its operation. The equipment configuration design is a stage in the technology calculations performed in the design of the new and conversion of existing industries for the manufacturing of new products. The main task of this stage is to select the equipment for each CES; i.e., select the geometrical dimensions (working vol 410
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umes, working surfaces) and the number of the main and auxiliary machines, and parameters of operation that ensure the desired output of each product. Upon the upper level problem for each CES is solved, the optimal parameters of the design and operation mode of each basic and auxiliary equipment unit of each processing stage are sought. As a result, some initial data for the upper level problem (material indexes, duration of operations in some stages of the CES) can be modified and the equipment selection problem can be solved again if needed. In the majority of statements of the equipment selection problem available in the literature, only some of the features of equipment operation are taken into account (all the features affecting the operation of equipment, including variations in the product batch size in the course of processing, are considered in the publications of the authors of the present paper). Furthermore, here we face a mixed integer nonlinear programming problem, for which no efficient methods are presently available. In the procedure proposed in this paper, the upper level problem is decomposed into the problem of finding the optimal parameters of the CES operation (nonlinear programming problem) and selection of equipment for each stage (integer programming problem). As a result of this decomposition, a threelevel hierarchy of problems is produced. At the upper level, the optimal parameters of the CES operation are found; at the middle level, the equipment for each stage of the technological system is selected; and the lower level, the optimal parameters of the design and operation mode of each basic and auxiliary equip ment unit of each processing stage are determined. In such a decomposition, the upper level problem requires for its solution some data obtained by solving the equipment selection problem; for that reason, these yet unknown data must be first predicted and then improved in the course of an iterative process. To select the predicted values, the problem solvability conditions and the procedure for their verification and enforcement developed by the authors (in the papers referenced below) are used. The proposed procedure can be used in chemical industry design organizations and in design units of existing enterprises to organize the change of manufactured range of products. 1. HIERARCHY OF PROBLEMS OF EQUIPMENT CONFIGURATION SELECTION OF MCPs AND INFORMATION RELATIONS BETWEEN THEM Beginning from the 1970s, many publications consider as the main equipment design problem the selection of geometrical dimensions and number of basic equipment units for each stage of the CES and the parameters of the system operation as a whole and of its individual equipment units that ensure the manufacturing of the prescribed amount of each product in the prescribed range in the course of the plan ning period (e.g., see [1–6]). The basic characteristics of the operation mode of each CES manufacturing each product are the product batch size (the mass of the batch after all the processing stages are com pleted) and period of the product manufacturing cycle (the time interval between the beginning and end of manufacturing two batches produced one after the other). Note that almost all mathematical formulations of the main equipment design problem found in the literature concern only the basic equipment of the CES stages (reactors, filters, and dryers), but they do not include relations for the selection of auxiliary equipment. However, the number of auxiliary equip ment units is by far greater than the number of the basic ones. The most numerous are measuring tanks for liquid raw materials, collectors of intermediate products and wastes, pumps, and heat exchangers (both external and built into the main machines (jackets, coil pipes). Auxiliary equipment is selected after the main equipment is chosen using the results (operational parameters of the main equipment) obtained at that stage. Even though there are significant differences in the statements of the main equipment design problem, all authors agree that this is a mixed integer nonlinear program. The main difficulty in its solution is that continuous parameters of the operational mode are to be found along with the discrete characteristics of the equipment used at the processing stages. Various approaches to the solution of this problem (see [1–9]) do not propose efficient methods for its solution in a sufficiently full statement that takes into account all important features of operation of chemical plants. We determined these features based on the more than 35 year experience of participating in the design and conversion of dozens of chemical plants. These fea tures are as follows: The structure of material flows in CESs is often different for different products and it can be branched (some stages of processing the raw materials and intermediate products can be performed simulta neously). The time needed to processes batches of products at certain processing stages (mainly equipped with continuous basic machines) depend on the batch size and the main dimensions of the basic equipment units, and they cannot be fixed in advance. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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BORISENKO, KARPUSHKIN Problem RF: determination of parameters of CES operational mode and equipment of its stages
а) product batch sizes; b) material indexes, cycle times of processing product batches at stages; c) basic operation times and specific performances of basic equipment units j ∈ JS i
. . .
i
Problem VAi: choice of main sizes and number of the basic equipment units stage j and processing methods for product batches
а) equipment unit type and its main size; b) list of implemented operations, their duration, and material indexes; в) physical and chemical characteristics of working media, types of heat agents and coolants; c) equipment elements and structural materials
. . .
а) number of basic equipment needed to process batches of each product; b) indicators of processing methods for product batches; c) sizes of the basic equipment units at stage j ∈ JS
. . .
а) operation implementation modes: temperature, pressure, and kinetic characteristics; b) improved material indexes, time of operations, specific energy expenditure, and specific performances; c) geometric sizes of elements and checking calculations
Problem OAjf: optimization of design parameters and operational mode of equipment unit f at stage j of the CES
. . .
Hierarchy of CES MCP equipment configuration problems and information links between them.
Operations of charging and discharging the basic equipment units of some processing stages can be per formed simultaneously with physical and chemical transformations (suspension supply into a filterpress, reception of filtrate in cleaning filtering). At some processing stages, the sizes of product batches may vary—a batch can be subdivided into sev eral identical portions for sequential or synchronous processing or several batches can be processed jointly. Of specific note is the fact that possible variations in batch product sizes in the course of their process ing are practically not studied in the available literature. However, such situations are typical for MCPs— they are caused by the desire to minimize the number of processing stages, that is, to execute similar pro cessing stages involved in manufacturing of different products on the same pieces of equipment even when the sizes of product batches and material indexes are quite different. Note that the optimization of the design and operational parameters of individual basic and auxiliary pieces of equipment is typically beyond the scope of the majority of studies. On the other hand, there are many publications in which these issues are considered but the fact that a piece of equipment is a part of a CES manufacturing certain product and the influence of this piece on other pieces of equipment are ignored. However, it is clear that the operational parameters of the CES and its required performance must be taken into account when the design and operational mode of individual pieces of equipment are con sidered. Furthermore, the optimization of the design and operational parameters of a piece of equipment can require the redesign of equipment of the CES containing this piece. Based on the reasoning above, we propose the threelevel hierarchical structure of the equipment design problem (see figure). At the upper level, the optimization problem denoted by the letters RF is solved to determine the CES operational parameters when manufacturing the required amounts Qi (i = 1, I ) of all the products in the given range I during the planning period Tp. At the middle level, the problems VAj ( j = 1, J ) of choosing the main dimensions and number of the basic and auxiliary equipment units at each of J processing stages and of choosing the processing techniques by the basic equipment units at each stage are solved. At the lower level, the problems OAjf (f ∈ Foj ∪ Fvj) of optimization of the design JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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parameters and operation mode of the equipment unit f at stage j are solved, where Foj and Fvj are the sets of indexes of the basic and auxiliary equipment units, respectively. The figure also shows the main information relationships between the problems of the three hierarchi cal levels; the middle level is fed not only with the results of solving the problem RF (product batch sizes, time of their processing cycles at processing stages, the time taken by the basic operations for the stages j ∈ Jsi (i = 1, I ), where they depend on the product batch sizes) but also with some initial data of the prob lem RF (material indexes of the processing stages for each product, specific performances of the basic equipment units of the stages j ∈ Jsi). The main source of the initial data is in the process regulations— technological regulations for the new plants and manufacturing regulations for the existing plants. One of the main requirements for the manufacturing technology of new plants is the compliance with the require ments of the state ecological examination service; these requirements can be checked using the system proposed in [10]. It is seen from the figure that the solution of the problem RF requires some results obtained by solving the problems VAj—the number of equipment units at each processing stage, the size of each product batch to be processed on the basic equipment units of each stage, and the dimensions of the basic equipment unit j ∈ Jsi. For that reason, the procedure of the joint solution of problems RF and VAj is to make a pre diction and iteratively improve the values of these parameters. The problems OAjf are classified into technological and mechanical design problems for the basic and auxiliary equipment units of stage j of the CES. The initial data for their solution include the equipment unit type and its main geometrical size, the list of supported operations (which may be different for differ ent output products), the time of those operations and their material indexes, recommendations for implementation of these operations (temperature, pressure, hydrodynamic conditions), physical and chemical characteristics of processed materials, types of thermic agents and coolants that can be used to organize heat exchange (if needed), list of components of the equipment unit to be designed and types of materials from which these components are made and the methods used to connect them. In the course of the technological design, the optimal mode of the working cycle operations for the manufacturing of each product is chosen. As a result, the material indexes of the processing stages for products, the time of processing operations of product batches at the stages, and the specific performances of equipment units can be improved. The optimality criteria for these problems can be chosen among the maximum output of the target product, minimum operation time, and minimum energy cost under the required efficiency level. The basic constraints are relationships of mathematical models of operations performed by the equipment units in the course of production. The mechanical design of a basic or auxiliary equipment unit assumes the determination of the geo metric sizes of its elements (wall thickness, shaft diameters) that ensure their strength, hardness, etc. con ditions in the execution of all processing operations of each product batch at the corresponding processing stage. Here, the optimality criterion is the minimum materials consumption of the equipment unit com ponents. For standard equipment units, verification calculations are performed. 2. STATEMENT OF THE UPPER AND MIDDLE LEVEL PROBLEMS AND THEIR SOLUTIONS In distinction from the decomposition strategies of the main equipment design problem proposed in [7–9], the strategy proposed in this paper is based on the principle of dividing the problem parameters depending on how they change—we separate the continuous parameters of CES operation mode from discrete characteristics of the equipment used in the processing stages. Taking into account the decisive influence of the operational parameters, we use the problem RF as the upper level problem; it is formulated as a nonlinear programming problem. At the middle level, we solve the VAj ( j = 1, J ) problems formulated as discrete optimization problems. As a result of solving the problem RF, the product batch sizes wi (i = 1, I ) and the start tosijkl and end tofijkl time of each operation l of each cycle k of the operation of the basic equipment units of each pro cessing stage j while producing each product i must be determined. This is the operationwise schedule of the operation cycles of the basic equipment units of all the CES stages in the course of one cycle of each product manufacturing. The values of these parameters must ensure that all the material flows needed for the production of the amount Qi of each product in time Tp while minimizing the energy needed for all processing operations. A detailed justification of the optimization criterion of the CES mode of operation and the equipment used in its stages can be found in [11]. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Given wi (i = 1, I ) and tosijkl, tofijkl (i = 1, I , j ∈ J i , k = 1, K ij , l = 1, Lijk ), the duration of production cycles tij (i = 1, I , j ∈ J i ) at the CES stages, the duration of cycles Tci (i = 1, I ) of manufacturing the products on CES stages, and the total time needed for their production Ti (i = 1, I ) are determined. Here, Ji is the set of indexes of the CES stages whose equipment is involved in processing batches of product i, Kij is the number of operation cycles of the equipment of stage j involved in the production of product i, and Lijk is the num ber of operations of cycle k of the operation of the equipment of stage j involved in the production of prod uct i. Thus, the problem RF can be generally formulated as follows:
Ze(W *) = min{Ze(W )| sω(W ,TOS,TOF ) ≥ 0, ω ∈ Ωr}; W
here, Ze is the cost of energy needed to produce the indicated amounts of all products in the period Tp, W = ( w1, ..., w I ) is the set of product batch sizes, TOS = {tosijkl|i = 1, I , j ∈ J i , k = 1, K ij , l = 1, Lijk } and TOF = {tofijkl|i = 1, I , j ∈ J i , k = 1, K ij , l = 1, Lijk } are the sets of start and finish times of processing operations of all products by the equipment of CES stages, and Ωr is the set of relations of the mathematical model of the design solution aimed at the choice of the operational parameters of the MCP CES and equipment of its stages. The relations sω(W, TOS, TOF) ≥ 0 are as follows (see [12]): constraints on the variation of the product batch sizes, constraints on CES performance for individual products, synchronization conditions of the operational cycles of equipment of adjacent CES stages. The algorithm for solving the problem RF is based on the wellknown constrained optimization tech nique called backtracking search [13]. A detailed description of this algorithm can be found in [14]. The purpose of solving the problems VАj is to choose the main geometrical sizes Xj and the number Nj of the basic equipment units of stage j, the number Nvjv and sizes Xvjv (v = 1, Gv j ) of the auxiliary equip ment units of this stage, and the coefficients rij of variations of the product batch sizes and indications pij of their processing method that ensure the processing of batches of all products while minimizing the cost of the technological equipment of the stage. Here, Gvj is the number of groups of auxiliary equipment units of the same type (measuring devices for liquid raw materials, product and waste collectors, pumps for charging and discharging the basic equipment units, and external and internal heat exchangers); pij = 1 if the basic equipment units of stage j synchronously process equal portions of product i batch and pij = 0 if the product i batches are processed entirely; the coefficient rij = 1 if the size of the product i batch does not change at stage j; if rij = γ > 1, then the batch is divided into γ equal parts, which are processed one after another; if rij = 1/γ, then γ batches are merged for joint processing; Nj are determined as a result of calcu lating the number of basic equipment units nij of stage j of the CES that are needed to process the batches of each product. In the general form, the problem VAj is formulated as
Zs j (N *j , X *j , NV j*, XV j*) =
min N j , X j ,NV j , XV j
{Zk j (N j , X j , NV j , XV j )| sω(N j , X j , NV j , XV j , P j , R j ) ≥ 0, ω ∈ Ωs j },
where Zsj are the depreciation charges of the cost of the basic and auxiliary equipment units of stage j for the period Tp (a more detailed discussion of the optimality criterion for the basic and auxiliary equipment of stage j can be found in [11, 15]); Pj = {pij | i = 1, I }; Rj = {rij | i = 1, I }; NVj = {Nvjv | v = 1, Gv j } and XVj = {Xvjv | v = 1, Gv j } are combinations of the number and sizes of the auxiliary equipment units at stage j; and Ωsj is the set of relations of the mathematical model of the equipment design solution at stage j. The rela tions sω(Nj, Xj, NVj, XVj, Pj, Rj) ≥ 0 are as follows (see [12, 15]): constraints on the variation of the main geometrical dimensions of the basic and auxiliary equipment units of different types; membership conditions of the chosen values Xj and Xvjv in the sets of main geometrical dimensions of the basic and auxiliary equipment units that can be used at the corresponding CES stage; relations for determining Nj and Nvjv and their integrality conditions. The algorithm for solving the problems VAj is reduced to finding the minimum feasible size of each equipment unit of stage j. If there are no feasible dimensions (some of the constraints sω(Nj, Xj, NVj, XVj, JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Pj, Rj) ≥ 0 are not satisfied), then the problem is reduced to the straightforward enumeration of the possible values of the parameters nij, pij, and rij (i = 1, I , j∈Ji) satisfying all the constraints and finding those of them that are preferable from the viewpoint of the criterion Zsj. Note that these problems are independent of each other; hence, they can be solved concurrently. As the problems VAj are solved, nij, pij, and rij for which the problem RF was solved can be modified; on the other hand, in order to determine the duration of batch processing operations at the stages j ∈ Jsi while solving this problem, one needs to fix the dimensions of the basic equipment units of these stages. For that reason, the joint solution of the problems RF and VAj includes an iterative procedure of improving nij, pij, rij, and Xj (j ∈ Jsi); the initial prediction of nij, pij, and rij can be improved by checking and satisfying the solvability conditions of the problems RF and VAj. The initial prediction is made using the following considerations: From the viewpoint of depreciation charges of the equipment units cost of CES stages, the values nij = 1 are preferable; therefore, pij = 0 because one unit is always cheaper than several units provided that they have the same total size. Splitting and merging of product batches at certain CES stages can increase the duration of production cycles; i.e., it can increase the energy consumption of processing and negatively affect the quality of prod ucts. For that reason, initially we set rij = 1. The set of existence conditions for the problems RF and VAj includes the following conditions: The product compatibility condition, which requires that batches of different products in the given range can be processed by the same equipment units:
min (wij ) ≤ max (wij ),
j =1,...,J
j =1,...,J
i = 1, I ,
(2.1)
where wij are possible values of the batch size of product i determined by the calculation of the lower and upper bounds on the main dimensions of the basic equipment units of the CES stages, by the upper bounds on the time needed to process the product batches at the stages j ∈ Jsi, and by the lower bounds on the duration of batch production cycles (Tc∗i, i = 1, I ). The existence condition of at least one basic equipment unit whose size makes it possible to process the batches of all products for each CES stage:
[X Lj , X Uj ] ∩ XS j ≠ ∅,
j = 1, J ,
(2.2)
where X Lj , X Uj are the minimum and maximum feasible values of the main size of the basic equipment units of stage j calculated using min ( wij ) and max ( wij ) and XSj are the sets of sizes of the standard basic j =1,..., J
j =1,..., J
equipment units suitable for the CES stages (they are formed preliminary based on equipment catalogs, price lists, and data about the spare equipment available in the existing plant). The desired performance condition I
∑ max (w ) ≤ Tp, i =1
QTc i *i
j =1,..., J
(2.3)
ij
that is the requirement that the amount Qi of each product can be produced in time Tp. For all the situations in which these conditions can be violated, recommendations on changing nij, pij, and rij that can satisfy these conditions are developed. To choose the most preferable recommendation, an advice of an experience expert (production engineer) or (and) solution of an auxiliary optimization prob lem can be used. Note that the satisfaction of these conditions does not guarantee that a feasible solution of the problems RF and VAj are obtained; however, it allows one to immediately reject infeasible combina tions of nij, pij, and rij. For example, the verification of these conditions in the equipment design of a CES consisting of 11 processing stages supposed to produce seven azo dyes in Tp = 6840 h provided that the total yearly output is 314 t showed that two basic equipment units were needed at a filtering stage and batches of some products were to be joined at the stages of semifinished products dissolution. A detailed description of the existence conditions of solutions and a procedure helping to satisfy them can be found in [14]. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Upon the completion of the verification and satisfaction procedure for solutions of the RF and VAj problems, the values Xj (j ∈ Jsi) are predicted; more precisely, the values X Uj (see condition (2.2)) are cho sen because they are associated with the minimum operation times and, as a consequence, the minimum values of Tci for which the planned performance of the CES with respect to all products is most probable. The iterative algorithm of the joint solution of the RF and VAj problems includes the following steps. 1. Preliminary prediction of nij, pij, and rij (i = 1, I , j ∈ J i ). 2. Verification and satisfaction of conditions (2.1)–(2.3) and possible correction of the parameters nij, pij, and rij. Prediction of Xj = X Uj (j ∈ Jsi). 3. Solution of the problem RF to determine wi (i = 1, I ) that ensure the production of the amount Qi of each product in the time Tp while minimizing the criterion Ze(w1,…,wI). 4. Solution of the problems VAj ( j = 1, J ) to determine the size and number of equipment units at each CES stage maximizing each function Zsj and the parameters nij, pij, and rij that ensure that the batches of each product can be processed by the equipment units of all CES stages involved in its production. 5. Return to Step 3 if nij, pij, rij, and Xj (j ∈ Jsi) obtained by solving VAj differ from their values for which the problem RF was solved. The algorithm for solving these problems is described in detail in [14, 15]. The backtracking search algorithm for solving the problem RF only guarantees that a local minimum (in a vicinity of the initial approximation) determined by the initial values of nij, pij, and rij is found. The validity of the choice of the initial approximation described above is confirmed by practical calculations—any changes in the product batches increase the energy consumption and the attempt to increase nij above the minimally possible value (by one, by pairs, by triples, etc.) at the stages restricting Tci does not result in decreasing the opti mality criterion in the majority (90%) of CESs of MCPs. An algorithm for solving the problems VAj based on an exhaustive search strategy guarantees that the global minimums of the optimality criteria are found. To test the proposed procedure for the choice of the basic equipment (solution of the problems RF and VAj), the company Ekokhimproet (Tambov, Russia) provided the initial data and the results of choosing the equipment for 23 CESs of reallife MCPs. For 11 CESs, two iterations of the joint solution of the prob lems RF and VAj were needed; for seven CESs, three iterations were needed; and from four to seven itera tions were needed for the other CESs. The application of this procedure for designing the conversion of five CESs in the company Pigment (Tambov, Russia) producing dispersible dyers, showed that the number of iterations did not exceed two, and in two cases a single iteration proved to be sufficient. This is explained by the fact that the sets XSj in the case of conversion rarely include more than one or two suitable equip ment unit sizes. These calculations demonstrated a high efficiency of the proposed procedure of designing the equip ment of MCP CESs compared with the conventional procedure. The expenses for the CES equipment are reduced by 15–25% (due to the application of the existence conditions for the problems RF and VAj and due to optimization algorithms), and the time needed for the design is reduced by a factor of several tens— from two–five working days for each CES to several minutes (due to the use of the system EquipDesign developed for solving the problems VAj [16] complemented with the modules for the verification and sat isfactions of conditions (2.1)–(2.3), solution of the problem RF, and the implementation of the iterative algorithm for the joint solution of the problems RF and VAj). 3. EXAMPLE OF LOWER LEVEL PROBLEM The main source of initial data for the problem OAjf of the technological and (or) mechanical design of a specific equipment unit number f of stage j are process regulations for the products in the manufacturing of which this equipment unit is involved; these are information about the physics of the underlying pro cesses, equations of chemical transformations, material balances of operations, recommendations on the conditions under which operations should be performed and on the type and versions of the equipment unit. The main geometrical sizes (working volume, working surface) and the number of the basic and aux iliary equipment units intended for the implementation of a specific process, and the volume or (and) mass of the materials processed by the unit are obtained by solving the problems RF and VAj. The problems OAjf are solved based on models of the processes implemented by the equipment unit or using methods rec ommended in the corresponding normative documents and standards. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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By way of example, consider the problem of designing and selection of mechanical agitators of vertical bulkcapacity storages. A mathematical statement of this problem was developed using the results of studying the influence of geometric characteristics of the body, physical, chemical, and thermophysical characteristics of the agitated medium, the agitation mode, the conditions for agitator shaft suitability and the sealing ability of its sealants, and heat exchange conditions in the unit on the following parameters of the agitator design: agitator type ta, its diameter da, and the number of agitators za on the shaft of the device, the diameter d and the rotation rate n of the shaft, parameters of the agitators used for mixing in the transient and laminar modes: the number of horizon tal crossarms of the gate type and helical ribbon agitators nca, step between the crossarms of the helical ribbon agitator lca and the number of its blades nb, the height of the anchor type, gate type, and screw agi tator Нa, and the lead of the helix tb. The mathematical statement of the problem is based on the design procedure recommended in [17, 18] for the calculation of the following quantities: agitation power expenditure in the vertical bulkcapacity storage in the turbulent, transient, and lami nar modes; parameters of circulation and turbulent diffusion in the medium being agitated; parameters describing the quality of agitation of various media in the turbulent mode—homogeniza tion time of mutually soluble liquids, distribution of concentration of the solid phase of suspensions in height and radius of the storage, the speed of the dispersed phase drops and their size when immiscible liquids are agitated, average specific gas content of the medium and the average bubble size when a gas liquid medium is agitated, and solid polydisperse material dissolution time; parameters of circulation and homogenization time of the agitated medium in the laminar and turbu lent modes; agitator shaft diameter (for the most widely used rigid console shaft of constant cross section) for which the vibrostability, rigidity, and strength conditions must be satisfied; heattransfer coefficient due from the agitated medium for different storage designs and agitation modes; heat flow that can be ensured by the heat exchanger. To calculate the specific leak of the working medium in the shaft seal zone of the agitator, the procedure recommended in [19] is used. This problem is important for all manufacturing plants (not only chemical ones) in which vertical bulk capacity storages can be involved in various CESs and used for executing various processes. The proposed statement of the problem combines the cases of laminar and turbulent agitation modes because it often happens that the turbulent mode becomes transient and then laminar as the technological process progresses. The generalized statement of the problem is
Za jf (t a*jf , pa*jf , da*jf , d *jf , n*jf ) = min {Za jf (t ajf , pajf , dajf , d jf , n jf )| sω(t ajf , pajf , d ajf , d jf , n jf ) ≥ 0, ω ∈ Ωa jf }, t ajf , pajf ,dajf ,d jf ,n jf
where Zajf is the deprecation cost of the device with the motor reductor, and electrical energy cost con sumed by the agitator in a year, tajf is the agitator type, pajf is the set of the agitator parameters depending on its type, agitation mode (for turbulent agitation and all types of agitators), pajf = (zajf)—for transient and laminar modes, pajf = (Нajf) for the anchor type agitator, pajf = (Нajf, ncajf) for the gate type agitator, pajf = (nbjf, ncajf, lcajf) for the helical ribbon agitator, pajf = (Нajf, tbjf) for the screw agitator, and Ωajf is the set of relations of the mathematical model of the design solution for the choice of standard parameters of the agitator design and commercial motor reductor. The inequalities sω(tajf, pajf, dajf, djf, njf) ≥ 0 are used as con straints on the following quantities: agitator parameters and the hydrodynamic conditions in the storage, rotation rate of the agitator shaft and its diameter, specific leak of the working medium in the shaft seal zone, heat flow provided by the heat exchanger if the agitated medium heats up or cools down, characteristics of the motor reductor, parameters of the agitation quality. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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The initial data for this problem are formed by solving the problem RF (the name and duration of oper ations concerning the work of the agitator in each operation cycle of each basic piece of equipment at stage j), the problem VAj (the working volume of this piece and its fill factor while processing batches of each prod uct), and the data from the process regulations (properties of the working media and data about changes in the temperature in the course of processing, the desired values of the mixture quality parameters, the equipment unit make, the diameter and height, the material and thickness of the body wall, the type of heat exchanger, the type of shaft seal, and the dimensions and position of internal devices). The equip ment unit make, the type of the shaft seal, and the dimensions and position of internal devices are included in the process regulations because these are the properties of the working media that determine the choice of the equipment unit make, the recommended agitation modes, the transportation technique, the techniques for controlling the parameters of the process implementation mode, and the fill factor). The algorithm for solving the problem determines the optimal design of the agitator by searching through the values of the parameters of commercial agitators and selecting a suitable motor reductor stan dard size with the minimal selling price. After selecting the optimal values of the agitator design parame ters (or upon checking the feasibility conditions of an existing device), the following initial data for the RF problem can be improved: the time needed for processing batches of products in bulkcapacity storages of the CES stages that are involved in agitating mutually soluble liquids and solution of solid granular raw materials; specific electrical energy cost (in W/kg) consumed by the agitators of vertical bulkcapacity storages when processing the batches of products and intermediate products. A detailed formulation of the problem, an algorithm for its solution, and an example of choosing an agitator for a reallife MCP can be found in [20]. CONCLUSIONS A new procedure for designing equipment of MCPs is proposed that forms a hierarchy of problems. At the upper level, the parameters of the operational mode of the CES are found (nonlinear programming problem); at the middle level, the basic and auxiliary equipment units for the processing stages of the sys tem are selected (discrete optimization problems); and the technological and mechanical design of indi vidual equipment units is performed at the lower level. Such a representation of the equipment design pro cedure makes it possible to avoid the difficulty that is often mention in the relevant literature—the need of solving the general mixed nonlinear integer programming problem; furthermore, it enables one to take into account the mutual influence of the operational mode of the CES and the parameters of individual equipment units. The mathematical formulations of the problems of finding the parameters of the operational mode and selecting the basic and auxiliary equipment units are based on the analysis of features of reallife multi product chemical plants; the features affecting the product manufacturing procedure and the design of equipment units, including the branched structure of material flows in CESs, the possibility of varying product batch size in the course of their processing, simultaneous execution of batch charging and dis charging operations with physical and chemical transformations, are taken into account. The efficiency of a CES operational mode is estimated based on the total cost of energy consumed in the planned pro duction period, and the results of the equipment design are estimated based on the deprecation cost of the basic and auxiliary equipment in the same period. A procedure for the simultaneous determination of parameters of the CES operational mode and the equipment design of its processing stages is developed. This procedure uses a prediction and iterative improvement of the number of basic equipment units that are needed for processing batches of products, size variation coefficients at the stages, indication of their processing methods, and the sizes of basic equipment units. The predicted values of the parameters are determined by checking and satisfying the existence conditions of the equipment design problems; the application of these conditions considerably reduces the amount of computations and time needed to solve the problems. By way of example of the optimization of the design and operational mode of an individual piece of equipment, the statement of the problem of designing a mechanical agitator for the most widely used equipment of MCPs—vertical bulkcapacity storage—is considered. REFERENCES 1. E. N. Malygin and S. V. Karpushkin, “Automated design of equipment for technologyintensive flexible plants,” Khim. Prom., No. 2, 118–123 (1985). 2. V. V. Kafarov and V. V. Makarov, Flexible CAM Systems in Chemical Industry (Khimiya, Moscow, 1990) [in Rus sian]. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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3. S. Parageorgaki and G. V. Reklaitis, “Optimal design of multipurpose batch processes. 1. Problem formulation,” Ind. Eng. Chem. Res. 29, 2054–2062 (1990). 4. V. T. Voudouris and I. E. Grossmann, “MILP model for scheduling and design of a special class of multipurpose batch plants,” Comp. Chem. Eng. 20, 1335–1360 (1996). 5. X. Lin and C. A. Floudas, “Design, synthesis and scheduling of multipurpose batch plants via an effective con tinuoustime formulation,” Comp.& Chem. Eng. 25, 665–682 (2001). 6. D. Mokeddem and A. Khellaf, “Optimal solutions of multiproduct batch chemical process using multiobjective genetic algorithm with expert decision system,” J. Autom. Meth. Manag. Chem. 2009 (2009). 7. S. Parageorgaki and G. V. Reklaitis, “Optimal design of multipurpose batch processes. 2. A decomposition solu tion strategy,” Ind. Eng. Chem. Res. 29, 2062–2073 (1990). 8. L. S. Gordeev, V. V. Makarov, Yu. V. Sboeva, et al., “Decomposition algorithm for optimization of multiproduct chemical manufacturing systems,” Program. Prod. Sist., No. 1, 2–10 (1997). 9. T. Pinto, A. BarbosaPovoa, and A. Novais, “Decomposition based algorithm for the design and scheduling of multipurpose batch plants,” in Poster Papers of the 16th European Symp. on Computer Aided Process Engineering, GarmischPartenkirchen, Germany, 2006, pp. 1051–1056. 10. V. A. Nemtinov and Yu. V. Nemtinova, “On an approach to designing a decision making system for state envi ronmental examination,” J. Comput. Syst. Sci. Int. 44, 389–398 (2005). 11. S. V. Karpushkin, M. N. Krasnyanskii, and A. B. Borisenko, “A procedure for estimating the efficiency of mul tiproduct chemical plant equipment,” Inform. Sist. Tekhnol., No. 5, 96–106 (2011). 12. S. V. Karpushkin, M. N. Krasnyanskii, and A. B. Borisenko, “A procedure for the selection of basic equipment in the design of multiproduct chemical plants. Part 1: Statement of problems and procedure for their simulta neous solution,” Inform. Tekhnol. Proektirovanii Proizvodstve, No. 3, 52–58 (2012). 13. A. I. Boyarinov and V. V. Kafarov, Optimization Methods in Chemical Technology (Khimiya, Moscow, 1969) [in Russian]. 14. S. V. Karpushkin, M. N. Krasnyanskii, and A. B. Borisenko, “A procedure for the selection of basic equipment in the design of multiproduct chemical plants. Part 2: Solvability conditions and algorithms,” Inform. Tekhnol. Proektirovanii Proizvodstve, No. 4, 34–40 (2012). 15. E. N. Malygin, S. V. Karpushkin, and P. G. Mikhailova, “Selection of auxiliary equipment for multiproduct chemical plants,” Vestn. Tambovsk. Gos. Techn. Univ. 17 (2), 483–487 (2011). 16. S. V. Karpushkin, M. N. Krasnyanskii, and A. B. Borisenko, “A system for the selection of multiproduct chem ical plant equipment,” Inform. Tekhnol., No. 10, 14–19 (2004). 17. Standardized Document RD 26019085: Mechanical Agitators: Design Method; effective from January 1, 1986 (RTP LenNIIKhimmash, Leningrad, 1985) [in Russian]. 18. Standardized Document RDRTM 26017282: Shafts of Vertical Apparatuses with Agitators: Design Methods; effective from July 1, 1983 (RTP LenNIIkhimmash, Leningrad, 1982) [in Russian]. 19. L. A. Kondakov and A. I. Golubev, Seals and Sealing Devices: Handbook (Mashinostroenie, Moscow, 1994) [in Russian]. 20. S. V. Karpushkin, M. N. Krasnyanskii, and A. B. Borisenko, “Techniques for the design and selection of mechanical agitators for vertical bulkcapacity storages,” Khim. Promyshl. Segodnya, No. 12, 48–55 (2011).
Translated by A. Klimontovich
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