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The synthesis of multistep process plant configurations
Computers and Chemical Engineering 23 (1999) 315—326
The synthesis of multistep process plant configurations James R. Phimister *, Eric S. Fraga, Jack W. Ponton Department of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA Department of Chemical Engineering, University College London, London WC1 7JE, UK Department of Chemical Engineering, University of Edinburgh, Edinburgh EH9 3JL, UK Received 8 November 1997; revised 27 July 1998; accepted 27 July 1998
Abstract A framework for generating and optimizing chemical plant process connectivity is presented. The method addresses plant synthesis and analysis for multistep processes, processes where multiple reactions occur in both series and parallel, and where interstitial separation operations may be required. The method is tailored for the initial stages of process design, whereupon limited development of the process flowsheet has occurred, and estimation of plant economic viability is required. The chemical process is modeled as an interconnected graph linking chemical component nodes to operating sections. A mixed integer non linear program is utilized to obtain ‘best’ flowsheet configurations with initial mass balances and estimations for reactor parameters. The method is illustrated by examining large scale continuous processes, with the hydrodealkylation of toluene (HDA) process and a multistage chloromethanes process presented as case studies. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Reactor-separator-recycle structures; MINLP; Multistep processes; Process synthesis; Optimization
1. Introduction In a review of process synthesis optimization approaches, Grossmann and Daichendt (1994) note the integration of mixed integer non-linear programming (MINLP) with other design methods remains a research challenge. In this paper we address Grossmann and Daichend’s challenge by integrating MINLP with the heirarchical decomposition approach of process synthesis. The approach benefits from a block representation of a process flowsheet, allowing for determination of plantwide connectivity and initial values for ‘dominant design variables’, without requiring formulation of a rich superstructure consisting of numerous interconnected unit operations. The approach is applied to multistep process representations. Two case studies are presented with results providing initial flowsheets and process parameters similar to those reported in both academia and industry.
*Corresponding author. Tel.: 215 898 8351; fax: 215 573 2093; e-mail: [email protected]
Douglas (1990) presents a heuristic and decomposition method for synthesizing multistep chemical processes. Multistep processes have individual reactor units in which successive and parallel reactions occur. By linking separate units in multistep processes raw feedstocks are converted to products through several reaction paths and intermediate chemicals. Douglas (1988, 1985) presents a method where flowsheets are derived by decomposition and analysis of successive process levels. Decomposition proceeds using heuristics and quantitative evaluation to select the best configuration at each level of design. The process flowsheet develops through each level where it is further decomposed into a number of other designs and the selection procedure repeated. The decomposition and heuristic approach has been included in an expert system (Kirkwood et al., 1988). Other synthesis expert systems are discussed by Crowe et al. (1992) and Engelmann et al. (1989). Optimization methods have been utilized to investigate many aspects of process synthesis. For reviews see Grossmann (1990), Grossmann and Daichendt (1990), and Nishida et al. (1981). For a thorough presentation of MINLP applications to chemical engineering refer to
0098-1354/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 1 3 5 4 ( 9 8 ) 0 0 2 7 6 - 2
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Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications (Floudas, 1995). A selection of papers shows optimization techniques used in synthesizing reactor networks (Achenie and Biegler, 1985, 1990), distillation trains (Fraga and McKinnon, 1994; Pibouleau et al., 1983; Pibouleau and Domenech, 1986) and integrated reactive-separation units (Balakrishna and Biegler, 1993). An algorithm for linking reaction and separation sections is presented by Floquet et al. (1985). Optimization techniques are used by Omtveit et al. (1994) and Kocis and Grossman (1989) to analyze the reaction separation HDA process presented by Douglas (1990). The former uses a two stage optimization algorithm while Kocis and Grossman employ an outer approximation/equality relaxation (OA/ER) algorithm for solving an MINLP formulation. Kokossis and Floudas (1991) address the reactor-separator synthesis problem by developing superstructures which accommodate all possible interconnections between reactors and distillation columns and using an MINLP solver to determine an optimal flowsheet. To solve problems where multistep reactionseparation processes require synthesis Fraga employs the dynamic programming CHiPS package (1996). Case studies using optimization techniques to address large-scale commercial practices relate an interactive and iterative experience. For recounted studies, an optimization model containing an objective function and constraints is formulated and solved. Using an appropriate solver, solutions are examined by experts in the field, the model subsequently refined, and the procedure repeated (Subramanian et al., 1994; Sinha et al., 1995). In our approach, a framework for applying mixed integer nonlinear optimization to the successive and iterative decomposition method of process design is presented. To accomplish this an MINLP is formulated to help address the synthesis of multistep processes at the initial stage of process design. At this first stage of process synthesis the process designer is considered to have limited information from which to make decisions. These decisions will greatly affect the eventual layout, operability and profitability of the plant. Initially, process information available may be limited to reaction paths with basic kinetic models, costs of raw materials, expected selling prices and the expected plant production. Our formulation encompasses this limited framework, while still allowing for detailed reactor models to be used if appropriate. The initial decisions which the designer encounters can be large in number. Both discrete (yes/no) decisions and selection of continuous parameters (e.g. feed flowrate) must be determined. Some decisions which might be encountered are: E How to ‘connect’ a number of plant sections, i.e. plant inter-connectivity and recycle configurations; E Which raw materials to select; E Whether some reaction paths should be avoided;
Which products to produce and in what quantities; Values for what Douglas terms ‘dominant design variables’ (1990) — those variable which will most significantly effect the economic viability of the process. To obtain useful decisions at this stage of process design and to gain initial values for some parameters, an MINLP is formulated, consisting of binary decision variables (y) specifying whether connections or plant sections exist (a value of 1 implies existence, 0 implies no existence) and continuous variables (x) representing flowrates and conversion. To address the connectivity and synthesis problem a profit based objective function is maximized: E E
Z"max f (x) VW subject to constraints on variables x and y. A superstructure is proposed for multistep processes which incorporates plant sections without requiring detailed specification of individual units. The formulation is illustrated for the well known HDA problem and results are presented for a multistep chloromethanes process.
2. Method In formulating the multi-section design problem as an MINLP the chemical process is viewed as a set of reactor—separator (RS) sections with interconnected component nodes matching ‘sources’ to ‘destinations’. A ‘source’ defines a location where a component is supplied and a ‘destination’ is where a component is required. Fig. 1 shows the superstructure for a process with five components and one reactor—separator section. Two arrays of component sources exist; those for the process feed and the nodes exiting the RS section. Likewise, two destination arrays exist; the reactor—separator entrance and the process exit array. The initial superstructure allows all possible interconnections. Note however, the grouping of individual components into streams is not specified, thereby simplifying problem formulation without reducing generality. 2.1. Formulating connections An interconnection is defined by the binary variable y where i a chemical component and is a member of G H I set Q, the components in the process. j is the source node and k, the destination node. j3S such that S,+ f, 1, 2, 2 , NS, where ‘f ’ represents the process feed, ‘I’ the source nodes for RS section 1, and ‘NS’, the total number of plant sections. Likewise k3D, where D,+1, 2, 2 , e,. Each member of D represents a node entering a given RS section or leaving the process (element ‘e’). For example, using this notation, the connection y "1 denotes that hydrogen is fed to the & D J
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Fig. 1. Initial superstructure for one RS section five component system.
Fig. 2. Reduced superstructure with integer constraints applied.
process and enters the first RS section, y "1 speci!& fies that methane is recycled in section one. 2.2. Component matching Siirola and Rudd (1971) suggest, during initial process design, some component nodes can be matched from sources to destinations, other sources ruled out, and the remainder are possible connections. For example, when a reactant A is fed to a process and is required in Section 1, the connection between this source and destination may exist. If A is not produced from any other section of the process then y "1 (1) D is specified as a constraint. Constraints can formalize heuristics, for example the heuristic that all components fed to the process are connected to at least one RS section may be desired. Therefore a feed source connecting directly to an exist
destination is forbidden. This is formalized by y "0, i3Q. (2) G D C G Additional heuristics could also be formulated. For instance, if set of reactants R containing +A, B,, a subset of all chemicals in the process (set Q), the heuristic ‘reactants do not exit the process’ would have the formulation: y "0, i3R, j3S. (3) G H C G H Eqs. (1)—(3) are represented in Fig. 2. In applying the logical connectivity constraints many initial connections may be eliminated and a second superstructure, a subset of the first superstructure, created. The intention in defining the logical constraints is to create a secondary superstructure which encompasses all possible processing configurations, while not including configurations which are nonsensical.
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2.3. NLP subproblem For each connection present a corresponding flow, F, exists. If a connection is not present then the flowrate is zero. Here, flowrates are denoted similarly to the connection variables. F represents a positive flowrate for G H I component i, connecting source j to destination k. Negative flowrates are not allowed. The constraints F !ºy )0, ∀i, j, k G H I G H I F *0, G H I ensures component flowrates meet this criteria. º is the largest process flowrate allowed in a connection between two locations. º is relaxed when solutions have flowrates on the upper bound. Likewise convergence can be improved by tightening the value of º. For each RS section a reactor model relates the chemical inflow to outflow. For instance for reaction: APB
(4)
occurring in RS1 and undergoing conversion X the reactor model F "(1!X) F , I H I H F " F #X F , j3S, k3D I H H I H H is defined.
For the HDA process there is one RS section (RS1), and five components. Using the definitions for sets described Q"+hyd, tol, benz, diph, meth,, S"+ f, 1,, D"+1, e,. The initial superstructure is shown in Fig. 1. ¸ogical constraints. For this example process bypasses will not be allowed: y "0, i3Q G D C G Benzene, diphenyl and methane are not fed to the process: y #y #y "0. D D
D The reactants toluene and hydrogen must connect the process feed to RS1. y "1, i3+tol, hyd,. G D Benzene, methane and diphenyl must be connected from RS1 to the process exit y "1, i3+benz, meth, diph,. G C Toluene cannot be connected to the process exist destination: y "0. H C H Lastly, all source nodes for RS1 must have a destination.
3. Example 1: Hydrodealkylation of toluene The HDA process (Douglas, 1988) provides an illustrative example for this approach. The process consists of feedstocks of toluene and hydrogen forming methane and benzene by reaction:
y *1, k3D, ∀i. G I I Applying the logical constraints gives the secondary superstructure shown in Fig. 3.
C H #H P C H #CH . A byproduct diphenyl is also formed reversibly:
(5)
RS1 Model. Letting X define the conversion of toluene, the selectivity t of toluene converted to form benzene is defined by
2C H C H #H .
(6)
0.0036 t"1! , (1!X)
Fig. 3. Reduced superstructure for HDA process.
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Fig. 4. Optimal configuration for HDA process. Mass balance shown (kmol/h).
where moles of benzene formed t" moles of toluene converted and X)0.97 and 0.2)t)1. Process specifications. The production requirement in 80 000 metric tonnes of benzene per year for an 8000 h year. The molar feed ratio entering RS1 is 5 : 1 hydrogen : toluene, and flowrates are zero where connections are not present. Objective function. The objective is to maximize profit defined by products sold minus raw material costs. Purchasing/Selling prices are supplied in Appendix A. A penalty of 1/20th of the purchasing price is incurred for every tonne transported between a given source and destination. The penalty prevents internal flowrates having zero shadow costs at the solution, thereby preventing multiplicity of solutions. HDA results. Using the DICOPT MINLP solver provided by GAMS (Brooke et al., 1988) the HDA problem is optimized. The optimal configuration with flowrates are shown in Fig. 4. The optimal conversion is 0.697 giving a selectivity of 0.977. This provides an objective function of $18.65 million per annum in a solution time of 0.12 seconds. All calculations reported are performed on an IBM RISC 6000. The solution provides a starting point from which further synthesis can proceed. As illustrated by Douglas this configuration is feasible, whereupon on decomposition a membrane separation system is used to separate methane from hydrogen.
together with byproduct HCl. All the chloromethanes were at the time saleable products, some of the HCl could be sold, and the rest disposed of cheaply. This route was soon eliminated in the developed world due to restrictions on dumping HCl, and a two section plant combining chlorination with hydrochlorination, described below as the first example, was used. This still produced all four chloromethanes, which remained as marketable products into the 1980s. However, a range of environmental and safety concerns have eliminated markets for monochloromethane (methyl chloride) as a precursor to leaded petrol additives, dichloromethane (methylene chloride) as a domestic solvent and tetrachloromethane (carbon tetrachloride) as an intermediate for CFCs. Only trichloromethane (chloroform), an intermediate to more ozone friendly HCFC refrigerants, and dichloromethane as an industrial solvent are likely to remain in demand in the longer term. For background information on the chloromethanes process refer to Kirk Othmer Encyclopedia of Chemical ¹echnology (Entry: Chloromethanes, 1993), Encyclopedia of Chemical Processing and Design (Entry: Chloromethanes, 1979) and Kurtz (1972). 4.1. Process description and chemistry Four processing sections are proposed in an attempt to eliminate less commercially useful products and to provide greater flexibility of product spectrum. Plants with two, three and four sections are investigated. In the first section, RS1, a feedstock of methanol reacts with hydrochloric acid to form monochloromethane. CH OH#HCl P CH Cl#H O.
(7)
Byproduct dimethyl ether is formed by the side reaction: 4. Example 2: Chloromethanes manufacture
2CH OH (CH ) O#H O.
The manufacture of chloromethanes is a multistep process of significant current interest. Historically, a simple single step direct chlorination of methane produced a mixture of mono- to tetrachlorinated products,
In the second plant section (RS2) successive chlorination steps result in a product distribution consisting of methylene chloride (CH Cl ), chloroform (CHCl ) and carbon tetrachloride (CCl ) (reactions (9)—(11)). Complete
(8)
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reaction of chlorine occurs in this section: CH Cl#Cl P CH Cl #HCl, (9) CH Cl #Cl P CHCl #HCl, (10) CHCl #Cl P CCl #HCl, (11) Additional sections can form chlorine and hydrogen through the electrolysis of HCl (RS3), and reduce carbon tetrachloride to chloroform (RS4). The four sections are shown in Fig. 5. Process specifications. The objective is to design a component flowsheet which maximized profits by producing 80 000 tonnes per year (8000 h) of the distributed products methyl chloride, methylene chloride, chloroform, carbon tetrachloride and the byproduct dimethyl ether. For this study, recycling of products of RS2 is limited to methyl chloride, HCl exists the process with a ‘removal’ penalty applied for each tonne produced. A maximum conversion of 0.97 is set for both sections. Models used in the chloromethanes study are provided in Appendix B. In forming the multiple products, it is desirable to have high conversions of methyl chloride to obtain the
profitable chloroform, yet this is countered by the low profitability of carbon tetrachloride. The amount of hydrochloric acid formed is also dependent on conversion in RS2. Since this byproduct is a reactant in RS1 greater production in RS2 will reduce the need to purchase HCl as a raw feedstock. 4.2. Two section plant This first study involves two sections, the hydrochlorination of methanol in RS1, and chlorination in RS2. The best component flowsheet solution determined by DICOPT for the two section plant is shown in Fig. 6. The solution is obtained in 0.23 s, giving a conversion of 0.97 in RS1 and 0.454 in RS2. The model contains 42 continuous variables, 40 binary variables, 33 equality constraints and 190 inequality constraints. The profit is determined to be $24 M/yr. Purchasing HCl affects the recycle structure and optimal design conditions. The conversion in RS2 is limited so as to supply all of the HCl required in RS1. The final solution is self-sufficient in HCl it is neither necessary to purchase HCl as a feedstock nor is HCl a product; therefore the design avoids any removal penalty.
Fig. 5. Potential plant sections for chloromethanes process.
Fig. 6. Component flowsheet solution for two section chloromethanes process. Mass balance shown (kmol/h).
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Fig. 7. Component flowsheet solution for four section chloromethanes process.
4.3. Four section plant Two potential plant sections, an electrolysis section and an hydrogenation section, are added to the study. Initially all plant sections are assumed to exist in the final solution and hence none are attributed binary variables. In the third section, RS3, HCl is electrolysed by reaction (12) to completion. 2HCl P Cl #H . (12) The fourth section (RS4) consists of a hydrogenation section. In RS4 carbon tetrachloride is hydrogenated to chloroform and hydrochloric acid by reaction (13). The maximum conversion for the reaction is set at 0.97. H #CCl P CHCl #HCl. (13) The best component flowsheet solution is shown in Fig. 7 with a mass balance provided in Table 1. Conversions in RS1, RS2, and RS4 are all on the upper bound of 0.97. The solution provides an objective function of $31.64 M/yr. The model consists of 250 binary variables, 253 continuous variables, 1018 inequality constraints and 43 equality constraints. Because the fourth section allows conversion of carbon tetrachloride to chloroform, maximum conversion in RS2 is favored. This configuration obtains maximum amounts of the most profitable component, chloroform, with all carbon tetrachloride formed in RS2 converted to chloroform in RS4. Chlorine is purchased as a feedstock with a partial makeup from the electrolysis section. 4.4. Plant section existence Decisions on the existence of whole plant sections are implemented by assigning binary variables for each plant section and modifying connectivity constraints to be valid when sections are not present. Provided section constraints are valid when only zero flowrates enter and leave a given section, section constraints do not require activation or deactivation (see Floudas, 1997, p. 239 for a discussion of constraint activation) when integer variables are applied to whole plant sections. Solving the MINLP with the restriction that only three plant sections are obtained in the final solution, the
Table 1 Process flowrates for 4 Section Chloromethanes Process (kmol/h) Comp. Source
Destinations 1
MeOH. feed MeOH. 1 out HCl. 2 out HCl. 4 out DME. 1 out H O.1 out Cl . feed Cl . 3 out CH Cl. 1 out CH Cl. 2 out CH Cl . 2 out CHCl . 2 out CHCl . 4 out CCl . 2 out CCl . 4 out H . 3 out
2
92.311 2.855 64.267 25.275
3
4
Exit
117.949 1.385 90.926 123.241 58.850 89.541 2.769 22.142 42.125 25.275 25.275 0.782 25.275
33.700
flowsheet shown in Fig. 8 is obtained. Table 2 provides the process mass balance. The conversion in RS1 and RS2 is 0.97 and 0.817 respectively providing an objective function value of $24.99 M/yr. In contrast to the two section process, where HCl is not purchased as a feedstock, HCl is purchased for use in both the reaction in RS1 and for electrolysis and formation of chlorine in RS2. The RS2 conversion shifts markedly upwards from the two section process because conversion in RS2 is not limited by the full consumption of HCl in RS1. However, in comparison to the four section process, a decrease in conversion results in RS2. This lower conversion avoids production of unprofitable carbon tetrachloride which previously had been converted back to chloroform. 4.5. Multiple solutions using integer cuts The hierarchical decomposition approach to designing process flowsheets suggests identifying multiple solutions at each level of design. These solutions can be returned
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Fig. 8. Component flowsheet solution for three section chloromethanes process.
Table 2 Process flowrates for 3 Section Chloromethanes Process (kmol/h) Comp. Source
Destinations 1
MeOH. feed MeOH. 1 out HCl. feed HCl. 1 out HCl. 2 out DME. 1 out H O. 1 out Cl . 3 out CH Cl. 1 out CH Cl. 2 out CH Cl . 2 out CHCl . 2 out CCl . 2 out H . 3 out
Table 3 Changes to chloromethanes component flowsheet shown in Fig. 7 resulting from successive integer cuts
2
3
Cut no.
Add connection Flow f G H I (i, j, k) (k mol/h)
1 2 3
H2.4.4 H2.4.5 H2.4.5 H2.4.5 HCl.4.3
Exit
128.543 3.975 340.917 216.235 216.235 1.928 126.610
4
0.78 268.8 0.78 268.8 202.2
Remove connection (i, j, k)
Obj. func. ($M)
31.64 31.64 31.64 HCl.4.1
31.63
Flowrates throughout plant change at successive solutions. Flowrate at lower bound.
216.235 124.682 27.922 58.098 41.615 24.969 216.235
for further screening if initial developed solutions prove unsatisfactory. By using integer cuts, alternative solutions providing similar objective function values, are obtained. Integer cuts successively add the constraint (1!yQ)# yQY*1 Q QY to the original formulation, and re-solve the problem. Vectors yQ and yQY are the binary variables having values of one and zero at the previous solution. The constraint F*ºQy is included to prevent solutions where connections exist with corresponding zero flowrates. Further, ºQ"Min(FQyQ) where FQ is the vector of nonzero flowrates at the previous solution, is added to the formulation to prevent trivial solutions with zero flowrates but values of one for the corresponding binary connection variable. Changes to the flowsheet shown in Fig. 7 which result from the first four integer cuts are supplied in Table 3. All solutions provide similar objective function values, close
to that obtained in the original solutions. The first three integer cut solutions provide all possible deviations in transporting the least expensive feedstock: hydrogen. The fourth integer cut solution changes the process connections of HCl. The next three higher cuts (not shown in Table 3) again explore deviations on transporting hydrogen, though now based on the component flowsheet obtained in the fourth integer cut solution.
5. Discussion The method described provides a framework for addressing the initial stages of design for continuous multistep reaction processes. Using a block conceptual representation of the chemical process, mixed integer optimization methods are employed to obtain initial solutions addressing the problems of process connectivity. The emphasis of this work has been to create a systematic framework which screens a large number of potential recycle structures and determines initial values for those variables which greatly affect process viability and profitability (‘dominant design variables’). For the ten component, four section chloromethanes example, there are on the order of 2 potential configurations. Although the majority of these configurations
J.R. Phimister et al. / Computers and Chemical Engineering 23 (1999) 315—326
are infeasible, a small fraction and hence a large number of configurations are feasible. Discerning from all of these possible solutions optimal structures and parameters by, for instance, a heuristic approach, may not be possible. In the approach discussed, binary variables represent potential connections between processing sections, or whether the section itself exists. Continuous variables represent flowrates along a connection, or sectional variables. Because raw material costs dominate total processing costs (35—85%, Douglas, 1990) it is essential to let process feed flowrates be continuous variables, rather than to synthesize structures based upon fixed feeds. As has been illustrated with the chloromethanes process, selection of feedstocks used changed according to plant sections are available. The results illustrate the validity of the approach, providing recycle structures and operating parameters with low waste and high product yield obtained as solutions. The recycle structure obtained in the HDA example is identified by Douglas as a potential recycle structure for further investigation. The desired reactor conversion of 0.697 compares favorably with 0.75 determined by Douglas (1988), and 0.675 determined by Omtveit et al. (1994). The two section chloromethane study shows a balance of HCl, whereby HCl formed in RS2 is completely consumed by the RS1, resulting in neither disposal nor purchasing of HCl required. This plantwide ‘self-sufficient’ balance on HCl has been reported in industry, and is identified as a motivating factor in selecting the reaction path (see Encyclopedia of Chemical Processing and Design, Entry: Chloromethanes). The results indicate a more complex model or objective function is not warranted for the case studies examined. When material costs dominate total processing costs, the objective function will prove insensitive to, for instance, energy costs or annualized fixed unit costs. Hence the simple objective function employed, products sold less materials purchased, is justified for obtaining initial solutions. Further, additional objective function or constraint complexity decreases the likelihood of finding a feasible and optimal solution, and increases uncertainty of whether a local minima obtained is a global minima. Because each section is incorporated into the MINLP as a ‘black box’, providing an output based on a given feed input, more accurate section modeling and costing can be implemented without loss of generality. The case study solutions provide an understanding of process motivations and sensitivities. As synthesis develops, more detailed models and optimizations based on initial results can ensue. To cater for market demand upper limits on feedstock inventories or production capacities on individual chemicals can be imposed as con-
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straints. Sensitivity investigations can be performed to observe changes in design as market demand an price changes occur. Further, as decomposition of sections occurs, improved sectional analysis proceeds with incorporation of unit operation capital costs and more detailed flowsheets and models. While not synthesizing the ‘complete process’ with unit operations, the proposed MINLP approach addresses process connectivity, provides upper bounds on profitability, and allows insight into the financial and strategic motivations for implementing large scale design modifications.
6. Conclusions In this paper we have endeavored to address Grossmann and Daichendt’s (1994) challenge integrating MINLP optimization procedures within a decomposition framework. Using a block diagram-graph representation for a multistep chloromethanes process, and by identifying component source-destination matches and implementing heuristics, an MINLP problem was formulated. Using the GAMS MINLP solver potential ‘best’ configurations were determined. Chloromethane process networks were optimized for three different models incorporating successive sectional operations to enhance product distribution. Results showed improved objective function values for the successive structures analyzed. Moreover, on adding processing sections the differing configurations offered insights into underlying processing strategies.
Acknowledgements James Phimister would like to thank the ECOSSE group of the Department of Chemical Engineering, University of Edinburgh for funding during his MS degree, and Dr. Guignard-Spielberg of the Wharton Business school, University of Pennsylvania, for assistance in making this work possible.
Appendix A Table 4 Selling/purchasing prices for HDA process Component
Sell/purchase price ($/tonne)
Toluene Hydrogen Benzene Methane Diphenyl
200 100 500 100 20
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Appendix B
B.1.4. Process constraints The following constraints apply to all chloromethane examples.
Table 5 Purchasing/selling prices for the chloromethanes process Component
Purchasing price $/tonne
Selling price $/tonne
Methanol Hydrogen chloride Dimethyl ether Water Chlorine Methyl chloride Methylene chloride Chloroform Carbon tetrachloride Hydrogen
150 100 300 100 100 400 500 600 600 100
150 !20 500 100 100 100 500 600 0 !100
F *0, G H I F !ºy )0 ∀i, j, k, G H I G H I y "0, where i3C G D C G (y #y )"0, K H C AJ H C F 8000 F RMM "80 000, G H C G 1000 G H where i3P, P"+c3, c2, c1, c, d,. B.2. RS1 model
B.1. Chloromethanes model B.1.1. Notation m"methanol, hcl"hydrogen chloride, d"dimethyl ether, h2o"water, cl2"chlorine, ch3"methyl chloride, ch2"methylene chloride, ch"chloroform, c"carbon tetrachloride, h2"hydrogen. C"+m, hcl, d, h2o, cl2, ch3, ch2, ch, c, h2, throughout all models for all of the constraints/inequalities for all models j3S, k3D. F is the flowrate of component i, connecting source G H I j, to destination k (kmol/h). y is a binary variable G H I representing the existence of a connection for component i, connecting source j, to destination k (1 implies existence). z represents plant section for section i. SummaG tions are over all elements in the respective set except where otherwise noted. B.1.2. Formulation Z"max f (x), VWX h (x, y, z)"0, g (x, y, z))0, x 3+X"+x "x 3RL, x*)x)x3,, y 3+½"+y "y 3+0, 1,,, z 3+Z"+z "z 3+0, 1,,.
B.2.2. Reaction model F "(1!X ) F , K I K H I H F " F #0.97X F , AF I AF H K H I H H F "0.015 * (1!X ) F , B I K H I H F " F !0.97X F , FAJ I FAJ H K H I H H F " F #0.985X F . FM I F H FM H I H H B.3. RS2 model
B.1.3. Objective function For all examples the objective function was
B.2.1. Section restrictions Letting S"+ f, 1, 2, D"+1, 2, e,, A1"+m, hcl,, B1" C5A1, G1"+m, hcl, h2o, d, ch3,, H1"C5G1, variable X is the conversion of methanol in RS1 with 0.2).X )1. Section restrictions are: y "z , K D y *z , FAJ H H y "0, where i3B1. G H G H y *z ∀i3G1, G I I y "0, where i3H1, G I G I
8000 Z" Sell * RMM * * F ! Cost G G G H C G 1000 G H G I 1 * RMM * * F ! Cost * RMM * * F , G G H I 20 G G G H I G H I where i3C.
B.3.1. Section restrictions A2"+ch3, cl2,, B2"C5A2, G2"+ch3, ch2, ch, c, hcl,, H2"C5G2, variable X is the conversion of methanol in RS2 with 0.2)X )1. y "z , AF D y *z , AJ H H
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y *z ∀i3G4, G I I
y "0, where i3B2, G H G H y *z , ∀i3G2, G I I y "0, where i3H2. G I G I
y "0, where i3H4. G I G I
B.3.2. Reaction model F "(1!X ) F , AF I AF H I H F " F ! (F #F #F ), AF I AFAJ H AF I AF I A I I H I
325
F "0.5 * X F ! F , AF I AF H K I I H I ! F , F "0.3 * X F AF H K I A I I H I F " F #2 F #3 F , FAJ I AF I AF I A I I I I I y " y . AJ H FAJ I H I B.4. RS3 model B.4.1. Section restrictions A3"+hcl,, B3"C5A3, G3"+cl2, h2,, H3" C5G3 y *z , FAJ H H y "0, where i3B3, G H G H y *z ∀i3G3, G I I y "0, where i3H3. G I G I B.4.2. Reaction model F "0.5 F AJ I FAJ H I H F "0.5 F F I FAJ H I H B.5. RS4 model B.5.1. Section restrictions A4"+h2, c,, B4"C5A4, G4"+ch, c, h2, hcl,, H4" C5G4, variable X is the conversion of carbon tet rachloride in RS4 with 0.2)X )1. y *z ∀i3A4, G H H y "0, where i3B4, G H G H
B.5.2. Reaction model F "(1!X ) F , A I A H I H F " F !X F , F I F H A H I H H F "X F , AF I A H I H F "X F . FAJ I A H I H References Achenie, L. K. E., & Biegler, L. T. (1985). Algorithmic synthesis of chemical reactor networks using mathematical programming. Ind. Engng. Chem. Fundam., 25, 621—627. Achenie, L. K. E., & Biegler, L. T. (1993). A superstructure based approach to chemical reactor network synthesis. Comput. Chem. Engng., 25, 23—40. Balakrishna, S., & Biegler, L. T. (1993). A unified approach for the simultaneous synthesis of reaction, energy and separation systems. Ind. Engng. Chem. Res., 32, 1372—1382. Brooke, A., Kendrick, D., & Meeraus, A. (1988). GAMS: A user’s guide. Redwood City, CA: The Scientific Press. Crowe, C., Douglas, P. L., Glasgow, J., & Mallick, S. K. (1992). Development of an expert system for process synthesis. ¹rans. Inst. Chem. Engrs., 70A, 110—117. Douglas, J. M. (1990). Synthesis of multistep reaction processes. In J. J. Siirola, I. E. Grossman, G. Stephanopoulos (Eds.), Foundations of computer-aided process design. Amsterdam: Elsevier. Douglas, J. M. (1988). Conceptual design of chemical processes. New York: McGraw—Hill. Douglas, J. M. (1985). A hierarchical. decision procedure for process synthesis, A.I.Ch.E. J., 31, 353—363. McKetta, J. J. (Ed.) (1979). Encyclopedia of chemical processing and design. New York: Marcel Dekker. Engelmann, H. D., Erdmann, H. H., Funder, R., & Simmrock, K. H. (1989). The solving of complex process synthesis problems using distributed expert systems. Comput. Chem. Engng., 13, 459—465. Floquet, P., Pibouleau, L., & Domenech, S. (1985). Reactors—separators sequences synthesis by a tree searching algorithm. Ind. Chem. Engng. Symp. Ser., 92, 415—426. Floudas, C. A. (1995). Nonlinear and mixed-integer optimization: fundamentals and applications. Oxford: Oxford University Press. Fraga, E. S. (1996). The automated synthesis of complex reaction/separation processes using dynamic programming. Chem. Engng. Res. Des., 74, 249—260. Grossmann, I. E. (1990). MINLP Optimization strategies and algorithms for process synthesis. In J. J. Siirola, I.E. Grossman, & G. Stephanopoulos (Eds.) Foundations of computer-aided process design. Amsterdam: Elsevier. Grossmann, I. E., & Daichendt, M. M. (1994). New trends in optimization-based approaches to process synthesis. Proc. PSE (pp. 95—109). Othmer, K. (1993). Kirk Othmer encyclopedia of chemical technology. (4th Ed) New York: Wiley.
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Kirkwood, R. L., Locke, M. H., & Douglas, J. M. (1988). A prototype expert system for synthesizing chemical process flowsheets. Comput. Chem. Engng., 12, 329—343. Koci, G. R., & Grossmann, I. E. (1989). A modelling and decomposition strategy for the MINLP optimization of process flowsheets. Comput. Chem. Engng., 13, 797—819. Kokossis, A. C., & Floudas, C. A. (1990). Optimization of complex reactor networks — I. Isothermal operation. Chem. Engng., 45, 595—614. Kokossis, A. C., & Floudas, C. A. (1991). Synthesis of isothermal reactor—separator-recycle systems. Chem. Engng. Sci., 46, 1361—1383. Kokosis, A. C., & Floudas, C. A. (1994). Optimization of complex reactor networks — II. Nonisothermal operations. Chem. Engng. Sci., 49, 1037—1051. Kurz, B. E. (1972). Homogeneous kinetics of methyl chloride chlorination, Ind. Engng. Chem. Process Des. Dev., 11, 332—337. Nishida, N., Stephanopoulos, G., & Westerberg, A. W. (1981). A review of process synthesis. A.I.Ch.E. J., 27, 321—349.
Omtveit, T., Wahl, P. E., & Lien, K. M. (1994). Decomposed algorithmic synthesis of reactor—separation-recycle systems. Comput. Chem. Engng., 18, 1115—1124. Pibouleau, L., Said, A., & Domenech, S. (1983). Synthesis of optimal and near optimal distillation sequences. Chem. Engng. J., 27, 9—19. Pibouleau, L., & Domenech, S. (1986). Discrete and continuous approaches to the optimal synthesis of distillation sequences, Comput. Chem. Engng., 10, 479—491. Siirola, J. J., & Rudd, D. F. (1971). Computer-aided synthesis of chemical process designs: from reaction path data to the process task networks. Ind. Engng. Chem. Fund., 10, 353—363. Sinha, G. P., Chandrasekaran, B. S., Mitter, N., Dutta, G., Singh, S. B., Choudhury, A. R., & Roy, P. N. (1995). Strategic and operational management with optimization at Tata Steel. Interfaces, 25, 6—19. Subramanian, R., Scheff Jr., R. P., Quillinan, J. D., Wiper, D. S., & Marsten, R. E. (1994). Coldstart: Fleet assignment at Delta Air Lines. Interfaces, 24, 104—120.
The synthesis of multistep process plant configurations James R. Phimister *, Eric S. Fraga, Jack W. Ponton Department of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA Department of Chemical Engineering, University College London, London WC1 7JE, UK Department of Chemical Engineering, University of Edinburgh, Edinburgh EH9 3JL, UK Received 8 November 1997; revised 27 July 1998; accepted 27 July 1998
Abstract A framework for generating and optimizing chemical plant process connectivity is presented. The method addresses plant synthesis and analysis for multistep processes, processes where multiple reactions occur in both series and parallel, and where interstitial separation operations may be required. The method is tailored for the initial stages of process design, whereupon limited development of the process flowsheet has occurred, and estimation of plant economic viability is required. The chemical process is modeled as an interconnected graph linking chemical component nodes to operating sections. A mixed integer non linear program is utilized to obtain ‘best’ flowsheet configurations with initial mass balances and estimations for reactor parameters. The method is illustrated by examining large scale continuous processes, with the hydrodealkylation of toluene (HDA) process and a multistage chloromethanes process presented as case studies. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Reactor-separator-recycle structures; MINLP; Multistep processes; Process synthesis; Optimization
1. Introduction In a review of process synthesis optimization approaches, Grossmann and Daichendt (1994) note the integration of mixed integer non-linear programming (MINLP) with other design methods remains a research challenge. In this paper we address Grossmann and Daichend’s challenge by integrating MINLP with the heirarchical decomposition approach of process synthesis. The approach benefits from a block representation of a process flowsheet, allowing for determination of plantwide connectivity and initial values for ‘dominant design variables’, without requiring formulation of a rich superstructure consisting of numerous interconnected unit operations. The approach is applied to multistep process representations. Two case studies are presented with results providing initial flowsheets and process parameters similar to those reported in both academia and industry.
*Corresponding author. Tel.: 215 898 8351; fax: 215 573 2093; e-mail: [email protected]
Douglas (1990) presents a heuristic and decomposition method for synthesizing multistep chemical processes. Multistep processes have individual reactor units in which successive and parallel reactions occur. By linking separate units in multistep processes raw feedstocks are converted to products through several reaction paths and intermediate chemicals. Douglas (1988, 1985) presents a method where flowsheets are derived by decomposition and analysis of successive process levels. Decomposition proceeds using heuristics and quantitative evaluation to select the best configuration at each level of design. The process flowsheet develops through each level where it is further decomposed into a number of other designs and the selection procedure repeated. The decomposition and heuristic approach has been included in an expert system (Kirkwood et al., 1988). Other synthesis expert systems are discussed by Crowe et al. (1992) and Engelmann et al. (1989). Optimization methods have been utilized to investigate many aspects of process synthesis. For reviews see Grossmann (1990), Grossmann and Daichendt (1990), and Nishida et al. (1981). For a thorough presentation of MINLP applications to chemical engineering refer to
0098-1354/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 1 3 5 4 ( 9 8 ) 0 0 2 7 6 - 2
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Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications (Floudas, 1995). A selection of papers shows optimization techniques used in synthesizing reactor networks (Achenie and Biegler, 1985, 1990), distillation trains (Fraga and McKinnon, 1994; Pibouleau et al., 1983; Pibouleau and Domenech, 1986) and integrated reactive-separation units (Balakrishna and Biegler, 1993). An algorithm for linking reaction and separation sections is presented by Floquet et al. (1985). Optimization techniques are used by Omtveit et al. (1994) and Kocis and Grossman (1989) to analyze the reaction separation HDA process presented by Douglas (1990). The former uses a two stage optimization algorithm while Kocis and Grossman employ an outer approximation/equality relaxation (OA/ER) algorithm for solving an MINLP formulation. Kokossis and Floudas (1991) address the reactor-separator synthesis problem by developing superstructures which accommodate all possible interconnections between reactors and distillation columns and using an MINLP solver to determine an optimal flowsheet. To solve problems where multistep reactionseparation processes require synthesis Fraga employs the dynamic programming CHiPS package (1996). Case studies using optimization techniques to address large-scale commercial practices relate an interactive and iterative experience. For recounted studies, an optimization model containing an objective function and constraints is formulated and solved. Using an appropriate solver, solutions are examined by experts in the field, the model subsequently refined, and the procedure repeated (Subramanian et al., 1994; Sinha et al., 1995). In our approach, a framework for applying mixed integer nonlinear optimization to the successive and iterative decomposition method of process design is presented. To accomplish this an MINLP is formulated to help address the synthesis of multistep processes at the initial stage of process design. At this first stage of process synthesis the process designer is considered to have limited information from which to make decisions. These decisions will greatly affect the eventual layout, operability and profitability of the plant. Initially, process information available may be limited to reaction paths with basic kinetic models, costs of raw materials, expected selling prices and the expected plant production. Our formulation encompasses this limited framework, while still allowing for detailed reactor models to be used if appropriate. The initial decisions which the designer encounters can be large in number. Both discrete (yes/no) decisions and selection of continuous parameters (e.g. feed flowrate) must be determined. Some decisions which might be encountered are: E How to ‘connect’ a number of plant sections, i.e. plant inter-connectivity and recycle configurations; E Which raw materials to select; E Whether some reaction paths should be avoided;
Which products to produce and in what quantities; Values for what Douglas terms ‘dominant design variables’ (1990) — those variable which will most significantly effect the economic viability of the process. To obtain useful decisions at this stage of process design and to gain initial values for some parameters, an MINLP is formulated, consisting of binary decision variables (y) specifying whether connections or plant sections exist (a value of 1 implies existence, 0 implies no existence) and continuous variables (x) representing flowrates and conversion. To address the connectivity and synthesis problem a profit based objective function is maximized: E E
Z"max f (x) VW subject to constraints on variables x and y. A superstructure is proposed for multistep processes which incorporates plant sections without requiring detailed specification of individual units. The formulation is illustrated for the well known HDA problem and results are presented for a multistep chloromethanes process.
2. Method In formulating the multi-section design problem as an MINLP the chemical process is viewed as a set of reactor—separator (RS) sections with interconnected component nodes matching ‘sources’ to ‘destinations’. A ‘source’ defines a location where a component is supplied and a ‘destination’ is where a component is required. Fig. 1 shows the superstructure for a process with five components and one reactor—separator section. Two arrays of component sources exist; those for the process feed and the nodes exiting the RS section. Likewise, two destination arrays exist; the reactor—separator entrance and the process exit array. The initial superstructure allows all possible interconnections. Note however, the grouping of individual components into streams is not specified, thereby simplifying problem formulation without reducing generality. 2.1. Formulating connections An interconnection is defined by the binary variable y where i a chemical component and is a member of G H I set Q, the components in the process. j is the source node and k, the destination node. j3S such that S,+ f, 1, 2, 2 , NS, where ‘f ’ represents the process feed, ‘I’ the source nodes for RS section 1, and ‘NS’, the total number of plant sections. Likewise k3D, where D,+1, 2, 2 , e,. Each member of D represents a node entering a given RS section or leaving the process (element ‘e’). For example, using this notation, the connection y "1 denotes that hydrogen is fed to the & D J
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317
Fig. 1. Initial superstructure for one RS section five component system.
Fig. 2. Reduced superstructure with integer constraints applied.
process and enters the first RS section, y "1 speci!& fies that methane is recycled in section one. 2.2. Component matching Siirola and Rudd (1971) suggest, during initial process design, some component nodes can be matched from sources to destinations, other sources ruled out, and the remainder are possible connections. For example, when a reactant A is fed to a process and is required in Section 1, the connection between this source and destination may exist. If A is not produced from any other section of the process then y "1 (1) D is specified as a constraint. Constraints can formalize heuristics, for example the heuristic that all components fed to the process are connected to at least one RS section may be desired. Therefore a feed source connecting directly to an exist
destination is forbidden. This is formalized by y "0, i3Q. (2) G D C G Additional heuristics could also be formulated. For instance, if set of reactants R containing +A, B,, a subset of all chemicals in the process (set Q), the heuristic ‘reactants do not exit the process’ would have the formulation: y "0, i3R, j3S. (3) G H C G H Eqs. (1)—(3) are represented in Fig. 2. In applying the logical connectivity constraints many initial connections may be eliminated and a second superstructure, a subset of the first superstructure, created. The intention in defining the logical constraints is to create a secondary superstructure which encompasses all possible processing configurations, while not including configurations which are nonsensical.
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2.3. NLP subproblem For each connection present a corresponding flow, F, exists. If a connection is not present then the flowrate is zero. Here, flowrates are denoted similarly to the connection variables. F represents a positive flowrate for G H I component i, connecting source j to destination k. Negative flowrates are not allowed. The constraints F !ºy )0, ∀i, j, k G H I G H I F *0, G H I ensures component flowrates meet this criteria. º is the largest process flowrate allowed in a connection between two locations. º is relaxed when solutions have flowrates on the upper bound. Likewise convergence can be improved by tightening the value of º. For each RS section a reactor model relates the chemical inflow to outflow. For instance for reaction: APB
(4)
occurring in RS1 and undergoing conversion X the reactor model F "(1!X) F , I H I H F " F #X F , j3S, k3D I H H I H H is defined.
For the HDA process there is one RS section (RS1), and five components. Using the definitions for sets described Q"+hyd, tol, benz, diph, meth,, S"+ f, 1,, D"+1, e,. The initial superstructure is shown in Fig. 1. ¸ogical constraints. For this example process bypasses will not be allowed: y "0, i3Q G D C G Benzene, diphenyl and methane are not fed to the process: y #y #y "0. D D
D The reactants toluene and hydrogen must connect the process feed to RS1. y "1, i3+tol, hyd,. G D Benzene, methane and diphenyl must be connected from RS1 to the process exit y "1, i3+benz, meth, diph,. G C Toluene cannot be connected to the process exist destination: y "0. H C H Lastly, all source nodes for RS1 must have a destination.
3. Example 1: Hydrodealkylation of toluene The HDA process (Douglas, 1988) provides an illustrative example for this approach. The process consists of feedstocks of toluene and hydrogen forming methane and benzene by reaction:
y *1, k3D, ∀i. G I I Applying the logical constraints gives the secondary superstructure shown in Fig. 3.
C H #H P C H #CH . A byproduct diphenyl is also formed reversibly:
(5)
RS1 Model. Letting X define the conversion of toluene, the selectivity t of toluene converted to form benzene is defined by
2C H C H #H .
(6)
0.0036 t"1! , (1!X)
Fig. 3. Reduced superstructure for HDA process.
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319
Fig. 4. Optimal configuration for HDA process. Mass balance shown (kmol/h).
where moles of benzene formed t" moles of toluene converted and X)0.97 and 0.2)t)1. Process specifications. The production requirement in 80 000 metric tonnes of benzene per year for an 8000 h year. The molar feed ratio entering RS1 is 5 : 1 hydrogen : toluene, and flowrates are zero where connections are not present. Objective function. The objective is to maximize profit defined by products sold minus raw material costs. Purchasing/Selling prices are supplied in Appendix A. A penalty of 1/20th of the purchasing price is incurred for every tonne transported between a given source and destination. The penalty prevents internal flowrates having zero shadow costs at the solution, thereby preventing multiplicity of solutions. HDA results. Using the DICOPT MINLP solver provided by GAMS (Brooke et al., 1988) the HDA problem is optimized. The optimal configuration with flowrates are shown in Fig. 4. The optimal conversion is 0.697 giving a selectivity of 0.977. This provides an objective function of $18.65 million per annum in a solution time of 0.12 seconds. All calculations reported are performed on an IBM RISC 6000. The solution provides a starting point from which further synthesis can proceed. As illustrated by Douglas this configuration is feasible, whereupon on decomposition a membrane separation system is used to separate methane from hydrogen.
together with byproduct HCl. All the chloromethanes were at the time saleable products, some of the HCl could be sold, and the rest disposed of cheaply. This route was soon eliminated in the developed world due to restrictions on dumping HCl, and a two section plant combining chlorination with hydrochlorination, described below as the first example, was used. This still produced all four chloromethanes, which remained as marketable products into the 1980s. However, a range of environmental and safety concerns have eliminated markets for monochloromethane (methyl chloride) as a precursor to leaded petrol additives, dichloromethane (methylene chloride) as a domestic solvent and tetrachloromethane (carbon tetrachloride) as an intermediate for CFCs. Only trichloromethane (chloroform), an intermediate to more ozone friendly HCFC refrigerants, and dichloromethane as an industrial solvent are likely to remain in demand in the longer term. For background information on the chloromethanes process refer to Kirk Othmer Encyclopedia of Chemical ¹echnology (Entry: Chloromethanes, 1993), Encyclopedia of Chemical Processing and Design (Entry: Chloromethanes, 1979) and Kurtz (1972). 4.1. Process description and chemistry Four processing sections are proposed in an attempt to eliminate less commercially useful products and to provide greater flexibility of product spectrum. Plants with two, three and four sections are investigated. In the first section, RS1, a feedstock of methanol reacts with hydrochloric acid to form monochloromethane. CH OH#HCl P CH Cl#H O.
(7)
Byproduct dimethyl ether is formed by the side reaction: 4. Example 2: Chloromethanes manufacture
2CH OH (CH ) O#H O.
The manufacture of chloromethanes is a multistep process of significant current interest. Historically, a simple single step direct chlorination of methane produced a mixture of mono- to tetrachlorinated products,
In the second plant section (RS2) successive chlorination steps result in a product distribution consisting of methylene chloride (CH Cl ), chloroform (CHCl ) and carbon tetrachloride (CCl ) (reactions (9)—(11)). Complete
(8)
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reaction of chlorine occurs in this section: CH Cl#Cl P CH Cl #HCl, (9) CH Cl #Cl P CHCl #HCl, (10) CHCl #Cl P CCl #HCl, (11) Additional sections can form chlorine and hydrogen through the electrolysis of HCl (RS3), and reduce carbon tetrachloride to chloroform (RS4). The four sections are shown in Fig. 5. Process specifications. The objective is to design a component flowsheet which maximized profits by producing 80 000 tonnes per year (8000 h) of the distributed products methyl chloride, methylene chloride, chloroform, carbon tetrachloride and the byproduct dimethyl ether. For this study, recycling of products of RS2 is limited to methyl chloride, HCl exists the process with a ‘removal’ penalty applied for each tonne produced. A maximum conversion of 0.97 is set for both sections. Models used in the chloromethanes study are provided in Appendix B. In forming the multiple products, it is desirable to have high conversions of methyl chloride to obtain the
profitable chloroform, yet this is countered by the low profitability of carbon tetrachloride. The amount of hydrochloric acid formed is also dependent on conversion in RS2. Since this byproduct is a reactant in RS1 greater production in RS2 will reduce the need to purchase HCl as a raw feedstock. 4.2. Two section plant This first study involves two sections, the hydrochlorination of methanol in RS1, and chlorination in RS2. The best component flowsheet solution determined by DICOPT for the two section plant is shown in Fig. 6. The solution is obtained in 0.23 s, giving a conversion of 0.97 in RS1 and 0.454 in RS2. The model contains 42 continuous variables, 40 binary variables, 33 equality constraints and 190 inequality constraints. The profit is determined to be $24 M/yr. Purchasing HCl affects the recycle structure and optimal design conditions. The conversion in RS2 is limited so as to supply all of the HCl required in RS1. The final solution is self-sufficient in HCl it is neither necessary to purchase HCl as a feedstock nor is HCl a product; therefore the design avoids any removal penalty.
Fig. 5. Potential plant sections for chloromethanes process.
Fig. 6. Component flowsheet solution for two section chloromethanes process. Mass balance shown (kmol/h).
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Fig. 7. Component flowsheet solution for four section chloromethanes process.
4.3. Four section plant Two potential plant sections, an electrolysis section and an hydrogenation section, are added to the study. Initially all plant sections are assumed to exist in the final solution and hence none are attributed binary variables. In the third section, RS3, HCl is electrolysed by reaction (12) to completion. 2HCl P Cl #H . (12) The fourth section (RS4) consists of a hydrogenation section. In RS4 carbon tetrachloride is hydrogenated to chloroform and hydrochloric acid by reaction (13). The maximum conversion for the reaction is set at 0.97. H #CCl P CHCl #HCl. (13) The best component flowsheet solution is shown in Fig. 7 with a mass balance provided in Table 1. Conversions in RS1, RS2, and RS4 are all on the upper bound of 0.97. The solution provides an objective function of $31.64 M/yr. The model consists of 250 binary variables, 253 continuous variables, 1018 inequality constraints and 43 equality constraints. Because the fourth section allows conversion of carbon tetrachloride to chloroform, maximum conversion in RS2 is favored. This configuration obtains maximum amounts of the most profitable component, chloroform, with all carbon tetrachloride formed in RS2 converted to chloroform in RS4. Chlorine is purchased as a feedstock with a partial makeup from the electrolysis section. 4.4. Plant section existence Decisions on the existence of whole plant sections are implemented by assigning binary variables for each plant section and modifying connectivity constraints to be valid when sections are not present. Provided section constraints are valid when only zero flowrates enter and leave a given section, section constraints do not require activation or deactivation (see Floudas, 1997, p. 239 for a discussion of constraint activation) when integer variables are applied to whole plant sections. Solving the MINLP with the restriction that only three plant sections are obtained in the final solution, the
Table 1 Process flowrates for 4 Section Chloromethanes Process (kmol/h) Comp. Source
Destinations 1
MeOH. feed MeOH. 1 out HCl. 2 out HCl. 4 out DME. 1 out H O.1 out Cl . feed Cl . 3 out CH Cl. 1 out CH Cl. 2 out CH Cl . 2 out CHCl . 2 out CHCl . 4 out CCl . 2 out CCl . 4 out H . 3 out
2
92.311 2.855 64.267 25.275
3
4
Exit
117.949 1.385 90.926 123.241 58.850 89.541 2.769 22.142 42.125 25.275 25.275 0.782 25.275
33.700
flowsheet shown in Fig. 8 is obtained. Table 2 provides the process mass balance. The conversion in RS1 and RS2 is 0.97 and 0.817 respectively providing an objective function value of $24.99 M/yr. In contrast to the two section process, where HCl is not purchased as a feedstock, HCl is purchased for use in both the reaction in RS1 and for electrolysis and formation of chlorine in RS2. The RS2 conversion shifts markedly upwards from the two section process because conversion in RS2 is not limited by the full consumption of HCl in RS1. However, in comparison to the four section process, a decrease in conversion results in RS2. This lower conversion avoids production of unprofitable carbon tetrachloride which previously had been converted back to chloroform. 4.5. Multiple solutions using integer cuts The hierarchical decomposition approach to designing process flowsheets suggests identifying multiple solutions at each level of design. These solutions can be returned
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Fig. 8. Component flowsheet solution for three section chloromethanes process.
Table 2 Process flowrates for 3 Section Chloromethanes Process (kmol/h) Comp. Source
Destinations 1
MeOH. feed MeOH. 1 out HCl. feed HCl. 1 out HCl. 2 out DME. 1 out H O. 1 out Cl . 3 out CH Cl. 1 out CH Cl. 2 out CH Cl . 2 out CHCl . 2 out CCl . 2 out H . 3 out
Table 3 Changes to chloromethanes component flowsheet shown in Fig. 7 resulting from successive integer cuts
2
3
Cut no.
Add connection Flow f G H I (i, j, k) (k mol/h)
1 2 3
H2.4.4 H2.4.5 H2.4.5 H2.4.5 HCl.4.3
Exit
128.543 3.975 340.917 216.235 216.235 1.928 126.610
4
0.78 268.8 0.78 268.8 202.2
Remove connection (i, j, k)
Obj. func. ($M)
31.64 31.64 31.64 HCl.4.1
31.63
Flowrates throughout plant change at successive solutions. Flowrate at lower bound.
216.235 124.682 27.922 58.098 41.615 24.969 216.235
for further screening if initial developed solutions prove unsatisfactory. By using integer cuts, alternative solutions providing similar objective function values, are obtained. Integer cuts successively add the constraint (1!yQ)# yQY*1 Q QY to the original formulation, and re-solve the problem. Vectors yQ and yQY are the binary variables having values of one and zero at the previous solution. The constraint F*ºQy is included to prevent solutions where connections exist with corresponding zero flowrates. Further, ºQ"Min(FQyQ) where FQ is the vector of nonzero flowrates at the previous solution, is added to the formulation to prevent trivial solutions with zero flowrates but values of one for the corresponding binary connection variable. Changes to the flowsheet shown in Fig. 7 which result from the first four integer cuts are supplied in Table 3. All solutions provide similar objective function values, close
to that obtained in the original solutions. The first three integer cut solutions provide all possible deviations in transporting the least expensive feedstock: hydrogen. The fourth integer cut solution changes the process connections of HCl. The next three higher cuts (not shown in Table 3) again explore deviations on transporting hydrogen, though now based on the component flowsheet obtained in the fourth integer cut solution.
5. Discussion The method described provides a framework for addressing the initial stages of design for continuous multistep reaction processes. Using a block conceptual representation of the chemical process, mixed integer optimization methods are employed to obtain initial solutions addressing the problems of process connectivity. The emphasis of this work has been to create a systematic framework which screens a large number of potential recycle structures and determines initial values for those variables which greatly affect process viability and profitability (‘dominant design variables’). For the ten component, four section chloromethanes example, there are on the order of 2 potential configurations. Although the majority of these configurations
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are infeasible, a small fraction and hence a large number of configurations are feasible. Discerning from all of these possible solutions optimal structures and parameters by, for instance, a heuristic approach, may not be possible. In the approach discussed, binary variables represent potential connections between processing sections, or whether the section itself exists. Continuous variables represent flowrates along a connection, or sectional variables. Because raw material costs dominate total processing costs (35—85%, Douglas, 1990) it is essential to let process feed flowrates be continuous variables, rather than to synthesize structures based upon fixed feeds. As has been illustrated with the chloromethanes process, selection of feedstocks used changed according to plant sections are available. The results illustrate the validity of the approach, providing recycle structures and operating parameters with low waste and high product yield obtained as solutions. The recycle structure obtained in the HDA example is identified by Douglas as a potential recycle structure for further investigation. The desired reactor conversion of 0.697 compares favorably with 0.75 determined by Douglas (1988), and 0.675 determined by Omtveit et al. (1994). The two section chloromethane study shows a balance of HCl, whereby HCl formed in RS2 is completely consumed by the RS1, resulting in neither disposal nor purchasing of HCl required. This plantwide ‘self-sufficient’ balance on HCl has been reported in industry, and is identified as a motivating factor in selecting the reaction path (see Encyclopedia of Chemical Processing and Design, Entry: Chloromethanes). The results indicate a more complex model or objective function is not warranted for the case studies examined. When material costs dominate total processing costs, the objective function will prove insensitive to, for instance, energy costs or annualized fixed unit costs. Hence the simple objective function employed, products sold less materials purchased, is justified for obtaining initial solutions. Further, additional objective function or constraint complexity decreases the likelihood of finding a feasible and optimal solution, and increases uncertainty of whether a local minima obtained is a global minima. Because each section is incorporated into the MINLP as a ‘black box’, providing an output based on a given feed input, more accurate section modeling and costing can be implemented without loss of generality. The case study solutions provide an understanding of process motivations and sensitivities. As synthesis develops, more detailed models and optimizations based on initial results can ensue. To cater for market demand upper limits on feedstock inventories or production capacities on individual chemicals can be imposed as con-
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straints. Sensitivity investigations can be performed to observe changes in design as market demand an price changes occur. Further, as decomposition of sections occurs, improved sectional analysis proceeds with incorporation of unit operation capital costs and more detailed flowsheets and models. While not synthesizing the ‘complete process’ with unit operations, the proposed MINLP approach addresses process connectivity, provides upper bounds on profitability, and allows insight into the financial and strategic motivations for implementing large scale design modifications.
6. Conclusions In this paper we have endeavored to address Grossmann and Daichendt’s (1994) challenge integrating MINLP optimization procedures within a decomposition framework. Using a block diagram-graph representation for a multistep chloromethanes process, and by identifying component source-destination matches and implementing heuristics, an MINLP problem was formulated. Using the GAMS MINLP solver potential ‘best’ configurations were determined. Chloromethane process networks were optimized for three different models incorporating successive sectional operations to enhance product distribution. Results showed improved objective function values for the successive structures analyzed. Moreover, on adding processing sections the differing configurations offered insights into underlying processing strategies.
Acknowledgements James Phimister would like to thank the ECOSSE group of the Department of Chemical Engineering, University of Edinburgh for funding during his MS degree, and Dr. Guignard-Spielberg of the Wharton Business school, University of Pennsylvania, for assistance in making this work possible.
Appendix A Table 4 Selling/purchasing prices for HDA process Component
Sell/purchase price ($/tonne)
Toluene Hydrogen Benzene Methane Diphenyl
200 100 500 100 20
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Appendix B
B.1.4. Process constraints The following constraints apply to all chloromethane examples.
Table 5 Purchasing/selling prices for the chloromethanes process Component
Purchasing price $/tonne
Selling price $/tonne
Methanol Hydrogen chloride Dimethyl ether Water Chlorine Methyl chloride Methylene chloride Chloroform Carbon tetrachloride Hydrogen
150 100 300 100 100 400 500 600 600 100
150 !20 500 100 100 100 500 600 0 !100
F *0, G H I F !ºy )0 ∀i, j, k, G H I G H I y "0, where i3C G D C G (y #y )"0, K H C AJ H C F 8000 F RMM "80 000, G H C G 1000 G H where i3P, P"+c3, c2, c1, c, d,. B.2. RS1 model
B.1. Chloromethanes model B.1.1. Notation m"methanol, hcl"hydrogen chloride, d"dimethyl ether, h2o"water, cl2"chlorine, ch3"methyl chloride, ch2"methylene chloride, ch"chloroform, c"carbon tetrachloride, h2"hydrogen. C"+m, hcl, d, h2o, cl2, ch3, ch2, ch, c, h2, throughout all models for all of the constraints/inequalities for all models j3S, k3D. F is the flowrate of component i, connecting source G H I j, to destination k (kmol/h). y is a binary variable G H I representing the existence of a connection for component i, connecting source j, to destination k (1 implies existence). z represents plant section for section i. SummaG tions are over all elements in the respective set except where otherwise noted. B.1.2. Formulation Z"max f (x), VWX h (x, y, z)"0, g (x, y, z))0, x 3+X"+x "x 3RL, x*)x)x3,, y 3+½"+y "y 3+0, 1,,, z 3+Z"+z "z 3+0, 1,,.
B.2.2. Reaction model F "(1!X ) F , K I K H I H F " F #0.97X F , AF I AF H K H I H H F "0.015 * (1!X ) F , B I K H I H F " F !0.97X F , FAJ I FAJ H K H I H H F " F #0.985X F . FM I F H FM H I H H B.3. RS2 model
B.1.3. Objective function For all examples the objective function was
B.2.1. Section restrictions Letting S"+ f, 1, 2, D"+1, 2, e,, A1"+m, hcl,, B1" C5A1, G1"+m, hcl, h2o, d, ch3,, H1"C5G1, variable X is the conversion of methanol in RS1 with 0.2).X )1. Section restrictions are: y "z , K D y *z , FAJ H H y "0, where i3B1. G H G H y *z ∀i3G1, G I I y "0, where i3H1, G I G I
8000 Z" Sell * RMM * * F ! Cost G G G H C G 1000 G H G I 1 * RMM * * F ! Cost * RMM * * F , G G H I 20 G G G H I G H I where i3C.
B.3.1. Section restrictions A2"+ch3, cl2,, B2"C5A2, G2"+ch3, ch2, ch, c, hcl,, H2"C5G2, variable X is the conversion of methanol in RS2 with 0.2)X )1. y "z , AF D y *z , AJ H H
J.R. Phimister et al. / Computers and Chemical Engineering 23 (1999) 315—326
y *z ∀i3G4, G I I
y "0, where i3B2, G H G H y *z , ∀i3G2, G I I y "0, where i3H2. G I G I
y "0, where i3H4. G I G I
B.3.2. Reaction model F "(1!X ) F , AF I AF H I H F " F ! (F #F #F ), AF I AFAJ H AF I AF I A I I H I
325
F "0.5 * X F ! F , AF I AF H K I I H I ! F , F "0.3 * X F AF H K I A I I H I F " F #2 F #3 F , FAJ I AF I AF I A I I I I I y " y . AJ H FAJ I H I B.4. RS3 model B.4.1. Section restrictions A3"+hcl,, B3"C5A3, G3"+cl2, h2,, H3" C5G3 y *z , FAJ H H y "0, where i3B3, G H G H y *z ∀i3G3, G I I y "0, where i3H3. G I G I B.4.2. Reaction model F "0.5 F AJ I FAJ H I H F "0.5 F F I FAJ H I H B.5. RS4 model B.5.1. Section restrictions A4"+h2, c,, B4"C5A4, G4"+ch, c, h2, hcl,, H4" C5G4, variable X is the conversion of carbon tet rachloride in RS4 with 0.2)X )1. y *z ∀i3A4, G H H y "0, where i3B4, G H G H
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