Reaction Invariants and Mole Balances for Plant Complexes

Ind. Eng. Chem. Res. 2002, 41, 3771-3783

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Reaction Invariants and Mole Balances for Plant Complexes† Sagar B. Gadewar, Michael F. Doherty, and Michael F. Malone* Department of Chemical Engineering, University of Massachusetts, Amherst Massachusetts 01003-9303

Many pharmaceuticals, specialty chemicals, pesticides, or monomers are manufactured in processes with a large number of reaction steps and separations between reactors. These coupled steps lead to the use of multiple plants consisting of reaction-separation systems, coupled by intermediate and recycle flows in a plant complex. Integrated plant complexes are also used for producing petrochemicals. Douglas (Synthesis of Multistep Reaction Processes. In Foundations of Computer-Aided Design; Siirola, J. J., Grossmann, I. E., Stephanopoulos, G., Eds.; CacheElsevier: Amsterdam, The Netherlands, 1990; pp 79-103) described a procedure for the conceptual design of such plant complexes that relies on heuristics and that can generate a number of alternative layouts. We describe a framework to automate the determination of mole balances around and between plants in such complexes. In this procedure, the plant complex is represented as a state task network and reaction invariants are used to formulate the component balances. Mole balances for mixing and splitting of component flows are identified automatically, which provides the structure of the coupling between plants and a systematic means to identify alternatives to make, buy, or make remotely for intermediates. The approach is demonstrated for production of Bisphenol A, adipic acid, and for an anticonvulsant drug. The methodology greatly reduces the engineering effort in formulating the overall balances for interconnected plants with many reactions. Introduction Conceptual design or “process synthesis” generates potentially profitable alternatives based on the experimental and mathematical analyses of chemical routes to produce a desired product from available raw materials. These alternatives consist of coupled reactor networks, distillation column sequences, gas recovery systems, and similar subsystems. The interaction between these subsystems is typically important in finding good designs among the many process alternatives. The synthesis of many individual subsystems themselves has been a subject of study, e.g., as reviewed by Hendry et al.,2 Nishida et al.,3 Westerberg,4,5 and Grossmann and Daichendt.6 Systematic approaches to process synthesis are the basis for computer-aided design programs. One of the earliest tools, AIDES (adaptive initial design synthesizer), was developed by Siirola et al.7,8 AIDES uses a set of available reactions and process equipment to assemble a flow sheet that reaches the products from raw materials using a linear programming approach. Motard and co-workers9,10 developed BALTAZAR, which allocates resources and matches goals stream-by-stream to produce a flow sheet. The economic objective of reducing costs is addressed indirectly by minimizing mass flows in the plant. Neither AIDES nor BALTAZAR considered equipment costs but used heuristics intended to emphasize better economic performance. Subsequent studies based on optimization methods, such as those described by Grossmann and co-workers,11,12 can be used to include more quantitative economic objectives. In optimization-based methods, a † This paper is dedicated to Professor Jim Douglas in recognition of his many outstanding contributions to chemical process engineering. * To whom correspondence should be addressed. Phone: (413) 545-0838. Fax: (413) 545-1133. E-mail: mmalone@ ecs.umass.edu.

superstructure is typically used as a basis to include process alternatives. The presence or absence of certain devices, as well as their interconnections, can be modeled using discrete binary variables. The solution of a mixed-integer nonlinear program gives the optimum flow-sheet structure and design variables for a defined objective. It is essential for this or similar approaches to determine the appropriate superstructure for a large class of process flow sheets. Douglas13-15 described a hierarchical approach for conceptual design, suitable for single sets of reactions. This approach is based on heuristics and short-cut materials balances and design models which are used to generate promising candidate process flow sheets and to estimate optimum operating conditions for each of these. This approach essentially develops a superstructure, beginning with a limited amount of detail and increasing engineering effort and data requirements only for promising alternatives. An economic analysis is performed at each level of the hierarchy, thereby allowing early termination of poor ideas. A prototype computer code for this procedure, PIP (process invention procedure), was described by Kirkwood et al.16 This hierarchical procedure is complementary to optimization-based approaches, as shown by Daichendt and Grossmann.12 Douglas1 developed an approach for products made using multiple sets of reactions based on additional heuristics and using the hierarchical procedure repeatedly. This approach provides conceptual designs for integrated “plant complexes” and includes a much wider class of processes such as speciality polymers and chemicals, pharmaceuticals, pesticides, and agricultural chemicals. Chemical routes for these processes are often characterized by incomplete knowledge of the reaction steps; sometimes only single reactions with yield less than unity are known or the reactions are not balanced stoichiometrically. Schultz and Douglas17 developed a procedure of introducing fictitious components to get an

10.1021/ie010877m CCC: $22.00 © 2002 American Chemical Society Published on Web 03/07/2002

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estimate of waste loads for systems with incomplete knowledge of the reaction chemistry. A surprisingly large number of process alternatives (104-109) can often be found14 using this procedure. The hierarchical procedure uses heuristics and models to make decisions that generate a base case design, and changing any of these decisions generates a process alternative. Levels 1 and 2 of the hierarchical procedure need the determination of the streams entering and leaving each of the individual plants. Estimation of the overall mass balances can require considerable engineering effort, especially for multistep reactions found in a plant complex. Two useful approaches normally used to formulate steady-state mole balances for a chemical process are18 (i) the sequential modular approach, which calculates the output flows of a unit operation for given input flows and equipment parameter values (a flow sheet is solved sequentially to determine all process flows), and (ii) the equation solving approach, which represents a flow sheet by a collection of equations which must be solved simultaneously (in this method too, all equipment parameters have to be specified to determine the process flows). At level 2 of Douglas’ hierarchical procedure, the equipment configuration is not known. We need an approach that is independent of equipment specification and also allows us to formulate the overall mole balances without specifying the extents of reaction for reacting systems. Recently, Gadewar et al.19 published a systematic method for formulating mole balances in complex reaction chemistries with multiple reactions in a single plant. That method uses reaction invariants which take the same values before, during, and after the reaction. This method does not need a priori specification of the extents of reactions to formulate the mole balances. For interconnected plants with many reactions, a systematic method that can be automated will significantly reduce the effort in formulating and solving the material balances. An automated procedure allows more process alternatives to be evaluated in a given time. In this paper we devise a framework for automating the level 2 material balances, arising in Douglas’ hierarchical procedure for plant complexes. Base Case Flow-Sheet Generation Level 1: Number of Plants and Plant Connections. A multistep reaction process can be divided into a number of simple plants, each defined by a group of reactions and the associated separations (for which there may be many alternatives). The hierarchical procedure for single plants can be applied for each of these once sufficient information on the input and output flows is decided. A base case flow sheet is generated by applying the following heuristics:1 (i) All reactions that occur at the same temperature, pressure, and phase and use the same catalyst are grouped together in the same plant. (ii) “Avoidable” reactions (those that do not lead to the desired product) can be done in a separate plant. (iii) Forbidden matches of components or reactions need additional plants. Level 2: Input-Output Structure for Individual Plants. Decisions at this level also change the structure of the base case flow sheet. The following rules and heuristics are applied for the base case flow sheet.1 (i) A fresh feed stream to each plant is created for every reactant that is not supplied either by a reaction

within that plant or by a plant connection. Decisions to buy or make remotely various intermediates generate alternatives. (ii) Reactants in the gas phase need vent or recycle and purge streams. Additional feed for the case of purge streams cannot be ignored. (iii) Reversible byproducts may be recycled to suppress undesired equilibrium reactions. (The various options generate alternatives.) (iv) The desired products and all other byproducts exit the plants. Make locally, make remotely, or buy decisions are important to determine the plant connections. The base case assumes that none of the intermediates or desired products are bought in lieu of making them. A process alternative can be generated for each decision of buying a product or intermediate as exemplified below. Alternative flow sheets are generated from the base case by changing the decisions made at levels 1 and 2. Further details on these heuristics and their impact on plant connections can be found in work by Schultz.20 Representation of Integrated Plants The efficient automation of the material balances in plant complexes requires a data structure. A convenient structure is the “state-task network” (STN) introduced by Kondili et al.21 The STN is a useful representation because it forms a graph with streams as vertices that are connected by tasks that transform one stream into another. There is also a one-to-one correspondence between the STN and the adjacency matrix representation of the plant complex, which helps in automating the mole balance methodology described below. Each component flow in the process is a state, and the level 2 tasks consist of reaction and separation in a single plant, mixing of flows, and splitting of flows. Although some flows may actually enter or leave a plant as mixtures in a single stream, the representation in terms of pure-component flows is particularly convenient in what follows and subsequent estimation of the overall flows is straightforward. In the example that follows, we describe the use of this representation to encode the structure of the flow sheet in the automatic procedure for overall balances. Mole Balances at Level 2 The basic units in a flow sheet of a plant complex are (1) a process block or “plant” that involves a reaction and separation system, (2) mixers to combine flows from different plants, and (3) splitters to divide a single flow to feed two or more plants. The overall mole balances for a plant complex consist of mole balances for each of these three basic blocks. First, we show how mole balances can be written systematically. Then, we demonstrate the automated implementation of this procedure on three examples. Process Block. Consider a reaction system consisting of c components undergoing R independent chemical reactions, as shown schematically in Figure 1. The feed(s) are represented by a c-dimensional column vector of inlet molar flow rates for each species, n0; the products are represented by a vector of outlet flow rates for each species, n. The R independent chemical reactions are written as

ν1,rA1 + ν2,rA2 + ... + νc,rAc h 0, r ) 1, 2, ..., R (1)

Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3773

A splitter, shown in Figure 3, consists of a single inlet and multiple outlet flows. The mole balances for the splitter is

Fi,a ) Fi,b + Fi,d

(6)

where Fi,a, Fi,b, and Fi,d are flows consisting of component i. At level 2, because each flow is a pure-component flow, a single mole balance (eq 6) can be written for each splitter. If there are cp species in plant p with Rp independent reactions, there are cp - Rp mole balances for the plant. Each mixer and splitter will have one mole balance. If the total number of streams in a flow sheet for a plant complex is f (including all external feeds and exit streams), the degrees of freedom for the overall balances for the flow sheet are

Figure 1. Block diagram of a single plant.

NP

DOF ) f Figure 2. Schematic diagram for a mixer.

where Ai is the reacting species and νi,r is the stoichiometric coefficient of component i in reaction r. The usual convention is used: νi,r > 0 if component i is a product, νi,r < 0 if it is a reactant, and νi,r ) 0 if component i is an inert. The reaction invariants for the system are given by the transformed mole numbers,19,22 which take the same value before, during, and after the reaction. These are

Ni0 ) ni0 - νiT(VRef)-1nRef 0, i ) 1, ..., c - R (2) Ni ) ni - νiT(VRef)-1nRef, i ) 1, ..., c - R

(3)

where νiT is the row vector of dimension R of the stoichiometric coefficients of component i in all of the R reactions and VRef is a square matrix of dimension R × R whose columns are the stoichiometric coefficients for R reference components. Details on a criteria for the choice of reference components are described by Ung and Doherty.22 The transformed mole numbers Ni0 are the reaction invariants based on the inlet molar flow rates, and Ni are the reaction invariants based on the outlet molar flow rates. The mole balances for the reacting system are simply

Ni0 ) Ni, i ) 1, ..., c - R

(4)

We note that the number of mole balances for R independent reactions is c - R. The reaction invariants are a linear transformation of the number of moles of species to give conservation relationships which depend only on the reaction chemistry. These linear transformations correspond exactly to the mole balances. A mixer, shown in Figure 2, consists of two or more inlet flows and a single outlet flow. The mole balances for the mixer is simply

Fi,a + Fi,b ) Fi,d

(5)

where Fi,a, Fi,b, and Fi,d are flows consisting of component i. At level 2, we treat each flow as a pure component, so the inlets and outlets of a mixer consist of a single component. Therefore, a single mole balance (eq 5) can be written for each mixer.

(ci - Ri) - m - s ∑ i)I

(7)

where NP is the number of plants in the complex and m and s are the number of mixers and splitters, respectively. Example 1: Bisphenol A Production Bisphenol A (BPA) is manufactured by reaction of phenol and acetone. An undesired isomer of BPA (i-BPA) is also produced and is subsequently converted to BPA via an intermediate, phenyl isopropyl hydroxide (PIPH). We develop the automatic procedure for overall mole balances using the BPA process as a demonstration. Level 0. The plant complex manufactures a single petrochemical, BPA. We choose to use a dedicated continuous process for this product. The physical properties of the components and the reactions with their conditions are determined at this level. Level 1: Number of Simple Plants and the Process Connectivity. Using the heuristics of Douglas,1 listed above, we group all reactions occurring at the same temperature and pressure, and that use the same catalyst. We assume that the number of plants is equal to the number of these groups. The eight species in the reactions are phenol, acetone, BPA, i-BPA, water, trisphenol-A (TPA), PIPH, and the dimer of PIPH (diPIPH). The reactions are

Plant I (75 °C, P ) 15 psia) 2(phenol) + acetone f BPA + H2O

(8a)

2(phenol) + acetone f i-BPA + H2O

(8b)

3(phenol) + 2(acetone) f TPA + 2H2O

(8c)

Plant II (250 °C, P ) 4 psia) i-BPA h phenol + PIPH

(9a)

2PIPH h di-PIPH

(9b)

Plant III (70 °C, P ) 15 psia) phenol + PIPH f BPA

(10)

Phenol and acetone are reactants, i-BPA and PIPH are intermediates, H2O, TPA, and di-PIPH are byproducts, and BPA is the desired product. Sometimes, byproducts

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Figure 3. Schematic diagram for a splitter.

that each outlet state can be achieved from each inlet state. Each component flow in the plant complex is a state in the STN. The structure of the STN or the plant complex can be represented algebraically by an adjacency matrix. The adjacency matrix is defined by ai,j ) 1 if a directed line joins flow i to flow j; otherwise, ai,j ) 0; the diagonal elements in the adjacency matrix are zero. Once the STN is established from the flow sheet, the adjacency matrix can be formulated automatically because there is a one-to-one correspondence between the entries in the matrix and the connection in the network. The adjacency matrix for Figure 5 is

Figure 4. Flow sheet for the BPA plant complex at level 2.

Automation of Reaction Invariants

Figure 5. State-task-network for the BPA plant complex.

The adjacency matrix represents the connection of subsystems or plants by (component) flows, and each flow has to be identified with the species it represents. The species corresponding to each flow is represented by a flow designation matrix, P, where pi,j ) 1 if flow i consists of species j; otherwise, pi,j ) 0. The rows of P represent flows, and the columns represent species. The species are acetone (S1), phenol (S2), water (S3), BPA (S4), TPA (S5), i-BPA (S6), DiPIPH (S7), and PIPH (S8).

can be valuable coproducts and their value has to be considered in the determination of the economic potential. The plant connections are shown in Figure 4 for which the plant connectivity is determined by examining the reaction chemistry. Plants I and II are connected by a flow of i-BPA because it is produced in the first reaction set (plant I) and consumed in the second reaction set (plant II). Phenol is a feed to plants I and III and is produced in plant II. Therefore, a feed of phenol enters plants I and III, whereas the phenol product from plant II is mixed with the fresh feed of phenol to plant III (or plant I). Further details on establishing plant connections are given in work by Schultz.20 Level 2: Input-Output Structure of the Flow Sheet. At this level, overall mole balances are formulated for each plant. One method of writing these mole balances involves specification of the extents of reactions. Gadewar et al.19 proposed an alternative method using reaction invariants. Figure 5 shows the STN23 for the BPA plant complex from Figure 4. The reactions in the plant transform the inlet streams into the outlet streams. We, therefore, say

Matrix P shows that flow F1 is acetone, F2, F7, F10, F12, and F13 are phenol, and so on. Identification of the Stoichiometric Matrix and Flows for Each Plant. There are six reactions within the plant complex. The stoichiometric matrix for the complete set of reactions is


where reactions r1-r6 are given by eqs 8a-10, respectively. We can write a reaction stoichiometry matrix for each plant as

to those based on the outlet molar flow rates. Because the structure of the flow sheet is encoded in the adjacency matrix, it is possible to automatically determine which species are absent from inlet or outlet streams, which occurs when feeds contain no product species and when the common heuristic is used that valuable ingredients are recovered and recycled within a plant. Of course, some species will have both inlet and outlet flows, e.g., species in plants with purge streams. The following analysis will identify such streams and formulate the flow vectors accordingly. Individual flow designation matrices for each plant are determined from eq 12 as

and

Species that do not participate in the reactions for a plant correspond to the zero rows in VI, VII, and VIII. To write the mole balances for a single plant, it is necessary to identify the flows entering and leaving the plant. Once a single flow is associated with a particular plant, it is easy to identify all flows associated with that plant from the adjacency matrix. This can be done easily during the formulation of the STN. For instance, in this example, F1 is an inlet to plant I; the adjacency matrix shows that flow F2 is also an inlet to plant I (because F1 and F2 have identical rows) and that flows F3, F4, F5, and F6 are outlets of plant I (unit entries in row 1). Inlets and outlets for plants II and III are similarly identified. A flow vector for each plant is written by using the following convention: all flows are positive (Fi > 0, ∀ i) and preceded by a negative sign for flows entering the plant and a positive sign for flows leaving the plant. The flow vectors for the three plants in this example are

[]

[]

-F1 -F2 -F6 -F9 F3 F7 -F FI ) F , FII ) F , FIII ) 10 4 8 F11 F5 F9 F6

[ ]

(15)

Mole balances are written by equating the reaction invariant molar flows based on the inlet molar flow rates

The modified flow vectors to be used in formulating the mole balances are

F/I ) PI‚PIT‚FI

(19)

F/II ) PII‚PIIT‚FII

(20)

F/III ) PIII‚PIIIT‚FIII

(21)

The transformations in eqs 19-21 account for components that enter and leave the same plant. For instance, if a component enters plant I via flow rate Fi and leaves the plant via flow rate Fj, the flow vector F/I will have an element Fj - Fi in place of element -Fi and also in place of element Fj (which are in flow vector FI). A caveat is that a redundant mole balance is generated for each component that enters and leaves the same plant (for instance, plant I in example 2), but the redundant balances can be easily identified automatically. In this example, there are no components that enter and leave the same plant and the modified flow vectors F/k (where k ) I, II, and III) are identical to the flow vectors given by eq 15. No redundant mole balances are generated. Mole Balances. In the mole balance methodology, reference components must be chosen, equal in number to the independent reactions in a single plant. The

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reaction stoichiometry matrix for each plant given in eq 14 is in terms of the species in the entire plant complex. Because we are interested in the component flows, the reaction stoichiometry matrix is expressed using the flow designation matrix, P, which relates flows to species. This is one of the reasons for specifying each flow as a pure-component flow. The modified stoichiometric matrices are

Using eqs 25-27, the mole balances for plant I are

-F1 ) -F4 - 2F5 - F6

(28a)

-F2 ) -2F4 - 3F5 - 2F6

(28b)

F3 ) F4 + 2F5 + F6

(28c)

Now, when flows F7 and F8 are chosen as the reference flows for plant II, the mole balances are

-F6 ) -F7

(29a)

F9 ) F7 - 2F8

(29b)

When flow F9 is chosen as a reference flow for plant III, the mole balances are where

V /k ) Qk‚P‚Vk, k ) I, II, and III

(23)

The matrix Qk selects flows that are connected to plant k. It is a subset of an identity matrix of dimension equal to the total number of flows, I (14 × 14). For instance, QI ) [q1, q2, q3, q4, q5, q6]T, where qi is the ith row of the identity matrix I (14 × 14), because flows F1, F2, F3, F4, F5, and F6 are connected to plant I. Rearranging eqs 2-4, the mole balances for a single plant are written as

ni - ni0 ) νTi (VRef)-1(nRef - nRef0), i ) 1, ..., c - R (24) If species i enters the plant and does not leave (e.g., valuable reactants), ni ) 0, and if i does not enter the plant but only leaves (e.g., many products or byproducts formed by an irreversible reaction), ni0 ) 0. This information is known from the STN and the flow designation matrix, P. This information is used to set the flow vectors given by eqs 19-21. Therefore, the general quantities ni - ni0 and nRef - nRef0 in eq 24 are related to a particular flow sheet by eqs 19-21. Equation 24 is written in terms of the flow vectors after the flow vector is divided into a set of reference and nonreference components. The mole balances in terms of process flows are

-F10 ) -F9

(30a)

F11 ) F9

(30b)

Note that the reference components must be chosen such that the stoichiometric matrix for the reference flows is invertible. Details on a criteria for choosing the reference components can be found in work by Ung and Doherty.22 Mixer and Splitter Balances. Figure 4 shows that there are two mixers (combining flows F4 and F11 and flows F7 and F13) and one splitter (dividing F12 into flows F2 and F13). From the STN we would like to determine these streams automatically, which can be accomplished using a modified adjacency matrix defined as

A* ) A + I

(31)

where I is the identity matrix. Multiplying the modified adjacency matrix by the flow designation matrix, P, gives a matrix H ) A*‚P, which represents the set of species to which each flow is connected.

/ / ) Vk,NR(Vk,Ref)-1Fk,Ref , k ) I, II, and III (25) Fk,NR

The number of reference components for each plant is equal to the number of independent reactions in that plant.19,22 Here, Vk,Ref and Vk,NR are the stoichiometric matrices for the reference and nonreference flows in / / and Fk,NR are the modified plant k, respectively; Fk,Ref flow vectors for the reference and nonreference flows in plant k, respectively. If flows F4, F5, and F6 are chosen as the reference flows for plant I,

[]

[ ]

F4 -F1 / ) -F2 F/I,Ref ) F5 , FI,NR F6 F3

(26)

If flows i and k containing species j are mixed, this corresponds to Hi,j ) Hk,j ) 2. (If flows i and k mix to form stream l, then Pi,j ) Pk,j ) 1 and Ai,m ) Ak,m ) 1.) For example, column 2 of H has a coefficient 2 in the 7th and 13th rows because F7 and F13 are mixed. Rows


7 and 13 of the adjacency matrix show that these flows combine to form F10. The matrix H identifies the inlets to a mixer, while the adjacency matrix identifies the outlet of the mixer. Similarly, from column 4 of H and the adjacency matrix, we determine that F4 and F11 mix to form flow F14. The mole balances are

F7 + F13 ) F10

(33a)

F4 + F11 ) F14

(33b)

The inlets to a plant are identified in the earlier section, where we determined the overall balances for single plants. For plants with purge streams, the species in the inlet to a plant also appear in the outlet. Therefore, flows representing such species will have a coefficient 2 in the matrix H even if they do not enter a mixer. Inlet flows to single plants are, therefore, disregarded in our analysis for mixers because they might unwittingly imply the presence of mixers. Row 12 of matrix H has a coefficient 3. This signifies that flow F12 is split into two flows (a coefficient 4 would indicate splitting into three flows). The adjacency matrix shows that F12 is split into F2 and F13. The splitter balance is

F12 ) F2 + F13

(34)

Equations 28a-30b together with eqs 33a-34 are the complete set of level 2 balances for the BPA plant complex. There are 10 equations in 14 variables leaving 4 degrees of freedom. These degrees of freedom are satisfied either by specifying at least one flow rate (e.g., production rate of BPA) and several selectivities.19 For instance, if the total production rate of BPA is 100 mol/ h, i.e., F14 ) 100 mol/h. For certain conditions, the ratio of the selectivities for i-BPA to BPA in plant I is Si-BPA/ SBPA ) 0.75; therefore, F6/F4 ) 0.75. For TPA to BPA, STPA/SBPA ) 0.1; therefore, F5/F4 ) 0.1. In plant II, the selectivity to PIPH is 100% and F9/F6 ) 1. The 4 degrees of freedom are satisfied. Solving eqs 28a-30b and eqs 33a-34 using a linear equation solver, we find F1 ) 111.42, F2 ) 217.14, F3 ) 111.42, F4 ) 57.14, F5 ) 5.71, F6 ) 42.85, F7 ) 42.85, F8 ) 0.0, F9 ) 42.85, F10 ) 42.85, F11 ) 42.85, F12 ) 217.14, and F13 ) 0.0 (all in mol/h). Methods for choosing the degrees of freedom to solve a set of linear equations are discussed by Westerberg et al.18 Any change in the operating conditions changes the selectivities; therefore, these numerical specifications will change and the solutions can be found. Algorithm for Mole Balances A. Characterization of the Flow Sheet. 1. From the reaction chemistry decide the number of plants, reactions in each plant, and plant interconnections using the heuristics above (or generate an alternative). 2. Find the STN corresponding to the plant connections. 3. Determine the adjacency matrix for the STN. 4. Identify the flow designation matrix. B. Mole Balances for Single Plants. 1. Determine the reaction stoichiometry matrix for each plant based on the reactions occurring and using the stoichiometry matrix from step A.1 above.

2. Determine the connections for all of the inlet and outlet flows for each plant from the adjacency matrix. 3. Use the method of reaction invariants to formulate the mole balances for each plant as follows. (a) Determine the flow designation matrix for each plant (eqs 16-18). (b) Determine the transformed flow vector for each plant from the flow vector and flow designation matrix (eqs 19-21). (c) Assign reference flows for each plant; the number of reference flows is equal to the number of independent reactions. (d) Partition the modified flow vector into a flow vector for the reference and nonreference components (as in eq 26). (e) Partition the reaction stoichiometry matrix into a reaction stoichiometry matrix for the reference and nonreference components (as in eq 27). (f) Use eq 25 to formulate the mole balances. C. Mixer and Splitter Balances. 1. Multiply the modified adjacency matrix by the flow designation matrix, H ) A*‚P. 2. Check for a column in H with a coefficient 2 in any of the rows. For this column, find all of the rows with a coefficient 2. Neglect the rows corresponding to the inlets of each plant. 3. Find a column from the adjacency matrix with unit entries for the rows identified in step 2. The rows correspond to the flows being mixed and the column of A represents the output. 4. Write down the mixer balance as sum of flows corresponding to the rows in step 2 ) flow corresponding to the column in step 3. 5. Repeat from step 2 for each column with a coefficient of 2. 6. Check for a row in H with a coefficient of 3 or greater. Note that a coefficient of 3 corresponds to a flow splitting into two streams and a coefficient of 4 to a flow splitting into three streams, etc. 7. Determine the columns of the adjacency matrix with unit entries for the row identified in step 6. The row corresponds to the inlet of a splitter, and the columns correspond to the outlets of the splitter. 8. Write down the splitter balance as flow corresponding to the row in step 6 ) sum of the flows corresponding to the columns in step 7. 9. Repeat steps 7 and 8 as necessary until all splitter balances are determined (all rows in H with a coefficient of 3 or greater). Example 2: Production of Adipic Acid Adipic acid is used in the production of nylon 6,6, and a proposed alternative to the traditional route is based on a two-step carbonylation of butadiene in the presence of methanol. 24-27 The reactions in the plant complex for this process are1,27

Plant I (400 °F, P ) 25 psia) 2Co(OAc)2 + 8CO + 2H2 h Co2(CO)8 + 4HOAc (35) Plant II (250 °F, P ) 8700 psia) butadiene + CO + MeOH h MPA

(36)

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Figure 6. Base case flow sheet for the ADA plant complex at level 2.

Plant III (350 °F, P ) 2200 psia) MPA + CO + MeOH h DMA

(37a)

MPA + CO + MeOH h DMG

(37b)

MPA + CO + MeOH h DMS

(37c)

2MPA h heavies

(37d)

Plant IV (212 °F, P ) 25 psia) Co(CO)8 + 5O2 + 4HOAc h 2Co(OAc)2 + 8CO2 + 2H2O (38) Plant V (212 °F, P ) 15 psia) DMA + 2H2O h ADA + MeOH

(39a)

DMG + 2H2O h MGA + MeOH

(39b)

DMS + 2H2O h ESA + MeOH

(39c)

Butadiene, carbon monoxide, methanol, and water are

reactants, methyl pentanoate (MPA) and dimethyl adipate (DMA) are intermediates, adipic acid (ADA) is the desired product, and dimethyl methyl glutarate (DMG), dimethyl ethyl succinate (DMS), methylglutaric acid (MGA), ethyl succinic acid (ESA), and heavies are byproducts. DMG and DMS are useful byproducts because they react with water to form methanol (in plant V), which is a reactant in the plant complex. When the byproducts are valuable coproducts, their value has to be considered in the determination of the economic potential. The catalyst generation and recovery consists of components cobalt acetate (Co(OAc)2) as a catalyst precursor, dicobalt octacarbonyl (Co2(CO)8) as the catalyst, oxygen, acetic acid (HOAc), and hydrogen as reactants, and carbon dioxide and water as byproducts. This is an example where the reaction chemistry is not completely known because some species are lumped together as heavies. Reactions (39b) and (39c) are avoidable reactions and can be carried out in a separate plant.


for this plant. Choosing water as the reference component, the mole balances are

F4 ) F5, F6 ) 2F5, F24 - F36 ) 0, F25 - F37 ) 0, 1 5 F26 - F38 ) 0, F27 ) F5, F31 - F35 ) F5, 2 2 F32 - F33 ) 0, F34 ) 4F5 (43) Plant V. The species are DMA, DMG, DMS, water, methanol, ADA, MGA, and ESA. There are eight components (c ) 8) and three reactions (R ) 3); therefore, five mole balances (c - R ) 5) can be written for this plant. When ADA, MGA, and ESA are chosen as the reference components, the mole balances are

F36 ) F40, F37 ) F41, F38 ) F42, F39 ) 2F40 + 2F41 + 2F42, F43 ) F40 + F41 + F42 (44) Figure 7. STN for the base case ADA plant complex.

Using the heuristics at level 1 and 2, one flow sheet is determined, as shown in Figure 6. The inlet and outlet flows for each process are identified in the figure, and the STN is shown in Figure 7. Applying the algorithm for mole balances, overall balances for single plants are as follows. Plant I. The species are cobalt acetate, carbon monoxide, carbon dioxide (as an inert), hydrogen, dicobalt octacarbonyl, water, and acetic acid. There are seven components (c ) 7) and one reaction (R ) 1); therefore, six mole balances (c - R ) 6) can be written for this plant. Choosing dicobalt octacarbonyl as the reference component, the mole balances are

F1 - F7 ) 8F11, F3 - F9 ) 2F11, F4 ) 2F11, F6 ) 4F11, F2 - F8 ) 0, F5 - F10 ) 0 (40) Plant II. The species are butadiene, carbon monoxide, methanol, MPA, dicobalt octacarbonyl, and carbon dioxide (as an inert). There are six components (c ) 6) and one reaction (R ) 1); therefore, five mole balances (c - R ) 5) can be written for this plant. Choosing MPA as the reference component, the mole balances are

F12 - F18 ) F19, F13 - F16 ) F19, F15 ) F19, F14 - F17 ) 0, F11 - F20 ) 0 (41) Plant III. The species are MPA, carbon monoxide, carbon dioxide (as an inert), methanol, DMA, DMG, DMS, heavies, and dicobalt octacarbonyl. There are nine components (c ) 9) and four reactions (R ) 4); therefore, five mole balances (c - R ) 5) can be written for this plant. Choosing DMA, DMG, DMS, and heavies as the reference components, the mole balances are

F19 ) F24 + F25 + F26 + 2F30, F21 - F28 ) F24 + F25 + F26, F23 ) F24 + F25 + F26, F20 - F27 ) 0, F22 - F29 ) 0 (42) Plant IV. The species are DMA, DMG, DMS, dicobalt octacarbonyl, cobalt acetate, oxygen, nitrogen (as an inert), carbon dioxide, acetic acid, and water. There are 10 components (c ) 10) and one reaction (R ) 1); therefore, nine mole balances (c - R ) 9) can be written

Mixer-Splitter Mole Balances. Using the algorithm for mixer-splitter balances, we find that the flow sheet has one mixer that mixes flows F44 and F45 to form flow F15 and one splitter that splits flow F43 into flows F23 and F44. Therefore, the mixer and splitter balances are

F44 + F45 ) F15

(45)

F43 ) F23 + F44

(46)

Equations 40-46 are the complete set of level 2 balances for the ADA plant complex. There are 32 equations in 45 variables leaving 13 degrees of freedom for solving them. These degrees of freedom are satisfied by specifying at least one flow rate (e.g., the production rate of ADA) and either 12 selectivities or 12 molar ratios.19 Process Alternatives Decisions at level 2 generate alternatives. A generally important decision at this level is whether to buy intermediates and products or whether to manufacture them at another location instead of manufacturing them on-site. This changes the structure of the flow sheet because one or more plants in the plant complex will not be needed. New waste streams might be generated or some existing waste streams might disappear, thereby changing the waste treatment costs in addition to the ingredients and investment costs. Once a new flow sheet is generated the methodology described here can be applied to determine the overall mole balances that enable the estimation of the new economic potential. The task of formulating the mole balances for single plants need not be repeated because they are known from the base case flow sheet. The mixer and splitter balances, however, may need to be reformulated. MPA is an intermediate in the manufacture of caprolactam. For manufacturers of both nylon 6, and nylon 6,6, it may be an option to transport MPA to the location of the ADA plant complex. In that case, plant II is not necessary in the ADA plant complex above and a flow sheet that corresponds to this alternative is shown in Figure 8. No new waste streams are generated, and the fresh feed streams of butadiene and methanol are eliminated. A byproduct stream of methanol, F44, is

3780


Figure 8. Flow sheet for the ADA plant complex corresponding to the decision to make MPA remotely.

generated. The mole balances for this process alternative are given by eqs 40, 42-44, and 46 along with a mole balance F11 ) F20 (as a result of removing plant II). (For simplicity, the nomenclature of the flows from the base case was retained.) One alternative that is generated using our methodology is to buy the intermediate DMA instead of manufacturing it. If this decision is made, the resulting flow sheet is shown in Figure 9. Plants I-IV in the base case flow sheet (Figure 6) are not needed to produce ADA in such a scenario. In the new flow sheet, the byproducts MGA and ESA are not produced because the feed does not contain DMG and DMS. Hence, the waste streams are reduced in this alternative. The mole balances for this alternative are

F36 ) F40, F39 ) 2F40, F43 ) F40

(47)

Although DMA is a commercially available product, the

Figure 9. Flow sheet for the ADA plant complex corresponding to the decision to buy DMA.

scale of ADA manufacture makes this alternative impractical. However, the alternative is technically feasible, and we use it to demonstrate our methodology.


Figure 10. Flow sheet for the CI-1008 pharmaceutical plant complex at level 2.

Our methodology is used to automatically write the mole balances at level 2 once a flow sheet is synthesized. It is possible to automate the generation of a flow sheet corresponding to a decision of buying intermediate(s) and/or product(s) (or making them remotely as opposed to locally) from the base case flow sheet. However, the decision of buying a certain intermediate or product depends on its availability and cost considerations, and if intermediate(s) and or product(s) are made remotely, the costs of transportation and inventory must be considered. Also, the waste streams are affected by these decisions, thus affecting the waste treatment costs. Example 3: Production of an Enantiomerically Pure Anticonvulsant (CI-1008) CI-1008 is an anticonvulsant and is a title assigned by Parke-Davis Co. to (S)-3-(aminomethyl)-5-methylhexanoic acid. CI-1008 is under development and is expected to give a better performance than Neurontin, a commercially available drug. Several synthetic routes to produce CI-1008 were studied by Hoekstra et al.28 We will apply our procedure to a synthetic route based on norephedrine, given by Scheme 1 in work by Hoekstra et al.28 The numbers in parentheses for species in the reaction chemistry listed below identifies the species with their corresponding chemical structure in Scheme 1 in Hoekstra et al.28 When Douglas’ heuristics are applied,1 the reactions can be carried out in the following plant complex:

Plant I C6H12O2 (2) + SOCl2 f C6H11OCl (3) + HCl + SO2 (48) Plant II C9H13ON (4) + CO(OEt)2 f C10H11O2N (5) + 2EtOH (49)

Plant III 3 + 5 f C16H21O3N (6) + HCl

(50)

Plant IV 6 + BrCH2CO2C4H9 f C22H31O5N (7) + HBr

(51)

Plant V 7 + H2O f C12H22O4 (8) + 5

(52)

8 + 2H2 f C12H24O3 (9) + H2O

(53a)

Plant VI 8 + 3H2 f C8H14O2 (12) + C4H10 + 2H2O

(53b)

Plant VII 9 + TsCl f C12H23O3Ts (10) + HCl

(54)

Plant VIII 10 + NaN3 f C12H23O2N3 (11) + NaOTs (55) Plant IX 11 + 5H2 f C8H17O2N (1 or CI-1008) + C4H10 + 2NH3 (56a) 11 + 5H2 f C8H15ON (13) + C4H10 + H2O + 2NH3 (56b) Components 2, SOCl2, 4, CO(OEt)2, BrCH2CO2C4H9, water, hydrogen, TsCl, and NaN3 are reactants, 3 and 5-11 are intermediates, HCl, SO2, EtOH, HBr, C4H10, NaOTs, and ammonia are byproducts, and 1 (i.e., CI1008) is the desired product. A flow sheet is shown in Figure 10 and the corresponding STN in Figure 11. Applying the algorithm for mole balances gives the overall balances for single plants in Table 1.

3782


Table 1. Mole Balances for Individual Plants in Example 3 plant

species

I II III IV V VI

2, SOCl2, 3, HCl, SO2 4, CO(OEt)2, 5, EtOH 3, 5, 6, HCl 6, BrCH2CO2C4H9, 7, HBr 7, H2O, 8, 5 8, H2, 9, H2O, 12, C4H10

VII VIII IX

9, TsCl, 10, HCl 10, NaN3, 11, NaOTs 11, H2, 1, 13, C4H10, NH3, H2O

mole balances F1 ) F5, F2 ) F5, F3 ) F5, F4 ) F5 F6 ) F8, F7 ) F8, F9 ) 2F8 F5 ) F12, F10 ) F12, F11 ) F12 F12 ) F15, F13 ) F15, F14 ) F15 F15 ) F20, F16 ) F20, F17 ) F20 F20 - 0.5F24 - 0.5F26 ) 0, F21 - 1.5F24 - 0.5F26 ) 0, 0.5F24 + F25 - 0.5F26 ) 0, F22 - 0.5F24 + 0.5F26 ) 0 F26 ) F29, F27 ) F29, F28 ) F29 F29 ) F32, F30 ) F32, F31 ) F32 F32 - 0.5F36 ) 0, F33 - F34 - 2.5F36 ) 0, F35 - 0.5F36 ) 0, F37 - F39 ) 0, 0.5F36 - F37 - F38 ) 0

process alternatives. The methodology simplifies the task of determining mole balances for interconnected reaction systems and allows rapid evaluation of process alternatives for their economic potential based on raw materials usage and wastes generated. The methodology is first demonstrated on the process to manufacture BPA. Also, results for a process producing ADA and a pharmaceutical process manufacturing an anticonvulsant CI-1008 are described. Automation of level 3 of the hierarchical procedure can further simplify the application of Douglas’ decision-making procedure. Acknowledgment We are grateful to the sponsors of the Process Design and Control Center, University of Massachusetts, Amherst, MA. Figure 11. STN for the CI-1008 pharmaceutical plant complex.

Mixer-Splitter Balances. Using the algorithm for mixer-splitter balances, we find that the flow sheet has one mixer that mixes flows F8 and F19 to form flow F10 and one splitter that splits flow F17 into flows F18 and F19. Therefore, the mixer and splitter balances are

F8 + F19 ) F10

(57)

F17 ) F18 + F19

(58)

Mole balances in Table 1 together with eqs 57 and 58 are the complete set of level 2 balances for the pharmaceutical plant complex manufacturing the anticonvulsant CI-1008. There are 33 equations in 39 variables leaving 6 degrees of freedom. Pharmaceutical process studies usually determine the yields of various components, and the mole balances can be reformulated in terms of yields. Using the procedure for mole balances, the economic potential at level 2 can be estimated for different chemical routes to produce CI-1008. Conclusions A systematic framework based on graph theory and reaction invariants is presented for formulating mole balances in integrated plant complexes. The flow sheet at level 2 of Douglas’ hierarchical procedure1,13,14,17 consists of interconnected blocks (single plants, mixers, and splitters). The concept of reaction invariants enables automated enumeration of mole balances for a single plant. The degrees of freedom for solving the balances are also identified. Decisions based on reaction chemistry as well as business decisions like make or buy, make locally, or make remotely, etc., all generate

Nomenclature A ) adjacency matrix A* ) modified adjacency matrix, A* ) A + I ai,j ) element of the adjacency matrix Ai ) chemical species c ) total number of reacting and inert components f ) total number of molar flows in the flow sheet F ) vector of molar flow rates F* ) modified vector of molar flow rates Fi ) molar flow rate of component i (mol/time) FRef ) vector of molar flow rates for the reference components H ) mixer-splitter identification matrix I ) identity matrix m ) number of mixers in the flow sheet nRef ) column vector of the outlet mole numbers for the R reference components n0Ref ) column vector of the inlet mole numbers for the R reference components ni ) number of moles of component i at the outlet ni0 ) number of moles of component i at the inlet Ni ) mole number transform of component i Ni0 ) mole number transform of component i based on inlet mole numbers NP ) number of single plants in a plant complex P ) flow designation matrix relating flows to species pi,j ) element of the flow designation matrix Qk ) subset of an identity matrix for the kth plant R ) number of independent reactions rj ) jth reaction s ) number of splitters in the flow sheet Si ) ith species V ) nonsquare matrix of dimension (c, R) of stoichiometric coefficients for the c components in the R reactions VRef ) square matrix of dimension (R, R) of the stoichiometric coefficients for the R reference components in the R reactions

Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3783 Greek Letters νi,r ) stoichiometric coefficient of component i in reaction r νiT ) row vector of the stoichiometric coefficients for component i in each reaction Subscripts and Superscripts 0 ) inlet -1 ) inverse of matrix I, II, and III ) plant number i ) components NR ) nonreference components r ) reactions Ref ) reference components

Note Added after ASAP Posting This article was released ASAP on 3/7/02. Some fonts were changed to clarify matrices. The corrected version was posted on 3/19/02. Literature Cited (1) Douglas, J. M. Synthesis of Multistep Reaction Processes. In Foundations of Computer-Aided Design; Siirola, J. J., Grossmann, I. E., Stephanopoulos, G., Eds.; Cache-Elsevier: Amsterdam, The Netherlands, 1990; pp 79-103. (2) Hendry, J. E.; Rudd, D. F.; Seader, J. D. Synthesis in the Design of Chemical Processes. AIChE J. 1973, 19, 1. (3) Nishida, N.; Stephanopoulos, G.; Westerberg, A. W. A Review of Process Synthesis. AIChE J. 1981, 27, 321-351. (4) Westerberg, A. W. A Review of Process Synthesis. In Computer Applications to Chemical Engineering: Process Design and Simulation; Squires, R. G., Reklaitis, G. V., Eds.; ACS Symposium Series 124; American Chemical Society: Washington, DC, 1980; p 53. (5) Westerberg, A. W. Synthesis in Engineering Design. Comput. Chem. Eng. 1989, 13, 365. (6) Grossmann, I. E.; Daichendt, M. M. New Trends in Optimization-Based Approaches to Process Synthesis. Comput. Chem. Eng. 1996, 20, 665-683. (7) Siirola, J. J.; Powers, G. J.; Rudd, D. F. Synthesis of System Designs: III. Toward a Process Concept Generator. AIChE J. 1971, 17, 677-682. (8) Siirola, J. J.; Rudd, D. F. Computer-Aided Synthesis of Chemical Process Designs. Ind. Eng. Chem. Fundam. 1971, 10, 353-362. (9) Mahalec, V.; Motard, R. L. Evolutionary Search for an Optimal Limiting Process Flowsheet. Comput. Chem. Eng. 1977, 1, 149-160. (10) Lu, M. D.; Motard, R. L. Computer-Aided Total Flowsheet Synthesis. Comput. Chem. Eng. 1985, 9, 431-445. (11) Papoulias, S. A.; Grossmann, I. E. A Structural Optimization Approach in Process SynthesissIII. Comput. Chem. Eng. 1983, 7, 723-734.

(12) Daichendt, M. M.; Grossmann, I. E. Integration of Hierarchical Decomposition and Mathematical Programming for the Synthesis of Process Flowsheets. Comput. Chem. Eng. 1997, 22, 147-175. (13) Douglas, J. M. A Hierarchical Decision Procedure for Process Synthesis. AIChE J. 1985, 31, 353-362. (14) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988; pp 4 and 16. (15) Douglas, J. M. Synthesis of Separation System Flowsheets. AIChE J. 1995, 41, 2522-2536. (16) Kirkwood, R. L.; Locke, M. H.; Douglas, J. M. A Prototype Expert System for Synthesizing Chemical Process Flowsheets. Comput. Chem. Eng. 1988, 12, 329-343. (17) Schultz, M. A.; Douglas, J. M. Stream CostssA First Screening of Reaction Pathways. Ind. Eng. Chem. Res. 2000, 39, 2410. (18) Westerberg, A. W.; Hutchison, H. P.; Motard, R. L.; Winter, P. Process Flowsheeting; Cambridge University Press: Cambridge, U.K., 1979; p 27. (19) Gadewar, S. B.; Doherty, M. F.; Malone, M. F. A Systematic Method for Reaction Invariants and Mole Balances for Complex Chemistries. Comput. Chem. Eng. 2001, 25, 1199-1217. (20) Schultz, M. A. Ph.D. Dissertation, University of Massachusetts, Amherst, MA, 1998. (21) Kondili, E.; Pantelides, C. C.; Sargent, R. W. H. A General Algorithm for Short-Term Scheduling of Batch OperationssI. MILP Formulation. Comput. Chem. Eng. 1993, 17, 211-227. (22) Ung, S.; Doherty, M. F. Vapor-Liquid Phase Equilibrium in Systems with Multiple Chemical Reactions. Chem. Eng. Sci. 1995, 50, 23. (23) Sargent, R. W. H. A Functional Approach to Process Synthesis and its Application to Distillation Systems. Comput. Chem. Eng. 1998, 22, 31-45. (24) Kummer, R.; Schneider, H.-W.; Platz, R.; Magnussen, P.; Weiss, F.-J. Manufacture of Butanedicarboxylic Acid Esters. U.S. Patent 4,171,451, Oct 16, 1979. (25) Platz, R.; Kummer, R.; Schneider, H.-W. Preparation of Butanedicarboxylic Acid Esters. U.S. Patent 4,258,203, Mar 24, 1981. (26) Kummer, R.; Schneider, H.-W.; Weiss, F.-J. Preparation of Butanedicarboxylic Acid Esters. U.S. Patent 4,259,520, Mar 31, 1981. (27) Isogai, N.; Hosokawa, M.; Okawa, T.; Wakui, N.; Watanabe, T. Process for Producing Adipic Acid Diester. U.S. Patent 4,404,394, Sep 13, 1983. (28) Hoekstra, M. S.; Sobieray, D. M.; Schwindt, M. A.; Mulhern, T. A.; Grote, T. M.; Huckabee, B. K.; Hendrickson, V. S.; Franklin, L. C.; Granger, E. J.; Karrick, G. L. Chemical Development of CI-1008, an Enantiomerically Pure Anticonvulsant. Org. Proc. Res. Dev. 1997, 1, 26-38.

Received for review October 22, 2001 Revised manuscript received January 11, 2002 Accepted January 15, 2002 IE010877M

Reaction Invariants and Mole Balances for Plant Complexes

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