Discrete Programming and Data Analysis for Heat-Integrated Process

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Discrete Programming and Data Analysis for Heat-Integrated Process Synthesis in Early Design Eric S. Fraga* Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K.

Ken I. M. McKinnon Department of Mathematics & Statistics, University of Edinburgh, Edinburgh EH9 3JZ, U.K.

Decisions made in the earliest stages of design can have a major impact on the final design. Therefore, these decisions should take into account as many of the important aspects of the problem as possible. One aspect that is critical to the economic and environmental performance of a process design is the energy utilization. However, it is common practice to choose a process design without considering any process integration until late in the design process. Such a sequential procedure is known to miss optimal heat-integrated solutions. This paper describes the combination of discrete programming and data analysis techniques for considering heat integration at the earliest stages of the design process. Through an iterative procedure, the engineer is able to identify process designs that are amenable to heat integration. 1. Introduction The automated synthesis of process flowsheets is a useful tool for the early, conceptual stages in the design of a chemical plant. The aim of such a tool is to identify good process structures that can subsequently be analyzed using rigorous simulation and other tools. The quality of the initial process generation and selection stage is enhanced if alternative processes can be compared using a variety of criteria. Specifically, including heat integration from the beginning of the design process can preclude or reduce the likelihood of dismissing good designs. Heat integration is necessary on the basis of both economic and environmental criteria. This is particularly true for distillation sequences, which are an essential part of most processes of interest to the chemical processing industry. However, heat integration might also be of interest in processes involving reactors and other special units. A process synthesis procedure that is able to handle the heat integration requirements of a wide range of processes is a valuable tool in early design. Automated procedures for heat-integrated process synthesis generally fall into two categories (see the recent review by Grossmann et al.1): (1) sequential approaches in which one or more process designs are identified without consideration of heat integration, followed by a heat exchanger network synthesis (HENS) step applied to each design, and (2) integrated approaches that use just one step to identify the optimal heat-integrated process design. The advantage of the first type of approach is that the initial process selection problem is smaller and can be tackled, in many cases, through the use of a superstructure approach. The main disadvantage is that there is no guarantee that the process identified in the first step will have good heat integration characteristics.2 An alternative for the first approach, suggested by some authors,3 is to generate * To whom correspondence should be addressed. E-mail: [email protected]. Fax: +44(0)20 7679 3817.

all valid process flowsheets and apply a heat exchanger network synthesis (HENS) method to each of them. Although effective for identifying the best heat-integrated design, this approach is only suitable for small problems because of the combinatorial growth in the number of alternative designs with the problem size. An approach that considers heat integration from the start is more likely to identify good designs.2 The majority of methods presented in the literature for this approach rely on the use of a superstructure representation that defines a mixed integer nonlinear programming (MINLP) problem. This problem is solved using one of a variety of general solution methods for MINLP problems,1-4 and the solution is one of the structures embedded in the superstructure. There are three difficulties with the explicit superstructure approach: The superstructure for even small problems is complex and difficult to generate, there can be numerical issues regarding initialization, and nonconvexities in the NLP relaxations can result in the global optimum being missed. Nevertheless, a combined approach appeals because of the greater likelihood of identifying energy efficient process designs (see recent work in this area,5-12 as well as the review by Grossmann et al.1), although most are limited in the size of problems they can handle or in the use of simplified models to ensure success in the solution procedures used. This paper describes a combined approach suitable for large problems that attempts to identify designs that have desirable integration characteristics. The approach is based on the use of discrete programming techniques for the simultaneous generation and evaluation of an implicit superstructure. It copes well with numerical problems and nonconvexities and poses no restrictions on the types of models used. Furthermore, the discrete approach lends itself to an efficient implementation through the reuse of previously calculated results and can be combined with data mining techniques to improve performance and provide insight into the problem, insight that can be used to support an iterative ap-

10.1021/ie010187i CCC: $27.50 © 2004 American Chemical Society Published on Web 11/22/2003

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Figure 2. Subgraph extracted from the full search graph (Figure 1) that represents one possible solution to the separation problem. Figure 1. Search graph generated by the implicit enumeration approach for a simple separation sequence synthesis problem, showing the reuse of computation through the use of dynamic programming. Rectangular boxes denote subproblem nodes. Rounded boxes represent specific unit designs, including feed tanks and separations, that generate one or more output streams used to define new subproblems and product tanks that have no outputs and, hence, correspond to leaf nodes of the search graph.

proach to design.13 The combination of data mining and discrete programming leads to the implementation of a method targetted at early, conceptual design that supports and interacts with the engineer. 2. The Jacaranda System The new approach for heat-integrated process synthesis has been incorporated within the Jacaranda system for automated design.14 Jacaranda is based on a combination of techniques including implicit enumeration, dynamic programming, and discretization. The problem definition consists of the available raw materials (feed streams), a list of processing technologies (units), and the desired product specifications. The basic algorithm consists of choosing one of the feed streams, identifying which of the processing technologies can process the chosen stream, generating alternative designs for each of these technologies, and then recursively processing the outputs of the units in the same manner. For a problem to be solvable by Jacaranda, it must be expressable in the following general recursive form

f(p) ) min {c(u) + u∈U(p)

∑

f(s)}

(1)

s∈S(u)

where p is the problem to be solved or one of its subproblems, f(p) is the cost of the best way of solving p, U(p) is the set of immediate actions that are applicable to p, c(u) is the cost of a specific action u, and S(u) is the set of subproblems resulting from action u. In a separation problem, p would be the stream to be separated, f(p) the minimum cost of the separation process, U(p) the set of designs of units that can take stream p as a feed, c(u) the cost of a unit design u, and S(u) the set of output streams of the unit design u. In the case of sharp distillation, all that is needed for the specification of a subproblem is the subsequence of components present in its feed stream. For example, Figure 1 can be interpreted as the full search graph for the separation of a four-component mixture using distillation columns with sharp separation. Square boxes indicate subproblems (represented by the composition of the stream associated with each subproblem), and rounded boxes represent individual unit designs (showing the separation performed or the use of feed and product tanks). The search graph, traversed by the evaluation of eq 1, is equivalent to a superstructure defined implicitly by problem definition indicated above. A subgraph, an example of which is shown in Figure 2, can be interpreted as a possible solution extracted from the full search graph. Square boxes now indicate

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the streams connecting specific instances of units, indicated by rounded boxes. In the search graph, any subproblem can be a subproblem of more than one parent problem. This observation can be used to save computation by introducing aspects of dynamic programming into the search procedure. For instance, once the subproblem CD in Figure 1 has been solved, say, in response to the output of the B/CD separation, the solution to this subproblem can be reused when the outputs of the AB/CD separation are considered. This is achieved by creating a dynamic programming table indexed by a key associated with each subproblem. For a sharp separation synthesis problem, the key is easily defined as simply consisting of the components found in the stream to be separated. More general synthesis problems, however, will involve continuous quantities such as component flows, stream enthalpies, unit design parameters, and unit operating conditions. In Jacaranda, all continuous quantities are discretized. The discretization provides a mapping from continuous space to discrete space, allowing one to define a key for each subproblem and, hence, make efficient reuse of computation through the dynamic programming table. The use of discretization results in a combinatorial search procedure. It is therefore important to choose the discretization to fit the properties of the problem. As it is often difficult to identify appropriate discretization parameters a priori, the Jacaranda system is intended to be used iteratively. Using engineering insight, the user chooses some discretization parameters, solves the problem, analyzes the results, and then repeats these steps with new parameters. Examples of such an iterative approach include the recent work by Kravanja et al.13 and Laing and Fraga.15 Jacaranda has been applied to a wide range of separation problems, including bioprocess synthesis16,17 and combined reaction/separation problems with potentially complex recycle structures.18 The latter require a more complex definition of a subproblem. In Jacaranda, subproblems are defined by two quantities: zero or more streams to be processed and a set of qualifiers that are needed to complete the subproblem definition. A qualifier specifies some special attribute, or quality, that the solution to the given subproblem must have, over and above the implicit problem definition specified by the set of streams associated with the subproblem. For instance, a qualifier could be that the solution to the subproblem must generate a given product in the solution obtained for processing the streams. A qualifier could also indicate that the solution can make use of some special unit, a unit that is available at zero cost (useful for retrofit problems, for instance). More generally, qualifiers provide a mechanism for enforcing extra global balance constraints. These are linear equality or inequality constraints that can involve any part of the solution. In these cases, each qualifier represents a single extra balance constraint on a subproblem. This constraint is rewritten as a sequence of balance constraints, one for each of the intermediate nodes in a potential solution to the subproblem. Let i be a node in the search graph that corresponds to a unit design, as indicated by rounded boxes in Figure 1. Let xi be the amount of some quantity consumed by the design node i. The amount xi could represent a limited resource (e.g., a utility or an existing processing unit that can be incorporated at no cost) or some

quantity that is exchanged between different subproblems (such as required for heat integration, as shall be described below). xi could represent a continuous or discrete value, and it could be positive, negative, or zero. Let yj be the amount of the quantity assigned for use by any solution to subproblem j (nodes indicated by square boxes in Figure 1). Now consider, by way of example, the design node AB/CD in the subgraph shown in Figure 2. This node leads to two subproblems, AB and CD, each of which generates one design node with corresponding final product subproblems. In this case, for an associated quantity assigned to subproblem ABCD, yABCD, described by a qualifier, we have

yABCD ) xAB/CD + yAB + yCD yAB ) xA/B + yA + yB yCD ) xC/D + yC + yD yA ) xA

(2)

yB ) xB yC ) xC yD ) x D In other words, the amount of the qualified resource, yABCD, assigned to subproblem ABCD is distributed between the design node AB/CD (xAB/CD) and the two subproblems AB (yAB) and CD (yCD). Similarly, the amount assigned to subproblem AB, yAB, is distributed to the A/B design node and subsequent subproblems A and B, and so on. For illustration, suppose that the actual qualifier, or linear balance constraint, for subproblem ABCD is yABCD ) 0. This implies a zero net amount, but such a result can be achieved by a combination of positive and negative values assigned to individual design nodes in the solution to ABCD. According to eq 2, the following must hold

xAB/CD + xA/B + xC/D + xA + xB + xC + xD ) 0 In other words, the consumption and generation of the particular quantity described by the qualifier must be balanced over the subgraph rooted at subproblem node ABCD, noting that the individual xi values could be positive, negative, or zero. This set of constraints allows the problem to be decomposed according to the original problem graph. For example, subproblem AB is now qualified by the requirement that any unit designs applied to this subproblem and the subproblems generated by this unit design together consume (or generate) an amount yAB of the conserved quantity. It is useful to use two characterizations to classify conserved quantities that will be described by these balanced constraints or qualifiers. First, a quantity can be discrete or continuous. An example of a discrete quantity would be the number of pieces of equipment of a given type available (or available at a reduced price). An example of a continuous quantity is the amount of high-pressure steam available for use anywhere in the site. In Jacaranda, such continuous quantities have to be discretized. Second, a quantity can be characterized according to whether it can be distributed or must be paired. In a distributed quantity, any

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number of the variables in the underlying balance constraint can be nonzero, whereas in a paired quantity, only two can be nonzero. The two examples given above are distributed quantities. An example of a paired quantity is the flag used to generated systems with recyles.18 A problem can have several different balance constraints for different quantities (indeed the number can be infinite). Often, these quantites can be grouped together, and only one of the group will have nonzero use in the solution. An example of this would be where there were alternative pressures (and so temperatures) for distributed steam and the model is required to choose one of them. It is convenient to think of such a group balance constraint for a set of quantities as a single balance condition for a quantity with a variable property (in this example, steam pressure). The focus of this paper is a description of how the qualifier mechanisms can be used to model heat integration in process synthesis. In particular, we will show that the distributed quanties with variable properties (in this case, temperatures) have a significant advantage in this context. 2.1. Support for Early Design. In early design, one of the main difficulties with the use of process synthesis software is the definition of the problem. Often, it is necessary to know the result in order to generate a welldefined problem specification. For instance, the definition of allowable waste products might not be possible until the range of suitable processing technologies has been identified. One of the aims of the work presented in this paper, and of the Jacaranda system as a whole, is to facilitate the use of process synthesis in early design. A variety of techniques are used to this end, of which the following are relevant to heat integration: (1) the use of data analysis tools for extracting information about the search space after the synthesis step has been completed and (2) a simple and easy to manage problem definition that does not require the definition of a superstructure. This paper describes the use of data analysis for extracting useful information from the data accumulated by the synthesis procedure. Although the analysis is useful in helping gain a view of the search space and its characteristics, it will also be used by the search procedure to automate parameter selection for heat integration. 3. Heat Integration in an Implicit Enumeration Procedure The introduction of qualifiers allows subproblems to be solved with extra constraints. In a previous paper,18 we described the use of qualifiers for the generation of process designs with recycle structures. The qualifiers were also used to limit the number of occurrences of any given unit, useful both for reaction-separation systems and for cases in which a special zero-cost unit might be considered to identify the need for novel technologies. The aim of this paper is to describe the virtual heat link (VHL) qualifier. Virtual heat links allow subproblems to assume that extra heat sources or heat sinks are available for use in the solutions to the subproblems. By ensuring that source and sink links are generated in heat balanced combinations, an overall heat balance is ensured. The rest of this section describes how these heat links are defined and how they are incorporated within the search procedure.

Figure 3. Sequence of representations of the same integrated process flowsheet, showing how the matching ends of a virtual heat link correspond to an integrated exchanger. (a) Actual integrated process. (b) Integrated via virtual stream. (c) Completely uncoupled integrated process.

3.1. Virtual Heat Link. The aim of heat links is to enable the solution to a given subproblem, part of a process flowsheet, to include heat exchange with other parts of the process flowsheet generated by Jacaranda. For instance, in the search graph in Figure 3, when addressing subproblem AB generated by the unit design AB/CD, we would like to consider possible exchanges with the process associated with the CD subproblem. Paired qualifiers, which describe matching source and sink heat exchanges, can be introduced at this point in the search graph. Each end of the link can be passed to each subproblem, and if both subproblems can simultaneously use matching ends, the result might be a process design that exchanges heat between the two branches.

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In its simplest form, a virtual heat link qualifier is defined by a temperature, T, and a duty, Q. If the duty is positive, the qualifier represents a source of heat; if the duty is negative, it represents a sink of heat that can be used for meeting cooling requirements. Matching heat links can be thought of as representing a virtual stream that transfers heat from a heat source (e.g., a condenser) to a heat sink (e.g., a reboiler). Figure 3 shows a sequence of representations for the same integrated process. The first representation, Figure 3a, is a simple integrated process flowsheet. The reboiler for unit 2 is integrated with the condenser for unit 3. The next representation, Figure 3b, replaces the single integrated exchanger with two exchangers, each exchanging with a single virtual stream, indicated by the dashed curved line. The virtual stream is transporting heat from the condenser to the reboiler. Finally, the third figure, Figure 3c, shows the completely uncoupled representation of the process in which the extra arrows indicate that the reboiler for unit 2 expects a heat source from elsewhere in the process and that the condenser for unit 3 will provide heat to elsewhere in the process. This process would have been generated by having passed the cold end of the virtual stream to the subproblem associated with the bottoms output of unit 1 and the hot end to the tops output subproblem. For the integration via the virtual stream to work, the temperature and duty of the virtual stream must be appropriate for the requirements of both units. The difficulty is that the temperatures and duties for potential process designs are not known ahead of time. We will return to this issue after discussing the implementation of virtual heat links in the search procedure. 3.2. Virtual Heat Link Algorithm. The objectoriented nature of the Jacaranda system means that heat links are responsible for generating their own allocation to the different outputs of a design node. A node is created for each unit design that is generated. This design node is responsible for estimating and extracting the relevant cost information from the unit design and for creating the new subproblems. When qualifiers are introduced, each original unit design now corresponds to a series of design nodes, each of which will have a different assignment of qualifiers to the outputs of the unit. In the context of heat integration, the targets of the heat link assignment procedure are the unit’s output streams and the heat-transfer requirements of the unit, such as the reboiler and the condenser for a distillation column. The allocation method is based on ensuring that an overall energy balance is achieved. Again, in the simplest case, the procedure is based on a multiple-digit base-3 counting algorithm in which each value corresponds to one of three cases: (1) no heat sink or source, (2) a heat source, or (3) a heat sink. The root problem definition will have associated with it a heat link qualifier that provides neither a sink nor a source of heat. When a unit design has been generated, the number of outputs to which heat link qualifiers should be assigned is determined. For example, for a simple distillation unit, there will be four assignment targets: the reboiler, the condenser, and the two output streams. All possible combinations of assignments are then enumerated. Table 1 shows the assignments for such a distillation column, where the minus sign indicates a sink of heat, an empty entry means neither sink nor source, and the plus sign indicates a source

Table 1. Heat Link Assignments for a Distillation Column Assuming a Feed Stream Assignment of Neither a Sink nor a Source unit output assignments potentially feasible?

index

reboiler

condenser

tops

bottoms

1 2 3 4 5 6

-

-

+

+

+ -

+

+

-

N N N N N N

-

Y Y Y Y Y

-

N N

+ + +

7 8 9 10 11

-

12 13

+ +

-

+

+ -

-

+ +

14 15 16 17 18

+ + + + +

-

19

+

+

-

-

Y Y Y Y Y

-

N

of heat. The total possible number of allocations is 34 ) 81, as there are four targets and three possible values for each. Of course, many of these allocations are infeasible as they do not satisfy the requirement of energy balance. The result is 19 different potentially feasible balanced assignments, shown in Table 1. Of these, some more can be discarded as infeasible because of the allocation of a sink of heat to a reboiler (requires heating) or a source of heat to a condenser. The entries in the table have been annotated to indicate those that can be so identified as infeasible. The result, after the first feasibility check (suitability of individual targets), is a total of 10 different assignments. Even now, there are entries that are likely to be infeasible but that cannot be so identified without checking a complete row instead of single qualifier to target assignments. For instance, integration between the condenser and the reboiler of a single distillation column (see entries 14 and 16) is not feasible. Note also that, if the feed stream would have had a different link associated with it (e.g., a sink of heat), the resulting assignment table would be different. However, the new table would be similar in size. 3.3. Case Study I: Five-Component Separation Problem. The discrete approach for heat-integrated process synthesis will be demonstrated through the use of a simple five-component distillation sequence synthesis example.19 The aim is to design the minimum annualized cost process that separates the stream into pure single-component products. We assume the use of semisharp distillation units, modeled using the FenskeUnderwood-Gilliland equations and correlations. Each distillation unit has three degrees of freedom: the separation point (heavy and light keys with the assumption that the two are adjacent in relative volatility order), the operating pressure, and the reflux ratio (through the use of a multiplier on the minimum reflux ratio). There is no assumption about the number of distillation units required. This problem is small enough that an alternative approach of generating all feasible structures (there are 14 in total) and optimizing each one individually is

Ind. Eng. Chem. Res., Vol. 43, No. 1, 2004 149 Table 2. Statistics for Multiple Discrete Heat Links for Case Study I number of links statistic

0

1

2

3

4

problems nodes elapsed time (s) unit search total

16 646

44 10 856

126 63 612

365 500 149

1081 5 238 217

3 1 3

6 6 12

18 21 38

98 161 259

1253 3192 4446

feasible. However, the individual optimization problems are themselves MINLPs because of both the design of individual distillation units and the design of the HEN. In any case, this example is suitable for the purpose of illustrating issues in parameter selection. The five-component separation problem has been solved with different numbers of heat links. The operating pressure of the distillation units has been discretized using eight points, log-uniform in [1, 34] atm. The reflux rate factor (used as a multiplier for the minimum reflux ratio to determine the actual reflux rate) has been discretized using four points, log-uniform in [1.1, 1.5]. Distillation units have been designed assuming that the feed is a saturated liquid at the boiling point for the mixture at the unit’s operating pressure (q ) 1 in Underwood’s equation for estimating the minimum reflux ratio), an assumption made by Rathore et al.20 Although this can lead to unrealistic results, the assumption is made for this particular case study to provide a valid comparison with the published results. This assumption will be relaxed in the subsequent case studies. The domain for the reflux rate factor was not specified in the source, however, so we have chosen an arbitrary yet reasonable domain. All other parameters, specifications, and cost models, including the utility cost models (using the second, more realistic costs), are specified in the source.20 As mentioned above, a heat link has two items of data, the temperature T and the duty Q of the virtual stream used for exchange. As these values are part of the problem definition, the user needs to know the values to choose before solving the problem. This is difficult to do in most cases. Therefore, the user will have to choose a number of alternative values to try. This leads to a large increase in the computer time required to solve the problem, as indicated by the timing results shown in Table 2. The results show that the amount of time required increases exponentially with the number of links. The exponential growth in time limits the efficacy of the approach, especially for early design, where many T and Q values are needed to cover potential solutions about which little is known. At this point, it should be noted that a discrete approach for heat-integrated process synthesis was first implemented by Dhallu and Johns21 in the GENPROC system. In this approach, the concept of heatbits was introduced. Heatbits acted as pseudocomponents and could be processed by units that were aware of them. Although equivalent to a single heat link, the implementation was difficult to extend, particularly to multiple heat links, because each unit had to be able to handle the special heatbits directly. Furthermore, the heat exchanges for which the technique was designed were limited to latent heats. An extension of this approach was implemented in the CHiPS package22 to cater for sensible heat exchange, to allow for in-series as well as in-parallel exchangers, and to make the

heatbit assignment independent of the unit models for easier extensibility. The new approach described is this paper is more generic and provides the ability to extend the definition of links, as described next, to apply more effectively to early design. 3.4. Multilevel Heat Link Approach. The exponential growth in time, as demonstrated in Table 2, is the result of having to consider all combinations of qualifier assignments for all of the heat links present. Instead of multiple heat links to consider a range of temperatures and duties, we now consider extending the definition of the heat link to include a choice of temperatures and duties. If we first consider the duty element of a heat link, the problem with the approach described above is that each link represents an amount of duty that is indivisable. Therefore, to allow for a range of possible duties for exchange, a large number of links might be required. Instead, we consider making the duty component of a heat link distributable. A heat link definition is modified to include a base amount of duty, Qb, and the number of levels, nq, used to calculate the actual duty for a given instance of the heat link. The duty represented by a link is then one of

{i × Qb, i ) -nq, ..., nq}

(3)

The allocation of heat links to the heat-transfer requests and outputs of a unit now cycles through all different distributions, again ensuring that the overall energy remains balanced. The amount of duty used in any exchange is now a multiple of Qb. The total amount, nqQb, can be distributed over a set of targets, both for heating and for cooling. The algorithm complexity is now a function of the number of heat links, as before, and the number of levels for each heat link. The number of subproblems should grow as nv

O[

(2nq,i + 1)] ∏ i)1

(4)

where nv is the number of links and nq,i is the number of distributable duties for the ith heat link. Assuming that the number of levels per link, l ) 2nq + 1, is the same for each link, this growth rate is O(lnv). The number of nodes in the search graph depends on the actual types of units used. If we again consider distillation, the number of different allocations is l4nv. As before, many of these can be discarded on the basis of energy balance considerations. As a function of the number of links, nv, these growth factors are the same as before. The difference is that it is easier (for the user) to describe a set of discrete duty amounts that cover the space of interest. The polynomial growth in l (for constant nv) makes predicting the amount of effort easier as well. Table 3 shows the results of running the same problem as before but this time with only one heat link. However, this heat link is of the new variety, so we have generated results for different values of nq, the number of multiples of the base duty for the heat link. The results show that the number of problems increases linearly with this value. The rate of growth for the number of nodes (and hence the CPU time required) is higher than linear although not as high as the O(l4) predicted. The lower rate of growth for the number of

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Table 3. Problem Size Statistics and Computer Time Case Study I with 1 Heat Link with Varying Numbers of Distributable Levels nq statistic

1

2

3

4

5

6

7

8

problems nodes (′000) elapsed time (s) unit search total

42 11

71 36

99 80

127 145

155 236

181 349

207 488

234 664

9 9 18

17 19 36

28 38 66

46 64 110

68 100 168

98 145 243

136 209 344

183 287 470

nodes is due to the number of infeasible combinations and the pruning procedures implemented in the search algorithm, as described in the next section. The results indicate that a single multiple-duty VHL with six levels takes approximately the same amount of computer time as three single-duty VHLs. With eight levels, the time is in the same range. This is not surprising, as three single-level heat links corresponds to 23 ) 8 different duty levels when used in combination. Three single-level heat links can theoretically represent 33 ) 27 different combinations if source and sink links can be combined arbitrarily. However, the efficiency of the overall implementation depends on easily identifying inappropriate matches, such as a heat source assigned to a cooling requirement and vice versa. Therefore, combinations of links can only consist of all sources or all sinks to make this identification step possible. Although the introduction of distributable duties makes user input easier, it has had little effect on the computational effort. However, only the duty element of a heat link has been extended. For a sufficient coverage of the search space, and in particular to address the problem of different needs in different areas of the process, we should not restrict the temperature to a single value. We therefore introduce multiple temperature levels into the definition of the VHL. The new definition consists of a set of discrete temperature levels, {Ti, i ) 1, ..., nt}, as well as the distributable duties, defined by the base duty Qb and the number of duty levels nq. The Cartesian product of these two sets of discrete values defines the number of distinct exchanges encoded in a single VHL

{(T1, -nqQb), [T1, -(nq - 1)Qb], ..., (Tnt, nqQb)} (5) for a total number of levels l ) nt(2nq + 1). However, there are nt different encodings in the Cartesian product that represent no heat transfer at all. For efficiency, these collapse to a single entry in the implementation, giving l ) 2ntnq + 1. The growth rate in the search space expected is again O(l4). We expect fewer feasible allocations as a function of l in this case because of the range in temperatures. With the introduction of temperature levels, we should be able to address problems in which the temperature varies significantly over the different units in the possible flowsheets, and we should be able to do this more efficiently than with single-level singletemperature heat links. The number of exchanges across any two subtrees in the solution space is limited to the number of heat links. In the single-level VHL approach, the number of heat links directly determines how many exchanges of heat are possible between any two subtrees in a solution. Although the number of links does not limit the actual

Figure 4. Representation of process flowsheet solution showing heat exchange (dotted lines) between two subtrees, one of which has an exchange within two of its own subtrees.

number of exchanges, this number does limit the number of links across two subtrees in a solution. For example, the use of one link means that any two subtrees can only have one integrated exchanger between them. Figure 4 shows how a single link can nevertheless lead to multiple exchanges in the whole process. The difficulty with the limitation on the number of cross-links is that it is seldom the case that different parts of the process can effectively integrate with the same amount of duty at a single temperature. Therefore, although multiple integrations are allowed with a single link, the actual matches found will likely be suboptimal. By adding the concept of distributable quantities and multiple temperature levels to a virtual heat link, the effects of the limitation of single cross-links per heat link are reduced. Although it is still the case that, between any two subtrees, the heat exchange possible is limited to the amounts specified by the links available, the actual number of exchanges between any two subtrees is given by the number of levels that can be represented by the distributed quantity. Any number of these links, all at the same temperature, might end up at the same node in either subtree, but there is no requirement that this be the case. The restriction now is that, across any two subtrees, the total amount of exchange is limited by the total amount associated with the heat links used, each at a single temperature. By including multiple temperature levels in each heat link, it is easier to explore the effect of the temperature choice. Multiple temperatures can be processed in polynomial time as compared with the exponential time taken if multiple single-temperature links are used. 3.5. Heat Exchange Matching and Costing. The decoupling of the hot and cold sides of integrated exchangers makes costing the exchangers difficult. The cost model for utility exchangers is based on estimating the area using the following equation

A)

Q U∆T

where ∆T is defined as the log-mean temperature difference and is often approximated using a variety of techniques to reduce numerical difficulties. In Jacaranda, there is no need to use any simplification for this expression; the log-mean difference is used directly. However, for integrated exchangers, the actual temper-

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atures of the two sides are not known because of the decoupling, as indicated in Figure 3c. Heat links represent a virtual stream, so temperature values are available for both ends of a virtual exchanger (see Figure 3b). However, two exchangers are designed for each integration, which would lead to an overestimate of the capital cost. This is handled by halving the area estimated for each virtual heat exchanger. This helps reduce the overestimate, but some problems still remain. The first is that the capital cost (Cc) models are often based on the following equation

Cc ) R + βAγ where γ < 1 and R * 0. Costing an integrated exchanger as two, smaller, virtual exchangers leads to an overestimate because of the shape of this cost function. The second issue is that the area will itself be an overestimate because the temperature of the virtual stream will lie between the temperatures of the two ends of an integrated exchanger. Therefore, ∆T will be smaller than expected, and this might lead to exchanges being dismissed even though they are fine in principle. Having a larger number of temperature levels to consider will reduce the likelihood of this happenning and is one of the motivations for the new procedure. As it is unlikely that any particular heating or cooling request will match the duty specified in a heat link exactly, the matching of requests is fuzzy. Specifically, if the temperature of the virtual heat link is appropriate, a suitable match is defined if the amount requested, Qr, is less than or equal to 2Q. If the amount requested falls into the range 1/2Q e Qr e 2Q, only the virtual exchanger is designed, based on the duty Qr. Any excess or shortfall is met by utility alone, which is costed as an operating cost. If, however, the amount requested is less than 1/2Q, another exchanger is designed, one that exchanges heat with an appropriate utility. This second exchanger will contribute to both capital and operating costs. Finally, if the request is greater than 2Q, the match is marked as infeasible. The result is that the area estimates are approximate, not only because of the unknown temperatures but also because of the duty amount. A final factor that leads to inaccuracies in the cost estimates comes from being able to have one end of a virtual link attached to a single target with the other end split among various targets. The single target link will correspond to a single virtual exchanger. If the final integration pattern requires multiple exchangers, one for each exchange from that target, the capital cost calculations will have been underestimated. The resulting heat exchanger capital costs are therefore approximate. The factors described above lead to under- and overestimates individually. Together, it is difficult to quantify the actual effect. In any case, the intention is to identify processes that are amenable to process integration; the heat exchanger network generated should be redesigned after the synthesis step. Because Jacaranda can provide the n best solutions according to the approximate model, there is a good chance that one of the solutions generated will be the best on the accurate model. 3.6. Implementation Efficiency. The size of the search graph is directly related to the discretizations used and exhibits combinatorial growth, as we have seen in the previous section. In this section, we describe

some of the details of the actual implementation that relate to efficiency. An efficient implementation can make it possible to solve significantly larger problems, even with a combinatorial growth algorithm. Although the implicit enumeration procedure is conceptually simple, an implementation that attempts to be as efficient as possible is less so. In particular, Jacaranda is written in the Java language and is strongly object oriented. The generic nature of the framework requires the creation and use of many objects. Object creation is a computationally expensive task in all object-oriented languages. An efficient implementation, therefore, is based on two main considerations: (1) ensuring the greatest amount of reuse of previous computation and (2) minimizing the number of new objects created during the enumeration procedure. The first point leads to the use of dynamic programming combined with the enumeration procedure, as described earlier. However, the introduction of heat links leads to other possible sources of computation reuse. A reduction in the number of design nodes generated in the search graph is achieved by more effective pruning of the search space. The introduction of heat links leads to the generation of a large number of infeasible branches in the search graph, and identifying these branches as early as possible is crucial in reducing the number of design nodes generated. Reducing design nodes reduces the number of subproblem objects that are created. How both of these points are considered in Jacaranda is described below. Unit and Heat Exchanger Design Caches. The solution of a subproblem requires us to generate a series of unit designs. Each different combination of heat link assignments corresponds to a new unit design node in the search graph. However, as can be seen from Table 1, a single base unit design is responsible for a large number of distinct assignments, each of which shares the base design. Furthermore, many of the heating and cooling requests appear more than once with the same heat link assignment. Therefore, the first step in increasing the efficiency of the overall method is to create a design cache for storing unit and heat exchanger designs. A not used recently algorithm23 is used to manage the cache. The maximum number of entries in the cache and the number to clear out when the cache is full are parameters that can be specified by the user. Each entry in the cache corresponds to a base unit design together with a table of heat exchanger designs for each heating or cooling request. The tables are indexed by the heat link assignment for the request. For the assignments shown in Table 1, three different heat exchanger combinations would be designed for the reboiler and the condenser. The dynamic programming aspects of the underlying algorithm take care of the duplicate entries found for the tops and bottoms products. However, there is further duplication that can be avoided. Each subproblem is defined by a stream and the set of qualifiers. The subproblems can be grouped according to the stream on the basis that all of the subproblems with the same stream will lead to the same unit designs. Therefore, the unit design cache allows unit design reuse globally. Single Heat Link Infeasibilities. As indicated in Table 1, many of the assignment combinations are infeasible when the needs of the target output are considered. Even if an assignment is potentially feasible,

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the actual heat link might not be appropriate because of the temperature or the duty amount. Identifying such infeasible pairings and eliminating these from the enumeration procedure as quickly as possible is useful and has been implemented. This is a quick test and precedes the test described next. Heat Link Combination Infeasibilities. Even if the assignment of a given heat link to a unit output (which includes both heat-transfer requests and unit output streams) is feasible, more than one heat link might be assigned to the same output. Combinations of links might therefore lead to infeasible matches, particularly with respect to the amount of duty. For each unit design alternative in each subproblem, a table is created. There is an entry for each output, and each nv entry consists of a vector of boolean values, of size ∏i)1 li, where li is the number of temperature and duty level combinations for the ith heat link. The elements of the vector are accessed through an index generated by the heat links assigned to the output, emulating an nvdimensional array. The elements are initially set to true, and if a particular combination of heat links assigned to an output of the unit design is found to be infeasible, the corresponding element is set to false. When the same combination comes up again, for the same design alternative, this element is checked. If it is false, the particular design node can be dismissed without further evaluation. In the results below, we shall see that all three efficiency methods have a significant effect on the computation times, particularly for large problems with multiple heat links. 4. Data Analysis for Early Design Multilevel heat links have been defined to ease the problem definition stage. The ability to specify a wide range of duties and temperatures for the links reduces the amount of knowledge about potential solutions required by the user in defining the problem. Nevertheless, it is still necessary to define the parameters for the different heat links. In this section, we describe an analysis procedure that can be used to identify appropriate values for the heat links. At the core of the procedure is the belief that the user should be able to control all of the discretizations but that the main input for a given synthesis run should relate to the time the user is willing to wait for results. The quality of the results is directly related to the discretizations used, but the quality required differs during the process of design, and it is reasonable to allow the user to control this quality level. The user can trade off quality and time. In this section, we describe an automated procedure, incorporated into the synthesis tool, that uses simple data analysis techniques to (1) collect and analyze heattransfer requirements in the non-heat-integrated synthesis run, (2) present these results to the user, (3) identify reasonable values for temperature and duty values for a single multilevel heat link, and (4) use these parameters automatically in an iterative procedure. Steps 1 and 2 are useful by themselves. In some cases, the automated procedure (step 4) might not generate the best results, and the user might be required to use his or her own insight into the problem, aided by the data analysis, to tune the heat link parameters for the best results. 4.1. Automated Parameter Selection. Three parameters are required to define a single multilevel

virtual heat link: a set of temperatures, a base duty amount, and the number of distributable levels for the duty. We will convert this specification into the simpler one of specifying two integer values: nt, the number of temperature levels, and nq, the number of duty levels. Given these two values, the procedure will identify a suitable temperature range that will be discretized uniformly using nt values and the base duty amount, Qb. It is reasonable to expect the user to specify the two integer values as these have a direct and predictable impact on the amount of time required to solve the problem. In cases where the engineer is using the synthesis tool to explore the search space, small values will be appropriate. When the engineer is attempting to identify the best solution possible with a discrete approach, much larger values will typically be used. The values for these integer parameters can, for the most part, be chosen without a priori knowledge of the search space. The automated parameter selection procedure is based on first solving the equivalent non-heat-integrated problem. As the synthesis procedure is based on discrete programming, a discrete set of unit designs is generated. In particular, each unit design encountered will have associated heat-transfer requirements (HTRs), for cooling or heating or both. These requirements are collected in a simple list by the search procedure. Figure 5a shows the HTRs recorded for case study I. The points plotted above the x axis are requests for cooling (an excess of heat is available,) and the points below the x axis represent heating requests. Any feasible heat exchange will connect a point above the axis with one below the axis and to the left of the first point. The points on the graph consist of all of the requirements for the part of solution space searched. They do not indicate the feasibility or concurrency of the different matches possible. Furthermore, some heat-transfer requirements that are part of the solution space are not included because of the pruning operations of the search procedure. Nevertheless, the premise is that the data are representative of the heating and cooling requirements for potential heat-integrated solutions of the synthesis problem. In itself, presenting these data to the user can be useful. However, suitable heat links can be defined automatically through an analysis of the data. The problem can be re-solved using the new heat link definitions. Assuming that the data described above have been generated, the procedure for choosing the parameters that define the heat links is as follows: (1) Identify the temperature domain. This is defined by the minimum and maximum temperatures represented by the complete list of requirements. Figure 5a shows the data for case study I. For this example, the temperature domain is approximately (225, 460) K. (2) Subdivide the temperature domain into equalsized subintervals. Twenty intervals are used for the subdivision process. (3) Count the number of heat-transfer requests in each temperature interval. Cooling and heating are treated separately. For the case study, the results of this step, in the form of histograms, are shown in Figure 5b and c, respectively. (4) Calculate the average number of heating and cooling requests per interval. (These values are indicated in the figures as horizontal lines.) These values are used in step 5 as cutoff values.

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Because of the approximate matching of heat links allowed in Jacaranda, the values for the heat link definition need not be precise. They need only be in approximately the right range for the heat links to be effective. 5. Results

Figure 5. Heat-transfer requests and analysis for a fivecomponent separation problem.

(5) Identify a reduced temperature domain by eliminating intervals in which heat exchange is unlikely or impossible. The left endpoint of the new domain is set to the midpoint of the first temperature interval for which the number of heating requests exceeds the cutoff value. The right endpoint is the value of the midpoint of the last temperature interval for which the number of cooling requests exceeds the cutoff. In this example, the reduced temperature interval is (306, 397) K. (6) Calculate the average heating and cooling duty requirements in the reduced temperature domain. The minimum of the two averages is chosen to represent the maximum amount of exchange that is likely (remembering that heat exchange matches for virtual heat links are feasible for duties up to 2Q). The result for the case study is a duty of just under 14 MW. (7) Define the heat link parameters using the userspecified values nt and nq. For the temperature, a uniform set of nt discrete temperatures in the range found in step 5 is defined; for the duty, the amount defined in step 6 is divided by nq to give Qb.

Three case studies have been selected to demonstrate the effectiveness of the new discrete programming approach for heat-integrated process synthesis. The first case study has already been described. The second is a similar problem, albeit larger, and the third uses more complex distillation units allowing for feed pretreatment. In all cases, we examine the use of the data analysis techniques for the automated selection of the heat link parameters. 5.1. Case Study I, Continued. Table 4 shows the results of attempting this problem using the automatic selection of heat links for different numbers of temperature, nt, and duty, nq, levels, as well as the base case with no heat exchange (the first row). Two costs are presented. The first is the cost of the flowsheet selected by the synthesis procedure. The second cost column presents the cost of the flowsheet with the heat exchanger network generated using a postsynthesis HENS method,24 keeping the flowsheet fixed. This second cost is a more accurate representation of the flowsheet cost, as it has correct heat exchanger costings for integrated exchangers. The exchanges identified by the synthesis procedure are shown in the table. The flowsheet identified as best without heat integration is the third solution presented by Rathore et al.19 in their Figure 8. The different experiments with heat integration all lead to improved solutions. Two alternative sequences of units are identified, one of which corresponds to the structure chosen without heat integration and the other the same as that identified by Rathore et al.19 as the best structure. Both structures, in the best cases, have the same cost, and there is little to distinguish them. Table 5 presents the design parameters, including the heating and cooling requirements, for the two structures. The synthesis procedure identifies two exchanges for each of these structures, and these same exchanges are confirmed in the postsynthesis HENS procedure. The best solutions show almost 20% reduction in cost when compared with the nonheat-integrated case and over 10% when compared with Rathore et al.'s best solution. The identification of two different structures, essentially equivalent in economic performance, is a good result for the early stage of design. Each of these structures can be subsequently analyzed using more rigorous tools. The synthesis procedure has acted as a sieve to reduce the choice of 14 different structures to just 2. It has also provided initial values for the design parameters, values that should make it easier to apply a continuous optimization procedure for tuning and to choose one of the designs on the basis of more rigorous analysis. Table 6 shows the efficiency of the various implementation features. The separation problem has been solved, in this case, using two single-level heat links, one with T ) 335 K and Q ) 6 MW and the other with T ) 343 K and Q ) 6 MW. These values are chosen as representative of the exchanges encountered above. The first column of the table, labeled none, has the statistics for the base case with all efficiency measures disabled.

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Table 4. Summary of Results Obtained for Case Study I Using the Automatic Heat Link Identification Procedure cost (k$/year)

process

nt

T (K)

nq

Qb (MW)

initial

post

structure

0 1 1 2

352 352 337, 367

0 1 8 8

13.8 1.7 1.7

386 377 360 333

385 325 354 314

[(A/B]/C]/(D/E) [(A/B]/C]/(D/E) [(A/B]/C]/(D/E) [A/(B/C)]/(D/E)

4

324, ..., 379

4

3.4

350

313

[(A/B]/C]/(D/E)

4

324, ..., 379

8

1.7

347

313

[(A/B]/C]/(D/E)

8 8

316, ..., 387 316, ..., 387

1 2

13.8 6.9

364 338

321 314

[A/(B/C)]/(D/E) [A/(B/C)]/(D/E)

8

316, ..., 387

4

3.4

338

314

[A/(B/C)]/(D/E)

exchanges

time (min) 0.1 0.2 5.0 15.0

13.8 at 352 AB/C:D/E 6.9 at 352 ABC/DE:D/E 5.2 at 337 ABC/DE:D/E 8.6 at 337 B/C:D/E 3.4 at 343 ABC/DE:D/E 10.3 at 343 AB/C:D/E 5.2 at 343 ABC/DE:D/E 8.6 at 343 AB/C:D/E 13.8 at 337 B/C:D/E 6.9 at 337 ABC/DE:D/E 6.9 at 337 B/C:D/E 6.9 at 337 ABC/DE:D/E 6.9 at 337 B/C:D/E

10.9 74.8 2.3 9.7 64.3

Table 5. Unit Design Information for the Two Structures Identified for Case Study I unit

pressure (atm)

reflux ratio

number of stages

reboiler Q (MW)

condenser

Tin (K)

Tout (K)

Q (MW)

Tin (K)

Tout (K)

1 2 3 4

ABC/DE AB/C A/B D/E

12.4 20.5 12.4 1.7

1.35 7.29 2.54 8.99

[(A/B)/C]/(D/E) Structure 32 6.8 399.5 90 9.7 382.2 26 1.0 346.2 81 13.7 324.4

400.3 382.2 346.4 324.4

6.3 9.6 0.9 13.7

351.5 363.1 310.4 316.2

345.5 356.4 308.0 316.1

1 2 3 4

ABC/DE A/BC B/C D/E

12.4 20.5 12.4 1.7

1.35 5.71 9.19 8.99

[A/(B/C)]/(D/E) Structure 32 6.8 399.5 28 1.8 377.3 85 8.9 358.1 81 13.7 324.4

400.3 378.1 358.1 324.4

6.3 1.8 8.9 13.7

351.5 332.0 347.0 316.2

345.5 329.6 346.9 316.1

Table 6. Demonstration of Effectiveness of Implementation Efficiency Techniques for Case Study I efficiency measure design reuse statistic

none

subproblems repeated subproblems implicit enumeration nodes (′000s) unit designs heat exchange (HEX) designs (′000s) unit cache hit rate (%) HEX cache hit rate (%) single-Q infeasibilities caught (′000s) multiple-Q infeasibilities caught (′000s)

123 1905 1246 6766 1553

speedup, relative to “none”

1

Subsequent columns improve on the previous column by enabling a specified feature. The last column represents the case where all of the efficiency features have been enabled. The statistics collected include the number of unit designs, the number of heat exchanger designs, and the number of implicit enumeration nodes created. The greatest effect comes from checking for feasible qualifier assignments, particularly the check for infeasible single qualifier assignments. The final orderof-magnitude speedup shows the benefit of the effort required in developing an efficient implementation. 5.2. Case Study II: Nine-Component Separation Sequence Synthesis Problem. As case study I is relatively small, it is possible to enumerate all of the different process structures and evaluate each of these structures directly with only a relatively small performance penalty. The ability to use synthesis as a sieve in early design is less necessary. The second case study is larger, with a nine-component feed stream, whose composition is shown in Table 7. This problem has two characteristics that make it difficult and interesting to solve using the new heat-integrated process synthesis method: (1) First, the size of the solution space pre-

unit 123 1905 1246 966 1553 85

feasibility check HEX

123 1905 1246 966 1181 85 23

1.03

1.26

single 123 1243 91 966 31 85 81 1155 8.21

combination 123 1209 75 966 29 85 79 1129 3 9.85

Table 7. Feed Stream Specification for Case Study II component

boiling point (K, 1 atm)

flow (kmol/h)

propane isobutane n-butane isopentane n-pentane hexane heptane octane nonane

230.8 263.0 272.4 301.0 309.3 341.7 371.4 398.6 423.8

50 70 150 80 60 100 350 200 50

total flow

1110

cludes the sequential approach of generating all feasible structures and optimizing each separately. A simultaneous approach to heat-integrated synthesis is required to investigate the effect of heat integration in process selection. (2) In addition, the nine components exhibit a wide range of boiling points, and the distillation columns are allowed to operate over a wide range of pressures, [1, 32] atm. This makes it difficult to choose suitable temperatures for heat links. Furthermore, as the design of a distillation column includes the reflux

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Figure 7. Grand composite curve for solution obtained for case study II, generated without consideration of heat integration.

Figure 6. Best structure identified, using the automatic parameter selection procedure, for case study II without heat integration.

rate, the range of duties is also unknown. Therefore, the automatic identification procedure will be required, at least initially. The synthesis problem specification consists of the definition of the feed, including the physical property data for the nine components, and the list of processing technologies. The list of processing units includes the feed tank, a distillation unit based on the FenskeUnderwood-Gilliland shortcut method, and a product tank that accepts pure component streams. The unit and cost models are based on those described by Rathore et al.,20 including the definition of a discrete set of utilities (four cold utilities, including cooling water and ammonia at three different temperature levels, and four hot utilities, all steam at different temperatures), instead of using continuous utility cost functions, as in case study I. To make the problem more realistic, we have removed the assumption of saturated feed conditions. The energy content of the feed to a unit will depend on the operating conditions of the upstream unit, and this will affect the calculation of both the minimum reflux ratio and the energy requirements for the reboiler. As this problem is larger, we have reduced the number of pressure levels to four, log-uniform in [1, 32] atm. Four values are considered for the reflux rate factor, log-uniform in [1.1, 2.0]. Stream states (essentially the combination of phase, temperature, and pressure) are also discretized using four discrete levels for each phase. These are chosen to match the pressure levels allowed for distillation column operating pressure to avoid errors introduced by a mismatch between the unit and stream discretizations. Given our experience with case study I, only one run is attempted using the automatic link parameter selection procedure, with nt ) 4 and nq ) 4, which leads to T ) {385, 396, 407, 419} and Qb ) 2.8 MW. The parameter selection procedure generates, in the first pass, the best flowsheet without heat integration, shown in Figure 6. The grand composite curve25,26 for this process structure is shown in Figure 7.

Figure 8. Best heat-integrated flowsheet identified using the automated VHL selection procedure for case study II.

The integrated process has a different structure, shown in Figure 8, with an improvement of approximately 12% in the initial cost estimates and approximately 3% when the postsynthesis optimization procedure is applied. Two heat exchanges are identified by the synthesis procedure, both for cooling unit 8, separating h from i, for a total of 5.6 MW at 385 K. The excess heat is used for units 1 and 5. Figure 9 shows all of the heat-transfer requests generated in the first pass of the synthesis procedure and the results of the analysis. Automated design tools, especially at the early stages, should be used iteratively, whereby the problem definition is updated, on the basis of the insight gained from previous runs, and solved again. In this case, we see that the heat exchanges occur at the low end of the temperature range identified by the automated procedure. This leads us to question the validity of this range. Therefore, we pose the problem again but this time specifying the VHL parameters directly. We shift the temperature range down to determine whether the original automatically chosen range was appropriate. Specifically, we define a VHL with T ∈{355, 365, 375, 385} K and Qb ) 2.8 MW with nq ) 4 to investigate the

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Figure 10. Best heat-integrated flowsheet after the iterative procedure for case study II. The heat exchanges indicated were found using two temperature levels and eight duty levels.

Figure 9. Heat-transfer requests and analysis for case study II.

dependence of the solution identified on the heat link parameters. This second run identifies a different process as the best. The new structure is shown in Figure 10, where only the sequence of units should be considered at this point (the exchanges indicated relate to a later run, described below). This second run identifies three exchanges: 8.4 MW from unit 6's condenser to the reboilers of units 2 (5.6 MW) and 5 (2.8 MW) and 5.6 MW from the condenser of unit 8 to the reboiler of unit 7. As the temperature of two of the heat exchanges is again at the low end of the temperature scale, we try again with a modified heat link definition. This time, given the results of the previous runs, we assume that we have narrowed down the duty range, so we consider trading off the range of duties with the temperature levels. The new heat link definition consists of T ∈ {Ti|Ti ) 350 + 5i, i ) 0, ..., 7} K and Qb ) 5 MW, nq ) 2. This explores the wider temperature range at the expense of fewer discrete duty levels. The result is the same process structure as before (Figure 10) with exchanges at the same temperatures as before. We now do another run to explore the amount of exchange, assuming that we have identified good temperature

Figure 11. Grand composite curve for solution obtained for case study II using automated heat integration procedure with two temperature levels and eight duty levels.

values. This next run has T ∈{355, 385} K with Qb ) 2000 MW and nq ) 8. The result is again the same structure with the exchanges as indicated in Figure 10. The corresponding grand composite curve for this process structure, along with associated operating conditions, is shown in Figure 11. When compared with Figure 7, we see an improvement in overall utility consumption and an increase in process integration for meeting cooling and heating costs. The statistics for this sequence of runs are shown in Table 8. This table shows both the cost identified in the synthesis runs (column heading initial) and the cost obtained using the postsynthesis optimization procedure

Ind. Eng. Chem. Res., Vol. 43, No. 1, 2004 157 Table 8. Statistics for Case Study II cost (M$/year) no heat integration auto heat link mode manual selection manual selection manual selection

nt

nq

problems

nodes

time (min)

initial

post

4 4 8 2

4 4 2 8

126 4151 4069 4045 4069

4053 8 679 091 8 575 067 7 950 685 10 038 082

0.3 66.7 54.2 50.0 72.2

1.28 1.12 1.09 1.07 1.07

1.06 1.03 1.02 1.02 1.01

Table 9. Unit Design Information for Case Study II, Best Solution reboiler

condenser

unit

pressure (atm)

number of stages

reflux ratio

Q (MW)

Tin (K)

Tout (K)

Q (MW)

Tin (K)

Tout (K)

1 2 3 4 5 6 7 8

3.2 10.1 32.0 3.2 3.2 1.0 1.0 1.0

32 86 32 31 95 33 28 38

0.75 5.53 1.76 1.60 6.54 1.21 3.29 1.32

9.1 5.4 1.1 3.9 4.5 7.2 3.9 4.8

390.4 349.1 393.4 416.1 346.5 402.1 371.3 423.1

423.1 349.1 393.5 429.6 346.5 405.2 371.4 423.4

3.1 4.9 0.8 2.8 4.5 9.5 3.8 4.8

298.6 325.9 350.1 341.9 337.5 366.9 342.6 398.6

289.5 316.9 349.8 341.1 337.5 362.2 342.0 398.6

described earlier (column heading post). The postsynthesis costs, in particular, show that there is little to distinguish the three different structures identified by the various synthesis runs. However, these runs do provide some degree of confidence in selecting these three structures for further analysis using more rigorous tools. Furthermore, the settings of the design parameters identified for each unit in the flowsheets can be used as good initial guesses for rigorous optimization procedures. The design parameters for the best structure (Figure 10) are presented in Table 9. This case study demonstrates how a single virtual heat link, using multiple temperatures and multiple duties, can lead to multiple exchanges at different temperatures. This allows a single heat link to be useful in a search space that covers a wide range of temperatures and amounts of heating or cooling. Furthermore, it shows how a single heat link can achieve multiple exchanges from a single unit, albeit all at the same temperature (cf. exchange between units 6 and 5 and units 6 and 2). To achieve equivalent results with singlelevel heat links would have required, at a minimum, three different links. Identifying the settings for these three links would have been onerous without the use of multilevel links. 5.3. Case Study III: Distillation with Feed Pretreatment. The two examples studied above include simple distillation units with one reboiler and one condenser. The second case study is more realistic as it incorporates the state of the feed stream in the design of the distillation unit. The third case study takes this a step further and considers the use of distillation units with preheating or precooling applied to the feed stream. The example is based on example 2 in the article by Novak et al.5 The models have been modified to remove simplifications introduced by those authors, particularly for dealing with heat exchanger designs. The log-mean temperature difference is used instead of an arithmetic mean. Furthermore, the temperature drop or increase across an exchanger is taken into account when identifying process heat exchanges or when costing utility use. The difference in inlet and outlet temperatures for multicomponent streams can be significant and can therefore have an effect on process selection or configuration, as we shall see below. Finally, the distillation unit models have been extended to consider heating or cooling feed streams prior to introduction into the

Figure 12. Q-T diagram for non-heat-integrated solution for case study III, indicating need for higher cost utilities than expected from results presented by Novak et al.5

column. The feed treatment adds another target (for a total of five: feed pretreatment, top product condenser, bottom product reboiler, and two output streams) for the heat link algorithm and demonstrates the generic nature of the heat link algorithms. The first attempt at this problem includes no heat integration so as to provide a base case. The problem is solved with eight levels of discretization for stream states; eight pressure levels, again log-uniform, over [1, 20] atm; and eight reflux ratio factors, log-uniform over [1.05, 2.0]. Without heat integration, the best solution is the same structure, with the same operating pressures, as identified by Novak et al. but with a higher cost of 10.1 M$/year rather than 9.02 M$/year.5 The reason for this discrepancy is the difference in inlet and outlet temperatures for both the first unit’s reboiler, [400, 465] K, and the second unit’s reboiler, [431.7, 488.4] K, due to the multicomponent composition of the bottoms products of each of these units. The effect of the temperature range can be seen in the Q-T27,28 diagram (Figure 12). The use of more realistic models for the reboilers has two effects on the process synthesis stage. First, the higher outlet temperatures, when compared with the results presented by Novak et al., require the use of medium-pressure steam for the first unit and high-pressure steam for the second unit as opposed to low-pressure steam and medium-pressure steam, respectively. Second, the use of higher-cost

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Figure 13. Best heat-integrated structure for case study III. Figure 15. Grand composite curve for structure identified by synthesis procedure for case study III generated using normal distillation columns and without considering heat integration during the initial design stage.

Figure 14. Heat exchanger network for best solution for case study III.

utilities leads to the choice of lower reflux rates in the columns (and corresponding increase in the number of stages). This can be deduced from the reduction in heating and cooling duties (see Figure 12) for these columns when compared with the results presented by Novak et al.5 The next step is to consider the use of feed pretreatment, without heat integration. The solution identified has the same structure and same operating pressures as the base case but suggests that heating be applied to the feeds of the second (1.3 MW) and third columns (1.8 MW). The cost is only marginally lower at 10.0 M$/ year and is therefore unlikely to prove a viable design given the extra complexity. However, feed pretreatment might prove useful with heat integration. Therefore, we now solve this problem using the automatic parameter identification mode. This returns a solution with the same structure but different operating pressures (see Figure 13). The cost is 8.83 M$/year before postprocessing and 8.76 M$/year after postprocessing.24 The flowsheet is shown in Figure 14. Heat exchanges are indicated by circles with crosses, labeled with the name of the other half of the integrated exchanger. For example, hot stream H2 (condenser of second unit) is cooled by exchanging heat with cold streams C4 (feed preheater for second unit) and C1 (reboiler of first unit). The heat exchanger network for this process is shown in Figure 5. The automated parameter selection, with nt ) 3 and nq ) 3, generated T ) {451, 468, 486} K and Qb ) 6.1 MW for the heat link parameters. The solution was generated in just over 3 h. Although this is longer than required for the previous case studies, the search space

Figure 16. Grand composite curve for structure identified by synthesis procedure for case study III generated using normal distillation columns and considering heat integration from the beginning of the design process using the automated heat link generation procedure.

is larger because of the combinatorial increase in potential heat integrations. In particular, no a priori decisions about the type of feed stream pretreatment are made, and a finer discretization is used for both unit designs and stream state representation. In any case, the solution identified required no user interaction and led to a reduction of 12.4% in annualized cost. This reduction in cost makes it worth considering the new design as a potentially viable alternative to the initial flowsheet chosen without feed pretreatment and heat integration. The improvement in integration possibilities can again be demonstrated through the use of grand composite curves. Figures 15-17 show the grand composite curves corresponding to the three attempts at this problem: without heat integration, with heat integration using normal distillation units, and finally with heat integration allowing for feed pretreatment. The three figures show a successive increase in the potential integration.29 This case study demonstrates the general applicability of the procedure. The algorithms make assumptions about the types of units considered, and there is no need for the user to develop complex superstructures. Although the unit models used are shortcuts, the procedure does not preclude the use of more complex models. In fact, the unit design cache ensures that the compu-

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The use of discrete programming also allows one to consider more realistic models. Simplifications required by other methods, for reasons of nonlinearity or nonconvexity, are not required here. We have demonstrated, for instance, how removing certain simplifications can lead to more realistic results, especially in the context of heat integration. The result is a tool that tackles the problem of heat-integrated process synthesis at the early, conceptual stages and that links well into the whole design process. Acknowledgment

Figure 17. Grand composite curve for structure identified by synthesis procedure for case study III generated using distillation columns with feed pretreatment allowed and with heat integration considered from the beginning using the automated heat link generation procedure.

tational effort required for unit designs, however complex, will be minimized.

The authors gratefully acknowledge the support of the U.K. Engineering and Physical Sciences Research Council, which provided support for this research (Grants B/95/AF/2011 and GR/K88958). We also thank Professor Danny Lewin for providing early access to his work on the application of genetic algorithms for heat exchanger network synthesis and Dr. Rama Lakshmanan for his advice on the use of grand composite curves and the pinch method. Literature Cited

6. Conclusions The Jacaranda system for automated design has been extended to incorporate, efficiently, the ability to consider heat integration at the earliest stages of design. Jacaranda is based on the use of discrete programming for solving the mixed integer nonlinear programming problem implicit in a graph-based representation of the search space. Jacaranda is intended for use at the early, conceptual stages of process design. This paper has presented a novel and efficient approach for incorporating heat integration into the discrete process synthesis procedure. The result is a method that is able to handle larger problems than other methods without the need for simplifications in the models used and that takes a global view in the choice of structure. The use of discretization to convert the MINLP to a discrete graph search procedure can lead to combinatorial explosion in the size of the graph. In particular, the first approach to the incorporation of heat integration into Jacaranda22 is limited in application by the growth because of the number of heat links required for larger problems. The new approach implicitly generates a different search space, but one that subsumes the previous approach. The new search space is appropriate for early design, and the result is an algorithm with polynomial time requirements. For early design, time is a particularly important human factor, as exploration must be encouraged. The polynomial time properties ensure that the implementation is suitable for larger design problems. Another difficulty with the use of discretization is the need to identify correctly the appropriate discretization parameters a priori. The design and implementation of simple data analysis techniques has allowed us to develop an automated procedure for parameter selection. This alleviates some of the difficulty in problem definition and furthermore helps provide insight into the potential solutions available. By using this insight, the engineer can apply the procedure iteratively, converging to a good heat-integrated structure. This structure can then form the input for more detailed or rigorous methods, providing a good initial guess for the optimization solver.

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Received for review February 28, 2001 Revised manuscript received October 1, 2003 Accepted October 15, 2003 IE010187I