A Customized MILP Approach to the Synthesis of Heat Recovery

Ind. Eng. Chem. Res. 1998, 37, 2479-2495

2479

A Customized MILP Approach to the Synthesis of Heat Recovery Networks Reaching Specified Topology Targets Marı´a Rosa Galli† and Jaime Cerda´ * Instituto de Desarrollo Tecnolo´ gico para la Industria Quı´mica, Universidad Nacional del LitoralsCONICET, Guemes 3450, 3000 Santa Fe, Argentina

A mathematical representation of a heat-exchanger network structure that explicitly accounts for the relative location of heat-transfer units, splitters, and mixers is presented. It is the basis of a mixed-integer linear programming sequential approach to the synthesis of heat-exchanger networks that allows the designer to specify beforehand some desired topology features as further design targets. Such structural information stands for additional problem data to be considered in the problem formulation, thus enhancing the involvement of the design engineer in the synthesis task. The topology constraints are expressed in terms of (i) the equipment items (heat exchangers, splitters, and mixers) that could be incorporated into the network, (ii) the feasible neighbors for every potential unit, and (iii) the heat matches, if any, with which a heat exchanger can be accomplished in parallel over any process stream. Moreover, the number and types of splitters being arranged over either a particular stream or the whole network can also be restrained. The new approach has been successfully applied to the solution of five example problems at each of which a wide variety of structural design restrictions were specified. Introduction Most of the research efforts in the heat-exchanger network synthesis area have recently been devoted to the development of simultaneous algorithmic methods through which the optimal stream approach temperature, the least utility requirement, and the best network configuration are all found at once (Yee and Grossmann, 1990; Ciric and Floudas, 1991; Daichendt and Grossmann, 1994). However, one of the major disadvantages of such mixed-integer nonlinear programming (MINLP) superstructure-based synthesis approaches has been the poor involvement and the passive role played by the design engineer during the synthesis task. This is the main reason the so-called pinch technology has so far gained many adepts in the chemical industry where it is now intensively applied (Linnhoff and Hindmarsch, 1983; Trivedi et al., 1989; Linnhoff, 1993). The MINLP simultaneous algorithmic methods can be grouped into two families: (i) those relying on a hyperstructure-based MINLP problem formulation involving nonlinear nonconvex terms in both the constraint set and the objective function (Floudas and Ciric, 1989; Ciric and Floudas, 1991) and (ii) the so-called MINLP stagewise technique featuring a linear feasible region by assuming an isothermal mixing of split streams and a nonconvex objective function (Yee and Grossmann, 1990). Both suffer from similar shortcomings. In each case, splitters and mixers are always required to embed series and parallel configurations. As a result, such units are to stay even if a series configuration is pursued, and consequently the designer loses control on the number and type of splitters and mixers to be incorporated in the heat-exchanger network synthesis (HENS) problem formulation. Moreover, ad* To whom the correspondence should be addressed. Telephone: 042-559175. Fax: 54-42-550944. E-mail: jcerda@ intec.unl.edu.ar. † E-mail: [email protected].

ditional structural conditions usually specified by the designer to get a less complex heat-exchanger network with better operability and controllability features are handled through the introduction of new constraints in the problem model. In turn, the number of binary variables remains the same (Yee and Grossmann, 1990) or rises even further (Floudas and Ciric, 1989). In other words, the topology constraints reduce the size of the feasible solution space by adding extra cuts rather than diminishing the problem dimensionality. This may be too costly from the computer time viewpoint, especially for large industrial problems. In addition, the model nonconvexities in both approaches may lead to poor suboptimal solutions. To lessen the dimensionality of the HENS problem formulation, Daitchendt and Grossmann (1994) proposed a preliminary screening procedure relying on the use of convex aggregate models to generate a reduced superstructure while preserving the optimality of the solution found. Larger simplicity and robustness are obtained in model-based targeting approaches. These methods allow the designer to evaluate aspects of complexity, operability, and controllability during the synthesis. The vertical MILP transshipment formulation proposed by Gundersen and Grossmann (1990) and later extended by Gundersen et al. (1996) and the transportation model version developed by Gundersen et al. (1997) are all solved in a sequential framework to reach the minimum number of units (the MNU target) and the heat load distribution providing a near-minimum overall heattransfer area. Galli and Cerda´ (1998a,b) introduced a mixed-integer linear programming (MILP) sequential optimization approach based on a neighbor-based network representation where the location of each equipment item is defined by choosing its preceding and succeeding units in the network. Thus, nonattractive sequences of heat matches over a process stream can easily be handled by properly defining the sets of feasible predecessors and

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2480 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998

successors for each unit. In this way, it was developed a MILP mathematical framework that allows the designer to participate in the synthesis task by providing such topology constraints from the start. By doing that, the dimensionality of the feasible region becomes smaller as the number of design options diminishes and the HENS problem formulation becomes easier to solve. The results showed that the topology constraints normally imposed by the designer on the network configuration to make it structurally simpler usually produce both an increase in the least utility consumption (the LUC target) and a nonzero heat flow across the pinch. To find the optimal network design reaching the LUC target, the notion of pseudo pinch is therefore required by a sequential HENS approach to first partition the network structure into a pair of noninteracting subnetworks (assuming that a single hot and single cold utility are available). In consequence, the optimal stream pseudo pinch points are to be determined before accomplishing the design synthesis task by going one-byone from the top to the bottom subnetwork. However, the topology representation proposed by Galli and Cerda´ (1998a,b) regards the heat-transfer units as the only network constituents. By ignoring splitters and mixers, it was necessary to allow multiple predecessors for an exchanger to also handle parallel arrangements of heat matches. This somewhat complicates the expressions of the problem constraints, especially those related to a parallel arrangement of heat matches over a process stream. The number of such constraints shows a rather combinatorial increase with the stream population. Moreover, the designer loses control on the number and types of splitters and mixers arranged over either a process stream or the whole network. Avoiding expensive stream splits may result in a substantial decrease in the capital cost for a fixed energy cost, representing millions of dollars in savings in industrial applications (Trivedi et al., 1989). In this paper, a new MILP sequential synthesis approach based on an improved network representation that also regards splitter and mixers as network constituents is presented. Topological constraints on the number and/or types of splitters and mixers usually specified by the designer can also be handled to further reduce the problem size and enhance the computational efficiency of the MILP solution algorithm. Problem Statement The heat-exchanger network synthesis (HENS) problem being addressed in this paper can be stated as follows: Given are (1) a set of hot process streams i ∈ H, their supply (THiS) and target (THiT) temperatures, and their heat-capacity flow rates (FCp)i; (2) a set of cold process streams j ∈ C, their supply (TCjS) and target (TCjT) temperatures, and their heat-capacity flow rates (FCp)j; (3) a set of hot utilities s ∈ S and a set of cold utilities w ∈ W, their corresponding temperatures and costs per unit of heat provided or removed; (4) a set of structural design specifications. The topology features imposed by the designer on the network configuration are intended for the following: (a) Disallowing the execution of some heat matches or bounding their heat loads. (b) Prohibiting network design options that involve undesirable sequences of heat matches and/or nonattractive locations for stream splitters over some process streams. To this end, the sets of feasible predecessors

and successors for each potential unit (an exchanger or a stream splitter) are to be provided as additional problem data. Moreover, the designer can specify the heat matches that can be involved in a parallel arrangement over a process stream. (c) Favoring the selection of some specified heat matches as candidates for being the input or the output unit over a particular process stream. (d) Limiting the number of splitters of b outlets as well as the total number of splitters over a particular process stream. By this means, one can also bound the number of branches into which a stream may be split. (e) Bounding the overall number of splitters over the whole heat-exchanger network. (f) Restraining the set of heat exchangers with which a potential heat match can be arranged in parallel over a specified process stream. The problem goal is to find a network configuration reaching all the selected design targets. In addition to the prescribed hot/cold stream exit temperatures (THiT), the following have been chosen: (1) The topology conditions specified by the designer, (2) the constrained least utility consumption (LUC), and (3) the minimum number of units (MNU) as the sequence of design targets to be achieved along the network synthesis task. Since a sequential optimization strategy has been adopted, the least utility consumption (LUC) under the specified topology constraints is first determined as long as the search for a MNU design is restrained to a solution space only including the topology-constrained networks reaching the LUC target. Next, the stream pseudo pinch points are to be found to decompose the network into a set of noninteracting subnetworks, each one requiring a single cold/hot utility. Finally, the resulting subnetworks are independently designed by proceeding from the top to the bottom one. Similar to other sequential algorithmic methods such as the transshipment or the transportation approach, almost the same mathematical model (to be presented in a next section) is applied at every step. Only a few changes are introduced between two consecutive HEN synthesis stages due to the use of a different objective function (minimization of the utility usage or the number of units) and/or the incorporation of a new linear constraint (for instance, the constrained LUC target). Model Assumptions To keep the problem mathematical formulation linear, the following assumptions have been made: (1) countercurrent heat exchangers; (2) constant heat-capacity flow rates; (3) forbidden hot-hot or cold-cold stream matches; (4) isothermal mixing of split streams; (5) no split stream flowing through two or more exchangers in series; (6) no stream bypasses. By neglecting network configurations where a split stream goes through several exchangers in series, one can expect to find designs requiring larger heat-transfer areas in small problems where there is not much flexibility in selecting alternative network structures (Yee and Grossmann, 1990). However, these authors have also concluded that such a model limitation is less important at large-size problems because of a greater flexibility in the selection of both matching and configuration. Motivating Example In Figure 1, a motivating example introduced by Linnhoff and Hindmarsh (1983) is presented. A single

Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2481

Figure 1. (a) Optimal solution to the utility usage problem; (b) optimal network design, for the motivating example.

heating and a single cooling utility are available for use in the network and the value of HRAT is assumed to be 20 °C. When the process heat recovery is maximized, a process pinch arises at 70/90 °C and consequently the HENS problem can be decomposed into a pair of pinchdefined networks (i.e., the upper and the lower network). In addition to the cold/hot utility, such networks are comprised of three and four process streams, respectively (see Figure 1a). The minimum utility usages provided by the heat-cascade procedure are QS° ) 107.5 kW and QW° ) 40 kW, respectively. Above the pinch, the process is a net heat sink demanding only hot utility. In turn, the process behaves as a net heat source under the pinch and consequently just a cold utility is required. Three structural restrictions have been imposed by the designer on the network configuration: (a) no stream splitter over the cold stream C1; (b) the sequence of exchangers (H1/C1)-(H2/C1) over C1 going from lower to higher temperatures is prohibited; (c) the match (H1/ C2) has been forbidden. Least Utility Usage under Topology Constraints. Recently, Galli and Cerda´ (1998a) proposed a rather simple computational scheme to determine the heating and cooling utility requirements under network topology conditions specified by the designer. Such a methodology can be applied to pinched and unpinched problems, assuming that the process/utility pinch temperatures, if any, have already been determined by using available techniques such as the heat-cascade procedure (Linnhoff and Hindmarsch, 1983) or the transshipment model (Papoulias and Grossmann, 1983). However, the specified design constraints on the network configuration may prevent, from fully recover-

ing, the available process heat above the pinch as assumed by the heat cascade. If so, a heat surplus usually crosses the pinch and further hot utility usage over the value provided by the heat cascade (Q0S) will be required. The pinch heat flow and the additional utility requirement will both be equal to a positive value EU to be determined through a mathematical formulation that accounts for all the prescribed topology targets a-c. As proposed by Galli and Cerda´ (1998a) such a problem model will just consider an “isolated” pinchdefined upper network where a cold utility w ∈ W is also available to eventually remove the heat surplus EU. The additional heating utility consumption is found by adopting the minimization of the overall utility usage as the problem objective. The total heat duty of the coolers will be providing the value of EU. If no cooler is needed, EU will be zero and the topology constraints on the upper network design will have no impact on the utility usage. Similarly, the restrictions placed by the designer on the network structure may limit the process heat recovery at the pinch-defined lower network in such a way that some heat demands are to be satisfied through an additional hot utility consumption. Such a heat shortage EL makes it necessary to have a hot utility s ∈ S also available at the “isolated” lower network. Its value can be determined by applying the problem formulation to the pinch-defined lower network and finding the optimal hot utility usage. Again, the minimization of the utility usage will be the problem objective. If QS° and QW° stand for the least heating and cooling utility usages, respectively, in the absence of topology constraints (i.e., the values provided by the heat-cascade procedure), then the constrained utility targets become

QS* ) QS° + max(EU, EL) QW* ) QW° + max(EU, EL)

(1)

and the heat flow crossing the pinch will be Qp* ) max(EU, EL). Equation 1 assumes that the heat surplus EU from the upper subnetwork can be fully applied to meet the heat shortage EL at the lower one. The MILP mathematical framework to be presented in another section was applied to determine the impact of the structural constraints on the least utility usage at the motivating example. It was found that no cooler is really needed at the pinch-defined upper network and therefore EU ) 0 (see Figure 1a). Nonetheless, the structural restrictions were effective at the isolated pinch-defined lower network where a heat shortage EL ) 4 kW arises since a heater featuring a similar heat load is needed (see Figure 1a). As a result, a heat flow as large as max(0,4 kW) ) 4 kW crosses the pinch. Then, the optimal hot and cold utility usages, QS* and QW*, both increase by the same amount (4 kW) with regards to the nonconstrained case (i.e., QS* ) 111.5 kW and QW* ) 44 kW). At nonpinched problems, the determination of the constrained utility target just requires consideration of a single network where both hot and cold utilities are available. Figure 1b shows a network configuration for the motivating example that reaches both the constrained LUC and MNU design targets. It was developed by accomplishing the design stage, assuming that the minimum utility consumption under the network topol-


ogy constraints is already known. Before the design stage is started, however, the stream pseudo-pinch temperatures decomposing the network into independent subnetworks are to be determined. When the HENS problem exhibits multiple pinch temperatures, the proposed computational procedure to calculate the LUC target under topology constraints can still be applied. Let us assume that a pair of pinch points arises at the heat cascade, and consequently three pinch-defined subnetworks rather than two are to be considered. Starting from the top subnetwork, each one will be independently tackled as explained before to sequentially establish the heat surplus (or heat shortage) at every upper (bottom) subnetwork. The largest of such values will be giving the increase in the least utility usage due to the topology constraints. Stream Pseudo-Pinch Temperatures. The notion of pseudo pinch at HENS problems has been introduced to partition a heat-exchanger network into a pair of noninteracting subnetworks even if a finite heat flow goes across the pinch (assuming that a single cold and a single hot utility are available). In this way, each subnetwork can be independently designed and smaller subproblems are to be sequentially solved. In contrast to the pinch idea normally associated to a set of process streams, the pseudo-pinch temperature can vary with the stream. By definition, the decomposition of a HENS problem at pseudo-pinch temperatures will generate a pair of subproblems with the following features (Trivedi et al., 1989): (i) a “sink” subproblem ranging from the pseudo pinch at every process stream to higher temperatures and only requiring a hot utility as much as the heating utility target. Moreover, the total heat flow available, including the one provided by the hot utility, is entirely allocated to the cold process streams and therefore the “sink” subproblem as a whole is in enthalpy balance; (ii) a “source” subproblem ranging from the pseudo pinch at every stream to lower temperatures and only requiring a cold utility as much as the cooling utility target. Obviously, the set of streams belonging to the “source” subproblem, including the cold utility, is in heat balance too. As mentioned before, the topology constraints generally bound the process heat recovery, thus preventing the HEN from reaching pinch conditions. A positive heat flow across the pinch then arises, and the pinchdefined upper and lower networks can no longer be independently designed. To get a zero heat flow between those networks, a new boundary separating the upper from the lower network is to be defined (see Figure 2). Contrary to what is always assumed by any pinch-based synthesis procedure, a cold (hot) process stream entering (leaving) the new upper network no longer features an input (output) temperature equal to the process pinch (the pinch increased by ∆Tmin). Instead, each stream exhibits its own temperature at the new boundary, also called the stream pseudo-pinch point. Figure 1b shows a different decomposition of the network structure yielding a pair of subnetworks with the following features: (i) no heat exchanger crosses the boundary (i.e., no heat interaction between the new upper and lower networks); (ii) only a hot utility is required in the upper network; (iii) a cold utility is merely needed in the lower network; (iv) the constrained LUC target has been reached. Therefore, the stream temperatures at the new boundary fully meet the

Figure 2. Stream pseudo pinch temperatures resulting from a pinch heat flow. (a) No pinch heat flow; (b) allocation of the pinch heat flow to broaden the range of the upper network; (c) broadening the lower network.

pseudo-pinch conditions. As shown by Figure 1b, the temperatures of the process streams {H1,C1}at the new border are no longer 90/70 °C but they decrease to 82 °C for H1 and 62 °C for C1. Both pseudo-pinch temperatures are below the pinch. In other words, the common boundary of the upper and lower subnetworks shifts itself under the pinch over the process streams {H1,C1} in such a way that the heat match (H1/C1) can be performed in a single unit rather than in two (see Figure 1b). Therefore, effective structural restrictions on the network design producing a positive pinch heat flow may even yield a reduction in the number of units by moving the boundary between the subnetworks down from the pinch to the pseudo-pinch temperature over each process stream. The temperature of C2 at the new boundary is still at the pinch (70 °C). To develop a sequential algorithmic approach able to cope with HENS problems exhibiting multiple pinch temperatures, a general notation is used in the paper. Those networks located at both sides of the nth pinch are referred to as the adjacent networks n and n + 1, respectively, where network n + 1 is at temperatures lower than network n. Moreover, the pseudo-pinch temperature at the common boundary is denoted by TCjI(n) for j ∈ C and THiO(n) for i ∈ H, respectively, since it stands for the input (output) temperature of a cold (hot) stream to (from) the network originated by the nth pinch (i.e., the upper network with regard to the nth


pinch). Its value may be greater than, equal to, or lower than the nth-pinch temperature. Upper Network Heat-Recovery Target. In general, many different sets of pseudo-pinch temperatures can alternatively be chosen to decompose a constrained HENS problem into a pair of noninteracting subproblems. However, a careful analysis of Figure 1b indicates that a proper selection of the pseudo-pinch points and the related boundary between the independent subnetworks favors the allocation of the pinch heat flow to stream matches already performed at the pinch-defined upper network. In this manner, their temperature ranges will extend further toward lower temperatures, bringing about the merger of units performing such heat matches at both sides of the process pinch. Consequently, fewer units would be required to accomplish the heat-recovery process. Therefore, the determination of the best pseudo-pinch temperatures can be regarded as the problem of optimally locating the boundary separating the upper from the lower network. The resulting subnetworks can not only be independently designed but also include fewer units overall. The problem target to be achieved at the optimal boundary location will then be the maximization of the heat recovery at the upper network subject to the following two conditions: (i) only hot utility is required at the upper network and (ii) the overall heat duty of the heaters never surpasses the constrained heating utility target found before (QS* ) 111.5 kW). In other words, the best network decomposition implies also reaching a maximum heat-recovery target at the resulting upper network in addition to the least heating utility usage. Sometimes, the displacement of the boundary between the subnetworks not only decreases the total number of heat exchangers but even produces the complete merging between the upper and the lower network. To calculate such a heat-recovery target, the mathematical framework will consider just the upper network with a free bottom boundary. Only the process streams partially or completely ranging above the pinch are taken into account (i.e., the stream set {H1,C1,C2} in the motivating example). Therefore, the temperature of any cold (hot) process stream at the boundary between the upper and lower networks is no longer a known parameter but a problem variable to be determined. Its value can vary within a temperature range defined by (a) the stream target (supply) temperature TCjT (THiS) as the upper limit and (b) the stream supply (target) temperature as the lower limit, for any cold (hot) process stream cj (hi). In the motivating example, the pseudo-pinch points at the process streams {H1,C1,C2} will belong to the intervals {(60,150), (20,125), (25,100)}, respectively (i.e., their entire temperature ranges). At HENS problems featuring a heat cascade with N pinch temperatures, the optimal locations of N free boundaries between subnetworks are to be sequentially established starting from the top one associated to the highest pinch. Moreover, the (n + 1)-pinch temperature located below the nth pinch is to be considered in the definition of the temperature range lower limit for the input (output) temperature of a cold (hot) stream to (from) the nth subnetwork. Network Design Stage. In general, several sets of pseudo-pinch temperatures permit reaching the heatrecovery target at the upper network, but not all are able to minimize the number of units. For this reason,

it is better to provide the heat-recovery target rather than the values of the pseudo-pinch temperatures to accomplish the upper network design stage. By doing that, all the alternative sets of pseudo-pinch temperatures reaching the heat-recovery target at the upper network will be simultaneously considered. The best one of them providing the minimum number of units will be selected together with the upper network configuration at the design stage. To find the best upper network design, the MILP mathematical framework will account just for the upper network with a free bottom boundary and use the minimization of the number of units as the problem objective. The problem solution space is further restrained by imposing both the least heating utility usage and the upper network heat-recovery target as additional problem constraints. Again, the cold (hot) stream temperature at the bottom boundary (i.e., the stream pseudo-pinch point) will be a problem variable varying within a temperature range similar to the one used at the problem providing the upper network heatrecovery target. At the motivating example, therefore, the input temperatures of the cold streams (C1,C2) to the upper network are continuous variables changing within the intervals (20,125) and (25,100), respectively. In turn, the output temperature of H1 from the upper subnetwork can vary from 60 to 150 °C. By solving the proper problem formulation, one can find simultaneously the network configuration and the pseudo-pinch temperatures. As shown in Figure 1b, the pseudo-pinch points over the streams {H1,C1,C2} are 82, 62, and 70 °C, respectively. After the pseudo pinch temperatures are determined, the set of process streams belonging to the new lower network and their temperature ranges in the “source” subproblem are all known. In other words, the input (output) temperature of any cold (hot) stream to (from) the lower network at the design stage is a model parameter. In the motivating example, the new lower network comprises the process streams {H1,H2,C1,C2} with temperature ranges given by {(60,82), (60,90), (20,62), (25,70)}, respectively. To find the lower network design, the mathematical formulation will include the cooling utility target as an additional constraint and use the least number of units as the problem objective. Figure 1b depicts the network structure provided by the proposed approach. It includes six units (i.e., one less than the number expected for a pinch design). If the HENS problem features a pair of pinch temperatures, it will then be decomposed into three independent subnetworks to be sequentially designed starting from the top one. Except for the bottom subnetwork, the other two are designed as they were upper networks (with regard to the top and the next-to-the-top pinch temperatures, respectively) since the input (output) temperatures of cold (hot) streams to (from) each of them are unknown before the design task is accomplished. Problem Mathematical Formulation As already mentioned, the decomposition strategy permits one to write the HENS problem formulation at the level of the subnetwork. For the sake of simplicity, the use of the index n to denote the nth network in the mathematical model given below has mostly been avoided.


Figure 3. Interpretation of the problem binary variables.

of problem constraints when both parallel and series arrangements are simultaneously considered. B. Network Topology Restrictions. (See Figure 3.) The mathematical constraints defining the feasible arrangements of units over cold process streams at a particular network are next introduced. Similar restrictions for hot streams are given in the Appendix. Existence of a Unit in a Network. There are several kinds of equipment items in a heat-exchanger network (i.e., heat exchangers, heaters, coolers, stream splitters, and mixers). The proposed mathematical model will assume that a single unit is available for executing every potential heat match hi/cj where i ∈ HSj and j ∈ CW. Moreover, a certain number of splitters and mixers of b branches, if required, may be arranged over any process stream, where b can vary from 2 to NBj. The parameter NBj denotes the maximum allowed number of split streams into which the jth process stream can be divided. To decide whether a particular equipment item will be included in the network, different sets of decision variables are to be defined for as many as the number of equipment types being considered. Heat Exchanger (hi/cj). The selection of the heat match (hi/cj) is handled through the binary variable Yij, for i ∈ HSj and j ∈ CW. The extended stream sets HS and CW include hot and cold utilities, respectively. A variable Yij will take a zero value if the heat exchange between hi and cj is not performed in the network and one otherwise. In the former case, no heat flow qij can be transferred from the hot stream hi to the cold stream cj. Then,

qij e UijYij Figure 4. Interpretation of the output temperature variables.

i ∈ HSj, j ∈ CW

(2)

where Uij is an upper bound on the value of the heat load qij. In turn, HSj is the subset of hot streams including heating utilities that can exchange heat with cj. A heat match (hi/cj) being performed in the network may be arranged either in series or in parallel over cj or hi. If qcijs (or qcijd) represents the energy load of the heat match (hi/cj) when performed in series (or in parallel right after the dth splitter) over the cold stream cj, then the variable qij can be expressed as follows:

qij ) qcijs +

qcijd ) qhijs + ∑ qhijd ∑ d ∈ SP d ∈ SP j

i

i ∈ HSj, j ∈ CW (3) and the constraint (2) becomes

qcijs +

qcijd e UijYij ∑ d ∈ SP

i ∈ HSj, j ∈ CW

(4)

j

Figure 5. Interpretation of series and parallel exchanger heat load variables.

A. Nomenclature. (See Figures 3-5.) Before the problem mathematical formulation is introduced, refer to the Nomenclature Section where subscripts, superscripts, sets, parameters, and variables are defined. We need to define a pair of continuous variables to stand for the heat load of the match (hi/cj) over the process stream hi (or cj) (i.e., qhijs (or qcijs) and qhijd (or qcijd)), to avoid a combinatorial increase in the number

for the unit (hi/cj) over the cold stream cj. Only one of the terms in the left-hand side of (4) will be allowed to be positive when Yij is equal to 1 through additional restrictions to be presented later on. A restriction similar to (4) can be written in terms of the variables qhijs and qhijd, d ∈ SPi as shown in the Appendix. As a result, the variable qij is no longer included in the problem formulation. Though the substitution of qij by {qcijs, qcijd, qhijs, qhijd} produces a significant rise in the number of continuous variables involved in the mathematical model, it has been introduced to avoid a combinatorial increase in the number of constraints when parallel arrangements are to be handled. When


applied, the computational efficiency of the MILP solution procedure shows a remarkable improvement for large-size problems. Stream Splitter. Let d ∈ SPj denote a cj stream splitter dividing the cold stream cj into two or more branches. The splitter d is incorporated into the network over the jth cold stream whenever the binary variable YCdj is equal 1. Since the continuous variable qcijd represents the energy load of (hi/cj) when arranged in parallel right after the splitter d over cj, then the heat flow qcijd becomes null whenever YCdj ) 0. Therefore,

∑

i∈

qcijd e M YCdj

d ∈ SPj, j ∈ C

(5)

HSj

where M is a sufficiently large number that can be adopted equal to the largest heat flow demanded by cj within the network under analysis. If YCdj ) 0 for any d ∈ SPj (no splitter at all over cj) but the match (hi/cj) is performed (Yij ) 1), then (hi/cj) is arranged in series and the restriction (4) will allow the energy load qcijs to take a positive value. Stream Mixer. The presence of a dth splitter dividing a cj stream into b branches in the network implies necessarily the existence of a companion mixer m ∈ MXj of b split streams over cj, where (d,m) ∈ PAIR. Therefore, the mixer m is to be incorporated in the network whenever YCdj is equal to 1 and (d,m) ∈ PAIR. Otherwise, it should be excluded from the configuration. In other words, no additional decision variable is required to denote the presence of the mixer m ∈ MXj in the network structure since such a role is already played by the variable YCdj associated to the companion splitter. Parallel Exchangers Are Always Preceded by a Splitter and Succeeded by a Mixer. Because of the model assumptions, the proposed MILP formulation does not account for network structures involving stream bypasses and/or split streams flowing through two or more exchangers in series. In a feasible network, therefore, parallel heat exchangers are assumed to be always preceded by a splitter d ∈ SPj and succeeded by the companion mixer m ∈ MXj, where (d,m) ∈ PAIR. Then,

XCij,m ) XCd,ij d ∈ SPj, (d,m) ∈ PAIR, i ∈ HSj, j ∈ C Therefore, the set of binary variables XCij,m is no longer necessary to formulate the problem mathematical model. Upper Bound on the Number of Predecessors of an Equipment Item. Except for a stream mixer, each equipment item arranged over a cold or a hot process stream features at most a single preceding unit in a feasible network configuration. On the contrary, a b inlet stream mixer is to be preceded by a set of b parallel heat exchangers. The mathematical constraint limiting the number of predecessors for each type of unit is given below. Heat Exchanger (hi/cj). If the heat match (hi/cj) is performed in the network (Yij ) 1), then an upper bound equal to 1 must be imposed on the number of predecessors over hi ∈ H and cj ∈ C. Therefore, (hi/cj) must either have a single predecessor (an exchanger, a splitter, or a mixer) or it is the input unit over cj. In the same way, (hi/cj) should be either the input unit or the successor to a single equipment item over hi. Moreover, the match

(hi/cj) may be the first on cj and a noninput unit over hi or vice versa.

XCFij +

∑

k ∈ PHij

XCkj,ij +

∑

XCd,ij +

d ∈ SPj

∑ XCm,ij ) Yij m ∈ MX

i ∈ HSj, j ∈ C (6)

j

By some sequences of units involving the match (hi/ cj) being disallowed for operating or design reasons through a proper specification of the set PHij, the related sequencing variables XC’s are excluded from the mathematical model, thus reducing the problem size. Instead of making the synthesis task more complex, the topology restrictions permit the number of both binary variables and constraints to be cut down and acceleration of the convergence to the best-constrained network design. Stream Splitter. If the splitter d ∈ SPj is included in the network over cj (YCdj ) 1), then an upper bound equal to 1 must be imposed on the number of predecessors of the splitter d. Therefore, the splitter d must either have a single preceding unit (an exchanger or a noncompanion mixer) or it is the first one over cj.

XCFdj +

∑

XCij,d +

i ∈ HSj

∑

m ∈ MXj (d,m) ∉ PAIR

XCm,d ) YCdj d ∈ SPj, j ∈ C (7)

By restraining the possible location of a stream splitter d ∈ SPj over the jth cold stream, one can achieve a further reduction in the number of sequencing variables involving the unit d. Stream Mixer. If the mixer m ∈ MXjb is required for mixing b branches of the cold stream cj in the network, then YCdj ) 1, where (d,m) ∈ PAIR. Moreover, an upper bound equal to b on the number of heat exchangers preceding the mixer m must hold. Therefore, the mixer m should always have as many preceding heat matches (hi/cj) as the number of split streams being merged into a single one in it. Since the set of matches preceding a mixer m are the successors to the companion splitter d, then the upper bound on the number of exchangers preceding a mixer is automatically considered by bounding the number of successors to the companion splitter. A top value on the number of successors to each type of unit is next considered. Upper Bound on the Number of Successors to an Equipment Item. Apart from a stream splitter d ∈ SP, every equipment item always features at most a single succeeding unit in a feasible network configuration. The last unit does not present any successor at all. On the contrary, an existent b outlet splitter is to be succeeded by a set of b parallel heat exchangers. The mathematical constraint limiting the number of successors for each type of unit is given below. Heat Exchanger (hi/cj). If the heat match (hi/cj) is performed in the network (Yij ) 1), then an upper bound equal to 1 must be imposed on the number of successors over both the hot stream hi and the cold stream cj. In other words, (hi/cj) should at most have a single succeeding unit (a heat exchanger, a splitter, or a mixer) over cj and hi, respectively. Then,

∑

k ∈ SHij

XCij,kj +

∑

d ∈ SPj

XCij,d +

∑

XCd,ij e Yij

d ∈ SPj

i ∈ HSj, j ∈ C (8)


Stream Splitter. If the splitter d ∈ SPbj is arranged over the cold stream cj in the network (YCdj ) 1), an upper bound equal to b must then be imposed on the number of successors (only heat exchangers) to the splitter d. Then, the splitter d must have as many succeeding exchangers as the number of branches into which it divides the cold stream cj.

∑

XCd,ij ) bYCdj

i ∈ HSj

d ∈ SPjb, b ) 2,3, ..., NBj, j ∈ C (9) Stream Mixer. If a mixer m ∈ MXj is placed over the cold stream cj in the network (i.e., YCdj ) 1, (d,m) ∈ PAIR), then an upper bound equal to 1 upon the number of successors to the mixer m must hold. Therefore, the mixer m should at most have a single successor (a heat exchanger or a noncompanion splitter) over the cold stream cj.

∑

XCm,ij +

i ∈ HSj

∑

n ∈ SPj n*d

XCm,n e YCdj

m ∈ MXj, (d,m) ∈ PAIR, j ∈ C (10) Upper Bound on the Number of Input Units over a Cold Process Stream. Among the equipment being arranged over cj, only a single heat exchanger or a splitter d ∈ SPj can be the input unit. On the contrary, a mixer m ∈ MXj will never arise first over cj. Therefore,

∑

XCFij +

i ∈ HSj

∑

d ∈ SPj

XCFdj ) 1

j∈C

(11)

Optional Upper Bound on the Number of Stream Splitters. To lessen the complexity of the network structure, the designer can limit the number of splitters over a jth cold stream by specifying a top value NDj.

∑ YCdj e NDj d ∈ SP

j∈C

(12)

j

In the same way, the designer can specify an upper bound equal to ND on the overall number of splitters in the network.

∑ ∑

i ∈ H d ∈ SPi

YHdi +

∑ ∑

j ∈ C d ∈ SPj

YCdj e ND

(13)

Restricted Parallel Arrangements over a Process Stream. If the hot stream hk does not belong to the set HSij, then it cannot exchange heat with cj in parallel with the match (hi/cj) over cj. Therefore,

XCd,kj + XCd,ij e 1 d ∈ SPj, k ∉ HSij, i ∈ HSj, j ∈ C (14) C. Heat Balances around a Single Unit or a Set of Parallel Units. (See Figures 4 and 5.) The value of the jth cold stream exit temperature (tcij) from the heat match (hi/cj), i ∈ HSj at the nth network (originated by the nth pinch) can change within the following range:

TCjI e tcij e TCjO

(15)

where TCjI/TCjO stands for the input/output tempera-

ture of cj to/from the nth network. As already discussed, TCjO is a model parameter given by the minimum of the following two values: the jth cold stream target temperature (TCjT) and the input temperature to the network (n - 1). In turn, the temperature TCjI at the boundary between networks n and n + 1 is a problem variable featuring the following range:

max(Tpinch(n+1), TCjS) e TCjI e min(TCjI(n-1), TCjT) j ∈ C (16) If the HENS problem exhibits a single process pinch, then the inequality (16) restraining the value of the input temperature of cj to the upper network to be designed reduces to TCjS e TCjI e TCjT. When the computational scheme proposed by Galli and Cerda´ (1998a) is used to determine the utility usage target under structural constraints, each pinch-defined network is to be sequentially considered in order to establish the impact of the topology conditions on the heat-recovery process. In such a case, the input temperature TCjI is no longer a problem variable but a model parameter just equal to max(TCjS,Tpinch) for the upper network and TCjS for the lower network, respectively. A similar situation arises at the time of designing the bottom subnetwork (below any pinch) since the input temperature TCjI is a model parameter equal to the jth stream supply temperature TCjS. jth Cold Stream Exit Temperature from an Input Exchanger (hi/cj). If the exchanger (hi/cj) is the input unit over cj (XCFij ) 1), it should then be performed in series and its heat load will be denoted by qcijs. Otherwise, a jth cold stream splitter d ∈ SPjb would arise first over cj preceding a set of b parallel heat matches (XCFdj ) 1). Therefore,

tcij e TCjI + (qcijs/FCpj) + M(1 - XCFij) i ∈ HSj, j ∈ C (17) The heat balance around (hi/cj) is a conditional restriction to be enforced only if such a match is performed in the network. Then, it should written as an inequality. The sign (e) is to be adopted because the variable tcij will always tend to take the highest value allowed by the right-hand side of (17) in order to lessen either the utility usage or the number of heattransfer units. At the same time, it should be noted that the value of the heat load qcijs is controlled by the overall heat balance over cj. jth Cold Stream Exit Temperature from a Noninput Exchanger (hi/cj). Three different cases are to be considered depending on the type of equipment item (another exchanger, a splitter, or a mixer) preceding the match (hi/cj). Preceding Unit over cj Is Another Heat Exchanger (hk/ cj). If the predecessor is the single heat exchanger (hk/ cj), then XCkj,ij ) 1 and the match (hi/cj) is performed in series over the cold stream cj. Therefore,

tcij e tckj + (qcijs/FCpj) + M(1 - XCkj,ij) k ∈ PHij, i ∈ HSj, j ∈ C (18) Preceding Unit over cj Is a Stream Splitter d ∈ SPj. If a heat match (hi/cj) is preceded by a stream splitter d ∈ SPjb over the cold stream cj (XCd,ij ) 1), then it is arranged in parallel with other (b - 1) exchangers (hk/ cj) and its heat load is represented by qcijd. Moreover,


the hot stream hk should belong to the set of hot streams that can exchange heat with cj in parallel with the match (hi/cj) (i.e., hk ∈ HSij). Therefore,

tcij e tcdj + (qcijd +

∑

qckjd)/FCpj + M(1 - XCd,ij)

k ∈ HSij

b parallel heat exchangers. Moreover, the exit temperature of cj from the mixer m (tcmj) is given by the temperature at which it leaves any of the preceding heat exchangers (see Figure 4).

tcmj e tcij + M(1 - XCd,ij)

j

d ∈ SPj, i ∈ HS , j ∈ C (19)

d ∈ SPj, (d,m) ∈ PAIR, i ∈ HSj, j ∈ C (26)

In addition, any heat match (hk/cj) being arranged in parallel with (hi/cj) over cj is also preceded by the stream splitter d ∈ SPj (i.e., XCd,kj ) 1). Since it is not arranged in series over cj, then qckjs ) 0. Such conditions are automatically applied by including the following restriction:

jth Cold Stream Exit Temperature from an Output Unit. An output unit over the cold stream cj can be either a heat exchanger or a stream mixer m ∈ MXj. Contrarily, a stream splitter will never be the last unit over a process stream. In any case, the exit temperature from an output unit over cj must reach the network exit temperature (TCjO). To impose such a condition on the mathematical formulation, the sign (g) rather than (e) is used in every inequality constraint controlling the temperature of cj from any potential output unit. Output Unit over cj Is a Heat Exchanger (hi/cj). The output exchanger over a jth cold stream presents two major features: (i) it has no successor at all over cj and (ii) it increases the temperature of cj up to the network exit temperature TCjO.

qckjd e M XCd,kj

d ∈ SPj, k ∈ HSj, j ∈ C

(20)

and

qckjs e M(1 -

∑

d ∈ SPj

XCd,kj)

k ∈ HSj, j ∈ C (21)

where M is an arbitrarily large number. Whenever the heat match (hk/cj) is performed right after a stream splitter d ∈ SPj, the inequality (21) drives the value of qckjs to zero. If its predecessor is not the splitter d, then the restriction (20) forces qckjd to be zero. Preceding Unit over cj Is a Stream Mixer m ∈ MXj. A heat exchanger (hi/cj) can be involved in a parallel arrangement over the cold stream cj only if it is preceded by a stream splitter d ∈ SPj. In any other case, it is arranged in series and its heat load is denoted by qcijs. Then,

tcij e tcmj + (qcijs/FCpj) + M(1 - XCm,ij) m ∈ MXj, i ∈ HSj, j ∈ C (22) jth Cold Stream Exit Temperature from an Input Stream Splitter d ∈ SPj. The exit temperature of the jth cold stream from an input splitter d ∈ SPj is given by the input temperature of cj to the network. Then,

tcdj e TCjI + M(1 - XCFdj)

d ∈ SPj, j ∈ C (23)

jth Cold Stream Exit Temperature from a Noninput Stream Splitter d ∈ SPj. A noninput stream splitter d ∈ SPj over the cold stream cj can only be preceded by either a heat exchanger or a noncompanion mixer m ∈ MXj, where (d,m) ∉ PAIR. Moreover, the output temperature tcdj from a splitter d ∈ SPj becomes established by the temperature at which the cold stream cj leaves the preceding unit. Preceding Unit Is the Heat Match (hi/cj).

tcdj e tcij + M(1 - XCij,d)

d ∈ SPj, i ∈ HSj, j ∈ C (24)

Preceding Unit Is a Stream Mixer m ∈ MXj.

tcdj e tcmj + M(1 - XCm,d) d ∈ SPj, m ∈ MXj, (d,m) ∉ PAIR, j ∈ C (25) jth Cold Stream Exit Temperature from a Stream Mixer m ∈ MXj. A stream mixer m ∈ MXjb can never be the input unit over any cold process stream j ∈ C. If it belongs to the network, it will be preceded by a set of

tcij g TCjO - M(

∑

k ∈ SHij

XCij,kj +

∑

n ∈ SPj

∑

d ∈ SPj

XCij,d +

i ∈ HSj, j ∈ C (27)

XCn,ij)

For an output unit, the term between parenthesis in the right-hand side is null and eq 27 will then hold. Let us now assume that the heater (s/cj) over cj, if required, is always placed both last and in series (i.e., at the hottest extreme). In such a case, the restriction (27) can be written as follows:

tcij g TCjO - (qcsjs/FCpj) - M(

∑

d ∈ SPj

XCij,d +

∑

n ∈ SPj

∑

XCij,kj +

k ∈ SHij#

XCn,ij)

i ∈ Hj, j ∈ C (27′)

where SHij# is derived from the set SHij by excluding the hot utilities. Contrarily to (27) the restriction (27′) regards the last heat exchanger different from a heater over cj as the pseudo output unit. For such a reason, it reduces the target temperature TCjO by an amount equal to the temperature increase through the heater over cj. In addition, the heater cannot precede any equipment item over a cold process stream. Then, the set of successors of (s/cj) is empty. Output Unit is a Mixer of b Split Streams. If the mixer m ∈ MXj is the output unit over the cold stream cj, then the exit temperature from the mixer m becomes established by the stream target temperature TCjO. Since a mixer m ∈ MXj can be succeeded by either a heat match (hk/cj) or a splitter d ∈ SPj, then

tcmj g TCjO - M(

∑

k ∈ HSj

XCm,kj +

∑

d ∈ SPj (d,m) ∉ PAIR

XCm,d)

m ∈ MXj, j ∈ C (28) If the heat match between cj and a heating utility is always placed last over cj, then the constraint (28) can be written as follows:


tcmj g TCjO - (qcsjs/FCpj) - M(

∑ d ∈ SP

∑ XCm,kj +

k∈

Hj

m ∈ MXj, j ∈ C (28′)

XCm,d)

thij - tckj g ∆Tmin - M(1 - XCkj,ij)

j

(d,m) ∉ PAIR

D. Overall Heat Balance for the Cold Stream cj at the Network. To strictly meet the prescribed outlet temperature for the cold stream cj from the network (TCjO), the overall heat balance over cj is to be satisfied. Then,

∑

i ∈ HSj

(qcijs

+

∑

d ∈ SPj

qcijd)

O

k ∈ PHij, i ∈ HSj, j ∈ C (36) Preceding Unit over cj Is the dth Splitter.

thij - tcdj g ∆Tmin - M(1 - XCd,ij) d ∈ SPj, i ∈ HSj, j ∈ C (37)

I

) FCpj(TCj - TCj )

Preceding Unit over cj Is the mth Mixer.

j ∈ C (29)

E. Topology-Constrained Least Utility Usage. When a MNU design for an upper network is sought, the minimum heating utility consumption at the network (QS,min) is to be enforced to only account for structures reaching the LUC target. Therefore, the problem formulation should also include eq 30 given by

qcsjd) ) QS,min ∑ (qcsjs + d ∑ ∈ SP

j∈C

At the Cold Extreme of an Intermediate Exchanger over cj. Preceding Unit over cj Is the Heat Exchanger hk/cj.

(30)

j

QS,min is assumed to be a model parameter at the time of designing an upper network. When the bottom subnetwork below any process pinch is synthesized, the least cooling utility usage QW,min rather than QS,min becomes a problem target. Then, eq 30 is to be substituted by eq A26 given in the Appendix. F. Minimum Temperature Approach Constraints. At the Hot Extreme of an Input Heat Exchanger (hi/cj) over hi.

THiI - tcij g ∆Tmin - M(1 - XHFij) j ∈ CWi, i ∈ H (31) where ∆Tmin is the minimum allowed temperature approach. Its value can change with the heat match at the design stage. At the Hot Extreme of an Intermediate Exchanger (hi/cj) over hi. Preceding Unit over hi Is the Heat Exchanger hi/cl.

thil - tcij g ∆Tmin - M(1 - XHil,ij) l ∈ PCij, j ∈ CWi, i ∈ H (32) Preceding Unit Is the dth Splitter over hi.

thij - tcmj g ∆Tmin - M(1 - XCm,ij) m ∈ MXj, i ∈ HSj, j ∈ C (38) G. Additional Design Target for an Upper Network. As already discussed, the maximum heat recovery at an upper network (QREC) has been chosen as an additional target for the design task. By doing so, the stream pseudo pinch temperatures at its bottom boundary are optimally located to further reduce the number of units. In addition to eq 30 enforcing the heating utility usage target, the upper network design problem formulation should also include eq 39 to only account for network structures reaching both the hot utility usage and the heat-recovery targets.

(qcijs + ∑ qcijd) g QREC ∑ j∑ ∈C d ∈ SP

i ∈ Hj

When the bottom subnetwork featuring a temperature range below any process pinch is being designed, the problem formulation includes neither eq 30 nor 39. Only the cooling utility usage target (A26) is to be considered. H. Objective Function. A different objective function is adopted depending on the type of problem being solved. Constrained Utility Usage Problem. To determine the least utility usage under the topology constraints specified by the designer, the hot/cold utility requirement at every pinch-defined network is first to be determined (Galli and Cerda´, 1998a). In each case, the set of process streams and their temperature subranges belonging to the network should only be considered. Since the minimum utility usage is the design target to be determined, then the problem objective function is given by

min[z )

thdi - tcij g ∆Tmin - M(1 - XHd,ij) d ∈ SPi, j ∈ CWi, i ∈ H (33) Preceding Unit Is the mth Mixer over hi.

thmi - tcij g ∆Tmin - M(1 - XHm,ij) m ∈ MXi, j ∈ CWi, i ∈ H (34) At the Cold Extreme of an Input Heat Exchanger (hi/cj) over cj.

thij - TCjI g ∆Tmin - M(1 - XCFij) i ∈ HSj, j ∈ C (35)

(39)

j

(qhiws + ∑ qhiwd) + ∑ i∈H d ∈ SP n

i

(qcsjs + ∑ qcsjd)] ∑ j∈C d ∈ SP n

(40)

j

subject to the set of constraints (4) -(14), (17)-(29), (31)-(38) (A1)-(A10), and (A13)-(A25). The subscript Cn stands for the set of cold streams belonging to the nth network. From the optimal heat surplus (or heat shortage) at every pinch-defined upper (bottom) network, the largest one of them is providing the additional utility usage caused by the specified topology constraints. Upper Network Heat-Recovery Problem. To find the set of pseudo-pinch temperatures defining the


optimal location of the bottom boundary of an upper network, the heat-recovery target QREC(n) is to be computed. Such an additional design target is determined by solving another MILP problem featuring the following objective function:

max[z )

(qcijs + ∑ qcijd)] ∑ j∑ ∈C d ∈ SP

i ∈ Hj

(41)

j

and a feasible region defined by the set of constraints (4)-(38) and (A1)-(A25), including the hot utility usage target (30). It is assumed that cooling utility is not available. Upper Network Design Problem. To synthesize a upper network design featuring a minimum number of units while reaching both the utility usage and the heat-recovery targets, the objective function for the upper network design problem is given by

min[z )

Ysj] ∑ ∑ Yij + j∑ ∈C

(42)

i ∈ Hj j ∈ C

and the feasible region is defined by the set of constraints (4)-(39) and (A1)-(A25), including the hot utility usage target (30) and the heat-recovery target (39). In addition, it is assumed that cooling utility is not available. Bottom Network Design Problem. The objective function for the design of the bottom network located below any process pinch is given by

min[z )

Yiw] ∑ ∑ Yij + i∑ ∈H

(43)

i ∈ Hj j ∈ C

The solution space is given by the set of constraints (4)(29), (31)-(38), and (A1)-(A26) that includes the cooling utility usage target (A26). Hot utility is not available. Proposed Sequential Synthesis Algorithm 1. Determine the pinch temperatures for the HENS problem to be tackled by using the heat-cascade procedure or the transshipment formulation in order to find the temperature range and the subset of process streams of every pinch-defined network. If the problem does not present a process pinch, just a single network is to be considered. 2. Find the constrained LUC target: Solve the constrained utility usage problem for each individual upper (or bottom) pinch-defined network to find the corresponding heat surplus (or heat shortage) caused by the specified topology conditions. The largest of such values will be providing the additional utility usage required to reach the target temperatures. The mathematical model is generated by a general modeling system like GAMS. 3. Find the network heat-recovery target: Let us assume that the process features a single pinch temperature. If the heat flow across the pinch is positive, determine the upper network heat-recovery target (QREC) by solving the upper network heat-recovery problem. If the pinch heat flow is zero or no pinch arises, go directly to step 4. 4. Find the upper network configuration: Solve the upper network design problem. If the pinch heat flow is null, the bottom boundary of the upper network is fixed at the related pinch. Otherwise, the heat-recovery target is enforced to determine the optimal set of stream

Table 1. Sizes of the MILP Formulations and Required CPU Times To Solve Examples 1-4 (Base Case) example network restrictions 1 2 3 4

UP LO UP LO UP LO

232 133 216 332 138 83 503

continuous binary CPU variables variables time (s) 64 35 62 83 40 29 118

65 37 65 95 37 15 136

5.3 0.3 18 2.2 1.6 0.3 97.6

pseudo pinch temperatures together with the upper network design. When the process exhibits multiple pinch temperatures, step 4 should be repetitively applied as many times as the number of pinch points. For a nonpinched problem, the whole network configuration has already been found after completing step 4. For a pinched problem, go to step 5. 5. Find the bottom network design: Solve the bottom network design problem to complete the network design task. Results and Discussion The proposed designer-driven HENS synthesis approach has been applied to the solution of four examples involving five to nine process streams. In every example, it was assumed that (i) a single hot and a single cold utility are available and (ii) heaters and/or coolers, if required, are placed last and in series. Different kinds of structural restrictions were imposed on the network design to show the way the reported method accounts for topology targets specified by the designer as well as the computational efficiency with which they are handled. The mathematical model was generated by using the GAMS modeling system and subsequently solved with the OSL solver (Brook et al., 1992). Size of the MILP formulation and the CPU time required to solve it on a SiliconGraphics Workstation for every example are given in Table 1. Example 1. Data for example 1 comprising three hot and two cold process streams are given in Figure 6a. If ∆Tmin ) 10 °F and no topology target is imposed on the network configuration, the optimal heat cascade does not include any pinch at all and consequently the process behaves as a net heat sink. The “unrestricted” utility requirements are given by: QS° ) 1040 Btu/h and QW° ) 0 Btu/h. Therefore, the problem formulation should only consider a single network. Four alternative design specifications were given to restrain the network solution space: a. Only cold stream C1 can be split (base case); b. The cold stream C1 can only be split into, at most, two stream branches. c. In addition to the network topology condition (a), the heat match H2/C1 can never be involved in a parallel arrangement over C1. d. In addition to (a), the heat match H2/C1 can never be performed in parallel with H1/C1 over C1. Since C1 is the process stream featuring the largest heat-capacity flow, the designer just allows the splitting of C1 with no bound on the number of split streams at the base case. As a result, the set of stream splitters for C1 (SPC1) will include splitters of two and three outlets because the problem involves three hot streams. For any other process stream, the set of feasible splitters SPi is assumed to be empty. Moreover, splitters and mixers are removed from the sets of allowed predecessors of and successors to every potential heat exchanger over any process stream different from C1. In this way,


Figure 6. Optimal network design for (a) example 1, base case; (b) example 1, case b; (c) example 1, case c; (d) example 1, case d.

the number of binary/continuous variables and linear constraints are both significantly diminished. Far from

increasing the model size, the proposed HENS synthesis approach makes use of the topology targets to eliminate variables and constraints related to forbidden design options. By applying the proposed utility usage problem formulation, it was found that the structural restrictions for the base case has no impact on the minimum utility requirements: QS* ) 1040 Btu/h and QW* ) 0. Since the process does not present any pinch, the determination of the heat-recovery target at the upper subnetwork (QREC) is not required. Afterward, the HENS mathematical model was solved to minimize the number of units required to reach both the utility and the topology targets yielding the network configuration depicted in Figure 6a. It was found in a CPU time of 5.3 s (see Table 1). It includes three heat exchangers and two heaters (i.e., five units overall). A three-outlet splitter allows the parallel heat exchanges of the cold stream C1 with (H1, H2, and H3). The topology conditions significantly cut down the number of linear constraints from 353 to 232 and the number of total/binary variables from 206/113 to 129/65 (see Table 1). If, at most, two split streams are permitted over C1 (case b), then a cooling utility also becomes necessary to reach the target temperature of H2 because a portion of the available process heat cannot be recovered. This indicates the need of a three-outlet splitter over C1 to maximize the process heat recovery. The set of splitters over C1 (SPC1) now comprises only two-outlet splitters. As long as the problem does not feature a process pinch, the utility usage problem formulation is still applied to the whole set of process streams at once. At the optimal solution, the overall heat removed at the coolers amounts to 22 Btu/h and the constrained utility usage rises by the same amount: QS* ) 1062 Btu/h and QW* ) 22 Btu/ h. The optimal network design for case b is shown in Figure 6b. Because it now requires both an additional heat exchanger (H3/C2) and a cooler over H2, the least number of heat-transfer units rises to seven (i.e., a single unit more than the one expected for a five-stream HENS problem requiring both heating and cooling utilities). A new type of topology target is considered in case c. The heat match (H2/C1) cannot be arranged in parallel with other exchangers over C1. In other words, it cannot be preceded by a stream splitter over C1, and therefore any type of splitter is to be removed from the set of predecessors of the match (H2/C1) over C1. Furthermore, the set of splitters for C1 (SPC1) will only include two-outlet splitters because C1 may exchange heat with at most two hot streams in parallel. The new design specification causes a decrease in both the number of binary variables and constraints. Moreover, the utility requirement rises by 67 Btu/h as long as a similar amount of process heat cannot be recovered. Therefore, QS* ) 1107 Btu/h and QW* ) 67 Btu/h. By next solving the proposed HENS formulation, the network design shown in Figure 6c was discovered. It includes a two-outlet splitter and six heat-transfer units (i.e., the number of units expected for this type of HENS problem). Finally, case d prohibits the parallel arrangement of the pair of matches H1/C1 and H2/C1 over C1. Therefore, H1 must be excluded from the set of hot streams HSH2,C1 with which, by definition, C1 can exchange heat in parallel with H2/C1. For the same reason, H2 is to be removed from the stream set HSH1,C1. The best network configuration accomplishing the topology target for case d is described in Figure 6d. It


Figure 7. (a) Optimal solution to the utility usage problem; (b) the best network structure, both for example 2.

comprises a pair of coolers over the hot streams H2 and H3, respectively, where a total of 20 kW is transferred to the cooling utility. Consequently, the number of heattransfer units rises to eight, two above the number expected for a problem involving five process streams and two utilities. Example 1 has permitted us to demonstrate the ability of the proposed HENS approach to handle any kind of topology target during the HEN design task. Example 2. This is a seven-stream problem comprising six hot streams and a single cold stream. Data for example 2 are all given in Figure 7. This example has been solved by Galli and Cerda´ (1998a,b) by first imposing a series configuration and then allowing stream splitting over the cold stream C1. To illustrate the reduction in the number of constraints achieved with the new approach, no structural constraint in addition to the isothermal mixing of split streams has been imposed on the network design. When HRAT ) 20 °F, the hot stream H4 determines the pinch temperature at 410/430 °F and the heat-cascade procedure provides the following least utility usages: QS° ) 8390 Btu/h and QW° ) 6618 Btu/h, respectively. The streams {H1,H2,H3,C1} all range partially or completely above the pinch. Then, the cold stream C1 can at most be split into three stream branches in the upper network (UP). By including splitters of two and three branches in the set of splitters of C1 (SPC1) and properly defining the sets of preceding and succeeding units for each potential heat match, the HENS problem mathematical formula-

tion for the upper network includes 133 restrictions and a total of 72 variables of which 37 are binary (see Table 1). Since a “fictitious” cooler/heater at the individual upper/lower network is to be included to determine the heat surplus/shortage caused by the structural constraints, the problem size usually shows a slight increase when the utility usage target is sought. A cooler on stream H1 with a heat load of 158 kW shown in Figure 7a is an indication that a similar amount of process heat cannot be recovered above the pinch, and consequently a further cooling utility requirement arises. Then, QS*(UP) ) 8548 Btu/h and QW*(UP) ) EU ) 158 Btu/h. Such a rise in the utility consumption is caused by the isothermal mixing condition of the split streams in the two-outlet mixer over C1. Below the pinch, the process is still a heat source (i.e., QS*(LO) ) EL ) 0). Therefore, the constrained utility targets are given by QS* ) QS° + max(EU,EL) ) 8548 Btu/h and QW* ) QW° + max(EU,EL) ) 6776 Btu/h. As the number of process streams becomes higher, the HENS model size for the lower network (LO) becomes significantly larger (i.e., 216 linear restrictions and 127/65 problem variables). For comparison, the number of constraints in the corresponding MILP formulation of Galli and Cerda´ (1998b) amounts to 945 (i.e., 4 times larger). Since a pinch heat flow equal to 158 kW arises, the optimal set of stream pseudo-pinch temperatures were determined by solving the upper network heat-recovery problem. In this way, a heat-recovery target equal to 5552 Btu/h was found. By sequentially solving the upper and lower network design problems, the optimal HEN structure depicted in Figure 7b has been encountered. It requires an overall CPU time of 18 s (see Table 1). The number of units included in the network is the one expected (i.e., (4 + 6) ) 10). A simple analysis of the problem data would have permitted us to specify some weak topology conditions for the lower network, thus reducing its problem size. Linnhoff and Hindmarsh (1983) demonstrated that the process streams involved in a heat match (hi/cj) starting at the pinch (i.e., an output exchanger over cj in the lower network) must satisfy the condition (FCp)i g (FCp)j. Otherwise, the cold stream cj should be split. H4/C1 is the only potential match fulfilling such a condition in the lower network. The other two matches H1/C1 and H3/C1 require first splitting the process stream C1. In such a case, the companion mixer will be the output unit over C1. Therefore, the heat match H4/C1 and a stream mixer are the only candidates for being the output unit over C1. To favor network structures with a low number of splitters, the designer may specify H4/C1 as the last unit over C1 at the lower network. By making use of this problem insight, the same optimal solution will be found by solving a smallersize HENS problem because the number of binary variables will decrease by 4 and, in addition, 12 fewer inequality constraints are needed. Surprisingly, the CPU time drops three times. Example 3. Example 3 is a seven-stream problem whose data are given in Figure 8. If HRAT ) 20 °C and no topology target is specified, then the problem features a pinch temperature at 160/180 °C being controlled by the supply temperature of H4 and a minimum utility usage given by QS° ) 4380 kW and QW° ) 19 010 kW. By a prior analysis of the problem data, two structural restrictions on the network configuration have been specified by the designer (base


Figure 9. Optimal network design for example 4.

Figure 8. Optimal network configuration for (a) example 3, base case; (b) example 3, case b.

case): (i) the heat match (H4/C3) is forbidden, since it is thermodynamically infeasible for the problem; (ii) stream splitters are only permitted over the cold streams C1 and C2 because of their larger heat-capacity flow rates. Since three hot streams are present in both the upper and lower networks, then two-outlet and three-outlet splitters were included in the sets of splitters SPC1 and SPC2. For the other process streams, SPi is an empty set. By considering the design specifications to properly define the set of preceding and succeeding units for each potential heat exchanger, the resulting utility usage problem formulation for the upper network (UP) involves 332 linear constraints and a total of 178 variables of which 95 are binary. In turn, the mathematical model for the lower network (LO) includes 138 linear restrictions and a total of 77 variables of which 37 are binary. To find the constrained utility usage target, the MILP models were solved to optimality in 22 and 4 s, respectively. Neither a cooler is required to remove the heat surplus at UP nor a heater is needed at LO to meet the heat shortage eventually caused by the design specifications. Therefore, the topology targets have no impact on the least utility usage. As a result, no heat flow crosses the pinch, and therefore the determination of the heat recovery target at the upper network is not necessary. The best network design under the specified structural constraints is shown in Figure 8a. It includes three stream splitters, two above and one below the pinch. In the upper network, there are a three-outlet splitter over C1, a two-outlet splitter over C2, and seven units, while another three-outlet splitter over C1 and six units are included in the lower network (i.e., three

over the expected number of units (6 + 4) ) 10). The required CPU time is reported in Table 1. To reduce the complexity of the network configuration, the designer has decided to bound the number of splitters to a single one at each network. The new topology restriction produces a slight increase in the utility usage target as long as a heat flow of 115 kW cannot be recovered at the upper network (EU ) 115 kW) but the lower one still behaves as a net heat sink (EL ) 0). At the same time, it brings about a moderate decrease in the size of the utility usage problem formulation especially at the upper network where all the process streams except H4 are present. Thus, the number of linear constraints drops from 332 to 273 and the number of variables from 178/95 to 130/69, both at the UP problem model. Since the lower network does not require an external heat source (EL ) 0), then the process heat recovered at the upper network decreases by 115 kW and consequently the overall heat flow removed in the coolers rises by the same amount. The optimal network design under the new structural specification is depicted in Figure 8b. It includes a single two-outlet splitter over C1 and seven units at the upper network, while the lower one comprises a single three-outlet splitter on C1 and seven units (i.e., a single unit more than that in the base case). Example 4. Example 4 is a nine-stream problem first reported by Hall et al. (1990) that is presented in Figure 9. If the exchangers are all made of titanium, such authors found that the optimal value for ∆Tmin is 27 °C. By adopting ∆Tmin ) 27 °C, the process pinch is located at the common supply temperature of the hot streams H3 and H5 (i.e., 135/108 °C). The process involves a set of four process streams {H4,C2,C3,C4} whose temperature ranges go across the pinch, while the other five {H1,H2,H3,H5,C1} completely run below the pinch. Therefore, there are a single hot and three cold process streams in the upper network and the whole set of process streams in the lower one. By an analysis of the problem data, the designer has decided to specify a set of “weak” design specifications: (i) heat matches between H2 with any of the cold streams C1, C2, and C4 are all prohibited, because of their nonoverlapping temperature ranges; (ii) the heat match between C3 with any of the hot streams {H3,H4,H5} cannot be chosen as an input unit over C3, since the supply temperature of C3 (TC3S ) 30 °C) is significantly lower than the target temperatures of hot streams H3 (TH3T ) 110 °C), H4 (TH4T ) 95 °C), and H5 (TH5T ) 105 °C).


(iii) any type of splitters over the streams {H1,H4,C4} is not permitted, accounting for their lower heatcapacity flow rates; (iv) splitters with, at most, two outlets are allowed over the remaining streams {H2,H3,C1,C2,C3}, based on the relative values of the heat-capacity flow rates for hot and cold process streams. The topology constraints are said to be weak since they almost certainly cause no increase in the utility usage target at all. At the same time, however, it produces a drastic reduction in the size of the problem formulation. Accounting for the topology restrictions, the utility usage problem for the upper network comprises 90 linear constraints and a total of 52 variables of which 19 are binary. As expected, the optimal solution found in 0.24 s indicates that the design specifications have no impact in the utility requirement (i.e., QS*(UP) ) 2300 kW and QW*(UP) ) 0 kW). At the lower network, the “weak” structural constraints imposed on the network produce a sizable decrease in the number of variables and restrictions. Thus, the number of linear constraints drops by a factor of 4 from 2172 to 542 while the number of binary variables diminishes by a factor of 5 from 755 to 151. Confirming our expectations, the optimal solution found in 39 s reveals that the process below the pinch temperature still behaves as a net heat sink. No finite heat flow goes across the pinch, and consequently the determination of the heat recovery-target at the upper network can be avoided. By sequentially solving the HENS problem formulation for the upper and lower networks, the optimal HEN design shown in Figure 9 has been encountered. It includes 13 heat-transfer units and a single two-outlet splitter over the hot stream H5. Since the upper and lower networks involve a number of streams, including utilities, equal to 5 and 10, respectively, then (4 + 9) ) 13 is the expected least number of units for the problem. Therefore, the “weak” topology restrictions neither have an impact on the number of units required to maximize the process heat recovery. Model sizes and required CPU times for the network design problems are all included in Table 1. Conclusions A MILP approach to the optimal synthesis of topologyconstrained heat-exchanger networks that regards stream splitters and mixers as additional problem units has been developed. It represents a new step toward a model-based HENS methodology that not only broadens the scope for user interaction but also effectively restrains the solution space by just including practical network structures meeting the desired topology features. Moreover, the designer can even take advantage of the problem insights reported in the literature to further reduce the number of binary variables and constraints without excluding the best options. The treatment of stream splitters and mixers at the level of problem units leads to a simpler mathematical model. Except for stream mixers (splitters), the other equipment items feature at most a single predecessor (successor) in a network structure. This greatly simplifies the expressions of the problem constraints related to parallel arrangements of heat exchangers over a process stream. In this way, a more attractive MILP framework has been derived, exhibiting a rather steadily increase in the model size with the number of process streams. Consequently, a good computational efficiency of the solution algorithm is observed, especially for large

problems. Compared with the formulation of Galli and Cerda´ (1998a,b) the new development also permits control of the location, type, and number of stream splitters over either a process stream or the whole network. Most importantly, the combinatorial increase in the number of constraints when parallel arrangements are permitted no longer arises. Five example problems involving from four to nine process streams were successfully solved in a reasonable CPU time. Generalization of the problem formulation to also consider networks involving nonisothermal mixers and/ or a sequence of heat matches over a split stream is currently underway and the results will be included in a future paper. Acknowledgment We are grateful to acknowledge financial support from “Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET)” and “Universidad Nacional del Litoral”. Nomenclature Subscripts i,k ) a hot stream j,l ) a cold stream S ) a hot utility stream W ) a cold utility stream d,n ) a stream splitter m ) a stream mixer Superscripts i ) a hot stream j ) a cold stream s ) a series arrangement of exchangers d ) a parallel arrangement of exchangers right after the dth splitter b ) the number of outlets from a stream splitter n ) the nth network Sets of Streams and Heat Matches H ) set of hot process streams C ) set of cold process streams S ) set of available heating utilities W ) set of available cooling utilities HS ) the whole set of hot streams ) {i|i ∈ H ∪ S} CW ) the whole set of cold streams ) {j|j ∈ C ∪ W} FHM ) set of forbidden matches ) {(i/j)|i ∈ HS, j ∈ CW, (i/j) is a forbidden heat match} HSj ) set of hot streams that can exchange heat with cj ) {i|i ∈ HS and (i/j) ∉ FHM} HSij ) set of hot streams that can be involved in parallel heat matches with (hi/cj) over cj ) {k ∈ HSj|(k/j) ∉ FHM can be arranged in parallel with (i/j) ∉ FHM, k * i} ⊂ HSj CWi ) set of cold streams that can exchange heat with hi ) {j|j ∈ CW and (i/j) ∉ FHM} CWij ) set of cold streams that can exchange heat with hi parallel with (hi/cj) ) {l ∈ CWi|(i/l) ∉ FHM can be arranged in parallel with (i/j), l * j} ⊂ CWi Sets of Stream Splitters and Mixers SP ) {the whole set of feasible splitters} MX ) {the whole set of feasible isothermal mixers} SPi ) {d ∈ SP|d is a feasible splitter over the process stream i} MXi ) {m ∈ MX|m is a feasible isothermal mixer over the stream i} SPib ) {d ∈ SPi|d is a b outlet splitter over the stream i} MXib ) {m ∈ MXi|m mixes b branches of the stream i into a single one}

2494 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 PAIR ) {(d,m)|m ∈ MXib and d ∈ SPib, (d,m) belongs to the same stream splitter/mixer set}

qhijs ) heat duty of the exchanger hi/cj when performed in series over hi

Sets of Predecessors and Successors for a Heat Exchanger hi/cj

Appendix

PCij ) set of cold streams that can exchange heat with hi just before hi/cj ) {l ∈ CWi|(i/l ) ∉ FHM is a feasible predecessor of (i/j) ∉ FHM over hi} ⊂ CWi PHij ) set of hot streams that can exchange heat with cj just before hi/cj ) {k ∈ HSj|(k/j) ∉ FHM is a feasible predecessor of (i/j) ∉ FHM over cj} ⊂ HSj SCij ) set of cold streams that can exchange heat with hi right after hi/cj ) {l ∈ CWi|(i/j) ∉ FHM is a feasible predecessor of (i/l) ∉ FHM over hi} ⊂ CWi SCij# ) {l ∈ Ci|(i/j) ∉ FHM is a feasible predecessor of the match (i/l) ∉ FHM over hi} ⊂ Ci SHij ) set of hot streams that can exchange heat with cj right after hi/cj ) {k ∈ HSj|(i/j) ∉ FHM is a feasible predecessor of (k/j) ∉ FHM over cj} ⊂ HSj SHij# ) {k ∈ Hj|(i/j) ∉ FHM is a feasible predecessor of the match (k/j) ∉ FHM over cj} ⊂ Hj

Network Topology Restrictions over Hot Streams. The mathematical constraints defining the feasible arrangements of units over hot process streams at a particular network are next introduced.

Problem Parameters (FCp)i ) ith-process stream heat-capacity flow rate THiS ) supply temperature of the hot stream i ∈ H THiT ) target temperature of the hot stream i ∈ H TCjS ) supply temperature of the cold stream j ∈ C TCjT ) target temperature of the cold stream j ∈ C ∆Tmin or HRAT ) minimum allowed temperature approach

qhijs +

qhijd e UijYij ∑ d ∈ SP

j ∈ CWi, i ∈ HS (A1)

i

∑

j∈

XHFij +

qhijd e M YHdi

d ∈ SPi, i ∈ H

∑

l ∈ PCij

XHil,ij +

∑

m ∈ MXi

∑

XHFdi +

∑

XHd,ij +

d ∈ SPi

j ∈ CWi, i ∈ H (A3)

XHm,ij ) Yij

XHij,d +

j ∈ CWi

∑

m ∈ MXi (d,m) ∉ PAIR

XHm,d ) YHdi d ∈ SPi, i ∈ H (A4)

∑

l ∈ SCij

XHij,il +

∑

d ∈ SPi

XHij,d +

∑

n ∈ SPi

XHn,ij e Yij j ∈ CWi, i ∈ H (A5)

Sets of Problem Decision Variables Yij, YCdj, YHdi ) binary variables denoting the existence of the heat match hi/cj and the dth splitter over the cold stream cj or the hot stream hi, respectively XCFij, XCFdj ) binary variables denoting that either the heat match hi/cj or the dth splitter is first placed over the jth cold stream, respectively XHFij, XHFdi ) binary variables denoting that either the heat match hi/cj or the dth splitter is first placed over the ith hot stream, respectively XCkj,ij, XCkj,d, XCkj,m ) binary variables denoting that hk/cj precedes either the heat match hi/cj, the dth splitter, or the mth mixer over the jth cold stream XHil,ij, XHil,d, XHil,m ) binary variables denoting that hi/cl precedes either the heat match hi/cj, the dth splitter, or the mth mixer over the ith hot stream Sets of Nonnegative Continuous Variables tcij, tcdj, tcmj ) jth cold stream exit temperature from either the exchanger hi/cj, the dth splitter, or the mth mixer over cj, respectively thij, thdi, thmi ) ith hot stream exit temperature from either the exchanger hi/cj, the dth splitter, or the mth mixer over hi, respectively THiO(n) ) ith hot process stream exit temperature from network n TCjI(n) ) jth cold process stream input temperature to network n qcijd ) heat duty of the exchanger hi/cj when performed in parallel right after a splitter d ∈ SPj over the cold stream cj qcijs ) heat duty of the exchanger hi/cj when performed in series over cj qhijd ) heat duty of the exchanger hi/cj when performed in parallel right after a splitter d ∈ SPi over hi

(A2)

CWi

∑

XHd,ij + bYHdi

j ∈ CWi

d ∈ SPib, b ) 2,3, ..., NBi, i ∈ H (A6)

∑

XHm,ij +

j ∈ CWi

∑

n ∈ SPi n*d

XHm,n e YHdi

m ∈ MXi, (d,m) ∈ PAIR, i ∈ H (A7)

∑

XHFij +

j ∈ CWi

∑

∑

XHFdi ) 1

i ∈ H (A8)

d ∈ SPi

i∈H

YHdi e NDi

(A9)

d ∈ SPi

XHd,il + XHd,ij e 1 d ∈ SPi, l ∉ CWij, j ∈ CWi, i ∈ H (A10) Heat Balances around Every Unit in the Network.

THiO e thij e THiI

(A11)

max(Tpinch(n+1) + ∆Tmin, THiT) e THiO e min(Tpinch(n) + Tmin, THiS)

i ∈ H (A12)

thij g THiI - (qhijs/FCpi) - M(1 - XHFij) j ∈ CWi, i ∈ H (A13)


thij g thil - (qhijs/FCpi) - M(1 - XHil,ij) l ∈ PCij, j ∈ CWi, i ∈ H (A14) thij g thdi -

(qhijd

+

∑ qhil )/FCpi - M(1 - XHd,ij) d

Overall Heat Balance for the Hot Stream hi at the Network.

∑

j∈

CWi

(qhijs +

qhijd) ) FCpi(THiI - THiO) ∑ d ∈ SP i

i ∈ H (A25)

l ∈ Cji

d ∈ SPi, j ∈ CWi, i ∈ H (A15) d ∈ SPi, l ∈ CWi, i ∈ H

qhild e M XHd,il s

qhil e M(1 -

∑ XHd,il) d ∈ SP

(A16)

i

l ∈ CW , i ∈ H (A17)

Topology-Constrained Least Cooling Utility Usage.

(qhiws + ∑ qhiwd) ) QW,min ∑ i∈H d ∈ SP

(A26)

i

i

Literature Cited

thij g thmi - (qhijs/FCpi) - M(1 - XHm,ij) m ∈ MXi, j ∈ CWi, i ∈ H (A18) thdi g THiI - M(1 - XHFdi)

d ∈ SPi, i ∈ H (A19)

thdi g thij - M(1 - XHij,d) d ∈ SPi, j ∈ CWi, i ∈ H (A20) thdi g thmi - M(1 - XHm,d) d ∈ SPi, m ∈ MXi, (d,m) ∉ PAIR, i ∈ H (A21) thmi g thij - M(1 - XHd,ij) d ∈ SPi, (d,m) ∈ PAIR, j ∈ CWi, i ∈ H (A22)

∑

thij e THiO + M(

l ∈ SCij

XHij,lj +

∑ XHn,ij) n ∈ SP

∑

XHij,d +

d ∈ SPi

j ∈ CWi, i ∈ H (A23)

i

∑

thij e THiO + (qhiws/FCpi) + M( XHn,ij) ∑ XHij,d + n ∑ d ∈ SP ∈ SP i

XHij,il +

l ∈ SCij#

j ∈ Ci, i ∈ H (A23′)

i

thmi e THiO + M(

∑

XHm,il +

l ∈ CWi

∑

d ∈ SPi (d,m) ∉ PAIR

XHm,d)

m ∈ MXi, i ∈ H (A24) thmi e THiO + (qhiws/FCpi) + M(

∑

d ∈ SPi (d,m) ∉ PAIR

∑ XHm,il +

l ∈ Ci

XHm,d)

m ∈ MXi, i ∈ H (A24′)

Brook A,; Kendrick, D.; Meeraus, A. GAMS: A User’s Guide, Release 2.25; Boyd and Fraser Publishing Co., 1992. Ciric, A. R.; Floudas, C. A. Heat Exchanger Network Synthesis Without Decomposition. Comput. Chem. Eng. 1991, 15, 385. Daichendt, M. M.; Grossmann, I. E. Preliminary Screening Procedure for the MINLP Synthesis of Process SystemssII. Heat Exchanger Networks. Comput. Chem. Eng. 1994, 18, 679. Floudas, C. A.; Ciric, A. R. Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Comput. Chem. Eng. 1989, 13, 1133. Galli, M. R.; Cerda´, J. Synthesis of Structural-Constrained Heat Exchanger Networks. I-Serial Networks. Comput. Chem. Eng. 1998a, in press. Galli, M. R.; Cerda´, J. Synthesis of Structural-Constrained Heat Exchanger Networks. II-Split Networks. Comput. Chem. Eng. 1998b, in press. Gundersen, T.; Grossmann, I. E. Improved Optimization Strategies for Automated Heat Exchanger Network Synthesis through Physical Insights. Comput. Chem. Eng. 1990, 14, 925. Gundersen, T.; Duvold, S.; Hashemi-Ahmady, A. An Extended Vertical MILP Model for Heat Exchanger Network Synthesis. Comput. Chem. Eng. 1996, 20S, S97. Gundersen, T.; Traedal, P.; Hashemi-Ahmady, A. Improved Sequential Strategy for the Synthesis of Near-Optimal Heat Exchanger Networks. Comput. Chem. Eng. 1997, 21S, S59. Hall, S. G.; Ahmad, S.; Smith, R. Capital Cost Targets for Heat Exchanger Networks Comprising Mixed Materials of Construction, Pressure Ratings and Exchanger Types. Comput. Chem. Eng. 1990, 14, 319. Linnhoff, B. Pinch AnalysissA State-of-the-Art Overview. Trans. IChemE 1993, 71 (part A), 503. Linnhoff, B.; Hindmarsh, E. C. The Pinch Design Method for Heat Exchanger Networks. Chem. Eng. Sci. 1983, 38, 745. Trivedi, K. K.; O’Neill, B. K.; Roach, J. R.; Wood, R. M. A New Dual-Temperature Design Method for the Synthesis of Heat Exchanger Networks. Comput. Chem. Eng. 1989, 11, 667. Yee, T. F.; Grossmann, I. E. Simultaneous Optimization Models for Heat IntegrationsII. Heat Exchanger Network Synthesis. Comput. Chem. Eng. 1990, 14, 1165.

Received for review August 18, 1997 Revised manuscript received March 16, 1998 Accepted March 17, 1998 IE9705564

A Customized MILP Approach to the Synthesis of Heat Recovery

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