- Email: [email protected]

Computers Chem. Engng Vol. 22, No. 7—8, pp. 1017—1035, 1998 ( 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain S0098-1354(98)00001-5 0098—1354/98 $19.00#0.00

Synthesis of structural-constrained heat exchanger networks –II Split Networks Marı´ a Rosa Galli and Jaime Cerda´* Instituto de Desarrollo Tecnolo´gico para la Industria Quı´ mica, Universidad Nacional del Litoral, CONICET, Gu¨emes 3450, 3000 Santa Fe, Argentina (Received 23 April 1996; revised 27 May 1997) Abstract To tackle constrained HENS problems featuring split networks as the best design options, the MILP framework introduced in Part I has been generalized in such a way that the new formulation remains linear. Parallel arrangements were considered by allowing multiple preceding and succeeding units for each potential heat match. The proposed approach allows the design engineer to specify the feasible predecessors and successors for every unit as well as the heat exchangers with which it can be arranged in parallel. Such topology constraints on the network design are considered from the beginning. To this purpose, new structural conditions have been incorporated into the MILP problem formulation so as to generate the larger solution space resulting from using stream splitting. Moreover, additional sets of restrictions are also included to define the common exit temperature of a hot/cold process streams coming from a set of parallel units. Similar to Yee and Grossmann (1990), the mathematical formulation assumes (a) isothermal mixers, (b) a single exchanger over a split stream and (c) no stream by-pass. The generalized MILP framework is the basis of a targetting algorithmic method that accounts for all topology conditions from the start to sequentially determine (a) the constrained utility usage target; (b) the stream pseudo-pinch temperatures to decompose the problem into smaller independent subproblems and (c) the network designs at the level of structure one-by-one going from the top to the bottom network. Four example problems involving five to seven process streams were succesfully solved in a reasonable CPU time. ( 1998 Elsevier Science Ltd. All rights reserved Keywords: heat exchanger network; sequential synthesis; topology constraints; MILP formulation; split networks 1. Introduction In Part I, a MILP sequential approach for the optimal synthesis of series heat exchanger networks (HEN) under additional topology conditions has been presented. Very often, however, the series structure imposes a very costly barrier to the heat recovery process. This usually happens when different populations and/or inadequate heat capacity flowrates of hot and cold process streams around the pinch arise (Linnhoff et al., 1982; Linnhoff and Hindmarsh, 1983). In such cases, the split of some process streams into two or more stream branches at the pinch becomes necessary to execute some otherwise infeasible heat matches. In this way, significant reductions in both the utility requirement and the HEN overall cost are achieved. However, the impact on the utility savings is generally less profound when structural restrictions, like a low bound on the number of split streams and

* Corresponding author. E-mail: [email protected]

splitters, are specified by the designer to simplify the network topology. Nonetheless, a finite heat flow goes across the pinch even if the maximum energy recovery (MER) target under the topology constraints has been achieved. Pseudo-pinch HENS problems were originally formulated to cut down the required number of units by allowing heat to flow across the pinch. To deal with those problems, the so-called dual- temperature approach method was presented (Colbert, 1982; Trivedi et al., 1989; Ciric and Floudas, 1990). Such a method requires the specification of two temperature approaches: the heat recovery approach temperature (HRAT) to determine the least utility usage levels and the exchanger minimum approach temperature (EMAT) to define the minimum allowed temperature difference in the heat exchangers, where EMAT4 HRAT. On the contrary, this work is concerned with the solution of pseudo-pinch HENS problems originated by network topology constraints specified by the designer causing a finite pinch flow even if EMAT"HRAT and the MER condition have both

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been achieved. Therefore, a major difference with regard to prior contributions is the use of the pseudopinch notion because of the specification of structural restrictions rather than (or in addition to) the adoption of a minimum approach temperature EMAT lower than HRAT during the design phase. To tackle such pseudo-pinch HENS problems featuring split network designs as the best options, the MILP basic framework introduced in Part I will be generalized in such a way that the new formulation still be linear. Similar to Yee and Grossmann (1990), the non-series MILP formulation to be developed will assume (a) isothermal mixing of stream branches, (b) no stream branch flowing through two or more exchangers in series and (c) no stream by-passes. Assumptions (a) and (b) can indeed be avoided but the problem formulation will no longer be linear. As before, the network structure will be described in terms of heat exchangers, heaters and coolers. To bound the number of problem variables, splitters and mixers are not explicitly regarded in this work as additional units in the network. So, they cannot preceed or succeed to a group of two or more parallel exchangers. Instead, the new MILP framework will permit an exchanger (i) to have several predecesors or successors over the process streams where it is located but less than the maximum number of parallel units allowed by the designer and (ii) to be the preceding or succeeding equipment to two or more units in the network. To handle multiple predecessors and/or successors for each potential heat match, it becomes necessary not only to modify the linear restrictions controlling the network topology introduced in Part I but also to add further constraints to the problem formulation. For instance, the exit temperature restrictions defining the temperature at which a hot or cold stream leaves a particular heat exchanger (h /c ) will not only i j involve the heat load of (h /c ) but also the duties i j of the units arranged in parallel with (h /c ) over i j h and/or c . i j Such an improved description of the network structure constitutes a major step towards a general mathematical representation, no longer linear, also embedding HEN configurations comprising non-isothermal mixers, several units rather than a single one over a stream branch and stream by-passes. Development of a MILP mathematical formulation accounting for (i) multiple shells in series to avoid temperature cross-overs and (ii) splitters and mixers as additional units, is currently under way and the results will be included in a future paper. By doing that, the maximum number and types of splitters and mixers over any particular process stream can be explicitly prescribed by the user. The new MILP basic framework presented in Part II is the cornerstone of a targetting algorithmic method for the synthesis of split network designs that accounts for the topology constraints specified by the designer from the beginning. In contrast to previous contributions featuring similar properties (Yee and

Grossmann, 1990; Daichendt and Grossmann, 1994; Ciric and Floudas, 1991), the proposed approach permits one to have some control on the neighboring units of every potential heat match. Moreover, it is a sequential one rather than a simultaneous HENS technique even for the determination of the topologyconstrained utility usage target. By proceeding in this way, the method can make use of (i) simple expressions derived in Part I to compute the constrained utility target from the least utility requirements at each ‘‘isolated’’ pinch-defined network and (ii) the notion of pseudo-pinch point to decompose the network design problem into two or more smaller independent subproblems and thereafter solve them one-by-one going from the top to the bottom network. By using a decomposition approach, a significant reduction in the problem size is generally achieved and some important design data like the pseudo pinch points become available to the process engineer. As already mentioned, pseudo-pinch temperatures rather than the process pinch are to be considered since the specified structural constraints usually prevent from reaching a null pinch heat flow at maximum energy recovery (MER). However, there are generally several alternative sets of feasible pseudo-pinch points each one associated to a different MER network design. Therefore, a tailor-made version of the proposed MILP basic framework has been developed to establish the optimal set of pseudo-pinch temperatures associated to the network design with the fewest units. Based on such pseudo-pinch points, slightly different MILP formulations are subsequently solved to find the optimal design of every network from the top to the bottom one. Two of the major achievements of the proposed sequential method is the control gained by the designer on the network topology and the information provided by the method to alert him about the consequences of the specified design conditions. 2. The problem definition The HENS problem being addressed in Part II of this two-part paper can be stated as follows: Given are (1) a set of hot process streams i3H, their supply (¹HS ) and target (¹HT ) temperatures, and their heat i i capacity flowrates (FCp) ; (2) a set of cold process i streams j3C, their supply (¹CS ) and target (¹CT ) j j temperatures, and their heat capacity flowrates (FCp) ; (3) a set of hot utilities s3S and a set of cold j utilities w3¼ and their corresponding temperatures and (4) a set of design specifications. Such specifications pursue the following purposes: f Disallow the execution of some heat matches or

limit their heat loads. f Prohibit certain network configurations by provid-

ing the sets of feasible predecessors and successors for each potential heat match and/or fixing the desired input or output unit on a particular stream.

Synthesis of heat exchanger networks f Limit the feasible temperature ranges for one or

more heat exchangers. f Restrain the number and types of units with which a potential heat match can be arranged in parallel over a particular hot or cold process stream. Such a number may change from one to another exchanger over a process stream. The problem goals are to determine: (a) the constrained least utility consumption (LUC) required by the set of process streams to reach the desired target temperatures while accounting for all design specifications and (b) the HEN configuration that satisfies the additional restrictions imposed on the network structure by the designer and simultaneously reaches the constrained LUC target for the problem through the least number of units. To preserve the linearity of the problem mathematical formulation, the following assumptions have been made: (1) countercurrent heat exchangers; (2) constant heat capacity flowrates; (3) forbidden hot—hot or cold—cold stream matches; (4) isothermal mixing of stream branches, (5) no stream branch flowing through two or more exchangers in series and (6) no stream by-passes. 3. The supporting stream sets To state the HENS problem formulation, basic sets of streams and heat matches were already defined in Part I. Based on those basic sets, we have also introduced a group of new ones to generate all the possible preceding or succeeding units for each allowed heat match. In fact, the set of feasible predecessors (successors) for a heat exchanger (h /c ) over the stream i j h (c ) has been given in terms of the set of cold (hot) i j streams that can exchange heat with h (c ) just before i j (after) the match (h /c ). Such supporting stream sets i j whose members are to be specified by the designer may include cold (hot) utilities in addition to process streams. In the general case, cooling water and heating steam will both be treated as further process streams (Case 1). When heaters (coolers) are restricted to be placed last, i.e. at the hottest (coldest) extreme of a cold (hot) process stream, different supporting stream sets are required to generate the restricted families of predecessors and successors for each allowed heat exchanger. A further assumption is that the full cold (hot) process stream will flow through the heater (cooler), i.e. the auxiliary equipment is arranged in series. In other words, the heater (cooler) cannot be succeeded by another equipment and/or placed in parallel with other units (Case 2). New sets of process streams (CiLC¼i,HjLHSj) and stream—stream heat matches (SSMLHM) are to be introduced before defining the supporting stream sets for Case 2: MPC*, PH*, SC* and SH*N. They are: ij ij ij ij SSM"set of allowed stream—stream matches "M(i/j)3HM D i3H, j3CN

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Ci"set of cold process streams that can exchange heat with i3H"M j3C D(i/j)3SSMN Hj"set of hot process streams that can exchange heat with j3C"Mi3H D (i/j)3SSMN C "M j3C D ¹CT'¹ N U j 1*/#) H "Mi3H D ¹HS'¹ #*¹ N U i 1*/#) .*/ C "M j3C D ¹CS(¹ N L j 1*/#) Cd"M j3C D ¹CS(¹C N L j PP,j H "Mi3H D ¹HT(¹ #*¹ N L i 1*/#) .*/ d H "Mi3H D¹HT(¹H N L i PP,i PC*"Mm3Ci D (i/m)3SSM is a feasible predecessor of ij (i/j)3HM over i3HN PH*"Mk3Hj D (k/j)3SSM is a feasible predecessor of ij (i/j)3HM over j3CN SC*"Mm3Ci D (i/m)3HM is a feasible successor to ij (i/j)3SSM over i3HN SH*"Mk3Hj D (k/j)3HM is a feasible successor to ij (i/j)3SSM over j3CN where the suscript PP in the definitions of Cd and L Hd refer to the pseudo-pinch point. In this paper, we L write the problem constraints in terms of the supporting stream sets PC , PH , SC and SH introduced ij ij ij ij in Part I to get a generalized problem formulation where the auxiliary equipment at each process stream can be arranged anywhere either in series or in parallel with one or more heat exchangers (Case 1). To restrain the problem modelling to Case 2, such supporting stream sets are to be replaced by the corresponding ones with an asterisk (PC* , PH* , SC* and ij ij ij SH*) as shown in the next Section. For Case 2, the ij number of problem variables decreases as long as the sets of binary variables XC’s and XH’s involving heaters or coolers as preceding units are no longer needed. 4. The problem decomposition In the general case, the same match (h /c ) can be i j performed at a heat exchanger network (HEN) as many times as the number of networks comprised by the HEN. By definition, every network becomes defined by a process or a utility pinch temperature, each one involving at most a single hot or cold utility. If (NS#N¼) is the number of available utilities, then the network will potentially include a number of networks equal to (NS#N¼). But we do not know a priori the subset of utilities to be effectively used and the resulting number of networks at the optimal design. As a result, though it provides the optimal solution at once the choice of a simultaneous approach (Yee and Grossmann, 1990; Ciric and Floudas, 1991) requires to account, in the worst case, for a potential number of units per heat exchange equal to (NS#N¼). This would force us to handle a much

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larger set of binary variables of the type XC l and ij XH l, for i3HSj, j3C and l"1,2, 2, (NS#N¼). ij If instead a problem decomposition scheme is applied, a sequence of subproblems of smaller size is to be solved. In this paper, the latter approach has been adopted. Let us now assume that the HEN has been partitioned at the process/utility pinches to give rise a set of networks l3PN. The knowledge of the process pinch and the utility temperatures will permit us to establish each network temperature range. In this way, the set of hot and cold process streams belonging to a particular one can be identified by simply picking up those members of HS and C¼ whose temperature ranges include at least a portion of the network range. But the structural constraints imposed on the network design usually bring about a non-zero heat flow across the pinch. Therefore, such restrictions introduce energy interactions between adjacent pinch-defined networks. By analyzing every network on an individual basis, it will be found that not only a hot utility but also a cold utility (or vice versa) may be required. For instance, a ‘‘fictitious’’ cold utility need at the ‘‘isolated’’ upper network (N ) is an indication U that the topology constraints have increased the hot utility consumption above the pinch and, consequently, the non-allocated process heat flow must be removed from N . This can be done by either transferU ring energy across the pinch to the lower network or using a hypothetical heat sink to eliminate it. In turn, the need of a ‘‘fictitious’’ hot utility below the pinch reveals that the structural restrictions make the ‘‘isolated’’ lower network (N ) also a heat sink. Such extra L heating requirements at N can often be completely L satisfied by a proper allocation of the heat surplus to be removed from the upper network. Otherwise, the utility usage target for the problem will rise even further. Expressions for the evaluation of the constrained LUC target in terms of the minimum utility requirements at the individual pinch-defined networks have already been presented in Part I. To get rid of the energy interactions between neighboring pinch-defined networks at the design stage, the upper one is allowed to move forward beyond the unique boundary temperature (the pinch point). This implies the allocation of the pinch heat flow in such a way that the heat duties of some boundary exchangers in the upper network becomes greater and their temperature ranges run across the pinch. Further increase in the size of such units is sometimes obtained at the expense of other boundary exchangers whose heat loads become smaller and simultaneously their temperature ranges stay above the pinch. In this way, a new set of boundary temperatures rather than a single one is defined since the value of each one usually changes with the process stream. They are called the stream pseudo-pinch points. In this paper, we initially assume that the set of process streams determining the process pinch also

defines the pseudo-pinch temperatures, i.e. the stream set PPS"H XC . In other words, those process U U streams partially or completely running above the pinch are the only members of the set PPS. If so, the effect of a non-zero pinch heat flow is the net increase of the overall heat duties exhibited by the boundary units in the upper network. Assuming that the pseudo-pinch points are known, the best topologyconstrained HEN can be synthesized by sequentially finding the optimal design of every network starting from the top and proceeding towards the bottom one. Tailor-made versions of the MILP basic framework for split HENS problems have been developed for the determination of: (1) the least utility usages at every ‘‘isolated’’ pinchdefined network l3PN under the specified topology conditions (problem P1); (2) the optimal heat recovery at the extended upper network (problem P2.1); (3) the set of pseudo-pinch temperatures and the best structural design for every pseudo-pinch-defined network assuming that the least hot (cold) utility requirement is only available for use (problems P2.2 and P2.3). 4.1. Updating the stream set PPS defining the pseudo-pinch points After finding the minimum utility usage at every network l where l3PN, the constrained LUC target for the problem and the pinch heat flow can both be determined by simply applying the expressions already presented in Part I. Sometimes, the optimal solution to problem P1 for the lower network N L indicates the need to incorporate one or more cold streams [email protected] featuring ¹CT4¹ to the set of L j 1*/#) process streams PPS. If so, the set PPS determining both the set of pseudo-pinch points and the design of the extended upper network is modified as follows: PPS"H XCd, where Cd"C [email protected] and [email protected] LC . U U U U L L L Cd is the set of process streams defining the stream U pseudo-pinch temperatures. The above situation will arise at Cases C and D (with E (E ) if a portion of U L the pinch heat flow must be allocated to cold streams entirely running below the pinch at MER condition. If so, a ‘‘fictitious’’ heater (S/c ) on the cold stream [email protected] j L appears at the optimal solution to P1 for the network N , thus indicating the need of an external heating L source. The inlet temperature of a cold stream [email protected] L to the ‘‘fictitious’’ heater will establish its highest inlet temperature to the extended upper network (¹Cmx ). PP,j The pseudo pinch point at such a stream c will be j lower or equal to ¹Cmx . PP,j As already mentioned, the proposed formulation only accounts for network structures where non- isothermal mixers, multiple units on a split stream and stream bypasses never arise. Such simplifying assumptions permit to eliminate the stream flowrate variables from the mathematical model to keep the linearity of the problem constraint set. Though linear structural

Synthesis of heat exchanger networks

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constraints can be written to also consider the possibility of using non-isothermal mixers and several units per branch, the exit temperature restrictions associated to them are always non-linear. For instance, from a structural viewpoint non-isothermal and isothermal mixers are handled in a similar way but the former ones require to explicitly include the split stream flowrates in the inequalities restricting the values of the stream outlet temperatures from the parallel exchangers. 5. The MILP basic framework The MILP basic framework mostly provides the constraint set, including the design specifications, that defines the solution space within which the search for both (1) the utility usage target and (2) the optimal network design are to be made. The MILP framework is posed at the level of network but the decomposition scheme changes with the purpose of the problem. For the constrained utility usage problem P1, the network partitioning is made at the process/utility pinch temperature. In other words, pinch-defined networks are to be considered. By doing that, we guarantee that the number of potential units considered by the model will never be smaller than the lowest one required to minimize the constrained utility usage. Then, an increase in the least utility consumption because of an insufficient number of units will never happen. On the contrary, the HEN is decomposed at the pseudopinch points to seek the best network design. Since the pseudo-pinch temperatures are initially unknown, the bottom (top) boundary temperatures of the upper (lower) network now stand for new problem variables rather than fixed data. (I) The network structural constraints In order to identify the relative position of each heat exchanger (h /c ) over both the hot stream h and i j i the cold stream c , a set of logical constraints is to be j imposed. To make this presentation more readable, only the structural constraints over cold streams will be developed. Analogous constraints over hot streams are included in Appendix A. (I.1) The existence of a heat exchanger (h /c ) in the i j network: q 4º ½ , i3HSj, j3CW (1) ij ij ij where º is an upper bound on the value of q . ij ij (I.2) Lower bound on the number of predecessors of a non-input exchanger (h /c ). If the exchanger (h /c ) is i j i j really performed at the network, it must either have at least a preceding unit or be an input exchanger over the streams where it is located (see Fig. 1). In other words, the exchanger may be the first one over c and j a non-input unit on h or vice versa. Alternatively, it i can be placed first on both streams like the unit (H1/C1) in the network configuration shown in Fig. 1a, i.e. XCF "XHF "1. In Fig. 1b, the [email protected] [email protected] changer (H1/C1) is the first unit over the hot stream H1

Fig. 1. Coordinate variables describing the topology of a split HEN. (a) parallel input units over a cold stream; (b) parallel non-input units over a cold stream.

(XHF "1) and a non-input unit over the cold [email protected] stream C1 (XCF "0, XC "1). [email protected] [email protected],[email protected] ever the exchange (h /c ) is performed (½ "1) in i j ij a non-input unit over c , a lower bound on the numj ber of predecessors of (h /c ) over c equal one must be i j j imposed. For an input exchanger (XCF "1), the ij number of predecessors is null. Then, + XC 5½ !XCF , i3HSj, j3C, (2) kj, ij ij ij k|PHij where the summation is extended to every hot stream h that may exchange heat with c just before h , i.e. k j i k3PH . When a heater (S/c ), if required, is to be ij j arranged both last and in series on the cold stream c (Case 2), the set PH should be replaced by PH* in j ij ij equation (2). (I.3) Upper bound on the number of predecessors of a non-input exchanger (h /c ). Let NB be the specii j j fied number of stream branches into which the cold stream c can at most be partitioned. If the heat j exchanger (h /c ) arises at the network as a non-input i j unit over c , it can then be preceded by at most NB j j parallel units. For instance, the heater (S/C1) is preceded by the parallel exchangers (H1/C1) and (H2/C1) over C1. Then, NB 52. On the contrary, the numC1 ber of predecessors for an existing input unit (½ "1, ij XCF "1) will be zero (Fig. 1). Therefore, ij + XC 4NB (½ !XCF ), i3HSj, j3C. kj, ij j ij ij k|PHij (3)

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For Case 2, the set PH should be replaced by PH* . If ij ij no stream splitting is allowed at any process stream, then NB "1 and the constraints (3) become strict j equalities as shown in Part I. (I.4) Upper bound on the number of successors to a heat exchanger (h /c ). If the heat match (h /c ) does i j i j exist in the network, it should at most have NB j succeeding units arranged in parallel over the cold stream c . For instance, the unit (H3/C1) in the netj work configuration shown in Fig. 1b is succeeded by the parallel exchangers (H2/C1) and (H1/C1) over the cold stream C1. Then, NB 52. To account for the C1 bound on the number of successors, the following constraint is incorporated to the problem model, + XC 4NB ½ , i3HSj, j3C. (4) ij,kj j ij k|SHij For Case 2, an existing heat match (h /c ) should at i j most have NB parallel successors unless it is followed j by a heater (s/c ). In such a case, it will only have j a single successor, (s/c ). Since any hot utility S has j been excluded from SH* , then the constraint (4) will ij take the following form for Case 2: + XC 4NB (½ !XC ), i3Hj, j3C. ij, kj j ij ij, Sj k|SH*ij When (h /c ) belongs to the network and is succeeded i j by a heater (s/c ), then XC equal 1. Consequently, j ij,sj the RHS drops to zero and no other succeeding unit to (h /c ) different from (s/c ) is permitted: i j j + XC "0. k| SH*ij ij,kj (I.5) Lower bound on the number of input units over a process stream. Among the heat exchangers through which a particular process stream flows, including the auxiliary equipment, at least one of them should be the first or input unit (see Fig. 1): + XCF 51 j3C. (5) ij i|HSj Constraints (5) remain the same for Case 2. (I.6) Upper bound on the number of input units. If a cold stream c can at most be partitioned into NB j j split streams, then the maximum number of parallel input units on c is also NB , unless Case 2 is being j j considered and the first unit is a heater (s/c ). In such j a case, (s/c ) must be the unique input unit on the j stream c . j + XCF 4NB , j3C. (6) ij j i|HSj For Case 2, the summation is restricted to the set Hj and the RHS of the inequality (6) must therefore drop to zero whenever a heater arises as an input unit over c . Then, j + XCF 4NB (1!XCF ), j3C. ij j sj i|Hj (I.7) Required conditions for the parallel arrangement of non-input heat exchangers. As discussed before, this model does not account for network structures involving stream by- passes and/or stream branches

flowing through two or more exchangers in series. Therefore, a pair of non-input units arranged in parallel over a process stream should always satisfy the following two conditions: 1. They should share the same sets of predecessors and successors. 2. They should not precede or succeed themselves. From these conditions, a set of four mathematical restrictions for the parallel arrangement of non-input units can be derived. In the example shown in Fig. 2, the exchangers (H2/C1) and (H3/C1) are arranged in parallel over C1. Both are preceded by the exchanger (H4/C1) and succeeded by (H1/C1). Moreover, (H2/C1) is neither a predecessor nor a successor to (H3/C1). The former condition does not hold if a split stream is permitted to go through two or more exchangers arranged in series. Such a case is depicted in Fig. 3, where the exchangers (H2/C1) and (H3/C1) have different predecessors (Fig. 3a) or different successors (Fig. 3b). To exclude such configurations from the network feasible space, the following structural constraints are applied, (1) If (h /c ) is the common predecessor of the nonn j input units (h /c ) and (h /c ) over a cold process i j t j stream c and additionally the match (h /c ) is j t j succeeded by (h /c ), then the network structure k j

Fig. 2. Multiple preceding and succeeding units in a split network.

Fig. 3. A series of exchangers over a stream branch — A prohibited network design.

Synthesis of heat exchanger networks will be feasible only if (h /c ) is also a successor to k j (h /c ): i j XC 5XC #XC #XC !2, ij,kj nj,ij nj,tj tj,kj

t3PH W SH , i3HSj, j3C ij ij (7)

For Case 2, the stream sets MSH , SH , PH , ij tj ij PH , HSjN are to be substituted by MSH* , SH* , tj ij tj PH* , PH* , HjN, respectively. ij tj (2) If (h /c ) is the common successor to the intermedik j ate exchangers (h /c ) and (h /c ) over the cold proi j t j cess stream c and additionally the match (h /c ) is j t j preceded by (h /c ), then the network structure will n j be feasible only if (h /c ) is also a predecessor of n j (h /c ). i j XC 5XC #XC #XC !2, nj,ij ij,kj tj,kj nj,tj n3PH W PH , k3SH W SH , ij tj ij tj i,t3HSj j3C

(8)

(3) If the non-input units (h /c ) and (h /c ) share i j t j a common predecessor (h /c ) over the cold stream n j c , then they cannot precede or succeed one anj other: XC #XC #XC #XC 42 nj,ij nj,tj ij,tj tj,ij n3PH WPH , i,t3HSj, j3C ij tj

(9)

(4) If the non-input units (h /c ) and (h /c ) share i j t j a common successor (h /c ) over the cold stream k j c , then they cannot precede or succeed one anj other: XC #XC #XC #XC 42, ij,kj tj,kj ij,tj tj,ij k3SH WSH , i,t3HSj, j3C ij tj

(10)

For Case 2, the stream sets MSH , SH , HSjN are to be ij tj substituted by MSH* , SH* , HjN, respectively. Restricij tj tions (7)—(10) are applied to any pair of potential parallel exchangers and therefore they should hold whatever the number of parallel units over c . Morej over, such constraints should be ignored if NB "1. j (I.8) Required conditions for the parallel arrangement of input heat exchangers. Since parallel input units over a cold process stream have no predecessor at all, then the number of mathematical conditions to be satisfied by every pair of them reduces from four to only two. (1) Any pair of parallel input units over a cold stream c should share the same set of successors. j XC 5XCF #XCF #XC !2, ij,kj ij tj tj,kj k3SH W SH , i,t3HSj, j3C ij tj

(2) Any pair of input exchangers over a cold stream c cannot precede or succeed themselves: j XCF #XCF #XC #XC 42, ij tj ij,tj tj,ij

k3SH W SH , n3PH W PH , ij tj ij tj i, t3HSj, j3C

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(11)

(12)

For Case 2, the set HSj is to be replaced by Hj and the supporting stream sets with asterisks are to be used. The constraints (7)—(12) must be included in the problem mathematical model only if NB is j greater than 1. (I.9) A predecessor of an exchanger (h /c ) cannot i j simultaneously be its successor. It is indeed a redundant constraint being considered to just improve the lower bound provided by the LP relaxation. By doing that, a faster convergence of the branch-and-bound algorithm to the MILP optimal solution is usually achieved. Indeed, the exchanger outlet temperature constraints to be given below prevent from having a predecessor being simultaneously a successor of a given heat exchanger: XC #XC 41, k3PH W SH , kj,ij ij,kj ij ij i3HSj, j3C

(13)

If the variable XC takes the value one because kj,ij (h /c ) precedes (h /c ) over the cold stream c , then k j i j j XC "0. For Case 2, the sets MPH , SH and ij,kj ij ij HSjN are to be replaced by MPH* , SH* and HjN, ij ij respectively. (I.10) A predecessor of the exchanger (h /c ) can i j never be arranged right after any one of its successors. It is a redundant constraint since the exchanger outlet temperature restrictions to be given in the next Section will prevent from synthesizing such a type of infeasible network. If the match (h /c ) is preceded by i j (h /c ) and followed by (h /c ) at the network, then p j k j (h /c ) can never be a successor to (h /c ): p j k j XC #XC #XC 42, p3PH W SH , ij,kj pj,ij kj,pj ij kj i, k3HSj, j3C

(14)

(II) The outlet temperature constraints The outlet temperatures of the process streams h and c from a unit (h /c ) is not only a function of the i j i j exchanger coordinates (XHF , XH , XCF , XC ) ij ij ij ij and the heat load (q ) but also depend on the heat ij duties of the other units arranged in parallel with (h /c ) over h and/or c . In fact, a different set of linear i j i j constraints restraining the values of such outlet temperatures is to be applied depending on the location of the unit in the network (input, intermediate or output unit) and the way it is arranged over a hot or a cold process stream (series or parallel configuration). Since one does not know a priori to which class a particular

1024

M.R. GALLI and J. CERDA¨

unit will belong, the mathematical model must then include the three types of temperature constraints for each potential exchanger but only one will be active at a particular network while the others will become redundant. Such constraints on the stream outlet temperatures from (h /c ) will account for the restrictions i j imposed by (a) the stream outlet temperature from either a single or a set of preceding units (for an intermediate exchanger); (b) the initial temperature at which a process stream is supplied to either a single or a set of input exchangers and (c) the target temperature to be reached by the process stream leaving either a single or a set of output exchangers. In each case, the temperature constraint will take a different form depending on whether the unit is arranged in series or in parallel with one or more exchangers. Only the outlet temperature constraints for cold streams will be developed below. Analogous constraints for hot streams are included in Appendix B. The jth-cold stream outlet temperature from the exchanger (hi /cj ). Let tc denote the outlet temperature ij of cold stream c from the exchanger (h /c ) whose j i j value belongs to the following range: ¹CI4tc 4¹CO, j ij j

j3C

(15)

where ¹CI is the jth-cold stream inlet temperature to j the network and ¹CO the jth-cold stream outlet temj perature from the network. The values of the model parameters ¹CI and ¹CO j j vary with the type of problem being solved since each one uses a different network decomposition scheme. They are listed below. for the pinch-defined upper network: ¹CIj"¹pinch ;

¹COj"¹C¹j ,

for the pinch-defined lower network: ¹CIj"¹CSj;

¹COj"¹pinch ,

for the pseudo pinch-defined upper network: ¹CIj"¹CPP, j ;

¹COj"¹C¹j ,

for the pseudo pinch-defined lower network: ¹CI "¹CS ; j j

¹CO"¹C , j PP,j

where ¹ is the pinch temperature for cold 1*/#) streams. To determine the temperature change caused by the heat exchange (h /c ) on the process stream h (or c ), it i j i j is important to know whether the match (h /c ) is i j arranged in series or in parallel over h (or c ), what are i j the heat exchanges, if any, being performed in parallel with (h /c ) and their heat duties as well as the location i j and the heat load of the exchanger (h /c ) at the neti j work. (II.1) From an input unit (h /c ) arranged in series. i j The exit temperature of a cold process stream c from j a single input exchanger (h /c ) over c is defined by i j j

both the stream inlet temperature to the network (¹CI) and the temperature increase caused by the j overall heat flow q transferred to c at the unit (h /c ): ij j i j tc 4¹CI#(q /FCp ) ij j ij j

A

B

#M 1!XCF # + XCF , ij kj k|HSj kEi i3HSj, j3C

(16)

where M is a very large positive number. The constraint (16) will become redundant if (h /c ) is not the i j first unit (XCF "0) and/or the network features ij additional input units arranged in parallel with (h /c ) i j over the cold stream c ( + XCF '0, with the sumj kj mation being extended to any k3HSj, kOi). For Case 2, the constraint (16) still holds. (II.2) From an input unit (h /c ) arranged in parallel i j with another exchanger. If the heat exchangers (h /c ) i j and (h /c ) are both input units arranged in parallel k j over the cold stream c , then the common outlet j temperature of the stream c from such units must j satisfy the following restriction: tc 4¹CI#(q #q )/FCp ij j ij kj j

A

B

#M 2!XCF !XCF # + XCF , ij kj nj n|HSj nEi,k i, k3HSj, j3C. (17) The above restriction will be binding only if (i) the units (h /c ) and (h /c ) are placed first over c and (ii) i j k j j there is no other input exchanger. If so, + XCF "0 nj n|HSj nEi,k Consequently, the last term on the RHS drops to zero and the restriction (17) does hold. Otherwise, such a last term takes a large positive number and the constraint (17) becomes redundant. For Case 2, the set HSj is to be replaced by Hj in the inequality (17). (II.3) From an input unit (h /c ) arranged in parallel i j with two or more exchangers. To account for networks featuring three or more parallel input units over a cold stream c , it is necessary that the model also j includes constraints restraining the value of the stream outlet temperature from a heat exchanger (h /c ) when it is arranged in parallel with a set of i j p"2,3, 2, (NB !1) units over c . Let us define j j a subset of p'1 hot streams different from h that can i exchange heat with c , i.e. a subset of HSj. The value of j ‘‘p’’ can range from 2 to (NB !1). There will be as j many subsets of such a type as the combination of ‘‘p’’ in (NHS) !1 hot streams, i.e. j p , HSj , for l"1,2, . . . , lpi NHS !1 j p"2,3, . . . , (NB !1), j XCF "XCF "1 and ij kj

A

B

Synthesis of heat exchanger networks where (NHS) is the cardinality of HSj. Then, the j outlet temperature of the cold stream c for a particuj lar set of p'1 input units arranged in parallel with (h /c ) over c must satisfy the following restriction: i j j

A

BN

tc 4¹CI# q # + q ij kj ij j k|HSjlpi

FCp j

A

#M 1#p!XCF ! + XCF ij kj k|HSjlpi

B

A

B

p # + XCF , l"1,2 , . . . , , nj NHS !1 j n|HSj j nbHS lpi nEi p"2, 3, 2, (NB !1), i3HSj, j3C. (18) j where M is an arbitrarily large number and the subset HSj LHSj includes any p members of the set HSj l1* different from h . Equation (18) will become active i only if (h /c ) is effectively arranged in parallel with i j a set of p'1 units (h /c ), where every h 3HSj . k j k lpi Otherwise, it should become redundant. In any case, at most only one of the alternative outlet temperature constraints for the unit (h /c ) will be binding at the i j optimum. If the heat exchanger (h /c ) or any match i j (h /c ), k3HSj is not an input unit over the cold k j lpi stream c , then XCF "0 and/or XCF "0 for some j ij kj k3HSj . Consequently, the restriction (18) for the lpi match (h /c ) accounting for the parallel arrangement i j given in terms of HSj will become redundant. A simlpi ilar conclusion is derived whenever an exchanger (h /c ), nNHSj (nOi) is also an input unit (XCF " n j lpi nj 1) because the number of parallel units would be greater than (p#1). For Case 2, the sets MHSj , HSjN lpi are to be replaced by MHj , HjN, respectively, and lpi NHS by the cardinality of Hj decreased by one i.e. j NH . j The model parameter NB sets an upper bound on j the number of units with which a particular exchanger (h /c ) can be arranged in parallel over the stream c . i j j Since the constraint (18) is written in terms of heat exchangers over a particular stream c , then a maxj imum number of units in parallel with (h /c ) over i j c (denoted by NBC ) can be easily handled. The superj ij script C denotes a bound on the number of parallel units over a cold stream. Of course, NBC 4NB . For ij j instance, it may happen that the designer specifies that the unit (h /c ) must be arranged in series over i j c (NBC "1) while (h /c ) can be placed in parallel j ij k j with at most a couple of exchangers (NBC "3). To kj make the model constraints simpler, however, a common value NB for all the units over the stream c has j j been assumed in equation (18). The analysis of the constraint (18) reveals that the model is able to handle not only an upper bound on the number of split streams (NBC ) but also a restricted ij set of parallel arrangements for a particular unit (h /c ) i j over the stream c . Let us suppose that HSj"Mh , h , j i k h , h N and NB "2 for every match over c . Then, n t j j (h /c ) can at most be arranged in parallel with a single i j

1025

unit. Though p41, the designer can still disallow the parallel arrangement of units (h /c ) and (h /c ) by i j k j excluding the hot stream Mh N from the family of sets k HSj spanned by l for p"1, i.e. MMh N, Mh NN. lpi n t (II.4) From a non-input exchanger (h /c ) arranged in i j series. The outlet temperature of a cold process stream c from a single non-input exchanger (h /c ) is deterj i j mined by both the exit temperature of c from the j preceding unit and the exchanger heat load q . If ij (h /c ) is a predecessor of (h /c ) and there is no exk j i j changer arranged in parallel with (h /c ) over c , then i j j the outlet temperature of c from (h /c ), tc , must j i j ij verify the following constraint: tc 4tc #(q /FCp ) ij kj ij j

A

B

, #M 1!XC # + XC kj,nj kj,ij n| HSj nEi,k k3PH , i3HSj, j3C. (19) ij The restriction (19) will be binding only if the exchanger (h /c ) uniquely features the unit (h /c ) as i j k j a predecessor (XC "1) and additionally no heat kj,ij match is placed in parallel with (h /c ) over c i j j (XC "0, ∀n3HSj, nOi, k). For Case 2, the sets kj,nj MPH , HSjN are to be replaced by MPH* , HjN, respecij ij tively. (II.5) From a non-input exchanger (h /c ) arranged i j in parallel with one or more exchangers. The outlet temperature of a cold process stream c from a nonj input exchanger (h /c ) is determined by the outlet i j temperature of c from the set of preceding units, the j exchanger heat load q and the overall heat duty of ij the set of units arranged in parallel with (h /c ). Wheni j ever (a) the unit (h /c ) is a predecessor of (h /c ) over k j i j c and (b) the match (h /c ) has been arranged in j i j parallel with other ‘‘p’’ units (h /c ), t3HSj , then the t j l1* outlet temperature of c from (h /c ) must satisfy the j i j following constraint:

A A

BN

tc 4tc # q # + q ij tj ij kj t|HSjlpi

FCp j

#M 1#p!XC kj,ij

B

! + XC # + XC , kj,tj kj,nj t|HSjlpi n|HSj j nbHS lpi nEi p l"1,2 , 2, , NHS !1 j p"1, 2 , 2, (NB !1), k3PH , i3HSj, j3C. j ij (20)

A

B

The restriction (20) for the parallel arrangement of the non-input unit (h /c ) with a set of ‘‘p’’ non-input i j exchangers defined by HSj will hold only if they all lpi feature the unit (h /c ) as a common predecessor k j (XC "1; XC "1,∀t3HSj ) and additionally kj,ij kj,tj lpi

M.R. GALLI and J. CERDA¨

1026

no further heat match is placed in parallel with (h /c ) i j over c . For Case 2, the sets MHSj , PH , HSjN are to j lpi ij be replaced by MHj , PH* , HjN, respectively, and the lpi ij model parameter NHS by NH . j j (II.6) From an output or last exchanger (h /c ). i j A heat exchanger (h /c ) is the last one over the process i j stream c if it has no successor at all. This definition j still holds even if (h /c ) is arranged in parallel with i j some other matches. Then, the output unit on a cold stream c can be identified based on the fact that the j sum of the binary variables XC over all possible ij,kj successors (h /c ) to the exchanger (h /c ) is null. To k j i j guarantee the achievement of the stream target temperature ¹CO, the outlet temperature constraint for j the output exchanger (h /c ) over c is to be written as i j j an (5) inequality. Then,

A

tc 5¹CO!M ij j

B

+ XC , i3HSj, j3C (21) ij,kj k|SHij

If (h /c ) is a last exchanger over c , the summation i j j over all its possible successors will be equal zero and the above restriction (21) will be binding. However, (h /c ) can either be an input or a non-input exchanger. i j Then, one of the constraints (16)—(20) will also apply. Therefore, the preceding unit outlet temperature and the stream target temperature are both simultaneously taken into account by imposing a couple of outlet temperature constraints on the value of tc ij from an output unit. At each network above the pinch, at most a single hot utility ‘‘s’’ is usually available. Assuming that the heater (s/c ) on any stream c at a particular network is j j always placed both last and in series (Case 2), the number of structural variables MXCF , XC N can be ij ij decreased as long as those ones involving a heater (s/c ) as a preceding unit are no longer necesssary. In j such a case, the achievement of the target temperature for the cold stream c is guaranteed by imposing the j following constraint on the outlet temperature tc ij from the last stream/stream exchanger (h /c ): i j

A

tc 5¹CO!(q /FCp )!M ij j S,j j

B

+ XC* , ij, kj k|SHij*

i3Hj, j3C, where the summation over SH* never includes a hot ij utility. In such a case, tc 5¹CO ½ , j3C. sj j sj (II.7) Overall heat balance for a cold stream c . The j set of restrictions imposed on the outlet temperature of cold stream c from an output unit guarantees that j its value always meets the temperature target, i.e. tc 5¹CO. The overall heat balance for the cold ij j stream c forces the solution to satisfy such constraints j as strict equalities. + q "FCp (¹CO!¹CI ), j3C ij j j j i|HSj

(22)

(II.8) Relationship between the outlet temperatures of c from two neighboring exchangers. If the heat match j (h /c ) is a successor to (h /c ) over the cold stream c , i j k j j then the outlet temperature of c from (h /c ), tc , must j i j ij be higher than tc , i.e. kj tc 5tc #M(XC !1), ij kj kj,ij k3PH , i3HSj, j3C (23) ij Restriction (23) is indeed a redundant constraint being included in the problem model simply to get a faster convergence to the optimal solution. For Case 2, the sets MPH , HSjN are to be substituted by ij MPH*, HjN, respectively. ij (III) The minimum temperature approach constraint. Since tc represents the jth-stream outlet temperature ij from the exchanger (h /c ), then the inlet temperature i j to such a unit will be equal to tc if the match (h /c ) kj k j precedes (h /c ) over the cold stream c . Similarly, th i j j im represents the inlet temperature of h to the exchanger i (h /c ) if (h /c ) is performed right before (h /c ) over h . i j i m i j i For an input unit over h , the inlet temperature of h is i i equal to ¹HI. Therefore, M¹HI, tc N and Mth , tc N i i ij im ij will stand for the process stream temperatures at the hot extreme of an input and a non-input exchanger (h /c ) over h , respectively. In turn, Mth , ¹CIN and i j i ij j Mth , tc N will represent the stream temperatures at ij kj the cold extreme of an input and a non-input exchanger (h /c ) over c , respectively. Then, the mini j j imum allowed temperature approach constraints at both the hot and cold extremes of (h /c ) can be written i j as shown below. (III.1) At the hot extreme of a heat exchanger (h /c ). i j For an input exchanger (hi/cj) over the hot stream hi: ¹HI!tc 5*¹ !M(1!XHF ), i ij .*/ ij j3C¼i, i3H,

(24)

where *¹ is the minimum allowed temperature .*/ approach for the match (h /c ). Its value can change i j with the heat match at the design phase. For an intermediate exchanger (h /c ) over the hot i j stream h : i th !tc 5*¹ !M(1!XH ), im ij .*/ im,ij m3PC , j3C¼i, i3H (25) ij For Case 2, the set PC is to be replaced by PC*. ij ij (III.2) At the cold extreme of a heat exchanger (h /c ). i j For an input exchanger (h /c ) over the cold stream c : i j j th !¹CI5*¹ !M(1!XCF ), ij j .*/ ij j3Ci, i3H (26) For Case 2, the set C¼i is to be substituted by Ci. For an intermediate exchanger (h /c ) over the cold i j stream c : j th !tc 5*¹ !M(1!XC ), ij kj .*/ kj,ij k3PH , j3Ci, i3H (27) ij For Case 2, the set PH is to be replaced by PH* . ij ij

Synthesis of heat exchanger networks 6. The constrained least utility usage problem (P1) To compute the constrained least utility usage for the problem, the hot/cold utility requirements at each individual pinch-defined network are first to be determined. Therefore, the MILP basic framework should be sequentially applied to every network l by considering the subset of process streams and their temperatures subranges belonging to the network l. Since the minimum utility usage is the design target to be determined, the problem objective function for Problem P1 is given by min + q # + q (28) iW sj i|H j|C where Hl and Cl indeed refer to the sets of hot and cold process streams pertaining to the pinch-defined network l. Therefore, the proposed MILP model for the determination of the minimum utility consumption under topology constraints at the network l consists in the minimization of the objective function (28) over the feasible space defined by the sets of linear restrictions (1)—(27), (A1)—(A13) and (B1)—(B8). All the continuous variables Mq , tc , th N are non-negative ij ij ij and the discrete variables MXC , XH , XCF , kj,ij im, ij ij XHF , ½ N are 0—1. Since the problem model has ij ij been posed at the level of network, the proposed MILP formulation should be repetitively applied going from the top to the bottom network. In each case, a single hot and a single cold utility will be available. Above the process pinch, a non-zero cooling utility usage in the network will denote a finite heat flow across the process (or utility) pinch. Below the process pinch, an additional heating utility requirement may arise though the process is a net heat source due to the specified structural constraints. The expressions introduced in Part I will permit to compute the problem least utility usage from the utility requirements at each individual pinch-defined network. 7. The heat recovery target at the upper network (Problem P2.1) To define the feasible sets of stream pseudo-pinch temperatures, a new problem called P2.1 must be solved. The inlet (outlet) temperature of any cold (hot) stream to (from) the upper network is no longer ¹ (¹ #*¹ ) but a problem variable denot1*/#) 1*/#) .*/ ing its pseudo-pinch point temperature. To lower the number of units, such pseudo-pinch points will be chosen in such a way that a maximum allocation of the pinch heat flow Q* to heat matches already perP formed above the pinch is achieved. In this manner, their heat loads will increase as much as possible. To reach such a goal, the maximum heat recovery at the extended upper network Q* is to be adopted as the UP problem target for P2.1. As discussed before, a portion of the pinch heat flow should sometimes be destinated to cold streams [email protected] featuring ¹CT4¹ . This L j 1*/#) will be considered by incorporating them to the set PPS initially involving the pinch-determining process

1027

streams as follows: PPS"H XCd, Cd"C [email protected] . U U U L U Therefore, the constraint set for problem P2.1 must include the MILP basic framework for the set of process streams PPS determining the pseudo-pinch points. In addition, the LUC target is to be enforced by limiting the available amount of hot utility to just Q* and disallowing the use of an external heat sink. S This is another design target further restraining the solution space within which the search for the stream pseudo-pinch temperatures is made. (P2.1) Max Q " + + q UP ij d j|CU i|HjU s.t. Eqs. (1)—(27), (A.1)—(A.13), (B.1)—(B.8), + q "Q*, S d Sj j|CU ¹HI"¹HS , ¹HO5¹HT, i3H , i i i i U ¹CS4¹CI4¹ , ¹CO"¹CT, j j 1*/#) j j j3C , U ¹CS4¹CI4¹Cmx , ¹CO"¹CT, j j PP,j j j [email protected] L where ¹HO and ¹CI are now problem variables and i j the set Cd denotes the set of streams for the extended U upper network. Cold utility is not available. The optimal values of ¹HO and ¹CI stand for feasible pseudoi j pinch point temperatures maximizing the heat recovery at the extended N . Nonetheless, the final U choice is left to the next problem (P2.2) because the heat recovery target Q* may be achieved through UP two or more alternative sets of pseudo-pinch points. 8. Optimal design of the extended upper network (Problem P2.2) To identify both the sets of pseudo-pinch points minimizing the number of units and the related design of the extended upper network, the following problem (P2.2) is to be solved: Nº " + + ½ UP ij d j|CU i|HSUj s.t. Eqs. (1)—(27), (A.1)—(A.13), (B.1)—(B.8), (P2.2) Min

+ q "Q*, Sj S d j|CU + + q 5Q* , ij UP d j|CU i|HUj ¹HI"¹HS , ¹HO5¹HT, i3H , i i i i U ¹CS4¹CI4¹ , ¹CO"¹CT, j3C , j j 1*/#) j j U ¹CS4¹CI4¹Cmx , ¹CO"¹CT, [email protected] j j PP,j j j L The optimal values for the variables ¹HO and ¹CI are i j the pseudo-pinch point temperatures at streams h and c , respectively, arising as model parameters in i j

M.R. GALLI and J. CERDA¨

1028

the formulation of problem (P2.3) shown in Appendix C. In turn, [email protected] is the subset of cold streams entirely L running below the pinch that requires an additional utility usage because of the topology constraints at the pinch-defined lower network. 9. Results and discussion The generalized neighbor-based targetting algorithmic approach to the topology-constrained HENS problem has been applied to the solution of four examples involving five to seven process streams. The first two were already solved in Part I assuming a series configuration and they are revisited to analyze the impact of using stream splits on both the value of the constrained MER target and the optimal network design. In every example, it was assumed that heaters and/or coolers are placed last and in series (Case 2). Sizes of the MILP formulations and the required CPU times on a SiliconGraphics Workstation are given in Table 1 for every example. Example 1. Example 1 is a five-stream problem first presented by Gundersen and Grossmann (1990). In Part I of this two-part paper, this example has been solved by imposing a series configuration on the sought network. Stream splitting is now allowed but the number of parallel units on each process stream is under designer control. Example 1 includes three hot and two cold process streams whose supply/target temperatures (°C) and heat capacity flowrates (kW/°C) are all shown in Fig. 4. Assuming HRAT"

10°C, it was found that the process pinch occurs at the supply temperature of stream H1 (149/159°C) and the unconstrained least utility usages amount to: QO" S 10645 kW and QO "8395 kW, respectively. Three of W the process streams MH2, H3, C2N feature temperature ranges that include the pinch temperature. The other two MH1,C1N run entirely below the process pinch. A triplet of desired structural conditions have been specified by the process designer: (1) the unit (H1/C1) is forbidden as a feasible successor to (H1/C2) over the hot stream H1; (2) arrangements involving at most a couple of parallel units are allowed over the hot streamsMH1, H2 and H3N and the cold stream C1, and (3) a series configuration over the cold stream C2 is only permitted. To establish the minimum utility consumption under these topology restrictions, the constrained utility usage problem formulation (P1) has been applied to each pinch-defined network. The upper network involves a couple of hot streams MH2,H3N and a single cold stream MC2N. Since C2 is the unique cold stream and parallel exchangers over C2 are not allowed by the designer, then the HENS problem modelling for series networks was applied to find the same results already shown in Part I. As before, additional heating needs caused by the specified series design are required since some available heat from the stream H3 is to be removed in a ‘‘fictitious’’ cooler featuring a heat duty equal to 605 kW. Then, Q*(N )" S U 11 250 kW and Q* (N )"E "605 kW. W U U

Table 1. Sizes and CPU times for the set of MILP-formulations being tackled to find the optimal networks for Examples 1—4 Example

Problem

Restrictions

Continuous variables

Binary variables

CPU-time (seg.)

1

P1 (N ) U P1 (N ) L P2.1 P2.2 P2.3

48 280 39 40 227

15 31 12 12 25

18 57 12 12 48

0.42 33.3 1.08 1.00 16.42

2

P1 (N ) U P1 (N ) L P2.1 P2.2 P2.3

180 1101 151 152 945

20 30 16 16 26

29 57 20 20 50

1.53 26.0 1.30 0.95 36.4

3

P1(N ) U P1(N ) L P2.2 P2.3

473 164 395 142

42 49 40 12

87 28 72 20

15.5 0.90 20.1 0.68

4

P1(N ) U P1(N ) L P2.1 P2.2 P2.3

449 80 801 802 142

42 17 49 49 12

87 20 96 96 20

51.3 0.5 4.24 3.01 0.68

Synthesis of heat exchanger networks

1029

"10 645#max(605, 3258) "13 903 kW Q* "QO #max(E , E ) W W U L "8395#max(605, 3258) "11 653 kW

Fig. 4. (a) The least utility usage at each constrained pinchdefined network and (b) The user-constrained optimal network design, both for Example 1.

However, the solution to problem (P1) for the pinch-defined lower network shows some changes. The removal of C1 from the set of cold streams with which the stream H1 can exchange heat right after (H1/C2) turns SC into an empty set. In other H1,C2 words, (H1/C2) does not present any feasible successor. If required, therefore, (H1/C2) must be performed last over H1. In addition, the units are to be connected in series over C2. Both restrictions limit the process heat recovery below the pinch so much that the process turns to be a heat sink, thus requiring energy from either the upper network or an external heat source. However, a parallel configuration resulting from splitting both H1 and C1 can now be synthesized to somewhat lower such additional heating needs from 3399 kW (series configuration) to 3258 kW (split configuration), i.e. 4.3% less. Then, Q*(N )"E "3258 kW. A CPU time equal to 33.3 s S L L was required by the GAMS/OSL solver (Brooke et al., 1992) to find the optimal solution to problem P1 for the pinch-defined lower network (see Table 1). By applying the expressions introduced in Part I, the constrained LUC target and the pinch heat flow can be evaluated as follows: Q*"QO#max(E , E ) S S U L

Q*"max(E , E )"max(605, 3258)"3258 kW P U L Since a heat shortage E arises at the cold stream L C2 (Case D with E (E ), the pseudo-pinch points U L are still defined by the pinch-determining process streams MH2, H3, C2N. By considering them in the formulation of Problem (P2.1), it was found a maximum heat recovery target at the upper network equal to 12 926 kW, slightly lower than the one obtained in Part I (13 001 kW). The incorporation of such a MER target in the definition of Problem (P2.2) and its subsequent solution permits to find the optimal upper network design (Fig. 4b). It can be observed that the new pseudo-pinch temperatures defining the ranges of both constrained networks are a little higher compared with Part I: ¹ "138.2°C, ¹ " PP,H2 PP,H3 151.6°C and ¹ "128.2°C. By next solving ProbPP,C2 lem (P2.3), it was found the optimal design of the lower network shown in Fig. 4b comprising one unit less than the best series structure. In fact, the lower network includes five units, the minimum number predicted for a pinched network design (Fig. 4b). Table 1 shows the number of continuous/binary variables, the number of restrictions and the computing time required by the GAMS/OSL solver on an Indy Silicon Graphics Workstation to find the optimal solution to each problem. A total CPU time of 18.5 s was needed to solve problems P2.1—P2.3 (see Table 1). The formulation proposed by Yee and Grossmann (1990) requires three stages and involves 151 constraints and 93 variables of which 29 are binary, plus the additional constraints imposing an upper bound on the number of parallel units. However, such a formulation is unable to forbid the exchanger (H1/C1) as a feasible successor to (H1/C2) over the stream H1. Example 2. Example 2 is a seven-stream problem involving six hot streams and a single cold stream. Heat capacity flow rates (BTU/h°F) and supply/target temperatures (°F) are all presented in Fig. 5. If the minimum allowed temperature approach HRAT is equal to 20°F, then the hot stream H4 determines the pinch temperature at 410/430°F and the unconstrained least hot and cold utility usages are given by: QO" S 8390 BTU/h and QO "6618 BTU/h, respectively. In W Part I this example has been solved by specifying a series configuration as the unique structural constraint. The results showed that the least utility usage in the upper network is the only one affected when split networks are excluded from the solution space. Therefore, the heat recovery target may be improved

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"8390#max(158, 0) "8548 BTU/h, Q* "QO #max(E , E ) W W U L "6618#max(158, 0) "6776 BTU/h, Q*"max(E , E )"max(158, 0)"158 BTU/h. P U L By sequentially solving Problems (P2.1)—(P2.3), the optimal network design depicted in Fig. 5b has been encountered. Compared with Part I, the heat recovery target Q* has increased to 5552 BTU/F since the UP split of the stream C1 allows the match (H1/C1) to be performed to a larger extent. As a result, the pseudopinch point at stream H1 goes down (¹ "441°F) PP,H1 while the boundary temperatures at streams H3 and C1 both remain at the pinch values 430/410°F. Problem sizes and computing times required are given in Table 1.

Fig. 5. (a) The least utility usage at each constrained pinchdefined network and (b) the user-constrained optimal network design, for Example 2.

by allowing the stream splitting of the cold stream C1. No further constraints disallowing preceding and/or succeeding units for some potential matches have been applied. The subset of streams MH1, H2, H3, C1N all feature temperature ranges partially or completely above the pinch. Then, the cold stream C1 can at most be split into three stream branches in the upper network. If such a condition is imposed, the mathematical formulation for problem P1 will involve 180 restrictions, 20 continuous variables and 29 binary variables. The optimal solution to P1 for N shown in Fig. 5a has U been found in 1.53 s (see Table 1). A ‘‘fictitious’’ cooler on stream H1 with a heat load of 158 BTU/h is an indication that further cooling utility is still needed though a parallel configuration is now permitted. Then, Q*(N )"8548 BTU/h and Q* (N )"E " S U W U U 158 BTU/h. Such a increase in the utility consumption is mainly caused by the required isothermal mixing of the stream branches. As found in Part I, the process is still a heat source below the pinch, i.e. Q*(N )"E "0. So, one can S L L expect to find an optimal lower network featuring a series structure. Therefore, the constrained utility target is given by Q*"QO#max(E , E ) S S U L

Example 3. Example 3 is a seven-stream problem first presented by Colberg and Morari (1990) and later solved by Yee and Grossmann (1990). The problem data are shown in Fig. 6. If HRAT is assumed to be equal to 20°F, then the process pinch is located at the supply temperature of the stream C1 (497/517°F) and the unconstrained least utility usages are given by QO"244 BTU/h and QO "173 BTU/h, respectively. S W Except for the cold stream C3 which run entirely below the pinch temperature, all the other process streams MH1, H2, H3, C1, C2, C4N feature temperature ranges running above or through the pinch. Then, the problem size reduction caused by the decomposition approach is less important. Before solving Example 3, however, the problem data were analyzed by the designer to derive structural conditions helping to get an optimal network design faster. Since the hot stream H3 features a supply temperature close to the pinch, the designer has then specified that any heat match involving H3 at the upper network must be performed first over any cold stream. This is equivalent to say that MPC , PC , PC N are all empty sets. H3,C1 H3,C2 H3,C4 In other words, there is no feasible preceding unit for the heat exchangers MH3/C1, H3/C2, H3/C4N, if required, over the corresponding cold stream. No additional constraint has been specified restricting the set of neighboring units for any potential match. As long as the upper network involves three hot and three cold streams, at most three units can be arranged in parallel along any process stream. However, the designer has observed that (i) the heat capacity flowrates of cold streams C2 and C4 are both very small and (ii) the available amount of heat above the pinch at H3 is rather low. Then, he/ she decided to better prohibit the split of the streams MH3, C2, C4N but obviously confining such a constraint for H3 just at the upper network. Moreover, the designer allows a maximum of two parallel units on the remaining

Synthesis of heat exchanger networks

Fig. 6. The user-contrained optimal network design for Example 3.

streams MH1, H2, C1N. The utility usage problem formulation (P1) for the upper network, including the structural conditions specified by the designer, involves 473 restrictions, 42 continuous variables and 87 binary variables. The optimal solution found in 51.3 s reveals that the upper network still requires Q* (N )"QO"244 BTU/h. Since the specified topolS U S ogy constraint has no impact on the least utility usage, then the heat leakage across the pinch remains null, i.e. E "0 (see Fig. 6). U In turn, the lower network involves the streams MH3, C2, C3, C4N and the sought design should include at most a couple of parallel heat matches over the hot stream H3 according to the designer’s specification. The optimal solution to problem (P1) for the lower network found in 0.9 s indicates that no further heating utility need is caused by the prescribed topology constraints (see Fig. 6). Then: Q* (N )"QO " W L W 173 BTU/h. No requirement of a ‘‘fictitious’’ cooler at the upper network and a ‘‘fictitious’’ heater at the lower one reveals that the heat flow across the pinch is also zero. Then, the heat recovery target for the unconstrained HENS problem still holds and the determination of the pseudo pinch temperatures and the UP-network heat recovery target are no longer necessary. By next solving Problems (P2.2) and (P2.3), it was confirmed that the minimum number of units in the upper and lower networks are equal to 8 and 4, respectively (see Fig. 6). This implies a couple of additional units at the upper network compared with the unconstrained problem because of the structural conditions specified by the designer. Example 4. Sometimes, cold process streams featuring target temperatures lower than or equal to ¹ are to be included in the stream set PPS deter1*/#) mining the pseudo-pinch points. In other words, they should be considered as possible destinations of the pinch heat flow transferred to the lower network. Example 4 is a particular case where such a situation arises. The problem data are those already given for Example 3 at Fig. 6 but the designer has now pro-

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Fig. 7. The least utility usage at each constrained pinchdefined network for Example 4.

hibited the match (H3/C3). The constrained least utility usage at each isolated pinch-defined network is depicted in Fig. 7. It can be observed that a ‘‘fictitious’’ heater over the cold stream C3 entirely running below the pinch is required. Therefore, the specified structural constraint brings about the need of an external heating source at the ‘‘isolated’’ lower network to meet the overall heat demanded by the cold stream C3. In this way, the pinch-defined lower network also becomes a heat sink, i.e. Q*(N )"E "458 BTU/h S L L and the constrained utility target rises as follows: Q*"QO#max(E , E ) S S U L "244#max(0, 458) "702 BTU/h, Q* "QO #max(E , E ) W W U L "173#max(0, 458) "631 BTU/h, Q*"max(E , E )"max(0, 458)"458 BTU/h. P U L Additional hot utility needs by forbidding the match (H3/C3) give rise to a finite pinch heat flow of 458 BTU/h to be destinated to the cold stream C3. Therefore, C3 should be included among the pseudopinch determining process streams being considered in the formulations of Problems (P2.1) and (P2.2). At the same time, the input temperature to the ‘‘fictitious’’ heater (¹mx ) is to be imposed as an upper PP,C3 bound on the inlet temperature of C3 to the extended N . The optimal solution to (P2.1) indicates that U a process heat flow equal to 756 BTU/h is to be recovered in the upper network. In turn, the solutions to (P2.2) and (P2.3) provide the network design shown in Fig. 8. Such a figure indicates that the optimal pseudo-pinch points are given by: ¹ "517°F, PP,H3 ¹ "497°F, ¹ "326°F and ¹ "497°F. PP,C2 PP,C3 PP,C4 The existence of a finite pinch heat flow reduces the number of units by 2.

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M.R. GALLI and J. CERDA¨

Fig. 8. The user-constrained optimal network design for Example 4.

10. Conclusions A general MILP sequential approach to the heat exchanger network synthesis problem has been presented. Its main purpose is to get a much greater involvement of the process engineer in the synthesis task by allowing him/her to initially specify some desired structural conditions effectively restraining the solution space within which the best HEN is sought. The proposed method permits the user to prescribe at the level of network the allowable preceding, succeeding or parallel units for any potential heat exchanger as well as the preferred input and/or output unit over any process stream. Since stream splitting is permitted, then a particular unit can now be preceded or succeeded by two or more heat exchangers. To handle multiple predecessors and successors and the notion of parallel units, a large set of new structural restrictions has been incorporated to the series MILP framework presented in Part I to produce an improved version also accounting for split networks. The increase in the number of problem constraints comes mostly from the sets of restrictions (7)—(12) and (17)—(20). In contrast to available HENS simultaneous approaches, however, the topology conditions specified by the designer produce a significant reduction on the size of the proposed MILP formulation. Because of the way it was conceived, the approach can be regarded as a target-based synthesis method that takes into account all the specified topology constraints from the beginning. Like any sequential synthesis method, the notion of ‘‘best design’’ is still given in terms of operating and fixed cost targets featuring different hierarchies, i.e. the least utility usage at the upper level and the minimum number of units at the lower one. The values of such constrained design targets usually differ from those obtained at pinch conditions. Nonetheless, they are determined by finding the least utility requirements at every individual pinch-defined network. Though lower than the one observed at series networks, the structural restrictions specified by the designer generally bring about a finite pinch heat flow even if the process energy

recovery is maximized under the topology conditions and HRAT"EMAT. As a result, a pseudo-pinch HENS problem is to be faced at the design stage. To simplify the search for the optimal network structure, the pseudo-pinch temperatures arising at MER conditions are first determined to decompose the HEN into independent networks. From the alternative pseudopinch points, we are interested in those favoring the merging of units through the pinch as much as possible. To this end, the maximum allocation of the pinch heat flow to heat matches already executed above the pinch is adopted as an additional design target. After finding the optimal pseudo-pinch points, the network structures are sequentially determined by proceeding from the top to the bottom one. In each case, the least number of units has been chosen as the problem objective. The example problems studied in Part I plus a new one involving seven streams have been solved but now allowing split designs. Despite the mathematical formulations present larger sizes, the optimal heat exchanger networks were still found in a reasonable CPU time by running the GAMS/ OSL solver on a SiliconGraphics Workstation. In a next paper, splitters and mixers will be handled as additional units to get control on the number and types of them over any process stream. Acknowledgements We are grateful to acknowledge financial support from ‘‘Consejo Nacional de Investigaciones Cientı´ ficas y Te´cnicas (CONICET)’’ and ‘‘Universidad Nacional del Litoral’’. References Brook, A., Kendrick, D. and Meeraus, A. (1992) GAMS: A ºser’s Guide, Release 2.25. Boyd and Fraser Publishing Company. Ciric, A.R. and Floudas, C.A. (1990) Application of the simultaneous match-network optimization approach to the pseudo-pinch problem. Comput. Chem. Engng 14, 241—250. Ciric, A.R. and Floudas, C.A. (1991) Heat exchanger network synthesis without decomposition. Comput. Chem. Engng 15, 385—396. Colbert, R.W. (1982) Industrial heat exchanger networks. Chem. Engng Prog. 78(1), 47—54. Colberg, R.D. and Morari, M. (1990) Area and capital cost targets for heat exchanger network synthesis with constrained matches and unequal heat transfer coefficients. Comput. Chem. Engng 14, 1—22. Daichendt, M.M. and Grossmann, I.E. (1994) Preliminary screening procedure for the MINLP synthesis of process systems—II. Heat exchanger networks. Comput. Chem. Engng 18, 679—709. Gundersen, T. and Grossmann, I.E. (1990) Improved optimization strategies for automated heat exchanger network synthesis through physical insights. Comput. Chem. Engng 14, 925—944. Linnhoff, B. and Hindmarsh, E.C. (1983) The pinch design method for heat exchanger networks. Chem. Engng Sci. 38, 745—763.

Synthesis of heat exchanger networks Linnhoff, B. et al. (1982) A ºser Guide on Process Integration for the Efficient ºse of Energy. U.K.: The Institute of Chemical Engineering. Trivedi, K.K., O’Neill, B.K., Roach, J.R. and Wood, R.M. (1989) A new dual-temperature design method for the synthesis of heat exchanger networks. Comput. Chem. Engng 11, 667—685. Yee, T.F. and Grossmann, I.E. (1990) Simultaneous optimization models for heat integration — II. Heat exchanger network synthesis. Comput. Chem. Engng 14, 1165—1184.

Appendix A. The network structural constraints over hot streams 1. ¸ower bound on the number of predecessors of a non-input exchanger (hi/cj) + XHim,ij5½ij!XHFij , j3C¼i, i3H (A.1) m|PCij where the summation is extended to every cold stream cm that may exchange heat with hi just before cj . If the cooler over hi, if any, is to be arranged last and in series (Case 2), then PCij must be substituted by PC*ij in (A.1). 2. ºpper bound on the number of predecessors of a non-input exchanger (hi/cj). Let NBi be the specified number of split streams into which the hot stream hi can at most be partitioned in the network. If the heat exchanger (hi/cj) arises at the network as a noninput unit over h , it can then be preceded by at most i NB units arranged in parallel. Otherwise, the neti work structure will be infeasible. Therefore, + XHip,ij4NBi (½ij!XHFij), j3C¼i, i3H. p|PCij (A.2) For Case 2, the set PCij is to be substituted by PC*ij . 3. ºpper bound on the number of successors to a heat exchanger (h /c ): i j + XHij,im4NBi ½ij, j3C¼i, i3H. (A.3) m|SCij For Case 2, the constraint (A.3) takes the following form: + XHij,im4NBi (½ij!XHij,iW), j3Ci, i3H. m|SCij* 4. ¸ower bound on the number of input units: + XHF 51, i3H. (A.4) ij j|CWi 5. ºpper bound on the number of input units. The maximum number of input units arranged in parallel over hi is NBi, unless Case 2 is being considered and the first unit is a cooler (hi/w). In such a case, (hi/w) must be the unique input unit on the stream h : i + XHFij4NBi , i3H. (A.5) j|CWi For Case 2, + XHFij4NBi (1!XHFiW), i3H. j|Ci

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6. Required conditions for the parallel arrangement of non-input heat exchangers. Similar mathematical conditions for the parallel arrangement of non-input units over a hot process stream hi can be written. (6.1) A pair of non-input units with a common predecessor over a hot stream hi will share the same set of successors: XH

5XH #XH #XH !2, ij,im ip,il ip,ij il,im m3SCij WSCil, p3PCijWPCil, j, l3C¼i ,

i3H.

(A.6)

(6.2) A pair of non-input units with a common successor over a hot stream h will share the same i set of predecessors: XHip,ij5XHil,im#XHij,im#XHip,il!2, p3PCij WPCil, m3SCij WSCil, j, l3C¼i,

i3H.

(A.7)

For Case 2, the stream sets MPCij, PCil, SCij, SCil, C¼iN are to be substituted by MPC*ij, PC*il , SC*ij, SC*il , CiN in both restrictions (A.6) and (A.7). (6.3) A couple of intermediate parallel exchangers with a common predecessor over a hot stream hi cannot precede or succeed themselves: XHip,ij#XHip,il#XHij,il#XHil,ij42, p3PCijWPCil, j,l3C¼i, i3H.

(A.8)

(6.4) A couple of non-input parallel exchangers with a common successor over a hot stream hi cannot precede or succeed themselves: XHij,im#XHil,im#XHij,il#XHil,ij42, m3SC WSC , j,l3C¼i, ij il

i3H.

(A.9)

For Case 2, C¼i is to be replaced by Ci and the supporting stream sets with an asterisk are to be used. Restrictions (A.6)—(A.9) are applied to any pair of parallel exchangers and therefore they hold whatever the number of parallel units over hi. 7. Required conditions for the parallel arrangement of input heat exchangers (7.1) Two parallel input units over a hot stream hi share the same set of successors: XHij,ik5XHFij#XHFil#XHil,ik!2, k3SCijWSCil, j,l3C¼i, i3H.

(A.10)

(7.2) A pair of input exchangers over a hot stream h cannot precede or succeed themselves: i XHFij#XHFil#XHij,il#XHil,ij42, l3PHijWSHij,

j3C¼i, i3H. (A.11)

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For Case 2, C¼i is to be replaced by Ci and the supporting stream sets with asterisks are to be used in both (A.10) and (A.11). 8. A predecessor of an exchanger (hi/cj) cannot be simultaneously its successor: XHij,im#XHim,ij41, m3PCijWSCij, j3C¼i, i3H.

th 5¹HI!(q /FCp ) ij i ij i (A.12)

For Case 2, the sets MPCij, SCi+ and C¼iN are to be replaced by MPC*, SC* and CiN, respectively. ij ij 9. A predecessor of the exchanger (h /c ) can never be i j arranged right after any one of its successors: XHim,ij#XHij,ir#XHir,im42, r3PCimWSCij, j,m3C¼i, i3H.

(A.13)

Appendix B. The ith-hot stream outlet temperature from the exchanger (hi/cj) Let th be the outlet temperature of a hot stream ij h from the exchanger (h /c ) whose value must belong i i j to the following range: ¹HO4th 4¹HI, i3H, (B.1) i ij i where ¹HI is the ith-hot stream inlet temperature to i the network and ¹HO is the ith-hot stream outlet i temperature from the network. for the pinch-defined upper network: ¹HIi"¹HSi;

¹HOi"¹pinch#*¹min,

for the pinch-defined lower network: ¹HIi"¹pinch#*¹min ;

¹HOi"¹H¹i ,

for the pseudo pinch-defined upper network: ¹HIi"¹HSi;

¹HOi"¹HPP, i ,

for the pseudo pinch-defined lower network: ¹HIi"¹HPP, i ;

temperature ¹HO can be guaranteed by writing the i outlet temperature constraints as (4) inequalities. To strictly meet THO, then the reverse sign (5) should be i used in the formulation of the outlet temperature restrictions for non-output units. Then,

¹HOi"¹H¹i ,

where *¹ is the minimum allowed temperature .*/ approach for any heat exchange and ¹ is the 1*/#) pinch temperature for cold streams. A common value of *¹ has been adopted for the determination of .*/ the utility usage target. However, it may change with the match during the design phase. Depending on the location of (h /c ) over the stream h , a different coni j i straint set will limit the value of th . Their expressions ij are given below. 1. From an input unit (h /c ) arranged in series. The i j outlet temperature of a hot process stream h from an i input exchanger (h /c ) arranged in series over h is i j i always defined by both the stream inlet temperature to the network (¹HI) and the temperature change i caused by the heat flow q exchanged with c . To ij j express such a relationship as an inequality, it becomes important to decide about the sign (4 or 5) to be used. The achievement of the stream target

A

B

!M 1!XHF # + XHF ij im m|CWi mEj j3C¼i, i3H.

(B.2)

For Case 2, the constraint (B.2) still holds. 2. From an input unit (h /c ) arranged in parallel with i j ‘‘p’’ exchangers:

A

BN

th 5¹HI# q # + q ij i ij im m|CWilpj

A

FCp !M 1#p i

B

!XHF ! + XHF # + XHF , im il ij i l | CWi m|CWlpj lbCWilpj lEj p l"1, 2, 2, , NCW !1 i p"1, 2, 2,(NB !1), j3C¼i, i3H. (B.3) i

A

B

where M is an arbitrarily large number and the subset C¼i LC¼i includes any p members of the set C¼i lpj different from c . Therefore, there will be as many j different subsets C¼i LC¼i as the combination of pj ‘‘p’’ in (NC¼) !1 streams, i.e. i p C¼i , for l"1, 2, 2, , lpj NC¼ !1 i p"1, 2, 2, (NB !1), i

A

B

where (NC¼) is the cardinality of the set C¼i. For i Case 2, the sets MC¼i , C¼iN are to be substituted by lpj MCi , CiN while the model parameter (NHS) is relpj j placed by NH . j 3. From a non-input exchanger (h /c ) arranged in i j series: th 5th #(q /FCp ) ij im ij i

A

B

!M 1!XH # + XH , im,ij im,il l | CWi lEj m3PC , j3C¼i, i3H. (B.4) ij For Case 2, the sets MPC , C¼iN are to be replaced by ij MPC*, CiN, respectively. ij 4. From a non-input exchanger (h /c ) arranged in i j parallel with one or more exchangers:

A

BN

th 5th # q # + q ij im ij il i l|CWlpj

A

FCp

i

!M 1#p!XH im,ij

B

! + XH # + XH , im,il im, in i l|CWlpj n | CWi i nb CWlpj nEi

Synthesis of heat exchanger networks

A

B

p , p"1, 2, 2,(NB !1), i NC¼ !1 i m3PC , j3C¼i, i3H. (B.5) ij For Case 2, the sets MC¼i , PC , C¼iN are to lpj ij substituted by MCi , PC*, CiN while (NC¼) is relpj ij i placed by NC . i 5. From an output or last exchanger (hi/cj). The last exchanger over a hot stream h does not feature any i successor at all. To ensure the achievement of the stream target temperature at the network ¹HO, therei fore, the outlet temperature constraint for the output unit (h /c ) is to be written as an (4) inequality in the i j following way: l"1, 2, . . . ,

th 4¹HO#M ij i

A

B

+ XH , j3C¼i, i3H, ij,im m|SCij (B.6)

where the summation over the successors to (h /c ) i j drops to zero for the last unit, and consequently the constraint (B.6) will hold. Otherwise, it becomes redundant. At each network below the pinch, a single cold utility is usually available. By assuming that the cooler on any stream h , if required, is always placed i both last and in series (Case 2), the number of structural variables MXHF , XH N can be decreased as ij ij long as all the variables involving the cooler as a preceding unit are no longer necessary. In such a case, the achievement of the target temperature for the hot stream h is guaranteed by imposing the following i constraint on the outlet temperature from the last stream/stream exchanger (h /c ): i j th 4¹HO#(q /FCp )#M ij i iW i

A

B

+ XH , ij,im m|SCij*

j3Ci, i3H, where the summation never includes a cooler as a possible successor. Moreover, the ith-stream outlet temperature from a cooler is always bounded by its target temperature at the network: th 4¹HO ½ , i3H. iW i iW 6. Overall heat balance for hot stream h . The set of i restrictions on the exit temperature of a hot stream h from an output unit (h /c ) must ensure the fulfili i j

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ment of the process specifications, i.e. th 4¹HO. The ij i overall heat balance for the hot stream h forces the i solution to satisfy such a constraint as a strict equality. + q "FCp (¹HI!¹HO), i3H. (B.7) ij i i i j|CWi It may happen that the designer only specifies an upper bound or a temperature range for the value of ¹HO. In such a case, the target temperature ¹HO i i becomes a problem variable but the proposed model still applies. Moreover, the constraint (B.7) remains unchanged. 7. Relationship between the outlet temperature of a hot stream h from two neighboring exchangers. If the i heat match (h /c ) is a successor to the heat match i j (h /c ) over the hot stream h , the outlet temperature of i m i h from (h /c ), th , is never greater than th : i i j ij im th 4th #M(1!XH ), ij im im,ij m3PC , j3C¼i, i3H. (B.8) ij It is indeed a redundant constraint being considered to get a faster convergence to the optimal solution. For Case 2, the sets MPC , C¼iN are to be replaced by ij MPC*, CiN. ij Appendix C. The mathematical modelling for the lower network design (P2.3) Min Nº " + + ½ ij LO d d i|H L j|CWL i s.t. equations (1)—(27), (A.1)—(A.13), (B.1)—(B.8) + q " + FCp (¹HI!¹HO) i i i d iw d i|H L i|H L ! + FCp (¹CO!¹CI), j j j d j|C L ¹HI"¹H ; ¹HO"¹HT; i3Hd, i PP,i i i L ¹CI"¹CS; ¹CO"¹C ; j3Cd, j j j PP,j L where the set Cd involves the cold streams featuring L ¹CS(¹C . In turn, Hd comprises the hot streams j PP,j L with target temperatures lower than ¹H . PP,i

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