- Email: [email protected]

Computers and Chemical Engineering 32 (2008) 2320–2332

Synthesis of multipass heat exchanger networks using genetic algorithms Jos´e Mar´ıa Ponce-Ortega a,b , Medardo Serna-Gonz´alez b , Arturo Jim´enez-Guti´errez a,∗ b

a Departamento de Ingenier´ıa Qu´ımica, Instituto Tecnol´ ogico de Celaya, Celaya, Gto. Mexico Facultad de Ingenier´ıa Qu´ımica, Universidad Michoacana de San Nicol´as de Hidalgo, Morelia, Mich. Mexico

Received 3 November 2006; received in revised form 28 November 2007; accepted 30 November 2007 Available online 1 February 2008

Abstract In this work, a methodology based on genetic algorithms (GAs) is developed for the optimal synthesis of multipass heat exchanger networks (HENs). The network model is based on a stagewise superstructure, and the problem of finding the optimum number of 1–2 shells in series of multipass heat exchangers is aided by an efficient optimization model that uses the standard FT design method. The proposed methodology allows for proper handling of the trade-offs involving energy consumption, number of units, number of 1–2 shells and network area to provide a network with the minimum total annual cost. The results of the examples show that the new approach is able to find more economical networks than those generated by other methods. © 2007 Elsevier Ltd. All rights reserved. Keywords: Heat exchanger networks; Multipass heat exchangers; Genetic algorithms; Superstructure

1. Introduction Research efforts on the synthesis of heat exchanger networks (HENs) have been significant in the last few decades (Furman & Sahinidis, 2002). As energy costs have continued to rise, the need for energy-efficient processes is an industrial priority. Most of the methods published for the HEN synthesis problem consider the use of single pass exchangers. In industrial practice, however, multipass heat exchangers are more commonly used in order to reduce the temperature cross in the exchangers and to meet space constraints. Multipass exchangers involve part countercurrent and part cocurrent flow, and each match within a network with multipass exchangers may require more than one shell. Ignoring the number of shells and the additional area required by such exchangers during the synthesis of HEN can lead to errors in capital cost estimations. The most common approach for the synthesis of multipass HEN has been based on the pinch point method, which decomposes the synthesis problem into two sequential steps that reduce

∗

Corresponding author. Tel.: +52 461 611 7575x139; fax: +52 461 611 7744. E-mail address: [email protected] (A. Jim´enez-Guti´errez).

0098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2007.11.012

the computational load of the problem (Ahmad, Linnhoff, & Smith, 1990; Ahmad & Smith, 1989). In the first step, targets are generated ahead of design for energy, heat transfer area, number of units and number of shells for the network for several values of the minimum allowed approach temperature, Tmin . After selecting an optimal or desired value for Tmin , a systematic method is used to design networks that meet the energy target at near-minimum capital cost. However, this approach assumes vertical heat transfer between the composite curves. As a result, the minimum area target is valid only if the heat transfer coefficients for all the streams are the same (Gundersen & Grossmann, 1990; Linnhoff & Ahmad, 1990; Serna & Jim´enez, 2004). If differences in heat transfer coefficients are significant, it is necessary to consider non-vertical heat exchanges to find a proper trade-off between network area and number of 1–2 shells. Furthermore, the use of composite curves does not allow the solution of problems with forbidden stream matches. Liu, Pehler, and Cahela (1985) and Trivedi, Roach, and O’Neill (1987) presented techniques for the synthesis of multipass HENs that have the same limitations as the pinch method and that do not improve the designs provided with the use of pinch concepts (Ahmad et al., 1990). Reddy, Rao, and Davies (1998) proposed a set of rules for designing multipass HEN

J.M. Ponce-Ortega et al. / Computers and Chemical Engineering 32 (2008) 2320–2332

Nomenclature a,b,c A Af Cexc Cexc cu Cexc hu CCU CHU CPS CU FCp fitness FT

cost function constants heat exchanger area annualization factor heat exchanger capital cost cooler capital cost heater capital cost cold utility cost hot utility cost {j|j process cold stream} cold utility heat capacity flowrate cost for the genetic algorithm correction factor for logarithm mean temperature difference g1, g2 feasibility for temperature differences for the hot and the cold side gcu feasibility for temperature differences for coolers ghu feasibility for temperature differences for heaters h film heat transfer coefficient HPS {i|i hot process stream} HU hot utility NOK total number of stages Ns number of 1–2 shells in series offspringqi,j,k offspring for the population in the inner GA offspringzi,j,k offspring for the population in the outer GA P heat exchanger thermal effectiveness P1,2 thermal effectiveness for a single 1–2 shell parentqi,j,k parent for the population in the inner GA parentzi,j,k parent for the population in the outer GA penalty penalty factor heat load for hot stream i qi qj heat load for cold stream j heat load for the match (i,j,k) qi,j,k p qi,j,k gene for the population qcui heat load for cold utility heat load for hot utility qhuj r penalty coefficient rr random number R heat capacity flowrate ratio ST {k|k stage in the superstructure, k = 1, . . ., NOK} ti,k temperature for stream i of the stage k temperature for stream j of the stage k tj,k T temperature difference TLM log-mean temperature difference Tmin minimum temperature difference TAC total annual cost TIN inlet temperature TOUT outlet stream temperature U overall heat exchanger transfer coefficient zi,j,k binary variable for process heat exchangers zcui binary variables for coolers zhuj binary variables for heaters

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Greek symbols α random number for the crossover operation τ random number for mutation operation Subscripts i process hot stream j process cold stream k subscripts for the stages 1, . . ., NOK max maximum min minimum

with a smaller number of 1–2 shells, while Galli and Cerda (2000) proposed a mixed integer linear programming model to minimize the number of shells of 1–2 type. These works do not necessarily provide networks with minimum total annual cost because they do not take into account the trade-offs among energy consumption, number of units, number of 1–2 shells, and network area. As noted by Furman and Sahinidis (2001), HEN problems are NP-hard. This property affects the use of deterministic techniques, such as the generalized Bender decomposition (Ciric & Floudas, 1991) and the outer-approximation method (Yee & Grossmann, 1990), because of their limitations to find the best solution for non-convex problems. Furthermore, when multipass exchangers are used, the situation becomes more complex because the determination of the number of shells requires the evaluation of the FT correction factor for every match in the network. One may consider alternative modeling strategies based on meta-heuristic (or stochastic) methods for this complex problem, such as simulated annealing (Athier, Floquet, Pibouleau, & Domenech, 1997), genetic algorithms (GAs) (Lewin, 1998; Ravagnani, Silva, Arroyo, & Constantino, 2005), and Tabu Search (Lin & Miller, 2004). The use of genetic algorithms has been reported for the synthesis of HENs based on 1–1 heat exchangers. Lewin (1998) used a binary GA for the structural optimization of the network, and each potential structure was optimized with a NLP model. The extension of that approach for multipass units is troublesome because the NLP model would be affected by the non-differentiable and non-convex equations needed for this case. Ravagnani et al. (2005) proposed a two-stage algorithm. In the first stage the Tmin is optimized using a continuous GA in combination with the problem table from the pinch point method (Linnhoff & Flower, 1978); in the second stage, the best value of Tmin is used to divide the network into two regions, and a continuous GA is used to get the optimal network for each of those regions. Although this approach yields maximum energy recovery networks using a sequential two-step procedure, it does not consider the trade-offs between the energy and capital cost of multipass HENs. The objective of this paper is to propose a new formulation for the synthesis of HENs with multipass heat exchangers based on genetic algorithms. The new formulation considers simultaneously utility and capital costs of the multipass heat exchanger units and overcomes the limitations of other proposed methods.

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The HEN representation is based on the stagewise superstructure model proposed by Yee and Grossmann (1990). The new formulation models the performance of 1–2 shell-and-tube heat exchangers with the standard FT design method (Kern, 1950). In addition, a new simple algorithm is used to calculate the optimum number of 1–2 shells in series, which can be solved with minor computational effort. The proposed method avoids the use of simplified formulations that could generate solutions not sufficiently close to the optimal one (see for example Ahmad, Linnhoff, & Smith, 1988) or procedures that require many repetitive calculations to design the individual multipass exchangers with minimum cost (see for example Moita, Fernandes, Matos, & Nunes, 2004). 2. Outline of the proposed algorithm For the HEN problem considered in this work, input data include the set of hot and cold streams with their supply and target temperatures and heat capacity flowrates, the hot and cold utility temperatures and their unit costs, and the film heat transfer coefficient for each stream. The problem then consists of determining the optimal heat exchanger network structure, the required hot and cold utilities, the heat loads for each unit and its number of shells. The objective is to minimize the total annualized investment and operating costs for the network. The solution to the HEN problem is based on GAs, which are meta-heuristic search techniques based on the mechanism of natural selection (Goldberg, 1989). They offer potential advantages for the solution of non-convex problems, both in terms of their robustness for the numerical solution and for the perspectives of getting a global optimum or a better near-optimal solution than algorithms based on gradient methods, at the expense of larger memory and CPU time requirements. One of the reasons for their effectiveness is that GAs can be directly applied to models with discontinuous, non-differentiable, or non-convex functions, and do not use any information on derivatives. In addition, because of their random nature, they are likely to escape from a local optimum point during the search process (Michalewicz, 1996). Thus, a complex problem such as the synthesis of HENs that includes multipass heat exchangers can be conveniently formulated within a GA framework. The HEN synthesis problem is modeled in this work with the superstructure approach suggested by Yee and Grossmann (1990) to include all possible combinations for energy integration. For a problem with NH and NC hot and cold process streams, the number of stages (NOK) required for the superstructure formulation is generally set as NOK = max{NH ,NC } (see Fig. 1). The problem is then to find the optimal network imbedded in such a superstructure.

thesis problem with multipass heat exchangers consists in the minimization of the total annual cost, which includes the annualized capital cost for heat exchangers between process streams, coolers and heaters, and the costs due to cooling and heating utilities. ⎛ Cexci,j,k + Cexc cui TAC = Af ⎝ i ∈ HPSj ∈ CPSk ∈ ST

+

⎞

Cexc huj ⎠ +

j ∈ CPS

i ∈ HPS

CCU qcui +

i ∈HPS

CHU qhuj

j ∈ CPS

(1) It should be noticed that the capital cost of the heat exchangers units explicitly takes into account the heat transfer area and the number of shells; such a detail contributes to the non-convexities of the model and has not been considered in previous works. Also, a simplified objective function has been used in other works, such as the minimization of the number of shells, which does not necessarily provide a network with minimum cost. The following constraints are needed as part of the model formulation (with all symbols defined in Nomenclature section). Energy balances for the process streams (TINi − TOUTi )FCpi = qi,j,k k ∈ STj ∈ CPS

+qcui ,

i ∈ HPS

(TOUTj − TINj )FCpj =

(2)

qi,j,k

k ∈ STi ∈ HPS

+qhuj ,

j ∈ CPS

Energy balance for each stage of the superstructure qi,j,k , k ∈ ST, i ∈ HPS (ti,k − ti,k+1 )FCpi =

(3)

(4)

j ∈ CPS

(tj,k − tj,k+1 )FCpj =

qi,j,k ,

k ∈ ST, j ∈ CPS

(5)

i ∈ HPS

Assignment of inlet temperatures TINi = ti,1 ,

i ∈ HPS

TINj = tj,NOK+1 ,

(6)

j ∈ CPS

(7)

Temperature feasibility 3. Mathematical model for networks with multipass heat exchangers The following sets are defined for the model formulation. HPS is a set that contains the hot process streams, CPS is a set for the cold process streams, and the set ST contains the stages of the superstructure. The objective function for the HEN syn-

ti,k ≥ ti,k+1 ,

k ∈ ST, i ∈ HPS

(8)

tj,k ≥ tj,k+1 ,

k ∈ ST, j ∈ CPS

(9)

TOUTi ≤ ti,NOK+1 , TOUTj ≥ tj,1 ,

i ∈ HPS

j ∈ CPS

(10) (11)

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Fig. 1. A heat exchanger network superstructure.

Heating and cooling duties

Constraints for the multipass heat exchangers:

(ti,NOK+1 − TOUTi )FCpi = qcui , (TOUTj − tj,1 )FCpj = qhuj , Logical constraints 1, for qi,j,k > 0 , zi,j,k = 0, for qi,j,k = 0

i ∈ HPS

j ∈ CPS

(12) (13)

Capital cost of multipass heat exchangers c Ai,j,k , Cexci,j,k = a + bNsi,j,k Nsi,j,k i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

i ∈ HPS, j ∈ CPS, k ∈ ST (14)

1, for qcui > 0 , i ∈ HPS 0, for qcui = 0 1, for qhuj > 0 zhuj = , j ∈ CPS 0, for qhuj = 0 zcui =

Heat transfer area qi,j,k , Ai,j,k = Ui,j FTi,j,k TLMi,j,k i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

(15)

(16)

Pi,j,k =

(23)

Heat capacity flowrate ratio (17) Ri,j,k =

ti,k − ti,k+1 , tj,k − tj,k+1 i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

g2i,j,k = (ti,k+1 − tj,k+1 ) − Tmin ≥ 0, (18)

(24)

Log-mean temperature difference TLMi,j,k =

gcui = (ti,NOK+1 − TOUTcu ) − Tmin ≥ 0, i ∈ HPS, zcui = 1

tj,k − tj,k+1 , ti,k − tj,k+1 i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

g1i,j,k = (ti,k − tj,k ) − Tmin ≥ 0,

i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

(22)

Thermal effectiveness

Temperature differences

i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

(21)

(ti,k − tj,k ) − (ti,k+1 − tj,k+1 ) ln[(ti,k − tj,k )/(ti,k+1 − tj,k+1 )]

for Ri,j,k = 1,

i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

(19)

(25)

and TLMi,j,k = ti,k − tj,k ,

ghuj = (TOUThu − tj,1 ) − Tmin ≥ 0, j ∈ CPS, zhuj = 1

(20)

for Ri,j,k = 1, i ∈ HPS, j ∈ CPS,

k ∈ ST, zi,j,k = 1

(26)

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Correction factor FTi,j,k

R2i,j,k + 1 ln[(1 − P1,2i,j,k )/(1 − Ri,j,k P1,2i,j,k )]

= , (Ri,j,k − 1) ln((2 − P1,2i,j,k (Ri,j,k + 1 − R2i,j,k + 1))/(2 − P1,2i,j,k (Ri,j,k + 1 + R2i,j,k + 1))) for Ri,j,k = 1, i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

and

(27)

√

FTi,j,k =

2P1,2i,j,k √ √ , (1 − P1,2i,j,k ) ln((2 − P1,2i,j,k (2 − 2))/(2 − P1,2i,j,k (2 + 2))) for Ri,j,k = 1, i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

Parameter P1,2 1 − Ri,j,k Pi,j,k 1/Nsi,j,k P1,2i,j,k = 1 − 1 − Pi,j,k 1 − Ri,j,k Pi,j,k 1/Nsi,j,k −1 Ri,j,k − , 1 − Pi,j,k

Because of the complexity of the model, which is highly nonlinear and non-convex, it is necessary to develop an efficient approach for its solution. 4. Solution methodology

for Ri,j,k = 1, i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

(29)

and Pi,j,k , (Pi,j,k − Nsi,j,k Pi,j,k + Nsi,j,k )

P1,2i,j,k =

for Ri,j,k = 1, i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

(30)

Feasible multipass constraint FTi,j,k ≥ 0.75,

i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1 (31)

Feasible number of shells Ns,mini,j,k < Nsi,j,k < ∞,

i ∈ HPS, j ∈ CPS, k ∈ ST,

zi,j,k = 1

(32)

Minimum number of shells in series Ns,mini,j,k = ln((1 − Ri,j,k Pi,j,k )/(1 − Pi,j,k ))

ln((1 − Ri,j,k + R2i,j,k + 1)/(Ri,j,k −1+ R2i,j,k + 1))

,

for Ri,j,k = 1, i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1 and Ns,mini,j,k

(33)

√ 2 Pi,j,k , = 2 1 − Pi,j,k for Ri,j,k = 1, i ∈ HPS, j ∈ CPS, k ∈ ST, zi,j,k = 1

(28)

(34)

Fig. 2 shows the major steps of the optimization procedure. The solution strategy decomposes the problem into two loops, each of them optimized with a GA. An outer loop consists of a binary GA, and is used to perform the structural optimization, since this problem involves the integer variables that define the positions of the heat exchangers within the network (zi,j,k ). The variable zi,j,k takes a value of 1 if the match between streams i and j is included in stage k of the network, and 0 otherwise. Fig. 3 illustrates the zi,j,k populations for two chromosomes (or individuals) for a problem with two hot and two cold streams. An inner-loop is used for the solution of a continuous GA for the qi,j,k variables that represent the amount of heat transferred between streams i and j in stage k. The proposed methodology begins by creating an initial population for the binary variables zi,j,k (set of network configurations); these initial networks can be randomly generated, or they can be given from the experience of the design engineer. For the implementations of this work, several population sizes were tried, and a size of I × J × K/3 individuals was finally used for the cases of study presented here, although this heuristic assignment could be reviewed for bigger problems, say more than ten streams, to produce a more manageable population size. Each individual (network) is analyzed for the maximum heat load that each exchanger can take, 0 if zi,j,k = 0 max qi,j,k = (35) min{qi , qj } if zijk = 1 where qi and qj are the overall head loads for any hot or cold process stream, respectively. For each proposed network, an inner GA is used to determine its optimal performance using the heat loads qi,j,k as search variables. An initial population for qi,j,k is generated using the limits imposed by Eq. (35). The population size used in the solution of the examples of this work for the inner GA was set as 100 individuals.

J.M. Ponce-Ortega et al. / Computers and Chemical Engineering 32 (2008) 2320–2332

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Fig. 2. Outline of the proposed algorithm.

Each network is then designed, for which a simple algorithm for the economic optimization of multipass 1–2 shell-and-tube heat exchangers in series is used. This optimization model is formulated using the FT design method (Kern, 1950), and includes a set of constraints that ensure feasible and practical heat exchangers. For each chromosome given by the genetic algorithm (i.e. a set of qi,j,k values), the evaluation of the fitness function is as follows.

1. The terminal temperatures of the HEN (ti,1 and tj,NOK+1 ) are fixed (Eqs. (6) and (7)). Intermediate temperatures (ti,k and tj,k ) are calculated from the heat loads given by the genetic operators and the heat balance for each match through Eqs. (4) and (5). 2. After all temperatures are known, the hot and cold utility loads (qhuj and qcui ) are calculated by the heat balance at the extremes of the superstructure using Eqs. (12) and (13).

Fig. 3. Examples for individual representation of populations zi,j,k .

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3. From the utilities heat loads (qhuj and qcui ), the binary variables zcui and zhuj are determined with Eqs. (15) and (16). 4. A feasibility test for temperature differences (g1i,j,k , g2i,j,k , gcui and ghuj ) for each match in the network is made using Eqs. (17)–(20). 5. If all feasibility constraints are satisfied go to step 6. Otherwise, at least one of the proposed exchangers is infeasible, in which case the multipass units are not designed and a penalty term is used, 2 penalty = r (g12i,j,k (T s)+g2i,j,k (T s)) i ∈ HPSj ∈ CPSk ∈ ST

(36)

where r is a penalty coefficient (which was set as 1 × 103 in the numerical applications of this work). The constraints g1i,j,k and g2i,j,k define the feasible temperature differences for the cold and hot side, respectively. Once all chromosomes of the population are tested, go to step 8. 6. Design the multipass heat exchangers of the network. Each unit is individually optimized to determine the number of shells in series that provides the minimum exchanger cost. The design algorithm consists of the following steps: a) Specify the initial capital cost of the multipass exchanger as Cexc1i,j,k = ∞ and Iflag = 1. b) Calculate the minimum number of shell in series (Ns,mini,j,k ) from Eq. (33) or (34). c) Choose the smallest integer value for the number of shells in series (Nsi,j,k ) satisfying Nsi,j,k > Ns,mini,j,k to ensure an initial feasible design. d) Calculate the heat exchanger thermal effectiveness (P1,2i,j,k ) from Eq. (29) or (30) for the current value of Nsi,j,k . e) Find the correction factor for the logarithm mean temperature difference (FTi,j,k ) from Eq. (27) or (28). f) If Iflag > 1, then go to step (h); else, go to step (g). g) Check condition given by Eq. (31). If condition does not hold, then set the number of shells in series as Nsi,j,k = Nsi,j,k + 1 and go to step (d). Else, a feasible design has been found. Set Iflag = 2 and go to step (h). h) Calculate the heat exchanger area (Ai,j,k ) with Eq. (22). i) Calculate the capital cost of the exchanger (Cexci,j,k ) with Eq. (21). If Cexci,j,k < Cexc1i,j,k , then set Nsi,j,k = Nsi,j,k + 1, Cexc1i,j,k = Cexci,j,k and go to step (d). Otherwise, the algorithm stops, with Cexc1i,j,k as the minimum cost for the heat exchanger with Nsi,j,k shells in series. 7. The total annual cost (TAC) associated with each chromosome is calculated (Eq. (1)). 8. The ﬁtness function (to be minimized) is calculated as follows, TAC, if g1i,j,k ≥ 0 and g2i,j,k ≥ 0 fitness = (37) TACmax + penalty, otherwise where TACmax is the maximum TAC value taken from all feasible solutions for the population. Thus, any infeasible solution will have a higher ﬁtness value that any feasible solution.

It should be mentioned that the design algorithm of step 6, which is applied to all units of a feasible network, is part of the contributions of this work, and has given excellent results in the numerical applications presented here. 5. Basic operation for the genetic algorithms The steps required by the GA to produce a new generation include the selection of the parent, and the operations of elite count, crossover and mutation. Some details of the implementation of the GAs are given below. For the inner loop (continuous GA to optimize qi,j,k ), the selection function for the parents for the next generation was based on the scaled values from the fitness scaling function using the roulette wheel proportional scheme (for details see Goldberg, 1989). A fitness scaling function converted the fitness scores returned by the fitness function to ranked values, such that individuals with lower scaled values had a higher probability to be selected. The elite count (i.e. the number of individual with the best fitness values in the current generation that are guaranteed to survive into the next generation) was set as 2 individuals. The crossover operation is defined as follows: p1 p1 p1 if parent1qi,j,k = (q1,1,1 , . . . , qi,j,k , . . . , qI,J,K ) and p2

p2

p2

parent2qi,j,k = (q1,1,1 , . . . , qi,j,k , . . . , qI,J,K ) parents,

and

represent

two

o1 , . . . , qo1 , . . . , qo1 ) offspring1qi,j,k = (q1,1,1 I,J,K i,j,k

o2 , . . . , qo2 , . . . , qo2 ) represent and offspring2qi,j,k = (q1,1,1 I,J,K i,j,k two offspring, then the genes of the offspring are produced p1 o1 = α through linear combinations such that qi,j,k i,j,k qi,j,k + (1 − p2

p2

p1

o2 = α αi,j,k )qi,j,k and qi,j,k i,j,k qi,j,k + (1 − αi,j,k )qi,j,k , where αi,j,k is a uniform random number between [−0.5,1.5] (see Michalewicz, 1996). The crossover fraction used in this work was 0.78 (Goldberg, 1989). For the mutation operation, a non-uniform scheme was p p p used. Let parentqi,j,k = (q1,1,1 , . . . , qi,j,k , . . . , qI,J,K ) be a chrop mosome and qi,j,k the element to be mutated (where p max ]) to produce chromosome offspring qi,j,k ∈ [0, qi,j qi,j,k = p p o (q1,1,1 , . . . , qi,j,k , . . . , qI,J,K ). Let τ be a random number that takes values of zero or one. Then, the mutated offspring was obtained by

p p max o = qi,j,k

qi,j,k + (number generation, qi,j − qi,j,k ), p qi,j,k

− (number generation,

p qi,j,k

− 0),

if τ = 0

if τ = 1

where (number generation, y) = y(1 − rr1−(number generation/100) ) where rr is a random number in the interval [0,1]. The mutation fraction used for the solution of the cases of study was 0.2 (Goldberg, 1989). As convergence criteria, a maximum number of 100 generations or the lack of improvement in the fitness function in 20 successive generations was used. After each solution of the inner loop, the algorithm checked the convergence criteria for the population of zi,j,k for the outer loop, which was no improvement in two successive genera-

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Fig. 4. Crossover scheme for the outer GA.

Fig. 5. Mutation scheme for the outer GA.

tions. If the stopping criteria was not satisfied, the algorithm produced a new populations of binary variables zi,j,k through a similar selection scheme as the inner GA but with an elite count of 1 (i.e. only the best network remained for the next generation). The position where the parents were divided to form a child was randomly generated. Fig. 4 shows a schematic representation of the crossover operation used in the outer loop from the two parents depicted in Fig. 3. The mutation scheme used in the outer GA is shown in Fig. 5. In this step, the algorithm generated random changes in the values of a particular position of one parent to generate a new offspring (i.e. a new network). 6. Numerical applications Four different example problems, which have been previously solved in the literature, are used to show the application of the proposed approach. Comparisons of the results using the proposed methodology with published solutions are included. The genetic algorithm was implemented using the MATLAB environment.

Example 1. Galli and Cerda (2000) reported this example, which is known as the 5SP1 problem in the HEN literature. The basic data are given in Table 1. The minimum temperature difference required at all points of the network is specified as 20 F. The cost function for heat exchangers is US$ 8600 + US$ 670Ns (A/Ns )0.83 (with A in ft2 ). The global heat transfer coefficient is 0.05 Btu/h ft2 F for heat exchangers and 1 Btu/h ft2 F for heaters. A maximum area of 50 ft2 was imposed on the shell size. The network obtained using the proposed algorithm is shown in Fig. 6. All the hot process streams are fully integrated with

Table 1 Stream data for Example 1 Stream

TIN (F)

TOUT (F)

FCp (Btu/h F)

H1 H2 C1 C2 C3 Steam

480 400 200 100 150 514

250 150 400 400 360 514

3.15 2.52 2.42 2.16 2.45

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Fig. 6. Optimum network obtained for Example 1. Table 2 Heat exchangers details for Example 1

Table 4 Stream data for Example 2

Exchanger

q (Btu/h)

Ns

A (ft2 )

Cost (US$ × 104 )

Stream

TIN (◦ C)

TOUT (◦ C)

FCp (kW/◦ C)

1 2 3 4 5 6

484 240.31 475.96 153.47 254.2 38.54

3 2 5 2 1 1

101.33 90.12 176.53 75.59 1.53 0.24

4.59 4.02 7.31 3.59 0.95 0.88

H1 H2 H3 C1 Steam

260.00 221.11 204.44 −3.889 226.00

43.33 110.00 43.33 215.556 226.00

10.55 26.376 15.823 36.9268

Table 3 Comparison of results for Example 1 Concept (ft2 )

Total area Number of units Number of shells Total capital cost (US$ × 105 )

Galli and Cerda (2000)

This work

433.84 8 13 2.247

445.34 6 14 2.134

the cold process streams. For the cold streams, C1 requires only one exchange with H1 to reach its target temperature. The other two cold streams need heating utilities to complete their thermal processing after the energy integration is accomplished. The external heating requirements amount to 292.74 Btu/h, which meets the minimum utility requirements. The HEN requires 6 units with 14 shells, for a total area of 445.3 ft2 . The investment required is US$ 2.135 × 105 ; the details of the heat exchangers are shown in Table 2. The configuration obtained in this work is different from that reported by Galli and Cerda (2000), who used a MILP formulation for this problem to find the minimum number of shells. As

shown in Table 3, the MILP model by Galli and Cerda (2000) yields a network structure with fewer shells (13 instead of 14) than the network in Fig. 6. However, their network has more units (8 instead of 6) and a higher total annual cost (5.29%) than the solution obtained in this work. In this case, the number of units has a higher weight on the total annual cost because of the contribution of fixed capital costs. This aspect compensates the increase in total area and number of shells of the network obtained in this work. In addition, the design of the individual units with the proposed methodology considers the optimal distribution of area among the feasible number of 1–2 shells in series. It should be noted that, in general, when the capital cost of each unit is given by an exponential cost law with a fixed capital cost and an exponent lower than 1 (i.e. a + bNs (A/Ns )c , with c < 1), designs that achieve maximum energy recovery with the lowest number of shells do not necessarily minimize the total annual cost. Example 2. This example was reported by Reddy et al. (1998), who used a heuristic approach to obtain a multipass HEN with the minimum number of shells. Table 4 gives the stream data for

Fig. 7. Optimum network for Example 2.

J.M. Ponce-Ortega et al. / Computers and Chemical Engineering 32 (2008) 2320–2332 Table 5 Details for the heat exchangers of Example 2

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Table 8 Details for the heat exchangers of Example 3

Exchanger

q (kW)

Ns

A (m2 )

Cost (US$ × 104 )

Exchanger

q (kW)

Ns

A (m2 )

Cost (US$ × 104 )

1 2 3 4 5 6 7 8

655.73 1198.3 1106.0 1732.1 925.02 1443.2 705.05 338.43

2 2 3 2 1 2 1 1

33.21 61.72 62.27 56.81 13.82 36.41 12.73 20.48

2.24 3.16 3.35 3.01 1.45 2.34 1.41 1.68

1 2 3 4 5 6 7

159.99 599.91 140.01 320.00 460.00 40.00 1340.10

1 1 1 1 1 1 1

74.26 170.48 37.59 50.13 100.75 4.48 178.29

3.25 5.62 2.21 2.58 3.94 1.09 5.80

Table 9 Comparison of results for Example 3

Table 6 Comparison of results for Example 2

Concept

Concept

Reddy et al. (1998)

This work

Total area (m2 ) Number of units Number of shells Total capital cost (US$ × 105 )

304.2 10 10 1.98918

297.49 8 14 1.8673

TIN

H1 H2 C1 C2 Hot utility Cold utility

120 160 10 80 180 10

Total area Number of units Number of shells Total cost (US$ × 105 )

Ahmad and Smith (1989)

This work

618.34 7 7 2.4587

616.10 7 7 2.4531

Table 10 Stream data for Example 4

Table 7 Stream data for Example 3 Stream

(m2 )

Stream (◦ C)

TOUT 60 40 100 115 179 20

(◦ C)

FCp

(kW/◦ C)

8.0 10.0 2.0 60.0

this case, which has been labeled as the 4SP2 problem in several papers. The capital cost function for each multipass exchanger is US$ 8600 + US$ 670Ns (A/Ns )0.83 (with A in m2 ). For area estimations, U values for exchangers and coolers is taken as 0.8517 kW/m2 ◦ C, and for heaters as 1.1356 kW/m2 ◦ C. This problem resembles a cooling network, in which one cold stream is used to process three hot streams. The network obtained using the proposed algorithm is shown in Fig. 7; to get this solution, it was necessary to increase the number of stages of the superstructure to 8; otherwise (local-optimum) solutions with higher values of the objective function would be obtained. The solution obtained in this work corresponds to an unpinched network with no splitting of the cold stream, which allows multiple matches between the cold stream and each of

H1 H2 H3 H4 H5 C1 C2 C3 C4 Hot utility Cold utility

TIN (◦ C) 120 80 135 220 135 65 75 30 60 250 15

TOUT (◦ C) 65 50 110 95 105 90 200 210 140 249 16

FCp (kW/◦ C) 25 150 145 10 130 75 70 50 25

h (kW/m2 ◦ C) 0.5 0.25 0.30 0.18 0.25 0.27 0.25 0.15 0.45 0.35 0.20

the hot streams. This leads to a configuration with three cyclic matches (H1–C1, H2–C1, and H3–C1). The details of the heat exchangers of the network are reported in Table 5. The optimum network for this case has 8 units and 14 shells, with a total area of 297.5 m2 and a total capital cost of US$ 1.867 × 105 . The network reported by Reddy et al. (1998) has a total capital cost of US$ 1.9892 × 105 . The solution obtained in this work provides savings in the investment for the network exchangers of 6.14% with respect to the solution by Reddy et al. (1998). Table 6 provides further information on the results for Example 2.

Fig. 8. Optimum network for Example 3.

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Table 11 Details for the exchangers of Example 4 Exchanger 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

q (kW) 794.33 1419.3 1307.3 898.4 3005.9 849.4 523.39 455.64 551.6 1650 300 2850 4950 4625 550

Table 13 CPU time required for the solution of each example

Ns

A (m2 )

Cost (US$ × 105 )

Example

CPU time (s)

1 1 2 2 2 1 1 1 2 1 1 1 1 1 1

167.07 250.12 794.96 247.20 1182.20 260.93 112.92 91.76 148.04 735.56 37.96 591.29 425.08 572.92 23.24

0.7818 0.9650 2.2202 1.0504 2.9452 0.9879 0.6530 0.5996 0.7980 1.8821 0.4506 1.6269 1.3175 1.5936 0.4039

1 2 3 4

1019 2251 337 4829

solution reported by Reddy et al. (1998). This is the result of the application of the design algorithm for individual multipass exchangers (step 6 of the methodology). The proposed methodology has provided for this problem the multipass network with the smaller total annual cost with the lowest number of units.

Table 12 Comparison of results for Example 4 Concept

Ahmad et al. (1990)

This work

Total area (m2 ) Number of units Number of shells Total capital cost (US$/yr × 105 ) Utility cost (US$/yr × 106 ) Total annual cost (US$/yr × 106 )

5907.07 15 20 4.344 1.145 1.579

5641.28 15 19 4.196 1.145 1.564

The total network area generally decreases as the number of units increases. However, in this case, the multipass network with the lowest number of units (8 units, Fig. 7) provides also the lowest area, although it requires four more shells than the

Example 3. The stream data of Table 7, taken from Ahmad and Smith (1989), were used for this problem. The film heat transfer coefficients for all streams are 0.2 kW/m2 ◦ C, and a minimum temperature difference of 20 ◦ C is specified. The cost function for heat exchangers is US$ 8600 + US$ 670Ns (A/Ns )0.83 (with A in m2 ) and the problem consists of minimizing the total capital cost for the exchanger subject to minimum utility consumption. Fig. 8 shows the network obtained with the application of the proposed algorithm and Table 8 shows the details for the heat exchangers. The network requires the splitting of the cold stream C2 (for matches with H1 and H2) to maximize the energy recovery. The HEN requires a total area of 616 m2 , with seven units, each one of them with one shell. The total capital cost for the exchangers is US$ 2.453 × 105 . As a comparison, the network reported by Ahmad and Smith (1989) has a total capital cost of US$ 2.459 × 105 , 0.22% higher than the one detected with GA. Table 9 shows a comparative summary of results. The minor savings provided by GA with respect to the solution reported by

Fig. 9. Optimum network for Example 4.

J.M. Ponce-Ortega et al. / Computers and Chemical Engineering 32 (2008) 2320–2332

2331

Fig. 10. Performance of the genetic algorithm for each example.

Ahmad and Smith (1989) can be related to the assumption of equal heat transfer coefficients for all streams, which explains why the method by Ahmad and Smith (1989) provided a solution quite close to the optimum one obtained in this work. Example 4. Ahmad et al. (1990) reported the stream data given in Table 10, which are taken as a basis to search for an optimal HEN with multipass 1–2 heat exchanger with the minimum total annual cost. The installed heat exchanger cost is calculated by US$ 30,800 + US$ 750Ns (A/Ns )0.81 , where A is in m2 . The plant life is 6 years, and the capital interest is 10% per year. The utility costs are US$ 110 and 10 kW−1 yr−1 for the hot and cold utilities, respectively. A minimum temperature difference of 17 ◦ C is specified. The network obtained using the proposed algorithm is reported in Fig. 9. Three hot streams (H3, H4 and H5) and one cold stream (C1) are fully integrated with process–process exchangers. From the hot streams that are fully integrated, two of them need to be split. H3 is split into three branches and H5 into two branches to reach their target temperatures. The network shows minimum consumption of utilities, and the details of the exchangers are given in Table 11. The network requires a total area of 5641 m2 (266 m2 less than the one obtained by Ahmad et al. (1990)), with 15 units and 19 shells (one less shell than the one obtained by Ahmad et al. (1990)). For the economics of the network, the total utility cost is US$ 1.145 × 106 yr−1 and the annualized capital cost is US$ 4.196 × 105 yr−1 , for a total annual cost of US$ 1.564 × 106 yr−1 . Ahmad et al. (1990) reported a total annual cost of US$ 1.579 × 106 yr−1 . As can be observed, the main cost for this example is due to utilities (73%); thus, total savings of the optimal network are only 0.96% with respect to the solution reported by Ahmad et al. (1990). Nonetheless, a reduction in area of 4.7% is observed from the

results of the GA application, along with one less shell. Table 12 summarizes the results for Example 4. The CPU time required for the solution of each example is reported in Table 13. The CPU times were obtained using a Pentium 4 processor with 2.8 GHz. Fig. 10 shows how the fitness of the GA changed with the number of generations for each case of study presented in this work. 7. Conclusions A new algorithm for the synthesis of heat exchanger networks with multipass exchangers has been presented. The algorithm considers simultaneously the optimization of the utility consumption and the capital cost of the multipass units. The capital cost of the multipass heat exchangers are considered without any simplification, taking into account the heat transfer area and the number of shells. In addition, operational constraints are incorporated into the model formulation. The algorithm uses two GAs for the optimization of the network, a binary GA for the structural optimization and a continuous GA for the optimization of the heat loads of the exchangers within the network. The algorithm based on GAs avoids the convergence difficulties associated with the introduction of non-convex equations. A new algorithm for the design of individual multipass heat exchangers with minimum cost is used within the inner GA. The proposed HEN algorithm avoids the problems associated with the search for a good initial point, or the dependence of such initial points to the experience of the design engineer, and its application to the examples presented here has shown better results than the ones previously reported using other methodologies. This work can be extended to include the detailed exchanger design as part of the search for the optimal network. Such an

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extension would require the incorporation of pumping costs on the network economics, and its solution would provide the optimal pressure drop for each process stream. References Ahmad, S., Linnhoff, B., & Smith, R. (1990). Cost optimum heat exchanger networks. 2: Targets and design for detailed capital cost models. Computers & Chemical Engineering, 14(7), 751–767. Ahmad, S., Linnhoff, B., & Smith, R. (1988). Design of multipass heat exchangers: An alternative approach. Journal of Heat Transfer-Transactions of The ASME, 110(2), 304–309. Ahmad, S., & Smith, R. (1989). Targets and design for minimum number of shells in heat exchanger networks. Chemical Engineering Research & Design, 67(5), 481–494. Athier, G., Floquet, P., Pibouleau, L., & Domenech, S. (1997). Synthesis of heat exchanger network by simulated annealing and NLP procedures. AIChE Journal, 43(11), 3007–3020. Ciric, A. R., & Floudas, C. A. (1991). Heat exchanger network synthesis without decomposition. Computers & Chemical Engineering, 15(6), 385–396. Furman, K. C., & Sahinidis, N. V. (2001). Computational complexity of heat exchanger network synthesis. Computers & Chemical Engineering, 25(9–10), 1371–1390. Furman, K. C., & Sahinidis, N. V. (2002). A critical review and annotated bibliography for heat exchanger network synthesis in the 20th century. Industrial & Engineering Chemistry Research, 41(10), 2335–2370. Galli, R. M., & Cerda, J. (2000). Synthesis of heat exchanger networks featuring a minimum number of constrained size shells of 1–2 type. Applied Thermal Engineering, 20(15–16), 1443–1467. Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Boston, MA: Addison Wesley Longman. Gundersen, T., & Grossmann, I. E. (1990). Improved optimization strategies for automated heat exchanger networks synthesis through physical insights. Computers & Chemical Engineering, 14(9), 925–944.

Kern, D. Q. (1950). Process heat transfer. USA: McGraw Hill. Lewin, D. R. (1998). A generalized method for HEN synthesis using stochastic optimization. II: The synthesis of cost-optimal networks. Computers & Chemical Engineering, 22(10), 1387–1405. Lin, B., & Miller, D. C. (2004). Solving heat exchanger network synthesis problems with Tabu Search. Computers & Chemical Engineering, 28(8), 1451–1464. Linnhoff, B., & Ahmad, S. (1990). Cost optimum heat exchanger networks. 1: Minimum energy and capital using simple models for capital cost. Computers & Chemical Engineering, 14(7), 729–750. Linnhoff, B., & Flower, J. R. (1978). Synthesis of heat exchanger networks. I: Systematic generation of energy optimal networks. AIChE Journal, 24(4), 633–642. Liu, Y. A., Pehler, F. A., & Cahela, D. R. (1985). Studies in chemical process design and synthesis. Part VII: Systematic synthesis of multipass heat exchanger networks. AIChE Journal, 31(3), 487–491. Michalewicz, Z. (1996). Genetic algorithms + data structures = evolution programs. Berlin: Springer-Verlag. Moita, R. D., Fernandes, C., Matos, H. A., & Nunes, C. P. (2004). A cost-based strategy to design multiple shell and tube heat exchangers. Journal of Heat Transfer-Transactions of The ASME, 126(1), 119–130. Ravagnani, M. A. S. S., Silva, A. P., Arroyo, P. A., & Constantino, A. A. (2005). Heat exchanger networks synthesis and optimization using genetic algorithm. Applied Thermal Engineering, 25(7), 1003–1017. Reddy, K. A., Rao, Ch. D. P., & Davies, G. S. (1998). Synthesis of multipass heat exchanger networks. AIChE Journal, 44(4), 999–1002. Serna, M., & Jim´enez, A. (2004). An area targeting algorithm for the synthesis of heat exchanger networks. Chemical Engineering Science, 59(12), 2517–2520. Trivedi, K. K., Roach, J. R., & O’Neill, B. K. (1987). Shell targeting in heat exchanger networks. AIChE Journal, 33(12), 2087–2090. Yee, T. F., & Grossmann, I. E. (1990). Simultaneous optimization models for heat integration. II: Heat exchanger network synthesis. Computers & Chemical Engineering, 14(10), 1165–1184.

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