Retrofit of Heat Exchanger Networks by a Hybrid Genetic Algorithm

Aug 29, 2014 - ABSTRACT: The retrofit of heat exchanger networks (HENs) is very important to ... proposed the application of the pinch concept in retr...
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Retrofit of Heat Exchanger Networks by a Hybrid Genetic Algorithm with the Full Application of Existing Heat Exchangers and Structures Xin-Wen Liu,*,†,‡ Xing Luo,†,§ and Hu-Gen Ma† †

School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China School of Chemical Engineering, Ningbo University of Technology, Ningbo 315016, China § Institute for Thermodynamics, Gottfried Wilhelm Leibniz Universität Hannover, D-30167 Hannover, Germany ‡

ABSTRACT: The retrofit of heat exchanger networks (HENs) is very important to high-energy consumption enterprises for saving energy, protecting the environment, and improving their market competitiveness. It is also an important branch of heat integration research. The retrofit of HENs is more difficult than grassroots design because of the greater number of constraints. In this paper, the novel mathematical model for HEN retrofit is based on the stepwise superstructure model from Yee et al. The existing heat exchangers and structures are applied fully in the retrofitted HEN, and the modification cost is decreased with energy saving. The hybrid genetic algorithm is used to solve the mixed integer nonlinear program problem. Two examples are given to show the improved effect of the novel retrofit method with the minimum of the total cost of new heat exchangers and newly added areas of the existing heat exchangers and repiping and utilities. This ΔTmin fixes the amount of energy recovery and predicts the additional amount of heat exchanger area. By following the design procedure, one can easily realize these targets. The drawback of this method is that the targets do not tell exactly where the additional areas are added and how many network restructure modifications such as repiping and rerouting are required. Asante and Zhu5 proposed a step-by-step interactive approach for heat exchanger network retrofit by combining the features of pinch and mathematical programming. They introduced a concept of network pinch that identified the bottleneck of the network and the most effective change. A MILP model is formulated for this purpose. Once a topology change has been accepted, be it the addition of a new exchanger or a new split or a relocation of an existing heat exchanger, the new topology will then be optimized as NLP. The procedure is repeated until the designer cannot find any more economical changes. The procedure identifies a single topology change at a time in a sequential manner that may in theory yield a suboptimal solution. Ma and Hui12 proposed a solution method based on mathematical programming for HEN retrofits. This is a two-step approach. The first step uses a constant approach temperature model to optimize the structure of the final HEN. A MINLP model is then used in the second step, which takes into account the actual approach temperatures, to finalize the design. Sorsak and Kravanja13 described a simultaneous MINLP optimization model for the retrofit of heat exchanger networks comprising different exchanger types. Li and Bai14 applied pinch theory, especially the optimal pinch approach temperature and grand composite curve, to predict the retrofit target and utilities for the retrofit of crude preheating HEN in Maoming Petrochemical Co. Li and Chang15 reported a novel pinch-based retrofit method for

1. INTRODUCTION The synthesis of heat exchanger networks (HENs) has been studied for more than 30 years. A variety of effective methods for the synthesis of HENs have been developed depending on different process demands. These methods can be classified into three categories, including the pinch technology, mathematical programming, and artificial intelligence methods. A new exciting direction for the mixed integer nonlinear program (MINLP) from mathematical programming is the logic-based method for optimization that promises to facilitate problem formulation and improve solution efficiency and robustness.1 To avoid being trapped into local optimal solutions, MINLP is simplified into linear programming (LP)2 or nonlinear programming (NLP) as described by Yee et al.3 Other attempts have been made to reduce the size of the problem by using “block”4 and “stage”5 concepts. In the search for the global optimal solution for MINLP problems, the stochastic method is more probable than the deterministic methods because of its random nature. Commonly, the algorithms used in the synthesis of HENs are genetic algorithms (GA)6−8 and simulated annealing (SA).9 With the development of the worldwide energy crisis and the greater requirements for energy conservation and environmental protection, more and more attention has been paid to the retrofit of an existing HEN. However, the infrastructure of the existing HEN in the retrofit scenario greatly complicates the design task.10 Therefore, the studies of the systematic and effective retrofit methods become more and more important. The understanding of the HEN retrofit has not yet reached a consensus in the academic world. Tjoe and Linnhoff11 first proposed the application of the pinch concept in retrofitting HENs. They suggested a two-step procedure, namely, target and design, which could guide engineers to conduct HEN retrofits systematically. Given the process stream data, costs and economical information, and current network conditions, such as energy consumption and exchanger surface areas, a target method generates a desired heat transfer approach temperature (ΔTmin). © 2014 American Chemical Society

Received: Revised: Accepted: Published: 14712

March 10, 2014 August 28, 2014 August 29, 2014 August 29, 2014 dx.doi.org/10.1021/ie501021m | Ind. Eng. Chem. Res. 2014, 53, 14712−14720

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2.1. Constraints. (1) Heat balance of each stream:

reducing the utility consumption rates in any given HEN design by determining the specific retrofit targets, i.e., the cross-pinch heat load by simple manual computation. Via the elimination of the heat load, the retrofitted HENs could be systematically produced with a revised version of the pinch design method. Smith et al.16 developed a two-level pinch approach for the optimization of all continuous variables, so that the existing HENs were exploited to make sure that the bottleneck was the network topology rather than the heat transfer area. Hu17 made a HEN retrofit by process simulation combined with the actual production and found the bottlenecks of heat exchanging, i.e., overly low exit temperature or high pressure, and then adjusted them, i.e., increasing the exit temperature demand to ensure steady operation and energy saving. Bochenek and Jezowski18 retrofitted the existing HENs with the standard heat exchangers by applying the genetic algorithms. The structures and parameters of an existing HEN were optimized by genetic operations. Regardless of the retrofit methods, their aims are all to acquire the best cost effectiveness while satisfying the inlet and exit temperature requirements of the cold and hot streams. Yee and Grossmann19 studied the redistribution of the existing heat exchangers based on the MILP model and obtained the matching schemes of streams and the distribution schemes of heat exchangers; however, not all the potential combinations are considered in this model, and the application of the existing heat exchangers is not considered in detail. Ciric and Floudas20 presented a two-stage retrofit method. In the first stage, the MILP model was developed for estimating the heat transfer area and ascertaining the process stream matching and heat exchanger matching schemes. In the second stage, on the basis of superstructure of heat exchanger networks, the NLP model was developed. Solution of the NLP model could yield the actual retrofit HEN, and the total modification cost reaches its minimum. Saboo21 presented an evolutionary strategy that is based on the nonlinear optimization, constrained MILP synthesis, and feasibility evaluation capabilities of RESHEX. Their procedure generates a number of successive retrofit design alternatives without the explicit consideration of economic data. Jones et al.22 presented a strategy for the retrofit of heat exchanger networks based on the generation of a number of alternative designs and their evaluation using simulation run. They selected the best design based on the full utilization of the existing equipment and the addition of area in some of the heat exchangers. In this paper, the retrofit of HEN is conducted with the existing heat exchangers and structures fully utilized. The objective function is the minimum of the total cost of the modifications and utilities in the retrofitted HEN. The elite strategy of the hybrid genetic algorithm is applied to acquire the full utilization of the existing heat exchangers and structure.

(THin, i − THout, i)fhi =

∑ ∑ qijk + qCU, i k

j

(i ∈ Nh , j ∈ Nc , k ∈ Ns) (TCout, j − TC in, j)fcj =

∑ ∑ qijk + qHU,j k

i

(i ∈ Nh , j ∈ Nc , k ∈ Ns)

where qCU,i and qHU,j are the loads of the cold and hot utilities, respectively, THin,i (thi,0) and TCin,j (tcj,Ns) are the inlet initial temperatures, THout,i and TCout,j are the target temperatures, f hi and fcj are the heat capacity flow rates of hot stream i and cold stream j, respectively, and qijk is the heat load of the corresponding heat exchanger. (2) Heat balance of each heat exchanger: (thi , k − thijk )fhijk = (tcijk − tcj , k + 1)fcijk = qijk (i ∈ Nh , j ∈ Nc , k ∈ Ns)

where thijk and tcijk are the exit temperatures of hot stream i and cold stream j, respectively, in the kth stage, f hijk and fcijk are the heat capacity flow rates, thi,k and thi,k+1 are the inlet temperatures, and tcj,k and tcj,k+1 are the exit temperatures. (3) Stream splits in the kth stage: Nc

∑ fhijk = fhi (i ∈ Nh , k ∈ Ns) j=1 Nh

∑ fcijk = fcj (j ∈ Nc , k ∈ Ns) i=1

(4) Energy balance calculation of each split in the kth stage: Nc

∑ thijkfhijk = thi , k+ 1fhi (i ∈ Nh , k ∈ Ns) j=1 Nh

∑ tcijkfcijk = tcj , kfcj (j ∈ Nc , k ∈ Ns) i=1

(5) Inlet temperature of each stream:

THin, i = thi ,0 (i ∈ Nh)

TC in, j = tcj , Ns (j ∈ Nc)

2. MATHEMATICAL MODEL FOR HEN RETROFIT The physical model for HEN retrofit in this paper is a stagewise superstructure model from Yee and Grossmann.23 It is assumed that the whole HEN is divided into Ns stages [5s = (k|k = 1, 2, ..., Ns)]. In general, Ns is the maximum of Nh and Nc; in other words, Ns = max{Nh, Nc}, where Nh and Nc represent the number of hot streams and the number of cold streams, respectively [ 5h = (i|i = 1, 2, ..., Nh), 5c = (j|j = 1, 2, ..., Nc)]. In each stage, the cold and hot streams are matched mutually by stream splits, so the maximal matching number is NhNc. Heaters and coolers are placed at the ends of the cold and hot streams, respectively.

(6) Feasible temperature constraints: thi , k ≥ thijk , tcj , k + 1 ≤ tcijk , THout, i ≤ thi , Ns , TCout, j ≥ tcj ,0 (i ∈ Nh , j ∈ Nc , k ∈ Ns)

where tcj,0 is the exit temperature of the jth cold stream in the first stage. (7) Load of the cold and hot utilities: (thi , Ns − THout, i)fhi = qCU, i (i ∈ Nh) (TCout, j − tcj ,0)fcj = qHU, j (j ∈ Nc) 14713

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zijk is also a discrete 0−1 variable and determines whether the corresponding stream should be split.

where thi,Ns is the exit temperature of the ith hot stream in the Ns-th stage. (8) Minimal temperature difference constraints of heat transfer. For a heat exchanger:

⎧ ⎪1, (fhijk < fhi) ∪ (fcijk < fcj) zijk = ⎨ (i ∈ Nh , j ∈ Nc , k ∈ Ns) ⎪ ⎩ 0, (fhijk = fhi) ∪ (fcijk = fcj)

thi , k − tcijk ≥ dtmin , thijk − tcj , k + 1 ≥ dtmin

The areas of the cooler (ACU,i) and heater (AHU,j) can be determined by the following equations:

(i ∈ Nh , j ∈ Nc , k ∈ Ns)

A CU, i

⎧ fhi(thi″ − tout, i) ⎪ , thi″ − tout, i > 0 ⎪ = ⎨ UCU, iΔtm,CU, i (i ∈ Nh) ⎪ ⎪ 0, ″ − ≤ th t 0 i out, i ⎩

AHU, j

⎧ fcj(tcout, j − tc″j ) ⎪ , tcout, j − tc″ > 0 ⎪ = ⎨ UHU, jΔtm,HU, j (j ∈ Nc) ⎪ ⎪ 0, tcout, j − tc″ ≤ 0 ⎩

For hot utilities: thu, j ,in − TCout, j ≥ dtmin , thu, j ,out − tcj ,0 ≥ dtmin (j ∈ Nc)

For cold utilities: thi , Ns − tc u, i ,out ≥ dtmin , THout, i − tu, i ,in ≥ dtmin (i ∈ Nh)

To determine the heat transfer areas Aijk and the heat capacity flow rates f hijk and fcijk, the exit stream temperature vector T″ = [t″1 ,t″2 ,...,t″Nh,t″Nh+1,t″Nh+2,...,t″Nh+Nc]τ is calculated by the explicit temperature solution of HENs proposed by Chen et al.,24 in which t1″, t2″, ..., tN″ h are the Nh exit stream temperatures of the hot process streams and t″Nh+1, t″Nh+2, ..., t″Nh+Nc are the Nc exit stream temperatures of the cold process streams. UCU,i and UHU,j are the total coefficients of heat transfer and are assumed to be constant. Constraint (3) is ensured with the corrections

where thu,j,in and thu,j,out are the temperatures of the hot utility for the jth cold stream at the inlet and outlet of the heater, respectively, tcu,i,in and tcu,i,out are the temperatures of the cold utility for the ith hot stream at the inlet and outlet of the cooler, respectively, and dtmin is the allowable minimal heat transfer temperature difference. (9) Other constraints: We use yijk, yCU,i, and yHU,j as discrete 0−1 variables to show whether heat exchangers, heaters, or coolers exist. ⎧ 1, Aijk > 0 ⎪ yijk = ⎨ (i ∈ Nh , j ∈ Nc , k ∈ Ns) ⎪ ⎩ 0, Aijk ≤ 0 yCU, i

yHU, j

⎧ ″ ⎪1, thi − tout, i > 0 (i ∈ Nh) =⎨ ⎪ ⎩ 0, thi″ − tout, i ≤ 0

fhijk =

fhik Nc * ∑ j = 1 fhijk

fcijk =

fcjk Nh * ∑i = 1 fcijk

(i ∈ Nh , k ∈ Ns)

(j ∈ Nc , k ∈ Ns)

in which a superscript asterisk denotes a parameter that should be modified. 2.2. Objective Function. To obtain the target HEN with optimal structure matches and the lowest level of consumption of utilities, the objective function is designed to include the cost of cold utilities and hot utilities, the cost of heaters and coolers, the cost of new heat exchangers, the cost of newly added areas of heat exchangers, and the cost of repiping. The cost of calculating the heat exchangers is

⎧ ″ ⎪1, tcout, j − tc j > 0 =⎨ (j ∈ Nc) ⎪ ⎩ 0, tcout, j − tc″j ≤ 0

where Aijk is the area of the heat exchanger matching the ith hot stream and the jth cold stream in the kth stage, thi″ and tcj″ are the exit temperatures of heat exchangers, and thout,i and tcout,j are the target temperatures. We further introduce a discrete 0−1 variable mijk to indicate whether the new heat exchanger should be bought. The discrete 0−1 variable nijk indicates whether the fixed cost of heat exchanger should be considered.

Cf + CAB

The first term, Cf, is the fixed cost of the heat exchanger, and the second term is the area cost of the heat exchanger. C, A, and B are coefficients of the area cost of the heat exchanger, the area of the heat exchanger, and the exponent of the area cost, respectively. CCU and CHU are the cost of the unit cold utility and hot utility, respectively. Cp is the cost of repiping a single stream. Therefore, the objective function is

e ⎧ ⎪1, A ijk > A ijk (i ∈ Nh , j ∈ Nc , k ∈ Ns) mijk = ⎨ e ⎪ ⎩ 0, Aijk ≤ Aijk

e ⎧ ⎪1, A ijk ≤ 0 (i ∈ Nh , j ∈ Nc , k ∈ Ns) nijk = ⎨ e ⎪ ⎩ 0, Aijk > 0

⎧ ⎪ min⎨∑ ∑ ∑ [(Cf nijk + CδAijk B)yijk mijk ] ⎪ ⎩ i j k

e e ⎧ ⎪ A ijk − A ijk , A ijk > A ijk (i ∈ Nh , j ∈ Nc , k ∈ Ns) δAijk = ⎨ e e ⎪ Aijk ≤ Aijk ⎩ Aijk ,

+

∑ [(Cf + CA CU, i B)yCU, i ] + ∑ [(Cf + CAHU, j B)yHUj ] i

where δAijk, Aijk, and are the areas of the needed added heat exchanger, the needed heat exchanger, and the existing heat exchanger at the node of ijk, respectively. Aeijk

+

⎫ ⎪

∑ CCUqCU, iyCU, i + ∑ C HUqHU, jyHU, j + ∑ ∑ ∑ (Cpzijk)⎪⎬ i

14714

j

j

i

j

k

⎭

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Figure 1. Flowchart of the hybrid genetic algorithm.

optimum and robustness.25 In this paper, the hybrid genetic algorithm is used to optimize the objective function of HEN retrofit. The flowchart of the hybrid genetic algorithm is given as Figure 1. In this work, we take a HEN as an individual and the genes are the heat transfer areas and the heat capacity flow rates of hot streams and those of cold streams of all possible heat exchangers in the HEN. The value of a gene as a heat transfer area can even be negative, which functions also as a binary variable; that is,

The retrofit of the existing HEN is conducted in this work on the basis of the mathematical model presented above.

3. HYBRID GENETIC ALGORITHM FOR HEN RETROFIT The mathematical model of HEN retrofit in this work is a complex MINLP problem, which is nonconvex, multimodal, and discrete. The traditional optimization methods have no way of solving it efficiently, while the hybrid genetic algorithm has been shown to have better capacity for global searching 14715

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When the main genetic operators are applied, the simple genetic algorithm is still not efficient for finding the global optimum. To improve the structural search ability of the algorithm, several strategies are combined with the genetic algorithm presented here, and some of them are introduced as described below. (1). Local Optimizing Strategy. Because the structure of a heat exchanger network is strongly coupled with the heat transfer areas and heat capacity flow rates, any random changes in these parameters would yield a large increase in the total annual cost. An individual with a good structure but unsuitable values of heat transfer areas and heat capacity flow rates would have a low fitness and would have lost the chance to survive. The genetic algorithm presented here is combined with a local optimization procedure to modify the individual parameters that is in fact a mutation toward a local optimum. Whether an individual parameter undergoes the modification is determined by a given probability PL ∈ [0, 1]. For each individual, if random (0,1) < PL and Aijk > 0, the classical Newton search method was used in the algorithm to modify the genes of an individual

a zero or negative value of heat transfer area means that there is no heat exchanger. Instead of the traditional binary system coding, float coding was used for the continuous variables. The fitness of each individual is determined with the reciprocal of its total annual cost and is adjusted with a linear transform as below f=

C −1 + Cmin −1 − 2(C −1)avg Cmin −1 − (C −1)avg

where C is the total annual cost of the individual, Cmin is the minimal total annual cost in the population, and (C−1)avg is the average value of the reciprocal of the total annual cost of the population. The main genetic operations used in this work are selection, crossover (structure crossover and parameter crossover), and mutation (structure mutation and parameter mutation). (1). Selection. Roulette wheel selection is used in this work. If the condition l−1

∑i = 1 fi N

∑i = 1 f

l

< random (0, 1) ≤

∑i = 1 fi N

∑i = 1 fi

(l = 1, 2, ..., N )

xi* = xi −

is satisfied, in which f i is the fitness of individual i, then individual l is selected. Then selection operator not only offers the parent individuals for crossover operator but also selects the individuals with a given selection probability of PS (e.g., PS = 0.2) and puts them into the offspring population directly. (2). Crossover. In the genetic algorithm, the structure crossover or parameter crossover was used by a stochastic choice according to the crossover probability PCX (PCX = 1 − PS). If random (0, 1) < PCX, the crossover operation will be performed. Two individuals p1 and p2 are selected as a pair of parent individuals. If random (0, 1) < PSC (e.g., PSC = 0.6), the structure crossover operation is conducted. The HEN structure will be randomly divided into two parts; the heat exchangers together with their parameters of one part will be taken from individual p1, and those of another part will be taken from individual p2. If random (0, 1) ≥ PSC, the parameter crossover operation is performed, which is a linear combination of the corresponding heat exchangers of two parent individuals. Each crossover operation will produce two child individuals, which are put into the offspring pool. The selection and crossover operations will be repeated until the number of the individuals in the offspring population reaches the original number of the population. (3). Mutation. The mutation operator is applied to the individuals in the offspring pool. If a randomly produced value SM = random (0, 1) < PSM (e.g., PSM = 0.005), an offspring individual undergoes a structure mutation. Each possible heat exchanger is checked with a probability of PGM (PGM = 0.01−0.1). When random (0, 1) < PGM, the sign of the area of the exchanger is changed, which means a binary value change. This binary value defines whether there is a heat exchanger. If PSM ≤ SM < PSM + PPM in which PPM is the probability of parameter mutation (e.g., PPM = 0.005), then this offspring individual undergoes a parameter mutation. Each gene is checked with the probability of PGM. When random (0, 1) < PGM, the value of the gene is changed randomly according to

dF(x)/dxi d2F(x)/dxi 2

(i = 1, 2, ..., n)

where xi is the ith positive gene to be modified, n is the number of positive genes, F(x) is the objective function, and xi* is the modified gene. The first and second derivatives of F(x) can be calculated numerically by d F (x ) ≈ [F(x1 , ..., xi + Δx , ..., xn) dxi − F(x1 , ..., xi − Δx , ..., xn)]/(2Δx)

d2F(x) ≈ [F(x1 , ..., xi + Δx , ..., xn) dxi 2 + F(x1 , ..., xi − Δx , ..., xn) − 2F(x1 , ..., xn)] /[(Δx)2 ]

(2). Simulated Annealing Algorithm. The simulated annealing algorithm can be applied to the population before the genetic operations of a generation. When the population comes into a new generation, each individual copies itself, and then, all genes of the copy are scattered by ⎧ s≥0 ⎪ x + (xmax − x)s , x* = ⎨ ⎪ ⎩ x − (xmin − x)s , s < 0

where s = random ( −1, 1)[1 − e−C1/(M + 1)]

and M is the number of generations. The copy will compete with its original. If the copy is better than the original, the copy will replace its original. Otherwise, whether the copy should be accepted is determined by the Metropolis acceptance rule. The copy will replace its original with a probability of P = e−(foriginal − fcopy )(M + 1)/ C2

x = xmin + random (0, 1) × (xmax − xmin)

where C1 and C2 are problem-dependent constants. They could be adjusted according to the number of accepted copies. (3). Structure Identification and Structure Control Strategy. In a simple genetic algorithm, the individuals are likely to have the same HEN structure and are trapped into

x ∈ ({A , fh , fc })

After the mutation operation, the offspring population becomes a new population in the next generation. 14716

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Table 1. Stream Data for Case 1 stream Fcp (kW/K) H1 H2 C1 C2 HU CU

30 15 20 40

Tin (K) Tout (K) 443 423 293 353 450 293

U (kW m−2 K−1)

$ kW−1 year−1

0.8 0.8 0.8 0.8 0.8 0.8

− − − − 80 20

333 303 408 413 450 313

Table 2. Relative Cost Data of Case 1 item

cost function ($)

cost of area for a new heat exchanger cost of area for an existing heat exchanger fixed cost for a new heat exchanger cost of moving an existing heat exchanger20 cost of repiping a single stream20

1300A0.6 1300X0.6 3000 300 50

Figure 3. Existing structure of HEN for case 1.

Table 3. Matching of the Existing Heat Exchangers in Case 1 heat exchanger

heat exchanger matches

area of heat exchanger (m2)

1 2 3 4 5

H2C2 H2C1 H1C1 H1CU HUC2

46.74 68.72 38.31 40.23 35.00

of HENs. With this strategy, multiple structures can be kept in a population. (4). Elite Strategy. The elite individual has the largest value of fitness function and the best adaptability. To make full use of the elite individual, after the production of a new population, the elite individual will be found and optimized locally by the steepest descent operator first and then the elite individual is sent into offspring directly. The elite strategy is useful for maintaining the good genes of the population from evolution.

4. RETROFIT OF A HEAT EXCHANGER NETWORK The diagram of HEN retrofitting is given as Figure 2. First, the structure of the existing HEN is analyzed. The heat exchangers in the existing HEN are numbered in sequence, and the areas of the corresponding heat exchangers are listed in the table. Second, the stage of the existing HEN is determined, and the number of stages is the maximum of the number of hot and cold streams. The stage of HEN is numbered from the left to right in sequence, and the maximal stage number is Ns. Third, the locations of the existing heat exchangers in HEN are represented by ijk values. Here

Figure 2. Diagram of HEN retrofit.

ijk = (k − 1)NhNc + (i − 1)Nc + j

a local optimum. Therefore, the maximal number of individuals in a population with the same structure can be fixed. If more individuals have the same structure, the worst ones will be replaced by the new individuals. This is in fact a kind of niche technique and was first used in the synthesis

The ijk value represents the heat exchanger matching hot stream i with cold stream j in stage k. Fourth, the existing heat exchanger Aeijk is introduced into the program of the hybrid genetic algorithm. The location of the existing heat exchanger 14717

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Table 4. Structure Match and Area Distribution of a Retrofit HEN for Case 1 match (stage)

Q (kW)

match of existing heat exchangers

existing area (m2)

modification area (m2)

cost ($)

H1C1 (1) H1C2 (1) H2C1 (1) H2C1 (2) H1CU HUC2

500 2231 1187 613 569 169

3 new 1 2 4 5

38.31 0 46.74 68.72 40.23 35

35.90 231.06 81.01 52.81 18.01 5.39

0 3.71 × 104 1.12 × 104 0 0 0

Figure 5. Existing structure of HEN for case 2.

Table 7. Matching of the Existing Heat Exchangers in Case 2

Figure 4. Retrofit structure of HEN for case 1.

Table 5. Stream Data for Case 2 stream H1 H2 H3 H4 H5 H6 H7 C1 C2 C3 HU CU

Fcp (kW/°C)

Tin (°C)

Tout (°C)

U (kW m−2 °C−1)

$ kW−1 year−1

470.00 825.00 42.42 100.00 357.14 50.00 136.36 826.09 500.00 363.64

140 160 210 260 280 350 380 270 130 20 500 20

40 120 45 60 210 170 160 385 270 130 499 40

0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

− − − − − − − − − − 60 5

item

cost function ($) 300A 300X 0 300 50

heat exchanger matches

area of heat exchanger (m2)

1 2 3 4 5 6 7 8 9 10 11

H7C2 H5C2 H4C2 H6C2 H2C3 H3C3 H1CU H4CU H7CU HUC1 HUC2

1.21 × 103 1.23 × 103 6.92 × 102 2.25 × 102 1.61 × 103 2.31 × 102 2.36 × 103 3.39 × 102 1.41 × 102 1.44 × 103 0.53 × 102

5. CASE STUDIES AND ANALYSIS Case 1. This example is taken from ref 19. The existing HEN contains two hot streams, two cold streams, and one group of hot and cold utilities. The initial and target temperatures of hot streams and cold streams, the coefficients of heat exchange, and the heat capacity flow rates are listed in Table 1. The relative cost data are listed in Table 2. The existing HEN structure is shown in Figure 3. It contains five matching units of cold and hot streams, including H2C2, H2C1, H1C1, H1CU, and HUC2, and the area distribution of heat exchangers is listed in Table 3. The 1.5 × 103 kW steam and 1.9 × 103 kW cooling water are needed in the existing HEN, and the cost of utilities is approximately $1.58 × 105 year−1. The result of the HEN retrofit demonstrates that the new structure of HEN contains six matching units, including H1C1, H1C2, and H2C1 in the first stage and H2C1, H1CU, and HUC2 in the second stage. The match of H1C2 in the first stage is the newly added heat exchanging unit. The match of H2C1 in the first stage is from the existing H2C2, and the area of the needed heat exchanger ranges from 46.74 to 81.01 m2. All the heat exchangers in the existing HEN are reused in the retrofit HEN. The modification cost including the addition of area, the movement of the heat exchanger, and the repiping is up to $1.12 × 104. The 169 kW steam and 569 kW cooling water are needed in the optimal retrofit HEN. The cost of utilities could be reduced to $2.49 × 104 year−1, and the total cost of modification is $4.83 × 104. Compared with the existing HEN, the retrofit HEN could be 84.24% more energy efficient. The payback period is ∼0.363 year, and the cost of retrofit and matches and area distribution are listed in Table 4. The retrofit HEN structure is illustrated in Figure 4.

Table 6. Relative Cost Data of Case 2 cost of area for a new heat exchanger cost of area for an existing heat exchanger fixed cost for a new heat exchanger cost of moving an existing heat exchanger20 cost of repiping a single stream20

heat exchanger

with an area Aeijk in the retrofitted HEN is determined by ijk. Fifth, the hybrid genetic algorithm is run, and the new heat exchanger or the new area does not need to be bought until the new individual Aijk is greater than Aeijk. That is to say, mijk = 0. If the new individual Aijk is greater than Aeijk, mijk = 1, and the needed added area is equal to δAijk. At the same time, if Aeijk exists, nijk = 0 and the fixed cost of the heat exchanger is not considered. Otherwise, nijk = 1, and the fixed cost of the heat exchanger needs to be considered. Finally, if the splits of streams exist, the cost of repiping should be considered, and if not, the cost of repiping should be zero. Therefore, the retrofitted HEN with the full use of the existing heat exchangers and structures could be obtained. 14718

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Table 8. Structure Match and Area Distribution of Retrofit HEN for Case 2 match (stage)

Q (kW)

H7C2 (1) H5C2 (2) H6C1 (3) H4C2 (3) H7C2 (3) H3C2 (4) H5C2 (4) H6C2 (4) H2C3 (5) H7C2 (5) H4C2 (6) H3C3 (6) H6C2 (7) H4C3 (7) H1CU H4CU H6CU H7CU HUC1

1.91 × 10 2.12 × 104 3.24 × 103 8.42 × 103 3.05 × 103 9.60 × 102 3.81 × 103 5.52 × 103 3.30 × 104 6.99 × 103 9.13 × 102 6.04 × 103 90.16 9.61 × 102 4.70 × 104 9.71 × 103 1.56 × 102 9.01 × 102 9.18 × 104

match of existing heat exchangers

existing area (m2)

modification area (m2)

cost ($)

1 2 new 3 new new new 4 5 new new 6 new new 7 8 11 9 10

1.21 × 10 1.23 × 103 0 6.92 × 102 0 0 0 2.25 × 102 1.61 × 103 0 0 2.31 × 102 0 0 2.36 × 103 3.39 × 102 53.3 1.41 × 102 1.44 × 103

1.21 × 103 1.36 × 103 2.13 × 102 6.92 × 102 1.44 × 102 53.48 1.57 × 102 2.25 × 102 1.61 × 103 3.46 × 102 56.86 2.27 × 102 5.92 17.02 2.36 × 103 3.39 × 102 2.3 16.20 1.40 × 103

0 3.89 × 104 6.40 × 104 0 4.32 × 104 1.61 × 104 4.71 × 104 0 0 1.04 × 105 1.71 × 104 0 1.78 × 103 5.11 × 103 0 0 300 0 0

4

3

Figure 6. Retrofit structure of HEN for case 2.

is $3.89 × 104. The cost of moving heat exchanger 11 to the match of H6CU is $300, so the total modification cost is $3.37 × 105. The 9.18 × 104 kW steam and 5.78 × 104 kW cooling water are needed in the optimal retrofit HEN, and the cost of utilities could reach $5.79 × 106 year−1. The cost of utilities could save $5.40 × 105 year−1. Compared with the existing HEN, the retrofit HEN could be 8.53% more energy efficient. The payback period is ∼0.62 year.

Case 2. This example is taken from ref 26. The existing HEN contains seven hot streams, three cold streams, and one group of hot and cold utilities. The initial and target temperatures of hot streams and cold streams, the coefficients of heat exchange, and the heat capacity flow rates are listed in Table 5. The relative cost data are listed in Table 6. The existing HEN structure is shown in Figure 5. It contains six matching units of cold and hot streams, including H7C2, H5C2, H4C2, H6C2, H2C3, H3C3, H1CU, H4CU, H7CU, HUC1, and HUC2, and the area distribution of heat exchangers is listed in Table 7. The 1.0 × 105 kW steam and 6.6 × 104 kW cooling water are needed in the existing HEN, and the cost of utilities is approximately $6.33 × 106 year−1. The result of the HEN retrofit demonstrates that the new structure of HEN contains 19 matching units (Table 8) and the matches of H6C1 and H7C2 in the third stage, H3C2 and H5C2 in the fourth stage, H7C2 in the fifth stage, H4C2 in the sixth stage, and H6C2 and H4C3 in the seventh stage are the newly added heat exchanging units (Figure 6). All the heat exchangers in the existing HEN are also reused in the retrofit HEN. The cost of the newly added heat exchangers and repiping is approximately $2.98 × 105. The area of H5C2 in the second stage added ranges from 1.23 × 103 to 1.36 × 103 m2, and its cost

6. CONCLUSIONS In this paper, a novel mathematical model has been proposed for the retrofit of HENs. It was obviously a MINLP problem. The mathematical model for HEN retrofit was built on the base of the stage-wise superstructure. The stream splits increased the probability of matching between cold and hot streams, so the costs of utilities would fall. In addition, the full application of the existing heat exchangers in the retrofit HEN also reduced the cost of buying new heat exchangers. To solve the MINLP problem, the hybrid genetic algorithm was introduced. A genetic algorithm could simulate the evolution phenomena in nature. It uses only the value of the objective function and therefore is suitable for solving the nondifferential problems. The genetic algorithm is combined with simulated 14719

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(15) Li, B.-H.; Chang, C.-T. Retrofitting heat exchanger networks based on simple Pinch Analysis. Ind. Eng. Chem. Res. 2010, 49, 3967− 3971. (16) Smith, R.; Jobson, M.; Chen, L. Recent development in the retrofit of heat exchanger networks. Appl. Therm. Eng. 2010, 30, 2281− 2289. (17) Hu, Y. X. The application of systematic optimization in the device of atmospheric and vacuum distillation and its HEN retrofit. Shanghai Huagong 2002, 17, 14−17. (18) Bochenek, R.; Jezowski, J. M. Genetic algorithms approach for retrofitting heat exchanger network with standard heat exchangers. 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering; Marquardt, W., Pantelides, C., Eds.; Elsevier: Amsterdam, 2006; pp 871−876. (19) Yee, T. F.; Grossmann, I. E. Optimization model for structural modifications in the retrofit of heat exchanger networks. Foundations of Computer-Aided Process Operations; 1987; pp 653−662. (20) Ciric, A. R.; Floudas, C. A. A mixed integer nonlinear programming model for retrofitting heat-exchanger networks. Ind. Eng. Chem. Res. 1990, 29, 239−251. (21) Saboo, A. N. RESHEX: An interactive software package for the synthesis and analysis of resilient heat exchanger networks I: Program description and application. Comput. Chem. Eng. 1986, 10, 577−589. (22) Jones, D. A.; Yilmaz, A. N.; Tilton, B. E. Synthesis techniques for retrofitting heat recovery systems. Chem. Eng. Prog. 1986, 82, 28−33. (23) Yee, T. F.; Grossmann, I. E. Simultaneous optimization models for heat integration-II. Heat exchanger network synthesis. Comput. Chem. Eng. 1990, 14 (10), 1165−1184. (24) Chen, D.-Z.; Yang, S.-S.; Luo, X. An explicit solution for thermal calculation and synthesis of superstructure heat exchanger networks. Chin. J. Chem. Eng. 2007, 15 (2), 296−301. (25) Luo, X.; Wen, Q.-Y.; Georg, F. A hybrid genetic algorithm for synthesis of heat exchanger networks. Comput. Chem. Eng. 2009, 33, 1169−1181. (26) Ahmad, S.; Petela, E. Supertarget: Applications software for oil refinery retrofit. AIChE Annual Meeting, March 29 to April 2, Houston, TX; 1987.

annealing algorithm, local optimizing strategy, structure control strategy, and elite strategy so that the structural search ability of the algorithm is conspicuously enhanced. The existing heat exchangers are introduced into the algorithm as elites. If the existing heat exchangers could be used in the corresponding matches, the cost of buying heat exchangers would be saved. If the areas of the existing heat exchangers could satisfy the demand of new matches, the areas of the corresponding heat exchangers need to be increased and the costs calculated as the modification costs. Therefore, in this algorithm, the existing heat exchangers and HEN structure could be used as much as possible.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: +86 574 87081240. Fax: +86 574 87081240. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Part of this project was supported by the Zhejiang Provincial Education Office Project (Grant Y201431687).

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REFERENCES

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