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Synthesis of flexible multi-stream heat exchanger networks based on stream pseudo-temperature with genetic/simulated annealing algorithms Ma Xiangkun a,1, Yao Pingjing a,*,2, Luo Xing b,3, Roetzel Wilfried b b

a Institute of Process Systems Engineering, Dalian University of Technology, Dalian 116012, China Institute of Thermodynamics, University of the Federal Armed Forces Hamberg, Hamberg D-22039, Germany

Received 7 November 2006; accepted 14 March 2007

Abstract The synthesis of flexible heat exchanger network (HEN) is generally regarded as an over-design of process units over a specified range of deviations in process parameters from their nominal values. The HEN obtained is more costly because of the over-design of HEN. The global solution to flexible design problems cannot be guaranteed because of the resulting non-differentiable, non-convex, max–min–max constraint of mixed integer nonlinear programming (MINLP) models. In this paper a new simultaneous two-stage strategy for synthesizing flexible multi-stream HEN (FMSHEN), optimized by genetic/simulated annealing algorithm (GA/SA), is presented. First, based on the pseudo-temperature enthalpy (T– H) diagram method, a new nonlinear programming (NLP) formulation involving all of the vertices of the polyhedral uncertainty region in the space of process parameters is proposed, with the supposition that the feasible region defined by the reduced inequality constraints is convex. An overdesign FMSHEN is obtained by optimizing the stream heat transfer temperature difference contribution. Secondly, the optimal structure of the over-design FMSHEN is retained and each heat exchanger area is modified in order to make the FMSHEN less costly. The total annual cost of MSHEN, obtained from the simulation of MSHEN according to the vertices of the polyhedral uncertain region, is regarded as an objective function, and GA/SA is adopted for optimizing the heat exchanger areas. The remarkable feature of the strategy is that the size and the complexity of the problem are reduced significantly and with more probability of locating the global solution. Finally, two examples are illustrated to demonstrate the performance of the strategy for the synthesis of flexible multi-stream heat exchanger networks. # 2007 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Flexibility; Multi-stream heat exchanger network; Genetic/simulated annealing algorithm; Pseudo-temperature; Heat exchanger area optimization

1. Introduction Heat exchanger networks (HEN) are mostly synthesized under the assumption of a specified operating condition and many methods have been developed in the last few decades (Zamora and Grossmann, 1998; Linnhoff and Flower, 1978; Wei et al., 2004; Yee and Grossmann, 1990). A detailed review on HEN synthesis methods proposed in the 20th century can be found in the recent literature (Furman and Sahinidis, 2002). HEN has to adapt to inevitable parameter variations because of the changes in the operating and economic environments of a process, such as supply temperatures, flow rates, seasonal variation, etc. Even for the optimally synthesized HEN based on a certain operation * Corresponding author. E-mail address: [email protected] (Y. Pingjing). 1 2 3

condition, these changes are to deprive the design of its thermodynamic and economic efficiency. In general, flexibility used to be defined as the ability to operate feasibly in the region spanned by the deviations in process parameters from their nominal values (Swaney and Grossmann, 1985). Therefore, in order to carry forward these methods into the plants that are actually implanted in practice, flexibility issues should be considered during the synthesis and design of HEN. The synthesis problem of flexible multi-stream HEN (FMSHEN) has been to find the network of heat exchangers which has the least annualized cost, which remains feasible against deviations in the space of uncertainty parameters and which brings each process stream from its given source temperature to a specified target temperature. Since a systematic method was presented by Marselle et al. (1982), the vast number of works are concentrated in the field of flexible analysis and retrofit, and so the investigation on the synthesis of FHEN is limited, though some methods have been developed (Aaltola, 2002; Cerda et al., 1990; Chen and Hung, 2004, 2005;

0368-1653/$ – see front matter # 2007 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jcice.2007.03.004

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Nomenclature heat transfer area (m2) exponent for area cost area cost coefficient per unit cost for cold utility ($/kW) per unit cost for hot utility ($/kW) fixed cost for heat exchanger ($) heat capacity flow rate of cold stream (kW/K) heat capacity flow rate of hot stream (kW/K) heat transfer film coefficient (kW/m2 K) sum of number of MSHEs and number of independent TSHEs in enthalpy interval k LMTDijk logarithmic mean temperature difference of match (i, j) in enthalpy interval k (K) Nk number of enthalpy intervals Nv number of vertex points NC number of cold process streams NH number of hot process streams qcu heat exchanged between hot stream i and cold utility cu (kW) qhuj heat exchanged between cold stream j and hot utility hu (kW) qijk heat exchanged between hot stream i and cold stream j in enthalpy interval k (kW) tcjk temperature of cold stream j at interval k (K) tciijk inlet temperature of cold stream j of heat exchanger between hot stream i and cold stream j at interval k (K) tcoijk outlet temperature of cold stream j of heat exchanger between hot stream i and cold stream j at interval k (K) thik temperature of hot stream i at interval k (K) thiijk inlet temperature of hot stream i of heat exchanger between hot stream i and cold stream j at interval k (K) thoijk outlet temperature of hot stream i of heat exchanger between hot stream i and cold stream j at interval k (K) TH,in TC,in inlet temperature of hot and cold stream, respectively (K) TH,out TC,out outlet temperature of hot and cold stream, respectively (K) C DTC; heat transfer temperature difference contribution j of cold stream j (K) H DTC;i heat transfer temperature difference contribution of hot stream i (K) U overall heat transfer coefficient (kW/m2 K) A B C Ccu Chu Cf cpc cph h Lk

Subscripts cu cold utility hu hot utility i hot process stream or hot utility j cold process stream or cold utility k index for enthalpy interval l MSHEs or independent TSHEs p index for vertex point

Sets CU HU LK NC NH NK NV

cold utility hot utility {ljl is a MSHE or a TSHE, l = 1, 2, . . ., Lk} {jjj is a cold process stream} {iji is a hot process stream} {kjk is a enthalpy interval on T–H diagram} {vjv is a vertex point of the polyhedral uncertainty region}

Floudas and Grossmann, 1986, 1987; Halemane and Grossmann, 1983; Konukman et al., 2002; Papalexandri and Pistikopoulos, 1994). To distinguish from the synthesis of HEN based on a certain operation condition, the FHEN must be feasible within the range of uncertainty parameters. In other words, the synthesis of FHEN can be regarded, in essence, as an over-design of process units over a specified range of deviations in process parameters from their nominal values. Cerda et al. (1990) and Li et al. (2004) developed a method of FHEN synthesis based on the operating point that has the maximum energy recovery. Based on the stage wise superstructure model proposed by Yee and Grossmann (1990) and Konukman et al. (2002) constructed the FHEN including the vertices of the polyhedral uncertainty parameters region with the assumption that the feasible region is convex. Chen and Hung (2004) introduced the maximum area consideration in the objective function to design the candidate of HEN. The HEN obtained remains feasible within the range of uncertainty parameters, but it is an over-design for the most operating points within the ranges of uncertainty parameters. That is to say, there are relatively excessive heat exchanger areas for most of the operating points and these can be substituted by the hot/cold utilities. So during the synthesis and design of FMSHEN a trade-off between investment cost and utility cost should be considered. Most of the work in the literature is concentrated on the over-design of HEN, and it appears that the literature lacks consideration of a more efficient design after the over-design step to obtain a lower minimum annual cost. Moreover, as the simultaneous MINLP model involving multiple operating point conditions proposed by Aaltola (2002), a global optimum cannot be guaranteed due to non-differentiable, non-convex and the possible presence of many local optima. As the problem size and the number of operating points increase the simultaneous MINLP model will become more complex and more difficult to solve and the calculation time will increase exponentially. In this paper, a new simultaneous two-stage strategy for the synthesis of FMSHEN is presented, which is optimized by GA/ SA. In the first stage, with the supposition that the feasible region defined by the reduced inequality constraints is convex, a new NLP formulation that involves all of the vertices of the polyhedral uncertainty region in the space of process parameters based on the pseudo-temperature enthalpy (T–H) diagram method, is used to obtain an optimal over-design MSHEN that remains feasible over a range of uncertainty

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parameters. In the second stage, the optimal structure of the over-design MSHEN is retained and each heat exchanger area is modified in order to make the FMSHEN less costly. The total annual cost of MSHEN, obtained from the simulation of heat exchanger network according to the vertices of the polyhedral uncertain region, is regarded as an objective function, and GA/SA is adopted for optimizing the heat exchanger areas. Two examples are illustrated to demonstrate the performance of the strategy for the synthesis and design of flexible multi-stream heat exchanger networks. 2. Definition of FMSHEN synthesis problem In this paper only the steady-state synthesis problem of HEN is taken into account. The following assumptions are made: the exchangers are of countercurrent type; no phase changes are allowed; constant heat capacities; utility duties can be adjusted. The FMSHEN synthesis problem can be stated as follows: A set of process streams are given and for each is specified a nominal source temperature, a nominal target temperature, a nominal heat capacity flow rate and a nominal heat transfer coefficient. Also specified are a set of hot utilities HU, a set of cold utilities CU, the nominal values of uncertainty parameters and their range of deviations from their nominal values. Cost data are also given for possible heat transfer equipment, including fixed and area dependent cost factors. The objective then is to determine the MSFHEN configuration with minimum annual cost. The solution defines the network by providing the following items: (1) minimum utility required that can be adjusted; (2) stream matches and the number of process units; (3) area of each process unit; (4) network configuration and flow rates for all branches.

323

Du are the scaled deviations from u0. The region D describes the scaled hyper-rectangle that is required to be within the feasible region defined by Eqs. (2) and (3). As was pointed out in the literature (Grossmann and Floudas, 1987; Halemane and Grossmann, 1983; Papalexandri and Pistikopoulos, 1994; Swaney and Grossmann, 1985) this problem is difficult to solve directly since it involves a max– min–max constraint that leads to a non-differentiable global optimization problem. For a HEN synthesis problem where the uncertain parameters are considered to be the source stream temperatures only, the feasible region defined by the reduced inequality constraints is convex. So the critical point that limits the solution lies at the vertices of the polyhedral region of uncertainty region (Chen and Hung, 2005; Floudas, 1995; Konukman et al., 2002). If the FHEN is valid on all of the vertices of the polyhedral uncertainty region, the FHEN will be valid in the over range of uncertainty parameters. For a HEN problem with N uncertainty parameters, there are Nv = 2N vertices in the polyhedral region of uncertainty parameters. Fig. 1 shows the geometric interpretation of the feasible region, polyhedral region of uncertainty, reduced inequalities, nominal and vertex operating points and critical vertex that limits flexibility for a two-dimensional (N = 2) hypothetical problem for which the total number of vertices, Nv, is four. So in this paper the source–stream temperatures are considered to be uncertainty parameters only, the over-design MSHEN is obtained involving all of the vertices of the polyhedral uncertainty region. The pseudo T–H diagram method proposed by Xiao et al. (2006) is extended for over-design of MSHEN and a new NLP model involving all of the vertices of the polyhedral uncertainty region is presented, in which the size and the complexity of the problem are reduced significantly.

3. Over-design of MSHEN The FMSHEN synthesis problem can be formulated as a MINLP. The general formulation for FMSHEN synthesis problem and flexibility can be stated as min f ðd; x; u; p; uÞ d;x;u

(1)

3.1. Stream heat transfer temperature difference contribution value Stream temperature is an important parameter in heat transfer and stream pseudo-temperature represents a heat transfer energy level that is determined in terms of its factual temperature and

s.t. hðd; x; u; p; uÞ ¼ 0

(2)

gðd; x; u; p; uÞ 0

(3)

u 2 D ¼ fujðu0 Du Þ u ðu0 þ uþ Þg

(4)

where d is the vector of exchanger area, x the vector of state variables (outlet temperatures of streams), u the vector of control (heater and cooler loads), p the vector of fixed parameters (heat-transfer coefficients), and u is the vector of uncertain parameters (in this work, source–stream temperatures). Eq. (1) is the objective function and can be formulated in terms of total annualized cost or total utility consumption. Eqs. (2) and (3) are the vectors of equality and inequality constraints. Eq. (4) describes the range of the uncertain parameters as deviations from nominal values, u0, where Du+ and

Fig. 1. Geometric representation of critical vertex point.

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corresponding heat transfer temperature difference contribution value. The stream pseudo-temperature is defined as follows: for hot stream i : for cold stream j :

H ; TiP ¼ T i DTC;i C T Pj ¼ T j þ DTC; j

(5)

where TiP is the pseudo-temperature of hot stream i, T Pj the pseudo-temperature of cold stream j, Ti the actual temperature H of hot stream i, Tj the actual temperature of cold stream j, DTC;i the heat transfer temperature difference contribution value of C hot stream i, and DTC; j is the heat transfer temperature difference contribution value of cold stream j. The traditional heat transfer temperature difference contribution value is deduced in terms of maximum principle and minimum heat transfer area of per unit heat load of HEN, the economic influence is considered but the thermodynamic influence is not. In the heat exchange process system, the exergy loss of heat transfer is different. The exergy loss at a high temperature level is smaller when compared to a loss at a low temperature level for the same stream heat transfer temperature difference contribution value. It also means that for the same exergy loss, the temperature drop at a high temperature level is larger than that at a low temperature level. Therefore, equal exergy loss was introduced into the stream temperature difference contribution value to make the distribution of exergy flow more reasonable (Sun and Yao, 2002): rﬃﬃﬃﬃﬃﬃﬃﬃ 2 ai hr T i DTCi ¼ DTCr ¼ Cai 1=2 hi 1=2 T i 2 (6) ar h i T r 2 where DTCi and DTCr refer to heat transfer temperature difference contribution of stream i and reference stream r, K, respectively. hi and hr refer to heat transfer film coefficient of stream i side (including the fouling factor) and reference stream r side, kW/m2K, respectively. ai and ar refer to cost per unit heat transfer area of stream i and reference stream r, $/m, respectively. Ti and Tr refer to thermodynamic mean temperature of stream i and reference stream r, K, respectively. In general, DTCr , ar and hr can be determined by the statistical method or by experience, ai and hi can also be accurately calculated or estimated by experience. The stream heat transfer temperature difference contribution value is an approximation because Eq. (6) is not rigorous and should span an indefinite interval, for example, 20% of its approximation value. Bad choices of the stream heat transfer temperature difference contribution value can make HEN costly, so in this paper the stream heat transfer temperature difference contribution values are taken as the decisive variables. The stream heat transfer temperature difference contribution values and the MSHEN structure are optimized simultaneously. 3.2. Pseudo T–H diagram method extended to over-designing MSHEN The hot and cold heat transfer temperature difference contribution values are the main factors that restrict the matches between hot streams and cold streams and have a great impact on the annual cost of HEN. If the suitable stream heat transfer

temperature difference contributions are determined, reasonable match for heat transfer between the streams can be achieved through the T–H diagram which can effectively use temperature levels and reasonably distribute heat transfer temperature differences and heat loads. Moreover, countercurrent heat transfer in heat exchangers can be achieved by the T–H diagram in principle. In other words, the minimum thermodynamic area network can be attained under the given heat loads. The extension of the pseudo T–H diagram method to overdesign MSHEN is straightforward. The MSHEN structure for each vertex point is constructed by pseudo T–H diagram method, then the over-design MSHEN is combined considering the MSHEN structure for each vertex point. The steps of the over-design HEN are as follows: (1) The mass flow-rates, start and target temperatures, thermal capacity, heat transfer film coefficients, heat transfer temperature difference contribution values, uncertainty parameters for the streams are provided. (2) The Nv operating points that can substitute the variety of the uncertainty parameters are obtained on the vertices of the polyhedral uncertainty parameters region. (3) For each vertex operating point: (a) The hot and cold stream temperatures of the operating points are shifted by their pseudo-temperatures, and their heat loads are unchanged. (b) As can be seen in Fig. 2, the hot and cold composite curves are drawn on the pseudo T–H diagram, and then both of them close up horizontally until they touch each other. The touch point is referred to as the pinch. Once the pinch is located, the hot and cold utility usages and the system heat recovery are determined simultaneously. (c) The composite curves are divided into enthalpy intervals and the smaller enthalpy intervals are combined according to the regulated constraints of minimum enthalpy interval, then the composite curves are divided into k enthalpy intervals. According to the reverse operation of the procedure of making composite curves, the vertical matches between the hot streams and cold streams in every enthalpy intervals are determined. Heat exchangers including the same hot stream (or cold stream) in every enthalpy intervals are merged into a multi-stream exchanger. Then the MSHEN for one vertex operating point is constructed completely. (4) The MSHENs for all vertex operating points are combined to construct the over-design MSHEN, that is to say, the overdesign MSHEN structure includes all possible matches in the MSHEN for all vertex operating point. Heat exchanger areas are fixed once the MSHEN is chosen. In order to guarantee the MSHEN is feasible for each vertex point condition the maximum heat exchanger area for the same match is chosen relative to the Nv MSHENs. The maximum area of heater/ cooler and the maximum utilities for the same stream is also chosen. So the over-design MSHEN is obtained. The objective function for the simultaneous NLP model will be defined as the sum of all capital costs for heat exchanger

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(3) Area for each exchanger for each vertex point: (10) Akl ¼ maxðAkl p Þ; p 2 NV (4) Heat loads for hot and cold utilities for each vertex point: qcu;i ¼ maxðqcu;i p Þ;

qhu; j ¼ maxðqhu; j p Þ;

(11)

i 2 NH; j 2 NC; p 2 NV (5) Areas for hot and cold utility equipment for each vertex point: Acu;i ¼ maxðAcu;i p Þ;

Ahu; j ¼ maxðAhu; j p Þ;

(12)

i 2 NH; j 2 NC; p 2 NV 4. Heat exchanger area optimization for over-design MSHEN

Fig. 2. T–H diagram based on pseudo-temperature.

areas and units and operating costs for utilities. Cost equation of heat exchanger is Cf + CAB. Then the objective function for the simultaneous NLP model (P1) is shown in Eq. (7). Objective function: X X X C cu;i qcu;i þ C hu; j qhu; j þ Cf cu;i Mincost ðP1Þ ¼ i

þ

X

j

Cf hu; j þ

XX

j

þ

X

k B Chu; j Ahu;hujj

þ

i

Cf kl þ

l

XX

j

k

X

cui C cu;i ABcu;i

i

C kl ABklkl

(7)

l

where Ai jk ¼

qi jk ; ðU i jk LMTDi jk Þ

LMTDi jk ¼ Ui j ¼

½thii jk tcoi jk ½thoijk tciijk ; lnððthii jk tcoi jk Þ=ðthoi jk tcii jk ÞÞ

hi h j ; hi þ h j

The reasonable matches in the structure of the over-design MSHEN which is feasible for the over range of uncertainty parameters are obtained as described in Section 3. The optimal MSHEN, although flexible, is more costly because each heat exchanger area is a maximum value relative to the Nv MSHEN (see Fig. 3). There are a great deal of operating points in the range of uncertain parameters during the operation of MSHEN. During the synthesis and design of MSHEN not only should the most common operating point be considered but also the most uncommon operating points as well. The Aovd of optimal FMSHEN is an over-design for the operating point in the range of uncertain parameters, whether or not such a MSHEN exits as in Fig. 3: the Aopt of MSHEN cannot satisfy the requirements of heat exchange for some operating points and the residual heat requirements are complemented by the hot and cold utilities, e.g. operating point a; the Aopt of MSHEN is close to the area of the optimal MSHEN of some operating point, e.g. operating point b; but the Aopt of MSHEN is an over-design for the operating points, e.g. operating point c; that is to say, the obtained over-design MSHEN is not the most optimal MSHEN and a trade-off between

i 2 NH; j 2 NC; k 2 NK; l 2 LK; cu 2 CU; hu 2 HU Constraints: (1) Overall heat balance for each stream for each vertex point: ðT H;in;i; p T H;out;i; p Þcphi X X ¼ qi jk p þ qcu;i p ;

i 2 NH; p 2 NV;

k 2 NK j 2 NC

ðT C;out; j; p T C;in; j; p Þcpc j X X ¼ qi jk p þ qhu; j p ;

j 2 NC; p 2 NV

(8)

k 2 NK i 2 NH

(2) Approach temperature for each exchanger for each vertex point: thii jk p tcoi jk p dtmin ; thoi jk p tcii jk p dtmin p 2 NV

(9)

Fig. 3. Total optimal area for each operating point in the range of uncertainty.

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equipment cost and utility cost after the over-design MSHEN should be considered. The area of each heat exchanger should be adjusted to minimize the annual cost of MSHEN. In this paper the structure of optimal over-design MSHEN is retained and each heat exchanger area is optimized by GA/SA to find the FMSHEN with lower minimum annual cost. Once the structure and heat exchanger area of MSHEN are chosen the equipment cost is fixed subsequently, but the utility cost of each operating point is not the same. The summation of the equipment cost and the utility cost is the total annual cost of one operating point. The operating proportion of each operating point is different and so the total annual cost can be obtained by Eq. (13): COSTtotal ¼

1 X COST p H p

(13)

p¼1

where COSTtotal refers to the total annual cost of FMSHEN, COSTp refers to the annual cost of operating point p, Hp refer to the operating proportion for 1 year for point p. Expensive calculations are needed when the total annual cost of FMSHEN is calculated through Eq. (13) and so a simplified method is proposed. In this paper the vertices of the polyhedral uncertainty region are used to represent the operating points in the range of uncertainty parameters and each operating point is assigned an operating proportion to describe the approximate operation of FMSHEN in 1 year. The total annual cost of FMSHEN is calculated by Eq. (14): X COSTtotal ¼ COST p H p (14) p 2 NV

The utility of each operating point is obtained by the simulation of each operating point, and the area of utility equipment is the maximum value relative to the Nv MSHEN. 4.1. Simulation of MSHEN The simulation problem of MSHEN can be stated as follows: supply temperatures, target temperatures, heat capacity flow-

rates and heat transfer coefficients of NH hot process streams and NC cold process streams, heat each exchanger area and the flow-rates of stream branches. A set of hot utilities HU and a set of cold utilities CU, and their corresponding temperatures are given. Then the objective is to determine the temperatures that enter into the heater/cooler and the hot/cold utility usages. A problem of two hot streams and two cold streams is adopted as an example and the structure of the optimal overdesign MSHEN is shown at Fig. 4. The MSHEN is divided into two enthalpy intervals and the vertical matches are achieved at every interval. The procedure of MSHEN simulation is as follows. Given are the cold temperatures that enter into the heaters tc1,0, tc2,0, then the outlet temperatures of exchangers in interval 1 are calculated with the known supply temperatures of hot streams. The stream branches are mixed to get the outlet temperature th1,1, th2,1, tc1,1, tc2,1 of interval 1. Then the outlet temperatures of interval 2 are calculated subsequently and the allowable error is calculated by Eq. (15): f ¼

X

2

ðT C;in; j; p tc j2 Þ ;

j 2 NC; p 2 NV

(15)

j 2 NC

If the limited error is less than the allowable error the simulation of MSHEN is accomplished. If the temperature of hot and cold streams cannot reach the target temperatures the heat loads are provided by hot/cold utility and the relevant utilities are determined. 4.2. Model for heat exchanger area optimization of MSHEN The main goal of heat exchanger area optimization is to delete the deficiencies of over-design and get the FMSHEN with the lower minimum total annual cost. The residual heat exchanger area is substituted by the utility loads that can be adjusted. The steps of area optimization for FMSHEN are as follows:

Fig. 4. The structure of an over-design MSHEN.

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(1) The mass flow rates, start and target temperatures, thermal capacities, heat transfer film coefficients for the streams, uncertainty parameters, and optimal MSHEN after P1 are provided. (2) The Nv operating points are obtained according to the vertices of the polyhedral uncertainty region and each operating point is assigned an equal operating period proportion. (3) The simulation of each operating point is done on the MSHEN by the method described in Section 4.1. The hot and cold utility usages for each operating point are also obtained. (4) The total annual cost of MSHEN is calculated by Eq. (16). The objective function of heat exchanger area optimization is similar to the objective function of over-design MSHEN. The model formulation (P2) is shown in Eq. (16). Objective function: XX XX Mincost ðP2Þ ¼ H p Ccu;i qcu;i p þ H p Chu; j qhu; j p p

þ

i

X

Cf cu;i þ

i

þ

X

X j

cui Ccu;i ABcu;i þ

þ

k

Cf hu; j þ

X

i

XX

p

j

XX k

Cf kl

l

B

Chu; j Ahu;hujj

j

Ckl ABklkl ;

i 2 NH; j 2 NC; k 2 NK;

l

p 2 NV

(16)

where

(3) Usages of hot and cold utilities for each vertex point: ðthoik p thoi Þ ¼ qcu;i p ; ðtco j tco j0 p Þ ¼ qhu; j p ;

i 2 NH; p 2 NV; j 2 NC; p 2 NV

(19)

(4) Utility areas of hot and cold streams for each vertex point: Acu;i ¼ maxðAcu;i p Þ; i 2 NH; p 2 NV; Ahu; j ¼ maxðAhu; j p Þ; j 2 NC; p 2 NV(20) 5. Genetic/simulated annealing algorithm Genetic/simulated annealing algorithm (GA/SA) starts with several groups of random individuals and is expected to have a good performance to find the global optimum solution. Simulated annealing algorithm (SA) is used to help limit the tendency for premature convergence and to help escape from local optima. Under this consideration, a GA/SA hybrid was developed (Wei et al., 2004; Yu et al., 2000) and demonstrated successfully in their studies. Moreover, in order to improve the accuracy and the convergence speed of GA/SA, float coding was used instead of the traditional binary system coding. Furthermore, OCX (orthogonal crossover) and EC (effective crowding) operators for improving the performance of GA were introduced in GA/SA in addition to the basic crossover and mutation operators. GA/SA shows that fast convergence and good results can be achieved as in the literature (Wei et al., 2004; Xiao et al., 2006; Yu et al., 2000). In this paper the proposed algorithm is also used to get the global solution. 6. Strategy for FMSHEN Using GS/SA

Ai jk ¼ qi jk ; U i jk LMTDi jk

The strategy aims to obtain the FMSHEN that remain feasible in the range of uncertainty parameters with the minimum annual cost. The procedures for the FMSHEN synthesis and design are as follows:

qi jk ¼ cphi jk ðthii jk thoi jk Þ ¼ cpci jk ðtcoi jk tcii jk Þ; LMTDi jk ¼ Ui j ¼

327

½thii jk tcoi jk ½thoi jk tcii jk ; lnððthii jk tcoi jk Þ=ðthoi jk tcii jk ÞÞ

6.1. Stage 1: over-design of MSHEN

hi h j hi þ h j

Constraints: (1) Overall heat balance for each stream for each vertex point: ðT H;in;i; p T H;out;i; p Þcphi X X ¼ qi jk p þ qcu;i p ;

i 2 NH; p 2 NV;

k 2 NK j 2 NC

ðT C;out; j; p T C;in; j; p Þcpc j X X ¼ qi jk p þ qhu; j p

j 2 NC; p 2 NV

(17)

k 2 NK i 2 NH

(2) Temperatures of hot and cold streams for each vertex point: thoik p T H;out;i; p ; i 2 NH; p 2 NV; tco j0 p T C;out; j; p ; j 2 NC; p 2 NV

(18)

(1) Generation of the stream heat transfer temperature difference contribution value: Heat transfer temperature difference contribution value for every stream is calculated by Eq. (6) based on the nominal temperature of hot/cold streams and is generated randomly in a given indefinite interval of its original temperature difference contribution value. For example, if H the original temperature difference contribution value DTC;i of stream i, calculated by Eq. (6), and the indefinite interval of temperature difference contribution value of stream i is H H ½0:8DTC;i ; 1:2DTC;i , then the equation of randomly generated temperature difference contribution value is H00 H DTC;i ¼ random ð0:8; 1:2Þ DTC;i . (2) Generation of initial population: The over-design MSHEN is obtained by a pseudo T–H diagram method as described in Section 3.2. The MSHEN can be taken as an initial individual. The stream heat transfer

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temperature difference contribution values are chosen repetitively and then the initial population is obtained. (3) Solution to over-design MSHEN: The stream heat transfer temperature difference contribution values are optimized by GA/SA to obtain the optimal MSHEN structure. 6.2. Stage 2: heat exchanger area optimization (1) Generation of the heat exchanger area:Each heat exchanger area is generated randomly in a given indefinite interval of its original exchanger area that is obtained in the overdesign MSHEN. For example, the assumed indefinite interval of heat exchanger area maybe [0.8Aijk,1.0Aijk], and the equation of randomly generated heat exchanger area is A00i jk ¼ random ð0:8; 1:0Þ Ai jk . Then a new MSHEN is obtained with the optimal structure of MSHEN. (2) Generation of initial population: The simulation of each vertex operation point is done on the MSHEN and the total annual cost is calculated as an initial individual fitness. Each heat exchanger area is chosen repetitively and then the initial population is obtained. (3) Solution to FMSHEN: The heat exchanger areas are optimized by GA/SA to get the optimal FMSHEN with the lower minimum annual cost. The outline of the two-stage strategy for the synthesis and design problem of FMSHEN is shown in Fig. 5. In over-design MSHEN formulation (P1), the optimization variables are the stream heat transfer temperature difference

contributions, so the number of optimization variables is (NH + NC). In addition, because MSHEN of each vertex operation is obtained by T–H diagram method, binary variables are eliminated. Compared with the stage-wise superstructure model of Yee et al. (1990) that has 3 NH NC NK optimization variables (NK is the number of stages in the superstructure model, NK = max(NH, NC)), the size and complexity of the problem are reduced significantly. Moreover, the initial population generated in both stages is a feasible solution, so the calculation efficiency of GA/SA can be enhanced remarkably. 7. Case study In this section, two examples are solved to describe the performance of the method and the search algorithms in this paper. The results are represented by the HEN graphs. Process exchanger are represented by vertical lines and circles and are matched with the streams, each heat exchanger area is underlined (m2), with the heat exchangers being represented by E. 7.1. Example 1 This example is taken from Cerda et al. (1990) and consists of two hot streams, two cold streams and one hot and one cold available utility. A range of uncertainty parameters is defined by allowing the source temperatures of the streams H1 and C1 to vary 10 K from their nominal values. Stream data are shown in Table 1.

Fig. 5. The outline of the two stage strategy for FMSHEN synthesis.

M. Xiangkun et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 321–331 Table 1 Stream data and optimal heat transfer temperature difference contributions for Example 1 Stream

Tin (K)

Tout (K)

Fcp (kW/K)

h (kW/m2K)

Optimized DTi (K)

H1 H2 C1 C2 S1 W1

512 10 512 379 10 399 600 293

393 421 423 523 600 313

7.032 8.440 6.096 10.000 – –

1 1 1 1 2.4 2.4

9.1076 9.7308 10.2712 9.4479 – –

Note: Heat transfer coefficient U = 1/(1/hh + 1/hc) (kW/m2K), for all matches, the exchanger costs are 8000 + 1000 [area (m2)]0.6 $/a; cost of hot utility = 80 $/kW a; cost of cold utility = 20 $/kW a.

Table 2 Source temperature (K) at the vertex operating point for Example 1

H1 C1

Point 1

Point 2

Point 3

Point 4

512 + 10 379 + 10

512 + 10 379 10

512 10 379 + 10

512 10 379 10

7.1.1. Stage 1: over-design of MSHEN With two uncertainty parameters in this example, the total number of vertices in the polyhedral region of uncertainty parameters is Nv = 22 = 4. The source temperatures for all the vertices are listed in Table 2. The over-design MSHEN model (P1) based on the four vertex operating points is formulated. Then the hot/cold stream temperature difference contribution values are

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Table 3 The result of the over-design MSHEN for Example 1 Exchanger

Aijk

Point 1

Point 2

Point 3

Point 4

Selected values

E1 E2 E3 E4 E5 HU1 CU1 CU2

A1.2.0 A2.2.0 A1.1.1 A2.1.1 A2.2.1 Ahu2 Acu1 Acu2

24.9436 31.9356 9.8321 3.1492 11.6240 3.6613 3.2226 0.8516

25.4160 28.2522 10.5312 3.3701 12.7679 4.8769 2.9701 0.5479

25.4793 31.9356 8.5856 0.6182 10.4311 3.6613 3.7973 1.5431

25.4160 28.2522 9.2701 0.6668 11.3329 4.8769 3.5596 1.2571

25.4793 31.9356 10.5312 3.3701 12.7679 4.8769 3.9536 1.5431

optimized by GA/SA (Table 1). The results for each vertex operating point are listed in Table 3. Process to process matches are defined by indexes of hot stream, cold stream and enthalpy interval, respectively (e.g. A1.2.0 stands for match H1–C2 in interval 0) and utility equipment by number of stream (e.g. Ahu2 refers to hot utility in C2). Bolded figures stand for the maximum areas in the relevant matches. The total annual cost of MSHEN is 97912.6722 $/a. The optimal structure of over-design MSHEN is shown in Fig. 6. For example, heat exchanger E1 and E2 can be merged as an MSHE. 7.1.2. Stage 2: heat exchanger area optimization for over-designed MSHEN The optimal structure of over-design MSHEN is retained and the heat exchanger area optimization is to overcome the deficiencies of over-design MSHEN. The obtained heat

Fig. 6. Over-design MSHEN structure for Example 1.

Fig. 7. Final optimal result of FMSHEN for Example 1.

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Table 4 Stream data and optimal heat transfer temperature difference contributions for Example 2 Stream

Tin (K)

Tout (K)

Fcp (kW/K)

h (kW/m2 K)

Optimized DTi (K)

H1 H2 H3 H4 H5 C1 S1 W1

[500 10, 500 + 5] 480 460 510 520 [290 10, 290 + 10] 600 293

320 380 360 490 460 550 600 313

6 4 6 20 12 18 – –

1 1 1 1 1 1 2.4 2.4

14.7050 16.5796 15.1552 25.9234 24.1176 14.8518 – –

Note: Heat transfer coefficient U = 1/(1/hh + 1/hc) kW/m2 K, for all matches, the exchanger costs are 8000 + 1000 [area (m2)]0.6 $/a; cost of hot utility = 80 $/ kW a; cost of cold utility = 20 $/kW a.

exchanger area Aijk is regarded as the original heat exchanger area while the assumed indefinite interval of heat exchanger area is taken as [0.8Aijk, 1.0Aijk]. Each heat exchanger area is generated randomly in the assumed indefinite interval and optimized by GA/SA to get an FMSHEN with the lower minimum annual cost. The total annual cost of MSHEN is 114336.5058 $/a. The final optimal structure of FMSHEN is shown in Fig. 7. Compared with the result of an over-design stage, the annual cost of MSHEN is reduced by 14.4%. The final optimal FMSHEN will remain feasible in the range of uncertainty parameters due to the consideration of vertex operating points of polyhedral region and the simulation of all vertex operating points.

7.2. Example 2 This example is taken from Chen and Hung (2005) and includes five hot streams and one cold stream. The uncertainty parameters are source temperature H1, C1 and their uncertainty ranges are [490, 505], [280, 300], respectively. Stream data are shown in Table 4. The optimal structure of over-designed MSHEN is shown in Fig. 8. The total annual cost of MSHEN is 219325.7955 $/a. The final result after heat exchanger area optimization is shown in Fig. 9 and the total annual cost of FMSHEN is 190094.7605 $/a. Although a cooler is added, the total annual cost of FMSHEN is reduced by 13.3%.

Fig. 8. Over-design MSHEN structure for Example 2.

Fig. 9. Final optimal result of FMSHEN for Example 2.

M. Xiangkun et al. / Journal of the Chinese Institute of Chemical Engineers 38 (2007) 321–331

8. Conclusion In this paper, a new two-stage strategy for the synthesis of FMSHEN is presented, which is the extension of FHEN synthesis and can improve the deficiency of most papers published in literature. In the first stage, the simultaneous NLP formulation for an over-design MSHEN is optimized by GA/ SA in order to select the optimal structure of the more costly MSHEN. So in the second stage, the model of heat exchanger area optimization is formulated to minimize the total annual cost of FMSHEN. For the two examples, the annual cost of FMSHEN is reduced by 13% compared with the annual cost of over-design MSHEN. The feasibility of solution by GA/SA procedure is assured by the pseudo T–H diagram method, hence the size and the complexity of the problem are reduced significantly and the calculation efficiency of GA/SA is enhanced remarkably with more probability of obtaining the global solution. In the second stage, however, the total annual cost of FMSHEN of each individual is obtained from the Nv vertex operating point simulations of MSHEN, that will result in expensive computational time. So further research work should be done to improve the strategy. Acknowledgement The authors acknowledge financial support provided for this research by the Deutsche Forschungsmeinschaft (DFG, No. RO 294/9). References Aaltola, J., ‘‘Simultaneous Synthesis of Flexible Heat Exchanger Network,’’ Appl. Thermal Eng., 22, 907 (2002). Cerda, J., M. R. Galli, N. Camussi, and M. A. Isla, ‘‘Synthesis of Flexible Heat Exchanger Networks. I. Convex Networks,’’ Comput. Chem. Eng., 14 (2), 197 (1990). Chen, C. L. and P. S. Hung, ‘‘Multicriteria Synthesis of Flexible HeatExchanger Networks with Uncertain Source–Stream Temperatures,’’ Chem. Eng. Process, 44, 89 (2005). Chen, C. L. and P. S. Hung, ‘‘Simultaneous Synthesis of Flexible HeatExchanger Networks with Uncertain Source–Stream Temperature and Flow Rates,’’ Ind. Eng. Chem. Res., 43, 5916 (2004). Floudas, C. A. and I. E. Grossmann, ‘‘Automatic Generation of Multiperiod Heat Exchanger Network Configurations,’’ Comput. Chem. Eng., 11 (2), 123 (1987). Floudas, C. A. and I. E. Grossmann, ‘‘Synthesis of Flexible Heat Exchanger Networks for Multiperiod Operation,’’ Comput. Chem. Eng., 10 (2), 153 (1986).

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