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Optimizing heat exchanger networks with genetic algorithms for designing each heat exchanger including condensers Benoît Allen, Myriam Savard-Goguen, Louis Gosselin * Département de Génie Mécanique, Université Laval, 1065, Avenue de la Médecine, Québec City, Québec, Canada G1V 0A6

a r t i c l e

i n f o

Article history: Received 21 July 2008 Accepted 7 June 2009 Available online 10 June 2009 Keywords: Heat exchanger network (HEN) Genetic algorithm (GA) Pinch analysis Design integration Shell-and-tube heat exchanger Condenser

a b s t r a c t This paper presents a procedure for the design of the components of a heat exchanger network (HEN). The procedure ﬁrst uses pinch analysis to maximize heat recovery for a given minimum temperature difference. Using a genetic algorithm (GA), each exchanger of the network is designed in order to minimize its total annual cost. Eleven design variables related to the exchanger geometry are considered. For exchanger involving hot or cold utilities, mass ﬂo w rate of the utility ﬂuid is also considered as a design variable. Partial or complete condensation of hot utility ﬂuid (i.e., water vapor) is allowed. Purchase cost and operational cost are considered in the optimization of each exchanger. Combining every exchanger minimized cost with the cost of hot utility and cold utility gives the total cost of the HEN for a particular DTmin. The minimum temperature difference yielding the more economical heat exchanger network is chosen as the optimal solution. Two test cases are studied, for which we show the minimized total cost as a function of the minimum temperature difference. A comparison is also made between the optimal solution with the cost of utilities and without it. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Heat exchanger networks (HEN) are required in applications involving heat exchange between two or more ﬂuids [1]. They are found in many industries such as crude oil distillation, furnace systems, multipurpose batch plants, cooling water systems and chemical plants [2–8]. These industries generally consume large amounts of energy. For example, in some batch plants, energy consumption can reach 10% of the total expenses [5]. Well-designed HENs can signiﬁcantly contribute to decrease energy consumption, and therefore energy expenses. Since ﬂuid match possibilities and design options for each exchanger of the network are numerous, several optimization techniques have been developed for the heat exchanger network design problem. A review of 461 papers on the topic was published in 2002, see Ref. [9]. Several methods have been developed to solve the HEN synthesis problem (e.g., tree searching algorithm method [10], neural networks [11], mixed integer nonlinear programming [12], etc.). Again, the reader is referred to Ref. [9] for more details. One of these methods is the pinch analysis. Although pinch analysis relies on heuristic rules and does not guarantee an ‘‘optimal” HEN, it is one of the most prominent approaches to design HENs and maximize heat recovery. Because of that, we used pinch analysis in this paper.

* Corresponding author. Tel.: +1 418 656 7829; fax: +1 418 656 7415. E-mail address: [email protected] (L. Gosselin). 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.06.006

During the HEN design procedure, the cost of the HEN is calculated, usually based on the required surface area for each heat exchanger with assumed heat transfer coefﬁcients. This approach has several limitations: it does not include the pumping power cost and provide little information on the design of the heat exchangers (HEs) themselves. Nevertheless, some authors have improved the approach. For example, Frausto-Hernandez et al. [13], Polley et al. [14], Silva and Zemp [15] included a pressure drop analysis to asses the pumping power cost. Optimization methods involving the design of the heat exchangers of the HEN have been studied by Ravagnani et al. [16], Polley and Panjeh Shahi [17], Markowski [18], Roque and Lona [19], and Ravagnani and Caballero [20]. However, the number of design variables considered for these exchangers are fairly limited. Furthermore, boiling and condensation are not considered. As mentioned above, here, we used pinch analysis (with splitting) to design or generate HENs. As the hot utility was assumed to produce water vapor, the HENs generated in this paper must include condensers. Then, a genetic algorithm (GA) is used to design in details each heat exchanger for minimizing total cost (i.e., purchase and operation costs). In the end, the optimal minimal temperature difference, HEN and HEs are determined. Among the innovative aspects of this work are the level of details for HE optimization in the context of HEN design, the consideration of condensers in the HEN and the optimization of utility ﬂuid mass ﬂow rates. Also, we demonstrate that GAs can perform these tasks. It is worth to mention that HEN synthesis has been performed with GAs in the past, e.g., Refs. [21–26]. Here, it is not the HEN that is

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Nomenclature A surface area, m2 CHU hot utility cost, $/year CCU cold utility cost, $/year COSTHU/CU hot/cold utility cost, $/W h heat capacity, J/kg K cp d tube diameter, m D shell diameter, m E pumping power, W H heat transfer rate, W latent heat, J/kg ifHU L length, m _ m mass ﬂow rate, kg/s OC operational cost, $/year PC purchase cost, $/year t time, h T temperature, °C TC total annualized cost, $/year x quality

q g

density, kg/m3 pump efﬁciency

Subscripts/superscripts b bafﬂe c, h cold, hot C condensation CU cold utility G gas phase HU hot utility i, o inlet, outlet L liquid phase max maximum min minimum s, t shell, tubes S, T supply, target sat saturation w window-ﬂow zone

Greek symbols l dynamic viscosity, Pa s

designed directly with GAs, but the heat exchangers themselves (delivered by the pinch analysis). We chose this approach for the following reasons: (i) pinch analysis being well established and employed widely, it seemed useful to show how one can design the entire ‘‘HE stock” for the HENs provided by this method, and how this procedure is coupled to the pinch analysis methodology; (ii) In order to pinpoint more sharply the outcomes of the GA on the HEs’ designs and how it affected the HEN optimization, we preferred at this point to use a deterministic HEN design procedure. This choice reduces the level of randomness and of complexity of the methodology, and helps analyzing the very impact of this work. Future work could combine the approach developed in this paper for HE design with HEN design based on GA in order to perform the overall and detailed a multi-level optimization of the system with GAs.

Since DTmin of the HEN is usually not prescribed, we can vary its value to minimize the global cost of the network. The optimal value of DTmin was found by designing networks with their HEs for several values of DTmin. We are thus able to compute an annualized cost (i.e., cost of the HEs and cost of the utilities) for each network and the more economical DTmin is identiﬁed by comparing each network total annualized cost.

2. HEN problem formulation and design procedure This paper relies on pinch analysis to determine the best ﬂuid matches. We will not describe the detailed pinch analysis procedure that is available elsewhere [1]. Only the main features are presented. Each ﬂuid involved must reach a target temperature (TT,h, TT,c) and is provided at a supply temperature (TS,h, TS,c). The mass ﬂow rates are known. Given a value of minimum temperature difference (DTmin), we match hot and cold ﬂuids in order to maximize heat recovery. Matches are allowed only between hot and cold ﬂuids. For each match, a heat exchanger has to be designed. Here, we considered shell-and-tube HEs. For the streams that could not reach their target temperature only by heat recovery, the cold and hot utilities are used. Stream splitting is allowed in order to increase match possibilities and maximize heat recovery. A list of required heat exchangers with their corresponding duty (i.e., heat transfer rate, ﬂuids and mass ﬂow rates involved) is established from that procedure. Each HE has to be designed so as to minimize the total global cost of the network. The cost of a HE includes its purchase cost and its operation cost (pumping power). It is expressed in this paper as an annualized cost. The total cost minimization is performed with a genetic algorithm. The cost estimation procedure is summarized in Fig. 1.

Fig. 1. Overall procedure for total cost estimation.

B. Allen et al. / Applied Thermal Engineering 29 (2009) 3437–3444

3. Description of the HE design problem and genetic algorithms 3.1. Objective function Our objective is to minimize the heat exchanger network total cost which is deﬁned by

TC ¼

n X

½PC j þ OC j þ CHU þ CCU

ð1Þ

j¼1

where n is the number of HEs in the network. PCj and OCj stand, respectively for the annualized purchase and operational costs. PC depends on the HE geometry. OC accounts for pumping powers required to operate the HE. Details relative to their calculation are given in [27–29]. CHU is the annual cost of the hot utility used in the process:

CHU ¼ t COST HU

a X _ HU;j ðcp;HU;G ðT HU;i T HU;sat Þ m j¼1

þ ifHU ð1 xo;j Þ þ cp;HU;L ðT HU;sat T HU;o;j ÞÞ

ð2Þ

where a stands for the number of exchangers involving hot utility and t is the annual operating period. We assumed that vapor was used as HU, and therefore, Eq. (2) accounts for the possible condensation (partial or total). The three terms in the summation of Eq. (2) represent the power given by the vapor to the HEs involving HU, and the power given by the condensing mixture and the sub-cooled liquid if applicable. Similarly, CCU stands for the total annual cost of the cold utility:

CCU ¼ t COST CU

b X

_ CU;j ðcp;CU ðT CU;i T CU;o;j ÞÞ m

_ CU;min ¼ m

_ h ðT h;i T h;o Þ cp;h m cp;CU ðT h;i DT min T CU;i Þ

3439

ð6Þ

There is no physical restriction on the maximum CU mass ﬂow _ CU;max ). However, a mass ﬂow rate interval had to be specirate (m ﬁed, so the available mass ﬂow rate was chosen as the maximum _ CU;max were considered. _ CU;min and m value and 128 values between m A hot stream of superheated water vapor is used as hot utility. This stream condensates inside HEs. The advantage of using vapor as HU is the high heat transfer coefﬁcients that characterize a process involving phase change [30]. Modeling of heat exchanger with condensation has been developed in a previous article [29]. However, we present in Section 4, an extension of this work for the case of partial condensation which was not considered. Supply temperature (THU,i) of the hot utility is known but there is no restriction on its outlet temperature (THU,o). Consequently, hot util_ HU ) is taken as a design variable. Its limit vality mass ﬂow rate (m ues are established in order to ensure that condensation takes place. However, hot utility ﬂuid is not required to completely condensate. Fig. 2a and b illustrate extreme situations from which mass ﬂow rate limits are established. It must be greater than a minimum value obtained when there is a difference DTmin between hot utility outlet temperature and cold ﬂuid inlet temperature:

T HU;o ¼ T c;i þ DT min

ð7Þ

From the energy balance between hot utility and cold ﬂuid, we have:

ð3Þ

j¼1

where b stands for the number of exchangers involving cold utility. Utility costs (i.e., COSTCU and COSTHU) are expressed in $/W h. We used a cost of 0.000015 $/W h for hot utility and 0.000010 $/W h for cold utility [1]. 3.2. HEs design variables Heat exchangers that make part of the HEN are separated in three categories: (1) cold ﬂuid to hot ﬂuid heat exchangers, (2) heat exchangers with cold utility and (3) heat exchangers with hot utility. Eleven design variables are common to every exchanger. They are related to the shell-and-tube heat exchanger geometry. These design variables can be found in [29]. A twelfth design variable is added for the side (shell or tubes) where each ﬂuid ﬂows. Since the outlet temperature of the cold utility stream in HEs of the second category is not predetermined, its mass ﬂow rate in each HE with CU can vary in order to obtain a minimum annualized cost. Therefore, an additional design variable is added to heat exchangers with CU: the mass ﬂow rate of the cold utility ﬂuid. _ CU;min ) in order The ﬂow rate must respect a minimum value (m not to violate the speciﬁed minimum temperature difference (DTmin) between the inlet temperature of the hot ﬂuid (T h;i ) and the outlet temperature of the cold utility (T CU;o ). The maximum value of TCU,o can be expressed by

T CU;o;max ¼ T h;i DT min

ð4Þ

With no heat loss to the environment, the heat transfer rate between the hot ﬂuid and the cold utility is determined by

_ h cp;h ðT h;i T h;o Þ ¼ m _ CU cp;CU ðT CU;o T CU;i Þ Q ¼m

ð5Þ

The minimum ﬂow rate of cold utility is calculated by combining Eqs. (4) and (5)

Fig. 2. (a) Temperature of cold and hot ﬂuids in a shell-and-tube heat exchanger without condensation. (b) Temperature of cold and hot ﬂuids in a shell-and-tube heat exchanger with condensation of the hot ﬂuid.

B. Allen et al. / Applied Thermal Engineering 29 (2009) 3437–3444

_ HU;min ðcp;HU;G ðT HU;i T HU;sat Þ þ ifHU _ c cp;c ðT c;o T c;i Þ ¼ m m þ cp;HU;L ðT HU;sat T HU;o ÞÞ

ð8Þ

Combining Eqs. (7) and (8), the minimum mass ﬂow rate can then be expressed as a function of known parameters:

_ HU;min ¼ m

_ c cp;c ðT c;o T c;i Þ m cp;HU;G ðT HU;i T HU;sat Þ þ ifHU þ cp;HU;L ðT HU;sat ðT c;i þ DT min ÞÞ ð9Þ

Maximum value occurs when the hot utility reaches its saturation temperature and just starts to condensate. This is represented in Fig. 2a. An energy balance leads to:

_ c cp;c ðT c;o T c;i Þ ¼ m _ HU;max ðT HU;i T HU;sat Þ m _ c cp;c ðT c;o T c;i Þ m _ HU;max ¼ m T HU;i T HU;sat

ð10Þ ð11Þ

3.3. Optimization of HEs using genetic algorithms Genetic algorithms (GAs) are an optimization tool inspired by the Darwinian natural selection. It has been proved that using genetic algorithms is a quick way to ﬁnd the best HE design among a large number of possibilities. A recent review on the utilization of GAs in heat transfer problems is available [35]. The GA is used for the purpose of designing low-cost heat exchangers that respect the heat duties imposed from the pinch analysis (Section 2). A priori, the geometry leading to the lowest cost is not easy to determine. An increase of the heat transfer area leads to a lower operating cost but a higher purchase cost. Moreover, millions of possible designs are feasible. Considering the design variables listed above (Section 3.2), 302 millions possible heat exchanger designs are possible for hot to cold ﬂuid heat exchangers, 38.7 billions for CU heat exchangers and 25.8 billions for HU heat exchanger. The main parameters of the binary GA used in this paper are the same as in Refs. [28,29]. GAs are probabilistic, and therefore two runs of the GA with the exact same settings could lead to two different results. Therefore, for each HE the GA optimization was performed seven times. The best result among these runs was retained for this speciﬁc HE.

4. Extension of Ref. [29] to HEs with partial condensation Calculation of the cost of a shell-and-tube condenser as a function of the design variables for a given heat duty was described in Ref. [29]. However, complete condensation was assumed. Therefore, we extend here the procedure to HEs with partial condensation. We ﬁrst determine if condensation of the hot utility occurs completely or partially by comparing the total heat transfer rate to the cold ﬂuid with the heat transfer required for a complete condensation. Two cases are possible. If

_ HU ifHU 6 m _ c cp;c ðT c;o T c;i Þ _ HU cp;HU;G ðT HU;i T HU;sat Þ þ m m

ð12Þ

then condensation is complete. Otherwise, when

_ HU ifHU > m _ c cp;c ðT c;o T c;i Þ _ HU cp;HU;G ðT HU;i T HU;sat Þ þ m m

ð13Þ

the condensation is partial. For the ﬁrst case, hot utility vapor will come out of the exchanger as sub-cooled ﬂuid. Then, condensation is complete and the method developed for optimization of shell-and-tube condensers in [29] is directly applied to design the corresponding heat exchanger. For case #2, hot utility at the outlet of the exchanger will be a mix of gas and liquid at saturation temperature (THU,o = THU,sat).

The heat transfer coefﬁcient will be calculated using correlation developed by Chato [32] if condensation occurs in tubes and Nusselt correlation if condensation occurs on the shellside. In order to be able to calculate pressure drop on the side where condensation occurs, mix quality (xo) at the heat exchanger outlet has to be determined. Isolating the quality from the energy balance on the HU side, we obtain:

xo ¼

_ HU cp;HU;G ðT HU;i T HU;o Þ Q m ifHU

ð14Þ

If condensation occurs in the tubes, total pressure drop can be separated in two terms

DPt ¼ DPt;G þ DPt;C

ð15Þ

where DPt,G and DPt,C stand, respectively for the pressure drop in the vapor section and for the pressure drop in the condensation section. We used the expression previously developed [28] for the ﬁrst term. For the condensation zone, it has been shown [29] that pressure drop can be expressed by the following formula: 9=5

DPt;C ¼ 0:046

_ HU ðlHU;G lHU;L Þ 32m 4

0:2

9=5 d24=5 i

p

1=5

LC

qHU;G qHU;L

Z

ð1 xÞqHU;G þ xqHU;L dx ðð1 xÞlHU;G þ xlHU;L Þ 0 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} xo

I

ð16Þ

The analytic resolution of the integral I yields to

I¼

5ðlHU;G xo lHU;G þ xo lHU;L Þ4=5 36ðlHU;G þ lHU;L Þ2

½ð4xo 9ÞqHU;G lHU;L

þ ð4xo þ 4ÞqHU;G lHU;G þ ð4xo þ 5ÞqHU;L lHU;G 4xo qHU;L lHU;L

ð17Þ

Table 1 Process requirements for test case #1. Stream

Stream ﬂuid

Supply temp. (°C)

Target temp. (°C)

Flow rate (kg/s)

H1 H2 C1 C2 HU CU

Crude oil Water Water Kerosene Steam Water

150 130 100 50 200 20

30 50 140 140 – –

7.2 3 10 3.6 – –

220 000 Total cost

200 000 180 000 160 000

Cost ($/year)

3440

Utility cost

140 000 120 000 100 000 80 000 60 000 40 000

Cost of HEs

20 000 0 0

2

4

6

8

10

ΔT

12

14

16

18

20

(°C)

min

Fig. 3. Minimum heat exchanger network total cost as a function of minimum temperature difference for test case #1.

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B. Allen et al. / Applied Thermal Engineering 29 (2009) 3437–3444

DPs;C;b

/

Cold / Hot Sream Heat echanger Pinch CU/HU Exchanger with cold / hot utility

Stream

CU

H1

C2

ð20Þ

where Y2 is the Chisholm parameter and LO refers to the total ﬂow having the liquid properties. No analytical solution is found for the integral in Eq. (20). Consequently, a numerical integral is performed to solve the problem. Window-ﬂow pressure drop is calculated with the following expression:

CU

H2

Z xo h i @p ¼ Lc 1 þ ðY 2 1Þðx x2 Þ0:815 þ x1:37 dx @z LO 0

HU

DPs;C;w ¼ C1

HU

Z xo h i @p LC 1 þ ðY 2 1Þx dx @z LO 0

ð21Þ

and the integration gives 20

40

60

80

100

120

140

160

DPs;C;w ¼

Temperature (ºC) Fig. 4. Optimal heat exchanger network design for test case #1.

HE #

Cold stream

Tc,i (°C)

Tc,o (°C)

_ c ðkg=sÞ m

Hot stream

Th,i (°C)

Th,o (°C)

_ h ðkg=sÞ m

1 2 3 4 5 6 7 8

C1 C1 C2 C2 C1 C2 CU CU

100 100 100 50 114.3 109.7 20 20

120.8 106 109.7 100 140 140 – –

5.6 4.4 3.6 3.6 10 3.6 – –

H1 H2 H2 H1 HU HU H1 H2

150 130 130 102 200 200 78 103

103 103 103 78 – – 30 50

7.2 2.2 0.8 7.2 – – 7.2 3

gt qHU;G

þ

_ HU ðxo qHU;G þ ð1 xo ÞqHU;L Þ DPt;C m 2gt xo ð1 xo ÞqHU;G qHU;L

ð18Þ

The shell-side pumping power is calculated as in [28]. If partial condensation occurs on the shell-side, total pressure drop is once again separated in two parts

DPs ¼ DPs;G þ DPs;C

Es ¼

_ HU DPs;G m

gs qHU;G

þ

_ HU ðxo qHU;G þ ð1 xo ÞqHU;L Þ DPs;C m 2gs xo ð1 xo ÞqHU;G qHU;L

ð23Þ

and tube side pumping power is calculated as in [28]. 5. Test cases

Pumping power is then calculated using following equation for tubes

_ HU DPt;G m

ð22Þ

Total pumping power for shell-side is calculated as follows:

Table 2 Stream data for shell-and-tube heat exchangers of the optimal heat exchanger network for test case #1.

Et ¼

" # @p Y2 1 2 L C xo þ xo @z LO 2

ð19Þ

The pressure drop for superheated gas (DPs,G) is calculated using Bell-Delaware method [33]. The entire procedure is explained elsewhere [27]. It has been shown in [34] that condensation sub-section pressure drop has two contributions: the cross ﬂow zone pressure drop (DPs;C;b ) and the window-ﬂow zone pressure drop (DPs;C;w ). From the Chisholm correlation [27], we obtained the following expression to calculate the cross ﬂow zone pressure drop

We considered two different test cases to show the ability and the versatility of the proposed optimization procedure. Every heat exchanger designed for the two test cases are assumed to operate 5000 h per year. Electricity cost is 0.10 $/kW h and pump efﬁciency is 85%. Each HE has a lifetime of 20 years and the annual interest rate is 5%. Thermophysical properties (density, heat capacity, thermal conductivity, and viscosity) of the ﬂuids used in the following examples are considered constant, except water for which properties are taken at the average temperature of the ﬂuid in the exchanger. Optimization of the heat exchanger network has been performed for 20 different values of the minimum temperature difference (DTmin) ranging from 1 °C to 20 °C.

Table 4 Process requirements for test case #2. Stream

Stream ﬂuid

Supply temp. (°C)

Target temp. (°C)

Flow rate (kg/s)

H1 H2 H3 C1 C2 C3 C4 HU CU

Kerosene Water HGO Crude oil Water BPA LGO Steam Water

393 160 354 72 62 120 147 372 10

60 40 60 356 210 370 284 – –

77 47 53 81 35 41 26 – –

Table 3 Optimal HE geometries as found by the GA for test case #1 optimal HEN. HE #

Pt

Tube conﬁg. (°)

Lbc

Bc (%)

dtb

dsb

Dotl

Ds (mm)

do (in)

Lbi Lbo

Fluid in tubes passes

Nb. of tube

Utility mass ﬂow rate (kg/s)

1 2 3 4 5 6 7 8

1.4do 1.5do 1.5do 1.5do 1.5do 1.5do 1.2do 1.5do

90 90 90 90 45 45 90 90

0.3Ds 0.2Ds 0.2Ds 0.2Ds 0.55Ds 0.2Ds 0.4Ds 0.55Ds

25 25 25 25 25 25 25 25

0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do

0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do

0.80(Ds dsb) 0.95(Ds dsb) 0.95(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.95(Ds dsb) 0.80(Ds dsb)

300 300 300 300 350 300 350 350

5/8 5/8 5/8 5/8 5/8 5/8 5/8 5/8

1.5Lbc Lbc Lbc 1.2Lbc Lbc Lbc 1.3Lbc 1.6Lbc

Cold Cold Cold Hot Cold Cold CU CU

1 4 4 4 1 1 1 4

– – – – 0.236 0.061 25.20 19.23

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B. Allen et al. / Applied Thermal Engineering 29 (2009) 3437–3444 2 500 000

Total cost

Cost ($/year)

2 000 000

Utility cost

1 500 000

1 000 000

500 000

Cost of HEs 0 0

2

4

6

8

10

ΔT

12

14

16

18

20

(°C)

min

Fig. 5. Minimum heat exchanger network total cost as a function of minimum temperature difference for test case #2.

/

Cold / Hot stream Heat exchanger Pinch CU/HU Exchanger with cold / hot utility CU

H3 CU

Stream

H2

CU

H1 C4

HU

C3

HU

C2 C1

0

50

100

150

200

250

300

350

400

Temperature (ºC) Fig. 6. Optimal heat exchanger network design for test case #2.

We ﬁrst considered a simple example that involves two hot streams and two cold streams. Process requirements for test case #1 are shown in Table 1. For each minimum temperature difference considered, a HEN has been designed and optimized to recover as much heat as possible as explained in Sections 2 and 3. Fig. 3 shows the annualized total cost, utility cost and HEs cost

as a function of DTmin. Utility costs are a way to gage heat recovery. The more we use hot and cold utilities, the less heat is recovered. Considering the cost of utilities, our results show that the optimal Tmin is 3 °C. Not considering utilities cost, the optimal solution is found at DTmin = 20 °C. Fig. 4 shows a schematic representation of the optimal heat exchanger network with the matches between cold and hot ﬂuids as well as the points from which utilities are used for each stream. Details about each heat exchanger inlet and outlet temperatures are given in Table 2. Optimal design geometries of the eight HEs are listed in Table 3. A second test case is studied. This example involves more streams and largest ﬂow rates. Data is presented in Table 4. The minimal cost as a function of DTmin is shown in Fig. 5. The global optimal solution is when DTmin 4 °C. This optimal value, as well as the one obtained for test case #1, is comparable with optimal DTmin values found in the literature for similar processes, see e.g. Refs. [31,36]. Fig. 6 shows a representation of the optimal network. Heat exchangers temperature and optimal design geometries are listed in Tables 5 and 6. As mentioned previously, Figs. 3 and 5 present the curves obtained from our simulations. When DTmin increases, less heat is recovered and therefore, HU and CU are more solicited. This explains why the utility costs increase with DTmin. We clearly see that utilities cost increase linearly with DTmin. On the other hand, the cost of the HEs themselves decreases when DTmin increases. For the two cases considered, this decrease is greater for low values of DTmin. As a result, combination of HEs cost and utility costs present an optimum. For test case #1, a total of eight heat exchangers were optimized for each value of DTmin. Hence, it is no surprise that the curves in Fig. 3 are smooth. On the other hand, for the second example, the total number of exchangers in the network varies between 13 and 16 depending on DTmin. However, we did not notice any considerable step of the curves in Fig. 5; they are all smooth. It is worth to recall that the optimization of each HE was performed seven times (see Section 3.3). The maximal variation of the HE cost between two runs of the GA was 5% for the ﬁrst test case and 1% for the second one. Even though these variations are relatively small, they were sufﬁcient to ‘‘disrupt” the curves of Figs. 3 and 5 and create ‘‘artiﬁcial” local minima, when only one run of the GA was performed for each HE. The procedure proposed here with seven runs of the GA per HE was found to be robust and to yield a good reproducibility. An interesting observation is the small number of HEs that were actually calculated to ﬁnd the best solution. For each test case

Table 5 Stream data for shell-and-tube heat exchangers of the optimal heat exchanger network for test case #1. HE #

Cold stream

Tc,i (°C)

Tc,o (°C)

_ c ðkg=sÞ m

Hot stream

Th,i (°C)

Th,o (°C)

_ h ðkg=sÞ m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

C3 C3 C4 C2 C1 C4 C4 C1 C3 C2 C1 C3 C4 CU CU CU

156 201.4 156 156 156 147 147 72 120 62 72 300.5 274.4 10 10 10

201.4 300.5 274.4 210 356 157 156 156 156 156 156 370 284 – – –

41 41 26 35 81 19.6 6.4 4 41 35 77 41 26 – – –

H1 H3 H3 H3 H1 H2 H3 H3 H3 H2 H1 HU HU H1 H2 H3

393 354 277.4 219.8 369.3 160 160 160 160 160 160 372 372 75 86.8 123.8

369.3 277.4 219.8 160 160 150.1 150 99.2 123 65.2 75 – – 60 40 60

77 53 53 53 77 12 6.4 5.6 41 35 77 – – 77 47 53

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B. Allen et al. / Applied Thermal Engineering 29 (2009) 3437–3444 Table 6 Optimal HE geometries as found by the GA for test case #2 optimal HEN. HE #

Pt

Tube conﬁg. (°)

Lbc

Bc (%)

dtb

dsb

Dotl

Ds (mm)

do (in)

Lbi Lbo

Fluid in tubes

Nb. of tube passes

Utility mass ﬂow rate (kg/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1.5do 1.5do 1.2do 1.4do 1.5do 1.5do 1.5do 1.5do 1.5do 1.5do 1.5do 1.5do 1.4do 1.2do 1.5do 1.5do

90 90 90 90 90 90 90 90 90 90 90 45 90 90 90 90

0.55Ds 0.5Ds 0.55Ds 0.45Ds 0.55Ds 0.55Ds 0.55Ds 0.55Ds 0.5Ds 0.35Ds 0.25Ds 0.55Ds 0.2Ds 0.55Ds 0.55Ds 0.45Ds

25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25

0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do

0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.01do 0.10do 0.01do 0.01do 0.01do 0.01do

0.80(Ds dsb) 0.95(Ds dsb) 0.90(Ds dsb) 0.95(Ds dsb) 0.95(Ds dsb) 0.95(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb) 0.80(Ds dsb)

1000 800 600 650 700 1050 650 650 400 300 300 650 300 800 1450 1450

5/8 5/8 5/8 5/8 5/8 5/8 5/8 5/8 5/8 5/8 5/8 5/8 1.5 5/8 7/8 5/8

1.2Lbc 1.3Lbc 1.6Lbc 1.3Lbc 1.1Lbc 1.4Lbc 1.2Lbc 1.3Lbc Lbc 1.2Lbc 1.1Lbc Lbc Lbc 1.6Lbc 1.6Lbc 1.6Lbc

Cold Hot Hot Hot Hot Hot Hot Cold Cold Cold Hot Cold Cold CU Hot Hot

1 4 1 1 1 1 1 1 2 4 4 1 1 1 4 4

– – – – – – – – – – – 0.908 0.063 105.8 341.4 348.7

studied in this paper, 0.00002% of every possible design has been calculated to converge. This proves that using GA for the problem studied in this paper results in an important saving of computational time. 6. Conclusions A procedure is proposed for designing in details the components of a HEN. For a given DTmin, an optimal HEN was determined based on pinch analysis. Then, each HE of the network was optimized with a GA. The HU and CU ﬂuids ﬂow rates were also optimized. The minimized total cost of the HEN was calculated. The procedure was repeated for different DTmin in order to ﬁnd the optimal value of DTmin. The procedure was validated with two test cases. GA rapidly identiﬁed the best design for each HE, including the condensers of the network. This yields a better estimate of the total HEN cost, by including the pumping power in the total cost, and by providing a detailed design for each HE. Further research could include other types of HEs, such as plate heat exchangers, and let the GA decide for each HE of which type it should be. Also, it would be interesting to let GAs design simultaneously both the HEN architecture (e.g., as in Refs. [21–26]) and the HEs. A multi-level optimization procedure could be implemented with nested or mega GAs. For example, a simple approach could be the determination of the optimal DTmin by a GA. However, more evolved procedures would take advantage of GAs to explore more thoroughly (i.e., without heuristic rules or simplifying assumptions) the design space of HEN problems. This paper is a step toward a multifaceted and detailed combined optimization of HENs and HEs. Acknowledgement Allen and Gosselin’s work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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