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Simultaneous synthesis of ﬂexible heat exchanger network Juha Aaltola Helsinki University of Technology, P.O. Box 4400, 02015 Hut, Finland Received 1 October 2001; accepted 24 November 2001

Abstract A framework is presented for generating ﬂexible heat exchanger networks (HENs) over a speciﬁed range of variations in the ﬂow rates and temperatures of the streams. The ﬂexible HEN is synthesised using a combination of a multiperiod simultaneous MINLP model and search algorithms, where the total annual costs due to utility duties, exchanger areas and selection of matches are optimised. The simultaneous HEN synthesis allows the data to be distributed according to a probability distribution and it does not rely on the concept of pinch point. Both search algorithms involve a developed multiperiod NLP/LP model where utility costs are minimised. The proposed procedures are explained through an example including variations resulting in a network with variable splits and bypasses. This framework results in a HEN working under variations without losing stream temperature targets while keeping an economically optimal energy integration. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Flexible heat exchanger network; Multiperiod optimisation

1. Introduction Methods for heat exchanger network (HEN) synthesis have been developed by Cerda et al. [3], Papoulias and Grossmann [11] and Yee and Grossmann [15] who aim to design a HEN that yields a reasonable trade-oﬀ between capital and operating cost, through sequential or simultaneous approaches. Recently Galli and Cerda [5,6] proposed a MILP framework with new decomposition strategy and Briones and Kokossis [1,2] presented synthesis method that combines thermodynamics and mathematical programming. When the environment introduces signiﬁcant changes in the operating conditions, a synthesised HEN must be ﬂexible. Many authors have discussed this subject e.g. Marselle et al. [8] identiﬁed a

E-mail address: [email protected]ﬁ (J. Aaltola). 1359-4311/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 2 ) 0 0 0 0 8 - X

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number of worst operating conditions: maximum heating, maximum cooling and maximum total heat exchange. Swaney and Grossmann [13] introduced a ﬂexibility index, which deﬁnes the maximum parameter range that can be achieved for feasible operation. Kotjabasakis and Linnhoﬀ [7] introduced sensitivity tables to ﬁnd which heat exchanger areas should be increased and which heat exchanger should be bypassed in order to make a nominal design suﬃciently ﬂexible. Floudas and Grossmann [4] introduced an active set strategy for automated solution of the ﬂexibility index of Swaney and Grossmann [13]. Papalexandri and Pistikopoulos [9,10] presented a framework for the synthesis and retroﬁt of ﬂexible and structurally controllable HENs considering HEN synthesis and ﬂexibility simultaneously using mathematical programming. Tantimuratha et al. [14] proposed a screening and targeting process for HEN design with ﬂexibility consideration in both grassroots and retroﬁt cases. The screening stage, which is based on the screening models of Briones and Kokossis [1,2] considers both economic and ﬂexibility aspects prior to network development. In this work a multiperiod optimisation model for ﬂexible HEN is developed based on the MINLP formulation of Yee and Grossmann [15] where the three-way trade-oﬀ between energy, units and area is considered simultaneously. Modelling of bypasses in multiperiod MINLP formulation is avoided by considering decrease of the area as heat capacity ﬂow increase in the bypass fraction. The elimination of the bypasses, a stagewise superstructure presentation and isothermal mixing assumption allows the feasible space of the problem to be deﬁned by a set of linear constraints. Also the online search algorithm with ﬁxed area targets is proposed for searching optimum set points of HEN under diﬀerent operating conditions.

2. Flexible heat exchanger network synthesis 2.1. Problem deﬁnition The problem to be addressed in this part of the work can be stated as follows: Given are a set of hot (H) and cold (C) process streams including a multiperiod stream data with a set of inlet and outlet stream temperatures and heat capacity ﬂow rates. Also a set of hot (HU) and cold (CU) utilities are speciﬁed. The objective then is to synthesise a heat exchanger network at minimum total annualised cost, able to operate feasibly under the speciﬁed set of periods (P). This model provides matches that take place (zi;j;k , zcui and zhuj ) and for every period, areas of each exchanger (Ai;j;k;p , Acui;p and Ahuj;p ), exchanger load required (qi;j;k;p , qcui;p and qhuj;p ) and possible bypass ﬂow for each exchanger. Indices are i ¼ hot process stream, j ¼ cold process stream, k ¼ index for stage and p ¼ index for period (see Fig. 1). The following assumptions are made: The exchangers are of countercurrent type, no phase changes are allowed, constant heat capacities, utility duties can be adjusted, variable bypass is available across each exchanger and split fractions can be varied. 2.2. Multiperiod simultaneous MINLP model A model for the synthesis of ﬂexible and structurally controllable HEN is developed based on the simpliﬁed superstructure presentation of Yee and Grossmann [15]. The simpliﬁed superstructure consists of a number of stages and in each stage many diﬀerent possibilities for stream matching are allowed to take place (see Fig. 1). This multiperiod MINLP formulation is robust

J. Aaltola / Applied Thermal Engineering 22 (2002) 907–918

909

Fig. 1. HEN superstructure is simpliﬁed by dividing it to stages.

and eﬃcient for small scale problems, primarily due to the non-linearities being solely in the objective function and not in the constraints. This is due to the assumption of an isothermal mixing junction at the outlets of the stages, which eliminates the need for the energy balances and reduces the size of the model. However the simpliﬁed HEN superstructure has few fundamental limitations. Utilities are located in the ends of the superstructure and the superstructure does not allow a split stream to have exchangers in series nor does it allow stream bypassing. The objective function is non-linear and non-convex and hence despite the linear set of constraints the solution of the resulting optimisation model represents a local optimum. However, it is possible to generate good HEN structures by performing several runs with diﬀerent bounds on the utility requirements. The estimates for minimum hot utility requirement can be achieved from the transhipment model [11]. The partial solutions from the multiperiod model are compared to solutions from the single period model and when these are close enough the hot utility duty mix is chosen. The multiperiod simultaneous HEN synthesis problem can be formulated as an optimisation problem introducing continuous variables for the: (i) heat load between hot and cold process streams qi;j;k;p (for each stage k), hot process stream and cold utility qcui;p , and cold process stream and hot utility qhuj;p in each period p, (ii) temperatures of hot thi;k;p and cold stream tcj;k;p at the hot ends of each stage k in each period p and temperature approaches for process and process utility matches at each stage k in each period p. Binary variables for the: (i) existence of process matches zi;j;k at each stage k and (ii) existence of process utility matches zcui and zhuj at the ends of the superstructure. The set of constraints consists of: (i) overall heat balance for each stream in each period, (ii) heat balance for each stream at each stage in each period,

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(iii) (iv) (v) (vi) (vii)

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assignment of inlet temperatures in each period, feasibility of temperatures in each period (maintain increase/decrease), energy balances for utility matches in each period, logical constraints for existence of matches in each period and calculation of temperatures for each temperature location in each period.

The objective function total annual costs (TAC) can be deﬁned as the summations of: Unit costs for matches, mean area costs for stream matches, mean area costs for hot utility matches, mean area costs for cold utility matches, mean hot utility costs and mean cold utility costs. The Paterson approximation [12] is used to calculate the log mean temperature diﬀerence (LMTD) to avoid numerical diﬃculties (Eq. (1)). 1=2 thi;k;p tcj;k;p þ thi;kþ1;p tcj;kþ1;p 2 þ thi;k;p tcj;k;p thi;kþ1;p tcj;kþ1;p LMTDi;j;k;p ¼ 6 3 ð1Þ The multiperiod simultaneous MINLP model provides the optimal HEN structure, which minimises costs when the heat load and LMTD of every match is allowed to change separately under each speciﬁed period. The areas are deﬁned as a relation of the heat load and LMTD, thus the optimal solution may represent more than one value of area considering one match. However there is only one investment decision to make and the maximum area is chosen. For those operating conditions with a smaller area Ai;j;k;p than maximum Amaxi;j;k , a bypass is needed to keep the stage temperatures at an optimal level. In order to calculate these bypass fractions, we need to establish ﬁrst the performance equations. Exchanger load that represent periods when according to optimal solution the required area is smaller than Amaxi;j;k , will be given by: qi;j;k;p ¼ Ai;j;k;p LMTDi;j;k;p Ui;j

ð2Þ

Because, the optimal heat load qi;j;k;p and stage temperatures must remain the same although the existing area is Amaxi;j;k instead of Ai;j;k;p , the area increase will decrease the value of LMTD. Thus the new log mean temperature diﬀerence LMTDbpi;j;k;p (when bypass open) is introduced. Now the optimal exchanger load will be given by: qi;j;k;p ¼ Amaxi;j;k LMTDbpi;j;k;p Ui;j

ð3Þ

Substituting Eq. (2) in Eq. (3) then yields the LMTD when bypass open: LMTDbpi;j;k;p ¼

Ai;j;k;p LMTDi;j;k;p Amaxi;j;k

ð4Þ

In order to derive the bypass fraction equation, the temperature after exchanger (before mixing bypass stream) thai;k;p needs to be computed. When considering the hot side bypass, thi;kþ1;p in Eq. (1) is replaced by thai;k;p and LMTDi;j;k;p is replaced by LMTDbpi;j;k;p . Now this expression is solvable for thai;k;p . The equation that represents heat balance for the combination of heat exchanger and bypass (Fig. 2) will be given by: ðFCphi;j;k;p FCphbi;j;k;p Þðthi;k;p thai;k;p Þ ¼ ðthi;k;p thi;kþ1;p Þ FCphi;j;k;p

ð5Þ

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Fig. 2. Bypass placed over hot side.

where FCphi;j;k;p is the hot side heat capacity ﬂow rate addressed to match zi;j;k and FCphbi;j;k;p is the hot side bypass fraction over exchanger zi;j;k . Finally the equation for calculating the hot side bypass fractions over exchanger zi;j;k can be written as follows: FCphbi;j;k;p ¼ FCphi;j;k;p

ðthi;k;p thi;kþ1;p ÞFCphi;j;k;p ðthi;k;p thai;k;p Þ

ð6Þ

The bypasses considering the cold side are calculated correspondingly. The objective function considers the area of one match to be the mean value of areas in diﬀerent periods, hence the model under-estimates total area costs and over-estimates exchanger areas. The exact way to calculate the real area costs is to add up the costs related to the maximum match areas. The methods for searching maximum area and to obtain real area investment costs in objective function, introduce non-linearities in constraints or non-linearities with discontinuous derivatives in the objective function. A model with these methods would be more diﬃcult to solve and less robust. Despite the area cost under-estimation, the multiperiod model ﬁnds good partial solutions for separate periods, which can be observed when comparing real costs of the multiperiod model to a single period model. The real TAC costs are calculated after the optimisation within a separate part of the model. 2.3. Multiperiod LP/NLP model and search algorithm Since the area cost under-estimation of the simultaneous MINLP model may result in too large exchanger areas it also tends to lead to smaller utility consumption than what is economical. That is why the search algorithm is developed to achieve an optimal trade-oﬀ between capital and operating cost leading to the real minimum TAC when the structure and the maximum areas of each match are ﬁrst deﬁned by the simultaneous MINLP model. The main idea of the synthesis strategy as follows: (i) (ii) (iii) (iv) (v)

solve the MINLP model to obtain the structure of the network and data considering the matches with maximum exchanger areas, change allowable exchanger area Ai;j;k;p by changing the value of Ki;j;k;p , solve the LP/NLP model to minimise utility costs taking account of limitations from step 2, if the solution is improving go back to step 2, if the solution is not improving move to the next Ki;j;k;p and go back to step 2.

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Fig. 3. Outline of the synthesis strategy.

Tightening or slackening inequality constraints for maximum exchanger areas (step ii) and minimising the utility costs each time (step iii) the minimum real TAC is found. This algorithm aims to decrease the exchanger areas and increase the utility load. It also aims to produce constant areas between diﬀerent conditions and thus have fewer bypasses. The multiperiod LP model is based on the multiperiod simultaneous MINLP model with few diﬀerences. The objective function minimises total utility costs when the structure is ﬁxed and maximum exchanger areas are limited. Maximum areas Amaxi;j;k are limited by the additional inequality constraint Eq. (7) where mean temperature diﬀerences DTminlpi;j;k and DTi;j;k;p are used instead of LMTD to maintain linear nature. In Eq. (7), qminlpi;j;k and DTminlpi;j;k are parameters considering match zi;j;k with Amaxi;j;k obtained from the MINLP model (see Fig. 3) and qi;j;k;p and DTi;j;k;p are variables in the LP model. qi;j;k;p 6

DTi;j;k;p qminlpi;j;k Ki;j;k;p DTminlpi;j;k

ð7Þ

The NLP search algorithm works as the LP search does but the multiperiod NLP model takes account also of the non-isothermal mixing of the streams after the parallel exchangers. Inputs (qsalpi;j;k and DTsalpi;j;k ) are obtained from the LP search algorithm.

3. Application example This example modiﬁed from Floudas and Grossmann [4] is an illustrative problem for describing the developed models and search algorithms. This example consists of six hot streams and one cold stream. A range of operating conditions is deﬁned by allowing the inlet temperatures of the streams to vary 10 K, and ﬂowrates 10%, both from their nominal values. The ﬂowrate variations of the streams are all fully correlated. The objective then, is to synthesise a HEN with minimum TAC to operate feasibly under these speciﬁed conditions. 3.1. Data The problem data for the nominal period and the selected three other periods are shown 4 in Table 1. Cost of furnace is $191:94qhu0:7 j;p , the fuel cost is $204:73 10 qhuj;p and the cost of 4 cooling water is $60:58 10 qcui;p . The cost equation for the exchangers is $4333A0:6 i;j;k;p . Overall heat transfer coeﬃcients for process matches Ui;j (kW/m2 K) are U1;1 ¼ 0:6, U2;1 ¼ 0:4, U3;1 ¼ 0:3, U4;1 ¼ 0:4, U6;1 ¼ 0:3, and for cold utility matches Ucui are Ucu1 ¼ 0:1, Ucu5 ¼ 0:3 and Ucu6 ¼ 0:4. Contrary to the original example the nominal condition represents 75% of a year and

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Table 1 Operating conditions for example Stream Period 1

Period 2

Period 3

Nominal

Tstart Ttarget FCp Tstart Ttarget FCp Tstart Ttarget FCp Tstart Ttarget FCp (°C) (°C) (kW/K) (°C) (°C) (kW/K) (°C) (°C) (kW/K) (°C) (°C) (kW/K) H1 H2 H3 H4 H5 H6 C1

640 560 540 480 460 420 300

460 480 480 400 310 350 650

9.90 7.15 3.30 39.60 7.70 79.20 29.70

620 540 520 460 440 400 300

460 480 480 400 310 350 650

9.90 7.15 3.30 39.60 7.70 79.20 29.70

620 540 520 460 440 400 300

460 480 480 400 310 350 650

8.10 5.85 2.70 32.40 6.30 64.80 24.30

630 550 530 470 450 410 310

460 480 480 400 310 350 650

9.00 6.50 3.00 36.00 7.00 72.00 27.00

lifetime used is four years and rate of interest is 15%. These changes have been made to bring out the beneﬁts of the simultaneous method. 3.2. Multiperiod simultaneous MINLP model The number of stages for this example was chosen to be 4. Solving the multiperiod MINLP model for the following hot utility limits (qhulimitp ¼ total utility consumption in period p): qhulimit1 ¼ 3600 kW, qhulimit2 ¼ 4100 kW, qhulimit3 ¼ 3500 kW and qhulimitNom ¼ 3500 kW, gives the results shown in Table 2. Process to process matches are deﬁned by indexes of hot stream, cold stream and stage respectively (e.g. 3.1.2 stands for match H3-C1 in stage 2) and utilities by number of stream and utility type (e.g. qcu5;1 is cold utility in H5). Bolded ﬁgures stand for the maximum areas. Bypass fractions are listed in Table 2 whether the bypass is placed over the hot (FCphbi;j;k;p ) or cold side (FCpcbi;j;k;p ). TAC after the MINLP model is $974,784 when the maximum area is 916.7 m2 . 3.3. LP search algorithm Basic data for the LP search algorithm is the same as for the MINLP model. The LP search uses also additional data obtained from the MINLP model involving the ratio of heat load and mean temperature diﬀerence of process to process match related to the maximum exchanger area matches (bolded in Table 2). Also the HEN conﬁguration obtained from the multiperiod MINLP model is ﬁxed in the search algorithm. For all process to process matches initial Ki;j;k;p ¼ 1. Solving the LP search algorithm gives the results shown in Table 3. TAC after the LP search algorithm is $969,413 when maximum total area is 910.5 m2 . 3.4. NLP search algorithm The NLP search algorithm is solved to overcome the isothermal mixing assumption. The NLP search uses the solution of the LP search as an initial point. Solving the NLP search algorithm gives the results shown in Table 4. After the NLP search algorithm the ﬁnal TAC is $963,051

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Table 2 Results from the multiperiod simultaneous MINLP model Match ijk

Aijkp (m2 ) qijkp (kW) FCphijkp (kW/K) FCpcijkp (kW/K) thinikp (°C) thoutikp (°C) tcinjkp (°C) tcoutjkp (°C) FCphbijkp (kW/K) FCpcbijkp (kW/K)

Aijkp (m2 ) qijkp (kW) FCphijkp (kW/K) FCpcijkp (kW/K) thinikp (°C) thoutikp (°C) tcinjkp (°C) tcoutjkp (°C) FCphbijkp (kW/K) FCpcbijkp (kW/K)

Period 1 ðqhu1;1 ¼ 3247, qcu5;1 ¼ 1155, qcu6;1 ¼ 4116Þ

Period 2 ðqhu1;2 ¼ 4001, qcu5;2 ¼ 1001, qcu6;2 ¼ 2088Þ

1.1.1

1.1.2

2.1.2

3.1.2

4.1.3

6.1.4

1.1.1

1.1.2

2.1.2

3.1.2

4.1.3

6.1.4

14.8 665 2.4 7.1 640 573 518 541 – –

87.6 1117 2.4 4.2 573 460 455 518 0.09 1.25

43.6 572 1.7 2.1 560 480 455 518 – –

28.2 198 0.8 0.7 540 480 455 518 0.05 0.04

214.1 3168 9.4 7.1 480 400 348 455 2.38 0.91

55.3 1428 18.9 7.1 420 402 300 348 15.42 3.95

0.0 0 0.0 0.0 620 620 515 515 – –

54.6 1584 2.4 5.2 620 460 443 515 0.23 2.8

35.2 429 1.7 1.4 540 480 443 515 0.29 0.16

27.9 132 0.8 0.4 520 480 443 515 0.09 0.01

231.4 2376 9.4 7.1 460 400 363 443 2 0.72

115.0 1872 18.9 7.1 400 376 300 363 – –

Period 3 ðqhu1;3 ¼ 3218, qcu5;3 ¼ 819, qcu6;3 ¼ 1652Þ

Nominal ðqhu1;4 ¼ 3008, qcu5;4 ¼ 980, qcu6;4 ¼ 2803Þ

11.3 534 1.9 5.8 620 554 496 518 0.57 3.77

14.0 572 2.1 6.4 630 566 517 539 0.18 1.92

40.1 762 1.9 3.6 554 460 445 496 0.26 1.86

22.3 351 1.4 1.7 540 480 445 496 0.44 0.67

12.3 108 0.6 0.5 520 480 445 496 0.28 0.16

209.3 1944 7.7 5.8 460 400 365 445 1.81 0.62

101.0 1588 15.4 5.8 400 375 300 365 6.03 0.71

138.1 958 2.1 3.9 566 460 460 517 – –

43.6 455 1.6 1.9 550 480 460 517 – –

30.8 150 0.7 0.6 530 480 460 517 – –

315.7 2520 8.6 6.4 470 400 366 460 – –

84.7 1517 17.1 6.4 410 389 310 366 10.68 1.85

when the ﬁnal maximum total area is 913 m2 . The resulting HEN conﬁguration is shown in Fig. 4, including ﬁve process to process units, two cold utility units, one furnace and three bypasses. 3.5. Comparison Floudas and Grossmann [4] introduced a sequential method for an automatic synthesis of multiperiod HEN, which they applied for this same example. Contrary to the above example they did not consider annual costs and they used three periods to describe the operation and the nominal condition was excluded. It is however possible to approximate the utility requirements under the nominal condition assuming a same minimum temperature approach of 10 K that they were using for all other periods. Therefore it is also possible to calculate the comparable TAC according to the results of sequential model (Table 5). The main diﬀerences in resulting HEN conﬁguration, when comparing the proposed simultaneous HEN synthesis framework with the sequential, are that the sequential introduces three cold utility and ﬁve bypasses when simultaneous results in two cold utility and three bypasses. Examining resulting investment costs shows that the simultaneous gives 72 m2 less exchanger area than sequential but 37 kW bigger furnace. Comparisons of the operating costs are shown in Table

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Table 3 Results from LP search algorithm Match ijk

Aijkp (m2 ) qijkp (kW) FCphijkp (kW/K) FCpcijkp (kW/K) thinikp (°C) thoutikp (°C) tcinjkp (°C) tcoutjkp (°C) FCphbijkp (kW/K) FCpcbijkp (kW/K)

Aijkp (m2 ) qijkp (kW) FCphijkp (kW/K) FCpcijkp (kW/K) thinikp (°C) thoutikp (°C) tcinjkp (°C) tcoutjkp (°C) FCphbijkp (kW/K) FCpcbijkp (kW/K)

Period 1 ðqhu1;1 ¼ 3200, qcu5;1 ¼ 1155, qcu6;1 ¼ 4069Þ

Period 2 ðqhu1;2 ¼ 3830, qcu5;2 ¼ 1001, qcu6;2 ¼ 1916Þ

1.1.1

1.1.2

2.1.2

3.1.2

4.1.3

6.1.4

1.1.1

1.1.2

2.1.2

3.1.2

4.1.3

6.1.4

14.6 647 2.4 7.1 640 575 520 542 – –

99.2 1135 2.4 4.2 575 460 456 520 0.04 0.66

46.2 572 1.7 2.1 560 480 456 520 – –

30.6 198 0.8 0.7 540 480 456 520 – –

224.0 3168 9.4 7.1 480 400 350 456 2.04 0.75

57.9 1475 18.9 7.1 420 401 300 350 15.52 3.97

11.0 527 2.4 7.1 620 567 503 521 0.86 4.97

58.4 1057 2.4 4.6 567 460 449 503 0.21 2.07

31.7 429 1.7 1.9 540 480 449 503 0.39 0.50

19.0 132 0.8 0.6 520 480 449 503 0.25 0.10

304.7 2376 9.4 7.1 460 400 369 449 – –

137.3 2044 18.9 7.1 400 374 300 369 – –

Period 3 ðqhu1;3 ¼ 3054, qcu5;3 ¼ 819, qcu6;3 ¼ 1488Þ

Nominal ðqhu1;4 ¼ 3025, qcu5;4 ¼ 980, qcu6;4 ¼ 2820Þ

5.2 265 1.9 5.8 620 587 513 524 1.31 5.14

13.2 551 2.1 6.4 630 569 518 538 0.32 2.82

57.8 1031 1.9 4.0 587 460 452 513 0.11 1.79

32.2 351 1.4 1.4 540 480 452 513 0.28 0.26

24.3 108 0.6 0.4 520 480 452 513 0.12 0.02

306.0 1944 7.7 5.8 460 400 372 452 – –

124.5 1752 15.4 5.8 400 373 300 372 4.60 0.44

118.3 979 2.1 4.0 569 460 459 518 – –

43.1 455 1.6 1.9 550 480 459 518 0.06 0.12

30.5 150 0.7 0.6 530 480 459 518 – –

305.0 2520 8.6 6.4 470 400 366 459 – –

83.1 1500 17.1 6.4 410 389 310 366 12.09 2.34

6 and ﬁnally when reviewing capital (CC) and operating costs (OC) the beneﬁt gained from simultaneous method is $20,300 annual savings (Table 7).

4. Conclusions A framework has been proposed to synthesise ﬂexible HEN and to ﬁnd its set points for different operating conditions so that annual costs will be minimised. This framework includes a multiperiod simultaneous MINLP model to synthesise a ﬂexible HEN conﬁguration and a search algorithm involving multiperiod LP/NLP model to overcome limitations related to the MINLP model. The proposed ﬂexible HEN conﬁguration synthesis ignores pinch considerations and it does not rely on a sequential decomposition of the problem nor on ﬁxed temperature approaches. This synthesis method is able to take into account weighted periods so that the most common operating condition can dominate while the uncommon one is still considered. The elimination of the bypass modelling, a stagewise superstructure presentation and isothermal mixing assumption allows the feasible space to be deﬁned by a set of linear constraints. These simpliﬁcations make the

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Table 4 Results from NLP search algorithm Match ijk

Aijkp (m2 ) qijkp (kW) FCphijkp (kW/K) FCpcijkp (kW/K) thinikp (°C) thoutikp (°C) tcinjkp (°C) tcoutjkp (°C) FCphbijkp (kW/K) FCpcbijkp (kW/K)

Aijkp (m2 ) qijkp (kW) FCphijkp (kW/K) FCpcijkp (kW/K) thinikp (°C) thoutikp (°C) tcinjkp (°C) tcoutjkp (°C) FCphbijkp (kW/K) FCpcbijkp (kW/K)

Period 1 ðqhu1;1 ¼ 3216, qcu5;1 ¼ 1155, qcu6;1 ¼ 4085Þ

Period 2 ðqhu1;2 ¼ 3832, qcu5;2 ¼ 1001, qcu6;2 ¼ 1918Þ

1.1.1

1.1.2

2.1.2

3.1.2

4.1.3

6.1.4

1.1.1

1.1.2

2.1.2

3.1.2

4.1.3

6.1.4

0.0 0 0.0 0.0 640 640 542 542 – –

108.9 1782 2.4 4.2 640 460 456 556 0.03 0.70

45.7 572 1.7 2.1 560 480 456 520 – –

30.8 198 0.8 0.7 540 480 456 521 – –

237.6 3168 9.4 7.1 480 400 349 456 2.16 0.81

57.0 1459 18.9 7.1 420 402 300 349 15.57 4.00

0.0 0 0.0 0.0 620 620 521 521 – –

67.5 1584 2.4 4.9 620 460 449 526 0.15 2.14

37.4 429 1.7 1.6 540 480 449 514 0.25 0.19

29.7 132 0.8 0.5 520 480 449 511 0.15 0.03

305.9 2376 9.4 7.1 460 400 369 449 0.15 0.04

137.0 2042 18.9 7.1 400 374 300 369 – –

Period 3 ðqhu1;3 ¼ 3051, qcu5;3 ¼ 819, qcu6;3 ¼ 1485Þ

Nominal ðqhu1;4 ¼ 3019, qcu5;4 ¼ 980, qcu6;4 ¼ 2814Þ

0.0 0 0.0 0.0 620 620 524 524 – –

0.0 0 0.0 0.0 630 630 538 538 – –

68.6 1296 1.9 3.6 620 460 452 539 0.09 1.32

40.5 351 1.4 1.8 540 480 452 498 0.36 0.72

28.5 108 0.6 0.4 520 480 452 516 0.05 0.01

309.2 1944 7.7 5.8 460 400 372 452 – –

129.4 1755 15.4 5.8 400 373 300 372 4.37 0.41

132.4 1530 2.1 4.1 630 460 459 549 – –

45.7 455 1.6 1.8 550 480 459 521 – –

30.8 150 0.7 0.6 530 480 459 518 – –

Fig. 4. HEN conﬁguration resulting from simultaneous HEN synthesis.

309.1 2520 8.6 6.4 470 400 366 459 – –

130.1 1506 17.1 6.4 410 389 310 366 12.02 2.31

J. Aaltola / Applied Thermal Engineering 22 (2002) 907–918

917

Table 5 Comparison of exchanger areas between simultaneous and sequential method A (m2 )

1.1.2

2.1.2

3.1.2

4.1.3

6.1.4

qcu1:1

qcu5:1

qcu6:1

Atot

Sequential Simultaneous

95 132

73 46

36 31

386 309

142 137

13 –

82 88

159 170

985 913

Table 6 Comparison of the resulting operational costs Furnace fuel

Cooling water

Sequential

Simultaneous

Sequential

Simultaneous

Prd1 (kW) Prd2 (kW) Prd3 (kW) PrdN (kW)

2992 3795 3105 3085

3216.21 3831.94 3050.83 3018.79

5016 2882 2358 3860

5240.21 2918.94 2303.83 3793.79

Cost ($/a)

562,800

556,988

198,974

197,255

Table 7 Comparison of capital (CC), operational (OC) and total annual costs (TAC) Sequential Simultaneous

CC ($/a)

OC ($/a)

TAC ($/a)

221,550 208,781

761,775 754,243

983,325 963,023

model more robust and eﬃcient. In order to partially overcome limitations due to these simpliﬁcations a search algorithm has been developed. It has been shown that the simpliﬁed superstructure presentation proposed by Yee and Grossmann [15] can be applied to consider generating ﬂexible HENs. The major diﬃculty in the proposed framework is to guarantee the network feasibility in the case where the number of uncertain parameters is large. Thus further research is needed in the ﬁeld of critical, or appropriate, operating conditions. Also reducing the problem size in the multiperiod MINLP is one of the future challenges.

References [1] V. Briones, A.C. Kokossis, Hypertargets: a conceptual programming approach for the optimisation of industrial heat exchanger networks. I. Grassroots design and network complexity, Chem. Eng. Sci. 54 (1999a) 519–539. [2] V. Briones, A.C. Kokossis, Hypertargets: a conceptual programming approach for the optimisation of industrial heat exchanger networks. Part III: industrial applications, Chem. Eng. Sci. 54 (1999b) 685–706. [3] J. Cerda, A.W. Westerberg, D. Mason, B. Linnhoﬀ, Minimum utility usage in heat exchanger network synthesis—a transportation problem, Chem. Eng. Sci. 38 (1983) 373–387. [4] C.A. Floudas, I.E. Grossmann, Synthesis of ﬂexible heat exchanger networks with uncertain ﬂowrates and temperatures, Comput. Chem. Eng. 11 (1987) 319–336.

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