CHEMICAL ENGINEERING TRANSACTIONS VOL. 81, 2020 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš Copyright © 2020, AIDIC Servizi S.r.l. ISBN 978-88-95608-79-2; ISSN 2283-9216 Hybrid Synthesis Method for Multi-period Heat Exchanger Networks Adeniyi J. Isafiadea,*, Michael Shortb aDepartment of Chemical Engineering, University of Cape Town, South Africa bDepartment of Chemical and Process Engineering, University of Surrey, Guildford, United Kingdom Aj.Isafiade@uct.ac.za Developing a flexible heat exchanger network, that will remain operable in the face of potential variations in stream parameters, or variables, around some nominal values, especially for large problems, are difficult to solve simultaneously. A hybrid synthesis approach that systematically combines sequential and mathematical programming techniques may be more suitable to adopt. The proposed method presented in this paper entails firstly generating two representative single period networks using the Pinch Technology Stream Temperature versus Enthalpy Plot (STEP) approach and the multi-period stage-wise superstructure model for heat exchanger network synthesis. Streams for the representative network are obtained using the largest stream heat demand as criteria. The second stage of the proposed method entails generating a reduced multi-period stage-wise superstructure, using a combination of the matches obtained in the single period representative networks of the first step as initialising matches. The solution of the reduced superstructure of the second step, which is solved as a Mixed Integer Non-Linear Programming (MINLP) model, is then selected as the best multi-period network. The newly developed method of this paper is tested using an example from the literature. One of the solutions obtained is only 0.9 % higher, in terms of total annual cost, than the best solution presented in the literature but has the benefit of a fewer number of units. 1. Introduction Most of the methods developed for heat exchanger network synthesis (HENS) have assumed constant stream supply/target temperatures and flowrates. These parameters may fluctuate due to issues such as changes in environmental conditions, changes in feed and product quality, process upsets. The few methods which did not assume fixed values for stream parameters, and which used mathematical programming, have shortcomings such as the inability to obtain globally optimal solutions due to the highly nonlinear nature of the model equations involved in HENS, especially for cases where detailed heat exchanger models are involved (Short et al., 2016). On the other hand, the use of sequential optimisation approaches, where the designer can influence the design process, such as Pinch Technology, does not adequately trade off competing variables in HENS, especially those involving multi-period profiles. Developing a flexible heat exchanger network (HEN), that will remain operable in the face of potential variations in stream parameters, especially for large problems, may be better suited to a hybrid synthesis approach that systematically combines sequential and mathematical programming techniques. Verheyen and Zhang (2006) adopted the stage-wise superstructure (SWS) model of Yee and Grossmann (1990), which was originally developed for single period networks, for the synthesis of multi-period networks. Verheyen and Zhang (2006) incorporated variability into the HEN design optimisation using the maximum heat exchanger area approach in the objective function. In this approach, heat exchangers are sized based on the maximum heat load available for same stream pairs in all periods. Isafiade and Odejobi (2016) also adopted the multi-period SWS model of Verheyen and Zhang (2006) in their work. The authors included the reduced superstructure synthesis approach where superstructures are initialised in a systematic way. The initialising approach adopted by Isafiade and Odejobi (2016) entails solving each sub-period’s single period problem and then populating the reduced superstructure with the matches selected in the optimal solutions for the sub-periods. Kang et al. (2015) used a representative period method and a stepwise simplifying method for the synthesis of multi-period HENs. DOI: 10.3303/CET2081199 Paper Received: 22/05/2020; Revised: 21/06/2020; Accepted: 24/06/2020 Please cite this article as: Isafiade A.J., Short M., 2020, Hybrid Synthesis Method for Multi-period Heat Exchanger Networks, Chemical Engineering Transactions, 81, 1189-1194 DOI:10.3303/CET2081199 1189 Solving large HENS problems using a simultaneous synthesis approach to global optimality in reasonable solution time is not a trivial task. The problem is compounded further for cases involving multi-period stream data due to increase in non-linearities. Pinch Technology, which is advantageous because the designer is involved in the synthesis process is also difficult to adopt for large HEN problems especially at the design stage where there are no explicit rules for pairing streams that are far from the pinch. The sequential nature of Pinch Technology makes it even more difficult to adopt for problems involving multi-period profiles. This is because, apart from the difficulty in identifying the pinch point, designing the heat exchangers to meet the operating and capital cost targets, while still ensuring optimal transfer of heat for stream pairs in all periods of operations, is not a trivial task. The work by Yoro et al. (2019) is among the few studies that have adopted pinch technology for synthesising multi-period networks. The approach of the authors did not include a robust capital cost targeting. This paper presents a hybrid approach for identifying initialisation matches that can be used to initialise a reduced simultaneous based superstructure for HENS involving multi-period profile. The proposed method of this paper makes provision for desirability features of HENS methods such as low total annual cost (TAC), minimum number of units and simple networks with few or no stream splits. 2. Problem statement The problem of this paper is stated as follows: Given is a set of hot, HP, and cold, CP, process streams, including their supply, Ts, and target, Tt, temperatures and heat capacity flowrates. The streams, whose temperatures and flowrates can change from one period, P, to another, are to be cooled and heated. Given are sets of hot utilities (HU) and cold utilities (CU) which can be used in any of the operational periods to cool and heat the process streams. Other parameters given are stream heat transfer coefficients, (h), heat exchanger area and installation costs, and duration of each period of operation. The goal is to synthesise a multi-period HEN having a minimum TAC. 3. Methodology The method adopted in this paper also involves generating a reduced superstructure, similar to that of Isafiade and Odejobi (2016). The key difference is that the approach of this paper uses a hybrid technique to identify the set of matches that would be used to initialise the reduced superstructure. The hybrid technique entails adopting the Stream Temperature Versus Enthalpy Plot (STEP) and Heat Allocation and Targeting diagram (HEAT) of Wan Alwi and Manan (2010), on one hand to generate a set of potential matches and using a representative single period SWS model to identify another set of potential matches on the other hand. The two sets of potential matches are then systematically used to initialise a reduced multi-period superstructure that is solved as a mixed integer non-linear (MINLP) model. The STEP method, rather than the traditional Pinch Technology approach, is adopted in the hybrid synthesis model of this paper because it establishes utility targets, and designs to meet these targets, in a single step. This is advantageous for reasons such as easy identification of direct potential one-to-one hot-cold stream matches without the need for stream splitting. This is unlike the traditional Pinch Technology approach where the process of establishing stream matches that meet utility and heat exchanger area targets require stream splitting because streams are represented as composites and not as individual streams. Figure 1 is a summary of how the proposed method of this paper works. The details of how the STEP and HEAT diagrams are generated for single period HENS problems can be found in Wan Alwi and Manan (2010), while the details of the mathematical model equations can be found in the works of Verheyen and Zhang (2006) and Isafiade and Short (2016). The expression for the maximum heat exchanger area approach and objective functions, are both shown in Eq(1) and Eq(2). 𝐴𝑖,𝑗,𝑘 ≥ 𝑞𝑖,𝑗,𝑘,𝑝 (𝐿𝑀𝑇𝐷𝑖,𝑗,𝑘,𝑝) (𝑈𝑖,𝑗 ) (1) 𝑚𝑖𝑛 {∑ ( 𝐷𝑂𝑃𝑝 ∑ 𝐷𝑂𝑃𝑝 𝑁𝑂𝑃 𝑝=1 ∑ ∑ ∑ 𝐶𝑈𝐶𝑗 ∙ 𝑞𝑖,𝑗,𝑘,𝑝 𝑘𝜖𝐾𝑗𝜖𝐶𝑈𝑖𝜖𝐻𝑃 + 𝐷𝑂𝑃𝑝 ∑ 𝐷𝑂𝑃𝑝 𝑁𝑂𝑃 𝑝=1 ∑ ∑ ∑ 𝐻𝑈𝐶𝑖 ∙ 𝑞𝑖,𝑗,𝑘,𝑝 𝑘𝜖𝐾𝑗𝜖𝐶𝑃𝑖𝜖𝐻𝑈 ) 𝑝𝜖𝑃 } + 𝐴𝐹 (∑ ∑ ∑ 𝐶𝐹𝑖𝑗 ∙ 𝑧𝑖,𝑗,𝑘 + 𝑘𝜖𝐾𝑗𝜖𝐶𝑖𝜖𝐻 ∑ ∑ ∑ 𝐴𝐶𝑖𝑗 ∙ 𝐴𝑖,𝑗,𝑘 𝐴𝐶𝐼 𝑘𝜖𝐾𝑗𝜖𝐶𝑖𝜖𝐻 ) ∀ 𝑖𝜖𝐻𝑃, 𝑗𝜖𝐶𝑃, 𝑘𝜖𝐾, 𝑝𝜖𝑃 (2) In Eq(1), qi,j,k,p represents the quantity of heat (kW) exchanged between hot stream i and cold stream j in interval k of the SWS and operational period p, Ui,j represents overall heat transfer coefficient between hot stream i and cold stream j (0.1 W/m2°C), LMTDi,j,k,p represents the logarithmic mean temperature difference (°C) between hot stream i and cold stream j in interval k of the SWS and operational period p, while Ai,j,k represents heat exchanger 1190 area (m2) transferring heat between hot stream i and cold stream j in interval k of the superstructure. In Eq(2), DOPp and NOP represent duration of operational period p and number of periods, CUCj and HUCi represent cost per unit of cold and hot utility (1.3 $/(kW∙y) for cold utility and 115.2 $/(kW∙y) for hot utility), AF (0.2) represents annualisation factor, CFi,j (8,333.3 $) represent heat exchanger installation cost, zi,j,k is a binary variable which is used to indicate the existence, or otherwise, of a heat exchanger between hot stream i and cold stream j in interval k of the superstructure, ACi,j (641.7 $/m2) represents cost per unit heat exchanger area while ACI is heat exchanger area cost exponent. Figure 1: Structure of solution procedure of proposed methodology 4. Case study The example used in this paper to illustrate the proposed synthesis method is the popular 3 hot streams and 4 cold streams multi-period HEN problem which was first presented by Verheyen and Zhang (2006). The problem, which has been solved by several authors, involves 3 periods of operations of equal durations. The stream data for the 3 operational periods is shown in Table 1. Applying the first step of the solution procedure shown in Figure 1 gives the representative streams shown in Table 2. Table 1: Stream data of case study for start of operation, middle of operation and end of operation Streams Period 1 Period 2 Period 3 Ts (°C) Tt (°C) F (kW∙°C) Ts (°C) Tt (°C) F (kW∙°C) Ts (°C) Tt (°C) F (kW∙°C) H1 390 60 201.6 406 60 205.0 420 60 208.5 H2 160 40 185.1 160 40 198.8 160 40 175.2 H3 354 60 137.4 362 60 136.4 360 60 134.1 C1 72 356 209.4 72 365 210.3 72 373 211.1 C2 62 210 141.6 62 210 141.0 62 210 140.5 C3 220 370 176.4 220 370 175.4 220 370 174.5 C4 253 284 294.4 250 290 318.7 249 286 271.2 HU 500 450 500 450 500 450 CU 0 10 0 10 0 10 Note that the representative stream data are obtained by identifying a fictitious stream to represent streams in all operational periods. As an example, the fictitious stream representing hot stream 1 in periods 1, 2 and 3, is a stream having the largest heat load. This is determined from Table 1 by identifying the following, the highest supply temperature (420 °C in period 3), the lowest target temperature (60 °C, same for all periods) and the largest heat capacity flowrate (208.5 kW∙°C in period 3). The same procedure is adopted for the cold streams, where the highest supply temperature, the lowest target temperature and the largest heat capacity flowrates are also identified. Doing this for all streams results in the data shown in Table 2. The second step of the solution procedure involves the generation of the STEP and HEAT diagrams on one hand, and the generation of an Generate representative single period stream data from multi-period data Generate shifted hot and cold STEPs and HEAT diagram Solve single period data as MINLP model using SWS Identify enthalpy intervals and potential matches Identify selected matches from network solution Generate reduced multi- period MINLP SWS superstructure and initialise using identified matches from the two single period Solve while systematically adding or removing initialising matches to improve solution 1191 SWS based network solution on the other hand. In both cases, the representative stream data of Table 2 are used. The resulting STEP diagram is shown in Figure 2. In this figure, the red lines represent the hot STEP while the green lines represent the cold STEP. Table 2: Representative stream data Stream Ts (°C) Tt (°C) F (kW∙°C) H1 420 60 208.5 H2 160 40 198.8 H3 362 60 137.4 C1 72 373 211.1 C2 62 210 141.6 C3 220 370 176.4 C4 249 290 318.7 The HEAT diagram, which is not shown here, is used to match streams on a one-to-one basis within each temperature interval. It is worth stating that future studies will involve the use of the HEAT diagram to set targets for heat exchanger design parameters such as pressure drop, number of shells and tubes. Such targets will then be used to initialise a reduced superstructure. In this paper, only the matches identified in the STEP diagram of Figure 2 are used as initialising matches for the reduced superstructure. In the STEP method, each enthalpy interval corresponds to a heat exchanger. In Figure 2, there are 20 enthalpy intervals, of which some involve direct overlap between the hot and cold STEPs, with the hot STEP being at a higher temperature interval compared to the cold. These intervals correspond to the region of process-to-process heat transfer. Other enthalpy intervals have either only the hot STEP, or even both hot and cold, but the cold STEP is at a higher temperature interval compared to the hot, while one interval has only the cold STEP. Intervals having only the hot STEP will be matched with cold utilities while regions having only the cold STEP will be matched with hot utilities. The shifted stream pinch temperature in the STEP diagram is 230 °C. Figure 2: STEP diagram for the case study Figure 3 shows the mapping of the targeted matches in the STEP diagram onto the stage-wise multi-period superstructure. Some targeted exchangers in the enthalpy intervals of the STEP diagram have been combined and represented as a single match in Figure 3. This is done to reduce the number of intervals from 20 in Figure 2 to 10 in Figure 3. The second aspect of step 2 of the synthesis procedure shown in Figure 1 involves generating an additional set of potential matches that will be used to initialise the reduced superstructure. This additional set of matches are obtained by solving the representative streams in Table 2 using the single period SWS model. The SWS at this stage is also solved using 10 intervals to ensure consistency with the 10 intervals of the STEP grid structure in Figure 3. 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000 180,000 1192 Figure 3: STEP matches populated in the stage-wise multi-period superstructure Figure 4: Reduced stage-wise multi-period superstructure comprising a combination of matches from the STEP and conventional single period SWS model. Figure 5: Final optimal solution network It is worth stating that the solution of the network, whose structure is not shown here, has a TAC of 6,364,069 $/y. Figure 4 represents the third step of the synthesis procedure of Figure 1, which involves the generation of the reduced superstructure. The reduced superstructure comprises a combination of the matches obtained in the STEP diagram and the single period conventional SWS. In Figure 4, the matches represented with dark shaded circles are those that are common to both the STEP grid structure of Figure 3 and the solution of the SWS single period model. The matches represented with thick lined circles are those that are only common to the STEP grid structure, while those represented with broken circles are only common to the single period SWS model solution structure. The matches in Figure 4 are then used as the initialising matches in the reduced superstructure which comprises the multi-period stream data of Table 1. The superstructure was solved as an MINLP model. Solving the model as a reduced superstructure is beneficial, as presented by Isafiade et al (2015), as it has shorter solution time due to the model having fewer binary variables. Using the STEP diagram as one of the structures from which 1193 promising matches are obtained is beneficial in that when the resulting solution network of the STEP diagram is converted to potential matches in the SWS, the ensuing superstructure will have many intervals and no stream splits. This is evident in Figure 3 where there are no stream splits. Networks without stream splits are preferable to those with stream splits because they are less complex in terms of layout and controllability. Also, other features of heat exchangers, such as pressure drop, number of shells and tubes, can be targeted at the STEP diagram stage, and then exported to the multi-period SWS model as initialising parameters. This is planned as follow-up work to this paper. Figure 5 shows the final solution obtained in this paper for the case study. The TAC of the solution which has 12 units is 6,268,587 $/y. This TAC comprises an annual operating cost of 3,079,861 $/y and an annualised investment cost of 3,188,726 $/y. It is worth noting that in the network of Figure 5, no matches are in intervals 6 and 9. The solution can be improved not in terms of TAC but in the number of units in the optimal solution by excluding the initialising matches present in these intervals in the reduced superstructure of Figure 4 and then solving the updated reduced superstructure. Doing this gives a solution having a TAC of 6,277,254 $/y and 11 units. This second, solution, which is just 0.14 % higher than the solution of Figure 5 in terms of TAC, would be preferred due to the fewer number of units. The best solution, in terms of TAC, presented in the literature is that of Pavao et al. (2018). Their solution, which is just 0.9 % lower than the second solution of this paper, has 13 units. 5. Conclusions This paper has presented a hybrid approach to identifying potential initialising matches for solving the multi- period HENS problem using a reduced SWS model. The method is beneficial in that promising initialising matches are obtained from the STEP method and SWS model for single period HENS problems. The result obtained from the case study considered compares favourably with those reported in the literature in that, apart from having a relatively low TAC, it also produces a simple network with no stream splits. The motivation for adopting the hybrid approach in this paper is that the STEP diagram serves as a good basis from which not only utilities and heat exchanger areas can be targeted and designed, but estimates for other heat exchanger design parameters such as pressure drop, number of shells and tubes, can be obtained and then exported to the reduced multi-period HENS SWS model as initialising parameters to solve the problem. Also, solving large- scale HENS problems is not a trivial task. The difficulty is compounded further for large-scale HENS problems involving multi-period profiles due to an increase in non-linearities. It is hoped that these issues will be addressed in future studies using an extended version of the methodology of this paper. Acknowledgments This study is supported by the National Research Foundation of South Africa (Grant number: 85536) and the Research Office of the University of Cape Town, South Africa. The funds are gratefully acknowledged. 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