Synthesis and Retrofit of Multiperiod Heat Exchanger Networks Considering Heat Transfer Enhancement and Environmental Impact

Heat transfer enhancement has been used to achieve lower heat transfer area and reduced utility requirements in heat exchanger network synthesis problems involving retrofit. This paper presents a technique for implementing heat transfer enhancements in both grassroot and retrofit heat exchanger network synthesis problems involving multiperiod operations. For the grassroot scenario, the approach adopted involves extending the stage-wise superstructure model for multiperiod heat exchanger network synthesis to accommodate various options of tube-side and shell-side heat transfer enhancement techniques. The extended model is then applied to a retrofit scenario using the reduced superstructure synthesis approach. The results obtained demonstrate the benefits of heat transfer enhancement through reduced operating costs in the grassroot and retrofit scenarios. For the grassroot scenario, the operating and capital costs obtained in this paper are 12.3% and 20% lower than solutions obtained using conventional non-enhanced synthesis methods, while for the retrofit scenario, solutions which compare favourably with literature solutions, and which has the potential to be more environment friendly by virtue of lower energy utilisation, are obtained.


Abbreviations
installation cost for heat exchanger for match i, j (€) CFTS i, ti installation cost for tube-side HTE of type ti for hot stream i (€) CFSS j, si installation cost for shell-side HTE of type si for cold stream j (€) DOP p duration of period p Ec i mass percentage of the pollutant of concern EI min lowest environmental impact (1/y) ef j, si enhancement factor for cold stream j with enhancement type si at the shell side ef i, ti enhancement factor for hot stream i with enhancement type ti at the tube side h i, p individual heat transfer coefficient for hot stream i in period p (kW/m 2 • ° C) h j, p individual heat transfer coefficient for cold stream j in period p (kW/m 2 • ° C) LHV i lower heating value of utility source (kWh/kg) FCp i, p heat capacity flowrate of hot stream i in period p (kW/ ° C) FCp j, p heat capacity flowrate of cold stream j in period p (kW/ ° C) R g factor used to allocate weights to objectives TAC min lowest total annual cost of network ($/y) T s i,p supply temperature of hot stream i in period p (°C) T t i,p target temperature of hot stream i in period p (°C) T s j,p supply temperature of cold stream j in period p (°C) T t j,p target temperature of cold stream j in period p (°C) SSE j, si shell-side area cost coefficient for cold stream j with enhancement type si (€/m 2 ) TSE i, ti tube-side area cost coefficient for hot stream i with enhancement type ti (€/m 2 ) U i, j, p overall heat transfer coefficient for match i, j in period p (kW/m 2 • ° C) η efficiency of boiler Ω upper bound for the quantity of heat that can be exchanged in match i, j, k Ω i, ti parameter to ensure that the tube-side heat transfer coefficient constraint for hot stream i and enhancement type ti is not violated Ω j, si parameter to ensure that the shell-side heat transfer coefficient constraint for cold stream j and enhancement type si is not violated ϕ upper bound for driving force in match i, j, k hot utility exchanger area that will be newly purchased for retrofitted network (m 2 ) ACU New∕Add i,j,k cold utility exchanger area that will be newly purchased for retrofitted network (m 2 ) AP New∕Add i,j,k process exchanger area that will be newly purchased for retrofitted network (m 2 )

Introduction
The energy crises of the 1970s necessitated the design of energy efficient industrial processes of which heat exchanger network synthesis (HENS) has over the last four decades been a classical conceptual approach to achieve such energy optimal systems.The study of HENS has over the years received significant attention from researchers using methods that can be classified under insight-based approaches, deterministic and non-deterministic mathematical programming approaches.In HENS, various scenarios which include single period operations, multiple period operations, grassroot and retrofit scenarios have been investigated, although the multiple period operations, especially when it includes retrofit, have received relatively far less attention.In individual heat exchanger designs, the concept of heat transfer enhancement (HTE) has been well studied (Maradiya et al. 2018); however, its extension to the network context has received far less attention.
According to Zhu et al. (2000), who adopted a targeting approach, the issues that must be addressed when HTE is incorporated in HEN retrofit are determination of exchangers that are most suitable for enhancement, level of heat transfer performance to be achieved and what enhancement techniques to adopt.Wang et al. (2012) developed a new model for calculating heat transfer coefficients and pressure drop of each individual exchangers in a HEN.This is then followed using heuristic rules to screen candidate heat exchangers to determine which is best for enhancement.According to the review paper presented by Klemeš et al. (2020), process intensification and process integration are the two approaches that have been used for improvement in energy recovery in the process industry.The authors investigated the methods developed for the retrofitting of HENs and the role played by heat transfer enhancements in the retrofits.They defined HTE as a technique that adopts the modification of equipment to increase heat transfer rate for a given area, thereby attaining higher heat load within the heat exchanger.The authors further classified HTE into the active and passive categories with the active requiring external energy and devices while the passive involves reconfiguration of heat exchangers and/or addition of inserts.Under the category of the passive for shell and tube heat exchangers, Klemeš et al. (2020) identified tube-side enhancement and shellside enhancement types.The passive HTE results in one or more of the following, change in flow pattern inside tubes or on the external surfaces of tubes or change in shape of tubes.The review paper of Klemeš et al. (2020) also discussed the pinch and thermodynamic approaches and mathematical programming approaches to HEN retrofit considering HTE.Under the category of pinch and thermodynamic approaches, Akpomiemie and Smith (2015) used dominant ratio to identify the best location within an existing HEN where HTE can be implemented.After identifying the best location, the authors then used sensitivity analysis to determine which exchangers to enhance and in what order.This is then followed by using a costbased optimisation approach to ensure that the retrofitted network is feasible.Akpomiemie and Smith (2016) extended the work of Akpomiemie and Smith (2015) by using sensitivity analysis and area ratio approach, through comparison of the two, to determine the best set of heat exchangers to enhance and to what extent.To ensure feasibility of the resulting HEN after enhancement, the authors used a nonlinear optimisation-based model.The authors discovered that the area ratio approach was better in that it was more robust and computationally less expensive.According to Akpomiemie and Smith (2017), HTE is beneficial in that it saves energy; however, it may impact negatively on the imposed pressure drop constraints on the enhanced retrofitted HEN.The authors then presented an approach to solve the pressure drop constraint issue.The method entails establishing the heat exchangers that violate imposed pressure drop constraints and then applying a systematic approach to determine the best modification technique to adopt for the exchangers.The last step entails using a nonlinear-based optimisation model with the modified heat exchangers to achieve the enhancement.
The papers reviewed so far have mostly applied HTE to HEN retrofit scenarios.In terms of the application of HTE to grassroot scenarios, the work by Odejobi et al. (2015) is one of the first in this area.The authors adopted the stagewise superstructure (SWS) of Yee and Grossmann (1990) for HENS by simultaneously optimising capital and operating costs with the capital cost including options of various tubeside and shell-side enhancements.The developed model determines the best tube-side or shell-side enhancements to adopt based on performance and cost.It is worth stating that although the implementation of HTE techniques in HEN has received some attention in the literature; however, most of the application presented are for retrofit scenarios with the work of Odejobi et al. (2015) being one of the few that considered grassroot scenario for HENS involving single period operations.The profile of process streams is usually multiperiod in nature due to issues such as changes in environmental conditions, start-ups and shutdowns, changes in feed quality or product demand and process upsets.Therefore, it will be worthwhile to extend the existing HTE methods developed for both single period grassroot and retrofit HEN problems to the multiperiod scenario which is the goal of this paper.
The synthesis of HENs for multiperiod operations has within the last four decades received attention in the literature with most methods developed from the SWS for HEN presented by Yee and Grossmann (1990) for single period problems.Prior to the SWS-based models, Floudas and Grossmann (1986) extended the linear programming and mixed integer linear programming transhipment models of Papoulias and Grossmann (1983) for single period HENs to multiperiod problems.The approach adopted by Floudas and Grossmann (1986) entails targeting the pinch points, minimum utility and minimum number of heat exchangers required in each period of operation.The final multiperiod network is then manually generated based on the targets obtained in the targeting step.This manual approach of generating the final multiperiod network, which according to Floudas and Grossmann (1986) is a drawback of the method, was improved by Floudas and Grossmann (1987) using a multiperiod version of the nonlinear programming automatic network generation technique presented by Floudas et al. (1986) for single period HENs.It is worth stating that the key shortcomings of the synthesis methods presented by Floudas and Grossmann (1986) and Floudas and Grossmann (1987) are found in the sequential nature of the optimisation wherein the solution of a step is dependent on the solution of the previous step which has the tendency to exclude the simultaneous trade-off of competing variables especially in more complex problems involving retrofit and HTE.This shortcoming then led to the development of the simultaneous synthesis methods for multiperiod problems.
The first paper to extend the SWS model of Yee and Grossmann (1990) to the multiperiod scenario is Aaltola (2002) where an average area approach was used in the objective function.Verheyen and Zhang (2006) later found that the average area approach of Aaltola (2002) does not produce optimally sized heat exchangers in the multiperiod network, so the authors developed the maximum area approach.The maximum area approach, which is based on the SWS model, was later extended by Isafiade et al. (2015) using the reduced superstructure synthesis approach.The work of Isafiade et al. (2015) was in turn extended by Isafiade and Short (2016) to multiperiod problems involving unforeseen changes in periodic durations.To ensure that key design parameters are accounted for in multiperiod HENS using mathematical programming, Short et al. (2016a) adopted the correction parameter approach which was adapted from the work of Short et al. (2016b) developed for single period HENS problems.In the aspect of retrofit of multiperiod HENs, far fewer papers have been presented.Kang and Liu (2014a) adopted a two-step approach for the retrofit of multiperiod networks wherein the first step entails setting a retrofit target by solving the multiperiod HEN problem while the second step entails using a reverse order approach to match required heat exchangers with existing ones.In another paper by Kang and Liu (2014b), the reverse order matching approach of Kang and Liu (2014a) was extended to handle problems involving restrictions on the operating pressures of the heat exchangers.The reverse order matching approach was again further extended by Kang and Liu (2015) to include addition of new and extra heat transfer areas as well as substitution of heat exchangers.It is worth stating that only the conventional retrofit approaches have been applied to multiperiod HENs, while the opportunities, and benefits of implementing HTE through simultaneous optimisation using mathematical programming has not been explored.So, this paper will be the first to extend the implementation of HTE to multiperiod HENS for both grassroot and retrofit scenarios using a simultaneous optimisation approach.

Problem Statement
The multiperiod HENS problem addressed in this paper entails two scenarios.The first involves developing a synthesis method for incorporating HTE in grassroot multiperiod HENS problems while the second involves extending the first scenario to multiperiod HENS retrofit problems.

Problem Statement 1 for the Grassroot Scenario
This problem statement involves a set H of hot streams (process and hot utilities) and a set C of cold streams (process and cold utilities).The hot and cold process streams have distinct stream parameters in specified periods of operations P. For each period, the hot streams have supply temperatures T s i,p , target temperatures T t i,p , heat capacity flowrates FCp i, p and operational period dependent individual stream heat transfer coefficients h i, p , while the cold streams also have supply temperatures T s j,p , target temperatures T t j,p , heat capacity flowrates FCp j, p and operational period dependent stream individual heat transfer coefficients h j, p .Hot and cold utilities having specified supply and target temperatures and unit costs are available at each of the operational periods.The heat exchangers available for heat transfer between the hot process streams and cold process streams are shell and tube exchangers with options of enhancement using a set of tubeside enhancements TI (such as coiled wire inserts, twisted tape inserts, internal fins, combination of twisted tape inserts and internal fins and combination of coiled wire inserts and internal fins) and a set of shell-side enhancements SI (such as external fins, helical baffles and a combination of both).Other parameters given are installation costs for heat exchanger areas, installation costs for tube-side and shellside enhancements, unit area costs, area cost exponents for the heat exchangers and the equivalent unit costs for tubeside and shell-side enhancements.The goal is to synthesise a heat exchanger network that optimally transfers heat at each period of operation within the network.The objective function minimises the sum of the annual operating and annual capital costs.

Problem Statement 2 for the Retrofit Scenario
Given an existing multiperiod HEN having a set H of hot streams (process and hot utilities) and a set C of cold streams (process and cold utilities), the hot and cold process streams have distinct stream parameters in specified periods of operations P. For each period, the hot streams have supply temperatures T s i,p , target temperatures T t i,p , heat capacity flowrates FCp i, p and individual heat transfer coefficients h i, p , while the cold streams have supply temperatures T s j,p , target temperatures T t j,p , heat capacity flowrates FCp j, p and individual stream heat transfer coefficients h j, p .The existing network, which involves a set of heat exchangers of known sizes, heat loads and stream pairs, uses known quantities of hot and cold utilities at each of the periods of operations.Other parameters given are installation and unit area costs for new heat transfer areas or heat exchangers, set of tubeside and shell-side enhancements (same as given in problem statement 1) for enhancing heat transfer in the retrofitted network.The objective is to generate an optimal retrofitted heat transfer enhanced network with relatively lower energy consumption compared with the original network, optimal utilisation of existing heat exchangers and minimal installation of new exchanger units.

Methodology
The contribution of this paper involves extending the existing SWS model for multiperiod HENS presented by Verheyen and Zhang (2006) and Isafiade et al. (2015) to capture the scenarios described in the two problem statements.This will involve adapting the HTE model for single period grassroot HENS problems presented by Odejobi et al. (2015) to multiperiod grassroot scenarios as stated in problem statement 1.The SWS model used by Odejobi et al. (2015) is based on the original version presented by Yee and Grossmann (1990) while the version used in this paper is based on the modified version presented by Bogataj and Kravanja (2012) and is shown in Fig. 1.The extended model equations for the multiperiod grassroot HENS involving HTE of problem statement 1 will then be combined with an improved version of the reduced superstructure retrofit synthesis approach for multiperiod HENs presented by Isafiade (2018) in problem statement 2 to obtain a new approach for simultaneously implementing HTE with the retrofit of multiperiod HENS.So, the first group of model equations in this paper (Eqs.(1) to (11)) are the conventional multiperiod SWS models adapted from Verheyen and Zhang (2006) and Isafiade et al. (2015).The second group of Eqs. ( 12) to (25) involves the newly developed equations for simultaneously implementing HTE in grassroot multiperiod HENS, while Eqs.( 26) to (38), which are also newly developed in this paper, involve equations for simultaneously implementing HTE in retrofit multiperiod HENS problems.To account for environmental impact, through simultaneous optimisation and while implementing HTE, Eq. ( 39), which describes the environmental impact objective function, and Eq. ( 40), which describes the implementation of multi-objective optimisation using the goal method, are presented.These model equations are described next.

Hot and Cold Stream Heat Balance
The overall enthalpy balance for every hot stream i and every cold stream j in period p must be satisfied through Eqs. ( 1) and (2).In these equations, q i, j, p, k is the quantity of heat (in kW) exchanged between hot stream i and cold stream j in period p and stage k of the multiperiod HEN SWS.

Hot and Cold Stream Stage Enthalpy Balance
Equations (3) and (4) describe the stage enthalpy balance for every hot stream and every cold stream in the multiperiod HEN SWS model.Equation (3) shows that the difference between temperature of hot stream i in period p at temperature location k (i.e.t i, p, k ) and the temperature (t i, p, k + 1 ) of the same stream in the same period at temperature location k + 1 multiplied by the heat capacity flowrate of the stream in period p is equated to the sum of the quantity of heat exchanged between the hot stream and every cold streams in that stage in the multiperiod SWS model.The equivalent equation for cold streams is illustrated by Eq. ( 4).

Monotonicity of Temperature Along the Superstructure
In the multiperiod SWS model, temperatures of hot and cold streams in every period of operations where they exist should decrease from the left-hand side of the superstructure to the right-hand side.This is ensured for both hot and cold streams through constraints ( 5) and ( 6).

Logical Constraint on Exchanger Heat Load
For every heat exchanger, the maximum heat that can be exchanged between hot and cold streams is specified by the logical expression in Eq. ( 7).This maximum heat load, represented by Ω, is determined by the minimum of the heat loads available in the hot and cold streams participating in the heat exchanger across all periods of operations.In Eq. ( 7), Ω is multiplied by a binary variable y i, j, k , which indicates the existence, or otherwise, of a heat exchanger match between hot stream i and cold stream j in stage k of the superstructure.

Heat Exchanger Approach Temperatures
Since the version of the multiperiod SWS model used in this paper involves the isothermal mixing assumption, then exchanger approach temperature is directly equivalent to the stage temperature location differences between the hot and cold streams participating in an exchanger within the stage of the superstructure concerned.This is illustrated by Eqs. ( 8) and ( 9) for temperature locations k and k + 1 of the superstructure.In these equations, dt i, j, p, k represents the 1 The modified stage-wise superstructure as presented by Bogataj and Kravanja (2012) exchanger approach temperature at temperature location k, ϕ is a parameter which can be set by the designer as the maximum of '0' and a match's hot end and cold end temperature differences.

Exchanger Minimum Approach Temperature (EMAT)
Equation ( 10) is used to specify a lower bound (EMAT) for the exchanger approach temperatures.

Logarithmic Mean Temperature Difference (LMTD) Calculation
The LMTD is calculated using the Paterson (1984) approximation as shown in Eq. ( 11).( 8) Like the work of Odejobi et al. (2015), the next set of model equations (Eqs.( 12) to ( 25)) account for the simultaneous optimisation of HTE techniques in a grassroot multiperiod HENS based on the SWS model.The approach adopted in this paper implements HTE only for the process heat exchangers, which according to Odejobi et al. (2015) ensures maximum heat recovery.For every heat exchanger, the fluid flowing through the tube side is assumed to be the hot stream while cold stream is assumed to flow in the shell side.

Overall Heat Transfer Coefficient
After HTE, the tube-side heat transfer coefficient TU i, p and the shell-side heat transfer coefficient SU j, p , which are both specified as variables to be optimised, are used to calculate each process exchanger's overall heat transfer coefficient UO i, j, p as shown in Eqs. ( 12) and ( 13).Since the problem is multiperiod, then the variables are operational period dependent.Note that in Eq. ( 12), it is assumed that the tube has relatively low thermal resistance.
(12) DU i,j,p = 1 Odejobi et al. (2015), where HTE was implemented for single period HENS involving grassroot design, this work aims to achieve relative increases in overall heat transfer coefficients of streams.Such increase will then result in relatively smaller heat exchanger areas.Since different types of tube-side and shell-side enhancements result in varying degrees of relative increase in the overall heat transfer coefficient, this work uses a set of constraints to ensure that the optimal enhancement type is selected for both tube side and shell sides while considering the multiperiod profile of the streams flowing through the exchangers.Parameters that differentiate the enhancement types used in this work include enhancement factor and enhancement area cost coefficient.The enhancement factor determines the maximum heat transfer coefficient that can be achieved.Equations ( 14) and ( 15) are used to ensure that the corresponding enhancement type and enhancement factor are systematically selected in the multiperiod optimisation to achieve optimal performance in the multiperiod network.In Eqs. ( 14) and ( 15), ef i, ti is (13) UO i,j,p = 1 DU i,j,p i ∈ H;j ∈ C; p ∈ P the enhancement factor for the tube-side enhancement and ef j, si is the enhancement factor for the shell-side enhancement.Ω i, ti and Ω j, si are arbitrarily set as large values.yt i, ti and yt j, si are binary variables indicating whether tubeside enhancement ti is implemented for hot stream i and whether shell-side enhancement si is implemented for cold stream j.
To simplify the model, TU i, p and SU j, p in Eqs. ( 12) and ( 13) are specified as only being stream dependent while yt i, ti and yt j, si in Eqs. ( 14) and ( 15) are specified as stream type and enhancement type dependent.The implication of this simplification is that if yt i, ti = 1 in Eq. ( 14) for a hot stream (e.g.H1), then a specific tube-side enhancement will be implemented for every heat exchanger through which H1 flows.However, the tube-side enhancement that will be implemented will be determined by the operational period that requires the maximum sized heat exchanger area which in turn is dependent on UO i, j, p .The same applies to shell-side enhancements.( 14)

Enhancement Area Cost Coefficients
While Eqs. ( 14) and ( 15) are used to ensure that the most suitable enhancements and optimal enhancement levels are selected to achieve optimal multiperiod network performance, Equations ( 16) and ( 17) are used to ensure that simultaneously, the cost optimal enhancement types are selected.In Eqs. ( 16) and ( 17), TSE i, ti and SSE j, si are the tube-side area cost coefficient and shell-side area cost coefficient, while ATUB i, ti and ASHEL j, si are the area cost coefficients implemented for the tube side and shell side.

Enhancement Logical Constraints
Like the work of Odejobi et al. (2015) for single period network, logical constraints are used to ensure that for each heat exchanger, only one tube-side and one shell-side enhancements can be implemented.This is illustrated by Eqs. ( 18) and ( 19).
The form in which Eqs. ( 18) and ( 19) are presented is such that one enhancement type must be selected for each hot process stream and each cold process stream.To accommodate the case where the option of non-enhancement is included in the search space, additional tube-side enhancement with dummy values of 0 for TSE i, ti and 1 for ef i, ti , as well as shellside enhancement with dummy values of 0 for SSE j, si and 1 for ef j, si , can be included in Eqs. ( 14), ( 15), ( 16) and ( 17).Also, in Eqs. ( 14) and ( 15), a lower bound, which is equal to h i, p and h j, p , must be set for TU i, p and SU j, p respectively, so that the option of as low as possible HTE can be implemented.

Heat Exchanger Area
The maximum area approach of Verheyen and Zhang (2006) is updated in this paper to capture the implementation of HTE in multiperiod grassroot problems.The maximum area heat exchanger, also known as the representative heat exchanger, ensures that the heat exchanger is sized based on the heat demand of the period requiring the largest heat load for same stream pairs that are matched in multiple periods.Equations ( 20) and ( 21) illustrate the ( 16) constraints for AHU i, j, k (area of hot utility exchanger) and ACU i, j, k (area of cold utility exchanger).In Eq. ( 20), U i, j, p is overall heat transfer coefficient for a match involving streams i and j in period p.Since in this paper, HTE is only implemented for process heat exchange, Eq. ( 20) applies only to hot utility heat exchangers, i.e. units exchanging heat between hot utilities and cold process streams while Eq. ( 21) applies only to cold utility exchangers.
The representative area for the process heat exchangers (AP i, j, k ) that will be selected for enhancement is illustrated in Eq. ( 22).Note that a process heat exchanger may be selected for HTE or not selected.If not selected, then UO i, j, p in Eq. ( 22) will become U i, j, p .This implies that TU i, p in Eq. ( 14) will become h i, p while SU j, p will become h j, p , and U i, j, p will then be computed using the equivalent expressions for Eqs. ( 12) and ( 13) for a non-enhanced process heat exchanger.
Equations ( 23) and ( 24) represent the areas that would be obtained for the tube-side HTE (APTS i, j, k ) and the shellside HTE (APSS i, j, k ).Since the HEN problem considered in this paper involves multiperiod, then the maximum area approach as used for the conventional multiperiod SWS HENS model (Verheyen and Zhang 2006) is adapted to the HTE scenario to determine the sizes of the representative enhancements that will be inserted in the heat exchangers to serve same stream pairs exchanging heat in multiple periods of operations.This implies that for every potential match that involve same stream pair that exist in multiple operational periods, the period requiring the largest tube-side and shell-side HTE is selected as the representative HTE with sizes APTS i, j, k and APSS i, j, k . (20)

Objective Function
The objective function that addresses problem statement 1 is shown in Eq. ( 25).The equation involves the minimisation of the sum of the annual operating cost (represented by the two terms in the first curly bracket) and annual capital cost (represented by all the terms in the second curly bracket).Since the problem is multiperiod, the operating cost component is weighted proportionately with the periodic durations as also done by Isafiade and Fraser (2010).In the equation, DOP p is the duration of each period p, NOP is the number of periods, CUC and HUC are costs per unit of cold utility and hot utility.The annual capital cost comprises the annualisation factor, AF, fixed charge, CF i, j , for the installation of a heat exchanger, fixed charges CFTS i, ti and CFSS j, si for the implementation of tube-side and shellside HTE, cost per unit area AC i, j , ACHUT i, j and ACCUT i, j for process heat exchanger, hot utility exchanger and cold utility exchanger, AE is the heat exchanger area cost component, while ATUB i, ti and ASHEL j, si are costs per unit area of tube-side and shell-side HTEs.
For problem statement 2, the next set of equations can be used to simultaneously trade-off operating cost and capital costs in the retrofit of multiperiod HENs involving HTE.The capital cost component involves the simultaneous trade-off of tube-side and shell-side HTEs with the optimal utilisation of existing heat exchangers and minimisation of additional newly purchased heat exchanger areas.Before describing the set of additional model equations for this retrofit scenario, the reduced superstructure synthesis approach for retrofit of multiperiod network presented by Isafiade (2018) is first discussed.
The workings of the reduced superstructure synthesis approach for retrofit of HENs presented by Isafiade (2018) entails three steps.In the first step, the problem data of the original network is extracted and solved for a grassroot scenario using the conventional SWS synthesis method for multiperiod HENs presented by Verheyen and Zhang (25) 2006).In this first step, the exchangers are costed using the costs for new heat exchangers.The set of matches selected in the optimal solution of the first step are identified.These matches will constitute a subset of the matches that will be used to initialise the reduced superstructure.The second set of subsets of matches that will be used to initialise the reduced superstructure are obtained in the second step.These matches are the units in the original network.The third step entails generating the reduced superstructure using the set of matches obtained in the first and second steps as initializing binary variables.The reduced superstructure is then solved as a mixed integer nonlinear program (MINLP) model to obtain the optimal retrofitted multiperiod network.A flowchart that illustrates the implementation of the reduced superstructure synthesis approach for retrofit problems is shown in Fig. 2.
The rationale behind the initialising binary variable in the reduced superstructure ensures that the search space for the potentially best solution is reduced by populating it with matches that have higher potential to contribute to the best retrofit network design.Since the problem is multiperiod, reducing the search space by initialising the reduced superstructure with fewer binary variables also helps in getting feasible and good solutions within reasonable computer time.
The set of equations for the retrofit scenario (Eqs.( 26) to (38)) will be used in combination with Eqs. ( 1) to ( 24) to address problem statement 2. Like Eqs. ( 20) to (24), Eqs. ( 26) to (30) adopt the maximum area approach to implement HTEs in retrofit scenarios.In Eqs. ( 26) and ( 27), AHU Select i,j,k and ACU Select i,j,k represent the maximum sized hot and cold utility exchangers, in Eq. ( 28), AP Select i,j,k represents the maximum sized process heat exchanger, while in Eqs. ( 29) and (30), APTS Select i,j,k and APSS Select i,j,k represent the maximum areas that would be obtained for the tube-side and shell-side HTEs.( 26) To determine the sizes of representative heat exchangers, or additional areas, that will be purchased, Eqs. ( 31) to ( 35) are adopted.
In Eqs. ( 31) to (33), and AP New∕add i,j,k represent the heat exchanger area that may have to be purchased.Since the reduced superstructure for the retrofit scenario is initialised with sets of matches that include existing matches in the original network, it then means that if an existing match is selected in the solution network of the reduced superstructure, the size of the additional exchanger area to be costed is determined by Eqs.(31) to (33).In this case, no installation cost will be included in the computation, only cost of extra exchanger area.On the other hand, if a match that is selected in the solution of the reduced superstructure did not exist in the original network, then Eqs. ( 31) to ( 33) is computed for new exchangers that will include installations costs, meaning that in the equations will be zero.It is worth stating that for Eqs. ( 34) and ( 35), APTS Exist,orig i,j,k and APSS Exist,orig i,j,k do not exist in the original network of the case study investigated, so, they are by default equal to zero.However, if the original network involves HTEs, then Eqs. ( 34) and ( 35) will be computed like Eqs. ( 31) to (33).

Objective Function
For the retrofit scenario, two kinds of objective functions are adopted.The first objective function, shown in Eq. ( 36), computes the minimum TAC of the retrofitted network, while the second objection function, shown in Eq. ( 37), computes the optimal net present value (NPV). (36) Fig. 2 Flowchart representation of the reduced superstructure synthesis approach for multiperiod problems 1a.Extract problem data of original network 1b.Using the extracted data of step 1a, and cost funcƟon for new heat exchangers, solve the problem for a grass-root scenario using the mulƟperiod SWS model of Verheyen and Zhang (2006).IdenƟfy the matches selected.
2. IdenƟfy the exisƟng heat exchangers in the original network 3a.Use the matches selected in step 1b and those idenƟfied in step 2 to generate a reduced SWS superstructure.
3b. Solve the reduced superstructure of step 3a as a mulƟperiod MINLP model using EquaƟons 1 to 24 and 26 to 38.
3c.For the soluƟon network generated in step 3b, if needed, reassign unused heat exchangers to obtain final retrofiƩed network.
In Eqs. ( 31 , then the excess area is reassigned elsewhere. In Eq. ( 36), TUC is the sum of the weighted hot and cold utility costs, shown in Eq. ( 37), N is the project's lifetime, AOC ON is the original network's annual operating cost and r is the interest rate.
To further evaluate various options of solutions considering environmental impact, Eq. ( 39) was included in the set of model equations.
Equation (39) was adapted from Shenoy (1995).The equation was modified in this paper to cater for multiperiod problems where periodic durations may be unequal.In Eq. (39), EI is environmental impact, Ec i is the mass percentage of the pollutant of concern (in this case, carbon dioxide) in the non-oxidised form, η is combustion efficiency in the boiler, LHV i is the lower heating value of the fuel source, 3.67 is obtained by dividing molecular weight of carbon dioxide by the atomic weight of carbon, while Nhours is the operational number of hours in a year.Evaluating HENS retrofit options using a critical criterion such as environmental impact will further help in attaining sustainable HENs.This is especially important in HENS retrofit problems where various options of HTE techniques are being traded-off as done in this paper.
The multi-objective optimisation approach adopted in this paper, shown in Eq. ( 40), was adapted from Gxavu and Smaill (2012).The equation is the goal method of multi-objective optimisation. (37

�
In Eq. ( 40), R g is a factor used to allocate weights to the two objectives which in this paper are TAC and EI.The sum of the weights allocated to each objective criterion must be equal to 1. TAC min and EI min in the multi-objective equation are the best possible (i.e.lowest) values of TAC and EI.

Case Study
Two case studies are investigated in this paper.The case studies are modified from existing problems in the literature for the purpose of demonstrating the newly developed synthesis techniques of this paper.The first case study is used to address problem statement 1 while the second case study is used to address problem statement 2.

Case Study 1
The data for the first case study is shown in Table 1.Data for the stream supply and target temperatures, heat transfer coefficients, unit costs for utilities and capital cost functions and parameters are taken from the vacuum gas oil hydrotreater unit of an oil refinery presented by Verheyen and Zhang (2006).The HTE data is taken from Odejobi et al. (2015) and is shown in Table 2.The intensification cost presented by Odejobi et al. (2015), which is in US dollars, was changed to Euros in this paper to ensure consistency with the utility costs and heat exchanger capital cost presented by Verheyen and Zhang (2006).In Table 2, intensification types, for tube side and shell side, as well as fixed charge (represented as A) and unit area cost (represented as B) for intensification, and maximum intensification for each intensification type, are all shown.In Table 2, numbers 1 to 5 are used to identify tube-side HTE while numbers 1 to 3 are used to identify shell-side HTE.
This problem, which involves three periods of operations, has been solved by various authors without considering either shell-side or tube-side enhancements.Such authors include Verheyen and Zhang (2006), Isafiade and Fraser (2010), Isafiade et al. (2015), Kang and Liu (2016), Jiang and Chang (2013), Isafiade and Short (2016), etc.The superstructure for this case study was modelled mathematically using Eqs.(1) to (24) as constraints and Eq. ( 25) as the objective function.The resulting model, which was developed and solved in the General Algebraic Modelling Systems (GAMS, version 24.4.6)mathematical optimisation environment (GAMS 2012), involved 2911 single equations, 1451 single variables and 144 discrete variables.The solution was obtained in 41 s of central processing unit time using a machine with specifications Intel core i7, 1.80GHz CPU with 16 GB of RAM.The solution obtained has a TAC of 5,269,438 €, which is about 15% lower than one of the best non-enhanced solutions presented in the literature by Pavão et al. (2018).The solution network obtained for this case study is shown in Fig. 3.In the figure, the area of the representative exchangers is shown above each exchanger while the supply and target temperatures as well as heat capacity flowrates for each stream in each period of operations are also shown.
To illustrate the benefits of HTE in multiperiod HEN problems using the approach developed in this paper, the details of the solution network obtained for case study 1 will be compared with the solution presented by Isafiade and Short (2016) for the non-enhanced scenario of the same problem.The parameters obtained in the solution of Isafiade and Short (2016) are shown in Table 3 while those of this paper are shown in Table 4.The two solutions were obtained using a multiperiod SWS model involving five temperature locations.The solution of Isafiade and Short (2016) involves 10 units while that of this paper involves 9 units.The solution of this paper and that of Isafiade and Short (2016) have the same set of matches except H1, C1, 2 which is present in the solution of Isafiade and Short (2016).Shown in Table 5 are the HTE types selected for process exchangers 3, 4, 5 and 6, as well as the breakdown of the HTE costs.Table 6 compares annual operating and annual capital costs of non-enhanced case of Isafiade and Short (2016), identified as the base case, with those obtained in this paper.It can be seen in the table that the implementation of HTE, using the method of this paper, results in energy saving to the tune of 407,086 €/year.This was possible due to the improved process heat recovery due to increased individual heat transfer coefficients brought about by the HTE.The utility and capital costs obtained in this paper is about 12.3 % and 20% lower than those of Isafiade and Short (2016).

Case Study 2
This case study is a modified version of case study 1 and is taken from Kang and Liu (2015).The problem, which entails retrofit scenario, has also been solved by Isafiade (2018) using the reduced superstructure synthesis approach for HENS retrofit problems.The problem data is the same as that presented for case study 1 except few differences.
The differences include individual stream heat transfer coefficient, which is the same for all streams, is 2000 W/ (m 2 •°C) in this case study, supply and target temperatures for hot utilities are 500 °C and 500 °C, while for cold utilities, they are 15 °C and 25 °C.Other differences are that the capital costs do not have an annualisation factor while the area cost exponent is 0.7.This case study is used in this paper to demonstrate how problem statement 2 can be addressed.To implement the integration of simultaneous optimisation of HTE with retrofit of multiperiod HENS problem, the reduced superstructure synthesis approach is first used to identify candidate binary variables that will be used to populate the reduced multiperiod superstructure.The reduced superstructure is generated by first solving the existing HEN problem data for a grassroot scenario, the matches obtained are then used, together with the existing matches in the original network, to populate the reduced superstructure.The original network is shown in Fig. 4, while the solution of Kang and Liu (2015) is shown in Fig. 5.The reduced superstructure, which is the same as obtained by Isafiade (2018), is shown in Fig. 6, while the final solution obtained by Isafiade (2018) is shown in Fig. 7.
For this case study, three scenarios are investigated.All three scenarios made use of the constraints shown in Eqs.(1) to (24) and Eqs. ( 26) to (35).For the objective function, scenario 1 is solved considering only economics as the objective on one hand and using both economic and environmental impact on the other hand.For the economic objective, Eq. ( 36) and the set of HTE types shown in Table 2 were used while for the multi-objective case, Eqs. ( 36), ( 39) and ( 40), as well as the set of HTE data shown in Table 2, were used.Scenario 2 also uses the objective function shown in Eq. ( 36) but uses the HTE data shown in Table 7.The only difference between the data shown in Table 7 and those in Table 2 is the unit area cost for the various enhancement types.The unit costs in Table 7 are 50% lower than those shown in Table 2.For scenario 3, the objective function shown in Eq. ( 37), which determines net present value, and HTE data shown in Table 2 are used.
In Table 8, which illustrates a break down in terms of the parameters obtained in this work for scenario 1, A exist is the area of a match that exists in the original network and is selected in the final network.In the table, for matches where A max − A exist is '0', it means that the area of the existing match A exist is fully utilised in the final network.For matches where A max − A exist is negative, it means that although the existing match was selected in the final network, however, it was not fully utilised which implies that the unused area can be reassigned for use elsewhere in the network in a postprocessing step.Note that to avoid raising negative numbers to a power, variables and APSS New∕add i,j,k in Eqs. ( 31) to (35) are specified as positive variables in the GAMS model.However, during the postprocessing step, the designer can identify whether A max − A exist is negative to determine the quantity of unused heat exchanger area to be reassigned.For matches where A max − A exist is positive, it means that although the existing match was selected, however, extra area had to be added to meet the required exchanger size.After reassigning the unused exchanger heat transfer areas, the capital cost of the network was then calculated considering the cost of HTE for the process exchangers and the cost of newly added exchanger heat transfer areas and a cost of € 453,897 was obtained.For the AOC, a cost of € 1,560,722 was obtained which then results in a TAC of € 2,014,620.
In terms of payback period (PBP), a value of 0.46 years was obtained.Shown in Table 9 is the breakdown of the heat transfer area, costs and payback period of the retrofitted network, which are compared with the results obtained by Isafiade (2018).Row 2 in Table 9 is for the scenario where the unused exchangers are reassigned for use elsewhere without considering reassignment cost, row 3 is for the case where there is no reassignment of the unused exchanger (the solution network is shown in Fig. 8) while row 4 in the table is for the case where there is no reassignment of unused exchangers and the HTE data shown in Table 7 are used to compute the enhancement costs.In Table 9, the payback period obtained by Isafiade (2018) is the lowest.However, in terms of total energy required by the network and cost of energy saved, the results obtained in this paper are better.This implies that in terms of environmental impact, the network solution of this paper has the potential to perform better.In terms of environmental impact, the multi-objective equation shown in Eq. ( 40) was solved using R g values of 0.4 and 0.5.In Eq. ( 39), it was assumed that the hot utility was generated from wood,  (2016).Still in Eq. ( 39), 0.8 was assumed for η, 4.28 kWh/kg was used as the lower heating value of wood, while the yearly operational hours, Nhours, was assumed to be 8160.Table 10 shows a breakdown of the solutions obtained at the two R g values.In the table, Z represents the optimal multi-objective value obtained.It can be seen in the table that when equal weightings are given to the two objective criteria, the number of units obtained is 10 while the set of HTE selected are 3, 1 and 1 for the tube side and 2, 1, 2 and 3 for the shell side.However, for the case where a higher weight of 0.6 was allocated to TAC, while a lower weight of 0.4 was allocated to EI, the number of units obtained is 13 while the HTE selected are 5, 5 and 5 for the tube side and 2, 2, 3 and 3 for the shell side.This shows that including EI as a second objective criteria influences the kind of HTE that is selected in the optimal solution.It is worth noting that the cost of HTE constitutes a significant portion of the total cost of retrofit.This implies that the lower the unit cost of enhancement, the better the solution network of the HTE scenario will perform relative to the conventional non-enhanced case; hence, the reason the lower HTE costs shown in Table 7 were also used to compute the network cost for scenario 2. The breakdown of solutions obtained for scenario 2 is shown in row 4 of Table 9. Considering the solution network shown in Fig. 8, there are four newly purchased heat exchangers, which are 3, 5, 9 and 10, and three existing heat exchangers, which are 1, 4 and 13, whose heat transfer areas were not fully utilised.Exchangers 3 and 12 in Fig. 8 are exchangers that existed in the original network and were fully utilised in the retrofitted network while the rest of the exchangers required additional heat transfer areas.
Solving the case study for scenario 3, where the objective function in Eq. ( 37) was used, gave a solution with an NPV of 2,639,754 €.For this NPV value, the AOC ON used is 2,554,959 €/y, the interest rate used is 20%, while the plant life is 5 years.The NPV comprises an AOC of 1,511,779 €/ year, an investment cost for additional area (which includes the cost of enhancement) of 484,990 €, an energy saving of 1,043,179 €/year and a PBP of 0.47 years.The resulting solution network involves 2 hot utility exchangers, 3 cold utility exchangers and 8 process exchangers that were all enhanced.

General Remarks
It is worth stating that for case study 1, although a linear costing term for heat exchangers does simplify the model, however, the linear costing term used for the case study is the same as used by Verheyen and Zhang (2006) and other authors that have solved the same problem.So, the same linear costing term was used in this paper for the purpose of fair comparison with the results presented in the literature.For case study 2, a nonlinear costing term was used, and feasible solutions obtained because the stage-wise superstructure model, especially when the isothermal mixing assumption is included as done in this paper, ordinarily involves few nonlinear terms so it usually gives feasible solutions even when nonlinear costing terms are used.Also, the reduced superstructure optimisation method used in this paper has the advantage of a reduced solution search space due to fewer binary variables which then improves the chances of obtaining feasible solutions.However, as with all nonlinear models, there is no guarantee of obtaining the global optimal solution especially in the context of retrofit of multiperiod HENs.
The oversizing of heat exchanger areas in one or more operational periods has the tendency to shift stream inlet/ outlet temperatures away from their individual operational period targets during practical implementation.Therefore, adequate control system must be implemented for the multiperiod network considering the potential of the heat exchangers to foul over time.

Conclusions
This paper has presented a new synthesis method for implementing enhanced heat transfer in grassroot and retrofit HEN problems involving multiple periods of operations.For the grassroot scenario presented in case study 1, the result obtained demonstrates how heat transfer area and quantity of external utility requirements are both reduced through the implementation of HTE.For       No of units Tube-side HTE Shell-side HTE 0.5 0.5 1.166 2,321,663 63,799,820 10 2-3; 3-1; 4-1 1-2; 2-1; 3-2,4-3 0.6 0.4 1.156 2,271,513 67,134,940 13 2-5; 3-5; 4-5 1-2; 2-2; 3-3,4-3 the retrofit scenario investigated in case study 2, although the solution obtained was not better than the solution presented in the literature when compared using PBP, however, a relatively larger quantity of energy is saved.The solution of this paper has the tendency to perform better than the solution produced by the non-enhanced retrofitted networks when the unit costs of the heat transfer enhancements are relatively lower than the values used in the second case study.The solution of the newly developed method was also evaluated by including environmental impact as a second objective and solving the resulting model using multi-objective optimisation.The solution obtained shows that different sets of HTE are selected at the different combinations of weightings given to the two objectives.This implies that in future studies when other key design parameters such as pressure drop are considered, as well as options of multiple sources of utilities, a more holistic approach must be adopted in evaluating the best solution networks.

Fig
Fig.4Original network for period 1 of case study 2

Fig
Fig.6Reduced superstructure for case study 2(Isafiade 2018) area of the existing match in the original network, is A max (maximum area exchanger) is the representative exchanger, A/A max is ratio of actual area required by each period compared to the area of the representative exchanger

Fig. 8
Fig. 8 Final solution network obtained in this paper for case study 2

Table 1
Stream and capital cost data for case study 1(Verheyen and B is TSE i, ti for tube-side HTE and SSE j, si for shell-side HTE.Maximum times of intensification (i.e.enhancement factor) is ef i, ti for tube-side HTE and ef j, si for shell-side HTE

Table 3
(Isafiade and Short 2016)changers in each period of operation for case study 1(Isafiade and Short 2016)

Table 5
Cost details for HTE types selected for tube side and shell side of process heat exchangers in case study 1

Table 6
(Isafiade and Short 2016)apital and operating costs for the base case HEN(Isafiade and Short 2016)and enhanced network (this paper)

Table 7
Enhancement types, costs and parameters used in scenario 2 of case study 2 B is TSE i, ti for tube-side HTE and SSE j, si for shell-side HTE.Maximum times of intensification (i.e.enhancement factor) is ef i, ti for tube-side HTE and ef j, si for shell-side HTE

Table 9
Comparison of the results of case study 2 withIsafiade (2018)