Synthesis of flexible heat exchanger networks:A review☆

Lixia Kang,Yongzhong Liu,2,*

1 Department of Chemical Engineering,Xi’an Jiaotong University,Xi’an 710049,China

2 Key Laboratory of Thermo-Fluid Science and Engineering,Ministry of Education,Xi’an 710049,China

ABSTRACT Dealing with uncertainty is one of practical issues in design and operation of heat exchanger networks(HENs),arising the problem of flexible HEN synthesis.This paper addresses the state-of-the-art methods for flexible HEN synthesis based on sensitivity analysis,resilience analysis,flexibility analysis and multiperiod synthesis techniques as well.Each of these methods is summarized by presenting their general procedures and recent developments on modeling,solving strategies and applications.Some current topics related to flexible process synthesis have been briefly presented to provide several future research possibilities.

Keywords:Heat exchanger network Sensitivity analysis Resilience analysis Flexibility analysis Multiperiod synthesis

1.Introduction

Heat exchanger network (HEN) synthesis has been one of the most important research fields in process industries because it enables rational utilization of energy and substantially improves the economic profitability and environmental efficiency of production plants [1].The HEN synthesis problems were first defined in 1969 [2].After more than fifty years,the methods for systematic synthesis of HEN have been developed by both Pinch analysis techniques and mathematical programming methods that can be further distinguished by their basic physical superstructures,transshipment model [3,4]or stage-wise superstructure (SWS)model[5].These methods have been treated as the standard methods for HEN synthesis owing to their great success in industrial practices [6].An extensive review of HEN methods can be found in Gundersen and Naess’s work [7].A comprehensive review of the major turning points and emerging trends in developing and improving of heat integration and HEN methods through the years 1975-2008 is given by Morar and Agachi [8].

It is worth noting that these methods address the HEN synthesis problems under the assumption of the fixed operating parameters at nominal conditions.However,in practice,it is possible that there are significant uncertainties in design and operation of HEN arisen either from random exogenous disturbances,such as variable feed qualities,changing product demands,and seasonal differences or from unexpected variations in the internal parameters,for example,heat transfer coefficients,reaction rate constants,and other physical properties.These uncertainties may lead to the situation that the operations of the devices deviate the optimal operational conditions and even fail to meet the production requirements.As a result,dealing with these uncertainties has become one of the practical issues in designing and operating HEN[1,9],which is known as the problem of flexible HEN synthesis and can be generally stated as follows:

Given is a set of hot streams that have to be cooled,and a set of cold streams that have to be heated.The variations of operating parameters,such as flowrates,inlet and outlet temperatures are specified for these streams.Auxiliary heating and cooling are available from a set of hot utilities and a set of cold utilities.The problem is to synthesize a HEN that remains feasible for the given changes in operating conditions,and which satisfies the certain criterions.

In this statement,the variations of inlet operating parameters can be described by a set of finite operating points,θk,k=1,2,...N,the bounded ranges,θL≤θ ≤θUand probability distribution functions θ～f(μ,σ) based on a statistical setting.The operating parameters,such as flowrates,inlet and outlet temperatures,and heat transfer coefficients of process streams,as well as the costs,productions and types of utilities fluctuate.The desired criteria for HEN synthesis could be a relatively narrow range for the targeted temperatures of process streams,the minimum energy consumption,the minimum number of heat transfer units,the least heat transfer areas,the lowest total annual cost (TAC) and/or the maximum flexibility in general.

Motivated by this problem,various methods for flexible HEN synthesis,mainly in steady sate cases have been developed to account for the different uncertainties,on the basis of the standard HEN synthesis methods with fixed operating parameters.In general,these methods can be divided into four categories,including the sensitivity analysis based methods,resilience analysis based methods,flexibility analysis based methods and multiperiod synthesis approaches,as shown in Fig.1.Although some excellent reviews have been published on the mathematical models [8,10],optimization algorithms [11]and solution methods [6,12]for HEN synthesis problems and the evolution of flexibility synthesis of chemical processes [13-15],a comprehensive review on the methods for flexible HEN synthesis has not be reported.

Subsequently,in this work,a review of the methods for flexible HEN synthesis is presented.The general procedure and recent development on modeling,solving strategies and applications of each method are reviewed.In addition,a brief introduction on some current topics that are relevant to flexible process synthesis have also been discussed to provide future research possibilities.

2.Sensitivity Analysis Based Synthesis of HENs

According to Fig.1,McGalliard and Westerberg[16]first incorporated the structural sensitivity issue into the HEN.A general procedure for flexible HEN synthesis based on sensitivity analysis mainly includes three steps:(1) sensitivity analysis,(2) process modifications and (3) performance evaluation,as shown in Fig.2.

Fig.1.An overview of flexible HEN synthesis.

Fig.2.An overview of flexible HEN synthesis based on sensitivity analysis.

In this procedure,for a given HEN design,sensitivity analysis is first used to explain the propagation of disturbance variables through the HEN and qualitatively determine the interactions between the disturbance variables and the controlled variables of the HEN.Based on these sensitivity information of HEN,possible process modifications,such as adjustments of heat exchangers,heat transfer areas,utility flowrates and bypass should be conducted to accommodate the disturbances so that a set of flexible HEN candidates is generated.Note that,after this step,checking the minimum approach temperature and pinch-matches is usually carried out to ensure the feasible operation of HEN.By comparing the energy and/ or cost criteria of these HEN candidates,the final flexible HEN design is determined.

In general,sensitivity analysis is characterized by the deviations of network outputs resulting from the given changes in operating conditions[17].The analysis results are very important for optimal operation of a given network,and it will also benefit the new design of a HEN for determining operation parameters,adjusting heat transfer area arrangement and maintaining the HEN at its best operating status.According to the methods for the generation of sensitivity information,the flexible HEN synthesis based on sensitivity analysis can be classified into three categories,and each of them will be reviewed as follows.

2.1.Downstream path methods

Linnhoff and Kotjabasakis [18]first introduced the concept of“downstream paths”in the HEN analysis,which is defined as the unbroken connection from a disturbance variable to a controlled variable in the grid representation of HEN.By analyzing the interactions between a disturbance and controlled variables along the downstream path,the sensitivity tables [19]among disturbance variables and controlled variables can be constructed to determine which heat exchanger areas should be increased,and which heat exchanger should be bypassed in order to make a nominal design that is sufficiently flexible.

Although the sensitivity table methods provide excellent physical understanding of structural HEN flexibility,and can be used to suggest design changes to eliminate or counteract these effects in the early stage of the structural design stage,it generates the sensitivity information by perturbing one parameter at a time and repeatedly solving the resulting network.This,on one side may ignore the interactions among the controlled variables and several simultaneous disturbances.On the other hand,it may become time consuming in view of the large-scale cases.In addition,this method ignores the nonlinearities resulting from heat capacity flowrates and heat transfer coefficients,which may cause very large errors in some cases,especially when the disturbances are large with different types at the same time.

To address these problems,extensive research activities have been focused on the improvement of efficiencies in downstream path identification,interaction analysis and sensitivity tables construction,with the consideration of simultaneous variations of disturbances and nonlinear physical properties of streams.Zhu et al.[20]proposed an efficient procedure to identify the heat load loops and downstream paths on the basis of graph theory.Li et al.[21]realized the simulation of downstream path analysis of HEN under simultaneous variations of temperatures,flowrates and physical properties of the streams in the HEN with splits and cycles.Based on the inherent linear relationship between the outlet and inlet temperatures of HEN,Jin et al.[22]introduced the temperature-change sensitivity coefficient to generate the sensitivity tables,and the transmission characteristics of temperature fluctuation in the HEN can thus be examined at the same time.In recent,the improved sensitivity tables have been extended to design an optimal,flexible HEN for a process with several periods of operations[23].

2.2.Simulation based methods

For the simulation based sensitivity analysis,a steady-state model or a dynamic simulation of HEN is commonly required so that the responses of HEN with respect to single disturbance and simultaneous disturbances can be examined.A list of the software tools that are widely used for process simulation in chemical industries is given by Farel and Bekhradi [24].

Papastratos et al.[25]pioneered a computer program,called CAD-HEN to conduct the dynamic simulation and response analysis of HEN to generate the sensitivity information.Wang and Sundén [26]investigated the HEN’s response to the disturbances of supply temperatures,flowrates and fouling factors through a detailed steady state simulation.de Oliveira Filho et al.[27]presented a steady state simulation scheme of HEN based on a matrix method,which allows for the evaluation of heat transfer coefficients,physical property variations related to temperatures and pseudo-stationary simulations.Bakar et al.[28]proposed a simulation based method on Aspen HYSIS to test the sensitivity of the HEN which is the minimum derivation of the output of the HEN with respect to the disturbances.

Based on the sensitivity information generated through the simulation,Picón-Nú?ez and Polley[29]used the network simulation to evaluate and modify the initial HEN design.Glemmestad et al.[30]analyzed the generated structural information of HEN to determine which bypass should be adjusted to compensate for a deviation in the output temperature so that the utility consumption is minimized.Picón-Nú?ez et al.[31]presented a steady state simulation model for de-bottlenecking of HEN so that the flexibility of the existing HEN can be improved by selecting the optimal combinations of possible retrofit strategies and the economic performances of HEN retrofit candidates can be calculated and compared more efficiently.Varga et al.[32]analyzed the feasibility of the HEN in a crude oil atmospheric distillation unit,aiming to increase the crude processing capacity,improve energy efficiency and feed flexibility.Ma et al.[33]adopted genetic algorithm/simulated annealing(GA/SA)to optimize the heat exchanger areas after sensitivity analysis based on steady-state simulation.

2.3.Equation oriented methods

Unlike the sensitivity analysis based on downstream path methods and the computer-aided simulation methods,the equation oriented methods perform the sensitivity analysis on the basis of rigorous steady-state or dynamic equations of heat exchangers or HEN.This method enables to handle more practical factors that influence the outputs of HEN,such as simultaneous disturbances,heat exchanger types,flow pattern and physical properties of the streams.

For sensitivity analysis based on heat exchanger equations,Ratnam and Patwardhan[17]conducted the sensitivity analysis of the HEN based on the logarithmic mean temperature difference(LMTD)design method,which can be applied even when the deviations in operating parameters are large and simultaneous.However,this method does not provide an explicit relationship between disturbances and target temperatures.To this end,Wang et al.[34]introduced an improved method that calculates the effects of all deviations from a base case simultaneously by solving a set of linear equations,based on the fact that any new modes of operation can be described as deviations from a base case.Yang et al.[35]developed a simplified system model for rapidly evaluating the propagation caused by even severe temperature disturbances and/or by moderate heat capacity flowrate fluctuations through a HEN.They approximated the LMTD by an arithmetic mean temperature difference(AMTD)in order to generate a set of linear algebraic equations.However,this approximation applied has limited the use of this procedure to HEN with complete counter-flow.Then,Heggs and Vizcaíno [36]used a rigorous method to evaluate the propagation of steady state disturbances through HENs,where each heat exchanger is described by the effectiveness-NTU(number of transfer unit)model so that all types of heat exchanger configurations and changes in the inlet temperatures and mass flows of streams,and fouling conditions in the heat exchangers can be handled.

For sensitivity analysis based on HEN models,Jezowski and Jezowska [37]pointed out that a target step should be embedded into the flexible design of HEN,which provides an insight into the pinch behavior of HEN with respect to the disturbances.Then,the transshipment model was extended [38]and the corresponding approach [39]was developed for calculating all possible pinch locations that can occur in minimum utility cost of HEN,accounting for the bounded variations on flowrates of process streams.A stochastic optimization approach was also developed to generate the sensitivity information based on their model [40].Persson and Berntsson [41]combined the pinch analysis and sensitivity analysis to study the influence of seasonal variations on energy saving of HEN in a pulp mill.Guha and Chaudhuri[42]investigated the transient behavior of HEN in case of any fluctuation of inlet temperature of hot or cold stream on the basis of a mathematical model and the corresponding numerical algorithm,which give an idea about the relative interaction between the different streams and the identification of the process streams that are more susceptible to inlet temperature change.Al-Mutairi and Odejobi [43]developed a HEN model to study the effects of inlet temperatures and flowrates changes on the design of HEN in a thermal power plant.Results show that the variations in process streams’ inlet temperatures and flowrates significantly affect the requirement of heat transfer areas,heating and cooling utilities and the investment cost.All these information can be used to guide a better design of the flexible HEN.

Note that the equation oriented methods can not only generate sensitivity information in a rigorous and efficient manner but also allow the simultaneous design of flexible HENs.To design a flexible HEN based on sensitivity analysis,Li and Motard [44]derived an equation to study the effect of minimum approach temperature on the economic savings.Based on this idea,Suaysompol and Wood[45]proposed a flexible pinch design method,where the minimum approach temperature is considered a variable.A computer-aided design package of this method was later developed,embedded with the A*heuristic search algorithm for locating cost effective solutions in the vast space of alternative designs[46].Osman et al.[47]defined a temperature flexibility range as the range of the temperature added to the minimum approach temperature by increasing the temperatures of hot streams and/or decreasing the temperatures of cold streams.In their method,the path combination approach was first used to identify the suitable candidate for retrofit solutions to be treated with the temperature flexibility,and the economic criteria,such as energy saving,investment cost and payback period were then calculated to select the desired retrofit.Payet et al.[48]addressed the ability of the system to absorb disturbances without changing utility flowrates.In their work,a mass equilibrium summation enthalpy non-linear model was used to identify the most frequent disturbances and critical streams whose output temperature needs to be kept within a strict interval.A mixed-integer linear programming(MILP)model was used for the HEN synthesis and a linear programming (LP) model was developed for the modeling of the HEN response to disturbances.

Recently,the retrofit of HEN can also be implementedby sensitivity analysis,especially for the flexible HEN retrofit with heat transfer enhancement.For an existing HEN with shell-and-tube heat exchangers,Wang et al.[49]proposed a heuristic rules based method for the retrofit of flexible HEN,where sensitivity analysis was adopted to identify the suitable heat exchangers for employing enhanced heat transfer techniques.This method can thus be used to handle the variable heat transfer coefficients,pressure drop constraints,varying stream thermal properties and detailed geometry of heat exchangers in HEN retrofit.Then,Jiang et al.[50]used an improved sensitivity analysis model to quickly select the most sensitive heat exchanger in HEN,accounting for different types of heat exchangers.However,the chosen sensitive heat exchangers should be enhanced one by one until the feasibility can no longer be maintained.Akpomiemie and Smith [51]then proposed a method for flexible HEN retrofit where the sensitivity analysis was used to find the sequence of the most effective heat exchangers to enhance so that the performance of HEN can be improved after identifying the locations to apply enhancement.An optimization model was then adopted for the feasibility check instead of a repeated procedure in previous work.It should be noted that,in these studies,the topological modification of the existing HEN is not allowed when sensitivity analysis is used for flexible HEN retrofit because the downstream paths,simulation models and equations will change once the topology of the HEN is modified.Of course this can be addressed by using an iterative approach until the feasibility check is passed.

3.Resilience Analysis Based HEN Synthesis

According to Fig.1,the research on resilience HEN synthesis in steady state was mainly active in 1980s because the term “resilience”refers in particular to a dynamic characteristics of a system.Marselle et al.[52]first established the concept of resilient in HENs that can tolerate uncertainties in temperatures and flowrates.Morari [53]categorized the process resilience into steady state resilience and dynamic state resilience,and treated them in different ways.In the steady state,process resilience was treated as process flexibility,i.e.the ability of a plant to handle different feedstock,product specifications,and operating conditions.Based on the definitions of the structural resilience and network resilience of HEN as follows,Saboo and Morari [54]proposed a general design procedure for resilient HEN synthesis,as shown in Fig.3.

· Structural resilience:A network structure is said to be resilient if it allows to bring all streams to the specified target temperatures and features maximum energy recovery without violating the minimum approach temperature for the whole disturbance range

· Network resilience:A network is said to be resilient if it is structurally resilient,and all heat exchanger areas are chosen so as to allow maximum energy recovery for the whole disturbance range.

In this procedure,for a set of given fundamental data,an initial HEN design is first obtained by using the standard HEN synthesis methods.A resilience analysis is then carried out to check the feasibility of the design.If infeasibility occurs,the initial HEN will be updated by merging the basic design with the designs at the newly added corner points that are the combinations of inlet temperature extremes.Otherwise,the HEN design is said to be resilient,where all target temperatures are within their bounded ranges and the minimum approach temperature is not violated so that the maximum energy recovery is achieved.

To further improve this procedure,significant efforts have been made to the quantitative measurement of resilience,efficient tools and models for automated synthesis and analysis of HEN,economic evaluation of resilience HEN by the optimization of heat transfer areas.Saboo et al.[55]proposed the resilience index to measure the largest disturbance that the network can tolerate without becoming infeasible.An interactive software package,RESHEX for the synthesis and analysis of resilient HENs was then developed on the basis of an MILP transshipment model where the heat transfer area is targeted via a modified LP formulation in order to avoid the decomposition of the problems into two parts by the pinch point.Saboo et al.[56]presented a nonlinear programming (NLP) formulation to test the resilience of HEN with simultaneous temperature and flowrates disturbances.This formulation becomes an MILP model for the temperature dependent heat capacities by using the piecewise linearization technique.This formulation was later extended to account for the cases of stream splits,large temperature disturbances,and flowrate variations in HEN [57].An excellent review on the systematic methods for HEN resilience analysis and the synthesis of resilient HEN in 1980s can be found in Colberg and Morari’s work [58].

According to the definition of structural resilience of HEN,Cerda et al.[59]proposed an MILP transshipment model and a corresponding Pinch analysis based procedure to design a structurally flexible HEN where the feasibility conditions are handled as constraints,simply accounting for the bounded ranges of inlet temperatures.This model and procedure were then generalized and modified to address the non-convexities caused by large temperature disturbances in HEN [60].Later,they presented a four-step method to account for the specified uncertainties in both inlet temperatures and flowrates of the streams [61].The computational efficiency of the proposed method can be guaranteed as the number of uncertainties increases.Based on the similar idea,Aguilera and Nasini [62]presented an MILP formulation for testing the structural flexibility of the HEN with respect to flowrate variations [62]and nonoverlapping inlet temperature variations [63].Li and Niemeyer[64]presented a adjustment scheme to maintain the flexible operation of HEN by adjusting the inlet parameters of the HEN,aiming at the maximum heat recovery.However,the arrangement of heat transfer areas is not addressed in their work.Thus,Colberg et al.[65]derived a NLP formulation based on the composite curves to target the resilience of HEN before looking into the detailed design.Konukman et al.[66]proposed a NLP model for the retrofit of HEN with the given resilience target constraints,aiming to minimize the TAC by adjusting the existing heat transfer areas and bypass fractions.Konukman et al.[67]proposed an MILP model for simultaneous synthesis of flexible HEN,where the variation ranges of uncertain parameters were considered as constraints in SWS models.Chen et al.[68]developed a two-stage method to synthesize the flexible HEN with fouling growth.An overdesigned HEN was first obtained based on pseudo-T-H diagram approach at the maximum fouling thermal resistance and was further simplified according to the variations of heat transfer coefficients.The heat transfer areas were then optimized through parallel GA/SA.

In short,the resilience index can be obtained through simple physical considerations or rigorous numerical algorithms,depending on its definition.Its application is to identify the bottlenecks in the existing networks or to guide the selection among alternative schemes in a new design.In addition,the resilience index provides a quantitative basis to compare different design and retrofit options of HEN.Unlike the procedure presented in Fig.2,a procedure for simultaneous synthesis and resilience analysis can be realized through trial and error method.

4.Flexibility Analysis Based HEN Synthesis

According to Fig.1,flexibility analysis on HEN first appeared in the 1970s [69].However,the modern theory of flexibility analysis were founded in the 1980s,mainly contributed by Grossmann and his coworkers [70-73].In this section,the flexibility of a HEN is defined as the ready capability to adapt to new,different,or changing requirements in the steady state [74].

Fig.3.An overview of flexible HEN synthesis based on resilience analysis.

4.1.Flexibility analysis

The first effort to embed design uncertainty as part of a mathematical formulation should be attributed to Halemane and Grossmann[70].In this formulation,a feasible test problem is involved to determine whether a given design specification performs feasibly over known ranges of uncertain parameters.To quantitatively describe the flexibility of a system,Swaney and Grossmann [71]defined a flexibility index that is characterized by the maximum allowable deviations of the uncertain parameters from their nominal values,by which feasible steady-state operation can be assured.Swaney and Grossmann [75]also pointed out that,under certain convexity assumptions,critical points that restrict flexibility lie on vertices or extreme values of uncertain parameter space.Based on this assumption,a flexibility index problem was posed to calculate the flexibility index.Realizing that the feasibility test problem and the flexibility index problem result in a bi-level formulation,Grossmann and Floudas[73]converted the bi-level problems into single level MILP or mixed-integer nonlinear programming(MINLP)problems by replacing the original lower-level problem by their Karush-Kuhn-Tucher(KKT)conditions.These reduced single level formulations have now become the most widely used or even the solely used model for flexibility analysis of HEN[14].

Due to the discontinuity and non-convexity of rigorous MINLP formulations,various procedures and algorithms have been developed to facilitate the solutions to the feasibility test problem and flexibility index problem,either in a sequential way or a simultaneous way.For the sequential methods,the flexibility index is usually calculated after a step of critical points(or active sets)identification whereas the efficient solving strategies or algorithms are usually adopted in simultaneous methods.Based on the assumption that the constraints of the HEN are convex,a vertex enumeration formulation and an iterative cutting plane algorithm were used to solve the feasibility test problem[71].Later,a direct search procedure and an implicit enumeration scheme were employed to solve the flexibility index problem[75].To avoid the tremendous vertex enumeration,an active strategy was used to solve the single level formulation of flexibility index problem,which is exempt from the assumption of convexity,and thus can be used in nonconvex cases [73].A recent summary of the methods for critical points identification was presented by Pintariand Kravanja[76].

Ostrovsky et al.[77]suggested to use the Branch and Bound(BB) algorithm to determine the lower and upper bounds of feasibility test problem proposed.Ostrovsky [78]proposed a BB-active algorithm for the efficient solution of feasibility test problem,which avoids the enumeration procedure.Floudas and Gümüs[79]proposed an α-BB global optimization algorithm for the solution to the feasibility test problem and flexibility index problem of a system with a given structure,which relies on a difference of convex functions transformation and a branch and bound framework.The feasibility test and the two-stage optimization problem were later extended by Ostrovsky et al.[80]in order to account for the possibility of accurately estimating some of the uncertain parameters while estimating with less accuracy the remaining uncertain parameters.Moon et al.[81]presented a new parallel hybrid algorithm based on GA in conjunction with a nearest constraint projection technique to numerically solve the flexibility index problem.The computational time can be dramatically reduced.Li et al.[82]introduced the direction matrix to describe the deviation directions of the uncertain parameters and the flexibility index can be effectively obtained by searching the critical directions via SA algorithm.Acevedo and Pistikopoulos [83]proposed an outerapproximation/equation relaxation (OA/ER) algorithm that involves the iterative solution of NLP sub-problems and a parametric MILP master problem for flexibility analysis under bounded uncertain parameters.Jiang et al.[84]also proposed an iterative OA algorithm to determine the flexibility index of HEN that is represented by quadratic inequalities,where a LP sub problem and MILP master problem need to be solved iteratively.

Nowadays,the flexibility index and flexible index problem have been extended to address the stochastic uncertain parameters of process systems[85].However,the flexibility analysis of HEN with stochastic uncertain parameters is rarely reported.More details on the evolution of flexibility index and the corresponding formulations can be found in Grossmann and his co-workers’ work [14].

4.2.Flexibility targeting

In the abovementioned methods,i.e.sensitivity analysis and resilience analysis,address the variations in the process parameters on the basis of either a fixed structure or a basic design in advance by using Pinch design method or superstructure methods.Tantimuratha et al.[86]presented a conceptual programming method for flexibility target of HEN based on the area target model [87-89],which can be used to provide flexibility targets prior to the design in detail.This method was then extended to assess trade-offs among operating cost,capital cost and flexibility[90].Similarly,the pinch design method and transshipment model have also been adopted to target the flexibility before the detailed HEN design is reached.Tan et al.[91]developed a floating pinch method for utility targeting in HEN,where the true pinch point of HEN and utility target can be obtained by identifying the corner points on the composite curves when the flowrates and temperatures of streams are variables.Later,this method was extended to target the utility of a heat integrated resource conversion network[92].Chen and Hung[93]constructed a multi-objective MINLP model to address the uncertain variations on source temperatures,aiming at the minimum utility consumption,least number of matches and greatest flexibility index simultaneously.A two-phase fuzzy multi-criteria decision-making method was then presented to attain a best compromised solution.

4.3.Flexible HEN synthesis

For flexible HEN synthesis based on flexibility analysis,a twostage strategy is usually adopted where a multiperiod synthesis step and a flexibility analysis step are carried out in an iterative manner,as shown in Fig.4.

In this procedure,for a set of given fundamental data,an initial HEN design can be determined by using the multiperiod synthesis technique at the nominal operating points or some chosen critical points.Then,a flexibility analysis is conducted in a sequential way to check the feasibility of the operation,where the critical points for flexibility are first identified and the feasibility test problem or flexibility index problem is then solved.When the flexibility requirement is not satisfied,a multiperiod synthesis will be implemented to update the initial design by increasing the critical points until the flexible operation can be guaranteed at all critical points.

According to this procedure,Grossmann and Sargent [69]proposed a new strategy to address the flexible HEN retrofit with bounded heat transfer coefficients,based on the assumption of the monotonicity of the inequality constraints.Thus,in this method,the critical points for flexibility was first determined by merging the active bounds of the constraints according to the signs of their gradients,and the heat transfer areas were designed by solving a NLP problem where the objective function was a weighted cost function that reflects the costs over the expected range of operation.Floudas and Grossmann [72]combined the transshipment model with flexibility analysis for flexible HEN synthesis.In their procedure,the potential stream matches were predicted at the first stage and the final network configuration was determined at the second stage.In each stage,the flexibility analysis was conducted to examine the feasibility of the design over the specified ranges of uncertain parameters.A NLP formulation for flexible HEN retrofit at a desired level of flexibility has also been developed aiming at the minimum cost for process modifications[94].

Fig.4.An overview of flexible HEN synthesis based on flexibility analysis.

In contrast to the procedure presented for resilience HEN synthesis,the critical point identification,the feasibility check and the network updating are all conducted by solving rigorous mathematical models instead of using heuristic rules.Thus,this procedure provides a more rigorous and systematic tool for flexible HEN synthesis.Extensive activities have been focused on the improvement of the model and algorithms involved in this procedure.Pintariand Kravanja[95]formulated an improved multiperiod model with the flexibility constraints to generate the HEN design,whose flexibility was further examined by stochastic Monte Carlo optimization and the flexibility index calculation.Bai et al.[96]improved this procedure by accounting for the uncertain operating conditions and the growth of the fouling resistances simultaneously.An improved direction matrix method was used to evaluate the flexibility index and identify the critical points for flexibility in the first step.A flexible synthesis method based on critical points was then used to minimize the TAC of the HEN.Zhang et al.[97]proposed a MINLP model to obtain a structurally flexible HEN.In their method,a structurally flexible HEN structure was first obtained by solving the SWS model and flexibility index problems,which was then modified to minimize the TAC by using particle swarm optimization(PSO)algorithm.Li et al.[98]adopted this procedure for flexible HEN synthesis with nonconvex feasible region,where both the structure flexibility and area flexibility were considered.In this method,an initial structure was renewed by the topological union with the structure of the critical point and the improved heat transfer loops disconnection strategy.The heat transfer areas were optimized by an iterative approach,including the determination of critical points and flexibility index via direction matrix method.

To handle the HEN synthesis problem with large number of uncertain parameters,a variety of simplified strategies have also been developed,focusing on model reduction and uncertain parameters discretization.Nishitani et al.[99]generated an initial HEN design under nominal operating points and evaluated the flexibility of the design at the vertex of the feasible region to redesign the HEN.A non-inferiority test was then performed to screen the final results with less heat transfer areas.This method was later extended to flexible HEN design with a large number of uncertain parameters by reducing the size of the problems [100].Pintariand Kravanja [101]used the direct optimization strategy to determine the design variables,and a reduced dimensional stochastic optimization method was then applied to calculate the feasibility and flexibility of HEN.This strategy can reduce the number of decision variables in two levels and thus reduce the size of the mathematical models.Rooney and Biegler [102]suggested to discretize the uncertain parameters based upon the principal components of their joint confidence region so that a multiperiod synthesis model can be constructed and solved to generate the design candidates.The feasibility of the resulting design was checked and the discretization of uncertain parameters was updated with critical points until the flexibility of the design was guaranteed.Zheng et al.[103]adopted the probability bound analysis theory to describe the operation uncertainties and a double-loop sampling method was used to generate the critical points of HEN.The Aspen Energy Analyzer software was then used to design the HEN under each critical point and the final solution was reached according to the economic and flexibility objectives determined by simulations.

Moreover,a similar procedure has been proposed for flexible HEN synthesis with stochastic uncertain parameters that are commonly described by their probability distribution functions.Pistikopoulos and Mazzuchi [85]introduced a stochastic flexibility index that measures the probability for that a given design is feasible to operate by explicitly considering the existence of operating degrees of freedom.By assuming a Gaussian distribution of the uncertain parameters,a LP model of HEN and a computational strategy was presented to conduct the flexibility analysis with any number of uncertain parameters,whatever they are correlated or not.Based on the general probability distribution functions of uncertain parameters,Pistikopoulos and Ierapetritou [104]formulated the problem of flexible HEN synthesis as a two-stage stochastic programming problem where the objective was to maximize an expected revenue or profit with measuring design feasibility simultaneously.Pintariand Kravanja[105]presented a sequential twostage strategy for flexible HEN synthesis with stochastic uncertain parameters.In the first stage,an overdesigned HEN with a flexibility-qualified structure was determined to assure feasibility of design for a fixed degree of flexibility.The structural alternatives and additional manipulative variables were then added to achieve the efficient control,aiming to minimize the expected cost function.

5.Multiperiod HEN Synthesis

When the uncertain parameters are described by a set of discrete operating points,a flexible HEN synthesis problem can be formulated as a multiperiod MILP or MINLP model.Various methods have been developed to solve this kind of problems,either in a sequential way or simultaneous way.

5.1.Sequential model based HEN synthesis

In this section,the multiperiod HEN synthesis based on transshipment model [3]are reviewed.As shown in Fig.5,a LP model,an MILP and a NLP model are usually solved in sequence to target energy consumption,amount of heat transfer units and capital cost accordingly in multiperiod HEN synthesis based on transshipment model.

For the specified variations in flowrates and inlet and outlet temperatures of process streams in HEN,Floudas and Grossmann[110]presented a multiperiod version of the LP and MILP transshipment model,aiming at the minimum utility cost and the number of units of HEN in each period,then a NLP model[111]was added to address the automatic synthesis of HEN structures and optimization of the heat transfer areas.Although this procedure can ensure the minimum utility cost for each period of operation and the fewest number of units for each un-pinched network,the fewest number of units for the whole network cannot be ensured.In this context,Lee[112]proposed a three-step procedure to synthesize a multiperiod HEN with minimum cost.In this method,a feasible network was synthesized at each period,which were combined to form a feasible superstructure that features minimum utility in each period and overall fewest number of units.The final structure with minimum cost was further generated via an enumeration scheme.

So far,this transshipment model based procedure has been applied to address various problems associated with multiperiod operation in the industries.Bagajewicz and Soto[113]adopted the multiperiod transshipment model to design the HEN of an atmospheric crude fractionation units,which enables to account for the variations in flowrates,inlet and outlet temperatures of streams in different time period.Ji and Bagajewiez [114]used this procedure to handle the multiperiod operation caused by different feeds.Main et al.[115]integrated the utility system into the multiperiod transshipment model in order to obtain a design with the minimum total cost.Miranda et al.[116]improved the multiperiod transshipment model by including the bypass streams in the superstructure model and taking the resulting subperiod information to initialize the heat transfer areas optimization.The HEN retrofit problem with multiperiod operation has also been addressed by using this procedure that can be automatically performed on TransGen software,and it can be applied to the large-scale problems[117].This method was then combined with HENSYN software to retrofit the HEN with economic performance considerations[118].

In addition,the multiperiod hyperstructure model has also been developed to design and retrofit the flexible HENs.Papalexandri and Pistikopoulos [119]first presented a multiperiod HEN model based on the hyperstructure model for the retrofit of HEN,where the flowrates,inlet temperatures,overall heat transfer coefficients of process streams are variable,aiming at the minimum operating and retrofit investment costs.Then,a flexibility analysis problem in an iterative scheme was integrated into the model to achieve a desired flexibility target in HEN retrofit,where the critical operating points were considered the operating periods[120].A simultaneous synthesis of HEN and mass transfer networks has also been addressed on the basis of this multiperiod MINLP model[121].However,this model can be decomposed into a NLP subproblem that derives a HEN for a set of matches and a MILP master problem that identifies the set of matches and heat loads that satisfy the transshipment model.This,of course,limits the HEN synthesis problem to a relatively small scale.

It can be concluded that,all these methods decompose the problem into different stages can significantly reduce the size of the problem and facilitate the solution generation of multiperiod HENs,which will surely benefit the efficient solution of large-scale HEN synthesis problems with multiperiod operation.However,these methods often ignore the interactions between the capital cost and operation cost.This drawback can be overcome by simultaneous model based methods that will be addressed in next section.

Fig.5.An overview of procedure for flexible HEN synthesis based on multiperiod transshipment model.

5.2.Simultaneous model based HEN synthesis

In principle,the SWS model base methods can take the trade-off between the capital cost and operation cost by solving a multiperiod SWS model[5]and thus obtain a better solution than the transshipment model.Aaltola [122]initially extended the SWS model to a multiperiod scenario,where an average area approach was adopted in their objective function.This approach,however,would underestimate the capital cost of the multiperiod HEN.To address this problem,Chen and Hung [123]introduced the maximum area consideration in the objective function by“max”function.Verheyen and Zhang[124]developed a multiperiod HEN synthesis method by adopting the maximum area consideration in the form of inequalities.This method includes a reduced multi-period MINLP model to design the HEN structure,and an NLP model to optimize the operating parameters.This formulation has become the most prevalent model.Fig.6 gives an overview of this method.

Base on this formulation,extensive research activities have been conducted on the improvement of this model and corresponding algorithms.Nemet et al.[125]adopted this model to address the variability of utility prices that are forecasted by the historical data in three scenarios.This method was then extended to the design of HEN with a stochastic fluctuation of utility price,aiming to maximize the expected net present value[126].Nemet et al.[127]used this model to account for flexible HEN synthesis where the multiple periods were constructed by different failure rates and different aspects of safety.Papalexandri et al.[128]utilized this framework to design optimal energy management schemes that were flexible to accommodate discrete variable demands and uncertainty of operating conditions.Novak-Pintaric and Kravanja [129]proposed a multiperiod HEN model to address the investment planning for the gradual retrofit of HEN,aiming to maximize the net present value.Ma et al.[130]applied the multiperiod model to synthesize the multi-plant HENs.For a fixed structure of HEN,Francesconi et al.[131]presented a multiperiod NLP model to accounts for the periodic variations on hydrogen production levels in an ethanol steam reforming process.Ma et al.[132]used this model to design a multiperiod HEN with multi-stream heat exchangers.Short et al.[133]improved this multiperiod HEN synthesis model by considering the detailed design of heat exchangers.Timmerman et al.[134]employed this multiperiod model to integrate the utility system and HEN,aiming at the maximum heat recovery.Isafiade et al.[135]integrated renewable energy into the multiperiod HEN model with multiple utilities,aiming at the highest economics and environmental benefits at the same time.

Owing to its strong nonlinearity and non-convexity nature,the stepwise strategies and efficient algorithms are usually adopted to facilitate the solution to the multiperiod HEN model.Lai et al.[136]proposed a step-wised method to synthesize the flexible HEN by combing the pinch technique and mathematical programming method.Friedler et al.[137]proposed a two-stage method to synthesize the multiperiod HEN,where the superstructure of the HEN was determined by P-graph method and the optimal solution was derived by the accelerated branch and bound algorithm (ABB).Escobar et al.[138]proposed a specialized heuristic algorithm on the basis of Lagrangean decomposition.This is an iterative scheme where feasible solutions are postulated from the Lagrangean decomposition sub-problems,and the Lagrangean multipliers are updated through a sub-gradient method.For a given structure of HEN,Varvarezos et al.[139]used the sequential quadratic programming (SQP) algorithm to facilitate the solution to the multiperiod NLP model for HEN synthesis.Zhang et al.[140]proposed an iterative algorithm to solve the multiperiod NLP model based on HEN simulation,aiming at the minimum total cost of HEN in a diesel hydrotreating units.Ahmad et al.[141]adopted the SA to solve the multiperiod MINLP model for HEN synthesis.Silva et al.[142]used the PSO algorithm to solve the multiperiod HEN synthesis model.Oliveira et al.[143]presented the multiperiod HEN model to address the periodically operating conditions of sugarcane biorefinery,aiming to minimize the TAC.This model was solved by a hybrid metaheuristic approach that combines the SA and rocket fireworks optimization.A post-optimization strategy was then added to further improve the results [144].

Fig.6.An overview of the procedure for multiperiod HEN synthesis.

Some simplified strategies have also been presented according to the characteristics of periods,aiming to reduce the number of binary variables in multiperiod HEN model.Isafiade et al.[145]presented an interval-based multiperiod model [145]where the stages of the SWS were determined by the intervals of stream temperatures so that the numbers of streams and possible stream matches in each stage were reduced.For the multiperiod HENs with larger number and significant difference durations of the subperiods,a representative subperiod method [146],where the subperiod with the longest duration was selected the representative subperiod and the corresponding optimal stream matches obtained through single period HEN synthesis were considered the final structure of the multiperiod HEN.This method was further improved to address the unforeseen changes in lengths of subperiods,where a worst-case operational scenario of the multiperiod HEN was chosen as the representative subperiod and the solution to this representative network was then used to initialize the multiperiod network [147].Likewise,Tangnanthanakana and Siemanond [10]suggested to obtain a representative network by solving the multiperiod HEN synthesis model for a chosen subperiods.By changing the chosen periods,the best network can be obtained by screening different multiperiod HEN candidates according to their TACs.For the multiperiod HENs with unequal durations of the subperiods,Kang et al.[148]developed a simplified model method where the single period HEN model was first solved to obtain the optimal stream matches,and those exist in all subperiods were selected to construct a simplified multiperiod HEN.By solving this reduced model,the multiperiod HEN was obtained.Based on this framework,Isafiade et al.[149]constructed a reduced multiperiod HEN model by introducing the stream matches obtained by solving multiperiod HEN models at different minimum approach temperatures and different number of stages in the established superstructure.The selection of multiple utilities was also addressed in their work.

In contrast to the above mentioned methods,the time sharing mechanism [150]is capable of dealing with the multiperiod HEN synthesis,which technically avoids the solution of the multiperiod HEN model.In this method,the optimal stream matches and heat transfer areas in each subperiod were determined through the synthesis of single period HEN,and the final structure and heat transfer areas of the multiperiod HEN were then achieved by using time sharing mechanism and the maximum area principle,aiming to reduce the capital cost and maintain the utilities demand at the minimum levels in each period [151].Recently,a corresponding algorithm has been developed for the automated implementation of this method [152],and an extension of this method has also been presented for correction of the inconsistent in calculation of the heat transfer areas [153].To further optimize the heat transfer areas of the resulting multiperiod HEN,a re-optimization scheme was recently developed by Pav?o et al.[154].This method is easy to implement and can ensure the optimal operation at each periods with relatively less heat transfer areas.However,the structure of the multiperiod HEN obtained by time sharing method turns out to be complex and additional cleaning scheduling would be required to avoid the streams mixing during the switching of the subperiods.

Moreover,on the basis of these frameworks of multiperiod HEN synthesis,the problem of multiperiod HEN retrofit has been also addressed.Kang and Liu [155]suggested to retrofit multiperiod HEN in two step,where the retrofit target was first determined by using a standard multiperiod HEN synthesis method and various retrofit strategies were then implemented to finalize the retrofit.In their method,the final retrofitted HEN was obtained by matching the required heat exchangers with the existing ones in reverse order,which is known as the reverse order match method.Later,a graph theory based method [156]was also proposed to account for some practical restrictions on retrofit,such as the restrictions on maximum additional areas,the operating pressure of heat exchangers as well as the cost and emission limitation on retrofit.In this method,the matching of heat exchangers was formulated as a bipartite matching problem and thus can be solved by graph algorithms efficiently.A mathematical formulation[157]and a multi-objective consideration [158]of this method have also been presented recently.

It is worthy of noting that the multiperiod synthesis techniques can not only be used to address the flexible HEN problems with discrete uncertain parameters but also be combined with flexibility analysis to address those problems with bounded and stochastic uncertain parameters.

6.Extensions on Flexible HEN Synthesis

In spite of the abovementioned methods for flexible HEN synthesis,a series of contributions has been made to the flexible process synthesis in process systems engineering community.In this section,several current topics associated with flexible process synthesis are briefly introduced to provide some research opportunities.

· Sensitivity analysis of HENs integrated with their background processes.In recent,sensitivity analysis of the HEN is conducted to explore the response of HEN with respect to the changing operating parameters of the reactors [159]and distillation columns[160,161],due to the fact that HEN links the reaction systems and separation systems and the parameter variations of these systems do have an impact on HEN operations.In this context,more disturbances,namely the operating conditions of the distillation columns and reactors,and more complicated interactions among these disturbances and outputs of HEN need to be studied in order to provide information for integration of HENs with their background processes in various process industries.

· Flexibility analysis addressing stochastic and time-variable uncertain parameters.In fact,the flexibility analysis formulation has been extended to address flexible design with conference intervals [95],stochastic flexibility [85]where the uncertain parameters vary randomly,the temporal flexibility [162]to describe the cumulative effects of temporary disturbances in finite time intervals,and dynamic flexibility[163,164]of process systems that involve time-variable uncertain parameters.The corresponding global optimization algorithms have also been applied to facilitate the efficient solution of feasibility test problem and flexibility index problem.Thus,although various HEN synthesis methods have been proposed in the literature,fundamental study on addressing uncertainties arisen from market fluctuations using stochastic variables or time-variable parameters deserve much efforts definitely.

· Flexible HEN synthesis with periodic and non-periodic variations.The multiperiod synthesis methods usually assume that the continuous changing operating conditions can be approximated by several discrete and finite operating periods so that the critical points for flexibility of HEN can be significant reduced.However,this is not always the case because the operating parameters in each period change continuously.In recent,the multiperiod synthesis techniques and flexibility analysis has been combined to address this problem where the multiperiod synthesis techniques are used for the synthesis with periodic variations and the flexibility analysis and debottlenecking strategy are adopted to deal with the non-periodic variations in each period[165].However,further research and analysis are desperately required,including the best selection of the periods for HEN in different industries[166],the reasonable description of variations in each period and more efficient methods for modeling and solution of these problems,especially in large scale systems.

· Flexible HEN synthesis with exogenous and endogenous uncertainties.In the process systems engineering,there are usually two types of uncertainties:the exogenous uncertainties whose values are revealed independently of decisions,and the endogenous ones whose realizations are influenced by the decisions.These two types of uncertainties should be treated in different ways [15].A comprehensive review on modeling and solving strategies for process synthesis with exogenous and endogenous uncertainties has been presented recently [167].This problem,however,has been paid less attention in HEN synthesis where both exogenous(i.e.utility types and prices,environmental conditions) and endogenous (i.e.temperatures,flowrates,heat transfer coefficients and physical properties of process streams etc.) uncertainties are involved.

7.Conclusions

In this paper,the state-of-the-art methods for the synthesis of flexible HENs are reviewed,including the sensitivity analysis based methods,the steady-state resilience analysis based methods,the flexibility analysis based methods and the multiperiod synthesis approaches.In retrospect,the general procedures and recent progress of these methods have been reviewed.Furthermore,some current topics that are related to flexible process synthesis have also been presented.It has been shown that the sensitivity analysis enables to quantitatively determine the interactions between the disturbance variables and controlled variables through downstream path methods,the simulation or equation oriented approaches,which provides guidelines for the retrofit and re-design of a given HEN design.Similarly,the steady-state resilience analysis can also provide useful information for the process modifications.However,the resilience analysis does not begin with a given design but integrated the design and analysis by an iterative manner so that a flexible HEN with specified performance criteria can be obtained.In contrast to these two methods that involve heuristic rules or physical investigations,the flexibility analysis based method provides a systematic and rigorous mathematical framework for both multiperiod synthesis and flexibility analysis.Moreover,the flexibility index and the corresponding mathematical formulations provide the basis for quantitative comparison among different HEN design and retrofit options,and thus have become the most widely used and even the only used formulation for flexible process synthesis in recent years.Specifically,when the uncertain parameters in HEN are described by several discrete operating periods,the multiperiod synthesis techniques can also be applied to synthesize flexible HENs,where only a multiperiod transshipment model or a multiperiod SWS model is needed to be solved by using a series of efficient algorithms and/or simplification strategies.All these analysis indicate that,on one hand,there has been significant progress in this area;on the other hand,there are still many challenges to embrace and obstacles to overcome,especially in the application of these methods in practical process industries.