Tan Qiang; Chen Yuting; Xu Yanyan,3; Ye Shuang,4; Xiao Hao; Huang Weiguang,3

(1. Shanghai Adνanced Research Institute, Chinese Academy of Sciences, Shanghai 201210;2. Uniνersity of Chinese Academy of Sciences, Beijing 100049;3 ShanghaiTech Uniνersity, Shanghai 201210;4. Dalian National Laboratory for Clean Energy, Dalian 116023;5. Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190)

Abstract: To simultaneously improve the quantity and quality of heat recovery in a heat exchanger network (HEN), this study conducts a theory analysis based on the first and second laws of thermodynamics. Under the premise of maximizing the heat recovery quantity of HEN, ?diss is used as an evaluation index to optimize the quality of heat recovery. Meanwhile,the total annual cost (TAC) is considered as another optimization objective to ensure the economic feasibility of the HEN. A superstructure-based multi-objective mixed integer non-linear programming approach is put forward to solve the trade-offbetween minimizing ?diss and minimizing TAC. This allows for the optimum HEN structure to be obtained. A well-studied example is solved to highlight the benefits of the proposed method.

Key words: heat exchanger network (HEN); heat recovery quantity; heat recovery quality; economic; superstructure

1 Introduction

A heat exchanger network (HEN) is a crucial component in process industries, since it can reduce the consumption of utilities and save energy by redistributing heat between process streams[1]. HEN optimization problems were first defined by Masso[2]in 1969. Since then, various approaches to HEN optimization have been put forward in the past few decades[3]. HEN optimization methods have evolved into two main groups, namely the pinch analysis (PA) method and the mathematical programming(MP) method[4-5]. The PA method originally emerged to set the minimum utility consumption target for HENs under the pinch point temperature difference (ΔTpinch),based on thermodynamic principles, and provided guidance for the design of HENs[6-7]. Liu et al.[8]adopt a gradual optimization integration strategy based on theT-Hdiagram, such that the maximum energy recovery is achieved under the principle of energy cascade utilization and stepwise matching. Compared with the PA method, which is usually sequential, the MP method can consider design factors (energy, areas, and units)both sequentially and simultaneously. Papoulias and Grossmann[9]formulated a linear programming (LP)problem to obtain the minimum energy consumption based on the transshipment model where variables,including the residual heat transferred between intervals and the flowrates of utilities, are all linear. The minimum number of required heat exchange units involving integer variables is then optimized as the second objective by mixed-integer linear programming (MILP) under the energy target[9]. Based on the network structure of the above model, and with the addition of nonlinear terms such as the logarithmic mean temperature difference and the area exponent in the solution process, a nonlinear programming (NLP) formulation proposed by Floudas et al.[10]was applied for solving the minimum capital cost,seeking the minimum heat exchanger area. Simultaneous MP methods primarily apply mixed-integer nonlinear programming (MINLP) formulations for HEN problems without decomposition of these three objectives. The

MINLP model based on stagewise superstructure[11]for HEN optimization was developed by Yee and Grossmann[12]to optimize the minimum total annual cost(TAC). Xiao et al.[13]present a node dynamic adaptive non-structural model taking TAC as the optimization objective for solving largescale HEN synthesis problems.Zirngast et al.[14]presents an improved algorithm for the MINLP synthesis on flexible HENs with a large number of uncertain parameters, regarding TAC as the objective to evaluate the optimal results. Regardless of model or algorithm improvements, the above-mentioned MP literature focuses primarily on the economic effect.

However, the urgency of carbon neutrality requires that more attention be payed to the original energy target to recover as much heat as possible, while still considering the economic cost. The concept of energy in HENs includes two aspects, heat quantity and heat quality.Generally, recovery of heat quantity is much easier to optimize than recovery of heat quality in HENs[15]because, while improvements in heat quantity recovery result in reduced utility consumption, the improvement of heat quality is usually difficult to evaluate. In the heat transfer process, heat can only be transferred from high to low temperature, and the existence of irreversible loss leads to the decrease of heat quality recovery(temperature). If the heat quality recovery is reduced without optimization, a lower temperature stream with a large quantity of energy will become useless in subsequent heat transfer process. Alternatively, higher quality streams may be used as a heat source in the thermal cycle for power generation[16]. Under the same heat quantity, heat with higher quality (temperature) is supposed to generate more power. Hence, the focus of this study is to take as the premise the maximum heat quantity recovery and then reduce the irreversible heat loss to improve heat quality recovery, while also considering the economic cost.

The maximum heat quantity recoveryQmaxcan be obtained by both PA[17]and MP[18]methods. The irreversible heat loss, which leads to the decrease of heat quality, can be intuitively depicted via the area enclosed by hot and cold curves in the temperature-heat flow (T-Q)diagram[19-20]. This area is equal to the entransy dissipation rate?dissproposed by Guo et al.[21]Under the condition of determiningQmax, better heat quality recovery of the HEN can be achieved by arranging heat transfer between varied hot and cold streams to reduce?diss. A heat exchanger characterized by smaller temperature differences between the streams generates a smaller irreversible loss (?diss) in a given heat transfer demand, but its area will increase.This means that reducing?dissunder a certain heat quantity recovery leads to increased capital cost in the HEN. This in turn leads to an increase in TAC including fixed operating costs. Designing the HEN with the only goal being to minimize?disswill lead to poor economic effects. Hence, it is necessary to consider both?dissand TAC to optimize the design of the HEN.

To deal with the trade-off between these two contradictory objectives, a multi-objective mixed integer non-linear programming (MOMINLP) model based on stagewise superstructure is presented in this paper. The ε-constraint method is adopted to solve the multi-objective optimization problem (trade-off between?dissand TAC),which can obtain the required Pareto solution set and finally help attain the best solution[22].

In summary, this article first describes the HEN optimization problem. Then, the heat quantity and quality recovery effects between hot and cold streams in HEN are analyzed and regarded as optimization objectives.After that, the trade-off between the enhancement of heat quality recovery and the reduction of TAC is solved by a superstructure-based MOMINLP method. Finally, a comprehensive case study is presented to demonstrate the proposed method, and a conclusion is made to summarize the significance of this paper.

2 Problem Statement

Consider a HEN that consists of a set of hot process streams (HPSs) {i|1…，H}, which need to be cooled from inlet temperaturesTh,i,into target outlet temperaturesTh,i,out, and a set of cold process streams (CPSs) {j|1…，C}, which need to be heated from inlet temperaturesTc,j,into target temperaturesTc,j,out. The heat capacity flow rates(FCph,iandFCpc,j) of both hot and cold process streams are also given. The hot and cold streams can transfer heat with each other, and the additional utility consumption can be reduced by improving the heat recovery between hot and cold streams. It should be noted that heat has the duality of quantity conservation and quality decline in the heat transfer process. In addition to reducing the utility consumption by increasing the heat recovery quantity, it is necessary to improve the heat transfer quality by reducing the irreversible loss in the HEN. The problem is to find a parameter to evaluate the effect of heat quality recovery in the entire HEN. Meanwhile, improving the heat quality recovery requires more heat transfer areas, which leads to an increase in capital cost. It is therefore also essential to comprehensively consider the trade-off between heat quality recovery and process economy.

3 Thermodynamic Analysis

3.1 Thermodynamic analysis of heat transfer quantity in HEN

In Figure 1, a composite curve (CC) diagram of the HEN is constructed that can illustrate the heat quantity recovery(horizontal axisQ) and heat quality recovery (vertical axisT)[23-24]. The red curve is a hot composite curve composed of hot process streams, and the blue curve is a cold composite curve composed of cold process streams.The HEN is divided into three parts in the CC diagram:the internal heat recovery sectionQhr, the remaining cold utility needed sectionQcu, and the remaining hot utility needed sectionQhu. The smallest vertical distance between the hot and cold composite curves is called the pinch point temperature differenceΔTpinch. Composite curves can be moved horizontally along theQdirection.As the hot and cold composite curves move toward each other, the value ofΔTpinchdecreases whileQhrincreases.The minimum allowed temperature differenceΔTminis the limit value forΔTpinchthat guarantees effective heat transfer and finite heat transfer area. WhenΔTpinchis equal toΔTmin,Qhrreaches the maximum value and the hot utility requirementQhuand cold utility requirementQcuboth achieve their minimum values. The heat quantity recovery of the HEN reaches its optimum by maximizing the recovery under the specifiedΔTmin.

3.2 Thermodynamic analysis of heat transfer quality in HEN

Heat quality recovery decreases due to the existence of irreversible loss in the HEN heat transfer processes. In Figure 2, the entransy dissipation rate has the physical meaning of the degree of irreversibility of heat transfer in a single heat exchanger. It can be calculated as[25]

Figure 1 Composite curve diagram of HEN

whereQtis the total heat transfer rate between the hot and cold streams as shown in Figure 2.ThandTcare,respectively, the temperature of the hot and cold streams during heat transfer across the differential element dQ.In Figure 2, the shaded area enclosed by the hot stream and cold stream curves in theT-Qdiagram can also be calculated asTherefore, the shaded area intuitively indicates the local?dissduring heat transfer between the hot and cold streams[26].

Figure 2 T-Q diagram of a heat transfer process occurring in single heat exchanger

For a HEN composed of multiple heat exchangers, the overall entransy dissipation rate?HENdissis the sum of?dissvalues across all heat exchangers[20], which can be described as

whereNrepresents the amount of the heat exchangers in the HEN.

It can be seen from Figure 2 and Eq. (2) that the irreversible loss is related to heat quantity recovery and temperature difference between the hot and cold streams.In the case of fixed heat quantity recovery, reducing the temperature difference between the hot and cold streams involved in the heat transfer process also reduces the irreversible lossin the heat recovery process,resulting in better heat quality recovery of hot and cold streams.

Consider as an example a single counterflow heat exchanger in which the hot stream is assumed to remain unchanged and different cold streams are selected to match the hot stream under the same heat transfer quantity. In Figure 2, the inlet temperatures of cold stream 1 and cold stream 2 are the same but theirFCpvalues are different. Meanwhile, theFCpof cold stream 2 and cold stream 3 are the same but their inlet and outlet temperatures are different. The inlet and outlet temperatures of cold stream 3 are closest to the hot stream. It can be seen that?diss(the shaded area enclosed by the hot and cold stream curves) between the hot stream and cold stream 3 is the best of the three cold streams because the heat transfer temperature difference between the hot stream and cold stream 3 is smallest.

A HEN is composed of multiple heat exchangers. Theof HEN can be reduced by arranging a proper match between hot and cold streams, which is reflected in the adjustment of the HEN structure. The inlet and outlet temperatures andFCpof streams in HEN can be changed by partitioning the streams into different temperature stages and splitting into different mass flow rates. In the case where the sum of the heat transfer quantity of each heat exchanger isQhr, the?disscan be reduced by selecting the proper HEN structure. In addition to improving heat quantity and quality recovery, the economic effect of the HEN must also be considered. To address the tradeoff between energy and economic factors, a MOMINLP model is established in the next section.

4 Superstructure-Based MOMINLP Formulation

Based on the thermodynamic analysis in the previous section and the demands of practical engineering, a MOMINLP model based on a stagewise superstructure[11]is proposed to improve heat quantity and quality recovery while considering the economics of the HEN.The optimum HEN structure can be selected by the optimization of the MOMINLP model proposed. First, the minimum utility consumption (Qu) of the HEN must be calculated to obtain the maximum heat quantity recovery under the specifiedΔTmin. Next, the heat recovery quality can be optimized by minimizing?diss. Finally, the tradeoff between the minimum?dissand the minimum TAC is solved through the ε-constraint method[27]. The detailed model formulations of the energy balances, the constraint conditions, and the objective functions are presented in the following subsections.

4.1 Energy balances

4.1.1Energy balances for each process stream in the HEN

The overall enthalpy change of hot process streamiisFCph,i(Th,i,in-Th,i,out), which is equal to the total heat exchanged with cold process streamj(qijk) and the external cold utilityqcu,irequired by hot process streami. Similarly, the overall enthalpy change of cold process streamjequals the sum of all the heat exchanged with hot process streami(qijk) and the external hot utilityqhu,jrequired at the end of the cold process streamj.

The model assumes thatFCph,iandFCpc,jare constant throughout the heat transfer process.

4.1.2Energy balance for each heat exchanger

To simplify the formula, it is assumed that all the streams are isothermally mixed at each stage. The energy balance on both sides of the heat exchanger is then

whereTh,i,kandTc,j,krepresent the temperature of the hot streamiand the cold streamjin thekth stage,respectively.Th,i,(k+1)andTc,j,(k+1)represent the temperature of the hot streamiand the cold streamjin the (k+1)th stage, respectively.

4.1.3Energy balance for utilities

whereTc,j,1is the temperature of the cold streamjat stage 1, and other symbols are introduced in the previous equations.

4.2 Constraint conditions

4.2.1Temperature feasibility constraints

The inequalities below provide the monotonic decrement of temperature at each successive stage from the hottest side to the coldest side of the superstructure:

4.2.2Logical constraints for the existence of heat exchange boundary

whereФis the upper bound of the heat load of each heat exchange unit andzis a binary variable that indicates whether there is a heat exchange unit (1 means there is a heat exchange unit, 0 indicates no heat exchange unit).

4.2.3Minimum temperature differences constraints in heat exchange units

For heat exchange between hot and cold streams:

For heat exchange between hot streams and cold utilities:

For heat exchange between cold streams and hot utilities:

Γrepresents the upper limit of the temperature difference,Thu,inandTcu,inrepresent, respectively, the hot and cold utility inlet temperatures, andThu,outandTcu,outrespectively represent the hot and cold utility outlet temperatures.

4.3 Objective functions

To optimize the quantity of heat recovery and evaluate the quality of the fixed heat recovery quantity,Qmaxneeds to be calculated under a specifiedΔTmin. Compared with theQmaxof the HEN, the minimum utility consumptionQuis easier to be obtained. Hence, P0 is presented as an original problem to solve for attaining the minimumQu.Then,Qmaxcan be calculated as the sum of heat transfer quantities of all heat exchangers in the HEN under the condition of obtaining the minimumQu.

The obtainedQmaxis then used as a prerequisite for optimizing the heat recovery quality. The objective for achieving the optimum heat recovery quality of HEN can be described by P1:

where?diss,ijkis the entransy dissipation rate of the heat transfer process in each heat exchanger of the HEN.Th,iandTc,jare the temperatures of the hot streamiand the cold streamjin each heat exchanger, respectively.

The other objective is to minimize total annual cost (TAC),which is the sum of the annualized capital cost and the annual operating cost presented in P2.

Costcainvolves the fixed investment cost and the area cost of the heat exchangers and utility units.

whereβ1is the fixed investment cost coefficient of the heat exchanger,β2is the area cost coefficient of the heat exchanger, andαrepresents the area index coefficient of the heat exchanger.Areaijkrepresents the heat transfer area between the hot streamiand the cold streamjin thekth stage, andAreahu,iandAreacu,represent, respectively,the heat transfer area of hot and cold utilities. The heat transfer area can be calculated as

where is the heat transfer coefficient of each heat exchange unit. LMTD represents the logarithmic mean temperature difference of the integral average over the entire heat exchanger area,

whereΔTmaxandΔTminrepresent the maximum and minimum heat exchange temperature difference between the two ends of the heat exchanger, respectively. Note that it is not mathematically tractable whenΔTmaxandΔTminare equal. To avoid numerical problems, Chen’s approximation for the LMTD is used, as shown in Eq. (23)[28]:

The operating cost includes the cost of cold and hot utilities, which can be expressed as:

whereβ3andβ4are the operating cost coefficients of cold and hot utilities, respectively. SinceQmaxis taken as the prerequisite for the other two optimization objectives, the TAC operating cost remains fixed in this study.

5 Methodology

The MOMINLP model in this study includes three optimization objectives: the minimumQu(P0), the minimum?diss(P1), and the minimum TAC (P2).To improve heat quantity and quality recovery simultaneously,P0 is the premise of the latter two objectives. Improving the heat quality recovery of the HEN requires more heat transfer area, increasing the capital cost, which makes P1 and P2 contradictory optimization objectives. The task of this methodology is to obtain a set of Pareto optimal solutions that solve the trade-off between minimizing?dissand minimizing TAC. First, the minimumQuis achieved, and operating cost remains unchanged. The minimumcan then be solved by adjusting the HEN structure, and the TAC under this HEN structure, referred to as TAC?diss, can be calculated. However, TAC?dissis expected to be greater than the TAC values under other HEN structures with largerdue to the conflict betweenand TAC. The ε-constraint method is used to solve this trade-off. Figure 3 is a graphic representation of the design metric space of the proposed problem. The red curve in the figure represents the Pareto frontier curve, which is composed of the Pareto optimal solution set.and TAC*are the two extreme points of the Pareto frontier curve, corresponding to the solutions of each single objective optimization problem.The ε-constraint method is a solving methodology to supplement the Pareto frontier curve along between the two extreme points. The horizontal distance betweenG*dissand TAC*is divided equally intoN-1 parts, and the TAC interval of each part (represented asΔe) can be calculated as

Figure 3 Graphic representation of the ε-constraint method for a bi-objective optimization.

The value of the relaxation coefficient ε is calculated as

The TAC value is relaxed under the different εn, and the corresponding minimum?disscan be solved with the different TAC value constrained. Finally, the Pareto frontier of?dissand TAC is attained.

6 Case Study

A diesel fraction hydrogenation unit of one petrochemical company[29]was selected to demonstrate the advantages of the methodology proposed here. The process stream data are listed in Table 1. The heat transfer coefficient of all heat exchangers is 1 kW/(m2?K). To compare with the previous study[29]under the same heat recovery quantity,theΔTminis also set as 10 K.

According to the stream data, the maximum quantity of heat recoveryQmaxis 116529.8 kW whenΔTpinchis equal toΔTmin. The corresponding minimum hot and cold utilities are 14213.28 kW and 21728.48 kW, respectively.These values are prerequisites for optimizing heat quality recovery and economy.

Due to the lack of economic analysis in literature[29], the economic parameters adopted from the work of Jianget al.[30]are used in this paper, as displayed in Table 2.Through solving P1, the minimum value of?dissand the corresponding TAC are obtained as 3077104.6 kW?K and 4999957.41 $/a, respectively. Through solving P2,the minimum TAC and the corresponding value of?dissare 3948501.42 $/a, and 4111644.2 kW?K, respectively.These are the two extreme points of the Pareto frontier curve. The step length Δeof the relaxation coefficient ε in each iteration is chosen as 0.02. After that, the relaxation coefficient εnis obtained by Eq. (26) and the relaxed TAC values are calculated by multiplying the minimum TAC by the relaxation coefficient εn. Finally, the corresponding values of?disscan be solved under the different relaxed TAC values, and the Pareto frontier curve is obtained as Figure 4. Note that the Pareto frontier is composed of a set of optimum solutions and that each point might represent the optimum HEN structure under certain conditions.

Figure 4 Pareto frontier of and TAC

Table 1 The data of streams for diesel hydrogenation unit

Table 2 The cost parameters of Case

To find a compromise between?dissand TAC, the rate of decline for?dissand the rate of increase for TAC are shown in Figure 5 for different relaxation coefficient εn. The difference between the rise rate of TAC and the decline rate of?dissreaches its maximum when the value of εnis 1.10, where the rise rate of TAC is 10.00% and the decline rate of?dissis 20.19%. Consequently, , the?dissand TAC atε= 1.10 are taken as the optimum?dissand

TAC. The values of?dissand TAC are 3281622.5 kW?K and 4339891.56 $/a, respectively. The corresponding structure is taken as the final optimum result of the multi objective optimization method, which is presented in Figure 6.

Figure 5 Changing rate of TAC and ?diss in different ε

Three different design schemes are compared:optimization through entransy theory as described in ref 29 (Scenario A), optimization with the best?diss(Scenario B), and optimization of the multi-objective optimization method (Scenario C). Their performance is compared in Table 3.

Table 3 Comparison of Performance Parameters for case

The configuration under scenario A is shown in Figure 7.It needs fourteen heat exchangers to realize heat transfer between process streams. Two heaters and nine coolers are needed to supplement the heat transfer between process streams. The scheme requires a total of 14213.28 kW of hot utility and 21728.48 kW of cold utility, and?dissis 4111644.2 kW?K. The TAC is 4568052.2 $/a,which contains the capital cost of 3606613 $/a and the operating cost of 961439.2 $/a.

For scenario B, the solution with the smallest?disson the Pareto frontier is selected. This network configuration scheme is shown in Figure 8. Compared with scenario A, scenario B has the same hot utility and cold utility requirements. Scenario B requires twenty-seven heat exchangers, nine coolers, and five heaters to meet the energy demand of all hot and cold streams. The TAC is 4999957.91 $/a, which is 9.45% higher than that of scenario A, while?dissis 3077104.6kW?K, which is 25.0% lower than that of scenario A. Arranging more heat exchangers can significantly improve how the HEN matches hot and cold streams, but the TAC surges as a result.

Figure 6 Structure of the optimum HEN as determined in this paper

Figure 7 Structure of the optimum HEN from Xia[29]

Figure 8 Structure of the HEN with optimal ?diss

The HEN corresponding to scenario C needs twentythree heat exchangers for heat recovery between process streams, plus nine coolers and five heaters to balance the heat. The cold utility and hot utility consumption are 14213.28 kW and 21728.48 kW, respectively, the same as for scenarios A and B. The value of?dissin this scenario is 3281622.5 kW?K, which is 20.0% lower than scenario A, while the TAC is 4339891.56 $/a, which is 4.9% lower than scenario A.

Under the condition of fixedQhr, the minimum TAC and the minimum?dissare two contradictory objectives -reducing?dissrequires more heat exchangers and heat exchange area, which increases TAC. Based on the comparison of the above three scenarios, optimization scenario B of this paper is slightly inferior based on its economic performance, even though it has an advantage in terms of heat quantity and quality recovery, which is essential for the reduction of energy consumption and carbon emissions. Scenario C can be selected as the the best overall optimization result to consider?dissand TAC simultaneously. Compared to scenario A in the literature,?dissdecreased by 20.0% while TAC decreased by 4.9%in scenario C.

7 Conclusion

This paper aims to improve heat quantity and quality recovery in a HEN when economically feasible. Under the premise that the quantity of heat recovery is of main importance,?dissis selected to evaluate the heat quality recovery of HEN; the smaller the value of?diss,the better the heat quality recovery. Since reducing?disswill lead to an increase in capital cost and TAC, a superstructure-based MOMINLP model is formulated and the ε constraint method is used to solve the tradeoff between the minimum?dissand the minimum TAC.The Pareto solution set obtained by this two-objective optimization can help decision-makers consider a comprehensive set of options that consider both heat recovery and economy. Through the case study provided here, this method demonstrates that?dissand TAC can be significantly reduced. In comparison with the optimization results in the previous studies,?dissand TAC of the optimal solution in this paper are reduced by 20.0% and 4.9% respectively.

Acknowledgment:The authors would like to gratefully acknowledge the financial support provided by the National Key R&D Program of China (2017YFE0116300, 2019YFE0122100),the Youth Innovation Promotion Association, CAS (2017353),and the DNL Cooperation Fund, CAS (DNL202023).