Article pubs.acs.org/IECR
Retrofit of Heat Exchanger Networks for Multiperiod Operations by Matching Heat Transfer Areas in Reverse Order Lixia Kang† and Yongzhong Liu*,†,‡ †
Department of Chemical Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, P.R. China Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, Xi’an, Shaanxi, 710049, P.R. China
‡
S Supporting Information *
ABSTRACT: This work presents a retrofitting approach for heat exchanger networks (HENs) for single-period and multipleperiod operations, aiming at the improvement of operation flexibility of HENs. A two-step method is proposed to solve the problems of redundant heat transfer areas in multiperiod HENs and the incomplete utilization of existing heat-transfer areas in HEN retrofit. In this approach, a multiperiod HEN design model is first solved to target the retrofit. Then, the final HEN is obtained by matching the required heat exchangers with the existing ones in reverse order. The proposed matching procedure is provided in detail and mathematically proven. Three case studies are employed to illustrate the advantages of the proposed method for solving the problems in the multiperiod HEN retrofit. In addition, the proposed method is extended to solving single-period HEN retrofit problems. Results indicate that the proposed method makes full use of the existing heat exchangers to reduce retrofit costs and increases the energy utilization efficiency of the HEN in multiperiod operations. The proposed method is simple and easy to implement. It can reduce the computational load and the difficulties in solving practical industrial problems. predicted energy price fluctuations in future based on energy prices in the past. The model was further improved by maximizing both the expected net present value with no risk assessment performed and the certainty equivalent with risk assessment regarding future utility prices and investment.6 Ma et al.7 explored the synthesis of multiperiod HEN with multistream heat exchangers. They obtained a feasible HEN structure by using the temperature−enthalpy diagram and modified the heat transfer areas and heat loads by using a genetic algorithm. Ahmad et al.8 also found a better solution to the multiperiod HEN model by using a simulated annealing algorithm. To make full use of the existing heat transfer area and reduce the capital costs, Sadeli and Chang9 proposed a design approach for multiperiod HENs based on heuristic principles. Jiang and Chang10 developed a strategy for synthesis of multiperiod HENs by using timesharing mechanisms. The multiperiod HEN structure was obtained on the basis of the simultaneous synthesis of single-period HENs. Their method ensures that the HEN works under optimal operating conditions in each operating period and that the total heat transfer area and the capital costs are significantly reduced, when compared with those obtained by conventional methods.3 However, it should be noted that the HEN generated by their method has a complicated structure and redundant heat transfer areas. However, the above-mentioned research mainly focuses on the synthesis and design of multiperiod HENs; little attention has been paid to multiperiod HEN retrofit. Moreover, studies on HEN retrofit has mainly focused on retrofitting single-period HENs. Klemeš et al.11 reviewed the recent developments in
1. INTRODUCTION In chemical production processes, demand and supply shocks, seasonal changes in operational conditions, and attenuation of catalyst activity may cause cyclic fluctuations of the heat exchanger networks (HENs). When these fluctuations occur, the HEN in the system may become insufficient and eventually infeasible to meet the requirements for periodic operations. In the present, for accommodation of the cyclic changes, the existing HEN is often retrofitted by increasing allowance for areas of heat exchangers, increasing utility consumptions, and bypassing streams. Nevertheless, the capital costs for these solutions tend to be high. To minimize the capital costs and utility consumption of the HEN, Aaltola1 developed a multiperiod mixed integer nonlinear programming (MINLP) model based on the superstructure model proposed in ref 2. In this simultaneous approach, a feasible HEN structure was initially obtained by solving the MINLP model, and then, the heat transfer areas and split ratios in the HEN were optimized to achieve the minimum total annual cost. However, the average heat transfer areas in the multiple periods are used in the MINLP model, which may underestimate the capital costs and the total annual costs. Zhang and Verheyen3 removed relaxation variables and weighed parameters from Aaltola’s model and used the maximum heat transfer area in the multiple periods to calculate the area cost in the objective function. Application of their improved model to industrial cases showed that their model can produce more reasonable solutions to the problem. Further studies have been done to improve Zhang and Verheyen’s model and the relevant algorithm. Isafiade and Fraser4 studied the effect of uneven period durations on multiperiod HEN design. In addition, a submodel for estimation of energy costs was embedded to calculate variable energy costs. Nemet et al.5 solved the multiperiod HEN model by taking the incremental net present value as the objective function and © 2014 American Chemical Society
Received: Revised: Accepted: Published: 4792
December 5, 2013 January 29, 2014 February 26, 2014 February 26, 2014 dx.doi.org/10.1021/ie4041143 | Ind. Eng. Chem. Res. 2014, 53, 4792−4804
Industrial & Engineering Chemistry Research
Article
The rest of this paper is organized as follows. In section 2, the multiperiod HEN retrofit problems are defined. The implementation procedure and the mathematical proof of the reverse order matching method are then described in sections 3 and 4. In section 5, multiperiod HEN retrofit problems are solved by using three different retrofit strategies and the results are discussed and analyzed. The extension of the proposed method to solving a single-period HEN retrofit problem is given in section 6. By comparing with the results available in the literature, the results of the single-period HEN retrofit further verify the effectiveness of the proposed method. Finally, the conclusions of this paper are presented in section 7.
analysis and design of HEN for retrofit situation. Furman and Sahinidis12 have reported that the MINLP model for HEN synthesis is difficult to solve due to its nonlinearity and nonconvexity, and the HEN retrofit problem is even more difficult to solve. The HEN retrofit problems are usually solved stepwise. Ciric and Floudas13 solved the HEN retrofit problem in two steps. They first determined the energy target and then retrofit the HEN by solving a MINLP model. Their HEN retrofit model involves a large number of feasible modification strategies, and the model can be easily expanded. However, the trade-off between capital costs and energy costs was not considered. For the HEN retrofit, Yee and Grossmann14 divided the procedure into two steps: selection and optimization. The economically feasible retrofit strategies were first determined, and then, the optimal HEN was obtained by solving a MINLP model. However, the scale of the retrofit problems is restricted due to excessive binary variables. Briones and Kokossis15 presented a two-step strategy for the HEN retrofit. The optimal modification strategy was obtained by solving a MILP model, and then, the optimum operating conditions were determined by solving the MINLP model. Ma et al.16 used a constant approach temperature model to optimize the structure of the HEN and, then, a MINLP model to finalize the structure by taking the actual approach temperatures into account. Rezaei and Shafiei17 addressed the HEN retrofit problem by decomposing the model into a NLP model and an ILP model. Genetic algorithm (GA) was used to modify the HEN structure, and the NLP and ILP models were used to calculate the maximum heat recovery and the minimum retrofit cost, respectively. Zhang and Rangaiah18 also used GA to solve the HEN retrofit problems. Kovač Kralj19 developed a simultaneous model for HEN retrofit based on Yee and Grossman’s stagewise model of superstructure representation.2 In this simultaneous model, the existing HEN structure and heat transfer area were considered by adding constraint equations and discrete variables. However, it fails to make full use of the existing heat exchangers and heat transfer areas. Although encouraging progress has been reported on HENs retrofit in the above-mentioned studies, there still exist limitations in practical applications of the retrofit strategies proposed in these studies: • The inlet and outlet temperatures and the heat capacity flow rates of streams are usually taken as constants in HEN retrofit models. Periodic fluctuation in operating conditions is commonly not considered, which deviates from practical situations. • The utilization of the existing heat exchangers is limited to increasing the heat transfer area, adding new heat exchangers, or removing existing heat exchangers, but rematching the existing heat exchangers is usually ignored. • For a multiperiod HEN retrofit problem in an industrial system, a HEN retrofit model involves a large number of discrete variables and constraints inevitably, leading to the exponential increase in the computational load and difficulties in searching feasible solutions. In this paper, to overcome these limitations, we propose a novel two-step method, referred to as the reverse order matching method, for retrofit of HENs operating in multiple periods. In the proposed method, the HEN retrofit target is first determined by using the multiperiod HEN design approach. Then, the existing heat transfer areas are rematched with the required heat transfer areas in reverse order to determine the final HEN structure.
2. PROBLEM STATEMENT Given are (1) structural parameters of the existing HEN, such as the number of the heat exchangers, the heat transfer areas, and the heat loads of heat exchangers; (2) operational parameters in each period, including inlet and outlet temperatures, heat capacity flow rates, and heat transfer coefficients of the streams; (3) utility parameters in each period, such as the types, inlet and outlet temperatures, heat transfer coefficients, and costs of the utilities; (4) cost parameters of the devices, including installation costs and heat transfer area costs of heat exchangers. The purposes of multiperiod HEN retrofit are to (1) meet the requirements of the multiperiod operational conditions; (2) maximize energy utilization efficiency; (3) make full use of the existing heat transfer areas by rematching the existing and required heat exchanger area; (4) minimize the HEN retrofit cost, which is the sum of installation cost of additional heat exchanger units, additional cost of heat transfer areas, utility cost, and electricity cost. 3. REVERSE ORDER MATCHING METHOD FOR MULTIPERIOD HEN RETROFIT The reverse order matching method for multiperiod HEN retrofit is mainly divided into two steps: (1) determining a target for the multiperiod HEN retrofit by solving the multiperiod HEN design model; (2) rematching the existing heat transfer areas with the required heat transfer areas in reverse order. 3.1. Determination of a Target for Multiperiod HEN Retrofit. In this paper, Zhang’s model3 is used to design the multiperiod HEN. The objective of the multiperiod HEN design model is to minimize the newly increased annual cost, including the increased capital costs and operating costs, shown as min TACad = Af(Cf Z ad + Ca(Aad )β ) ad ad ad + (C hu + Ccu + Cele )
(1)
For the capital costs, Af is the annualized factor, which can be expressed in the form of Af = r(1 + r)y/(1 + r)y − 1; r is the rate of return; and y is the plant time. C stands for cost; Zad and Aad are the numbers of the newly added heat exchangers and heat transfer areas, respectively, which are defined as Z ad =
∑ ∑ ∑
zi , j , k +
i ∈ HP j ∈ CP k ∈ ST
∑
zcu, i +
i ∈ HP
∑
zhu, j − Z ex
j ∈ CP
(2)
Aad ≥
∑ ∑ ∑ i ∈ HP j ∈ CP k ∈ ST
− Aex
(Aireq ,j,k)
+
∑ i ∈ HP
req (Acu, i) +
∑
req (Ahu, j)
j ∈ CP
(3)
where 4793
dx.doi.org/10.1021/ie4041143 | Ind. Eng. Chem. Res. 2014, 53, 4792−4804
Industrial & Engineering Chemistry Research
Z ex =
Article
∑ znex
(4)
n∈N
Aex =
will undergo steps 1 and 2 until all unmatched Aex n are located below Areq . i,j,k Step 3: Number-Based Matching of the Existing and Required Heat Exchangers. All heat exchangers unmatched in step 2 are sorted out for matching. If the number of the required heat exchangers is larger than that of the existing heat exchangers, zeros are used to represent the absent heat exchangers; otherwise, remove the smallest Aex n in turn until the number of the required heat exchangers is the same as that of the existing heat exchangers. Step 4: Rematching of Heat Transfer Areas in Reverse Order. All Areq i,j,k in step 3 are placed on the left side in descending order and Aex n on the right side in ascending order. The corresponding heat exchangers on the two sides are matched. In this case, the additional heat transfer area is calculated as ΔAi,j,k = ex Areq i,j,k − An and the heat transfer area obtained after retrofit is ex expressed as Ai,j,k = Areq i,j,k > An . The procedure will be further exemplified in detail by case studies in the subsequent sections. It should be noted that the proposed procedure is implemented after the minimum number of heat exchangers requiring additional areas is guaranteed. Hence, the prerequisites of the proposed method to obtain the least cost of the increased ex heat transfer areas are (1) Areq min ≥ Amax and (2) the number of the required heat exchangers equals to that of the existing heat exchangers. In the proposed procedure, the preparatory work is carried out in the first three steps. Step 1 is the fundamental of the retrofit, and step 2 is to ensure that all Areq i,j,k that can be directly substituted by Aex are picked out. In this way, the number of heat exchangers n requiring additional heat transfer areas is minimized. In addition, the heat transfer areas after step 2 automatically satisfy (Areq i,j,k)min > (Aex n )max. Step 3 intends to make sure that the number of required heat exchangers is the same as that of the existing heat exchangers.
∑ A nex
(5)
n∈N
where z is a binary variable indicating the existence of a match between a hot stream and a cold stream. Aex is the heat transfer area of the existing heat exchanger. In the present HEN retrofit model, the required area of the heat exchanger is obtained by using Chen’s approximation20 to calculate the logarithmic mean temperature difference. For the operating costs, heating utility, cooling utility, and electricity costs can be written as ⎛ DPp ad C hu = C hu⎜⎜ ∑ ⎝ p ∈ PR NP
∑
qhu, j , p −
j ∈ CP
⎛ DPp ad Ccu = Ccu⎜⎜ ∑ ∑ qcu, i , p − ⎝ p ∈ PR NP i ∈ HP ad Cele = Cele
−
⎞ ex ⎟ qhu, j⎟ ⎠ j ∈ CP
∑
⎞ ex ⎟ qcu, i⎟ ⎠ i ∈ HP
∑
⎛ DPp Y ΔP ⎜ out in ⎜ ∑ 1000ηpρc p(tcu − tcu) ⎝ p ∈ PR NP
⎞ ex ⎟ qcu, i⎟ i ∈ HP ⎠
(6)
∑
(7)
qcu, i , p
i ∈ HP
∑
(8)
where qhu and qcu are utility consumptions for heating and ex cooling in each period of the required HEN; qex hu and qcu are utility consumptions for heating and cooling of the existing HEN. DP is the duration of each period, and NP is the duration of all the HEN periods. Y is the hours of operation. ΔP and ηp are the pressure drop and the efficiency of the pump. The terms ρ and cp are the density and the specific heat capacity. Here, tincu and tout cu stand for the inlet and outlet temperatures of the cooling utility. The constraints of the model are given in Appendix A, and calculations of electricity consumption are presented in Appendix B. 3.2. Implementation of Reverse Order Matching Method. In the existing HEN, there are N heat exchangers, ex ex ex marked as 1, 2, ..., N. Aex 1 , A2 , ..., An , ..., and AN denote the corresponding heat transfer areas of each heat exchanger. Areq i,j,k is the required heat transfer area obtained by the multiperiod HEN design model, and Ai,j,k is the heat transfer area after HEN retrofit. The subscript (i,j,k) stands for the position of a heat exchanger in HEN, wherein i = 1, 2, ..., HP, HU; j = 1, 2, ..., CP, CU; k = 1, 2, ..., NK + 1. After the target for the multiperiod HEN retrofit is obtained, we can rematch the existing heat transfer areas Aex n with the required heat transfer areas Areq i,j,k according to the following steps: req Step 1: Sorting of the Heat Transfer Areas. Aex n and Ai,j,k are sorted in descending order of their values. Step 2: Matching and Replacement of Heat Transfer Areas. req If Aex n is adjacent to and located above Ai,j,k among the ordered req heat transfer areas, Ai,j,k can be substituted by Aex n directly. They req are linked by a dashed arrow, starting from Aex n and ending at Ai,j,k. Then, the heat transfer area of the heat exchanger located at (i,j,k) ex after retrofit is Aex n , the heat transfer area after retrofit is Ai,j,k = An req > Ai,j,k, and the newly added heat transfer area ΔAi,j,k is zero. After ex Areq i,j,k is replaced by its adjacent An , the rest of the heat exchangers
4. MATHEMATICAL PROOF OF THE REVERSE ORDER MATCHING METHOD The target of the multiperiod HEN retrofit is obtained by solving the multiperiod HEN design model. Once the target is determined, the operating costs saved by the retrofitted HEN and the extra capital costs due to adding new heat exchangers are known. Nevertheless, the costs of the increased heat transfer areas could be different when the required heat transfer areas are matched with the existing heat transfer areas in different ways, as well as the cost of repiping and construction work. In general, the effect of the number of heat exchangers on additional investment cost, which is related to the installation, repiping, and construction work cost, is greater than that of additional heat transfer areas. To reduce the subsequent investment cost, we proposed to ensure the minimum number of heat exchangers requiring additional heat transfer areas at first and then match the required heat transfer areas with the existing ones. It is on this premise that as long as the minimum cost of the increased heat transfer areas is achieved, the corresponding HEN retrofit cost will be the lowest. Therefore, the problem can be restated as follows. There are M heat exchangers after the HEN retrofitting. ΔAm is the increased heat transfer area of the mth heat exchanger, m = 1, 2, ..., M. The increased area cost can be formulated as Ca(ΔAm)β, where Ca is the area cost coefficient and β (β > 0) is the area cost exponent. The total cost of the increased heat β transfer areas can be expressed as Ca∑M m=1(ΔAm) . {ΔAm} is a 4794
dx.doi.org/10.1021/ie4041143 | Ind. Eng. Chem. Res. 2014, 53, 4792−4804
Industrial & Engineering Chemistry Research
Article
denumerable set and ∑M m=1ΔAm = c, where ΔAm > 0, m = 1, 2, ..., M. The problem is to find conditions that {ΔAm} satisfies to render
minimum total cost of the increased heat transfer areas can be req achieved if Aex m and Ai,j,k are matched in reverse order. From the discussion above, we may summarize that the minimum total cost of the increased heat transfer areas can be req attained by matching the heat transfer areas Aex m with Ai,j,k in reverse order because the area cost exponent β generally falls within this scope in practice.
M
f=
∑ (ΔA m)β m=1
to reach its minimum f*. It is worth noting that the nature of the objective function f is dependent upon the value of β. Hence, we discuss f according to the three cases of β.
5. CASE STUDIES FOR MULTIPERIOD HEN RETROFIT There are several alternative revamp options for HEN retrofit, such as by adjusting the heat transfer areas, increasing numbers of heat exchangers, and rematching streams. In the following sections, three cases of multiperiod HEN retrofit are performed by using the proposed method to exemplify three retrofit strategies. The original HEN for single-period operation is shown in Figure 1. There are five heat exchangers in the HEN. The heat
β=1 In this case, the objective function reduces to the following expression M
f=
M
∑ (ΔA m)β
=
m=1
∑ ΔA m = c m=1
(9)
This means that f is independent upon the distribution of the set {ΔAm}. That is to say that the increased area cost is independent upon the area distribution when the exponent of the area cost is equal to one.
β>1 In this case, the objective function f becomes a concave function. For an arbitrary ΔAm ∈ [amin, amax], where amin = min{ΔAm, m = 1, 2, ..., M} and amax = max{ΔAm, m = 1, 2, ..., M}, f satisfies ⎛ ∑M ΔA ⎞ β 1 β ∑ (ΔA m) ≥ ⎜⎜ m= 1 m ⎟⎟ M m=1 M ⎝ ⎠
Figure 1. Existing HEN structure for single-period operation (multiperiod case).
M
ex transfer areas Aex n and heat capacities qn are also presented in Figure 1. The multiperiod operating conditions come from Jiang and Chang’s work.10 The capital cost of the heat exchanger is expressed as Chx = Cf + CaAβ. Other required parameters are presented in Table 1. All calculations are carried out using the GAMS software package. The global solver is DICOPT, and the MIP and NLP solvers are CONOPT and CPLEX.
(10)
Substituting the expression of f into eq 10, we obtain M
M
f=
∑ (ΔA m)β m=1
≥
(∑m = 1 ΔA m)β M β−1
(11)
It is obvious that if and only if ΔA1 = ΔA2 = ... = ΔAM, f gets its minimum f * = cβ/Mβ−1. This means that the closer the values of the increased areas, the greater possibility there is for the total cost of the increased areas to reach its minimum.
Table 1. Parameters Used in This Work parameter Y ΔP ηp ηe ηt ρ cp Chu Ccu Cf Ca β
0
