Synthesis of Flexible Heat Exchanger Networks Considering

Article Cite This: Ind. Eng. Chem. Res. 2019, 58, 12124−12136

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Synthesis of Flexible Heat Exchanger Networks Considering Gradually Accumulated Deposit and Cleaning Management Linlin Liu, Yiyuan Bai, Lei Zhang, Siwen Gu, and Jian Du* Institute of Chemical Process Systems Engineering, School of Chemical Engineering, Dalian University of Technology, Dalian, 116024 Liaoning, China

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S Supporting Information *

ABSTRACT: This paper presents a flexible heat exchanger network (HEN) synthesis methodology to cope with (i) the fluctuations of uncertain process parameters and (ii) the gradually accumulated deposit, which have never been simultaneously considered yet. First, a called all-cycle flexibility analysis method is developed to perform the network evaluation throughout entire operation horizon. Then at Part (I) of the methodology, an optimization-based sequential framework is proposed for the design of flexible HENs. The three major steps are (i) initial HEN synthesis; (ii) all-cycle flexibility analysis; and (iii) network improvement toward qualified flexibility and cost expectation via a mixed-integer nonlinear programming (MINLP) model-based iteration between network optimization and flexibility analysis. Furthermore, the optimization of the cleaning schedule cooperates with that of the heat transfer areas in Part (II), striving for lower network redundancy and total cost. At last, two cases are studied to demonstrate the application of the methodology. Results indicate that the fouling in heat exchangers and the parametric fluctuations in operation act synergistically on the flexibility of HEN. And the flexible HEN solutions with and without periodic cleaning both can be obtained by employing the proposed framework with sufficient flexibility guaranteed. Pintarič and Kravanja4 presented a sequential two-stage strategy for the stochastic synthesis of flexible HEN, where the initial feasible network structure was achieved at the first stage and then bypass steams, coolers, and heaters were optimized for flexibility concerns. Extension and utilization of this method on a large-scale uncertain problem was presented recently.5 In the stepwise method developed by Li et al.,6 the problem was solved with two major procedures in sequence: structure synthesis at nominal operating condition and structure and area optimization at critical points. Iteration is unavoidable in a typical flexible HEN synthesis problem, so in order to lessen the iteration times for desirable results, Chen and Hung7 employed an integer cuts strategy to reduce the search space in their study. And in a very recent study, the problem was extended into dynamic flexible HENs with considering ranges of variations in stream output temperatures.8 The parametric variations of stream temperatures and flow rates are the major uncertainties taken into account in above studies. In addition to the mentioned contributions, the flexible multiperiod HEN synthesis studies were also launched in recent years, aiming at accommodating parametric fluctuations in all subperiods. The purpose of multiperiod HEN synthesis remains

1. INTRODUCTION Heat exchanger network (HEN) synthesis is widely recognized as the most effective approach for intensive utilization of energy in process industry, via identifying the best heat exchange matches of process streams and heat load distributions. However, most of the contributions for HEN synthesis were merely subject to the nominal conditions of fixed operating parameters. In the real industrial world, production would face various disturbances from both process operation and external environment, and thus the design merely referring to a single condition would fail to meet process expectation and even lead to safety issues. This unsteady nature requires the designed HEN systems to be operable and flexible while chasing the best economy performance. An effective way to address the mentioned problem is to implement flexible HEN synthesis, wherein the sequential steps of initial network design, flexibility analysis, and network improvement are usually executed. The most attractive characteristic of flexible synthesis is that it could output a system with sufficient flexibility over a specified range of variations, without asking to know the precise distributions of concerned fluctuations, which are hard to predict commonly. The early works of Floudas and Grossmann,1 Kotjabasakis and Linnhoff,2 and Papalexandri and Pistikopoulos3 have built a great foundation for this topic, and an increasing attention has been gained since the new century. © 2019 American Chemical Society

Received: Revised: Accepted: Published: 12124

March 26, 2019 May 25, 2019 June 6, 2019 June 7, 2019 DOI: 10.1021/acs.iecr.9b01672 Ind. Eng. Chem. Res. 2019, 58, 12124−12136

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Industrial & Engineering Chemistry Research

optimum cleaning schedule of HEN while maintaining network capability of heat exchange becomes an important issue. Xiao et al.28 investigated the cleaning schedule problem in their HEN synthesis study, that a topological union of the structures at several fouling periods was organized as an initial HEN, and then the structure and heat exchanger areas of this oversynthesized network were simultaneously optimized with the scheduling of the cleaning operation. Nevertheless, parametric fluctuations were not included in their study, and the initial network was redundant in a high degree. Ishiyama et al.29 proposed a twolayer model to present the deposit, as the coke/aging layer was removable by mechanical cleaning and the gen/fouling layer by chemical cleaning. A mathematical model was built to manage the cleaning measures, however, only for a single heat exchange device rather than the more complicated network.30 Liu et al.31 launched the study for HEN. They investigated the employment of online chemical and off-line mechanical cleaning for fouling and aging deposit, respectively. The heat-transfer area, cleaning schedule, and cleaning measure for each heat exchanger were simultaneously optimized in their study. Later, to compensate the heat gap of online cleaning, spare devices were used,32 and the utilization of spare devices including their location and areas, the area margin of other exchangers, the bypass stream flow rates, and the new involved utilities were further studied.33 From the above-mentioned introduction it is concluded that the dynamic growth of deposit and the factor of time have never been considered in flexible HEN synthesis in addition to the uncertain fluctuation of operational parameters, even though their joint occurrence is unavoidable in practice. The difficulty mainly lies in how to combine the two kinds dynamic factors and how to realize the synthesis, since the problem presents as nonconvex if the deposit is considered. In particular, a flexibility analysis covering the entire operating cycle is required to appraise the results against time-related variations (dynamic growth of deposit), and a new network synthesis and improvement strategies are also in demand to make the optimal trade-off between the flexibility criterion and the cost criterion. This paper takes the challenge and proposes a sequential methodology for the synthesis of a desired flexible HEN. The methodology is devised based on our previous work on flexibility analysis24 and cleaning management;31 however, the content is very different from either the problem or the method. At the beginning of the study, a full-cycle flexibility analysis method accounting for quantitative evaluation of flexibility throughout the whole operating cycle is developed by considering the effect of deposit growth in addition to parametric fluctuations (temperatures and flow rates). On this basis, the methodology is established; on one hand, it allows solving the HEN synthesis problem targeting to minimum TAC criteria while guaranteeing qualified flexibility of the solution, and on the other hand, the overdesign problem and the adverse influence of deposits on HEN flexibility are able to be relieved by scheduling periodic cleaning for exchangers simultaneously with the further optimization of heat transfer areas.

to ensure the network capacity against variations, and several contributions9−13have been reported in this area; however, to the authors’ understanding, the multiperiod research focuses more on several discrete operating scenarios while the flexible synthesis research orients at a variation space. Thus, the methods for the multiperiod problem can be either step-by-step or one-step, but for flexible problem, the methods were performed commonly in a sequential framework, as mentioned before. Considering the possible parametric fluctuations in each subperiod, Escobar et al.14 and Kang and Liu15 designed flexible multiperiod HEN under the basic framework of typical flexible HEN synthesis. In the steps of flexible synthesis, flexibility analysis is an optimization-based procedure for bottleneck identification of the system against uncertainty, for which the flexibility index is a widely used quantitative indicator due to its adequate ability to characterize a system accommodating uncertain disturbances and the critical points, which play vital role in network improvement, are combinations of those uncertain parameters limiting variations, or saying the worst-case of every possible realization of uncertainty.16 In the early studies of flexible analysis, Swaney and Grossmann17 established the classical maximum−minimum−maximum model for the determination of flexibility index. Search algorithms for the convex problem18 and nonconvex problem19 were both performed to assist the determination of flexibility index and critical points. On basis of these pioneer contributions, extending studies continued. Bansal et al.20 studied the problem for linear systems by using parametric programming and then extended into nonlinear systems.21 Floudas et al.22 raised a global optimization algorithm, aiming to solve flexibility test and flexibility index problems of nonconvex systems. Pintarič and Kravanja23 developed three analysis methods, namely, the Karush− Kuhn−Tucher formulation, iterative two-level method, and approximate one-level method. These methods were proved capable of handling both convex and nonconvex problems. A similar work was also launched by Li et al.,24 wherein the direction matrix was proposed to describe the deviations of uncertain parameters. To solve the problem containing a large number of uncertain parameters, a solution strategy was exploited by Pintarič et al.25 to reduce the critical scenarios by simple sensitivity analyses. And for the problems having quadratic or linear inequalities, a new flexibility index algorithm based on iterative calculation was reported by Jiang et al.26 in a recent study. From the presented works it is found that the modeling and solving are central issues and also difficulties in flexibility analysis, due to the complexity caused by fluctuations of uncertain parameters. Thus, evolution of flexibility analysis is imperative if a more complicated problem, for example, taking time into consideration, has to be settled. The continual thickening of deposit on heat transfer surface contributes the most to the degraded performance of heat exchangers, due to the sustained decrease of the heat transfer coefficient. It is feasible to compensate the negative effect of the deposit growth on heat exchange performance by offering more heat exchanger area. So, the common strategy of HEN design with adequate flexibility guaranteed is to use the maximal thermal resistance instead of considering its dynamic characteristic; however, the resultant network solution is often oversynthesized with high capital cost. With this concern, efforts have been paid to mitigate the effect of fouling by adjusting operating parameters;27 however, only regular cleaning can solve this problem from the root. Therefore, determining the

2. PROBLEM STATEMENT The conventional HEN synthesis is performed under nominal conditions with fixed operating parameters, such as constant stream supply/target temperatures and heat capacity flow rates. However, the obtained network configuration of precise positioning would fail to maintain the steady production once encountering fluctuant operations and gradually accumulated deposits, resulting in the stream outlet temperatures deviating 12125

DOI: 10.1021/acs.iecr.9b01672 Ind. Eng. Chem. Res. 2019, 58, 12124−12136

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Figure 1. Schematic flowchart of flexible HEN synthesis process.

universal time-related asymptotic fouling model, so the parameters of each stream in this model are also given. The objective of the work is to determine a flexible HEN that features the minimum total annualized cost (TAC) meanwhile maintaining the system operable and feasible under fluctuant operating conditions. The fouling-involved flexible HEN synthesis study and the scheduling study that follows for exchanger cleaning are both launched to solve the mentioned problem, with giving the network configurations and the cleaning management of heat exchange units. To simplify the synthesis process, pressure drop and further fluid dynamics considerations are not taken into account in this study.

seriously from the targets due to the inherent control and operation limitation of the configuration. Therefore, this work focuses on developing a systematic methodology for the synthesis of flexible HEN, particularly taking the deposit growth, which will cause time-related variation of heat exchange performance, into account in addition to the parametric fluctuations. The synthesis problem to be addressed in this paper can be briefly stated as follows: There are a set of hot process streams to be cooled and a set of cold process streams to be heated. Given are (i) the nominal operating conditions, including heat capacity flow rates, supply, and target temperatures of all streams; (ii) the available heating and cooling utilities with their heat exchange parameters; (iii) the unit costs of utilities and the cost coefficients of heat transfer units, including the fixed cost coefficients, area cost coefficients, and area cost exponents; and (iv) the range of expected parametric disturbances, e.g., the lower and upper bounds of temperatures and flow rates for inlet streams. Furthermore, in order to investigate the effect of fouling on heat transfer performance during HEN system design, we assume all heat exchange units are well cleaned before service; while in operation, the fouling grows gradually following a

3. METHODOLOGY AND MODEL FORMULATON 3.1. Methodology Framework. The framework of the proposed methodology is presented in Figure 1. Overall, two synthesis stages constitute into the whole methodology, the Flexible heat exchanger network synthesis at Stage I, followed by the Simultaneous optimization for heat exchange areas and cleaning schedule at Stage II. As indicated, there are three major steps in the methods of both the stages, and the model-based optimizations are both based on internal iterations. Here it is 12126


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Industrial & Engineering Chemistry Research worth mentioning that qualified flexible HEN solutions are available just with the sole execution of Stage I. However, the generated network would be oversynthesized for some (most) of the operating period, resulting in high expenditure on heat exchanger investment, since the progressive growth of deposit would lead to redundant heat exchange areas in even units, such that in Stage II, we take the cleaning schedule of heat exchangers into account, seeking for a better management of the system. As indicated in the framework, flexibility analysis is an important step (Step I-3 and Step II-3) in both synthesis stages, so at the beginning of this study, the computing method developed for the flexibility index by Li et al.24 is adopted and improved by importing new constraints to present the variation resulted from deposit growth, thus to perform the all-cycle flexibility analysis covering the concerned operating cycle. In this procedure, the effect of deposit growth on heat transfer is reflected in terms of thermal resistance in the heat exchange area calculation, by resulting in an impact on the heat transfer coefficient. On the basis of all-cycle flexibility analysis, the two stages are implemented according to the proposed routes. The method, procedure, and mathematical model for each mentioned part are detailed in subsequent sections: Section 3.2 is for all-cycle flexibility analysis, Section 3.3 presents the method for Stage I flexible HEN synthesis, and Section 3.4 shows the method for Stage II simultaneous optimization for heat exchange areas and the cleaning schedule. 3.2. All-Cycle Flexibility Analysis and Mathematical Model. The procedure of flexibility analysis is shown on the right side of Figure 1. In flexibility analysis theory, flexibility index F is a most widely used indicator for evaluating flexibility degree of an existent network or an initial network obtained via any HEN synthesis method. Flexibility index F is defined as the maximum scaled deviation of that expected in positive and negative directions of fluctuant parameters, according to which the critical points are identified as the worst-case scenarios that hinder the design to accommodate the whole variation space. So as indicated, flexibility analysis is an optimization problem. It has been proved that the feasible region of design is convex if uncertain parameters refer merely to source−stream temperatures (constant heat capacity flow rates and heat transfer coefficients, etc.), and the corresponding critical points must locate at the vertices of the hyper-rectangle as shown in Figure 2. Nevertheless, most chemical process problems are nonconvex, leading to the critical points being on any side of the hyperrectangle, as Figure 3 indicates. In this study, the growth of deposit is newly involved in addition to the conventional uncertain parameters such as

Figure 3. Critical point in the nonconvex feasible region.

stream flow rates and temperatures, contributing to the nonconvexity of the flexible analysis problem on account of the coupling effect among these variations. Li et al.24 have developed a method to deal with the nonconvexity problem in flexibility analysis, however, only limited to the fluctuations of flow rates and temperatures of inlet streams. Deposit growth will cause sustained reduction to the overall heat transfer coefficients, so this paper reformulates the mathematical formula presented in Li et al.24 by introducing area inequality constraints, such that to make it capable of handling the more complicated flexibility analysis problem concerned by this study. The detailed explanation of the original theory can be founded in Li et al.;24 nevertheless, the fundamental content of the model and the modified mathematical formulas for flexibility index (F) calculation are given as follows: F = min δ Dj j

(1)

Dj

δ = max δ δ ,z

h(z , x , θ ) = 0

s.t.

(2) (3)

g (z , x , θ ) ≤ 0

(4)

a(z , x , r(t ), θ ) − a N ≤ 0

(5)

θ = θ N + δDj × Δθ

(6)

δDj

Wherein represents the maximum deviation allowed in direction Dj, h indicates the equality constraints for mass balances and heat balances, g represents the inequality constraints related to conventional uncertain parameters (flow rate, temperature, etc.), such as heat transfer temperature difference constraints, heat load constraints, etc., x indicates the state variables that could be deduced from other variables, z are the control variables, commonly referring to heat load of coolers and heaters, and θ implies the uncertain parameters that vary between lower and upper bounds, Δθ+ and Δθ−. The impact of deposit growth on heat exchange is reflected through fouling resistance r(t) in heat exchange area formula a, requiring the time-related heat exchange area a to be no more than its initial value aN as eq 5 shows. As most deposits in petrochemical production grow asymptotically with time, a universal asymptotic fouling formation model34 displayed in eq 7 is adopted to describe how resultant thermal resistance r varies with time t. −t/ τ r (t ) = r ∞ ) f (1 − e

Figure 2. Critical point in the convex feasible region. 12127

(7) DOI: 10.1021/acs.iecr.9b01672 Ind. Eng. Chem. Res. 2019, 58, 12124−12136

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Figure 4. Sketch of network improvement: (a) initial HEN and (b) network superstructure.

in which r∞ f , the asymptotic fouling resistance, and τ, the time constant, are both empirical parameters. Thus, apparently, the flexibility analysis considering deposit growth should be implemented along with the course of time, and it should be an all-cycle flexibility analysis. We decompose the entire operating horizon into multiple short time periods, use the terminal moment of each to present thermal resistance (deposit condition) of the period with eq 7, and then execute flexibility analysis to judge the flexibility degree of the system. In this study, flexibility analysis of an existent HEN is executed day by day for accuracy, while of course the other step size is permitted. The calculation procedure is given in the right part of Figure 1, wherein the random algorithm of Parallel Simulated Annealing (PSA) is employed to determine the feasibility index and critical points, due to the computation difficulty resulted from nonconvexity of the problem. In the calculation along time, feasibility index F ≥ 1 means the HEN is able to operate satisfactorily over the entire disturbance range; otherwise, the critical point(s) indicating the worst operating condition(s) is (are) determined by searching critical directions according to the minimum−maximum presentation of allowed deviation shown in eqs 1 and 2. In this study, each critical point is a collection of the information on all uncertainty parameters, including flow rate, temperature, and thermal resistance of deposit. The identified points are important for flexible HEN synthesis because once their operations are satisfied by improving existing HEN, the updated network is flexibility qualified. 3.3. Flexible HEN Synthesis and Mathematical Model. To ensure the HEN instinctive ability of resisting uncertainty, this paper proposes a flexible HEN synthesis method consisting of an all-cycle flexibility analysis step and network improvement step. A stage-wise superstructure-based MINLP model is formulated to obtain HEN solutions with simultaneous achievement of desired flexibility and cost target, the minimum TAC. Therein, flexibility of the resulting network is satisfied by increasing the margin area of the original units in priority and then deploying new heat exchangers, heaters, and/or coolers. The synthesis procedure is presented in Stage I of Figure 1, mainly including three steps: Step I-1: Initial HEN Synthesis. For retrofit purposes, the initial HEN should be existent in operation, while for design purposes, it should be provided by designers. In this study, the MINLP model proposed by Yee and Grossmann35 for single period HEN synthesis is adopted to design the initial network with breaking the isothermal mixing assumption. The solution

requiring the lowest TAC is expected at nominal values of all uncertain parameters and no deposit. Taking Figure 4a, for example, EX(i, j, k) represents the matching between hot stream i and cold stream j at stage k, and through optimization accordingly, the identified units EX(2,2,1), EX(1,1,2), EX(2,1,2), and CU1 constitute the initial network configuration. Step I-2: Flexibility Analysis. Calculate flexibility index F of initial HEN using the proposed all-cycle flexibility analysis method. Here, considering the increasing of the deposit, the flexibility index F is calculated for each of the days that comprise an operational period. The model composed by eqs 1−6 is employed for the determination. Therefore, we can achieve a set of F along a timeline within which the minimum value, Fmin, is picked out to feature the flexibility of the network. If Fmin < 1, identify the critical points with their operating information because these points are the cruxes for flexibility unsatisfaction, and then go to Step I-3 for further network improvement; otherwise, Fmin ≥ 1 implies the network is capable of handling the concerned fluctuations, and the procedure of the method stops. Step I-3: Flexible HEN Synthesis. Retain the information on the initial network (stream pairs of matches and heat exchanger areas), and use the data (temperatures, flow rates, and thermal resistances) of critical points identified in Step I-2 to improve the network structure by increasing margined degree of existent unit areas in priority, for the sake of reducing complexity. However, if any in-feasibility is observed against the constraints, new heat exchangers, heaters, and/or coolers are allowed to be added into network. This is a complementary strategy in case of the initial network is deficient or facing severe variations. Its applicational embodiment would be case dependent, initial network dependent and fluctuation dependent. The potential locations of new heat exchange devices are implied by dotted lines in the stagewise superstructure presentation35 of Figure 4b. This step is an optimization process, which forwards to a HEN holding qualified flexibility against arbitrary fluctuations of operation, meanwhile reaching the best economic expectation. The flexible HEN synthesis under uncertainties can be generally formulated into the following mathematical model. The economic objective of minimum TAC consists of two expense items, the total annualized capital cost (TACC) regarding the use of heat exchangers, heaters, and coolers, and the total annualized operating cost (TAOC) for the consumption of hot and cold utilities, as indicated in eqs 8−11.

min TAC 12128

(8) DOI: 10.1021/acs.iecr.9b01672 Ind. Eng. Chem. Res. 2019, 58, 12124−12136

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determining whether the network is flexibility satisfiable only by increasing the margined areas of original units. So the units taken into calculation are identical with those servicing in initial HEN. Area calculations for auxiliary coolers at the outlet stage of hot streams are performed as follows:

(9)

TAC = TACC + TAOC fix ∑ ∑ ∑ z(i , j , k) TACC = CHE i

+

M CHE

j

∑ ∑ ∑ z (i , j , k ) · A (i , j , k ) β i

+

fix CHU

k

j

∑ zHU(j) +

M CHU

∑ zHU(j)·AHU (j)

j

ACU (i) =

β

i

(18)

qCU (i) ≤ zCU (i) ·q M , i ∈ NH

i

TAOC = CHU ∑ zHU (j)·qHU (j) + CCU ∑ zCU (i)·qCU (i)

(20)

i

LMTDCU (i) =

(11)

out in [Th(i , NOK + 1) − TCU ] − [Thout (i) − TCU ] , i ∈ NH out out in ln[[Th(i , NOK + 1) − TCU ] /[Th (i) − TCU ]]

Wherein subscripts HE, CU, and HU stand for heat exchangers and cold and hot utilities, respectively. The general calculation formula for capital cost can be expressed as Cf ix + CM·Aβ, where the first term Cfix represents fixed installation cost, the second term represents heat transfer area cost, and CM, A, and β indicate capital cost parameter, heat transfer area, and capital cost area exponent, respectively. CCU and CHU are unit cost parameters of utilities, and q represents heat load of utilities. Binary variable z is used to indicate the existence of a heat exchange match and the unit, where if a match exists if a match doesn’t exist

(21) N ACU (i) ≥ ACU (i), i ∈ NH

AHU (j) =

(12)

LMTD(j) =

(23) (24)

out in [THU − Tc(j , 1)] − [THU − Tc out(j)] , j ∈ NC out in ln[[THU − Tc(j , 1)] /[THU − Tc out(j)]]

(26) N AHU (j) ≥ AHU (j), j ∈ NC

(27)

(ii) Stream outlet temperature constraints: |Thout (i) − Thout , N (i)| ≤ ζ , i ∈ NH

(28)

|Tc out(j) − Tc out , N (j)| ≤ ζ , j ∈ NC

(13)

out

out

out,N

(29) out,N

In which Th (i), Tc (j) and Th (i), Tc (j) denote the output temperatures at critical condition and nominal condition, respectively, to ensure the solutions are feasible for implementation. Within this optimization, if no feasible solution can be achieved merely by increasing the heat exchange area, new exchangers should be allowed to join in by relieving the strict match constraints on z in the model. 3.4. Simultaneous Optimization for Heat Exchange Areas and Cleaning Schedule. In order to perform all heat exchange duties in uncertain conditions, the areas of heat exchangers and heaters/coolers obtained in Section 3.3 must be large enough. There is no doubt that the generated networks are feasible and flexible for operation; however, the heat exchanger area would be redundant because thermal resistance increases with deposit. Reflected in the flexibility index, the situation would be Fmin ≥ 1 for entire operation cycle, but F ≫ 1 for most of time. So in association with the definition of the flexibility index, it is deduced that the heat transfer areas must be

(14)

(15)

[Th(i , k) − Tc(j , k)] − [Th(i , k + 1) − Tc(j , k + 1)] , i ∈ NH , j ln[[Th(i , k) − Tc(j , k)] /[Th(i , k + 1) − Tc(j , k + 1)]]

A(i , j , k) ≥ AN (i , j , k)i ∈ NH , j ∈ NC , k ∈ NK

, j ∈ NC

(25)

LMTD(i , j , k) =

∈ NC , k ∈ NK

UHU (j) ·LMTDHU (j)

1 1 1 critical , j ∈ NC = + + r critical(j) + rHU UHU (j) h(j) hHU

1 1 1 = + + r critical(i) + r critical(j), i ∈ NH , j U (i , j) h(i) h(j) ∈ NC , k ∈ NK

qHU (j)

qHU (j) ≤ zHU (j) ·q M , j ∈ NC

q(i , j , k) A (i , j , k ) = , i ∈ NH , j ∈ NC , k U (i , j) ·LMTD(i , j , k)

q(i , j , k) ≤ z(i , j , k) ·q M , i ∈ NH , j ∈ NC , k ∈ Nk

(22)

And for auxiliary heaters:

At this step, the heat exchange matches in the initial network are enforced in utilization, so the zs for these units are set to be “1” before model implementation. The commonly used stage-wise superstructure-based model is modified for the HEN synthesis in this step. The regular constraints are shown in the Supporting Information, and the additions and modifications are detailed as below. (i) Heat transfer area constraints: Heat exchanger areas are obtained as follows:

∈ NK

(19)

1 1 1 critical , i ∈ NH = + + r critical(i) + rCU UCU (i) h(i) hCU

(10)

j

, i ∈ NH

UCU (i) ·LMTDCU (i)

j

fix M ∑ zCU(i) + CCU ∑ zCU(i)·ACU (i)β + CCU

l o1 z=m o0 n

qCU (i)

k

(16) (17)

Wherein qM is a large heat load constant used for big-M presentation for heat load of each exchanger and A and AN denote the heat transfer areas at critical condition and nominal condition, respectively. The adverse influence of deposit on flexibility is reflected via these constraints, with the purpose of 12129


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Industrial & Engineering Chemistry Research overdesigned if F > 1, leaving space for further reduction as well as the capital cost. Periodic cleaning is an effective strategy for deposit removal; as a result, the required heat exchange areas would reduce because of the restored thermal performance of heat exchangers with cleaning. Figure 5 shows the variation profile of thermal

Step II-2: Calculate the frequency of cleaning with formula Ncl = tend/Tcy. Once the network operation completes a cleaning cycle, the heat transfer units should be removed for off-line cleaning. Step II-3: Implement simultaneous optimization. The iterative procedure is executed by progressively lengthening the cleaning cycle Tcy, for example, by 1 day for each calculation. Meanwhile within each calculation, heat transfer areas are optimized to guarantee the minimum flexibility index Fmin = 1. So, the all-cycle flexibility analysis method is also employed in this step but for each cleaning cycle rather than the entire procedure. Objective of the optimization remains to pursue a qualified network costing the minimum TAC, including (i) TACC, the capital cost for heat exchanger installation and area; (ii) TAOC, detailed in eq 31), the annual utility cost aggregated from the daily (or other time step) utility usage, where the integral form is used to present the utility cost summation of each day in the entire operation horizon and NY is the operation horizon in yearly unit; and (iii) TACLC, detailed in eq 32), the annual cleaning cost regarding the total cleaning times Ncl,n and operation horizon in the yearly unit, NY. Among the solutions generated from the lengthening of the cleaning cycle, the one (including areas and cleaning schedule) requesting the lowest cost is picked out to be the output.

Figure 5. Diagram of periodic cleaning.

TAC = TACC + TAOC + TACLC ÄÅ ÅÅ Å TAOC = ÅÅÅÅCHU ∑ zHU (j) ·qHU (j , t ) ÅÅ j ÅÅÇ ÉÑ ÑÑ Ñ + CCU ∑ zCU (i) ·qCU (i , t )ÑÑÑÑ dt /NY ÑÑ i ÑÑÖ

resistance along with time by adopting a universal asymptotic fouling formation model, where the solid line represents no cleaning operation and the dotted line having a periodic cleaning operation. In a whole operation cycle, tend, as indicated, the network cost will increase if the cleaning cycle is lengthened from Tcy to Tcy′, because of the positive impact of higher thermal resistance on capital cost and utility cost; however, the cleaning cost will decrease because of fewer cleaning times. So in the second part of the methodology, the simultaneous optimization of heat exchange areas and cleaning schedule is carried out to improve the management of the network by making trade-offs among TACC, TAOC, and total annualized cleaning cost (TACLC). The optimization of the cleaning schedule refers to determining when and which heat transfer unit should be cleaned along the operation horizon, while the optimization of areas accounts for how small of a size the exchanger could reach. To simplify the optimization process, the following assumptions are applied: (i) Normally, there is strong correlation effect among the heat exchange units in HEN, and any shutdown of single unit for cleaning will seriously undermine the network flexibility. Hence, it is assumed that all heat transfer units are removed off-line for cleaning at the terminal of each cleaning cycle. This assumption is set to highlight the thought of proposed method rather than to increase complexity of the problem. The constraints can be released by involving more variables into model to optimize unique cleaning operation for each exchanger. (ii) Areas of heat exchangers, heaters and coolers are decreased by the same magnitude in optimization. The optimization is implemented in an iterative procedure, illustrated in Part II of Figure 1. Step II-1: Obtain a flexible HEN without cleaning action considered, e.g., the flexible network solution achieved in Stage1. Hold the network configuration, and keep the original area An of each heat exchanger, heater, and cooler as the initial value for the optimization. Set the duration of initial cleaning cycle Tcy (Tcy ≤ tend), and then the desired cycle is optimized with subsequent steps by gradually lengthening the cycle duration.

(30)

∫

ij j TACLA = Ccl·jjj∑ ∑ ∑ z(i , j , k) ·A(i , j , k) jj k i j k +

(31)

yz

∑ zHU(j)·AHU (j) + ∑ zCU(i)·ACU (i)zzzzz· j

Ncl /NY

i

z {

(32)

3.5. Implementation of the Models. As mentioned, the HEN synthesis for network construction and the flexibility analysis for network evaluation are both optimization problems. In this study, the solutions of HEN synthesis steps (Step I-1, Step I-3) are found by handling the models in the General Algebraic Modeling System (GAMS) with solver BARON,36 and the models of flexibility analysis (Step I-2) and area optimization on that basis (Step II-3) are coded with the PSA algorithm in C++ language.

4. CASE STUDY Two cases taken from literature are studied to illustrate the application of the proposed methodology. Case 1 focuses on producing a flexible HEN without involving a cleaning operation, while in Case 2, the two parts presented in the methodology framework are both performed. Case 1: Flexible HEN Synthesis without Cleaning Operation. This flexible HEN synthesis case consists of three hot process streams, three cold process streams, and one cold and one hot utility. The basic process data are absorbed from 12130


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Figure 6. Initial HEN structure at nominal condition for Case 1.

Figure 7. Flexibility index variation for initial HEN over operating horizon for Case 1.

trend of the flexibility index, but the locational variation of the touch point is not deterministic, depending on the shape of the feasible region, which is fickle, causing small distributions on the profile. In Figure 7, the lowest flexibility index along the time horizon is obtained at 0.02, implying the failure of initial HEN at some possible conditions within variation ranges; thus, network improvement (Step I-3) is further required to enhance the network capacity against fluctuations. In Step I-2, critical points are determined in company with the calculation of flexibility index. With the element sequence (Thin1 , Thin3 , Tcin1 , Tcin2 , r(t)), in which thermal resistance collection r(t) refers to (rH1(t), rH2(t), rH3(t), rC1(t), rC2(t), rC3(t), rHU(t), rCU(t)) in detail, two critical points are identified at the 683rd day and the 696th day, as the parameter groups are (456.65, 558.15, 393.78, 327.66, 0.267, 0.306, 0.369, 0.250, 0.328, 0.291, 0.0680, 0.150) and (453.18, 558.15, 392.25, 323.93, 0.269, 0.309, 0.372, 0.252, 0.331, 0.294, 0.0687, 0.151). Based on the critical point results, some insights over uncertain parameters can be summarized. First as we known, the feasible region is convex, and the critical directions should be located at upper/lower values of the variation ranges if merely stream inlet temperatures fluctuate. However, due to the involvement of deposit growth, the temperature results of critical points are collected at (456.65, 558.15, 393.78, 327.66) and (453.18, 558.15, 392.25, 323.93) rather than the boundaries. Second, although the terminal moment of the operation horizon was commonly taken as the worst case in

Xiao et al.,28 with the nominal operating parameters of process streams given in Table S1. The uncertain parameters taken into account by this case are the inlet temperatures of streams H1, H3, C1, and C2, varying within the range of ±5 K, where accordingly, the variations of the parameters are summarized in Table S2. Other information includes the following: fixed investment cost of the heat transfer unit is calculated by unified formula COST = 8000 + 1000A0.8 $·y−1; unit costs of hot and cold utilities are 70 and 10 $·kW−1·y−1, respectively; the concerned operating cycle is 2 years (720 days); the minimum heat transfer temperature difference ΔTmin is set to be 10 K. Following the procedure of Stage I, the initial HEN shown in Figure 6a is achieved under nominal operating conditions toward minimum TAC target at first. Then at Step I-2, all-cycle flexibility analysis is implemented to evaluate this network. The flexibility index is daily calculated via the model presented in Section 3.2. Figure 7 shows the flexibility index profile shaped by the result of each day. As indicated, the flexibility index throughout the entire operation horizon presents a downward trend in general; meanwhile, local fluctuations are observable on the profile. This appearance is the result of coupling effects between parametric disturbances and deposit growth. Like that indicated in Figure 3 for the nonconvex problem, the flexibility index is determined by the first touch point of the uncertainty region and the feasible region of a network. The feasible region of a given HEN will vary with the fouling condition, which in general shrinks with deposit growth, leading to the downward 12131


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263791 $·y−1, 513004 $·y−1, and 776796 $·y−1, respectively. Figure 9 shows the profile of flexibility index along time, wherein the minimum flexibility index Fmin is obtained at 1.06, close to 1, indicating the network adequate capacity of accommodating fluctuations. The “1” approximate flexibility index implies the fitness of the result, and no further endeavor is required to retrofit the network. However, in some cases, the flexibility index of “1” is hard to closely approach, and network overdesign is inevitable. Case 2: Flexible HEN Synthesis Considering Cleaning Management. Process stream data of the second example is still from Xiao et al.28 Two hot and two cold process streams and one hot and one cold utility are given with their nominal operating condition tabulated in Table S3. In addition to inlet temperatures, fluctuations of heat capacity flow rates are also taken into account by this case. The uncertain parameter data are collected in Table S4, varying in ranges of ±10K and ±0.5 kW·K−1. The unit costs of hot and cold utilities are 80 and 20 $· kW−1·y−1, respectively, and the cleaning cost is 10 A0.7$·y−1·per time. Other data, such as operation horizon, minimum heat transfer temperature difference, and capital cost formula are identical with Case 1. Figure 10 shows the initial HEN solution, wherein 5 heat exchange units are employed to accomplish the heat exchange task at nominal conditions. Implementing flexibility analysis to initial HEN, the minimum flexibility index over the entire cycle is received at 0.13, implying inflexibility of the initial HEN to resist uncertainties, and therefore, further improvement is demanded. Stipulating the critical point in the element sequence (Thin1 , Tcin2 , FCPh1, FCPc2, r(t)), in which r(t) refers to r(t) = (rH1(t), rH2(t), rC1(t), rC2(t), rHU(t), rCU(t)) in detail, two critical points are identified within flexibility analysis at (546.73, 336.40, 15.50, 15.34, 0.199, 0.278, 0.245, 0.306, 0.0662, 0.146) and (563.15, 333.25, 15.22, 14.67, 0.204, 0.285, 0.252, 0.314, 0.0684, 0.151), and the positions are shown in Figure 11. Then at the flexible HEN synthesis step, the resulting network achieved against critical points is presented in Figure 12. The allcycle flexibility indexes are profiled in Figure 13 with the minimum flexibility index Fmin found at 1.19, indicating that the improved network has an overdesigned margin to tolerate the process uncertainties. Therefore, in order to lessen the redundancy of the heat transfer area, periodic cleaning is considered for further optimization. From the flexibility synthesis result shown in Figure 12, we collect a group of initial areas of (104.1, 57.1, 112.8 52.7, 103.4) in sequence of (EX(1,1,1), EX(1,2,2), EX(2,1,2), CU2, HU1). These areas will be optimized simultaneously with the periodic cleaning schedule. The optimization objective consists of two parts: (i) the minimum flexibility index Fmin = 1 in each cleaning

previous studies, the critical points of this case are picked out at t = 683 day and t = 696 day rather than the last day. These appearances indicate that the synergistic effect of parametric fluctuation and growth of deposit results in nonlinearity of the constraints and nonconvexity of the feasible region of the problem. Afterward, the flexible HEN synthesis approach proposed in Step I-3 is implemented, aiming at an improved network solution with qualified-flexibility (Fmin ≥ 1) and minimum TAC. The information on critical points is used for synthesis. The increment of original heat transfer area and the addition of heat exchangers, heaters, and/or coolers are regarded as optimization variables in the modeling of flexible HEN design. Figure 8 shows the final network for this case, without a new heat exchange device added into the initial network structure.

Figure 8. Final flexible HEN structure for Case 1.

Cost composition of the final network solution is given in Table 1. The annual capital cost, operation cost, and the total are Table 1. Cost Composition for Case 1 area (m2)

unit EX(2,1,1) EX(1,3,2) EX(3,2,2) CU1 CU2 HU1 HU2 HU3 total TAC ($·y−1)

121.420 149.506 90.384 32.675 93.683 67.592 18.951 36.885 611.10

TACC ($·y−1)

TAOC ($·y−1)

Q (kW)

45198.22 51935.63 37374.01 21015.55 824.40 38228.64 2656.90 31281.25 3806.90 16417.74 1350.00 22340.68 1674.40 263791.72 776796

8244.0 26569.0 266483 94500 117208 513004

Table 2. Comparison of Costs before optimization 2

after optimization −1

2

unit

area (m )

TACC ($·y )

unit

area (m )

EX(1,1,1) EX(1,2,2) EX(2,1,2) CU2 HU1 Total TAOC ($·y−1) TAC ($·y−1)

104.1 57.1 112.8 52.7 103.4 430.1

49111.2 33427.8 51837.7 31847.8 48889.9 215114.4

EX(1,1,1) EX(1,2,2) EX(2,1,2) CU2 HU1 Total TAOC ($·y−1) TAC ($·y−1)

97.8 50.8 106.5 46.5 97.1 398.7

214549.1 429663.5

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TACC ($·y−1) 47108.5 31157.5 49867.8 29575.6 46884.4 204593.8 209400.3 418230.9

TACLC ($·y−1) 989.2 625.4 1050.0 587.9 984.3 4236.8


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Figure 9. Flexibility index variation for the final HEN over operating horizon for Case 1.

Figure 12. Flexible HEN structure for Case 2. Figure 10. HEN structure at nominal condition for Case 2.

the savings of utility even involving the cleaning expenditure; thus, 2.5% reduction of TAC is finally achieved, demonstrating the virtue of considering the cleaning problem in flexible HEN design.

cycle Tcy and (ii) the minimum TAC over the whole operation horizon. The iterative process proposed in Stage II is applied to generate results by progressively extending the cleaning cycle (by 1 day) from the 30th day. The curve in Figure 14 depicts the variation of TAC over the cleaning cycle where the minimum TAC is positioned at 378230.9 $·y−1 when the cleaning cycle Tcy is 145 days. As such, four cleaning times are required in the 720day operating cycle, the 145th, 290th, 435th, and 580th day. The variation of the flexibility index after simultaneous optimization is depicted in Figure 15. It is observed that the minimum flexibility index Fmin in each cleaning cycle is equal to “1”, guaranteeing the solution sufficient process flexibility. The detailed costs before and after the optimization are listed in Table 2, wherein the proposed method produces 4.9% reduction for capital investment cost because of the lower requirement of heat exchanger area and 2.4% operating cost reduction due to

5. CONCLUSIONS This article proposes a new optimization-based framework to perform the synthesis of flexible heat exchanger networks. Two kinds of uncertainties, the fluctuating operation parameters (varying in known ranges) and the gradually accumulated deposits (increasing with time) are simultaneously considered for the first time in the determination of flexible HENs. At the beginning of the study, an all-cycle flexible analysis method is developed from its classical theory, helping to evaluate the capacity of a heat exchange system against uncertainties over an entire operating period. On this basis, the proposed framework is composed of two consecutive parts: (Part I) Flexible HEN

Figure 11. Flexibility index variation for initial HEN over operating horizon for Case 2. 12133


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Figure 13. Flexibility index variation for flexible HEN over operating horizon for Case 2.

Two examples are adopted to illustrate the procedure of the proposed methodology. The fluctuating operation parameters in case 1 refer only to inlet temperatures, while in case 2 flow rates are included. Within the Part I procedure, flexibility indexes of network solutions are improved from 0.02 to 1.06 and 0.02 to 1.19 for the cases, respectively, demonstrating the capability of the method to accommodate not only nominal condition but also arbitrary parametric fluctuations accompanied with gradual growth of deposits. Furthermore, a 2.5% cost savings of the final network solution is achieved by Case 2 with the Part II procedure, showing us the advantage of considering cleaning operation for heat exchange area reduction as well as cost reduction concerns. In the future, the two assumptions stipulated in Part II will be released to investigate the unique cleaning schedule of each heat exchange unit, and more effective solving algorithms such as the decomposition strategy will be studied to extend the method to a larger scale problem.

Figure 14. Diagram of TAC variation.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b01672. Mathematical model for HEN synthesis and Tables S1− S4 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(J.D.) Tel.: +86 411 84986301. Fax: +86-411-84986201. Email: [email protected]. ORCID

Lei Zhang: 0000-0002-7519-2858 Jian Du: 0000-0001-7667-4835

Figure 15. Variation of flexibility index.

Notes

The authors declare no competing financial interest.

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synthesis, in which the steps of initial HEN synthesis, flexibility analysis, and flexible HEN synthesis constitute the whole procedure. Strategies of increasing areas of existent heat exchange units and involving new units are both applied for network improvement. (Part II) Simultaneous optimization of heat exchange areas and cleaning schedule, aiming at a more economical flexible HEN. This part is launched to deal with the overdesign problem resulted from the stepwise nature of flexible HEN synthesis, which is very common in flexible HEN synthesis study but normally ignored. We proposed an iteration-based method to lower the heat exchange areas by integrating periodic cleaning into optimization, such as to make trade-offs among capital cost, utility cost, and cleaning cost regarding the flexibility requirement of the system.

ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from Natural Science Foundation of China (No. 21878034 and 21776035) and the Fundamental Research Fund for Central Universities of China (DUT18LAB11).

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NOMENCLATURE A heat exchange area for process stream exchanger, m2 ACU heat exchange area for cooler, m2 AHU heat exchange area for heater, m2 CCU unit price for cold utility, $·kW−1·y−1 CHU unit price for hot utility, $·kW−1·y−1 fix CCU fixed installation cost for cooler, $·y−1 12134


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Industrial & Engineering Chemistry Research ix CfHE f ix CHU CM CU CM HE CM HU Ccl D d dt F FCp h LTMD NOK q/Q qM r r∞ f Th Tc TAC TACC TAOC TACLC tend Tcy ΔTmin t U x z

δ τ Δθ+ Δθ− θ β

fixed installation cost for heat exchanger, $·y−1 fixed installation cost for heater $·y−1 area cost coefficient for cooler, $·y−1 area cost coefficient for heat exchanger, $·y−1 area cost coefficient for heater, $·y−1 unit cost for cleaning, m2·$−1 direction matrix design variable heat transfer temperature difference flexibility index heat capacity flow rate, kW·K−1 heat transfer film coefficient, kW·m−2·K−1 Log heat transfer temperature difference, K the number of stages in the superstructure heat load, kW heat load constant with large value, kW fouling resistance of stream, m2·K·kW−1 asymptotic fouling resistance of stream, m2·K·kW−1 temperature of hot stream, K temperature of cold stream, K total annulized cost, $·y−1 total annulized capital cost, $·y−1 total annulized operating cost, $·y−1 total annulized cleaning cost, $·y−1 Entire operating horizon, day cleaning cycle, day minimum heat transfer temperature difference, K time, day overall heat transfer coefficient, W·m−2·K−1 state variable binary variable indicating a heat exchanger/cooler/ heater existent or not maximum scaled deviation asymptotic fouling rate model parameter, day the upper value of the expected deviation the lower value of the expected deviation uncertain operating parameter exponent for area cost

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Superscripts

critical in N out

critical point inlet nomial point outlet

Subscripts

CU HE HU i j k min

cold utility or heater using cold utility heat exchanger for two process stream hot utility or heater using hot utility hot stream cold stream stage number in HEN superstructure minimum value

Sets

NH set of cold streams NH set of hot streams NK set of the stages of the superstructure

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REFERENCES

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