An Enhanced Stage-wise Superstructure for Heat Exchanger

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An enhanced stage-wise superstructure for heat exchanger networks synthesis with new options for heaters and coolers placement L. V. Pavão, Caliane Bastos Borba Costa, and Mauro Ravagnani Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03336 • Publication Date (Web): 12 Jan 2018 Downloaded from http://pubs.acs.org on January 12, 2018

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Industrial & Engineering Chemistry Research

An enhanced stage-wise superstructure for heat exchanger networks synthesis with new options for heaters and coolers placement Leandro V. Pavão, Caliane B. B. Costa, Mauro A. S. S. Ravagnani1 Department of Chemical Engineering, State University of Maringá Av. Colombo, 5790, Bloco D90, CEP 87020900, Maringá, PR, Brazil Abstract Several methods for heat exchanger networks (HEN) design are based on the use of superstructures. The models they give rise to can lead to different design options to be explored in HEN synthesis. In this work, a stage-wise superstructure with new features for the optimal placement of heaters and coolers, including the possibility of the use of multiple utilities, is presented. In the model, those units can be placed in different stream split branches in all stages, differing from the usual allocation at stream ends. Such possibilities yield a more complex mathematical model. To solve it, an improved version of a previously presented hybrid meta-heuristic method was used. Three examples from the literature were studied. The use of the superstructure with the additional utility-related options, as well as the enhanced meta-heuristic solution method, led to configurations with lower associated total annual costs than those reported in previous works. Keywords: optimization; heat exchanger networks; meta-heuristics; mathematical modeling; process synthesis

1

Introduction

Mathematical programming is fundamental in numerous approaches for heat exchanger networks (HEN) synthesis. Many formulations are derived from superstructures. These comprise possible matching and piping options for yielding cost-optimal HEN when mathematical optimization models derived from them

1

Corresponding author. Tel: +55 (44) 3011-4774, Fax: +55 (44) 3011-4793

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are solved. Fundamental works which presented essential concepts for the further development of such approaches must be highlighted. The superstructure of Floudas et al.1 entailed, for a set of matches selected a priori with a minimum number of matches criterion, aspects such as cross-flow and by-passing. Such complex features led to a thorough model, which required the use of a sequential mathematical formulation for facilitating solutions obtaining. Final HEN configurations were achieved after the solution of three optimization problems: linear programming (LP), mixed-integer linear programming (MILP) and non-linear programming (NLP), which aimed at, respectively, finding the minimal utilities requirement for the HEN, defining the matches that should be present and the heat exchanger area. That superstructure was further revamped to the hyperstructure2, where all possible matches were present .The hyperstructure originated a mixed-integer nonlinear programming (MINLP) formulation, able to find feasible solutions in a single-step optimization problem. In that work, the authors also evaluated flexibility features of the HENs synthesized. A model which became rather popular in the literature given its simplicity is the stage-wise superstructure (SWS) of Yee and Grossmann.3 The SWS is certainly a major contribution and serves as basis for the development of new models to this day. Structurally uncomplicated and with coherent simplifications, the model derived from it was able to efficiently lead to near-optimal solutions, despite not having all the structural options from the super and hyperstructure models of Floudas and co-workers.1,2 Within each stage of the SWS, all matches between hot and cold stream were possible in different split stream branches. Heaters and coolers were placed respectively at cold and hot streams ends. The original mathematical model developed from the SWS by Yee and Grossmann3 also assumed simplifications such as the isothermal mixing in order to avoid nonlinear energy balances, making the model less complex to solve. Later, several models have been proposed, many based on the hyperstructure and the SWS and their derived mathematical formulations. Zamora and Grossmann4 presented a model based on the SWS with constraints forbidding stream splits for yielding HENs structurally simpler. The work of Lewin et al.5 presented an incidence matrix as a form of representing HEN structure that could straightforwardly 2 ACS Paragon Plus Environment

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encode solutions for optimization with Genetic Algorithms. The study was presented in two parts. In the first part,5 the incidence matrix comprised no stream splits, whereas in the second one6 such feature was present. It is worth mentioning that Lewin’s6 model did not assume isothermal mixing. Considering only models derived from the SWS, the first one that did not assume isothermal mixing was proposed by Björk and Westerlund.7 Their strategy was to create approximate convexified subproblems, facilitating the achievement of near-optimal solutions with the non-isothermal consideration. Ravagnani et al.8 used a two-row matrix for representing HEN structures. Such data arrangement was well-suited for their Genetic Algorithm approach. Each column represented a heat exchanger, while the first and second rows described the hot and cold streams passing through that piece of equipment. Isafiade and Fraser9 presented a superstructure based on temperature intervals from which an MINLP formulation was derived. Their model also comprised the allocation of utilities in a stream split branch and was also adaptable to the use of multiple utilities by considering those as process streams. The temperature intervals used allowed, in some cases, that heaters/coolers were placed before heat exchangers in the HEN structure. In contrast, Yee and Grossmann’s3 SWS allows the use of utilities only at streams outlet. Toffolo10 used a graph representation. Structural modifications were performed during the optimization procedure, and the final HEN topology could present features such as serial heat exchangers in a single stream split branch and branches re-splitting. Ponce-Ortega et al.11 also presented a superstructure where heaters/coolers were placed in stream split branches and could be located in intermediate positions. A disjunction within their mathematical model was used to select the optimal utility to be used in problems with multiple utilities. Bogataj and Kravanja12 presented an augmented form of the SWS, where matches present in the SWS were transferred to an aggregated sub-structure. Such sub-structure allowed for the reduction of the number of nonconvex terms in the model. Huo et al.13 presented a flexible form of the SWS. The model used could comprise splits in all stages (with non-isothermal mixing), as well as it could present no splits. In addition, the structure could be configured a priori as a hybrid one, i.e., some stages had splits while others did not. According to those authors, that feature was the main aspect of their model, given that optimal solutions have splits only in few stages, and that the “splitless” model was 3 ACS Paragon Plus Environment


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numerically simpler. In that work, splits forbidding was handled differently from the method of Zamora and Grossmann.4 Instead of constraining the original SWS to have only one unit per stream per stage, all stage units were placed sequentially. In that way, fewer stages could be used. Inspired both by the hyperstructure and the SWS, Huang and Karimi14 developed a superstructure which had stages, but also comprised complex features such as crossflows. He and Cui15 used a “chessboard” representation without stream splitting. In their model, streams order were rearranged so that optimal solutions might be found with a model equivalent to using a single stage in a no-splits SWS. Isafiade et al.16 extended the augmented SWS proposed by Bogataj and Kravanja12 to include multiple utilities in outlet stages, each placed on a branch of a split stream. Peng and Cui17 used a no-splits SWS. Their solution algorithm, however, allowed replacing infeasible matches with utilities. With such features, utilities might be placed in intermediate stages. Zhang et al.18 presented an improved “chessboard” model where more than one stage might be considered. A no-splits SWS model similar to those from Huo et al.13 and Peng and Cui17 was employed by Pavão et al.19 However, in that work, the SWS was only a form of deciding matches that were present, and, once the topology was defined, a reduced mathematical formulation with smaller matrices was used in the continuous optimization to facilitate calculations. Recently, Kim et al.20 presented an extended version of the SWS with sub-stages which allowed the placement of serial heat exchangers in different branches of stream splits. Other aspects in HEN synthesis can be considered in formulations derived from superstructures, especially the SWS. The rigorous design of heat exchangers was included to the HEN synthesis task in the model of Mizutani et al.21 Following, Ravagnani and Caballero22 presented an improved model where parameters such as tube lengths and baffles spacing were also considered as optimization variables and which rigorously followed the tubular exchangers manufacturers association (TEMA) standards. Onishi et al.23 proposed a model for achieving the optimal integration between heat and work with pre-fixed pressure manipulating steps, which was later improved and applied to more complex cases.24 The model was also adapted to handle retrofit cases.25 Further, Onishi and co-workers developed a larger superstructure with several work-exchange options.26 The incorporation of environmental impacts to the 4 ACS Paragon Plus Environment

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HEN synthesis problem was assumed in the works of Pavão et al.27, who handled that objective along with economic aspects via multi-objective optimization (MOO), Mano et al.28, who approached the problem converting environmental impacts into monetary units, and Onishi et al.29, who also employed MOO techniques for assessing environmental and economic performances in work and heat integration. Verheyen and Zhang30 considered uncertain temperatures and flowrates in HEN, presenting a model for multiperiod HEN synthesis based on the SWS of Yee and Grossmann3. Pavão et al.31 employed metaheuristics to solve the multiperiod HEN synthesis problem using the SWS as well, achieving promising results. Jiang and Chang32 used the SWS as basis for a multiperiod HEN synthesis framework with the possibility of changing working fluids in the units depending on the operation period (timesharing mechanism). The method was later improved by Miranda et al.33, who were able to achieve slightly better results. A rearrangement/re-sizing stage was coupled to the timesharing mechanisms32 by Pavão et al.,34 leading to even better results. Besides temperatures and flow rates, other uncertainties such as fluctuations in costs of utilities were also considered in the HEN literature via scenarios evaluation methods35 and the use of stochastic variables and risk metrics.36 Given the wide range of piping and matching arrangement possibilities, there are still options to be covered in superstructure models. Such features increase the difficulties in solving the problem by leading to a greater number of possible combinations and nonlinearities in the formulations they give rise to. Consequently, this work aims at presenting an enhanced version of the SWS where different utilities might be used in all streams and stages and at employing an efficient solution approach to the model generated in order to attain near-optimal solutions. It is expected that the combination of such new structural options along with an adequate solution strategy are able to lead to solutions with lower costs than those achieved by previous approaches. Similar alternatives for the allocation of utilities have been proposed in previous works in the literature. For instance, the model of Ponce-Ortega et al.11 comprised an extra branch of stream split with a utility heat exchanger. The mentioned strategy of Peng and Cui17 of replacing a “process stream to process stream” unit by a utility unit is another example. The extra branch concept of Ponce-Ortega et al.11 certainly inspired the development of the present superstructure to some 5 ACS Paragon Plus Environment


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extent. The novelty in the present work resides in the fact that multiple extra stream split branches are used, one for each type of utility, and an “auxiliary” stage is put at the end of the streams containing only heaters or coolers, similarly to the idea of Isafiade et al.16 Moreover, non-isothermal mixing is assumed in the derived model. Another major motivation for this work is the fact that, although meta-heuristics have proven to be a promising alternative for the HEN synthesis problem, most works using such approaches employ the original SWS or the non-split SWS. Hence, given the results attained by meta-heuristics so far, an attempt to apply such sort of methodology to an extended SWS form may represent a contribution to the field. The hybrid Simulated Annealing (SA) – Rocket Fireworks Optimization (RFO) method presented in Pavão et al.37 was applied to large-scale HEN synthesis cases and achieved promising results. It is worth mentioning that RFO is a combination of an adaptation of Simulated Annealing for continuous spaces (abbreviated as CSA, for Continuous Simulated Annealing) with Particle Swarm Optimization (PSO). The SA-RFO method took great advantage from insights obtained in two previous meta-heuristic based studies, which hybridized Genetic Algorithm (GA) and PSO38 and SA and PSO.19 SA-RFO is here reworked to handle the new variables that arise from the new enhanced SWS. Three industrial-sized case studies are investigated to demonstrate the possibilities of the new formulation, as well as the optimization method efficiency. 2 2.1

Methodology Problem statement

A set of I hot process streams and J cold process streams is given, with known inlet and target temperatures, total heat capacities and heat transfer coefficients. Those streams must reach their target temperatures. Multiple types of utilities with known inlet and outlet temperatures and heat transfer coefficients can be available in a plant, such as steam at different pressures, hot oil, flue gas, air cooling and cooling water. Process streams must reach their goal temperatures by the heat exchange between cold process stream and hot utilities, between hot process stream and cold utilities or between two process streams (one hot and one cold). Such task is performed in heat exchanger units. Economic functions are

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available for calculating the costs of heat exchangers according to their area. Utility costs data for each type of utility are also available. With the provided process information, a combination of heat exchangers performing heat integration among process streams and auxiliary heating/cooling via utilities (i.e., a heat exchanger network) must to be found with minimal total annualized costs (TAC). It means that an optimal trade-off between heat exchanger areas and utility costs must be attained. In order to achieve such objective, an optimization problem is tailored and presented in Section 2.2. 2.2

Mathematical model

The formulation takes advantage from concepts proposed by Yee and Grossmann3 in their stage-wise superstructure (SWS) model. The concept of “stages” in that model was fundamental. In a stage, a process stream could be split into a number of branches corresponding to the number of process streams that it could exchange heat with. That is, for instance, in a problem with two hot and three cold streams, each of the formers are split into three and each of the latter into two. A single heat exchanger is placed in each branch, so that all possible matches are present in a stage. Figure 1 presents a simple case with two hot and two cold streams modeled with the SWS and the continuous variables in accordance with the mathematical formulation employed by Pavão et al.37 Note that serial heat exchangers in a stream are possible if the number of stages is greater than one. In the SWS, heaters and coolers are placed at the end of the streams, which is a simplifying assumption.



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Figure 1. Stage-wise superstructure with variables representation as in the formulation of Pavão et al.37 The enhanced SWS here proposed comprises heaters and coolers at intermediate stages, placed individually in stream split branches. If the problem contains more than one utility type, more split branches are used. Figure 2 presents a case comprising two hot and two cold streams and two types of hot and two types of cold utilities modeled with the enhanced SWS here proposed. The continuous variables from the mathematical model associated to each point of the new superstructure are also depicted for better elucidation.

Figure 2. Enhanced stage-wise superstructure and its associated variables In this section, the mathematical formulation derived from the new superstructure is also presented. Models derived from the SWS have been mathematically written in different forms in the literature. The nomenclature and mathematical representations here used generally follow the forms used in Pavão et


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al.37 Some of the equations and constraints are similar to those previously presented. Hence, more detailed comments are addressed mainly to the model additions and modifications.

Optimization model:

min

{TAC}

s.t.

Eqs.(3) − (58)

(1)

The total annualized costs (TAC) calculation is the sum of operating costs related to heaters and coolers heat loads, capital costs regarding the deployment of heat exchangers (pairing process streams) and heaters/coolers. These costs are calculated for equipment placed both at intermediate and outlet stages, as follows:

TAC =

∑∑ Ccu ⋅ FhQcuoutlet ⋅ Qcuoutlet + ∑∑∑ Ccu ⋅ Qcuinter + ∑∑ Chu ⋅ FcQhuoutlet ⋅ Qhuoutlet + ∑∑∑ Chu ⋅ Qhuinter ∑∑∑ z ⋅ ( B + C ⋅ A β ) + ∑∑∑ zcuinter ⋅ ( B + C ⋅ Acuinter ∑∑∑ zhuinter ⋅ ( B + C ⋅ Ahuinter β ) + ∑∑ zcuoutlet ⋅ ( B + C ⋅ Acuoutlet β ) + β ∑∑ zhuoutlet ⋅ ( B + C ⋅ Ahuoutlet ), i,n

n

i

n

m

j

i ,n

m, j

m

n

i

j

k

m

j

k

i

n

m

j

i , j ,k

m

n

m, j ,k

+

k

i , n ,k

β

)+

k

m , j ,k

m, j

j

i ,n ,k

i

i ,n

k

m, j

m

i, j ,k

i ,n ,k

n

i

(2)

m , j ,k

i ,n

m, j

i ∈ N H , m ∈ N HU , j ∈ N C , n ∈ N CU , k ∈ N S

where cu and hu suffixes stand for cold and hot utilities and inter and outlet mean intermediate and outlet stages. For instance, variable A is related to heat exchanger areas, hence Ahuoutlet is the area of heaters at outlet stages. The binary variable z represents the presence/absence of a unit. Qhuoutlet and Qcuoutlet represent the total heat available, while FhQcuoutlet and FcQhuoutlet represent fractions of such available/required heat that are exhausted/received at a given cooler/heater. The parameters Ccu and Chu are for operating costs of a given utility, B is capital fixed cost, C is a capital cost parameter and β is the



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capital cost area exponent. The subscripts m and n represent hot and cold utility types, while i, j and k stand for hot and cold process streams and stages. In the original mathematical formulation tailored by Yee and Grossmann,3 isothermal mixing was assumed. That is, in a given stage, for a given stream, heat exchanger outlet temperatures were assumed equal, so that no mixers energy balances were necessary, facilitating solutions obtaining. That assumption is not here made, and mixers energy balances are performed as follows:

∑ z i, j,k ⋅ Qi, j ,k + ∑ zcuinteri,n,k ⋅ Qcuinter i,n,k j

Tmixh i ,k = Tmixh i ,k −1 −

n

CPh i

,

(3)

i ∈ N H , j ∈ N C , n ∈ N CU , k ∈ N S

∑z Tmixc j , k = Tmixc j , k +1 +

j

i , j ,k

⋅ Qi , j , k + ∑ zhuinter m , j , k ⋅ Qhuinter m , j , k m

CPc j

,

(4)

i ∈ N H , m ∈ N HU , j ∈ N C , k ∈ N S

Outlet temperatures of heat exchangers (matching process streams) are calculated as follows: Thout i , j ,k = Tmixh i ,k −1 −

Tcout i , j , k = Tmixc j , k +1 +

z i , j ,k ⋅ Q i , j ,k Fh i , j ,k CPh i

,i ∈ N H , j ∈ NC , k ∈ N S

z i , j , k ⋅ Qi , j , k Fc i , j , k CPc j

,i ∈ N H , j ∈ NC , k ∈ NS

(5)

(6)

It should be noted that stages are numbered from left to right, which means Tmixhi,k-1 regards the temperature of stream i at the outlet of mixer in stage (k-1). That is the inlet temperature of the hot stream in the (i,j,k) exchanger. Analogously, for cold streams, the inlet temperature in the (i,j,k) exchanger is the mixer outlet temperature from the stage to the right of that unit, i.e., the outlet temperature from the mixer on stage (k+1), stream j (Tmixcj,k+1). In case that the unit being evaluated is in the first stage, the original


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hot stream inlet temperature is used. In the same manner, when that unit is at the last stage, the cold inlet temperature considered is that from the original cold stream data. Energy balances for heaters/coolers placed in intermediate stages were added to the model for calculating these units outlet temperatures, as follows:

Thoutcuint eri ,n , k = Tmixhi , k −1 −

zcuinteri , n , k ⋅ Qcuinteri , n ,k (7)

Fhcuinteri ,n ,k CPhi i ∈ N H , j ∈ N C , n ∈ N CU , k ∈ N S

Tcouthuint erm , j , k = Tmixc i , k +1 +

zhuinter m , j , k ⋅ Qhuinter m , j , k

Fchuinter m , j , k CPc j i ∈ N H , m ∈ N HU , j ∈ N C , k ∈ N S

,

(8)

Energy balances are performed for heaters/coolers placed at outlet stages as well. Firstly, total heat available in such stages (Qcuoutlet and Qhuoutlet) is obtained, as follows:

Qcuoutleti = CPhi ⋅ (Thi0 − Thi final ) −    ∑∑ z i , j ,k ⋅ Qi , j ,k + ∑∑ zcuinteri ,n,k Qcuinteri ,n,k ,   n k  j k  i ∈ N H , j ∈ N C , n ∈ N CU , k ∈ N S

(9)

Qhuoutlet j = CPc j ⋅ (Tc jfinal − Tc 0j ) −    ∑∑ z i , j ,k ⋅ Qi , j ,k + ∑∑ zhuinterm , j ,k ⋅ Qhuinterm , j ,k , m k  i k  i ∈ N H , m ∈ N HU , j ∈ N C , k ∈ N S

(10)

Then, outlet temperatures for such units are obtained as follows:

Thoutcuout let i ,n = Tmixh i , K −

zcuoutlet i , n ⋅ FhQcuoutle t i , n ⋅ Qcuoutlet i , n

i ∈ N H , n ∈ N CU

Fhcuoutlet i , n CPh i

,

(11)



Tcouthuout let m , j = Tmixc j ,1 +

zhuoutlet m , j ⋅ FcQhuoutle t m , j ⋅ Qhuoutlet m , j

m ∈ N HU , j ∈ N C

Fchuoutlet m , j CPc j

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,

(12)

Note that FhQcuoutlet, FcQhuoutlet, Fhcuoutlet and Fchuoutlet are fraction variables. However, the former type (FhQcuoutlet and FcQhuoutlet) regards the fraction of the non-exhausted energy requirement that is handled via a given utility, while the latter type (Fhcuoutlet and Fchuoutlet) are stream flow split fractions. Heat exchanger areas are obtained as follows (Eqs.(13)-(16)):

LMTDi , j ,k =

(Tmixhi ,k −1 − Tcout i , j ,k ) − (Thout i , j ,k − Tmixc j ,k +1 )  Tmixhi ,k −1 − Tcout i , j ,k   ln  Thout i , j ,k − Tmixc j ,k +1    i ∈ N H , j ∈ NC , k ∈ N S

, (13)

When temperature differences at both ends of the heat exchanger are equal, it can be noted that the formal LMTD equation (as in Eqs. 13, 17, 21 25 and 28) yields an indeterminate form (both the numerator and denominator are zero). Computationally, that leads the present algorithm to assign a “NaN” (not a number) to the corresponding position in LMTD matrices and, consequently, to all further areas and costs calculations. It must also be noted that the formal logarithmic mean approaches the same value as the arithmetic mean for temperature differences that are nearly equal, which means that when such situations occur, the arithmetic mean can be used. On the other hand, there are widely used approximations for the LMTD in the literature that are single equations, such as Chen’s39 and Paterson’s,40 which do not yield undetermined forms and are continuous (i.e., there are no disjunctions in their calculations), making them simple to implement in an optimization environment such as GAMS. However, those equations might also be slightly inaccurate in some cases, leading to over or underestimating heat exchanger areas. For a more complete evaluation on those approximations accuracy, the reader is referred to the work of Huang et al.,41 who present an interesting comparison of several LMTD approximations applied over typical 12 ACS Paragon Plus Environment

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literature benchmark problems. Those authors point out, for instance, that using Chen’s approximation might cause an overestimation on capital costs, and that Paterson’s, on the other hand, yields a slight underestimation. Given those possible inaccuracies, the formal LMTD was used in the present work. The situation of equality between temperature differences can be handled with a simple conditional statement (if/else) within the objective function evaluation in the meta-heuristic code. Such manner of handling that sort of numerical issue is an attractive feature in meta-heuristic methods. Hence, if the condition that (Tmixhi,k-1 – Tcouti,j,k) = (Thouti,j,k – Tmixcj,k+1) is true, the program is instructed to calculate the LMTD with:

LMTDi, j ,k =

(Tmixhi,k −1 − Tcouti, j ,k ) + (Thouti, j ,k − Tmixc j ,k +1 ) i ∈ N H , j ∈ NC , k ∈ N S

2

,

(14)

Area calculations proceed as follows:

U i, j =

1 1 1 + hhi hc j

Ai , j , k =

,i ∈ N H , j ∈ NC

z i , j , k Qi , j , k U i , j LMTD i , j ,k

(15)

,i ∈ N H , j ∈ NC , k ∈ N S

(16)

At intermediate stages, area calculations for coolers are performed as follows:

LMTDcuinteri ,n,k =

(Tmixhi ,k −1 − Tcuout n ) − (Thoutcuinteri ,n,k − Tcuinn )  Tmixhi ,k −1 − Tcuout n ln  Thout i ,n,k − Tcuinn i ∈ N H , n ∈ N CU , k ∈ N S

   

, (17)

If (Tmixhi,k-1 – Tcuoutn) = (Thoutcuinteri,n,k – Tcuinn), LMTDcuinter is obtained with:

LMTDcuinte ri ,n ,k =

(Tmixh i ,k −1 − Tcuout n ) + (Thoutcuint eri ,n ,k − Tcuin n ) i ∈ N H , n ∈ N CU , k ∈ N S

2

,

(18)



Ucui ,n =

1 1 1 + hhi hcu n

Acuinteri =

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, i ∈ N H , n ∈ N CU

zcuinteri ,n ,k Qcuinteri ,n ,k Ucu i ,n LMTDcuinte ri ,n ,k

(19)

, i ∈ N H , n ∈ N CU , k ∈ N S

(20)

And, for intermediate stages heaters:

LMTDhuinte rm , j , k =

(Thuout m − Tmixc j , k +1 ) − (Thuin m − Tcouthuint erm , j , k )  Thuout m − Tmixc j , k +1 ln   Thuin − Tcouthuint er m m, j ,k  m ∈ N HU , j ∈ N C , k ∈ N S

   

, (21)

If (Thuoutm – Tmixcj,k+1) = (Thuinm – Tcouthuinterm,j,k), LMTDhuinter is obtained with:

LMTDhuinte rm , j ,k =

(Thuout m − Tmixc j ,k +1 ) + (Thuin m − Tcouthuint erm , j ,k ) m ∈ N HU , j ∈ N C , k ∈ N S

Uhu m, j =

1 1 1 + hhu m hc j

Ahuinter j =

2

, j ∈ N C , m ∈ N HU

zhuinter m , j , k Qhuinter m , j , k Uhu m , j , k LMTDhuinte rm , j , k

,

(22)

(23)

, m ∈ N HU , j ∈ N C , k ∈ N S

(24)

Area calculations for coolers at outlet stages are performed as follows:


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LMTDcuoutlet i , n =

(Tmixhi , K − Tcuout n ) − (Thoutcuoutlet i ,n − Tcuin n )  Tmixhi , K − Tcuout n ln  Thoutcuoutlet − Tcuin i ,n n  i ∈ N H , n ∈ N CU

   

, (25)

If (Tmixhi,K – Tcuoutn) = (Thoutcuoutleti,n – Tcuinn), LMTDcuoutlet is obtained with:

LMTDcuoutleti ,n =

(Tmixhi , K − Tcuoutn ) − (Thoutcuoutleti ,n − Tcuinn ) 2

i ∈ N H , n ∈ N CU

Acuoutlet i ,n =

zcuoutlet i ,n FhQcuoutlet i ,n Qcuoutlet i Ucu i LMTDcuoutlet i ,n

,

(26)

, i ∈ N H , n ∈ N CU

(27)

And, for heaters:

LMTDhuoutl et m , j =

(Thuin m − Tmixc

j ,1 )

− (Thuout m − Tcouthuout let m , j )

 Thuin m − Tmixc j ,1 ln   Thuout m − Tcouthuout let m , j  m ∈ N HU , j ∈ N C

   

, (28)

If (Thuinm – Tmixcj,1) = (Thuoutm – Tcouthuoutletm,j), LMTDhuoutlet is obtained with:

LMTDhuoutlet m, j =

(Thuinm − Tmixc j ,1 ) − (Thuoutm − Tcouthuoutlet m, j ) m ∈ N HU , j ∈ N C

Ahuoutlet m , j =

2

zhuoutlet m , j FcQhuoutle t m , j Qhuoutlet Uhu j LMTDhuoutl et m , j

j

,

, m ∈ N HU , j ∈ N C

(29)

(30)

Feasibility constraints are necessary to ensure that solutions will be physically possible to implement. First, regarding fraction variables, those must be positive and their sum must be one, as follows:



∑ Fh

i , j ,k

j

+ ∑ Fhcuinteri ,n ,k = 1.0, i ∈ N H , j ∈ N C , n ∈ N CU , k ∈ N S n

Page 16 of 52

(31)

Fhi , j ,k ≥ 0.0, i ∈ N H , j ∈ N C , k ∈ N S

(32)

Fhcuinteri ,n ,k ≥ 0.0, i ∈ N H , n ∈ N CU , k ∈ N S

(33)

∑ Fhcuoutlet

(34)

= 1.0, i ∈ N H , n ∈ N CU

i ,n

n

∑ FhQcuoutle t

i ,n

= 1 .0, i ∈ N H , n ∈ N CU

(35)

n

∑ Fc i

i , j ,k

+ ∑ Fchuinterm , j ,k = 1.0, i ∈ N H , m ∈ N HU , j ∈ N C , k ∈ N S

(36)

m

Fci, j ,k ≥ 0.0, i ∈ NH , j ∈ NC , k ∈ NS

(37)

Fchuinterm, j ,k ≥ 0.0, m ∈ N HU , j ∈ N C , k ∈ N S

(38)

∑ Fchuoutlet

(39)

m, j

= 1.0, m ∈ N HU , j ∈ N C

m

∑ FcQhuoutle t

m, j

= 1.0, m ∈ N HU , i ∈ N H

(40)

m

Temperature constraints imposed are as follows, for heat exchangers pairing process streams: 16 ACS Paragon Plus Environment

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Tmixh i ,k −1 ≥ Tcout i , j ,k + EMAT , i ∈ N H , j ∈ N C , k ∈ N S

(41)

Thout i , j ,k ≥ Tmixc i , k +1 + EMAT , i ∈ N H , j ∈ N C , k ∈ N S

(42)

For coolers in intermediate stages:

Tmixh i , k −1 ≥ Tcuout n + EMAT , i ∈ N H , n ∈ N CU , k ∈ N S

(43)

Thoutcuint eri ,n, k ≥ Tcuin n + EMAT , i ∈ N H , n ∈ N CU , k ∈ N S

(44)

For heaters in intermediate stages:

Thuoutm ≥ Tmixcj ,k +1 + EMAT, m ∈ N HU , j ∈ NC , k ∈ N S

(45)

Thuin m ≥ Tcouthuint erm , j ,k + EMAT , m ∈ N HU , j ∈ N C , k ∈ N S

(46)

For coolers in outlet stages:

Tmixh i , K ≥ Tcuout n + EMAT , i ∈ N H , n ∈ N CU , k ∈ N S

(47)

Thoutcuout let i ,n ≥ Tcuin n + EMAT , i ∈ N H , n ∈ N CU

(48)

For heaters in outlet stages:

Thuinm ≥ Tmixc j ,1 + EMAT , m ∈ N HU , j ∈ N C , k ∈ N S

(49)

Thuout m ≥ Tcouthuout let m , j + EMAT , m ∈ N HU , j ∈ N C

(50)

New maximum heat load constraints were added to the model for the new variables. Note in Eqs. (52) and (53) that maximum heat loads of heaters/coolers are those present in the streams they are placed on.



(

)

Page 18 of 52

Qmax i0, j ,k = min CPhi ⋅ (Thi0 − Thi final ), CPc j ⋅ (Tc jfinal − Tc 0j ) , i ∈ N H , j ∈ NC , k ∈ N S

(51)

CPh i ⋅ (Th i0 − Th i final ), i ∈ N H , n ∈ N CU , k ∈ N S

(52)

Qmaxcuinte ri0, n , k =

CPc j ⋅ (Tc jfinal − Tc 0j ), m ∈ N HU , j ∈ N C , k ∈ N S

(53)

0 ≤ Qi , j ,k ≤ Qmaxi0, j ,k , i ∈ N H , j ∈ N C , k ∈ N S

(54)

0 ≤ Qcuinter i ,n ,k ≤ Qmaxcuinte ri0,n ,k , i ∈ N H , n ∈ N CU , k ∈ N S

(55)

0 ≤ Qhuinter m , j ,k ≤ Qmaxhuinte rm0, j , k , m ∈ N HU , j ∈ N C , k ∈ N S

(56)

0 ≤ Qcuoutlet i ≤ CPhi ⋅ (Thi0 − Thi final ), i ∈ N H

(57)

0 ≤ Qhuoutlet j ≤ CPc j ⋅ (Tc jfinal − Tc 0j ), j ∈ N C

(58)

Qmaxhuinte rm0 , j , k =

2.3

Solution Approach

The methodology here used to obtain near-optimal solutions is a reworked version of SA-RFO.37 The method consists in a two-level scheme where the binary variables of the HEN synthesis problem are handled with Simulated Annealing and the continuous ones with Rocket Fireworks Optimization. The latter is a combination of an adaptation of SA to handle continuous variables (Continuous Simulated


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Annealing, CSA) and Particle Swarm Optimization. The SA-RFO optimization routine consists, in summary, in adding a random heat exchanger to the HEN structure, identifying its group (see sub-section 2.3.2), optimizing the heat exchangers sizes and stream split flows according to that structure and evaluating the acceptance of the new solution according to SA acceptance rules. Figure 3 (a) outlines the main SA-RFO scheme, which will be here called OPT1. Figure 3 (b) depicts a graph with current solution costs during OPT1 and OPT2 (improvement step) procedures. The latter regards a supplementary optimization strategy that will be described further in this section (Sub-section 2.3.4).

Figure 3. (a) Block diagram of SA-RFO (OPT1 procedure) and (b) Solutions TAC evolution during OPT1 and OPT2 procedures Much of the algorithm code was re-written in order to handle the newly added variables and features of the enhanced SWS. Besides the changes performed to the objective function calculation procedure, the main adjustments regard the following features: moves of heater/coolers addition in SA, match groups identification, moves related to heat loads and stream fraction in heaters/coolers in CSA and equations/variables adaptation in PSO. Moreover, the new OPT2 optimization step (see sub-section 2.3.4) is performed to improve solutions quality. 19 ACS Paragon Plus Environment


2.3.1

Page 20 of 52

New moves in SA

The moves performed with SA are the simple addition of a unit every iteration. Using the original SWS formulation, that means replacing a random zero in the z matrix by one. Former utilities presence/absence matrices, zcu and zhu, were actually, in the original SA-RFO code, output variables of the model, as were their associated heat load variables, Qcu and Qhu. That is, the variables manipulated in the continuous space were Q, Fh and Fc. Other variables could be calculated explicitly and when Qcu and Qhu achieved zero, zcu and zhu were enforced to assume zero in order to avoid the fixed capital costs parameter (B) being summed up to the final TAC. When a heat exchanger achieved a zero heat load value during continuous optimization, a zero value was imposed to its associated position in z, i.e., it was removed from the HEN structure. In the re-worked SA step of the algorithm, the manipulated variables in the new model are z, zcuinter, zhuinter, zcuoutlet and zhuoutlet. All units have the same probability of addition. Note that in the new model, binary variables related to utilities in outlet stages (zcuoutlet and zhuoutlet) are no longer an output as were zcu and zhu in the former formulation. When a value in zcuoutlet or zhuoutlet is set to one that means that a fraction of the remaining energy in a given stream is to be exhausted/received in that respective cooler/heater. Handling such energy with a fraction ensures that at least one of the coolers/heaters in outlet stages will present some heat load, since that quantity is divided among those units existing in the zcuoutlet/zhuoutlet matrices. Analogously to the z case, when one of those heat load fractions achieves zero, its respective position in the binary matrix is set to zero as well. 2.3.2

Match groups identification

The match groups identification strategy was presented in Pavão et al.19 with a non-split SWS formulation and adapted to the SWS comprising splits used in Pavão et al.37 In summary, a “match group” is a set of heat exchanger units whose associated variables (e.g., heat loads, outlet temperatures, and areas) depend on each other’s and will not affect other groups’ equipment temperatures or sizes when manipulated. That strategy is well-suited with the SA scheme used, given that it performs a single unit addition move at a 20 ACS Paragon Plus Environment

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time. Hence, in every iteration the group to which the new unit belongs can be identified, and only that very group has to be optimized regarding its associated continuous variables. The initial solution used in RFO at each iteration is the optimizing group with all units having zero heat loads and equally distributed stream fractions. Other groups’ associated variables are retained from previous solutions. With utility related variables being outputs in the models used in both afore-referred works, the match groups identification were performed taking into account only heat exchangers pairing process streams. In the new model, however, there are manipulated variables related to utilities at both intermediate and outlet stages. In that sense, the adapted version used in this work considers heaters and coolers when identifying the groups, as illustrated in Figure 4.

Figure 4. Re-worked match groups identification strategy 2.3.3

RFO modifications

The new variables added to the model also required some new features to be added to the continuous optimization strategy used (RFO). As previously stated, RFO consists in two optimization steps: CSA and PSO. CSA can achieve a promising solution, which is then transferred to the initial population of random solutions used by PSO. In this way, the CSA solution can be improved by PSO. In RFO, the initial solution contains, for the units in the match group of the newly added HE, zero heat loads and equally divided stream split fractions, which means that required/exceeding heat is provided/removed via utilities. That solution is generally feasible with some exceptions. For instance, if the utility chosen violates thermodynamic constraints because of its operating temperature range, the solution is infeasible.



Page 22 of 52

In CSA, new moves performed to continuous variables were included. Moves related to heat loads of heaters/coolers in intermediate stages (Qhuinter and Qcuinter) are analogous to those of process streams heat exchangers (Q). Those performed to new fraction variables (Fhcuinter, Fchuinter, Fhcuoutlet, Fchuoutlet, FhQcuoutlet and FcQhuoutlet) are similar to the ones carried out to stream fractions in branches where process streams heat exchangers are placed (Fh and Fc). Equal probabilities of perturbations being performed are assigned to all variables to be optimized, which are determined previously in the match group identification step. Mathematically, the new set of moves can be represented as follows:

Qmove = ϕ ⋅ rand ( −1.0,1.0) ⋅ Qmax i0, j ,k , i ∈ N H , j ∈ N C , k ∈ N S

(59)

where rand(-1.0,1.0) is a random number between -1.0 and 1.0 and φ is a slowing factor presented by Pavão et al.,37 which depends on the CSA temperature parameter. That factor acts to make moves smaller in final CSA iterations, when continuous variables are expected to be closer to optimal solutions.

Qi, j ,k ← Qi, j ,k + Qmove, i ∈ N H , j ∈ N C , k ∈ N S

(60)

Qcuinteri,n,k ← Qcuinteri,n,k + Qmove, i ∈ N H , n ∈ N CU , k ∈ N S

(61)

Qhuinterm, j ,k ← Qhuinterm, j ,k + Qmove, m ∈ N HU , j ∈ N C , k ∈ N S

(62)

Fmove = ϕ ⋅ rand (−1.0,1.0)

(63)

Fhi, j ,k ← Fhi, j ,k + Fmove, i ∈ N H , j ∈ N C , k ∈ N S

(64)


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Fci, j ,k ← Fci, j ,k + Fmove, i ∈ N H , j ∈ N C , k ∈ N S

(65)

Fhcuinteri,n,k ← Fhcuinteri,n,k + Fmove, i ∈ N H , n ∈ N CU , k ∈ N S

(66)

Fchuinterm, j ,k ← Fchuinterm, j ,k + Fmove, m ∈ N HU , j ∈ N C , k ∈ N S

(67)

Fhcuoutleti,n ← Fhcuoutleti,n + Fmove, i ∈ N H , n ∈ N CU

(68)

FhQcuoutle t i , n ← FhQcuoutle t i ,n + Fmove , i ∈ N H , n ∈ N CU

(69)

Fchuoutletm, j ← Fchuoutletm, j + Fmove, m ∈ N HU , j ∈ N C

(70)

FcQhuoutle t m , j ← FcQhuoutle t m , j + Fmove , m ∈ N HU , j ∈ N C

(71)

It is important to highlight that no bounds, apart from those described in the model constraints, in Section 2.2, are imposed on any of the variables during the optimization. The PSO stage of RFO required the inclusion of new matrices representing the new variables. That implies additional routines for generating random values to the new variables and their associated velocities during the initial swarm creation step. The equations for updating PSO solutions now also include the new variables. The heat reallocation strategy for constraints handling of Pavão et al.37 now includes intermediate stages heaters/coolers as well. That strategy can be briefly outlined as the following: if the sum of heat loads of all units at a stream exceeds the maximal exchangeable heat in that stream, such surpassing quantity must



Page 24 of 52

be removed from a random unit in that stream. The revised strategy is mathematically described as follows. Set of equations for updating available heat in each stream:

Qhi ← Qhi − Qmove, i ∈ N H

(72)

Qc j ← Qc j − Qmove, j ∈ N C

(73)

where Qh and Qc are the current values of available or required energy in a given hot or cold stream. Maximal heat load for each unit:

Qmaxi , j ,k ← min(Qhi , Qc j ), i ∈ N H , j ∈ N C , k ∈ N S

(74)

Qmaxcuinte ri ,n ,k ← Qhi , i ∈ N H , n ∈ N CU , k ∈ N S

(75)

Qmaxhuinterm, j ,k ← Qc j , m ∈ N HU , j ∈ N C , k ∈ N S

(76)

It is worth noting that, when a move leads to a quantity that exceeds the maximum heat load for a match, its respective Qmax, Qmaxcuinter or Qmaxhuinter assumes a negative value. Hence, exceeding heat load removal is performed as follows:

Qi, j ,k ← Qi, j ,k + Qmaxi, j ,k , i ∈ N H , j ∈ N C , k ∈ N S

(77)

Qcuinter i ,n ,k ← Qcuinter i ,n ,k + Qmaxcuinte ri ,n ,k , i ∈ N H , n ∈ N CU , k ∈ N S

(78)

Qhuinter m , j ,k ← Qhuinter m , j ,k + Qmaxhuinte rm , j ,k , m ∈ N HU , j ∈ N C , k ∈ N S

(79)


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The strategy for handling fractions is similar. However, since the sum of fractions needs to be equal to one, and not lower or equal to the maximum, as in heat loads case, the move value may be directly removed from a random branch. The mathematical description is as follows:

Fh i , j , k ← Fh i , j , k − Fmove , i ∈ N H , j ∈ N C , k ∈ N S

(80)

Fhcuinteri,n,k ← Fhcuinteri,n,k − Fmove, i ∈ N H , n ∈ N CU , k ∈ N S

(81)

Fhcuoutlet i ,n ← Fhcuoutlet i ,n − Fmove, i ∈ N H , n ∈ N CU

(82)

FhQcuoutle t i ,n ← FhQcuoutle t i ,n − Fmove , i ∈ N H , n ∈ N CU

(83)

Fc i , j , k ← Fc i , j , k − Fmove , i ∈ N H , j ∈ N C , k ∈ N S

(84)

Fchuinter m , j ,k ← Fchuinter m , j , k − Fmove , m ∈ N HU , j ∈ N C , k ∈ N S

(85)

Fchuoutlet m , j ← Fchuoutlet m , j − Fmove , m ∈ N HU , j ∈ N C

(86)

FcQhuoutle t m , j ← FcQhuoutle t m , j − Fmove , m ∈ N HU , j ∈ N C

(87)

It is worth noting that the afore-described method (Eqs. (72)-(87)) is applied at every move performed during CSA, as well as during the update of each particle position in PSO. All constraints related to temperature are handled via penalty functions, as in Pavão et al.37



2.3.4

Page 26 of 52

Solutions improvement step

In some SA-RFO optimization runs, the current solution at the end of the algorithm execution had higher TAC than the best solution found during the whole experiment. Such aspect is commonly observed in SA based approaches, given the algorithm solutions updating rule. For the sake of a more wide exploration, there is a probability of a new solution with higher costs being accepted (see Pavão et al.37 for more details on the implementation in SA-RFO). Such strategy is of great aid in avoiding premature local minima stagnation. During the algorithm execution, when the referred method is applied, the current solution can be replaced by new ones with slightly worse costs. In some cases, when the current solution is also the best solution found so far, that solution can be replaced with a worse one, and the algorithm is not able to achieve again that best solution (which is evidently stored by the method), or one with equivalent costs. In order to give the best solution found by SA-RFO a greater chance to be improved, a subsequent procedure (OPT2) is here proposed. That solution is taken as starting point in OPT2, and a slightly modified version of SA-RFO is applied to it. In this modified version, the temperature parameter of SA is set to a value that is sufficiently small so that no solutions with worse TAC are accepted. With that setup, 500 outer level moves are performed in SA-RFO. Moreover, the continuous variables of the best solution are retained and used in the new structure. Note that it differs from the original strategy, in which all heat loads of the match group being optimized begin with zero heat load and equally divided stream fractions. Figure 3 (b) presents a graph illustration of this strategy.

3

Numerical examples

The proposed mathematical formulation and the re-worked SA-RFO optimization method are here applied to three case studies taken from the literature in order to evaluate the proposed methodology efficiency. The algorithm was written in C++ and tests were run on a computer with a 3.50 GHz Intel® Core™ i5-4690 processor with 8.00 GB of RAM. In each example, results are presented in figure forms containing data for heat loads, areas and, for heaters and coolers, the type of utility used. Tables containing results reported in the literature are also included in each example. Parameters related to SA-


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RFO method were tuned according to recommendations of Pavão et al.37 The heat exchanger minimum approach temperature parameter (EMAT) used in all cases was 1 K. 3.1

Example 1

The first example is the well-studied aromatics plant heat integration problem. It was proposed by Linnhoff and Ahmad42 and has nine streams, being four hot and five cold (data is presented in Table S1, in the Supporting Information). It uses one type of hot and one cold utility (hot oil and cooling water). Applying the present methodology to this case is important especially for benchmarking purposes. Several authors have applied different optimization schemes to this example. Recent attempts were performed by Fieg et al.43 with an SWS-based mathematical model solved with a monogenetic algorithm, Peng and Cui17 with a no-splitting SWS formulation solved with an SA-based approach, and Pavão et al.37 with SA-RFO applied to an SWS-based formulation. Although larger cases have already been addressed in the literature, no method, to the best of the authors’ knowledge, was yet able to ensure global optimization to the aromatics plant problem. Hence, it is important to compare results achieved in the present study to solutions reported previously. The problem was solved with six stages in the SWS. That implies the method handles 486 continuous and 183 binary variables. Among continuous variables, there are heat loads (Q, Qhuinter, Qcuinter), intermediate stage stream fractions (Fh, Fc, Fhcuinter, Fchuinter), outlet remaining/required heat and split fractions (FcQhuoutlet, FhQcuoutlet, Fhcuoutlet, Fchuoutlet). Among binary ones, there are those related to the presence of “process stream to process stream” heat exchanger (z), intermediate heater/cooler (zhuinter, zcuinter), and outlet heater/cooler (zhuoutlet, zcuoutlet). For comparison, in the typical SWS model used by Pavão et al.,37 there were 360 continuous and 120 binary handled variables. The OPT1 procedure took 4256 s, achieving TAC of 2,909,955 $/yr. OPT2 was performed in 1086 s and improved TAC slightly to 2,909,906 $/yr. That configuration has lower associated TAC than those previously reported in the literature and is presented in Figure 5. A comparison to results reported by other authors is presented in Table 1. The new model led to a noteworthy result, especially given the number of authors that have investigated this problem. Note that



Page 28 of 52

the configuration achieved presents higher utility requirements than several others reported in the literature. However, the present methodology led to less heat exchange area, yielding a final TAC lower than that of previous works, meaning that not only the new model has good heat integration trading-off potentialities, but also the solution approach was able to perform a satisfactory optimal solution search. It is important to draw attention to the fact that the achieved configuration presents a heater placed on a cold stream (C1) split branch, which is a key feature available in the superstructure presented in this work. If placed at the stream end, like in the original SWS, the solution would be infeasible. That is because the cold stream would reach the heater at nearly 289 °C, while the hot utility outlet temperature was fixed by Linnhoff and Ahmad42 at 250 °C, making that match thermodynamically prohibitive.

Figure 5. HEN configuration obtained for Example 1

Table 1. Solutions comparison for Example 1

Linnhoff and Ahmad42 Lewin6 Toffolo10 Fieg et al.43 Huo et al.13 Peng and Cui17 Pavão et al.37 This work

Units 13 12 14 14 13 15 12 14

Hot util. (MW) 25.31 25.09 23.60 23.61 24.22 24.50 23.85 24.76

Cold util (MW) 33.02 32.81 31.40 31.33 31.94 32.22 31.57 32.48

TAC ($/yr) 2.960×106 2.936×106 2,920,130 2,922,298 2,922,585 2.935×106 2,919,675 2,909,906


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3.2

Example 2

As in Example 1, this case also regards HEN synthesis in an aromatics plant. However, the number of process streams for this problem is larger. Process streams suitable for heat integration and their operating conditions data (Table S2 in Supporting Information) were extracted by Khorasany and Fesanghary44 from the Bandar Imam aromatics plant. Such industrial facility is located in the Persian Gulf. It should be noted that, according to those authors, not all streams from the plant were extracted for heat integration given the fact that some of them present restrictions regarding process conditions and control. This is a 16 streams problem (six hot and ten cold) with two types of hot utility (steam and flue gas) and one cold utility (cooling water). After the optimization attempt by Khorasany and Fesanghary44 with a Harmony Search algorithm, other authors have approached the case. Huo et al.13 used a flexible splitting/nosplitting SWS solved via GA and PSO. Pavão et al.19 applied their SA-PSO method with Parallel Processing to solve a no-splits model. Zhang et al.18 employed the chessboard representation with streams rearrangement and no stream splits, whose mathematical model was solved with a random walking algorithm with compulsive evolution. In the solution presented by Zhang et al.,18 cooler areas were slightly underestimated. We revised such areas values according to heat load data presented by the authors and recalculated TAC. Despite such deviation, the solution achieved by Zhang et al.18 has the lowest TAC among those presented in the literature to this day. Four stages were used in the enhanced SWS for solving this case, leading to a model with 980 continuous and 370 binary variables manipulated by the solution approach. In OPT1, a solution with TAC of 6,801,324 $/yr was found after 9584 s and it was improved in OPT2 to 6,801,261 $/yr after 1718 s. That configuration is depicted in Figure 6, and the comparison to literature solutions is presented in Table 2. Note that a cooler is placed in an intermediate stage in H1 stream. If it were at the end of that stream, TAC would increase to 6,806,002 $/yr. The relatively large improvement in TAC, when compared to solutions previously reported in the literature, is greatly due to the substantial reduction in hot utility requirements. In relation to the original real-world plant, a relative reduction of 91.4% on the heating demand was achieved. It is also worth noting that, in



Page 30 of 52

the solution achieved, all auxiliary heating on the streams extracted by Khorasany and Fesanghary44 is performed via flue gas.



Existing Plant44 Khorasany and Fesanghary44 Huo et al.13 Pavão et al.19 Zhang et al.18 This work a Revised value

3.3

Units 18 18

Hot util. (MW) 122.16 66.07

Cold util (MW) 524.72 469.62

TAC ($/yr) 8,856,412 7,435,740

16 17 19 19

38.80 34.21 23.79 10.47

442.37 437.78 427.36 414.03

7,361,190 7,301,437 7,212,115a 6,801,261

Example 3

This is one of the largest case studies present in the literature. It is composed of 41 process streams (23 hot and 18 cold), and 6 types of hot utilities and 4 types of cold utilities are available. It was originally proposed by Shethna et al.45 and modified by Fieg et al.,43 whose problem data is here used (Table S3 in Supporting Information). The solution method proposed by Fieg et al.43 divided the problem into several 30 ACS Paragon Plus Environment

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smaller sub-networks, which are more easily optimized. Moreover, they used the original SWS with nonisothermal mixing assumption. Those authors were able to achieve TAC of 18,839,067$/yr. Here, all matching possibilities are always considered and the problem is treated as a whole. Three stages were used in the enhanced SWS, rendering a model with 5326 continuous and 2042 binary variables. The configuration found in OPT1 procedure had TAC of 18,639,197 $/yr, improved to 18,622,063 $/yr after OPT2. Those procedures took 50,634 s and 4667 s, respectively. The processing times are significantly larger than in previous examples. However, given the size of this case and also considering that the approach is to be used at preliminary plant design stages, a computational time of around 15 hours is still satisfactory. The TAC achieved after OPT2 is 1.15% lower than that of the solution reported by Fieg et al.43 with the same number of units and a slightly lower requirement of both heating and cooling. Similarly to the aforementioned authors’ solution, most of the auxiliary heating is performed via flue gas, while, regarding auxiliary cooling, most of the task is carried out via air cooling. The configuration is presented in Figure 7. Other features comparison can be found in Table 3. Note in the final configuration (Figure 7) that a cooler is placed in an intermediate stage in stream H23. If the cooler used to exhaust the heat of H23 were placed in its outlet, TAC would increase by 227$/yr. The placement of utilities in intermediate stages did not yield much TAC improvement in this case solution. However, it should be noted that the enhanced SWS and the optimization of its associated mathematical model via SA-RFO were able to lead to efficient utility choices and a good configuration even with a large number of utility options and process streams. Consequently, a lower TAC than that reported in the literature to this rather challenging HEN synthesis case was obtained. Despite being an interesting example given the number of process streams and available utilities, this case was scarcely studied in the literature. Hence, the present work aims to draw attention to this example. Further studies and application of both deterministic and non-deterministic methods would certainly be of interest of the community for benchmarking purposes.



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Fieg et al.43 This work

Units 44 44

Hot util. (MW) 163.07 161.93

Cold util (MW) 50.48 49.33

TAC ($/yr) 18,839,067 18,622,063


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4

Conclusions

A stage-wise superstructure with different options for heaters and coolers placement was presented. In the model, such units can be located in intermediate stages, differently from the usual approach of performing auxiliary heating or cooling at streams ends. The energy not exchanged among process streams or exhausted/received to/from utilities in intermediate stages is exhausted/received in outlet stages. In those, only heaters/coolers are present. Moreover, in cases where multiple utilities are available, instead of using decision variables to select which utility will be used in a unit, each utility option is placed in one stream split branch. Hence, more than one type of utility may be used in a stream. Results obtained for three cases reported in the literature demonstrated that the placement of utility units at intermediate stages instead of only at stream ends may lead to slight total annual costs reduction. Solutions with lower TAC than those reported in the literature and that would not be reproducible with the original stage-wise superstructure were achieved. For Example 1, the methodology achieved a solution with 0.33% lower TAC than the best solution previously reported. Although it may suggest only a marginal gain, in absolute values that means an additional annual saving of nearly $10,000. Besides, given the fair number of methodologies applied to that problem, that result has significant benchmarking importance. In Example 2, which is a real-world case, a substantial reduction was achieved on utilities requirement. Remarkably, TAC is 5.7% lower than the best previously reported solution. The hot utility requirement reduction is also noteworthy, being more than 90% lower than in the original plant and more than 50% lower than the best literature solution. The third case study is certainly one of the larger HEN synthesis examples ever tackled. The present method was able to outperform the literature solution by 1.15%. That is a challenging case which, given its scale and complexity, certainly deserves more attention from the HEN synthesis academic community. The improvements made to the hybrid meta-heuristic approach used, SA-RFO,37 turn the method capable of solving the model arisen from the new superstructure. It is important to note that, compared to previous investigations where SA-RFO was applied, several additional variables to be handled were present. The enhanced method, however, was able to achieve promising near-optimal solutions in all case studies. The 33 ACS Paragon Plus Environment


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increased difficulty of the new model led to processing times relatively higher than when SA-RFO was applied to a model based on the original SWS. However, such times are still reasonable at a preliminary HEN design stage. With the promising results achieved, the new superstructure model can be further expanded to handle other aspects of HEN synthesis, such as operation under multiple periods, optimization of multiple criteria or the integration with work exchange. Moreover, despite encompassing more structural options than the original SWS, the new superstructure is also not able to produce solutions with sequential units in a single stream split branch or solutions with crossflows in the HEN. Such structural aspects might as well be considered for further development in future works.

5

Acknowledgements

The authors gratefully acknowledge the financial support from Coordination for the Improvement of Higher Education Personnel – CAPES (Brazil) and the National Council for Scientific and Technological Development – CNPq (Brazil).

6

Nomenclature

Variables A

[m2]

Heat exchanger area

Acuinter

[m2]

Intermediate stage cooler area

Acuoutlet

[m2]

Outlet stage cooler area

Ahuinter

[m2]

Intermediate stage heater area

Ahuoutlet

[m2]

Outlet stage heater area

Fc

[-]

Cold stream fraction for a heat exchanger branch

Fchuinter

[-]

Cold stream fraction for a heater branch in an

intermediate stage Fchuoutlet

[-]

Cold stream fraction for a branch in an outlet stage

FcQhuoutlet

[-]

Fraction of the required heat in outlet stage of a cold

stream provided with a given hot utility 34 ACS Paragon Plus Environment

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Fh

[-]

Hot stream fraction

Fhcuinter

[-]

Hot stream fraction for a cooler branch in an

intermediate stage Fhcuoutlet

[-]

Hot stream fraction for a branch in an outlet stage

FhQcuoutlet

[-]

Fraction of the available heat in outlet stage of a hot

stream exhausted with a given cold utility LMTD

[K or °C]

Heat exchanger logarithmic mean temperature difference

LMTDcuinter

[K or °C]

Intermediate stage cooler logarithmic mean temperature

LMTDcuoutlet [K or °C]

Outlet stage cooler logarithmic mean temperature

difference

difference LMTDhuinter [K or °C]

Intermediate stage heater logarithmic mean temperature

difference LMTDhuoutlet [K or °C]

Outlet stage heater logarithmic mean temperature

difference Q

[kW]

Heat exchanger heat load

Qcuinter

[kW]

Heat load of an intermediate stage cooler

Qcuoutlet

[kW]

Available heat in outlet stage of a hot stream

Qhuinter

[kW]

Heat load of an intermediate stage heater

Qhuoutlet

[kW]

Required heat in outlet stage of a cold stream

Qmax0

[kW]

Maximum exchangeable heat in a match

Qmaxcuinter0 [kW]

Maximum available heat for an intermediate stage cooler

Qmaxhuinter0 [kW]

Maximum required heat for an intermediate stage heater

TAC

[$/yr]

Total annualized cost

Tcout

[K or °C]

Heat exchanger cold stream outlet temperature

Tcouthuinter

[K or °C]

Intermediate stage heater cold stream outlet temperature 35 ACS Paragon Plus Environment


Tcouthuoutlet [K or °C]

Page 36 of 52

Outlet stage heater cold stream outlet temperature

Thout

[K or °C]

Heat exchanger hot stream outlet temperature

Thoutcuinter

[K or °C]

Intermediate stage cooler hot stream outlet temperature

Thoutcuoutlet [K or °C]

Outlet stage cooler hot stream outlet temperature

Tmixc

[K or °C]

Cold stream mixer outlet temperature

Tmixh

[K or °C]

Hot stream mixer outlet temperature

z

[-]

Heat exchanger existence binary variable

zcuinter

[-]

Intermediate stage cooler existence binary variable

zcuoutlet

[-]

Outlet stage cooler existence binary variable

zhuinter

[-]

Intermediate stage heater existence binary variable

zhuoutlet

[-]

Outlet stage heater existence binary variable

B

[$]

Fixed heat exchanger cost

C

[$/(m2β)]

Heat exchanger cost factor

Ccu

[$/(kWyr) or $/(kWh)]

Cold utility cost

Chu

[$/(kWyr) or $/(kWh)]

Hot utility cost

CP

[kW/K or kW/°C]

General process stream total heat capacity

CPc

[kW/K or kW/°C]

Cold stream total heat capacity

CPh

[kW/K or kW/°C]

Hot stream total heat capacity

EMAT

[K or °C]

Heat exchanger minimal temperature approach

h

[kW/(m2K) or kW/(m2°C)]

General heat transfer coefficient

hc


Cold stream heat transfer coefficient

hcu


Cold utility heat transfer coefficient

hh


Hot stream heat transfer coefficient

hhu


Hot utility heat transfer coefficient

Parameters


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Tc0

[K or °C]

Initial cold stream temperature

Tcfinal

[K or °C]

Final (target) cold stream temperature

Tcuout

[K or °C]

Outlet cold utility temperature

Th0

[K or °C]

Initial hot stream temperature

Thfinal

[K or °C]

Final (target) hot stream temperature

Thuout

[K or °C]

Outlet hot utility temperature

Tin

[K or °C]

General process stream inlet temperature

Tout

[K or °C]

General process stream outlet temperature

U


Global heat transfer coefficient

Ucu


Cooler global heat transfer coefficient

Uhu


Heater global heat transfer coefficient

β

[-]

Capital cost exponent

min

[-]

Returns the minimum value along a given set

rand

[-]

Returns a random number within the defined interval

i

[-]

Hot stream

j

[-]

Cold stream

k

[-]

Stage

K

[-]

Final stage

m

[-]

Hot utility

n

[-]

Cold utility

Functions

Indexes:

Sets: 37 ACS Paragon Plus Environment


NC

[-]

Cold streams

NCU

[-]

Cold utilities

NH

[-]

Hot streams

NHU

[-]

Hot utilities

NS

[-]

Stages

Page 38 of 52

SA-RFO related terms: Fmove

[-]

Stream fraction continuous move

Qc

[kW]

Required heat in a cold stream after each CSA move

Qh

[kW]

Available heat in a hot stream after each CSA move

Qmax

[kW]

Maximum exchangeable heat in a match after each CSA

[kW]

Maximum available heat for an intermediate stage cooler

move Qmaxcuinter

after each CSA move Qmaxhuinter

[kW]

Maximum required heat for an intermediate stage heater

after each CSA move Qmove

[kW]

Heat load continuous move

φ

[-]

Slowing factor

7

Supporting Information

A supporting document is provided, containing Tables S1, S2 and S3 with streams and economic data regarding each numeric example tackled in this work. This information is available free of charge via the Internet at http://pubs.acs.org/.

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Page 44 of 52

For Table of Contents Only


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Stage-wise superstructure3 with variables representation as in the formulation of Pavão et al.28 69x70mm (300 x 300 DPI)

ACS Paragon Plus Environment


Enhanced stage-wise superstructure and its associated variables 115x131mm (300 x 300 DPI)


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(a) Block diagram of SA-RFO (OPT1 procedure) and (b) Solutions TAC evolution during OPT1 and OPT2 procedures 171x90mm (300 x 300 DPI)



Re-worked match groups identification strategy 68x46mm (300 x 300 DPI)


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HEN configuration obtained for Example 1 108x41mm (300 x 300 DPI)





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For Table of Contents Only 82x44mm (300 x 300 DPI)


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An Enhanced Stage-wise Superstructure for Heat Exchanger

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