Heat-exchanger network synthesis

Heat-exchanger network synthesis

Heat-exchanger network synthesis 18.1 18 Introduction We will address heat-exchanger network (HEN) design problem in two chapters: HEN synthesis in...

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Heat-exchanger network synthesis 18.1

18

Introduction

We will address heat-exchanger network (HEN) design problem in two chapters: HEN synthesis in this chapter and HEN retrofit in the next one. Such separate treating of these topics does not mean that they have to be solved by different techniques. In fact, there are some methods that can deal with both problems in practically common framework. For instance, simultaneous approaches addressed in this chapter require minor changes in order to be applicable to revamp design. The division into synthesis and retrofit results from the two following reasons: l

l

Retrofit design problem is of significantly higher combinatorial character than synthesis. Also, ‘nonlinearity’ level is higher. Due to these features, both insight-based and simultaneous methods have problems with efficient and robust solution, particularly if larger-scale problems are to be solved. Also, they aren’t able, at present, to account for such variables as discrete parameters of standard heat exchangers. In the chapter on HEN retrofit, we will describe application of genetic algorithms to retrofitting HEN comprising standard heat exchangers. Existing general approaches to HEN retrofit are most often based on synthesis techniques. Hence, to explain the latter, one should get knowledge of the former.

Even in case of only HEN synthesis problem, numerous approaches have been suggested to date. Explanation of all of them is beyond the scope of this book. We decided to present in more detail certain selected representative techniques from two wide classes: sequential and simultaneous. Sequential methods can be performed by insight-based techniques that do not use optimization at all or in limited scope (as auxiliary tool). Alternatively, all or some key stages can be performed with extensive use of optimization. Insight-based techniques usually aren’t reliable tools to attack larger-scale problems. Nevertheless, they are quite efficient to solve smaller ones. Also, they provide understanding of optimization models. Hence, we will address both insight-based and optimization sequential approaches for HEN grass-root design in the next two sections. Then, selected contributions to simultaneous approaches will be addressed.

18.2

Sequential approaches

18.2.1 Pinch technology-based methods Pinch technology (PT)-based approaches to design HEN are widely used in industry. They are simple, employ a lot of graphical tools that visualize solution algorithm, and, thus, give insights and a feeling of deep understanding. Also, they give the designer Energy Optimization in Process Systems and Fuel Cells. https://doi.org/10.1016/B978-0-08-102557-4.00018-9 Copyright © 2018 Elsevier Ltd. All rights reserved.

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Energy Optimization in Process Systems and Fuel Cells

control on designing procedure. However, the usefulness and simplicity of calculating procedures are most often demonstrated on simple cases that do not show several potential pitfalls and, also, a great computational load of iterative, error-and-trial procedure needed for larger-scale problems. The PT approaches do not guarantee optimal solution or even near-optimal one. They, however, lead to better HEN than existing ones or those designed by intuition and designing experience. A user has to be aware that they are not systematical tools, and deep understanding of basic PT concepts is necessary to use them in practice. The final pinch design method (PDM) developed by Ahmad (1986) and also presented later in Linnhoff and Ahmad (1990) and Ahmad et al. (1990) had evolved from the first version by Linnhoff and Hindmarsh (1983). The PDM allows calculating network structure and basic parameters of heat exchangers and splitters. Because the PDM employs some heuristic elements and, also, due to inherent simplifications hidden in PT concepts, the second design stage is often necessary. This is the evolution stage where the design is improved by minimizing further HEN total annual cost (TAC), particularly by improving trade-off: number of matches—cost of utilities. Also, NLP optimization of heat-exchanger areas and split fractions can finally be performed for network of fixed topology. Targeting stage is of crucial importance in the PDM. It provides the targets and the optimum HRAT value (calculated by supertargeting) that is applied in the design. Also, targeting techniques are often a basis of designing procedures, and thus, their knowledge is fundamental for performing the design stage. Here, we limit the description of the PDM to a brief presentation of basic steps. Readers interested in more thorough presentation are referred to Linnhoff et al. (1982), Shenoy (1995), Smith (2005), papers by Linnhoff and Ahmad (1990), and Ahmad et al. (1990). After the targeting stage, the designer knows the optimal ΔTmin value (ΔTmin,opt), pinch temperature/temperatures, and values of all targets. To meet the ‘no heat exchange via pinch’ rule, the division at pinches must be performed. Most often, a single pinch exists, and we will explain the method for this case. If there are more pinches, some subtasks are ‘between-pinches’ problems. They are very restricted as for structure and heat loads of matches. Though the basic rules of PT are valid for between-pinches problems, a special care should be paid to ensure that decisions on splits at one pinch are consistent with those valid for the adjacent one. Advices on design HEN with multiple pinches can be found in Jezowski (1992a) who addressed alternative approach to that suggested by Trivedi et al. (1989a). The upper subtask (above the pinch) and lower one (below the pinch) are solved separately since they are treated as independent problems in the first stage of the PDM. Evolution stage is, then, applied to integrate them properly into one network, though simple merging is often treated sufficient. The design, that is, structure development by matching streams, starts at the pinch and proceeds away from it as shown in Fig. 18.1. Thus, the streams at the pinch (at-pinch streams having pinch temperature) must be matched first. For the above-pinch task, all hot at-pinch process streams above the pinch must be matched with some cold at-pinch streams, in order to eliminate cold 

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Tp Above-pinch problem

Below-pinch problem

Fig. 18.1 ‘Directions’ of solution procedure in the PDM.

utility usage above the pinch. Likewise, below the pinch, all cold at-pinch streams must be matched with chosen hot at-pinch streams to avoid hot utility usage below the pinch. Hence, a number of branches of streams have to be properly adjusted at the pinch to engage all hot at-pinch streams (above the pinch) and all cold at-pinch streams (below the pinch) in matches. Notice that proper matching of at-pinch streams is a crucial point of the PDM since it decides on reaching MER target. This target is of utmost importance since it influences greatly the total cost of final solution. To determine a match of at-pinch streams, that is, to find proper branching scheme and to calculate heat load, the PDM offers the following rules: (1) Adjustment of CP values of streams to be matched (called CP rule in the following). (2) Adjustment of temperature difference profiles (called Δtm rule in the following).

The both rules are generalization of earlier rules on stream branching and ‘tick-off’ heuristic; see Linnhoff and Hindmarsh (1983). They are aimed at reaching all three targets. However, as we will show in the following, this requires tedious calculations, and it is doubtful whether the goal can be achieved rigorously. The rules are as follows: (Ad.1) Streams chosen for heat exchange should have a ratio of CP values close to the ratio of CP values of appropriate segments of the composite curves, that is, the segments that contain the streams. (Ad.2) Temperature profiles in a match should feature a close resemblance to temperature profiles of the appropriate segments of the composite curves.

Though the rules are simple, their application is often not. The designer usually faces the necessity of employing an error-and-trial procedure. She/he has to choose at-pinch streams for a match and check the rules. If they are not satisfied, the designer has two options: (a) Change one stream in the match (b) Split a hot or cold stream or both

Additionally, when calculating the match, the designer has to view it in the context of all other possible at-pinch matches. Also, the rules are of qualitative character. The rules for branching streams h and c to be matched are as follows: (A) For a choice of CP value for branches (A1) above pinch: CPh  CPc (A2) below pinch: CPh  CPc

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Energy Optimization in Process Systems and Fuel Cells

(B) For the number of branches at the pinch (a branch can be an original stream or a branch of original stream) (B1) above pinch: no. of hot branches  no. of cold branches (B2) below pinch: no. of hot branches  no. of cold branches

After decision on branching and choosing split ratio, the designer has to determine heat load of the match. To minimize number of matches, the use of CP rule is advised. Also, ‘tick-off’ rule, which says that a single residual stream can be created, is helpful (the heuristic results from the reasoning in Section 16.2 for achieving MNM target by creating a single residual in each match). All the rules aren’t sharp tools. In the case of complex problems with many streams, one may expect many trials to satisfy them. There is, however, a useful technique to control the structure development and match calculation. This is called remaining problem analysis (RPA). The concept is based on extensive use and comparison of targets at steps of sequential choosing and rating matches. To explain it, let us assume that two streams (hi and cj) had been selected for the match at current step of sequential procedure. We assume also that the matches calculated to the point are valid, that is, they ensure a possibility of satisfying finally all targets. Heat load (Qij) and heattransfer area of the match (Aij) can be, then, calculated. Let AT1, Qu1, and N1 denote, respectively, the minimum total area, the maximum heat recovery, and the minimum number of match targets for overall set of streams at the current design step, that is, streams unmatched and, also, streams hi and cj chosen for the match. AT2, Qu2, and N2 denote the same targets for streams not matched and residual stream from the match hi and cj.

The match hi with cj is ‘good’ (acceptable) if   AT 1  AT 2 + Aij  εA

(18.1)

  N 1  N 2 + 1  εN

(18.2)

  Qu1  Qu2 + Qij  εQ

(18.3)

where εA, εN, and εQ are tolerances of the designer. To explain the PDM, we present here an example of calculating the at-pinch matches for below-the-pinch subproblem in example 13.1 (data in Table 13.1). The at-pinch streams are shown in Fig. 18.2A. First, let us explain the logic behind branching rules (B). Assume, for a moment, that stream c2 has been split. Let the matches between both hot streams and some two of three cold streams be applied as shown in Fig. 18.2B. In result, one cold at-pinch stream is left, and application of heating utility would be necessary. It is important to notice that split of stream c2 does not spoil rule (A). Hence, both rules have to be satisfied simultaneously. We should start with the selection of streams using CP rule. The ratio of CP for the composite curves just below the pinch is 1.82. Any possible match h1–c1, h1–c2, h2–c1,

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Parameters of solutions to example 18.1 according to Jezowski et al. (2001a,b)

Table 18.1 

Solution no. and control parameters

Number of matches

Energy penalty (kW)

1{η > 0.001; EMAT ¼ 10 K} 2{η > 0.001; EMAT ¼ 6 K} 3{η > 0.001; EMAT ¼ 6 K} 4{η < 0.001; EMAT ¼ 1 K}

8 8 8 7

522.07 375.00 32.80 749

Tp = 90/70°C

Tp = 90/70°C

Tp = 90/70°C

CP h1

2.0

h1

h1

h2

8.0

h2

h12

c1

h22

2.5

c12

c1

3.0

c22

c12

c1 c2

(A)

(B)

(C)

Fig. 18.2 Illustration of the PDM for matching at-pinch streams in example 13.1: (A) at-pinch streams; (B) illustration for branching rule (B); (C) possible arrangement of at-pinch matches with many splits.

c22

and h2–c2 does not feature such the CP ratio. Hence, we must proceed to branching applying CP rule and, also, following rules (A) and (B). Stream h2 is the candidate for splitting since it has the highest CP value. This gives three hot branches and two cold branches, which satisfies rule (B) and does not spoil rule (A). However, there is the question of estimating CP values of the branches. CP rule gives us also very crude advice since CP ratio changes with temperature and composite curve segments aren’t equivalent to segments of streams in a match. The PDM provides us also with Δtm rule. It requires drawing a plot of temperature difference between the composite curves versus temperature of cold streams. Then, similar plot for each potential match has to be drawn. The profiles of two plots: for composite curves and for a match should be similar in proper temperature ranges. A close resemblance ensures vertical heat transfer in a match, thus, the minimum area (but only for identical heat-transfer coefficient of streams as we pointed out in Chapter 16). Notice that one can consider additional split of stream c2 since the number of hot and cold branches satisfies rule (B). Also, it is valid as to rule (A). The branches are shown in Fig. 18.2C. The matches shown in Fig. 18.2C h1  c12, h12  c1, and h22  c22 give the minimum number according to N  1 rule. The additional split gives more freedom in estimating CP values and, thus, has a potential to reach MTA target more closely. However, the designer should follow also the rule of thumb: try to keep the number of splitters at a minimum. Even such simple example illustrates the calculation load and the necessity of problem understanding with the PDM approach. Nevertheless, there are numerous applications of the PDM or some techniques from PT to industrial cases; for instance, more

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€ recent are contributions by Markowski (2000), Ozkan and Dincˇer (2001), Matijasevic and Otmaievic (2002), Noureldin and Hasan (2006), and Yoon et al. (2007). Due to some limitations of the PDM, solutions obtained are rarely optimal, and further improvements are possible. The most substantial effect on TAC can be achieved by minimizing number of matches via eliminating loops, called ‘loop breaking’. The general concept was explained in Section 16.2 within the discussion on the minimum number of matches—see, for example, Fig. 16.3. Because the final network is achieved in the PDM by simple merging of two or more subnetworks calculated independently, it most often has some loops across pinch/pinches. Also, a HEN can feature additional loops, not across pinch, since there is inadequacy of the PDM matching rules, particularly for matches of streams away from the pinch. The latter loops can be broken by a simple method developed by Su and Motard (1984). However, the technique should not be employed to breaking loop via pinch since it has no control mechanism for heat recovery decrease—see, for example, Engel and Morari (1988). Notice that by breaking such loop, one increases utility usage if temperature approaches in matches are limited by HRAT value, as is the case in the PDM. The increase is called energy penalty. Hence, an improvement of TAC can be achieved only if deleting a match causes a saving higher than an increase of utility cost. However, the use of DTA concept can result in smaller utility cost penalty than for single ΔTmin. Even no additional cost of energy can be achieved. This effect is identical to that explained in Section 16.2 on match number minimization. Relaxation of outlet temperature conditions can also increase chances of minimizing energy penalty. The objective of loop breaking is minimizing the number of matches at minimum energy penalty. There are several more or less heuristic approaches to loop breaking, for instance, contributions by Trivedi et al. (1990), Zhu et al. (1993, 1999). The works of Pethe et al. (1989), Gundersen et al. (1991), and Han et al. (1998) are also the contributions to loop-breaking identifications. Here, we present a systematic method developed by Jezowski et al. (2001a,b) and addressed shortly by Jezowski et al. (2000a). The approach is based on solution of MILP optimization model for HEN of given structure, that is, the HEN from the structure-development stage of the PDM. Notice that the structure is fixed, but in the loop-breaking approach, it can subject to changes opposite to, for instance, parameter NLP optimization. To formulate loop-breaking model, we need to model HEN of given topology. We will apply here also the model of Jezowski et al. (2001a,b) since it seems compact, numerically efficient, and easy to code. Notice that it can be employed also to other problems like HEN simulation, parameter optimization, or HEN retrofit with ‘network pinch’ approach. The latter issue will be addressed in Chapter 19. HEN consists of the following elements: matches (i.e. process–process matches, heaters, and coolers), splitters, mixers, and substreams. A substream is a part of a stream (process stream and utility as well) between two apparatuses, between inlet to a HEN and an apparatus, and between an apparatus and outlet from a HEN, where by apparatus we mean a match, a splitter, or a mixer. 





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Fig. 18.3 Illustration for HEN mathematical model applied in the loop-breaking method by Jezowski et al. (2001a,b). 

An index is assigned to each element. The following index sets are created: K ¼ {k j k is match (kE, process-process stream match; kH, heaters; kC, coolers)} L ¼ {l j l is substream (lH, hot substream; lC, cold substream)} M ¼ {m j m is mixer} N ¼ {n j n is splitter }

The scheme of numbering substreams is explained in Fig. 18.3. Also, matrices and vectors for this network (defined in the following) are inserted in this figure. Substreams are numbered in ascending order from 1 starting with the most upper stream in grid representation. In the case of hot stream, the numbers increase from left side while for cold streams in opposite direction, that is, from right to left side of grid representation. This convention has to be obeyed for mixers and splitters, too. It is important to note that a branch of a split is treated as a stream in the numbering scheme. Other elements, that is, apparatus, are sequentially numbered from 1, though this is not strict requirement. Similar to the models for MNU and MTA, target index sets for streams are as follows: H ¼ {i j i is hot stream} C ¼ {j j j is cold stream}

It is convenient for purposes of the model to classify substreams into inlet and outlet substreams. Under the assumption that CP values of all streams do not depend on temperature, each substream has uniquely assigned two parameters: T and CP. In order to simplify presentation of the model, it is assumed in the following that maximum two branches can exist. This limitation can be easily removed since each branch can be further split into two and so on. Let us define the following vectors and matrices based on the index sets and the numbering scheme defined above. The reader is also referred to Fig. 18.3 for explanation.

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Energy Optimization in Process Systems and Fuel Cells

(1) E—matrix for assigning substreams to matches. It has as many rows as many matches exist in a HEN and two columns. Elements of E have the following meaning: e(k,1) ¼ lH—number of inlet hot substream to kth match e(k,2) ¼ lC—number of inlet cold substream to kth match Notice that from the convention of numbering substreams, an outlet hot/cold substream has a number greater by one than hot/cold inlet substream from match k. Hence, matrix E provides sufficient information on assignments of match to substreams since numbers of outlet substreams are known from numbers of inlet substreams to the match. (2) S—matrix for assigning substreams to splitters. It has as many rows as many splitters exist in a HEN and two columns. Elements of S have the following meaning: s(m,1) ¼ lS—number of inlet substream to mth splitter s(m,2) ¼ lb—number of outlet substream to mth splitter such that lb 6¼ lS + 1 (it is called type ‘b’ substream) By convention of numbering, the second outlet substream has always number lS + 1 and wasn’t to be included into S. (3) MI—matrix for assigning substreams to mixers. It has as many rows as many mixers occur in a HEN and two columns. Elements of MI have the following meaning: mi(n,1) ¼ lM—number of outlet substream from mixer n mi(n,2) ¼ lb—number of inlet substream to nth mixer such that lb 6¼ lM  1 (it is called type ‘b’ substream) By convention of numbering, the second inlet substream has always number lS  1 and wasn’t to be included into matrix MI. (4) ISH—vector for numbers of inlet hot substreams to a HEN, with elements ish(i) ¼ lH; i 2 H (5) OSH—vector for numbers of outlet hot substreams from a HEN, with elements osh(i) ¼ lH; i 2 H (6) ISC—vector for numbers of inlet cold substreams to a HEN, with elements isc( j) ¼ lC; j 2 C (7) OSC—vector for numbers of outlet cold substreams from a HEN, with elements osc( j) ¼ lC; j 2 C (8) TIH—vector of given inlet temperatures of hot streams, with elements tih(i) ¼ Tiin; i 2 H (9) TIC—vector of given inlet temperatures of cold streams, with elements tic( j) ¼ Tjin; j 2 C (10) TOH—vector of given outlet temperatures of hot streams, with elements toh(i) ¼ Ti out ; i 2 H (11) TOC—vector of given outlet temperatures of cold streams, with elements toc( j) ¼ Tj out ; j 2C (12) CP—vector of given CPs of substreams in a HEN, with elements CP(l); l 2 L (since the assumption of constant CP of stream assignment of CP values to substreams from stream data is straightforward) (13) Q—vector of heat loads of matches in a HEN, with elements Q(k); k 2 K (14) T—vector of temperatures of substreams in a HEN, with elements T(l); l 2 L

The vectors and matrices are sufficient to define uniquely a HEN. Though there are many matrices and vectors, they are very compact. In order to define existence or nonexistence of a match in a HEN, it is necessary to use binary variables y(k); k 2 K, such that  1 if match k is in y ðk Þ ¼ (18.4) 0 if not

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In order to ensure linearity of optimization model, the following assumptions have to be imposed: (1) Heat-transfer areas of heat exchangers will not be accounted for in optimization. (2) CP values of substreams have to be fixed. In consequence, CP values for branches of split streams are fixed, too. Also, the number of branches can’t change. It is forbidden to eliminate an existing branch. Hence, one can’t eliminate all matches at a branch since Jezowski et al. (2001a,b) suggested that such ‘empty’ branch has no physical meaning except for control purposes as a bypass. 

For the assumptions, the model is of MILP type with following decision variables: T(l), Q(k), and y(k); l 2 L and k 2 K Due to assumptions (1) and (2), the goal function cannot include area and split fractions. Hence, the goal function is given by (18.5) and comprises utility cost and the term corresponding to fixed cost of matches.

min

X k2K

"

X   X   y ðk Þ + η Q kH + Q kC kH 2K

#! (18.5)

kC 2K

Parameter η in (18.5) is used to convert both terms: number of matches and utility load to common basis. The value of η has to be assumed by the user. Based on typical relations, investment cost of matches versus utility cost, Jezowski et al. (2001a,b) recommend value of η of order 0.01–0.001. It is important to note that the lower value is given for η and the higher preference is given for reducing number of matches but less for saving energy. The designer can account for some specific features of economic scenario by choosing η value. Before presenting a detailed optimization model, we will first discuss its important features. In order to ensure a possibility of minimizing the energy penalty due to crosspinch loop breaking, Jezowski et al. (2001a,b) proposed to 



(a) use DTA concept, that is, to apply EMAT  HRAT, (b) apply ‘small’ superstructure instead of existing structure.

Notion of ‘small’ superstructure is similar to that applied by Asante and Zhu (1996a) for ‘network pinch’ approach to retrofitting HEN (see Chapter 19). Such superstructure involves some slack matches with zero heat load seeded by the designer into existing structure. Slack matches can be activated by optimization subroutine. Thus, some structural changes are possible additionally to deletion of exchangers. Let us note that such structural changes are necessary to achieve an optimum since Sagli et al. (1990) and Gundersen et al. (1991) observed that it may be impossible to reach optimal HEN from a local optimum by standard evolutionary approach only (i.e. without new matches and/or relocations of existing heat exchangers). There are no strict rules on how to build a superstructure. Jezowski et al. (2001a,b) advised to add empty 

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Energy Optimization in Process Systems and Fuel Cells

heaters and coolers at the ‘end’ of streams and, also, into such locations that can result in independent subnetworks. Constraints of the basic loop-breaking optimization model (called LB model) are given in the following (the goal function is defined by Eq. 18.5): Constraints of LB model (A) Assignments of inlet temperatures to substreams

T ðishðiÞÞ ¼ tihðiÞ; i 2 H

(18.6)

T ðiscð jÞÞ ¼ ticð jÞ; j 2 J

(18.7)

(B) Heat balances for matches (process exchangers, heaters, and coolers) B1hot streams T ðeðk, 1ÞÞ  T ðeðk, 1Þ + 1Þ ¼

QðkÞ ; k2K CPðeðk, 1ÞÞ

B2cold streams T ðeðk, 2Þ + 1Þ  T ðeðk, 2ÞÞ ¼

Q ðk Þ ; k2K CPðeðk, 2ÞÞ

(18.8)

(18.9)

Constraints (18.8) ensure also that outlet temperatures of hot substreams are not higher than inlet temperatures. The same effect has conditions (18.9) for cold substreams. (C) Heat balances for splitters

T ðs ðm, 1Þ + 1Þ ¼ T ðs ðm, 1ÞÞ T ðs ðm, 2ÞÞ ¼ T ðs ðm, 1ÞÞ

 m2M

(18.10)

(D) Heat balances for mixers T ðmiðn, 1Þ + 1Þ ¼

T ðmiðn, 1ÞÞ  CPðmððn, 1ÞÞ + T ðmiðn, 2ÞÞ  CPðmiðn, 2ÞÞ ;n 2 N CPðmiðn, 1Þ + 1Þ

(18.11)

(E) Thermodynamic constraints for matches T ðeðk, 1ÞÞ  T ðeðk, 2Þ + 1Þ  EMAT + DT ðyðkÞ  1Þ; k 2 K

(18.12)

T ðeðk, 1Þ + 1Þ  T ðeðk, 2ÞÞ  EMAT + DT ðyðkÞ  1Þ; k 2 K

(18.13)

The above inequalities deactivate also thermodynamic conditions for nonexistent matches. Notice that for yk equals 1, the left-hand sides become negative if parameter DT is given sufficiently large number. Similar conditions will be applied in Chapters 19 and 20 in modelling HEN design optimization approaches.

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(F) Conditions on outlet temperatures from HEN

tohðiÞ ¼ T ðoshðiÞÞ; i 2 H

(18.14)

tocð jÞ ¼ T ðoscð jÞÞ; j 2 C

(18.15)

(G) Logical constraints for heat loads of matches (G1) Constraints ensuring that deleted matches feature zero heat loads

QðkÞ  QðkÞmax yðkÞ; k 2 K

(18.16)

Parameter Q(k)max can be replaced by large number Qmax identical for all k 2 K. (G2) Logical conditions to prevent elimination of a branch

X

yðk0 Þ > 0

(18.17)

k0 2K 0

where K0 is the set of matches placed at the same branch. (H) Explicit constraints on variables

QðkÞ  0; k 2 K

(18.18a)

yðkÞ ¼ 0, 1; k 2 K

(18.18b)

T ðlÞ  0; l 2 L

(18.18c)

Other problem-specific constraints can be also included. For instance, one can insert conditions (18.19) to prevent deleting a ‘must-be’ match from a HEN. yðkkÞ ¼ 1; kk 2 KK

(18.19)

where KK is subset of must-be matches. Another restriction that can be helpful in optimization is to impose a lower bound on final number of matches in a HEN. Constraint has the simple form X

y ðk Þ  N l

(18.20)

k2K

where Nl is the lower bound on the number of matches. Multiple global optima can exist for MILP optimization models. In order to generate all optimal solutions, one has to add integer cuts that prevent solutions from previous runs to be computed once more. Here, we present example 18.1 taken from Jezowski et al. (2001a,b) to show possibilities of generating alternative solutions. The problem has been also investigated in Trivedi et al. (1990), Zhu et al. (1993), and Zhu et al. (1999). Initial HEN is given in 

588

H1 H2 H3 H4 C1 C2 C3

Energy Optimization in Process Systems and Fuel Cells

160°C

4 230°C 172°C 2 3

249°C

150°C

5

227°C

7

271°C 5 150°C 183 3 4 492 70

217°C 250°C

H1

231

2 161

216°C

150°C

9 8

150°C

6

160°C

CP 143°C 110°C 11 C1 7.032 231 138°C 8.440

150°C

140°C

10 8

140°C 403 6 678

7 910

6

106°C 11.816 C2 118 146°C 5.600 96°C 9.144

126°C 123°C 10 11 9 102 22 51

116°C 140°C

7.296 18

12

Fig. 18.4 Initial HEN for example 18.1: H1, heater; C1 and C2, coolers; CP values in bold; heat load beneath heat exchanger.

Fig. 18.4. It contains 13 matches: 10 process heat exchangers, one heater, and two coolers. Heat load of the heater is equal to 231 kW. The HEN meets the basic ‘standards’ of PT, that is, divided at pinch (Tp ¼ 150/140, HRAT ¼ 10); consumes the minimum utility load; and features the minimum number of matches according to (16.2). As we have mentioned in the preceding parameter, η in goal function (18.5) can be employed to control trade-off: number of unit-energy penalty. EMAT influences also this trade-off since the initial HEN had been designed with single ΔTmin equal to HRAT ¼ 10. By varying parameters η and EMAT (HRAT is fixed at 10), Jezowski et al. (2001a,b) calculated certain alternative networks. Table 18.1 gathers main parameters (number of matches and energy penalty) for the alternatives. Notice that the approach was able to reduce number of units by five and even six in one case. Network no. 3 seems to be the best solution since it has small energy penalty. It is shown in Fig. 18.5. However, solution no. 4 from Table 18.1 can also be of interest if capital expenditure is high in relation to energy cost. It is interesting to observe the reason of achieving number of units below the minimum from (16.1), that is, below (N  1) rule; notice that total number of streams (N) in this example is 9. The network is shown in Fig. 18.6. Analysis reveals that seven matches results from division into two subnetworks as Fig. 18.7 illustrates. In order to choose the best solution generated by loop-breaking approach, additional calculations are needed to determine total cost of networks. Additionally, the solutions aren’t optimal in terms of TAC. This is the effect of model linearity. To account for TAC optimal solutions, the assumption imposed by Jezowski et al. (2001a,b) as for fixed values of CP has to be removed. Also, it is necessary to include capital cost of heat exchangers into the goal function and add design equations for surface area of matches. The model will become more general but, likely, difficult to solve since it would be MINLP. 



Heat-exchanger network synthesis

H1 H2 H3 H4 C1 C2 C3

589

CP

160°C

110°C

249°C 227°C

7

271°C

5 141.0°C

8

C1 7.032 351.6 138°C 8.440

9

106°C 11.816 C2 28.3 146°C 5.600 96°C 9.144

108.4°C

6 138.1°C 5 8 199.9 385.3

160°C 217°C 250°C

225.3°C

235.3 256.7

H1 263.8

6 700 224.7°C 7 1016.2

116°C

9 736.9

140°C

7.296 18

Fig. 18.5 Solution no. 3 from Table 18.1 to example 18.1 (symbols as in Fig. 18.4). CP

H1 H2 H3 H4 C1 C2 C3

160°C

110°C

249°C

225.3°C

2

227°C

8

271°C

C1 7.032 361.60 138°C 8.440

9 177.4°C

106°C C2 11.816 844.52 146°C 5.600 96°C 9.144

6

160°C

8 586.21

217°C 250°C

189.9°C H1 1080.05 162.2°C

217.7°C

6 700

116°C

9 736.89

140°C

7.296 18

2 199.99

Fig. 18.6 Solution no. 4 from Table 18.1 to example 18.1 (symbols as in Fig. 18.4).

C2 h3

h1

h2

C1

9 c2

8 cu

h4

c1

hu

6 c3

2

Fig. 18.7 Two independent subnetworks in the network from Fig. 18.6.

H1

590

Energy Optimization in Process Systems and Fuel Cells

18.2.2 Sequential design with optimization approaches Some optimization targeting approaches are able to give heat load distribution (HLD) optimal in regard to all or some targets, for example, TAC-Transp model in Chapter 17. As we have shown in that chapter, HLD does not define uniquely HEN structure. The designer has to develop arrangements for all matches in the HLD. For small-scale problems, particularly such as those that do not require splitters and mixers, this can be performed by inspection with relatively few trials—see examples in Chapter 17. However, for more complex problems, a systematic procedure is needed. HEN consists not only of heat exchangers but also of splitters and mixers. This fact had not been given sufficient consideration in early stages of HEN design approach development. The paradigm of ‘homogenous networks’ had dictated to pay interest to heat exchange only. In consequence, splitters and mixers had not been included explicitly into conceptual models and into calculation procedures. For instance, even in insight-based methods, such as the PDM, stream branching causes serious difficulties. Also, there are no targeting approaches for determining location and the number of splitters and mixers since they are assumed to have no cost. Notice, however, that there are some more recent approaches that account for topological features—they will be mentioned in the next section. To deal with HEN synthesis problem in a systematic manner, a superstructure concept is often employed as the rigorous and general technique. The concept found a wide application in simultaneous design methods for various systems. First, a superstructure is created, and then, optimal structure is calculated by optimization. Simultaneous approaches to HEN synthesis are explained in the next section. However, the superstructure concept can be also applied to design a HEN with known HLD, that is, in sequential procedure for HEN synthesis. A superstructure, in general, must involve all possible structures of HEN. Due to the fact that matches and their loads are fixed by HLD, the superstructure can be substantially reduced. This results in simplification of optimization problem since it is sufficient to solve NLP problem instead of the MINLP one. There are two well-established propositions of superstructures for HEN synthesis: this by Floudas et al. (1986) and Ciric and Floudas (1990a,b) and that by Yee et al. (1990a) and Yee and Grossmann (1990). The latter is described in full in the next section. Here, we will concentrate on the use of superstructure by Floudas (Ciric and Floudas), abbreviated to C–F superstructure in the following, to sequential HEN synthesis. For such design mode, the superstructure contains only those matches that corresponds to given HLD. C–F superstructure consists of superstructures for each match in the HLD. To allow for all possible arrangements, the match superstructure has the following features: (a) There is a mixer at the inlet stream to heat exchanger. The number of streams entering the mixer is equal to the number of heat exchangers associated with the stream. (b) There is a splitter at the outlet stream from the heat exchanger. The number of streams from the splitter equals the number of heat exchangers associated with the stream.

Heat-exchanger network synthesis

591

2

Fs2 s2

h1

T

1 s2

F M3

Fh1, Th1in

1

Ts1

M1

1

1

Fs1

TM1, FM1

Tcin

s1

1 s3

F

Fs32

Th1out

T2

s3

h2

2

Fh2, Th2in

TM2 FM2

FC

c

2

Fs1 Ts1

M2

Th2out

Fig. 18.8 C–F superstructure for the illustrative example: S1 and S2, splitters; M1 and M2, and M3, mixers.

Fig. 18.8 shows the example of the superstructure for the illustrative case of two matches. This illustrative example corresponds to the HLD with two matches: h1-c and h2-c of loads qh1,c and qh2,c. Note that the superstructure embeds all possible arrangements of two matches as shown in Fig. 18.9, including two arrangements such as those discussed in Section 16.2 (see Fig. 16.5) called ‘parallel pinch-crossing arrangement’. These connections have been given criticism as being impractical. In the sequential HEN synthesis, utility loads are fixed. Also, the number of heat exchangers is given. Hence, these parameters aren’t optimized. In consequence, no binary variables for heat exchangers are necessary. Due to the fact that no cost (including fixed charge) is given to splitters and mixers and, also, to piping sections, binary variables aren’t needed for this equipment. The optimization problem does not include discrete parameters in contrast to optimization problem, which is to be solved in h1

h1

1

c h2

h1

h2

1

1

2

c 2

h2 h1 c

2

h2 h1

1

1

2

c h2

2

Fig. 18.9 HEN structures embedded in the superstructure from Fig. 18.8.

c

592

Energy Optimization in Process Systems and Fuel Cells

simultaneous methods for HEN synthesis. To delete splitter, mixer, and the associated branches from the superstructure, continuous variables are sufficient, namely, if flow rate through a connection is zeroing the branches and the apparatus it goes into is deleted. Here, we begin the explanation of optimization model for C–F superstructure for the illustrative example from Fig. 18.8. Parameters of the model Fc—flow rate of stream c cpc—heat capacity of stream c Fh1 and Fh2—flow rate of stream h1 and h2 (respectively) EMAT—minimum temperature approach in matches (notice that EMAT can be matchdependent) qh1,c —heat load of match h1-c in exchanger 1 qh2,c—heat load of match h2-c in exchanger 1 Tc in and Tc out —inlet and outlet temperature (respectively) of stream c Th1in and Th1in—inlet and outlet temperature (respectively) of stream h1 in exchanger 1 Th2in and Th2in—inlet and outlet temperature (respectively) of stream h2 in exchanger 1 Uh1,c and Uh2,c—overall heat-transfer coefficients in the matches β and γ—cost coefficients for heat-exchanger investment cost

Heat-transfer surface area is calculated for 1-1 heat exchangers from the formula A¼

q LMTD  U

(18.21)

Logarithmic mean temperature differences are most often calculated in HEN optimization models from Chen (1987) or Paterson (1984) approximations—Eqs (18.22), (18.23), respectively—to eliminate difficulties with standard formula (18.24) for LMTD, particularly in cases where both temperature differences are identical. Chen equation was employed in this model. It slightly overestimates heat-transfer area, while Paterson’s equation underestimates. Thus, Chen’s approximation should provide ‘safer’ design. Also, this formula gives LMTD equal to zero if both temperature differences are zero, too. Generally, the influence of approximation applied is minor and is usually discussed only when comparing results of synthesis methods. However, modelling HEN using equations with LMTD parameters can make some numerical troubles. Hence, Akman et al. (2002) suggested applying ‘LMTD-free’ model. Similar heat-exchanger model (though less general) will be presented in Chapter 19 in presentation of retrofit approach by Bochenek (2003): LMTD ¼ ½Δt1  Δt2  0:5ðΔt1 + Δt2Þ1=3

(18.22)

   2 1=2 1 Δt1 + Δt2 LMTD ¼ ðΔt1 + Δt2Þ  3 3 2

(18.23)

Heat-exchanger network synthesis

LMTD ¼

593

Δt1  Δt2 Δt1 ln Δt2

(18.24)

where Δt1 and Δt2 are temperature differences at ends of a match. The optimization model for HEN synthesis with known HLD is as follows: Synthesis-HLD-C–F model (A) Goal function (includes area-dependent cost of heat exchangers)

8 <

0 qh1,c

1γh1, c

A i min βh1,c @h : Δt11 Δt21 0:5ðΔt11 + Δt21 Þ1=3 Uh1,c 0 1γh2, c ) qh2,c A i + βh2, c @h Δt12 Δt22 0:5ðΔt12 + Δt22 Þ1=3 Uh2,c

(18.25)

(B) Mass balances for splitters

Fc ¼ F1S1 + F2S1

(18.26a)

FM1 ¼ F1S2 + F2S2

(18.26b)

FM2 ¼ F1S3 + F2S3

(18.26c)

(C) Mass and heat balances for mixers

FM1 ¼ F1S1 + F1S3

(18.27a)

FM1 TM1 ¼ F1S1 Tcin + F1S3 T 2

(18.27b)

FM2 ¼ F2S1 + F1S2

(18.27c)

FM1 TM2 ¼ F1S2 Tcin + F1S2 T 1

(18.27d)

Fc ¼ F2S2 + F2S3

(18.27e)

FM1 Tcout ¼ F2S2 T 1 + F1S3 T 2

(18.27f)

(D) Heat balances for exchangers

  qh1, c ¼ Fc cpc T 1  TM1

(18.28a)

594

Energy Optimization in Process Systems and Fuel Cells

  qh2, c ¼ Fc cpc T 2  TM2

(18.28b)

(E) Definitions of temperature differences in heat exchangers in Δt11 ¼ Th1  T1

(18.29a)

out  TM1 Δt21 ¼ Th1

(18.29b)

in  T2 Δt12 ¼ Th2

(18.29c)

out Δt22 ¼ Th2  TM2

(18.29d)

(F) Constraints on temperature differences

Δt11  EMAT

(18.30a)

Δt21  EMAT

(18.30b)

Δt21  EMAT

(18.30c)

Δt22  EMAT

(18.30d)

(G) Constraints on variables

F1Sk , F2Sk  0; k ¼ 1, 2,3 Fm1 , Fm2  0

(18.31)

The above model is nonlinear due to nonlinear goal function and heat balances of mixers. Biegler et al. (1997) suggested that it should not cause troubles with calculating the global optimum. However, certain literature references show that NLP optimization of HEN with fixed structure is not a trivial task, even for small problems. Some evidence is that the small optimization model for two exchangers has been included into benchmark of test tasks for global optimizers. Quesada and Grossmann (1993) proposed advanced global optimization algorithm with spatial branch and bound procedure and convex underestimators. Interestingly, a simple random-search approach of Luus and Jaakola (see Chapter 1) was able to solve such NLP problems to global optima as reported in Luus (1993) though for small-scale tasks. Brief discussion of optimization robustness issue will be given at the end of the chapter.

18.3

Simultaneous approaches to HEN synthesis

In this section, we will address mainly the approaches applying superstructure developed by Yee et al. (1990a,b). The synthesis method of Ciric and Floudas (1990a,b), with C–F superstructure, is the alternative; in fact, it is even more rigorous.

Heat-exchanger network synthesis

595

However, the former attracted greater attention and followers who proposed various extensions and modifications. Such wider application is, most likely, caused by lower ‘level’ of nonlinearity of MINLP optimization model that should allow finding at least good local optima with available solvers. The key feature of the model is that only goal function is nonlinear, while model constraints are all linear. The superstructure of Yee et al. (1990a,b), called Y–G superstructure in the following, is based on the following key concepts: l

l

The superstructure is organized in a number of stages—stage-wise superstructure. Notice that temperatures of streams exiting stages are not fixed—they are decision variables in optimization in contrast to the division into temperature or enthalpy intervals. Streams entering each stage are split so as to achieve all possible matches between hot and cold process streams. Then, the streams are mixed at the outlet of stages.

The following assumptions are imposed: l

l

l

l

Heater is placed at the highest-temperature region of cold process stream. One heater suffices for one process stream. Cooler is located at lowest-temperature region of hot process stream. One cooler services one hot process stream. Mixing of streams is isothermal, that is, each branch of stream to a mixer has identical temperature. Heat exchangers are modelled as matches (1-1 units).

The illustration of the superstructure with two stages and for two hot (H1 and H2) and two cold process streams (C1 and C2) is shown in Fig 18.10. Here, we present the optimization model following the explanation in Biegler et al. (1997). The formulation uses only one hot utility and one cold utility. Because steam is commonly applied, heating utility index s will be used instead of general index m.

Stage k = 1 s

TC1out H1

tC1, 1

H1-C1

tH1, 1

Stage k = 2

tC1, 2 tH1, 2

H1-C2

H1-C1

tC1, 3 c1 tH1, 3

H1-C2

tC2, 3 s

TC2out

H2-C1

tC2, 1

H2

tH2, 1

tC2, 2

c2

H2-C1

tH2, 3 w

tH2, 2 H2-C2

H2-C2 Temperature k=1 locations

TH1out

k=2

Fig. 18.10 Illustration of Y–G superstructure for HEN synthesis.

k=3

out TH2

596

Energy Optimization in Process Systems and Fuel Cells

Likewise, index w (from cooling water) will denote cooling utility instead of general index n. Indices and index sets H ¼ {i j i is hot process stream} C ¼ {j j j is cold process stream} k—stage or temperature location; stages are numbered from 1 to K with descending temperature; for stage k, there are two temperature locations, k at inlet and k + 1 at outlet s—heating utility w—cooling utility

Model parameters CPi/CPj—CP of stream i/j EMAT—minimum temperature approach in matches K—number of stages phus—unit cost of heating utility s pcuw—unit cost of cooling utility w Qmax and D—sufficiently large numbers Ti in =Tjin —inlet temperature of stream i/j to the superstructure Ti out =Tjout —outlet temperature of i/j to the superstructure Ts in =Twin —inlet temperature of s/w Ts out =Twout —outlet temperature of s/w Uij, Usj, and Uiw—overall heat-transfer coefficients in matches α, β, and γ—parameters for investment cost of a heat exchanger

Model variables LMTDijk—logarithmic mean temperature difference of match i with j in stage k LMTDsj—logarithmic mean temperature difference of match s with j LMTDiw—logarithmic mean temperature difference of match i with w qijk—load of match i with j in stage k qiw—load of match i with w qsj—load of match s with j tik—temperature of i at location k tjk—temperature of j at location k yijk—binary variable for match i with j in stage k; yijk ¼ 1 means that the match exists yciw—binary variable for match of i with w yhsj—binary variable for match of s with j Δtijk—temperature difference for match i with j at temperature location k Δtiw—temperature difference in match i with w Δtsj—temperature difference in match s with j

Notice that Biegler et al. (1997) defined temperature differences in matches with utilities (Δtiw) and (Δtsj) for outlet temperatures of utilities only. Thus, they assumed that inlet temperatures of utilities are sufficiently high for utility s and sufficiently low for utility w to ensure appropriate temperature differences. This assumption can be easily removed. Similarly to NLP model of Ciric and Floudas (Synthesis-HLD-C–F model), LMTD is approximated by Chen’s equation (18.22).

Heat-exchanger network synthesis

597

Synthesis-Simult-Y–F model 82 39 2 3 = < X K XXX X X X min 4 qsj phus + qiw pcuw 5 + 4 yijk αij + yhsj αsj + yciw αiw 5 ; : 2

i2H

j2C

K XXX βij +4 i2H j2C k¼1

qijk Uij LMTDij

i2H j2C k¼1

!γ ij +

X j2C

βsj

qsj Usj LMTDsj

i2H

j2C

!γ sj +

X i2H

 βiw

qiw Uiw LMTDiw

γ iw

3 5

(18.32)

The first term defines total cost of both utilities; the second defines fixed charge for heat exchangers, heaters, and coolers; and the last one defines area-dependent cost of all heat exchangers. Here, only for brevity’s sake, we inserted LMTD parameters into the goal function. It should be noticed, however, that this spoils the linearity of the model constraints. Hence, parameters LMTD should be replaced in (18.32) by right-hand sides of LMTDijk ¼



   1=3 Δtijk Δtij, k + 1 0:5 Δtijk + Δtij, k + 1

    1=3 LMTDiw ¼ Δtiw Tiout  Twin 0:5 Δtiw + Tiout  Twin h



i1=3 LMTDsj ¼ Δtsj Tsin  Tjout 0:5 Δtsj + Tsin  Tjout The constraints of the model are as follows: (1) Overall heat balances of streams The balances should also ensure that required outlet temperatures of process streams would be reached. For hot process streams, the required enthalpy changes must be satisfied by matches with cold process streams and utility w. Likewise, enthalpy change of each cold stream must be equal to loads on matches with cold process streams and utility s: K X   X CPi Tiin  Tiout ¼ qijk + qiw ; i 2 H

(18.33a)

K X

X CPj Tjout  Tjin ¼ qijk + qsj ; j 2 C

(18.33b)

k¼1 j2C

k¼1 i2H

(2) Heat balances for stages In addition to overall balances, we have to include heat balances for each stage. Due to the assumption of isothermal mixing, heat balances of mixers are unnecessary. Also, CP parameters for branches aren’t required: ðtik  ti, k1 ÞCPi ¼

X j2C

qijk ; i 2 H; k ¼ 1,…,K

(18.34a)

598

Energy Optimization in Process Systems and Fuel Cells



X  tjk  tj, k + 1 CPj ¼ qijk ; j 2 C; k ¼ 1, …,K

(18.34b)

i2H

(3) Assignments of inlet temperatures to the superstructure The following equalities are necessary to assign given inlet stream temperatures to hot streams entering the first stage and to cold streams entering the last stage: ti1 ¼ Tiin ; i 2 H

(18.35a)

tjK ¼ Tjin ; j 2 C

(18.35b)

(4) Constraints ensuring feasibility of temperatures in the superstructure The constraints enforce that temperatures at outlet from successive stages decrease. Since heat load on a certain stream in a stage can be zero, the weak inequalities should be used: tik  ti, k + 1 ; i 2 H; k ¼ 1, …,K

(18.36a)

tjk  tj, k + 1 ; j 2 C; k ¼ 1, …,K

(18.36b)

Also, it is necessary to impose a limit on outlet temperature from each stage in respect to fixed outlet temperature. The constraints have to account for the fact that some or all streams will be heated up in heaters and cooled down in coolers. Hence, weak inequalities have to be used. ti, K + 1  Tiout ; i 2 H

(18.36c)

tj, 1  Tjout ; j 2 C

(18.36d)

(5) Definitions of heat loads of heaters and coolers



qsj ¼ CPj Tjout  tj1 ; j 2 C

(18.36c)

  qiw ¼ CPi ti, K + 1  Tiout ; i 2 H

(18.36d)

(6) Logical conditions on heat loads The conditions ensure that matches that do not exist, that is, have binary variables equal to zero, must have also zero heat load. For simplicity’s sake, single parameter Qmax is used in all constraints: qijk  Qmax yijk  0; i 2 H; j 2 C; k ¼ 1, …,K

(18.37a)

qiw  Qmax yciw  0; i 2 H; k ¼ 1,…,K

(18.37b)

qsj  Qmax yhsj  0; j 2 C; k ¼ 1,…, K

(18.37c)

Heat-exchanger network synthesis

599

(7) Calculations of approach temperatures at temperature locations For each match within the superstructure, temperature approaches at both sides have to be calculated in the model. Note that they would be determined for nonexistent matches, too. Hence, logical conditions have to be applied to deactivate the thermodynamic conditions for nonexistent matches and to ensure that temperatures of matches yield positive temperature differences at both sides. Biegler et al. (1997) formulated the conditions as weak inequalities. Minimization of HEN cost will cause maximization of temperature differences. Hence, constraints for active matches with binaries equal to 1 will be active:     Δtijk  tik  tjk + D 1  yijk ; i 2 H; j 2 C; k ¼ 1, …,K 

Δtij, k + 1  ti, k + 1  tj, k + 1





 + D 1  yijk ; i 2 H; j 2 C; k ¼ 1,…, K

(18.38a) (18.38b)



  Δtsj  Tsout  tj, 1 + D 1  yhsj ; j 2 C; k ¼ 1,…,K

(18.38c)

    Δtiw  ti, K + 1  Twout + D 1  yciw ; i 2 H; k ¼ 1,…, K

(18.38d)

The above constraints ensure that temperature approaches aren’t negative for active matches. Usually, however, additional condition is imposed that they should be larger than given EMAT value. Hence, we should also add Δtijk  EMAT; i 2 H; j 2 C; k ¼ 1,…, K + 1

(18.38e)

Δtiw  EMAT; i 2 H

(18.38f)

Δtsj  EMAT; j 2 C

(18.38g)

Notice that match-dependent EMAT can be applied. (8) Conditions on variables qijk ,qiw , qsj ,tik , tjk  0

(18.39a)

yijk ,yc iw , yh sj ¼ 0, 1

(18.39b)

The above model can be easily reduced to a targeting model for the minimum total surface area. Such reduced model has been referred to in Chapter 16 as Yee and Grossmann method. The Y–F superstructure and, in consequence, the model involve some simplifications. First, isothermal mixers are assumed at each stage. Second, all heaters and coolers are placed only at the ends of process streams, that is, at lowest temperatures of hot streams and at highest temperatures of cold streams. Note also that the superstructure does not allow placing more than one heat exchanger at one branch. At last, the number of stages is heuristic parameter. Yee et al. (1990a,b) suggested that setting it at the maximum of hot and cold process streams will result in sufficiently accurate results. However, Daichendt and Grossmann (1994a,b) observed that to achieve a high accuracy the number of stages has to be equal to the number of temperature intervals created for fixed EMAT value. This, however, can lead to a huge number of variables

600

Energy Optimization in Process Systems and Fuel Cells

and prevent efficient solution. The excellent review on computational issues is given in Grossmann and Daichendt (1996). Sorsak and Kravanja removed the limitation to 1-1 units—first in short conference paper (Sorsak and Kravanja, 1999) and, then, in journal paper (Sorsak and Kravanja, 2002). More importantly, they included in the latter work a possibility of heatexchanger-type selection from double-pipe, shell-and-tube (U-tube), and plate-andframe. The selection is performed on the basis of the following: (a) Capital cost via various cost coefficients (b) Permissible surface area range (c) Operational pressure and temperature limitations

Table 18.2 shows the appropriate parameters suggested in Sorsak and Kravanja (2002). The model of heat exchanger does not include pressures, and hence, they are treated as fixed parameters. Initial prescreening procedure suffices for elimination of heatexchanger types that do not satisfy pressure conditions. Permissible surface area ranges were accounted for by cost coefficients α and β. Sorsak and Kravanja (2002) applied linearized investment cost function CAP ¼ α + βA

(18.40)

Thus, the coefficients in Table 18.2 are not identical to those applied in standard formula (14.2). The linearization accounts also for surface area ranges. Due to this, logical conditions for modelling area surface limitations were eliminated from optimization model. However, conditions for operational temperatures have to be used. In order to omit numerical difficulties with standard formula (18.24) for correction factor Ft of shell-and-tube heat exchangers, Sorsak and Kravanja suggested the following approximation: 0

1

B C B C πP B C tan Ft ¼ 1  2 B C 6 X @ X i jA ai ½ lgðRÞ bj ½ lgðRÞ 2 1

i¼0

(18.41)

j¼0

Parameters for heat-exchanger-type selection according to Sorsak and Kravanja (2002)

Table 18.2

Type Double pipe Shell and tube Plate and frame

pmax (MPa)

Tmin ÷ Tmax (°C)

Amin ÷ Amax (m2)

α ($/y)

β ($/y)

30.7 30.7 1.6

100  600 200  600 25  250

0.25  200 10  1000 1  1200

1937 21,615 17,034

201 93 61

Heat-exchanger network synthesis

601

Table 18.3 Parameters a and b in (18.41) according to Sorsak and Kravanja (2002) i

ai

bj

0 1 2 3 4 5 6

15.210806 1.603737 42.337642 – – – –

0.623393 0.698824 0.250187 0.434342 0.301423 0.171389 0.141410

Parameters P and R are defined by Eqs (16.43), (16.44). Coefficients a and b from the original paper are listed in Table 18.3. To present the model of Sorsak and Kravanja (2002), we have to define some new indices, parameters, and variables in reference to Synthesis-Simult-Y–F model. Notice that symbols, which will not appear in the following, have the same meaning as in Yee and Grossman approach. In order to account for heat-exchanger-type selection, Y–G superstructure has to be extended, and also, additional index set is necessary. Each match in Y–G superstructure shown in Fig. 18.10 is modelled as heat-exchanger-type superstructure that comprises all types of interest. To account for possibility that no available exchanger type satisfies all limitations of area, pressure, and temperature ranges, the superstructure has to include bypass, too. Fig. 18.11 illustrates this match-type superstructure. The index set of heat-exchanger types is as follows: HET ¼ {l j l is heat-exchanger type or bypass}

Index 1 denotes double-pipe heat exchanger; 2, plate-and-frame; 3, shell-and-tube; and 4 bypass. Heaters and coolers are modelled as 1-1 units. The overall model keeps the assumption that single heating utility (s) and single cooling utility (w) are used. To shorten equations of the model, we will use index set for stages: ST ¼ {k j k is stage of Y–G superstructure} Double pipe; l = 1

Fig. 18.11 Illustration of match-type superstructure according to Sorsak and Kravanja (2002).

H1–C1 Plate & frame; l = 2 H1–C1 H1

tH1, k+1

tH1, k

Shell & tube; l = 3 tC1, k

H1–C1 bypass: l = 4

tC1, k+1

C1

602

Energy Optimization in Process Systems and Fuel Cells

In addition to parameters of Synthesis-Simult-Y–F model, we need also the following: DFt—sufficiently large number Dlo—negative number (should be large as to absolute value) Tlmax—upper limit on permissible temperature for heat exchanger of type l (l 6¼ 4) Tlmin—lower limit on permissible temperature for heat exchanger of type l (l 6¼ 4) αs, βs —parameters for investment cost of heater αw, βw—parameters for investment cost of cooler αl, βl—parameters for investment cost of process heat exchanger of type l ¼ 1, 2, 3

Also, there are additional variables in the model. At first, additional index l has to be added to the variables associated with matches LMTDijkl, qijkl, and yijkl. The new variables are the following: qlijkl—heat load of heat exchanger A B tiijkl /tiijkl —heat-exchanger inlet/outlet temperature of stream i A B —heat-exchanger inlet/outlet temperature of stream j tjijkl /tjijkl A A Δtijkl /Δtijkl —temperature difference in heat exchanger at side A/B (hot/cold)

The model of Sorsak and Kravanja (2002) is presented in the following in the way similar to that of Yee and Grossman model. Synthesis-Simult-S–K model (" min " +

j2C

qsj phus +

XX X

" +

X

X i2H

X

qiw pcuw

yijkl αl +

i2H j2C k2ST l2HET , l6¼4

XX X

X

i2H j2C k2ST l2HET , l6¼4

+

#)

X  βs j2C

X j2C

 βl

yhsj αsj

+

X i2H

qijkl Uij Ftijkl LMTDijkl

# yciw αiw



 X   qsj qiw +  βw Usj LMTDsj Uiw LMTDiw i2H

(18.42)

The goal function is similar to that from Yee–Grossmann approach. Also, parameters LMTD should be replaced as follows: LMTDijkl ¼

h



i1=3 ΔtAijkl ΔtBij, k + 1,l 0:5 ΔtAijkl + ΔtBij, k + 1,l

    1=3 LMTDiw ¼ Δtiw Tiout  Twin 0:5 Δtiw + Tiout  Twin h



i1=3 LMTDsj ¼ Δtsj Tsin  Tjout 0:5 Δtsj + Tsin  Tjout

Heat-exchanger network synthesis

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The constraints of the model are listed in the following: (1) Overall heat balances of streams

  XX CPi Tiin  Tiout ¼ qijk + qsj ; i 2 H

(18.43a)



XX CPj Tjout  Tjin ¼ qijk + qiw ; j 2 C

(18.43b)

k2ST j2C

k2ST i2H

(2) Heat balances for stages

ðtik  ti, k1 ÞCPi ¼

X

qijk ; i 2 H; k 2 ST

(18.44a)

X  qijk ; j 2 C; k 2 ST tjk  tj, k + 1 CPj ¼

(18.44b)

j2C



i2H

(3) Assignments of inlet temperatures to the superstructure

ti1 ¼ Tiin ; i 2 H

(18.45a)

tjK ¼ Tjin ; j 2 C

(18.45b)

(4) Constraints ensuring monotonic decrease of temperatures and conditions on outlet temperatures

tik  ti, k + 1 ; i 2 H; k 2 K

(18.46a)

tjk  tj, k + 1 ; j 2 C; k 2 K

(18.46b)

ti, K + 1  Tiout ; i 2 H

(18.46c)

tj, 1  Tjout ; j 2 C

(18.46d)

(5) Definitions of heat loads



qsj ¼ CPj Tjout  tj1 ; j 2 C

(18.47a)

  qiw ¼ CPi ti, K + 1  Tiout ; i 2 H

(18.47b)

qijk ¼

X l2HE, l6¼4

qlijkl ; i 2 H; j 2 C; k 2 ST

(18.47c)

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Energy Optimization in Process Systems and Fuel Cells

(6) Logical conditions on heat loads

qijk  Qmax

X

yijkl  0; i 2 H; j 2 C; k 2 ST

(18.48a)

qlijkl  Qmax yijkl  0; i 2 H; j 2 C; k 2 S; l 2 HE

(18.48b)

qiw  Qmax yciw  0; i 2 H; k 2 ST

(18.48c)

qsj  Qmax yhsj  0; j 2 C; k 2 ST

(18.48d)

l2HE, l6¼4

(7) Calculations of approach temperatures at temperature locations with logical conditions

    Δtijk  tik  tjk + D 1  yijk ; i 2 H; j 2 C; k ¼ 1, …,K

(18.49a)

    Δtij, k + 1  ti, k + 1  tj, k + 1 + D 1  yijk ; i 2 H; j 2 C; k ¼ 1, …, K

(18.49b)



  Δtsj  Tsout  tj,1 + D 1  yhsj ; j 2 C; k ¼ 1,…, K

(18.49c)

    Δtiw  ti, K + 1  Twout + D 1  yciw ; i 2 H; k ¼ 1,…, K

(18.49d)

Up to this point, the constraints were almost identical to those in Synthesis-Simult-Y–G model. The following conditions are necessary due to new features of Sorsak and Kravanja model. (8) Conditions modelling feasible heat exchange for shell-and-tube heat exchangers (l ¼ 3) Approximation (18.41) requires additional conditions to ensure feasible heat exchange and, likely, numerical stability. For better understanding, let us define once more parameters P and R using symbols of the model:



tik  ti, k + 1 tj, k + 1  tjk



tj, k + 1  tjk tik  tj, k + 1

The constraints are

  tjk  tj, k + 1 + D 1  yijk3  Dft

(18.50a)

  tik  ti, k + 1 + D 1  yijk3  Dft

(18.50b)

  tik  tj, k + 1 + D 1  yijk3  Dft

(18.50c)

Notice that additional condition necessary to fully satisfy feasible heat exchange is included also in point (7).

Heat-exchanger network synthesis

605

(9) Constraints modelling influence of operating temperatures on heat-exchanger-type selection Hot streams:

X

tik ¼

l2HET

ti, k + 1 ¼

tiAijkl ; i 2 H; j 2 C; k 2 ST

X l2HET

tiBij, k + 1,l ; i 2 H; j 2 C; k 2 ST

(18.51a)

(18.51b)

tiAijkl  yijkl Tlmax ; i 2 H; j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.51c)

tiBij, k + 1,l  yij, k + 1,l Tlmax ; i 2 H; j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.51d)

tiAijkl  yijkl Tlmin ; i 2 H; j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.51e)

tiBij, k + 1,l  yij, k + 1,l Tlmin ; i 2 H; j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.51f)

Cold streams:

X

tjk ¼

l2HET

tj, k + 1 ¼

tjAijkl ; i 2 H; j 2 C; k 2 ST

X l2HET

tjBij, k + 1,l ; i 2 H; j 2 C; k 2 ST

(18.52a)

(18.52b)

tjAijkl  yijkl Tlmax ; i 2 H; j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.52c)

tjBij, k + 1,l  yij, k + 1,l Tlmax ; i 2 H; j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.52d)

tjAijkl  yijkl Tlmin ; i 2 H, j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.52e)

tjBij, k + 1,l  yij, k + 1,l Tlmin ; i 2 H;j 2 C; k 2 ST; l 2 HET, l 6¼ 4

(18.52f)

(10) Definitions and logical conditions for temperature differences

Δtijk ¼

X l2HET

Δtij, k + 1 ¼

ΔtAijkl ; i 2 H; j 2 C; k 2 ST

X l2HET

X l2HET

X l2HET

ΔtBijkl ; i 2 H; j 2 C; k 2 ST

(18.53a)

(18.53b)

ΔtAijkl  yijkl D; i 2 H; j 2 C; k 2 ST; l 2 HET

(18.53c)

ΔtBijkl  yijkl D; i 2 H; j 2 C; k 2 ST; l 2 HET

(18.53d)

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Energy Optimization in Process Systems and Fuel Cells

(11) Logical conditions for bypasses The following conditions allow for negative temperature differences for bypasses:

Δtijk4  yijk4 Dlo; i 2 H; j 2 C; k 2 ST

(18.54a)

Δtij, k + 1, 4  yijk4 Dlo; i 2 H; j 2 C; k 2 ST

(18.54b)

(12) Logical conditions to ensure that single heat-exchanger type or bypass can be selected

X

yijkl ¼ 1; i 2 H; j 2 C; k 2 ST

(18.55)

l2HET

(13) Conditions on variables

qijk , qiw , qsj ,tik , tjk , tiAijkl , tiBijkl , tjAijkl , tjBijkl  0

(18.56a)

yijkl , yc iw , yh sj ¼ 0, 1

(18.56b)

To cope with complex MINLP, Sorsak and Kravanja applied advanced optimization method—significantly modified outer approximation with equality relaxation (OA/ ER) coded in software MIPSYN. Multilevel strategy has been developed where a space of integer variables increases from level to level. Additionally, several ‘tricks’ have been applied, such as integer-infeasible-path optimization, to enhance optimization. The largest example given in Sorsak and Kravanja (2002) contained 13 hot streams and 7 cold streams. They applied 21 stages in Y–G superstructure. The number of binaries amounts to 5753. To cope with such huge number of binaries, they had to divide the problem at pinch. Nevertheless, programme execution was stopped at the second level since too long CPU time (more than 47 h) is required for the second level (with larger binary space). The question arises on whether large MINLP synthesis model are solvable for even medium-scale industrial problems. Notice that the model by Sorsak and Kravanja retains many assumptions of Y–G superstructure and optimization model, for instance, that of isothermal mixing. The elimination of the latter can improve significantly the results as Bj€ ork and Westerlund (2002) have shown for a small example solved by a slightly modified Synthesis-Simult-Y–F model. However, they also noticed a higher CPU time in comparison with the case of isothermal mixing. It is also interesting to note that Bj€ ork and Westerlund claimed that solution of SynthesisSimult-Y–G model is easier if goal function is linear but constraints are not (even with isothermal mixers). Namely, they proposed to applied parameters A* in the last term of goal function (18.32). Hence, it will have the following form: K XXX i2H j2C k¼1

βijk A∗ijk +

X j2C

βsj A∗sj +

X i2H

βiw A∗iw

(18.57)

Heat-exchanger network synthesis

607

In result, area equations have to be added to the constraints, such as (18.58)—here shown for matches of process streams only: qijk ðA∗ Þ1=γij

 Uij LMTDijk  0; i 2 H; j 2 C; k ¼ 1,…,K

(18.58)

The goal function becomes linear at cost of introducing nonlinear constraints into the MINLP model. Such ‘trick’ improves solution if convexification of signomial terms is applied in such a way as in Bj€ ork and Westerlund (2002). The conclusion is that seemingly small changes in problem formulation can enhance solution. They, however, have to be tailored for optimization subroutine. There are still many limitations of the simultaneous approaches in view of industrial needs. They suffer, for instance, from the lack of stream segmentation, application of standard heat exchangers, calculation of stream heat-transfer coefficients within optimization model, and accounting for certain structural limitations. The latest issue has been addressed in series of papers by Galli and Cerda (1998a,b,c,d, 2000). The important features they included are, for instance, as follows: Inserting of restricted matches Disallowing of certain sequence of matches Disallowing certain locations of splitters over some streams Forcing locations of specified matches at inlet or outlet of specified streams Limitations on splitter number and number of branches over particular streams and/or within a network (f ) Restraining a set of matches with which a potential heat match can be arranged in parallel over a given process stream (a) (b) (c) (d) (e)

To model such conditions, Galli and Cerda suggested several logical conditions. The point is that they require additional binaries and, of course, additional constraints. The number of binary variables drastically increases. Despite the generality and rigorousness of superstructure concept for HEN synthesis, the approach is unable, at present, to cope with large-scale problems or even smaller ones if all industrial needs are to be taken into consideration. The main reason is in existing optimization techniques for MINLP (the problem of superstructure completeness has not been discussed in HEN synthesis). There are at least two ways to cope with difficulties: (a) Development of robust efficient optimization technique that can guarantee global minimum (though usually under some conditions) (b) Application of other stochastic/metaheuristic strategies, problem decomposition, and problem linearization

A significant progress can be noticed in respect to MINLP deterministic solvers. One example is solution technique of Synthesis-Simult-S–R model proposed by Sorsak and Kravanja (2002). The literature on this problem is ample, for instance, Floudas et al. (1986), Floudas and Ciric (1989a,b), Floudas et al. (1989), Quesada and Grossmann (1993), Daichendt and Grossmann (1994a,b), Grossmann and Daichendt (1996), Zamorra and Grossmann (1998a,b), and Lee and Grossmann

608

Energy Optimization in Process Systems and Fuel Cells

(2001, 2003). However, the convergence to the global optimum can be guaranteed only for simple model of heat exchangers, and, also other typical limitations. The problem of scale is still unresolved. Stochastic/metaheuristic approaches have been proposed in order to eliminate the necessity of solving complex MINLP model in equation-oriented form or in form of logical notation in disjunctive programming. It is worth noting that even pure random generation results in ‘good’ solutions. Chakraborty and Ghosh (1999) first showed this for HENs with no splits, and then, Pariyani et al. (2006) contributed with extension on HENs with splitters. Random-search optimization approach was tried to solve simple HEN problem even in the 1970s—see Kelahan and Gaddy (1977). Simulated annealing has been applied in Dolan et al. (1989, 1990). Genetic algorithms were employed by Lewin et al. (1998), Lewin (1998), and, also, Ravagnani et al. (2005). The more recent tabu search metaheuristic was tried with success by Lin and Miller (2004). Also, hybrid two-level approaches found application to HEN design. Stochastic optimization is used at first level for structure optimization while deterministic optimization at the second to solve NLP problem of parameter optimization. Such strategy, with SA at first level, was proposed in Athier et al. (1997) for HEN synthesis and, then, by Athier et al. (1998) for retrofit. We will describe in more detail similar two-level method in Chapter 19. Promising effects can be observed since some solutions reported in the works on stochastic programming are better than those calculated by solution of MINLP model with deterministic solvers. However, the methods developed to date are able to cope, in reasonable CPU time, with at most medium-size problems with typical simplifications. It is important, however, that SA-based approach by Dolan et al. (1989, 1990) was able to deal with standard heat exchangers where deterministic method would, most likely, fail. The retrofit approach for such apparatus will be presented in Chapter 19. HEN problem decomposition is the alternative, particularly for large-scale problems. Zhu et al. (1995a,b) and Zhu (1997) proposed decomposition into blocks with some heuristic rules. This wasn’t rigorous decomposition. Bj€ork and Petersson (2003) developed sequential two-stage method. The main idea was to divide the entire initial set of streams into independent subsets to reduce problem scale for synthesize subsets in the second stage. The grouping has been performed with genetic algorithms. Division into independent subsets has been applied also in a sequential approach by Pettersson (2005). The steps of the procedure are as follows: l

l

l

l

Solution of LP with goal function that does not include fixed cost of heat exchangers Solution of MILP model to reduce number of matches by inserting fixed charges into goal function Solution of MILP to identify independent subsets of matches Solution of MILP models for each subset

Notice that investment costs have been linearized, and also, multishell apparatus are not considered. For medium-size problem with nine (9) process streams, Pettersson (2005) reached the best solution reported to date in the literature. This is the evidence that some relaxation of the sequential procedure is not crucial. CPU time depends largely on parameter setting (mainly number of intervals) and ranges for the problem

Heat-exchanger network synthesis

609

from 1000 to about 2700 s for good solutions. Much larger problem with 39 process streams required CPU time up to 60,826 s though solutions of similar ‘quality’ have been obtained in very short time of order of minutes. However, the best network reported consumes much more utilities than the minimum according to MER target. Finally, Barbaro and Bagajewicz (2002, 2005) developed MILP model for both synthesis and retrofit (though retrofit has not been addressed in full). They applied linearized goal function similarly to approaches of Sorsak and Kravanja and Pettersson. The MILP model constraints are based on transhipment/transportation formulation for temperature intervals. The model is very complex and requires numerous binary variables. Nevertheless, it is MILP and, hence, much easier to solve than MINLP. The CPU time required to solve medium-size problems presented in this work is very short in comparison with other approaches. The results are at least close to optimal solutions in the literature. This simultaneous approach can deal with multishell heat exchangers. Hence, the concept of model linearization seems very promising way for further works. Also, hybrid approaches may help in a quest for solving large-scale problems. There are several examples of embedding ‘intelligent’ heuristics that largely reduce solution space such as Zhu et al. (1995a,b), Zhu (1997), Trivedi et al. (1989b), Jezowski (1990b, 1992b), and Gundersen et al. (2000). At last, tools from artificial intelligence may also enhance the solution.