- Email: [email protected]

Vol. 3, pp. 295.302,

1979

0098-1354/79/040295-08~02.00/0 Pergamon Press Ltd.

Paper 5D.2 UNDERSTANDING

HEAT EXCHANGER

NETWORKS

BODO LINNHOFF,~ DAVID R. MASONS and IAN WARDLE~ ICI Corporate

Laboratory,

Runcorn,

WA7 4QF, England

(Received 28 November 1979) Abstract-Over the past ten years or so, various systematic methods have been described in the literature for the preliminary design of heat exchanger networks. These methods have been applied to simplified example problems but there is little evidence of applications in industry. Hopefully this paper will show the reason for this situation by giving a fundamental understanding of the basic concepts involved in heat exchanger network design, both in simplified and practical contexts. We do not give much detailed description of the methods used in ICIfi but merely comment on certain aspects of these methods as well as other published work so that we can highlight the attributes which we believe successful methods must have. In particular, there are a number of physical facts and phenomena relating to networks in their entirety rather than to isolated unit operations. It is demonstrated that design methods can be immensely improved when these phenomena are properly taken account of. Conclusions and Significance-Research into the systematic design of networks for heat exchange has somehow gone astray in the last ten years or so. Too much consideration was given to the improvement of problem solving techniques such as tree searching, branch and bound, mixed integer optimisation, heuristic learning, etc. while too little attention was paid to the intrinsic characteristics of the problem itself. The present paper outlines a number of fundamental facts relating to the problem itself. Most of these facts relate to networks in their entirety rather than to individual unit operations. In detail, the questions discussed are (1) the significance of AT,,, in networks, (2) the role of multiple utilities, (3) the number of ‘units’ (i.e. pieces of equipment), (4) stream splitting and multiple matching, (5) the role of constraints, and (6) the role of uncertain data. These questions are basic to the successful design of industrial networks but have, in spite of this, received very little attention in the literature to date. Worse still, it is probably fair to say that most work published in heat exchanger network design is based on assumptions that demonstrate ignorance of the fundamental facts discussed in the present paper. The methods currently employed in ICI are based on our understanding of these facts and have achieved considerable success both in the context of simplified example problems as well as in real life applications. To demonstrate this, an example is given of a practical application and the global optimum network for the ‘unsolvable’ literature problem lOSP1 is presented. Another point is important. Since our methods are based on physical insight rather than firm procedures, they encourage a blend of formal algorithm and interaction by the user, giving the flexibility in application so essential in industrial design. We believe that synthesis methods designed to ‘replace’ the user have little chance of practical application for the time being. Therefore, our methods do not attempt to ‘think for the user’ but they ‘help the user think’ for himself, or even force him to think. This is especially true when the methods are used in the mode t Bodo Linnhoff was educated at the TU Hanover, ETH Zurich (MSc in mechanical engineering), and Leeds University (PhD in Chemical Engineering). During 1973, he taught thermodynamics at the ETH and worked for Holderbank AG on new technology evaluation and process efficiency studies. In 1974 he came to Britain to research into the use of thermodynamic concepts in process design. He joined ICI in 1977, initially to apply his research to projects in Petrochemicals Division. In 1978, he joined the Company’s Corporate Laboratory where he is now Section Manager with responsibility for R&D in network design aids. Based on his work in thermodynamics and process design, he has been awarded several scholarships and prizes. He has published a number of papers discussing energy efficiency and network simplicity in integrated-design. Author to whomcorrespondence should be addressed. t David Mason is a graduate of Cambridge University, where he obtained the MA and PhD degrees in Pure Mathematics. From 1972 to 1976 he-was a Research Fellow at Gonville and Caius College, Cambridge, working in Finite Group Theory. He has written about a dozen papers on this subject. He joined ICI in 1978 and is now a consultant in Mathematics and Computing to the Design Systems Group at the Company’s Corporate Laboratory. His work for ICI has been mainly in the design of executive systems for ICI’s flowsheeting package, and in the area of systematic network design aids (process synthesis). §Ian Wardle graduated from the Department of Fuel Technology and Chemical Engineering at Sheffield University with a BScTech, MScTech and PhD in 1975. His research topics included many aspects of computer aided design such as capital cost estimation, flowsheeting systems and process synthesis. Having left university he joined Design Systems Group in ICI’s Corporate Laboratory working initially on Advanced Flowsheeting Techniques and from 1976 on Process Synthesis. He now works for ICI Mond Division as a process engineer. TThese methods were previously described by Linnhoff [l], Linnhoff & Flower [2], Flower & LinnholT[3] and Boland & Linnhoff [4]. In the present text they will be referred to as the TI-method, the ED-method and the TC-method. CACE

3: 1.4

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et al.

of simply predicting design targets (such as maximum energy recovery and minimum number of units), challenging the engineer to use his own skills and intuition to achieve the targets and only if not successful to resort to actual design methods. INTRODUCTION

The systematic design of chemical processes, commonly called ‘Process Synthesis’, has received much attention in the chemical engineering literature over the past ten years or so. One part of the process, the heat exchanger network, has been subjected to the most concentrated effort via a series of simple example problems referred to as 4SP1, 4SP2, etc. the first of which were introduced by Masso & Rudd [S]. These problems have been discussed extensively elsewhere (e.g. Linnhoff & Flower [2]) and, quite clearly, any synthesis method proposed which cannot solve these simple problems will have little chance of acceptance in an industrial environment. The methods currently employed in ICI do solve the literature problems and, beyond this, have achieved considerable success in real life applications. In the course of these applications, it has become clear that the main virtue of the methods is the understanding of some fundamental physical facts. Based on this understanding, we will highlight the key features which we believe that methods for providing better designs must have. What is a ‘better design’? Consider the simplified flowsheet in Fig. l(a) which represents the traditional design for the front end of a speciality chemicals process. Six heat transfer ‘units’ (i.e. heaters, coolers and exchangers) are used with utility heating and cooling requirements of 1722 and 654 units, respectively. The alternative design in Fig. l(b) was produced using our systematic techniques. It uses only four heat transfer ‘units’ and utility heating is reduced by about 40% with cooling no longer required. The design is as safe and as operable as the traditional one but is inherently

better.

This example illustrates what we mean by a ‘better’ design. We do not mean a design that is ‘properly optimised’ but one which prior to any optimisation is more elegant, inherently simpler, and more to the

point. As suggested by Fig. 1, there might be more scope for ‘better’ designs in this sense than is commonly thought and it is hoped that the techniques for heat exchanger networks are only a starting point for future techniques which will assist engineers to design more elegant entire process networks (see Flower & Linnhoff [6]). However, we will discuss no more practical applications here, nor the working of our methods in any detail, but just set out our understanding of heat exchanger networks, an understanding which we believe is essential for ‘better’ designs to be found. UNDERSTANDING

THE PROBLEM

For convenience, we have structured in_to six separate subject areas :

(1) AT,, and energy recovery Donaldson [7] varied AT,, in some of the literature problems (5SP1, 6SP1, 7SP1, 7SP2) between 20 and 50’F. For some problems, he obtained better solutions with increased AT,i” while for others, he obtained worse solutions. He concluded that the effect was problem dependent but did not suggest causes. The reasons for his observation will become clear below. Firstly there is a maximum feasible degree of energy recovery which may or may not be sensitive to AT,, for every problem. Figure 2 shows a simplified diagram in which AQu represents the utility heat supplied to a network and AQ, the utility cooling. By simple heat balance, (AQ,-AQ,) will always correspond to the difference between the heat loads of the process streams to be heated and those to be cooled. The maximum feasible degree of energy recovery corresponds to the situation when both AQ, and AQn are minimal. Figure 3 shows AQu and QC as a function of AT,,, for a hypothetical problem. Below AT,, = 20°F the utility heat loads are independent of ATmi, whereas

I I

1

“z-:‘:“,”

I

I__PRODUCT

FEED

(4

the discussion

(b)

Fig. 1. Traditional and alternative design for front-end of speciality chemicals process.

Understanding heat exchanger networks

Process Streams to be Cooled

NETWORK

(COLDUTILITIES

(

‘z,AH, Process

Streams

Fig. 2. Maximum energy recovery. above 20°F the loads on both utilities increase. Since (AQ, - AQn) is a function of the process stream heat loads which are not subject to change, AQn and AQ, increase with the same sensitivity.

t

UTILITY

HEAT

LOADS

297

to be realised in a design would have to be smallest. Thus, a design constraint expressed in the form of a value for AT,, would come to bear on the design nowhere else than at this point forcing the curves in Fig. 4 apart and necessitating larger values for both AQ,., and AQ,. Figure 3 and Fig. 4 between them illustrate that there are two types of problems. In the first type, on which Fig. 3 is based, the degree of heat recovery will only be sensitive to A7’,‘,i, at some value of AT,, > 0. Below this threshold value, utility consumption is constant with either heating or cooling required but not both. In the second type of problem, on which Fig. 4 is based, the degree of heat recovery is always sensitive to AT,,. Also, heating and cooling are always required. The first type of problem may be symbolised by just one very hot process stream to be cooled and one very cold process stream to be heated. In a design for this ‘network problem’, the use of the AZ& concept would be nonsensical. The second type of problem would be symbolised by a moderately hot stream to be cooled and a high target temperature cold stream to be heated. Table 1 shows the threshold values of A&” for the literature problems. From this table, problems 4SP2,5SPl, 6SP1,7SPl, 7SP2, and lOSP1 emerge as being of the first kind. The remaining problems (i.e. TCl, TC2,4SPl and lOSP2) are of the second type. Table 1. Maximum values for AT,,i, if energy recovery is to remain unimpaired

1

:

0

IO

:

20

30

40

:

50

:

_

AT

min

(OFI

60

Fig. 3. Utility requirements as a function of Atmin. Consider the graph in Fig. 4 which shows, for another hypothetical problem, streams to be cooled (upper curve) plotted with those to be heated (lower curve) on common temperature versus enthalpy axes. Temperature

t

AOe

’ 1 Enthalpy

Fig. 4. Locating the heat recovery pinch. The horizontal distance between the curves at the hot ends corresponds to AQu and that at the cold ends to AQ,. We can see that there is a pinch in the system and it is at this point that the actual temperature differences

Problem _____ TCl TC2 4SPl 4SP2 5SPl 6SPl 7SPl 7SP2 lOSP1 lOSP2 ~_

Introduced by Linnhoff & Flower [2] Linnhoff & Flower [2] Lee et al. [8] Ponton & Donaldson [9] Masso & Rudd [S] Lee ef al. [S] Masso & Rudd [S] Masso & Rudd [S] Pho & Lapidus [lo] Wells & Hodgkinson [l l]

Sensitivity threshold lies at AT,, = 0 0 0 -46’F - 43°F - 65°F - 49’F -5l’F - 72’F 0

The above discussion throws some new light on the A.T,i, constraint. Clearly, in the case of problems like 4SP2, SSPl, 6SP1, 7SP1, 7SP2 and lOSP1 where minimum utility solutions with either heating or cooling can be obtained with actual temperature differences far in excess of 20°F (the usual value chosen for AT,,), the constraint cannot possibly affect the trade-off situation between energy and capital ! All the constraint can do is prevent feasible and perhaps attractive solutions from being generated. In the other cases (i.e. TCl, TC2, 4SPl and lOSP2), a priori definition of a figure for AT,, is probably far too crude an approach to handle the important question of optimising energy versus capital. In these cases, the pinch as shown in Fig. 4 should be located and the economically correct degree of energy recovery established by optimisation with respect to the specific streams and the equipment involved in the pinch region. What had happened in Donaldson’s ‘experiment’ referred to above should now be clear. Donaldson tackled problems of the ‘first kind’ only and used a hill

298

B.

hJNHOF’F

climbing branch and bound procedure which, with different AT,,-values, will climb different hills. Attributing his results to any influence that AT,, could have on the trade-off situation between capital and energy costs would be a gross misinterpretation of his results. (2) The use ofutilities As has been stated many times in the literature, the overall costs of solutions for the literature problems are heavily biased towards utility costs. Ithas also been frequently stated that this is due to the particular costing parameters chosen by Masso & Rudd[5]. However, there is another somewhat more fundamental reason that explains the dominant role of utility costs in these examples. The solutions to the literature problems must always incur a double penalty if they do not achieve the maximum feasible degree of energy recovery. Any increase in utility heating requirement, and thus in annual network cost, will be amplified by a corresponding increase in utility cooling requirement due to the fact that (AQ, - AQ”) is a constant. Conversely of course any saving is a double saving. Consider Fig S(a) which is based on a network shown by Linnhoff & Flower [2]. If we reduce the load on the heater in this network by x units this will also reduce the load on the cooler by x units, i.e. we achieve a double saving. However, if we had a situation where two levels of hot utility exist, as in Fig. 5(b), we can reduce the load on the hotter utility heater by x units and increase the load on the lower temperature utility heater with no effect on the cooler. In this case no double saving occurs. This latter case is somewhat more realistic than the situation encountered in the literature problems and quite obviously leaves room for decisions affecting equipment lay-out, as well as level and amount of utilities used, without the effect of double savings or double penalties in operating costs. As a result, real life problems are less dominated by operating costs than the literature problems.

I 60

-x

60+x

6oof-JCl

240

(4

et al

(3) The number of units In a practical context, a heat exchanger network should incorporate as few heat transfer ‘units’ (i.e. heaters, coolers and exchangers as possible). This usually ensures a cheaper network in terms of equipment charges, pipework, foundations, maintenance and control, see also Boland & Linnhoff[4]. Inspection of the highly simplified cost equations used in the literature problems reveals, despite the simplifications, a similar justification for networks which incorporate the minimum number of units, see Linnhoff & Flower [2]. Given the justification for the smallest possible number of units in a network can we identify what the minimum number will be? The answer is ‘yes’ and has previously been given by Hohmann[l2] and by Linnhoff & Flower[2]. However, there is more to be said to this than a simple ‘yes’. Applying mathematics, Euler’s network relation teaches us :

U=N+L-S

U = the number of units N = the total number of streams (process and utility) L = the number of ‘independent’ loops (see appendix) and S = the number of separate components in a network. This is described in more detail in the Appendix and leads to a very important understanding of some fundamental concepts in network design. For our purpose we need to realise that there are three types of situations : (1) In Fig. 6(a) we see six distinct streams, four process streams and two utilities (i.e. steam and cooling water). The heat load of each stream is shown above or below the circle representing the stream. The lines between circles represent matches, also with heat loads shown. When introducing a first match in the design, the heat load can never exceed that of the smaller of the two streams matched, leaving a residual on the larger stream. On matching the residual, another residual is produced and so on. However, due to the overall heat balance between hot and cold streams the last match cannot produce a residual. Thus, it is found that the number of matches (i.e. the number of heat transfer ‘units’ in the network) is one less than the number of streams. In terms of Eq. (l), L=OandS= lsothat:

U=N-1.

Fig. 5. The effect of more than one utility level. Note: This representation of streams and exchangers in a grid is adopted from Linnhoff & FlowerC2). Hot streams are shown running to the right at the top of the diagram and cold streams running to the left at the bottom. Stream supply and target temperatures are noted in “F. Heat loads may be noted underneath each match.

(1)

where

(2)

This case might be termed the ‘base case’ and establishes what might usefully be referred to as the ‘minimum number of units’. (2) The data used in Fig. 6 represents a special case. Namely, it is possible to identify subsets of hot streams and subsets of cold streams with heat balance existing not only for the overall problem but also for the subsets. This situation may give rise to a design such as shown in Fig. 6(b) where the network consists of more than one separate component and features less than

299

Understanding heat exchanger networks 90

70

30

9 Jy

S = 1 and L = 1, leading to :

ST

II

30

U = N. so

40

C, b 40

cw

g. 100

SO

NUMBER OF UNITS IS ONE LESS THAN THE NUMBER OF STREAMS

40

50

100

ONE UNIT LESS IN TWO SEPARATE

70

90

HI

Ii2

c,30k &I ST

x

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II0

160-X1

Cl

c2

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COMPONENTS

40

+x1

50

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ONE UNIT MORE IN A NETWORK CONTAINING CYCLE

A

Fig. 6. Understanding the number of units.

the ‘minimum number of units’ as defined above. In Fig. 6(b), L = 0 and S = 2 so that : U=N-2.

(2b)

The understanding of this special case is important because one does not want to overlook the chance to reduce the number of units in a practical context. Also, it is often possible to deliberately modify heat loads of streams and group streams together so as to create situations such as in Fig. 6(b). (3) Finally, consider Fig. 6(c). This topology is identical to that in Fig. 6(a) except for an additional match between the hot utility (i.e. steam) and cold stream No. 2. The additional match introduces a loop into the design (compare the remarks made about so that we have ‘cycles’ in the Appendix)

115.3

(2c)

The load on the additional match may be chosen (‘X’ in Fig. 6c) leading to corresponding changes in the heat loads of all matches situated in the loop. We can now draw several extremely important conclusions : (a) It is possible to predict, prior to any design and simply based on an analysis of problem data, the minimum number of units. (b) In a network incorporating the minimum number of units the heat loads are uniquely defined by the topology. There is no continuous optimisation problem ! (c) Continuous variables only exist in networks containing loops and thus incorporating more than the minimum number of units. However, while there may be a considerable number of exchangers the heat loads of which could be chosen (four in Fig. 6c) the number of independent variables is given by the number of independent loops which has to be much lower (one in Fig. 6~). Perhaps the most important point in connection with solving the literature problems is that far fewer networks exist with the minimum number of units than with more than the minimum number. To illustrate this consider problem 1OSPl. Using a simple formula suggested by Ponton & Donaldson [9] to estimate the problem size this problem requires looking at 1.55 * 1025 candidate networks for complete solution. However, we now know that only 843 of these networks feature the minimum number of units ! What is more, these 843 networks feature no more than 10 units each and no continuous variables to optimise ! (The networks implied by Ponton and Donaldson’s formula feature up to 25 units each with up to 15 continuous variables to consider for optimisation). In the past, the global optimisation of lOSP1 has been considered to be impossible with more or less any amount of computer time. Using the TC-method previously described by Flower & Linnhoff [3] which is based on the above insights regarding the number of units and continuous variables, the global optimum has now been identified, see Fig. 7.

411.7

Fig. 7. The global optimum for lOSP1 (Annual cost: $43857).

B. LINNHOWet al.

300

240’

Fig. 8. A system

that cannot be solved with the minimum

(4) Stream splitting and cyclic networks (The expression cyclic is used to describe a network incorporating more than one match between the same two streams.) The design strategies to use cyclic arrangements or parallel splitting of streams are closely inter-related. Both are used to -achieve improved energy recovery, or to -introduce flexibility into a design. In addition, stream splitting may be used to reduce the number of units while at the same time maintaining, or even improving, energy recovery. Cyclicity by contrast, and by definition, introduces additional matches. An extensive discussion is given by Linnhoff & Flower [2]. However, an additional comment ought to be made here which refers to a statement made by Hohmann [12]. Hohmann and later Hohmann & Lockhart [13] stated that a minimum number of units network could be found for any stream system if parallel stream splitting was used. This view was challenged by Linnhoff [l] who provided the counter example shown in Fig. 8. This stream system cannot be solved with less than three units (i.e. one more than the minimum number), whatever stream splitting or other arrangements are made. However, the fact that Hohmann’s statement stood undisputed for about 8 years may be taken as evidence that systems such as shown in Fig. 8 are few and far between. Inspection of real life problems in ICI has supported this view. In other words, most practical problems are amenable to solution with the minimum number of units and the understanding of stream splitting as a technique to achieve the minimum number is of the utmost practical importance. (5) Constraints The literature problems have often been quoted as having an exceedingly large number of solutions. However, once we have recognised some of the fundamentals of network design described above we can identify constraints that reduce the problem size. As an example, consider a case in which we have predicted prior to design that no heating from utilities will be required. We can then say that any design which is to achieve full heat recovery will have to feature certain matches to bring the hottest cold streams to their target temperatures without heaters. A similar argument can be applied to problems for which utility cooling is not required. Practical problems feature even more constraints. In a practical context, we may have streams which must not be matched with each other because, e.g. it may be hazardous or because the streams may be physically distant. Alternatively, requirements for start-up might dictate where utility heaters are to be placed.

Incorporating considerably

cp=

number

I-O

of units.

such constraints into the problem data reduces the number of feasible solutions.

(6) Data modijication Another aspect often exhibited by practical problems is the possibility of data modification to give rise to markedly better solutions. Very slight modifications in, say, temperatures often suffice to make desirable solutions possible. As an example, Fig. 9 shows a solution for lOSP1 which has one less than the minimum number of units by virtue of the fact that one stream fails to meet its target temperature by 2.75”F. (This slight modification of data creates a situation such as described in Fig. 6b). The structure shown in Fig. 9 was first presented by Linnhoff & Flower [Z].

527.0 Fig.

9. ‘Special offer’ for 1OSPl.

APPLYING THE UNDERSTANDING

Let us now look at published design methods and see whether or not they exhibit the understanding described above : (I) ATmi, and energy recovery The most important single aspect of design is perhaps the correct trade-off between energy and capital costs. Most published methods achieve very little indeed in this respect. Only Hohmann [12] with the ‘feasibility table’ and Linnhoff & Flower [Z] with the ‘problem table’ have produced algorithms which correctly predict the maximum feasible degree of energy recovery. Claims by Rathore & Powers [14] and Nishida et al. [15] to have done so were proved

Understanding heat exchanger networks incorrect by Linnhoff & Flower [2]. Further, as previously discussed, the degree of energy recovery may or may not be sensitive to AT,,,. Useful methods should be able to predict correctly the upper bound on energy recovery, to discuss its dependence on ATmi,, and to locate any ‘pinches’ such as shown in Fig. 4. If sensitivity exists for lowish values of ATmi,, they should offer a facility for optimising the trade-off between heat recovery and equipment cost in a fashion less crude than by simply imposing a standard figure for AT,i,. A full description of a suitable procedure is given by Linnhoff [ 11. (2) Multiple utilities Another reason why published methods often fail to attain the economically correct degree of energy recovery lies in the fact that they are ‘tuned’ too strongly to the use of single utilities. An interesting example of multiple utilities was given by Wells & Hodgkinson [ 1 l] who suggested raising steam rather than rejecting high level surplus heat into cooling water. This introduces a situation where network costs can decrease as a design develops (namely, when steam raising boilers are implemented) and it would be interesting to see how, for instance, a branch and bound algorithm would cope with this situation. (3) The number ofunits Almost all published methods are based on generation trees that are in line with the assumptions made by Ponton & Donaldson[9] which, as outlined above, lead to approx. 1.55 * 10z5 networks to be examined for a problem such as lOSP1. Furthermore, these methods have to cope with the so-called mixed integer optimisation problem (e.g. Kelahan & Gaddy [16]) that is introduced by the presence of continuous variables. Since the continuous variables themselves are the consequence of an unnecessary evil (namely, the presence of more heat transfer units than necessary, see Fig. 6), these methods essentially solve a non-existent problem. Clearly, they tackle the task from the ‘wrong end’, making it appear to be far more complex than it really is. With the insight described in this paper, most algorithmic methods could now be improved so as to first look for minimum number of units solutions and only if not successful to increase the allowed number of units. However, even if such modifications were made, virtually all published methods could still not be guaranteed to produce the optimum. What is worse, they could sometimes be guaranteed not to produce the optimum. The reason for this is that virtually all published methods rely on the so-called stream termination criterion (e.g. Kelahan & Gaddy[16]): a match is ended either when one of the two streams matched reaches its target temperature or when the AT,, constraint is violated. A glance at Fig. 7 (i.e. the global optimum for lOSPl), reveals that this structure could not possibly have been found using this criterion: the network contains two matches whose terminations contradict the criterion. Essentially, this criterion reflects the philosophy of design of unit operations but not of networks : once the two streams to be matched are identified, a match is designed with the remainder of the network being ignored. This seems to be a rather poor state of the art. After all, is it not the very essence of network design to

301

deal with unit operations not in isolation? The TCmethod described by Flower & Linnhoff [3] is probably the only method known that will terminate each match from an overall network point of view and, as said before, it is this method by which the structure of Fig. 7 was found. (4) Stream splitting and cyclicity Often, conventions adopted for network representation exclude certain design options. The matrix representation proposed by Pho & Lapidus [lo], for instance, can represent neither cyclic matches not split streams. A discussion of this point is given by Westerberg [ 171. Having discussed the particular importance of stream splitting as an option in design, it can now be safely said that methods which exclude this option can at best be useful for a very limited range of problems in practice. Similar, but often more subtle, is the way in which some representations might exclude certain options other than stream splitting or cyclicity. As an example consider Fig. 10. The network from Fig. 5(a) is drawn in the ‘cross-grid’ representation used by Wells & Hodgkinson [ 111. This was only possible after streams had been ordered out of sequence. Certain networks incorporating loops cannot be shown at all using this representation. (5) Constraints As we have stated above, realistic problems will often have constraints attached to them. The strategy recommended in most published work to handle this situation is to -solve the unconstrained problem, and to -select those solutions which happen to obey the constraints. Clearly, this strategy is inappropriate. More sensibly, a strategy of ‘selective generation’ should be employed. In other words, the constraints should be incorporated into the synthesis procedure so that only solutions which will obey the constraints are generated and no time is wasted with the generation and evaluation of solutions that must by necessity be useless. Again, the TC-method is probably the only method known that employs such ‘selective generation’. Experience in ICI to date suggests that the complexity of a problem is not so much described by the number of streams but more accurately by a

Fig. 10. The network from Fig. S(a) in another representation.

B. LINNHOFF et al.

302

combination of both the number number of constraints.

of streams

and the

(6) Data modi$cation A method of any realism should tell the user any slight modifications of data which could lead to superior solutions. This aspect has so far been given little attention with notable exceptions by Hohmann [12], Donaldson [7] and Linnhoff & Flower [2]. The TC-method will show up solutions of the type shown in Fig. 9. However, there seems to be considerable scope left for development to tackle this particular aspect of the problem satisfactorily. Acknowledgements~Bodo Linnhoff expresses his thanks to Dr. J. R. Flower who supervised his research at Leeds University. All three authors would like to thank ICI Ltd. for determined support of ‘process synthesis’ and for the permission to publish this paper. Also, they would like to acknowledge the contribution made by Dr. K. Carpenter.

REFERENCES analysis in the design of 1. B. Linnhoff. Thermodynamic process networks. PhD thesis, The University of Leeds, England (1979). (1978). 2. B. Linnhoff & J. R. Flower. AIChE J. 24,633-654 & B. Linnhoff, A thermodynamic3. J. R. Flower combinatorial approach to the design of optimum heat exchanger networks. AIChE J. 26, l-9 (1980). design of 4. D. Boland & B. Linnhoff, The preliminary networks for heat exchange by systematic methods. The Chemical Engineer, 222-228 (1979). 5. A. H. Masso & D. F. Rudd, AIChE J. 15, 10-17 (1969). analysis in 6. J. R. Flower & B. Linnhoff, Thermodyanamic the design of process networks. This issue, paper 5B.2. 7. R. A. B. Donaldson, Studies in the computer aided design of complex heat exchange networks. PhD thesis, University of Edinburgh, Scotland (1976). 8. K. F. Lee, A. H. Masso & D. F. Rudd, I&EC Fundamentals 9,48-58 (1970). 9. J. W. Ponton & R. A. B. Donaldson, Chem. Engng. Sci. 29, 2375-2377 (1974). 10. T. K. Pho & L. Lapidus, AIChE J. 19,1182-l 189 (1973). Process Engineering, 11. G. L. Wells & M. G. Hodgkinson, pp. 59-63 (1977). 12. E. C. Hohmdnn, Optimum networks for heat exchange. PhD thesis, University of Southern California (1971). & F. J. Lockhart, Optimum heat 13. E. C. Hohmann exchanger network synthesis. Paper No. 22a, AIChE Nat. Meet., Atlantic City, N.J. (1976). 14. R. W. S. Rathore & G. J. Powers, Ind. Engng. Chem. Process Des. Develop. 14, 175-181 (1975). 15. N. Nishida, Y. A. Liu & L. Lapidus, AIChE J. 23,77-92 (1977). 16. R. C. Kelahan & J. L. Caddy, AIChE J. 23, 816822 (1977). 17. A. W. Westerberg, Synthesis of heat exchanger systems. In: Process Synthesis, AIChE Today Course, Philadelphia, PA (1978). Graph Theory. Addison-Wesley, Reading, 18. F. Harary, Massachusetts (1972).

APPENDIX In this appendix we list some results in Graph Theory which are used in the present paper. A graph is any collection of points, some pairs of which are connected by lines. Figures (A) and (B) give two examples of a graph. Note that the lines BG and EC in Fig. (A) are not supposed to cross, that is, the diagram should be drawn in three dimensions. If a graph has p points and 4 lines it is called a (p,q) graph[Harary[18], p. 391. In our applications, the points correspond to process and utility streams, and the lines to heat exchangers. A path is a sequence of distinct lines, each one starting where the previous one ends. For example, in Fig. (A) AECGD is a path. A graph is connected if any two points can be joined by a path. Thus, Fig. (A) is connected, but Fig. (B) is not. More generally, points which are connected by paths to some fixed point are said to form a component. Thus, Fig. (B) has two components and Fig. (A) has only one. A cycle is a path which begins and ends at the same point, like CGDHC in Fig. (A). If two cycles have a line in common, they can be linked to form a third cycle by deleting the common line. In Fig. (A) for example, BGCEB and CGDHC can be linked to give BGDHCEB. In this case, this last cycle is said to be dependent on the other two. (For a precise definition of dependence see Harary[18], p. 39). The maximum number of independent cycles is called the cycle rank of the graph. In our application, the cycle rank is the number of degrees of freedom in the heat loads on the exchangers. The main result which we need is proved elsewhere (Harary[lX], Theorem 4.5). Namely, the cycle rank of a (p,q) graph with k components is q-p+ k. An important special case occurs when the graph is connected and cyclefree. In this case 4 = p - 1, or, in terms of our application, the number of heat exchangers required is one less than the number of streams (including utility streams). Further, the general result means that the number of degrees of freedom in the heat loads on the exchangers is equal to the number of exchangers minus the number of streams (including utilities), plus the number of components into which the network can be divided.

Fig. A.

A

B

C

D

E

F

G

H

lax Fig. B.

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