Expert Systems With Applications. Vol.5, pp. I 1I-I 19, 1992 Printedin the USA.
0957--4174/92 $5.00+ ,00 © 1992PergamonPressLtd.
Rule-Based System for the Synthesis of Heat Exchanger Networks J. A. SOUTO AND J. J. CASARES Empresa Nacional de ElectricidadS.A., Spain A. R O D R [ G U E Z Universidad de Santiago, Spain
Abstract--Heuristic methods have been recognized as simple and effective procedures for the synthesis of heat exchanger networks. These problems and techniques are especially adequate for the application of Artificial Intelligence techniques for their solution. In the present work, a heuristic synthesis method is presented and incorporated in a rule-based system using PROLOG, and is applied to a standard problem very satisfactorily. 1. I N T R O D U C T I O N
thesis problem that may not guarantee the economic optimum, but may solve many problems rather satisfactorily. The philosophy behind the synthesis of a Heat Exchanger Network ( H E N ) is to optimize the energy usage in industrial processes by integration of process streams. Process streams that need to be cooled (hot streams) reduce their energy content by heating other process streams that need to be heated (cold streams). Accordingly, the first objective is to save energy. Therefore, the synthesis method must produce a good energy recovery through the process streams. As a characteristic parameter the maximum amount of energy to be exchanged between process streams ( M E R ) is adopted. This quantity will be limited by the energy requirements and the thermal levels of the process streams. Also, the installation cost of the network must be taken into account in two directions: • The area necessary to exchange a given amount of heat Q between two streams is:
RULE-BASED SYSTEMS(Bafiares-Alcfintara et al., 1985) are computer codes capable of dealing with complex, ill-structured problems of variable dimension. Their flexible organization, in the process of searching for solutions as well as in the data structures they can handle, is a key aspect for their application possibilities. The present paper discusses the implementation of a well-known heuristic method used for the synthesis of heat exchanger networks in a rule-based system, RICPERT, using PROLOG. Heat exchanger networks are energy recovery systems that allow for a reduction in the operating costs of an industrial plant. A number of heuristic (Rathore & Powers, 1975), thermodynamic (Linnhoff& Hindmarsh, 1983), and mathematical (Rodriguez, Souto, & Casares, 1990) methods have been developed for the synthesis of these networks. The problem analysis by mathematical or thermodynamic methods has been able to identify some characteristic parameters in the evaluation of the network (Linnhoff, 1982): the minimum approach temperature ( M A T ) , and the maximum energy recovery (MER). Nevertheless the synthesis problem still is a creative task, with a great number of alternatives to consider, and therefore impossible to structure in a sequential algorithm that avoids an exhaustive search in a short computation time. On the other hand, unstable energy prices make it very difficult to fix definite and absolute rules. Heuristic methods offer an approach for the syn-
A = Q/UAT
where A T is an average temperature difference between the streams, and U is the overall heat transfer coefficient. To avoid extremely large heat transfer areas, a minimum approach temperature is usually defined for any section of the heat exchanger. Thus a minimum value of A T is guaranteed for all the exchanges of the network. The number of installed heat exchangers must be reduced as much as the optimum heat recovery allows, because of the associated increase in cost of the system and the added control problems.
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J. A. Souto et al.
2. S Y S T E M D E V E L O P M E N T Of the various heuristic methods, Ponton and Donaldson's method (Ponton & Donaldson, 1974), is possibly the most popular for the synthesis of heat exchanger networks. Its major limitation is the set up of exchange alternatives, and hence the method has been modified (Souto, Rodriguez, & Casares, 1990) and its heuristics improved. For this task, a system based on rules that incorporated such modifications is developed and implemented.
Stage 1. Preliminary Considerations
The first step to be followed is the definition of a generic synthesis method with two objectives: • Implementation of the inference engine for the management of the heuristic rules. • Definition of the data structures o f the program. The basic scheme for this generic synthesis method is shown in Fig. 1. The synthesis o f any network is the most complex and important step and requires particular consideration, that is, • to search for the streams to match, • the exchange of energy, • depletion of the residual streams, • generation of alternatives to the synthesis, • rearrangement of the network achieved. Therefore, the steps to follow are very much problem-dependent and so the inference engine developed
FIGURE 2. Inference engine.
controls the application of the heuristics for each step. A scheme of this inference engine is shown in Fig. 2. For the data structures used by the program, three types of objects are considered: • Process streams (hot and cold), with their physical properties. • Heat exchanges between process streams, resulting in the internal network. • Heat exchanges with utilities, defining the external network. Because the objective for the implementation of the synthesis by means of heuristics was the improvement of the rules and the definition of an optimum synthesis method, the simplest knowledge representation was selected with data structures flexible enough to allow for further modifications in the future.
Stlearns 1 Dill
Stage 2. Basic Heuristics
FIGURE 1. Heat exchanger network synthesis method.
Of the five steps to follow in the synthesis, only the first three are essential for finding a solution. The other two allow improvements in the solution achieved but could be avoided in the first stage of the development of heuristics. To search for the streams to match, the heuristic H / H , from Ponton and Donaldson's method, was implemented selecting the streams to exchange energy based on their upper temperatures: the initial temperature for the hot stream and the target temperature for the cold stream. When a match between a hot and a cold stream is considered, two problems arise: 1. The type of exchange to take place. Only countercurrent heat exchange will be considered because it offers the best use of the thermal levels of the streams. 2. The computation of the amount of energy to be exchanged; the tick-off heuristic decides to exchange as much energy as possible. This maximum amount of energy is limited by the following constraints:
R ule-Based System for H E N s
• Heat availability of the streams. For the hot stream, the heat transfer will be, AHh = CPh( Ths - Th,)
For the cold stream, the heat transfer will be, AH~ : CP~( T . - T , )
• The minimum approach temperature MAT that reduces the possibility of energy exchange between two streams. The tick-off heuristic tries to exchange the maxim u m amount of energy between the two selected streams considering these constraints. The depletion of the residual streams is the simplest of the heuristics as it is only necessary to determine the energy to be supplied to the cold streams or removed from the hot streams to achieve their final temperatures. The H / H and tick-off heuristics are shown in Fig. 3. In these rules it can be observed that the heuristic H / H uses the backwards search method to find two streams that could exchange energy. The tick-off heuristic is more complex to apply as five possible exchange situations can occur; therefore, two rules had to be defined. The first one selects the present exchange situation whilst the other computes the energy exchanged.
The tick-offheuristic applies the basic reasoning for each match, to eliminate one of the streams from the problem, if possible. But the application of this rule requires a detailed study of the exchange cases and a solving strategy for each of the cases. If, for each stream its highest temperature is taken as a reference, the energy exchanged will be a function of the temperature as follows. For a hot stream, A H = CPh( Ths - T ) , T=
Ths - ( I / CP~ ) A H .
For a cold stream, T = Tct- (I/CPc)AH.
In any case when using the actual temperatures, the value for MAT has to be applied as the practical limit for the temperature gradient between the hot and cold streams. The problem can be simplified using the reduced temperatures ( Thr or Tcr), expressed as, 1. For the hot streams Thr = Th -- M A T / 2 . 2. For the cold streams Tcr = T c -~- M A T / 2 .
/* S e a r c h i n g f o r t h e h o t e s t c o l d s t r e a m */
/ . Narmgement o f the exchange 1/
search_cold(Nc):ncolddin(Ncold), Nc > N c o l d , ! .
T i c k _ o f f ( I , Qh, Qc): great(Qh, Qc), hotsmx(Nh), coldlax(Nc), e x t r a c t ( N h , _ , T i h , _ , CPh, d in , h o t ) , T i h o t : Tih - Qc/CPh, retract(hotdin(Nh, tin,_), dindata), assert(hotdin(Nh, tin, Tlhot), dindata, e x t r a c t ( N c , _ , T i c , Toc,_, d i n , c o l d ) , s t o r e _ e x c h a n g e ( I , [p(Nh), p ( N c ) ] , [ Tih , T i h o t , Tic, Toc, q c ] ) , d e p l e t i o n ( c o l d ) , !.
search_cold(Nc):coldmax(Nx), c o l d d i n ( N x , t o u t , Tx), c o l d d i n ( N f , t o u t , Tc), g r e a t ( T c , Tx), retract(coldmax(Nx), select), aeeerta(coldmax(Nc), s e l e c t ) , fail, search c o l d ( N c ) : Ncold = Nc + 1, search cold(Hcold). /* S e a r c h i n g f o r the hot s t r e a m */ search_hot(Nh):nhotdin(ghot),!. search_hot(l/h):hotlmx(Nx), h o t d l n ( N x , t i n , Tx), h o t d i n ( N h , t i n , Th), g r e a t ( T h , TX), retract(hotlmx(Nx), select), essert(hotsmx(Nh), s e l e c t ) , fail.
search_hot(Nh):Nhot = Nh + 1, search_hot(Nhot).
T i c k _ o f f ( I , Qh, Q c ) : l e s s ( Q h , Qc), hotmax(Nh), coldmax(Nc), e x t r a c t ( N c , _ , _ , Toc, CPc, d l n , c o l d ) , Tocold = Toc - Qh/CPc, retract(colddin(Nc, tout,_), dindata), a s s e r t ( c o l d d i n ( N c , tout, Tocold), dindata), e x t r a c t ( N h , _ , Ti h, Toh,_, d i n , h o t ) , store_exchange(I, [p(Nh), p ( N c ) ] , [Tth, Toh, Tocold, Toc, Qh]), d e p l e t i o n ( h o t ) , !. T i c k _ o f f ( I , Qh, Qc): equal(Qh, Qc), hotltax(Nh), coldlax(Nc), e x t r a c t ( N h , _ , Tih, Toh,_, d in , h o t ) , e x t r a c t ( N c , _ , Tic, Toc,_, d in , h o t ) , store_exchange(I, [p(Nh), p ( N c ) ] , [Ti h, Toh, T i c , Toc, Qh]), d e p l e t i o n ( b o t h ) , !.
FIGURE 3. H/H and Uck-off heuristics.
J.A. Souto et al.
With these criteria in mind, if two streams reach at any point within the exchanger a temperature gradient equal to MAT, this will be equivalent to saying that both reduced temperatures are the same. This change of variables does not imply any variation in the energy balances as all the initial and target temperatures of the hot streams are reduced in M A T / 2 and all the initial and target temperatures of the cold streams are increased in M A T / 2 . Therefore, the increments of temperature (Ts - Tt) are the same and the energy available for each stream remains unaltered. The balance equations using the reduced temperatures are of the same type, Tr = T h s r - ( I / C P h ) A H , Tr = T c t r - ( I / C P c ) A H .
The evaluation of the possibilities for a match requires a systematic classification of the different alternatives. This analysis starts identifying the degrees of freedom in the problem and the computation of the possible matches. This step involves comparison of the extreme temperatures of the streams considering a countercurrent exchange: input temperature of the hot stream The, with the outlet temperature of the cold stream Tot; outlet temperature of the hot stream Tht, with the inlet temperature of the cold stream Tcs, and the heat capacities of both streams CPh and CPc. There are, then, three degrees of freedom for each match (greater than, less than or equal) and three relationships: the result is 33, or 27 theoretical possibilities for an exchange between two streams. Of the 27 alternatives, some can be considered equivalent when evaluating the amount of heat to be exchanged, while others cannot be considered without violating the limit imposed by MAT. A detailed study of all the other alternatives classifies them in five groups, always with the aim to exchange the maximum amount of energy and keeping the limit of MAT. A small detail of the code in P R O L O G of the heuristic of classification of these five groups is shown in Fig. 4. Wood et al. (1989) established a series of simple rules to determine the amount exchanged for each match. In the procedure presented here, five cases of exchange can be identified that allow a better use of the energy contents of the streams. The five situations are as follows. Case 1. The energy exchange is equal to the capabilities of the stream which can exchange less energy. Under this condition, 1. If one of the streams can exchange more energy than the other, the latter is removed from the problem, remaining as the residual stream a stream with a temperature span equal to that not covered by the other stream.
/* Decision of the exchanse */ Pinch(casel,
Pinch(Case, hot, cold):tout(T), hotmax(Nh), h o t d i n ( N h , t i n , Th), l e s s ( T , Th), Case = case4, Case = case1,
tcut(T), hotmax(Nh), hotdin(Nh, tout,Toh}, less equal(T, Toh), Case = casel, !; tout(T),
coldmax(Nc), colddin(Nc, tin, Tic), lessequal(T, Tic), Case = e a s e l , !; Case = case3 , ! . Pinch(case1,
Pinch(Case, cold, cold):hotmax(Nh), hotdin(Nh, tout, Toh),
coldmax(Nc), colddin(Nc, tin, Tic), great_equal(Tic, Toh), Case = c a s e 2 , !; Case = case5, !. Pinch(Case, e q u a l , c o l d ) : hotmax(Nh), hotdin(Nh, tout, Toh), coldmax(Nc), c o l d d i n ( N c , tin, T i c ) , g r e a t _ e q u a l ( T i c , Toh), Case = case2, !; Case : caseS, !. Pinch(Case,
hotmax(Nh), hotdin(Nh, coldmax(Nc, colddin(Nc, great_equal(Tic, Toh), Case = case2, !; Case = caseS, !. Pinch(casel,_,
tout, Toh), tin, Tic),
FIGURE 4. S e l e c t i o n of e x c h a n g e a l t e r n a t i v e s .
2. If both streams have the same capability, both are removed from the problem. Case 2. Energy exchange between the hot stream and the cold stream in the temperature intervals allowed by the value of MAT, trying to use always the highest thermal level of the hot stream in the match and that the cold stream be consumed from its initial temperature till where it is possible. The AT between the streams will be equal to MAT in the lowest thermal level of the match. As a result there will be two residual temperature intervals: 1. The residual interval of the hot stream will become a residual stream, and will be used in the following steps of the synthesis. 2. The cold stream may need external heating to attain its target temperature. Case 3. Energy exchange between the hot and cold
streams from their highest thermal levels (initial and target temperatures, respectively) till the limit imposed
Rule-Based System for HENs
by MAT. As a result there will be two residual streams that are incorporated subsequently to the synthesis process. Case 4. Similarly to Case 2, both streams will be utilized at their highest thermal levels. But, in this case ATwill be equal to MAT at the highest thermal level of the exchange. The residual streams will generate a Case l, and thus the stream with less heat availability will be exhausted. Case 5. Very similar to Case 2, but now the hot stream has to be used completely, and there will be a residual cold stream in the lower temperatures section that is incorporated subsequently to the synthesis. The systematic application of the previously developed heuristic rules has allowed the elaboration of a heat exchanger network synthesis method, that generates a network for every problem set, with a good energy integration and a reasonably low number of units. To this point, the system was ready for the initial tests that consisted in its use for the solution of simple problems with two to four process streams. The aim was: • to verify that the system provided the expected results. • to validate that the heuristic rules implemented were thermodynamically sound. Once these two requisites were checked and the system performed adequately, more complex problems were tried and the results now were not so satisfactory: • In some instances, the networks generated were very complex with a great number of small heat exchangers and so it was decided to include a rule to deal with this type of problem. • In general, a low energy recovery was attained, therefore the system was modified implementing rules that provide for more exchanges.
2. an excessive number of exchanges as the amount of energy exchanged at each match is very small. This parameter, that is defined for each problem as required, will limit the matches to those where the amount of energy exchanged is greater than this minimum value. B. Exchange Alternatives. From the results of the initial tests on the system the need to improve energy recovery became clear. The answer has been to develop a heuristic for stream splitting. Stream splitting consists in the physical separation of a stream, hot or cold, in two streams, with the same thermal levels as the original stream, but reduced heat capacities. It is a technique that implies more exchange possibilities at the expense of a greater number of units: There will be more streams to use in the internal network (more energy integration), but the number of matches will, probably, be more as the streams available will have a smaller heat content than the original problem (smaller heat capacities). The selection of the stream to split will be based on the network of a previous synthesis. The cold stream with the highest heat capacity and heated with external utilities is selected, as no hot stream was capable of exhausting it completely. The objective of splitting is to obtain two cold streams with less energy requirements than the original stream, to allow more exchanges with the hot streams, that is, more integration. The procedure followed to split the selected cold stream is defined by the ratio R, between the heat capacities of the resulting two streams (a and b). To establish this value, the possibilities of exchange of these new cold streams are considered with respect to the existing hot streams. Following the H / H heuristic the first hot stream to try to exchange is the one with the highest initial temperature: Therefore, we will select the two hot streams with the highest initial temperature and will compute the ratio of their heat capacities (CPh l / CPh2). Then, this ratio will be applied to the splits of the cold streams,
Stage 3. Added Heuristic Rules CPc = CPca + CP~b, A. M i n i m u m Amount o f Energy to be Exchanged. The maximum amount of energy to be exchanged will be limited by the heat availability of the streams. The match will try to bring one of the streams (either the hot or the cold) to its final temperature. Such a stream will not need to be considered any more. On the other hand, it is also interesting to set a lower limit to the amount of energy to be exchanged in each match to avoid a complex and uneconomical network as a consequence of: 1. the manufacture of very small exchangers of cumbersome design increasing the construction cost of the network without a significant energy recovery,
R = CPhl/CPh2, R = CPca/CPcb.
Combining these three equations, CPca = (];Pc/( 1 + CPh2/CPhl), CPcb =
( ; P c - CPca.
The objective is to reduce the external heating of the cold stream by allowing it to exchange with other hot streams of the internal network.
J. A . S o u t o
e t al.
......: ........:.~ .: ...:: ..... ,. i,i.,.•:ii..iii,. • .:........: .i!.1117.111....11 ? .......: .........: .: ..........i•i:i) ~
. . 'u
i i ! ii I
. . . . : " . . . . . . . .
i i E o.ng.,,
( " =nWobanchoo "
FIGURE 5. Rule-based system RICPERT.
Again, tests were conducted to assure the possibilities of the method implemented, obtaining as a result high energy recovery at the cost of not so simple final networks, also, more frequently than before external utilities had to be used for heating and these exchanges could, in general, be grouped in a single exchange with very few alterations to the network. With this problem in mind a new heuristic was implemented for the grouping of matches that operates over the network that achieves the desired energy recovery. The grouping of heating exchanges for each cold stream is always possible if these exchanges correspond to the same stream, so the cold stream will achieve the same temperature as with the exchange at the highest temperature. Thus, the restriction imposed by MAT is obeyed. To group exchanges between streams, the loops have to be identified: A loop is a closed path of energy transfer between process streams. To open a loop implies the grouping of two exchanges in one without any extra energy consumption. If the loop is composed of two exchanges, the opening is simple. The heuristic developed here identifies this type of loops and then groups the two exchanges of each loop. The implementation, in PROLOG, of all the heuristics discussed previously resulted in a rule-based system called RICPERT. The structure of this system is shown in Fig. 5. A certain structure of this type, with well-defined modules and at the same time working flexibility, presents major advantages: • Modular programming which, with the heuristic rules defined independently (though included in the knowledge base), permits a rapid modification of C.
almost any aspect of the synthesis method followed by the rule-based system. The addition of new rules is also simple, it only implies the inclusion of the new rule in Prolog in the knowledge base of the rulebased system. • The first-in-depth search sets automatically various alternatives for the problem solution, whilst the heuristic rules determine the direction of the search to limit the region of analysis to areas possibly close to the optimum. • The dynamic structure of the data of variable dimension adapting itself at all stages to the needs of each problem. It behaves, like an instant memory in which the dynamic structure of the information about the process streams evolves towards the energy exhaustion of each one.
TABLE 1 Design Specifications of the Example Problem. Streams Data in K and kW/K
Hot Streams Stream
1 2 3 4
516.3 505.2 460.8 427.4
433.0 388.6 338.6 366.3
11.8 9.2 15.0 10.6
1 2 34
366.3 338.6 358.0 333.0
488.6 477.4 438.6 421.9
8.8 12.2 18.5 9.1
Rule-Based System for HENs
8SP1 K and kW/K
NAT = 1 0 . 0
Energy r e c o v e r y = 4539
C o l d Water Q
F I G U R E 6. User's screen in R I C P E R T .
3. THE NUMERICAL PROBLEM The rule-based system developed (RICPERT) allows the analysis of a great variety of heat exchanger network synthesis problems. As an example of its possibilities, a problem known in the specialized literature (Grossmann, & Sargent, 1978), as 8SP1, is solved. The problem consists of 4 hot process streams and 4 cold process streams with flow and thermodynamic data as shown in Table 1. The accepted value for the minimum approach temperatures in this case is l 0 K and, initially there is no limit in the minimum energy to be exchanged. From all this information, RICPERT computes the minimum amount of external utilities (MER utilities) using the problem table algorithm (Linnhoff, 1982). With this procedure, the MER is obtained for the considered value of the MAT and this figure will be used later to compare with the energy recovery obtained for the network synthesized. A typical user's session will generate a screen as shown in Fig. 6. The network obtained is presented in table form, which can be easily transported to a diagram as that of Figure 7. RICPERT does the synthesis following the heuristic rules previously mentioned, and starting with the
streams with highest temperatures. For each match the adequate exchange case is applied (see Table 2) until no more exchanges between process streams can be arranged. The residual streams are exhausted with external utilities. As a result of the synthesis, the heat exchanger network shown in Fig. 7 is obtained. From the energy saving point of view, this network recovers 99.5% of the MER; this value indicates that energy recovery is good and therefore it will be not necessary to split streams, as this procedure will imply a greater number
TABLE 2 Applied Exchange Cases in First Solution RICPERT Exchange Number
1 2 3 4 5 6
1 2 3 2 4 3
1 2 3 2 4 3
1 2 3 5 4 2
J. A Souto et al. (kW/K)
g;' 4K "
( 2 " 1 4 2 ~'S~\'ll2.°W 4 ""
(=)_ 3,,=#-,).~ 3,,= ~8,_
FIGURE 7. Problem 8 S P I : First Solution of RICPERT.
of heat exchangers and a more complex network and no significant energy recovery. As energy recovery is satisfactory, the work of RICPERT will concentrate in grouping the exchanges of the network shown in Fig. 7: First, RICPERT groups heating matches SI and $5, and, $2 and $6. Thus the number of heating matches is reduced to 4. Then, RICPERT finds 2 first-order loops: between streams H2 and C2 (matches 2 and 4), and between streams H3 and C3 (matches 3 and 6). The grouping of exchanges is done, two by two, obtaining as a result the network shown in Fig. 8. It is a rather simple network with only 11 exchanges (6 matches between process streams and 5 exchanges with external utilities), but still with a high energy recovery.
It was not necessary to limit the minimum amount of energy to be exchanged as the number of exchanges is small and none of them exchanges an extremely low amount of energy. All this synthesis process was carried out in less than 4 seconds in an 80286 IBM PC AT running at 8 MHz, and therefore the solution of problems with higher number of process streams will not suppose a prohibitive computing time. 4. C O N C L U S I O N S The methods for heat exchanger networks synthesis applied nowadays try to achieve, usually, a high energy recovery. This condition can, in certain cases, lead to
(kW/K) 11.8 9.2 15.0 10.6
_ 71.3kW _ 4 1 9K 4/
FIGURE 8. Problem 8 SPI: Final solution of RICPERT.
Rule-Based System for HENS
highly complex networks that must be corrected by the designer based on his experience. The system developed here, based on Ponton and Donaldson's algorithm (1974), includes some of the heuristics usually considered by the designer after the synthesis, 1. Stream splitting allows higher energy recovery in the final network, with lower operating costs as a consequence. 2. The constraint on the m i n i m u m a m o u n t of heat to be exchanged avoids networks with a large n u m b e r of rather small heat exchangers. The final network will present less heat exchangers (lower investment) and a system easier to control. The two heuristics added to the basic algorithm give a n u m b e r of alternatives for the synthesis. The designer would only need to carry out the economic assessment o f each of the networks proposed, according to the available cost data and the forecast for the prices of energy and then to select that of m i n i m u m cost. Development work in the synthesis method is presently taking place in two directions: • I m p r o v e m e n t of the user interface, with the presentation of the network as a grid diagram similar to that o f Fig. 8 and the use of the mouse pointer for interactive working on the screen. • Addition of the design rules for the heat exchangers to be used at any match, to evaluate the final cost of the network from the individual cost of each unit. As the system in its present state is able to achieve the o p t i m u m solution in a few seconds, the use of design routines for the heat exchangers, once the final network is defined, should not imply unacceptable
computing times and the design tool generated could be extremely useful for the design of heat exchanger networks and flexible enough to adapt to the new design technologies.
Bafiares-AIcfintara,R., Sriram, D., Venkatasuhramaniam, V., Westerherg, A.W., & Rychener, M. ( 1985). Knowledge-basedexpert systems for CAD. Chemical Engineering Progress, 81 (9), 2530. Grossmann, I.E., & Sargent, R.W.H. (1978). Optimum design of heat exchangernetworks. Computers and Chemical Engineering, 2(1), 1-7.
Linnhoff, B. (1982). Process integrationfor the efficient use of energy. Rugby: The Institution of Chemical Engineers. Linnhoff, B., & Hindmarsh, E. (1983). The pinch design method for heat exchanger networks. Chemical Engineering Science, 38(5), 745-763. Ponton, J.W., & Donaldson, R.A.B. (1974). A fast method for the synthesis of optimal heat exchanger networks. Chemical Engineering Science, 29(12), 2375-2377. Rathore, R.N.S., & Powers,G.J. (1975). A forwardbranchingscheme for the synthesis of energy recovery systems.Industrial & Engineering Chemistry Process Design and Development, 14(2), 175181. Rodriguez, A., Souto, J.A., & Casares, J.J. (1990). Sintesis de redes de intercambiadores.Anfilisiscomparativode dos procedimientos basados en la programaci6n lineal. M6todos num~ricos en Ingenieda, 6(4), 527-541. Souto, J.A., Rodriguez, A., & Casares, J.J, (1990). PROLOG-based expert system for CAT in chemical engineering. CATS'90: International Conference on Computer Aided Training in Science and Technology, (pp. 557-563), Barcelona: Pineridge Press.
Wood, R.M., Trivedi, K,K., O'Neill, B.K., & Roach, J.R. (1989). A new dual-temperature design method for the synthesis of heat exchanger networks. Computers and Chemical Engineering, 13(6), 667-685.