Compurersc/tern.Engng.Vol. 14, No. 2, pp. 197211, 1990 Printed in Great Britain. All rights reserved
00981354/90 $3.00 + 0.00 Copyright C 1990 Pergamon Press plc
SYNTHESIS OF FLEXIBLE HEAT EXCHANGER NETWORKSI. CONVEX NETWORKS J. CERDA~, INTEClnstituto Litoral (U.N.L.)
(Received
M.
R. GALLIS,
N.
CAMUSSI§ and
M.
A.
ISLAT
de Desarrollo Tecnolbgico para la Industria Quimica, Universidad National del and Consejo National de Investigaciones Cientificas y Tkcnicas (CONICET), Casilla de Correo No. 91, 3000 Santa Fe, Argentina
9 January
1989; final revision received 7 Jul_y 1989; received
for
publication
29 Augus/
1989)
AbstractA new methodology for the optimal synthesis of structurally flexible heat exchanger networks is presented. Since the stream supply temperatures arc specified by ranges rather than fixed values, the sought network design must be able to reach the target temperatures for any realization of the uncertain parameters. Moreover, it should always require a minimum utility consumption. To this end, an optimal heat recovery strategy is derived by using the notion of transient and permanent process streams. Based on it, a novel version of the heat cascade which assumes a continuous pinch behavior is developed to determine: (i) the heat recovery targets to be achieved by the network; and (ii) the dominant pinch temperatures constraining the heat exchanges and defining the problem subnetworks. They constitute the building blocks of an MILP mathematical formulation through which a structurally ftexible network featuring the least number of units is found. Unlike prior attempts, a feasibility condition handled as a model restriction eliminates the need for flexibility tests. A couple of examples involving convex pinch domains have been successfully solved in a short computer time. This shows a slight increase with the number of uncertainties. The method has been extended to the synthesis of nonconvex networks in Part II.
INTRODUCTION Recent works on heat exchanger network synthesis (HENS) have taken a step forward to further improve current HENS techniques. Changing plant operating conditions usually imposes additional requirements on the sought design. Marselle et al. (1982) first presented examples where the best networks provided by available HENS methods (Linnhoff and Hindmarsh, 1983; Cerd& and Westerberg, 1983; Papoulias and Grossmann, 1983; Floudas et al., 1986) failed to meet process specifications whenever small variations in suppry temperatures and flowrates occurred. Then, a new definition of the HENS problem is introduced where each process stream is now characterized by supply temperatures and heat capacity flowrate ranges rather than fixed values (Marselle et al., 1982). The problem goal is still the search of networks featuring maximum energy recovery (MER) and the minimum number of units (MNU). In this case the network must assure MER while reaching all stream target temperatures at any point of the problem disturbance range. For this reason, it is said that a feasible network for the new HENS problem has to be structurally flexible. A network is
tMembers of CONICET’s Research Staff and Professors at U.N.L. $Fellowship of CONICET. 4Member of CONICET’s Research Assistant Staff. CACC142F
structurally flexible if it guarantees the accomplishment of all process specifications and the MER target for any realization of the uncertain operating parameters. From the set of structurally flexible networks, the one comprising the minimum number of units (MNU) is sought. Saboo and Morari (1984) have classified flexible HENS problems into two classes, I and II. Class I problems only account for inlet temperature variations, but not all the problems considering supply temperature changes are of class I. According to Saboo and Morari (1984), there are two additional requisites to be satisfied. First, the pinch should always be defined by the same stream and the limits of the pinch domain where the process pinch is confined become determined by two extreme situations. In one of the limiting cases, the heating utility usage is maximized while the cooling utility need reaches a maximum value on the other (Saboo and Morari, 1984). The second requisite specifies that the supply temperature range for any stream different from the one determining the pinch does not overlap the pinch domain. The requisites are so strong that a few problems belong to the class I group. Class II problems are those ones not fulfilling class I problem requisites. As pointed out by Calandranis and Stephanopoulos (1986), new synthesis tools are needed for solving HENS problems with varying pinch determining streams, i.e. with “pinch jumps”. Marselle et al. (1982) proposed a heuristic technique to synthesize structurally flexible networks. 197
198
J.
CERDA
These are found by manually combining network structures developed for a selected set of four extreme points through “standard” HENS algorithms, i.e. HENS techniques which assume known, fixed values for inlet temperatures and flowrates. Such vertices are chosen on the basis that they are the four worst operating conditions. In other words, feasibility at them is assumed to ensure the same property at any other operating point. Saboo and Morari (1984) have shown that the assumption does not even hold for some class I problems to which the Corner Point Theorem applies. The theorem ensures that the best flexible network can be found by only considering the whole set of vertices of the disturbance region (Saboo and Morari, 1984). Consequently, they presented an evolutionary approach for the solution of class 1 HENS problems which includes a feasibility test at all corner points. More recently, Floudas and Grossmann (1987a) developed a systematic approach to tackle HENS problems involving temperature and flowrate ranges. Although neither of the examples solved in the paper exhibited nonconvex behavior, e.g. a pinch “jump”, simply because the specified flowrate variations have no effect on the pinch location. In one of the examples there is no pinch and the other shows a temperaturedriven continuous pinch displacement. The algorithm proceeds in two steps to find: (i) the set of heat matches through a multiperiod MILP transshipment model (Floudas and Grossmann, 1986); and (ii) the network configuration by means of a multiperiod NLP formulation (Floudas and Grossmann, 1987b). Feasibility tests are run at both stages to iteratively identify critical operating points claiming for further heat matches or changes in the network topology. By using an active constraint strategy, they derived feasibility tests which do not assume that the critical points are vertices and by so doing eliminate the need for an exhaustive vertex search procedure (Grossmann and Floudas, 1987). This requires development of the current network structure at each iteration. The algorithm ends up with a network design that features both MER and MNU at the critical points. The feasibility tests imply the solution of a rather significant number of NLPs. As the number of uncertain parameters rises, the computer time for running the tests could grow rapidly and the efficiency of the method would become questionable for problems of moderate size. Depending on whether or not they exhibit a noncontinuous behavior (a “pinch jump”), HENS problems are going to be classified in this work into two types called convex and nonconvex, respectively. Convex HENS problems are those where the pinch never jumps even though certain stream supply temperatures change within some specified limits. Otherwise, they are nonconvex. The inlet temperature range specification is particularly important for the convexity of a given HENS problem. Frequently, a convex problem becomes nonconvex when some
et al.
temperature disturbances grow beyond certain values or new ones are considered. The turning sizes and the nature of the disturbances causing nonconvexities vary with the problem. Part I of this paper presents a smallsize MILP mathematical formulation for the optimal synthesis of convex flexible networks. Therefore, inlet temperature variations are only assumed. The set of pinch temperatures to be considered by the problem formulation to always maximize the process heat recovery is provided by a new version of the heat cascade diagram. By solving the proposed model a structurally flexible HEN at the level of matches can be found without using a feasibility test. A special rocedure is also given to derive the network config..ration. DEFINITION
OF
PERMANENT
PROCESS
AND
TRANSIENT
STREAMS
Let h, be a hot process stream whose supply within the interval temperature can vary (f’), f’: + ATT) that includes the nominal operating temperature T;, N. Such a stream h, can be described in terms of:
(i> a “permanent”
hot stream A; featuring a temperature range (F;, T:) and a heat capacity flowrate (FC,,), at any plant operating condition (see Fig. 1). hot stream 4 that exhibits a (ii) a “transient” temperature range (F’: + v, AT:, If’:) where the parameter vi can vary from 0 to 1 depending on the actual operating condition. (FC,,); is also the heat capacity flowrate of &. Figure 1 shows the transient stream g, when v, = 1, i.e. the hottest transient stream h’,. Therefore, the notion of a transient stream is introduced to handle the stream’s uncertain parameters. Similarly, a cold stream c, whose supply temperature can vary from f;  AT; to f; is equivalent to a stream set comprising: (a) a “permanent” cold stream 2, having a steady temperature range (i;‘, Tj); cold stream cj featuring a and (b) a “transient” temperature range given by (Tj ~ 0, AT;, ?=j) where the parameter 0, can take any value from 0 to 1 according to the actual operating condition (see Fig. 1). Both streams 2, and F, have a heat capacity flowrate equal to (FC,,)j. Figure 1 shows the transient cold stream ?, when 19~ = 1, i.e. the coldest transient stream Z,. REFORMULATION
OF
THE
RECOVERY
TARGET
When steady changes in supply temperatures might occur during plant operation, the real problem is the synthesis of a HEN that reaches not only process specifications but also minimizes the utility consumption at any expected operating condition. However, the selection of such a target as the building block of a mathematical model for the flexible HENS problem
Synthesis of flexible heat exchanger
i HOT
Fig.
I. Modified
stream
STREAM
partitioning
COLD STREAM
hi 1
procedure
would seem inappropriate. Since the disturbance region includes an infinite number of operating points, an extremely complex formulation unsuitable for handling practicalsize problems would then be required to ensure the network feasibility anywhere. Prior attempts have circumvented this problem on the basis of computational experience. This has indicated that a structurally flexible network can be synthesized by taking into account a small set of critical operating points. By ensuring feasibility at those stream conditions one can guarantee such a property along the entire disturbance range. Rather than follow this type of approach, it has been found more convenient to reformulate the heat recovery target. In fact, permanent hot and cold streams as previously defined are present in the process for any stream conditions. If the MER target is to always be achieved, special attention must then be focussed on both the fulfilment of permanent energy needs and the allocation of permanent heat supplies. As permanent cold streams are steady heat users, an optimal heat recovery policy should intend to meet their energy demands by making use of the most reliable process heat sources, e.g. the permanent hot streams. Only if these are (in quality or quantity) insufficient to satisfy all permanent energy requirements, one must take advantage of the process heat from transient hot streams for that purpose. An optimal heat recovery strategy follows the same approach to meet transient energy demands. After implementation of permanent/permanent stream heat matches, some lowerquality permanent energy supplies remain for use in the plant. If so, they should
networksI
199
cj
used to develop the heat flow diagram.
be considered prior to transient hot streams for the accomplishment of transient energy requirements. After completion of the heat recovery process, all residual energy needs are to be satisfied through hot utilities while any process heat surplus will be transferred to cooling water. In this way, a minimum usage of both cold and hot utilities is attained at any operating point. Therefore, this design goal requires to reach the following ordered sequence of heat recovery targets. First level heat recovery “Maximize permanent
target
(HR T I)
the heat flow being transferred hot and cold streams”.
between
This can be readily attained by constructing the permanent heat cascade, i.e. the heat cascade involving only permanent process streams, as shown later. By doing so, the permanent pinch temperature can be found. Secondlevel
heat recovery
target
(HRT2)
“Maximize both (i) the heat flow provided by transient hot streams to residual permanent cold streams and (ii) the allocation of permanent heat surplus to transient streams”. Target HRT2 can be better understood by recalling the physical meaning of the permanent pinch point. Above the permanent pinch, the set of permanent process streams can be regarded as a net heat sink. To meet still unsatisfied permanent heat demands, target HRT2 favors the use of transient
J.
200
CERDA
heat sources rather than heating utilities. Furthermore, priority is given to the permanent heat surplus below the permanent pinch to satisfy energy needs of transient cold streams. Thirdlevel
heat recovery
target
(HRT3)
“Maximize the heat flow between residual transient streams”.
hot
and
cold
Target HRT3 stresses the economical advantage of using transient process heat instead of heating utilities. Thus, both heating and cooling utilities can be simultaneously saved. Implementation of the proposed heat recovery strategy leads to a new form of the heat cascade diagram. To this end, the stream partitioning procedure of Cerda et al. (1983) must be adequately modified as shown in the next section. The new heat cascade will provide important insights for the optimal synthesis of flexible HENS. In particular, it gives a complete characterization of the related pinch domain within which the process pinch is located at any stream condition. THE
STREAM
PARTITIONING
PROCEDURE
In order to account for the minimum temperature approach, a stream partitioning procedure is usually performed to generate a set of hot and cold substreams (Cerda et al., 1983). These are defined in such a way that a heat match between any pair of hot and cold substreams is either fully allowed or forbidden. Development of the heat cascade diagram for a HENS problem involving inlet temperature ranges requires a suitable modification of the standard partitioning procedure. Every transient heat source or sink featuring its maximum heat duty is assumed to be in the process for the construction of the problem heat cascade. The cascade is therefore developed for the operating point where supply temperatures are at their top or bottom values depending on whether the streams are hot or cold, respectively. Although the heat cascade gives a complete picture of the pinch temperature range exhibited by the process along the disturbance region. Consequently, the partitioning algorithm of Cerda er al. (1983) is going to be applied to the set of permanent and fullrange transient streams just for deriving the problem heat cascade. For fullrange cold and hot transient streams, the parameters t?,, jEC and v,, io H are both equal to 1. As usual, the supply temperatures of the streams to be partitioned define the problem temperature intervals. Since they are: T;,
F;AT;,
v,,~c
and P’:,
F;+AT:,
then the temperature problem are no longer
V?EH,
levels at the new HENS established by the nominal
et
al.
values but by the upper and lower bounds on the supply temperatures of every process stream. Of course, hot stream temperatures should be decreased by the minimum temperature approach AT,,. Briefly, the proposed partitioning technique implies that each process stream is first decomposed into permanent and fullrange transient streams and subsequently by temperature levels. Table 1 lists the data for a fourstream flexible HENS problem (Example 1). The inlet temperature of stream Cl is expected to vary from 96 to 116°C while T; can change from 229 to 249°C. The set of permanent and fulltransient streams for the problem is shown in Table 2. The enlarged array of supply temperatures is given by: (>239,
239,229,
219,
126, 116, 96),
assuming AT,,,i, = 10°C (see Fig. 2). By applying the partitioning algorithm of Cerda et al. (1983) to permanent and fullrange transient streams, the set of hot and cold substreams included in Table 3 is obtained. Uncertainties in both T; and T; have introduced two additional temperature levels and five new substreams. THE
HEAT
FLOW
DIAGRAM
A novel version of the heat cascade that provides the pinch domain for convex HENS problems involving supply temperatures ranges is introduced in this section. The construction of the new heat cascade requires consideration of the complete set of fullrange transient heat sources and sinks present at the process all over the disturbance range. Moreover, the proposed sequence of heat recovery targets HRT1 to HRT3 constitutes its fundamental basis. The primary target is the maximization of the heat flow transferred from hot to cold permanent streams. A practical means of reaching that is the construction of the permanent heat flow diagram. To this end, the deficit or surplus of permanent heat at each
Table
I.
6.096 7.032 10.000 8.440
Cl H2 c3 H4
2.
Set
of data
for
Example
(kvs)
streams
Table
Set
of’
permanent
(,Td)
e1 el Hz R2 53 A4
,kM% 6.096 6.096 7.032 7.032 10.0 8.44
1)
,z,
96/l 16 2491229 I26 239
and fullrange Example 1 Temperature
StrealllS
I
(‘C) 116150 96116 229l 20 249229 12G250 239148
150 120 2.50 148
transient
streams
for
range 0% + + 
207.3 121.9 766.5 140.6 + 1240.0  768.0 7. Q, =

105.9
Synthesis of flexible heat exchanger networksI
201
PERMANENT HEAT CASCADE
250°
&139.7 250*
1 TRANSIENT HEAT SOURCES
2390
_l
2290 PINCH DOMAIN

it
2190
126” E
g6”_
TRANSIENT HEAT SINKS
116’
_L El
_“C Fig. 2. Example 1. 4
temperature level must be evaluated (see Fig. 3). By adding a minimum heat flow at the top of the cascade to just meet every energy requirement, the permanent pinch can be found. However, the secondary target favors the use of cheaper heat sources present in the process, i.e. the transient hot streams. If transient heat above the permanent pinch is available, it should then be allocated to the permanent heat cascade before using heating utilities. Therefore, the amount of heat provided by transient hot streams at each temperature level must be computed (see Fig. 3). From these it follows that: (1) the proposed heat remarks, cascade handles transient heat sources and sinks separately; and (2) transient heat is never used in the heat flow diagram to fulfil permanent energy needs below the permanent pinch. Similarly, the excess of permanent heat beneath the permanent pinch, if any, is the energy source’s first choice to meet the demands of transient heat sinks. This requires the evaluation of the heat flow demanded by the set of transient cold streams at each temperature interval (see Fig. 3). By following the sequence of heat recovery targets, permanent heat supplies are never transferred to transient cold streams at temperatures higher than the permanent pinch. Table 3. Set of hot and cold Substreams
IkW
FC,
substreams
for
I cj
Cm’1
1
Example
Temperature range
Q
IkW>
10.0 7.032 10.0
239250 249239 229239
+
7.032 10.0 8.44 6.096 7.032 10.0 8.44 6.096 7.032 6.096 7.032
239229 219229 239229 12&150 229i 36 126219 229148 l&12.5 136126 9&l 16 126120
70.3 + 100.0 84.4 + 146.3 654.0 + 930.0 683.6 +61.0  70.3 + 121.9 42.2
I
110.0  70.3 + 100.0
Fig.
3. The
new heat cascade diagram for Example
547
I
During execution of the procedure, it may happen that a certain surplus of Ithlevel transient heat above the permanent pinch still remains after all permanent energy needs at level I have been fulfilled. If so, the exceeding heat is transferred downwards to the transient heat source just below. On the other hand, transient energy supplies can be used to meet transient heat demands at both sides of the permanent pinch after the amount of permanent heat still available below the permanent pinch has been allocated. This is suggested by the thirdlevel heat recovery target which favors in such a case the use of transient heat. Hot utility is always the last choice. To meet process specifications, the remaining permanent and/or transient heat demands are satisfied through heating utilities. At the same time, heat surplus is transferred to cooling utilities. E_sample
I (a convex
HENS
problem)
Supply temperature ranges are specified for both the cold stream Cl and the hot stream H2 (see Fig. 2). As a result, there are two transient heat sources at levels 2 and 3 and a unique transient heat sink at level 6. The problem heat cascade is shown in Fig. 3. It can be obtained either by hand or through solving the utility usage problem (Pl) presented in the next section when some heat matches are forbidden. At the permanent heat cascade one can observe two pinch points located at 2191229 and 229/239”C, respectively, within the uncertainty range of 2i. Each of them can be reached by replacing one or both hot transient sources with a heating utility. Since the set of permanent process streams is a net heat sink above the related “permanent” pinch, this one takes place at 219/229”C, i.e. at the lowest value of the uncertain parameter T3. As the value of T; rises, the process pinch moves continually upwards to steadily occur at
J. CERDA
202
the supply temperature of stream H2. While this happens, the heating utility consumption diminishes because of the larger surplus of transient heat above the permanent pinch. Displacement of the pinch just stops when T; becomes equal to the fixed inlet temperature of stream H4 (T; = 239°C). At this point, there is a change of the pinchdetermining stream. For higher values of T;, the process pinch remains at 2291239°C and H4 becomes the new pinchcontrolling stream. Moreover, no transient heat flow crosses the pinch at 2291239°C either. Therefore, the problem exhibits and infinite number of pinch points confined to a continuous region bounded by the two pinch points at 2191229 and 229/239”C, respectively. For convex HENS problems, therefore, the controllingstream transition occurs without a discontinuity in the pinch shifting, i.e. a pinch jump does not happen. When the pinchdetermining stream changes in Example 1, the process pinch is tied to the common supply temperatures of process streams H2 and H4 simultaneously. This is a particular instance of a general behavior.
PI al.
C C 4(l;ilY?jkk) + C C q(h3 ?,k) iE HI
/E LPI
iE”l
IELP
I$k
f
+q(S,E,,)=&,
VkEKTj,
(ii) allocation of the available temperature level:
(iii) nonnegative
vjeC;
heat
flow
(2)
at each
heat flows.
(5)
Since the overall heat flow exchanged cold permanent streams is given by:
by hot and
and similarly, THE
CONSTRAINED
UTILITY
USAGE
PROBLEM
(Pl)
In order to develop the heat flow diagram for constrained HENS problems involving temperature uncertainties, the utility usage problem (Pl) is going to be defined. It is desired that the proposed formulation for (Pl) accounts for process design constraints preventing the execution of certain heat matches. To handle such constraints, the hot stream set W associated to cj (j = 1, 2, . , C) and the cold streams set C’ related to h, (i = 1,2, _ , H) are so defined, W = {h, 1 i E n stream h, C’ = {c,\j E C stream hi
and the heat match and cold stream cj is and the heat match and cold stream c, is
between hot permitted), between hot permitted},
the minimum temperature approach Similarly, restrictions are implicitly taken into account by introducing a temperature level set for every permanent and transient stream: LP’ = LT’ = KPj= KT’ = A level heat (i) each
{I {Z (1 {I
1 permanent hot stream A; is in level f}, 1 transient hot stream 6 is in level I}, 1permanent cold stream cj is in level k}, 1 transient cold stream F, is in level k}.
feasible solution to the HENS problem at the of matches should satisfy the following set of balances: fulfilment of cold stream heat requirements at temperature level:
(7)
then, the proposed sequence of heat recovery targets HRT1 to HRT3 can be achieved by maximizing the following expression: max Q(&,
c)
+ 0.75
+ 0.75
Q(fi,
Q(A,
c)
c)
+ 0.50
Q(r?,
c),
(10)
chosen as the problem objective function, subject to constraints (l5). Values of the coefficients in the problem objective only reflect the priority levels of the three heat recovery targets HRT1 to HRT3. Their magnitudes are not really important provided that they are set in the proper order, i.e. the higher the priority level the larger the coefficient. The same results can be found by setting them equal to 3, 2 and 1 instead of I, 0.75 and 0.50, respectively. By expressing the heat flows included in the objective function (10) in terms of permanent and transient utility usages through equations (14) the solution sought can be found by minimizing: nin ,$] {kE;p, 4(X
+
5
c
i1 { Ir‘P,
h)
+ o5
4(&,,
wt
c q(S, a> kekm 1 0.5
1
,eLP
q(&,, W)
>
Synthesis of flexible heat exchanger Therefore, targets HRTI to HRT3 tend to penalize the use of heating/cooling utilities for the fulfilment of permanent process needs by assigning to them higher unit costs. The magnitudes of the penalties depend upon the set of coefficients in (IO) but it has no effect on the optimal solution to (Pl). Fundamental ideas for the formulation of the transportation problem has been taken from studies on inexact transportation problems (Kacprzyk and Krawczak, 1979). The minimum conditions
utility usage at some extreme
operating
The optimal heat recovery strategy adopted as the design target for problem (Pi) leads indeed to the minimum utility usage at any operating condition. Then, one can expect that the least utility consumption at some important extreme operating points be known after solving problem (PI). This is still possible at some intermediate values of one or more uncertain parameters only if they are used to define the problem temperature levels, i.e. at extreme supply temperatures for other process streams. For instance T; = 229°C is an intermediate inlet temperature for H2 and an extreme one for H4. Usually, the authors pay a greater attention to the following stream conditions: Point A
Point S
Point C
Point
b
Supply temperatures are the highest expected for all hot process streams and the lowest ones for all cold process streams (Maximum Heat Recovery Condition). Supply temperatures are the highest expected for all hot and cold process streams (Maximum Cooling Condition). Supply temperatures are the lowest expected for all hot and cold process streams (Maximum Heating Condition). Supply temperatures are the lowest expected for all hot process streams and the highest ones for all cold process streams (Minimum Heat Recovery Condition).
Points B, C and D can be considered as particular cases of Point A because either the set of transient heat sinks, the set of transient heat sources or both have vanished at them. Since the overall heat flows exchanged by permanent and transient cold(hot) streams with heating(cooling) utilities are, respectively, given by: (11) c O(S,C)=
1 j=l
c AbKT!
q(S,<,),
(12)
networksl
203
Q(&
W) = 5 c q(&,> i I IeLP,
V>
(13)
Q(&
BY=
B’),
(14)
5
c
q(K;,,
I = I IELP
then, the least utility usages at operating Points AD can be derived from the heat cascade [or the optimal solution to (Pl)] by applying the following expressions: Point
Point
A: Q:
= Q*(S,
Q$
= Q*(Z?,
Q:
= Q*(X
c)
+ Q*(.S,
W) + Q*(fi,
W).
c),
(16)
(17)
W)+Q*(A,
+Q*(fi,
c)+
Transient heat sinks vanish the cooling utility.
W) &*(A,
c).
and they are replaced
(IS) by
C: Qs = Q*(S,
P)
+ Q*(A, Q& = Q*(fi, Transient heat by the heating Point
(15)
B:
Q$=Q*@.
Point
c),
sources utility.
+ Q*(S,
L?)
c’> + Q*(fi,
c),
W). vanish
(19) (20)
and
they
are replaced
D: Q;
= Q*(S,
QZ
= Q *(A,
c)
+ Q*(fi,
W) + Q *(I?,
6) P).
(21) (22)
Both transient heat sources and sinks vanish and they are replaced by heating and cooling utilities, respectively. By substituting the optimal heat flow values shown in Fig. 3 into equations (1522), one can obtain the least utihty usages at operating Points AD for Example I. They are shown in Table 4. To illustrate the way the minimum utility usage at an interior point can be derived from the optimum of (PI), it will be assumed that T; = 239°C and r; = 116’C (Point E). Therefore, there is no cold transient stream in the process like at Point B. Equation (18) still holds but equation (17) must be replaced by:
where the additional term Q&((ct, c) stands for the heat flow provided by the transient heat source at level 2 which is no longer in the process, e.g. it amounts to 70.3 kW (see Table 4).
J.
204 Table
4.
Minimum
utility
usages (Example
at
the
operating
Points
CERDA AE
I)
*
Point Point Point Point Point
THE
A B C D E
PROBLEM
cl%,
c%,
139.7 139.7 225.6 225.6 210.0
245.5 367.5 190.8 312.8 367.5
DOMINANT
PINCH
POINTS
Convex flexible HENS problems are those featuring a continuous pinch transition whenever the pinchcausing stream changes. The pinch domain for convex problems can comprise single or multiple temperature intervals. At each one, the pinch is usually tied to a particular stream inlet temperature whose uncertainty range does include such a temperature level. Therefore, the generating stream giving rise to a portion of the pinch domain is the one controlling the pinch location along that portion. For instance, there is a singlelevel pinch domain generated by stream H2 in Example 1. In general, the generating stream varies with the however, temperature level. This explains why a multilevel pinch domain often exhibits two or more generating streams all hot or all cold. It might even happen that convex problems featuring a singlelevel pinch domain presents several alternative pinchdetermining streams all hot or all cold (see Example 2). When one generating stream is cold and the other is hot, a pinch jump usually occurs at some operating conditions and the problem is nonconvex (see Part II). Saboo and Morari (1984) have only regarded as Class I HENS problems those featuring a continuous pinch domain generated by a single process stream. Starting from either the lowest or the highest pinch temperature, the pinch points relevant for the HEN synthesis task are those crossing new process stream temperature ranges. In this paper they are called the problem dominant pinch points. The stream crossings represent thermodynamic constraints upon the heat recovery process to be satisfied if one must always reach the MER target. Since new stream crossings can only arise at leveldefining temperatures within the pinch domain, then such temperatures can be regarded as dominant pinch candidates. A proper formulation of the flexible HENS problem should embed them to assure the achievement of the MER at any operating condition. Identification of the dominant pinch temperatures is therefore a key step carried out with the help of the heat cascade diagram. A zero heat flow is observed at any of them. When the stream supply temperatures are not fixed values, determination of the minimum number of units requires accounting for all dominant pinch temperatures while the related stream conditions at which they arise become important for the synthesis of the optimal network structure. This leads to introducing the notion of network design conditions
et
al.
as those defining configuration.
by
themselves
the
best
network
In Example 1, streams H4 and C3 are crossed by the lowest boundary pinch temperature (see Fig. 2). A zero heat flow at the permanent heat cascade is telling us that the set of permanent streams (H4, C3) at level 3 are being cut by the pinch at 219/229”C (see Fig. 3). Moreover, the top boundary pinch crosses streams H2 and C3 since the overall heat flow across the temperature 229/239”C is null. As a result, a pair of units for the heat matches between C3 and H2 at both sides of the top pinch are needed. Similarly, all the heat available at stream H4 above the lower boundary pinch must be used to meet process heat demands at levels higher than 219/229”C. Consequently, heat exchanges between streams H4 and C3 above and beneath the lower boundary pinch must be accomplished in two different units. The two dominant pinch candidates are really so in Example 1. The process pinch is at the top location, 229/239”C for any value of T; in the interval 239249°C. In particular, it is there at the extreme operating Points A and B. On the other hand, the bottom dominant pinch arises for T; = 229°C. i.e. at the extreme Points C and D. Then, the stream conditions AD (or Bc) are to be considered for the search of the network topology as shown in the following sections. It should be stressed that a convex flexible HENS problem always presents, at most, a single pinch somewhere between the top and bottom pinch temperatures. Therefore, the HEN can still be regarded as being composed of two subnetworks placed at both sides of the process pinch, i.e. the upper and lower subnetworks. Any heat match (h,, c,) taking place above the dominant pinch crossing streams h, and cj is carried out in the upper subnetwork. Otherwise, it belongs to the lower subnetwork. Some exchanger heat duties can vary with the stream conditions if the units start from the pinch or involve transient substreams. This is the case for the exchangers executing the heat match HZ/C3 above and below the upper boundary pinch in Example 1. The knowledge of the dominant pinch temperatures facilitates the determination of the heat matches to be considered at each subnetwork by the problem formulation. Table 5 gives that information for Example 1. A significant number of heat matches identified by the symbol I+ have been discarded beforehand because they cross a problemdominant pinch. In this way, the number of qvariables is drastically diminished. THE
MLNIMUM
NUMBER
OF
UNITS
In addition to reaching the MER target for any operating condition, an optimal flexible network structure features a minimum number of units. This is another design target to be achieved by the network sought through a proper definition of the synthesis
Synthesis of flexible heat exchanger networksI
205
Table 5. Set of heat matches to be considered at each subnetwork (Example 1) H
C
fi22
s32 s33
(U) I’
s14 s34 e15 e‘14 W
I+ I+ I+ I+ I+
fi23
fi43
A24
%I4
fi25
ti26
I
I
I
I
w (L) (L) (L) (L1
(L) (L) (L) (L) CL1
I (L) u) (L) (L) IL)
I+ I+ I+ If 1+
I
I
S
iui
I
W)
I I
(L) (I) (L) (LI
I+ I+ I+ I* I+
(t (LI
I = infeasible heat match; I’ = heat match crossing a dominant pinch; U = upper subnetwork; I_ = lower subnetwork.
procedure. In any case, the development of simple expressions to predict the least number of units in a particular HENS problem involving temperature uncertainties will permit us to know whether the MNU target has been reached. It is always true that a heating utility is needed to meet process specifications above the pinch only. However, the pinch location varies with the inlet temperature of the controlling stream at flexible HENS problems. As a result, the temperature range and the number of units in the upper subnetwork where heating utility is required both change. The number of exchangers needed can be highest predicted by using the widely known expression: NZ,,, = N:+N,U+N,1, where Ns and NY stand for the number of hot and cold process streams, respectively, whose temperature ranges are either crossed by any of the dominant pinch temperatures or run above all of them. By the definition of a dominant pinch, no further stream crossing can occur. As a result, more units than N&, will ever be necessary. Ns represents the number of different heating utilities available in the network. Uncertainties in stream supply temperatures generally bring about an increase in the number of units. An additional exchanger is required whenever the temperature range of a transient stream, either hot or cold, is crossed by a dominant pinch. In Example 1, besides the pair of streams (H4, C3) being crossed by the bottom dominant pinch, stream H2 is cut by the top pinch (see Fig. 2). Then, the least number of units in the upper subnetwork is given by: N:,n
= 2+1+l1=3,
when only one heating utility is available. Since a single transient stream like fi2 is cut by a dominant process pinch, the number of units is increased by one due to temperature uncertainties. Moreover, allocation of a heater to any cold stream receiving heat from stream H2 above the upper boundary pinch at 229/239”C will always permit achievement of the process specifications. A similar expression can be applied to find the minimum number of exchangers at the lower subnetwork where only the cooling utility is required. In
this case, the values of Nk and Ni represent the number of hot and cold streams whose temperature ranges are either crossed by any of the dominant pinch temperatures or run below them: N&=N:,+N$+N:,1. In Example 1, cold stream Cl running below the two dominant pinch temperatures should be added to those involved in the evaluation of NEi,. Then, N;,, Therefore,
= 2+2+11=4.
the MNU N ml” = N:,,
THE
FLEXIBLE
HENS
target for Example + N&,
1 is given by:
= 3 + 4 = 7.
PROBLEM
FORMULATION
(PZ)
A feasible HEN is now sought at the level of matches that reaches the least utility usage through the fewest number of exchangers for any operating condition. Therefore, the proposed mathematical formulation for the flexible HENS problem (P2) must guarantee the accomplishment of both the MER and MNU targets. (i)
The MER
target
The MER target is achieved by avoiding any heat match (h,, c,) implying a heat flow through a dominant pinch crossing streams h, and cj. This is automatically attained by imposing the three recovery targets HRT1 to HRT3 found by solving problem (PI) since they embody the knowledge about the problem pinch domain. Then, any feasible solution to (P2) should satisfy the following constraints: 
HRTI
target:

HRT2
target:
Q(&

HRT3
e> > Q*(fi,
6>,
(23)
Q(&
2;) L Q ‘(A,
c),
(24)
Q(&
c)
6),
(25)
c),
(26)
target: Q (&
to always
> Q*(&
c) 2 Q *(a,
reach the MER
target.
J. CERDA
206 (ii)
The MiVU
farget
To accomplish the MNU target, a set of binary variables yiis for each problem subnetwork is introduced into the mathematical modelling of problem (P2). Each variable yijs indicates whether or not a heat exchanger is actually required to perform the heat match (h,, c,) at subnetwork s. By doing this, the MNU target can be achieved by choosing the foIlowing expression as the objective for problem (P2):
min,ESN c ,5 5 Y,% + = I /EC,
be replaced by the heating utility at some operating conditions. Similarly, an additional amount of hot utility is required to occasionally meet heat demands from cold transient streams. By imposing y,,, = 1 (atlocation of a heater to cold stream cj) whenever Ui, > 0, one can guarantee that the specified outlet temperature Tj is always reached. This condition is mathematically expressed through the following constraint: W < Ys.,Ms 7
I2
,= I
ef al.
Ys., +
;
,=I
Y,.,,
(27)
where the set of subnetworks SN is given by SN = {upper( lower(l)}. Since Y,,~ must be zero if the heat match hi/c, is not performed at subnetwork s, then:
J%.,=O,
C’={jlj~C
~EC,,
iEH,,
s ESN,
Tj>
T,,}
and M, is an upper bound on the value of U/,, j E C’. In the same way, the maximum cooling utility usage for hot stream h, is given by:
c c c rtc
q(L
5kk)
kee‘,
(28)
where the summations include the temperature levels involved in the subnetwork s. General definitions for the stream set H,, C, and the level sets LP;, LT:, KPJ’ and KT:’ are given in the Appendix. (iii) Structural feasibility
1,
and
IELP,
for any
(30)
where
+ yij, = 0, 1
“, E C’.
condition
Below the lowest pinch temperature, the process does not require the heating utility at any operating condition. Above the highest pinch, the cooling utility is not necessary. For the set of cold process streams running partially or entirely above the bottom pinch, the heating utility requirement usually increases as some cold or hot transient streams arise or vanish, respectively. Let ITp and I,, stand for the temperature levels just above the highest and the lowest pinch temperatures, respectively. The maximum heating utility need of a cold stream cI is then given by:
To reach process specifications, hot transient streams supplying heat above the lowest process pinch are to
(31) Then,
,v,.~=O,~, t’i~H’,
(32)
where H’={ili~H and M,
and
is an upper bound
r:
on the value of LIW.
TheoremRestrictions (2932) ensure network structure provided by problem reach process specifications at any condition.
that the (P2) can operating
ProofAs previously discussed, the fulfilment of the new heat recovery targets HRT1 to HRT3 permit reaching the process pinch at any stream conditions. the minimum utility usage at each Therefore, operating point can be obtained by substituting in the heat cascade: (a) nonexisting transient hot streams supplying heat above the lowest process pinch by heating steam; and (b) nonexisting transient cold streams receiving process heat below the highest process pinch by cooling water. The whole set of such transient hot and cold streams are taken into account by equations (29) and (31). If each hot(cold) stream exchanging heat with transient cold(hot) streams below(above) the top process pinch has available a cooler(heater), it can then always reach the target temperature. But this is
Synthesis of flexible heat exchanger networksI the condition imposed through inequalities (30) and (32). Therefore, the network structure guarantees the achievement of all process specifications at any operating condition, and consequently the theorem has been proved. In this way. the flexible HENS problem (P2) has been modelled through an MILP formulation having an objective function given by (27) subject to the set of constraints (l5), (2326) and (2832). THE
NETWORK
DESIGN
CONDITIONS
From a structure viewpoint, a process pinch becomes dominant if the set of process streams around it includes at least one not present in the surroundings of any other pinch. This is a particular feature of the pinch points arising at leveldefining temperatures, i.e. the dominant pinch candidates. From the group of pinch candidates, only those ones showing a different stream population are to be considered. Therefore, the dominant pinch temperatures not only rule the number of units in each subnetwork but also their structural configurations. Because of that, the structural feasibility at the stream conditions giving rise to the dominant pinch temperatures automatically implies the same property all over the disturbance range. Since there is a single pinch at each operating point, then there will be as many design conditions as the number of problem dominant pinches. However, there is not a onetoone correspondence between pinch temperatures and operating points. Indeed, an infinite number of operating points can usually lead to the same pinch. Any of them can therefore be chosen as one of the network design conditions. Of course, it is better selecting an operating point for which the units and their heat duties at each subnetwork can be readily derived from the optimal solution to problem (P2). In Example 1, the upper and lower subnetwork configurations have been determined by first picking up as design points operating conditions A and D shown in Table 6. They have been chosen because the top and bottom dominant pinch temperatures occur at Points A and D, respectively. An alternative choice is given by the pair of Points B and C. Then, the optimal network is Table 6. Selected design points for the upper and lower subnetworks at Examples I and 2 Lower streams
Design point
I
network Design point
Upper Design poinl
2
I
network Design point 2
synthesized best HEN conditions. sidered the alternative of Fioudas THE
207
through a proper combination of the structures found for the selected design In each subnetwork, it should be conset of units provided by problem (P2). An means is the multiperiod NLP algorithm and Grossmann (1987b).
FLEXIBLE
HEN
Development of a require accomplishment steps:
SYNTHESIS
PROCEDURE
flexible network structure of the following sequence of
1. Define the set of permanent and transient process streams. 2. Partition all permanent and transient streams at the extreme supply temperatures of every process stream. In this manner, the set of substreams to be considered in the development of the heat cascade is obtained. 3. Construct the problem heat cascade and determine at once both (a) the heat recovery targets, and (b) all the temperatures at which zero permanent heat flows arise, i.e. the dominant pinch candidates. 4 Starting from the lowest pinch, discard those dominant pinch candidates not crossing a new pair of process streams (hi, cj). Then, establish for each hot stream the set of heat matches to be accomplished in different units to avoid crossing a dominant pinch. Choose a network design condition for each dominant pinch. 5. Solve the flexible HENS problem (P2) and find the minimum set of units needed at each subnetwork. From the optimal solution, evaluate the exchanger heat duties at the design points (see Appendix). 6. Develop the optimal configuration for each subnetwork by combining manually the structures found at the design points or applying the multiperiod NLP algorithm of Floudas and Grossmann (1987b). RESULTS
AND
DISCUSSION
To illustrate how it proceeds and the computer time it requires to run, the proposed synthesis procedure has been applied to find the optimal network structure at two convex example problems. A specific discussion of each example is given below. In all cases, the computer time required on a VAX 1l/780 to find the optimal solution is mentioned. Example
ll6I50 229l 20 126219 229148
Example 96150 239I 20 126229 239l 48
I
C1 HZ c3 H4
II&150 229l 20 126219 229148
Example 106150 239l 20 126229 239148
2
Cl HZ c3 H4
Example 1 is a convex HENS problem featuring a singlelevel convex pinch region (see Fig. 2). The optimal solution shown in Fig. 4A has been found in 10.2 s. It comprises seven units, including a heater and a cooler in complete agreement with the value predicted by the proposed MNUexpressions. Next to each unit, the related permanent and transient heat
219250 239229
219250 239229
249239 229250
249239 229250
I
J.
208
(Al
CERDli
et a/.
Optimal solution to problem P2
(8)
Optimol ot each
set of units subnetwork
Fig. 4. Optimal network structure at the level of units for Example 1
duties are shown (see Fig. 4A). Transient heat loads only arise at those exchangers involving cold and/or hot transient streams. Moreover, the pair of heat matches inside the pinch region are given separately to facilitate the determination of the exchanger heat duties at operating Points AD (see Fig. 4B). In fact, the heat duty of unit 2 reduces to zero and that of unit 4 increases up to 768 kW when the pinch is at the top of the pinch domain. On the contrary, the heat match H2/C3 accomplished inside the pinch domain is always carried out at the lower subnetwork in unit 3. Values of the heat duties for operating Points AD are readily obtained by: (a) moving the heat loads ?490
12390
from the upper to the lower unit or vice versa according to the pinch location; and (b) replacing transient heat sources and sinks by heating and cooling utilities, respectively, when they vanish (see Appendix). As mentioned before, the exchanger heat loads for operating Points A and D are of particular interest since they can play the role of design conditions for the upper and lower subnetworks (see Table 6). Alternatively, one can choose the extreme conditions B and C. Overlapping of the corresponding configurations leads to the optimal network structure (see Fig. 5). By narrowing the uncertainty range in the value of T; to 229239°C 201.7”
(55.00
9S0 llS"
Fig. 5. Optimal network configuration for Examples 1 and 2 (temperature values correspond to Example 1).
Synthesis of flexible heat exchanger networksl exchanger 1 can be saved. This is the fixed cost network flexibility. associated to the required Another way of saving the investment in unit 1 consists in no longer reaching the MER target when T; is greater than 239°C. In other words, no process heat above 239°C is recovered. In contrast, the uncertain parameter T; does not have any impact upon the network structure. E.xampL
209
PERMANENT HE AT CASCADE
2500
16,
=25.6
2
Two further uncertain parameters are introduced in Example 2 by also specifying ranges for the inlet temperatures of cold stream C3 and hot stream H4 (see Fig. 6). The heat cascade shown at Fig. 7 reveals that the process pinch can vary from 219/229 to 239/249”C tied to the uncertain temperatures T”, and Ts . In fact, the supply temperature ranges of streams H2 and H4 both overlap the pinch domain at level 3 (see Fig. 6). Although Example 2 can still be regarded as a convex problem because no pinch jump happens. The convex pinch zone ranging from 2291239 to 239/249X can be generated by either H2 or H4. This implies a shifting of the pinch tied to T; or T; depending upon the stream conditions. Despite the transient heat flow across the permanent pinch at 229/239”C is nonzero in Fig. 7, some transient stream crossings can be taking place there. Therefore, a slight modification of the heat cascade should be made as follows. When several transient heat sources are present at the same temperature level, they should be represented separately in the heat flow diagram. If the heat flow leaving the temperature interval is lower than the heat availability of a particular transient stream hj at that level, then the lower boundary temperature can cross 4. by
kW/K
TRANSIENT HEAT SINKS
Fig. 7. The heat cascade for Example 2.
the lack of the other hot streams. At level 3, the heat flow (54.7 kW) is lower than the heat load of any of the two transient energy sources, i.e. 70.3 kW (H2) and 84.4 kW (H4). Therefore, such hot transient streams can be alternatively crossed by the lower boundary pinch. Crossing of stream H2 is quite relevant because it adds a new unit to the upper subnetwork. The information provided by the heat cascade upon the stream crossings is illustrated through intersection points at Fig. 6. It is used to define the set of heat matches that can be carried out at both the upper and lower subnetworks for the formulation of the flexible HENS problem (P2). The optimal solution to (P2) found in a CPUtime of 13.29 s is depicted in Fig. 8. Since the upper boundary pinch does not introduce additional streams crossings, only the other two dominant pinch candidates at 219/229 and 229/239”C are really dominant. The related design conditions at which the dominant pinch points arise, given in Table 6, are quite similar to those selected for Example 1. As a result, the network topology is identical in both cases (see Fig. 5). However, the exchanger heat duties are slightly different.
kW/K
CONCLUSIONS
@
Fig. 6. Example
2.
A pinchbased synthesis procedure of convex flexible heat exchanger networks accounting for supply temperature variations has been developed. Minimum utility usage is guaranteed at any point of the disturbance region. Optimal flexible network structures are still found even if certain heat matches are forbidden. The proposed technique has very much in common with some synthesis methods currently used
J. RD.&
et al.
the NLP multiperiod formulation of Floudas and Grossmann (1987b), the optimal network configuration is developed. Several examples have been efficiently solved through using rather smallsize MILP formulations. In all cases, the network structure includes a minimum number of units. Further studies, however, are still required to develop a general algorithmic approach to deal with synthesis problems involving pinch domains of higher complexity. AcknowledgementsThis work was carried out under support provided by Consejo National de Investigaciones Cientificas y Tecnicas (CONICET) of Argentina and Universidad National de1 Litoral (Santa Fe, Argentina). The financial aid from Secretaria de Ciencia y Tecnica (SECyT) of Argentina is also gratefully acknowledged.
NOMENCLATURE
Fig. 8. Optimal network structure at the level of units for Example 2.
for fixed inlet temperatures (Cerda and Westerberg,
1983; Papoulias and Grossmann, 1983). They all are modelled as fixedcost transportation or transhipment problems and solved by running an MILP computational algorithm. In contrast to prior approaches, no feasibility test is required as long as it is imbedded in the problem formulation. Treatment of the disturbance heat loads requires the introduction of transient cold and hot substreams and their corresponding heat balances. Additionally, the formulation should this time account for multiple dominant pinch temperatures rather than a single one to always reach the least utility usage through the fewest number of units. Such pinch temperatures constrain the heat exchanges and bring the need for further units. They are all identified in a single step through a new version of the heat cascade implementing a properlydefined heat recovery policy to ensure the minimum utility usage at any operating point. At convex HENS problems, the dominant pinch points always occurs at extreme inlet temperatures. Moreover, the network structure comprises two subnetworks requiring a single unit for each heat match. Simple expressions have been derived to predict beforehand the number of units required in each subnetwork. Dominant pinch temperatures also rule the network topology. All of them should be taken into account during the synthesis of the HEN configuration. Then, there will be as many design points for each subnetwork as the number of problemdominant pinch points. At each design condition, a different pinch must occur. By combining manually the structures found at the design conditions or using
c, = Process cold stream J’ c,* =jthcold substream at temperature level k C = Set of process cold streams h, = Process hot stream i h,, = ithhot substream at temperature level I N = Set of process hoi streams Q;,= Ithlevel heat flow available at hot stream i Q,: 1 ~;h~e~~o~~; flow required by cold stream .i
S r: r: AT:
= = = =
AT,,, = Us = iYw = W =
Hot utility ithstream supply temperature ithstream target temperature Maximum variation of ithstream supply perature Minimum allowed temperature approach Pinch hot utility usage maximum value Pinch cold utility usage maximum value Cold utility
tem
A = Permanent stream or substream  = Transient stream or substream * = Optimal value Subscripts L M s U
= = = =
Lower subnetwork Middle subnetwork Subnetwork s Upper subnetwork
REFERENCES
Calandranis J. and G. Stephanopoulos,
Structural opera
bility analysisof heat exchangernetworks.Chem. Engng Res. Des. 64, 347 (1986).
Cerda J. and A. W. Westerberg, Synthesizing heat exchanger networks having restricted stream/stream matches using transportation problem formulations. Chem. Engng Sci. 38, 1723 (1983). Cerda J., A. W. Westerberg, D. Mason and B. Linnhoff, Minimum utility usage in heat exchanger network synthesis. Chem. Engng Sci. 38, 373 (1983). Floudas C. A. and I. E. Grossmann, Synthesis of flexible heat exchanger networks for multiperiod operation. Comput. &em. Engng 10, 153 (1986). Floudas C. A. and I. E. Grossmann, Synthesis of flexible heat exchanger networks with uncertain flowrates and temperatures. Comput. them. Engng 11, 319 (1987a).
Synthesis of flexible heat exchanger networksI Floudas C. A. and I. E. Grossmann, Automatic generation of multiperiod heat exchanger network configuration. Comput. them. Engng 11, 123 (1987b). Floudas C. A., A. R. Ciric and I. E. Grossmann, Automatic synthesis of optimum heat exchanger network configurations. AIChE JI 32, 276 (1986). Grossmann I. E. and C. A. Floudas, Active constraint strategy for flexibility analysis in chemical processes. Compur. them. Engng 11, 675 (1987). Kacprzyk J. and M. Krawczak, On an inexact transportation problem. Proc. ZFIP Con$ on Optimization Techniques, Warsaw, 4S September (1979). Linnhoff B. and E. Hindmarsh, The pinch design method for heat exchanger networks. Chem. Engng Sci. 38, 745
In contrast, the upper and lower level sets for a cold streamj E C depends upon the heat match being considered. For the match (h,, cl):
(1982).
Papoulias S. A. and I. E. Grossmann, Structural optimization approach in process synthesisII. Heat recovery networks. Comput. them. Engng 7, 707 (1983). Saboo A. K. and M. Morari, Design of resilient processing plantsIV. Some new results on heat exchanger network synthesis. Chem. Engng Sci. 39, 579 (1984).
and
Temperature
Sets for

LPI,={lI/eLP’
1I E LT’ LP;
LT;=
If
ieH1,
) I E LP’
(1 IIELT’ LP;=LP’
AD
Design Point A: 4*(/;,~,),+q*(~i,~,J)S
W) = 4*&
i,), + s’(ii,, W) + q’(2,
q.. W),
P+YS, C/) = q*(s, 2,) + qYS, Z,),
/
and
I < I,+ ) c LT’. and

1 >I,?}cLP’,
and
1 >I,C)cLT’. and
LT;s
LT’.
(A31
Design Point B: (A4)
PB(h,,C,)r=q*(~i;ir~,)s+4*(/ill~j),,
QB(h,, W) = q*(fi,. W) + q*(/;,. W)

+s*(&q)l’
(A5) ([email protected]
Design Point C: QC(h,. c,), = s*(&
E,), +q%,,
c,).,
QC(h,, I+? = q*(&, W).
(A7) (A8)
QC(S, c,) = q*(S, E,) + q*(S, eji) + r. [s*(t;, ,EH,
LTiu = LT’.
and
(Al) (A2)
where the asterisk denotes an optimal value for problem (P2) and iEH, jECand seSN.
QYS. c,) = q*(S, t,).
and
s LP’
VisH+.
Points
+ 2 [9*(&q) ,E0
In the same way, the lower level sets for h, E H+ containing the temperature intervals corresponding to the lower subnetwork are given by: LP; = (I
heat duties at operating
= HfuHT,
Let 1: be the temperature level just above the dominant pinch crossing the stream i E H +. Therefore, the upper permanent and transient level sets LP; and LTI, for stream iGHf are given by:
i E Ht.
k>II,f},
QYh,.
HL = H+uHl.
If
k>I:)
and
Values of the exchanger heat duties at operating Points AD can be determined from the optimal solution to (P2) by using the following expressions:
As already mentioned, the sets of hot streams involved in the upper and lower subnetworks, respectively, are given by:
= {I
and
Exchanger
Each
H = H+uHtuHl.
ViEH+.
If i E HI, the temperature level f;t should be replaced by max(l,+ ), i.e. the lowest dominant pinch level.
Subnetwork
LT;
KP,,‘=(k/ksKPJ
f4’(lj;.
Level
k
KTf={k/kEKTJ
Q”(h,~c,)~=
Let’s define the sets of hot process streams being crossed by a dominant pinch (H + ), running above of all dominant pinch temperatures (H’) or below them (Hi), respectively. Since they do not intersect each other. then:
H,
and
KT’;={k)kEKT’
kdi,+}
If i E H?, the temperature level I,? should be replaced by min(l,+ ), i.e. the highest dominant pinch level. Whenever KPJA = $, the heat match (h,, c,) is not performed at the upper subnetwork and the related binary variable yy can be deleted from the formulation of (P2). Moreover:
APPENDIX Stream
and
KP/:={klkcKPJ
(1983).
Marselle D. F., M. Morari and D. F. Rudd. Design of resilient processing plantsII. Design and control of energy management systems. Chem. Engng Sci. 37, 259
21 I

e,) + q*(h;. ?,,I.
(A9)
Design Point D: QD(b,.c,),=q’(A,,~,,),,
(AI(J)
Qo(h,, I.+‘) = q*(G,. @‘) + C s*(&, e,), ,ec,
(All) (A12)