Ind. Eng. Chem. Res. 2006, 45, 4629-4636
4629
Near Independent Subnetworks in Heat Exchanger Network Design Hiren K. Shethna† and Jacek M. Jez3 owski*,‡ Aspen Technology, Inc., Suite 900, 125 Ninth AVenue SE, Calgary, Alberta AB T2G OP6, Canada, and Rzeszo´ w UniVersity of Technology, 35-959 Rzeszow, ul. W. Pola 2, Poland
A minimum number of matches in a heat exchanger network (HEN) can be achieved by maximizing the number of independent subnetworks. The mixed-integer linear programming (MILP) transshipment model of Papoulias and Grossmann (Comput. Chem. Eng. 1983, 7, 707) allows for reaching the maximum number of perfectly balanced subnetworks, i.e., all streams are forced to reach given outlet temperatures. We have developed a method that maximizes the number of near-independent subnetworks, in which the streams achieve a certain specified outlet temperature within a tolerance. This conforms better to an industrial scenario and potentially generates a higher number of subnetworks. The approach is based on a MILP transshipment model formulation solved in a recursive procedure. The method ensures the minimal number of matches and minimal deviations from the nominal outlet temperatures. 1. Motivation and Significance In a heat exchanger network (HEN), several hot process streams and heating utilities exchange heat with several cold process streams and cooling utilities. The overall stream set, where the heat load on each utility is completely defined, is heat balanced. In some cases, the set can be decomposed into smaller heat-balanced subsets. Each subset forms an independent subnetwork. Hence, after such a division, a HEN consists of several independent subnetworks. Identifying a number of subnetworks (Nsub) is necessary to accurately determine the global minimum number of heat exchange units according to eq 1 developed by Linnhoff et al.3
Nglob ) Ns - Nsub
(1)
The formula does not account for the division at a pinch, which is applied in order to reach maximum heat recovery. Wood et al.4 developed a branching scheme at pinches that allows for reaching the global minimum number of units. There is also a possibility of applying cross-pinch matches, i.e., exchanger minimum approach temperature (EMAT) less than heat recovery approach temperature (HRAT), to achieve a global minimum number of units along with utility load minimization, e.g., ref 5. However, recently Furman and Sahinidis6 have shown by an illustrative example that there exist cases where Nglob from eq 1 cannot be achieved even if there is no pinch. Nevertheless, eq 1 gives a good lower bound on the minimum number of matches. Commonly, the number of subnetworks (Nsub) is assumed to be unity. In most cases, this is a good assumption, since exactly heat-balanced subsets are rare in the case of fixed values of stream outlet temperatures. Such fixed values are used in the majority of HEN design approaches. However, in an industrial scenario, some process streams are often allowed to vary outlet temperature within a certain range. A stream to a tank is a good example. Hence, the requirement of reaching fixed outlet temperatures by all streams is superfluous. Let us assume that a designer specifies a nominal outlet temperature of all streams and, also, provides ranges of the * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (48) 178651380. Fax: (48) 178543655. † Aspen Technology, Inc. ‡ Rzeszo´w University of Technology.
temperature variation for some streams (some of them can have fixed outlet temperatures). Such tolerances can be estimated on the basis of knowledge of downstream processes. Obviously, tolerances will give additional freedom to create more subnetworks, particularly in large-scale industrial problems. For fixed nominal outlet temperatures, such subsets/ subnetworks are not in heat balance, and therefore, we will refer to them as nearbalanced or near-independent. A possibility of designing HENs having several nearindependent subnetworks gives the following advantages: (1) It reduces the total number of heat exchangers in a HEN according to eq 1 and, in consequence, can decrease its total annual cost (TAC) (2) It makes operability and controllability problems easier to manage (3) It makes the detailed design stage easier, since subnetworks have a small number of streams The objective of this work is to develop a systematic approach for generating near-independent subnetworks. The approach yields a HEN featuring the largest number of subnetworks (the smallest number of matches) and, simultaneously, ensures the smallest increase of utility loads in comparison with the fixed outlet temperature case. A given set of outlet temperature tolerances (or tolerances in terms of heat loads) is also met. Heat loads of matches in all subnetworks are known from the solution, i.e., the method calculates heat load distribution (HLD) in all subnetworks. Additionally, the method can be used to generate several HENs that have a smaller number of subnetworks than the maximum but can feature a lower TAC or better conform to specific conditions of an industrial problem. 2. Literature Approaches to Date To calculate the maximum number of perfectly heat-balanced subnetworks (i.e., for fixed outlet temperatures), one can apply a transshipment model developed by Papoulias and Grossmann.1 The main objective of this mixed-integer linear programming (MILP) formulation is to find the optimal heat load distribution that minimizes the number of matches in a HEN for given minimal loads of utilities. However, the model does not account for near-independent subnetworks, since it requires fixed values of outlet temperatures of all streams, i.e., no outlet temperature tolerances are allowed.
10.1021/ie0504250 CCC: $33.50 © 2006 American Chemical Society Published on Web 05/23/2006
4630
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
Table 1. Data for the Illustrative Examplea stream
TS (K)
TT (K)
CP (kW/K)
∆H (kW)
H1 H2 H3 C1 C2 C3
395.0 405.0 520.0 288.6 353.0 278.0
343.0 288.0 519.0 493.0 383.0 288.0
4.0 6.0 528.0 5.0 6.9333 20.8
-208.0 -702.0 -528.0 1022.0 208.0 208.0
a
HRAT ) 10 K, hot utility target ) 0 kW, and cold utility target ) 0
kW.
Mocsny and Govind7 were the first who have shown the importance of generating near-independent subnetworks. To the best of our knowledge, there are no other attempts which follow this idea. However, the approach developed by Mocsny and Govind7 has some shortcomings. First, it requires an exhaustive search of all possible (2Nh 2Nc) permutations of generated subsets and does not ensure reaching a maximum number of subsets. Second, the method ensures only that the additional total heat loads on utilities are not greater than given values. There is no control of outlet temperature of individual streams or their heat loads (in relation to the nominal parameters). Therefore, a solution can feature unacceptable violations of outlet temperature of some streams. While generating perfectly balanced or near-balanced subnetworks, both the first and second rules of thermodynamics have to be accounted for. The approach by Mocsny and Govind7 did not account for the second rule. To show the consequences, we introduce here an illustrative examplessee data in Table 1. The streams are balanced, i.e., neither hot nor cold utilities are needed, though there is the pinch at 363/353 K for HRAT ) 10 K. It is easy to find by inspection two solutions, A and B, with two subsets that are balanced in regards to the first rule of thermodynamics:
A. {H1, C3} and the complement {H2, H3, C1, C2} B. {H1, C2} and the complement {H2, H3, C1, C3} However, in solution A the complement is pinched and requires 128 kW of hot and cold utilities. In the case of solution B, subset {H1, C2} features a pinch and the load of 80 kW of both types of utilities has to be added. Hence, the method of Mocsny and Govind7 requires additional screening of generated subsets. An important conclusion is that any reliable and efficient approach for creating near-balanced subnetworks has to account for all conditions of maximum energy recovery. In this work, we develop a systematic approach for generating near-independent subnetworks, which meets the following conditions: C1. Number of independent subnetworks making up the overall HEN is maximal (to minimize number of heat exchangers in overall HEN). C2. Given tolerances on outlet temperatures or tolerances on heat loads of individual streams are met. C3. Heat loads of utilities are minimized (i.e., the loads necessary for compensation of deviations from fixed nominal outlet temperature case). The method also provides information on heat loads of matches in subnetworks. Additionally, the method can be applied to produce all HENs consisting of various numbers of nearindependent subnetworks fulfilling conditions C2 and C3. Such networks differ in regards to total annual cost, operability, and flexibility. After some detailed calculations, the designer can choose one or a few designs that are the best for specific industrial requirements.
Figure 1. Illustration of recursive procedure.
The rest of the paper is organized as follows. In the next section, we present the general description of a method of calculating subsets that meet conditions C1, C2, and C3. It consists of recursive solution of the MILP optimization model for generation of a minimal subnetwork and its complement. Next, the MILP model of calculating both subnetworks is described. The examples of the method application follow, and a summary section ends the presentation. 3. General Description of the Approach Our approach consists of a recursive procedure where, at each step, two near-independent subsets are calculated from an initial set (S) of streams that are in heat balance for given HRAT. One subset (S1) is minimal, while the second (S2) is its complement. The minimal subset includes the minimal number of streams. Both subsets meet conditions C2 and C3 and also fulfill the following conditions:
S 1 ∪ S2 ) S
(2)
S 1 ∩ S2 ) L
(3)
In successive steps of the procedure, complement set S2 from the previous step is further divided into two subsets: the minimal subset and its complement. The procedure ends up at step k when no further division is possible, i.e., set S2 at step (k + 1) cannot be divided into the minimal subset and the complement subset (S1 at step k + 1 is an empty set). The tree shown in Figure 1 illustrates this recursive procedure. The set of minimal subsets from steps 1, ..., k and the complement from step k forms an overall HEN consisting of k + 1 near-independent subnetworks (see also Figure 1). Because this HEN consists of minimal subsets, it contains the maximum number of near-independent subnetworks, i.e., the HEN meets condition C1. Additionally, since all the generated subsets meet conditions C2 and C3, the overall HEN also fulfills them. The generation method does not account for total area of subnetworks and, in consequence, for their TAC. Hence, we propose a method that produces several HENs with various nearindependent subnetworks. The method described in Section 5 consists of recursive execution of the procedure described above. Additionally, conditions have to be added to prevent generation of identical solutions. 4. Calculation of the Minimal Subset and Its Complement 4.1. Overview of the Approach. For given initial set S, both subsets are calculated by solving the MILP optimization model formulated in the framework of the transshipment formulation. The formulation requires a division into temperatures intervals (TIs) similarly to other approaches for energy recovery maximization, for instance, optimization methods1,2,8,9 or pinch technology methods, e.g., refs 10 and 11. The objective of the approach is to create the minimal subset (S1) with the minimal number of streams and its complement (S2) from an initial set (S). The subsets have to meet conditions C1 and C2.
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4631
For each stream s in set S, a binary variable is defined as follows:
ys ) 1 if s is in S1; ys ) 0 if s is in S2
(4)
Hence, by minimizing ∑ys, we ensure the minimal number of streams in the minimal subset. To ensure feasible heat exchange in both subsets, two independent transshipment formulations have to be applied: one for the minimal subset and the second for its complement. To discriminate streams in the subsets, we multiply given enthalpy change of stream s by ys for each s ∈ S1 and by (1 - ys) for each s ∈ S2. To account for tolerances and to control and minimize deviations from the given outlet temperatures, two types of variables are used: (1) residual heat flow from the last TI of each hot process stream and (2) loads on slack heaters placed in the last TI of each cold process stream. The former variables control the temperature deviations for hot process streams, and the latter do so for cold process streams. Notice that, to allow for such deviations, we have to apply nonzero residual from the last interval in contrast to the transshipment models for fixed temperatures.1,2,8,9 Additionally, heat balances for the last intervals of hot and cold process streams have to be relaxed into inequalities. The minimal deviations are achieved by minimizing total heat loads of slack heaters and total residual flows from the last intervals of hot streams. For a given initial set of streams S, there are the following sets:
H ) {i | i is a hot stream}; C ) {j | j is a cold stream} After creating temperature intervals (TIs), the following sets can be defined:
Figure 2. Notations and heat exchange in the last interval of (a) the hot stream and (b) the cold stream.
• minimum and maximum values of outlet temperatures (i.e., tolerances in terms of temperatures) or tolerances (tol) in terms of heat loads of streams min max min [TTmax ] or [tol+/, tol+/]; i , TTi , TTj , TTj i j
i ∈ H, j ∈ C
• enthalpy changes for streams in intervals (except the last ones) for temperature range of TIs
∆QHm ) enthalpy change of i ∈ Hm; m ∈ M, m ∉ Mi ∆QCm ) enthalpy change of j ∈ Cm; m ∈ M, m ∉ Mj • minimal and maximal permissible enthalpy changes of streams in last intervals (here in terms of temperature tolerances)
(∆QHm)max ) CPi(-TTmin + Tlast i i )
(5)
+ Tlast (∆QHm)min ) CPi(- TTmax i i )
(6)
(∆QHm)min ) CPj(TTmin - Tlast j j )
(7)
Cm ) {j | j ∈ Cj and j is present in interval m}
- Tlast (∆QHm)max ) CPj(TTmax j j )
(8)
Let us notice that application of HRAT used to calculate heat loads of utilities is not required to create intervals. Instead, an EMAT value such that 0.0 < EMAT e HRAT should be used similar to the other transshipment models for heat recovery1 or HLD calculation.1,8,9 It is convenient to distinguish the last temperature interval for each stream by defining the following sets:
is the highest temperature of the last TI for i ∈ H where Tlast i is the lowest temperature of the last TI for j ∈ C. and Tlast j Figure 2 illustrates notations and heat exchange in the last interval of hot stream (Figure 2a) and cold stream (Figure 2b). The variables applied in the model are as follows: qijm ) heat loads of matches for i ∈ H, j ∈ C in intervals m ∈ M; Rim ) residual heat of i ∈ Hm that leaves interval m (and enters interval m + 1); qsjm ) heat loads of slack heaters, j ∈ Cm, m ∈ Mj; yi ) binary variables for i ∈ H, yi ) 1 if stream i is in subset S1 and 0 if not; yj ) binary variables for j ∈ C, yj ) 1 if stream j is in subset S1 and 0 if not. The linear optimization problem for determining near-independent subsystems is formulated in the next section. 4.2. MILP Optimization Model for Creating the NearIndependent Subsets.
M ) {m | m is a number of the TI} Hm ) {i | i ∈ H and i is present in interval m or can cascade heat to interval m}
Mi ) {mi, i ∈ H is present in mi but is not present in any n > misi.e., mi is the last interval for i} Mj ) {mj, j ∈ C is present in mj but is not present in any n < mjsi.e., mj is the last interval for j} The data for the MILP optimization model of creating the subsets are as follows:
min{(
yj + ∑yi) + W( ∑ qsjm + ∑ Rim)} ∑ j∈C i∈H j∈C ,m∈M i∈H ,m∈M m
j
s.t.
m
i
(P1)
4632
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
Rim - Rim-1 +
∑ qijm ) yi∆QHm;
i ∈ H, m ∈ M, m ∉ Mi
i ∈ H, m ∈ M, m ∉ Mi (P6)
pairs: (P2, P6), (P3, P7), (P4, P8), and (P5, P9)), but they are active only for streams in the complement subset S2. Constraints P2-P9 altogether form two separate, relaxed transshipment models for both subsets. Inequalities P10 and P11 have to be included to ensure that the minimization of the number of streams in subset S1 will not generate an empty set. In the formulation above, we put lower bound equals 2 on the number of streams in the minimal subset (i.e., at least one hot stream and one cold stream). It is possible to apply user-specified lower limits on the number of hot and cold streams in P10 and P11. 4.3. Remarks and Comments to the Model. (a) For clarity of presentation, we did not consider forbidden matches. However, they can be simply included similarly to the transshipment model for multiple utilities1,2,8,9 or HLD determination.1,2 Furthermore, there is a possibility of disallowing a choice of certain streams i, j in a minimal subset by imposing the following constraint:
j ∈ C, m ∈ M, m ∉ Mj (P7)
yi + yj )1
j∈Cm
(P2)
qijm ) yj∆QCm; ∑ i∈H
j ∈ C, m ∈ M, m ∉ Mj
yi(∆QHm)min e Rim - Rim-1 +
(P3)
qijm e yi(∆QHm)max; ∑ j∈C m
i ∈ Hm, m ∈ Mi (P4) yj(∆QCm)min e qsjm +
∑ qijm e yj(∆QCm)max;
j ∈ C m, m ∈ M j
i∈Hm
Rim - Rim-1 +
(P5)
qijm ) (1 - yi)∆QHm; ∑ j∈C m
qijm ) (1 - yj)∆QCm; ∑ i∈H m
(1 - yi)(∆QH)min e -Rim-1 +
∑ qijm + Rim e (1 - yi) j∈C m
(∆QHm)max; i ∈ Hm, m ∈ Mi (P8) (1 - yj)(∆QCm)min e qsjm +
qijm e (1 - yj)(∆QCm)max; ∑ i∈H m
j ∈ Cm, m ∈ Mj (P9) yi g 1 ∑ i∈H
(P10)
yj g 1 ∑ j∈C
(P11)
Rim g 0; i ∈ Hm, m ∈ M, m * 1
(P12)
qijm g 0; i ∈ Hm, j ∈ Cm
(P13)
qsjm g 0; j ∈ Cm
(P14)
yi ) {0, 1}; i ∈ H
(P15)
yj ) {0, 1}; j ∈ C
(P16)
The objective function P1 minimizes the number of matches in the minimal subset. Also, it minimizes loads on slack heaters, i.e., temperature relaxation of cold streams and residuals from last intervals of hot streams, i.e., temperature relaxation of hot streams. A weighting factor W is used to scale the heat load on the residual and slack heaters in the objective function. The factor should ensure that an optimization procedure minimizes the number of streams in the minimal subset. We have chosen the scale to be the inverse of the net heat load on all hot streams including the hot utilities. This ensures that the net value of the sum of all residual and slack heat loads is dimensionless and less than unity, thus guaranteeing the uniqueness of the objective function value. Equality constraints P2 and P3 are heat-balance equations for streams in subset S1 for temperature intervals except the last ones. To account for the tolerances on the outlet temperatures of streams in S1, heat balances for the last intervals of streams are modeled as inequalities P4 and P5. Constraints P6P9 are identical to those mentioned above (see the following
(b) The model can be directly extended in order to account for segmented streams, i.e., streams divided into segments of constant CP value in order to account for temperature-variable CP of streams. Such an extended model was applied to solve example 5 in this work. (c) We assumed in the optimization model that the higher temperature Tlast in eqs 5-8 for heat balances in the last intervals is lower than TTmax for hot streams and higher than TTmin for cold ones. It is usually the case, since tolerances are relatively small; nevertheless, some exceptions can occur. They can be taken into account by creating an additional TI based on a tolerance and by the use of inequality constraints for this interval and the last one as well. (d) In the case of a pinched initial set, a possibility of generating near-balanced subsets is limited to some extent, particularly in the case of small tolerances. The reason is that the near-independent subset is pinched and its complement is pinched, too. Hence, relatively large tolerances can be necessary to generate near-balanced subsystems. However, the application of the concept of EMAT e HRAT can increase chances of creating near-balanced subsets. Loads of utilities are, thus, determined for HRAT. A smaller EMAT allows cross-pinch heat exchange. In consequence, there is more freedom for creating near-independent subsets. The model does not require any changes to account for EMAT e HRAT, since it is sufficient to apply EMAT while dividing into temperature intervals. (e) Calculation of heat loads of matches in both subsets is straightforward. We can discriminate the following: (1) streams from subset S1, since for them yi )1 and yj )1 for i ∈ H, j ∈ C; and (2) streams from subset S2, since for them yi ) 0 and yj ) 0 for i ∈ H, j ∈ C. Hence, we can calculate heat loads of matches in the subsets by simply summing up values of qijm. However, we do not insist on applying these match loads to synthesize subnetworks, since they do not account for area and TAC. Usually, as we will show by examples, there are several small subsets of very few streams. Hence, a simple pinch technology method, such as that from refs 12 and 13, can be applied to design subnetworks featuring small TAC. 5. Method for Generating Alternative Solutions A collection of near-balanced subsets of stream set S that meet conditions C2 and C3 will be referred to as solution subsets. For a certain set S, there are more than single solution
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4633 Table 2. Data for Example 1
Table 3. Data for Example 2
stream
TS (C)
TT (C)
CP (kW/K)
stream
TS (°C)
TT (°C)
CP (kW/°C)
∆H (kW)
H1 H2 H3 C1 C2 C3 ST CU
300 200 190 160 180 190 350 30
200 190 170 180 190 230 349 50
10 100 50 50 100 25
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11
323 389 311 355 366 311 350 352 366 311 422 339 433 522 510 544 472 505 544 475 583 517 511
423 495 494 450 478 490 450 468 478 489 478 411 366 411 349 422 339 339 420 339 478 366 339
7.62 6.08 8.44 17.28 13.90 8.44 17.28 13.90 8.44 8.44 21.78 13.84 8.79 10.55 14.77 12.56 17.73 14.77 12.56 17.73 12.53 8.32 6.96
762.00 644.48 1544.5 1641.60 813.90 1510.8 1728.00 1612.40 945.28 489.00 1219.7 996.48 588.93 1171.10 2378.00 1532.3 2358.10 2451.80 1557.44 2411.30 1315.70 1256.30 1197.1
subsets. This means that for given S and specified tolerances, it is possible to determine a number of various HENs composed of near-independent subnetworks. It is preferable to generate them all to allow a choice of such one (or a few ones) that is optimal in terms of TAC. Notice that some solutions can feature the same number of subsets because of linearity of the optimization problem, which results in multiple global optima. There are also such solutions that feature a lesser number of subsets than the maximum. However, even those should not be eliminated beforehand, since the optimization problem formulation does not account for area of a HEN. Thus, solution subsets with a high number of subsets can require a larger heat transfer area and, in consequence, higher total cost. It is possible to estimate an area target for each subset by applying available targeting methods, e.g., refs 14-16. This will give a smaller collection of possible solution subsets that are cost attractive and should be subjected to final rigorous design procedure. To generate all possible solutions, one can apply the procedure outlined in Section 4. To prevent generation of identical solutions, integer cuts have to be added to optimization problem formulation in the second and all successive runs. 6. Examples We have used DASH Optimization’s XPRESS solver (version 12b) to solve the MILP model. The recursive solution for generating a HEN has been coded into the computer program. With the solver, we found a solution in the order of minutes for medium to large problems (with 10 or more process streams). Example 1. At first, we present a solution for a small, “academic” task from Gundersen and Grossmann,17 Gundersen et al.,18 and Briones and Kokossis19ssee data in Table 2. The aim is to show that the method yields perfectly balanced subsets if they exist. For HRAT ) 20 K, it is pinched problem with Qhu ) Qcu ) 1000 kW and a pinch at 200/180 °C. Notice that all streams including utilities have the same heat load of 1000 kW, giving a possibility of creating perfectly balanced subsets. For EMAT ) HRAT ) 20 K, the method yielded the following heat-balanced subsets: above pinch, {ST, C3} and {H1, C2}; and below pinch, {H2, C1}, {H3, CU}. With EMAT ) 10 K, we calculated additional solution subsets, including, among others, one that is area and total cost optimal according to ref 18, i.e., {H1, C1}, {H2, C2}, {H3, CU}, (C3, ST}. The same solutions can be reached with the standard model for HLDs generation.1,2 It is, however, worth noting that the former model requires Nh × Nc binary variables, while our model requires only (Nh + Nc) binaries. This is of importance, since Furman and Sahinidis6 showed that the MILP model from ref 1 could not be solved for some large-scale problems. Example 2. As example 2, we employed a problem from Mocsny and Govind.7 The problem involves 12 cold streams and 11 hot onesssee Table 3. For values of HRAT up to about 38 K, this is a threshold problem and requires 2553.67 kW of cold utility (Mocsny and Govind7 applied HRAT ) 10 K).
Therefore, we added cold utility CU of temperature range 273 up to 283 °C and of load 2553.67 kW to the set of process streams. With maximal tolerances of 1.0% in terms of temperature tolerances, Mocsny and Govind7 were able to generate the four following near-balanced subsets (see also Figure 1 in ref 7):
S1 ) {H5, H8, H11, C4, C7}, S2 ) {H4, H6, H10, C5, C6, C11, C12} S3 ) {H1, H2, H3, H9, C1, C3, C8, C10}, S4 ) {H7, C2, C9} Hence, the overall HEN will consist of 20 matches according to eq 1. To compare violations of utility heat loads for fixed nominal final temperatures in both methods, we calculated heat loads of utilities for the subsets that would be necessary to add in order to reach such temperatures. The results for subsets S1S4 are as follows:
S1: Qhu ) 0.0 kW, Qcu ) 43.22 kW; S2: Qhu ) 43.22 kW, Qcu ) 0.0 kW S3: Qhu ) 0.0 kW, Qcu ) 32.3 kW; S4: Qhu ) 32.3 kW, Qcu ) 0.0 kW Total additional loads are Qhu ) 75.5 kW and Qcu ) 75.5 kW. We have applied the same tolerances as Mocsny and Govind7 and also EMAT ) HRAT ) 10 K. Our approach was able to generate 7 subsets:
{H7, C5}, {H4, C3}, {H3, C8, C1}, {H2, H1, C7}, {H6, C12, C10}, {H11, H10, C6, C9}, and {H8, H5, H9, C11, C4, C2, CU} Thus, a total HEN will consist of 17 units in 7 separate subnetworks (vs 20 units in 4 subnetworks in the solution from ref 7) and requires additional loads of Qhu ) 61.7 kW (vs 75.5 kW in7) and Qcu ) 61.7 kW (vs 75.5 kW in ref 7) if nominal outlet temperatures are to be met. The relaxation of heat balance is less than that in the solution of Mocsny and Govind,7 and additionally, the method calculated
4634
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
Table 4. Data for Example 3 stream
TS (°C)
TT (°C)
CP (kW/°C)
∆H (kW)
H1 H2 H3 H4 H5 C1 C2 C3 C4 C5 C6 C7 C8 C9 HU CU
149.4 137.8 118.3 124.4 117.8 22.8 30.0 56.1 112.8 113.3 38.9 217.2 137.8 648.9 1500.0 10.0
148.9 30.0 117.8 35.0 40.0 40.0 112.8 112.8 113.3 648.9 75.0 217.8 138.3 760.0 1490.0 20.0
3840.0 1.447 1982.0 1.264 1.581 0.866 1.473 4.321 2268.0 5.530 8.033 3416.632 2022.0 6.679
1920.00 155.99 991.00 113.00 123.00 14.90 121.96 245.00 1134.00 2854.70 289.99 2050.00 1011.00 742.04
Table 5. Data for Example 4 stream
TS (°C)
TT (°C)
CP (Mcal/h/°C)
∆H (Mcal/h)
H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 C1 C2 C3 C4 C5 C6 C7 C8 HU CU
123 162 211 261 307 316 340 148 125 137.5 106 40 137.5 181 39 147 37 124 62 1800 5
98.5 49 147.5 49 182 64 91 28 29 90 39 142 185.5 354 97 148 67 125 107 308 30
510.204 36.216 150.322 43.596 13.315 6.0436 108.165 22.633 19.563 34.107 110.217 254.011 287.75 247.2486 50.745 4414.394 33.813 2754.545 33.434
12.500 4.092 4.092 9.242 1.664 1.523 26.933 2.716 1.878 1.620 7.385 25.909 13.812 42.774 2.943 4.414 1.014 2.755 1.505
three subsets more, thus reducing the unit number by an additional three matches in comparison with their HEN. Example 3. This problem (data in Table 4) is pinched and involves only 6 hot streams and 10 cold ones, including utilities. For HRAT ) 10 K, the minimum utility loads amount to the following: Qhu ) 5861.91 kW and Qcu ) 701.279 kW, and these values were used for utilities HU and CU in Table 4. The same tolerances as in example 2 were used. The problem is of moderate size, and one should not expect many subsets for such small tolerances. For HRAT ) EMAT ) 10 K, we were able to calculate only two solution subsets.
1st solution: {H4, H3, H2, C3, C6, CU} and {H5, HU, H1, C4, C8, C7, C2, C1, C5, C9} 2nd solution: {H1, HU, C4, C8, C7, C5, C9} and {H2, H3, H4, H5, C1, C2, C3, C6, CU} Example 3 has also been solved for EMAT ) 5 K. In this case, as many as 14 various solution subsets were generated, including those two calculated for EMAT ) 10 K. One solution has three subsets, {H5, C2}, {HU, C7, C3, C5, C9}, and {H1, H2, H3, H4, C1, C4, C6, C8, CU}, and all others have two subsets. Let us analyze thoroughly the solution with three subsets. For fixed outlet temperatures of all streams, the first subset {H5, C2} needs a hot utility load of 1.0374 kW, the second one needs
a load of 20.85 kW, and the third subset requires a cold utility of 28.718 kW in order to reach the temperatures. The designer can allocate temperature violations to some streams, taking into account control and flexibility issues. Simple validation shows that, in the case of the second and third subsets, there are practically no temperature violations, because of the large CP values of streams C7, H1, H3, C4, and C8. In the first subset, one can accept temperature change by
