A Recursive Synthesis Method for Heat Exchanger Networks. II. Case

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Ind. Eng. Chem. Res. 2001, 40, 1176-1185

A Recursive Synthesis Method for Heat Exchanger Networks. II. Case Studies Y. Ren, B. K. O’Neill,* and J. R. Roach Department of Chemical Engineering, University of Adelaide, Adelaide 5005, South Australia, Australia

In Part I of this series, a new synthesis procedure for heat exchanger networks was proposed. The method is recursive and based on a tree-decomposition strategy with a simple match-selection procedure. This paper presents several case studies to illustrate elements of the proposed method. Case studies are categorized into three groups: capital-dominant problems, energy-cost-dominant problems, and problems featuring constraints. The results from a series of case studies confirm that the new method is efficient and the effort required to achieve the final design is significantly reduced when compared with that required by alternative methods. 1. Introduction In the preceding paper, a recursive synthesis method for cost-optimal network design was presented. The principal features of the method are a binary tree decomposition strategy, a simple superstructure, and a match-selection model. Using the proposed decomposition strategy, the method is capable of handling problems featuring substantially different film heat-transfer coefficients as well as those with equal or nearly equal film-heat transfer coefficients using an identical set of design rules. This approach differs from the traditional methods in which different design rules are normally required.5,12 The match-selection model is a binary (0-1) assignment problem and is easily solved (even for extremely large problems) by the well-documented Hungarian algorithm.2 It is simple and efficient. Furthermore, a systematic design procedure for selection of the streams located away from the partition point (or pinch) is provided. The difficulties and inefficiencies (in evolution) inherent in the heuristic rules for the pinch design method are avoided by using the proposed matchselection model. To reduce the complexity and the number of variables, the partition temperatures of the streams in the simplified superstructure are initially fixed for hot and cold streams within each subsystem or node. This constraint is subsequently relaxed, and optimization of partition temperatures is then performed as part of network synthesis. Optimization of a fixed-topology network directly addresses the tradeoff between the capital investment and the energy cost. Five case studies will be presented to demonstrate all facets of the proposed design method. 2. Capital-Cost-Dominant Problems For capital-dominant problems, the energy cost contribution is less than half of the total annual cost. The solution tree for such problems consists of a single root node. 2.1. Case Study 1. This problem was proposed by Gundersen and Grossmann5 and later studied by Zhu et al.12 A key feature of this problem is the large * Author to whom correspondence should be addressed.

difference in film heat-transfer coefficients (an order of magnitude). Traditional methods experience difficulties in solving this problem. The original problem did not include utility cost data, plant life, or an interest rate. To undertake a costoptimal design, a set of typical utility cost data, plant life, and interest rate has been assumed. exchanger cost ($) cost of hot utility cost of cold utility

10,000 + 1000A0.8 $110/kW year $10/kW year

To undertake a pinch design, the value of the ∆Tmin (HRAT) is set equal to 20 K. Because no match is permitted to cross the pinch, the resulting design is shown in Figure 1. It requires a total transfer area of 674 m2 with an annualized capital investment of $256,751. Annual utility consumption is 1000 kW of hot utility and 1000 kW of cold utility. The optimal network devised by Gundersen et al.5 is presented in Figure 2. The total area for the network is reduced from 674 to 494 m2, and the total annualized capital investment falls from $256,750 to $211,120. Unfortunately, the design in Figure 1 cannot be evolved to that in Figure 2.5 Clearly, pinch decomposition is not able to produce the optimal design for this problem. Gundersen and Grossmann5 solved this problem by applying mathematical programming techniques. Unfortunately, the optimal design could not be directly discovered from their problem formulation. Zhu et al.12 proposed an alternative strategy. First, they employed the diverse pinch approach8 to generate a set of modified composite curves. Streams c1 and c3 were shifted across the pinch. Based on this decomposition, an initial design was discovered (Figure 3). The optimal design was deduced by evolving this design using a MINLP solver. By contrast, the design shown in Figure 2 can be easily deduced by application of the proposed recursive design method. Following the design logic of the recursive synthesis method, this problem is identified as a capital-dominant problem, and a single design for the root node is required. The match matrix for the root node produces an initial design involving matches h1-c1, h2-c2, and h3-c3. The match h3-c3 is interesting as the supply temperature of the cold stream c3 equals the supply temper-

10.1021/ie000366k CCC: $20.00 © 2001 American Chemical Society Published on Web 01/26/2001

Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001 1177

Figure 4. $106,640.

Figure 1. Design for case study 1 with HRAT ) 20 K.

Figure 2. Crisscross design for case study 1 with HRAT ) 20 K.

Figure 3. Initial design from Zhu et al. (1995).

ature of the hot stream h3. Clearly, energy cannot be transferred between the two streams, and as a consequence, h3 and c3 must reach their target temperatures by using a cold or hot utility (Figure 2). The remaining sections of the initial design include the hot stream h3 and the cold stream c3. Further matching is not possible. The design incurs a total cost of $339,210 with an energy cost of $120,000 (less than 40% of the total annual cost). Hence, further decomposition of the root node is unnecessary, and this initial design constitutes the final design for this problem. It is identical to the optimal design illustrated in Figure 2. It is interesting to observe that the optimal design is discovered in a single step by the new algorithm. 3. Energy-Dominant Problems In energy-cost-dominant cases, energy cost contributes more than half of the total annual cost. To improve

Design for the root node; total annual cost )

energy recovery, the root node is decomposed into two child nodes. These child nodes can be further decomposed. Three case studies including the well-known aromatics plant are presented to demonstrate the features and application of the recursive synthesis strategy. For all three case studies, the recursive synthesis method generates a result that is better than (or at least identical to) the best designs proposed by other researchers. Normally, the proposed synthesis method requires a significantly reduced effort when compared to current methods such as the pinch design method. 3.1. Case Study 2. This simple example has been studied by Linnhoff and Hindmarsh,6 Yee and Grossmann,11 Ciric and Floudas,4 and Zhu et al.12 The match matrix for the root node produces an initial design involving the matches h1-c2 and h2-c1 (Figure 4). Further savings cannot be achieved after considering the remaining sections of the problem. The design costs $106,640. The hot and cold utility consumptions are 500 kW and 900 kW, respectively. Utility cost ($58,000) contributes 54% (d > 0.5) of the total annual cost, and further decomposition is required. The system is decomposed into two child nodes (above and below the pinch) using a HRAT ) 5.6 K. This value was determined by applying the well-established Supertarget algorithm. The match matrix for node 1 (above the pinch) produces an initial design containing the matches h1c2 and h2-c1. The remaining portion of the design contains the single hot stream h1 and the single cold stream c1, thereby producing the single match h1-c1. Node 2 (below the pinch) is composed of a single cold stream and two hot streams. Hence, the cold stream is split to match with the two hot streams. The resulting design is combined with that for node 1. The total annual cost is $80,900. The partition temperatures and split ratio for c1 are next optimized, thereby reducing the cost to $80,130. A small heater is eliminated during this process (Figure 5). Yee and Grossmann11 discovered the solution shown in Figure 6 using a two-stage superstructure. Their MINLP formulation involved 62 constraints and 50 variables. Nine variables were binary. The solution obtained by Linnhoff and Hindmarsh6 is shown in Figure 7. Its annual cost is $89,830. Figure 8 illustrates the design proposed by Zhu et al.12 This design is obtained by optimizing the heat loads of their initial design. The design was further optimized to yield a final design identical to that proposed by Yee and Grossmann.11 The design resulting from this recursive algorithm achieves a slightly improved result with a significantly reduced effort. 3.2. Case Study 3: The Aromatics Plant. This problem is a widely studied industrial-scale problem.7,9,12 It is based on the simplified flowsheet of one

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Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001

Figure 5. Optimal design for the case study; annual cost ) $80,130.

Figure 9. Design for child node 1-1 (above the pinch). Figure 6. Optimal design of Yee and Grossmann (1990); total annual cost ) $80,270.

Figure 7. Design from Linnhoff and Hindmarsh (1983), total annual cost ) $89,830.

Figure 8. Optimal design using NLP3 model (Zhu et al., 1995), total annual cost ) $81,770.

of the largest aromatic complexes in Europe. The reaction section is not available for heat integration because of constraints imposed by startup and safety.1,7 For the root node design, a single hot utility is required to balance the match matrix. An initial design produced using the match-selection model involves matches h1-c1, h2-c5, h3-c2, and h4-c3. Cold stream c4 is heated by a hot utility. A match between hot stream h1 and cold stream c4 is chosen for the design

Figure 10. Level-two design for node 1-2.

of the remaining sections. No further match opportunities exist at this level. The network’s annual cost is $2,548,530, including a utility cost of $1,983,050. The utility cost contribution is roughly 78% of the total cost. Clearly, further decomposition is required. 3.2.1. Design of the Level-Two Nodes. The optimal HRAT equals 19 K (Supertarget), producing a hot pinch at 160 °C and a cold pinch at 141 °C. The root node is decomposed into two child (level-two) nodes, one above and one below the pinch.


Figure 11. Final design for the level-two decomposition; annual cost ) $2,333,360.

Figure 12. Final level-two design; total annual cost ) $2,315,720.

The match matrix for child node 1 (above the pinch) produces an initial design with matches h1-c1, h2-c5, and h3-c4. Cold stream c2 is heated to its target temperature by a hot utility. The design for the remaining part involves matches h1-c5 and h3-c2. No further matches can be found. The resulting design for node 1

is shown in Figure 9. The utility cost for the design is $1,468,410. It contributes more than 80% to the total annual cost of $1,770,520. The match matrix for child node 2 produces an initial design involving matches h1-c2, h3-c4, and h4-c3. Cold streams c1 and c5 are heated to their target

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Figure 13. Final design; total annual cost ) $2,286,820.

Figure 14. Design of Ahmad and Linnhoff (1990); total annual cost ) $2,363,770.

temperatures by a hot utility. The design for the remaining part is a sequential match problem. To increase flexibility, the match between hot stream h4 and cold stream c3 is broken. Cold streams c1, c3, and c5 then sequentially match with hot stream h4 in the order of the enthalpy changes of the streams. The resulting design for node 2 is shown in Figure 10. The energy cost is $208,800. This contribution is roughly 34% of the total annual cost $621,870.

If the designs were terminated here, then the combination of the two designs would cost $2,392,390. After merging the coolers in hot stream h3, matching hot stream h3 and cold stream c4, this cost is reduced to $2,375,200. The total annual cost is further reduced to $2,333,360 by the optimization of partition temperatures (Figure 11). 3.2.2. Design of the Level-Three Nodes. Given that the first (above pinch) design for the level-two nodes (Figure


Figure 15. Final design. Table 1. Summary of Different Designs design method

units

hot utility (MW)

new method (two branches) new method (three branches) new method (four branches) Ahmad and Linnhoff (1990) Suaysompol and Wood (1991)

13 15 17 17 13

20.1 19.1 19.0 21.2 21.2

total annual cost ($) 2,333,360 2,315,720 2,286,820 2,363,770 2,462,580

9) is dominated by the energy cost, further decomposition is required. To minimize the number of the streams on both sides of node 1-1 and node 1-2, the partition

temperatures for child node 1-1 and child node 1-2 are set at 220 and 201 °C for the hot and cold streams, respectively. Given the new partition temperatures, the match matrix for node 1-1 now produces a design with the match h1-c5, and the match matrix for node 1-2 produces an initial design with matches h1-c1, h2-c5, and h3-c4. The remaining section of node 1-2 constitutes a sequential matching problem. The match between hot stream h3 and cold stream c4 is relaxed. Cold streams c2, c4, and part of c5 then sequentially

Table 2. Hot Stream Data stream

h1

h2

h3

h4

h5

h6

h7

h8

mp

hp

Tin (°C) Tout (°C) CP (kW/K) h (W/m2K)

182 93 47.19 1020

70 49 1438.1 850

147 146 28700 1250

146 93 73.33 1420

174 38 27.00 1480

54 48 1683.3 1080

48 38 65.00 910

112 38 57.03 1140

126 126

198 198

3750

3690

Table 3. Cold Stream Dataa,b stream

c1

c2

c3

c4

c5

c6

c7

Tin (°C) Tout (°C) CP (kW/K) h (W/m2K)

50 160 47.27 1480

50 93 107.67 1020

101 102 24 000 1590

49 82 100 1360

101 118 94.12 1020

174 175 22 100 1820

112 113 12 800 1360

water 25 40 1250

Data: exchanger cost ($) ) 218.2A, plant lifetime ) 5 years, interest ) 10%, HP steam cost ) $123.84/(kW year), MP steam cost ) $92.16/(kW year), water cost ) $0.264/(kW year). b Note: Exchanger cost and cooling water costs in this example appear low relative to steam costs, but these values were not corrected to retain the possibility for comparison with the initial solution. a

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Figure 16. Design provided by Aspen.

match with h3 in the order of the target temperatures of the cold streams. The heater H3 in the node 1-2 can be merged with the heater H2 in node 1-1. Two partition temperatures in the pre-optimal design of node 1-1 must now be optimized. The optimal design for node 1-1 costs $1,735,010. The design for node 1-2 (shown in Figure 10) is a capital-cost-dominant problem. Decomposition is not required. The designs for the two nodes can then be merged into the design for the problem, and the resulting optimal design costs $2,315,720 (Figure 12). Although the decomposition for node 1-2 is not required according to the algorithm, it will be considered to investigate the impact of this further decomposition on the final design. Node 1-2 was divided at 119 and 100 °C for the hot and cold streams, respectively. The match matrix for node 2-1 produces an initial design with matches h1c1, h3-c4, and h4-c3. The remaining parts of node 2-1 constitute a sequential problem. The match matrix for node 2-2 produces an initial design that involves matches h1-c2, h3-c4, and h4-c3. No further feasible matches can be found. This new design for node 2 merges with design for node 1. The final optimal design is shown in Figure 13. The design provided by Ahmad and Linnhoff1 is illustrated in Figure 14. Compared to this designs, all designs generated by the new method (Figures 11-13)

achieve a lower total annual cost. A summary of designs is presented in Table 1. The designs for this example demonstrate that increasing the node level for decomposition can reduce the total annual cost and utility consumption for preoptimal design, until all branches require only a hot or cold utility. In this example, three levels of decomposition with three branches achieve a design with 15 units and a total annual cost of $2,315,720. Three levels of decomposition with four branches provides a design with 17 units and a total annual cost of $2,288,430. Although the four-branch designs cost is 1.2% less than the threebranch design, the increase in the number of units makes the network more complicated. 3.3. Case Study 4. This case study is taken from the practical example provided on the Aspen Plus Web home page. The relevant stream and cost data are presented in Tables 2 and 3. The supply temperatures for hot streams h6 and h7 are below 54 °C. None of the available cold streams can exchange energy with these streams. These streams must be cooled with a cold utility and will not be considered in the design. The match matrix for the root node produces an initial design involving matches h1-c7, h2-c4, h3-c3, h4c1, h5-c5, and h8-c2. The design for the remaining parts involves matches h1-c2, h3-c7, and h5-c4. Further matches cannot be found. The design for the


Figure 17. Design with imposed MP steam generation constraint.

root node costs $4,106,840. The utility cost is $3,618,690, roughly 88% of the total annual cost. Hence, decomposition of the root node is required. The system is partitioned at 70 °C for the hot streams and at 50 °C for the cold streams. The design for subsystem 1 (above the partition temperature) is utilitycost-dominated with an energy-cost contribution of roughly 90%. Subsystem 1 is further decomposed at 121 and 101 °C. Stream c6 is heated by HP steam, and it does not participate in any process-process matches. The initial design for subsystem 1-1 involves matches h1-c1, h3c3, h4-c7, and h5-c5. The initial design for its remaining portions matches h1-c5 and h3-c7. All hot streams are “ticked off”, and further matching is not possible. The initial design for subsystem 1-2 involves matches h4-c2, h5-c4, and h6-c1. Design of its remaining sections generates the single match h1-c4. All hot streams are ticked off. Optimization of the partition temperatures for subsystem 1 produces a design with a total cost of $3,786,020. The design for subsystem 2 involves a single match (h5-c4). Optimization of the partition temperatures produces a final design costing $4,064,530. This cost excludes costs for streams h6, h7, and c6, which are heated or cooled solely by utilities (Figure 15). The design proposed by Aspen’s designers is shown in Figure 16. It costs $6,759,420, but hot steam h3 is used to generate a hot utility (MP steam) of 28.7 MW,

thereby generating an energy credit of $2,645,000. Hence, the design costs $4,190,680. To compare solutions on an equal basis, the stream h3 is constrained to provide this MP generation duty, and a new design is evolved by the recursive algorithm with a total cost of $6,721,400 (Figure 17), ignoring the energy credit of the MP steam. Again, this design compares favorably with that proposed by the Aspen designers. 4. Problems Involving Constraints Considerations such as safety or layout can impose constraints on certain matches. For example, a match between a specific hot stream j and a specific cold stream i might be forbidden or imposed. Such constraints are readily handled using the recursive design algorithm. Forbidden matches are assigned to a large arbitrary cost that is sufficient to ensure that the match will not be selected. By contrast, a match is assured of selection if its cost index in the match matrix is assigned to zero. 4.1. Case Study 5. This problem was studied by Cerda and Westerberg,3 Trivedi et al.,10 and Suaysompol and Wood.9 In the original problem, streams had different capacity-flowrate (CP) values for different temperature ranges. To simplify the data, the value of CP for each stream is set to the average CP. Hot stream h2 cannot match above the bubble point of cold stream c1. The utility cost for the root node design (Figure 18) is $7,270, a contribution of about 49% of the total annual

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Figure 18. Root-node design ignoring constraints.

Figure 21. Design for node 1 with constraints.

Figure 19. Design for the root node including constraints. Figure 22. Final design neglecting constraints; total cost ) $12,530.

Figure 20. Design for node 1 without constraints.

cost of $14,530. The d value is close to the critical value, and decomposition of the root node is warranted given that a level of fuzziness is inherent in the critical value. Hot stream h2 cannot match stream c1 above its bubble point. Imposing this constraint results in the design for the root node shown in Figure 19. The utility cost for this design is $8,985, 57% of the total annual cost of $14,530. Hence, further decomposition of the root node is required. 4.1.1. Design for the Level-Two Nodes. The root node is decomposed into nodes 1-1 and 1-2. The partition temperatures are 160 and 140 °C for the hot and cold streams, respectively. The match matrix for node 1-1 without constraint produces an initial design with matches h1-c1 and h2-c2. The remaining parts of the network contain the single hot stream h2 and the single cold stream c1. A match between h2 and c1 is possible (Figure 20). The utility cost for the design is $2,930, 31% of the total annual cost of $9,520, and further decomposition is not required. If hot stream h2 is forbidden from providing a match above the bubble point of cold stream c1 (temperature range of 180-250 °C), then the cost index for a potential match h2-c1 is arbitrarily set to 106 (a sufficiently large cost penalty to ensure no matching). The match matrix for node 1 with constraints produces an initial design with matches h1-c1 and h2-c2 (Figure 21). Because this match is forbidden, no matches are available in the remaining parts of the problem. The utility cost for the design is $4,240 (∼39% of the total annual cost of $10,690). Clearly, further decomposition for the root node is not required. Node 2 contains an unconstrained problem. Its design includes the match h2-c1. The combination of the designs for node 1 and node 2 without constraints achieves a total annual cost of $13,390. Three partition

Figure 23. Final design including constraints.

temperatures must be optimized, and the final design shown in Figure 22 results. The total annual cost is reduced to $12,530. The combination of the designs for node 1 and node 2 with constraints costs $14,555. Three partition temperatures are next optimized, and the final design shown in Figure 23 is produced. The total annual cost is reduced to $13,020. 5. Conclusions A new method, the recursive synthesis method for cost-optimal heat exchanger networks, has been proposed in a companion paper, Part I. Its applicability to the network synthesis task has been demonstrated by application to a number of case studies. The method is simple and straightforward. It significantly reduces the difficulties and uncertainties associated with evolutionary methods such as the pinch design algorithm. In addition, complex mathematical programming is not required for match selection. The new method readily handles problems with different cost equations, a range of film transfer coefficients, as well as forbidden and imposed matches. The design procedure is straightforward and easy to implement. Literature Cited (1) Ahmad S.; Linnhoff, B.; Smith, R. Cost optimum heat exchanger networks. 2. Target and design for detailed capital-cost models. Comput. Chem. Eng. 1990, 14, 751-767.

Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001 1185 (2) Andersen D. R.; Sweeney, D. J.; Williams, T. A. An Introduction to Management Science: Quantative Approaches to Decision Making, 6th ed.; West Publishing Co.: St. Paul, MN, 1991. (3) Cerda, J.; Westerberg, A. W. Synthesizing heat exchanger networks having restricted stream/stream matches using transportation problem formulations. Chem. Eng. Sci. 1983, 38, 17231740. (4) Ciric, A. R.; Floudas, C. A. Application of the simultaneous match-network optimisation approach to the pseudo-pinch problem. Comput. Chem. Eng. 1990, 14, 241-250. (5) Gundersen T.; Grossmann, I. E. Improved optimisation strategies for automated heat exchanger network synthesis through physical insights. Comput. Chem. Eng. 1990, 14, 925-944. (6) Linnhoff, B.; Hindmarsh, E. The pinch design method for heat exchanger networks. Chem. Eng. Sci. 1983, 38, 745-763. (7) Linnhoff, B.; Ahmad, S. Cost optimum heat exchanger networks. 1. Minimum energy and capital using simple capitalcost models. Comput. Chem. Eng. 1990, 14, 729-750. (8) Rev, E.; Fonyo, Z. Diverse pinch concepts for heat exchange network synthesis: The case of different heat transfer conditions. Chem. Eng. Sci. 1991, 46, 1623-1634.

(9) Suaysompol, K.; Wood, R. M. The flexible pinch design method for heat exchanger networks. I. Heuristic guidlines for free hand designs. Trans. Inst. Chem. Eng. Chem. Eng. Res. Des. 1991, 69, 458-464. (10) Trevidi, K. K.; O’Neill, B. K.; Roach, J. R.; Wood, R. M. A new dual-temperature design method for the synthesis of heat exchanger networks. Comput. Chem. Eng. 1989, 14, 601-611. (11) Yee, T. F.; Grossmann, I. E. Simultaneous optimisation models for heat integration. II. Heat exchanger network synthesis. Comput. Chem. Eng. 1990, 14, 1165-1184. (12) Zhu, X. X.; O’Neill, B. K.; Roach, J. R.; Wood, R. M. A new method for heat exchanger network synthesis using area targeting procedures. Comput. Chem. Eng. 1995, 19, 197-222.

Received for review March 31, 2000 Accepted November 17, 2000

IE000366K

A Recursive Synthesis Method for Heat Exchanger Networks. II. Case

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