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A hybrid methodology for detailed heat exchanger design in the optimal synthesis of heat exchanger networks J.M. Garcia, J.M. Ponce and M. Serna Facultad de Ingenieria Quimica - Universidad Michoacana de San Nicolds de Hidalgo. Morelia, Michoacdn, Mdxico

Abstract This paper presents a hybrid method for the synthesis and optimization of heat exchanger networks, which includes detailed design of heat exchangers. This task is achieved by combining the pinch design method with mathematical programming techniques, together with an optimal design algorithm of shell and tube heat exchangers based on the rigorous Bell-Delaware method. As result, the stream pressure drops are treated as optimization variables. Thus, the capital cost of the pumping devices and the electricity cost to run these equipments are considered in this problem together with the costs for heat exchanger area and utility consumption. The problem is decomposed as a binary tree, where each node is categorized as either capital-dominant or energydominant problem. Subsequent decomposition of each node is determined by this dominance. The final design is obtained recursively applying a design algorithm from child nodes to their parent node. The match-selection procedure is a hybrid method that exhibits some of the features of both evolutionary and mathematical programming methods. The method starts allocating matches using an IP assignment model. This step is then followed by an evolutionary procedure in which the remaining selections of the design are treated as new problems. The process is repeated until no savings can be discovered. The method avoids the solution of complex MINLP models, and consequently it is possible to solve large problems. Furthermore, it readily copes with typical constraints, such as forbidden matches and imposed matches. Therefore, safety and layout considerations are easily incorporated into the design.

Keywords: Heat exchanger network synthesis, shell and tube heat exchangers, BellDelaware method, detailed design. 1. Introduction The synthesis of heat exchanger networks (HENs) has been the subject of significant amount of research over the last three decades, promoted heavily during the 1970s because of the rising costs of energy observed. The two most common methods for grassroots HENs designs are broadly classified in two categories, namely, the pinch design method and mathematical programming methods (Furman and Sahinidis, 2002). In the pinch design method the system is normally partitioned using a pinch decomposition strategy, and, matches are selected by application of heuristic rules (Linnhoff and Hindmarsh, 1983). This approach is time consuming and the quality of the results is determined by the designer's experience. In the second kind of methods, a mixed-integer nonlinear programming (MINLP) model is applied for the development of a HEN based on a network superstructure solving for the minimum total annual cost.

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Thus, mathematical programming methods have a more rigorous approach to find a solution, but lack of user interaction. Until recently, however, with exception of an attempt to synthesize heat and power integrated HENs using detailed MINLP models that was done by Sorsak and Kravanja (1999), both methods have neglected the effects of detailed heat exchanger design on HENs synthesis; for example, film heat transfer coefficients for the streams are assumed constant. To outline appropriately interactions between the consumption of power, the capital cost and the energy recovery level of the network, recently Mizutani et al. (2003) proposed a MINLP model for the HENs synthesis including detailed design considering shell and tubes units. This formulation represents great advance in the appropriate treatment of the problem. However, since the heat exchangers are represented rigorously using the Bell-Delaware method (Taborek, 1983), it is expected that the function of the total annual cost to minimize is large and irregular. In addition, the complexity of the resultant MINLP model is of such magnitude that Mizutani et al. (2003) had to impose certain geometric limitations to the units in order to facilitate the obtaining of feasible solutions. Consequently, these two aspects of the approach of these authors reduce their possibilities to find a good solution. This paper presents a hybrid method for HENs synthesis and optimization including detailed design of heat exchangers. This task is achieved by combining a tree decomposition method with a match-selection model, together with an optimal design algorithm of shell and tube heat exchangers based on the rigorous Bell-Delaware method. Thus, the capital cost of the pumping devices and the electricity cost to run these equipments are considered in this problem together with the costs for heat exchanger area and utility consumption. This approach simplifies notably the problem of detailed HENs design, reduces the computation effort significantly and, therefore, it allows solving large-scale problems.

2. Recursive method for the synthesis of heat exchanger networks The network structure is generated using a recursive method based on a tree decomposition strategy and a match-selection model proposed by Ren et al. (2001). This approach decomposes the synthesis problem as a binary tree and calculates the cost of each node, which is then categorized as either a capital-dominant or an energydominant problem. Subsequent decomposition of the system depends on this dominance. For problems dominated by capital cost, any reduction in capital investment will decrease the overall cost. To reduce the investment capital, it is possible to use fewer units and/or improve heat exchanger driving forces. When reducing capital costs, energy consumption usually is increased and, therefore, energy costs; however, the total annual cost will fall because the capital cost is the dominant factor. Hence, such systems are best treated as a single entity, thereby avoiding further decomposition. An inverse case is presented in problems where the main component of the total annual cost is the utility cost; here energy saving becomes important, and decomposition might reduce the utility consumption. The decomposition is carried out in the pinch point; then the pinch point design rules are used to generate maximum energy recovery networks. The fractional contribution of utility cost to the total annual cost is defined as: d=

utilitycost total annual cost

(1)

The value of d for a network can be used to determine whether energy or capital dominates the total annual cost. According to the experience, Ren et al. (2001) suggest

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that a fractional value of 0.5 represents the transition point. Therefore, if d > 0.5, the utility cost is the main contribution for the total cost; otherwise the main contribution for the total annual cost is the capital cost. Usually, the dominant-cost component for the problem is unknown prior to design; consequently, the root node (the initial design) is considered as capital-dominant problem. For such problems, decomposition to secondary nodes is unnecessary. A single root node exists in the binary tree; however, it is possible that the design of the root node is dominated by its utility cost, then decomposition is required, which allows an expansion of the binary tree in two branches, or secondary nodes, and the process is repeated.

2.1. Match selection model A simplified superstructure was proposed to determine a basic topology for the initial design in each node assuming that the matches between a hot stream i and a cold stream j is independent of any other one and, consequently, that follows the pattem shown in Fig. 1. This simplifying assumption identifies all possible matches. It should be noticed that in this approach the interactions among the diverse matches are rejected. Thus, it is easy to obtain the optimum detailed design of each match independently using the algorithm of Serna and Jim6nez (2005) and, as a result, the total annual costs associated to all possible matches. Therefore, the match selection model developed by Ren et al. (2001) is enhanced through the incorporation of the optimum detailed design of individual heat exchangers. Aa

HOTSTREAM

A~ (~

APi

APe [email protected]

COLD UTILITY

A~

COLDSTREAM )

s~

AP,'/ COOLER

Ahj " HOTUTILITY

~ APh

APnj HEATEXCHANGER

+ HEATER

Fig. 1. Matching model for single hot and single cold process streams For each match, the optimal cost (Costij) can be defined by:

Costgj = CostHX o + CoStHhu + CostCcu

(2)

where the term CostHXij is the optimal cost for the detailed heat exchanger, which include the capital cost for the exchanger and the capital and operational costs for the two pumps. The others two terms, CostHhu and CostCcu, are the heater and cooler optimal costs, respectively, which are defined in similar form as the heat exchanger, only it is necessary to add the utility costs. It is important to highlight that the pumping costs for the hot utility service is considered only for the case that this is in liquid phase. A fundamental part of the recursive method is the inclusion of a match matrix, where the optimal total annual costs associated to all possible matches are stored. The optimal cost of the match between a hot stream i and a cold stream j is stored as the element a/j

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of the matrix. In the simplified superstructure, the match matrix is always square; when the number of hot or cold streams is different, utility streams are included to balance the matrix. Therefore, the model for the optimal matches selection in the initial design settles down as an integer programming problem (IP) as: l

1

min Z Z auxu i=1 j = l

'

(3)

s.t. ~--~x/j=1 Vj= I,I i=1 1

~--~xu = 1 Vi=l,l j=l

1 = max(m, n,) The solution of this problem can be easily obtained even for large size problems. Using this methodology, the difficulties associated to the solution of complex NLP, MINLP or MILP problems are avoided.

2.2. Remainder section analysis In the initial design each stream is restricted to participate in a single match; however, following the evaluation of the remainder sections several cold streams (or hot) can be coupled with more than one hot stream (or cold). When this situation is presented, it is possible to analyze the convenience of stream splitting to modify the driving forces and, consequently, to reduce the total annual cost. To simplify the match selection mathematical model, the partition temperatures in the simplified superstructure are assumed constant. Hence, an optimization of partition temperatures is performed following the combination of secondary nodes. This optimization repeats every time that a node breaks down.

3. Shell and tubes heat exchanger rigorous design In order to determine the elements of the match matrix, it is necessary to carry out the optimal design of the heat exchangers which result of proposing the transfer of heat between each hot with each cold stream according to the model shown in Fig. 1. To achieve this task, this work proposes networks constituted only by shell and tubes heat exchangers. The algorithm proposed by Serna and Jimdnez (2005) provides the detailed optimal design of these units. The development of this algorithm is based on two compact formulations combined with the basic design equation to represent the thermohydraulic behavior of the fluids. The function to minimize consists of the capital cost of the exchanger and the two pumps, plus the power cost necessary to operate these pumps, and can be defined as: (4)

+~

4

44

~~

84J t, rl#~r~ )t.4~

84J

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Here the values for the parameters KT and Ks depend on the exchanger geometry, the physical properties and the volumetric flow for the tube and shell fluids, respectively (Serna and Jim6nez, 2005).

4. Results The example used for the illustration of the hybrid method is a problem reported by Mizutani et al. (2003). The stream data for this problem is presented in Table 1. A minimum temperature difference of 10°K is also specified. Mizutani et al. (2003) give the cost data and the physical properties for the streams. The hybrid method gives the final network shown in Fig. 2. The performance of the network is given in Table 2 together to that of the network obtained by Mizutani et al. (2003) for the purpose of comparison. From Table 2, we can see clearly that the proposed methodology reduces significantly (72%) the total annual cost of the final design. It is appreciated, therefore, that the MINLP model of Mizutani et al. (2003) fails in the search of the best solution, due to the enormous complexity that the detailed design of the shell and tubes heat exchangers incorporate to the synthesis of heat exchanger networks problem. Thus, the result from the example demonstrates that the hybrid methodology is suitable for optimal synthesis of HENs that include the detailed design of heat exchanger. Table 1. Streams data for Example

H1

Stream

m (kg/s)

Tin (°K)

Tout (°K)

H1 H2 H3 C1 C2 C3 CW S

16.3 65.2 32.6 20.4 24.4 65.2

426 363 454 293 293 283 300 700

333 333 433 398 373 288 320 700

E4 ~x/)

426 °K

333 °K

"-r"

E2 H2

363°K

_

@ ! [

E1 @

1-13 454°K

358°K

o E6 ,¢ [email protected]

I 385"083 °K_

646.629 kW 373°K.

i !344"943°K . .

288 °K -,

.

@

.

.

1680.008 kW

j

E3 @

i 338"561 °Ki~ '

~

E5 @

i

433 °K

@ 3720.019 kW @. 3110.2 kW

333°K .

310"775°K @

293°K C1

889.82 kW 293°K C2 283 °K C3

800.0004 kW

Figure. 2. Final network for Example

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Table 2. Results comparison Mizutani et al.3

Thiswork

Exchangers investment cost

$12,388/yr

$10,920.88/yr

Utility cost

$173,456/yr

$38,797.74/yr

Pumping costs

$17,076/yr

$6,236.96/yr

Total annual cost

$202,920/yr

$55,955.58/yr

5. Conclusions A combined heuristic and mathematical programming approach is presented for the detailed design of shell and tube heat exchangers in the optimal synthesis of HENs. Thus, the proposed hybrid methodology considers the pressure drop together with area and utility costs in the network design. The synthesis problem is recursively solved using a tree decomposition strategy and a match-selection model (Ren et al., 2001). Using the assumption of independent matches, the optimum detailed design of individual heat exchangers was obtained by the algorithm proposed by Sema and Jim6nez (2005). The heat exchanger design model is based on the Bell-Delaware method for the shell-side fluid flow (Taborek, 1983). A very important feature of the hybrid method is that it reduces significantly the computation effort required to solve the problem of detailed design of heat exchanger networks. This is due to the good selection of matches among streams in the diverse stages of the recursive method by using an IP model instead of a MINLP model, which is more complex and more difficult to solve. For the example solved in this work, the hybrid method improves significantly the results obtained previously by Mizutani et al. (2003).

Acknowledgements The authors would like to acknowledge the support received from CIC-UMSNH, M6xico (grant 20.1) for the development of this work.

References Furman, K.C., Sahinidis, N.V., 2002. A critical review and annotated bibliography for heat exchanger network synthesis in the 20th Century, Industrial & Engineering Chemistry Research 41, 2335-2370. Linnhoff, B., Hindmarsh, E.C., 1983. The pinch design method for heat exchanger networks. Chemical Engineering Science 38, 745-763. Mizutani, F.T., Pessoa, F.L.P., Queiroz, E.M., Hauan, S., Grossmann, I.E., 2003. Mathematical programming model for heat-exchanger network synthesis including detailed heat exchanger designs, 2. Network synthesis. Industrial & Engineering Chemistry Research 42, 4019-4027. Ren, Y., O'Neill, B. K., Roach, J.R., 2001. A recursive synthesis method for heat exchanger networks, I. Algorithm. Industrial & Engineering Chemistry Research 40, 1168-1175. Serna, M., Jim6nez, A., 2005. A compact formulation of the Bell-Delaware method for heat exchanger design and optimization. Chemical Engineering Research & Design 83(5), 539-550. Sorsak, A., Kravanja, Z., 1999. Simultaneous MINLP synthesis of heat and power integrated heat exchanger networks. Computers & Chemical Engineers 23(Supplement), S 143-S 147. Taborek, J., 1983. Shell-and-tube exchangers: Single-phase flow. Heat exchangers design handbook, (E. U. Schlunder, ed.), Vol. 3, Section 3.3. Hemisphere Publishing Corp., Washington, DC.

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