On Optimaltime-Sharing Schemes Formulti-Period HEN Designs

On OptimalTime-Sharing Schemes forMulti-Period HEN Designs

Da Jiang

Chuei-Tin Chang*

Department of Chemical Engineering

National Cheng Kung University

Tainan, Taiwan 70101, ROC

*Corresponding author

Abstract

A mathematical programming model is formulated in this work to generate time-sharing schemesto obtain minimum area cost (capital costs) for a given multi-period heat-exchanger network.Many researchers have develop various methods to optimize the heat-exchanger network to reduce the totaloperation and investment cost. Periodical changes in operating conditions result in different optimization structure of the heat-exchanger network where the capital investments usually takes account the maximum of the areas over all periods of operation. Obviously, its operation is inefficient that not all periods require so large areas to exchange the heat. Here, taken no account of how to synthesize multi-period heat-exchanger network, an equipment sharing strategy is use to decrease the capital cost of total heat-exchanger area. A mathematical programming approach is used to automatically generate the best equipment sharing structure.

Keywords: Sharing strategy, multi-period, heat-exchanger network

1. Introduction

HEN synthesis is a well-studiedpractical issue. Many effectivedesign methods have already been developed on the basis offixed process parameters.To account for seasonalvariations in these parameters, Aaltola (2002) modified the MINLP model proposed by Yee et al. (1990) so as to solve the corresponding multi-period optimization problem.In a subsequent study, Chen and Hung (2004) proposed a three-step optimization procedure to generate flexible HEN designs that are operable in more than one period.Verheren and Zhang (2006) also improved the available models for the same purpose. A common feature in these studies is that a single unit wasutilizedto satisfy all possible heat-exchange needs of the same match.This practice is feasible only when the heat-transfer area of the selected exchanger is sufficient for the largest duty and also its operating conditions do not change significantly from one period to another. If the latter condition cannot be met, this exchanger may be inoperablein periods with much smaller duties and, furthermore, itscapital investmentcould be unnecessarily high.

The aforementioned drawbacks of the traditional multi-period HEN designs have been circumvented in this study with time-sharing schemes. In particular, a chosen set of exchangers are allowed to be shared by more than one match in multiple periods. With this approach, it is possible not only to reduce the capital investment of a conventional network but also to improve it operability.A simple mathematical programming model has been formulated to automatically generate the best sharing structure. The proposed implementation procedure is summarized in the following sections.

1. Cost Reduction Options(Can you rewrite this section?)

Let us first assume that, in order to generate the best time-sharing scheme of a multi-period HEN design,the optimal solution of the conventional MINLP model (Aaltola, 2002; Chen and Hung, 2004; Verheren and Zhang, 2006) can be obtained in advance, e.g., see Table 1 and Table 2. While the conceptual structure of this HEN and also the operating conditions of its embedded matches are kept intact, the total capital cost can be reduced with two types of time-sharing options:

Table 1 Feasible reorganization structure case 2 (Table 4 in Aaltola 2002)

Match / Required Area in Period 1(m2) / Required Area in Period 2(m2) / Required Area in Period 3(m2) / Required Area in Period 4(m2) / Max Area
(1,1,1) / 0 / 0 / 0 / 0 / 0
(1,1,2) / 108.9 / 67.5 / 68.6 / 132.4 / 132.4
(2,1,2) / 45.7 / 37.4 / 40.5 / 45.7 / 45.7
(3,1,2) / 30.8 / 29.7 / 28.5 / 30.8 / 30.8
(4,1,3) / 237.6 / 305.9 / 309.2 / 309.1 / 309.2
(6,1,4) / 57 / 137 / 129.4 / 130.1 / 137

Table 2Unfeasible reorganization structure case 1 (Chen et. al 2005)

Match / Required Area in Period 1(m2) / Required Area in Period 2(m2) / Required Area in Period 3(m2) / Required Area in Period 4(m2) / Max Area
(1,2,1) / 6.68 / 82.67 / 6.49 / 68 / 82.67
(2,2,1) / 323.32 / 325.32 / 249.51 / 340 / 340
(1,1,2) / 240 / 240 / 240 / 153.75 / 240

2.1 Heat duty swap

The important condition to reduce the total heat exchanger area is that twoor more areasin different match and in different periods can be swapped.Itcontains two conditions simultaneously, one is thatthe largest required areas are in two different periodsfor two matches, the other is the required area in a match is not always larger than that in the other match among the above two periods.We give an original structure of the multi-period HEN as shown in Table 3. Assume A(a, c) and A(b, d) are the largestrequired areas among all periods in the Matcha and Match b, respectively, A(a, c) > A(b, c) and A(b, d) A(a, d).Obviously, if we have two exchangers with the areas of B(1)= max(A(a, c), A(b, d)), and B(2)= max(A(b, c),A(a, d)), we can reorganize these two exchangers to meet the demands of all periods for these two matches. For period c, B(1) and B(2) meet the demands of A(a, c) and A(b, c), respectively. And for period d, B(2) and B(1) meet the demands of A(a, d) and A(b, d), respectively. This simply net can reduce the area of min(A(a, c), A(b, d))- max(A(b, c),A(a, d)) for these two matches.In Table 4, two matches [(1,1,2), (6,1,4)] have the space to exchange to reduce area. However, it only reduces 132.4-130.1=2.3(m2) according to the above computer equation. If other period’s required area in Match a or Match b is larger than B(2)except for Period c and Period d, it will become more complicated to compute the reduced area, which need mathematical programming model to solve as shown in the next section.

Unfeasible reorganization structure:Any two matches has one of the following states: 1)both the largest required areas are in the same period among all the matches;or 2)if the largest required areas are not in the same period among all the matches but both the required areas in one match are larger than that of the other in the same period. Let’s take two examples to illustrate the unfeasible sharing structure.

The first case is obtained from Chen et. al (2005) as shown in Table 1. There three combinations in this HEN. Twomatches[(1,2,1), (1,1,2)] meet the first state. Twomatches[(1,2,1), (2,2,1)] meet the second state, and twomatches[(1,2,1), (1,1,2)] meet the second state, too. Therefore this HEN is an unfeasible sharing structure.

The second case is obtained from the Table 2 in Aaltola (2002). Totally, there are 15 combinations. [(1,1,1),(2,1,2)], [(1,1,2),(3,1,2)], [(1,1,2), (4,1,3)], [(1,1,2), (6,1,4)], [(3,1,2), (4,1,3)], [(3,1,2), (6,1,4)], [(4,1,3), (6,1,4)] meet the first state. And, [(1,1,1), (1,1,2)], [(1,1,1), (3,1,2)],[(1,1,1), (4,1,3)],[(1,1,1), (6,1,4)],[(1,1,2), (2,1,2)], [(2,1,2), (3,1,2)], [(2,1,2),(4,1,3)], [(2,1,2),(6,1,4)] meet the second state. So the HEN in Table 2 is not a reorganization structure, too.

Table 2 Unfeasible reorganization structure case 2 (Table 2 in Aaltola 2002)

Match / Required Area in Period 1(m2) / Required Area in Period 2(m2) / Required Area in Period 3(m2) / Required Area in Period 4(m2) / Max Area
(1,1,1) / 14.8 / 0 / 11.3 / 14 / 14.8
(1,1,2) / 87.6 / 54.6 / 40.1 / 138.1 / 138.1
(2,1,2) / 43.6 / 35.2 / 22.3 / 43.6 / 43.6
(3,1,2) / 28.2 / 27.9 / 12.3 / 30.8 / 30.8
(4,1,3) / 214.1 / 231.4 / 209.3 / 315.7 / 315.7
(6,1,4) / 55.3 / 115 / 101 / 84.7 / 115

2.2 Exchanger area decomposition

Table4 A scheme to illustrate the structure of original required area of heat exchanger net.

… / Period c / … / Period d / …
…
Match a / A(a, c) / A(a, d)
…
Match b / A(b, c) / A(b, d)
…

1. Mathematical Programming Model

As mentioned previously, a mathematical programming approach can be used to automatically generate the best sharing structure.The model formulation is given below:

(1)

where,Am,p is the required heat-transfer areaof match min period p;xe represents the actual heat-transfer area of exchanger e; yeis a binary variable reflecting whether or not exchanger e is present in the time-sharing scheme; zm,p,q is another binary variable reflecting whether or not exchanger e is selected to facilitate match m in period p; φis a function of exchanger area for computing the capital cost of exchanger e;is a model parameter used to impose realistic upper bound on the heat-transfer area for match m in period p.M is the set of all match. P is the set of all period. E is the existence of the heat exchanger.

1. Examples

The capital investments of exchangers in this example are computed with a cost model adopted from literature, i.e.

(2)

where xeis the heat-transfer area in m2 and is a cost coefficient (4333 USD/m2-yr). It is also assumed that . The sharing model equations for the examples presentedin this paper have been solved using the solver baronin the GAMS environment.The objectivefunction is non-linear and non-convex and hence the solutionof the resulting optimization model represents a local optimum.The initial value will influence the solving path and willlead the program towards a set of different local minima. It is possible to generate good sharing structures by performing several runs withdifferent initial value. Then,the best local minimum is chosen and is presentedas the solution to the heat exchanger design problem for thiscase.

Example 1

This example obtained from Isafiade and Fraser (2010) is an illustrative problem for describingthe sharing model. This example consists of six matches and three periods.Every match requires a largest heat exchanger area to construct the flexible HEN. Therefore, five heat exchangers are required and the total area is 111.95 m2. We find that this HEN has a feasible sharing structure. Solving this sharing MINLP model gives the results shown in Table 3. In this sharing structure, some larger required areas are given larger base heat exchanger, and vice versa. For example, the required area at Period 2 and Period 3 in the same Match (1,2,1) are 32.1 and 11.71 m2. In order to satisfy those requests, two base heat exchangers with areas of 32.1 and 21.67 m2 are set up as show in Table 3. And then base heat exchanger with the area of 32.1 m2 is used at Period 3 in Match (2,3,4). Those adjustments will reduce the total area. Contrary to the original example the total area is 96.77 m2. All the sharing heat exchangers are suitable for the original required areas.

Table 2. The exchanger areas in Case I (Obtain from Table 4 of Isafiade and Fraser)

Match / Required Area in Period 1(m2) / Required Area in Period 2(m2) / Required Area in Period 3(m2) / Max Area
(1,2,1) / 31.2 / 32.1 / 11.71 / 32.1
(2,2,3) / 5.12 / 5.12 / 5.12 / 5.12
(1,1,4) / 8.07 / 14.39 / 1.96 / 14.39
(2,3,4) / 29.58 / 8.3 / 31.8 / 31.8
(1,3,5) / 21.67 / 2.39 / 28.54 / 28.54

Total Cost=134630.34

Figure 1 Original HEN

Table 3. Assign base exchangers to the corresponding matches in suitable periods (Case 1)

Area (m2) / S(1)
32.1 / S(2)
21.67 / S(3)
29.58 / S(4)
5.12 / S(5)
8.3
(1,2,1).1
(1,2,1).2
(1,2,1).3
(2,2,3).1
(2,2,3).2
(2,2,3).3
(1,1,4).1
(1,1,4).2
(1,1,4).3
(2,3,4).1
(2,3,4).2
(2,3,4).3
(1,3,5).1
(1,3,5).2
(1,3,5).3

Total Cost=122199.5654

Figure 2 Period 1

Figure 3 Period 2

Figure 4 Period 3

3.2 Second example

This example is also obtained from Isafiade and Fraser (2010). Solving this sharing MINLP model will give more complex results shown in Table 5. Some required areas need more than one heat exchanger to be combined together in series as shown in Table 5 with the underline mark, most of which are redundant. For example, the required area of (2,2,2).2in Table 5 needs two heat exchanger swith areas of 4.15 and 23.023 m2. But actually, the heat exchanger with area 23.023 already satisfies the (2,2,2).2, which only requires 27.173 m2. Therefore, we find out the redundant combination manually and we delete the redundant combination to simply the switch of the heat exchanger over different periods. All are marked withstrikethrough.

Table 4. The exchanger areas in Case 2 (Obtain from Table 10 of Isafiade and Fraser)

Match / Required Area in Period 1(m2) / Required Area in Period 2(m2) / Required Area in Period 3(m2) / Required Area in Period 4(m2) / Max Area
(2,3,1) / 0.377 / 0 / 4.947 / 0 / 4.947
(2,2,2) / 24.849 / 27.173 / 22.111 / 24.975 / 27.173
(1,2,3) / 0 / 4.604 / 4.527 / 5.746 / 5.746
(3,1,3) / 0 / 0 / 0 / 3.801 / 3.801
(3,2,3) / 0 / 0 / 0 / 0 / 0
(1,1,5) / 49.07 / 54.596 / 53.833 / 47.82 / 54.596
(1,3,5) / 16.488 / 25.953 / 25.891 / 0.338 / 25.953

Total Cost=143091.16

Table 5. Assign base exchangers to the corresponding matches in suitable periods (Case 2)

Area (m2) / S(1) / S(2) / S(3) / S(4) / S(5) / S(6)
4.947 / 25.891 / 54.596 / 0.377 / 4.15 / 23.023
(2,3,1).1
(2,3,1).3
(2,2,2).1
(2,2,2).2
(2,2,2).3
(2,2,2).4
(1,2,3).2
(1,2,3).3
(1,2,3).4
(3,1,3).4
(1,1,5).1
(1,1,5).2
(1,1,5).3
(1,1,5).4
(1,3,5).1
(1,3,5).2
(1,3,5).3
(1,3,5).4

Total Cost=130637.46

Table 6. The exchanger areas in Case 3

Match / Required Area in Period 1(m2) / Required Area in Period 2(m2) / Required Area in Period 3(m2) / Max Area
(1,1,1) / 14.8 / 15.7 / 14 / 15.7
(1,2,2) / 87.6 / 54.6 / 52.3 / 87.6
(2,1,2) / 28.2 / 27.9 / 30.8 / 30.8
(2,2,3) / 214.1 / 231.4 / 304.6 / 304.6
(2,3,4) / 0 / 143.2 / 0 / 143.2
(1,CU,5) / 0 / 0 / 56.6 / 56.6
(2,CU,5) / 100.9 / 108.3 / 105.2 / 108.3

Total Cost=459923.51

Table 7. Assign base exchangers to the corresponding matches in suitable periods (Case 3)

Area (m2) / S(1) / S(2) / S(3) / S(4) / S(5) / S(6)
108.3 / 30.8 / 28.2 / 304.6 / 54.6 / 183.3
(1,1,1).1
(1,1,1).2
(1,1,1).3
(1,2,2).1
(1,2,2).2
(1,2,2).3
(2,1,2).1
(2,1,2).2
(2,1,2).3
(2,2,3).1
(2,2,3).2
(2,2,3).3
(2,3,4).2
(1,CU,5).3
(2,CU,5).1
(2,CU,5).2
(2,CU,5).3

Total Cost =418570.00

Table 8. The exchanger areas in Case 4

Match / Required Area in Period 1(m2) / Required Area in Period 2(m2) / Required Area in Period 3(m2) / Max Area
(1,,1,2) / 134.11 / 134.11 / 134.11 / 134.11
(2,1,2) / 16.59 / 0 / 0 / 16.59
(2,2,2) / 41.14 / 0 / 0 / 41.14
(3,2,1) / 0 / 53.88 / 0 / 53.88
(1,CU,3) / 14.08 / 14.08 / 14.08 / 14.08
(2,CU,3) / 37.05 / 0 / 0 / 37.05
(3,CU,3) / 0 / 61.02 / 0 / 61.02
(HU,1,0) / 19.73 / 29.3 / 29.03 / 29.3
(HU,2,0) / 0 / 0 / 8.71 / 8.71

Total Cost =351804.15

Table 9. Assign base exchangers to the corresponding matches in suitable periods (Case 3)

Area (m2) / S(1) / S(2) / S(3) / S(4) / S(5) / S(6)
61.02 / 34.15 / 37.05 / 19.73 / 134.11 / 14.08
(1,1,2).1
(1,1,2).2
(1,1,2).3
(2,1,2).1
(2,2,2).1
(3,2,1).2
(1,CU,3).1
(1,CU,3).2
(1,CU,3).3
(2,CU,3).1
(3,CU,3).2
(HU,1,0).1
(HU,1,0).2
(HU,1,0).3
(HU,2,0).3

Total Cost =253963.2

Comparison

Comparisons of the capital cost are shown in Table 10. This table show the capital cost both for original and sharing HEN. The table also gives the saving percent of the total capital cost after the implement the sharing strategy.The benefits gained from sharing heat exchanger structure are 9.23%, 8.70%, 8.99% and 27.81% for those four examples.

Table 10Comparisons of the capital cost for original and sharing HEN

Example 1 / Example 2 / Example 3 / Example 4
Captical Cost, Original / 134630.34 / 143091.16 / 459923.51 / 351804.15
Captical Cost, Sharing / 122199.57 / 130637.46 / 418570.00 / 253963.20
Saving Percent,% / 9.23 / 8.70 / 8.99 / 27.81

1. Conclusion

Taken no attention of how tocalculate the minimumheating and cooling requirements for a heat-exchanger network,this study presents a sharing strategy for the design of multi-period heat-exchanger network where the required heat exchanger area are known.For a fixed match in different periods, the required heat-exchanger areas are not same.Within the overall objective of investment cost optimization of a multi-period industrial process, it is of great importance to improve the efficiency of recombining heat-exchanger network.This work gives a mathematical programming approach to automatically generate the best equipment sharing structure when there are significant changes in the environment of a plant.This paper shows the several criteria to discern the feasible heat-exchanger network to be recombined. Based on the extensive case studies performed so far, it can be observed that this proposed approach is especially effective for multi-period HEN design problems in which the process conditions vary significantly.

Sharing the available units is useful for the largetemperature changesduring different periodsofheat exchangenetwork. However, whenthe network structureis not same for different periods, it needs to lay down more pipelines to connect the HEN so that the pipeline investment costs will be increased. So in the next work, we shall take account that pipeline investment costs to find out the sharingHEN which is not only toreduce investmentcosts but also to easier switch.

Reference

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