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An effective layer pattern optimization model for multi-stream plate-fin heat exchanger using genetic algorithm
International Journal of Heat and Mass Transfer 60 (2013) 480–489
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An effective layer pattern optimization model for multi-stream plate-fin heat exchanger using genetic algorithm Min Zhao, Yanzhong Li ⇑ School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 12 September 2012 Received in revised form 21 December 2012 Accepted 23 December 2012 Available online 9 February 2013 Keywords: Multi-stream plate-fin heat exchanger Layer pattern Optimization Genetic algorithm
a b s t r a c t For multi-stream plate-fin heat exchangers (MPFHEs), a crucial factor affecting performance is the layer pattern used to distribute hot and cold streams. However, for the design of MPFHE layer pattern, so far it still remains at semi-qualitative, empirical or trial-and-error stage, lacking efficient approaches to obtain the optimum. In this paper, an effective layer pattern optimization model using genetic algorithm (GA) is developed in detail. It includes the chromosome of binary string that represents hot and cold layers alternatively arranged, the dual fitness functions that act on chromosome individuals alternatively, the constraint that requires the spacing layers between hot and cold streams to be no less than 1 and no greater than 2, the selection of optimization tool and the corresponding setting of GA parameters. When used to optimize the layer pattern of actual MPFHE, this model exhibits well. The average thermal efficiency of exchanger reaches up to 98% of that obtained under the ideal common wall temperature assumption. Furthermore, the study of this paper make it clear the association of previous classical deviation from ideal of layer pattern zigzag curve with MPFHE thermal performance, which is not a strict monotonic decreasing relation as people thought to be, but a multiple vs. multiple relation. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Multi-stream plate-fin heat exchangers (MPFHEs) are widely used in the cryogenic and chemical engineering industries. For them, whether the performance is good or not closely relates with the layer pattern that is used to distribute hot and cold streams. When the layers are not arranged properly, it will lead to the significant reduction of exchanger performance and the accompanying unexpected thermal stresses. In severe cases, even if the exchanger has sufficient backup heat transfer areas, it could not make up for this performance reduction. However, for the design of MPFHE layer pattern, so far it still remains at semi-qualitative, empirical or trial-and-error stage, lacking efficient approaches to obtain the optimum. Fan [1] first proposed the segregated pattern in which a single hot and cold layer are arranged alternatively. It seems that this sandwiched pattern allows hot and cold streams to contact fully so that the exchanger can achieve high thermal efficiency. However, the fact is not that. Firstly, it implies that exchanger ought to have comparable hot and cold layers in total number, which is difficult to meet for most actual MPFHEs. Secondly, such complete alternative arrangement of single hot and cold layers very likely causes the imbalance of heat between layers. ⇑ Corresponding author. Tel.: +86 29 82668725; fax: +86 29 82668789. E-mail addresses: [email protected] (M. Zhao), [email protected] (Y. Li). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.041
In the 15th International Congress of Refrigeration, Suessman and Mansour [2] pointed out that the performance degradation of MPFHE arising from improper layer pattern is because they form and intensify this heat imbalance, i.e., there is an excess of heat accumulated in some layers and an excess of cold accumulated in others, as shown in the zigzag curve of Fig. 1. Ideally, the zigzag curve of layer pattern oscillates between positive and negative values regularly. Therefore, by checking the deviation of its zigzag curve from ideal, a layer pattern can be evaluated roughly. In addition, to study the impact of this heat imbalance on exchanger performance, Suessman and Mansour divided MPFHE into several zones called ‘Local Heat Balance Zone’ along the layer stacking direction. According to the Fourier theorem, if the less the number of layers contained in these zones is, the more uniform the transverse wall temperature of exchanger will be, and the less the performance of exchanger will be affected. This is the wellknown Local Heat Balance principle which has been playing an important role in guiding MPFHE layer arrangement today. However, because MPFHE usually has up to 100 or more layers and therefore the corresponding possible layer patterns could be numerous (i.e., combinatorial explosion), it is impractical for us to check all the possible layer patterns’ zigzag curve deviation to find the optimum. Also, because of this combinatorial explosion feature, traditional optimization methods are not suitable to solve the MPFHE layer pattern problem. So till today people have to use the trial-and-error procedure to get the relatively satisfactory layer arrangement. Firstly, according to the experience or the Local Heat
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Nomenclature Cphi Cpcj cj Ei hi l Li _ hi m _ cj m n nb nc nh N Nhi N cj Nmin phi pcj Phi P cj
specific heat of hot stream i specific heat of cold stream j cold stream j, j = 1,2, . . . , nc excess heat load of each hot/cold layer string in layer pattern ring in Eq. (7) hot stream i, i = 1,2, . . . , nh length of chromosome x each line length of the layer pattern chromosome zigzag curve, i = 1,2, . . . , n mass flowrate of hot stream i mass flowrate of cold stream j number of xi = 1 in the chromosome x number of binary bits that represents a layer in the chromosome of MPFHE layer pattern number of different cold streams in a MPFHE number of different hot streams in a MPFHE total number of layers in a MPFHE number of layers of hot stream i number of layers of cold stream j Pnh Pnc minimum of N and i.e., j¼1 N cj , i¼1 hi P Pnc nh in Fig. 2 N min ¼ min N ; N c h j j¼1 i i¼1 outlet pressure of hot stream i outlet pressure of cold stream j inlet pressure of hot stream i inlet pressure of cold stream j
Balance principle give the layer pre-arrangement, and then through the simulation means check whether the performance of MPFHE achieves design requirements under this layer pattern. If not, adjust it and repeat the above procedure till the required performance is met. Obviously, this design method is not only timeand labor-consuming, but also could not ensure the quality of final results. Recently in solving combinatorial explosion optimization problems of real world, genetic algorithm (GA) has got wide applications owing to its small dependence on the studied problems and powerful and robust searching ability [3–16]. Many authors have also used GA to the field of heat exchanger optimization design [17–23]. For example, Ponce-Ortega et al. [21] use it to optimize shell-and-tube heat exchanger parameters such as the number of tube-passes, the diameter and pitch of tube, the baffle cut, etc. Xie et al. [18] use it to optimize the structure size of plate-fin heat
Fig. 1. Zigzag curve of a multi-stream plate-fin heat exchanger.
P1nh ; P2nh qhi qcj
number of permutation from nh elements to pick up 1 and 2, respectively heat load per layer of hot stream i heat load per layer of cold stream j
Q string ; Q string heat load of hot or cold layer string i, ci hi i ¼ 1; 2; . . . ; Nmin Si vertical coordinate (i.e., cumulative heat load) of each bit of xi = 1 in the layer pattern chromosome zigzag curve, i = 1,2, . . . , n Smid vertical coordinate (i.e., cumulative heat load) of midi point of each line in the layer pattern chromosome zigzag curve, i = 1,2, . . . , n Smid arithmetic-mean of Smid in Eq. (6) i t hi outlet temperature of hot stream i outlet temperature of cold stream j t cj T hi inlet temperature of hot stream i T cj inlet temperature of cold stream j weighting in Eq. (6), i = 1,2, . . . , n wi Wi weighting in Eq. (7), i ¼ 1; 2; . . . ; N min x = [x1, x2, . . . , xl] chromosome of MPFHE layer pattern xi bit of chromosome x, i = 1,2, . . . , l Greek symbols e allowable tolerance
exchanger, etc. However, people seldom use GA to optimize the layer pattern of MPFHE. So far only Ghosh et al. [24] have done the related work. In their GA model, the optimization objective is to maximize MPFHE heat transfer rate estimated by their own rating program. The layer pattern chromosome is adopted sequential coding, i.e., from left to right each layer of the pattern is in turn represented by a certain fixed number of bits, nb. When the total number of layers is N, a string of nb N bits joined together will represent the whole layer pattern. For the value of nb, authors give two methods N to determine, nb ¼ log2 and nb = N 1. The former is used for some specific cases as a means of verification, and the latter is applied for all tested cases. For the main GA parameters, authors select tournament selection, uniform crossover and uniform mutation, respectively. This model obtains satisfactory results when used to optimize the layer pattern which is composed of 3–8 different fluid stream layers. However, their tested cases are far from the actual MPFHE layer pattern optimization situation, in which each stream has their own number of layers constraint, both number and type of individuals in the GA population pool increase dramatically as a result of multiple streams and tens or hundreds of layers in exchanger, single and cascade arrangement are mixed in a layer pattern, etc. all these together increase the difficulty of GA to meet constraints and obtain good optimal results greatly. This paper is committed to solving the above practical problems using special chromosome coding, fitness function and constraints so that the better layer pattern solution can be obtained. An important feature of our GA model is the use of dual fitness functions. During the optimization process, it is found that the minimization of the deviation of zigzag curve from ideal is not enough. Its optimal result’s MPFHE thermal efficiency is only 92.4% on the average. When another upgrade fitness function is combined with it to alternatively optimize the layer pattern individuals (see Sections 2 and 3 of this paper), the optimal result of GA has got an evident improvement, whose MPFHE thermal efficiency has increased from 92.4% up to 98% and some even close to 99% (98.7%).
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Another important feature of our model is that the related heat transfer calculations (by MUSE software) are not embedded into genetic algorithm as a part of its fitness function and therefore the computational time is reduced greatly [19,25]. In addition, through the study of this paper, the relation of previous classical deviation from ideal of layer pattern zigzag curve with MPFHE thermal performance is made clear, which, on the whole, is that the heat transfer rate of MPFHE increases with the deviation decreasing. But they do not follow a strict monotonic decreasing relation as people thought to be, but a multiple vs. multiple relation. Below let us give the detailed GA optimization model first, and then case study is followed and results are discussed. Finally, the conclusions are drawn. 2. Modelling 2.1. Problem presentation Suppose there is a MPFHE in which nh hot streams exchange heat with nc cold streams. For hot stream hi (i = 1,2, . . . , nh), its flowrate, inlet/outlet temperature and pressure, and number of layers _ hi , T hi , t hi , Phi , ph , N hi , respectively. For cold stream cj are m i (j = 1,2, . . . , nc), its flowrate, inlet/outlet temperature and pressure, _ cj , T cj , t cj , P cj , pc , N cj , respectively. and number of layers are m j Now the problem is how to arrange these hot and cold layers, Pnh Pn c i.e., N ¼ i¼1 N hi þ j¼1 N cj , to make MPFHE heat transfer rate maximal. 2.2. Genetic algorithm optimization model To solve the above optimization problem using GA, it is required to encode the layer pattern solution first. It is natural for us to think of sequential coding [24]. However, with the number of layers increasing, this coding will become extremely difficult and cumbersome in treating objective function and constraints. So we adopt the following chromosome of binary string x = [x1, x2, . . . , xl] to represent the layer pattern. Encoding of chromosome Firstly, according to the above optimization conditions calculate Pnh the total number of hot and cold layers of MPFHE, i¼1 N hi and Pnc Pnh Pnc j¼1 N cj . If j¼1 N cj , the first nh bits of binary string i¼1 N hi > x = [x1, x2, . . . , xl] will represent hot stream hi, i = 1,2, . . . , nh, respectively. Then the followed nc bits will represent cold stream cj, j = 1,2, . . . , nc, respectively. Next, the followed nh bits, i.e., xnh þnc þ1 ; xnh þnc þ2 ; . . . ; x2nh þnc , will again represent hot stream hi (i = 1,2, . . . , nh) to form the substring of h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh that represents hot streams surrounding cold Pc streams. Finally, this substring repeats nj¼1 N cj times to form the entire chromosome x = [x1,x2, . . . , xl]. The above forming process can be illustrated as follows,
Clearly, the length of this layer pattern chromosome is Pc Pn h Pnc l ¼ ð2nh þ nc Þ nj¼1 N cj . Similarly, when j¼1 N cj , the i¼1 N hi 6 above substring will change from h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh to c1 c2 . . . cnc h1 h2 . . . hnh c1 c2 . . . cnc , and its repeating time will Pnc Pn h change from pattern j¼1 N cj to i¼1 N hi . So the length of layer Pc chromosome x will change from l ¼ ð2nh þ nc Þ nj¼1 N cj to Pnh l ¼ ð2nc þ nh Þ i¼1 N hi . In the above layer pattern chromosome x = [x1,x2, . . . , xl], xi = 1 (i = 1,2, . . . , l) means the stream represented is selected, while xi = 0 means the corresponding stream is not selected. Therefore, orderly listing all the streams of xi = 1 in the chromosome can obtain the layer pattern solution. For example, when Pnc Pnh j¼1 N cj , the layer pattern of chromosome x = [10 . . . i¼1 N hi > 110 . . . 010 . . . 001 . . . 110 . . . 111 . . . 0 . . . . . . . . . 10 . . . 000 . . . 110 . . . .0] is h1 . . . hnh c1 . . . h1 . . . h2 . . . hnh c1 . . . cnc h1 h2 . . . . . . h1 . . . . . . cnc h1 , where means no stream on this position. For such chromosome coding, people may question the compoPn h Pc sition of substring. For example, when i¼1 N hi > nj¼1 N cj , why use the substring of h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh instead of h1 h2 . . . hnh c1 c2 . . . cnc which can also form hot streams surrounding cold streams in the chromosome. The simplest reason is that the substring of h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh forms the double string of h1 h2 . . . hnh between two adjacent strings of c1 c2 . . . cnc in the chromosome, which can provide all the possible combination for any two hot streams, while substring h1 h2 . . . hnh c1 c2 . . . cnc could not. Of course, such double string of h1 h2 . . . hnh could not offer all the possible combination for any greater than two hot streams. However, because these multi-fold arrangements hinder the hot stream in the inner layer effectively giving off the heat to the cold streams, they are to be avoided as far as possible in the actual layer pattern of MPFHE. As for the cold stream cj, because Pnc Pnh j¼1 N cj , it is not necessary to use double c1 c2 . . . cnc to i¼1 N hi > obtain all the possible combinations for any two cold streams. For the double string h1 h2 . . . hnh , people may further question ðP 1n þP 2n Þ
why not use shorter bits, for example, nb ¼ log2 h h , to represent all the possible combinations of no greater than two hot streams, where P1nh and P 2nh are the number of permutation from nh elements to pick up 1 and 2, respectively. This is because, first of all, the vaðP1n þP 2n Þ
lue of log2 h h is not always integer and therefore illegal chromosomes inevitably exist in the population of GA. These illegal chromosomes can not be evaluated normally by GA because they have no actual meaning. So additional strategies are needed to consider to get rid of them or convert them into qualified ones. Secondly, for this shorter bits coding, the corresponding program of realizing fitness function and constraints is much more complicated. Thirdly, our case study shows that the average time of each GA running of present coding is only 1–2 min when used to optimize the layer pattern of MPFHE with 108 layers, 3 hot fluids and 4 cold fluids, whose length of layer pattern chromosome is 429 bits and the population size of GA is 2000–4000 (see Section 3 of Case study in this paper). Moreover, with the number of layers or population size of GA decreasing, the running time of GA will be further reduced. Fitness Function After encoding, it is required to define the fitness function. As mentioned before, dual fitness functions are adopted in this GA model. The first one is called main whose objective is to minimize the deviation from zero of each line midpoint in the layer pattern zigzag curve, whose weighting is the line length. For a certain layer
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
(
pattern chromosome x = [x1,x2, . . . , xl], its calculation procedure is as follows: 1. According to the known optimization conditions, calculate the heat load per layer of each stream in MPFHE.
_ h Cph ðt hi T hi Þ m i qhi ¼ i ; i ¼ 1; 2; . . . ; nh N hi _ cj Cpc ðtcj T cj Þ m j qcj ¼ ; j ¼ 1; 2; . . . ; nc N cj
where qhi is of hot stream hi whose sign is ‘‘’’, while qcj is of cold stream cj whose sign is ‘‘+’’. 2. In terms of the following program, calculate the vertical coordinate (i.e., cumulative heat load) of each bit of xi = 1 in the layer pattern chromosome zigzag curve, Si, i = 1,2, . . . , n, where n is the total number of xi = 1 in the chromosome.
i ¼ 0; % counter S0 ¼ 0; % the origin of zigzag curve For j ¼ 1; 2; . . . ; l If xj ¼ 1
ð3Þ
i ¼ i þ 1; Si ¼ 0; % initialize the accumulator For k ¼ 1; 2; . . . ; j Si ¼ Si þ qxk ;
where qxk ¼ qhi or qcj of Step 1 depending on the position of xk in the chromosome. 3. Calculate each line length of the above layer pattern chromosome zigzag curve and their midpoints’ vertical coordinate, Li and Smid ; i ¼ 1; 2; . . . ; n. i
Li ¼ jSi Si1 j;
¼0 Smid i
ð4Þ
i¼1
ð5Þ
4. Finally, with the Li and Smid of Step 3, the main fitness function i can be expressed as,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n 2 u 1 X min:fmain ðxÞ ¼ t Smid Smid wi i x n 1 i¼1
ð1Þ ð2Þ
Smid ¼ 12 ðSi1 þ Si Þ i – 1 i
483
ð6Þ
where
Smid ¼ wi ¼
n 1 X Smid n 1 i¼1 i
1 Li n X Li i¼1
Now let us discuss the second fitness function, upgrade fitness function, whose definition is related to a layer pattern ring which is formed by linking both ends of layer pattern chromosome. When Pnc Pnh j¼1 N cj , the corresponding layer pattern ring is as shown i¼1 N hi > in Fig. 2. From the figure it can be seen that the double hot layer string of h1 h2 . . . hnh and the cold layer string of c1 c2 . . . cnc are alterP P nc nh natively arranged for N min ¼ min times. Simij¼1 N cj i¼1 N hi ; Pnc Pnh larly, when i¼1 N hi 6 j¼1 N cj , the layer pattern ring will turn into the double cold layer string of c1 c2 . . . cnc and the hot layer string being alternatively arranged for of h1 h2 . . . hnh P Pnc nh times. Nmin ¼ min j¼1 N cj i¼1 N hi ; P nc Pnh P nc P nh Ideally, whether for j¼1 N cj case or j¼1 N cj i¼1 N hi > i¼1 N hi 6 case, every hot/cold layer string in the layer pattern ring is in thermal equilibrium with half of two cold/hot layer strings on its both sides. However, in fact the deviation from ideal always exists. The objective of upgrade fitness function is to minimize this deviation.
Fig. 2. Schematic of layer pattern ring when
Pnh
i¼1 N hi
>
Pnc
j¼1 N cj .
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Pnh P nc Pnh Pnc For both j¼1 N cj and j¼1 N cj cases, the i¼1 N hi > i¼1 N hi 6 expression of upgrade fitness function can be unified as,
string
Qc
W i ¼ P i string at each cold layer string as analytical object. As to Qc
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uN min uX min fupgrade ðxÞ ¼ t ðEi Þ2 W i x
each hot layer string of layer pattern ring as analytical object or
ð7Þ
i¼1
where Ei is the excess heat load of each hot or cold layer string with half of its adjacent two cold or hot layer strings, Appendix A gives Ei detailed calculation procedure for layer pattern chromosome x, in which the heat load of each hot and cold layer string, Q string and hi Q
Q string , are calculated in Appendix B. The weighting W i ¼ P ci
string hi string Qh i
at
i
which layer strings as analytical objects, it will depend on the MPFHE to be optimized. For example, for the MPFHE in Case study of this paper, the effect of upgrade fitness function is very apparent when cold layer strings as study objects, while that of hot layer strings is trivial. Fig. 3 shows GA optimization procedure with the above two fitness functions. Firstly, under the main fitness function GA obtains the first optimal solution from initial population generated randomly, and then it is put into MUSE software to check whether the thermal performance of MPFHE arrives design requirements. If
Fig. 3. GA optimization procedure with dual fitness functions.
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Table 1 Setting of GA parameters on ga solver of Matlab 2010b. Item
Value or function name
Population size
800–2000 (Main fitness function) 2000–4000 (Upgrade fitness function) Uniform Rank Stochastic uniform 2 Scattered 0.8 Uniform 0.01 Forward 0.2 20 100 Inf Inf 50 Inf 106
Creation function Scaling function Selection function Elite count Crossover function Crossover fraction Mutation function Mutation rate Migration direction Migration fraction Migration interval Stopping generations Stopping time limit Stopping fitness limit Stopping stall generations Stopping stall time limit Stopping function tolerance
not, GA will use this solution as initial population to go on the next optimization process under the upgrade fitness function. Similarly, the second obtained result is sent into MUSE software to test the MPFHE performance. If the performance is still not satisfactory, GA will again use the second optimal result as initial population to continue optimization process under the main fitness function. So, like this repeat the above alternative procedure of dual fitness functions on the chromosome individuals until the satisfactory solution is obtained. In addition, because GA is a probability algorithm, its optimal solution is often inconsistent each time even under the same conditions. Therefore, to obtain the steady solution of each fitness function in the above GA optimization process, the following strategy is adopted. Firstly, GA uses the initial population generated randomly to obtain the first optimal solution, and then it uses this solution as initial population to optimize again to obtain the second optimal solution. If the difference of these two optimal solutions’ MPFHE thermal performance is greater than the allowable tolerance e, it will repeat the above procedure until the tolerance is met. Constraints Firstly, the demanded layer number of each stream in MPFHE should be met. Secondly, to ensure the efficiency of exchanger, the spacing layers between hot and cold streams should be 62 and simultaneously P1. Of course, this constraint premise is that the ratio of the total number of hot and cold layers is P1:2 or 62:1. Otherwise, it will not be satisfied for ever. Thirdly, the number of hot or cold layers on both ends of MPFHE is required to be 1. Appendix C gives the corresponding expressions of constraints above. Optimization tool and Setting of GA parameters The ga solver of Matlab 2010b is selected to perform the above optimization strategy. Table 1 gives the setting of GA parameters in the optimization process. 3. Case study and results discussion Below let us see the effects of the above optimization model through a typical example.
Fig. 4. Schematic of a practical MPFHE in air separation plant.
This is a MPFHE used in the air separation plant. Fig. 4 shows the schematic, in which three hot streams No. 1–3 are, respectively, high-, middle-, and low-pressure air, which flow from the top of exchanger to the bottom, and four cold streams, i.e., lowpressure nitrogen No. 4, low-pressure waste nitrogen No. 5, lowpressure oxygen No. 6, and middle-pressure liquid oxygen No. 7, flow from the bottom of exchanger to the top to exchange heat with the above hot streams. Table 2 lists the layer pattern optimization conditions. According to these conditions, the heat load per layer of each hot and cold stream, qhi ði ¼ 1; 2; 3Þ and qcj ðj ¼ 1; 2; 3; 4Þ, are figured out listed in the last row of the table. It can be seen that they are different much from each other. So if the layers are not arranged properly, the heat imbalance between layers will be easy to form and intensify, thereby leading to the performance of MPFHE degrading greatly. Therefore, for this MPFHE obtaining the optimal layer pattern is especially significant. Fig. 5A–D gives four representative results obtained by the present GA optimization model, in which the serial number of GA running on the horizontal axis corresponds to the times of the main and the upgrade fitness function acting on the layer pattern chromosome individual alternatively, the discontinuous points on the top is the value change of the main and the upgrade fitness function, and the bottom curve is the corresponding variation of resulting MPFHE thermal efficiency. Obviously, with the times increasing of dual fitness function alternatively acting on the layer pattern chromosome individuals the main and the upgrade fitness function value both decrease till the change tends small. While for the corresponding MPFHE thermal efficiency, some increase slowly first
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Table 2 Optimization conditions of MPFHE layer pattern.a Items
Units
_i Flowrate, m
kg h K K MPa MPa
Inlet temperature Ti Outlet temperature ti Inlet pressure Pi Outlet pressure pi No. of layers Fin geometry Fin type Fin height Fin thickness Fin frequency Fin porosity Fin serration length Heat load/layer a
mm mm m1 mm kW
1
Stream no. 1
2
3
4
5
6
24161
22611
21319
27754
23622
92.88
14285
313.0 110.0 5.485 5.445 8
313.0 158.0 2.456 2.440 8
287.0 102.0 0.580 0.567 23
96.0 309.7 0.129 0.114 34
96.0 309.7 0.127 0.114 30
93.5 309.7 0.137 0.128 1
96.0 309.7 3.140 3.1 4
Perforated 6.5 0.5 714.28 0.043
Serrated 6.5 0.3 714.28
Serrated 6.5 0.2 714.28
Serrated 9.5 0.2 714.28
Serrated 9.5 0.2 714.28
Serrated 9.5 0.2 714.28
Perforated 6.5 0.2 714.28 0.05
3 133.00
3 49.00
3 +49.95
3 +45.72
3 +4.96
301.70
7
+393.93
The sign of ‘‘’’ in the last low represents the released heat load per layer from hot stream, and ‘‘+’’ represents the absorbed heat load per layer by cold stream.
Fig. 5. Four representative optimal results of present GA model.
and then increase fast such as Fig. 5A, some increase gradually such as Fig. 5B, and so on. However, no matter what their optimization trail is, their final results can always reach 98% of MPFHE thermal efficiency through several times of main and upgrade fitness function alternatively working. This strongly confirms the effectiveness of present GA optimization model.
From the figure it can be also observed such points which have smaller fitness values but their MPFHE thermal efficiencies are not higher as expected. So they are called retrograde points. Their emergence has broken our conventional concept from Suessman and Mansour in optimizing MPFHE layer pattern. From Fig. 5D, it can also be seen that the retrograde degree of the
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Fig. 6. MPFHE heat transfer rate vs. fitness function values.
upgrade fitness function is severer than that of the main fitness function. To find the relation of MPFHE thermal performance with the main and upgrade fitness function, MPFHE heat transfer rate vs. the values of fitness function is plotted in Fig. 6. From the figure it can be clearly seen that the relation of MPFHE heat transfer rate with each fitness function is multiple vs. multiple, that is, for a certain MPFHE heat transfer rate, there are several main or upgrade fitness function values to correspond, and vice versa. However, there is no denying that on the whole the heat transfer rate of MPFHE is increasing with the decrease of the values of main or upgrade fitness function. And the degree of MPFHE heat transfer rate increasing with the decrease of the main fitness function is faster than that with the decrease of upgrade fitness function, especially in the range of large fitness function values. With the value of fitness function decreasing, this difference will reduce till they coincide, i.e., ultimately the heat transfer rate of MPFHE achieves the ideal value under the common wall temperature condition.
Appendix A. Calculation procedure of Ei for the layer pattern chromosome x = [x1,x2, . . . , xl] Pnh Pc When i¼1 N hi > nj¼1 N cj for each hot layer string, the excess heat load Ei is calculated as, For i ¼ 1; 2; . . . ; If i = 1
0
Ei ¼ Q string hi
1
B string C string C 0:5B nc @Q c1 þ Q c P A Nc
else Ei ¼
Q string hi
for each cold layer string, the excess heat load Ei is calculated as, nc P
N cj
j¼1
If i ¼
nc P
N cj
j¼1
0
1
B string string C C Ei ¼ Q string 0:5B ci @Q h1 þ Q h P A nc Nc
else Ei ¼ Q string ci
j
j¼1 string 0:5 Q h þ Q string h i
iþ1
end end Pc Pnh N hi 6 nj¼1 N cj When i¼1 for each hot layer string, the excess heat load Ei is calculated as, For i ¼ 1; 2; . . . ;
nh P j¼1
If i ¼
nh P j¼1
N hj
N hj 0
1
B string C string C Ei ¼ Q string 0:5B nh @Q c1 þ Q c P A hi j¼1
This research was supported by the Chinese State Science and Technology Support Program (Grant No. 2012BAA08B03).
j
j¼1 0:5 Q string þ Q string ci1 ci
end end
4. Conclusions
Acknowledgements
N cj
j¼1
For i ¼ 1; 2; . . . ;
In this paper, in order to effectively obtain the optimal layer pattern of MPFHE, a novel GA optimization model is developed in detail. It includes the chromosome of binary string that represents hot and cold layers alternatively arranged, the dual fitness functions that act on chromosome individuals alternatively, the constraint that requires the space layers between hot and cold streams to be no less than 1 and no greater than 2, and the selection of optimization tool as well as the corresponding setting of GA parameters. This model exhibits well when used to optimize the layer pattern of actual MPFHE. The average thermal efficiency of exchanger reaches up to 98%. In addition, through the study of this paper, the relations of previous classical deviation of layer pattern zigzag curve from ideal with MPFHE performance is made clear, which is not a strict monotonic decreasing relation as people thought to be, but a multiple vs. multiple relation.
nc P
Nh j
(continued on next page)
488
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
else Ei ¼ Q string 0:5 Q string þ Q string ci ciþ1 h
nc P
N cj
X
j¼1
i
end end
xk1 þði1Þð2nh þnc Þ þ xnh þnc þk1 þði1Þð2nh þnc Þ ¼ Nhk
1
i¼1 nc P
for each cold layer string, the excess heat load Ei is calculated as, nh P
For i ¼ 1; 2; . . . ;
j¼1
If i = 1 Ei ¼
0
Q string ci
X
xk2 þnh þði1Þð2nh þnc Þ ¼ Nck
j¼1 else Ei ¼ Q string 0:5 Q string þ Q string ci h h i1
2
j
xk1 þnc þði1Þð2nc þnh Þ ¼ Nhk1
i¼1 nh P
Nh j
when
nh nc X X N hi 6 N cj i¼1
!
j¼1
Nh
X
j¼1
i
end end
j¼1
Nh
X
j¼1
B string string C C 0:5B @Q h1 þ Q h P A nh
i¼1
!
j¼1
nh P
1
nh nc X X Nhi > N cj
N cj
i¼1
N hj
when
j
xk2 þði1Þð2nc þnh Þ þ xnc þnh þk2 þði1Þð2nc þnh Þ ¼ N ck2
i¼1
and Q string for the Appendix B. Calculation procedure of Q string ci hi layer pattern chromosome x = [x1, x2, . . . , xl]
where k1 = 1,2, . . . , nh;k2 = 1,2, . . . , nc. For the second and the third constraints, nh nc P P N hi > N cj when i¼1
j¼1
nc P
For i ¼ 1; 2; . . . ; Pnc
Pnh
When i¼1 N hi > j¼1 N cj , each hot and cold layer string and Q string are calculated as, ci nc P
For i ¼ 1; 2; . . . ;
Q string hi
If i = 1 nh P xm ¼ 1; m¼1
N cj
j¼1
If i = 1 l¼ð2nh þnc Þ
Q string ¼ h i
nh P m¼1
P
qxm xm þ
nc P
l¼ð2nh þnc Þ
nc P
X j¼1
N cj
N cj
xm ¼ 1;
j¼1
qxm xm ;
m¼lnh þ1
else Q string ¼ hi
m¼lnh þ1 3nP h þnc
else 1 6
3nP h þnc m¼nh þnc þ1
qxmþði2Þð2n
h þnc Þ
m¼nh þnc þ1
xmþði2Þð2nh þnc Þ ;
nhP þnc
16
end
m¼nh þ1
Q string ¼ ci
nhP þnc m¼nh þ1
N cj
j¼1
qxmþði1Þð2n
h þnc Þ
xmþði1Þð2nh þnc Þ ;
xmþði2Þð2nh þnc Þ 6 2;
xmþði1Þð2nh þnc Þ 6 2;
end End
end When
Pnh
i¼1 N hi
6
Pn c
j¼1 N cj ,
and Q string are calculated as, ci
each hot and cold layer string Q string hi
when
nh P i¼1
nc P
N hi 6
j¼1
For i ¼ 1; 2; . . . ;
N cj
nh P j¼1
nh P
For i ¼ 1; 2; . . . ;
j¼1
N hj
If i = 1 nc P xm ¼ 1;
N hj
m¼1
If i = 1 l¼ð2nc þnh Þ
¼ Q string ci
nc P m¼1
P
qxm xm þ
nh P j¼1
m¼lnc þ1
Nh
j
l¼ð2nc þnh Þ
qxm xm ;
else
nh P
X j¼1
j
xm ¼ 1;
m¼lnc þ1
Q string ¼ ci
nhP þ3nc m¼nh þnc þ1
qxmþði2Þðn
h þ2nc Þ
xmþði2Þðnh þ2nc Þ ;
else 1 6
¼
nhP þnc m¼nc þ1
qxmþði1Þðn
h þ2nc Þ
xmþði1Þðnh þ2nc Þ ;
End
Appendix C. Constraints expression for the layer pattern chromosome x = [x1,x2, . . . , xl] For the first constraint,
nhP þ3nc m¼nh þnc þ1
end Q string hi
Nh
nhP þnc
16
m¼nc þ1
xmþði2Þðnh þ2nc Þ 6 2;
xmþði1Þðnh þ2nc Þ 6 2;
end End
References [1] Y.N. Fan, How to design plate-fin heat exchangers, Hydrocarbon Processing 45 (11) (1966) 211–217.
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489 [2] W. Suessman, A. Mansour, passage arrangement in plate-fin exchanges, in: Proceedings of 15th International Congress of Refrigeration, 1979, pp. 421– 429. [3] M. Sakawa, K. Kato, T. Mori, Flexible scheduling in a machining center through genetic algorithms, Comptuters and Industrial Engineering 30 (4) (1996) 931– 940. [4] D. Tate, A. Smith, A genetic approach to the quadratic assignment problem, Computers and Operations Research 22 (1995) 73–83. [5] M. Gen, G. Zhou, An approach to the degree-constrained minimum spanning tree problem using genetic algorithm, Report No. ISE95-3, Ashikaga Institute of Technology, Ashikaga, Japan, 1995. [6] F. Croce, R. Tadei, G. Volta, A genetic algorithm for the job shop problem, Computers and Operations Research 22 (1995) 15–24. [7] C. Palmer, An approach to a problem in network design using genetic algorithms, Ph.D. thesis, Polytechnic University, New York, 1994. [8] M. Gen, Y. Tsujimura, E. Kubota, Solving job-shop scheduling problem using genetic algorithms, in: M. Gen, T. Kobayashi (Eds.), Proceedings of the 16th International Conference on Computers and Industrial Engineering, Ashikaga, Japan, 1994, pp. 576–579. [9] A. Kershenbaum, Telecommunications Network Design algorithms, McGrawHill, New York, 1993. [10] C. Holsapple, V. Jacob, R. Pakath, J. Zaveri, A genetics-based hybrid scheduler for generating static schedules in flexible manufacturing contexts, IEEE Transactions on Systems, Man and Cybernetics 23 (1993) 953–971. [11] H. Fang, P. Ross, D. Corne, A promising genetic algorithm approach to job-shop scheduling, rescheduling, and open-shop scheduling problems, in: S. Forrest (Ed.), Proceedings of the Fifth International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, San Mateo, CA, 1993, pp. 375–382. [12] R. Albrecht, C. Reeves, N. Steele, Artificial Neural Nets and Genetic Algorithms, Springer-Verlag, New York, 1993. [13] R. Nakano, T. Yamada, Conventional genetic algorithms for job-shop problems, in: R. Belew, L. Booker (Eds.), Proceedings of the Fourth International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, San Mateo, CA, 1991, pp. 477–479. [14] J. Hesser, R. Manner, O. Stucky, Optimization of Steiner trees using genetic algorithms, in: J. Schaffer (Ed.), Proceedings of the Third International
[15]
[16] [17]
[18] [19]
[20]
[21]
[22]
[23]
[24]
[25]
489
Conference in Genetic Algorithms, Morgan Kaufmana Publishers, San Mateo, CA, 1989, pp. 231–236. J. Grefenstette, R. Gopal, B. Rosmaita, D. Gucht, Genetic algorithms for the traveling salesman problem, in: J. Grefenstette (Ed.), Proceedings of the First International Conference on Genetic Algorithms, Lawrence Erlbaum Associates, Hillsdale, NJ, 1985, pp. 160–168. I. Shrode, T. Nishida, Y. Namasuya, Stochastic spanning tree problem, Discrete Applied Mathematics 3 (1981) 263–273. H. Peng, X. Ling, Optimal design approach for the plate-fin heat exchangers using neural networks cooperated with genetic algorithms, Applied Thermal Engineering 28 (5-6) (2008) 642–650. G.N. Xie, B. Sunden, Q.W. Wang, Optimization of compact heat exchangers by a genetic algorithm, Applied Thermal Engineering 28 (8-9) (2008) 895–906. L. Gosselin, M. Tye-Gingras, F. Mathieu-Potvin, Review of utilization of genetic algorithms in heat transfer problems, International Journal of Heat and Mass Transfer 52 (9-10) (2009) 2169–2188. J. Guo, L. Cheng, M. Xu, Optimization design of shell-and-tube heat exchanger by entropy generation minimization and genetic algorithm, Applied Thermal Engineering 29 (14-15) (2009) 2954–2960. J.M. Ponce-Ortega, M. Serna-Gonzalez, A. Jimenez-Gutierrez, Use of genetic algorithms for the optimal design of shell-and-tube heat exchangers, Applied Thermal Engineering 29 (2-3) (2009) 203–209. S. Sanaye, H. Hajabdollahi, Thermal-economic multi-objective optimization of plate fin heat exchanger using genetic algorithm, Applied Energy 87 (6) (2010) 1893–1902. M. Yousefi, R. Enayatifar, A.N. Darus, Optimal design of plate-fin heat exchangers by a hybrid evolutionary algorithm, International Communications in Heat and Mass Transfer 39 (2) (2012) 258–263. S. Ghosh, I. Ghosh, D.K. Pratihar, B. Maiti, P.K. Das, Optimum stacking pattern for multi-stream plate-fin heat exchanger through a genetic algorithm, International Journal of Thermal Sciences 50 (2) (2011) 214–224. N. Queipo, R. Devarakonda, J.A.C. Humphrey, Genetic algorithms for thermosciences research-application to the optimized cooling of electronic components, International Journal of Heat and Mass Transfer 37 (6) (1994) 893–908.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
An effective layer pattern optimization model for multi-stream plate-fin heat exchanger using genetic algorithm Min Zhao, Yanzhong Li ⇑ School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 12 September 2012 Received in revised form 21 December 2012 Accepted 23 December 2012 Available online 9 February 2013 Keywords: Multi-stream plate-fin heat exchanger Layer pattern Optimization Genetic algorithm
a b s t r a c t For multi-stream plate-fin heat exchangers (MPFHEs), a crucial factor affecting performance is the layer pattern used to distribute hot and cold streams. However, for the design of MPFHE layer pattern, so far it still remains at semi-qualitative, empirical or trial-and-error stage, lacking efficient approaches to obtain the optimum. In this paper, an effective layer pattern optimization model using genetic algorithm (GA) is developed in detail. It includes the chromosome of binary string that represents hot and cold layers alternatively arranged, the dual fitness functions that act on chromosome individuals alternatively, the constraint that requires the spacing layers between hot and cold streams to be no less than 1 and no greater than 2, the selection of optimization tool and the corresponding setting of GA parameters. When used to optimize the layer pattern of actual MPFHE, this model exhibits well. The average thermal efficiency of exchanger reaches up to 98% of that obtained under the ideal common wall temperature assumption. Furthermore, the study of this paper make it clear the association of previous classical deviation from ideal of layer pattern zigzag curve with MPFHE thermal performance, which is not a strict monotonic decreasing relation as people thought to be, but a multiple vs. multiple relation. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Multi-stream plate-fin heat exchangers (MPFHEs) are widely used in the cryogenic and chemical engineering industries. For them, whether the performance is good or not closely relates with the layer pattern that is used to distribute hot and cold streams. When the layers are not arranged properly, it will lead to the significant reduction of exchanger performance and the accompanying unexpected thermal stresses. In severe cases, even if the exchanger has sufficient backup heat transfer areas, it could not make up for this performance reduction. However, for the design of MPFHE layer pattern, so far it still remains at semi-qualitative, empirical or trial-and-error stage, lacking efficient approaches to obtain the optimum. Fan [1] first proposed the segregated pattern in which a single hot and cold layer are arranged alternatively. It seems that this sandwiched pattern allows hot and cold streams to contact fully so that the exchanger can achieve high thermal efficiency. However, the fact is not that. Firstly, it implies that exchanger ought to have comparable hot and cold layers in total number, which is difficult to meet for most actual MPFHEs. Secondly, such complete alternative arrangement of single hot and cold layers very likely causes the imbalance of heat between layers. ⇑ Corresponding author. Tel.: +86 29 82668725; fax: +86 29 82668789. E-mail addresses: [email protected] (M. Zhao), [email protected] (Y. Li). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.041
In the 15th International Congress of Refrigeration, Suessman and Mansour [2] pointed out that the performance degradation of MPFHE arising from improper layer pattern is because they form and intensify this heat imbalance, i.e., there is an excess of heat accumulated in some layers and an excess of cold accumulated in others, as shown in the zigzag curve of Fig. 1. Ideally, the zigzag curve of layer pattern oscillates between positive and negative values regularly. Therefore, by checking the deviation of its zigzag curve from ideal, a layer pattern can be evaluated roughly. In addition, to study the impact of this heat imbalance on exchanger performance, Suessman and Mansour divided MPFHE into several zones called ‘Local Heat Balance Zone’ along the layer stacking direction. According to the Fourier theorem, if the less the number of layers contained in these zones is, the more uniform the transverse wall temperature of exchanger will be, and the less the performance of exchanger will be affected. This is the wellknown Local Heat Balance principle which has been playing an important role in guiding MPFHE layer arrangement today. However, because MPFHE usually has up to 100 or more layers and therefore the corresponding possible layer patterns could be numerous (i.e., combinatorial explosion), it is impractical for us to check all the possible layer patterns’ zigzag curve deviation to find the optimum. Also, because of this combinatorial explosion feature, traditional optimization methods are not suitable to solve the MPFHE layer pattern problem. So till today people have to use the trial-and-error procedure to get the relatively satisfactory layer arrangement. Firstly, according to the experience or the Local Heat
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
481
Nomenclature Cphi Cpcj cj Ei hi l Li _ hi m _ cj m n nb nc nh N Nhi N cj Nmin phi pcj Phi P cj
specific heat of hot stream i specific heat of cold stream j cold stream j, j = 1,2, . . . , nc excess heat load of each hot/cold layer string in layer pattern ring in Eq. (7) hot stream i, i = 1,2, . . . , nh length of chromosome x each line length of the layer pattern chromosome zigzag curve, i = 1,2, . . . , n mass flowrate of hot stream i mass flowrate of cold stream j number of xi = 1 in the chromosome x number of binary bits that represents a layer in the chromosome of MPFHE layer pattern number of different cold streams in a MPFHE number of different hot streams in a MPFHE total number of layers in a MPFHE number of layers of hot stream i number of layers of cold stream j Pnh Pnc minimum of N and i.e., j¼1 N cj , i¼1 hi P Pnc nh in Fig. 2 N min ¼ min N ; N c h j j¼1 i i¼1 outlet pressure of hot stream i outlet pressure of cold stream j inlet pressure of hot stream i inlet pressure of cold stream j
Balance principle give the layer pre-arrangement, and then through the simulation means check whether the performance of MPFHE achieves design requirements under this layer pattern. If not, adjust it and repeat the above procedure till the required performance is met. Obviously, this design method is not only timeand labor-consuming, but also could not ensure the quality of final results. Recently in solving combinatorial explosion optimization problems of real world, genetic algorithm (GA) has got wide applications owing to its small dependence on the studied problems and powerful and robust searching ability [3–16]. Many authors have also used GA to the field of heat exchanger optimization design [17–23]. For example, Ponce-Ortega et al. [21] use it to optimize shell-and-tube heat exchanger parameters such as the number of tube-passes, the diameter and pitch of tube, the baffle cut, etc. Xie et al. [18] use it to optimize the structure size of plate-fin heat
Fig. 1. Zigzag curve of a multi-stream plate-fin heat exchanger.
P1nh ; P2nh qhi qcj
number of permutation from nh elements to pick up 1 and 2, respectively heat load per layer of hot stream i heat load per layer of cold stream j
Q string ; Q string heat load of hot or cold layer string i, ci hi i ¼ 1; 2; . . . ; Nmin Si vertical coordinate (i.e., cumulative heat load) of each bit of xi = 1 in the layer pattern chromosome zigzag curve, i = 1,2, . . . , n Smid vertical coordinate (i.e., cumulative heat load) of midi point of each line in the layer pattern chromosome zigzag curve, i = 1,2, . . . , n Smid arithmetic-mean of Smid in Eq. (6) i t hi outlet temperature of hot stream i outlet temperature of cold stream j t cj T hi inlet temperature of hot stream i T cj inlet temperature of cold stream j weighting in Eq. (6), i = 1,2, . . . , n wi Wi weighting in Eq. (7), i ¼ 1; 2; . . . ; N min x = [x1, x2, . . . , xl] chromosome of MPFHE layer pattern xi bit of chromosome x, i = 1,2, . . . , l Greek symbols e allowable tolerance
exchanger, etc. However, people seldom use GA to optimize the layer pattern of MPFHE. So far only Ghosh et al. [24] have done the related work. In their GA model, the optimization objective is to maximize MPFHE heat transfer rate estimated by their own rating program. The layer pattern chromosome is adopted sequential coding, i.e., from left to right each layer of the pattern is in turn represented by a certain fixed number of bits, nb. When the total number of layers is N, a string of nb N bits joined together will represent the whole layer pattern. For the value of nb, authors give two methods N to determine, nb ¼ log2 and nb = N 1. The former is used for some specific cases as a means of verification, and the latter is applied for all tested cases. For the main GA parameters, authors select tournament selection, uniform crossover and uniform mutation, respectively. This model obtains satisfactory results when used to optimize the layer pattern which is composed of 3–8 different fluid stream layers. However, their tested cases are far from the actual MPFHE layer pattern optimization situation, in which each stream has their own number of layers constraint, both number and type of individuals in the GA population pool increase dramatically as a result of multiple streams and tens or hundreds of layers in exchanger, single and cascade arrangement are mixed in a layer pattern, etc. all these together increase the difficulty of GA to meet constraints and obtain good optimal results greatly. This paper is committed to solving the above practical problems using special chromosome coding, fitness function and constraints so that the better layer pattern solution can be obtained. An important feature of our GA model is the use of dual fitness functions. During the optimization process, it is found that the minimization of the deviation of zigzag curve from ideal is not enough. Its optimal result’s MPFHE thermal efficiency is only 92.4% on the average. When another upgrade fitness function is combined with it to alternatively optimize the layer pattern individuals (see Sections 2 and 3 of this paper), the optimal result of GA has got an evident improvement, whose MPFHE thermal efficiency has increased from 92.4% up to 98% and some even close to 99% (98.7%).
482
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
Another important feature of our model is that the related heat transfer calculations (by MUSE software) are not embedded into genetic algorithm as a part of its fitness function and therefore the computational time is reduced greatly [19,25]. In addition, through the study of this paper, the relation of previous classical deviation from ideal of layer pattern zigzag curve with MPFHE thermal performance is made clear, which, on the whole, is that the heat transfer rate of MPFHE increases with the deviation decreasing. But they do not follow a strict monotonic decreasing relation as people thought to be, but a multiple vs. multiple relation. Below let us give the detailed GA optimization model first, and then case study is followed and results are discussed. Finally, the conclusions are drawn. 2. Modelling 2.1. Problem presentation Suppose there is a MPFHE in which nh hot streams exchange heat with nc cold streams. For hot stream hi (i = 1,2, . . . , nh), its flowrate, inlet/outlet temperature and pressure, and number of layers _ hi , T hi , t hi , Phi , ph , N hi , respectively. For cold stream cj are m i (j = 1,2, . . . , nc), its flowrate, inlet/outlet temperature and pressure, _ cj , T cj , t cj , P cj , pc , N cj , respectively. and number of layers are m j Now the problem is how to arrange these hot and cold layers, Pnh Pn c i.e., N ¼ i¼1 N hi þ j¼1 N cj , to make MPFHE heat transfer rate maximal. 2.2. Genetic algorithm optimization model To solve the above optimization problem using GA, it is required to encode the layer pattern solution first. It is natural for us to think of sequential coding [24]. However, with the number of layers increasing, this coding will become extremely difficult and cumbersome in treating objective function and constraints. So we adopt the following chromosome of binary string x = [x1, x2, . . . , xl] to represent the layer pattern. Encoding of chromosome Firstly, according to the above optimization conditions calculate Pnh the total number of hot and cold layers of MPFHE, i¼1 N hi and Pnc Pnh Pnc j¼1 N cj . If j¼1 N cj , the first nh bits of binary string i¼1 N hi > x = [x1, x2, . . . , xl] will represent hot stream hi, i = 1,2, . . . , nh, respectively. Then the followed nc bits will represent cold stream cj, j = 1,2, . . . , nc, respectively. Next, the followed nh bits, i.e., xnh þnc þ1 ; xnh þnc þ2 ; . . . ; x2nh þnc , will again represent hot stream hi (i = 1,2, . . . , nh) to form the substring of h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh that represents hot streams surrounding cold Pc streams. Finally, this substring repeats nj¼1 N cj times to form the entire chromosome x = [x1,x2, . . . , xl]. The above forming process can be illustrated as follows,
Clearly, the length of this layer pattern chromosome is Pc Pn h Pnc l ¼ ð2nh þ nc Þ nj¼1 N cj . Similarly, when j¼1 N cj , the i¼1 N hi 6 above substring will change from h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh to c1 c2 . . . cnc h1 h2 . . . hnh c1 c2 . . . cnc , and its repeating time will Pnc Pn h change from pattern j¼1 N cj to i¼1 N hi . So the length of layer Pc chromosome x will change from l ¼ ð2nh þ nc Þ nj¼1 N cj to Pnh l ¼ ð2nc þ nh Þ i¼1 N hi . In the above layer pattern chromosome x = [x1,x2, . . . , xl], xi = 1 (i = 1,2, . . . , l) means the stream represented is selected, while xi = 0 means the corresponding stream is not selected. Therefore, orderly listing all the streams of xi = 1 in the chromosome can obtain the layer pattern solution. For example, when Pnc Pnh j¼1 N cj , the layer pattern of chromosome x = [10 . . . i¼1 N hi > 110 . . . 010 . . . 001 . . . 110 . . . 111 . . . 0 . . . . . . . . . 10 . . . 000 . . . 110 . . . .0] is h1 . . . hnh c1 . . . h1 . . . h2 . . . hnh c1 . . . cnc h1 h2 . . . . . . h1 . . . . . . cnc h1 , where means no stream on this position. For such chromosome coding, people may question the compoPn h Pc sition of substring. For example, when i¼1 N hi > nj¼1 N cj , why use the substring of h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh instead of h1 h2 . . . hnh c1 c2 . . . cnc which can also form hot streams surrounding cold streams in the chromosome. The simplest reason is that the substring of h1 h2 . . . hnh c1 c2 . . . cnc h1 h2 . . . hnh forms the double string of h1 h2 . . . hnh between two adjacent strings of c1 c2 . . . cnc in the chromosome, which can provide all the possible combination for any two hot streams, while substring h1 h2 . . . hnh c1 c2 . . . cnc could not. Of course, such double string of h1 h2 . . . hnh could not offer all the possible combination for any greater than two hot streams. However, because these multi-fold arrangements hinder the hot stream in the inner layer effectively giving off the heat to the cold streams, they are to be avoided as far as possible in the actual layer pattern of MPFHE. As for the cold stream cj, because Pnc Pnh j¼1 N cj , it is not necessary to use double c1 c2 . . . cnc to i¼1 N hi > obtain all the possible combinations for any two cold streams. For the double string h1 h2 . . . hnh , people may further question ðP 1n þP 2n Þ
why not use shorter bits, for example, nb ¼ log2 h h , to represent all the possible combinations of no greater than two hot streams, where P1nh and P 2nh are the number of permutation from nh elements to pick up 1 and 2, respectively. This is because, first of all, the vaðP1n þP 2n Þ
lue of log2 h h is not always integer and therefore illegal chromosomes inevitably exist in the population of GA. These illegal chromosomes can not be evaluated normally by GA because they have no actual meaning. So additional strategies are needed to consider to get rid of them or convert them into qualified ones. Secondly, for this shorter bits coding, the corresponding program of realizing fitness function and constraints is much more complicated. Thirdly, our case study shows that the average time of each GA running of present coding is only 1–2 min when used to optimize the layer pattern of MPFHE with 108 layers, 3 hot fluids and 4 cold fluids, whose length of layer pattern chromosome is 429 bits and the population size of GA is 2000–4000 (see Section 3 of Case study in this paper). Moreover, with the number of layers or population size of GA decreasing, the running time of GA will be further reduced. Fitness Function After encoding, it is required to define the fitness function. As mentioned before, dual fitness functions are adopted in this GA model. The first one is called main whose objective is to minimize the deviation from zero of each line midpoint in the layer pattern zigzag curve, whose weighting is the line length. For a certain layer
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
(
pattern chromosome x = [x1,x2, . . . , xl], its calculation procedure is as follows: 1. According to the known optimization conditions, calculate the heat load per layer of each stream in MPFHE.
_ h Cph ðt hi T hi Þ m i qhi ¼ i ; i ¼ 1; 2; . . . ; nh N hi _ cj Cpc ðtcj T cj Þ m j qcj ¼ ; j ¼ 1; 2; . . . ; nc N cj
where qhi is of hot stream hi whose sign is ‘‘’’, while qcj is of cold stream cj whose sign is ‘‘+’’. 2. In terms of the following program, calculate the vertical coordinate (i.e., cumulative heat load) of each bit of xi = 1 in the layer pattern chromosome zigzag curve, Si, i = 1,2, . . . , n, where n is the total number of xi = 1 in the chromosome.
i ¼ 0; % counter S0 ¼ 0; % the origin of zigzag curve For j ¼ 1; 2; . . . ; l If xj ¼ 1
ð3Þ
i ¼ i þ 1; Si ¼ 0; % initialize the accumulator For k ¼ 1; 2; . . . ; j Si ¼ Si þ qxk ;
where qxk ¼ qhi or qcj of Step 1 depending on the position of xk in the chromosome. 3. Calculate each line length of the above layer pattern chromosome zigzag curve and their midpoints’ vertical coordinate, Li and Smid ; i ¼ 1; 2; . . . ; n. i
Li ¼ jSi Si1 j;
¼0 Smid i
ð4Þ
i¼1
ð5Þ
4. Finally, with the Li and Smid of Step 3, the main fitness function i can be expressed as,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n 2 u 1 X min:fmain ðxÞ ¼ t Smid Smid wi i x n 1 i¼1
ð1Þ ð2Þ
Smid ¼ 12 ðSi1 þ Si Þ i – 1 i
483
ð6Þ
where
Smid ¼ wi ¼
n 1 X Smid n 1 i¼1 i
1 Li n X Li i¼1
Now let us discuss the second fitness function, upgrade fitness function, whose definition is related to a layer pattern ring which is formed by linking both ends of layer pattern chromosome. When Pnc Pnh j¼1 N cj , the corresponding layer pattern ring is as shown i¼1 N hi > in Fig. 2. From the figure it can be seen that the double hot layer string of h1 h2 . . . hnh and the cold layer string of c1 c2 . . . cnc are alterP P nc nh natively arranged for N min ¼ min times. Simij¼1 N cj i¼1 N hi ; Pnc Pnh larly, when i¼1 N hi 6 j¼1 N cj , the layer pattern ring will turn into the double cold layer string of c1 c2 . . . cnc and the hot layer string being alternatively arranged for of h1 h2 . . . hnh P Pnc nh times. Nmin ¼ min j¼1 N cj i¼1 N hi ; P nc Pnh P nc P nh Ideally, whether for j¼1 N cj case or j¼1 N cj i¼1 N hi > i¼1 N hi 6 case, every hot/cold layer string in the layer pattern ring is in thermal equilibrium with half of two cold/hot layer strings on its both sides. However, in fact the deviation from ideal always exists. The objective of upgrade fitness function is to minimize this deviation.
Fig. 2. Schematic of layer pattern ring when
Pnh
i¼1 N hi
>
Pnc
j¼1 N cj .
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Pnh P nc Pnh Pnc For both j¼1 N cj and j¼1 N cj cases, the i¼1 N hi > i¼1 N hi 6 expression of upgrade fitness function can be unified as,
string
Qc
W i ¼ P i string at each cold layer string as analytical object. As to Qc
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uN min uX min fupgrade ðxÞ ¼ t ðEi Þ2 W i x
each hot layer string of layer pattern ring as analytical object or
ð7Þ
i¼1
where Ei is the excess heat load of each hot or cold layer string with half of its adjacent two cold or hot layer strings, Appendix A gives Ei detailed calculation procedure for layer pattern chromosome x, in which the heat load of each hot and cold layer string, Q string and hi Q
Q string , are calculated in Appendix B. The weighting W i ¼ P ci
string hi string Qh i
at
i
which layer strings as analytical objects, it will depend on the MPFHE to be optimized. For example, for the MPFHE in Case study of this paper, the effect of upgrade fitness function is very apparent when cold layer strings as study objects, while that of hot layer strings is trivial. Fig. 3 shows GA optimization procedure with the above two fitness functions. Firstly, under the main fitness function GA obtains the first optimal solution from initial population generated randomly, and then it is put into MUSE software to check whether the thermal performance of MPFHE arrives design requirements. If
Fig. 3. GA optimization procedure with dual fitness functions.
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
485
Table 1 Setting of GA parameters on ga solver of Matlab 2010b. Item
Value or function name
Population size
800–2000 (Main fitness function) 2000–4000 (Upgrade fitness function) Uniform Rank Stochastic uniform 2 Scattered 0.8 Uniform 0.01 Forward 0.2 20 100 Inf Inf 50 Inf 106
Creation function Scaling function Selection function Elite count Crossover function Crossover fraction Mutation function Mutation rate Migration direction Migration fraction Migration interval Stopping generations Stopping time limit Stopping fitness limit Stopping stall generations Stopping stall time limit Stopping function tolerance
not, GA will use this solution as initial population to go on the next optimization process under the upgrade fitness function. Similarly, the second obtained result is sent into MUSE software to test the MPFHE performance. If the performance is still not satisfactory, GA will again use the second optimal result as initial population to continue optimization process under the main fitness function. So, like this repeat the above alternative procedure of dual fitness functions on the chromosome individuals until the satisfactory solution is obtained. In addition, because GA is a probability algorithm, its optimal solution is often inconsistent each time even under the same conditions. Therefore, to obtain the steady solution of each fitness function in the above GA optimization process, the following strategy is adopted. Firstly, GA uses the initial population generated randomly to obtain the first optimal solution, and then it uses this solution as initial population to optimize again to obtain the second optimal solution. If the difference of these two optimal solutions’ MPFHE thermal performance is greater than the allowable tolerance e, it will repeat the above procedure until the tolerance is met. Constraints Firstly, the demanded layer number of each stream in MPFHE should be met. Secondly, to ensure the efficiency of exchanger, the spacing layers between hot and cold streams should be 62 and simultaneously P1. Of course, this constraint premise is that the ratio of the total number of hot and cold layers is P1:2 or 62:1. Otherwise, it will not be satisfied for ever. Thirdly, the number of hot or cold layers on both ends of MPFHE is required to be 1. Appendix C gives the corresponding expressions of constraints above. Optimization tool and Setting of GA parameters The ga solver of Matlab 2010b is selected to perform the above optimization strategy. Table 1 gives the setting of GA parameters in the optimization process. 3. Case study and results discussion Below let us see the effects of the above optimization model through a typical example.
Fig. 4. Schematic of a practical MPFHE in air separation plant.
This is a MPFHE used in the air separation plant. Fig. 4 shows the schematic, in which three hot streams No. 1–3 are, respectively, high-, middle-, and low-pressure air, which flow from the top of exchanger to the bottom, and four cold streams, i.e., lowpressure nitrogen No. 4, low-pressure waste nitrogen No. 5, lowpressure oxygen No. 6, and middle-pressure liquid oxygen No. 7, flow from the bottom of exchanger to the top to exchange heat with the above hot streams. Table 2 lists the layer pattern optimization conditions. According to these conditions, the heat load per layer of each hot and cold stream, qhi ði ¼ 1; 2; 3Þ and qcj ðj ¼ 1; 2; 3; 4Þ, are figured out listed in the last row of the table. It can be seen that they are different much from each other. So if the layers are not arranged properly, the heat imbalance between layers will be easy to form and intensify, thereby leading to the performance of MPFHE degrading greatly. Therefore, for this MPFHE obtaining the optimal layer pattern is especially significant. Fig. 5A–D gives four representative results obtained by the present GA optimization model, in which the serial number of GA running on the horizontal axis corresponds to the times of the main and the upgrade fitness function acting on the layer pattern chromosome individual alternatively, the discontinuous points on the top is the value change of the main and the upgrade fitness function, and the bottom curve is the corresponding variation of resulting MPFHE thermal efficiency. Obviously, with the times increasing of dual fitness function alternatively acting on the layer pattern chromosome individuals the main and the upgrade fitness function value both decrease till the change tends small. While for the corresponding MPFHE thermal efficiency, some increase slowly first
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Table 2 Optimization conditions of MPFHE layer pattern.a Items
Units
_i Flowrate, m
kg h K K MPa MPa
Inlet temperature Ti Outlet temperature ti Inlet pressure Pi Outlet pressure pi No. of layers Fin geometry Fin type Fin height Fin thickness Fin frequency Fin porosity Fin serration length Heat load/layer a
mm mm m1 mm kW
1
Stream no. 1
2
3
4
5
6
24161
22611
21319
27754
23622
92.88
14285
313.0 110.0 5.485 5.445 8
313.0 158.0 2.456 2.440 8
287.0 102.0 0.580 0.567 23
96.0 309.7 0.129 0.114 34
96.0 309.7 0.127 0.114 30
93.5 309.7 0.137 0.128 1
96.0 309.7 3.140 3.1 4
Perforated 6.5 0.5 714.28 0.043
Serrated 6.5 0.3 714.28
Serrated 6.5 0.2 714.28
Serrated 9.5 0.2 714.28
Serrated 9.5 0.2 714.28
Serrated 9.5 0.2 714.28
Perforated 6.5 0.2 714.28 0.05
3 133.00
3 49.00
3 +49.95
3 +45.72
3 +4.96
301.70
7
+393.93
The sign of ‘‘’’ in the last low represents the released heat load per layer from hot stream, and ‘‘+’’ represents the absorbed heat load per layer by cold stream.
Fig. 5. Four representative optimal results of present GA model.
and then increase fast such as Fig. 5A, some increase gradually such as Fig. 5B, and so on. However, no matter what their optimization trail is, their final results can always reach 98% of MPFHE thermal efficiency through several times of main and upgrade fitness function alternatively working. This strongly confirms the effectiveness of present GA optimization model.
From the figure it can be also observed such points which have smaller fitness values but their MPFHE thermal efficiencies are not higher as expected. So they are called retrograde points. Their emergence has broken our conventional concept from Suessman and Mansour in optimizing MPFHE layer pattern. From Fig. 5D, it can also be seen that the retrograde degree of the
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M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
Fig. 6. MPFHE heat transfer rate vs. fitness function values.
upgrade fitness function is severer than that of the main fitness function. To find the relation of MPFHE thermal performance with the main and upgrade fitness function, MPFHE heat transfer rate vs. the values of fitness function is plotted in Fig. 6. From the figure it can be clearly seen that the relation of MPFHE heat transfer rate with each fitness function is multiple vs. multiple, that is, for a certain MPFHE heat transfer rate, there are several main or upgrade fitness function values to correspond, and vice versa. However, there is no denying that on the whole the heat transfer rate of MPFHE is increasing with the decrease of the values of main or upgrade fitness function. And the degree of MPFHE heat transfer rate increasing with the decrease of the main fitness function is faster than that with the decrease of upgrade fitness function, especially in the range of large fitness function values. With the value of fitness function decreasing, this difference will reduce till they coincide, i.e., ultimately the heat transfer rate of MPFHE achieves the ideal value under the common wall temperature condition.
Appendix A. Calculation procedure of Ei for the layer pattern chromosome x = [x1,x2, . . . , xl] Pnh Pc When i¼1 N hi > nj¼1 N cj for each hot layer string, the excess heat load Ei is calculated as, For i ¼ 1; 2; . . . ; If i = 1
0
Ei ¼ Q string hi
1
B string C string C 0:5B nc @Q c1 þ Q c P A Nc
else Ei ¼
Q string hi
for each cold layer string, the excess heat load Ei is calculated as, nc P
N cj
j¼1
If i ¼
nc P
N cj
j¼1
0
1
B string string C C Ei ¼ Q string 0:5B ci @Q h1 þ Q h P A nc Nc
else Ei ¼ Q string ci
j
j¼1 string 0:5 Q h þ Q string h i
iþ1
end end Pc Pnh N hi 6 nj¼1 N cj When i¼1 for each hot layer string, the excess heat load Ei is calculated as, For i ¼ 1; 2; . . . ;
nh P j¼1
If i ¼
nh P j¼1
N hj
N hj 0
1
B string C string C Ei ¼ Q string 0:5B nh @Q c1 þ Q c P A hi j¼1
This research was supported by the Chinese State Science and Technology Support Program (Grant No. 2012BAA08B03).
j
j¼1 0:5 Q string þ Q string ci1 ci
end end
4. Conclusions
Acknowledgements
N cj
j¼1
For i ¼ 1; 2; . . . ;
In this paper, in order to effectively obtain the optimal layer pattern of MPFHE, a novel GA optimization model is developed in detail. It includes the chromosome of binary string that represents hot and cold layers alternatively arranged, the dual fitness functions that act on chromosome individuals alternatively, the constraint that requires the space layers between hot and cold streams to be no less than 1 and no greater than 2, and the selection of optimization tool as well as the corresponding setting of GA parameters. This model exhibits well when used to optimize the layer pattern of actual MPFHE. The average thermal efficiency of exchanger reaches up to 98%. In addition, through the study of this paper, the relations of previous classical deviation of layer pattern zigzag curve from ideal with MPFHE performance is made clear, which is not a strict monotonic decreasing relation as people thought to be, but a multiple vs. multiple relation.
nc P
Nh j
(continued on next page)
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M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489
else Ei ¼ Q string 0:5 Q string þ Q string ci ciþ1 h
nc P
N cj
X
j¼1
i
end end
xk1 þði1Þð2nh þnc Þ þ xnh þnc þk1 þði1Þð2nh þnc Þ ¼ Nhk
1
i¼1 nc P
for each cold layer string, the excess heat load Ei is calculated as, nh P
For i ¼ 1; 2; . . . ;
j¼1
If i = 1 Ei ¼
0
Q string ci
X
xk2 þnh þði1Þð2nh þnc Þ ¼ Nck
j¼1 else Ei ¼ Q string 0:5 Q string þ Q string ci h h i1
2
j
xk1 þnc þði1Þð2nc þnh Þ ¼ Nhk1
i¼1 nh P
Nh j
when
nh nc X X N hi 6 N cj i¼1
!
j¼1
Nh
X
j¼1
i
end end
j¼1
Nh
X
j¼1
B string string C C 0:5B @Q h1 þ Q h P A nh
i¼1
!
j¼1
nh P
1
nh nc X X Nhi > N cj
N cj
i¼1
N hj
when
j
xk2 þði1Þð2nc þnh Þ þ xnc þnh þk2 þði1Þð2nc þnh Þ ¼ N ck2
i¼1
and Q string for the Appendix B. Calculation procedure of Q string ci hi layer pattern chromosome x = [x1, x2, . . . , xl]
where k1 = 1,2, . . . , nh;k2 = 1,2, . . . , nc. For the second and the third constraints, nh nc P P N hi > N cj when i¼1
j¼1
nc P
For i ¼ 1; 2; . . . ; Pnc
Pnh
When i¼1 N hi > j¼1 N cj , each hot and cold layer string and Q string are calculated as, ci nc P
For i ¼ 1; 2; . . . ;
Q string hi
If i = 1 nh P xm ¼ 1; m¼1
N cj
j¼1
If i = 1 l¼ð2nh þnc Þ
Q string ¼ h i
nh P m¼1
P
qxm xm þ
nc P
l¼ð2nh þnc Þ
nc P
X j¼1
N cj
N cj
xm ¼ 1;
j¼1
qxm xm ;
m¼lnh þ1
else Q string ¼ hi
m¼lnh þ1 3nP h þnc
else 1 6
3nP h þnc m¼nh þnc þ1
qxmþði2Þð2n
h þnc Þ
m¼nh þnc þ1
xmþði2Þð2nh þnc Þ ;
nhP þnc
16
end
m¼nh þ1
Q string ¼ ci
nhP þnc m¼nh þ1
N cj
j¼1
qxmþði1Þð2n
h þnc Þ
xmþði1Þð2nh þnc Þ ;
xmþði2Þð2nh þnc Þ 6 2;
xmþði1Þð2nh þnc Þ 6 2;
end End
end When
Pnh
i¼1 N hi
6
Pn c
j¼1 N cj ,
and Q string are calculated as, ci
each hot and cold layer string Q string hi
when
nh P i¼1
nc P
N hi 6
j¼1
For i ¼ 1; 2; . . . ;
N cj
nh P j¼1
nh P
For i ¼ 1; 2; . . . ;
j¼1
N hj
If i = 1 nc P xm ¼ 1;
N hj
m¼1
If i = 1 l¼ð2nc þnh Þ
¼ Q string ci
nc P m¼1
P
qxm xm þ
nh P j¼1
m¼lnc þ1
Nh
j
l¼ð2nc þnh Þ
qxm xm ;
else
nh P
X j¼1
j
xm ¼ 1;
m¼lnc þ1
Q string ¼ ci
nhP þ3nc m¼nh þnc þ1
qxmþði2Þðn
h þ2nc Þ
xmþði2Þðnh þ2nc Þ ;
else 1 6
¼
nhP þnc m¼nc þ1
qxmþði1Þðn
h þ2nc Þ
xmþði1Þðnh þ2nc Þ ;
End
Appendix C. Constraints expression for the layer pattern chromosome x = [x1,x2, . . . , xl] For the first constraint,
nhP þ3nc m¼nh þnc þ1
end Q string hi
Nh
nhP þnc
16
m¼nc þ1
xmþði2Þðnh þ2nc Þ 6 2;
xmþði1Þðnh þ2nc Þ 6 2;
end End
References [1] Y.N. Fan, How to design plate-fin heat exchangers, Hydrocarbon Processing 45 (11) (1966) 211–217.
M. Zhao, Y. Li / International Journal of Heat and Mass Transfer 60 (2013) 480–489 [2] W. Suessman, A. Mansour, passage arrangement in plate-fin exchanges, in: Proceedings of 15th International Congress of Refrigeration, 1979, pp. 421– 429. [3] M. Sakawa, K. Kato, T. Mori, Flexible scheduling in a machining center through genetic algorithms, Comptuters and Industrial Engineering 30 (4) (1996) 931– 940. [4] D. Tate, A. Smith, A genetic approach to the quadratic assignment problem, Computers and Operations Research 22 (1995) 73–83. [5] M. Gen, G. Zhou, An approach to the degree-constrained minimum spanning tree problem using genetic algorithm, Report No. ISE95-3, Ashikaga Institute of Technology, Ashikaga, Japan, 1995. [6] F. Croce, R. Tadei, G. Volta, A genetic algorithm for the job shop problem, Computers and Operations Research 22 (1995) 15–24. [7] C. Palmer, An approach to a problem in network design using genetic algorithms, Ph.D. thesis, Polytechnic University, New York, 1994. [8] M. Gen, Y. Tsujimura, E. Kubota, Solving job-shop scheduling problem using genetic algorithms, in: M. Gen, T. Kobayashi (Eds.), Proceedings of the 16th International Conference on Computers and Industrial Engineering, Ashikaga, Japan, 1994, pp. 576–579. [9] A. Kershenbaum, Telecommunications Network Design algorithms, McGrawHill, New York, 1993. [10] C. Holsapple, V. Jacob, R. Pakath, J. Zaveri, A genetics-based hybrid scheduler for generating static schedules in flexible manufacturing contexts, IEEE Transactions on Systems, Man and Cybernetics 23 (1993) 953–971. [11] H. Fang, P. Ross, D. Corne, A promising genetic algorithm approach to job-shop scheduling, rescheduling, and open-shop scheduling problems, in: S. Forrest (Ed.), Proceedings of the Fifth International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, San Mateo, CA, 1993, pp. 375–382. [12] R. Albrecht, C. Reeves, N. Steele, Artificial Neural Nets and Genetic Algorithms, Springer-Verlag, New York, 1993. [13] R. Nakano, T. Yamada, Conventional genetic algorithms for job-shop problems, in: R. Belew, L. Booker (Eds.), Proceedings of the Fourth International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, San Mateo, CA, 1991, pp. 477–479. [14] J. Hesser, R. Manner, O. Stucky, Optimization of Steiner trees using genetic algorithms, in: J. Schaffer (Ed.), Proceedings of the Third International
[15]
[16] [17]
[18] [19]
[20]
[21]
[22]
[23]
[24]
[25]
489
Conference in Genetic Algorithms, Morgan Kaufmana Publishers, San Mateo, CA, 1989, pp. 231–236. J. Grefenstette, R. Gopal, B. Rosmaita, D. Gucht, Genetic algorithms for the traveling salesman problem, in: J. Grefenstette (Ed.), Proceedings of the First International Conference on Genetic Algorithms, Lawrence Erlbaum Associates, Hillsdale, NJ, 1985, pp. 160–168. I. Shrode, T. Nishida, Y. Namasuya, Stochastic spanning tree problem, Discrete Applied Mathematics 3 (1981) 263–273. H. Peng, X. Ling, Optimal design approach for the plate-fin heat exchangers using neural networks cooperated with genetic algorithms, Applied Thermal Engineering 28 (5-6) (2008) 642–650. G.N. Xie, B. Sunden, Q.W. Wang, Optimization of compact heat exchangers by a genetic algorithm, Applied Thermal Engineering 28 (8-9) (2008) 895–906. L. Gosselin, M. Tye-Gingras, F. Mathieu-Potvin, Review of utilization of genetic algorithms in heat transfer problems, International Journal of Heat and Mass Transfer 52 (9-10) (2009) 2169–2188. J. Guo, L. Cheng, M. Xu, Optimization design of shell-and-tube heat exchanger by entropy generation minimization and genetic algorithm, Applied Thermal Engineering 29 (14-15) (2009) 2954–2960. J.M. Ponce-Ortega, M. Serna-Gonzalez, A. Jimenez-Gutierrez, Use of genetic algorithms for the optimal design of shell-and-tube heat exchangers, Applied Thermal Engineering 29 (2-3) (2009) 203–209. S. Sanaye, H. Hajabdollahi, Thermal-economic multi-objective optimization of plate fin heat exchanger using genetic algorithm, Applied Energy 87 (6) (2010) 1893–1902. M. Yousefi, R. Enayatifar, A.N. Darus, Optimal design of plate-fin heat exchangers by a hybrid evolutionary algorithm, International Communications in Heat and Mass Transfer 39 (2) (2012) 258–263. S. Ghosh, I. Ghosh, D.K. Pratihar, B. Maiti, P.K. Das, Optimum stacking pattern for multi-stream plate-fin heat exchanger through a genetic algorithm, International Journal of Thermal Sciences 50 (2) (2011) 214–224. N. Queipo, R. Devarakonda, J.A.C. Humphrey, Genetic algorithms for thermosciences research-application to the optimized cooling of electronic components, International Journal of Heat and Mass Transfer 37 (6) (1994) 893–908.