Design Optimization and Performance Prediction of Compact Heat Exchangers

Transcription

1 CHAPTER 9 Design Optimization and Performance Prediction of Compact Heat Exchangers Q. Wang 1, G. Xie 2 & Bengt Sunden 3 1 State Key Laboratory of Multiphase Flow in Power Engineering, Xi an Jiaotong University, China. 2 School of Mechantronics, Northwestern Polytechnical University, China. 3 Department of Energy Sciences, Lund University, Sweden. Abstract Applications of genetic algorithms (GAs) and artificial neural networks (ANNs) in thermal engineering have received much attention. Compact heat exchangers (CHEs) have special merits of high-performance and compactness and are widely used in various applications of energy and power engineering. This chapter demonstrates optimization design of a plate-fin heat changer and performance prediction of three kinds of tube-fin heat exchangers by a GA and an ANN, respectively. The successful applications of optimization and prediction of CHEs suggest that GAs and ANNs have strong ability of intelligent design and prediction of CHEs. Keywords: ANN; compact heat exchanger; GA; optimization; prediction 1 Introduction Compact heat exchangers (CHEs), including two types of heat exchangers namely, plate-fin types and fin-and-tube tube-fin types, are widely used for gas gas or gas liquid applications as shown in Fig. 1. CHEs are used in many industrial processes in chemical and petroleum engineering, refrigeration and cryogenics, heating and ventilation, air-conditioning heating, ventilation, air-conditioning (HVAC), aeronautics and astronautics, automotive, electric and electronic equipments, etc. CHEs own merits of compactness, small volume, low weight, high effectiveness, and low cost. CHEs have high thermal effectiveness, because some augmented doi: / /009

2 302 EMERGING TOPICS IN HEAT TRANSFER Figure 1: Compact heat exchangers: plate-fin type, tube-fin type. devices such as fins, ribs, dimples are used on primary surfaces to interrupt boundary layer growth due to the disturbance of the flow, and has high thermal conductivity due to the thin thickness of surface walls. Many high-performance augmented structures have been developed and applied to industrial engineering applications successfully. Heat exchanger optimization is an important field and full of challenges. The task of optimization may be considered as a design process, in which any possible candidates will be evaluated based on requirements. Savings of materials or energy, as well as capital cost and operating cost, are common objectives for industrial applications of heat exchangers. By contrast, heat exchanger design involves complex processes, including selection of geometrical parameters and operating dynamic parameters for the design, cost estimation, and optimization. Generally, the design task is a complex trial-and-error process, because all possible values of the geometrical parameters are combined and in turn the designed heat exchanger is evaluated with respect to the conditions of specified requirements, that is, outlet temperature, heat duty, and pressure drop. In this sense, there is always the possibility that the designed results are not the optimum. Therefore, researchers try to optimize thermal devices by means of optimization techniques, and many interesting and well-organized works have been presented. In recent years, much additional literature has appeared on optimization of heat exchangers, for example [1 13]. In recent years, applications of various genetic algorithms (GAs) in thermal engineering have received much attention for solving real-world problems [14]. Applications of GAs into heat exchangers optimizations have suggested that GAs have a strong ability of search and combined optimization and can successfully optimize and predict thermal problems. Thus, applications of GAs in the field of thermal engineering are new challenges. At this point, the GA technique may be used in the geometrical optimization of heat exchangers in order to obtain optimal results under specified design objectives within the allowable pressure drops. Several examples of optimizations of heat exchangers are presented in [15 25].

3 DESIGN OPTIMIZATION AND PERFORMANCE PREDICTION OF COMPACT HEAT EXCHANGERS 303 By contrast, to evaluate the heat exchanger performances, efficient and accurate methods for prediction of heat transfer and pressure drop have to be developed. Artificial neural networks (ANNs) have been developed for about two decades and are now widely used in various application areas such as performance prediction, pattern recognition, system identification, and dynamic control and so on, because ANNs provide better and more reasonable solutions. ANNs have more attractive advantages. It can approximate any continuous or nonlinear function using certain network configuration. It can be used to learn complex nonlinear relationship from a set of associated input output vectors. It can be implemented to dynamically simulate and control unknown or uncertain processes. In recent years, ANNs have been used in thermal systems for heat transfer analysis, performance prediction, and dynamic control of heat exchangers [26 50]. This chapter demonstrates optimization design and performance prediction of CHEs using a GA and an ANN, respectively. Such demonstrations intend to encourage researchers to extend relevant works through GAs and ANNs. 2 Design Optimization of a Plate-Fin Heat Exchanger In this section, geometrical optimization of a plate-fin under different objectives using GA is described. More details can be found in [23]. 2.1 Physical model A microturbine recuperator is considered [51]. A typical core of a plate-fin heat exchanger is shown in Fig. 2. For such a plate-fin compact heat exchanger (PFCHE), two streams are in cross-flow with different mass flow rates under specified performances, for example, given duty, allowable drops. There are many geometrical parameters, which may be taken as optimization variables such as shape length L 1, L 2, L 3, fin pitch fin density F p, fin height plate height F s, fin thickness d f, and number of flow arrangements N. In this work, the thickness of the plate and fins are assumed to be constant, and thus are not to be optimized as they satisfy initial requirements, and almost do not affect the performance of the heat exchanger. The Figure 2: A plate-fin heat exchanger core.

4 304 EMERGING TOPICS IN HEAT TRANSFER materials of plate and fin are aluminum with a thermal conductivity of 190 W/m K, density of 2,790 kg/m 3. Plain triangular fins and offset strip fins are used on each side, and the fin parameters are taken from Kays and London [52]. 2.2 Brief description of GA and optimization process The GA is maintained by a population of parent individuals that represent the latent solutions of a real-world problem. For example, the designer might encode the design parameters into corresponding binary strings, and then all the binary strings are connected into a binary string, which is represented as an individual. A certain number of sets of design parameters accordingly become a population of parent individuals. Figure 3 shows the flow chart of the GA for optimizing heat exchangers. Each individual is assigned a fitness based on how well each individual fits a given environment and is then evaluated by survival of the fitness. Fit individuals go through the process of survival selection, crossover, and mutation, resulting in creating next generation, called child individuals. A new population is therefore formed by selecting good individuals from parent and child individuals. After some generations, the algorithm converges to a best individual, which probably represents the best solution of the given problem. Once the geometrical structure sizes have been obtained by the GA-based individuals decoded, rating performance can be carried out under the specified Figure 3: Flowchart of optimization process.

5 DESIGN OPTIMIZATION AND PERFORMANCE PREDICTION OF COMPACT HEAT EXCHANGERS 305 requirements. The heat exchanger efficiency versus number of transfer units (e-ntu) method is used. For the rating performance of a heat exchanger, the data of heat transfer, j factors, and friction factors, f factors, are required for various Reynolds numbers. This is because in the optimization process, the shape parameters vary as GA searches and combines, in return leading to changing Reynolds numbers, fixed flow rate, while cross-section of passage is changed. Therefore, it is difficult and inconvenient to look for j and f from the figure of the reference as there are many possible combinations in the search process of GA. Fortunately, ahead of GA search relations among j and f versus Re are in-house obtained by curve-fitting experimental data from Kays and London [52]. In this study, a fiveorder polynomial relation is used to correlate the data. Thus, once the relations have been built, the computer-aided search and acquisition can be conducted. 2.3 Optimization results and discussion Minimum annual cost The objective function of minimum annual cost is expressed as Total annual cost = annualized cost of heat transfer area + operating cost of pump/compressor C op = k TAC( xi)= Cin+ Cop (1) n C = C A (1a) el in ΔPV t h t A h + k el ΔPV t h t c (1b) Here C A and k el are the price per unit area and electrical energy, respectively, n and s are the exponent of nonlinear increase with area increase and the hours of operation per year, respectively. DP, V t, and h are pressure drop, volumetric flow rate, and pump/compressor efficiency, respectively. Because in practical applications the heat exchangers are operated under specified requirements, and consumption of pumping power is necessary to transfer the fluid flow through the passages in the heat exchangers, pressure loss/drop will be inevitable. In addition, in industrial applications, some constraints of the pumping power are critical and must be considered. The pressure drop must be below a specified maximum value. Thus, the heat exchanger optimization is a constrained optimization process with the following inequality conditions: ΔPh< ΔPh,max, ΔPc < ΔPc Constraints: xmin< xi < xmax Here, ΔP h,max and ΔP c, max are the maximum allowable pressure drop on the hot and cold sides, respectively. In order to compare with previous works, two such values are set, namely, 0.3 and 2 kpa. x refers to the optimization design variable,max (2)

6 306 EMERGING TOPICS IN HEAT TRANSFER to be optimized. x max and x min refer to the upper and lower bounds of the design variables, respectively. By contrast, for cross-flow heat exchangers, it is economic that the effectiveness e should not be less than a certain value e min, generally say For constrained optimization, when the geometrical sizes, which generated by GA, cannot satisfy the specified performance through a rating routine, a penalty function has to be imposed/conducted on the objective function. In this paper, the step-wise penalty factor is defined as follows: 1 pf = 0 ΔP < ΔP max and e e min otherwise Thus, the fitness of the individuals should be adjusted into following equation: Fitness = pf (TAC max -TAC) (4) C max is a constant assigned to The case of invalid design, that is, minimum fitness is zero, will be removed from the possible combinations, not being updated in the GA evolution. On the contrary, in some applications, the pressure drop of the heat exchanger is not the most critical part of the total pressure drop of the thermal plant. It might well be that the gain by the optimized performance is evident over the degree of penalty by the scaled pressure drop. Therefore, it is interesting that the optimized results are different if the pressure drop constraint is removed. Here the GA optimization with pressure drop constraints is called GA1 while GA optimization without pressure drop constraints is called GA2. The evolution process for minimum cost is shown in Fig. 4a. At the beginning of the evolution process, for GA1 less than 300 generations while for GA2 less than 50 generations, the individuals with higher fitness are saved, and the individuals with small fitness are removed. After certain generations, larger than 700,100 generations, respectively, the differences between every individual are relatively large, in turn the variation of fitness for minimum cost is small, and finally a level-off value is found. Compared with results from the references, the optimized results for minimum cost are listed in Table 1. From the table, it is seen (3) (a) Minimum cost (b) Minimum volume Figure 4: Evolution process of minimum cost and volume.

7 DESIGN OPTIMIZATION AND PERFORMANCE PREDICTION OF COMPACT HEAT EXCHANGERS 307 Table 1: Optimized results of minimum annual cost. Geometrical parameters Performance L 1, m L 2, m L 3, m V, m 3 Cost, $ ΔP h kpa ΔP c, kpa Ref , GA , GA , that under pressure drop constraints the total cost decreases by about 15%, and the total volume also decreases by about 27%, while without pressure drop constraints the total cost decreases by about 16.5%, and the total volume also decreases by about 38%. For GA1 optimization, the allowable pressure drops of the two sides are almost fully utilized. For GA2 optimization, due to the without pressure drop constraints, the air-side pressure drop is beyond/over the allowable pressure drop, increasing by about 25.5%, while the gas-side pressure drop decreases by 20%. Therefore, if the total pressure drop is allowed to slightly exceed the permitted limit in some practical applications, the benefits from the optimized results might be considerable Minimum volume The objective function of minimum volume is expressed as Volume = hot stream flow length cold stream flow length no-flow length V = L1 L2 L3 (5) Fitness = pf*(v max /V) (6) Similar trend of the evolution process for minimum total volume is shown in Fig. 4b. The optimized results for minimum total volume are listed in Table 2. It is seen that with pressure drop constraints, the total volume decreases by about 30%, and the total cost also decreases by about 13.8%, while without pressure drop constraints the total volume decreases by about 49%, and the total cost also decreases by about 13.7%. For GA1 optimization, the allowable pressure drop of air side is almost fully utilized. For GA2 optimization, the pressure drops of both sides are over the allowable pressure drops, increasing by about 22% and 63%. Table 2: Optimized results of minimum volume. Geometrical parameters Performance L 1, m L 2, m L 3, m V, m 3 Cost, $ ΔP h, kpa ΔP c, kpa Ref , GA , GA ,

8 308 EMERGING TOPICS IN HEAT TRANSFER Thus, even if total pressure drop is allowed to slightly exceed the limit in some practical applications, the benefits from the optimized results might not be so considerable Minimum cost and volume When the moderate cost and volume are considered in the optimization process, a multi-objective problem may occur. With respect to multi-objective optimizations, there many advanced methods may exist, and it is possible that there exists one novel algorithm or methodology matching one kind of real-world problem. However, considering the sum of weighted objectives as the total objective function has gained a remarkable popularity in optimization problem, as it is relatively simple to be implemented in engineering applications, and one can adjust weighting factors to represent the degree of importance between all objectives. Thus, in this case the objective function in GA is now set, as follows: Cmax Vmax CV = α1 + α 2 (7) C V Fitness = pf CV (8) The constraints are similarly handled. The sum of the weighting factors is equal to unity, that is, α 1 +α 2 = 1. C max and V max are the scales for cost and volume, respectively, by which C and V are made compatible. This is because C and V have different physical units, and a direct summation will introduce numerical errors, in which the low value V, order of 0.1 will be eated by the high value C, order of The two weighting factors for cost and volume are equal to 0.5, which is CV = 0.5 C max /C V max /V. The optimized results are listed in Table 3. As expected, the results of CV are between C and V. With pressure drop constraints, the total cost decreases by about 14.5%, and the total volume also decreases by about 28%, while without pressure drop constraints the total cost decreases by about 15.2%, and the total volume also decreases by about 46.2%. For GA1 optimization, the allowable pressure drop of the air side is almost fully utilized. For GA2 optimization, the pressure drops of both sides are over the allowable pressure drops, increasing by about 19.5% and 53.2%, respectively. The results of this case can be helpful to a designer in order to complete the optimization of heat Table 3: Optimized results of minimum cost and volume. Geometrical parameters Performance L 1, m L 2, m L 3, m V, m 3 Cost, $ ΔP h, kpa ΔP c, kpa Ref , GA , GA ,

9 DESIGN OPTIMIZATION AND PERFORMANCE PREDICTION OF COMPACT HEAT EXCHANGERS 309 exchangers, in which the designer can change the weighting factors according to the degree of importance between cost and volume. Obviously, the results may be included between the results of two extreme cases: minimum cost and minimum volume. Based on practical requirements and constraints, one can change the upper and lower bounds of the geometrical parameters, and change the weight factors, to achieve the corresponding optimizations. 3 Performance Prediction of Tube-Fin Heat Exchangers In this section, performance prediction of some kinds of tube-fin heat exchangers using ANN is described. More details can be found in [50]. 3.1 Physical model and experimental database Tube-fin heat exchanger is one of the successful improvements of the tubular heat exchanger. As shown in Fig. 1, the hot air or flue gas flows across a finned tube bundle while cold water or refrigerant flows inside the round tubes that are arranged staggered. The heat is transmitted through the tube wall and finned surfaces. Fin patterns are diversified as varying as their geometries. The common types are plain fins, wavy fins, slotted fins, louvered fins, and fins with longitudinal vortex generators longitudinal vortex generators (LVGs). On the other side, the common types of tubes are round, flat, and oval. The heat transfer and pressure drop characteristics on the air side can be obtained by experimental measurements or exact numerical computations. Fortunately, much literature has contributed to tube-fin heat exchangers and established most useful correlations. The fin structures are schematically shown in Fig. 5. Experiments have been conducted to measure convective heat transfer and pressure drop of tube-fin heat exchangers with large number of large diameter tube Figure 5: Fin structures for tube-fin heat exchangers.

10 310 EMERGING TOPICS IN HEAT TRANSFER rows by Tang et al. [53 55] at Xi an Jiaotong University, China. Nine samples of three kinds of fin-and-tube heat exchangers are tested, the types of fins are plain fin, slit fin, fin with longitudinal vortex generations LVGs, and the number of tube rows are six, nine, and twelve, which may not commonly appear in HVAC&R. It should be emphasized that the outside diameter of the tube is 18 mm, which also is not used generally in refrigeration engineering. All tubes and fins are made of copper. The tubes are in staggered arrangement. The thickness of fin is 0.3 mm. Experiments were performed for Reynolds number ranging from 4,000 to 10,000 on the air side where the flow might be considered in transition to turbulent flow. The experimental apparatus and procedures are described in detail in [53 55]. Ninety-six sets of experimental data were obtained and divided into two parts: one is for training data, the rest is for testing data. The typical uncertainties of friction factor and Nusselt number are 8.5% and 6.9%, respectively. A total of 96 sets of data were run in the network, of which 75 sets of experimental data were used to train the network, while the rest of 21 sets of data were used to test the network. Note that 78% of the experimental data were used for training the network. The selection of test data from each heat exchanger may be somewhat arbitrary. However, these data are based on approximate uniform variation of Re and based on total number of data points from each heat exchanger. 3.2 Brief description of ANN Figure 6 illustrates a typical full-connected network configuration. Such an ANN consists of a series of layers with a number of neurons circle points in Fig. 6, which in this chapter is called nodes referring to figure. Each connection between two neurons with a real value is called weight. Neurons are gathered together into a column called a layer. Among various types of ANNs, the feed-forward or multilayer perception neural network is widely used in engineering applications. The input information is propagated forward through the network, while the output error is back propagated through the networks for updating the weights. As shown in Fig. 5, the first layer with six neurons and last layer with two neurons are called input layer and output layer, respectively, while the others in the middle are called hidden layers. The configuration in Fig. 5 has one hidden layer with four neurons and such type is briefly written as in this paper. There are many ways to design and implement ANNs. However, it is difficult to find an optimal network, considering the uniqueness of a real problem. Thus, a priori choice, such as selection of network topology, training algorithm, and network size should be made based on experience in order to keep the task to a manageable proportion. It is a very common way to use the back-propagation (BP) algorithm to train ANNs. The main idea of this algorithm is to minimize the cost function by the steepest descent method to add small changes in the direction of minimization. It simply consists of back-propagating the output errors to the network by modifying the weight matrices, that is, adding a correction weight dw to a synaptic weight w. Varying the learning rate dynamically or using momentum terms can improve the convergence speed. Although the BP algorithm needs long time to converge, the algorithm has gained a remarkable popularity in the neural network community, because it is relatively easy to implement

11 DESIGN OPTIMIZATION AND PERFORMANCE PREDICTION OF COMPACT HEAT EXCHANGERS 311 Figure 6: A fully connected feed-forward neural network. in engineering applications, as well as in thermal and energy applications. In addition, the BP algorithm might provide solutions to large and difficult problems. Thus, in this study the BP is implemented to train the network. 3.3 Prediction of turbulent flow and heat transfer By the aid of searching relatively good configuration for prediction, 10 different ANN configurations were studied, as shown in Table 4. The four evaluation parameters, Er, rms, R, and s are defined by Er = A e A A e p 100% (7) rms = 1 M M i= 1 e A A e A p 2 (8) Note: The error is the maximum value among the errors of the two output variables. N N e 1 1 A R = Ri N = N P (9a) A i= 1 i= 1 s = N 2 ( R R i ) N i= 1 (9b)

12 312 EMERGING TOPICS IN HEAT TRANSFER Table 4: Comparison of errors by different ANN configurations. Configuration Train error Test error Er % rms % R σ Correlations R and s are the maximum values that were determined from R and s of the three output variables, respectively. R reflects the average accuracy of the prediction, while s reflects the scatter of the prediction. Both quantities are important for an assessment of the relative success of the ANN analysis [26, 27]. For three layers, when the number of hidden nodes is increased from 6 to 8, R and s of the former are smaller than those of the latter. For four layers, when the number of hidden nodes of the first hidden layer is increased to 10 and the number of hidden nodes of the second hidden layer is increased to 5, R and s become larger. This indicates that adding more hidden nodes may not improve the predicted results. From Table 2, network own smaller Er and rms than those of ; however, the R and s of the former are larger than those of the latter. At this point, the configurations with four layers have higher prediction accuracy than those with five layers. It is also noted that adding more hidden layers may not make the prediction better. Thus, in this case, configuration is selected for testing, with smallest R = and s = and the maximum relative error is about 4% with most of them being less than 2%. To develop such a tool, the ANN is now trained with the combined database including the previous experimental data used for selecting the ANN and the numerical data by CFD simulations [56, 57]. The prediction is shown in Fig. 7. From the figures, the predicted data are close to the experimental and numerical data. The R and r of the ANN with the specific weights are and Also, the majority of predicted data points are less than 5% from the combined data. Therefore, based on the presented prediction so far, the ANN trained with the combined database of experimental and numerical data appears to be the best available prediction tool for obtaining turbulent flow and heat transfer for heat exchangers with large number of tube rows and large tube diameter. Therefore, for convenient use by engineers or researchers, the weights and biases under the ANN are listed below. By inserting the sets of

13 DESIGN OPTIMIZATION AND PERFORMANCE PREDICTION OF COMPACT HEAT EXCHANGERS 313 Figure 7: Predictions of Nu and f by ANN with turbulent model. weights and biases, the turbulent flow and heat transfer of heat exchangers can be optimistically obtained. 3.4 Prediction of laminar flow and heat transfer The desire to predict laminar convective heat transfer implies the need for updating the above developed ANN a second time. In a similar way as above, using a new database for training and testing the neural network is created by all the databases for turbulent and laminar flow and heat transfer [57, 58]. Totally, 277 sets of data points are obtained. Among these, 190 sets of data points are for training the secondly while the remaining ones are for testing the generality. The prediction of the friction factor and Nusselt number for the heat exchangers by the secondly updated ANN is shown in Fig. 8. The straight line means that the prediction is perfect and the dotted lines mean a deviation of 10%. From the figure, it is found that all the predictions of the Nusselt number are within a deviation of 10% and the majority 92% of the predictions of the friction factor is also within a deviation of 10%. By careful examination, it is found that most of the predictions are within 4% deviation from the database. The scatter distribution of the deviations shows good agreement with an accepted error less than 5%. It must be pointed out that in some situations the measurements and hence Figure 8: Predictions of Nu and f by ANN with laminar model.

14 314 EMERGING TOPICS IN HEAT TRANSFER Figure 9: Predictions of Nu and f by ANN and correlations. the uncertainties or computations have certain errors, sometimes up to 10%, or 20% for turbulent computations. The predictions here reflect this inaccuracy in the database because the predictions are of course not better than the perfect measurements or computations. Consequently, the secondly updated is now used for prediction of laminar and turbulent flow and heat transfer performance of the presented heat exchangers having large number of tube rows and tube diameter. In order to again show the confidence that the ANN approach is superior to correlations for predictions, the comparison is plotted in Fig. 9. Because the comparison based on the database for turbulent flow and heat transfer was shown in previous section, only the database from laminar flow and heat transfer is used here for this comparison. It can be seen that the predictions by ANN are better than those by multiple correlations. The average errors of f and Nu are 3.84% and 1.85%, respectively. Therefore, the conclusion that the ANN is superior compared with correlations for prediction of the presented heat exchanger is now supported again. In addition, for convenient use by engineers or researchers, the weights and biases for secondly updated ANN are listed below. The sets of weights and biases can be directly read into the network by an encoded program so that the turbulent and laminar convective heat transfer of the heat exchangers can be obtained. Thus two sets of weights and biases are provided here for future research works. 4 Concluding Remarks This chapter demonstrates successful application of a GA for optimization of a plate-fin CHE. A generalized procedure has been developed to carry out the optimization to find the minimum volume and/or annual cost of the heat exchanger, respectively, based on the e-ntu and the GA technique. It is concluded that the GA can provide a strong ability of auto-search and combined optimization in the optimization design of heat exchangers compared with the traditional designs in which a trial-and error process may be involved. By application of the GA in the optimal design, the heat exchanger configurations/structures can be optimized according to different design objectives such as minimum surface area and cost.

15 DESIGN OPTIMIZATION AND PERFORMANCE PREDICTION OF COMPACT HEAT EXCHANGERS 315 In addition, this chapter demonstrates successful application of an ANN for performance prediction of tube-fin CHEs. Various neural network configurations have been tested based on experimentally measured databases of Nusselt number and friction factor of three kinds of heat exchangers. The ANN architecture has strong ability to predict the heat exchanger performance of turbulent and laminar heat transfer and fluid flow with an accepted deviation close to the measurement uncertainty/error. The presented ANN yields the superior prediction of heat transfer and flow friction compared with power law or multiple correlations. The limitation of ANN is that it cannot describe the unknown physical phenomena directly. References [1] Martin, H., Economic optimization of compact exchangers. First International Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, ed. R.K. Shah, Banff, Canada, pp , [2] Muralifrishna, K. & Shenoy, U.V., Heat exchanger design targets for minimum area and cost. Transactions of the Institution of Chemical Engineers, 78, pp , [3] Jafari, Nasr M.R. & Polley, G.T., An algorithm for cost comparison of optimized shell-and-tube heat exchangers with tube inserts and plain tubes. Chemical Engineering Technology, 23, pp , [4] Wang, L., Performance analysis and optimal design of heat exchangers and heat exchanger networks, PhD thesis, Division of Heat Transfer, Department of Heat and Power Engineering, Lund Institute of Technology, [5] Wang, L. & Sunden, B., Design methodology for multistream plate fin heat exchangers in heat exchanger networks. Heat Transfer Engineering, 22, pp. 3 11, [6] Jia, R., Sunden, B. & Xuan, Y., Design and optimization of compact heat exchangers. Third International Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, ed. R.K. Shah, Davos, Switzerland, pp , [7] Jia, R. & Sunden B., Optimal design of compact heat exchangers by an artificial neural network method. Proceedings of HT2003, ASME Summer Heat Transfer Conference, Paper No. HT , [8] Unuvar, A. & Kargici, S., An approach for the optimum design of heat exchangers. International Journal of Energy Research, 28, pp , [9] Traverso, A. & Massardo, A.F., Optimal design of compact recuperators for micro-turbine application. Applied Thermal Engineering, 25, pp , [10] Reneaume, J.M. & Niclout, N., MINLP optimization of plate fin heat exchangers. Chemical and Biochemical Engineering Quarterly, 17, pp , 2003.

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