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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Multi-objective optimization design of condenser in an organic Rankine cycle for low grade waste heat recovery using evolutionary algorithm☆ Jiangfeng Wang ⁎, Man Wang, Maoqing Li, Jiaxi Xia, Yiping Dai Institute of Turbomachinery, 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

Available online 26 April 2013 Keywords: Condenser Genetic algorithm Low grade waste heat Multi-objective optimization Organic Rankine cycle Plate heat exchanger

a b s t r a c t The optimum design of a condenser is signiﬁcant in an organic Rankine cycle to achieve higher waste heat utilization efﬁciency. Based on the mathematical model of a condenser using plate heat exchanger (PHE), some key geometric parameters on the total heat transfer surface area and pressure drop of the condenser are examined. In order to obtain geometric parameters of a plate heat exchanger, a multi-objective optimization of the condenser in organic Rankine cycle is conducted to achieve the optimal geometry design. The total heat transfer surface area and pressure drop are selected as two objective functions to minimize both total heat transfer surface area and pressure drop under the constant heat transfer rate and LMTD conditions. The plate width, plate length and plant distance are selected as the decision variables. Non-dominated sorting generic algorithm-II (NSGA-II) which is an effective multi-objective optimization method is employed to solve this multi-objective optimization design of PHE. The results show that an increase in channel distance or plate width increases the total heat transfer surface area and decreases pressure drop in the condenser. It is noted that the plate length of PHE has a positive effect on the optimization design of PHE. By multi-objective optimization design of the PHE, a Pareto optimal point curve is obtained, which shows that a decrease in total heat transfer surface area of a condenser can increase the pressure drop through the condenser. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction The demand for energy is growing faster due to rapid industrialization and social growth, and energy shortage become one of the most serious issues for people. Conventional primary sources, such as coal, oil and natural gas, have limited reserves that are hard to meet the sustainable development. Moreover, there is a great deal of waste heat being released into the environment from industrial plant, which leads to serious energy loss and environmental pollution. In addition, there are also abundant geothermal resources and solar energy available in the world. Most of these heat sources, such as solar energy and geothermal energy, are classiﬁed as low grade heat sources. Recovering these low grade heat sources can achieve energy saving, low emissions and sustainable development. The organic Rankine cycle (ORC) is an effective method to recover the low grade heat source because it employs the organic working ﬂuids instead of water. Much research has been devoted to the organic Rankine cycle [1–3] to achieve higher energy conversion efﬁciency. The components in the ORC system have recently attracted attention due to their performances inﬂuencing the overall system performance.

☆ Communicated by: W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (J. Wang). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.04.014

Besides much work on the rotating part, such as expander and turbine [4–6], the heat exchangers, such as evaporator and condenser, are also the very important components in ORC system as the effective heat transfer reﬂects on the overall energetic and exergetic efﬁciency. The condenser regarding the heat sink of thermodynamic cycle has a signiﬁcant effect on the overall system performance of ORC. Therefore, Li et al. [7] conducted an exergoeconomic analysis and performance optimization of a condenser for a binary mixture of vapors in the ORC system. The objective function was deﬁned as the annual total cost per unit heat transfer rate considering the capital cost and the exergy loss cost. The minimum annual total cost had been obtained. However, they didn't deﬁne the type of the heat exchanger. Plate heat exchangers as compact heat exchangers are one of the most efﬁcient classes of heat exchangers and they are applied extensively in the ﬁeld of industries like food, sulfuric acid, salt solution, chemical engineering and so on. Due to compactness, high thermal efﬁciency, easy maintenance, less fouling and ﬂexibility, PHE is preferred to be used as a condenser in the ORC system. Much attention has been focused on the optimization design of plate heat exchanger. Lee et al. [8] examined the pressure drop and heat transfer characteristics of a plate heat exchanger conducted with staggered pins and conducted an optimization to minimize a global objective function consisting of the correlation between the Nusselt number and the friction factor. Wang et al. [9] presented a design method for PHEs with and without pressure drop speciﬁcations. In the case of

48

J. Wang et al. / International Communications in Heat and Mass Transfer 45 (2013) 47–54

Nomenclature A b Bo cp C D f G h K L m N Nu P Pr q Q rfg Re t T U W x

Cross-sectional area, m 2; heat transfer surface area, m2 Channel distance, m Boiling number Speciﬁc heat, J · kg −1 · K −1 Wetted perimeter of the cross-section, m Diameter, m Friction factor Mass velocity, kg · m −2 · s −1 Enthalpy, kJ · kg −1, convection heat transfer coefﬁcient, W · m −2 · K −1 Constant Plate length, m Mass ﬂow rate, kg · s −1 Number Nusselt number Pressure, MPa Prandtl number Average imposed wall heat ﬂux, W · m −2 Heat transfer rate, kW Enthalpy of vaporization, J · kg −1 Reynolds number Temperature, °C Temperature, K Overall heat transfer coefﬁcient, W · m −2 · K −1 Plate width, m Quality

Greek letters β Chevron angle, ° δ Thickness, cm ρ Density, kg · m −3 ΔT Log mean temperature difference between hot side and cold side, °C η Viscosity, kg · s −1 · m −1 λ Thermal conductivity, W · m −1 · K −1

Subscripts c Cold ﬂuid ch Channel cr Critical eq Equivalent f Working ﬂuid h Hydraulic; hot ﬂuid in Inlet l Liquid m Mean max Maximum min Minimum out Outlet sp Single-phase tp Two-phase v Vapor w Cooling water

design with pressure drop speciﬁcation, full utilization of allowable pressure drop was taken as the design objective. In the case of no pressure drop speciﬁcation, allowable pressure drops were determined through economical optimization. Pinto et al. [10] presented a screening

method to select the optimum conﬁgurations for plate heat exchangers with the minimum the heat transfer area that satisﬁed constraints on the number of channels, the pressure drops, ﬂow velocities and thermal effectiveness, as well as the exchanger thermal and hydraulic models. Examples showed that the screening method could successfully select a group of optimal conﬁgurations with a reduced number of exchanger evaluations. Park et al. [11] carried out an optimization of a plate heat exchanger with staggered pin arrays for a ﬁxed volume using the sequential linear programming method. The weighting method was employed to solve the multi-objective problem. Zhu et al. [12] discussed the integrated optimal design of the materials, placement, size and ﬂow rate of a plate heat exchanger. By optimization design, the PHE was effectively smaller than the real example given, with a consequent reduction in cost. Kanaris et al. [13] developed a general method for the optimal design of a plate heat exchanger with undulated surfaces. The PHE was optimized by means of response surface methodology with an objective function that linearly combined heat transfer augmentation with friction losses, using a weighting factor that accounted for the cost of energy. New correlations were provided for predicting Nusselt number and friction factor in such PHEs. As discussed above, most of the optimization problems for plate heat exchanger are based on a single objective function. However, in most of engineering problems, we usually encounter more than one objective and these objectives generally are in conﬂict with each other. Therefore, multi-objective optimization is required to achieve the optimization design of a plate heat exchanger. Genetic algorithm, based on the theory of evolution, is an effective method to solve the multiobjective optimization problem. Srinivas and Deb [14] proposed a genetic algorithm based on non-dominated sorting to perform multi-objective optimization, which was called non-dominated sorting generic algorithm (NSGA). Later, it was modiﬁed by Deb et al. [15] which eliminated higher computational complexity, lack of elitism and the need for specifying the sharing parameter, which was is called NSGA-II. In the present study, optimization design of a plate heat exchanger as the condenser in an organic Rankine cycle is conducted under the calculated conditions of heat transfer rate and mass ﬂow rate. A sensitive analysis is carried out to examine the effects of some geometric parameters of the PHE on the total heat transfer surface area and pressure drop. The multi-objective optimization of the condenser is carried out with two objective functions including the total heat transfer surface area and pressure drop by means of NSGA-II. Several geometric parameters including plate length, plate width and channel distance are considered as decision variables. 2. Thermodynamic and hydraulic modeling Fig. 1 shows the schematic diagram of an organic Rankine cycle for low grade waste heat recovery and Fig. 2 shows the T–s diagram of the organic Rankine cycle. Liquid organic working ﬂuid is compressed by a booster pump and fed to the heat recovery vapor generator (HRVG), where it is evaporated and becomes superheated vapor. The superheated vapor enters a turbine and expands to a low pressure to generate power output. Afterwards, the turbine exhaust is condensed to liquid in the condenser by rejecting heat to the environment. In the ORC system, the condenser is a very important component which can inﬂuence the overall system performance regarding the heat sink of thermodynamic cycle. A plate heat exchanger is adopted as the condenser in organic Rankine cycle for its high efﬁciency and compact structure. The considered stream in the PHE is supposed to follow the m × n arrangement, which implies the stream has n channels and m paths. The passage arrangement of the PHE is conﬁgured as countercurrent single-pass ﬂow as shown in Fig. 3. Fig. 4 displays the geometric characteristics of a plate heat exchanger. R134a is selected as the working ﬂuid in ORC

J. Wang et al. / International Communications in Heat and Mass Transfer 45 (2013) 47–54

49

Fig. 1. Schematic diagram of an organic Rankine cycle for waste heat recovery. Fig. 4. Geometric characteristics of plate heat exchanger.

Waste heat source

T Waste heat exhaust

3

2

R134a

4

1 w1

w2

for its good thermodynamic properties and low environmental impacts and water is used to cool down the R134a in the condenser. It is assumed that the state of working ﬂuid at the condenser outlet is saturated. The condenser in the present study is divided into two different regions: a two-phase condensation region and a single-phase superheat region as shown in Fig. 5. The heat balance equation in the condenser between the hot side and cold side is also represented as follows: Superheated region Q sp ¼ mf ðh4 −hd Þ ¼ mw ðhw2 −hwv Þ

ð1Þ

Two-phase condensation region

Water

s

Q tp ¼ mf ⋅ðhd −h1 Þ ¼ mw ⋅ðhwv −hw1 Þ

ð2Þ

Fig. 2. T–s diagram of an organic Rankine cycle for waste heat recovery.

t

Condensation region

Superheated region

t4 Qi t1

tw2 twv

tw1

i i+1

Q Fig. 3. Countercurrent single-pass ﬂow in plate heat exchanger.

Fig. 5. Temperature proﬁles in the condenser.

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J. Wang et al. / International Communications in Heat and Mass Transfer 45 (2013) 47–54

within the plant channels due to friction loss. The pressure drop in the single ﬂow region is given by

To simplify the theoretical models of PHE, a set of assumptions are introduced in the following: (1) The heat exchanger operates under steady-state conditions. (2) Heat losses to or from the surroundings and kinetic and potential energy changes are negligible. (3) Fouling effects are negligible. (4) The ﬂow in all channels is fully developed.

Δpsp;ch ¼

Q sp ¼ U sp Asp ΔT m;sp

ð3Þ

where Usp is the overall heat transfer coefﬁcient, Asp is the heat transfer surface area, ΔTm,sp is the log mean temperature difference (LMTD) between hot ﬂuid and cold ﬂuid. The log mean temperature difference is determined by ΔT m;sp ¼

ΔT max −ΔT min lnðΔT max =ΔT min Þ

ð4Þ

The overall heat transfer coefﬁcient is given by 1 1 δ 1 ¼ þ þ U sp hf;h λ hw;c

ð5Þ

f ¼

K Rez

The heat balance equation for each section in a counter-ﬂow conﬁguration is given by: Q i ¼ mf ⋅ hf;iþ1 −hf;i ¼ mw ⋅ hw;iþ1 −hw;i

0:646 6β 0:583 1=3 Nu ¼ 0:724 Re Pr π

Q i ¼ U i Ai ΔT i;m

ð7Þ

m G¼ Nch ⋅b⋅W

ð8Þ

4A 4Wb ¼ Dh ¼ C 2ðW þ bÞ

ð9Þ

ð10Þ

The convention heat transfer coefﬁcient for the single-phase ﬂow is expressed as h¼

λ⋅Nu Dh

The log mean temperature difference for each section is determined

ΔT i;m ¼

ð13Þ

It is assumed that the pressure drop due to the potential energy lost is neglected for the compact size of the heat exchanger and pressure drop with the inlet and outlet manifolds and ports is also neglected. Therefore, the pressure drop in the plate heat exchanger is mainly focused on the pressure drop

ΔT max;i −ΔT min;i ln ΔT max;i =ΔT min;i

ð18Þ

The overall heat transfer coefﬁcient for each section is given by 1 1 δ 1 ¼ þ þ U i hf;i λ hw;i

ð19Þ

The condensation heat transfer coefﬁcient on the hot ﬂuid for each section in the condenser is expressed as [18] Nui ¼

The Prandtl number is given by cp η λ

ð17Þ

by

where G denotes the mass velocity through the plate channels and Dh denotes respectively the hydraulic diameter of the ﬂow channel, being expressed by

Pr ¼

ð16Þ

The heat transfer rate for each section is given by

ð6Þ

The Reynolds number is given by GDh η

ð15Þ

(2) Two-phase ﬂow In the two-phase condensation region, the ﬂuid properties such as density, speciﬁc heat, viscosity and thermal conductivity are observed to suffer from dramatic variations with the quality variation of organic working ﬂuid. As a consequence, the widely used LMTD method based on constant properties might be inapplicable under such circumstances. Therefore, a modiﬁed LMTD method is employed here so as to achieve more accurate results. The heat transfer process in the two-phase condensation region is divided into relatively small sections, with so slight property variations in each section that constant properties can be assumed. Fig. 5 shows the discretization of the two-phase condensation region.

where hf,h and hw,c are respectively the convection heat transfer coefﬁcients for the hot ﬂuid and cold ﬂuid. The Chisholm and Wanniarachchi correlation is employed to calculate the Nusselt number for both hot ﬂuid and cold ﬂuid, being expressed as [16]

Re ¼

ð14Þ

The friction factor is given by [17]

The heat transfer processes for single-phase ﬂow and two-phase ﬂow are respectively discussed in the following. (1) Single-phase ﬂow The heat transfer rate is given by

4f N ch G2 L ⋅ Dh 2ρ

hf;i Dh 0:4 1=3 ¼ 4:118Reeq;i Prl λi

ð20Þ

where Pr1 is the Prandtl number of saturated liquid for the organic working ﬂuid and Reeq,i is the equivalent Reynolds number, which is Reeq;i ¼

Geq;i Dh ηl

ð21Þ

in which 0:5 ρ Geq;i ¼ G 1−xm;i þ xm;i l ρv

ð22Þ

where x m,i is the vapor quality for each section. ρ1 and ρv are respectively the density of saturated liquid and saturated vapor.

J. Wang et al. / International Communications in Heat and Mass Transfer 45 (2013) 47–54 Table 1 Waste heat conditions of a kiln.

51

Table 4 Performance of the organic Rankine cycle.

Term

Unit

Value

Parameter

value

Temperature of exhaust gas Mass ﬂow rate of exhaust gas Mass fraction of exhaust gas

°C kg · s−1 % %

130 15 51.2 48.8

Absorption heat in the HRVG/kW Turbine power/kW Pump power/kW Heat rejection in condenser/kW Net power output/kW Thermal efﬁciency/%

907.660 85.636 7.825 829.850 77.810 8.57

CO2 N2

Table 2 Simulation conditions of the organic Rankine cycle. Term

Value

Environment temperature/°C Environment pressure/MPa Turbine inlet pressure/MPa Turbine inlet temperature/°C Turbine isentropic efﬁciency/% Pinch temperature difference/°C Approach temperature difference/°C Pump isentropic efﬁciency/%

15.0 0.1013 2.5 130 70 8 6 70

subject to g i ðX Þ≤0; i ¼ 1; …; m hj ðX Þ≤0; j ¼ 1; …; n xk; min ≤xk ≤xk; max

The pressure drop relations for the working ﬂuid in the condenser are different in the different regions. The friction factor on the hot ﬂuid for each section are expressed as [18] 0:4

f tp;i Re

0:5

Bo

−0:8 pm −0:0467 ¼ 94:75Reeq;i P cr

ð23Þ

where Pcr is the critical pressure of working ﬂuid, Bo is the boiling number, which is deﬁned as Bo ¼

q Gr fg

ð24Þ

The heat transfer coefﬁcient on the single-phase region for each section is also expressed as [11] 0:646 6β 0:583 1=3 Rei Pri Nui ¼ 0:724 π

ð27Þ

The pressure drop in the two-phase ﬂow region is also calculated using Eq. (14). The total pressure drop through the condenser can be given by Δp ¼ Δpsp;ch þ Δptp;ch

ð28Þ

where X denotes the vector of decision variables to be optimized, F(X) is the vector of objective functions, gj(X) and hi(X) are the inequality constraints, and equality constraints, respectively, xk,min and xk,max are the range to which the decision variables belong. In a multi-objective optimization problem, the solutions which do not dominate each other are called non-dominated solutions or Pareto optimal solutions and they are the solutions of the multi-objective optimization problem. We choose the total heat transfer surface area and pressure drop as two objective functions to minimize both the heat transfer area and pressure drop under the constant heat transfer rate and LMTD conditions. The channel distance, plate length and plate width of the PHE are selected as the decision variables which have signiﬁcant effects on the heat transfer area and pressure drop of the condenser. 3.2. Optimization algorithm NSGA-II is employed to achieve the multi-objective optimization of the ORC system. Based on the theory of the natural selection in the biological genetic progress which was developed by Charles Darwin, NSGA-II features a fast sorting procedure and an elitism preservation mechanism while it does not require any tunable parameters, and it could search for all the Pareto optimal solutions simultaneously. Details about the procedure adopted by NSGA-II can be found in Ref. [15,19]. 4. Result and discussion The optimum design condition is obtained from a case calculation of an ORC system to recover the low grade exhaust gas from a typical

3. Multi-objective optimization of PHE 3.1. Objective functions and decision variables The multi-objective optimization problem can be described as minF ðX Þ ¼ ½f 1 ðX Þ; f 2 ðX Þ; f 3 ðX Þ…; f n ðX Þ

Τ

ð29Þ

Table 3 Simulation results of state points in the ORC. State

t/°C

p/kPa

h/kJ · kg−1

s/kJ · kg−1 · K−1

_ m/kg · s−1

1 2 3 4 w1 w2

25 26.47 120 77.30 15.00 24.92

665.38 2500.00 2500.00 665.38 120.00 120.00

234.546 236.711 487.888 464.190 63.095 104.587

1.120 1.122 1.847 1.877 0.224 0.366

3.61 3.61 3.61 3.61 20.00 20.00

Table 5 Initial calculation conditions of the condenser (PHE). Parameter

value

Mass ﬂow rate of hot ﬂuid (R134a)/kg · s−1 Mass ﬂow rate of cold ﬂuid (water)/kg · s−1 Heat transfer rate/kW Inlet temperature of hot ﬂuid (R134a)/°C Outlet temperature of hot ﬂuid (R134a)/°C Inlet temperature of cold ﬂuid (water)/°C Outlet temperature of cold ﬂuid (water)/°C Plate thickness/mm Chevron angle/° Thermal conductivity of plates/W · m−1 · K−1 Plate length/m Plate width/m Channel distance/m

3.61 20.00 829.850 77.30 25.00 15.00 24.92 0.4 60 16.3 0.46 0.22 0.0034

J. Wang et al. / International Communications in Heat and Mass Transfer 45 (2013) 47–54

180

60

160

50

140

40

120

30

100

20

80

10

85

80

Heat transfer surface area Pressure drop

70

80 75

60

70 50

65 60

40

55

60 0 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2

50 30 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62

Channel distance / mm

Plate length / m

Fig. 6. Effect of channel distance on the total heat transfer surface area and pressure drop.

kiln as shown in Table 1. The thermophysical properties of R134a are calculated by REFROP 9.0 [20] developed by the National Institute of Standards and Technology of the United States. The system simulation was carried out using Matlab software and iterative relative convergence error tolerance was 0.02%. Table 2 lists the simulation conditions of the organic Rankine cycle, and Tables 3 and 4 show the simulation results of the ORC system. Since some key geometric parameters of the PHE can inﬂuence the total heat transfer surface area and pressure drop, it is necessary to examine the effects of each geometric parameter on the total heat transfer surface area and pressure drop before optimization design. Table 5 summarizes the initial calculation conditions of the condenser (PHE). In the sensitive analysis of geometric parameters, when one geometric parameter is varied, others are kept constant as those in Table 5. Fig. 6 shows the effect of channel distance of the PHE on the total heat transfer surface area and pressure drop. It can be observed that as the channel distance of PHE increases, the total heat transfer surface area increases and the pressure drop decreases. A rise in channel distance of the PHE causes the overall heat transfer coefﬁcient to decline, resulting in an increase in the total heat transfer surface area of the condenser with unchanged inlet and temperature for both hot side and cold side. In addition, due to the decreasing total

90

80

Heat transfer surface area Pressure drop

85

70

80 75

60

70 50

65 60

Pressure drop / kPa

Heat transfer surface area / m2

Heat transfer surface area / m2

Heat transfer surface area Pressure drop

Pressure drop / kPa

Heat transfer surface area / m2

90

70

200

40

55 50 0.20

0.22

0.24

0.26

0.28

0.30

30

Plate width / m Fig. 7. Effect of plate width on the total heat transfer surface area and pressure drop.

Pressure drop / kPa

52

Fig. 8. Effect of plate length on the total heat transfer surface area and pressure drop.

heat transfer surface area and an unchanged surface area of the single plate, the number of plate drops accordingly. Thus, the mass velocity through the plate channels decreases. Moreover, the hydraulic diameter of ﬂow channel also rises with an increase in channel distance of PHE. Therefore, the pressure drop through the ﬂow channel decreases. Fig. 7 shows the effect of plate width on the total heat transfer surface area and pressure drop. It can also be seen that as the plate width of the PHE increases, the total heat transfer surface area rises slightly, but the pressure drop decreases signiﬁcantly. The reason for this is the same as the effect of channel distance does. An increase in plate width of PHE causes the hydraulic diameter of ﬂow channel to decline more slightly compared with channel distance. These results imply that plate width has a little effect on the total heat transfer surface area, but a signiﬁcant effect on the pressure drop. Fig. 8 shows the effect of plate length on the total heat transfer surface area and pressure drop. It can be observed that as the plate length increases, the total heat transfer surface area decreases and the pressure drop increases. An increase in plate length of the PHE has no effect on the hydraulic diameter of ﬂow channel, but causes the overall heat transfer coefﬁcient to increase, resulting in a decrease in the total heat transfer surface area of condenser. An increase in plate length raises the pressure drop directly according to Eq. (14). These ﬁndings imply that the plate length of the PHE has a positive effect on the optimization design of the PHE. In addition, the hydraulic diameter of the ﬂow channel is a key parameter which can inﬂuence the performance of the PHE signiﬁcantly. The ﬁndings from Figs. 6 to 8 conﬁrm that an increase in the total heat transfer surface area can raise the pressure drop in the condenser. Therefore, it is necessary to conduct a multi-objective optimization of the PHE to ﬁnd the optimum geometric parameters for the optimization design of the condenser. The two objective functions considered here for this multi-objective optimization problem are the total heat transfer surface area and pressure drop. To minimize both total heat transfer surface area and pressure drop, three geometric parameters including channel distance, plate width and plate length are selected as the decision variables. The decision variables and their ranges are listed in

Table 6 Decision variables and their lower and upper boundaries. Decision variables

Lower bound

Upper bound

Plate length /m Plate width /m Channel distance /m

0.4 0.2 0.003

0.6 0.3 0.005

J. Wang et al. / International Communications in Heat and Mass Transfer 45 (2013) 47–54

53

is mostly carried out based on engineering experiences and the importance of each objective for the decision-maker.

Table 7 Control parameters in NSGA-II. Tuning parameters

Value

Population size Maximum generations Crossover probability Mutation probability Selection process

30 150 0.7 0.05 Tournament

5. Conclusions

Table 6. Non-dominated sorting genetic algorithm-II (NSGA-II), which is an effective multi-objective optimization method, is employed to solve this multi-objective optimization problem. The control parameters in NSGA-II are listed in Table 7. Fig. 9 shows the Pareto frontier solution of geometric parameter for the condenser via multi-objective optimization, which clearly reveals the conﬂict between two objectives. Each geometric parameter that decreases the total heat transfer surface area causes an increase in pressure drop through the condenser. Table 8 lists the optimum design value and their corresponding decision variables for the condenser. There is no combination of decision variables which can optimize all objectives simultaneously. The ﬁnal solution among the optimum points should be chosen from the Pareto solutions. The selection process

In the present study, based on calculated design conditions of a condenser used in ORC system, a sensitive analysis is carried out to examine the effects of some geometric parameters on the total heat transfer surface area and pressure drop. Next, the condenser was optimally designed using multi-objective optimization. The design geometric parameters (decision variables) are channel distance, plate length, and plate width. The overall heat transfer surface area and pressure drop are selected as two objective functions. NSGA-II was employed to achieve the multi-objective optimization problem. As discussed, an increase in channel distance or plate width increases the total heat transfer surface area and decreases pressure drop in the condenser. The variation of hydraulic diameter of ﬂow channel can inﬂuence the performance of PHE signiﬁcantly. It is noted that the plate length of the PHE has a positive effect on the optimization design of the PHE. By a multi-objective optimization design of the PHE, a Pareto optimal point curve is obtained and shows that the optimum design solution with their corresponding decision variables is needed to be selected by a process of decision-making.

140

Pressure drop / kPa

Acknowledgment

R134a

120

The authors gratefully acknowledge the ﬁnancial support by the National Natural Science Foundation of China (Grant No. 51106117), the National Key Technology R&D Program (No. 2011BAA05B03) and the National High-tech Research and Development Program (Grant No. 2012AA053002).

100 80 60

References

40 20 0 40

60

80

100

120

140

160

180

200

220

Heat transfer surface area / m2 Fig. 9. Pareto frontier (Pareto optimal solutions) for the total heat transfer surface area versus pressure drop using NSGA-II.

Table 8 Optimum design values and their corresponding decision variables. Atot/m2

Δp/kPa

L/m

W/m

b/m

47.67 214.13 47.95 58.99 178.58 150.85 66.09 48.53 48.36 163.77 126.17 101.89 214.13 47.67 141.63 55.83 201.76 47.84

131.86 5.43 111.51 57.88 7.56 10.14 49.14 87.39 108.54 9.55 14.01 24.06 5.43 131.86 11.30 66.19 6.17 125.81

0.5996 0.4007 0.5996 0.5707 0.4481 0.4587 0.5252 0.5971 0.5994 0.4264 0.4881 0.5086 0.4007 0.5996 0.4531 0.5736 0.4183 0.5996

0.2067 0.2997 0.2379 0.2973 0.2897 0.2931 0.2809 0.2926 0.2405 0.2719 0.2921 0.2488 0.2997 0.2067 0.2957 0.2914 0.2903 0.2135

0.0030 0.0050 0.0030 0.0033 0.0049 0.0046 0.0033 0.0030 0.0030 0.0046 0.0044 0.0041 0.0050 0.0030 0.0045 0.0032 0.0050 0.0030

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