Effect of discontinuous helical turbulators on heat transfer characteristics of double pipe water to air heat exchanger

Energy Conversion and Management 118 (2016) 75–87

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Effect of discontinuous helical turbulators on heat transfer characteristics of double pipe water to air heat exchanger M. Sheikholeslami ⇑, M. Gorji-Bandpy, D.D. Ganji Department of Mechanical Engineering, Babol University of Technology, Babol, Iran

a r t i c l e

i n f o

Article history: Received 5 January 2016 Received in revised form 13 March 2016 Accepted 26 March 2016

Keywords: Discontinuous helical turbulators Air to water heat exchanger Double pipe Friction Heat transfer NSGA II

a b s t r a c t Effect of typical and perforated discontinuous helical turbulators on flow and heat transfer in an air to water double pipe heat exchanger is experimentally studied. Experimental analysis is conducted for different values of open area ratio (0–0.0625), Reynolds number (6000–12,000) and pitch ratio (1.83–5.83). According to experimental data, correlations for Nusselt number, friction factor and thermal performance are presented as functions of variable parameters. Non-dominated Sorting Genetic Algorithm II (NSGA II) is applied to find the optimal designs of high efficiency heat exchanger. Pareto front is presented as set of multiple optimum solutions. Results show that friction factor and Nusselt number reduce with rise of open area ratio and pitch ratio. Thermal performance is an increasing function of open area ratio while it is a decreasing function of pitch ratio. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Growth heat transfer improvement methods have been happened due to significance of improving the thermal performance of heat exchangers. These approaches improve convective heat transfer by decreasing the thermal resistance but at the cost of improve in pressure loss. A novel shell and tube heat exchanger with plate baffles was proposed by Yang and Liu [1]. They analyzed the temperature field, pressure field and path lines to demonstrate the advantage of their new heat exchanger. The heat transfer performance of porous-microchannels was studied by Dehghan et al. [2]. They found that porous inserts are more effective in the slip flow regime. Chen et al. [3] investigated performances of trisection helical baffled heat exchangers. They showed that both the shell side heat transfer coefficient and pressure drop increase. A new design of a finned double-pipe heat exchanger with longitudinal fins was presented by Syed et al. [4]. Their results indicated that the ratio of tip to base angles has proved to play significant role in the design of a double-pipe heat exchanger in reducing the cost, weight and frictional loss. Effect of porous baffles and flow pulsation on a double pipe heat exchanger performance was studied by Targui and Kahalerras [5]. They proved that the addition of oscillating components to the mean flow affects the flow structure, ⇑ Corresponding author. E-mail addresses: [email protected], m.sheikholeslami@stu. nit.ac.ir (M. Sheikholeslami), [email protected] (D.D. Ganji). http://dx.doi.org/10.1016/j.enconman.2016.03.080 0196-8904/Ó 2016 Elsevier Ltd. All rights reserved.

and enhances the heat transfer. Improvement of Nusselt number in a pipe with circular ring has been studied by Ozceyhan et al. [6]. They showed that the maximum overall improvement is 18% which is obtained at Re ¼ 15; 600. Effects of combined ribs and winglet type vortex generators (WVGs) on forced convection heat transfer and friction loss behaviors were investigated by Promvonge et al. [7]. They concluded that the values of Nusselt number and friction factor for utilizing both the rib and the WVGs are found to be considerably higher than those for using the rib or the WVGs alone. Experimental investigation of thermal performance of solar water heater system fitted with helical and Left–Right twist has been performed by Jaisankar et al. [8]. They showed that the helical twisted tape induces swirl flow inside the riser tubes unidirectional over the length. Bayrak et al. [9] analyzed exergy and energy of solar air heaters with permeable baffles. They showed that maximum performance is obtained for thickness of 6 mm. Influence of rib height and inlet temperature of fluid on thermal performance was presented by Ma et al. [10]. They showed that the style of flow has no variation with changing of inlet temperature. Ibrahim [11] used helical screw-tape inserts in order to augmentation of laminar flow and heat transfer in flat tubes. They showed that averaged Nusselt number enhances with the rise in Reynolds number and with the decrease in twist ratio and spacer length. Sheikholeslami et al. [12] studied about swirl flow devices effect on fluid flow and heat transfer. To improve heat transfer, combined swirl generator and conical-nozzle inserts were used by Promvonge and

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Nomenclature A d; D f ‘ L Nu N Pr P PR Q Rea T U

heat transfer area inner and outer pipe diameter Darcy friction factor (dimensionless) length of pipe length of test section Nusselt number number of perforated hole Prandtl number pressure pitch ratio (=P/Do) flow rate of water air flow reynolds number fluid temperature coefficient of overall heat transfer

Eiamsa-ard [13]. They showed that this method can improve rate of heat transfer up to 316%. Vibration behavior of conical ring was utilized by Yakut and Sahin [14] for increasing thermal performance. They concluded that maximum heat transfer occurs at the smallest pitch ratio. Water to air heat exchangers are one of the significant kinds of heat exchangers. Some of the uses of them are dehumidification, air conditioning, residential heating and apartment buildings. Sheikholeslami et al. [15] studied the turbulent flow and heat transfer in water to air double pipe heat exchanger. Sheikholeslami et al. [16] used agitator in water side to improve rate of heat transfer. They showed that effects of agitator are more pronounced for low Reynolds number. Recently, several authors studied about thermal enhancement methods [17–35]. The aim of this article is to study the effects of discontinuous helical turbulators on pressure drop and heat transfer improvement in an air to water double pipe heat exchanger. Experimental set up and formulas for measuring of g; Nu and f are presented. The impacts of open area ratio, pitch ratio and Reynolds number on pressure drop and heat transfer rate are studied. Also Nondominated Sorting Genetic Algorithm II (NSGA II) is used to find the optimal designs of high efficiency heat exchanger.

Greek symbols thermal diffusivity    2 2 k open area ratio ¼ Nds = ðDo þ 2hÞ  D2o l dynamic viscosity of nanofluid q density g thermal performance

a

Subscripts i inner o outer a air w water s smooth pipe

transfer enhancement (Fig. 2). Schultz and Cole method [36] is used for uncertainty analysis:

UR ¼

"  n X @R i¼1

@V i

2 #1=2 UV i

ð1Þ

where U R is the total error, U V i is the error of each independent parameter and n is the number of total parameters. Table 3 shows the uncertainties of the experimental parameters. The uncertainty analysis showed that the measuring errors were less than 10% for all the experiments presented in this study. 3. Measurement of coefficients of pressure loss and heat transfer rate The method for determination friction factor and Nusselt number is summarized as follows: Q a is heat transferred to the cold fluid:

Q a ¼ ðT a;out  T a;in ÞC p;a ma

ð2Þ

C p;a ; ma

where are specific heat and the rate of mass flow for air, respectively. Q w is heat transferred from the water:

2. Experimental technique

Q w ¼ ðT w;in  T w;out ÞC p;w mw

Experimental set up depicts in Fig. 1(a). In this set up: Di ¼ 2:8 cm; Do ¼ 3 cm; di ¼ 5 cm, do ¼ 6 cm. The length of the pipe is ‘ ¼ 2 m and the length of test section is L ¼ 1:2 m. Hot water and cold air are passed through the inner and outer pipes, respectively. Three heaters are used in the upper tank with the capacity of 2 kW, 2 kW and 3 kW. The inner tube is made from copper ðk ¼ 300 kcal=ðm h  CÞÞ, while the outer tube is made

where C p;w ; mw are the specific heat and the rate of mass flow for water, respectively. Q av e is average heat transfer rate which is defined as follows:

from Plexiglas ðk ¼ 5  104 kcal=ðm h  CÞÞ. T 1 ; T 2 ; T air1 ; T air2 , T w1 ; . . . ; T w6 and T a1 ; . . . ; T a4 were measured with Sheathe type thermocouples (element C.A; class 0.75) (Fig. 1(b)). An ST-8920 differential pressure is used to obtain the pressure drop in air side. It can measure the pressures in ±5000 Pa with 1 Pa resolution. In order to transfer the water from the lower tank to upper tank, a pump with the head of 5.5 m, is used. The 0:75 kW blower directed the air with T air1 ¼ 28  C to orifice meter. SV008iG5A-2 inverter is utilized to adjust air flow rate by changing the motor speed. Water flow rates are controlled with valves and measured with rotameter. The experimental work is done for counter flow state. The physical properties of air and water are variable with temperature as illustrated in Tables 1 and 2. In the test section, typical and perforated discontinuous helical turbulators are used in order to heat

Q av e ¼ ðT w;av e  T s;av e Þhi Ai

Q av e ¼ ðQ a þ Q w Þ=2

ð3Þ

ð4Þ

hi is the water side coefficient of heat transfer which is defined as follows:

ð5Þ

where Ai ¼ pDi L. U i is coefficient of overall heat transfer can be defined as follows:

Q av e ¼ U i Ai DT LMTD

ð6Þ

where DT LMTD is ð¼ ðDT 1  DT 2 Þ=LnðDT 1 =DT 2 ÞÞ. ho is air side coefficient of heat transfer, obtained from:

1 1 lnðDo =Di Þ 1 ¼ þ þ U i Ai h o Ao 2pkCu L h i Ai

ð7Þ

Average Nusselt number along the air side of inner pipe was calculated as follows:

M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87

77

Fig. 1. (a) Experimental setup; (b) pump; (c) blower; (d) orifice; (e) inverter; (f) schematic diagram; (g) test section and thermocouples.

Nu ¼

ho DH kair

ð8Þ

In order to calculate Nusselt number: hydraulic diameter is 4pðd2i D2o Þ d2 D2 ¼ i Do o . DH ¼ 4 pðD oÞ

where DH is hydraulic diameter. The friction factor can be calculated from

f ¼

DP DH ðLq u2 =2Þ

In order to calculate pressure loss: hydraulic diameter is 4pðd2 D2 Þ DH ¼ p4 ðDio þdioÞ ¼ di  Do .

ð9Þ

where DP is the pressure loss. The general definition of hydraulic diameter is DH ¼ 4A where A P is the cross sectional area and P is the wetted perimeter of the cross-section.

Thermal performance factor defined as follows [37]:

g¼

ðNu=Nus Þ 1

ðf =f s Þ3

ð10Þ

where Nus and f s are Nusselt number and friction factor of smooth pipe (without turbolentor), respectively.

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Fig. 1 (continued)

4. Results and discussion Effect of discontinuous helical turbulators on pressure loss and rate of heat transfer in an air to water double pipe heat exchanger is studied. In order to validate the experimental procedure, friction factor and Nusselt number for smooth heat exchanger are compared with those of obtained from previous correlations [38]: Correlation of Gnielinski,

Nu ¼

ðRe  1000Þðf =8ÞPr  2 ; 0:5 1 þ 12:7ðf =8Þ Pr3  1

3000 < Re < 5  106

ð11Þ

Correlation of Petukhov,

f ¼

1 ð1:64 þ 0:79LnðReÞÞ2

;

3000 6 Re 6 5  106

ð12Þ

The obtained results have good agreement with those of obtained from previous correlations (see Fig. 3). Effects of Reynolds number, open area ratio and pitch ratio on friction factor are shown in Fig. 4. Friction factor has reverse relationship with Reynolds number and pitch ratio. This phenomenon is due to reduction of reverse flow with enhancing of these parameters. Also as open area ratio increases pressure loss reduces and in turn friction factor has reverse relationship with open area ratio. This shows that presence of the hole in the turbulators can be used as a method for reduction of friction in the heat exchanger. Besides it can be indicated that friction factors in the heat exchanger equipped with the perforated turbulators with the smallest pitch ratio, 4.751131–5.135954 times smaller

than those in heat exchanger with typical helical turbulators, relying on Reynolds number. Fig. 5 depicts the effects of Reynolds number, open area ratio and pitch ratio on Nusselt number. Nusselt number increases as turbolentor insert in plain heat exchanger. This is due to this fact that using turbolentor provides stronger turbulence intensity or mixing leading to destruction of thermal boundary layer and generates vortex flow to delay the residence time of flow in the tube. Nusselt number increases with increase of Reynolds number due to decrease in thermal boundary layer thickness while it decreases with augment of open area ratio and pitch ratio. Nusselt number reduces with rise of open area ratio and pitch ratio due to reduction in turbulent intensity. Effects of Reynolds number, open area ratio and pitch ratio on friction factor ratio and Nusselt number ratio are shown in Figs. 6 and 7. As open area ratio and pitch ratio augment, both of the friction factor ratio and Nusselt number ratio reduce. Friction factor ratio increases with increase of Reynolds number while opposite trend is observed for Nusselt number ratio. This observation is due to this fact that in high Reynolds number, thermal boundary layer thickness is thinner than that of low Reynolds number. So inserting turbolentor is more pronounced in low Reynolds number. Fig. 8 shows the effects of Reynolds number, open area ratio and pitch ratio on thermal performance. Thermal performance has direct relationship with open area ratio while it has reverse relationship with Reynolds number and open area ratio. Fig. 9 shows the effect of open area ratio in different Reynolds number and pitch ratio. As Reynolds number increases, Nur and fr decreases but gr increases. This effect on gr are more pronounced for higher values of pitch ratio.

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87

Fig. 1 (continued)

Table 1 Temperature-dependent properties of air. Coefficient

A1 A2 A3 A4 A5

A1 þ A2  T þ A3  T 2 þ A4  T 3 þ A5  T 4 Properties of air

q (kg/m3)

Cp (J/(kg K))

l (kg/(ms))

k (W/(m K))

4.5399557047065677 2.3244292640615217E2 5.6404522707476041E5 6.2803748539876179E8 2.3678170919661321E11

1.0540764984602797E + 3 3.5067618164922393E1 5.8416753365658986E4 3.0329858178609656E7 5.2479296621138882E10

9.4680032779877928E5 1.0222587861878098E6 4.7054455296163551E9 9.1119064881185846E12 6.5461225665736524E15

1.8028147194179223E2 1.6851766935888901E4 1.3838388187738584E6 3.2630462746304979E9 2.7514584927209003E12

Corresponding polynomial representations of Nu; f and g are as follow:

Nu ¼ a12 þ a22 PR þ a32 Y 1 þ a42 PR2 þ a52 Y 21 þ a62 PRY 1 Y 1 ¼ a11 þ a21 Rea þ a31 k þ a41 Re2a þ a51 k2 þ a61 Rea k

ð13Þ

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87

Table 2 Temperature-dependent properties of water. Coefficient

A1 A2 A3 A4 A5

A1 þ A2  T þ A3  T 2 þ A4  T 3 þ A5  T 4 Properties of water

q (kg/m3)

Cp (J/(kg K))

l (kg/(ms))

k (W/(m K))

1.6622104933785317E+002 1.2256322983468429E+001 4.6535103004960353E002 7.7101273744096163E005 5.0319235543371908E008

1.2201774895976883E+004 9.2961742884825355E+001 4.0724280562804471E001 8.033901613887863E004 6.0554273200519027E007

4.5563422230298373E1 5.266709499675417E3 2.293722836497707E5 4.45178676075212E8 3.2451565252286636E11

2.1117772306964272E1 4.0615080360954991E3 4.0530952053441623E5 9.5665206133793231E8 6.7722130490045319E11

f ¼ b13 þ b23 X 1 þ b33 X 2 þ b43 X 21 þ b53 X 22 þ b63 X 1 X 2 2

2

X 1 ¼ b11 þ b21 k þ b31 PR þ b41 k þ b51 PR þ b61 kPR

ð14Þ

X 2 ¼ b12 þ b22 Rea þ b32 PR þ b42 Re2a þ b52 PR2 þ b62 Rea PR

g ¼ c13 þ c23 Z 1 þ c33 Z 2 þ c43 Z 21 þ c53 Z 22 þ c63 Z 1 Z 2 Z 1 ¼ c11 þ c21 Rea þ c31 PR þ c41 Re2a þ c51 PR2 þ c61 Rea PR Z 2 ¼ c12 þ c22 Rea þ c32 k þ

c42 Re2a

2

þ c52 k þ c62 Rea k

ð15Þ

Also aij ; bij and cij can be found in Tables 4–6, respectively. Comparison of the f ; Nu and g between experimental data and those of obtained from the current correlations is illustrated in Fig. 10. This figure indicates that maximum error of all data falls within ±7%, ±2% and ±6%, for f ; Nu and g, respectively. In order to find the optimal design of heat exchanger, Nondominated Sorting Genetic Algorithm II (NSGA II) has been applied. Open area ratio, Reynolds number and pitch ratio were considered as three design parameters. The objective functions are maximiz1

ing ðNu=Nus Þ and minimizing ðf =f s Þ3 . Results for Pareto-optimal curve are shown in Fig. 11, which clearly reveal the conflict between the two responses. All solutions in Pareto front are optimized solutions. For example, if thermal performance considered as a criteria for selection. This best design is obtained when Re ¼ 6000; k ¼ 0:0625; PR ¼ 1:83, which belongs to the case of 1

Nu=Nus ¼ 1:6259 and ðf =f s Þ3 ¼ 1:1262. 5. Conclusion An experimental procedure has been conducted to investigate turbulent flow and heat transfer in an air to water heat exchanger equipped with typical and perforated discontinuous helical turbulators. Effects of the Reynolds number, open area ratio and pitch ratio on pressure loss and heat transfer improvement are examined. Correlations of Darcy factor, Nusselt number and thermal performance are presented. Results show that Nusselt number reduces with rise of pitch ratio and open area ratio while it augments with rise of Reynolds number. Friction factor is a decreasing function of Reynolds number, open area ratio and pitch ratio. Thermal performance decreases with increase of Reynolds number and open area ratio while it increases with rise of open area ratio. Also multi objective optimization provides a Pareto front of optimal solution. Appendix A. NSGA-II approach NSGA-II differs from a simple genetic algorithm only in the way the selection operator works. The efficiency of NSGA-II lies in the way multiple objectives are reduced to a single fitness measure by the creation of number of fronts, sorted according to non-

Table 3 Uncertainties of the experimental parameters.

Fig. 2. Discontinuous helical turbulators.

Parameter

Absolute uncertainty

Relative uncertainty

Tube diameter Tube length Temperature Pressure Air flow rate Water flow rate Uncertainty in reading values of table (q, cp, l, k, etc.)

±0.05 mm ±0.5 mm ±0.7  C – – – –

±0.3% ±1% ±2% ±0.001–0.1%

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87 60

0.05

50 0.04

40

+12%

30

f Experimental

NuExperimental

+12% -12%

0.03

-12% 0.02

20 0.01

10

0

0

10

20

30

40

50

0

60

0

0.01

0.02

NuGnielinski

0.03

0.04

0.05

f Petukhov

(a)

(b)

Fig. 3. Verification of friction factor and Nusselt number for smooth heat exchanger.

λ = 0.0625

λ=0

0.06

0.28

PR = 5.83

PR = 5.83 0.26

PR = 2.92

0.055

PR = 2.92

PR = 1.83

PR = 1.83 0.05

0.22

0.045

f

f

0.24

0.2

0.04

0.18

0.035

0.16

0.03

0.14 6000

8000

10000

0.025 6000

12000

8000

10000

12000

Rea

Rea 0.1

0.4

Re = 6000

Re = 6000 Re = 8000 0.3

Re = 8000

0.08

Re = 12000

Re = 12000

0.2

f

f

0.06

0.04

0.1 0.02

0

PR=1.83

PR=2.92

PR=5.83

0

PR=1.83

PR=2.92

Fig. 4. Effects of Reynolds number, open area ratio and pitch ratio on friction factor.

PR=5.83

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87

λ=0

λ = 0.0625

90

90 PR = 5.83

80

PR = 5.83

PR = 2.92

80

PR = 1.83

PR = 1.83 70

Nu

Nu

70

60

60

50

50

40

40

30 6000

PR = 2.92

8000

10000

30 6000

12000

8000

10000

12000

Rea

Rea 100

100 Re = 6000

Re = 6000

Re = 8000 80

Re = 8000 80

Re = 12000

60

Re = 12000

Nu

Nu

60

40

40

20

20

0

PR=1.83

PR=2.92

0

PR=5.83

PR=1.83

PR=2.92

PR=5.83

Fig. 5. Effects of Reynolds number, open area ratio and pitch ratio on Nusselt number.

1.6

9

8

PR = 5.83

PR = 5.83

PR = 2.92

PR = 2.92

1.5

PR = 1.83

PR = 1.83 1.4

f/fs

f/fs

7 1.3

6 1.2 5

4 6000

1.1

8000

10000

12000

1 6000

8000

10000

Rea

Rea

(a) λ = 0

(b) λ = 0.0625

Fig. 6. Effects of Reynolds number, open area ratio and pitch ratio on friction factor ratio.

12000

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87 2

2

1.9

PR = 5.83

PR = 5.83

PR = 2.92

PR = 2.92

1.9

PR = 1.83

PR = 1.83 1.8

Nu/Nus

Nu/Nus

1.8

1.7

1.7

1.6

1.6

1.5

1.5

1.4 6000

8000

10000

1.4 6000

12000

8000

10000

Rea

Rea

(a) λ = 0

(b) λ = 0.0625

12000

Fig. 7. Effects of Reynolds number, open area ratio and pitch ratio on Nusselt number ratio.

λ = 0.0625

λ=0 1.1

1.6 PR = 5.83

PR = 5.83

PR = 2.92

PR = 2.92

PR = 1.83

PR = 1.83

1

η

η

1.5

0.9

0.8 6000

1.4

8000

10000

1.3 6000

12000

8000

Rea

10000

12000

Rea

2

2 Re = 6000

Re = 6000

Re = 8000 1.6

Re = 8000 1.6

Re = 12000

1.2

Re = 12000

η

η

1.2

0.8

0.8

0.4

0.4

0

PR=1.83

PR=2.92

PR=5.83

0

PR=1.83

PR=2.92

Fig. 8. Effects of Reynolds number, open area ratio and pitch ratio on thermal performance.

PR=5.83

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87

0.36

1.6 Re = 6000

Re = 6000

Re = 8000

Re = 8000

Re = 12000

1.2

Re = 12000

fr

Nur

0.24

0.8

0.12 0.4

0

PR=1.83

PR=2.92

0

PR=5.83

PR=1.83

(a)

PR=2.92

PR=5.83

(b) 2.8 Re = 6000 Re = 8000 Re = 12000

ηr

2.1

1.4

0.7

0

PR=1.83

PR=2.92

PR=5.83

(c) k¼0:0625 Fig. 9. Effects of Reynolds number and pitch ratio on (a) f r ¼ f k¼0:0625 ; (b) Nur ¼ NuNu ; (c) gr ¼ gk¼0:0625 . f k¼0 g k¼0 k¼0

Table 4 Constant coefficient for using Eq. (13). aij

i=1

i=2

i=3

i=4

i=5

i=6

j=1 j=2

1.83E06 23.52492

0.007928 3.51715

5.5E07 0.452103

1.5E07 0.518815

4.9E08 0.006017

0.00556 0.04036

Table 5 Constant coefficient for using Eq. (14). bij

i=1

i=2

i=3

i=4

i=5

i=6

j=1 j=2 j=3

0.228467654 1.66641E06 0.026380754

4.18307 3.07E05 0.047913

0.02002 5.89E05 0.27103

27.2533 1.8E09 1.103538

0.001889 0.000527 1.267871

0.050773 7.9E07 6.687298

domination. NSGA-II is a computationally efficient algorithm implementing the idea of a selection method based on classes of dominance of all the solutions. It incorporates an elitist and a rule for adaptation assignment that takes into account both the rank

and the distance of each solution regarding others (sharing mechanism for solution diversification). Fig. A1(a) shows what is meant by rank in a minimization case. The value of adaptation is equal to its rank. When comparing two solutions belonging to the same

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87 Table 6 Constant coefficient for using Eq. (15).

1.8

i=1

i=2

i=3

i=4

i=5

i=6

j=1 j=2 j=3

1.997454 1.997454 1.997454

1.997454 1.997454 1.997454

1.997454 1.997454 1.997454

1.997454 1.997454 1.997454

1.997454 1.997454 1.997454

1.997454 1.997454 1.997454

1.75 1.4

1.7

Nu/Nus

cij

1.6

rank, extreme solutions prevail over not extreme ones. If both solutions are not extreme, the one with the bigger crowding distance (i.e. the perimeter of the cuboid calculated between the two nearest neighbors) wins (Fig. A1(b)). This way extreme solutions and less crowded areas are encouraged [39]. The NSGA-II algorithm used in this work may be stated as:

1.65 1.2 η 1.6 1.55

1 Pareto

1.5 1.45

(1) Generation t = 0. (2) Generate a uniformly distributed parent population of size P. (3) Evaluate the individuals and sort the population based on the non-domination.

η 1

1.2

1.4

1.6

0.8 1.8

( f/fs )1/3 Fig. 11. The distribution of Pareto-optimal solutions and thermal performance factor.

80

0.25

0.2

0.15

+7%

NuPredicted

fPredicted

60

-7%

+2% 40

-2%

0.1

20 0.05

0

0

0.05

0.1

0.15

0.2

0

0.25

0

20

40

f Experimental

60

80

NuExperimental

(a)

(b) 1.6

η Predicted

1.2

+6% 0.8

-6%

0.4

0

0

0.4

0.8

1.2

1.6

η Experimental

(c) Fig. 10. Comparison of experimental data with those calculated from the correlation for (a) friction factor; (b) Nusselt number; (c) thermal performance.

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M. Sheikholeslami et al. / Energy Conversion and Management 118 (2016) 75–87

Fig. A1. Concepts used by NSGA-II [39].

(4) Assign each solution a rank equal to its non-domination level (minimization of fitness is assumed). (5) Use the usual tournament selection. (6) Use the simulated crossover operator and mutation to create an off spring population of size P. (7) Combine the offspring and parent population to form extended population of size 2P. (8) Sort the extended population based on non-domination. (9) Fill new population of size P with the individuals from the sorting fronts starting from the best. (10) Invoke the crowding-distance method to ensure diversity if a front can only partially fill the next generation. The crowding-distance method maintains diversity in the population and prevents convergence in one direction. (11) Update the number of generations, t = t + 1. (12) Repeat the steps (3) to (11) until a stopping criterion is met.

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