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Pareto based multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes
Accepted Manuscript Title: Pareto based multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes Author: Hamed Safikhani, Smith Eiamsa-ard PII: DOI: Reference:
S1359-4311(15)01269-7 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.033 ATE 7314
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
8-4-2015 9-11-2015
Please cite this article as: Hamed Safikhani, Smith Eiamsa-ard, Pareto based multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes, Applied Thermal Engineering (2015), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.033. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Pareto based multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes Hamed Safikhani 1, and Smith Eiamsa-ard 2 1
Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-88349, Iran 2
Department of Mechanical Engineering, Faculty of Engineering, Mahanakorn University of Technology, Bangkok 10530, Thailand
Multi-objective optimization of turbulent flow in corrugated tubes is investigated. The objective functions are experimentally derived correlations of heat transfer and pressure drop.
The design variables are two geometrical parameters of tubes (
P DH
and
e
) and Re and
DH
Pr numbers. The Pareto front comprises important thermal design principles.
Corresponding author: Tel.: +98 21 64543425, Fax: +98 21 66419736, Email: [email protected],
[email protected] (Hamed Safikhani). 1 Page 1 of 16
Graphical Abstract
Abstract In this study, by using the empirical correlations for heat transfer and pressure drop and also employing a multi-objective optimization (MOO) approach, the best possible combinations of heat transfer and pressure drop in helically corrugated tubes will be discovered. The design variables are two geometrical parameters of helically corrugated tubes, namely pitch to diameter ratio (
P DH
) and rib height to diameter ratio (
e
), Reynolds number (Re) and Prandtl number
DH
(Pr). The objectives are maximizing the non-dimensional heat transfer coefficient (Nu) and 2 Page 2 of 16
minimizing the non-dimensional pressure drop (f Re) in turbulent flow in helically corrugated tubes. It will be shown in the results that by using the multi-objective optimization approach, very important relations can be obtained which can be used in the thermal design of fluid flow in helically corrugated tubes. Keywords: helically corrugated tubes, multi-objective optimization, NSGA II, turbulent tube flows, heat transfer enhancement.
Nomenclature D
Tube diameter [m]
e
Corrugated rib height [m]
Density [kg m-3]
f
Friction factor [-]
Dynamic viscosity [N s m-2]
h
Heat transfer coefficient [Wm-2K-1]
k
Thermal conductivity [W m-1 K-1]
L
Length of tubes [m]
Greek symbol
Subscripts H
Hydraulic
Nu Nusselt Number ( hD h / k ) [-] P
Pitch of corrugated tubes [m]
Abbreviations
Po
Poiseuille number (= f Re)
NSGA Non-dominated Sorting Genetic Algorithms
Pr
Prandtl number ( / ) [-]
Re
Reynolds number ( UD h / ) [-]
U
Velocity [m s-1]
1. Introduction
3 Page 3 of 16
It is highly important to enhance the amount of heat transfer in various engineering fields. In general, the heat transfer enhancing methods can be divided into active and passive approaches; which the passive method, due to its ease of use and lower cost, has greatly attracted the attention of the researchers and engineers. A highly efficient technique for passive increasing the amount of heat transfer is to modify the tube surface so that the flow near the tube wall mixes more intensively [1-20]. Some examples of these modified tubes include the corrugated tubes, finned tubes, helical tubes, elliptical axis tubes, micro finned tubes, etc., as can be seen in Fig. 1. Among these modified tubes, corrugated tubes enjoy excellent thermal properties, because the amount of heat transfer enhancement in them is more than the pressure drop penalty of them. Because of the good thermal behavior of these tubes, numerous experimental, numerical and analytical investigations have been conducted on their characteristics. Rainieri and Pagliarini [6] investigated the heat transfer behavior of corrugated tubes with different pitch ratios. They showed that by using this type of tubes, flow swirl and vorticity increases extensively. Vicente et al. [7] explored the thermal behavior of corrugated tubes with different roughness geometries. They found out that the use of corrugated tubes can increase the heat transfer coefficient and the friction factor by 250 and 300%, respectively. Barba et al. [8] studied the flow of ethylene glycol in these tubes. Their results indicated that the friction factor in these tubes can increase up to 2.45 times related to the friction factor in plain tubes. Rozzi et al. [9] investigated the use of corrugated tubes with different Newtonian and non-Newtonian fluids. Through a set of experiments, Vicente et al. [10] used corrugated tubes to study the flows of water as well as ethylene glycol in laminar and turbulent flow regimes. Their results indicated that by using corrugated tubes, the amount of heat transfer and friction factor in these tubes can increase up to 30 and 25%, respectively. Dong et al. [11] investigated the flow of water and oil in spirally
4 Page 4 of 16
corrugated tubes. Their results showed that the heat transfer coefficient and friction factor of the spirally corrugated tube are higher than those of the plain tube up to 120% and 160%, respectively. Laohalertdecha and Wongwises [12] examined the flow of R-134a refrigerant in corrugated tubes. They discovered that by using these tubes, the amounts of heat transfer and friction coefficient increase considerably. Employing simple analytical models, Zimparov [13, 14] calculated the heat transfer and friction coefficients in corrugated tubes containing twisted tapes. Their results indicated the high accuracy of analytical models in comparison with empirical data. Through a set of experiments, Pethkool et al. [19] used helically corrugated tubes to investigate the effect of pitch and rib-height of corrugated tubes on the thermal behavior of fluid flow. Based on our information, no research has been carried out so far on the optimization of fluid flow in helically corrugated tubes. Although the use of helically corrugated tubes considerably enhances the heat transfer rate, it also increases the fluid pressure drop. So to achieve the optimal thermal behaviors in these tubes, a multi-objective optimization (MOO) approach should be used to discover the best possible design points with appropriate heat transfer and pressure drop values. NSGA II algorithm is one of the best and most complete multi-objective optimization algorithms, which will be used in this paper as well. This algorithm was first proposed by Deb [21], and it has been used in recent years in various engineering-related applications [22-25]. In this paper, experimentally derived correlations of Nusselt number (Nu) and friction factor (f) are used in a Pareto based multi-objective optimization approach to find the best possible combinations of heat transfer and pressure drop in helically corrugated tubes, known as the Pareto front. It will be shown that by using the multi-objective optimization approach, very
5 Page 5 of 16
important relations can be obtained which can be used in the thermal design of turbulent flow in helically corrugated tubes.
2. Defining the design variables In the present study there are four independent design variables: pitch to diameter ratio (
P
),
DH
rib height to diameter ratio (
e
), Reynolds number (Re) and Prandtl number (Pr). The
DH
graphical definitions of
P DH
and
e
are shown in Fig. 2; and the variation ranges of each
DH
design variables are shown in Table 1. According to Re values of Table 1, it is obvious that the flow regime is turbulent. Moreover the variation range of Pr is related to water as working fluid.
3. Defining the objective functions In heat exchangers tubes, heat transfer coefficient and pressure drop should be maximized and minimized respectively. Pethkool et al. [19] investigated the effects of various parameters such as
P DH
,
e
, and Re number on the heat transfer performance and pressure drop of helically
DH
corrugated tubes in a series of experimental tests. The schematic arrangement of experimental apparatus of them is shown in Fig. 3. In the test runs, the chilled water at constant flow rate was used as the cooling medium in the cooler. Then constant temperature of the water in storage tank was maintained by circulating chilled water through a cooling tower. In the test runs, the water tank was filled with cold water and then the water in the tank was pumped to a boiler. Temperatures of the inlet and outlet of the cold and the hot waters were recorded at the steady state condition and pressure drop across the test tube was measured by two static pressure taps. At the end, they presented two correlations indicating Nusselt number (Nu) and friction factor (f) 6 Page 6 of 16
P
as two functions of
e
,
DH
, Re and Pr , which are used in the present study for multi-
DH
objective optimization of heat transfer and pressure drop of turbulent flow in helically corrugated tubes. The empirical correlation of Pethkool et al. [19] for Nu
hD H k
and f
P L 1 2 D 2 U H
are as follows: Nu 1 . 579 (
P
)
0 . 35
e
(
DH
f 1 . 15 (
P
)
)
0 . 46
Re
0 . 639
Pr
0 .3
(1)
DH
0 . 164
DH
e
(
)
0 . 179
Re
0 . 239
(2)
DH
It should be noted that the behavior of f with respect to changing Re is not similar to that of P , for example increasing Re leads to decrease in f but to increase in P . Hence to solve this problem, instead of f, another parameter, namely f Re
P / L U / D
2
(Poiseuille number), which
has the same behavior with P with respect to changing Re, should be investigated in optimization process [26, 27]. Multiplying a Re on both side of Eq. (2) yields the f Re number as follows: f Re 1 . 15 (
P DH
)
0 . 164
(
e
)
0 . 179
Re
0 . 761
(3)
DH
Therefore the objective functions in the present paper is maximizing Nu (non-dimensional heat transfer coefficient) and minimizing f Re (non-dimensional pressure drop) which are presented in Eqs. (1) and (3) respectively.
4. Multi-objective optimization results
7 Page 7 of 16
In order to investigate the optimal thermal performance of helically corrugated tubes, the experimentally derived correlations which were presented in section 3 are now employed in a multi-objective optimization procedure using NSGA II algorithms [21]. In all runs a population size of 60 has been chosen with crossover probability (Pc) and mutation probability (Pm) as 0.7 and 0.07 respectively. The two conflicting objectives are Nu (non-dimensional heat transfer coefficient) and f Re (non-dimensional pressure drop) that should be optimized simultaneously with respect to the design variables
P DH
,
e
, Re and Pr (Table 1). The multi-objective
DH
optimization problem can be formulated in the following form:
Maximize Minimize Subject to :
Nu f 1 (
P DH
f Re f 2 (
P
0 . 18
e
, P
DH
, Re, Pr)
DH e
,
, Re)
DH
0 . 27
(4)
DH 0 . 02
e
0 . 06
DH 5500 Re 60000 3 Pr 6
Fig. 4 shows the Pareto front obtained for the two mentioned objective functions. As this figure indicates, the points have no dominancy over one another; i.e., no two points could be found with one of their objective functions equal to each other and another of their objective functions different from each other. In other words, if we move from one optimal point to another, one objective function will definitely improve and another objective function will certainly get worse. Although all the points in this Pareto front are optimal points, four points 8 Page 8 of 16
with special and unique characteristics are also observed, which have been designated as Points A, B, C and D. The details of the design variables, objective function values and geometry of these four optimal points have been shown in Table 2 and Fig. 5 respectively. Points A and D display the least f Re (pressure drop) and the highest Nu (heat transfer coefficient), respectively. Point B, which is known as the breaking point, has an interesting behavior. Actually, as one moves from Point A to Point B, f Re increases slightly (about 5.89%), while the value of Nu increases more considerably (about 13.22%). In general, an optimal point in thermal design is a point where both of the objective functions are equally satisfied. In this paper the mapping method [23] is used to compute and find such a point. In this method, both objective functions are mapped between 0 and 1 and then the norm of their sum is calculated. A point with the highest norm is the point at which both objective functions have been optimized to the same value. In this paper, Point C has been determined by using the mapping methods, and both objective functions are equally satisfied at this point. The changes of the optimum design variables with respect to the each objective function can be useful for the optimal thermal designing of helically corrugated tubes. Figs. 6 and 7 illustrate the changes of Nu and f Re respectively, versus the input design variables from point A to D. It is obvious from these figures that
P DH
and
e
vary almost linearly from A to B and are almost
DH
constant from Point B to D. Similarly, Re is constant from A to B and varies almost linearly from Point B to D. These useful and valuable relations that exist between the design variables of helically corrugated tubes cannot be extracted without using the multi-objective optimization approach presented in this paper.
9 Page 9 of 16
To make an interesting and useful comparison, the optimal data obtained from the Pareto front are compared and illustrated along with the existing experimental data of Pethkool et al. [19]. Fig. 8 shows the overlap of the Pareto front and the related empirical data. According to this figure, the Pareto front distinguishes the best boundary of the experimental data with regards to the lowest f Re and the highest Nu; which confirms the validity of the multi-objective optimization approach presented in this paper.
5. Conclusion In this paper, appropriate empirical correlations and also the NSGA II algorithm were employed to successfully achieve a multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes. The design variables were
P DH
,
e
, Re and Pr and the
DH
ultimate goal was to simultaneously increase the non-dimensional heat transfer coefficient (Nu) and reduce the non-dimensional pressure drop (f Re) in turbulent flow in helically corrugated tubes. It was shown that very important relations regarding the thermal performance of helically corrugated tubes were discovered, which could not be extracted without using the multiobjective optimization approach presented in this paper. At the end, the Pareto front obtained in this paper was compared and overlaid with the existing empirical data and it was found that the obtained Pareto front can distinguish the best boundary of the experimental data with regards to the minimum f Re and the maximum amount of Nu number which confirms the validity of the multi-objective optimization approach presented in this paper.
References
10 Page 10 of 16
[1]P.G. Rousseau, M. van Eldik, G.P. Greyvenstein, Detailed simulation of fluted tube water heating condensers. Int. J. Refrig. 26 (2003) 232–239. [2]Y. Qi, Y. Kawaguchi, Z. Lin, M. Ewing, R.N. Christensen, J.L. Zakin, Enhanced heat transfer of drag reducing surfactant solutions with fluted tube-in-tube heat exchanger. Int. J. Heat Mass Transf. 44 (2001) 1495–1505. [3]Y.T. Kang, R. Stout, R.N. Christensen, The effects of inclination angle on flooding in a helically fluted tube with a twisted insert. Int. J. Multiphase Flow 23 (1997) 1111–1129. [4]Z. Dengliang, X. Hong, S. Yan, Q. Baojin, Numerical heat transfer analysis of laminar film condensation on a vertical fluted tube. Appl. Therm. Eng. 30 (2010) 1159–1163. [5]L. Wanga, D. Suna, P. Liangb, L. Zhuangb, Y. Ta, Heat transfer characteristics of carbon steel spirally fluted tube for high pressure preheaters. Energy Convers. Manage. 41 (2000) 993– 1005. [6]S. Rainieri, G. Pagliarini, Convective heat transfer to temperature dependent property fluids in the entry region of corrugated tubes. Int. J. Heat Mass Transf. 45 (2002) 4525–4536. [7]P.G. Vicente, A. Garc, A. Viedma, Experimental investigation on heat transfer and frictional characteristics of spirally corrugated tubes in turbulent flow at different Prandtl numbers. Int. J. Heat Mass Transf. 47 (2004) 671–681. [8]A. Barba, S. Rainieri, M. Spiga, Heat transfer enhancement in a corrugated tube. Int. Commun. Heat Mass Transfer 29 (2002) 313–322. [9]S. Rozzi, R. Massini, G. Paciello, G. Pagliarini, S. Rainieri, A. Trifiro, Heat treatment of fluid foods in a shell and tube heat exchanger: comparison between smooth and helically corrugated wall tubes. J. Food Eng. 79 (2007) 249–254.
11 Page 11 of 16
[10] P.G. Vicente, A. García, A. Viedma, Mixed convection heat transfer and isothermal pressure drop in corrugated tubes for laminar and transition flow. Int. Commun. Heat Mass Transfer 31 (2004) 651–662. [11] Y. Dong, L. Huixiong, C. Tingkuan, Pressure drop, heat transfer and performance of singlephase turbulent flow in spirally corrugated tubes. Exp. Therm. Fluid Sci. 24 (2001) 131–138. [12] S. Laohalertdecha, S. Wongwises, The effects of corrugation pitch on the condensation heat transfer coefficient and pressure drop of R-134a inside horizontal corrugated tube. Int. J. Heat Mass Transf. 53 (2010) 2924–2931. [13] V. Zimparov, Prediction of friction factors and heat transfer coefficients for turbulent flow in corrugated tubes combined with twisted tape inserts. Part 1: friction factors. Int. J. Heat Mass Transf. 47 (2004) 589–599. [14] V. Zimparov, Prediction of friction factors and heat transfer coefficients for turbulent flow in corrugated tubes combined with twisted tape inserts. Part 2: heat transfer coefficients. Int. J. Heat Mass Transf. 47 (2004) 385–393. [15] J.A. Meng, X.G. Liang, Z.J. Chen, Z.X. Li, Experimental study on convective heat transfer in alternating elliptical axis tubes. Exp. Therm. Fluid Sci. 29 (2005) 457–465. [16] Z. Zhnegguo, X. Tao, F. Xiaoming, Experimental study on heat transfer enhancement of a helically baffled heat exchanger combined with three dimensional finned tubes. Appl. Therm. Eng. 24 (2004) 2293–2300. [17] S. Al-Fahed, L.M. Chamra, W. Chakroun, Pressure drop and heat transfer comparison for both microfin tube and twisted-tape inserts in laminar flow. Exp. Therm. Fluid Sci. 18 (1999) 323–333.
12 Page 12 of 16
[18] R.L. Webb, R. Narayanamurthy, P. Thors, Heat transfer and friction characteristics of internal helical-rib roughness, Transactions of the ASME. J. Heat Transf. 122 (2000) 134– 142. [19] S. Pethkool, S. Eiamsa-ard, S. Kwankaomeng, P. Promvonge, Turbulent heat transfer enhancement in a heat exchanger using helically corrugated tube. Int. Commun. Heat Mass Transfer 38 (2011) 340–347. [20] A. Durmus, M. Ese, Investigation of heat transfer and pressure drop in a concentric heat exchanger with snail entrance. Appl. Therm. Eng. 22 (3) (2002) 321–332. [21] K. Deb, S. Agrawal, A. Pratap and T. Meyarivan, T., A fast and elitist multi-objective genetic algorithm: NSGA-II”. IEEE Trans Evolutionary Computation, 6 (2002) 182-97. [22] H. Safikhani, M. A. Akhavan-Behabadi, N. Nariman-Zadeh and M. J. Mahmoodabadi, Modeling and multi-objective optimization of square cyclones using CFD and neural networks. Chem. Eng. Res. Des. 89 (2011) 301-309. [23] H. Safikhani, A. Hajiloo, M. A. Ranjbar, Modeling and multi-objective optimization of cyclone separators using CFD and genetic algorithms. Comput. Chem. Eng. 35 (6) (2011) 1064–1071. [24] H. Safikhani, A. Abbassi, A. Khalkhali, M. Kalteh, Multi-objective optimization of nanofluid flow in flat tubes using CFD, artificial neural networks and genetic algorithms. Adv. Powder Technol. 25 (5) (2014) 1608-1617. [25] N. Amanifard, N. Nariman-Zadeh, M. Borji, A. Khalkhali and A. Habibdoust, Modeling and Pareto optimization of heat transfer and flow coefficients in micro channels using GMDH type neural networks and genetic algorithms. Energy Convers. Manage. 49 (2008) 311-325.
13 Page 13 of 16
[26] S. W. Churchil, Viscous Flows: The Practical Uses of Theory, Butterworth, Boston, 1988, pp. 9, 40. [27] A. Bejan, Convection heat transfer, Wiley, 2004, pp. 107.
Figure Captions Figure 1: Geometry of different tube surfaces.
Figure 2: Schematic definition of
P DH
and
e
.
DH
Figure 3: Experimental set up for investigation of thermal performance of helically corrugated tubes [19]. Figure 4: Multi-objective Pareto results for Nu and f Re of optimal design points. Figure 5: Geometry of optimal points which are derived from Pareto front. Figure 6: Optimal variations of Nu with respect to design variables. Figure 7: Optimal variations of f Re with respect to design variables.
14 Page 14 of 16
Figure 8: Overlap graph of the obtained optimal Pareto front with the related experimental data [19].
15 Page 15 of 16
Table 1: Design variables and their range of variations. Design Variables From P DH e DH
To
0.18
0.27
0.02
0.06
5500 60000
Re
3
Pr
6
Table 2: The values of objective functions and their associated design variables of the optimum points. P
e
DH
DH
A
0.185
B
Point
Re
Pr
Nu
f Re
0.02
5500
6
61.76
305.85
0.237
0.06
6365
6
120.68
430.70
C
0.258
0.06
25396
6
299.71
1249.85
D
0.268
0.06
60000
6
507.56
2423.96
16 Page 16 of 16
S1359-4311(15)01269-7 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.033 ATE 7314
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
8-4-2015 9-11-2015
Please cite this article as: Hamed Safikhani, Smith Eiamsa-ard, Pareto based multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes, Applied Thermal Engineering (2015), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.033. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Pareto based multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes Hamed Safikhani 1, and Smith Eiamsa-ard 2 1
Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-88349, Iran 2
Department of Mechanical Engineering, Faculty of Engineering, Mahanakorn University of Technology, Bangkok 10530, Thailand
Multi-objective optimization of turbulent flow in corrugated tubes is investigated. The objective functions are experimentally derived correlations of heat transfer and pressure drop.
The design variables are two geometrical parameters of tubes (
P DH
and
e
) and Re and
DH
Pr numbers. The Pareto front comprises important thermal design principles.
Corresponding author: Tel.: +98 21 64543425, Fax: +98 21 66419736, Email: [email protected],
[email protected] (Hamed Safikhani). 1 Page 1 of 16
Graphical Abstract
Abstract In this study, by using the empirical correlations for heat transfer and pressure drop and also employing a multi-objective optimization (MOO) approach, the best possible combinations of heat transfer and pressure drop in helically corrugated tubes will be discovered. The design variables are two geometrical parameters of helically corrugated tubes, namely pitch to diameter ratio (
P DH
) and rib height to diameter ratio (
e
), Reynolds number (Re) and Prandtl number
DH
(Pr). The objectives are maximizing the non-dimensional heat transfer coefficient (Nu) and 2 Page 2 of 16
minimizing the non-dimensional pressure drop (f Re) in turbulent flow in helically corrugated tubes. It will be shown in the results that by using the multi-objective optimization approach, very important relations can be obtained which can be used in the thermal design of fluid flow in helically corrugated tubes. Keywords: helically corrugated tubes, multi-objective optimization, NSGA II, turbulent tube flows, heat transfer enhancement.
Nomenclature D
Tube diameter [m]
e
Corrugated rib height [m]
Density [kg m-3]
f
Friction factor [-]
Dynamic viscosity [N s m-2]
h
Heat transfer coefficient [Wm-2K-1]
k
Thermal conductivity [W m-1 K-1]
L
Length of tubes [m]
Greek symbol
Subscripts H
Hydraulic
Nu Nusselt Number ( hD h / k ) [-] P
Pitch of corrugated tubes [m]
Abbreviations
Po
Poiseuille number (= f Re)
NSGA Non-dominated Sorting Genetic Algorithms
Pr
Prandtl number ( / ) [-]
Re
Reynolds number ( UD h / ) [-]
U
Velocity [m s-1]
1. Introduction
3 Page 3 of 16
It is highly important to enhance the amount of heat transfer in various engineering fields. In general, the heat transfer enhancing methods can be divided into active and passive approaches; which the passive method, due to its ease of use and lower cost, has greatly attracted the attention of the researchers and engineers. A highly efficient technique for passive increasing the amount of heat transfer is to modify the tube surface so that the flow near the tube wall mixes more intensively [1-20]. Some examples of these modified tubes include the corrugated tubes, finned tubes, helical tubes, elliptical axis tubes, micro finned tubes, etc., as can be seen in Fig. 1. Among these modified tubes, corrugated tubes enjoy excellent thermal properties, because the amount of heat transfer enhancement in them is more than the pressure drop penalty of them. Because of the good thermal behavior of these tubes, numerous experimental, numerical and analytical investigations have been conducted on their characteristics. Rainieri and Pagliarini [6] investigated the heat transfer behavior of corrugated tubes with different pitch ratios. They showed that by using this type of tubes, flow swirl and vorticity increases extensively. Vicente et al. [7] explored the thermal behavior of corrugated tubes with different roughness geometries. They found out that the use of corrugated tubes can increase the heat transfer coefficient and the friction factor by 250 and 300%, respectively. Barba et al. [8] studied the flow of ethylene glycol in these tubes. Their results indicated that the friction factor in these tubes can increase up to 2.45 times related to the friction factor in plain tubes. Rozzi et al. [9] investigated the use of corrugated tubes with different Newtonian and non-Newtonian fluids. Through a set of experiments, Vicente et al. [10] used corrugated tubes to study the flows of water as well as ethylene glycol in laminar and turbulent flow regimes. Their results indicated that by using corrugated tubes, the amount of heat transfer and friction factor in these tubes can increase up to 30 and 25%, respectively. Dong et al. [11] investigated the flow of water and oil in spirally
4 Page 4 of 16
corrugated tubes. Their results showed that the heat transfer coefficient and friction factor of the spirally corrugated tube are higher than those of the plain tube up to 120% and 160%, respectively. Laohalertdecha and Wongwises [12] examined the flow of R-134a refrigerant in corrugated tubes. They discovered that by using these tubes, the amounts of heat transfer and friction coefficient increase considerably. Employing simple analytical models, Zimparov [13, 14] calculated the heat transfer and friction coefficients in corrugated tubes containing twisted tapes. Their results indicated the high accuracy of analytical models in comparison with empirical data. Through a set of experiments, Pethkool et al. [19] used helically corrugated tubes to investigate the effect of pitch and rib-height of corrugated tubes on the thermal behavior of fluid flow. Based on our information, no research has been carried out so far on the optimization of fluid flow in helically corrugated tubes. Although the use of helically corrugated tubes considerably enhances the heat transfer rate, it also increases the fluid pressure drop. So to achieve the optimal thermal behaviors in these tubes, a multi-objective optimization (MOO) approach should be used to discover the best possible design points with appropriate heat transfer and pressure drop values. NSGA II algorithm is one of the best and most complete multi-objective optimization algorithms, which will be used in this paper as well. This algorithm was first proposed by Deb [21], and it has been used in recent years in various engineering-related applications [22-25]. In this paper, experimentally derived correlations of Nusselt number (Nu) and friction factor (f) are used in a Pareto based multi-objective optimization approach to find the best possible combinations of heat transfer and pressure drop in helically corrugated tubes, known as the Pareto front. It will be shown that by using the multi-objective optimization approach, very
5 Page 5 of 16
important relations can be obtained which can be used in the thermal design of turbulent flow in helically corrugated tubes.
2. Defining the design variables In the present study there are four independent design variables: pitch to diameter ratio (
P
),
DH
rib height to diameter ratio (
e
), Reynolds number (Re) and Prandtl number (Pr). The
DH
graphical definitions of
P DH
and
e
are shown in Fig. 2; and the variation ranges of each
DH
design variables are shown in Table 1. According to Re values of Table 1, it is obvious that the flow regime is turbulent. Moreover the variation range of Pr is related to water as working fluid.
3. Defining the objective functions In heat exchangers tubes, heat transfer coefficient and pressure drop should be maximized and minimized respectively. Pethkool et al. [19] investigated the effects of various parameters such as
P DH
,
e
, and Re number on the heat transfer performance and pressure drop of helically
DH
corrugated tubes in a series of experimental tests. The schematic arrangement of experimental apparatus of them is shown in Fig. 3. In the test runs, the chilled water at constant flow rate was used as the cooling medium in the cooler. Then constant temperature of the water in storage tank was maintained by circulating chilled water through a cooling tower. In the test runs, the water tank was filled with cold water and then the water in the tank was pumped to a boiler. Temperatures of the inlet and outlet of the cold and the hot waters were recorded at the steady state condition and pressure drop across the test tube was measured by two static pressure taps. At the end, they presented two correlations indicating Nusselt number (Nu) and friction factor (f) 6 Page 6 of 16
P
as two functions of
e
,
DH
, Re and Pr , which are used in the present study for multi-
DH
objective optimization of heat transfer and pressure drop of turbulent flow in helically corrugated tubes. The empirical correlation of Pethkool et al. [19] for Nu
hD H k
and f
P L 1 2 D 2 U H
are as follows: Nu 1 . 579 (
P
)
0 . 35
e
(
DH
f 1 . 15 (
P
)
)
0 . 46
Re
0 . 639
Pr
0 .3
(1)
DH
0 . 164
DH
e
(
)
0 . 179
Re
0 . 239
(2)
DH
It should be noted that the behavior of f with respect to changing Re is not similar to that of P , for example increasing Re leads to decrease in f but to increase in P . Hence to solve this problem, instead of f, another parameter, namely f Re
P / L U / D
2
(Poiseuille number), which
has the same behavior with P with respect to changing Re, should be investigated in optimization process [26, 27]. Multiplying a Re on both side of Eq. (2) yields the f Re number as follows: f Re 1 . 15 (
P DH
)
0 . 164
(
e
)
0 . 179
Re
0 . 761
(3)
DH
Therefore the objective functions in the present paper is maximizing Nu (non-dimensional heat transfer coefficient) and minimizing f Re (non-dimensional pressure drop) which are presented in Eqs. (1) and (3) respectively.
4. Multi-objective optimization results
7 Page 7 of 16
In order to investigate the optimal thermal performance of helically corrugated tubes, the experimentally derived correlations which were presented in section 3 are now employed in a multi-objective optimization procedure using NSGA II algorithms [21]. In all runs a population size of 60 has been chosen with crossover probability (Pc) and mutation probability (Pm) as 0.7 and 0.07 respectively. The two conflicting objectives are Nu (non-dimensional heat transfer coefficient) and f Re (non-dimensional pressure drop) that should be optimized simultaneously with respect to the design variables
P DH
,
e
, Re and Pr (Table 1). The multi-objective
DH
optimization problem can be formulated in the following form:
Maximize Minimize Subject to :
Nu f 1 (
P DH
f Re f 2 (
P
0 . 18
e
, P
DH
, Re, Pr)
DH e
,
, Re)
DH
0 . 27
(4)
DH 0 . 02
e
0 . 06
DH 5500 Re 60000 3 Pr 6
Fig. 4 shows the Pareto front obtained for the two mentioned objective functions. As this figure indicates, the points have no dominancy over one another; i.e., no two points could be found with one of their objective functions equal to each other and another of their objective functions different from each other. In other words, if we move from one optimal point to another, one objective function will definitely improve and another objective function will certainly get worse. Although all the points in this Pareto front are optimal points, four points 8 Page 8 of 16
with special and unique characteristics are also observed, which have been designated as Points A, B, C and D. The details of the design variables, objective function values and geometry of these four optimal points have been shown in Table 2 and Fig. 5 respectively. Points A and D display the least f Re (pressure drop) and the highest Nu (heat transfer coefficient), respectively. Point B, which is known as the breaking point, has an interesting behavior. Actually, as one moves from Point A to Point B, f Re increases slightly (about 5.89%), while the value of Nu increases more considerably (about 13.22%). In general, an optimal point in thermal design is a point where both of the objective functions are equally satisfied. In this paper the mapping method [23] is used to compute and find such a point. In this method, both objective functions are mapped between 0 and 1 and then the norm of their sum is calculated. A point with the highest norm is the point at which both objective functions have been optimized to the same value. In this paper, Point C has been determined by using the mapping methods, and both objective functions are equally satisfied at this point. The changes of the optimum design variables with respect to the each objective function can be useful for the optimal thermal designing of helically corrugated tubes. Figs. 6 and 7 illustrate the changes of Nu and f Re respectively, versus the input design variables from point A to D. It is obvious from these figures that
P DH
and
e
vary almost linearly from A to B and are almost
DH
constant from Point B to D. Similarly, Re is constant from A to B and varies almost linearly from Point B to D. These useful and valuable relations that exist between the design variables of helically corrugated tubes cannot be extracted without using the multi-objective optimization approach presented in this paper.
9 Page 9 of 16
To make an interesting and useful comparison, the optimal data obtained from the Pareto front are compared and illustrated along with the existing experimental data of Pethkool et al. [19]. Fig. 8 shows the overlap of the Pareto front and the related empirical data. According to this figure, the Pareto front distinguishes the best boundary of the experimental data with regards to the lowest f Re and the highest Nu; which confirms the validity of the multi-objective optimization approach presented in this paper.
5. Conclusion In this paper, appropriate empirical correlations and also the NSGA II algorithm were employed to successfully achieve a multi-objective optimization of turbulent heat transfer flow in helically corrugated tubes. The design variables were
P DH
,
e
, Re and Pr and the
DH
ultimate goal was to simultaneously increase the non-dimensional heat transfer coefficient (Nu) and reduce the non-dimensional pressure drop (f Re) in turbulent flow in helically corrugated tubes. It was shown that very important relations regarding the thermal performance of helically corrugated tubes were discovered, which could not be extracted without using the multiobjective optimization approach presented in this paper. At the end, the Pareto front obtained in this paper was compared and overlaid with the existing empirical data and it was found that the obtained Pareto front can distinguish the best boundary of the experimental data with regards to the minimum f Re and the maximum amount of Nu number which confirms the validity of the multi-objective optimization approach presented in this paper.
References
10 Page 10 of 16
[1]P.G. Rousseau, M. van Eldik, G.P. Greyvenstein, Detailed simulation of fluted tube water heating condensers. Int. J. Refrig. 26 (2003) 232–239. [2]Y. Qi, Y. Kawaguchi, Z. Lin, M. Ewing, R.N. Christensen, J.L. Zakin, Enhanced heat transfer of drag reducing surfactant solutions with fluted tube-in-tube heat exchanger. Int. J. Heat Mass Transf. 44 (2001) 1495–1505. [3]Y.T. Kang, R. Stout, R.N. Christensen, The effects of inclination angle on flooding in a helically fluted tube with a twisted insert. Int. J. Multiphase Flow 23 (1997) 1111–1129. [4]Z. Dengliang, X. Hong, S. Yan, Q. Baojin, Numerical heat transfer analysis of laminar film condensation on a vertical fluted tube. Appl. Therm. Eng. 30 (2010) 1159–1163. [5]L. Wanga, D. Suna, P. Liangb, L. Zhuangb, Y. Ta, Heat transfer characteristics of carbon steel spirally fluted tube for high pressure preheaters. Energy Convers. Manage. 41 (2000) 993– 1005. [6]S. Rainieri, G. Pagliarini, Convective heat transfer to temperature dependent property fluids in the entry region of corrugated tubes. Int. J. Heat Mass Transf. 45 (2002) 4525–4536. [7]P.G. Vicente, A. Garc, A. Viedma, Experimental investigation on heat transfer and frictional characteristics of spirally corrugated tubes in turbulent flow at different Prandtl numbers. Int. J. Heat Mass Transf. 47 (2004) 671–681. [8]A. Barba, S. Rainieri, M. Spiga, Heat transfer enhancement in a corrugated tube. Int. Commun. Heat Mass Transfer 29 (2002) 313–322. [9]S. Rozzi, R. Massini, G. Paciello, G. Pagliarini, S. Rainieri, A. Trifiro, Heat treatment of fluid foods in a shell and tube heat exchanger: comparison between smooth and helically corrugated wall tubes. J. Food Eng. 79 (2007) 249–254.
11 Page 11 of 16
[10] P.G. Vicente, A. García, A. Viedma, Mixed convection heat transfer and isothermal pressure drop in corrugated tubes for laminar and transition flow. Int. Commun. Heat Mass Transfer 31 (2004) 651–662. [11] Y. Dong, L. Huixiong, C. Tingkuan, Pressure drop, heat transfer and performance of singlephase turbulent flow in spirally corrugated tubes. Exp. Therm. Fluid Sci. 24 (2001) 131–138. [12] S. Laohalertdecha, S. Wongwises, The effects of corrugation pitch on the condensation heat transfer coefficient and pressure drop of R-134a inside horizontal corrugated tube. Int. J. Heat Mass Transf. 53 (2010) 2924–2931. [13] V. Zimparov, Prediction of friction factors and heat transfer coefficients for turbulent flow in corrugated tubes combined with twisted tape inserts. Part 1: friction factors. Int. J. Heat Mass Transf. 47 (2004) 589–599. [14] V. Zimparov, Prediction of friction factors and heat transfer coefficients for turbulent flow in corrugated tubes combined with twisted tape inserts. Part 2: heat transfer coefficients. Int. J. Heat Mass Transf. 47 (2004) 385–393. [15] J.A. Meng, X.G. Liang, Z.J. Chen, Z.X. Li, Experimental study on convective heat transfer in alternating elliptical axis tubes. Exp. Therm. Fluid Sci. 29 (2005) 457–465. [16] Z. Zhnegguo, X. Tao, F. Xiaoming, Experimental study on heat transfer enhancement of a helically baffled heat exchanger combined with three dimensional finned tubes. Appl. Therm. Eng. 24 (2004) 2293–2300. [17] S. Al-Fahed, L.M. Chamra, W. Chakroun, Pressure drop and heat transfer comparison for both microfin tube and twisted-tape inserts in laminar flow. Exp. Therm. Fluid Sci. 18 (1999) 323–333.
12 Page 12 of 16
[18] R.L. Webb, R. Narayanamurthy, P. Thors, Heat transfer and friction characteristics of internal helical-rib roughness, Transactions of the ASME. J. Heat Transf. 122 (2000) 134– 142. [19] S. Pethkool, S. Eiamsa-ard, S. Kwankaomeng, P. Promvonge, Turbulent heat transfer enhancement in a heat exchanger using helically corrugated tube. Int. Commun. Heat Mass Transfer 38 (2011) 340–347. [20] A. Durmus, M. Ese, Investigation of heat transfer and pressure drop in a concentric heat exchanger with snail entrance. Appl. Therm. Eng. 22 (3) (2002) 321–332. [21] K. Deb, S. Agrawal, A. Pratap and T. Meyarivan, T., A fast and elitist multi-objective genetic algorithm: NSGA-II”. IEEE Trans Evolutionary Computation, 6 (2002) 182-97. [22] H. Safikhani, M. A. Akhavan-Behabadi, N. Nariman-Zadeh and M. J. Mahmoodabadi, Modeling and multi-objective optimization of square cyclones using CFD and neural networks. Chem. Eng. Res. Des. 89 (2011) 301-309. [23] H. Safikhani, A. Hajiloo, M. A. Ranjbar, Modeling and multi-objective optimization of cyclone separators using CFD and genetic algorithms. Comput. Chem. Eng. 35 (6) (2011) 1064–1071. [24] H. Safikhani, A. Abbassi, A. Khalkhali, M. Kalteh, Multi-objective optimization of nanofluid flow in flat tubes using CFD, artificial neural networks and genetic algorithms. Adv. Powder Technol. 25 (5) (2014) 1608-1617. [25] N. Amanifard, N. Nariman-Zadeh, M. Borji, A. Khalkhali and A. Habibdoust, Modeling and Pareto optimization of heat transfer and flow coefficients in micro channels using GMDH type neural networks and genetic algorithms. Energy Convers. Manage. 49 (2008) 311-325.
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[26] S. W. Churchil, Viscous Flows: The Practical Uses of Theory, Butterworth, Boston, 1988, pp. 9, 40. [27] A. Bejan, Convection heat transfer, Wiley, 2004, pp. 107.
Figure Captions Figure 1: Geometry of different tube surfaces.
Figure 2: Schematic definition of
P DH
and
e
.
DH
Figure 3: Experimental set up for investigation of thermal performance of helically corrugated tubes [19]. Figure 4: Multi-objective Pareto results for Nu and f Re of optimal design points. Figure 5: Geometry of optimal points which are derived from Pareto front. Figure 6: Optimal variations of Nu with respect to design variables. Figure 7: Optimal variations of f Re with respect to design variables.
14 Page 14 of 16
Figure 8: Overlap graph of the obtained optimal Pareto front with the related experimental data [19].
15 Page 15 of 16
Table 1: Design variables and their range of variations. Design Variables From P DH e DH
To
0.18
0.27
0.02
0.06
5500 60000
Re
3
Pr
6
Table 2: The values of objective functions and their associated design variables of the optimum points. P
e
DH
DH
A
0.185
B
Point
Re
Pr
Nu
f Re
0.02
5500
6
61.76
305.85
0.237
0.06
6365
6
120.68
430.70
C
0.258
0.06
25396
6
299.71
1249.85
D
0.268
0.06
60000
6
507.56
2423.96
16 Page 16 of 16