Parametric study and optimization of H-type finned tube heat exchangers using Taguchi method

Accepted Manuscript Parametric study and optimization of H-type finned tube heat exchangers using Taguchi method Heng Wang, Ying-wen Liu, Peng Yang, Ren-jie Wu, Ya-ling He PII: DOI: Reference:

S1359-4311(16)30325-8 http://dx.doi.org/10.1016/j.applthermaleng.2016.03.033 ATE 7901

To appear in:

Applied Thermal Engineering

Received Date: Accepted Date:

23 October 2015 5 March 2016

Please cite this article as: H. Wang, Y-w. Liu, P. Yang, R-j. Wu, Y-l. He, Parametric study and optimization of Htype finned tube heat exchangers using Taguchi method, Applied Thermal Engineering (2016), doi: http://dx.doi.org/ 10.1016/j.applthermaleng.2016.03.033

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Parametric study and optimization of H-type finned tube heat exchangers using Taguchi method Heng Wang, Ying-wen Liu*, Peng Yang, Ren-jie Wu, Ya-ling He Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an

Jiaotong University, Xi’an , shaanxi 710049, P R China

Abstract In this paper, a three-dimensional numerical model of H-type finned tube heat exchangers has been built. In order to optimize its structure and improve its performance, Taguchi method is applied to investigate the influence of seven geometric parameters including slit width, row number, fin height, spanwise tube pitch, longitudinal tube pitch, fin thickness and fin pitch. Numerical simulations are performed on eighteen cases with different combinations of seven geometric parameters and the overall thermal-hydraulic performance and its characteristics of heat transfer and flow friction are discussed in detail. The results show that fin pitch and fin height play a significant role on the heat transfer characteristics while the flow friction characteristics are mainly affected by fin height, spanwise tube pitch and fin pitch. The intuitive analysis and analysis of variance show that fin pitch, fin height and fin thickness have much stronger influence on its overall thermal-hydraulic performance than the rest four parameters. Finally, the optimal parametric combination of the H-type finned tube is obtained with an improvement of overall thermal-hydraulic performance by 11.4% to 16% for Re in range from 9000 to 24000.

Keywords: H-type finned tube, Taguchi method, numerical optimization, thermal-hydraulic performance

*Corresponding author. Tel.: (86)13087588436 Email address: [email protected]

1. Introduction Heat exchangers are widely used in various industries, such as space heating, refrigeration, air conditioning, power stations, chemical plants, petrochemical plants, petroleum refineries and waste heat recovery process. A heat exchanger is a device used to transfer heat between one or more fluids, which may be separated by a wall to prevent mixing or may be in direct contact. In order to optimize heat exchangers, it is necessary to enhance heat transfer, decrease fan power and reduce volume of heat exchangers at the same time. For many heat exchangers, air is often chosen as one of the working fluids while the other fluid is often water with high heat transfer coefficient. Hence it is especially important to improve the heat transfer at the air side of heat exchangers where the major thermal resistance lies. One of the important approaches is to use the extended heat transfer surface, such as plate fin [1], wave fin [2], louver fin [3], slit fin [4], etc. Otherwise, in order to solve the problem of energy shortage and environment pollution, plenty of research has been focused on the waste heat recovery. Although the low temperature waste heat in exhaust gas stream has lower capacity to do work as compared to the original hot gas according to the second law of thermodynamics, the amount of such low-utility waste heat is enormous in various industries. Hence the waste heat recovery is necessary in order to save energy and increase the efficiency of the system. Because of the advantages of anti-wear and anti-fouling, H-type finned tube heat exchangers have been widely used in boilers and waste heat recovery units to enhance the gas-side convection heat transfer. In recent years, a number of experimental and numerical studies have been reported on the air side heat transfer and flow friction characteristics of H-type finned tube. Liu et al. [5] carried out an experimental study on heat transfer and flow friction characteristics of H-type finned tube and proposed the correlations of Nusselt number (Nu) and Euler number (Eu). Wang et al. [6] 2

numerically examined the heat transfer and resistance characteristics as well as the comprehensive performance of two kinds of H-type finned tube (single and double). Zhang et al. [7] employed the one-factor-at-a-time method to optimize the geometric parameters of H-type finned tube, including fin height, fin pitch, transverse pitch and longitudinal pitch, in order to achieve the best comprehensive performance. Jin et al. [8] numerically studied the effects of some parameters on heat transfer and pressure drop characteristics of H-type finned tube bank with 10 rows. Chen et al. [9] experimentally investigated the effects of fin height, fin width, fin pitch and air velocity on fin efficiency, convective heat transfer coefficient, integrated heat transfer capacity and pressure drop. Wu et al. [10] numerically studied the heat transfer characteristics of H-type finned tube in both staggered and in-line arrangement. Jiang et al. [11] proposed a novel H-type finned tube with longitudinal vortex generators and employed CFD method to study the external flow and heat transfer characteristics. Zhao et al. [12] conducted numerical simulations to explore heat transfer and erosion characteristics for H-type finned oval tube with longitudinal vortex generators and dimples. He et al. [13] performed a three-dimensional numerical study on sulfuric acid deposition characteristics of H-type finned tube bank with 10 rows. Wang et al. [14-15] conducted numerical studies on the H-type finned oval tube with bleeding dimples and longitudinal vortex generators, where parametric effects of gas temperature, acid vapor concentration, water vapor concentration and Reynolds number on heat transfer and acid condensation were investigated. The previous studies provide crucial understanding and design guidelines for H-type finned tube. However, all of them adopted the one-factor-at-a-time method which required great efforts and did not consider the interactive effects between multiple geometric parameters of heat exchangers. Therefore, no reliable optimal parametric combination had been given. Due to the enormous costs, it is not always desired or practical to conduct a great number of 3

experiments and numerical studies to optimize the process or products with separate treatment for each of the parameters. Therefore, it is meaningful to develop a rational and proper method which can tremendously reduce costs and time requirement. Taguchi method, which is developed by Genichi Taguchi to improve the quality of products, is one of the optimizing design methods based on statistical principle. Taguchi method can simplify the process of optimization through uniformly combining levels of different factors and thus decreasing the number of experiments [16, 17]. Taguchi method has been widely and successfully applied to the study of heat exchangers [18-24]. For example, Sahin [18] experimentally studied the effects of Reynold number, fin height and pitch on heat transfer performance of a heat exchanger having cylindrical pin fins positioned in a rectangular channel. The optimal values of these three parameters were determined by Taguchi method. Zeng et al. [19] numerically optimized some parameters of a vortex-generator fin-and-tube heat exchanger using Taguchi method. Turgut et al. [20] experimentally determined the optimal parameter for the concentric heat exchanger with injector turbulators using Taguchi method. However, no study has been done to make a comprehensive parametric optimization of H-type finned tube heat exchangers using Taguchi method, which can simplify the design process in terms of time and effort. In this paper, a three dimensional numerical model of the H-type finned tube heat exchanger is performed. Based on Taguchi method, geometric parameters of the H-type finned tube are optimized. Seven parameters (slit width, number of tube rows, fin height, spanwise tube pitch, longitudinal tube pitch, fin thickness and fin pitch) are selected as the independent variables. The optimization range of each factor is specified to three levels. Heat transfer and flow friction characteristics are two main aspects of heat exchanger performance so they are regarded as the objectives to be optimized. In addition, the overall thermal-hydraulic performance is set to be another objective in this study that takes both heat 4

transfer and flow friction characteristics into account. The objective of the optimization is to reach the highest heat transfer and overall thermal-hydraulic performance as well as the lowest flow friction through combining optimal levels of the seven geometric parameters. In addition, the contribution ratios of each factor to the thermal hydraulic performance are obtained and analyzed. The validation test is performed to prove the additivity of factorial effects and the thus-obtained optimal parameter combination is justified. This study can provide useful guidelines and practical methodologies for design and optimization of the H-type finned tube heat exchangers.

2. Model description and numerical method 2.1. Physical model

(a) Physical model

(b) Top view of the computational domain

5

(c) Side view of the computational domain

Fig. 1 Schematic diagram of the H-type finned tube heat exchanger

A schematic diagram of an H-type finned tube heat exchanger with 5 rows is shown in Figure 1. Fig. 1(b) and Fig. 1(c) present the top view and the side view of the heat exchanger, respectively. Due to the geometric symmetry, the computational domain can be reduced to half of the finned tube with half of the spacing, which is denoted by the area enclosed by the rectangular frame. The top and bottom edges of the computational domain in Fig. 1(c) (shown in shaded area) are symmetrical planes of the two adjacent fins.

2.2. Governing equations and boundary conditions The fluid is assumed to be incompressible gas with constant thermo-physical properties at the average temperature of the flow field and the gas flow is steady. Because thermal resistance at water side and thermal resistance across the tube wall are much smaller due to the high convective heat transfer coefficient at the water side and the high thermal conductivity of tube wall, the outer tube wall can be assumed to be at constant temperature. Nevertheless the temperature variation on the fin surface cannot be neglected due to its relatively large area. Thus the computation of the fin domain is conjugated where the temperature at both solid fin surface and fluid is determined simultaneously [25]. In the numerical 6

model, the material for fins and tubes is set to be carbon steel with constant thermal conductivity. The governing equations include mass, momentum and energy conservation equations. The dimensionless form in tensor notation is presented below:


∇ ⋅ ( ρV φ ) = ∇ ⋅ (Γφ gradφ ) + Sφ

(1)

In the equation, the dependent variable φ can stand for the velocity components, temperature, turbulent kinetic energy k and dissipation rate ε . Γφ and Sφ represent the diffusion coefficient and the source term, respectively. The RNG turbulence model [26] is adopted in the numerical simulation which is shown as follows:

ρ

Dk ∂ = Dt ∂xi

ρ

 ∂k  α p µeff  + µ t − ρε ∂xi  

(2)

Dε ∂  ∂ε  ε ε2 2 α µ µ ρ = + C S − C −R t 2ε  p eff  1ε Dt ∂xi  ∂xi  k k

where the effective viscosity

µ eff = µ + µt

(3)

2 and µt = ρ Cµ k with C µ = 0.0845 . ε

The rate of strain term R is given by R=

Cµ ρη 3 (1 − η / η0 )ε 2

(4)

(1 + βη 3 )k

where η = Sk / ε , η 0 = 4.38 and β = 0.012 . The modulus of the rate of strain tensor S is given by

S 2 = 2Sij Siij

(5)

 ∂u  where Si j = 1  ∂ui + j  . The turbulence model constants are given by the RNG model. C1ε and  2  ∂x j ∂xi 

C2ε are set as 1.42 and 1.68, respectively.

7

Fig. 2 boundary conditions for the numerical simulation

The computational domain includes inlet extended region, heat transfer region and outlet extended region. As is shown in Fig. 1(b), the inlet extended region is formed by extending original heat transfer region upstream by 3 times of the tube diameter to ensure inlet uniformity. Similarly the outlet extended domain is formed by extending the original heat transfer region downstream by 10 times of the tube diameter so as to avoid backflow. At the entrance of the computational domain, the velocity inlet boundary condition is adopted assuming that dry air at the inlet has uniform velocity uin (10 m/s), temperature Tin (450 K) and turbulent intensity I (5%) with zero velocity components in the y and z direction. Outflow condition is set for the exit of the computational domain. Symmetry condition is assumed for the top, bottom, front and back boundary surfaces. The heat flux, normal velocity component and the normal first order derivatives of other variables are zero for the fluid symmetrical plane, while all velocity components and the normal first order derivatives of temperature are zero for the solid symmetrical plane. No-slip condition is adopted for the fluid near the solid surface. Because of the minor thermal resistance at water side, the temperature of outer tube wall can be assumed to be equal to that of 8

water (350 K). Heat convection from fluid to fins and heat conduction from fins to the tube wall are coupled to be solved, where the temperatures of fin and fluid at the interface are determined simultaneously. The settings of boundary conditions in this study are illustrated by Fig. 2. In this work, the three-dimensional numerical simulations are implemented using the commercial CFD FLUENT 6.3. The SIMPLE algorithm is used to solve the pressure-velocity coupling. The criteria for

convergence are set to be 10-6 for energy conservation equation and 10-4 for the other variables.

2.3. Data Reduction In order to quantify thermal-hydraulic performance of H-type finned tube, the following characteristic and non-dimensional parameters are defined. Re =

Nu =

ρ umax D µ

(6)

hD

(7)

λ

 p (Tin − Tout ) Q = mc

(8)

2∆p

(9)

Eu =

ρu 2m N

∆p = pin − pout ∆T =

h=

(10)

(Tout − Tw ) − (Tin − Tw )  T −T  ln  out w   Tin − Tw 

(11)

Q A∆T

(12)

In the equations above, umax is the air velocity in the minimum flow cross-section of the tube buddle. pin and pout represent the pressure at the inlet and outlet of the heat exchanger, respectively. Tin and Tout stand for the area weighted average temperature at the inlet and outlet cross section, respectively. Tw is the average temperature of total heat transfer area including fins and tube walls. The characteristic length for Nu, Eu and Re is outside diameter of the tube. A is the total air side heat transfer area. The friction 9

factor f and Colburn factor j are defined as: f =

j=

pin − pout D × 2 1 4L 2 ρ u max

(13)

Nu RePr1/3

(14)

where L represents the length of heat transfer domain as shown in Fig. 1.

2.4. Grid independency and model validation The geometric parameters of the H-type finned tube model used for grid-independency test are listed in Table 1. Table 1 Geometric parameters of the H-type finned tube for grid-independency test H/mm

W/mm

Fp/mm

Ft/mm

D/mm

S1/mm

S2/mm

N

40

6

6

1

34

90

90

3

92.5 92.0 91.5

Nu

91.0 90.5 90.0 89.5 89.0 150000

300000

450000

600000

750000

grid number

Fig. 3 Variation of Nu number with different grid system (Re = 21715)

In order to verify the grid-independency of the numerical solution, five mesh systems are investigated including 51000, 149000, 243000, 372000, 559000 and 757000 cells. The obtained Nu numbers for the six grid systems are shown in Fig. 3, which indicates that the grid dependency can be neglected for grid systems with 243000 cells or above. Therefore, the grid system with 243000 cells is adopted considering both precision and computing time. Similar tests are also performed for other cases.

10

Table 2 Geometric parameters of the H-type finned tube for model validation H/mm

W/mm

Fp/mm

Ft/mm

D/mm

S1/mm

S2/mm

N

84

15

20

2

38

90

90

10

100

0.28

Eu-numerical Eu-correlation

90

Nu-numerical Nu-correlation

0.26

80 0.24 60

0.22

50

0.20

Eu

Nu

70

40 0.18 30 0.16

20 10

0

5000

10000

15000

20000

25000

0.14

Re

Fig. 4 Nu and Eu comparisons between numerical simulation and correlation

Numerical simulation is also conducted to validate the reliability of the computational model and numerical method. The predicted Nusselt number and Euler number obtained by the numerical method are compared with the experimental correlations provided by Ref. [27] as shown in Fig. 4. The geometric parameters of the H-type finned tube for model validation are tabulated in Table 2. Air flows into the heat exchanger at the velocity ranging from 1 m/s to 10 m/s, with the corresponding Re number ranging from 2430 to 24309. Fig. 4 demonstrates that the mean deviation for Nusselt number is 10.5% and the mean deviation for Euler number is 8.1% in comparison with the experimental correlation. The good agreement between our prediction and experimental results validates the numerical model.

3. Taguchi method theory Taguchi method is one of the robust design and optimization methods, which is employed to determine the values of control factors discretely to reach the optimal value of objectives. A general optimization process by Taguchi method is given below. Firstly, the objectives to be optimized are selected. Then the control factors and their levels are specified. Afterwards, the numerical tests are 11

arranged by the orthogonal array and SN analysis is carried out. Main-effect plots are always employed to show the influence of control factors on the objective and find the optimal settings for control factors. Confirmation tests are performed to prove that the interactive effects between different factors are negligible. Finally, the selected optimal settings have to be validated to meet the required target. The specific procedures of optimization by Taguchi method are discussed below.

3.1. Selection of objectives The first step of optimization by Taguchi method is to select proper objectives to be optimized (minimized or maximized). Heat transfer and flow friction are both important characteristics to describe the performance of heat exchangers. For the current problem, j and f are employed as two targets to quantify the heat transfer and flow friction performance for H-type finned tube heat exchangers, respectively. Meantime, it is well recognized that flow friction usually increases with the enhancement of heat transfer and it is difficult to simultaneously intensify the heat transfer and reduce the flow friction. In order to evaluate the overall thermal-hydraulic performance of heat exchangers, Yun et al. [28] derived a new dimensionless number JF, where j and f are both taken into account. It is expressed as follows: JF =

where jref and

j / jref ( f / f ref )1/3

(15)

fref represent the j and f factor of the reference heat exchanger, respectively. In

conclusion, three objectives are selected to facilitate optimization by Taguchi method. j and f, utilized to describe the two components of the thermal-hydraulic performance, assist to clarify how the heat and flow characteristics of the heat exchanger are influenced by various factors. JF is used to optimize the overall thermal-hydraulic performance as a comprehensive evaluation criterion.

12

3.2. Factors and levels The control factors examined in this paper are seven geometric parameters of the H-type finned tube, i.e. slit width, number of tube rows, fin height, spanwise tube pitch, longitudinal tube pitch, fin thickness and fin pitch. The levels of each factor are tabulated in Table 3. These levels are selected uniformly within the rational ranges of control factors, which are given by Ref. [8]. It is noted that Taguchi method can only optimize the H-type finned tube by discretely selecting the optimal value of each control factor. However, Taguchi method is necessary and advantageous when designing and optimizing H-type finned tube roughly in the first step due to its low cost of time and strong robustness. The discrete optimization by Taguchi method lays foundation for further detailed and specific optimizations conducted continuously around the optimal levels in order to refine the optimization result. In this study, inlet velocity for all numerical cases is set to be 10 m/s with corresponding Re being 24309 and the tube diameter D is set as a constant value of 34 mm. Table 3 Levels of each factor Code

Factors(unit)

Level 1

Level 2

Level 3

A

Slit width W (mm)

6

12

18

B

Row number N

3

6

9

C

Fin height H (mm)

40

60

80

D

Spanwise tube pitch S1 (mm)

90

110

130

E

Longitudinal tube pitch S2 (mm)

90

110

130

F

Fin thickness Ft (mm)

1

2

3

G

Fin pitch Fp (mm)

6

12

18

3.3. Orthogonal array Orthogonal arrays are a vital part of Taguchi method. It can effectively reduce experimental runs 13

through compounding different levels of different factors uniformly. The general expression for orthogonal arrays is Ld (ak), where d is the total number of experimental or simulation runs in Taguchi method, a represents the number of levels for each factor and k is the number of factors examined. In this study, an orthogonal array of L18 (37) is adopted and its test layout is shown in Table 4 [17], where the numbers 1, 2 and 3 mean the level 1, level 2 and level 3. Because of seven factors with three levels in this study, there are 37 (=2187) level combinations in total from which the optimal one is selected. It is formidable to carry out all 2187 tests to find the optimal combination. However, the orthogonal array can reduce the huge number of test runs to 18 to obtain the optimal combination. It is because in the orthogonal array, the levels of each factor are equally combined (i.e., same number of repetitions) with the levels of the other factors and thus it is fair to compare the effects of seven factors by the main-effect plot. Therefore, accordingly it is reasonable to regard the combination of the optimal level of each factor as the optimal level combination. Table 4 The orthogonal array of L18 (37) Control factors Case No. A

B

C

D

E

F

G

Case 1

1

1

1

1

1

1

1

Case 2

1

2

2

2

2

2

2

Case 3

1

3

3

3

3

3

3

Case 4

2

1

1

2

2

3

3

Case 5

2

2

2

3

3

1

1

Case 6

2

3

3

1

1

2

2

Case 7

3

1

2

1

3

2

3

Case 8

3

2

3

2

1

3

1

Case 9

3

3

1

3

2

1

2

Case 10

1

1

3

3

2

2

1

14

Case 11

1

2

1

1

3

3

2

Case 12

1

3

2

2

1

1

3

Case 13

2

1

2

3

1

3

2

Case 14

2

2

3

1

2

1

3

Case 15

2

3

1

2

3

2

1

Case 16

3

1

3

2

3

1

2

Case 17

3

2

1

3

1

2

3

Case 18

3

3

2

1

2

3

1

3.4. SN analysis When the raw data are obtained through experiments or numerical simulations, they are usually transformed into signal-to-noise ratios (SNR) based on a logarithmic data transformation in order to improve statistical properties for optimization purpose [17]. There are three types of transformation formulae, namely nominal-the-best, larger-the-better and smaller-the-better. Nominal-the-best formula is applied to cases where the objective is expected to be kept as a specific value to gain the best performance. Larger-the-better formula and Smaller-the-better formula are employed in situations where the objective is expected to be as large as possible and as small as possible, respectively. For the optimization of heat exchangers, it is well recognized that j and JF are expected to be as large as possible while f is expected to be as small as possible. Hence, the Larger-the-better formula is used to transform the raw data of j and JF while the Smaller-the-better formula assists to transform the raw data of f. The Larger-the-better formula and the Smaller-the-better formula are presented below [17]:

SNR L = -10log(

1 n 1 ∑ ) (dB) n i =1 Yi

(16)

SNR S = -10log(

1 n ∑ Yi ) (dB) n i =1

(17)

where SNRL and SNRS represent the performance criteria for the Larger-the-better and 15

Smaller-the-better objectives respectively, Y is the raw data obtained from the numerical simulation (i.e. j, f and JF in this study) and n represents repeated times of each test which is taken to be one in this study. It is noted that the largest SNR is always preferred whether the target is of the Larger-the-better type or the Smaller-the-better type after the data transformation.

4. Results and discussion Based on numerical simulations of 18 cases and the calculating formulas mentioned above, the raw data and SNRs for j, f and JF are obtained as shown in Table 5. In order to identify the significance of the geometric factors, the intuitive analysis and analysis of variance are to be performed. Finally, the optimal geometric structure of the H-type finned tube is obtained. Table 5 Raw data and corresponding SNRs for each case Case No.

j

SNR-j

f

SNR-f

JF

SNR-JF

Case 1

0.003832

-48.33

0.05618

25.01

1.000

0

Case 2

0.003937

-48.10

0.03332

29.55

1.223

1.748

Case 3

0.002632

-51.59

0.02470

32.15

0.903

-0.882

Case 4

0.004420

-47.09

0.02905

30.74

1.437

3.150

Case 5

0.002708

-51.35

0.02340

32.61

0.946

-0.478

Case 6

0.003776

-48.46

0.08732

21.18

0.851

-1.403

Case 7

0.004545

-46.85

0.05151

25.76

1.221

1.734

Case 8

0.001929

-54.29

0.15961

15.94

0.355

-8.984

Case 9

0.003211

-49.87

0.01034

39.71

1.473

3.366

Case 10

0.003213

-49.86

0.08969

20.94

0.717

-2.885

Case 11

0.003444

-49.26

0.02857

30.88

1.126

1.030

Case 12

0.003043

-50.33

0.02062

33.71

1.109

0.900

Case 13

0.004641

-46.67

0.05272

25.56

1.237

1.847

Case 14

0.003724

-48.58

0.04531

26.88

1.044

0.375

16

Case 15

0.002649

-51.54

0.01683

35.48

1.033

0.284

Case 16

0.004367

-47.20

0.03851

28.29

1.293

2.229

Case 17

0.003769

-48.48

0.01366

37.29

1.576

3.950

Case 18

0.002013

-53.92

0.07989

21.95

0.467

-6.609

4.1. Parametric analysis of j and f In Taguchi method, intuitive analysis is usually selected to study factorial effects and contribution ratios of every control factor on the target. Based on the raw data in Table 5 and the calculating formulas mentioned above, factorial effects on SNR-j and SNR-f are obtained as shown in Table 6. In intuitive analysis, average SNR, which represents the output response of that level, needs to be firstly calculated. Average SNR means the arithmetic mean value of SNRs for each level of each factor. For example, in Table 6, the average SNR for level 1 of factor A is obtained by calculating the average of SNR responses of case 1, 2, 3, 10, 11 and 12, which all involve level 1 of factor A (see Table 4). R is the range of three level average SNRs for each factor and is defined as:

R = SNR max,i − SNR min,i

(18)

The ranks, which mean the orders of factors according to the values of R or contribution ratio, will be selected to evaluate the effectiveness of control factors.

Table 6 Factorial effects for SNR-j and SNR-f Level

A

B

C

D

E

F

G

1

-49.578

-47.667

-49.095

-49.233

-49.427

-49.277

-51.548

2

-48.948

-50.010

-49.537

-49.758

-49.570

-48.882

-48.260

3

-50.102

-50.952

-49.997

-49.637

-49.632

-50.470

-48.820

R (max-min)

1.154

3.285

0.902

0.525

0.205

1.588

3.288

Rank

4

2

5

6

7

3

1

Average SNR (dB)

j

17

1

28.707

26.050

33.185

25.293

26.465

31.035

25.322

2

28.758

28.858

28.190

28.952

28.295

28.383

29.212

3

28.157

30.713

24.247

31.377

30.862

26.203

31.088

R (max-min)

0.601

4.663

8.938

6.084

4.397

4.832

5.766

Rank

7

5

1

2

6

4

3

Average SNR(dB)

f

The contribution ratio of each factor represents its effect on the target and is defined as:

Contribution ratio(i ) =

SNR max,i − SNR min,i

(19)

k

∑ (SNR

max,i

− SNR min,i )

i =1

35

30

30

25

25

Contribution ratio/%

Contribution ratio/%

where i represents the control factor. k is the total number of control factors.

20 15 10

20

15

10

5

5

0

0

A

B

C

D

E

F

A

G

B

C

D

E

F

G

Factors

Factors

(a)

(b)

Fig. 5 Contribution ratio of each factor for SNR-j (a) and for SNR-f (b)

For a heat exchanger, j-factor and f-factor describes the heat transfer characteristics and the flow friction characteristics, respectively. It is very important and helpful to know the influence of every factor on them when designing and optimizing a heat exchanger. Based on the Table 6 and above-mentioned formula, factorial effects and contribution ratios for j and f are further presented in Fig. 5. The corresponding main-effect plots are also drawn to illustrate the effects of control factors according to the calculated average SNRs of each factor. Main-effect plots for SNR-j and for SNR-f are shown in Fig. 6. From Fig. 5(a), it can be seen that the order of the parametric effectiveness for j is G>B>F>A>C>D>E. Row number (B) and Fin pitch (G) have dominant influence on j with contribution ratios of 30.00% and

18

30.04% respectively. The reason is that the local heat transfer characteristic of each tube decreases along the flow direction, which contributes to a major decrease of average heat transfer characteristic of the whole H-type finned tube heat exchanger when row number increases as shown in Fig. 6(a). This result agrees with that presented in Ref [8]. The reason why the fin pitch has strong influence on heat transfer characteristic can be explained as follows. When fin pitch decreases, the heat transfer area per unit mass flow rate increases. Although the total heat transfer rate is increased too, they may not decrease proportionally, which leads to the major variation of heat transfer coefficient as illustrated in Fig. 6(b). The heat transfer characteristic does not vary monotonically with fin pitch as presented in Ref. [8]. It is because the one-factor-at-a-time method adopted in [8] neglects the interactive effects between different factors. The sum of contribution ratios of those two parameters are more than 60%, which renders effects of other parameters of minor importance for SNR-j. Hence, Row number and fin pitch should be considered first when optimizing the heat transfer characteristic. From Fig. 5(b), the order of the parametric effectiveness for SNR-f is C>D>G>F>B>E>A, which is different from that for j. Fin height (C) exerts the strongest influence on SNR-f with a contribution ratio of 25.33%. This is because the fluid-fin contact area increases significantly when fin height increases, which contributes to a sharp increase of the surface friction as shown in Fig. 6(b). In addition, spanwise tube pitch (D) and fin pitch (G) also have great influence on SNR-f both with contribution ratios of 17.24% and 16.34%, respectively. This can be explained as follows. The velocity of fluid across the tube increases drastically when spanwise tube pitch or fin pitch decreases, which causes a significant increase of flow friction as presented in Fig. 6(b). The sum of contribution ratios of those three factors amounts up to 58.91%, which indicates the major significance of the three factors for the flow friction characteristic of H-type finned tube heat exchangers. The variation trend of flow friction characteristic with these three 19

major factors agrees with that demonstrated in Ref. [8]. It is shown that slit width shows trivial effect on flow friction characteristics. -47

34

32

-48

SNR-f/ dB

SNR-j / dB

30

-49

-50

-51

-52

28

26

24

A1A2A3

B1B2B3

C1C2C3

D1D2D3

E1E2E3

F1F2F3

22

G1G2G3

A1A2A3

B1B2B3

C1C2C3

D1D2D3

Factors

Factors

(a)

(b)

E1E2E3

F1 F2 F3

G1G2G3

Fig. 6 Main-effect plots for SNR-j (a) and for SNR-f (b)

It has been known that the level that has the largest SNR is the optimal level for the factor. If interaction effects between all factors can be negligible, the combinations of levels showing the largest SNR-j and SNR-f for each factor are regarded as the optimal level combination for heat transfer characteristic and flow friction characteristic, respectively. From Fig. 6, the optimal combination for SNR-j is determined as A2B1C1D1E1F2G2 while that for SNR-f is A2B3C1D3E3F1G3. The difference between the optimal level combinations for SNR-j and SNR-f expresses that the heat transfer characteristic and flow friction characteristic cannot be optimized simultaneously. Therefore, a comprehensive target JF, which takes both heat transfer and flow friction characteristics into account, is employed to evaluate and optimize the overall thermal-hydraulic performance of the H-type finned tube.

4.2. Parametric analysis and optimization of JF 4.2.1 Intuitive analysis JF is adopted in this study as an evaluation criterion for the comprehensive economic performance of H-type finned tube heat exchangers. Factorial effects of control factors and contribution ratios for

20

SNR-JF are presented in Table 7 and Figure 7, respectively. The rank of parametric effectiveness for SNR-JF is D>F>C>A>G>E>B.

Table 7 Factorial effects for SNR-JF Level

A

B

C

D

E

F

G

1

-0.015

1.013

1.963

-0.812

-0.615

1.065

-3.112

2

0.629

-0.393

-0.143

-0.112

-0.143

0.571

1.470

3

-0.719

-0.724

-1.925

0.820

0.653

-1.741

1.538

R (max-min)

1.348

1.737

3.888

1.632

1.268

2.806

4.650

Rank

6

4

2

5

7

3

1

Average SNR (dB)

30

Contribution ratio/%

25

20

15

10

5

0

A

B

C

D

E

F

G

Factors

Fig. 7 Contribution ratio of each factor for SNR-JF

From Fig. 7, it shows that fin pitch has the strongest influence on SNR-JF while row number and longitudinal tube pitch have minor effects. The sum of contribution ratios of fin pitch, fin height and fin thickness amounts up to 65.46%, which renders these three parameters of major significance in terms of design and optimization for H-type finned tube heat exchangers. 3 2

SNR-JF/ dB

1 0 -1 -2 -3 -4

A1A2A3

B1B2B3

C1C2C3

D1D2D3

E1E2E3

F1F2 F3

Factors

Fig. 8 Main-effect plot for SNR-JF 21

G1G2G3

Fig. 8 is the main-effect plot for SNR-JF. By means of comparing the SNRs values under different levels, it is easy to obtain the optimal combination (A2B1C1D3E3F1G3) for SNR-JF, whose overall performance will be better than 18 cases. Compared Fig. 8 to Fig. 6, the influence of each factor on SNR-JF and its varying rationality can be proven. In Fig.8, SNR-JF increases quickly when fin pitch (G) increases from G1 to G2 while SNR-JF stays nearly unvaried when fin pitch further increases to G3. The reason is that heat transfer characteristic and flow friction characteristic are simultaneously improved when fin pitch increases from G1 to G2. But when fin pitch further increases to G3, heat transfer characteristic deteriorates and thus counteracts the merit of flow friction reduction. Because SNR-j increases slightly while SNR-f decreases sharply along with the increase of fin height (C) in Fig. 6, SNR-JF decline quickly. When fin thickness (F) increases from F1 to F2, the increase of SNR-j and the decrease of SNR-f cause the minor decrease of SNR-JF. When fin thickness further increases to F3, the decrease of both SNR-j and SNR-f leads to a sharp decrease of SNR-JF. Because the variation of j and f along with the change of slit width (A) is similar, SNR-JF similarly varied. From Fig. 8, SNR-JF increases gradually along with the increase of spanwise tube pitch (D), because SNR-f increase quickly and the variation of SNR-j is too small. In Fig. 6, it shows that the varying trend of SNR-f and SNR-j for row number (B) and longitudinal tube pitch (E) is similar. However, in Fig. 8, the varying trend of SNR-JF for them is opposite. The reason is that the variation of SNR-j for longitudinal tube pitch (E) is less than that for row number (B) and the variation of SNR-f for them is similar. It shows that the variation of SNR-JF for longitudinal tube pitch (E) is mainly affected by f factor, but that for row number (B) will be determined by f and j factor together.

4.2.2. Analysis of variance (ANOVA) 22

Table 8 Analysis of variance for JF Degree of

Sum of square

Factors

Contribution ratio Variance (V)

F-test

F0.05

freedom (df)

(SS)

A

2

5.456

2.728

3.771

9.55

2.91

B

2

10.202

5.101

7.052

9.55

5.44

C

2

45.463

22.732

31.43a

9.55

24.22

D

2

8.042

4.021

5.559

9.55

4.28

E

2

4.926

2.463

3.405

9.55

2.62

F

2

26.940

13.47

18.62a

9.55

14.35

G

2

85.232

42.616

58.92a

9.55

45.41

Error

3

2.17

0.723

Total

17

188.431

a

(%)

0.77

significant at 95% confidence level The statistical significance of the relative effects of seven geometric parameters on JF can be

obtained through analysis of variance (ANOVA). The results of ANOVA are listed in Table 8. Variance of a factor represents the response of the target (i.e. JF in this study) to the corresponding factor. A larger variance means a more effective factor. The contribution ratio of each factor is calculated according to variance and the results show that the order of factorial effectiveness is the same as that obtained from intuitive analysis where contribution ratios are calculated based on the range. It is shown that fin pitch (G) has the dominant influence on JF with a contribution ratio of 45.41% while slit width (A), row number (B), spanwise tube pitch (D) and longitudinal tube pitch (E) have minor effects. To verify the statistical reliability of the obtained results, F test is carried out at the 95% confidence level. The calculated F values are compared with standard tables of proper F test to determine which factor has significant effects on the performance characteristics. If the calculated F value for a factor is larger than the value from the standard F test table, that factor is regarded as significant. The results show 23

that fin pitch (G), fin height (C) and fin thickness (F) are statistically significant to the overall thermal-hydraulic performance of the H-type finned tube. Hence, these three parameters must be firstly considered and paid more attention during the design and optimization of H-type finned tube heat exchangers. Analysis of variance also further validates that intuitive analysis is correct.

4.2.3 Additivity by confirmation tests The optimal level combination for JF is obtained by Taguchi method in the part of intuitive analysis. However, note that we assume that there exists the additivity of effects of different factors and thus the combination of the optimal levels of seven control factors is the optimal combination for JF. The premise for this assumption is that interaction effects among factors are not significant. Therefore, additivity confirmation tests are conducted to assess the interaction effects. The procedure of this method is to compare the predicted optimal SNR-JF based on main effects and the real tested value under the determined optimal combination. If the difference between them is within +/-2 dB, interaction effects can be considered insignificant. According to analysis of variance, fin pitch (G), fin height (C) and fin thickness (F) are statistically significant for JF. The predicted SNR-JF is calculated based on the main effects of these three parameters by the following equation: SNR p =SNR + (SNR G − SNR) + (SNR C − SNR) + (SNR F − SNR )

(20)

where SNR is the grand average of SNR-JF of all 18 cases while SNRG, SNRF and SNRC are the optimal level average SNR-JF of factor G, F and C, respectively. Based on the above formula and the simulating result of optimal combination, the predicted SNR-JF and that of the optimal combination are 4.67 dB and 4.90 dB, respectively. The difference between them is only 0.23 dB, which is less than the criterion of +/-2 dB. Therefore, it proves that there exists the additivity of main effects of control factors and optimization by Taguchi method is reasonable. 24

4.2.4 Validation of the optimal combination 4.5

optimal combination case 17 case 9 random case

4.0 3.5

JF

3.0 2.5 2.0 1.5 1.0 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000

Re Fig. 9 Variations of JF with Re for H-type finned tubes with different parametric combinations

The optimal combination has been determined as A2B1C1D3E3F1G3. However, the optimal combination is selected at the condition of a fixed velocity of 10 m/s. Hence, the optimal combination has to be validated in a wide range of Re in order to gain reliability. To guarantee the representativeness, case 17 (A3B2C1D3E1F2G3) and case 9 (A3B3C1D3E2F1G2) are selected for comparison that have the largest JF output and the second largest JF output among all 18 cases, respectively. Furthermore, another case (A1B3C1D3E3F1G3) is also selected for universality, the level combination of which is randomly chosen and is not included in the 18 cases. The variations of JF with Re for H-type finned tubes with different parametric combinations are presented in Fig. 9. It is shown that the obtained optimal combination has higher JF than all three cases in the wide range of Re. Specifically, as compared with JF of case 17 which is the largest in the 18 numerical cases, JF of the optimal combination is improved by 11.4% to 16% with Re ranging from 9000 to 24000. The obvious advantage of the determined optimal combination proves the reliability of it. It can be seen in Fig. 8 that the JF of case 17 is very close to that of the random case. The reason is that they have the same parametric values in terms of fin pitch and fin 25

height, which are the most and the second most effective factors for JF. This phenomenon further proves the significance of fin pitch and fin height for overall thermal-hydraulic performance.

5. Conclusion In this paper, a parametric study is performed to investigate the effects of geometric parameters on the heat transfer characteristic, the flow friction characteristic and overall thermal-hydraulic performance of H-type finned tube heat exchangers. The effectiveness of various parameters on the three objectives is evaluated and the geometric structure of the H-type finned tube is optimized for the best overall thermal-hydraulic performance using Taguchi method. The main conclusions are drawn as follows: 1. The geometric parameters considered in the model include slit width, row number, fin height, spanwise tube pitch, longitudinal tube pitch, fin thickness and fin pitch. The results show that row number and fin pitch have dominant influence on heat transfer characteristic with contribution ratios of 30.00% and 30.04% respectively, while other five parameters have minor effects. As far as flow friction characteristic is concerned, fin height, spanwise tube pitch and fin pitch show strong influence with a combined contribution ratio of 58.91% while the effect of slit width is negligible. 2. The effects of various parameters on overall thermal-hydraulic performance are investigated. The results show that the dominant parameters are fin pitch, fin height and fin thickness. The sum of contribution ratios of these three parameters is 83.98% based on the analysis of variance. Hence, these three parameters should be considered with priority in the design and optimization of H-type finned tube heat exchangers when time and investment are limited. 3. The geometric structure of the H-type finned tube is optimized using Taguchi method with JF as a comprehensive evaluation objective. The optimal geometric parameter combination is determined as 26

A2B1C1D3E3F1G3. The confirmed additivity demonstrates that the interactive effects between different geometric factors are trivial and the optimization using Taguchi method is reasonable. The optimal structure can improve JF by 11.4% to 16% for Re in the range from 9000 to 24000, compared with the best case in the 18 cases.

Acknowledgements This work is supported by Chinese National Natural Science Foundation (No.51576150) and National Key Basic Research Program of China (973 Program) (2013CB228304).

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29

Nomenclature: A: Area, m2 C1ε C2ε Cµ : Turbulence model constants

cp: Specific heat at constant pressure, J kg-1 K-1 D: Outside diameter of tube, m df: Degree of freedom Eu: Euler number f: Fanning friction factor F: value of statistics Fp: fin pitch, m Ft: Fin thickness, m h: Heat transfer coefficient, W m-2 K-1 H: Fin height, m j: Colburn factor JF: JF factor k: Turbulent kinetic energy, m2 s-2 L: Length of heat transfer domain, m

m : Mass flow rate, kg s-1 N: Row number Nu: Average Nusselt number p: Pressure, Pa Pr: Prandtl number

∆p : Pressure drop, Pa 30

Q: Heat transfer rate, W R: Range of SNR Re: Reynolds number S1: Spanwise tube pitch S2: Longitudinal tube pitch S: Modulus of the mass rate-of-stress tensor, s-1 Sij: Mean stress rate, s-1 SNR: Signal-to-noise ratio SS: Sum of squares T: Temperature, K

∆T : Log mean temperature difference, K u: Velocity, m s-1 V: Variance W: Slit width, m

Greek symbols:

φ : General variable λ : Thermal conductivity, W m-1 K-1 ρ : Fluid density, kg m-3 ε : Turbulent energy dissipation rate, m2 s-3

µ : Viscosity, kg m-1 s-1 Γφ : Generalized diffusion coefficient 31

Sφ : Generalized source term

Subscripts: d: The total number of numerical simulations f: Fluid i: The ith factor in: Inlet k: The number of levels for each factor max: Maximum min: Minimum out: Outlet r: Reference s: Solid w: Wall

32

Highlights


H-type finned tube heat exchangers are optimized using Taguchi method.




Parametric influence is investigated by intuitive analysis and analysis of variance.




Effective design parameters for overall thermal-hydraulic performance are fin pitch, fin height and fin thickness.




The optimal parametric combination is validated to have better overall thermal-hydraulic performance.

33