Parametric optimization of H-type finned tube with longitudinal vortex generators by response surface model and genetic algorithm

Applied Energy 239 (2019) 908–918

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Parametric optimization of H-type finned tube with longitudinal vortex generators by response surface model and genetic algorithm

T

⁎

Song-Zhen Tang, Fei-Long Wang, Ya-Ling He , Yang Yu, Zi-Xiang Tong Key Laboratory of Thermo-Fluid Science and Engineering of Ministry of Education, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

H I GH L IG H T S

finned tube heat exchanger is proposed for waste heat utilization. • AThenovel mechanism of heat transfer enhancement is revealed. • Response surface model is established by the central composite design. • Pareto optimal solution set is obtained by multi-objective genetic algorithm. • The overall performance of optimized structure is improved by 48–55%. •

A R T I C LE I N FO

A B S T R A C T

Keywords: H-type finned tube Longitudinal vortex generators Response surface Genetic algorithms Waste heat utilization

Low-low temperature electrostatic precipitator technology is one of the important ways for energy saving and emission reduction of coal-fired power plants. In order to further improve the dust removal efficiency and enhance waste heat recovery performance, a novel H-type finned elliptical tube heat exchanger with longitudinal vortex generators is proposed. To achieve the maximum heat transfer enhancement with the minimum friction factor augmentation, the response surface model and multi-objective genetic algorithm are adopted to optimize the design parameters. Firstly, combined with the finite volume method and the central composite design method, the second-order response surface model between the design parameters (the length, height, angle, and position of longitudinal vortex generators) and the objective functions (Nusselt number and friction factor) is established. Then, based on the response surface model, the Pareto optimal solution set is obtained by the multi-objective genetic algorithm. Finally, by comprehensively comparing Nusselt number, friction factor and performance evaluation criteria of Pareto optimal solutions, the optimal combination is determined. Compared with the H-type finned tube heat exchanger, the performance evaluation criteria of the optimized novel heat exchanger is improved by 48–55%, which contributes to improve the overall performance of low-low temperature electrostatic precipitator system. The findings of this paper may provide practical guidelines for researchers and designers to develop efficient heat exchangers.

1. Introduction Energy is the important material basis for human survival and development, and the rational development and utilization of energy is of great significance to human life and society development. China's energy resources status determines that coal-based energy structure will still need to be maintained for quite some time to come [1]. The promotion of clean and efficient use of coal is the only way for the sustainable development of the coal industry [2]. Therefore, the technology route of synergetic treatment of flue gas based on low-low

⁎

temperature electrostatic precipitation technology is adopted to achieve ultra-low emissions from coal-fired power plants [3]. In the low-low temperature electrostatic precipitator technology, the flue gas heat exchanger is installed in front of the electrostatic precipitator for enhancing the performance of waste heat and increasing the efficiency of electrostatic precipitator [4]. Therefore, realizing high-efficiency heat transfer of flue gas heat exchangers is the premise of electrostatic precipitation technology to achieve high dust removal efficiency. In order to reduce the thermal resistance on the air side, a large number of enhanced surfaces are developed, such as annular finned

Corresponding author. E-mail address: [email protected] (Y.-L. He).

https://doi.org/10.1016/j.apenergy.2019.01.122 Received 5 October 2018; Received in revised form 20 November 2018; Accepted 17 January 2019 Available online 10 February 2019 0306-2619/ © 2019 Published by Elsevier Ltd.

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Nomenclature long axis diameter, mm short axis diameter, mm position of LVGs, mm friction factor friction factor ratio fin pitch, mm fin thickness, mm height of LVGs, mm length of LVGs, mm longitudinal vortex generators Nusselt number Nusselt number ratio performance evaluation index, (Nu/Nu0)/(f/f0)1/3. pressure, Pa transverse pitch, mm longitudinal pitch, mm temperature, K fin length, mm

fin width, mm velocity, m·s−1 slit width, mm

Greek symbols

α ρ μ λ

angle of LVGs, ° density, kg·m−3 dynamic viscosity, Pa·s thermal conductivity, W·m−1·K−1

Subscripts c in m out p w 0

calculated value inlet maximum outlet predicted value wall H-type finned circular tube

W1

a2

W2

a1

Velocity inlet

S1

Outflow

Symmetry S2

(a) Top view

Periodic

`

(b) Front view H b

h2

h1

b

w

a1 a2 b, h1, h2 f f/f0 Fp Ft H L LVGs Nu Nu/Nu0 PEC p S1 S2 T W1

W2 u w

(c) Finned tube unit Fig. 1. Geometry details of the physical model. 909

Ft

Fp

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tubes, spiral finned tubes, H-type finned tubes and so on. Among them, H-type finned tubes have been widely concerned and used in waste heat recovery applications due to their good heat transfer, anti-wear and anti-fouling performance. Jin and Tao et al. [5] investigated the flow and heat transfer characteristics of 10 rows of H-finned tube bundles, and obtained the correlations of heat transfer and resistance coefficient. Li and Zhu et al. [6] conducted a set of experiments to research the thermal-hydraulic performance of H-type finned tube bundles with in line layouts. Zhang and Wang et al. [7] proposed a novel heat exchanger with H-type fins and twisted-tape inserts in high temperature zone and longitudinal corrugated fins in low temperature zone, which has obvious advantages than traditional finned tube heat exchanger. In the study of fouling, wear and corrosion characteristics, He et al. [8–11] studied the fouling characteristics of H-type finned tubes, and revealed the effects of key parameters on the fouling growth mechanism. Li and Sun et al. [12] experimentally investigated the effect of fouling on thermal-hydraulic performance of H-type finned elliptical tubes under dusty flue gas conditions. Jin and Tao et al. [13] investigated the wear characteristics of H-type finned tube surfaces, and found that the antiwear performance of H-type finned elliptical tube is better than that of H-type finned circular tube. He et al. [14] numerically predicted the condensation of sulfuric acid vapor on 10 rows H-type finned tube surfaces, and the correlation of Sherwood number was developed. Through the above literature review, it is shown that H-type elliptical finned tube has better overall performance. However, with the deep cooling of flue gas, the performance of H-type finned tube still needs to be further enhanced. For heat transfer enhancement, the use of longitudinal vortex generators (LVGs) is widespread in compact heat exchangers because they can generate intensive longitudinal vortices with some pressure drop penalties [15,16]. Lei and He et al. [17], Chu and He et al. [18], He et al. [19], Wang and He et al. [20] systematically studied the effect of the vortex generators including delta winglet and rectangular winglet on heat transfer and flow resistance performance of the finned tube heat exchanger. On this basis, the trapezoidal winglet and curved angle winglet are proposed by Dang and Wang et al. [21] and Mohand et al. [22] to generate longitudinal and transverse vortices. Meanwhile, the thermal-hydraulic characteristics of wavy fin-and-elliptical tube heat exchanger with rectangular trapezoidal winglet, angle rectangular winglet and curved angle rectangular winglet were compared by Lotfi and Wang et al [23]. In addition, Luo and Sundén et al. [24] simulated the thermal performance of a solar receiver heat exchanger with deltawinglet vortex generators and semi-cylinder grooves. Ma and Wang et al. [25] explored the effect of LVGs on the performance of a thermoelectric power generator with a plate-fin heat exchanger. Zhao and Tang et al. [26] numerically studied the heat transfer characteristics for the single H-type finned oval tube with LVGs, and indicated that using LVGs contributes to effectively improve the heat transfer performance of the leeward side of the tubes. However, the application of LVGs in waste heat recovery needs further consideration. Firstly, according to the literature research, there in little research in the past was investigated to examine the effect of LVGs on fouling characteristics of tube bundle heat exchangers. It is necessary to verify the feasibility of using LVGs to enhance heat transfer and reduce ash fouling. Secondly, the fouling is mainly located on the leeward side and in the 20-60° area on the windward side of the tubes [27–29]. Therefore, it is necessary to arrange the LVGs in the main fouling location for reducing fouling and enhancing heat transfer. In addition, the geometry parameters of LVGs has a great influence on the fouling and thermal-hydraulic characteristics of the heat exchangers, so it is necessary to optimize the geometry parameters of LVGs. The genetic algorithm and response surface methodology [30,31] are adopted in this paper. Therefore, in this paper, a novel heat exchanger with H-type finned elliptical tubes and LVGs is proposed. The response surface model is established to identify the relation of geometry parameters and

thermal-hydraulic performance. Based on the obtained response surface model, the optimal combination of structural parameters is determined by multi-objective genetic algorithm. The comparison between the optimum novel heat exchanger and H-type finned circular tube heat exchanger is carried out for the design of high efficiency heat exchangers. 2. Computational methodology 2.1. Physical model A 6-row finned tube bundle is selected in this study with H-type fins for waste heat recovery utilization. The heat exchange tubes are arranged in aligned arrangement along the direction of the gas flow. The direction of the flow of gas is parallel to the surface of the fin. The finned tube heat exchanger is shown in Fig. 1. Fig. 1(a) is the top view of the heat exchanger. Due to the symmetry of the fin structure in the ydirection, half of the fin width is taken as the computational domain in the z-direction. In the mainstream direction (x-direction), the computational domain is extended 5 times and 10 times of longitudinal pitch at the inlet and outlet to ensure the uniform incoming flow distribution condition and outflow condition, respectively. Fig. 1(b) is a front view of the heat exchanger. In the z-direction, due to the periodic nature of the fin geometry, a fin pitch is selected as the computational unit. Fig. 1(c) is a schematic diagram of a finned tube unit. Four rectangular longitudinal vortex generators (LVGs) are arranged symmetrically on the surface of the fins. Geometric dimensions of H-type finned tube are presented in Table 1. Since the variation of the flue gas temperature in the passage of the finned tube heat exchanger is small, so the fluid property is assumed constant and the fluid is assumed to be incompressible and steady. At the inlet boundary, the fluid temperature is 420 K, the velocity is 5 m/s and the turbulent intensity is 5%. Since the fluid in the tube is water, the convection heat transfer coefficient of the water is much higher than that of flue gas, therefore it is safe to assume this constant wall temperature problem, the temperature of the tube wall is taken as a constant at 350 K. The boundary conditions of the computational domain are shown in Fig. 1(a). The uniform velocity boundary is imposed at the inlet, and the outflow boundary condition is set at the outlet. Symmetrical boundary conditions are imposed on the front and rear surfaces, while periodic boundary conditions are imposed on the upper and lower surfaces. No slip condition is applied to the fin and tube surfaces. The convective heat transfer process of the finned tube heat exchanger is a typical fluidsolid conjugate heat transfer problem, so the heat transfer process of the fluid and the fins is simultaneously calculated in this paper. 2.2. Numerical methods The gas-phase governing equations including mass, momentum and energy conservation equations are expressed as Eqs. (1)–(3). Table 1 Geometric dimensions of H-type finned tube. Parameters tube diameter D, mm long axis diameter a1, mm short axis diameter a2, mm transverse pitch S1, mm longitudinal pitch S2, mm fin width W2, mm fin length W1, mm fin pitch Fp, mm

Values *

38 50 25 95*, 62.5 76*, 100 70*, 50 70 17

Parameters

Values

fin thickness Ft, mm slit width w, mm length of LVGs L, mm height of LVGs H, mm position of LVGs b, mm position of LVGs h1, mm position of LVGs h2, mm angle of LVGs α, °

2 10 6–10 6–12 19–25 25 25 20–40

Note: When two values appear in the value column. * Circular tube, the other indicates elliptical tube. 910

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Continuity equation:

∂ (ρu i ) = 0 ∂x i

(1)

Momentum equation:

∂u j ⎞ ⎤ ∂p ∂ ⎡ ⎛ ∂u i ∂ (ρu i u j) = − μ⎜ + + ⎟ ∂x i ∂x j ⎢ ⎝ ∂x j ∂x i ⎠ ⎥ ∂x j ⎣ ⎦

Fig. 2. Grid system.

(2)

Energy equation:

ΔT =

∂ ⎛ ∂T ⎞ ⎜ρu j cp T − k ⎟ = 0 ∂x j ⎝ ∂x j ⎠

Tout − Tw Tin − Tw

)

(13)

(3)

ρ

Dk ∂ ⎛ ∂k ⎞ αk μeff = + μt S 2 − ρε Dt ∂x i ⎝ ∂x i ⎠

(4)

ρ

Dε ε ε2 ∂ ⎛ ∂ε ⎞ αε μeff = + C1ε μt S 2 − C2ε ρ − Rε Dt k k ∂x i ⎝ ∂x i ⎠

(5)

⎜

(

ln

h=

where k, ρ , μ , cp are the thermal conductivity, density, dynamic viscosity, and specific heat, respectively. Also, T represents the temperature; p is the pressure, and ui represents the component of velocity in the direction of xi. For the turbulence model, Tian and He et al. [32] predicted the complicated separated flow over wave finned tube with LVGs by comparing the laminar model, standard k-ε model and RNG k-ε model. The results of the RNG k-ε model are in best agreement with experimental data. Therefore, the RNG k-ε turbulence model is chosen in the present study. The RNG k-ε turbulence model: ⎜

(Tout − Tw ) − (Tin − Tw )

Φ̇ AΔT

PEC =

(14)

Nu/ Nu 0 (f / f0 )1/3

(15)

where D is the equivalent diameter; um is the average velocity at the ̇ and Ḣ out are the enthalpy minimum cross section; Φ̇ is the heat flux; Hin rates of fluid at the inlet and outlet; pin and pout are the average pressure at the inlet and outlet respectively; Tin and Tout are the average temperature at the inlet and outlet respectively; Tw is the temperature of tube wall; N is the number of tube rows; At is the sectional area at the minimum cross section; A is the total area; PEC is the overall performance evaluation criteria, which represents the heat transfer performance under unit pump work. Note that the subscript ‘0’ refers to the Htype finned circular tube.

⎟

⎟

2.4. Grid generation and independence test

where Rε is the rate of strain term described as follows:

Cμ ρϕ3 (1 − ϕ/ ϕ0 ) ε 2 Rε = 1 + βϕ3 k

The computational model and grid are established by software Gambit 2.4. A multi-block hybrid grid approach is used to generate the required calculation grids. Fine unstructured grids are used around the heat exchange tubes and vortex generators, and the structured grids are adopted to other regions. The grid is presented in Fig. 2. Fig. 3 gives the calculation results of Nu number corresponding to different grid numbers. It can be seen that when the number of grids increases over 912,000, the Nu number changes very little. Compared with the Nu number when the grid number is 1,432,000, the variation of Nu number is only 0.25%. It shows that the influence of the grid number on the calculation result can be ignored at this point, and the independence requirement can be satisfied. In the calculation of this paper, the final number of grid cells adopted is 1,432,000.

(6)

where μeff = μ + μt and μt = ρCμ k 2/ ε with Cμ = 0.0845, ϕ = sk / ε , ϕ0 = 4.38, β = 0.012, C1ε = 1.42 andC2ε = 1.68. αk and αε are the inverse effective Prandtl numbers for k and ε , respectively, αk =αε = 1.393 are used in the simulation. In this paper, the double precision pressure based solver is applied for numerical simulation. The standard scheme is adopted to discretize the pressure term and the second-order upwind scheme is used to discretize the momentum and energy terms. The SIMPLE algorithm is applied to solve the coupling between the velocity and the pressure fields. The solution is considered to be converged when all the residues are less than a prescribed value of 10-6.

52

2.3. Parameter definitions 51

The definitions of main parameters are given as follows:

Re =

ρu m D μ

Nu =

hD λ

(8)

̇ Φ̇ = Ḣ out − Hin

(9)

Δp = pin − pout

(10)

Eu =

f=

2Δp ρu m2 N

2Δp At ρu m2 A

Nu

(7) 50

49

48

(11)

0.4

0.8

1.2

1.6

2.0

Grid number / million (12)

Fig. 3. Calculation results of Nu under different grid numbers. 911

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90

0.42 Nu - Numerical Nu - Correlation

Eu - Numerical Eu - Correlation

Eu numbers, respectively. As can be seen from Fig. 4, the numerical simulation results agree well with the values from the correlation equation. The maximum deviation of the Nu number is within 9%, the average deviation is about 7%. The maximum deviation of the Eu number does not exceed 5%, and the average deviation is 3%. Consistency and correlations indicate that the calculation models and numerical methods in this paper are correct and reliable.

0.40

80

70 0.36 60

Eu

Nu

0.38

Fp 0.389 Ft 0.165 S1 −1.108 S2 0.293 H −0.624 W 0.029 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Nu = 1.66Re 0.585 ⎛ ⎞ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠ (16) ⎜

0.34

⎟

Fp −0.693 Ft 0.375 S1 −3.026 S2 0.388 H 1.835 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Eu = 11.63Re−0.157 ⎛ ⎞ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝D⎠ W −0.002 ⎛ ⎞ ⎝D⎠ ⎜

50

12000

15000

18000

21000

24000

27000

0.32

Re

⎟

(17)

Fig. 4. Comparison between numerical results and literature correlation.

2.6. Optimization methodology 2.5. Validation of numerical results

Fig. 5 depicts a flow chart of the optimization process that associates the proposed modeling method with an NSGA II optimization algorithm. First of all, the input and output variables and their respective ranges of values are defined. Then, the central composite design method is adopted to carry out the experimental design. The finite volume method is used to numerically calculate the corresponding

In order to verify the validity of the calculation model and numerical method, the numerical calculation of the Nu number and the Eu number is compared with the correlation in literature [5], as shown in Fig. 4. Eqs. (16) and (17) give the correlation between Nu numbers and

Fig. 5. The flow chart of the optimization process. 912

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the model coefficients are tested by analysis of variance (ANOVA). Tables 4 and 5 show the variance analysis results for Nu/Nu0 and f/f0, respectively. According to the analysis of variance in Table 4, it can be seen that the probability P of the model being false is 0, which is less than 0.0001, indicating that the model has a significant influence on Nu/Nu0 and a higher degree of confidence. In the linear item of the model, the effects of single factors L, H, α and b on Nu/Nu0 are very significant (P < 0.0001); in the quadratic terms of the model, the surface effects of L * L and b * b on Nu/Nu0 are significant; in the interaction term, the effects of L * H and a * b on Nu/Nu0 are significant (P < 0.05); but other items are not significant. It can be seen that the linear term has the most significant influence on Nu/Nu0. At the same time, as can be seen from Table 2, the value of R2, about 96.5%, indicates that the fitting effect of the model is better; and R2 (Adjusted) is 93%, indicating that the relevant independent variable in the model has a larger proportion of the variable that can be explained by the dependent variable, and the closer to 1, the greater the accuracy of these relations. Therefore, Nu/Nu0 can be analyzed and predicted using this model. According to the analysis of variance in Table 5, it is clear that the model has a significant influence on f/f0 and with a higher degree of confidence. In addition, the effects of linear terms and interaction terms on f/f0 are very significant. But the effect of interaction term on f/f0 is not obvious. Simultaneously, it can be seen R2 and R2 (Adjusted) are 96% and 92% respectively, showing that the regression model is effective. Therefore, f/f0 can be predicted by this model. After modeling with RSM, Eqs. (20) and (21) representing the functions of the Nu/Nu0 and f/f0 are obtained as follows.

Nusselt number and friction factor based on the different parameter combinations. On this basis, a second-order response surface model between the input variables and the objective functions is established to estimate the error of the RSM model. If the accuracy requirements are met, the RSM is associated with the NSGA II during the optimization phase. If the accuracy of the established RSM model is low, the RSM model needs to be improved until the inspection criteria are met. Next, according to the parameter analysis of the optimization process, the constraint conditions corresponding to the different parameters are determined. Through the multi-objective genetic algorithm, the initial population generation, selection, intersection, mutation, and loop iteration are realized until the Pareto optimal solution set is obtained. Finally, by comparing the corresponding response values of different parameters in the Pareto optimal solution set, the optimal satisfactory solution is determined. The thermal-hydraulic performance of the novel heat exchanger with the optimal combination of input parameters is compared with that of the reference heat exchanger. If the comprehensive performance is significantly improved, the optimization process is terminated. Otherwise, the design parameters continue to be optimized. 3. Response surface methodology modeling 3.1. Response surface model First of all, the central composite design (CCD) method is used to explore the entire design space and obtain sample data points. Then the response surface method (RSM) is used to approximate the objective function of the sample data points obtained, and an initial approximation model between the objective function and the design variables is established. Response surface model is established by statistical data, and the relationship between independent parameters and responses and the effects of interactions are highlighted. The Response surface model can be described as.

y = f (L, H , α, b) + ε

Nu/ Nu 0 = 3.60 + 0.026L - 0.1123H + 0.0078α − 0.080b + 0.00051L2 − 0.00362H 2 − 0.000175a2 + 0.00082b2 + 0.00027LH + 0.000849La − 0.00177Lb + 0.000816Ha + 0.00438Hb

f / f0 = 0.78 + 0.066L − 0.108H + 0.0217a + 0.029b − 0.0029L2 + 0.00593H 2 − 0.000161a2 + 0.00220b2 + 0.01174L ∗ H

(18)

+ 0.00422L ∗ a − 0.00827L ∗ b + 0.002479H ∗ a − 0.00591H ∗ b (21) − 0.001893a ∗ b

where ε is the error in response. Here, the second-order RSM model is used to construct a parameterized model. The second-order RSM model considering the linear terms, quadratic terms, and all interactions is selected to fit the response surface, and the general form of the model is described as: n

y = N0 +

The above response surface models can be used to predict different combinations of design variables. Fig. 6 shows the comparison between the calculated values and the predicted values of different optimization objectives. The subscript ‘p’ indicates the predicted values, and the subscript ‘c’ indicates the calculated values. It can be seen that the deviations between the calculated values and the predicted values are less than 10%, which indicates that the currently fitted correlation is sufficiently accurate and can be used to predict the heat transfer and flow resistance characteristics.

n

∑ Ni xi + ∑ Nii xi2 + ∑ ∑ Nij xi xj + ε i=1

i=1

i
(20)

+ 0.000040ab

(19)

where y represents the estimated response; N0 represents the constant item; Ni represents the primary coefficient of the independent variable xi; Nij represents the interaction coefficient between the independent variable xi and xj; and the Nii represents the quadratic coefficient corresponding to the independent variable xi. In this paper, the input variables are the angle α, length L, height H, and position b of LVGs, and the output variables are Nu/Nu0 and f/f0. The input variables are chosen in three levels. The central composite design (CCD) is used to design the experiments. A CCD method includes the factorial, central, and axial points, which are used to estimate linear terms and interaction terms, error terms, and quadratic terms, respectively. Table 2 presents the different levels of input variables based on the CCD.

4. Multi-objective optimization In the optimization of heat exchangers, it is necessary to consider the heat transfer and resistance performance of the heat exchanger comprehensively. However, due to the conflict between these two objectives, the method of using the weight coefficient to convert it into a Table 2 The different levels of input variables. Factors

3.2. Model estimation Length of LVGs, L (mm) Height of LVGs, H (mm) Angle of LVGs, α (°) Position of LVGs, b (mm)

Through the numerical calculation of the input variables generated by the central composite design method, sample data points are obtained, as shown in Table 3. Then, the second-order response surface model is used to fit the sample data. The accuracy and significance of 913

Uncoded/coded value −1

0

1

6 6 20 19

8 9 30 22

10 12 40 25

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Table 3 The design of experiment. Run order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Real value

response

Run order

L

H

α

b

Nu/Nu0

f/f0

10 8 8 6 6 8 6 8 10 6 6 6 8 10 6 8

9 9 9 6 9 9 12 9 6 12 6 6 9 12 12 9

30 30 30 20 30 30 40 30 40 40 40 40 30 40 30 30

22 22 25 25 22 22 25 22 25 19 25 19 19 25 22 22

2.27 2.26 2.21 2.16 2.31 2.26 2.12 2.26 2.43 2.12 2.40 2.50 2.41 2.30 2.07 2.26

1.55 1.48 1.35 1.12 1.46 1.48 1.46 1.48 1.54 1.80 1.44 1.57 1.71 1.96 1.43 1.48

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Table 4 The results of ANOVA for Nu/Nu0.

Real value

response

L

H

α

b

Nu/Nu0

f/f0

10 10 10 8 8 10 8 8 6 8 6 10 6 8 10

12 6 6 9 9 12 9 9 12 6 6 6 12 9 12

20 20 20 30 30 20 20 30 20 30 20 40 20 40 40

25 19 25 22 22 19 22 22 25 22 19 19 19 22 19

1.93 2.45 2.27 2.26 2.26 1.96 2.15 2.26 1.91 2.43 2.33 2.65 1.90 2.42 2.33

1.25 1.35 1.22 1.48 1.48 1.54 1.24 1.48 1.14 1.48 1.24 1.91 1.32 1.75 2.74

F-Value

P-Value

Table 5 The results of ANOVA for f/f0.

Source

DF

Adj SS

Adj MS

F-Value

P-Value

Source

DF

Adj SS

Adj MS

Model

14

0.9054

0.0647

31.6300

0.0000

Model

14

2.6591

0.1899

26.3400

0.0000

Linear L H a b

4 1 1 1 1

0.8370 0.0379 0.4659 0.2717 0.0477

0.2093 0.0379 0.4659 0.2717 0.0477

102.3300 18.5200 227.8400 132.8800 23.3000

0.0000 0.0010 0.0000 0.0000 0.0000

Linear L H a b

4.0000 1.0000 1.0000 1.0000 1.0000

2.2151 0.3605 0.2181 1.2568 0.4052

0.5538 0.3605 0.2181 1.2568 0.4052

76.7900 49.9900 30.2400 174.2800 56.1900

0 0 0 0 0

Square L*L H*H a*a b*b

4 1 1 1 1

0.0120 0.0000 0.0021 0.0008 0.0002

0.0030 0.0000 0.0021 0.0008 0.0002

1.4600 0.0000 1.0100 0.4100 0.0700

0.2600 0.9450 0.3310 0.5290 0.7900

Square L*L H*H a*a b*b

4.0000 1.0000 1.0000 1.0000 1.0000

0.0183 0.0003 0.0055 0.0007 0.0011

0.0046 0.0003 0.0055 0.0007 0.0011

0.6300 0.0500 0.7700 0.1000 0.1500

0.646 0.834 0.394 0.756 0.703

Interaction L*H L*a L*b H*a H*b a*b

6 1 1 1 1 1 1

0.0410 0.0000 0.0046 0.0018 0.0096 0.0249 0.0000

0.0068 0.0000 0.0046 0.0018 0.0096 0.0249 0.0000

3.3400 0.0200 2.2600 0.8800 4.6900 12.180 0.0100

0.0250 0.8860 0.1520 0.3610 0.0460 0.0030 0.9170

Interaction L*H L*a L*b H*a H*b a*b

6.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.4205 0.0820 0.1137 0.0394 0.0885 0.0453 0.0516

0.0701 0.0820 0.1137 0.0394 0.0885 0.0453 0.0516

9.72 11.3700 15.7700 5.4600 12.2700 6.2900 7.1500

0 0.004 0.001 0.033 0.003 0.023 0.017

Error Lack-of-Fit Pure Error Total

16 10 6 30

0.0327 0.0327 0.0000 0.9381

0.0020 0.0033 0.0000

*

*

Error Lack-of-Fit Pure Error Total

16.0000 10 6 30

0.1154 0.1154 0.0000 2.7745

0.0072 0.0115 0.0000

*

R2 = 96.51%, R2 (Adjusted) = 93.46%.

R2 = 95.84%, R2 (Adjusted) = 92.20%.

single-objective problem is not accurate enough. The method based on Pareto optimal solution can solve such multi-objective problems better. Therefore, to obtain the optimum combination of the input variables for the maximum Nu/Nu0 and the minimum f/f0, a multi-objective optimization with genetic algorithm is adopted. The objective functions and constraints are as follows.

optimal solution set according to the importance of each objective function. To select the most compromising solution from the Pareto front for industrial applications, the performance comparison of Nu/Nu0, f/f0, and PEC is shown in Fig. 8. According to Fig. 8, the PEC values of case4, case 13 and case23 are relatively high. In the case of rounded values for each variable, these three schemes have the same L, H, and b. The value of parameter α is between 21 and 24°. To obtain better overall performance under equal flow and equal pumping power, case 13 (L = 10 mm; H = 6 mm; α = 24°; b = 19 mm) is chosen.

Maximum Nu/Nu0 = f1(L, H, α, b) Minimum f/f0 = f2(L, H, α, b) Constraints: 6 mm < L < 10 mm; 20° < α < 40°; 19 mm < b < 25 mm.

6 mm < H < 12 mm;

5. Comparison of thermal-hydraulic performance In this paper, the NSGA-II method is used to optimize the objective functions obtained by RSM. NSGA II has been performed in this paper for a population size of 100, number of generations is 1000, a crossover fraction of 0.8, and a Pareto front population fraction of 0.3. The Pareto optimal solution set can be obtained by NSGA II, as shown in Fig. 7. It can be seen that since the Pareto optimal solution is not unique, the designer can select the most satisfactory solution from the Pareto

In order to reveal the physical mechanism of the heat transfer enhancement of H-type finned tube with LVGs, it is necessary to compare and analyze the streamlines distribution of different structures, as shown in Fig. 9. Fig. 9(a) shows the distribution of the streamlines on the x-y cross section (parallel to the fins) at Re = 10,676. It can be seen that the size 914

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Fig. 6. The comparison between the calculated values and the predicted values: (a) Nu/Nu0; (b) f/f0.

2.2

narrower. After using elliptical tubes and LVGs, the wake vortex area becomes very small and the detachment of fluid in the wake region is delayed. It can be found that the hydraulic characteristics of H-type finned tube with LVGs are significantly better than the reference fins. Fig. 9(b) depicts the streamlines distribution over different y-z cross sections (perpendicular to the direction of the main flow) at Re = 10,676. As can be seen from Fig. 9(b), compared with the H-type finned circular and elliptical tubes, clear vortexes are generated at the cross section of the rear side of H-type finned tube with LVGs. This is mainly because, under the effect of pressure difference before and after LVGs, a part of the fluid turns over the LVGs to form the main vortex. At the same time, a portion of the fluid bypasses the LVGs and flow separation occurs at the trailing edge of the LVGs, which create the angular vortexes. With the combination of the main vortex and the angular vortex, the secondary flow intensity of the fluid increases significantly. Fig. 9(c) shows the streamlines distribution between different Htype finned tube bundles. It can be seen that after using the LVGs, the longitudinal vortexes are generated at the cross-section of the rear side of the LVGs, and interact with the fluid in the main flow direction to form a strong three-dimensional spiral flow, which intensifies the degree of fluid mixing, effectively inhibits the formation of the boundary layer, and contributes to further intensifying heat transfer performance. The Nu number versus Re number is presented in Fig. 10. The Nu number of H-type finned elliptical tube bundle with LVGs is significantly higher than that of the original H-type finned tube bank by 88–95%. Fig. 11 shows the f factor versus the Re number. Compared with the H-type finned circular tube bundle, the f factors of H-type finned elliptical tube bundle and the finned tube bundle with LVGs are increased by 29% and 100–104% respectively. The PEC versus the Re number is given in Fig. 12. The PEC of the original H-type finned circular tube bundle is regarded as a reference for comparison (the baseline). The PEC of H-type finned elliptical tube bundle is increased by 31–33% and that of H-type finned elliptical tube bundle with LVGs is improved by 48–55%.

Pareto front

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Fig. 7. The optimal solution set.

0

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6. Conclusions

Fig. 8. Performance comparison of Nu/Nu0, f/f0, and PEC.

In this paper, the H-type finned elliptical tube bundle with LVGs is proposed. The aim of this paper is to obtain the optimum parameter combinations based on the RSM and multi-objective genetic algorithm. The thermal-hydraulic performance of optimal fin and reference fin is compared and analyzed. From the outcome of this study, the following

of the trailing vortex area of the H-shaped finned tube is very large, and the strong backflow seriously prevents fluid flowing to the downstream. Using elliptical tubes, the wake vortex area is suppressed and becomes 915

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Fig. 9. Streamlines distribution of different structures: (a) x-y cross section; (b) y-z cross sections; (c) three-dimensional streamlines.

L = 10 mm, H = 6 mm, α = 24° and b = 19 mm. (3) The physical mechanism of the heat transfer enhancement of Htype finned tube with LVGs is revealed. It can be seen that after using the LVGs, the strong three-dimensional spiral flow is formed and the thermal-hydraulic performance is enhanced. Compared with the original H-type finned circular tube bundle, the PEC of Htype finned elliptical tube bundle with LVGs is improved by 48–55%.

conclusions are drawn: (1) The central composite design method and the second-order response surface model are used to establish the approximation model between the objective function and the design variables. It can be seen that the deviations between calculated values and predicted values are less than 10%, which indicates that the currently fitted correlation is sufficiently accurate. (2) The multi-objective optimization with genetic algorithm is adopted to obtain the optimum combination of the input variables for the maximum Nu/Nu0 and the minimum f/f0. The optimal solution set is obtained. By comprehensively comparing Nu/Nu0, f/f0 and PEC of different schemes, the optimal combination is determined as

The enhanced heat transfer technology proposed in this paper can significantly improve the thermal-hydraulic performance of the flue gas heat exchanger and efficiently recover the waste heat, which contributes to increasing the dust removal efficiency of the electrostatic 916

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100 90 80

precipitator, and promotes energy-saving & emission reduction and the progress of energy utilization technologies.

Finned circular tube Finned elliptical tube Finned elliptical tube with LVGs

Acknowledgments

60

This work was supported by the National Key R&D Program of China (2018YFB0605901) and the National Natural Science Foundation of China (No. 51806165).

50

References

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30 20 10000

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0.040 Finned circular tube Finned elliptical tube Finned elliptical tube with LVGs

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Re Fig. 11. f factor versus Re number for different tube bundles.

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Re Fig. 12. PEC versus Re number for different tube bundles.

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