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Thermo-hydraulic performance optimization of wavy fin heat exchanger by combining delta winglet vortex generators
Applied Thermal Engineering 163 (2019) 114343
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Thermo-hydraulic performance optimization of wavy fin heat exchanger by combining delta winglet vortex generators
T
⁎
Chao Luoa, Shuai Wua, Kewei Songa, , Liang Huab, Liangbi Wanga a b
School of Mechanical Engineering, Key Laboratory of Railway Vehicle Thermal Engineering of MOE, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China Institute of Locomotive and Vehicle, Nanjing Institute of Railway Technology, Nanjing, Jiangsu 210031, China
H I GH L IG H T S
novel combination of wavy fin and vortex generators is numerically studied. • AConsiderable improvement of thermal performance is reported. • Optimum attack angle of 45° has the best thermal performance in laminar flow. • Thermal performance factor increases by up to 26.4% owing to the novel combination. •
A R T I C LE I N FO
A B S T R A C T
Keywords: Wavy fin Vortex generator Corrugation angle Attack angle Thermal performance
Wavy fins and vortex generators are effective methods for enhancing heat transfer of heat exchangers. In general, wavy fins and vortex generators are studied separately in the literature. A novel combination of a wavy fin and vortex generators is proposed in the present study. The effects of different corrugation angles of the wavy fin and different attack angles of the vortex generators are numerically studied to optimize the thermo-hydraulic performance of the novel combination. The Nusselt number, friction factor, and thermal performance factor are compared between different corrugation angles and attack angles and between wavy fins with and without vortex generators. The heat transfer performance is improved considerably by the novel combination. The maximum thermal performance factor increases by 26.4% owing to the combination of the vortex generator and wavy fin. The optimum attack angle of the vortex generator, which yields the highest heat transfer performance of the wavy fin plate heat exchanger, is reported as 45°.
1. Introduction Heat exchangers, which are energy transfer equipment, are applied widely in industrial fields, such as power engineering, chemical engineering, and air-conditioning [1–3]. The enhancement of the heat transfer efficiency of heat exchangers is of immense significance for saving energy. Both wavy fins and vortex generators are effective for enhancing heat transfer and are widely studied for compact heat exchangers. The effect of wavy configurations on the thermo-hydraulic performance of wavy fin is studied widely [4–12]. Naphon [4] and Ali et al. [5] experimentally studied the thermo-hydraulic performance of a wavy fin. Their results demonstrated that the heat transfer of wavy channel is significantly enhanced by a corrugated surface. Wang et al. [6] numerically reported that both heat transfer and pressure loss increase as the amplitude–wavelength ratio increases. Pehlivan et al. [7] ⁎
experimentally illustrated that the heat transfer rate of a wavy fin increases with an increase in the corrugation angle. Kwon et al. [8] and Mereu [9] reported that the corrugation angle significantly impacts the transition of the flow state in a wavy channel and that the transition to turbulent flow occurs earlier as the corrugation angle increases. Dong et al. [10,11] experimentally and numerically revealed that an increase in the wave amplitude can improve the heat transfer performance, whereas an increase in the fin length and fin pitch decreases the heat transfer performance. A wavy fin with surface modification can further improve the heat transfer performance. Zhang et al. [12] reported that a humped wavy fin can effectively improve the thermo-hydraulic performance. Hwang et al. [13] experimentally and numerically investigated the influence of secondary motions on a wavy channel and determined that secondary flows exert an important effect on heat transfer and pressure loss. Kim et al. [14] numerically revealed that the cross-cut wavy fin can yield a 21.71% increment in Nusselt number
Corresponding author. E-mail address: [email protected] (K. Song).
https://doi.org/10.1016/j.applthermaleng.2019.114343 Received 4 April 2019; Received in revised form 26 July 2019; Accepted 4 September 2019 Available online 05 September 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature A b C Dh f Gk GCI h H JF k L Lx N Nu p P PL r Re S tp T Ts
u, v, w X x, y, z
cross-sectional area (m2) half the transverse distance between two VGs (m) a quarter of the wavelength (m) hydraulic diameter (m) Fanning friction factor generation of turbulent kinetic energy grid convergence index grid spacing height of vortex generator (m) thermal performance factor or surface goodness factor turbulent kinetic energy base length of vortex generator (m) wavy fin length (m) number of wavy Nusselt number pressure loss (Pa) or order of accuracy for GCI wetted perimeter (m) width of the simulation domain (m) grid refinement ratio Reynolds number fin surface area (m2) fin spacing (m) temperature (K) bulk temperature of the cross-section (K)
components of velocity vector (m/s) non-dimensional distance, x/4C coordinates
Greek symbols β δS(x) ε θ λ λt μ μt ρ φ
attack angle of VG (°) fin strip area at position x (m2) turbulent dissipation rate or relative error for GCI corrugation angle (°) fluid thermal conductivity (W/(m·K)) turbulent thermal conductivity (W/(m·K)) viscosity (kg/(m·s)) turbulent dynamic viscosity (kg/(m·s)) density (kg/m3) numerical solution
Subscripts in local out s t w
inlet local value outlet span average value turbulent wall surface
generators is likely to exhibit a good thermal performance. Lotfi et al. [27,28] numerically studied the effect of different novel VGs on the thermo-hydraulic performance of a wavy fin-and-elliptical tube heat exchanger. The best thermo-hydraulic performance is obtained for the curved angle rectangular VG. Tian et al. [29] and Ke et al. [30] numerically studied the thermo-hydraulic performance of wavy fin-andtube heat exchangers with VGs on the fin. Considerable enhancement in the heat transferred by the wavy fin-and-tube heat exchanger is obtained owing to the longitudinal vortices generated by the VGs. The foregoing researches demonstrate that both the wavy fin and vortex generators are effective for heat transfer enhancement. Wavy fin and vortex generator have generally been studied separately. There are few researches on combinations of wavy fin and vortex generators. In this paper, a novel wavy fin with vortex generators is proposed for a plate heat exchanger. The thermo-hydraulic performance of the novel combination of wavy fin and vortex generators is numerically investigated by considering the optimal combination of the corrugation angle of the wavy fin and the attack angle of the vortex generators. The results have significant importance in the design of wavy fin plate heat exchanger for improving the thermo-hydraulic performance.
with a 3.95% increment in the friction factor compared with the typical wavy fin at Re = 300. Apart from wavy fin, winglet vortex generator (VG), which can generate longitudinal vortices, is also effective for heat transfer enhancement [15–26]. Fiebig et al. [16] and Biswas et al. [17] reported that longitudinal vortices generated by VG can enhance the heat transfer with less pressure penalty. A simple rectangular VG with an attack angle of 45° was reported to exhibit the highest heat transfer enhancement in a triangular channel [18]. Li et al. [19] compared the performance of a wavy fin and a plain fin with VGs. Their experimental results revealed that a plain fin with five-row tubes, enhanced with VGs, exhibits a better performance than that of a wavy fin with six-row tubes. The heat transfer enhancement is also affected by the arrangement configurations of VGs, such as the common-flow-up and commonflow-down configurations [17,20–22]. Sinha et al. [20] compared the performance of a plate-fin heat exchanger with two-row VG pairs arranged in different configurations. Their results revealed that the VGs with common-flow-up configurations exhibit higher heat transfer and flow characteristics than those with the other configurations. Longitudinal vortices with different rotating directions can significantly improve the heat transfer performance of a fin-and-tube heat exchanger [21]. Song et al. [22,23] quantitatively studied the interaction between longitudinal vortices and reported that the highest heat transfer performance can be obtained at a suitable transverse pitch of VGs. In addition to the normal plane VGs, wavy VG and curved VG have been recently reported to exhibit a better heat transfer performance than the normal plane VGs. Gholami et al. [24] reported that a wavy rectangular VG can enhance the heat transfer with a smaller pressure drop compared with the normal plane rectangular VG. Song et al. [25] revealed that the concave curved VG is more favourable to heat transfer performance than the convex curved VG and the plane VG. Song et al. [26] experimentally reported that a smaller curved VG is effective for heat transfer enhancement of circular tube-fin heat exchanger at low Re values. Moreover, a larger VG is suitable for heat transfer enhancement at large Re values. As either the wavy fin and vortex generators can effectively improve heat transfer performance, a combination of a wavy fin and vortex
2. Physical model A schematic view of the studied wavy fin with delta winglet vortex generators is shown in Fig. 1. The delta winglet vortex generators are mounted on the wavy fin surface in pairs. The geometric size of the fin, as shown in Fig. 2, is identical to that in the experiment conducted by Ali and Ramadhyani [5]. The simulation domain is prolonged by 20.69 mm in both the inlet and outlet regions. The three corrugation angles (θ) considered are 10°, 20°, and 30°. The wavy fin pitch is tp = 6.9 mm, and wavy fin wavelength is 4C = 45.7 mm. C is the distance from the trough of the wavy fin to the front point of the vortex generators. The region sketched by dashed lines in Fig. 2 (b) is selected as the simulation domain. A schematic view of the simulation domain is shown in Fig. 2 (c). The width of the simulation domain is PL = 44.45 mm. The geometric parameters of a delta winglet VG are employed based on previous studies [19,29,30]. The height of the delta 2
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Table 1 Parameters of simulated domain. Fin pitch (tp) Hydraulic diameter (Dh) Wavy fin wavelength (4C) Simulated domain (PL) Prolonged length at inlet (Ei) Prolonged length at outlet (Eo) Length of VG (L) Height of VG (H) Transverse VGs pitch (2b) Corrugation angles (θ) Attack angles of VGs (β) Number of wavy (N) Inlet temperature Wavy fin temperature
Fig. 1. Schematic view of wavy fin with delta winglet VGs.
winglet VG is two-third the height of the flow channel, and the length of the VG is twice the height of VG. The transverse distance of the VGs is 2b = 35.56 mm, and the attack angles (β) of the VG are 30°, 45°, 60°, and 75°. The length and height of each VG are L = 9.2 mm and H = 4.6 mm, respectively. The geometric parameters and boundary temperature conditions are summarized in Table 1.
Table 2 Grid independence study.
3. Mathematical formulation The operating fluid is water, which is considered as an incompressible fluid. As the inlet temperature of water is 300 K and the fin temperature is 310 K, the maximum temperature difference is 10 K. Thus, the physical properties of water are considered to be constant for this marginal temperature difference. The constant values of the physical properties of water are calculated at the mean temperature of 305 K. The flow in the computational domain is assumed to be threedimensional, steady-state, and without viscous dissipation. In laminar flow, the governing equations can be expressed as follows: Continuity equation:
∂ ρui = 0 ∂x i
6.9 mm 13.8 mm 45.7 mm 44.45 mm 20.69 mm 20.69 mm 9.2 mm 4.6 mm 35.56 mm 10°, 20°, 30° 30°, 45°, 60°, 75° 5 Tin = 300 K Tw = 310 K
Grid number
Nu
f
400,602 926,250 1,100,061 1,648,886 3,808,154 Maximum Relative error
13.26 13.10 13.14 13.09 12.98 2.2%
0.1440 0.1425 0.1419 0.1416 0.1404 2.6%
Momentum equation:
∂p ∂ ∂ ⎛ ∂uj ⎞ (ρui uj ) = μ − ∂x j ∂x i ∂x i ⎝ ∂x i ⎠ ⎜
⎟
(2)
Energy equation:
∂ ∂ ⎛ ∂T ⎞ (ρCp ui T ) = λ ∂x i ∂x i ⎝ ∂x i ⎠ ⎜
(1)
⎟
Fig. 2. Parameters of wavy fin and VG, (a) front view, (b) top view, (c) schematic view of simulation domain, (d) VG. 3
(3)
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In turbulent flow, the compact forms of the momentum and energy equations are
∂uj ⎞ ∂p ∂ ∂ ⎛ (ρui uj ) = (μ + μt ) − ∂x i ∂x i ⎝ ∂x i ⎠ ∂x j
(4)
∂ ∂ ⎛ ∂T ⎞ (ρCp ui T ) = (λ + λt ) ∂x i ∂x i ⎝ ∂x i ⎠
(5)
⎜
⎟
⎜
⎟
The RNG k–ε turbulent model [31]:
∂uj ⎞ ∂ ∂ ⎛ + Gk − ρε (ρkuj ) = (μ + μt ) λ ∂x i ⎠ ∂x i ∂x i ⎝
(6)
∂uj ⎞ ε ε2 ∂ ∂ ⎛ (ρεuj ) = (μ + μt ) λ + C1ε Gk − C2ε ρ k k ∂x i ⎠ ∂x i ∂x i ⎝
(7)
⎜
⎟
⎜
μt = ρCμ Fig. 3. Grid system, (a) 3D view, (b) grid around the VG.
⎟
k2 ε
(8)
where C1ε, C2ε, and Cμ are constants and are equal to 1.42, 1.68, and 0.0845, respectively. The velocity-inlet boundary condition is applied at the inlet, where the fluid velocity is assumed to be uniform with a constant temperature. According to the turbulence intensity correlation reported in [32], the turbulence intensity is determined based on Re as I = 0.16Re−1/8. The outflow boundary condition is applied at the outlet. The no-slip conditions are applied at the wavy fin and vortex generator surfaces. The fin surfaces of the prolonged regions are adiabatic. Constant temperature boundary conditions are applied on the vortex generators and the wavy fin surfaces except the prolonged regions. Symmetry boundary conditions are applied at the side surfaces of the simulation domain. The boundary conditions of the simulation domain are indicated in Fig. 2(c). At the inlet:
u (x , y, z ) = uin , v (x , y, z ) = 0, w (x , y, z ) = 0, T (x , y, z ) = Tin
(9)
At the outlet:
∂ ∂ ∂ ∂ u (x , y, z ) = 0, v (x , y, z ) = 0, w (x , y, z ) = 0, T (x , y, z ) = 0 ∂x ∂x ∂x ∂x (10) On the symmetric surfaces:
∂ ∂ ∂ u (x , y, z ) = 0, v (x , y, z ) = 0, w (x , y, z ) = 0, T (x , y, z ) = 0 ∂z ∂z ∂z (11) On the solid surfaces:
u (x , y, z ) = 0, v (x , y, z ) = 0, w (x , y, z ) = 0, T = Tw
(12)
Reynolds number is
Re =
ρuin Dh μ
(13)
Dh is the hydraulic diameter:
Dh =
4A P
(14)
The Fanning friction factor is
f= Fig. 4. Comparisons between numerical and experimental results, (a) Nu, (b) f.
Δp / L x Dh (ρuin2 /2) 4
(15)
The local Nusselt number is determined as
Nulocal = −Dh
∂T ∂n
/(Tw − Ts (x )) w
The cross-sectional bulk temperature Ts(x) is 4
(16)
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Fig. 5. Positions of Sections I, II, and III. (a) Front view, (b) top view.
Fig. 6. Velocity on Sections I and II with Re = 150, (a) θ = 10°, β = 45°, (b) θ = 20°, β = 45°, (c) θ = 30°, β = 45°, (d) θ = 20°, β = 30°, (e) θ = 20°, β = 45°, (f) θ = 20°, β = 60°, (g) θ = 20°, β = 75°.
∬ u (x , y, z ) T (x , y, z ) dA
4. Numerical method and grid independence
A
Ts (x ) =
∬ u (x , y, z ) dA A
The range of Re for laminar flow varies with the corrugation angles of the wavy fin. According to Dong et al. [10,11], the flow pattern is confined to steady laminar for Re ≤ 1000 and θ = 10°. Ali and Ramadhyani [5] reported that the flow is laminar for θ = 20° and Re ≤ 400. Comini et al. [36] found that the flow is laminar for θ = 30° and Re ≤ 300. Therefore, the turbulence model is applied when Re exceeds 1000, 400, and 300 for θ = 10°, 20°, and 30°, respectively. The commercial software FLUENT 17.0 is used for solving the governing equations. ICEM is used for generating the mesh for the simulation domain. The simulated domain is meshed with structured hexahedral elements. The governing equations along with the boundary conditions were solved using the finite volume method. The SIMPLEC algorithm was used to perform the coupling of velocity and pressure. The second order upwind scheme is used for the convection terms and central difference scheme for the diffusion terms. The first order upwind scheme was adopted for the turbulent kinetic energy and turbulent dissipation rate term. The least squares cell based method was used for the gradient term, and the second order accuracy was used for the pressure and energy terms. The normalized residual values are set as 108 and 10-5 for the energy equation and the other equations, respectively. The enhanced wall treatment is adopted for the turbulence model. The grid near-wall function requires y+ ≈ 1 according to the user
(17)
The span-average Nusselt number of the bottom and top fin surfaces is
Nus (x ) =
1 δS (x )
∬ Nuloacl dS δS (x )
(18)
The average Nu is calculated by averaging Nulocal on the entire fin surface area:
Nu =
1 S
∬ Nuloacl dS S
(19)
The thermal performance factor or surface goodness factor JF [33–35] based on identical pump power is
JF =
Nu/ Nu 0 (f / f0 )1/3
(20)
where the subscript zero implies the corresponding value of the wavy channel without VGs at an identical Re. 5
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Fig. 7. Contour plots of absolute vorticity on Sections I and II with Re = 150, (a) θ = 10°, β = 45°, (b) θ = 20°, β = 45°, (c) θ = 30°, β = 45°, (d) θ = 20°, β = 30°, (e) θ = 20°, β = 45°, (f) θ = 20°, β = 60°, (g) θ = 20°, β = 75°.
Fig. 8. Temperature on the sections I and II at Re = 150, (a) θ = 10°, β = 45°, (b) θ = 20°, β = 45°, (c) θ = 30°, β = 45°, (d) θ = 20°, β = 30°, (e) θ = 20°, β = 45°, (f) θ = 20°, β = 60°, (g) θ = 20°, β = 75°.
combination of wavy fin and vortex generators. The corrugation angle of the wavy fin is θ = 20°, and the attack angle of the vortex generator is β = 60°. Five grid systems with different grid numbers (400602, 926250, 1100061, 1648886, and 3808154) are presented in Table 2. The grid number of the fine grid is approximately 9.5 times larger than that of the coarse grid. The maximum relative errors of Nu and f are
manual of Fluent 17.0 [32]. As it is challenging to generate a good grid density for y+ ≥ 1 when the Re is small under a turbulent flow, y+ = 0.5 was adopted in the present study in order to obtain a fine grid in the studied range of Re. The near wall mesh spacing is gradually increased by 1.2 times until the mesh spacing tend to be balanced. The grid independence study was carried out at Re = 300 for the 6
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differences in Nu and f between the experimental results and the RNG k–ε model are less than 13.4% and 10.7%, respectively. Thus, the prediction accuracy of the numerical results can be guaranteed, and the numerical results are reliable. 5. Results and discussions 5.1. Temperature, velocity, and vorticity on the sections In order to study the effects of β and θ on the heat transfer and flow characteristics, three sections are selected in the third period of the wavy fin, as shown in Fig. 5. Section I, which is on the y–z plane, is located at an equal distance from the ends of the VGs for the different β and θ. Section II, which is also on the y–z plane, is fixed at the midpoint of the third period of the wavy fin. Section III is selected so as to pass through the midpoint of VG along the x–y plane in the third wavy period in order to illustrate the recirculation behind the VG. Fig. 6 shows the distribution of longitudinal vortices on the selected sections I and II for different β and θ at Re = 150. The attack angle is β = 45°, and θ ranges from 10° to 30° in Fig. 6(a)–(c). It is evident that the longitudinal vortices generated by VGs with an identical β are similar for different θ. The longitudinal vortices increase slightly as θ increases. On section II, there are two longitudinal vortices: the main vortex and corner vortex generated by the VG located in front of section II. The vortices on section II also increase with the increase in θ. Fig. 6(d)–(g) show the longitudinal vortices on sections I and II for different β ranging from 30° to 75° with identical θ (20°). The main vortex on section I first increases as β increases from 30° to 45° and then decreases as β increases from 45° to 75°. Thus, the main vortex is the largest for β = 45°. The velocity on section I becomes complex as β increases to a large value owing to the formation of recirculation behind the VG. The difference in the main vortices on section II is slight and the corner vortices generally increase marginally as β increases. Fig. 7 shows the contour plot of vorticity on sections I and II corresponding to Fig. 6. The vorticity on section I is apparently larger than that on section II owing to the attenuation of longitudinal vortices along the mainstream. The vorticity caused by the vortices generated by VGs with identical β increases as θ increases, on both sections I and II in Fig. 7(a)–(c). It can be determined that a wavy fin with a large θ can increase the heat transfer. For different β under an identical θ as show in Fig. 7(d)–(g), the zones of vorticity with the large value gradually separate on section I when β varies from 30° to 75°. Moreover, the vorticity of the main vortex for β = 30° is apparently smaller than those for the other β values. On section II, the difference in vorticity for the main vortex is marginal, and the vorticity for the corner vortex generally increases as β increases. It is apparent that the distributions of vorticity on Sections I and II are consistent with the distributions of longitudinal vortices. The large vorticity in Fig. 7 implies a high intensity of longitudinal vortices in Fig. 6. Fig. 8 shows the contour plot of temperature on sections I and II at Re = 150. The temperature on the cross sections can reflect the mixing of fluid behind the VGs. The temperature on Sections I and II increases gradually with the increase in θ under an identical attack angle, as shown in Fig. 8(a)–(c). This is because the mixture of the fluid is enhanced owing to the increase of the longitudinal vortices and corrugation angle. Fig. 8(d)–(g) show the effect of β on the distribution of temperature under identical θ = 20°. The temperature on Section I apparently increases with the increase of β. The temperature in the region behind the VG in Fig. 8(g) is obviously the highest. This is because the recirculation which is disadvantage to heat transfer forms behind the VG for large β. Obvious recirculation exists behind the VG when β is greater than 45°, and the recirculation is the largest for β = 75°, as shown in Fig. 9. The recirculation keeps the fluid with high temperature stay in the recirculation zone, which suppresses the heat transfer between the fin surface and fluid. On section II in Fig. 8(d)–(g), the temperature in the region behind the VG first increases slightly as β
Fig. 9. Recirculation behind the VG on Section III at θ = 20° and Re = 150, (a) β = 30°, (b) β = 45°, (c) β = 60°, (d) β = 75°.
2.2% and 2.6%, respectively. In order to ensure the quality of the grid and simultaneously save computer resource, all the results in this study are obtained using the medium-size grid number 1100061. The grid is adjusted marginally with the variation in the attack angle of VGs. The details of the grid around the wavy wall and the VG are shown in Fig. 3. The Grid Convergence Index (GCI) based on the Richardson extrapolation [37,38] is used to evaluate the mesh quality. GCI is defined as
GCI = FS
ε rp − 1
(21)
Here, FS is the safety factor and is equal to 1.25 [37], r is the grid refinement ratios, p is the formal order of accuracy of the algorithm, and ε is the relative error. The parameters of p, ε, and r are expressed as
p=
ε=
r=
|ln |(ϕ3 − ϕ2)/(ϕ2 − ϕ1)|| ln(r )
ϕ1 − ϕ2 ϕ1
(23)
hcoarse h fine
⎡1 h=⎢ N ⎣
N
(24) 1/3
∑ ΔVi ⎤⎥ i=1
(22)
⎦
(25)
φ1, φ2, and φ3 are numerical solutions on the fine, medium, and coarse grids with grid spacing h1, h2, and h3, respectively. The fine, medium-size and coarse grids with grid numbers 3808154, 1100061, and 400602 are selected for calculating of GCI. The results of GCI for the temperature and axial velocity are 0.15% and 0.36%, respectively. The accuracy of the numerical results is evaluated by comparison with the experimental results reported by Ali and Ramadhyani [5] under identical geometrical configuration and experimental conditions, as shown in Fig. 4. According to the experimental results, the flow is laminar when Re is less than 400 and is turbulent when Re is higher than 400 for the wavy fin with a corrugation angle of 20°. The numerical results of the laminar model agree well with the experimental results in laminar flow, and the maximum differences in Nu and f are less than 13% and 9.4%, respectively. Three turbulence models (RNG k–ε model, Realizable k–ε model, and SST k–ω model) combined with enhanced wall treatment are employed for turbulent flow. The results indicated that the difference in Nu between the experimental and numerical results is the smallest for the RNG k–ε model. Thus, the RNG k–ε turbulence model is adopted for turbulent flow. The maximum 7
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Fig. 10. Distribution of local Nu on bottom and top fin surfaces at Re = 150, (a) θ = 20°, smooth fin without VGs, (b) θ = 10°, β = 45°, (c) θ = 20°, β = 45°, (d) θ = 30°, β = 45°, (e) θ = 20°, β = 30°, (f) θ = 20°, β = 45°, (g) θ = 20°, β = 60°, (h) θ = 20°, β = 75°.
increases from 30° to 45°, and then decreases from β = 45° to 75°. This is because the main vortex is the largest for β = 45°, and the temperature in the flow channel is high owing to enhancement of heat transfer by the longitudinal vortices. As the recirculation forms and increases as β increases from 45° to 75°, the temperature on section II decreases owing to the suppression of heat transfer in the recirculation region.
on the pressure side is apparently larger than that on the corresponding suction side. The local Nu is the largest at the inlet and decreases gradually along the mainstream from the inlet owing to the development of the thermal boundary from the inlet. The local Nu is the smallest in each trough of the channel owing to the dead zone in the trough region of each wavy period. Moreover, the local Nu in the crest is significantly larger than that in the corresponding trough of the wavy channel. The local values of Nu on both the bottom and top fins are apparently enhanced in the region behind the VGs owing to the longitudinal vortices generated by the VGs. The local Nu on the fin surface except the trough region also increases with the increase in θ for an identical β of VG, as shown in Fig. 10(b)–(h). In the trough region, the local Nu decreases as θ increases owing to the increase in the dead zone in the trough region. Fig. 10(e)–(h) show the distributions of the local Nu for four values of β with identical θ (20°). On the bottom fin surface, the local Nu in the region around the VGs increases as β varies from 30° to 45° and then decreases as β varies from 45° to 75° owing to the formation of recirculation behind the VG as has discussed in Fig. 9. In the region
5.2. Local Nu on fin surfaces The distributions of local Nu on the bottom and top fin surfaces at Re = 150 are shown in Fig. 10. The distribution of local Nu on the smooth wavy fin without VGs is also shown for comparison. The pressure side and suction side of the wavy fin channel are marked in red and blue, respectively, in Fig. 5(a). The flow in the channel formed by the wavy fins is frequently interrupted by the corrugated surface, and the fluid is induced to flow toward the pressure sides of the wavy fin. The thermal boundary layer on the pressure side is thinned, and the local Nu 8
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Fig. 12. Distributions of Nu and f for different values of β at θ = 20°, (a) Nu, (b) f.
Fig. 11. Comparisons of Nus for different θ and β at Re = 150, (a) β = 45°, (b) θ = 20°.
between the trough and the front point of VG, Nus for β = 45° is smaller than that for β = 30°. In other regions, the difference in Nus between β = 30° and β = 45° is marginal. The value of Nus with β = 75° is the smallest, and there is a large difference in the region behind the VG owing to the formation of recirculation. The value of Nus for the wavy fin with β = 45° increases by up to 33% from that of the smooth wavy fin.
behind the VG, the value of the local Nu in the recirculation decreases, and the region with a marginal value of the local Nu increases as β increases, as shown on the bottom fin in Fig. 10(f)–(h). Fig. 11 shows the distributions of the span-averaged Nus for different θ and β. The wavy channel without VGs is also shown in Fig. 9(b) for comparison. Nus is the largest at the inlet and decreases from the inlet owing to the development of the thermal boundary layer. Fig. 11(a) shows the distributions of Nus for three θ values with β = 45°. The peak values of Nus are obtained in the region between the crests or troughs owing to the wash flow toward the pressure sides and the longitudinal vortices generated by the VGs. As the smallest value of local Nu is obtained in the trough, a marginal Nus is obtained at the trough. Nus increases as θ increases. The difference in Nus between the cases with θ = 30° and 20° is distinctly larger than that between the cases with θ = 20° and 10°. The maximum difference in Nus is approximately 48% for different θ. Fig. 11(b) shows the distributions of Nus for different values of β with θ = 20°. Nus for the wavy fin with VGs is apparently larger than that of the smooth wavy fin without VGs. The values of Nus for β = 30° and 45° are apparently larger than those for β = 60° and 75°. In the region around and behind the VGs of the wavy fin, the value of Nus for β = 45° is larger than that for β = 30°. Meanwhile, in the region
5.3. Effect of β on Nu, f, and JF Fig. 12 shows the distributions of the average Nu and f as a function of Re for different β at θ = 20°. Nu and f for the wavy fin without VGs are also shown for comparison. Nu increases with the increase in Re. Both Nu and f of the wavy fin with VGs are larger than that of the smooth wavy fin without VGs. The difference in Nu between the wavy fin with VGs and the wavy fin without VGs increases as Re increases owing to the increasing influence of longitudinal vortices on the heat transfer. The differences in Nu between wavy fin with different values of β are highly marginal, and the values of Nu are nearly identical for Re < 300. This is because the boundary layer is thick, and the longitudinal vortices generated by VGs are very weak for small Re. The value of Nu increases smoothly from Re = 100 to Re = 400 and then 9
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Fig. 13. Distributions of JF for different values of β, (a) θ = 10°, (b) θ = 20°, (c) θ = 30°.
Fig. 14. Distributions of Nu, f, and JF for different values of θ at β = 45°, (a) Nu, (b) f, (c) JF.
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each of the other cases. When β is larger than 45°, variations in β affect the values of Nu marginally for large values of Re in the turbulent flow regime. For the Re range between 300 and 500, Nu generally increases marginally as β increases from 30° to 75°. The maximum difference in Nu is approximately 34% at Re = 400 between the wavy fin with VGs and the smooth wavy fin without VGs. f generally increases as β increases. The difference in f between the wavy fins with β = 30° and 45° is larger than that with β = 45° and 60°. The difference in f between β = 60° and 75° is more or less small, and f for β = 75° is the largest. The maximum difference in f is approximately 30% between the wavy fin with VGs and the smooth wavy fin without VGs. The thermal performance factor JF can reflect the heat transfer performance of VGs based on the identical pump power. Fig. 13 shows the distributions of JF as a function of Re for different β and θ. JF first increases with the increase in Re in the laminar flow regime and then decreases with the increase in Re in the turbulent flow regime. JF is higher than 1.0 for most cases in the laminar flow regime. It decreases from the maximum value to approximately 1.0 when Re > 1000 in the turbulent flow regime for different θ. The largest value of JF is obtained at the largest Re of the laminar flow regime for the wavy fin with different θ. This is because the longitudinal vortices generated by VGs can destroy the flow boundary layer and significantly influences the heat transfer enhancement in laminar flow. Whereas the flow boundary layer is thin and the flow is already unsteady in turbulent flow, the effect of the longitudinal vortices generated by the VGs on the heat transfer is weakened. For wavy fin with different θ, the maximum JF increases as β increases. JF attains the maximum value at β = 45° and the minimum value at β = 75° when Re is small in the laminar flow regime. However, when Re is large under laminar flow, JF generally increases as β increases and attains the maximum value at β = 75°. The maximum value of JF can attain up to 1.26. This implies that the heat transfer performance of the proposed wavy fin with VGs is significantly higher than that of the smooth wavy fin under the performance evaluation criteria with identical pump power. Therefore, combining wavy fin with VGs is effective for enhancing the heat transfer performance of a wavy fin in laminar flow and in turbulent flow with Re < 1000.
5.4. Effect of θ on Nu, f, and JF The distributions of Nu, f and JF as a function of Re for different θ at β = 45° are shown in Fig. 14. Nu increases with the increase in both Re and θ. There is an apparent difference in Nu in the Re range between 300 and 1000 because the transition values of Re are different for different θ. The transition values of Re are 900, 400, and 300 for θ = 10°, 20°, and 30°, respectively. All the cases with different θ are in laminar flow when Re ≤ 300 and in turbulent flow when Re ≥ 1000. The flow is already turbulent for θ = 30°, whereas the flow is still laminar for θ = 10° at Re = 900. There is a large difference in Nu; it increases by 116% when θ ranges from 10° to 30° at Re = 900. f decreases as Re increases and increases as θ increases. The difference in f between different θ increases as Re increases. The difference in f between θ = 30° and 20° is apparently larger than that between θ = 20° and 10°. f of the wavy fin with θ = 30° increases by 159% and 415% compared with that of the wavy fin with θ = 20° and 10°, respectively, at Re = 3000. Thus, the increase in θ causes an exceptional increase in the pressure loss. Fig. 14(c) shows a comparison of the distributions of JF for different θ. JF first increases in the laminar flow and then decreases in the turbulent flow for the studied cases with different θ. JF for θ = 30° attains the largest value when the flow is laminar with Re < 300. In the Re range between 300 and 1000, the value of JF for laminar flow is generally larger than that for turbulent flow for different values of θ. This figure shows that θ has an apparent effect on the comprehensive heat transfer performance.
Fig. 15. Distributions of Nu, f, and JF as a function of β for different values of θ at Re = 150, (a) Nu, (b) f, (c) JF.
increases abruptly as the flow transforms from laminar to turbulent. Among the values of Nu for the different β of VGs, Nu for β = 30° is the smallest. There are large differences between the case with β = 30° and 11
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5.5. Effect of combining θ and β on Nu, f, and JF
500–514. [4] P. Naphon, Heat transfer characteristics and pressure drop in the channel with Vcorrugated upper and lower plates, Energy Convers. Manag. 48 (2007) 1516–1524. [5] M.M. Ali, S. Ramadhyani, Experiments on convective heat transfer in corrugated channels, Exp. Heat Transf. 5 (1992) 175–193. [6] C.C. Wang, K.Y. Chen, Forced convection in a wavy wall channel, Int. J. Heat Mass Transf. 45 (2002) 2587–2595. [7] H. Pehlivan, I. Taymaz, Y. İslamoğlu, Experimental study of forced convective heat transfer in a different arranged corrugated channel, Int. Commun. Heat Mass Transf. 46 (2013) 106–111. [8] H.G. Kwon, S.D. Hwang, H.H. Cho, Flow and heat/mass transfer in a wavy duct with various corrugation angles in two dimensional flow regimes, Heat Mass Transf. 45 (2008) 157–165. [9] R. Mereu, E. Colombo, F. Inzoli, Numerical analysis of fluid dynamics and thermal characteristics inside a wavy channel, Int. J. Numer. Meth. Heat Fluid Flow 23 (2013) 1049–1062. [10] J.Q. Dong, J.P. Chen, W.F. Zhang, J.W. Hu, Experimental and numerical investigation of thermal-hydraulic performance in wavy fin-and-flat tube heat exchangers, Appl. Therm. Eng. 30 (2010) 1377–1386. [11] J.Q. Dong, L. Su, Q. Chen, W.W. Xu, Experimental study on thermal-hydraulic performance of a wavy fin-and-flat tube aluminium heat exchanger, Appl. Therm. Eng. 51 (2013) 32–39. [12] X.L. Zhang, Y.C. Wang, M. Li, S.D. Wang, X.L. Li, Improved flow and heat transfer characteristics for heat exchanger by using a new humped wavy fin, Appl. Therm. Eng. 124 (2017) 510–520. [13] S.D. Hwang, I.H. Jang, H.H. Cho, Experimental study on flow and local heat/mass transfer characteristics inside corrugated duct, Int. J. Heat Fluid Flow 27 (2006) 21–32. [14] G.W. Kim, G.H. Rhee, Optimization study of cross-cut flow control for heat transfer enhancement in wavy fin heat exchangers: Concept of cross-cut reference length, Appl. Therm. Eng. 134 (2018) 527–536. [15] A.M. Jacobi, R.K. Shah, Heat transfer surface enhancement through the use of longitudinal vortices a review of recent progress, Exp. Therm Fluid Sci. 11 (1995) 295–309. [16] M. Fiebig, P. Kallweit, N. Mitra, S. Tiggelbeck, Heat transfer enhancement and drag by longitudinal vortex generators in channel flow, Exp. Therm. Fluid Sci. 4 (1991) 103–114. [17] G. Biswas, H. Chattopadhyay, A. Sinha, Augmentation of heat transfer by creation of streamwise longitudinal vortices using vortex generators, Heat Transf. Eng. 33 (2012) 406–424. [18] M. Samadifar, D. Toghraie, Numerical simulation of heat transfer enhancement in a plate-fin heat exchanger using a new type of vortex generators, Appl. Therm. Eng. 133 (2018) 671–681. [19] M.J. Li, H. Zhang, J. Zhang, Y.T. Mu, E. Tian, D. Dan, X.D. Zhang, W.Q. Tao, Experimental and numerical study and comparison of performance for wavy fin and a plain fin with radiantly arranged winglets around each tube in fin-and-tube heat exchangers, Appl. Therm. Eng. 133 (2018) 298–307. [20] A. Sinha, K.A. Raman, H. Chattopadhyay, G. Biswas, Effects of different orientations of winglet arrays on the performance of plate-fin heat exchangers, Int. J. Heat Mass Transf. 57 (2013) 202–214. [21] A. Sinha, H. Chattopadhyay, A.K. Iyengar, G. Biswas, Enhancement of heat transfer in a fin-tube heat exchanger using rectangular winglet type vortex generators, Int. J. Heat Mass Transf. 101 (2016) 667–681. [22] K.W. Song, L.B. Wang, Effects of longitudinal vortex interaction on periodically developed flow and heat transfer of fin-and-tube heat exchanger, Int. J. Therm. Sci. 109 (2016) 206–216. [23] K.W. Song, T. Tagawa, The optimal arrangement of vortex generators for best heat transfer enhancement in flat-tube-fin heat exchanger, Int. J. Therm. Sci. 132 (2018) 355–367. [24] A.A. Gholami, M.A. Wahid, H.A. Mohammed, Heat transfer enhancement and pressure drop for fin-and-tube compact heat exchangers with wavy rectangular winglet-type vortex generators, Int. Commun. Heat Mass Transf. 54 (2014) 132–140. [25] K.W. Song, T. Tagawa, Z.H. Chen, Q. Zhang, Heat transfer characteristics of concave and convex curved vortex generators in the channel of plate heat exchanger under laminar flow, Int. J. Therm. Sci. 137 (2019) 215–228. [26] K.W. Song, Z.P. Xi, M. Su, L.C. Wang, X. Wu, L.B. Wang, Effect of geometric size of curved delta winglet vortex generators and tube pitch on heat transfer characteristics of fin-tube heat exchanger, Exp. Therm. Fluid Sci. 82 (2017) 8–18. [27] B. Lotfi, B. Sundén, Q.W. Wang, An investigation of the thermo-hydraulic performance of the smooth wavy fin-and-elliptical tube heat exchangers utilizing new type vortex generators, Appl. Energy 162 (2016) 1282–1302. [28] B. Lotfi, M. Zeng, B. Sundén, Q.W. Wang, 3D numerical investigation of flow and heat transfer characteristics in smooth wavy fin-and-elliptical tube heat exchangers using new type vortex generators, Energy 73 (2014) 233–257. [29] L.T. Tian, Y.L. He, Y.B. Tao, W.Q. Tao, A comparative study on the air-side performance of wavy fin-and-tube heat exchanger with punched delta winglets in staggered and in-line arrangements, Int. J. Therm. Sci. 48 (2009) 1765–1776. [30] H.B. Ke, T.A. Khan, W. Li, Y.S. Lin, Z.W. Ke, H. Zhu, Z.J. Zhang, Thermal-hydraulic performance and optimization of attack angle of delta winglets in plain and wavy finned-tube heat exchangers, Appl. Therm. Eng. 150 (2019) 1054–1065. [31] A. Sadeghianjahromi, S. Kheradmand, H. Nemati, Developed correlations for heat transfer and flow friction characteristics of louvered finned tube heat exchangers, Int. J. Therm. Sci. 129 (2018) 135–144. [32] Fluent Inc, FLUENT 17.0 User’s Manual, 2015. [33] J.Y. Yun, K.S. Lee, Influence of design parameters on the heat transfer and flow
In order to analyse the influence of β of VGs and θ on the heat transfer and fluid flow characteristics of the wavy fin with VGs, the steady laminar flow with Re = 150 (at which the β of VGs and θ significantly affect the thermal performance of the wavy fin) is selected. Fig. 15 shows the distributions of Nu, f, and JF as a function of β for different θ at Re = 150. β = 0° implies the smooth wavy channel without VGs. Nu, f, and JF increase as θ increases. Nu first increases as β increases from 30° to 45° and then decreases as β increases from 45° to 75°. Nu attains the maximum at β = 45° for all θ. The maximum values of Nu of the proposed wavy fin for θ = 10°, 20°, and 30° are 5.8%, 8.5%, and 12.6% larger than the corresponding Nu of the smooth wavy fin, respectively. f generally increases as β increases. The maximum value of f of the proposed wavy fin is higher by 7%, 8.2%, and 7.7% than that of the smooth wavy fin for θ = 10°, 20°, and 30°, respectively. The differences in Nu and f between θ = 30° and 20° are larger than those between θ = 20° and 10°. Thus, the increasing in θ significantly affects Nu and f when θ is larger than 20°. JF first increases as β increases from 30° to 45° and then decreases as β increases from 45° to 75°. Thus, JF attains the highest value at β = 45° for different θ. The values of JF for β = 30° and 45° are higher than those for β = 60° and 75°. Therefore, the optimum β of VG exists in terms of the highest heat transfer performance of the wavy fin in laminar flow. 6. Conclusions The thermo-hydraulic performance of the novel combination of a wavy fin and vortex generators was numerically studied for different corrugation angles of the wavy fin and different attack angles of the vortex generators. The primary conclusions are summarized as follows: 1. The heat transfer performance of the novel wavy fin is apparently enhanced by combining vortex generators on the wavy fin. The span averaged Nus for the wavy fin with θ = 20° and β = 45° can increase by up to 33% compared with the smooth wavy fin. 2. The thermal performance factor first increases as Re increases in the laminar flow regime and then decreases as Re increases in the turbulent flow regime. Good performance can be obtained by an appropriate corrugation angle at different Re. The maximum thermal performance factor increases by 26.4%. 3. The optimum attack angle of vortex generators exists for the highest heat transfer performance in laminar flow. An attack angle of 45° is significantly more effective for enhancing thermo-hydraulic performance. Acknowledgement This research was funded by the National Natural Science Foundation of China (51866007); the Gansu Provincial Natural Science Foundation (17JR5RA092); Collaborative Innovation Team Project of Gansu (2018C-13); and the Foundation of a hundred youth talents training program of Lanzhou Jiaotong University. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2019.114343. References [1] K. Manjunath, S.C. Kaushik, Second law thermodynamic study of heat exchangers: A review, Renew. Sustain. Energy Rev. 40 (2014) 348–374. [2] W.M. Kays, A.L. London, Compact heat exchangers, third ed., McGraw-Hill, New York, 1984. [3] Z. Wang, Y.Z. Li, Layer pattern thermal design and optimization for multistream plate-fin heat exchangers-a review, Renew. Sustain. Energy Rev. 53 (2016)
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[36] G. Comini, C. Nonino, S. Savino, Convective heat and mass transfer in wavy finnedtube exchangers, Int. J. Numer. Meth. Heat Fluid Flow 12 (2002) 735–755. [37] P.J. Roache, Quantification of uncertainty in computational fluid dynamics, Annu. Rev. Fluid Mech. 29 (1997) 123–160. [38] A. Mansour, E. Laurien, Numerical error analysis for three-dimensional CFD simulations in the two-room model containment THAI+: Grid convergence index, wall treatment error and scalability tests, Nucl. Eng. Des. 326 (2018) 220–233.
friction characteristics of the heat exchanger with slit fins, Int. J. Heat Mass Transf. 43 (2000) 2529–2539. [34] K.W. Song, S. Liu, L.B. Wang, Interaction of counter rotating longitudinal vortices and the effect on fluid flow and heat transfer, Int. J. Heat Mass Transf. 93 (2016) 349–360. [35] R.K. Shah, Compact heat exchanger surface selection methods, Heat Transf. 4 (1978) 234–238.
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Thermo-hydraulic performance optimization of wavy fin heat exchanger by combining delta winglet vortex generators
T
⁎
Chao Luoa, Shuai Wua, Kewei Songa, , Liang Huab, Liangbi Wanga a b
School of Mechanical Engineering, Key Laboratory of Railway Vehicle Thermal Engineering of MOE, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China Institute of Locomotive and Vehicle, Nanjing Institute of Railway Technology, Nanjing, Jiangsu 210031, China
H I GH L IG H T S
novel combination of wavy fin and vortex generators is numerically studied. • AConsiderable improvement of thermal performance is reported. • Optimum attack angle of 45° has the best thermal performance in laminar flow. • Thermal performance factor increases by up to 26.4% owing to the novel combination. •
A R T I C LE I N FO
A B S T R A C T
Keywords: Wavy fin Vortex generator Corrugation angle Attack angle Thermal performance
Wavy fins and vortex generators are effective methods for enhancing heat transfer of heat exchangers. In general, wavy fins and vortex generators are studied separately in the literature. A novel combination of a wavy fin and vortex generators is proposed in the present study. The effects of different corrugation angles of the wavy fin and different attack angles of the vortex generators are numerically studied to optimize the thermo-hydraulic performance of the novel combination. The Nusselt number, friction factor, and thermal performance factor are compared between different corrugation angles and attack angles and between wavy fins with and without vortex generators. The heat transfer performance is improved considerably by the novel combination. The maximum thermal performance factor increases by 26.4% owing to the combination of the vortex generator and wavy fin. The optimum attack angle of the vortex generator, which yields the highest heat transfer performance of the wavy fin plate heat exchanger, is reported as 45°.
1. Introduction Heat exchangers, which are energy transfer equipment, are applied widely in industrial fields, such as power engineering, chemical engineering, and air-conditioning [1–3]. The enhancement of the heat transfer efficiency of heat exchangers is of immense significance for saving energy. Both wavy fins and vortex generators are effective for enhancing heat transfer and are widely studied for compact heat exchangers. The effect of wavy configurations on the thermo-hydraulic performance of wavy fin is studied widely [4–12]. Naphon [4] and Ali et al. [5] experimentally studied the thermo-hydraulic performance of a wavy fin. Their results demonstrated that the heat transfer of wavy channel is significantly enhanced by a corrugated surface. Wang et al. [6] numerically reported that both heat transfer and pressure loss increase as the amplitude–wavelength ratio increases. Pehlivan et al. [7] ⁎
experimentally illustrated that the heat transfer rate of a wavy fin increases with an increase in the corrugation angle. Kwon et al. [8] and Mereu [9] reported that the corrugation angle significantly impacts the transition of the flow state in a wavy channel and that the transition to turbulent flow occurs earlier as the corrugation angle increases. Dong et al. [10,11] experimentally and numerically revealed that an increase in the wave amplitude can improve the heat transfer performance, whereas an increase in the fin length and fin pitch decreases the heat transfer performance. A wavy fin with surface modification can further improve the heat transfer performance. Zhang et al. [12] reported that a humped wavy fin can effectively improve the thermo-hydraulic performance. Hwang et al. [13] experimentally and numerically investigated the influence of secondary motions on a wavy channel and determined that secondary flows exert an important effect on heat transfer and pressure loss. Kim et al. [14] numerically revealed that the cross-cut wavy fin can yield a 21.71% increment in Nusselt number
Corresponding author. E-mail address: [email protected] (K. Song).
https://doi.org/10.1016/j.applthermaleng.2019.114343 Received 4 April 2019; Received in revised form 26 July 2019; Accepted 4 September 2019 Available online 05 September 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature A b C Dh f Gk GCI h H JF k L Lx N Nu p P PL r Re S tp T Ts
u, v, w X x, y, z
cross-sectional area (m2) half the transverse distance between two VGs (m) a quarter of the wavelength (m) hydraulic diameter (m) Fanning friction factor generation of turbulent kinetic energy grid convergence index grid spacing height of vortex generator (m) thermal performance factor or surface goodness factor turbulent kinetic energy base length of vortex generator (m) wavy fin length (m) number of wavy Nusselt number pressure loss (Pa) or order of accuracy for GCI wetted perimeter (m) width of the simulation domain (m) grid refinement ratio Reynolds number fin surface area (m2) fin spacing (m) temperature (K) bulk temperature of the cross-section (K)
components of velocity vector (m/s) non-dimensional distance, x/4C coordinates
Greek symbols β δS(x) ε θ λ λt μ μt ρ φ
attack angle of VG (°) fin strip area at position x (m2) turbulent dissipation rate or relative error for GCI corrugation angle (°) fluid thermal conductivity (W/(m·K)) turbulent thermal conductivity (W/(m·K)) viscosity (kg/(m·s)) turbulent dynamic viscosity (kg/(m·s)) density (kg/m3) numerical solution
Subscripts in local out s t w
inlet local value outlet span average value turbulent wall surface
generators is likely to exhibit a good thermal performance. Lotfi et al. [27,28] numerically studied the effect of different novel VGs on the thermo-hydraulic performance of a wavy fin-and-elliptical tube heat exchanger. The best thermo-hydraulic performance is obtained for the curved angle rectangular VG. Tian et al. [29] and Ke et al. [30] numerically studied the thermo-hydraulic performance of wavy fin-andtube heat exchangers with VGs on the fin. Considerable enhancement in the heat transferred by the wavy fin-and-tube heat exchanger is obtained owing to the longitudinal vortices generated by the VGs. The foregoing researches demonstrate that both the wavy fin and vortex generators are effective for heat transfer enhancement. Wavy fin and vortex generator have generally been studied separately. There are few researches on combinations of wavy fin and vortex generators. In this paper, a novel wavy fin with vortex generators is proposed for a plate heat exchanger. The thermo-hydraulic performance of the novel combination of wavy fin and vortex generators is numerically investigated by considering the optimal combination of the corrugation angle of the wavy fin and the attack angle of the vortex generators. The results have significant importance in the design of wavy fin plate heat exchanger for improving the thermo-hydraulic performance.
with a 3.95% increment in the friction factor compared with the typical wavy fin at Re = 300. Apart from wavy fin, winglet vortex generator (VG), which can generate longitudinal vortices, is also effective for heat transfer enhancement [15–26]. Fiebig et al. [16] and Biswas et al. [17] reported that longitudinal vortices generated by VG can enhance the heat transfer with less pressure penalty. A simple rectangular VG with an attack angle of 45° was reported to exhibit the highest heat transfer enhancement in a triangular channel [18]. Li et al. [19] compared the performance of a wavy fin and a plain fin with VGs. Their experimental results revealed that a plain fin with five-row tubes, enhanced with VGs, exhibits a better performance than that of a wavy fin with six-row tubes. The heat transfer enhancement is also affected by the arrangement configurations of VGs, such as the common-flow-up and commonflow-down configurations [17,20–22]. Sinha et al. [20] compared the performance of a plate-fin heat exchanger with two-row VG pairs arranged in different configurations. Their results revealed that the VGs with common-flow-up configurations exhibit higher heat transfer and flow characteristics than those with the other configurations. Longitudinal vortices with different rotating directions can significantly improve the heat transfer performance of a fin-and-tube heat exchanger [21]. Song et al. [22,23] quantitatively studied the interaction between longitudinal vortices and reported that the highest heat transfer performance can be obtained at a suitable transverse pitch of VGs. In addition to the normal plane VGs, wavy VG and curved VG have been recently reported to exhibit a better heat transfer performance than the normal plane VGs. Gholami et al. [24] reported that a wavy rectangular VG can enhance the heat transfer with a smaller pressure drop compared with the normal plane rectangular VG. Song et al. [25] revealed that the concave curved VG is more favourable to heat transfer performance than the convex curved VG and the plane VG. Song et al. [26] experimentally reported that a smaller curved VG is effective for heat transfer enhancement of circular tube-fin heat exchanger at low Re values. Moreover, a larger VG is suitable for heat transfer enhancement at large Re values. As either the wavy fin and vortex generators can effectively improve heat transfer performance, a combination of a wavy fin and vortex
2. Physical model A schematic view of the studied wavy fin with delta winglet vortex generators is shown in Fig. 1. The delta winglet vortex generators are mounted on the wavy fin surface in pairs. The geometric size of the fin, as shown in Fig. 2, is identical to that in the experiment conducted by Ali and Ramadhyani [5]. The simulation domain is prolonged by 20.69 mm in both the inlet and outlet regions. The three corrugation angles (θ) considered are 10°, 20°, and 30°. The wavy fin pitch is tp = 6.9 mm, and wavy fin wavelength is 4C = 45.7 mm. C is the distance from the trough of the wavy fin to the front point of the vortex generators. The region sketched by dashed lines in Fig. 2 (b) is selected as the simulation domain. A schematic view of the simulation domain is shown in Fig. 2 (c). The width of the simulation domain is PL = 44.45 mm. The geometric parameters of a delta winglet VG are employed based on previous studies [19,29,30]. The height of the delta 2
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Table 1 Parameters of simulated domain. Fin pitch (tp) Hydraulic diameter (Dh) Wavy fin wavelength (4C) Simulated domain (PL) Prolonged length at inlet (Ei) Prolonged length at outlet (Eo) Length of VG (L) Height of VG (H) Transverse VGs pitch (2b) Corrugation angles (θ) Attack angles of VGs (β) Number of wavy (N) Inlet temperature Wavy fin temperature
Fig. 1. Schematic view of wavy fin with delta winglet VGs.
winglet VG is two-third the height of the flow channel, and the length of the VG is twice the height of VG. The transverse distance of the VGs is 2b = 35.56 mm, and the attack angles (β) of the VG are 30°, 45°, 60°, and 75°. The length and height of each VG are L = 9.2 mm and H = 4.6 mm, respectively. The geometric parameters and boundary temperature conditions are summarized in Table 1.
Table 2 Grid independence study.
3. Mathematical formulation The operating fluid is water, which is considered as an incompressible fluid. As the inlet temperature of water is 300 K and the fin temperature is 310 K, the maximum temperature difference is 10 K. Thus, the physical properties of water are considered to be constant for this marginal temperature difference. The constant values of the physical properties of water are calculated at the mean temperature of 305 K. The flow in the computational domain is assumed to be threedimensional, steady-state, and without viscous dissipation. In laminar flow, the governing equations can be expressed as follows: Continuity equation:
∂ ρui = 0 ∂x i
6.9 mm 13.8 mm 45.7 mm 44.45 mm 20.69 mm 20.69 mm 9.2 mm 4.6 mm 35.56 mm 10°, 20°, 30° 30°, 45°, 60°, 75° 5 Tin = 300 K Tw = 310 K
Grid number
Nu
f
400,602 926,250 1,100,061 1,648,886 3,808,154 Maximum Relative error
13.26 13.10 13.14 13.09 12.98 2.2%
0.1440 0.1425 0.1419 0.1416 0.1404 2.6%
Momentum equation:
∂p ∂ ∂ ⎛ ∂uj ⎞ (ρui uj ) = μ − ∂x j ∂x i ∂x i ⎝ ∂x i ⎠ ⎜
⎟
(2)
Energy equation:
∂ ∂ ⎛ ∂T ⎞ (ρCp ui T ) = λ ∂x i ∂x i ⎝ ∂x i ⎠ ⎜
(1)
⎟
Fig. 2. Parameters of wavy fin and VG, (a) front view, (b) top view, (c) schematic view of simulation domain, (d) VG. 3
(3)
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In turbulent flow, the compact forms of the momentum and energy equations are
∂uj ⎞ ∂p ∂ ∂ ⎛ (ρui uj ) = (μ + μt ) − ∂x i ∂x i ⎝ ∂x i ⎠ ∂x j
(4)
∂ ∂ ⎛ ∂T ⎞ (ρCp ui T ) = (λ + λt ) ∂x i ∂x i ⎝ ∂x i ⎠
(5)
⎜
⎟
⎜
⎟
The RNG k–ε turbulent model [31]:
∂uj ⎞ ∂ ∂ ⎛ + Gk − ρε (ρkuj ) = (μ + μt ) λ ∂x i ⎠ ∂x i ∂x i ⎝
(6)
∂uj ⎞ ε ε2 ∂ ∂ ⎛ (ρεuj ) = (μ + μt ) λ + C1ε Gk − C2ε ρ k k ∂x i ⎠ ∂x i ∂x i ⎝
(7)
⎜
⎟
⎜
μt = ρCμ Fig. 3. Grid system, (a) 3D view, (b) grid around the VG.
⎟
k2 ε
(8)
where C1ε, C2ε, and Cμ are constants and are equal to 1.42, 1.68, and 0.0845, respectively. The velocity-inlet boundary condition is applied at the inlet, where the fluid velocity is assumed to be uniform with a constant temperature. According to the turbulence intensity correlation reported in [32], the turbulence intensity is determined based on Re as I = 0.16Re−1/8. The outflow boundary condition is applied at the outlet. The no-slip conditions are applied at the wavy fin and vortex generator surfaces. The fin surfaces of the prolonged regions are adiabatic. Constant temperature boundary conditions are applied on the vortex generators and the wavy fin surfaces except the prolonged regions. Symmetry boundary conditions are applied at the side surfaces of the simulation domain. The boundary conditions of the simulation domain are indicated in Fig. 2(c). At the inlet:
u (x , y, z ) = uin , v (x , y, z ) = 0, w (x , y, z ) = 0, T (x , y, z ) = Tin
(9)
At the outlet:
∂ ∂ ∂ ∂ u (x , y, z ) = 0, v (x , y, z ) = 0, w (x , y, z ) = 0, T (x , y, z ) = 0 ∂x ∂x ∂x ∂x (10) On the symmetric surfaces:
∂ ∂ ∂ u (x , y, z ) = 0, v (x , y, z ) = 0, w (x , y, z ) = 0, T (x , y, z ) = 0 ∂z ∂z ∂z (11) On the solid surfaces:
u (x , y, z ) = 0, v (x , y, z ) = 0, w (x , y, z ) = 0, T = Tw
(12)
Reynolds number is
Re =
ρuin Dh μ
(13)
Dh is the hydraulic diameter:
Dh =
4A P
(14)
The Fanning friction factor is
f= Fig. 4. Comparisons between numerical and experimental results, (a) Nu, (b) f.
Δp / L x Dh (ρuin2 /2) 4
(15)
The local Nusselt number is determined as
Nulocal = −Dh
∂T ∂n
/(Tw − Ts (x )) w
The cross-sectional bulk temperature Ts(x) is 4
(16)
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Fig. 5. Positions of Sections I, II, and III. (a) Front view, (b) top view.
Fig. 6. Velocity on Sections I and II with Re = 150, (a) θ = 10°, β = 45°, (b) θ = 20°, β = 45°, (c) θ = 30°, β = 45°, (d) θ = 20°, β = 30°, (e) θ = 20°, β = 45°, (f) θ = 20°, β = 60°, (g) θ = 20°, β = 75°.
∬ u (x , y, z ) T (x , y, z ) dA
4. Numerical method and grid independence
A
Ts (x ) =
∬ u (x , y, z ) dA A
The range of Re for laminar flow varies with the corrugation angles of the wavy fin. According to Dong et al. [10,11], the flow pattern is confined to steady laminar for Re ≤ 1000 and θ = 10°. Ali and Ramadhyani [5] reported that the flow is laminar for θ = 20° and Re ≤ 400. Comini et al. [36] found that the flow is laminar for θ = 30° and Re ≤ 300. Therefore, the turbulence model is applied when Re exceeds 1000, 400, and 300 for θ = 10°, 20°, and 30°, respectively. The commercial software FLUENT 17.0 is used for solving the governing equations. ICEM is used for generating the mesh for the simulation domain. The simulated domain is meshed with structured hexahedral elements. The governing equations along with the boundary conditions were solved using the finite volume method. The SIMPLEC algorithm was used to perform the coupling of velocity and pressure. The second order upwind scheme is used for the convection terms and central difference scheme for the diffusion terms. The first order upwind scheme was adopted for the turbulent kinetic energy and turbulent dissipation rate term. The least squares cell based method was used for the gradient term, and the second order accuracy was used for the pressure and energy terms. The normalized residual values are set as 108 and 10-5 for the energy equation and the other equations, respectively. The enhanced wall treatment is adopted for the turbulence model. The grid near-wall function requires y+ ≈ 1 according to the user
(17)
The span-average Nusselt number of the bottom and top fin surfaces is
Nus (x ) =
1 δS (x )
∬ Nuloacl dS δS (x )
(18)
The average Nu is calculated by averaging Nulocal on the entire fin surface area:
Nu =
1 S
∬ Nuloacl dS S
(19)
The thermal performance factor or surface goodness factor JF [33–35] based on identical pump power is
JF =
Nu/ Nu 0 (f / f0 )1/3
(20)
where the subscript zero implies the corresponding value of the wavy channel without VGs at an identical Re. 5
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Fig. 7. Contour plots of absolute vorticity on Sections I and II with Re = 150, (a) θ = 10°, β = 45°, (b) θ = 20°, β = 45°, (c) θ = 30°, β = 45°, (d) θ = 20°, β = 30°, (e) θ = 20°, β = 45°, (f) θ = 20°, β = 60°, (g) θ = 20°, β = 75°.
Fig. 8. Temperature on the sections I and II at Re = 150, (a) θ = 10°, β = 45°, (b) θ = 20°, β = 45°, (c) θ = 30°, β = 45°, (d) θ = 20°, β = 30°, (e) θ = 20°, β = 45°, (f) θ = 20°, β = 60°, (g) θ = 20°, β = 75°.
combination of wavy fin and vortex generators. The corrugation angle of the wavy fin is θ = 20°, and the attack angle of the vortex generator is β = 60°. Five grid systems with different grid numbers (400602, 926250, 1100061, 1648886, and 3808154) are presented in Table 2. The grid number of the fine grid is approximately 9.5 times larger than that of the coarse grid. The maximum relative errors of Nu and f are
manual of Fluent 17.0 [32]. As it is challenging to generate a good grid density for y+ ≥ 1 when the Re is small under a turbulent flow, y+ = 0.5 was adopted in the present study in order to obtain a fine grid in the studied range of Re. The near wall mesh spacing is gradually increased by 1.2 times until the mesh spacing tend to be balanced. The grid independence study was carried out at Re = 300 for the 6
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differences in Nu and f between the experimental results and the RNG k–ε model are less than 13.4% and 10.7%, respectively. Thus, the prediction accuracy of the numerical results can be guaranteed, and the numerical results are reliable. 5. Results and discussions 5.1. Temperature, velocity, and vorticity on the sections In order to study the effects of β and θ on the heat transfer and flow characteristics, three sections are selected in the third period of the wavy fin, as shown in Fig. 5. Section I, which is on the y–z plane, is located at an equal distance from the ends of the VGs for the different β and θ. Section II, which is also on the y–z plane, is fixed at the midpoint of the third period of the wavy fin. Section III is selected so as to pass through the midpoint of VG along the x–y plane in the third wavy period in order to illustrate the recirculation behind the VG. Fig. 6 shows the distribution of longitudinal vortices on the selected sections I and II for different β and θ at Re = 150. The attack angle is β = 45°, and θ ranges from 10° to 30° in Fig. 6(a)–(c). It is evident that the longitudinal vortices generated by VGs with an identical β are similar for different θ. The longitudinal vortices increase slightly as θ increases. On section II, there are two longitudinal vortices: the main vortex and corner vortex generated by the VG located in front of section II. The vortices on section II also increase with the increase in θ. Fig. 6(d)–(g) show the longitudinal vortices on sections I and II for different β ranging from 30° to 75° with identical θ (20°). The main vortex on section I first increases as β increases from 30° to 45° and then decreases as β increases from 45° to 75°. Thus, the main vortex is the largest for β = 45°. The velocity on section I becomes complex as β increases to a large value owing to the formation of recirculation behind the VG. The difference in the main vortices on section II is slight and the corner vortices generally increase marginally as β increases. Fig. 7 shows the contour plot of vorticity on sections I and II corresponding to Fig. 6. The vorticity on section I is apparently larger than that on section II owing to the attenuation of longitudinal vortices along the mainstream. The vorticity caused by the vortices generated by VGs with identical β increases as θ increases, on both sections I and II in Fig. 7(a)–(c). It can be determined that a wavy fin with a large θ can increase the heat transfer. For different β under an identical θ as show in Fig. 7(d)–(g), the zones of vorticity with the large value gradually separate on section I when β varies from 30° to 75°. Moreover, the vorticity of the main vortex for β = 30° is apparently smaller than those for the other β values. On section II, the difference in vorticity for the main vortex is marginal, and the vorticity for the corner vortex generally increases as β increases. It is apparent that the distributions of vorticity on Sections I and II are consistent with the distributions of longitudinal vortices. The large vorticity in Fig. 7 implies a high intensity of longitudinal vortices in Fig. 6. Fig. 8 shows the contour plot of temperature on sections I and II at Re = 150. The temperature on the cross sections can reflect the mixing of fluid behind the VGs. The temperature on Sections I and II increases gradually with the increase in θ under an identical attack angle, as shown in Fig. 8(a)–(c). This is because the mixture of the fluid is enhanced owing to the increase of the longitudinal vortices and corrugation angle. Fig. 8(d)–(g) show the effect of β on the distribution of temperature under identical θ = 20°. The temperature on Section I apparently increases with the increase of β. The temperature in the region behind the VG in Fig. 8(g) is obviously the highest. This is because the recirculation which is disadvantage to heat transfer forms behind the VG for large β. Obvious recirculation exists behind the VG when β is greater than 45°, and the recirculation is the largest for β = 75°, as shown in Fig. 9. The recirculation keeps the fluid with high temperature stay in the recirculation zone, which suppresses the heat transfer between the fin surface and fluid. On section II in Fig. 8(d)–(g), the temperature in the region behind the VG first increases slightly as β
Fig. 9. Recirculation behind the VG on Section III at θ = 20° and Re = 150, (a) β = 30°, (b) β = 45°, (c) β = 60°, (d) β = 75°.
2.2% and 2.6%, respectively. In order to ensure the quality of the grid and simultaneously save computer resource, all the results in this study are obtained using the medium-size grid number 1100061. The grid is adjusted marginally with the variation in the attack angle of VGs. The details of the grid around the wavy wall and the VG are shown in Fig. 3. The Grid Convergence Index (GCI) based on the Richardson extrapolation [37,38] is used to evaluate the mesh quality. GCI is defined as
GCI = FS
ε rp − 1
(21)
Here, FS is the safety factor and is equal to 1.25 [37], r is the grid refinement ratios, p is the formal order of accuracy of the algorithm, and ε is the relative error. The parameters of p, ε, and r are expressed as
p=
ε=
r=
|ln |(ϕ3 − ϕ2)/(ϕ2 − ϕ1)|| ln(r )
ϕ1 − ϕ2 ϕ1
(23)
hcoarse h fine
⎡1 h=⎢ N ⎣
N
(24) 1/3
∑ ΔVi ⎤⎥ i=1
(22)
⎦
(25)
φ1, φ2, and φ3 are numerical solutions on the fine, medium, and coarse grids with grid spacing h1, h2, and h3, respectively. The fine, medium-size and coarse grids with grid numbers 3808154, 1100061, and 400602 are selected for calculating of GCI. The results of GCI for the temperature and axial velocity are 0.15% and 0.36%, respectively. The accuracy of the numerical results is evaluated by comparison with the experimental results reported by Ali and Ramadhyani [5] under identical geometrical configuration and experimental conditions, as shown in Fig. 4. According to the experimental results, the flow is laminar when Re is less than 400 and is turbulent when Re is higher than 400 for the wavy fin with a corrugation angle of 20°. The numerical results of the laminar model agree well with the experimental results in laminar flow, and the maximum differences in Nu and f are less than 13% and 9.4%, respectively. Three turbulence models (RNG k–ε model, Realizable k–ε model, and SST k–ω model) combined with enhanced wall treatment are employed for turbulent flow. The results indicated that the difference in Nu between the experimental and numerical results is the smallest for the RNG k–ε model. Thus, the RNG k–ε turbulence model is adopted for turbulent flow. The maximum 7
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Fig. 10. Distribution of local Nu on bottom and top fin surfaces at Re = 150, (a) θ = 20°, smooth fin without VGs, (b) θ = 10°, β = 45°, (c) θ = 20°, β = 45°, (d) θ = 30°, β = 45°, (e) θ = 20°, β = 30°, (f) θ = 20°, β = 45°, (g) θ = 20°, β = 60°, (h) θ = 20°, β = 75°.
increases from 30° to 45°, and then decreases from β = 45° to 75°. This is because the main vortex is the largest for β = 45°, and the temperature in the flow channel is high owing to enhancement of heat transfer by the longitudinal vortices. As the recirculation forms and increases as β increases from 45° to 75°, the temperature on section II decreases owing to the suppression of heat transfer in the recirculation region.
on the pressure side is apparently larger than that on the corresponding suction side. The local Nu is the largest at the inlet and decreases gradually along the mainstream from the inlet owing to the development of the thermal boundary from the inlet. The local Nu is the smallest in each trough of the channel owing to the dead zone in the trough region of each wavy period. Moreover, the local Nu in the crest is significantly larger than that in the corresponding trough of the wavy channel. The local values of Nu on both the bottom and top fins are apparently enhanced in the region behind the VGs owing to the longitudinal vortices generated by the VGs. The local Nu on the fin surface except the trough region also increases with the increase in θ for an identical β of VG, as shown in Fig. 10(b)–(h). In the trough region, the local Nu decreases as θ increases owing to the increase in the dead zone in the trough region. Fig. 10(e)–(h) show the distributions of the local Nu for four values of β with identical θ (20°). On the bottom fin surface, the local Nu in the region around the VGs increases as β varies from 30° to 45° and then decreases as β varies from 45° to 75° owing to the formation of recirculation behind the VG as has discussed in Fig. 9. In the region
5.2. Local Nu on fin surfaces The distributions of local Nu on the bottom and top fin surfaces at Re = 150 are shown in Fig. 10. The distribution of local Nu on the smooth wavy fin without VGs is also shown for comparison. The pressure side and suction side of the wavy fin channel are marked in red and blue, respectively, in Fig. 5(a). The flow in the channel formed by the wavy fins is frequently interrupted by the corrugated surface, and the fluid is induced to flow toward the pressure sides of the wavy fin. The thermal boundary layer on the pressure side is thinned, and the local Nu 8
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Fig. 12. Distributions of Nu and f for different values of β at θ = 20°, (a) Nu, (b) f.
Fig. 11. Comparisons of Nus for different θ and β at Re = 150, (a) β = 45°, (b) θ = 20°.
between the trough and the front point of VG, Nus for β = 45° is smaller than that for β = 30°. In other regions, the difference in Nus between β = 30° and β = 45° is marginal. The value of Nus with β = 75° is the smallest, and there is a large difference in the region behind the VG owing to the formation of recirculation. The value of Nus for the wavy fin with β = 45° increases by up to 33% from that of the smooth wavy fin.
behind the VG, the value of the local Nu in the recirculation decreases, and the region with a marginal value of the local Nu increases as β increases, as shown on the bottom fin in Fig. 10(f)–(h). Fig. 11 shows the distributions of the span-averaged Nus for different θ and β. The wavy channel without VGs is also shown in Fig. 9(b) for comparison. Nus is the largest at the inlet and decreases from the inlet owing to the development of the thermal boundary layer. Fig. 11(a) shows the distributions of Nus for three θ values with β = 45°. The peak values of Nus are obtained in the region between the crests or troughs owing to the wash flow toward the pressure sides and the longitudinal vortices generated by the VGs. As the smallest value of local Nu is obtained in the trough, a marginal Nus is obtained at the trough. Nus increases as θ increases. The difference in Nus between the cases with θ = 30° and 20° is distinctly larger than that between the cases with θ = 20° and 10°. The maximum difference in Nus is approximately 48% for different θ. Fig. 11(b) shows the distributions of Nus for different values of β with θ = 20°. Nus for the wavy fin with VGs is apparently larger than that of the smooth wavy fin without VGs. The values of Nus for β = 30° and 45° are apparently larger than those for β = 60° and 75°. In the region around and behind the VGs of the wavy fin, the value of Nus for β = 45° is larger than that for β = 30°. Meanwhile, in the region
5.3. Effect of β on Nu, f, and JF Fig. 12 shows the distributions of the average Nu and f as a function of Re for different β at θ = 20°. Nu and f for the wavy fin without VGs are also shown for comparison. Nu increases with the increase in Re. Both Nu and f of the wavy fin with VGs are larger than that of the smooth wavy fin without VGs. The difference in Nu between the wavy fin with VGs and the wavy fin without VGs increases as Re increases owing to the increasing influence of longitudinal vortices on the heat transfer. The differences in Nu between wavy fin with different values of β are highly marginal, and the values of Nu are nearly identical for Re < 300. This is because the boundary layer is thick, and the longitudinal vortices generated by VGs are very weak for small Re. The value of Nu increases smoothly from Re = 100 to Re = 400 and then 9
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Fig. 13. Distributions of JF for different values of β, (a) θ = 10°, (b) θ = 20°, (c) θ = 30°.
Fig. 14. Distributions of Nu, f, and JF for different values of θ at β = 45°, (a) Nu, (b) f, (c) JF.
10
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each of the other cases. When β is larger than 45°, variations in β affect the values of Nu marginally for large values of Re in the turbulent flow regime. For the Re range between 300 and 500, Nu generally increases marginally as β increases from 30° to 75°. The maximum difference in Nu is approximately 34% at Re = 400 between the wavy fin with VGs and the smooth wavy fin without VGs. f generally increases as β increases. The difference in f between the wavy fins with β = 30° and 45° is larger than that with β = 45° and 60°. The difference in f between β = 60° and 75° is more or less small, and f for β = 75° is the largest. The maximum difference in f is approximately 30% between the wavy fin with VGs and the smooth wavy fin without VGs. The thermal performance factor JF can reflect the heat transfer performance of VGs based on the identical pump power. Fig. 13 shows the distributions of JF as a function of Re for different β and θ. JF first increases with the increase in Re in the laminar flow regime and then decreases with the increase in Re in the turbulent flow regime. JF is higher than 1.0 for most cases in the laminar flow regime. It decreases from the maximum value to approximately 1.0 when Re > 1000 in the turbulent flow regime for different θ. The largest value of JF is obtained at the largest Re of the laminar flow regime for the wavy fin with different θ. This is because the longitudinal vortices generated by VGs can destroy the flow boundary layer and significantly influences the heat transfer enhancement in laminar flow. Whereas the flow boundary layer is thin and the flow is already unsteady in turbulent flow, the effect of the longitudinal vortices generated by the VGs on the heat transfer is weakened. For wavy fin with different θ, the maximum JF increases as β increases. JF attains the maximum value at β = 45° and the minimum value at β = 75° when Re is small in the laminar flow regime. However, when Re is large under laminar flow, JF generally increases as β increases and attains the maximum value at β = 75°. The maximum value of JF can attain up to 1.26. This implies that the heat transfer performance of the proposed wavy fin with VGs is significantly higher than that of the smooth wavy fin under the performance evaluation criteria with identical pump power. Therefore, combining wavy fin with VGs is effective for enhancing the heat transfer performance of a wavy fin in laminar flow and in turbulent flow with Re < 1000.
5.4. Effect of θ on Nu, f, and JF The distributions of Nu, f and JF as a function of Re for different θ at β = 45° are shown in Fig. 14. Nu increases with the increase in both Re and θ. There is an apparent difference in Nu in the Re range between 300 and 1000 because the transition values of Re are different for different θ. The transition values of Re are 900, 400, and 300 for θ = 10°, 20°, and 30°, respectively. All the cases with different θ are in laminar flow when Re ≤ 300 and in turbulent flow when Re ≥ 1000. The flow is already turbulent for θ = 30°, whereas the flow is still laminar for θ = 10° at Re = 900. There is a large difference in Nu; it increases by 116% when θ ranges from 10° to 30° at Re = 900. f decreases as Re increases and increases as θ increases. The difference in f between different θ increases as Re increases. The difference in f between θ = 30° and 20° is apparently larger than that between θ = 20° and 10°. f of the wavy fin with θ = 30° increases by 159% and 415% compared with that of the wavy fin with θ = 20° and 10°, respectively, at Re = 3000. Thus, the increase in θ causes an exceptional increase in the pressure loss. Fig. 14(c) shows a comparison of the distributions of JF for different θ. JF first increases in the laminar flow and then decreases in the turbulent flow for the studied cases with different θ. JF for θ = 30° attains the largest value when the flow is laminar with Re < 300. In the Re range between 300 and 1000, the value of JF for laminar flow is generally larger than that for turbulent flow for different values of θ. This figure shows that θ has an apparent effect on the comprehensive heat transfer performance.
Fig. 15. Distributions of Nu, f, and JF as a function of β for different values of θ at Re = 150, (a) Nu, (b) f, (c) JF.
increases abruptly as the flow transforms from laminar to turbulent. Among the values of Nu for the different β of VGs, Nu for β = 30° is the smallest. There are large differences between the case with β = 30° and 11
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5.5. Effect of combining θ and β on Nu, f, and JF
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In order to analyse the influence of β of VGs and θ on the heat transfer and fluid flow characteristics of the wavy fin with VGs, the steady laminar flow with Re = 150 (at which the β of VGs and θ significantly affect the thermal performance of the wavy fin) is selected. Fig. 15 shows the distributions of Nu, f, and JF as a function of β for different θ at Re = 150. β = 0° implies the smooth wavy channel without VGs. Nu, f, and JF increase as θ increases. Nu first increases as β increases from 30° to 45° and then decreases as β increases from 45° to 75°. Nu attains the maximum at β = 45° for all θ. The maximum values of Nu of the proposed wavy fin for θ = 10°, 20°, and 30° are 5.8%, 8.5%, and 12.6% larger than the corresponding Nu of the smooth wavy fin, respectively. f generally increases as β increases. The maximum value of f of the proposed wavy fin is higher by 7%, 8.2%, and 7.7% than that of the smooth wavy fin for θ = 10°, 20°, and 30°, respectively. The differences in Nu and f between θ = 30° and 20° are larger than those between θ = 20° and 10°. Thus, the increasing in θ significantly affects Nu and f when θ is larger than 20°. JF first increases as β increases from 30° to 45° and then decreases as β increases from 45° to 75°. Thus, JF attains the highest value at β = 45° for different θ. The values of JF for β = 30° and 45° are higher than those for β = 60° and 75°. Therefore, the optimum β of VG exists in terms of the highest heat transfer performance of the wavy fin in laminar flow. 6. Conclusions The thermo-hydraulic performance of the novel combination of a wavy fin and vortex generators was numerically studied for different corrugation angles of the wavy fin and different attack angles of the vortex generators. The primary conclusions are summarized as follows: 1. The heat transfer performance of the novel wavy fin is apparently enhanced by combining vortex generators on the wavy fin. The span averaged Nus for the wavy fin with θ = 20° and β = 45° can increase by up to 33% compared with the smooth wavy fin. 2. The thermal performance factor first increases as Re increases in the laminar flow regime and then decreases as Re increases in the turbulent flow regime. Good performance can be obtained by an appropriate corrugation angle at different Re. The maximum thermal performance factor increases by 26.4%. 3. The optimum attack angle of vortex generators exists for the highest heat transfer performance in laminar flow. An attack angle of 45° is significantly more effective for enhancing thermo-hydraulic performance. Acknowledgement This research was funded by the National Natural Science Foundation of China (51866007); the Gansu Provincial Natural Science Foundation (17JR5RA092); Collaborative Innovation Team Project of Gansu (2018C-13); and the Foundation of a hundred youth talents training program of Lanzhou Jiaotong University. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2019.114343. References [1] K. Manjunath, S.C. Kaushik, Second law thermodynamic study of heat exchangers: A review, Renew. Sustain. Energy Rev. 40 (2014) 348–374. [2] W.M. Kays, A.L. London, Compact heat exchangers, third ed., McGraw-Hill, New York, 1984. [3] Z. Wang, Y.Z. Li, Layer pattern thermal design and optimization for multistream plate-fin heat exchangers-a review, Renew. Sustain. Energy Rev. 53 (2016)
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