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Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach
ICHMT-03244; No of Pages 10 International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
3Q1
Yue-Tzu Yang ⁎, Yi-Hsien Wang, Jen-Chi Hsu Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
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a r t i c l e
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Available online xxxx
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Keywords: Liquid jet impingement Conjugate heat transfer Turbulent flow Optimization Response surface methodology Genetic algorithm method
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Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach☆
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The fluid flow and heat transfer characteristics of a free-surface liquid jet impingement cooling have been investigated numerically. The slot jet with water impinging normally on a flat plate is employed. To describe the turbulent structure, the turbulent governing equations are solved by a control-volume-based finitedifference method with a power-law scheme and the well-known turbulence model, which are associated with wall function. Numerical computations have been conducted with variations of jet exit Reynolds number (11,000 ≤ Red ≤ 17,000), dimensionless jet-to-surface distance (3 ≤ H/d0 ≤ 12), dimensionless jet width (1 ≤ B/d0 ≤ 2), and the heat flux (140 kW/m2 ≤ q″ ≤ 280 kW/m2). The theoretical model developed is validated by comparing the numerical predictions with available experimental data in the literature. Under the studied ranges, the variations of local Nusselt numbers by hydraulic diameter Nud of the dimensionless jetto-surface distance 3 ≤ H/d0 ≤ 12 along the flat plate decrease monotonically from its maximum value at the stagnation point. In addition, the shape of the inlet area and jet-to-surface distance are optimized by using the response surface methodology (RSM) and the genetic algorithm (GA) method after solutions are carefully validated with available experimental results in the literature. Based on the optimal results, the optimum condition is in H/d0 = 7.86 and B/d0 = 2 for this physical model. © 2015 Published by Elsevier Ltd.
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1. Introduction
36
With high heat and mass transfer enhancement, the impingement cooling technique has been widely applied in industries to dry papers, anneal metals and hot plates application, and cool electrical equipments. A number of studies have been studied in the field of freesurface liquid jets. However, the turbulent performance and threedimensional free jet impingement could explain better in comparisons with the difference between the numerical predictions and experimental data. Therefore, this present study focuses on the performance of turbulent flow field, thermal behaviors, the characteristics of jet-to-surface distance, and the optimum design results. Many studies have examined the thermal and hydraulic characteristics of different turbulent models on impingement jet. Martin [1] presented extensively on submerged jet impingement and its heat transfer characteristics by compiling experimental data. Chou and Hung [2] conducted an analytical study for cooling of an isothermal heated surface with a confined slot jet. By transporting from small diameter, the fully developed liquid jets impinging normally on a constant heat flux surface has been investigated by Elison and Webb [3]. This study focused on jet Reynolds numbers spanning the laminar,
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☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y.-T. Yang).
transitional, and turbulent flow regimes. Seyedein et al. [4] reported the results of numerical simulation of a two-dimensional flow field and heat transfer for a turbulent single-heated slot jet discharging normally into a confined channel. Low and high Re versions of turbulence models were used to model the turbulent jet flow. Laschefski et al. [5] numerically analyzed the velocity field and heat transfer in rows of rectangular impinging jets in a transient state. Cziesla et al. [6] simulated turbulent flow issuing from a slot jet array by using a sub grid stress model. Yang and Shyu [7] presented numerical predictions on the fluid flow and heat transfer characteristics of multiple impinging slot jets with an inclined confinement surface. The numerical results have shown that the maximum local Nusselt number and maximum pressure on the impinging surface moved downstream while the inclination angle increased. For impinging jets, comparisons between low Reynolds versions of the standard with other types of eddy viscosity models were presented by Behnia et al. [8]. Numerical databases of the impinging jets usually focused on the prediction of heat transfer rates with one or two turbulent models compared. The flow and heat transfer experimental characteristics of a turbulent air jet impinging on a semi-concave surface was presented by Yang et al. [9]. The experimental study has been carried out for jet impingement cooling on a semicircular concave surface when the jet flows were ejected from round-shaped nozzle, rectangularshaped, and 2D contoured nozzle. The flow field induced by a single circular jet exhausting perpendicularly from a flat plate into a cross-flow
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010 0735-1933/© 2015 Published by Elsevier Ltd.
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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List of symbols Inlet cross-section area (m2) Ac B Change the nozzle width with fixing the cross-sectional area Ac (mm) C1, C2, Cμ Closure coefficients for the turbulence equation d0 Diameter of the inlet nozzle (mm) Dh Hydraulic diameter (mm) E Energy (J) Fvol Source term of surface boundary for momentum equation ! g Gravitational acceleration (m/s2) H Nozzle-to-plate spacing (mm) Hb Base plate thickness (mm) h Heat transfer coefficient (W/m2·K) i Turbulence intensity k Thermal conductivity (W/m.K) k Turbulence kinetic energy (m2/s2) L Total width of plate (mm) l Turbulence length scale lb Length of plate (mm) lc Length of the measured temperature (mm) ld Width of plate (mm) o Nuj Jet local Nusselt number, ðT w q}d −T in Þk f ″ Nud Hydraulic local Nusselt number, q Dh q}Dh ðT w −T in Þk f Nu Average Nusselt number, ðT−T in Þk f p Pressure (kPa) P Wetted perimeter (mm) q″ Wall heat flux (W/m2) R1,R2 Surface curvature Red Reynolds number T Temperature (K) Tin Inlet temperature (K) u, v, w Velocity components (m/s) ! v Velocity vector vin Inlet velocity (m/s)
U
N
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Greek symbols α Volume fraction δ Delta function ε Turbulent energy dissipation rate (m2/s3)s) Κ Curvature of a surface μl, μt Laminar and turbulent viscosity (Pa·s) ν Kinematic viscosity ( m2/s) ρ Density (kg/m3) σ Surface tension coefficient σ l; σ t ; Empirical constants in turbulence model equations σ k; σ ε ϕ Volume fraction function Subscripts in Inlet s Solid w Wall − Average
80 81 82 83 84 85 86 87
Prandtl number fluid on a solid disk of finite thickness was considered by Rahman et al. [13]. Computed results included the velocity, temperature, and pressure distributions in the fluid and the local and average heat transfer coefficients at the solid–fluid interface. The conjugate heat transfer from discrete heat source to a two-dimensional jet of a high Prandtl number fluid discharging from a slot nozzle was performed by Bula and Rahman [14]. It was found that in addition to jet Reynolds number, plate thickness and its thermal conductivity had a significant impact on both temperature distribution and the average Nusselt number. The flow and heat transfer characteristics of a turbulent submerged circular air jet impinging on a horizontal flat surface was presented by Siba et al. [15]. Brescianini and Delichatsios [16] tested four high-Re turbulence models to assess their suitability in turbulent buoyant jets and plumes. An inverse methodology was used to determine the turbulent component of the heat transfer coefficient in the stagnation region and wall-jet region. The heat transfer characteristics of a planar free water jet normally or obliquely impinging onto a flat substrate were investigated experimentally by Ibuki et al. [17]. It was found that in oblique collisions, the profiles of the local Nusselt numbers were asymmetric. The locations of the peak Nusselt numbers did not coincide with the geometric center of the planar jet on the surface. The effect of nozzle-plate spacing in plane impinging jets using the direct numerical simulation was covered in Hattori and Nagano [18]. It was found that the second peak of the local heat transfer rate (Nusselt number) in the wall jet developing region appeared in the lower nozzle-plate spacing case. The simulation indicated that proper inlet conditions were essential in the case of a swirling inlet, although some discrepancies remained between the simulation results and the experimental data. The detailed flow field characteristics of a turbulent air slot jet impinging on a semicircular convex surface were investigated numerically by Yang and Hwang [19]. This study was useful in helping researchers to understand the flow field characteristics in stagnation regions of impingement surfaces. Three-dimensional numerical simulations of fluid flow and heat transfer characteristics for an inclined jet with cross-flow impinging on a heating plate was presented by Yang and Wang [20]. The generation of a pair of counter-rotating longitudinal vortices was clearly observed from the computations. The flow and heat transfer in two planar impinging jets using large-eddy simulation and experiments was demonstrated by Akiyama et al. [21]. For both the unforced and forced flows, the large-eddy simulation predictions have shown that the jet develops streamwise vortices, and the turbulent inflow condition provided disturbances that the jets can be efficiently amplified. Chen et al. [22] provided detailed theoretical solutions on laminar flow for free-surface slot jet impinging onto horizontal surfaces under arbitrary-heat-flux conditions. The thermal and hydraulic boundary layers of laminar flow were divided into four regions of flow along heat transfer surfaces including a stagnation zone and three wall jet zones. Numerical predictions of heat transfer coefficients under jet impingement from an array of nozzles have been made by Salamah and Kaminski [23]. Yang and Tsai [24] presented numerical study of transient conjugate heat transfer in a high turbulence air jet impinging over a flat circular disk. The turbulent governing equations are resolved by the control-volume based finite-difference method with a powerlow scheme, and the low-Re turbulence model successfully described the turbulent structure. It was shown that, with the inclusion of the conjugate heat transfer, the numerical results were better compared with the available experimental results. Five different low Reynolds k − ε turbulent models were compared by Wang and Mujumdar [25] to investigate the numerical accuracy with the experiment on confined jet impingement. They found that none of the five models was validated with the experiment. Zu and Yan [26] investigated the performance of seven different turbulence models to predict the thermal and flow characteristic of confined impinging jet. The SST k ‐ ω model was suggested to better perform on calculation efficiency and accuracy. Five different turbulence models were developed by Sharif and Mothe [27] for the prediction of heat transfer due to slot jet on plane and concave surface.
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Nomenclature
has been investigated numerically by Chochua et al. [10]. The validation of the diffusivity tensor heat transfer model was made by showing the St prediction with the relevant experiment. Furthermore, high and low Reynolds standard models were extensively checked by Roy [11]. Concerning the second-order models, some of them have been made by Shih et al. [12] and showed that second-order models only performed slightly better results than the eddy viscosity models. Transient conjugate heat transfer during the impingement of a free jet of high
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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Y.-T. Yang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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2.1. Governing equations
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The quarter schematic diagram of the geometry and the computational domain is shown in Fig. 1(a) and the whole geometric diagram is shown in Fig. 1(b). The nozzle inlet is on the top of the system, and the working fluid is water. Green area is the part of water to show the free-surface of the calculations. The dimensions of the computational domain are referred to the experimental study by Ibuki et al. [17]. Detailed dimensions are shown in Table 1 and Fig. 2. The density of the Inconel alloy plate is 8137 kg/m3 conductivity is 10.1 W/m·K, and CP is 460 J/kg·K. The turbulent threedimensional Navier–Stokes and energy equations are solved numerically by a finite volume scheme to simulate the thermal and turbulent flow fields. The following assumptions are used to simulate the stationary and rotational pin-fin heat sink:
189 190 191 192 193 194 195 196 197 198 199 200 201
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three-dimensional turbulent flow, 202 steady state, 203 incompressible fluid, 204 constant fluid properties, 205 negligible radiative heat transfer, and 206 surface temperature is maintained below liquid boiling 207 temperature. 208
P
(1) (2) (3) (4) (5) (6)
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Nozzle-to-plane distance was found to be the most important key factor of predicting Nu. When the nozzle-to-distance was longer than the length of potential core, the results revealed a higher prediction. The authors suggested the use of the RNG k − ε model and the SST k ‐ ω model to give a better prediction in the computation. Thirteen different turbulence models in the prediction of round-jet impingement heat transfer from an isothermal flat surface were evaluated by Herbert et al. [28]. The SST-k ‐ ω with a transitional flow option activated was found to have the best performance. In addition, when the impingement was within the jet potential core, it came out the best performance in the prediction of the secondary peak of the surface Nusselt number. Yang et al. [29] analyzed impingement cooling on the semicircular concave surface. They found that when the Reynolds number was fixed, the effect of the impingement distance on the average Nusselt was not significant except for the effect of low impingement distance. Li and Kim [30] performed numerical optimizations of elliptic and stepped circular pin-fin arrays using surrogate-based methods. Kim and Moon [31] obtained the similar results. They presented optimal shapes for which thermal performance was greater than for reference shapes. An intelligent design to optimize the configuration of a twodimensional slotted fin was presented by Wang et al. [32]. Two kinds of strip configuration were optimized by GA based on two different fitness functions, and the comparisons of the fluid flow structures and heat transfer characteristics were explored. Based on the above literature, three-dimensional turbulent impingement jet cooling application can be achieved further for future study. The objective of the present study is to compare the numerical simulations between four different turbulence models and to carry out a comprehensive investigation of thermal characteristics. The optimization of the impingement cooling by genetic algorithm combined with CFD has also been discussed. In addition, integration of the response surface models (RSM) and genetic algorithm (GA) with a CFD simulation into a system is implemented.
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(1) Volume fraction equation
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Fig. 1. Physical coordinate system and boundary conditions: (a) quarter system and (b) whole system.
D
With the assumptions described above, the governing equations are 210 as follows: 211
213
! v q ∇α q ¼ 0
ð1Þ 215
(2) Continuity equation 216
! ∇ ρv ¼0
ð2Þ 218
(3) Momentum equation 219
! !! ! ! ∇ ρ v v ¼ ρ g þ ρ F vol −∇p þ ∇ μ∇ v ! ! g ¼ g j only; F vol ¼ δσ κ∇ϕ
ð3Þ
! where F vol represent the surface tension force by the CSF (continuum 221 surface force) model provided by Brackbill et al. [33]. (4) Energy equation for the fluid
222 223
h i ! ∇ v ðρEÞ ¼ ∇ k f ∇T n X
where E ¼
ð4Þ
225 226
α q ρq Eq
q¼1 n X
ð5Þ α q ρq
q¼1
228
(5) Energy equation for the solid
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
4 t1:1 t1:2
Y.-T. Yang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
Table 1 Detailed dimensions of present computational domain.
t1:3
H
D
d0
ld
L
lb
lc
t1:4
10 mm
3 mm
1.62 mm
30 mm
40 mm
100 mm
60 mm
229
∇ ðks ∇T Þ ¼ 0
ð6Þ
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ð8Þ
k f ¼ kwater α q þ kair 1−α q
ð9Þ
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αq is introduced to represent the volume fraction of liquid in a computational cell. If the control volume is filled with water only, the function is unity. When both liquid and air existed, αq is between 0 and 1.
242
2.2. Turbulence models
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The standard k − ε model is a semi-empirical linear eddy viscosity model based on the model transport equations for the turbulence kinetic energy k and its dissipation rate ε. The transport equation for k is
derived from the exact equation, whereas ε is obtained from physical 246 reasoning and has little resemblance to its mathematically exact coun- 247 terpart. The transport equation for k and ε are given as follows: 248
∂k ∂ ¼ ∂xi ∂x j
"
249
μl þ
T
μ t ∂k ∂ui ∂u j þ þ μt σ k ∂x j ∂x j ∂xi
!
∂ui −ρε ∂x j
ð10Þ 252
(2) Transport equation for ε
C E R R O C Fig. 2. Geometrical dimensions: (a) front view, (b) top view, (c) change the nozzle width with fixing the cross-sectional area Ac.
250
#
E
ρu j
N
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Fig. 3. Computational grid distributions (H/d0 = 6.17, B/d0 = 1, Mesh = 142 × 48 × 110).
(1) Transport equation for k
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μ ¼ μ water α q þ μ air 1−α q
ð7Þ
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ρ ¼ α water α q þ ρair 1−α q
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The VOF (volume of fluid) technique is applied to track the free liquid surface. The density, viscosity, and thermal conductivity are given by
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253
ρu j
∂ε ∂ ¼ ∂x j ∂x j
"
#
μl þ
μ t ∂ε ε ∂ui ∂u j þ þ C1μ t k ∂x j ∂xi σ ε ∂x j
!
∂ui ε2 −C 2 ρ k ∂x j
ð11Þ 255 256
2
Where μ t ¼ ρC μ
k ε
ð12Þ 258
The closure coefficients that appear in the above equations are given by the following values: 259 C 1 ¼ 1:44; C 2 ¼ 1:92; C μ ¼ 0:09; σ k ¼ 1:0; σ ε ¼ 1:3:
ð13Þ 261 t2:1 t2:2
Table 2 Calculated results with RSM. No.
x1
x2
H/d0
B/d0
vin (m/s)
Nu
t2:3
1 2 3 4 5 6 7 8 9 10 11 12 13
−1 1 −1 1 −1.414 1.414 0 0 0 0 0 0 0
−1 −1 1 1 0 0 −1.414 1.414 0 0 0 0 0
4.31 10.68 4.31 10.68 3 12 7.5 7.5 7.5 7.5 7.5 7.5 7.5
1.14 1.14 1.85 1.85 1.5 1.5 1 2 1.5 1.5 1.5 1.5 1.5
5.42 5.42 3.62 6.61 4.25 4.25 6.1 3.41 4.25 4.25 4.25 4.25 4.25
122.63 132.85 151.4 154.91 138.54 147.04 138.46 159.02 147.44 147.44 147.44 147.44 147.44
t2:4 t2:5 t2:6 t2:7 t2:8 t2:9 t2:10 t2:11 t2:12 t2:13 t2:14 t2:15 t2:16
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
Y.-T. Yang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx t3:1 t3:2
(5) Outlet boundary (pressure boundary)
Table 3 Optimum results and comparison with CFD using RSM full factors design.
t3:3
H
t3:4
7.86
B
d0
d0
2
5
Nu from Eq. (35)
CFD prediction Nu
difference
160.2351
159.11337
0.7%
p ¼ patm
ð24Þ 291
∂T ∂k ∂ε ¼ ¼ ¼0 ∂n ∂n ∂n
ð25Þ
292
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2.3. Boundary conditions
ð15Þ
kin ¼
274
276 277
3 ði vin Þ2 2
ð16Þ F vol ¼ σ
C μ3=4
kin l
ð17Þ
where i is the turbulence intensity, i = 0.03, and l is the turbulence length scale, l = 0.07B. (2) Symmetric boundary ∂u ∂w ∂k ∂ε ¼ ¼v¼ ¼ ¼ 0 ∂n ∂n ∂n ∂n
ð18Þ
∂T f ¼0 ∂n
ð19Þ
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ð20Þ
C
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u¼v¼w¼0 T f ¼ Ts kf
∂T f ∂T s ¼ ks ∂n ∂n
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306 307 308
Red ¼
jvin jDh ν
309
ð30Þ 311
Where vin is the inlet velocity, Dh is the nozzle diameter, Dh ¼ 4AP c ; P is the wetted perimeter. 312 (2) Jet local Nusselt number 313 Nu j ¼
ð22Þ
hdo q″ do ¼ kf ðT w −T in Þk f
ð31Þ
Where Tw is the wall temperature, and d0 is the diameter of nozzle. 315 (3) Hydraulic local Nusselt number ð23Þ
Nud ¼
hDh q″ Dh ¼ kf ðT w −T in Þk f
ð32Þ
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∂T s ∂n
305
317
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q″ ¼ −ks
R
(4) Constant heat flux boundary
303
ð29Þ
Data reduction (1) Reynolds number The Reynolds number of the impinging jet is defined as
ð21Þ
E
282
R
281
1 ðρ þ ρair Þ 2 water
300
ð28Þ 302
2.4. Numerical computations
T
(3) Conjugate boundary
ρκ∇α water
296 297
ð27Þ 299
κ ¼∇n
3=2
εin ¼ 273
n ¼ ∇α water =j∇α water j
F
T ¼ T in ¼ 298K
ð26Þ
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ð14Þ
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v ¼ vin ; u ¼ w ¼ 0
1 1 þ R1 R2
P
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p2 −p1 ¼ σ
D
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(1) Inlet boundary
E
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(6) Surface tension boundary
Fig. 4. Effect of grid refinement on the jet local Nusselt number. . (Red = 17,000, Y = 0, Z = 0, H/d0 = 6.17, q″ = 280 kW/m2).
Fig. 5. Jet local Nusselt number distributions for different Red (H/d0 = 6.17, B/d0 = 1, Y = 0, Z = 0, q″ = 280 kW/m2).
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
Y.-T. Yang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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Fig. 6. Temperature contours at the interface under different nozzle-to-plate height. (a) H/d0 = 3, B/d0 = 1, Red = 17,000, q″ = 280 kW/m2. (b) H/d0 = 9, B/d0 = 1, Red = 17,000, q″ = 280 kW/m2. (c) H/d0 = 12, B/d0 = 1, Red = 17,000, q″ = 280 kW/m2.
(4) Average Nusselt number
319
320 321 322 323 324 325 326
hDh q″ Dh Nu ¼ ¼ kf T w −T in k f
ð33Þ
The calculations are solved by CFD to simulate the numerical results. A non-uniform and staggered grid system is employed to describe the performance of impingements. A staggered grid arrangement is used in which the velocities are stored at a location on the control-volume faces. Computational grid distribution with H/d0 = 6.17, B/d0 = 1, Mesh = 142 × 48 × 110 is shown in Fig. 3. The numerical computations are carried out by solving the governing conservation equations with
the boundary conditions. The numerical method used in the present study is based on the SIMPLE algorithm of Patankar [34], which is an iterative solution procedure where the computation begins by guessing the pressure field. The converged criterion is less than 10−3.
327
2.5. Optimal process
331
328 329 330
The optimum process is implemented by the combination of re- 332 sponse surface methodology (RSM) and genetic algorithms (GA). The 333 progress is described as follows. 334 a) Use the DOE method (RSM) to decide the system factor to reduce the 335 calculations. 336
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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Y.-T. Yang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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b) Implement the RSM design models by using CFD. c) Approach regression analysis by the raw data of Nusselt number and the design model. d) Use GA model to approach the regression model and analyze the results.
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Fig. 7. Velocity vectors (B/d0 = 1, Red = 17,000, Z = 0). (a) H/d0 = 6.17, (b) H/d0 = 3, (c) H/d0 = 6, (d) H/d0 = 9.
342 344 345 346 347 348
The response surface methodology is a multi-regression design for numerical model approaches. It can find out the mix effect of parameters and create an objective function for optimization. RSM is originally developed for the model fitting of physical experiments and later adopted in other fields. The objective function created by RSM is formulated as
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f ¼ a0 þ
n X i¼1
350 351 352 353 354
ai xi þ
n X n X
ai j xi x j þ …
ð34Þ
i¼1 j¼1
Where a0, ai, and aij are tuning parameters and n is the number of parameters. The studied DOE parameters are used as shown in Table 2 which is arranged by CCD (central composite design) method. In order to approach the optimum geometries, the Reynolds number is kept at 17,000. The optimum parameters are the dimensionless distance of
nozzle H/d0(3 ≤ H/d0 ≤ 12), and the dimensionless width of nozzle 355 B/d0(1 ≤ B/d0 ≤ 2). 356 2.6. Regression and objective functions fitting
357
The contrast of the CCD variables and bounded variables H/d0(3 ≤ H/ 358 d0 ≤ 12) and B/d0(1 ≤ B/d0 ≤ 2) is listed in the Table 2. From Table 2 of 359 CCD predicted results, the objective function is shown in Eq. (35). 360 Nu ¼ 48:18−8:61ðH=do Þ þ 56:08ðB=do Þ−1:49ðH=do Þ ðB=do Þ 2
2
−0:36ðH=do Þ −5:56ðB=do Þ
ð35Þ 362
After creating objective function, the genetic algorithms (GA) are used to find out the optimal geometries. The GA codes are used by MATLAB tool box code. The mutation function is set as 0.01. Crossover is set as scattered. GA solves the optimization problem iteratively based on the biological evolution process in nature. In the solution procedure, a set of parameter values is randomly selected. Set is ranked based on fitness values (i.e. performance factor in this study). A best combination of parameters leading to minimum or maximum fitness values will be determined. A new combination of parameters is generated from the best combination by simulating biological mechanisms of offspring, crossover, and mutation. The predicting average Nusselt
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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number Nu of regression function is in close agreement with those from the CFD computational results within a 0.7% difference as shown in Table 3.
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Fig. 8. Velocity contours at Red = 17,000, Z = 0. (a) H/d0 = 6.17, B/d0 = 1.3; (b) H/d0 = 6.17, B/d0 = 1.6. (c) H/d0 = 12, B/d0 = 1.3 (d); H/d0 = 12, B/d0 = 1.6.
Fig. 9. Comparison of hydraulic local Nusselt number (B/d0 = 1, q″ = 280 kW/m2).
3. Results and discussion
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The validation of the present study is to compare the different performances with the available experimental data by Ibuki et al. [17]. Fig. 4 shows the effect of grid refinement on the jet local Nusselt number between the experimental data and the present study. The condition is under Red = 17,000, Y = 0, Z = 0, H/d0 = 6.17, q″ = 280 kW/m2, and the grid numbers are 138 × 45 × 100, 142 × 48 × 110, and 144 × 50 × 114. The results show that the error between experimental data and the present study is within 10%. The mesh of 142 × 48 × 110 is grid independent although the center impingement side of the error is larger than other areas. It can be seen, at the impingement area, the variation of jet local Nusselt number is larger than the others. The results also show that the accuracy increases with the enhancement of the grid and 142 × 48 × 110 meshes provide satisfactory numerical accuracy and efficiency. In Fig. 5, the jet local Nusselt number distributions for different Red (H/d0 = 6.17, B/d0 = 1, Y = 0, Z = 0, q″ = 280 kW/m2) are employed to compare the numerical results with the available data. The stagnation point area has good approaches with the experimental data from the present study. However, the error becomes larger between the transition areas. The same results were obtained from Yang et al. [29].
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Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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The thermal effects and flow field status have been discussed in the present study with variations of jet exit Reynolds number (11,000 ≤ Re d ≤ 17,000), dimensionless jet-to-surface distance (3 ≤ H/d 0 ≤ 12), dimensionless jet width (1 ≤ B/d 0 ≤ 2) and the heat flux (140 kW/m2 ≤ q″ ≤ 280 kW/m 2). The results show that the stagnation points vary with the nozzle diameters that keep occurring at the center of the nozzle no matter under whichever condition the nozzle is. The optimized geometry data show better cooling results than the original proposed data with the implementation of response surface methodology (RSM) and genetic algorithm (GA) method. Based on the optimal results, the optimum condition is H/d0 = 7.86 and B/d0 = 2 under concerned constraints with experimental data. The overall average Nusslet number of optimization is around 10% higher than the original condition.
References
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In Fig. 6, the cross-section temperature contours of the interface under different nozzle-to-plate height are presented. Fig. 6(a) shows the results of H/d0 = 3, the plate temperature around impingement is kept under 300 K, which presents high cooling ability and the jet thermal profile is not obvious on the plate. From Fig. 6(b) and (c), the jet thermal profiles are clear and obvious from the temperature contours, indicating that the lower height of jet impingement makes higher heat transfer rate as well as takes away more heat fluid by the working fluid. It is such good cooling ability that the temperature gradients are not high in the center area. However, in Fig. 6(b) and (c), the temperature gradients are high and the jet thermal profile is obvious in these areas. The cooling ability and capacity need to be enhanced or added to the cooling flow rate. Fig. 7(a) shows the velocity distribution performance. Fig. 7(b) shows the higher value at the outlet area than the others due to the lower impingement height. From Fig. 8, it shows the velocity contours of different nozzle diameters with the same nozzle cross area. The results show that the stagnation points vary with the nozzle diameters that keep occurring at the center of the nozzle no matter under whichever condition the nozzle is.
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From the optimization process, the optimization parameter is listed in Table 3. The conditions are constrained as Red = 17,000, Tin = 289 K, q″ = 280 kW/m2, 3 ≤ H/d0 ≤ 12, and 1 ≤ B/d0 ≤ 2, and Ac is kept at constant. The final condition of H/d0 = 7.86 and B/d0 = 2 is the optimization combination for the target function. The difference from regression equation and CFD prediction is within 0.7%. The approached function can easily calculate and quickly get the performance of the flow thermal status. From Fig. 9, the comparison of the hydraulic local Nusselt number from original and optimal condition is shown. The overall average Nusslet number of optimization is around 10% which is higher than the original condition. From Fig. 10, better performance can be found in the optimal result.
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The fluid flow and heat transfer characteristics of a free-surface liquid jet impingement cooling have been investigated numerically.
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[1] H. Martin, Heat and mass transfer between impinging gas jets and solid surfaces, Adv. Heat Tran. 13 (1977) 1–60. [2] Y.J. Chou, Y.H. Hung, Impingement cooling of an isothermally heated surface with a confined slot jet, ASME J. Heat Transf. 116 (1994) 479–482. [3] B. Elison, B.W. Webb, Local heat transfer to impinging liquid jets in the initially laminar, transitional, and turbulent regimes, Int. J. Heat Mass Transf. 37 (1994) 1207–1216. [4] S.H. Seyedein, M. Hasan, A.S. Mujumdar, Modeling of a single confined turbulent slot jet impingement using various k- ε turbulence models, Appl. Math. Model. 18 (1994) 526–537. [5] H. Laschefski, T. Cziesla, G. Biswas, N.K. Mitra, Numerical investigation of heat transfer by rows of rectangular impinging jets, Numer. Heat Transfer 30 (1996) 87–101. [6] T. Cziesla, E. Tandogan, N.K. Mitra, Large eddy simulations of heat transfer from impinging slot jets, Numer. Heat Transfer, Part A 32 (1997) 1–17. [7] Y.T. Yang, C.H. Shyu, Numerical study of multiple impinging slot jets with an inclined confinement surface, Numer. Heat Transfer, Part A 33 (1998) 23–37. [8] P. Behnia, S. Parneix, Y. Shabany, P. Durbin, Numerical study of turbulent heat transfer in confined and unconfined impinging jets, Int. J. Heat Fluid Flow 20 (1999) 1–9. [9] Geunyoung Yang, Mansoo Choi, Joon Sik Lee, An experimental study of slot jet impingement cooling on concave surface: effects of nozzle configuration and curvature, Int. J. Heat Mass Transf. 42 (1999) 2199–2209. [10] G. Chochua, W. Shyy, S. Thakur, A. Brankovic, J. Lienau, L. Porter, D. Lischinsky, A computational and experimental investigation of turbulent jet and crossflow interaction, Numer. Heat Transfer, Part A 38 (2000) 557–572. [11] S. Roy, Numerical investigation of the blade cooling effect generated by multiple jets issuing at an angle into a incompressible horizontal crossflow, Numer. Heat Transfer, Part A 38 (2000) 701–718. [12] Y. Shih, M. Ray, S. Mujemdar, Computational study of impingement heat transfer under a turbulent slot jet, Ind. Eng. Chem. Res. 41 (2002) 4643–4651. [13] Muhammad M. Rahman, Antonio J. Bula, John E. Leland, Analysis of transient conjugate heat transfer to a free impinging jet, J. Thermophys. Heat Transf. 14 (2000) 330–339. [14] Antonio J. Bula, Muhammad M. Rahman, Numerical modeling of conjugate heat transfer during impingement of free liquid jet issuing from a slot nozzle, Numer. Heat Transfer, Part A 38 (2000) 45–66. [15] Erick A. Siba, M. Ganesa-Pillai, Kendall T. Harris, A. Haji-Sheikh, Heat transfer in a high turbulence air jet impinging over a flat circular disk, ASME J. Heat Transfer 125 (2003) 257–265. [16] C.P. Brescianini, M.A. Delichatsios, New evaluation of the turbulence model for free buoyant plumes, Numer. Heat Transfer, Part A 43 (2003) 731–751. [17] K. Ibuki, T. Umeda, H. Fujimoto, H. Takuda, Heat transfer characteristics of a planar water jet impinging normally or obliquely on a flat surface at relatively low Reynolds numbers, Exp. Thermal Fluid Sci. 33 (2009) 1226–1234. [18] Hirofumi Hattori, Yasutaka Nagano, Direct numerical simulation of turbulent heat transfer in a plane impinging jet, Int. J. Heat Fluid Flow 25 (2004) 749–758. [19] Y.T. Yang, C.H. Hwang, Numerical simulations on the hydrodynamics of a turbulent slot jet impinging on a semi-cylindrical convex surface, Numer. Heat Transfer, Part A 46 (2004) 995–1008. [20] Y.T. Yang, Y.X. Wang, Three-dimensional numerical simulation of an inclined jet with cross-flow, Int. J. Heat Mass Transf. 48 (2005) 4019–4027. [21] T. Akiyama, K. Yamamoto, K.D. Squires, K. Hishida, Simulation and measurement of flow and heat transfer in two planar impinging jets, Int. J. Heat Fluid Flow 26 (2005) 244–255. [22] Y.C. Chen, C.F. Ma, M. Qin, Y.X. Li, Theoretical study on impingement heat transfer with single-phase free-surface slot jets, Int. J. Heat Mass Transf. 48 (2005) 3381–3386. [23] Samir A. Salamah, Deborah A. Kaminski, Modeling of turbulent heat transfer from an array of submerged jets impinging on a solid surface, Numer. Heat Transfer, Part A 48 (2005) 315–337. [24] Y.T. Yang, S.Y. Tsai, Numerical study of transient conjugate heat transfer of a turbulent impinging jet, Int. J. Heat Mass Transf. 50 (2006) 799–807. [25] S.J. Wang, A.S. Mujumdar, A comparative study of five low Reynolds number k-ε models for impingement heat transfer, Appl. Therm. Eng. 25 (2005) 31–44.
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[30] P. Li, K.Y. Kim, Multi-objective optimization of a staggered elliptical pin-fin arrays, Numer. Heat Transfer, Part A 53 (2008) 418–431. [31] K.Y. Kim, M.A. Moon, Optimization of a stepped circular pin-fin array to enhance heat transfer performance, Heat Mass Transf. 46 (2009) 63–74. [32] Y. Wang, Y.L. He, D.H. Mei, W.Q. Tao, Optimization design of slotted fin by numerical simulation coupled with genetic algorithm, Appl. Energy 88 (2011) 4441–4450. [33] J.U. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (1992) 335–354. [34] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.
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[26] Y.Q. Zu, Y.Y. Yan, Numerical study on stagnation point heat transfer by jet impingement in a confined narrow gap, ASME J. Heat Transf. 131 (2009) 094504. [27] M.A.R. Sharif, K.K. Mothe, Evaluation of turbulence models in the prediction of heat transfer due to slot jet impingement on plane and concave surfaces, Numer. Heat Transfer, Part B 55 (2009) 273–294. [28] M.H. Herbert, K. Rafael, K. Matthias, M. Holger, Calculation of steady and pulsating impinging – an assessment of 13 widely used turbulence models, Numer, Numer. Heat Transfer, Part B 51 (2007) 565–583. [29] Y.T. Yang, T.C. Wei, Y.H. Wang, Numerical study of turbulent slot jet impingement cooling on a semi-circular concave surface, Int. J. Heat Mass Transf. 54 (2010) 482–489.
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Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
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Yue-Tzu Yang ⁎, Yi-Hsien Wang, Jen-Chi Hsu Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
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Keywords: Liquid jet impingement Conjugate heat transfer Turbulent flow Optimization Response surface methodology Genetic algorithm method
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Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach☆
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The fluid flow and heat transfer characteristics of a free-surface liquid jet impingement cooling have been investigated numerically. The slot jet with water impinging normally on a flat plate is employed. To describe the turbulent structure, the turbulent governing equations are solved by a control-volume-based finitedifference method with a power-law scheme and the well-known turbulence model, which are associated with wall function. Numerical computations have been conducted with variations of jet exit Reynolds number (11,000 ≤ Red ≤ 17,000), dimensionless jet-to-surface distance (3 ≤ H/d0 ≤ 12), dimensionless jet width (1 ≤ B/d0 ≤ 2), and the heat flux (140 kW/m2 ≤ q″ ≤ 280 kW/m2). The theoretical model developed is validated by comparing the numerical predictions with available experimental data in the literature. Under the studied ranges, the variations of local Nusselt numbers by hydraulic diameter Nud of the dimensionless jetto-surface distance 3 ≤ H/d0 ≤ 12 along the flat plate decrease monotonically from its maximum value at the stagnation point. In addition, the shape of the inlet area and jet-to-surface distance are optimized by using the response surface methodology (RSM) and the genetic algorithm (GA) method after solutions are carefully validated with available experimental results in the literature. Based on the optimal results, the optimum condition is in H/d0 = 7.86 and B/d0 = 2 for this physical model. © 2015 Published by Elsevier Ltd.
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1. Introduction
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With high heat and mass transfer enhancement, the impingement cooling technique has been widely applied in industries to dry papers, anneal metals and hot plates application, and cool electrical equipments. A number of studies have been studied in the field of freesurface liquid jets. However, the turbulent performance and threedimensional free jet impingement could explain better in comparisons with the difference between the numerical predictions and experimental data. Therefore, this present study focuses on the performance of turbulent flow field, thermal behaviors, the characteristics of jet-to-surface distance, and the optimum design results. Many studies have examined the thermal and hydraulic characteristics of different turbulent models on impingement jet. Martin [1] presented extensively on submerged jet impingement and its heat transfer characteristics by compiling experimental data. Chou and Hung [2] conducted an analytical study for cooling of an isothermal heated surface with a confined slot jet. By transporting from small diameter, the fully developed liquid jets impinging normally on a constant heat flux surface has been investigated by Elison and Webb [3]. This study focused on jet Reynolds numbers spanning the laminar,
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☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y.-T. Yang).
transitional, and turbulent flow regimes. Seyedein et al. [4] reported the results of numerical simulation of a two-dimensional flow field and heat transfer for a turbulent single-heated slot jet discharging normally into a confined channel. Low and high Re versions of turbulence models were used to model the turbulent jet flow. Laschefski et al. [5] numerically analyzed the velocity field and heat transfer in rows of rectangular impinging jets in a transient state. Cziesla et al. [6] simulated turbulent flow issuing from a slot jet array by using a sub grid stress model. Yang and Shyu [7] presented numerical predictions on the fluid flow and heat transfer characteristics of multiple impinging slot jets with an inclined confinement surface. The numerical results have shown that the maximum local Nusselt number and maximum pressure on the impinging surface moved downstream while the inclination angle increased. For impinging jets, comparisons between low Reynolds versions of the standard with other types of eddy viscosity models were presented by Behnia et al. [8]. Numerical databases of the impinging jets usually focused on the prediction of heat transfer rates with one or two turbulent models compared. The flow and heat transfer experimental characteristics of a turbulent air jet impinging on a semi-concave surface was presented by Yang et al. [9]. The experimental study has been carried out for jet impingement cooling on a semicircular concave surface when the jet flows were ejected from round-shaped nozzle, rectangularshaped, and 2D contoured nozzle. The flow field induced by a single circular jet exhausting perpendicularly from a flat plate into a cross-flow
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List of symbols Inlet cross-section area (m2) Ac B Change the nozzle width with fixing the cross-sectional area Ac (mm) C1, C2, Cμ Closure coefficients for the turbulence equation d0 Diameter of the inlet nozzle (mm) Dh Hydraulic diameter (mm) E Energy (J) Fvol Source term of surface boundary for momentum equation ! g Gravitational acceleration (m/s2) H Nozzle-to-plate spacing (mm) Hb Base plate thickness (mm) h Heat transfer coefficient (W/m2·K) i Turbulence intensity k Thermal conductivity (W/m.K) k Turbulence kinetic energy (m2/s2) L Total width of plate (mm) l Turbulence length scale lb Length of plate (mm) lc Length of the measured temperature (mm) ld Width of plate (mm) o Nuj Jet local Nusselt number, ðT w q}d −T in Þk f ″ Nud Hydraulic local Nusselt number, q Dh q}Dh ðT w −T in Þk f Nu Average Nusselt number, ðT−T in Þk f p Pressure (kPa) P Wetted perimeter (mm) q″ Wall heat flux (W/m2) R1,R2 Surface curvature Red Reynolds number T Temperature (K) Tin Inlet temperature (K) u, v, w Velocity components (m/s) ! v Velocity vector vin Inlet velocity (m/s)
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Greek symbols α Volume fraction δ Delta function ε Turbulent energy dissipation rate (m2/s3)s) Κ Curvature of a surface μl, μt Laminar and turbulent viscosity (Pa·s) ν Kinematic viscosity ( m2/s) ρ Density (kg/m3) σ Surface tension coefficient σ l; σ t ; Empirical constants in turbulence model equations σ k; σ ε ϕ Volume fraction function Subscripts in Inlet s Solid w Wall − Average
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Prandtl number fluid on a solid disk of finite thickness was considered by Rahman et al. [13]. Computed results included the velocity, temperature, and pressure distributions in the fluid and the local and average heat transfer coefficients at the solid–fluid interface. The conjugate heat transfer from discrete heat source to a two-dimensional jet of a high Prandtl number fluid discharging from a slot nozzle was performed by Bula and Rahman [14]. It was found that in addition to jet Reynolds number, plate thickness and its thermal conductivity had a significant impact on both temperature distribution and the average Nusselt number. The flow and heat transfer characteristics of a turbulent submerged circular air jet impinging on a horizontal flat surface was presented by Siba et al. [15]. Brescianini and Delichatsios [16] tested four high-Re turbulence models to assess their suitability in turbulent buoyant jets and plumes. An inverse methodology was used to determine the turbulent component of the heat transfer coefficient in the stagnation region and wall-jet region. The heat transfer characteristics of a planar free water jet normally or obliquely impinging onto a flat substrate were investigated experimentally by Ibuki et al. [17]. It was found that in oblique collisions, the profiles of the local Nusselt numbers were asymmetric. The locations of the peak Nusselt numbers did not coincide with the geometric center of the planar jet on the surface. The effect of nozzle-plate spacing in plane impinging jets using the direct numerical simulation was covered in Hattori and Nagano [18]. It was found that the second peak of the local heat transfer rate (Nusselt number) in the wall jet developing region appeared in the lower nozzle-plate spacing case. The simulation indicated that proper inlet conditions were essential in the case of a swirling inlet, although some discrepancies remained between the simulation results and the experimental data. The detailed flow field characteristics of a turbulent air slot jet impinging on a semicircular convex surface were investigated numerically by Yang and Hwang [19]. This study was useful in helping researchers to understand the flow field characteristics in stagnation regions of impingement surfaces. Three-dimensional numerical simulations of fluid flow and heat transfer characteristics for an inclined jet with cross-flow impinging on a heating plate was presented by Yang and Wang [20]. The generation of a pair of counter-rotating longitudinal vortices was clearly observed from the computations. The flow and heat transfer in two planar impinging jets using large-eddy simulation and experiments was demonstrated by Akiyama et al. [21]. For both the unforced and forced flows, the large-eddy simulation predictions have shown that the jet develops streamwise vortices, and the turbulent inflow condition provided disturbances that the jets can be efficiently amplified. Chen et al. [22] provided detailed theoretical solutions on laminar flow for free-surface slot jet impinging onto horizontal surfaces under arbitrary-heat-flux conditions. The thermal and hydraulic boundary layers of laminar flow were divided into four regions of flow along heat transfer surfaces including a stagnation zone and three wall jet zones. Numerical predictions of heat transfer coefficients under jet impingement from an array of nozzles have been made by Salamah and Kaminski [23]. Yang and Tsai [24] presented numerical study of transient conjugate heat transfer in a high turbulence air jet impinging over a flat circular disk. The turbulent governing equations are resolved by the control-volume based finite-difference method with a powerlow scheme, and the low-Re turbulence model successfully described the turbulent structure. It was shown that, with the inclusion of the conjugate heat transfer, the numerical results were better compared with the available experimental results. Five different low Reynolds k − ε turbulent models were compared by Wang and Mujumdar [25] to investigate the numerical accuracy with the experiment on confined jet impingement. They found that none of the five models was validated with the experiment. Zu and Yan [26] investigated the performance of seven different turbulence models to predict the thermal and flow characteristic of confined impinging jet. The SST k ‐ ω model was suggested to better perform on calculation efficiency and accuracy. Five different turbulence models were developed by Sharif and Mothe [27] for the prediction of heat transfer due to slot jet on plane and concave surface.
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has been investigated numerically by Chochua et al. [10]. The validation of the diffusivity tensor heat transfer model was made by showing the St prediction with the relevant experiment. Furthermore, high and low Reynolds standard models were extensively checked by Roy [11]. Concerning the second-order models, some of them have been made by Shih et al. [12] and showed that second-order models only performed slightly better results than the eddy viscosity models. Transient conjugate heat transfer during the impingement of a free jet of high
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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2.1. Governing equations
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The quarter schematic diagram of the geometry and the computational domain is shown in Fig. 1(a) and the whole geometric diagram is shown in Fig. 1(b). The nozzle inlet is on the top of the system, and the working fluid is water. Green area is the part of water to show the free-surface of the calculations. The dimensions of the computational domain are referred to the experimental study by Ibuki et al. [17]. Detailed dimensions are shown in Table 1 and Fig. 2. The density of the Inconel alloy plate is 8137 kg/m3 conductivity is 10.1 W/m·K, and CP is 460 J/kg·K. The turbulent threedimensional Navier–Stokes and energy equations are solved numerically by a finite volume scheme to simulate the thermal and turbulent flow fields. The following assumptions are used to simulate the stationary and rotational pin-fin heat sink:
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Nozzle-to-plane distance was found to be the most important key factor of predicting Nu. When the nozzle-to-distance was longer than the length of potential core, the results revealed a higher prediction. The authors suggested the use of the RNG k − ε model and the SST k ‐ ω model to give a better prediction in the computation. Thirteen different turbulence models in the prediction of round-jet impingement heat transfer from an isothermal flat surface were evaluated by Herbert et al. [28]. The SST-k ‐ ω with a transitional flow option activated was found to have the best performance. In addition, when the impingement was within the jet potential core, it came out the best performance in the prediction of the secondary peak of the surface Nusselt number. Yang et al. [29] analyzed impingement cooling on the semicircular concave surface. They found that when the Reynolds number was fixed, the effect of the impingement distance on the average Nusselt was not significant except for the effect of low impingement distance. Li and Kim [30] performed numerical optimizations of elliptic and stepped circular pin-fin arrays using surrogate-based methods. Kim and Moon [31] obtained the similar results. They presented optimal shapes for which thermal performance was greater than for reference shapes. An intelligent design to optimize the configuration of a twodimensional slotted fin was presented by Wang et al. [32]. Two kinds of strip configuration were optimized by GA based on two different fitness functions, and the comparisons of the fluid flow structures and heat transfer characteristics were explored. Based on the above literature, three-dimensional turbulent impingement jet cooling application can be achieved further for future study. The objective of the present study is to compare the numerical simulations between four different turbulence models and to carry out a comprehensive investigation of thermal characteristics. The optimization of the impingement cooling by genetic algorithm combined with CFD has also been discussed. In addition, integration of the response surface models (RSM) and genetic algorithm (GA) with a CFD simulation into a system is implemented.
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(1) Volume fraction equation
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With the assumptions described above, the governing equations are 210 as follows: 211
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(2) Continuity equation 216
! ∇ ρv ¼0
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(3) Momentum equation 219
! !! ! ! ∇ ρ v v ¼ ρ g þ ρ F vol −∇p þ ∇ μ∇ v ! ! g ¼ g j only; F vol ¼ δσ κ∇ϕ
ð3Þ
! where F vol represent the surface tension force by the CSF (continuum 221 surface force) model provided by Brackbill et al. [33]. (4) Energy equation for the fluid
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h i ! ∇ v ðρEÞ ¼ ∇ k f ∇T n X
where E ¼
ð4Þ
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α q ρq Eq
q¼1 n X
ð5Þ α q ρq
q¼1
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(5) Energy equation for the solid
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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Table 1 Detailed dimensions of present computational domain.
t1:3
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10 mm
3 mm
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30 mm
40 mm
100 mm
60 mm
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∇ ðks ∇T Þ ¼ 0
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ð8Þ
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αq is introduced to represent the volume fraction of liquid in a computational cell. If the control volume is filled with water only, the function is unity. When both liquid and air existed, αq is between 0 and 1.
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The standard k − ε model is a semi-empirical linear eddy viscosity model based on the model transport equations for the turbulence kinetic energy k and its dissipation rate ε. The transport equation for k is
derived from the exact equation, whereas ε is obtained from physical 246 reasoning and has little resemblance to its mathematically exact coun- 247 terpart. The transport equation for k and ε are given as follows: 248
∂k ∂ ¼ ∂xi ∂x j
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μl þ
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∂ui −ρε ∂x j
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(2) Transport equation for ε
C E R R O C Fig. 2. Geometrical dimensions: (a) front view, (b) top view, (c) change the nozzle width with fixing the cross-sectional area Ac.
250
#
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ρu j
N
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Fig. 3. Computational grid distributions (H/d0 = 6.17, B/d0 = 1, Mesh = 142 × 48 × 110).
(1) Transport equation for k
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μ ¼ μ water α q þ μ air 1−α q
ð7Þ
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ρ ¼ α water α q þ ρair 1−α q
P
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The VOF (volume of fluid) technique is applied to track the free liquid surface. The density, viscosity, and thermal conductivity are given by
D
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253
ρu j
∂ε ∂ ¼ ∂x j ∂x j
"
#
μl þ
μ t ∂ε ε ∂ui ∂u j þ þ C1μ t k ∂x j ∂xi σ ε ∂x j
!
∂ui ε2 −C 2 ρ k ∂x j
ð11Þ 255 256
2
Where μ t ¼ ρC μ
k ε
ð12Þ 258
The closure coefficients that appear in the above equations are given by the following values: 259 C 1 ¼ 1:44; C 2 ¼ 1:92; C μ ¼ 0:09; σ k ¼ 1:0; σ ε ¼ 1:3:
ð13Þ 261 t2:1 t2:2
Table 2 Calculated results with RSM. No.
x1
x2
H/d0
B/d0
vin (m/s)
Nu
t2:3
1 2 3 4 5 6 7 8 9 10 11 12 13
−1 1 −1 1 −1.414 1.414 0 0 0 0 0 0 0
−1 −1 1 1 0 0 −1.414 1.414 0 0 0 0 0
4.31 10.68 4.31 10.68 3 12 7.5 7.5 7.5 7.5 7.5 7.5 7.5
1.14 1.14 1.85 1.85 1.5 1.5 1 2 1.5 1.5 1.5 1.5 1.5
5.42 5.42 3.62 6.61 4.25 4.25 6.1 3.41 4.25 4.25 4.25 4.25 4.25
122.63 132.85 151.4 154.91 138.54 147.04 138.46 159.02 147.44 147.44 147.44 147.44 147.44
t2:4 t2:5 t2:6 t2:7 t2:8 t2:9 t2:10 t2:11 t2:12 t2:13 t2:14 t2:15 t2:16
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
Y.-T. Yang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx t3:1 t3:2
(5) Outlet boundary (pressure boundary)
Table 3 Optimum results and comparison with CFD using RSM full factors design.
t3:3
H
t3:4
7.86
B
d0
d0
2
5
Nu from Eq. (35)
CFD prediction Nu
difference
160.2351
159.11337
0.7%
p ¼ patm
ð24Þ 291
∂T ∂k ∂ε ¼ ¼ ¼0 ∂n ∂n ∂n
ð25Þ
292
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2.3. Boundary conditions
ð15Þ
kin ¼
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3 ði vin Þ2 2
ð16Þ F vol ¼ σ
C μ3=4
kin l
ð17Þ
where i is the turbulence intensity, i = 0.03, and l is the turbulence length scale, l = 0.07B. (2) Symmetric boundary ∂u ∂w ∂k ∂ε ¼ ¼v¼ ¼ ¼ 0 ∂n ∂n ∂n ∂n
ð18Þ
∂T f ¼0 ∂n
ð19Þ
279
ð20Þ
C
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u¼v¼w¼0 T f ¼ Ts kf
∂T f ∂T s ¼ ks ∂n ∂n
287
306 307 308
Red ¼
jvin jDh ν
309
ð30Þ 311
Where vin is the inlet velocity, Dh is the nozzle diameter, Dh ¼ 4AP c ; P is the wetted perimeter. 312 (2) Jet local Nusselt number 313 Nu j ¼
ð22Þ
hdo q″ do ¼ kf ðT w −T in Þk f
ð31Þ
Where Tw is the wall temperature, and d0 is the diameter of nozzle. 315 (3) Hydraulic local Nusselt number ð23Þ
Nud ¼
hDh q″ Dh ¼ kf ðT w −T in Þk f
ð32Þ
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∂T s ∂n
305
317
U
q″ ¼ −ks
R
(4) Constant heat flux boundary
303
ð29Þ
Data reduction (1) Reynolds number The Reynolds number of the impinging jet is defined as
ð21Þ
E
282
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281
1 ðρ þ ρair Þ 2 water
300
ð28Þ 302
2.4. Numerical computations
T
(3) Conjugate boundary
ρκ∇α water
296 297
ð27Þ 299
κ ¼∇n
3=2
εin ¼ 273
n ¼ ∇α water =j∇α water j
F
T ¼ T in ¼ 298K
ð26Þ
O
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ð14Þ
R O
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v ¼ vin ; u ¼ w ¼ 0
1 1 þ R1 R2
P
265
p2 −p1 ¼ σ
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(1) Inlet boundary
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(6) Surface tension boundary
Fig. 4. Effect of grid refinement on the jet local Nusselt number. . (Red = 17,000, Y = 0, Z = 0, H/d0 = 6.17, q″ = 280 kW/m2).
Fig. 5. Jet local Nusselt number distributions for different Red (H/d0 = 6.17, B/d0 = 1, Y = 0, Z = 0, q″ = 280 kW/m2).
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
Y.-T. Yang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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Fig. 6. Temperature contours at the interface under different nozzle-to-plate height. (a) H/d0 = 3, B/d0 = 1, Red = 17,000, q″ = 280 kW/m2. (b) H/d0 = 9, B/d0 = 1, Red = 17,000, q″ = 280 kW/m2. (c) H/d0 = 12, B/d0 = 1, Red = 17,000, q″ = 280 kW/m2.
(4) Average Nusselt number
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320 321 322 323 324 325 326
hDh q″ Dh Nu ¼ ¼ kf T w −T in k f
ð33Þ
The calculations are solved by CFD to simulate the numerical results. A non-uniform and staggered grid system is employed to describe the performance of impingements. A staggered grid arrangement is used in which the velocities are stored at a location on the control-volume faces. Computational grid distribution with H/d0 = 6.17, B/d0 = 1, Mesh = 142 × 48 × 110 is shown in Fig. 3. The numerical computations are carried out by solving the governing conservation equations with
the boundary conditions. The numerical method used in the present study is based on the SIMPLE algorithm of Patankar [34], which is an iterative solution procedure where the computation begins by guessing the pressure field. The converged criterion is less than 10−3.
327
2.5. Optimal process
331
328 329 330
The optimum process is implemented by the combination of re- 332 sponse surface methodology (RSM) and genetic algorithms (GA). The 333 progress is described as follows. 334 a) Use the DOE method (RSM) to decide the system factor to reduce the 335 calculations. 336
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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b) Implement the RSM design models by using CFD. c) Approach regression analysis by the raw data of Nusselt number and the design model. d) Use GA model to approach the regression model and analyze the results.
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R
Fig. 7. Velocity vectors (B/d0 = 1, Red = 17,000, Z = 0). (a) H/d0 = 6.17, (b) H/d0 = 3, (c) H/d0 = 6, (d) H/d0 = 9.
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The response surface methodology is a multi-regression design for numerical model approaches. It can find out the mix effect of parameters and create an objective function for optimization. RSM is originally developed for the model fitting of physical experiments and later adopted in other fields. The objective function created by RSM is formulated as
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f ¼ a0 þ
n X i¼1
350 351 352 353 354
ai xi þ
n X n X
ai j xi x j þ …
ð34Þ
i¼1 j¼1
Where a0, ai, and aij are tuning parameters and n is the number of parameters. The studied DOE parameters are used as shown in Table 2 which is arranged by CCD (central composite design) method. In order to approach the optimum geometries, the Reynolds number is kept at 17,000. The optimum parameters are the dimensionless distance of
nozzle H/d0(3 ≤ H/d0 ≤ 12), and the dimensionless width of nozzle 355 B/d0(1 ≤ B/d0 ≤ 2). 356 2.6. Regression and objective functions fitting
357
The contrast of the CCD variables and bounded variables H/d0(3 ≤ H/ 358 d0 ≤ 12) and B/d0(1 ≤ B/d0 ≤ 2) is listed in the Table 2. From Table 2 of 359 CCD predicted results, the objective function is shown in Eq. (35). 360 Nu ¼ 48:18−8:61ðH=do Þ þ 56:08ðB=do Þ−1:49ðH=do Þ ðB=do Þ 2
2
−0:36ðH=do Þ −5:56ðB=do Þ
ð35Þ 362
After creating objective function, the genetic algorithms (GA) are used to find out the optimal geometries. The GA codes are used by MATLAB tool box code. The mutation function is set as 0.01. Crossover is set as scattered. GA solves the optimization problem iteratively based on the biological evolution process in nature. In the solution procedure, a set of parameter values is randomly selected. Set is ranked based on fitness values (i.e. performance factor in this study). A best combination of parameters leading to minimum or maximum fitness values will be determined. A new combination of parameters is generated from the best combination by simulating biological mechanisms of offspring, crossover, and mutation. The predicting average Nusselt
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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number Nu of regression function is in close agreement with those from the CFD computational results within a 0.7% difference as shown in Table 3.
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R
Fig. 8. Velocity contours at Red = 17,000, Z = 0. (a) H/d0 = 6.17, B/d0 = 1.3; (b) H/d0 = 6.17, B/d0 = 1.6. (c) H/d0 = 12, B/d0 = 1.3 (d); H/d0 = 12, B/d0 = 1.6.
Fig. 9. Comparison of hydraulic local Nusselt number (B/d0 = 1, q″ = 280 kW/m2).
3. Results and discussion
376
3.1. Grid independence
377
The validation of the present study is to compare the different performances with the available experimental data by Ibuki et al. [17]. Fig. 4 shows the effect of grid refinement on the jet local Nusselt number between the experimental data and the present study. The condition is under Red = 17,000, Y = 0, Z = 0, H/d0 = 6.17, q″ = 280 kW/m2, and the grid numbers are 138 × 45 × 100, 142 × 48 × 110, and 144 × 50 × 114. The results show that the error between experimental data and the present study is within 10%. The mesh of 142 × 48 × 110 is grid independent although the center impingement side of the error is larger than other areas. It can be seen, at the impingement area, the variation of jet local Nusselt number is larger than the others. The results also show that the accuracy increases with the enhancement of the grid and 142 × 48 × 110 meshes provide satisfactory numerical accuracy and efficiency. In Fig. 5, the jet local Nusselt number distributions for different Red (H/d0 = 6.17, B/d0 = 1, Y = 0, Z = 0, q″ = 280 kW/m2) are employed to compare the numerical results with the available data. The stagnation point area has good approaches with the experimental data from the present study. However, the error becomes larger between the transition areas. The same results were obtained from Yang et al. [29].
378
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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The thermal effects and flow field status have been discussed in the present study with variations of jet exit Reynolds number (11,000 ≤ Re d ≤ 17,000), dimensionless jet-to-surface distance (3 ≤ H/d 0 ≤ 12), dimensionless jet width (1 ≤ B/d 0 ≤ 2) and the heat flux (140 kW/m2 ≤ q″ ≤ 280 kW/m 2). The results show that the stagnation points vary with the nozzle diameters that keep occurring at the center of the nozzle no matter under whichever condition the nozzle is. The optimized geometry data show better cooling results than the original proposed data with the implementation of response surface methodology (RSM) and genetic algorithm (GA) method. Based on the optimal results, the optimum condition is H/d0 = 7.86 and B/d0 = 2 under concerned constraints with experimental data. The overall average Nusslet number of optimization is around 10% higher than the original condition.
References
399
417
In Fig. 6, the cross-section temperature contours of the interface under different nozzle-to-plate height are presented. Fig. 6(a) shows the results of H/d0 = 3, the plate temperature around impingement is kept under 300 K, which presents high cooling ability and the jet thermal profile is not obvious on the plate. From Fig. 6(b) and (c), the jet thermal profiles are clear and obvious from the temperature contours, indicating that the lower height of jet impingement makes higher heat transfer rate as well as takes away more heat fluid by the working fluid. It is such good cooling ability that the temperature gradients are not high in the center area. However, in Fig. 6(b) and (c), the temperature gradients are high and the jet thermal profile is obvious in these areas. The cooling ability and capacity need to be enhanced or added to the cooling flow rate. Fig. 7(a) shows the velocity distribution performance. Fig. 7(b) shows the higher value at the outlet area than the others due to the lower impingement height. From Fig. 8, it shows the velocity contours of different nozzle diameters with the same nozzle cross area. The results show that the stagnation points vary with the nozzle diameters that keep occurring at the center of the nozzle no matter under whichever condition the nozzle is.
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3.3. Optimal results and discussion
419 420
429 430
From the optimization process, the optimization parameter is listed in Table 3. The conditions are constrained as Red = 17,000, Tin = 289 K, q″ = 280 kW/m2, 3 ≤ H/d0 ≤ 12, and 1 ≤ B/d0 ≤ 2, and Ac is kept at constant. The final condition of H/d0 = 7.86 and B/d0 = 2 is the optimization combination for the target function. The difference from regression equation and CFD prediction is within 0.7%. The approached function can easily calculate and quickly get the performance of the flow thermal status. From Fig. 9, the comparison of the hydraulic local Nusselt number from original and optimal condition is shown. The overall average Nusslet number of optimization is around 10% which is higher than the original condition. From Fig. 10, better performance can be found in the optimal result.
431
4. Conclusions
432
The fluid flow and heat transfer characteristics of a free-surface liquid jet impingement cooling have been investigated numerically.
411 412 413 414 415 416
421 422 423 424 425 426 427 428
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3.2. Temperature contours and flow field velocity distribution
T
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Fig. 10. Comparison of average Nusselt number (B/d0 = 1, q″ = 280 kW/m2.
400 401
[1] H. Martin, Heat and mass transfer between impinging gas jets and solid surfaces, Adv. Heat Tran. 13 (1977) 1–60. [2] Y.J. Chou, Y.H. Hung, Impingement cooling of an isothermally heated surface with a confined slot jet, ASME J. Heat Transf. 116 (1994) 479–482. [3] B. Elison, B.W. Webb, Local heat transfer to impinging liquid jets in the initially laminar, transitional, and turbulent regimes, Int. J. Heat Mass Transf. 37 (1994) 1207–1216. [4] S.H. Seyedein, M. Hasan, A.S. Mujumdar, Modeling of a single confined turbulent slot jet impingement using various k- ε turbulence models, Appl. Math. Model. 18 (1994) 526–537. [5] H. Laschefski, T. Cziesla, G. Biswas, N.K. Mitra, Numerical investigation of heat transfer by rows of rectangular impinging jets, Numer. Heat Transfer 30 (1996) 87–101. [6] T. Cziesla, E. Tandogan, N.K. Mitra, Large eddy simulations of heat transfer from impinging slot jets, Numer. Heat Transfer, Part A 32 (1997) 1–17. [7] Y.T. Yang, C.H. Shyu, Numerical study of multiple impinging slot jets with an inclined confinement surface, Numer. Heat Transfer, Part A 33 (1998) 23–37. [8] P. Behnia, S. Parneix, Y. Shabany, P. Durbin, Numerical study of turbulent heat transfer in confined and unconfined impinging jets, Int. J. Heat Fluid Flow 20 (1999) 1–9. [9] Geunyoung Yang, Mansoo Choi, Joon Sik Lee, An experimental study of slot jet impingement cooling on concave surface: effects of nozzle configuration and curvature, Int. J. Heat Mass Transf. 42 (1999) 2199–2209. [10] G. Chochua, W. Shyy, S. Thakur, A. Brankovic, J. Lienau, L. Porter, D. Lischinsky, A computational and experimental investigation of turbulent jet and crossflow interaction, Numer. Heat Transfer, Part A 38 (2000) 557–572. [11] S. Roy, Numerical investigation of the blade cooling effect generated by multiple jets issuing at an angle into a incompressible horizontal crossflow, Numer. Heat Transfer, Part A 38 (2000) 701–718. [12] Y. Shih, M. Ray, S. Mujemdar, Computational study of impingement heat transfer under a turbulent slot jet, Ind. Eng. Chem. Res. 41 (2002) 4643–4651. [13] Muhammad M. Rahman, Antonio J. Bula, John E. Leland, Analysis of transient conjugate heat transfer to a free impinging jet, J. Thermophys. Heat Transf. 14 (2000) 330–339. [14] Antonio J. Bula, Muhammad M. Rahman, Numerical modeling of conjugate heat transfer during impingement of free liquid jet issuing from a slot nozzle, Numer. Heat Transfer, Part A 38 (2000) 45–66. [15] Erick A. Siba, M. Ganesa-Pillai, Kendall T. Harris, A. Haji-Sheikh, Heat transfer in a high turbulence air jet impinging over a flat circular disk, ASME J. Heat Transfer 125 (2003) 257–265. [16] C.P. Brescianini, M.A. Delichatsios, New evaluation of the turbulence model for free buoyant plumes, Numer. Heat Transfer, Part A 43 (2003) 731–751. [17] K. Ibuki, T. Umeda, H. Fujimoto, H. Takuda, Heat transfer characteristics of a planar water jet impinging normally or obliquely on a flat surface at relatively low Reynolds numbers, Exp. Thermal Fluid Sci. 33 (2009) 1226–1234. [18] Hirofumi Hattori, Yasutaka Nagano, Direct numerical simulation of turbulent heat transfer in a plane impinging jet, Int. J. Heat Fluid Flow 25 (2004) 749–758. [19] Y.T. Yang, C.H. Hwang, Numerical simulations on the hydrodynamics of a turbulent slot jet impinging on a semi-cylindrical convex surface, Numer. Heat Transfer, Part A 46 (2004) 995–1008. [20] Y.T. Yang, Y.X. Wang, Three-dimensional numerical simulation of an inclined jet with cross-flow, Int. J. Heat Mass Transf. 48 (2005) 4019–4027. [21] T. Akiyama, K. Yamamoto, K.D. Squires, K. Hishida, Simulation and measurement of flow and heat transfer in two planar impinging jets, Int. J. Heat Fluid Flow 26 (2005) 244–255. [22] Y.C. Chen, C.F. Ma, M. Qin, Y.X. Li, Theoretical study on impingement heat transfer with single-phase free-surface slot jets, Int. J. Heat Mass Transf. 48 (2005) 3381–3386. [23] Samir A. Salamah, Deborah A. Kaminski, Modeling of turbulent heat transfer from an array of submerged jets impinging on a solid surface, Numer. Heat Transfer, Part A 48 (2005) 315–337. [24] Y.T. Yang, S.Y. Tsai, Numerical study of transient conjugate heat transfer of a turbulent impinging jet, Int. J. Heat Mass Transf. 50 (2006) 799–807. [25] S.J. Wang, A.S. Mujumdar, A comparative study of five low Reynolds number k-ε models for impingement heat transfer, Appl. Therm. Eng. 25 (2005) 31–44.
Please cite this article as: Y.-T. Yang, et al., Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.010
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[30] P. Li, K.Y. Kim, Multi-objective optimization of a staggered elliptical pin-fin arrays, Numer. Heat Transfer, Part A 53 (2008) 418–431. [31] K.Y. Kim, M.A. Moon, Optimization of a stepped circular pin-fin array to enhance heat transfer performance, Heat Mass Transf. 46 (2009) 63–74. [32] Y. Wang, Y.L. He, D.H. Mei, W.Q. Tao, Optimization design of slotted fin by numerical simulation coupled with genetic algorithm, Appl. Energy 88 (2011) 4441–4450. [33] J.U. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (1992) 335–354. [34] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.
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[26] Y.Q. Zu, Y.Y. Yan, Numerical study on stagnation point heat transfer by jet impingement in a confined narrow gap, ASME J. Heat Transf. 131 (2009) 094504. [27] M.A.R. Sharif, K.K. Mothe, Evaluation of turbulence models in the prediction of heat transfer due to slot jet impingement on plane and concave surfaces, Numer. Heat Transfer, Part B 55 (2009) 273–294. [28] M.H. Herbert, K. Rafael, K. Matthias, M. Holger, Calculation of steady and pulsating impinging – an assessment of 13 widely used turbulence models, Numer, Numer. Heat Transfer, Part B 51 (2007) 565–583. [29] Y.T. Yang, T.C. Wei, Y.H. Wang, Numerical study of turbulent slot jet impingement cooling on a semi-circular concave surface, Int. J. Heat Mass Transf. 54 (2010) 482–489.
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