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Heat transfer optimization of twin turbulent sweeping impinging jets
International Journal of Thermal Sciences 146 (2019) 106064
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International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Heat transfer optimization of twin turbulent sweeping impinging jets Amirsaman Eghtesad 1, Mohammadamin Mahmoudabadbozchelou 1, Hossein Afshin * School of Mechanical Engineering, College of Engineering, Sharif University of Technology, Tehran, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Optimization Uniform cooling Artificial neural network Genetic algorithm Twin turbulent sweeping impinging jets
In this study, a numerical investigation is carried out to reveal the potential of obtaining uniform cooling on a planar surface by impinging jets in a turbulent flow regime using the SST k ω turbulence model. Twin turbulent sweeping impinging jets (TTSIJ), as an idealization of an array of jets, are considered for heat transfer with the planar target surface. Unlike other studies that focused on a few effective parameters or limited the impinging jets to certain conditions, in this study, a more general investigation is performed to clarify the effects of all influential parameters on heat transfer. To this end, the effects of 8 design variables namely Reynolds number, the jet-to-target distance, the phase shift between the two jets, the frequency of pulsations of jets, the frequency of sweeping motion, the jet-to-jet separation distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angle of the nozzles on Nusselt distribution on the target surface are evaluated. For the purpose of elimination of thermal stress on the target surface, it is attempted to optimize the design variables to reach a uniformly distributed Nusselt number on the target surface for 9 design uniform Nusselt distribution. The optimization of the design variables is performed using artificial neural network (ANN) combined with genetic algorithm (GA) to minimize the discrepancy between the ideally uniform and optimized Nusselt distribution. Results show that within the assumed bounds of the design variables, the applied method is successful in obtaining uniform cooling with the accuracy of more than 98% and 93% in the best and worst situation, respectively.
1. Introduction Impinging jets have been used to improve heat transfer rate in many engineering applications. They could be used to increase either the cooling or heating rate of the impingement surface by disturbing the boundary layer. Based on the impinging jets configurations and the flow domain, authors have investigated the effects of selected parameters such as Reynolds number [1–3], jet arrangements, multiple impinging jets [4], jet-to-target distance and nozzle shapes [5] on heat transfer characteristics. Afroz and Sharif [6], determined the effects of the impingement surface slope on local Nusselt distribution by utilizing twin impinging slot jets in turbulent flow regime. They showed that the location of the maximum Nusselt number corresponded to the jets entrance angle. To reveal the outcome of Reynolds number and the frequency of oscilla tions, Demircan and Turkoglu [7] numerically analyzed a uni-direction pulsatile oscillating impinging jet. They showed that transient jets enforced higher heat transfer rate compared to steady state flows. By
applying a phase shift to a couple of oscillating jets with the same fre quency that were fixed in a given position, Hewakandamby [8] studied the effects of the separation distance between the two jets, the frequency of oscillations and Reynolds number. It was demonstrated that the pe riodic disturbance in the boundary layer lead to improvement of heat transfer over conventional steady flow jets. Xu et al. [9] explored the effects of unsteady intermittent pulsations on heat transfer of multiple impinging jets. The intermittent pulsations together with uniform ve locity, same frequency and different phase angles were applied for adjacent jets. They claimed that the transient flow could enhance the local Nusselt number along the impingement wall. Park et al. [10] experimentally investigated the heat transfer of a sweeping impinging jet, undergoing periodic oscillations for several Reynolds numbers and nozzle-to-plate spacing. They showed that near the center of the sweeping jet, a high Nusselt number region was formed and away from this region, the Nusselt number monotonically decreased. The flow dy namics of a sweeping jet was experimentally investigated using time-resolved particle image velocimetry by Wen and Liu [11]. They
* Corresponding author. E-mail address: [email protected] (H. Afshin). 1 These authors contributed equally to this work. https://doi.org/10.1016/j.ijthermalsci.2019.106064 Received 23 April 2019; Received in revised form 2 August 2019; Accepted 26 August 2019 Available online 4 September 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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International Journal of Thermal Sciences 146 (2019) 106064
Nomenclature
Gω h H I k L Lplate LSTD n Nu Nu N P p0 Pr q"
specific heat capacity hydraulic diameters of the nozzles cross-diffusion term frequency of pulsations frequency of oscillations generation of turbulence kinetic energy generation of ω heat transfer coefficient jet-to-target distance Identity matrix turbulence kinetic energy jet-to-jet separation distance length of the plate local standard deviation surface normal vector Nusselt number averaged Nusselt number number of points along the target surface pressure outside pressure Prandtl number ν =α surface heat flux
Sk Sω
source terms of k source terms of ω
cp Dh Dω fR fs ~k G
Re
Reynolds number Re ¼
STD t T T∞ T’ ui ðu ; vÞ u’i Uinlet w wTarget xi ðx; yÞ Yk Yω
standard deviation time temperature inlet temperature fluctuating temperature velocity (velocity components) fluctuating velocity inlet velocity of the flow slot width width of the target surface direction coordinate (x coordinate, y coordinate) dissipation of k dissipation of ω
Greek symbols α thermal diffusivity β maximum of swinging angle Γk effective diffusivity of K Γω effective diffusivity of QUOTE ω λ dimensionless jet to target distance μ dynamics viscosity ν kinematic viscosity ω specific rate of dissipation ρ fluid density Φ phase shift between the two jets τ period of swinging motion ζ thermal conductivity
2ρU∞ w
μ
concluded that at higher Reynolds numbers, the larger bending angle of the jet induced higher fluctuations. According to the above discussion, it could be concluded that impinging jets could improve heat transfer rate on the impinging sur face, but they could also generate thermal stress. In order to eliminate this issue, achieving uniform heat flux possess a significant challenge and importance. In order to obtain uniform heat flux on the target sur face, geometric parameters and flow properties must be designed appropriately. For example, Forouzanmehr et al. [12], adopted four steady laminar slot jets in order to obtain uniform cooling over an isothermal plate by optimizing the volumetric flow rate, the jet sepa ration distance, the slot widths, and jet-to-plate distance. Using twin slot
impinging jets, Farahani et al. [13] performed both numerical and experimental study to optimize the design variables (nozzle widths, jet-to-jet distance, jet-to-plate distance, the frequency of pulsating jets, and the flow rate) to achieve uniform cooling in laminar flow regime. Bijarchi and Kowsary [14] presented a laminar and steady co-axial jet to reach uniform Nusselt distribution along an isothermal surface. They concluded that the ratio of nozzle width to target length, had to be set at least equal to 0.6 to accomplish uniformity. Bijarchi et al. [15] applied a laminar swinging slot impinging jet (SSIJ) to obtain uniform cooling along a constantly-heated plate. They investigated 15 cases (5 Nusselt numbers and 3 nozzle to target surface ratios) and optimized Reynolds number, the jet-to-target distance, the frequency of swinging motion and
Fig. 1. The geometry configuration. 2
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International Journal of Thermal Sciences 146 (2019) 106064
of the plate, Lplate , is assumed to be 3wTarget to avoid the outlet effects on heat transfer. The target surface is heated by a heater with constant heat flux of q”.
Table 1The constant parameters used in the current study. Property
Symbol
Value
kg density 3 m
ρ
1.1614
kJ kgK thermal conductivity W =mK
cp
1007
ζ
0.0263
dynamics viscosity Pa:s
μ
1.84e-5
q”
1100
T∞
300
specific heat capacity
W surface heat flux m inlet temperature K
2.1. Governing equations and boundary conditions The working fluid is considered to be air with constant thermo physical properties and together with other constant parameters are given in Table 1. The continuity, Reynolds averaged mass, momentum and energy equations are given as:
the maximum of swinging angel to achieve uniform cooling on a planar surface. They stated that the SSIJ was successful in acquiring uniform cooling for Nusselt numbers below 20. Although there have been attempts to obtain uniform heat transfer by applying laminar impinging jets, achieving uniformity in turbulent flow regime, regarding their higher cooling potential, has drawn very little interest. As stated previously, so far authors limited their focus on some selected parameters that could influence heat transfer. In this research, however, a more general investigation is carried out to study the potential of uniform cooling using turbulent impinging jets. To this aim, twin turbulent sweeping impinging jets (TTSIJ) are considered as an idealization of an array of jets. The advantages of this research is due to three reasons: (1) exploring the possibility of attaining uniform cooling in turbulent flow regime (2) performing a comprehensive study on all the influential parameters that could affect heat transfer charac teristics (3) taking simultaneous advantage of the positive effects of the frequency of pulsations and the frequency of sweeping motions of the jets (unlike other studies that focused on only one of these factors). Regarding the 2nd point, the effects of the design variables Reynolds number, jet-to-target distance, the phase shift between the two jets, the frequency of pulsations of jets, the frequency of sweeping motion, the jet-to-jet separation distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angle of the nozzles, are evaluated. These parameters, based on their natures, were divided into geometric group and non-geometric group. The jet-to-jet separation distance, the jet-totarget distance and the hydraulic diameter of the nozzles are classified into the geometric group and Reynolds number, the phase shift between the two jets, the frequency of pulsations of the jets, the frequency of swinging motion and the maximum of sweeping angle of the nozzles are assigned to the non-geometric group. At first, the governing equations and the boundary conditions are presented. Secondly, parametric studies are performed to determine the effects of each of the design variable on Nusselt distribution. Finally, Artificial Neural Network (ANN) combined with Genetic Algorithm (GA) are used to obtain uniform cooling along the target surface.
∂ui ¼0 ∂xi
(1) � �
ρ
∂ui ∂ui þ ρuj ¼ ∂t ∂xj
∂P ∂ ∂ui ∂uj þ μ þ ∂xi ∂xj ∂xj ∂xi
ρ
∂T ∂T ∂ μ ∂T þ ρu j ¼ ∂xj Pr ∂xj xj ∂t
�
�
�
ρu’i u’j
(2)
� (3)
ρT’ u’j
where u is the velocity component in x and y directions (ui and uj ), x is the direction coordinate (xi and xj ), ρ is the fluid density, P is the pressure, μ is the dynamic viscosity, T is the temperature, Pr is Prandtl number, t is the time and finally T’ and u’ ðu’i and u’j Þ are the fluctuating
temperature and velocity components, respectively. The SST-kω model is employed [6] to study heat transfer with the following transport equations: � � ∂ ∂ ∂ ∂k ~ k Yk þ Sk ðρkui Þ ¼ (4) Γk þG ðρkÞ þ ∂t ∂xi ∂xj ∂xj �
�
∂ ∂ ∂ ∂ω Γω þ Gω ðρkui Þ ¼ ðρωÞ þ ∂t ∂xi ∂xj ∂xj
Yω þ Dω þ Sω
(5)
~ k represents the generation of turbulence kinetic energy, Gω represents G the generation of ω, Γk and Γω refer to effective diffusivity of k and ω, Yk and Yω represent the dissipation of k and ω, Dω refers to the crossdiffusion term, and finally Sk and Sω are source terms. No slip condition is assumed at the upper and lower surfaces: u ¼ 0jAD;
(6)
BC
The jets’ velocity in x and y directions are given by: � � � φ�� � 1 u1x ¼ Uinlet 1 þ sinð2πf 1R tÞ � sinβsin 2πf s t þ 2 2
2. Problem formulation As mentioned, the aim of this study is to investigate the heat transfer characteristics of TTSIJ and then to impose uniform cooling on the target surface by optimizing the design variables. Fig. 1 shows the schematic diagram of TTSIJ configurations and the flow domain which is confined between two parallel plates. The jet-to-jet separation distance, the jet-totarget distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angel, are denoted by L, H, Dh and β, respectively. Two modes of oscillations are considered for TTSIJ, a sweeping motion with steady periodic angular motion similar to a pendulum motion referred to as “the frequency of sweeping motion” denoted by fs, and steady period pulsating motion, resulting in fluctuations in the jets’ volumetric flow, referred to as “the frequency of pulsations” and denoted by fR. Reynolds number and fs are set equal for the two jets to simplify the experimental applications setups, and the phase shift between the jets, φ, is set to remain constant at all times. The ratio of the hydraulic diameter of the nozzles to the target width (Dh =wTarget ) is 0.1 for each jet and the length
(7)
ffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffi��� ffiffiffiffiffiffiffiffiffiffi �rffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� � 2 1 φ u1y ¼ Uinlet 1 þ sinð2πf 1R tÞ 1 sinβ sin 2πf s t þ 2 2
(8)
� � � � 1 u2x ¼ Uinlet 1 þ sinð2πf 2R tÞ � sinβsin 2πf s t 2
(9)
φ�� 2
ffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffi��� ffiffiffiffiffiffiffiffiffiffi �rffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� � 2 1 φ u2y ¼ Uinlet 1 þ sinð2πf 2R tÞ (10) 1 sinβ sin 2πf s t 2 2 � � In the above sets of equations, the terms Uinlet 1 þ12 sinð2πf 2R tÞ and � � Uinlet 1 þ12 sinð2π f 1R tÞ correspond to the fluctuations in the jets’ volu � � �� � � �� metric flow and the terms sinβsin 2πf s t þφ2 and sinβsin 2πf s t φ2
reflect the sweeping motion of the jets. Therefore the velocity magnitude � � � of the jets would be Uinlet 1 þ12 sinð2πf 1R tÞ and Uinlet 1 þ � 1 sinð2πf 2R tÞ . AB and CD are open to atmosphere resulting to the 2
3
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 2. The results of mesh independency analysis.
following equation: � �� pI þ μ ru þ ruT :n ¼
p0 n
Table 2 The design variables for mesh analysis.
(11)
Constant heat flux is applied to the target surface (EF) and is expressed in the form of: n:
� q” αf rT ¼ ρcp
(12)
(13)
By obtaining temperature and velocity distribution over the region, the local Nusselt number on the target surface based on the slots width and heat transfer coefficient is computed by the following equation: hðTi
Ts Þ ¼
Nu ¼
hð2wÞ ζ
ζ
∂T ∂y
(14) (15)
As the Nusselt distribution varies with time and position due to the unsteady nature of TTSIJ, a time-averaged Nusselt number within a cycle is defined to determine the averaged heat transfer on the target surface, Z tþτ 1 NuðxÞ ¼ Nuðx; tÞdt (16)
τ
λ
φ
fs
f1R
f2R
L
Dh
β
6000
0.6
π =4
5.5
5.5
5.5
0.45
0.15
π =8
“1”, “2” and “3”, respectively). As it is intended to study the heat transfer with the target surface and then to perform the optimization process to reach uniform cooling on the target surface, near the target regions demand precise investigation and require sufficiently accurate CFD computation and relatively the finest grid sizes. Also, as the importance of entrance regions where the air flows into the domain, the grid sizes are sufficiently small in these areas. On the other hand, the grid sizes far enough from these regions could be relatively coarse. The discussed grid structure was implemented to the domain by assigning the finest grid sizes to near the target regions, relatively fine grid sizes to entrance zones, and coarse grid sizes in areas other than the mentioned areas. Therefore, the SST k ω turbulence model is compatible with the dis cussed mesh structure for its accuracy in both viscous sub-layer and inviscid regions, and was adopted. The density of the mesh structure is adjusted to increase mono tonically towards the target surface because of higher velocity and temperature gradients to ensure yþ�1 that is necessary for the SST k ω model. Fig. 2 shows the Nusselt distribution for the three meshes. The relative error between “2” and “1” is less than 2%, and “2” is chosen for the domain. Table 2 shows the values of the design variables considered for mesh analysis. The jet-to-target distance is non-dimensionalized with H respect to the width of the target surface and is denoted by λ ¼ wTarget .
Finally, adiabatic surfaces are assumed at AD, BF and CE: n:ðrTÞ ¼ 0
Re
t
where τ is the period of sweeping motion (τ ¼ f1s ). It should be noticed that calculations are performed after a few cycles of sweeping to elim inate initial-condition-dependent results.
3. Verification To check the validity of the present study, the results by Xu et al. [9] are reproduced where the effects of unsteady intermittent pulsations of multiple impinging jets on heat and mass transfer were explored. Computational fluid dynamic approach is employed to obtain local heat transfer rate of twin intermittent pulsating jets and effects of Reynolds number and the frequency of pulsations are investigated. For the pur pose of validation, in the case of H/w ¼ 5 (H denotes the nozzle-to-plate-distance and w represents the slot width as in Ref. [9]), Re ¼ 5500, f ¼ 41 Hz and velocity amplitude of A ¼ 17%, the time-averaged local Nusselt number along the impingement wall was
2.2. Numerical method & mesh analysis SIMPLE algorithm is employed for the pressure-velocity coupling and the discretization of momentum and energy equations are based on second-order upwind method. Pressure values at faces are interpolated using momentum equation coefficients [16] and least square cell-based gradient is utilized so that the solution could vary linearly. Grid-independency of the results is checked under three structured grid densities by analysis of Nusselt distribution on the target surface with the number of elements 246 � 48, 180 � 40 and 141 � 32 (labeled as 4
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 3. The comparison of Nusselt distribution of the present work with those of [9,17].
Fig. 4. The classification of the design variables into 4 groups.
compared to experimental results [17]. In the present study, the same case is modeled and the result is depicted in Fig. 3. Compared to the experimental and numerical studies [9,17], the relative error is calcu lated to be 1% and 1.5%, respectively, which confirms the reliability of our numerical scheme. The next step is to investigate the effects of all the design variables and then the optimization of these variables to achieve uniform cooling on the target surface. It should be noticed that from here on, STD, stands for the standard deviation of Nusselt distribution with respect to the averaged Nusselt distribution:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u �2 N � u1 X Nui Nui STD ¼ t � 100 N i¼1 Nui
(17)
The 8 design variables are classified into 4 groups according to Fig. 4; Group1 contains those variables which increase heat transfer with the target surface while increasing, Group2 contains those which decrease heat transfer rate with increase in their values and Group3 contains those variables that could both increase and decrease heat transfer rate when their values change within the assumed bounds. Finally, Group 4 5
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 5. The effects of Reynolds number on Nusselt distribution on the target surface.
is defined based on the inlet velocity of TTSIJ and the slots width, Re ¼
Table 3 The design variables for parametric studying of Reynolds number.
2ρU∞ w
μ
λ
Φ
fs
f1R
f2R
L
Dh
β
0.6
π =4
5.5
5.5
5.5
0.45
0.15
5π =24
. It could be seen in Fig. 5 that for fixed values of other parameters
according to Table 3, increase in Reynolds number from 5400 to 6600 leads to the growth of averaged Nusselt number from 41.1 to 46.2 on the target surface. In other words, as the fluid momentum enhances, heat transfer coefficient raises mutually that results in increase in Nusselt number, thus, to achieve higher heat transfer rate, larger Reynolds number would be required. Also, as the jets impinge on the target sur face and turn upward after collision, and creates vortex in the domain, heat transfer would be affected. Without any phase shift, the jets sweeping motion to the left and right sides are similar to each other but with a phase shift angle, the jets oscillations produce periodic vortex generations that affect the heat transfer near the target region. It must be noticed, however, that the phase shift between TTSIJ, could shift the maximum of heat transfer from central regions to side regions. Ac π , the maximum Nusselt cording to Fig. 6, at the phase shift angle of 12 number is about 47 and the averaged Nusselt number is 43.0 with the 5π STD of 2.6 and at the phase shift angle of 12 the maximum Nusselt
includes those variables which any alteration in their values within the assumed bounds do not affect heat transfer noticeably. Regarding this, Reynolds number, the phase shift between TTSIJ, and the hydraulic diameter of the nozzles belong to Group1, the jet-to-target distance and the maximum sweeping angle belong to Group2, the frequency of pul sations and the frequency of oscillations situate in Group3 and finally the jet-to-jet separation distance would not affect heat transfer rate significantly and would go to Group 4. 3.1. Group1 At first, effects of Reynolds number is investigated. Reynolds number
Fig. 6. The effects of phase shift between the two jets on Nusselt distribution on the target surface. 6
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International Journal of Thermal Sciences 146 (2019) 106064
surface. For example, in the case of D/wTarget ¼ 0.18 the averaged Nusselt number is about 45.8, that demonstrates 16.8% increase in heat transfer coefficient compared to D/wTarget ¼ 0.12 which is 39.2 on average. For a specific Reynolds number, increase in the hydraulic diameter of the nozzles deduces to higher flowrate of air to the domain and any point on the target surface receives more cooling and overally, the whole target surface achieves higher cooling rate. It should be noticed, however, that depending on the values of other parameters, the STD of heat transfer on the target surface might also increase which is not desirable due to the purpose of attaining uniform cooling. Thus, depending on the target cooling rate, the hydraulic diameter of the
Table 4 The design variables for parametric studying of phase shift between TTSIJ. Re
λ
fs
f1R
f2R
L
Dh
β
6000
0.6
5.5
5.5
5.5
0.45
0.15
5π =24
number is 52, however, the Nusselt number is decreased to the minimum of 38 in the opposite side, resulting in the averaged value of 44 and STD of 4.8 on the target surface. Therefore, phase shift between TTSIJ could change the maximum and STD of the Nusselt distribution. The values of other design variables are specified according to Table 4. The last var iable in Group1 is the hydraulic diameter of the nozzles. The hydraulic diameter of the nozzles is proportional to the air flowrate to the domain. The larger hydraulic diameter of the nozzles results in the entrance of more volumes of air at any moment to the domain and impacts larger segments of the target surface. As could be seen in Fig. 7, increase in the hydraulic diameter of the nozzles, results in significant increment of heat transfer on the target
Table 5 The design variables for parametric studying of hydraulic diameter of the nozzles. Re
λ
Φ
fs
f1R
f2R
L
β
6000
0.6
π =4
5.5
5.5
5.5
0.45
5π =24
Fig. 7. The effects of non-dimensional hydraulic diameter of the nozzles on Nusselt distribution on the target surface.
Fig. 8. The effects of non-dimensional jet-to-target distance on Nusselt distribution on the target surface. 7
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International Journal of Thermal Sciences 146 (2019) 106064
relative error with respect to perfect uniformity under this circumstance could not be acceptable. Thus the idea of attributing a wider range of angles was implemented thereafter. As a result, in this study, the authors
Table 6 The design variables for parametric studying of dimensionless jet-to-target distance. Re
Φ
6000
1 wTarget L=2 Dh =2 to H 1 wTarget L=2 Dh =2 tan H
did not limit the maximum of sweeping angle to tan
fs
f1R
f2R
L
Dh
β
5.5
5.5
5.5
0.45
0.15
5π =24
obtain more uniformity on the target surface where shows the angle that the jets would meet the target ends assuming to move in a straight path. Considering other parameters according to Table 7, effects of maximum of sweeping angle on Nusselt distribution on the target surface is depicted in Fig. 9. For the considered values of jet-to-target distance, the hydraulic diameter of nozzles, and jet-to-jet separation distance, the angle between the mid-point of the nozzles and the mid-point of the target surface is π =8. For the maximum sweeping angle less than π =8, each impinging jet mostly affect the un derneath segments of the target surface and the central area lose effec tive cooling. This could be seen as the two peaks of Nusselt distribution in the side regions and a minimal of Nusselt number in the central re gions in the case of β ¼ π=12. Beyond β ¼ π=8 the jets mingle near the target surface and increase Nusselt number in central regions and reduce the difference between the maximum and minimum of Nusselt numbers as in the case of β ¼ 2π=12. However, considering a fixed value of the frequency of sweeping, further increase in maximum angle of sweeping results in faster sweeping of the TTSIJ, and the adequate time of heat transfer would not be attained. In other words, for a fixed value of fs, TTSIJ have to sweep a longer distance for higher β and therefore the heat transfer exchange with any point decreases. On the other side, all points on the target surface receive more uniform cooling and the STD of Nusselt distribution decreases as the case of β ¼ 3π=12. Finally, further increase of maximum angle of sweeping, noticeably decreases heat transfer time with each point on the target surface and the effective cooling decreases.
Table 7 The design variables for parametric studying of the maximum sweeping angle. Re
λ
Φ
fs
f1R
f2R
L
Dh
6000
0.6
π =4
5.5
5.5
5.5
0.45
0.15
nozzles should be designed suitably. The values of other design variables could be found in Table 5. 3.2. Group2 As introduced, the jet-to-target distance and the maximum of sweeping angle serves as important parameters that could influence heat transfer characteristics and require appropriate considerations. As the intensity of collision of TTSIJ with the target surface would absolutely influence heat transfer rate, the jet-to-target distance was regarded as the second variable in the “geometric variables”, which would decrease the collision intensity and the consequent magnitude of Nusselt number for larger distances and results in higher collision intensity and conse quently higher Nusselt number for lower distances. As depicted in Fig. 8, for fixed values of other parameters (see Table 6), as the dimensionless jet-to-target distance change from 0.48 to 0.72, the maximum Nusselt number changes from about 51 to about 46 (9.8% reduction) and makes the Nusselt distribution possess a flatter profile. Furthermore, the loca tion of maximum Nusselt number alters as the jet-to-target distance gets different values. Therefore, the jet-to-target distance alters the maximum Nusselt number and changes the location of maximum heat transfer. Finally, the idea of attributing different maximum angles of sweeping was presented in Ref. [15]. In this paper, at first, it was tried to reach uniform Nusselt distribution by setting the maximum of swinging w w angle equal to tan 1 Target 2H , where w was the nozzle width and wtarget was the target surface length and H was the jet-to-target distance. w w tan 1 Target referred to the angle that the jet would collide with the 2H target ends assuming to move in a straight path. It was found that the
3.3. Group3 In order to investigate the effects of the frequency of pulsations on Nusselt distribution f 1R is varied from 2 to 8 Hz, and f 2R is set equal to 0. Also, fs is assigned a small number, to provide the oscillations only due to pulsations in the inlet flow. The assigned values of other parameters are according to Table 8. According to Fig. 10, the right sides of the target surface receive more cooling of about 11–27% due to the pul sating motion of the overhead impinging jet than the left sides where the frequency of pulsations of the overhead impinging jet is zero, thus the right hand sides experience higher Nusselt numbers. As the frequency of
Fig. 9. The effects of the maximum sweeping angle on Nusselt distribution on the target surface. 8
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International Journal of Thermal Sciences 146 (2019) 106064
STD of the Nusselt distribution from 2.04 to 0.69 by eliminating the peak in Nusselt distribution, and providing more uniform cooling on the target surface. The reduction of maximum Nusselt distribution at midpoints on the target surface is due to lower interaction between the flow and the target surface caused by faster sweeping motion of the jets and less time for heat transfer. However, the side segments experience more cooling because of the faster disturbance of boundary layer. Therefore, in spite of decreasing the maximum Nusselt number from about 46 to 43.5, increase in the frequency of oscillations has a positive impact on uniformity. The averaged Nusselt distribution and the
Table 8 The design variables for parametric studying of the frequency of pulsations. Re
λ
Φ
fs
f2R
L
Dh
β
6000
0.6
0
0.1
0
0.45
0.15
5π =24
pulsations of the right impinging jet increases, the mixing of the un derneath fluid enhances, which absorbs more heat from the target sur face, and reaches higher temperature. Later, by mixing with the left impinging jet, decreases the cooling rate of the left sides of the target surface. According to Fig. 11 and considering the other variables ac cording to Table 9, as fs increases from 0.5 to 2.5, both the maximum and averaged Nusselt distribution increases due to higher fluctuations of the flow over the target surface. In other words, increase in frequency of pulsations, causes higher rates of boundary layer disturbance and en hances the mixing of flow near the target surface which raises the heat transfer rate. On the other side, increasing fs beyond 4 to 7, reduces the
Table 9 The design variables for parametric studying of the frequency of oscillations. Re
λ
Φ
f1R
f2R
L
Dh
β
6000
0.6
0
0.1
0
0.45
0.15
5π =24
Fig. 10. The effects of frequency of pulsations on Nusselt distribution on the target surface.
Fig. 11. The effects of phase shift between the two jets on Nusselt distribution on the target surface. 9
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Fig. 12. The effects of non-dimensional jet-to-jet separation on Nusselt distribution on the target surface.
Nusselt distribution at each τ/8 interval. The peaks of the Nusselt dis tribution can be found where the temperature gradient has the maximum values.
Table 10 The design variables for parametric studying of jet-to-jet separation distance. Re
λ
Φ
fs
f1R
f2R
L
Dh
β
6000
0.6
0
5.5
5.5
5.5
0.45
0.15
5π =24
4. Optimization In previous sections, parametric study on each of the design variables were performed to reveal their impacts on Nusselt distribution. It was declared that increase in Reynolds number and the jet-to-target distance have opposite effects on heat transfer rate. On the other hand, the phase shift between the two jets changes the magnitude and location of maximum heat transfer. The frequency of pulsations proved to impose higher heat transfer rate than steady jets. The maximum angle of sweeping showed to have a threshold beyond which the uniform cooling would diminish. Also, alteration in jet-to-jet separation distance showed not to affect the maximum of heat transfer, but could change its position. Finally, it was noticed that increase in the hydraulic diameter of the nozzles could significantly enhance heat transfer rate on the target surface. So far, it was indicated that each of the design variables could either enhance or diminish heat transfer rate and could even change the dis tribution of Nusselt number on the target surface. However, in order to attain uniform cooling, that is the main purpose of this paper, the design variables should be optimized to enable reaching a uniform design Nusselt distribution on the target surface. Therefore, 9 design Nusselt distributions with equal intervals namely 25, 30, 35, 40, 45, 50, 55, 60 and 65 are considered on the target surface and the 8 design variables are to be specified to attain the design distributions. In order to optimize these parameters, artificial neural network (ANN) combined with ge netic algorithm are employed. An Artificial neural network is an interconnected group of nodes that could be applied to model a computational structured framework where the complex relations between the inputs and outputs reveal as a func tion. To minimize the discrepancy between the ANN outputs and the real values for a specific task, the ANN should be trained with adequate number of data, firstly. To this aim, the 8 aforementioned design vari ables and the discrepancy between the obtained Nusselt distribution and the design Nusselt distribution are fed into the network for training. The discrepancy between the two sets of Nusselt distributions are defined as follow:
maximum Nusselt value decrease further by elevating the pace of oscillation in the case of fs ¼ 8.5, because any point on the target surface would have less time for heat transfer with the jets, but the uniformity is enhanced. Finally, for the frequencies between 2.5 and 4, the positive effects of faster disturbance of the boundary layer nullifies the negative effects of lower interaction between the flow and the target, and the Nusselt distribution remains nearly constant. 3.4. Group 4 According to Fig. 12, the averaged Nusselt number on the target surface does not change remarkably with changes in the jet-to-jet sep aration distance, and the changes could be seen only in the location of maximum Nusselt number on the target surface. Other design variables are according to Table 10. In order to better depict the contribution of the jets to heat transfer with the target surface streamlines, temperature contours and Nusselt distributions are presented at every τ/8 interval within a cycle. Highvelocity gradients near the surfaces due to no-slip conditions at the surfaces, lead to the exertion of shear stress between the layers of the fluid and together with the upward flow after the collision with the target surface, creates vortexes that results in better mixing of the flow and would influence the flow behavior and increases heat transfer with the target surface. As could be seen in Fig. 13, the location of the vor texes changes continuously as the sweeping angle of the jets vary in time. The continuous shift of the vortexes’ locations, affects all segments of the target surface and therefore higher Nusselt number is expected along all points on the target surface compared to laminar flow. Ac cording to the temperature contours, it could be concluded that the collision of the jets with any point on the target surface, enhances the cooling rate of the point due to high momentum of the flow, therefore, for all the temperature contours, lower temperature is expected at the collision points and as moving away from the colliding segments, the temperature would increase as the result of constant heat flux applied to the target. Furthermore, the right column of the figure, shows the 10
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 13. The streamlines and temperature contours in every τ/8 interval within a cycle.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u �2 N � u1 X NuD Nui LSTD ¼ t � 100 N i¼1 NuD
with respect to the real values. Therefore, the criterion considered for judging the validity of the outputs of the ANN, is that the test values need to reach higher than 99%. As demonstrated in Fig. 14, the ANN proves to present reliable results. Subsequently, the 9 output evaluated functions of the ANN, are regarded as the objective functions of the optimization process. For example, in the case of NuD ¼ 40, the LSTD corresponding to the design variables are fed into the network, and the evaluated function is then regarded as the objective function of the
(18)
Consequently, the ANN is trained 9 times to disclose the relative function between the design variables and the LSTD for each of the design Nusselt distributions. In ANN, the test values determine the performance of the network which estimate the accuracy of the model 11
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 13. (continued).
generation. Finally, over successive generations, the populations evolve towards the optimal solution. Fig. 15 depicts the best and the mean fitness variations with respect to the number of generations in the case of NuD ¼ 40. The best fitness value denotes the best individual found in the population while the mean fitness value simply refers to the average of fitness values across the entire population. It could be seen that around the 60th generation, these two values approach to each other until they become exactly equal, which means that the algorithm has reached the best fitness and the optimal state is achieved. Therefore, it could be concluded that the LSTD function value has a relative error of 1.3% with respect to perfectly uniform Nusselt distribution on the target surface. The relevant optimization steps is summarized in Fig. 16: One of the main advantages of the above procedure is the short run time required for optimization of any other design Nusselt numbers, since the objective function of the GA reveals apparent by ANN. Therefore, the trained ANN could be used to disclose the objective function of any other design Nusselt numbers only if the discrepancy between the obtained Nusselt distribution and the design Nusselt dis tribution are fed into the network for training. The optimization process is performed 9 times to obtain the optimal design variables for the 9 considered design Nusselt distributions that would minimize the LSTD function within the upper and lower bounds of the design variables. Table 12 represents the optimal values of the design variables for each design uniform Nusselt distribution on the target surface and the value of STD function that refers to the discrep ancy between the optimized and perfectly uniform Nusselt distribution. The optimized Nusselt distribution of all 9 NuD are illustrated in Fig. 17. It should be noticed that a perfectly uniform Nusselt distribution occur when all “Nui”s in the right hand side of equation (16) equal to NuD. Although it is expected to be unrealistic to obtain a perfectly uni form Nusselt distribution on the target surface, it is possible to approach a uniform distribution with acceptable accuracy. According to Table 12, it could be concluded that in all cases, TTSIJ could achieve uniform cooling with the relative error of less than 6.5%, but as the design Nusselt distribution decreases, the potential of TTSIJ for obtaining uniform cooling increases from 6.5% in the case of NuD ¼ 65 to 1.2% in the case of NuD ¼ 35. In order to assess the maximum uniform cooling potential of TTSIJ, the design Nusselt number was further increased to NuD ¼ 70, but the error increased to about 10%, therefore, the ability of the TTSIJ to obtain uniform cooling is limited to around NuD ¼ 65. In this case, the local STD of the optimized and design uniform Nusselt distributions increased to 6.5%, therefore, it could be deduced than within the considered bounds for the 8 design variables, the potential of obtaining uniform cooling decreases beyond NuD ¼ 65 and it is required to extend the bounds of these variables to reach higher cooling
Fig. 14. The test results of ANN. R ¼ 0.99914 demonstrates the accuracy of ANN outputs with respect to the real values. Table 11 The upper and lower bounds of the design variables. Variable
Re
λ
Φ
fs
f1R
f2R
L
Dh
β
Lower bound Upper bound
5000 7000
0.4 0.8
0
π =2
1 10
1 10
1 10
0.3 0.6
0.1 0.2
0 5π 12 =
genetic algorithm. For the purpose of optimization, the upper and lower bounds of the design variables are assigned according to Table 11. Genetic algorithm begins with a set of individuals called population that are characterized by a set of parameters (here the 8 design vari ables). The parameters are then joined into a string to form the solution. In other words, the genetic algorithm repeatedly modifies a population of individual solutions. At the beginning of each step, individuals are selected from the population randomly and then produce the next gen erations by crossover rules. Meanwhile, mutation rules are applied to enable the involvement of all population in producing the next 12
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 15. The best and mean fitness values of the genetic algorithm.
Fig. 16. The optimization process.
capacities. For the purpose of using the optimization process for geometricconstraint cases, the optimal values of the design variables are calcu lated through the same procedure considering different jet-to-target distances or in other words, different channel heights. Jet-to-target distance is selected since other parameters could be manipulated simpler to achieve their optimal values, but probably channel height would have some space restrictions. Therefore, the optimization process is performed for λ ¼ 0:4; 0:5; 0:6; 0:7 and 0:8 and one could choose the
optimal values according to the existing condition. The optimal values of the other design variables and the corresponding error of the Nusselt distribution compared to NuD ¼ 40 is presented in Table 13. 5. Conclusion In this paper, a comprehensive study of heat transfer characteristics of twin turbulent impinging jets (TTSIJ) was performed numerically to reveal the effects of Reynolds number, the jet-to-target distance, the 13
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International Journal of Thermal Sciences 146 (2019) 106064
Table 12 The optimal values of the design variables for each design uniform Nusselt distribution. Re
λ
Φ
fs
f1R
f2R
L
Dh
β
Error %
NuD ¼ 25
7000
0.4
0
1
1
10
0.6
0.1
5π =12
3.9
NuD ¼ 30
7000
0.8
0
1
10
1
0.3
0.2
5π =12
2.4
NuD ¼ 35
7000
0.8
0
1
10
10
0.6
0.2
5π =12
1.2
NuD ¼ 40
6958
0.73
π =6:3
7.1
4.8
6.7
0.3
0.1
π =4:8
1.3
NuD ¼ 45
5559
0.73
π =2.4
2.7
9.8
3.3
0.5
0.2
π =8
1.8
NuD ¼ 50
6480
0.79
π =15
4.2
2.7
7.4
0.4
0.2
π =235
2.1
NuD ¼ 55
5804
0.71
0
7.7
1.6
1.3
0.4
0.2
π =13:7
3.4
NuD ¼ 60
6926
0.68
π =3:9
6.6
1.4
8.6
0.5
0.2
π =12
3.7
NuD ¼ 65
6911
0.45
π =10
3.9
1.1
4.9
0.3
0.2
π =12:8
6.5
Fig. 17. The optimized Nusselt distribution for a) NuD ¼ 25, 30, 35, 40, 45 b) NuD ¼ 50, 55, 60 and 65.
14
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International Journal of Thermal Sciences 146 (2019) 106064
Table 13 The optimal values of the design variables for different values of jet to target distance. Re
λ
Φ
fs
f1R
f2R
L
Dh
β
Error %
6961
0.80
π =78
7.18
2.81
2.75
0.46
0.14
π =4:76
2.49
6958
0.73
π =5:8
7.07
4.81
6.75
0.35
0.13
π =4:8
1.37
6807
0.70
π =7:5
3.32
6.56
9.65
0.6
0.14
π =5:5
2.14
6956
0.60
π =18
7.20
2.85
6.77
0.46
0.13
π =0:5
1.40
5066
0.50
π =2:3
6.70
10.00
2.86
0.53
0.19
π =8:5
2.24
6845
0.40
0
1.00
1.00
9.00
0.60
0.20
π =2:4
2.95
phase shift between the two jets, the frequency of pulsations of each jet, the frequency of sweeping motion, the jet-to-jet separation distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angle of the nozzles and then optimizations of this variables were performed to obtain uniform Nusselt distribution on the target surface. Results showed that, as the Reynolds number increases, the fluid momentum enhances which raises the heat transfer coefficient on the target surface. Also, the phase shift between the two jets produce periodic vortex generations that augments the heat transfer on the target surface and could shift the maximum of heat transfer from central regions to side regions. For fixed values of the other variables, increase in the hydraulic diameter of the nozzles deduces to higher heat transfer rate due to applying higher flowrate of air to the domain that results in more cooling. In opposite to the positive effects of the three above-mentioned pa rameters on heat transfer, as the jet-to-target distance increases, the intensity of collision of TTSIJ with the target surface decreases and consequently reduces the maximum Nusselt number. Furthermore, the location of maximum Nusselt number alters and the Nusselt distribution profile gets flatter. Increasing the maximum angle of sweeping results in faster sweeping motion of the TTSIJ, and the adequate time for heat transfer would not be attained. On the other side, all points on the target surface receive more uniform cooling because the jets do not focus on a small particular area. Applying frequency of oscillation showed to impose higher heat transfer rate by disturbing the boundary layer over the impingement region. As f s increases to 2.5, both the maximum and averaged Nusselt distribution increase due to higher fluctuations of the flow and boundary layer disturbance over the target surface at first, but further increase of f s from 2.5 to 4 keeps the Nusselt distribution almost constant because the positive effects of faster disturbance of the boundary layer nullifies the negative effects of lower interaction between the flow and the target. Increasing fs beyond 4 to 7, reduces the STD by eliminating the peak in Nusselt distribution, and providing more uniform cooling on the target surface. Further increase in fs forces the TTSIJ to move faster which reduces the heat transfer time with the target surface and finally leads to smoother Nusselt profile. The frequency of pulsations showed to improve the heat transfer with the underneath segments of the target surface because of the more mixing of the flow and disturbance of the boundary layer. The jet-to-jet separation distance, could not remarkably alter the averaged Nusselt number magnitude on the target surface, but shifted the location of maximum Nusselt number on the target surface. For the purpose of optimization, 9 design uniform Nusselt distribution with equal intervals from NuD ¼ 25 to NuD ¼ 65 were considered on the target surface and the 8 design variables were optimized by subsequent use of artificial neural network (ANN) and genetic algorithm (GA). The discrepancy between the obtained Nusselt distribution and the design
Nusselt distribution, and the design variables were fed into the ANN and the output evaluated functions of the ANN, were regarded as the objective functions of the optimization process. Since the objective function of the GA reveals apparent by ANN, the run time of the opti mization process decreases noticeably in compared to optimization without using ANN. It was found that within the assumed bounds of the design variables, TTSIJ were successful to achieve uniform Nusselt dis tribution with the accuracy of more than 98% and 93% in the best and worst situation, respectively. The utilized procedure could be used for obtaining higher uniform Nusselt distribution by changing the bounds of the design variables. References [1] H.J. Poh, K. Kumar, A.S. Mujumdar, Heat transfer from a pulsed laminar impinging jet, Int. Commun. Heat Mass Transf. 32 (10) (2005) 1317–1324. [2] M.J. Remie, et al., Heat-transfer distribution for an impinging laminar flame jet to a flat plate, Int. J. Heat Mass Transf. 51 (11–12) (2008) 3144–3152. [3] W. Zhou, L. Yuan, Y. Liu, D. Peng, X. Wen, Heat transfer of a sweeping jet impinging at narrow spacings, Exp. Therm. Fluid Sci. 103 (January) (2019) 89–98. [4] M. Faris, R. Zulkifli, Z. Harun, S. Abdullah, Heat Transfer Augmentation Based on Twin Impingement Jet Mechanism, 2018 no. February 2019. [5] M. Attalla, H.M. Maghrabie, A. Qayyum, A.G. Al-Hasnawi, E. Specht, Influence of the nozzle shape on heat transfer uniformity for in-line array of impinging air jets, Appl. Therm. Eng. 120 (2017) 160–169. [6] F. Afroz, M.A.R. Sharif, Numerical study of heat transfer from an isothermally heated flat surface due to turbulent twin oblique confined slot-jet impingement, Int. J. Therm. Sci. 74 (2013) 1–13. [7] T. Demircan, H. Turkoglu, The numerical analysis of oscillating rectangular impinging jets, Numer. Heat Transf. A 58 (2) (2010) 146–161. [8] B.N. Hewakandamby, A numerical study of heat transfer performance of oscillatory impinging jets, Int. J. Heat Mass Transf. 52 (1–2) (2009) 396–406. [9] P. Xu, S. Qiu, M. Yu, X. Qiao, A.S. Mujumdar, A study on the heat and mass transfer properties of multiple pulsating impinging jets, Int. Commun. Heat Mass Transf. 39 (3) (2012) 378–382. [10] T. Park, K. Kara, D. Kim, International Journal of Heat and Mass Transfer Flow structure and heat transfer of a sweeping jet impinging on a flat wall, Int. J. Heat Mass Transf. 124 (2018) 920–928. [11] X. Wen, Y. Liu, Lagrangian analysis of sweeping jets measured by time-resolved particle image velocimetry (a) the jet ’ s maximum deflected positions (b) the jet ’ s maximum spreading angle, Exp. Therm. Fluid Sci. 97 (April) (2018) 192–204. [12] M. Forouzanmehr, H. Shariatmadar, F. Kowsary, M. Ashjaee, Achieving heat flux uniformity using an optimal arrangement of impinging jet arrays 137, June, 2015, pp. 1–8. [13] S.D. Farahani, M.A. Bijarchi, F. Kowsary, M. Ashjaee, Optimization arrangement of two pulsating impingement slot jets for achieving heat transfer coefficient uniformity, J. Heat Transf. 138 (10) (2016) 102001. [14] M.A. Bijarchi, F. Kowsary, Inverse optimization design of an impinging co-axial jet in order to achieve heat flux uniformity over the target object, Appl. Therm. Eng. 132 (2018) 128–139. [15] M.A. Bijarchi, A. Eghtesad, H. Afshin, M. Behshad, Obtaining uniform cooling on a hot surface by a novel sweeping slot impinging jet 150 (June 2018) (2019) 781–790. [16] C.M. Rhie, Numerical study of the turbulent flow past an airfoil with trailing edge separation 21 (11) (1983) 1525–1532. [17] E.C. Mladint, Local convective heat transfer to submerged pulsating jets 40 (1997) 14.
15
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Heat transfer optimization of twin turbulent sweeping impinging jets Amirsaman Eghtesad 1, Mohammadamin Mahmoudabadbozchelou 1, Hossein Afshin * School of Mechanical Engineering, College of Engineering, Sharif University of Technology, Tehran, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Optimization Uniform cooling Artificial neural network Genetic algorithm Twin turbulent sweeping impinging jets
In this study, a numerical investigation is carried out to reveal the potential of obtaining uniform cooling on a planar surface by impinging jets in a turbulent flow regime using the SST k ω turbulence model. Twin turbulent sweeping impinging jets (TTSIJ), as an idealization of an array of jets, are considered for heat transfer with the planar target surface. Unlike other studies that focused on a few effective parameters or limited the impinging jets to certain conditions, in this study, a more general investigation is performed to clarify the effects of all influential parameters on heat transfer. To this end, the effects of 8 design variables namely Reynolds number, the jet-to-target distance, the phase shift between the two jets, the frequency of pulsations of jets, the frequency of sweeping motion, the jet-to-jet separation distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angle of the nozzles on Nusselt distribution on the target surface are evaluated. For the purpose of elimination of thermal stress on the target surface, it is attempted to optimize the design variables to reach a uniformly distributed Nusselt number on the target surface for 9 design uniform Nusselt distribution. The optimization of the design variables is performed using artificial neural network (ANN) combined with genetic algorithm (GA) to minimize the discrepancy between the ideally uniform and optimized Nusselt distribution. Results show that within the assumed bounds of the design variables, the applied method is successful in obtaining uniform cooling with the accuracy of more than 98% and 93% in the best and worst situation, respectively.
1. Introduction Impinging jets have been used to improve heat transfer rate in many engineering applications. They could be used to increase either the cooling or heating rate of the impingement surface by disturbing the boundary layer. Based on the impinging jets configurations and the flow domain, authors have investigated the effects of selected parameters such as Reynolds number [1–3], jet arrangements, multiple impinging jets [4], jet-to-target distance and nozzle shapes [5] on heat transfer characteristics. Afroz and Sharif [6], determined the effects of the impingement surface slope on local Nusselt distribution by utilizing twin impinging slot jets in turbulent flow regime. They showed that the location of the maximum Nusselt number corresponded to the jets entrance angle. To reveal the outcome of Reynolds number and the frequency of oscilla tions, Demircan and Turkoglu [7] numerically analyzed a uni-direction pulsatile oscillating impinging jet. They showed that transient jets enforced higher heat transfer rate compared to steady state flows. By
applying a phase shift to a couple of oscillating jets with the same fre quency that were fixed in a given position, Hewakandamby [8] studied the effects of the separation distance between the two jets, the frequency of oscillations and Reynolds number. It was demonstrated that the pe riodic disturbance in the boundary layer lead to improvement of heat transfer over conventional steady flow jets. Xu et al. [9] explored the effects of unsteady intermittent pulsations on heat transfer of multiple impinging jets. The intermittent pulsations together with uniform ve locity, same frequency and different phase angles were applied for adjacent jets. They claimed that the transient flow could enhance the local Nusselt number along the impingement wall. Park et al. [10] experimentally investigated the heat transfer of a sweeping impinging jet, undergoing periodic oscillations for several Reynolds numbers and nozzle-to-plate spacing. They showed that near the center of the sweeping jet, a high Nusselt number region was formed and away from this region, the Nusselt number monotonically decreased. The flow dy namics of a sweeping jet was experimentally investigated using time-resolved particle image velocimetry by Wen and Liu [11]. They
* Corresponding author. E-mail address: [email protected] (H. Afshin). 1 These authors contributed equally to this work. https://doi.org/10.1016/j.ijthermalsci.2019.106064 Received 23 April 2019; Received in revised form 2 August 2019; Accepted 26 August 2019 Available online 4 September 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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International Journal of Thermal Sciences 146 (2019) 106064
Nomenclature
Gω h H I k L Lplate LSTD n Nu Nu N P p0 Pr q"
specific heat capacity hydraulic diameters of the nozzles cross-diffusion term frequency of pulsations frequency of oscillations generation of turbulence kinetic energy generation of ω heat transfer coefficient jet-to-target distance Identity matrix turbulence kinetic energy jet-to-jet separation distance length of the plate local standard deviation surface normal vector Nusselt number averaged Nusselt number number of points along the target surface pressure outside pressure Prandtl number ν =α surface heat flux
Sk Sω
source terms of k source terms of ω
cp Dh Dω fR fs ~k G
Re
Reynolds number Re ¼
STD t T T∞ T’ ui ðu ; vÞ u’i Uinlet w wTarget xi ðx; yÞ Yk Yω
standard deviation time temperature inlet temperature fluctuating temperature velocity (velocity components) fluctuating velocity inlet velocity of the flow slot width width of the target surface direction coordinate (x coordinate, y coordinate) dissipation of k dissipation of ω
Greek symbols α thermal diffusivity β maximum of swinging angle Γk effective diffusivity of K Γω effective diffusivity of QUOTE ω λ dimensionless jet to target distance μ dynamics viscosity ν kinematic viscosity ω specific rate of dissipation ρ fluid density Φ phase shift between the two jets τ period of swinging motion ζ thermal conductivity
2ρU∞ w
μ
concluded that at higher Reynolds numbers, the larger bending angle of the jet induced higher fluctuations. According to the above discussion, it could be concluded that impinging jets could improve heat transfer rate on the impinging sur face, but they could also generate thermal stress. In order to eliminate this issue, achieving uniform heat flux possess a significant challenge and importance. In order to obtain uniform heat flux on the target sur face, geometric parameters and flow properties must be designed appropriately. For example, Forouzanmehr et al. [12], adopted four steady laminar slot jets in order to obtain uniform cooling over an isothermal plate by optimizing the volumetric flow rate, the jet sepa ration distance, the slot widths, and jet-to-plate distance. Using twin slot
impinging jets, Farahani et al. [13] performed both numerical and experimental study to optimize the design variables (nozzle widths, jet-to-jet distance, jet-to-plate distance, the frequency of pulsating jets, and the flow rate) to achieve uniform cooling in laminar flow regime. Bijarchi and Kowsary [14] presented a laminar and steady co-axial jet to reach uniform Nusselt distribution along an isothermal surface. They concluded that the ratio of nozzle width to target length, had to be set at least equal to 0.6 to accomplish uniformity. Bijarchi et al. [15] applied a laminar swinging slot impinging jet (SSIJ) to obtain uniform cooling along a constantly-heated plate. They investigated 15 cases (5 Nusselt numbers and 3 nozzle to target surface ratios) and optimized Reynolds number, the jet-to-target distance, the frequency of swinging motion and
Fig. 1. The geometry configuration. 2
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International Journal of Thermal Sciences 146 (2019) 106064
of the plate, Lplate , is assumed to be 3wTarget to avoid the outlet effects on heat transfer. The target surface is heated by a heater with constant heat flux of q”.
Table 1The constant parameters used in the current study. Property
Symbol
Value
kg density 3 m
ρ
1.1614
kJ kgK thermal conductivity W =mK
cp
1007
ζ
0.0263
dynamics viscosity Pa:s
μ
1.84e-5
q”
1100
T∞
300
specific heat capacity
W surface heat flux m inlet temperature K
2.1. Governing equations and boundary conditions The working fluid is considered to be air with constant thermo physical properties and together with other constant parameters are given in Table 1. The continuity, Reynolds averaged mass, momentum and energy equations are given as:
the maximum of swinging angel to achieve uniform cooling on a planar surface. They stated that the SSIJ was successful in acquiring uniform cooling for Nusselt numbers below 20. Although there have been attempts to obtain uniform heat transfer by applying laminar impinging jets, achieving uniformity in turbulent flow regime, regarding their higher cooling potential, has drawn very little interest. As stated previously, so far authors limited their focus on some selected parameters that could influence heat transfer. In this research, however, a more general investigation is carried out to study the potential of uniform cooling using turbulent impinging jets. To this aim, twin turbulent sweeping impinging jets (TTSIJ) are considered as an idealization of an array of jets. The advantages of this research is due to three reasons: (1) exploring the possibility of attaining uniform cooling in turbulent flow regime (2) performing a comprehensive study on all the influential parameters that could affect heat transfer charac teristics (3) taking simultaneous advantage of the positive effects of the frequency of pulsations and the frequency of sweeping motions of the jets (unlike other studies that focused on only one of these factors). Regarding the 2nd point, the effects of the design variables Reynolds number, jet-to-target distance, the phase shift between the two jets, the frequency of pulsations of jets, the frequency of sweeping motion, the jet-to-jet separation distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angle of the nozzles, are evaluated. These parameters, based on their natures, were divided into geometric group and non-geometric group. The jet-to-jet separation distance, the jet-totarget distance and the hydraulic diameter of the nozzles are classified into the geometric group and Reynolds number, the phase shift between the two jets, the frequency of pulsations of the jets, the frequency of swinging motion and the maximum of sweeping angle of the nozzles are assigned to the non-geometric group. At first, the governing equations and the boundary conditions are presented. Secondly, parametric studies are performed to determine the effects of each of the design variable on Nusselt distribution. Finally, Artificial Neural Network (ANN) combined with Genetic Algorithm (GA) are used to obtain uniform cooling along the target surface.
∂ui ¼0 ∂xi
(1) � �
ρ
∂ui ∂ui þ ρuj ¼ ∂t ∂xj
∂P ∂ ∂ui ∂uj þ μ þ ∂xi ∂xj ∂xj ∂xi
ρ
∂T ∂T ∂ μ ∂T þ ρu j ¼ ∂xj Pr ∂xj xj ∂t
�
�
�
ρu’i u’j
(2)
� (3)
ρT’ u’j
where u is the velocity component in x and y directions (ui and uj ), x is the direction coordinate (xi and xj ), ρ is the fluid density, P is the pressure, μ is the dynamic viscosity, T is the temperature, Pr is Prandtl number, t is the time and finally T’ and u’ ðu’i and u’j Þ are the fluctuating
temperature and velocity components, respectively. The SST-kω model is employed [6] to study heat transfer with the following transport equations: � � ∂ ∂ ∂ ∂k ~ k Yk þ Sk ðρkui Þ ¼ (4) Γk þG ðρkÞ þ ∂t ∂xi ∂xj ∂xj �
�
∂ ∂ ∂ ∂ω Γω þ Gω ðρkui Þ ¼ ðρωÞ þ ∂t ∂xi ∂xj ∂xj
Yω þ Dω þ Sω
(5)
~ k represents the generation of turbulence kinetic energy, Gω represents G the generation of ω, Γk and Γω refer to effective diffusivity of k and ω, Yk and Yω represent the dissipation of k and ω, Dω refers to the crossdiffusion term, and finally Sk and Sω are source terms. No slip condition is assumed at the upper and lower surfaces: u ¼ 0jAD;
(6)
BC
The jets’ velocity in x and y directions are given by: � � � φ�� � 1 u1x ¼ Uinlet 1 þ sinð2πf 1R tÞ � sinβsin 2πf s t þ 2 2
2. Problem formulation As mentioned, the aim of this study is to investigate the heat transfer characteristics of TTSIJ and then to impose uniform cooling on the target surface by optimizing the design variables. Fig. 1 shows the schematic diagram of TTSIJ configurations and the flow domain which is confined between two parallel plates. The jet-to-jet separation distance, the jet-totarget distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angel, are denoted by L, H, Dh and β, respectively. Two modes of oscillations are considered for TTSIJ, a sweeping motion with steady periodic angular motion similar to a pendulum motion referred to as “the frequency of sweeping motion” denoted by fs, and steady period pulsating motion, resulting in fluctuations in the jets’ volumetric flow, referred to as “the frequency of pulsations” and denoted by fR. Reynolds number and fs are set equal for the two jets to simplify the experimental applications setups, and the phase shift between the jets, φ, is set to remain constant at all times. The ratio of the hydraulic diameter of the nozzles to the target width (Dh =wTarget ) is 0.1 for each jet and the length
(7)
ffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffi��� ffiffiffiffiffiffiffiffiffiffi �rffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� � 2 1 φ u1y ¼ Uinlet 1 þ sinð2πf 1R tÞ 1 sinβ sin 2πf s t þ 2 2
(8)
� � � � 1 u2x ¼ Uinlet 1 þ sinð2πf 2R tÞ � sinβsin 2πf s t 2
(9)
φ�� 2
ffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffi��� ffiffiffiffiffiffiffiffiffiffi �rffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� � 2 1 φ u2y ¼ Uinlet 1 þ sinð2πf 2R tÞ (10) 1 sinβ sin 2πf s t 2 2 � � In the above sets of equations, the terms Uinlet 1 þ12 sinð2πf 2R tÞ and � � Uinlet 1 þ12 sinð2π f 1R tÞ correspond to the fluctuations in the jets’ volu � � �� � � �� metric flow and the terms sinβsin 2πf s t þφ2 and sinβsin 2πf s t φ2
reflect the sweeping motion of the jets. Therefore the velocity magnitude � � � of the jets would be Uinlet 1 þ12 sinð2πf 1R tÞ and Uinlet 1 þ � 1 sinð2πf 2R tÞ . AB and CD are open to atmosphere resulting to the 2
3
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 2. The results of mesh independency analysis.
following equation: � �� pI þ μ ru þ ruT :n ¼
p0 n
Table 2 The design variables for mesh analysis.
(11)
Constant heat flux is applied to the target surface (EF) and is expressed in the form of: n:
� q” αf rT ¼ ρcp
(12)
(13)
By obtaining temperature and velocity distribution over the region, the local Nusselt number on the target surface based on the slots width and heat transfer coefficient is computed by the following equation: hðTi
Ts Þ ¼
Nu ¼
hð2wÞ ζ
ζ
∂T ∂y
(14) (15)
As the Nusselt distribution varies with time and position due to the unsteady nature of TTSIJ, a time-averaged Nusselt number within a cycle is defined to determine the averaged heat transfer on the target surface, Z tþτ 1 NuðxÞ ¼ Nuðx; tÞdt (16)
τ
λ
φ
fs
f1R
f2R
L
Dh
β
6000
0.6
π =4
5.5
5.5
5.5
0.45
0.15
π =8
“1”, “2” and “3”, respectively). As it is intended to study the heat transfer with the target surface and then to perform the optimization process to reach uniform cooling on the target surface, near the target regions demand precise investigation and require sufficiently accurate CFD computation and relatively the finest grid sizes. Also, as the importance of entrance regions where the air flows into the domain, the grid sizes are sufficiently small in these areas. On the other hand, the grid sizes far enough from these regions could be relatively coarse. The discussed grid structure was implemented to the domain by assigning the finest grid sizes to near the target regions, relatively fine grid sizes to entrance zones, and coarse grid sizes in areas other than the mentioned areas. Therefore, the SST k ω turbulence model is compatible with the dis cussed mesh structure for its accuracy in both viscous sub-layer and inviscid regions, and was adopted. The density of the mesh structure is adjusted to increase mono tonically towards the target surface because of higher velocity and temperature gradients to ensure yþ�1 that is necessary for the SST k ω model. Fig. 2 shows the Nusselt distribution for the three meshes. The relative error between “2” and “1” is less than 2%, and “2” is chosen for the domain. Table 2 shows the values of the design variables considered for mesh analysis. The jet-to-target distance is non-dimensionalized with H respect to the width of the target surface and is denoted by λ ¼ wTarget .
Finally, adiabatic surfaces are assumed at AD, BF and CE: n:ðrTÞ ¼ 0
Re
t
where τ is the period of sweeping motion (τ ¼ f1s ). It should be noticed that calculations are performed after a few cycles of sweeping to elim inate initial-condition-dependent results.
3. Verification To check the validity of the present study, the results by Xu et al. [9] are reproduced where the effects of unsteady intermittent pulsations of multiple impinging jets on heat and mass transfer were explored. Computational fluid dynamic approach is employed to obtain local heat transfer rate of twin intermittent pulsating jets and effects of Reynolds number and the frequency of pulsations are investigated. For the pur pose of validation, in the case of H/w ¼ 5 (H denotes the nozzle-to-plate-distance and w represents the slot width as in Ref. [9]), Re ¼ 5500, f ¼ 41 Hz and velocity amplitude of A ¼ 17%, the time-averaged local Nusselt number along the impingement wall was
2.2. Numerical method & mesh analysis SIMPLE algorithm is employed for the pressure-velocity coupling and the discretization of momentum and energy equations are based on second-order upwind method. Pressure values at faces are interpolated using momentum equation coefficients [16] and least square cell-based gradient is utilized so that the solution could vary linearly. Grid-independency of the results is checked under three structured grid densities by analysis of Nusselt distribution on the target surface with the number of elements 246 � 48, 180 � 40 and 141 � 32 (labeled as 4
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 3. The comparison of Nusselt distribution of the present work with those of [9,17].
Fig. 4. The classification of the design variables into 4 groups.
compared to experimental results [17]. In the present study, the same case is modeled and the result is depicted in Fig. 3. Compared to the experimental and numerical studies [9,17], the relative error is calcu lated to be 1% and 1.5%, respectively, which confirms the reliability of our numerical scheme. The next step is to investigate the effects of all the design variables and then the optimization of these variables to achieve uniform cooling on the target surface. It should be noticed that from here on, STD, stands for the standard deviation of Nusselt distribution with respect to the averaged Nusselt distribution:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u �2 N � u1 X Nui Nui STD ¼ t � 100 N i¼1 Nui
(17)
The 8 design variables are classified into 4 groups according to Fig. 4; Group1 contains those variables which increase heat transfer with the target surface while increasing, Group2 contains those which decrease heat transfer rate with increase in their values and Group3 contains those variables that could both increase and decrease heat transfer rate when their values change within the assumed bounds. Finally, Group 4 5
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 5. The effects of Reynolds number on Nusselt distribution on the target surface.
is defined based on the inlet velocity of TTSIJ and the slots width, Re ¼
Table 3 The design variables for parametric studying of Reynolds number.
2ρU∞ w
μ
λ
Φ
fs
f1R
f2R
L
Dh
β
0.6
π =4
5.5
5.5
5.5
0.45
0.15
5π =24
. It could be seen in Fig. 5 that for fixed values of other parameters
according to Table 3, increase in Reynolds number from 5400 to 6600 leads to the growth of averaged Nusselt number from 41.1 to 46.2 on the target surface. In other words, as the fluid momentum enhances, heat transfer coefficient raises mutually that results in increase in Nusselt number, thus, to achieve higher heat transfer rate, larger Reynolds number would be required. Also, as the jets impinge on the target sur face and turn upward after collision, and creates vortex in the domain, heat transfer would be affected. Without any phase shift, the jets sweeping motion to the left and right sides are similar to each other but with a phase shift angle, the jets oscillations produce periodic vortex generations that affect the heat transfer near the target region. It must be noticed, however, that the phase shift between TTSIJ, could shift the maximum of heat transfer from central regions to side regions. Ac π , the maximum Nusselt cording to Fig. 6, at the phase shift angle of 12 number is about 47 and the averaged Nusselt number is 43.0 with the 5π STD of 2.6 and at the phase shift angle of 12 the maximum Nusselt
includes those variables which any alteration in their values within the assumed bounds do not affect heat transfer noticeably. Regarding this, Reynolds number, the phase shift between TTSIJ, and the hydraulic diameter of the nozzles belong to Group1, the jet-to-target distance and the maximum sweeping angle belong to Group2, the frequency of pul sations and the frequency of oscillations situate in Group3 and finally the jet-to-jet separation distance would not affect heat transfer rate significantly and would go to Group 4. 3.1. Group1 At first, effects of Reynolds number is investigated. Reynolds number
Fig. 6. The effects of phase shift between the two jets on Nusselt distribution on the target surface. 6
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International Journal of Thermal Sciences 146 (2019) 106064
surface. For example, in the case of D/wTarget ¼ 0.18 the averaged Nusselt number is about 45.8, that demonstrates 16.8% increase in heat transfer coefficient compared to D/wTarget ¼ 0.12 which is 39.2 on average. For a specific Reynolds number, increase in the hydraulic diameter of the nozzles deduces to higher flowrate of air to the domain and any point on the target surface receives more cooling and overally, the whole target surface achieves higher cooling rate. It should be noticed, however, that depending on the values of other parameters, the STD of heat transfer on the target surface might also increase which is not desirable due to the purpose of attaining uniform cooling. Thus, depending on the target cooling rate, the hydraulic diameter of the
Table 4 The design variables for parametric studying of phase shift between TTSIJ. Re
λ
fs
f1R
f2R
L
Dh
β
6000
0.6
5.5
5.5
5.5
0.45
0.15
5π =24
number is 52, however, the Nusselt number is decreased to the minimum of 38 in the opposite side, resulting in the averaged value of 44 and STD of 4.8 on the target surface. Therefore, phase shift between TTSIJ could change the maximum and STD of the Nusselt distribution. The values of other design variables are specified according to Table 4. The last var iable in Group1 is the hydraulic diameter of the nozzles. The hydraulic diameter of the nozzles is proportional to the air flowrate to the domain. The larger hydraulic diameter of the nozzles results in the entrance of more volumes of air at any moment to the domain and impacts larger segments of the target surface. As could be seen in Fig. 7, increase in the hydraulic diameter of the nozzles, results in significant increment of heat transfer on the target
Table 5 The design variables for parametric studying of hydraulic diameter of the nozzles. Re
λ
Φ
fs
f1R
f2R
L
β
6000
0.6
π =4
5.5
5.5
5.5
0.45
5π =24
Fig. 7. The effects of non-dimensional hydraulic diameter of the nozzles on Nusselt distribution on the target surface.
Fig. 8. The effects of non-dimensional jet-to-target distance on Nusselt distribution on the target surface. 7
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relative error with respect to perfect uniformity under this circumstance could not be acceptable. Thus the idea of attributing a wider range of angles was implemented thereafter. As a result, in this study, the authors
Table 6 The design variables for parametric studying of dimensionless jet-to-target distance. Re
Φ
6000
1 wTarget L=2 Dh =2 to H 1 wTarget L=2 Dh =2 tan H
did not limit the maximum of sweeping angle to tan
fs
f1R
f2R
L
Dh
β
5.5
5.5
5.5
0.45
0.15
5π =24
obtain more uniformity on the target surface where shows the angle that the jets would meet the target ends assuming to move in a straight path. Considering other parameters according to Table 7, effects of maximum of sweeping angle on Nusselt distribution on the target surface is depicted in Fig. 9. For the considered values of jet-to-target distance, the hydraulic diameter of nozzles, and jet-to-jet separation distance, the angle between the mid-point of the nozzles and the mid-point of the target surface is π =8. For the maximum sweeping angle less than π =8, each impinging jet mostly affect the un derneath segments of the target surface and the central area lose effec tive cooling. This could be seen as the two peaks of Nusselt distribution in the side regions and a minimal of Nusselt number in the central re gions in the case of β ¼ π=12. Beyond β ¼ π=8 the jets mingle near the target surface and increase Nusselt number in central regions and reduce the difference between the maximum and minimum of Nusselt numbers as in the case of β ¼ 2π=12. However, considering a fixed value of the frequency of sweeping, further increase in maximum angle of sweeping results in faster sweeping of the TTSIJ, and the adequate time of heat transfer would not be attained. In other words, for a fixed value of fs, TTSIJ have to sweep a longer distance for higher β and therefore the heat transfer exchange with any point decreases. On the other side, all points on the target surface receive more uniform cooling and the STD of Nusselt distribution decreases as the case of β ¼ 3π=12. Finally, further increase of maximum angle of sweeping, noticeably decreases heat transfer time with each point on the target surface and the effective cooling decreases.
Table 7 The design variables for parametric studying of the maximum sweeping angle. Re
λ
Φ
fs
f1R
f2R
L
Dh
6000
0.6
π =4
5.5
5.5
5.5
0.45
0.15
nozzles should be designed suitably. The values of other design variables could be found in Table 5. 3.2. Group2 As introduced, the jet-to-target distance and the maximum of sweeping angle serves as important parameters that could influence heat transfer characteristics and require appropriate considerations. As the intensity of collision of TTSIJ with the target surface would absolutely influence heat transfer rate, the jet-to-target distance was regarded as the second variable in the “geometric variables”, which would decrease the collision intensity and the consequent magnitude of Nusselt number for larger distances and results in higher collision intensity and conse quently higher Nusselt number for lower distances. As depicted in Fig. 8, for fixed values of other parameters (see Table 6), as the dimensionless jet-to-target distance change from 0.48 to 0.72, the maximum Nusselt number changes from about 51 to about 46 (9.8% reduction) and makes the Nusselt distribution possess a flatter profile. Furthermore, the loca tion of maximum Nusselt number alters as the jet-to-target distance gets different values. Therefore, the jet-to-target distance alters the maximum Nusselt number and changes the location of maximum heat transfer. Finally, the idea of attributing different maximum angles of sweeping was presented in Ref. [15]. In this paper, at first, it was tried to reach uniform Nusselt distribution by setting the maximum of swinging w w angle equal to tan 1 Target 2H , where w was the nozzle width and wtarget was the target surface length and H was the jet-to-target distance. w w tan 1 Target referred to the angle that the jet would collide with the 2H target ends assuming to move in a straight path. It was found that the
3.3. Group3 In order to investigate the effects of the frequency of pulsations on Nusselt distribution f 1R is varied from 2 to 8 Hz, and f 2R is set equal to 0. Also, fs is assigned a small number, to provide the oscillations only due to pulsations in the inlet flow. The assigned values of other parameters are according to Table 8. According to Fig. 10, the right sides of the target surface receive more cooling of about 11–27% due to the pul sating motion of the overhead impinging jet than the left sides where the frequency of pulsations of the overhead impinging jet is zero, thus the right hand sides experience higher Nusselt numbers. As the frequency of
Fig. 9. The effects of the maximum sweeping angle on Nusselt distribution on the target surface. 8
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International Journal of Thermal Sciences 146 (2019) 106064
STD of the Nusselt distribution from 2.04 to 0.69 by eliminating the peak in Nusselt distribution, and providing more uniform cooling on the target surface. The reduction of maximum Nusselt distribution at midpoints on the target surface is due to lower interaction between the flow and the target surface caused by faster sweeping motion of the jets and less time for heat transfer. However, the side segments experience more cooling because of the faster disturbance of boundary layer. Therefore, in spite of decreasing the maximum Nusselt number from about 46 to 43.5, increase in the frequency of oscillations has a positive impact on uniformity. The averaged Nusselt distribution and the
Table 8 The design variables for parametric studying of the frequency of pulsations. Re
λ
Φ
fs
f2R
L
Dh
β
6000
0.6
0
0.1
0
0.45
0.15
5π =24
pulsations of the right impinging jet increases, the mixing of the un derneath fluid enhances, which absorbs more heat from the target sur face, and reaches higher temperature. Later, by mixing with the left impinging jet, decreases the cooling rate of the left sides of the target surface. According to Fig. 11 and considering the other variables ac cording to Table 9, as fs increases from 0.5 to 2.5, both the maximum and averaged Nusselt distribution increases due to higher fluctuations of the flow over the target surface. In other words, increase in frequency of pulsations, causes higher rates of boundary layer disturbance and en hances the mixing of flow near the target surface which raises the heat transfer rate. On the other side, increasing fs beyond 4 to 7, reduces the
Table 9 The design variables for parametric studying of the frequency of oscillations. Re
λ
Φ
f1R
f2R
L
Dh
β
6000
0.6
0
0.1
0
0.45
0.15
5π =24
Fig. 10. The effects of frequency of pulsations on Nusselt distribution on the target surface.
Fig. 11. The effects of phase shift between the two jets on Nusselt distribution on the target surface. 9
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 12. The effects of non-dimensional jet-to-jet separation on Nusselt distribution on the target surface.
Nusselt distribution at each τ/8 interval. The peaks of the Nusselt dis tribution can be found where the temperature gradient has the maximum values.
Table 10 The design variables for parametric studying of jet-to-jet separation distance. Re
λ
Φ
fs
f1R
f2R
L
Dh
β
6000
0.6
0
5.5
5.5
5.5
0.45
0.15
5π =24
4. Optimization In previous sections, parametric study on each of the design variables were performed to reveal their impacts on Nusselt distribution. It was declared that increase in Reynolds number and the jet-to-target distance have opposite effects on heat transfer rate. On the other hand, the phase shift between the two jets changes the magnitude and location of maximum heat transfer. The frequency of pulsations proved to impose higher heat transfer rate than steady jets. The maximum angle of sweeping showed to have a threshold beyond which the uniform cooling would diminish. Also, alteration in jet-to-jet separation distance showed not to affect the maximum of heat transfer, but could change its position. Finally, it was noticed that increase in the hydraulic diameter of the nozzles could significantly enhance heat transfer rate on the target surface. So far, it was indicated that each of the design variables could either enhance or diminish heat transfer rate and could even change the dis tribution of Nusselt number on the target surface. However, in order to attain uniform cooling, that is the main purpose of this paper, the design variables should be optimized to enable reaching a uniform design Nusselt distribution on the target surface. Therefore, 9 design Nusselt distributions with equal intervals namely 25, 30, 35, 40, 45, 50, 55, 60 and 65 are considered on the target surface and the 8 design variables are to be specified to attain the design distributions. In order to optimize these parameters, artificial neural network (ANN) combined with ge netic algorithm are employed. An Artificial neural network is an interconnected group of nodes that could be applied to model a computational structured framework where the complex relations between the inputs and outputs reveal as a func tion. To minimize the discrepancy between the ANN outputs and the real values for a specific task, the ANN should be trained with adequate number of data, firstly. To this aim, the 8 aforementioned design vari ables and the discrepancy between the obtained Nusselt distribution and the design Nusselt distribution are fed into the network for training. The discrepancy between the two sets of Nusselt distributions are defined as follow:
maximum Nusselt value decrease further by elevating the pace of oscillation in the case of fs ¼ 8.5, because any point on the target surface would have less time for heat transfer with the jets, but the uniformity is enhanced. Finally, for the frequencies between 2.5 and 4, the positive effects of faster disturbance of the boundary layer nullifies the negative effects of lower interaction between the flow and the target, and the Nusselt distribution remains nearly constant. 3.4. Group 4 According to Fig. 12, the averaged Nusselt number on the target surface does not change remarkably with changes in the jet-to-jet sep aration distance, and the changes could be seen only in the location of maximum Nusselt number on the target surface. Other design variables are according to Table 10. In order to better depict the contribution of the jets to heat transfer with the target surface streamlines, temperature contours and Nusselt distributions are presented at every τ/8 interval within a cycle. Highvelocity gradients near the surfaces due to no-slip conditions at the surfaces, lead to the exertion of shear stress between the layers of the fluid and together with the upward flow after the collision with the target surface, creates vortexes that results in better mixing of the flow and would influence the flow behavior and increases heat transfer with the target surface. As could be seen in Fig. 13, the location of the vor texes changes continuously as the sweeping angle of the jets vary in time. The continuous shift of the vortexes’ locations, affects all segments of the target surface and therefore higher Nusselt number is expected along all points on the target surface compared to laminar flow. Ac cording to the temperature contours, it could be concluded that the collision of the jets with any point on the target surface, enhances the cooling rate of the point due to high momentum of the flow, therefore, for all the temperature contours, lower temperature is expected at the collision points and as moving away from the colliding segments, the temperature would increase as the result of constant heat flux applied to the target. Furthermore, the right column of the figure, shows the 10
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 13. The streamlines and temperature contours in every τ/8 interval within a cycle.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u �2 N � u1 X NuD Nui LSTD ¼ t � 100 N i¼1 NuD
with respect to the real values. Therefore, the criterion considered for judging the validity of the outputs of the ANN, is that the test values need to reach higher than 99%. As demonstrated in Fig. 14, the ANN proves to present reliable results. Subsequently, the 9 output evaluated functions of the ANN, are regarded as the objective functions of the optimization process. For example, in the case of NuD ¼ 40, the LSTD corresponding to the design variables are fed into the network, and the evaluated function is then regarded as the objective function of the
(18)
Consequently, the ANN is trained 9 times to disclose the relative function between the design variables and the LSTD for each of the design Nusselt distributions. In ANN, the test values determine the performance of the network which estimate the accuracy of the model 11
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International Journal of Thermal Sciences 146 (2019) 106064
Fig. 13. (continued).
generation. Finally, over successive generations, the populations evolve towards the optimal solution. Fig. 15 depicts the best and the mean fitness variations with respect to the number of generations in the case of NuD ¼ 40. The best fitness value denotes the best individual found in the population while the mean fitness value simply refers to the average of fitness values across the entire population. It could be seen that around the 60th generation, these two values approach to each other until they become exactly equal, which means that the algorithm has reached the best fitness and the optimal state is achieved. Therefore, it could be concluded that the LSTD function value has a relative error of 1.3% with respect to perfectly uniform Nusselt distribution on the target surface. The relevant optimization steps is summarized in Fig. 16: One of the main advantages of the above procedure is the short run time required for optimization of any other design Nusselt numbers, since the objective function of the GA reveals apparent by ANN. Therefore, the trained ANN could be used to disclose the objective function of any other design Nusselt numbers only if the discrepancy between the obtained Nusselt distribution and the design Nusselt dis tribution are fed into the network for training. The optimization process is performed 9 times to obtain the optimal design variables for the 9 considered design Nusselt distributions that would minimize the LSTD function within the upper and lower bounds of the design variables. Table 12 represents the optimal values of the design variables for each design uniform Nusselt distribution on the target surface and the value of STD function that refers to the discrep ancy between the optimized and perfectly uniform Nusselt distribution. The optimized Nusselt distribution of all 9 NuD are illustrated in Fig. 17. It should be noticed that a perfectly uniform Nusselt distribution occur when all “Nui”s in the right hand side of equation (16) equal to NuD. Although it is expected to be unrealistic to obtain a perfectly uni form Nusselt distribution on the target surface, it is possible to approach a uniform distribution with acceptable accuracy. According to Table 12, it could be concluded that in all cases, TTSIJ could achieve uniform cooling with the relative error of less than 6.5%, but as the design Nusselt distribution decreases, the potential of TTSIJ for obtaining uniform cooling increases from 6.5% in the case of NuD ¼ 65 to 1.2% in the case of NuD ¼ 35. In order to assess the maximum uniform cooling potential of TTSIJ, the design Nusselt number was further increased to NuD ¼ 70, but the error increased to about 10%, therefore, the ability of the TTSIJ to obtain uniform cooling is limited to around NuD ¼ 65. In this case, the local STD of the optimized and design uniform Nusselt distributions increased to 6.5%, therefore, it could be deduced than within the considered bounds for the 8 design variables, the potential of obtaining uniform cooling decreases beyond NuD ¼ 65 and it is required to extend the bounds of these variables to reach higher cooling
Fig. 14. The test results of ANN. R ¼ 0.99914 demonstrates the accuracy of ANN outputs with respect to the real values. Table 11 The upper and lower bounds of the design variables. Variable
Re
λ
Φ
fs
f1R
f2R
L
Dh
β
Lower bound Upper bound
5000 7000
0.4 0.8
0
π =2
1 10
1 10
1 10
0.3 0.6
0.1 0.2
0 5π 12 =
genetic algorithm. For the purpose of optimization, the upper and lower bounds of the design variables are assigned according to Table 11. Genetic algorithm begins with a set of individuals called population that are characterized by a set of parameters (here the 8 design vari ables). The parameters are then joined into a string to form the solution. In other words, the genetic algorithm repeatedly modifies a population of individual solutions. At the beginning of each step, individuals are selected from the population randomly and then produce the next gen erations by crossover rules. Meanwhile, mutation rules are applied to enable the involvement of all population in producing the next 12
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Fig. 15. The best and mean fitness values of the genetic algorithm.
Fig. 16. The optimization process.
capacities. For the purpose of using the optimization process for geometricconstraint cases, the optimal values of the design variables are calcu lated through the same procedure considering different jet-to-target distances or in other words, different channel heights. Jet-to-target distance is selected since other parameters could be manipulated simpler to achieve their optimal values, but probably channel height would have some space restrictions. Therefore, the optimization process is performed for λ ¼ 0:4; 0:5; 0:6; 0:7 and 0:8 and one could choose the
optimal values according to the existing condition. The optimal values of the other design variables and the corresponding error of the Nusselt distribution compared to NuD ¼ 40 is presented in Table 13. 5. Conclusion In this paper, a comprehensive study of heat transfer characteristics of twin turbulent impinging jets (TTSIJ) was performed numerically to reveal the effects of Reynolds number, the jet-to-target distance, the 13
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Table 12 The optimal values of the design variables for each design uniform Nusselt distribution. Re
λ
Φ
fs
f1R
f2R
L
Dh
β
Error %
NuD ¼ 25
7000
0.4
0
1
1
10
0.6
0.1
5π =12
3.9
NuD ¼ 30
7000
0.8
0
1
10
1
0.3
0.2
5π =12
2.4
NuD ¼ 35
7000
0.8
0
1
10
10
0.6
0.2
5π =12
1.2
NuD ¼ 40
6958
0.73
π =6:3
7.1
4.8
6.7
0.3
0.1
π =4:8
1.3
NuD ¼ 45
5559
0.73
π =2.4
2.7
9.8
3.3
0.5
0.2
π =8
1.8
NuD ¼ 50
6480
0.79
π =15
4.2
2.7
7.4
0.4
0.2
π =235
2.1
NuD ¼ 55
5804
0.71
0
7.7
1.6
1.3
0.4
0.2
π =13:7
3.4
NuD ¼ 60
6926
0.68
π =3:9
6.6
1.4
8.6
0.5
0.2
π =12
3.7
NuD ¼ 65
6911
0.45
π =10
3.9
1.1
4.9
0.3
0.2
π =12:8
6.5
Fig. 17. The optimized Nusselt distribution for a) NuD ¼ 25, 30, 35, 40, 45 b) NuD ¼ 50, 55, 60 and 65.
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A. Eghtesad et al.
International Journal of Thermal Sciences 146 (2019) 106064
Table 13 The optimal values of the design variables for different values of jet to target distance. Re
λ
Φ
fs
f1R
f2R
L
Dh
β
Error %
6961
0.80
π =78
7.18
2.81
2.75
0.46
0.14
π =4:76
2.49
6958
0.73
π =5:8
7.07
4.81
6.75
0.35
0.13
π =4:8
1.37
6807
0.70
π =7:5
3.32
6.56
9.65
0.6
0.14
π =5:5
2.14
6956
0.60
π =18
7.20
2.85
6.77
0.46
0.13
π =0:5
1.40
5066
0.50
π =2:3
6.70
10.00
2.86
0.53
0.19
π =8:5
2.24
6845
0.40
0
1.00
1.00
9.00
0.60
0.20
π =2:4
2.95
phase shift between the two jets, the frequency of pulsations of each jet, the frequency of sweeping motion, the jet-to-jet separation distance, the hydraulic diameter of the nozzles, and the maximum of sweeping angle of the nozzles and then optimizations of this variables were performed to obtain uniform Nusselt distribution on the target surface. Results showed that, as the Reynolds number increases, the fluid momentum enhances which raises the heat transfer coefficient on the target surface. Also, the phase shift between the two jets produce periodic vortex generations that augments the heat transfer on the target surface and could shift the maximum of heat transfer from central regions to side regions. For fixed values of the other variables, increase in the hydraulic diameter of the nozzles deduces to higher heat transfer rate due to applying higher flowrate of air to the domain that results in more cooling. In opposite to the positive effects of the three above-mentioned pa rameters on heat transfer, as the jet-to-target distance increases, the intensity of collision of TTSIJ with the target surface decreases and consequently reduces the maximum Nusselt number. Furthermore, the location of maximum Nusselt number alters and the Nusselt distribution profile gets flatter. Increasing the maximum angle of sweeping results in faster sweeping motion of the TTSIJ, and the adequate time for heat transfer would not be attained. On the other side, all points on the target surface receive more uniform cooling because the jets do not focus on a small particular area. Applying frequency of oscillation showed to impose higher heat transfer rate by disturbing the boundary layer over the impingement region. As f s increases to 2.5, both the maximum and averaged Nusselt distribution increase due to higher fluctuations of the flow and boundary layer disturbance over the target surface at first, but further increase of f s from 2.5 to 4 keeps the Nusselt distribution almost constant because the positive effects of faster disturbance of the boundary layer nullifies the negative effects of lower interaction between the flow and the target. Increasing fs beyond 4 to 7, reduces the STD by eliminating the peak in Nusselt distribution, and providing more uniform cooling on the target surface. Further increase in fs forces the TTSIJ to move faster which reduces the heat transfer time with the target surface and finally leads to smoother Nusselt profile. The frequency of pulsations showed to improve the heat transfer with the underneath segments of the target surface because of the more mixing of the flow and disturbance of the boundary layer. The jet-to-jet separation distance, could not remarkably alter the averaged Nusselt number magnitude on the target surface, but shifted the location of maximum Nusselt number on the target surface. For the purpose of optimization, 9 design uniform Nusselt distribution with equal intervals from NuD ¼ 25 to NuD ¼ 65 were considered on the target surface and the 8 design variables were optimized by subsequent use of artificial neural network (ANN) and genetic algorithm (GA). The discrepancy between the obtained Nusselt distribution and the design
Nusselt distribution, and the design variables were fed into the ANN and the output evaluated functions of the ANN, were regarded as the objective functions of the optimization process. Since the objective function of the GA reveals apparent by ANN, the run time of the opti mization process decreases noticeably in compared to optimization without using ANN. It was found that within the assumed bounds of the design variables, TTSIJ were successful to achieve uniform Nusselt dis tribution with the accuracy of more than 98% and 93% in the best and worst situation, respectively. The utilized procedure could be used for obtaining higher uniform Nusselt distribution by changing the bounds of the design variables. References [1] H.J. Poh, K. Kumar, A.S. Mujumdar, Heat transfer from a pulsed laminar impinging jet, Int. Commun. Heat Mass Transf. 32 (10) (2005) 1317–1324. [2] M.J. Remie, et al., Heat-transfer distribution for an impinging laminar flame jet to a flat plate, Int. J. Heat Mass Transf. 51 (11–12) (2008) 3144–3152. [3] W. Zhou, L. Yuan, Y. Liu, D. Peng, X. Wen, Heat transfer of a sweeping jet impinging at narrow spacings, Exp. Therm. Fluid Sci. 103 (January) (2019) 89–98. [4] M. Faris, R. Zulkifli, Z. Harun, S. Abdullah, Heat Transfer Augmentation Based on Twin Impingement Jet Mechanism, 2018 no. February 2019. [5] M. Attalla, H.M. Maghrabie, A. Qayyum, A.G. Al-Hasnawi, E. Specht, Influence of the nozzle shape on heat transfer uniformity for in-line array of impinging air jets, Appl. Therm. Eng. 120 (2017) 160–169. [6] F. Afroz, M.A.R. Sharif, Numerical study of heat transfer from an isothermally heated flat surface due to turbulent twin oblique confined slot-jet impingement, Int. J. Therm. Sci. 74 (2013) 1–13. [7] T. Demircan, H. Turkoglu, The numerical analysis of oscillating rectangular impinging jets, Numer. Heat Transf. A 58 (2) (2010) 146–161. [8] B.N. Hewakandamby, A numerical study of heat transfer performance of oscillatory impinging jets, Int. J. Heat Mass Transf. 52 (1–2) (2009) 396–406. [9] P. Xu, S. Qiu, M. Yu, X. Qiao, A.S. Mujumdar, A study on the heat and mass transfer properties of multiple pulsating impinging jets, Int. Commun. Heat Mass Transf. 39 (3) (2012) 378–382. [10] T. Park, K. Kara, D. Kim, International Journal of Heat and Mass Transfer Flow structure and heat transfer of a sweeping jet impinging on a flat wall, Int. J. Heat Mass Transf. 124 (2018) 920–928. [11] X. Wen, Y. Liu, Lagrangian analysis of sweeping jets measured by time-resolved particle image velocimetry (a) the jet ’ s maximum deflected positions (b) the jet ’ s maximum spreading angle, Exp. Therm. Fluid Sci. 97 (April) (2018) 192–204. [12] M. Forouzanmehr, H. Shariatmadar, F. Kowsary, M. Ashjaee, Achieving heat flux uniformity using an optimal arrangement of impinging jet arrays 137, June, 2015, pp. 1–8. [13] S.D. Farahani, M.A. Bijarchi, F. Kowsary, M. Ashjaee, Optimization arrangement of two pulsating impingement slot jets for achieving heat transfer coefficient uniformity, J. Heat Transf. 138 (10) (2016) 102001. [14] M.A. Bijarchi, F. Kowsary, Inverse optimization design of an impinging co-axial jet in order to achieve heat flux uniformity over the target object, Appl. Therm. Eng. 132 (2018) 128–139. [15] M.A. Bijarchi, A. Eghtesad, H. Afshin, M. Behshad, Obtaining uniform cooling on a hot surface by a novel sweeping slot impinging jet 150 (June 2018) (2019) 781–790. [16] C.M. Rhie, Numerical study of the turbulent flow past an airfoil with trailing edge separation 21 (11) (1983) 1525–1532. [17] E.C. Mladint, Local convective heat transfer to submerged pulsating jets 40 (1997) 14.
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