3. Part I: Cooling characteristics of single and array impinging jets
3.2. Results and discussion
3.2.2. Effect of L/D
The cooling characteristics of single and array jets with various nozzle-to-plate distances (L/D) are investigated. In previous study, the heat transfer characteristics of the single impinging jet with varying L/D and 𝑅𝑒𝐷 was investigated by San et al. [44]. In general, at low L/D and high Reynolds number means that high momentum flows can impinge on the target plate, so leads to the enhanced heat transfer rates. In this experiment, this tendency is shown in the stagnation heat transfer coefficients, but slightly different for certain conditions of L/D and 𝑅𝑒𝐷.
Fig. 9 shows stagnation heat transfer coefficients of single impinging jet. As L/D increases, the heat transfer coefficients are decreases, but there is a peak value of the heat transfer coefficient at the specific value of L/D. The value of L/D with maximum heat transfer coefficient tends to increase as the 𝑅𝑒𝐷 increases. In the case of 𝑅𝑒𝐷= 104,159, the value of L/D with maximum heat transfer coefficient is 3.0 and when the 𝑅𝑒𝐷 is lower than 78,119 the peak value has at L/D= 1.0 or L/D= 2.0, as shown in Fig. 9 (a). The relations between heat transfer characteristics and vortices have been studied by Popiel and Fox et al. [45,46]. After the jet impinges the target surface, the jet flow grows to the wall jet region and jet vortices developed by confined wall and nozzle body. These vortices are beneficial to heat transfer enhancement and as 𝑅𝑒𝐷 increases, it means the more flow rate and lower temperature jet impinge the target plate. Fig. 9(b) illustrates the trend of heat transfer coefficients of N2 impinging jet with varying L/D, but the maximum value of heat transfer coefficient is not shown clearly. As previously discussed, the temperature drops of N2 is less sensitive than CO2 by pressure drop, thus resulting in the peak value are not clearly distinguished at high a 𝑅𝑒𝐷.
Fig. 10 shows the heat transfer coefficient with varying L/D for array impinging jets by using CO2 and N2 jet. Compared to the single impinging jet, the stagnation heat transfer coefficient of array impinging jets linearly decreases as L/D increases. As 𝑅𝑒𝐷 increases, the stagnation heat transfer coefficient is increases. At the high 𝑅𝑒𝐷, the heat transfer coefficients are more increases rapidly than low 𝑅𝑒𝐷. As can be seen Fig. 10(a) and (b), the heat transfer coefficients of N2 are less sensitive with varying of L/D. These results are considered to be due to the difference of Joule- Thomson coefficient between CO2 and N2.
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Fig. 9. Stagnation heat transfer coefficient distributions: (a) single 𝐂𝐎𝟐 impinging jet (b) single 𝐍𝟐 impinging jet.
Fig. 10. Stagnation heat transfer coefficient distributions: (a) array 𝐂𝐎𝟐 impinging jet (b) array 𝐍𝟐 impinging jet.
17 3.2.3. Effect of 𝑹𝒆𝑫
Fig. 11 depicts the stagnation heat transfer characteristics of single CO2 and N2 impinging jet with varying 𝑅𝑒𝐷 at L/D= 2.0, 4.0, 8.0, and 10.0. As Reynolds number increases, the larger flow rates and higher nozzle inlet pressures are guaranteed, so the low-temperature jets can be ejected from the nozzle. Therefore, the high 𝑅𝑒𝐷 enhances the heat transfer coefficient, both cases of CO2 and N2 impinging jet. As mentioned above, the small L/D ensures that jets with high momentum impinge the target plate, resulting in a high heat transfer rate. Compared to Fig. 11(a) and (b), the heat transfer coefficient of single CO2 impinging jet is more sensitive to 𝑅𝑒𝐷 increasing than N2.
Fig. 12 shows the heat transfer coefficient of array CO2 and N2 impinging jet with varying 𝑅𝑒𝐷 with the value of L/D = 2.67, 4.67, 8.67, and 12.67. Like the single impinging jet, the high 𝑅𝑒𝐷 and low L/D guarantee the enhanced heat transfer coefficient. The heat transfer coefficient of array impinging jet increases more rapidly at the high 𝑅𝑒𝐷 than low 𝑅𝑒𝐷. These result shows that the high 𝑅𝑒𝐷 yield the jet-to-jet interaction to beneficial the heat transfer improvement. Compared to array CO2 and N2 impinging jet, the heat transfer coefficients of CO2 jet are more sensitive to the value of L/D than N2 jet as shown Fig. 12(a) and (b). These results are due to that the lowered temperature of CO2 jet by the Joule-Thomson effect at low L/D condition.
Fig. 13 shows a comparison of the stagnation heat transfer coefficient between single and array impinging jet with various 𝑅𝑒𝐷 and volume flow rate at L/D= 2.0. For the condition of the same 𝑅𝑒𝐷, the stagnation heat transfer coefficient of N2 impinging jet is higher than that of CO2 impinging jet as shown in Fig. 13(a) and (b). The heat transfer coefficient of single N2 impinging jet from 𝑅𝑒𝐷= 55,256 is nearly twice higher that of single CO2 impinging jet from 𝑅𝑒𝐷= 52,079. However, as shown in Fig. 13(c), if the reference value of the flow rate is chosen as the volume flow rate, CO2 jet has a higher heat transfer coefficient than N2 jet under all conditions. This phenomenon is due to the difference in the thermal properties that affect the 𝑅𝑒𝐷, such as the viscosity of the CO2 and N2. As the volume flow rate increases, the pressure drops of the gas ejected from the nozzle increases and the difference of the heat transfer coefficient increases because CO2 has a higher Joule-Thompson coefficient than N2 . The stagnation heat transfer coefficient of the single impinging jet is always higher than that of the array impinging jet due to concentrated flow rates to the stagnation point.
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Fig. 11. Stagnation heat transfer coefficients varying with Reynolds number: (a) single 𝐂𝐎𝟐 impinging jet (b) single 𝐍𝟐 impinging jet.
Fig. 12. Stagnation heat transfer coefficients varying with Reynolds number: (a) array 𝐂𝐎𝟐 impinging jet (b) array 𝐍𝟐 impinging jet.
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Fig. 13. Stagnation heat transfer coefficients varying with Reynolds number at fixed L/d=2: (a) single impinging jet (b) array impinging jet (c) effect of working fluid with varying volume flow
rate.
20 3.2.4. Temperature non-uniformity
For the evaluation of the cooling performance of temperature uniformity, the maximum temperature difference (𝑇𝑑𝑖𝑓) of six thermocouples is chosen as the indication of the temperature non- uniformity, which is defined in Eq. (7). As the value of temperature non-uniformity is higher, the cooling uniformity of the target surface is deteriorated.
𝑇𝑑𝑖𝑓 = 𝑇𝑚𝑎𝑥− 𝑇𝑚𝑖𝑛 (7)
Fig. 14 shows comparison of temperature non-uniformity and average temperature of test section between single and array impinging jet with varying volume flow rate at L/D= 2.0. As shown in Fig. 14(a), array jet has a higher performance of cooling uniformity than that of single impinging jet under all volume flow rate condition. Fig.14 (b) shows a comparison of array impinging jet between CO2 and N2 with varying volume flow rate. The average temperature is the mean temperature of the eight thermocouples inserted into the test section. Under all volume flow rate conditions, the temperature non-uniformity of the test section by the array impinging jet cooling is little different, which means that it has a similar level of cooling uniformity. However, there is a big difference in the average temperature of the test section and the largest temperature difference is about 8.45 ℃. Therefore, the array impinging jet has a more uniform cooling performance than a single impingement jet, and the array CO2 impinging jet guarantees a lower surface temperature than the array N2 impinging jet.
Fig. 14. Comparative plots showing the temperature non-uniformity of (a) single and array impinging jet (b) array impinging jet with 𝐂𝐎𝟐 and 𝐍𝟐.
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4. Part II: Geometry optimization and confirmation experiments
4.1. Computational details 4.1.1. Governing equations
The numerical modeling for the optimization study is performed by the commercial CFD codes, ANSYS Fluent 19.0. ANSYS Fluent uses the cell-centered finite volume method (FVM) to discretize and solve the governing equations based on Reynolds averaged Navier-Stokes (RANS) approach. The working fluid (CO2) is set as real gas and state of the flow is steady and fully turbulent. The second- order upwind scheme is used to discretize the diffusive term. The pressure-velocity coupling is achieved by a coupled algorithm, which has been proved a robust and efficient scheme. The coupled algorithm solves the momentum and pressure-based continuity equation together, so it has a superior performance compared to the segregated solution algorithm. In the simulation, the iteration of solutions is continued until the convergence criterion which set as the normalized residuals of mass, momentum, and energy equations decrease below the 10−4 , 10−4 , and 10−9 at each iteration, respectively. Also, the average temperature of the impinging surface is monitored, and the solutions are considered to be converged when the monitored value was not changed within the 100 iterations.
The governing equations are as follows:
Continuity:
𝜕(𝜌𝑈𝑖)
𝜕𝑥𝑖 = 0 (8)
Momentum:
𝜕(𝜌𝑈𝑖𝑈𝑗)
𝜕𝑥𝑗 = −𝜕𝑃
𝜕𝑥𝑖+ 𝜇 𝜕
𝜕𝑥𝑗(𝜕𝑈𝑖
𝜕𝑥𝑗+𝜕𝑈𝑗
𝜕𝑥𝑖) + 𝜕
𝜕𝑥𝑗(−𝜌𝑢𝑖′𝑢𝑗′) (9) Energy:
𝐶𝑝𝜕(𝜌𝑈𝑖𝑇)
𝜕𝑥𝑖 = 𝜕
𝜕𝑥𝑖(𝜆𝜕𝑇
𝜕𝑥𝑖− 𝜌𝐶𝑝𝑢𝑖′𝑇′) (10)
Where 𝑈, 𝑇, 𝑃 are the time-averaged velocity, temperature, and pressure of the fluid. 𝑢′ and 𝑇′ are the velocity and temperature fluctuation. 𝑢𝑖′𝑇′ and 𝑢𝑖′𝑢𝑗′ are the turbulence heat flux vector and Reynolds stress term which based on the Boussinesq assumption and these values are defined as:
−𝜌𝑢𝑖′𝑢𝑗′= 𝜇𝑡(𝜕𝑈𝑖
𝜕𝑥𝑗+𝜕𝑈𝑗
𝜕𝑥𝑖) −2
3(𝜌𝑘 + 𝜇𝑡𝜕𝑈𝑖
𝜕𝑥𝑖) 𝛿𝑖𝑗 (11)
−𝜌𝑢𝑖′𝑇′ = 𝜇𝑡 𝑃𝑟𝑡(𝜕𝑈𝑖
𝜕𝑥𝑖) (12)
Where 𝜇𝑡 = 𝜌𝐶𝜇(𝑘2⁄ ) is the turbulent viscosity, 𝜀 𝛿𝑖𝑗 is the Kronecker delta.
22 4.1.2. Turbulence and near-wall modeling
The flow region of the impinging jet lies in the fully turbulent flow, so the realizable k-ε turbulence model is adapted in this study. The standard k-ε turbulence model was proposed by Launder and Spalding [47]. This model involves the transport equations for turbulent flow such as the turbulence kinetic energy (k) and the rate of dissipation (𝜀). Compared to the standard k-ε turbulence model, the realizable k-ε turbulence model contains an alternative equation for the eddy viscosity and a modified transport equation for the dissipation rate. As a result, the realizable k-ε turbulence model accurately predicts the jet spreading and near-wall turbulence effects than the standard k-ε turbulence model [48-50]. The detailed transport equations as follows:
𝜕(𝜌𝑘𝑢𝑗)
𝜕𝑥𝑗 = 𝜕
𝜕𝑥𝑗[(𝜇 +𝜇𝑡 𝜎𝑘)𝜕𝑘
𝜕𝑥𝑗] + 𝐺𝑘+ 𝐺𝑏− 𝜌𝜀 − 𝑌𝑚+ 𝑆𝑘 (13)
𝜕(𝜌𝜀𝑢𝑗)
𝜕𝑥𝑗 = 𝜕
𝜕𝑥𝑗[(𝜇 +𝜇𝑡 𝜎𝜀) 𝜕𝜀
𝜕𝑥𝑗] + 𝜌𝐶1𝑆𝜀 − 𝜌𝐶2
𝜀2
𝑘 + √𝜈𝜀+ 𝐶1𝜀
𝜀
𝑘𝐶3𝜀𝐺𝑏+ 𝑆𝜀 (14) Where 𝐺𝑘 and 𝐺𝑏 are referred to the production of turbulence kinetic energy due to the mean velocity gradients and buoyancy. 𝑆𝑘 and 𝑆𝜀 are the user-defined source terms. 𝜎𝑘 and 𝜎𝜀 are referred to the turbulent Prandtl numbers based on 𝑘 and 𝜀 , respectively. 𝐶1 , 𝐶2 , 𝐶1𝜀 , and 𝐶3𝜀 are the empirical constants. 𝑌𝑚 represents the contribution of the fluctuating dilatation.
One of the most important considerations to the numerical study of jet impingement heat transfer is in the vicinity of the impinging surface, especially the near-wall viscosity affected layer.
The enhanced wall treatment (EWT) is a two-layer methodology for wall treatments and used to evaluate the eddy viscosity in the near-wall. In EWT model, the whole domain is subdivided into a viscosity affected region (𝑅𝑒𝑦< 200) and fully turbulent region (𝑅𝑒𝑦> 200) depending on turbulent Reynolds number (𝑅𝑒𝑦). Therefore, the more accurate results near-wall region can obtain by applying the EWT. The turbulent Reynolds number (𝑅𝑒𝑦) defined as:
𝑅𝑒𝑦=𝜌𝑦√𝑘
𝜇 (15)
Where y denotes the wall-normal distance calculated at the cell centers. For the viscosity affected region, the one equation of Wolfstein [51], the turbulent eddy viscosity and dissipation are evaluated by algebraic expressions. For the fully turbulent region, the realizable k-ε turbulence model is employed.
In EWT for momentum and energy equations, the law of the wall formulated to extend its applicability throughout the near-wall region by blending the linear (laminar) and logarithmic (turbulent) using a
23
function suggested by Kander [52]. The temperature and velocity of the near-wall region are obtained by blending the laminar linear sublayer and turbulence logarithmic laws. The blended law of the wall function as follows:
𝑢+= 𝑒Γ𝑢+𝑙𝑎𝑚𝑖𝑛𝑎𝑟+ 𝑒1/Γ𝑢+𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 (16) Γ = − 𝛼(𝑦+)4
1 + 𝛽𝑦+ (17)
𝑇+= 𝑒Π𝑇+𝑙𝑎𝑚𝑖𝑛𝑎𝑟+ 𝑒1/Π𝑇+𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 (18) Π = − 𝛼(𝑦+𝑃𝑟)4
1 + 𝛽𝑃𝑟3𝑦+ (19)
Where 𝑦+= 𝑦𝑝𝑢𝜏
⁄𝜈 is the dimensionless height of the first layer cells near the wall, 𝑦𝑝 is the actual height of the first layer cell, 𝑢+ is the dimensionless velocity which defined as velocity over the shear velocity, 𝜏𝑤 is the wall shear stress, and 𝑢𝜏 is the shear velocity. Γ and Π are blending functions, and the value of constants is specified as 𝛼 = 0.01 and 𝛽 = 5.
4.1.3. Computational domain and boundary conditions
The schematic drawings of the computational domain and boundary conditions for the array impinging jet cooling simulation are shown in Fig. 15. The inlet boundary condition is set as the temperature 294 K and velocity 15.78 m/s, which is equivalent to the value of volume flow rate 5×10-4 m3⁄s and outlet is set as pressure outlet boundary condition. The nozzle-to-plate distance (L), nozzle diameter (d), and jet-to-jet spacing (S) are set as 4.0 mm, 1.5 mm, and 10.5 mm, respectively.
The bottom of solid domain is set as constant heat flux condition. Fig. 15(c) illustrates the CAD rendering image of impinging jet array. The structure of the outlet is replaced as the square based on the same hydrodynamic diameter to ease of grid generation. The nine nozzle holes (3×3) are arranged at equal intervals on the impinging jet array. For the purpose of reducing the computational cost, the entire domain is simplified to the one-quarter domain by applying the symmetry plane. Although the numerical simulations are performed for a small part of the domain, symmetry planes facilitated the computation which considering the 3×3 array nozzle holes. The adiabatic and no-slip boundary conditions are applied to walls except for the impinging surface.
24
Fig. 15. Schematics of the computational domain and boundary conditions: (a) front view (b) top view (c) CAD rendering image of impinging jet array.
25 4.1.4. Grid independence test
The computational domain is generated as a hexahedral grid system with refined meshing near the walls by using ANSYS meshing, as shown in Fig. 16. Due to the enormous gradients of temperature and thermo-physical properties near the impinging wall, much finer grids are required in the vicinity of the impingement surface to resolve the near-wall region with EWT. To ensure the accurate predictions of the near-wall region, the dimensionless distance from the wall (𝑦+) is kept less than unity in this study. To check the grid independency on the numerical results, four different grid numbers are used in the calculation. Fig. 17 shows the local temperature distributions depending on the grid numbers. The maximum differences between the used and fine grid are within the 0.1 K, so the grid number 2.78×106, which ensures the grid-independent results, is used in this study.
Fig. 16. The computational domain and grid system.
26
Fig. 17. The grid independence test.
27 4.2. Optimization strategy
First of all, the fixed and design parameters should be determined in order to optimization study. The fixed parameters are the nozzle arrangement and number of nozzles. Due to the previous study revealed that the inline arrangement has outperformed the staggered arrangement [18], so the nozzle arrangement is selected as the inline and radial is selected as a comparison arrangement. The design parameters are selected except those that can be changed by experimental method, such as 𝑅𝑒𝐷 and L/D. The jet-to-jet spacing, and nozzle size is the major parameters for affecting the temperature uniformity and local temperature distributions, respectively [17,56]. For array impinging jet, it is important to consider the effects of nozzle size at the central and outer of jet array due to the non- uniform distribution of mass flow rate. Therefore, jet-to-jet spacing (S), diameter of central nozzle (𝑑𝑐), diameter of outer nozzle (𝑑𝑜), minor axis length (a), and aspect ratio (AR = 𝑏 𝑎⁄ ) are selected as the design parameters, as can be seen in Fig. 18.
The geometry of nozzles is selected as the circular and elliptic nozzles based on the prior studies. The conventional circular jet has outperformed the triangle and rectangular jet, but elliptic jet with specific aspect ratio has outperformed the circular jet and yield the uniform heat flux of target surface [57,58]. Unlike the circular jets that grow in axisymmetric, the elliptic jets produce the azimuthal curvature variation which evolves in different ways [24]. Therefore, the flow patterns and cooling characteristics of array elliptic and circular jets will be different. Namely, the enhanced cooling performance can be expected through the stronger jet-to-jet interactions by placing the circular nozzle and elliptic nozzle at the central and outer of the jet array, respectively. The circular nozzle arranged inline and radial are named CJ1 and CJ2. Also, the jet array consisting of the outer elliptic nozzle arranged in a different direction and central circular nozzle are named EJ1 and EJ2, respectively.
The objectives of optimization study and range of design parameters are shown below:
Multi-Objective:
Minimize - 𝑇𝑎𝑣𝑒 (𝑆, 𝑑𝑐, 𝑑𝑜, 𝑎, 𝐴𝑅) Minimize - 𝑇𝑑𝑖𝑓 (𝑆, 𝑑𝑐, 𝑑𝑜, 𝑎, 𝐴𝑅) Subject to:
6.0 𝑚𝑚 ≤ 𝑆 ≤ 15.0 𝑚𝑚, 1.0 𝑚𝑚 ≤ 𝑑𝑐 ≤ 2.0 𝑚𝑚, 1.0 𝑚𝑚 ≤ 𝑑𝑜 ≤ 2.0 𝑚𝑚, 0.4082 𝑚𝑚 ≤ 𝑎 ≤ 0.8165 𝑚𝑚, 1.5 ≤ 𝐴𝑅 ≤ 3.0
28
The optimization study aims to find the optimal geometry of jet array that satisfies the objective function which minimize the average temperature (𝑇𝑎𝑣𝑒) and maximum temperature difference (𝑇𝑑𝑖𝑓) of the target surface. The range of geometric parameters about the nozzle size such as 𝑑𝑐, 𝑑𝑜, 𝑎, and AR are selected based on the same area.
The optimization process is divided into three steps, design of experiment (DoE), response surface methodology (RSM), and optimization. Fig. 19 shows the overall flow chart of the optimization procedure. Firstly, 29 DoE points are generated using the central composite design (CCD) method. The CCD method was established by Box and Wilson [53] and widely used in due to its performance of ideal solution for a second-order response surface fitting by selecting corner, axial and center points. Also, the RSM is used in various applications due to its merit that easy to numerically by approximating a complex design space using average values and deviations.
The response surfaces are build based on the CFD simulation results of DoE points using the Kriging model. The Kriging model is an interpolating technique and has several advantages over other interpolation methods. Firstly, the Kriging model simulates an unknown point using information from some estimated points. In addition, the Kriging model can analyze both global and local statistical properties [54]. As such, the Kriging model is suitable for CFD calculation which has a high strength of non-linearity. If the quality of the response surface is poor, the optimization process returns to the first step and adds the number of DoE points and then builds the response surface again until the quality meets the criteria. The final step is the optimization of design parameters using the multi-objective genetic algorithm (MOGA). The MOGA is the modification algorithm of genetic algorithm (GA) which based on the law of survival of fittest in order to find the multiple non-dominated solutions in a single run [55]. The MOGA based on the non-dominated sorted genetic algorithm-II (NSGA-II) that uses the elitism concept to find the Pareto-front, which is classified as non-dominated solutions. In order to verify the candidate for optimization result, the CFD calculation (real solve) is performed. If the difference between the CFD calculation and the candidate of optimization result is less than 5 %, select the candidate to optimal solution and return to the first step, DoE, if exceed than 5 %.
29
Fig. 18. Schematics of design parameters: (a, b) circular nozzle arranged inline and (c, d) radial outer elliptic nozzle arranged in a different direction.
30
Fig. 19. The flow chart of the optimization procedure.
31 4.3. Results and discussion
4.3.1. Optimization results
In general, there are no solutions to optimize all objective functions simultaneously in multi- objective optimization problems. One of the promising ways to solve these problems requires the sacrifice of other objective functions to improve some objective functions, and in the same way, the MOGA approach finds the optimal solutions that are not dominated by the other optimal candidates through NSGA-II. The non-dominated solutions are called the Pareto optimal solutions, and the set of non-dominated optimal solutions is called the Pareto-front. Fig. 20 shows the trade-off chart in the optimization process. The best set of optimal solutions is the blue, and gradually turn to red as the solutions became deteriorate. The objective functions of this study are the minimize the average temperature (𝑇𝑎𝑣𝑒) and maximum temperature difference (𝑇𝑑𝑖𝑓) of the target surface. Therefore, the optimal design can be obtained near the left bottom corner in the coordinate system. Among the Pareto-front, the candidate of optimum design is sorted by the decision support process (DSP) which based on the direct Monte Carlo method. The DSP takes into account the weighted objective functions and ranks the candidate points using a uniform distribution. The weighted functions of the objective are set as default, which means that the importance of the output parameters is identical. Table 3 shows the best designs among the candidates selected by the MOGA approach for the CJ1, CJ2, EJ1, and EJ2, respectively. To verify the optimization results, the real solve (CFD calculation) is performed to examine them and the predicted error between optimization result and CFD simulation is less than 5 % in all cases. It is observed that the nozzle diameter at the outer of jet array (𝑑𝑜) is slightly larger than nozzle diameter at the central of jet array (𝑑𝑐) due to it can be cover the larger cooling area. Compared with the initial design, it is observed that in the case of EJ2, the 𝑇𝑎𝑣𝑒 and 𝑇𝑑𝑖𝑓 (temperature non- uniformity) is improved by about 7.6 K and 7.7 %. Among the optimum designs, the elliptic jet array has outclassed of cooling performance the circular jet array, the best case is the EJ2 and the worst case is the CJ2.