Enhanced performance of a microchannel with rectangular vortex generators

Abstract

Keywords: Flow Control; Genetic Algorithm; Heat Transfer Enhancement; Microchannel; Vortex Generators

Introduction

The rapid advancements of electronic devices, the shrinking of chip sizes, the rapid increase in performance, and the increase in processor speeds with the development of computer technologies have also increased the heat density generated by these devices. The increase in heat generation intensity has also necessitated the development of appropriate thermal management mechanisms for these devices to operate properly. The development of micro-manufacturing methods has paved the way for the development of heat transfer enhancement methods. First, Tuckerman and Pease [1] demonstrated that chips could be effectively cooled by opening microchannels on the circuit board where they are mounted. However, due to narrower channels, pressure drops per unit length also increased considerably. As a result of comparison with various experimental results, it has been proved that the flow in the microchannel can be predicted by conventional NavierStokes equations [2]. Consequently, numerous researchers employed experimental [3–12] and numerical methods [12-28] to examine the flow structure and heat transfer phenomena in various types of microchannels. Microchannels have higher heat transfer than conventional channels, but on the other hand, various heat transfer enhancement methods have been applied in microchannels for more efficient cooling of electronic devices [30–32]. One of them is the vortex generator (VG), which was primarily used in conventional ducts or channels [33–39] and heat exchangers [40–44]. VGs can provide higher heat transfer enhancement and lower corresponding pressure loss compared to turbulators [45]. In contrast to wavy [46], corrugated [47], and louvered fins [48–51], VGs [45,52] increase the fluid mixing and temperature gradient normal to the wall and subsequently increase the convective heat transfer by forming secondary flow structures such as vortices in the flow without changing the direction of the main flow. By inserting VGs of varied numbers and angles into a microchannel, heat transfer can be significantly enhanced. Rectangular VGs implemented in a microchannel decreased the critical Reynolds number, which accelerated the onset of turbulence [53,54]. Similarly, the impact of rectangular VGs positioned in a microchannel on flow and heat transfer was studied numerically by considering the thermophysical fluid properties depending on temperature [55]. Fluid mixing in a T-shaped microchannel via diffusion and convection was significantly enhanced by adding rectangular winglet pairs type VGs [56]. In addition to these, VGs of different shapes, for example, cylindrical-type VGs with quarter-circle and half-circle cross-sections, were employed in a microchannel [57]. In order to increase the heat transfer capacity of microchannels, VG designs should be done properly, and pressure losses should be taken into account to reduce energy

consumption as well as heat transfer enhancements. Thus, the highest heat transfer capacity will be achieved with the least possible pressure loss. However, the optimal design may differ depending on the channel type, boundary conditions, or operating conditions. Here, we have used the computational fluid dynamics (CFD) method to predict the flow and heat transfer characteristics of a microchannel with rectangular VGs. The objective of the research is to control the flow structure, maximize heat transfer, and minimize pressure loss by varying the relevant parameters. Using response surface methods with genetic algorithm, we have asserted the optimum design of VGs and determined the effects of design and operating parameters. We believe that this study will shed light on the heat transfer enhancement with VGs and guide thermal management engineers in the design of cooling systems.

Theory

Physical and numerical models The microchannel with rectangular VGs is depicted in Fig. 1 as a schematic. Dimensionally, the microchannel is 20000 mm in length, 1500 mm in width, and 100 mm in height, with a 1000 mm gap from the channel inlet to the first set of VGs pairs (L1). As presented in Fig. 1, LVG, SVG and W1 represent VG height, the longitudinal pitch of VG pairs and the transverse pitch of VGs in VG pairs, respectively. In addition, the value of SVG indicates the longitudinal distance between VG pairs. Five distinct row numbers (three, four, five, six, and seven) of the VG pairs are studied in this investigation, and five sequential VG pairs put in the microchannel are presented in Fig. 1. Besides, α1 and α2 depict the attack angles of two different rectangular VGs layouts (see Fig. 1). Here, we have conducted numerical analysis for different fluid velocities, attack angles, the transverse pitch between VGs in a pair, number of VG pairs, and VG length. Table 1 demonstrates the considered ranges of geometrical parameters employed in response surface methodology and sensitivity analysis. Besides, the VGs have a fixed width of 40 mm. The ANSYS Fluent 19.1 is utilized to solve three dimensional-turbulent flow and heat transfer domain in the microchannel. Since heat transfer in a microchannel with VGs relies on the conjugate heat transfer mechanism, the energy equations for both solid domain conduction and convection between solid and fluid zones are solved. The time average of the transport and governing equations yields the RANS equations. In steady-state conditions, continuity, momentum, and energy equations for fluid and solid zones are as below, respectively: ∂ui =0 ∂x j

Table 1. The considered range of input parameters Geometrical parameters

The transverse pitch between VGs in a pair, W1 (μm) The angle of attack, α1 (º)

1 ∂P ∂   ∂ui ∂u j 2 ∂u j  + v + − δij  ρ ∂xi ∂x j   ∂x j ∂xi 3 ∂x j    (2) ∂ + −ui′u ′j ∂x j

nent, u´i are the fluctuating components. −ui′u ′j represents the effect of turbulence and is known as the Reynolds stress tensor. It is defined as below;  ∂u  ∂u j   2  −  k + vt ∂uk dij −ui′u ′j = vt  i +  ∂x j ∂xi   3 ∂x k 

The Reynolds-averaged method for simulating turbulence requires that the Reynolds stresses in Eq. 2 be accurately modeled. A common method for determining the relationship between Reynolds stresses and mean velocity gradients uses the Boussinesq hypothesis [60,61]. In ANSYS Fluent, turbulent heat transport is modeled employing the concept of the Reynolds analogy to turbulent momentum transfer. The modeled energy equation for fluid domain is given as below:

c p µt  ∂T  p  1 ∂  ∂      = u E + λ +   i ∂xi   ρ  ρ ∂xi  Prt  ∂xi 

In Eq. 4, E is the total energy per unit mass consisting of internal, kinetic, and potential energies (E = cvT + u2/2, the potential energy is omitted). Also, Prt is the turbulent Prandtl number. The energy equation for solid domain: ∂ 2T ls 2 = 0 ∂x j

SST k-ω turbulence model has some advantages over the Standard form of k-ω and k-ε turbulence models, and it generally increases the accuracy of the model for predicting free shear flows. In this model, the cross-diffusion term and mixing function in the ω equation can provide better results in the near-wall and far-field regions. The transport equation of this model is as follows: ∂ (rkui )

Table 2. Silicon and deionized water thermophysical properties [58,59]

is used for better accuracy considering mesh imperfections in terms of orthogonality. It is confirmed that the residuals of the continuity, momentum and energy equations are less than 10-5, 10-7, and 10-9, respectively. Fig.2 depicts the applied boundary conditions and the created mesh details on flow and solid domains. Some parameters are offered below to determine the thermal and flow performances of microchannels with VGs. The Reynolds number is determined as follows: Re =

In these equations, Gk denotes the production of kinetic energy due to turbulence. Gω symbolizes the generation of ω. Γk and Γω denote the effective diffusivities of k and ω, while Yk and Yω symbolize the dissipation of k and ω due to turbulence. Dω represents the term for cross-diffusion. User-defined source terms are omitted from the modeling as they are not utilized. Ref. [60] can be examined for information on the associated turbulence model. The velocity inlet boundary condition is implemented at the inlet, while the pressure boundary condition is performed at the outlet (0 Pa gauge pressure). In order to eliminate entrance and exit influences, the slip wall boundary condition is implemented at the flow domain walls positioned at 20Dh and 40Dh from the inlet and outlet, respectively. In contrast, the no-slip boundary condition is implemented on the walls of the test domain. The symmetry boundary condition is used at the middle of in the microchannel to reduce the solution cost. The constant surface temperature (323.15 K) is applied to the bottom wall of the test domain. Silicon and deionized water are utilized as the coolant fluid and VG material, respectively. The thermophysical properties of silicon and deionized water as a function of temperature are listed in Table 2. The pressure-based solver is utilized while the coupled algorithm is preferred for velocity-pressure coupling. Spatial discretization for governing equations is realized by a second-order upwind scheme. The least squares gradient

where Dh is the hydraulic diameter, and it is calculated as follows: Dh =

The dimensionless average Nu, a key indicator of heat transfer performance, is defined as follows: Nu =

where h indicates the average heat convection coefficient, and it is calculated as follows: h=

. where Q , Ah, and ΔTm signify the heat transfer rate, heat transfer surface area (base area), and logarithmic mean temperature difference (LMTD), respectively. LMTD is described as follows: ∆Tm =

JF, which indicates the relationship in both pressure loss and heat transfer with and without VGs, is

Figure 2. Boundary conditions and mesh detail of computational domain.

utilized to more accurately represent the thermal-hydraulic performance. JF =

Mesh independency and validation of the numerical model Based on the experimental findings of Liu et al. [53] four turbulence models (k-ε Realizable EWT, k-ε RNG EWT, k-ω SST, and Reynolds Stress EWT) are used to solve 3D thermal-hydraulic behaviors in microchannels with VGs. Initially, mesh independence of the solution is evaluated for all turbulence models evaluating Nu and f values (Fig. 3).

The solutions obtained with 2.3 million and 11 million elements differ by approximately 0.2% in both Nu and f values. Considering this situation and staying in the safe zone, the model with the number of 4.2 million elements has been used in all analyzes. Fig. 4 depicts the results of the numerical analysis performed in the experimental investigation of Ref. [53], taking into account all specifications of the “G1” channel configuration. The channel’s dimensions and characteristics under consideration such as the length, width, the angle of attack, and number of LVG pairs are 400 μm, 50 μm, 30°, and 5, respectively. Fig. 4a and 4b depict comparison of numerical and experimental results for Nu and f values, respectively. The numerical results for various Reynolds numbers are in strong agreement with the results of the experiments. The maximum deviations between numerical and experimental

Figure 3. Mesh Independence Studies for Microchannel with VGs (a) Nu, (b) f.

Figure 4. Comparability between experimental and numerical results of (a) Nu and (b) f. results are 14.5% for Nu and 13.9% for f values, respectively. As a result, the turbulence model that predicts the most accurate results is SST k-ω and has been employed in all analyses.

Results And Discussion

Thermal-Flow Characteristics In Fig. 5a, for microchannels with various numbers of VG pairs, the variation of the heat transfer characteristic j factors with the angle of attack of VG is shown. The angle of attack has a significant impact on the j factor. The j factors

increase as the angle of attack approaches 90°. The j factors begin to fall at the same rate as the angle of attack increases. While the impact of the number of VG pairs is noticeable at angles of attack both smaller and larger than 90°, the impact is significantly diminished at the attack angle of 90°. However, the j factors are significantly lower at a1=a2=90° when there are 3 VG pairs unlike other angles of attack. VGs positioned at a1=a2=90° provide about a 66% increase in heat transfer characteristics for the lowest VG pair number (NVG=3). The improvement rate in the j factor rises to 76% when NVG=7.

Figure 5. Effects of (a) (a1=a2=a), (b) NVG on j, f and JF (Vf = 5 m/s, LVG = 0.4 mm, W1 = 0.2 mm). VGs boost heat transfer, but they also increase pressure losses. The VGs add to the flow’s pressure loss because of the local contractions they cause, the energy required to create the vortices, and the rise in friction losses brought on by the vortices’ effect on the velocity profile of the flow. In Fig. 5a, the pressure losses caused by the VGs depending

on the attack angle are presented as the friction factor f. Friction factors increase until the angle of attack reaches 90°, then decline in conjunction with changes in heat transfer. Increasing the number of VGs in the microchannel increases the f-values significantly. When placed at a 90° angle of attack and NVG=7, the VGs increase the pressure

drop 4.65 times compared to the empty channel. This is a significant increase, far exceeding the augmentation in heat transfer. The fact that VGs increase both heat transfer and pressure loss suggests that their use may be subject to some restrictions. An increase in pressure loss will result in an increase in pumping power, and also there is a possibility of the formation of local cavitation zones. Nonetheless, the need for cooling in electronic systems necessitates the enhancement of heat transfer. Therefore, the primary objective is to provide the desired heat transfer capacity with as little pressure loss as possible. A criterion is required to evaluate the effectiveness of the employed heat transfer enhancement method. In this study, we have used the JF factor to evaluate the enhancement in heat transfer caused by VGs while taking into account pressure loss (Eq. 12). Fig. 6b depicts variations in the JF factor based on the angle of attack for a fixed Re and the number of different VG pairs. Although the augmentation in heat transfer and the increase in pressure loss have a similar trend depending on the angle of the VG, the amounts of change are quite different. Consequently, the change in JF depends on the angle of attack differently than the change in j and f. The JF produces two peaks for all VG pair numbers, indicating that there are two optimal angles of attack for each VG pair. The maximum and minimum JF values are obtained with NVG = 4 and 3, respectively. These two microchannels have optimal attack angles of 45° and 135°. 30° and 150° are the optimal angles for NVG = 7 and 6. The optimal angles are 45° and 150° for NVG= 5. For microchannel with NVG = 5 the different optimal attack angles based on the number of VGs are caused by several factors. Changing the number of VGs results in a change in the distance between each VG pair. Therefore, the vortex-vortex interaction and vortexVG interaction differ. The structure of vortices traveling downstream influences the structure of vortices generated by other VGs. It is concluded that the number of vortex generator pairs has a significant effect on heat and flow characteristics. For a more in-depth observation of this effect, the j, f, and JF values in Fig. 5b have been utilized depending on the number of VGs. Angles of attack of 45° and 135° are chosen because they produce the best results. Furthermore, the cases where the VG pairs are placed at different angles (a1 and a2) are investigated. The j factor increases rapidly as the number of VGs increases from 3 to 4, while the rate of increase in j decreases as the number of VGs increases further. The number of VGs causes a constant increase in the friction factor f. Contrarily, JF increases sharply until there are four VG pairs. While NVG=6 and NVG=7 yield the highest JF at a1=135° and a2=45°, NVG=4 yields the highest JF at a1= a1=45°. Fig. 6 illustrates how the spacing between the VGs affects the characteristics of pressure loss and heat transfer. A counter-rotating vortex pair is produced by each VG (Fig. 7).

The domains of the generated vortex pairs are also affected by the transverse pitch between the VGs. While the Wl/W ratio causes an increase with a little effect on heat transfer, it has less effect on pressure loss. With a rising Wl/W ratio, a gradually declining increase in JF is observed (Fig. 6a). As the Reynolds number rises, the intensity of the vortices changes. The interaction between the vortices is altered when the vortex intensity is changed. Therefore, the distance between the VGs to be placed in the microchannel should be taken into account depending on the Reynolds number . However, the optimum NVG remains the same in the current study despite the fact that j, f, and JF decrease with Reynolds numbers (Fig. 6b). We have stated that the heat transfer and flow characteristic vary based on the structure of the vortices. However, it would be insufficient to reach this conclusion without illustrating the flow structures caused by the vortices. Visualizing the flow structure and heat transfer within the microchannel is an effective method for this purpose. Fig. 7 illustrates streamline results based on VG angles. At a 0° angle of attack, flow separation does not occur as expected, but as the angle of attack increases, flow separation begins to exist around the VGs. Behind each VG, counter-rotating vortex pairs are formed. When the attack angle is 90°, the length of the vortex formation increases rapidly. As the angle of attack of VG exceeds 90°, the vortex formation lengths begin to shorten. The magnitudes of fluid velocity on the streamlines are represented by colors. Especially at the angles of attack close to 90°, significant increases in local flow velocities occur due to the local blocking effect of the microchannel flow cross-section by VGs. This situation increases the fluid mixing and causes the pressure loss to increase considerably. The vortex formation length is highly effective in influencing the characteristic of the flow in the microchannel. The VG geometry and the angle of attack can be used to control how much the effective vortex zone expands. But this is a limited method when considering the entire microchannel. Therefore, by increasing the number of VGs placed in the microchannel, more heat transfer surfaces can interact with the vortex structures. Fig. 8 presents the variation of streamlines depending on the number of VGs. When we consider the lowest and highest number of VG pairs, it is seen that the vortex formation lengths are shorter than the distance between the VGs. However, these are average values, as the structures formed by the vortex movements that continue periodically throughout the flow dampen each other. An inspection of the instantaneous flow features will show that the vortices are moving downwards. With the increase in the number of VGs, the total number of the swirling flow region within the wakes formed behind the VGs also increases. Flow activities occurring in this region intensify the mixing of cold and hot fluids in the downstream direction, as well as increase the number of pressure loss zones, so they are also very effective on the total

Figure 6. Effects of (a) W1/W ratio (Vf = 7.5 m/s, NVG = 5, LVG = 0.4 mm, α1= α2) (b) Re (Vf = 7.5 m/s, LVG= 0.4 mm, W1= 0.2 mm, α1= α2= 45°) on j, f and JF.

pressure loss. The length of the VG is also highly influential in the formation of vortices. The changes in the flow structure caused by the length of the VGs with 45° attack angles are depicted as streamlines in Fig. 10. Increasing the lengths of the VGs at a constant angle expands the vortex sizes and formation lengths.

Fig. 9a depicts the velocity contours obtained from parallel and perpendicular planes to the flow. The chosen planes contain different regions downstream from the VGs. Low-velocity flow regions are noticeable in the wake region, whereas high-velocity flow regions form close to the leading and trailing edges of the VGs. As the flow moves

Figure 7. Streamline of different attacks angles (Vf = 5 m/s, NVG = 5, LVG = 0.4 mm, HVG = 0.1 mm, W1 = 0.2 mm, α1 = α2 = 45°).

downstream, the velocity differences in the cross-section plane decrease as well. Fig. 9b depicts the temperature contours on the identical planes. In regions of high fluid velocity, the thermal boundary layer is also observed to be thinner. In regions of the wake with low fluid velocity, the thermal boundary layer is thicker. As the downstream flow

progresses, the thermal boundary layers in the wake region also begin to thin. Fig. 9c depicts TKE results taken on parallel planes. TKE values are quite high close to the leading and trailing edges, where flow separations and dominant shear layers occur. As the plane moves downstream, the turbulence values first spread over a larger area and then

Figure 8. Velocity contours of different number of VG pairs (a) NVG = 3, (b) NVG = 4, (c) NVG = 5, (d) NVG = 6, (e) NVG=7, (Vf = 5 m/s, LVG = 0.4 mm, HVG = 0.1 mm, W1 = 0.2 mm, α1 and α2 = 45°).

Figure 9. CFD results in different planes (a) velocity contours and streamlines, (b) temperature contours, (c) TKE contours (Vf = 5 m/s, NVG = 5, LVG = 0.4 mm, HVG = 0.1 mm, W1 = 0.2 mm, α1 and α2 = 45°). decrease due to viscous damping. The increase in turbulence value is also one of the factors contributing to the enhancement of heat transfer. Response Surface Methodology and Sensitivity Analysis In the current work, the mathematical approach to be employed for compatibility and sensitivity is created by using the Response Surface Method (RSM). Fig. 11 depicts the flowchart of the applied model. This method is applicable to thermal systems including heat exchangers,

microchannels, etc. [52,62]. The first phase is the fulfillment of an appropriate DOE approach for mathematical background and sensitivity analysis as illustrated in the flowchart [62]. The primary objective of the DOE is to generate a data set for examining the input-output relationships of design parameters. In this study, we have employed the Latin Hypercube Sampling(LHS)[63] DOE in consideration of the ranges of input parameters provided in Table

1. Since NVG values ranged from 3 to 7 and are constant,

Figure 10. Streamlines for microchannels (a) LVG = 0.175 mm, (b) LVG = 0.3375 mm, (c) LVG = 0.50 mm (Vf = 5 m/s, NVG = 5, HVG = 0.1 mm, W1 = 0.2 mm, α1 and α2 = 45°, Plane 1).

Figure 11. Flow chart of application of Response Surface Methodology.

Figure 12. Compatibility between numerical data and GARS results of (a) j and (b) f.

Figure 13. (a) W1 and LVG, α1, SVG versus JF (b) Local Sensitivity Analysis for j, f, and JF (Ref. values Vf = 7.5 m/s, LVG= 0.4 mm, NVG= 5, W1= 0.2 mm, α1= α2= 45°).

input parameters for each NVG. In this case, within the scope of DOE studies, a total of 150 analyses, 30 for each case, have been conducted.

The RSM seems to be a highly effective method for determining the impacts of all parameters on the obtained results. The RSM evaluates not only the individual

interactions between factors, but also their relations with one another. The Genetic Aggregation Response Surface (GARS) technique is considered as the RSM [64]. Due to its capacity to reach multiple response surface solutions and cross-validation, the Genetic Aggregation technique yields more trustworthy results than conventional metamodel [65]. When estimating j, f, and JF values with the GARS method using all the studied parametric and DOE results, the coefficients of determination are reported to be 0.987, 0.989, and 0.953, respectively. Fig.12 demonstrates concurrently the compatibility of j and f values derived from numerical analysis and the GARS method. Both values can be estimated with a very small margin of error using the GARS method. Both j and f values produce results within the error margin of ±5%. In this respect, the GARS method is regarded as a highly effective tool for estimating and analyzing the j and f values in the considered problem. In Fig.13a, the effects of the changes in W1 and LVG, α1, and SVG (NVG) on the JF factor are analyzed. In the description of Fig. 13, the reference values of the considered parameters are provided. In general, as depicted in Fig. 13a, the JF factor yields the best results at W1 values in the middle range. On the other hand, it is comprehended that low and high JF values produce relatively fewer results. It is observed that increasing the VG generally improves thermal-flow performance. In contrast, the JF factor is maximized at the middle values of W1 and the middle values of LVG. The JF factor augments with increasing length at low W1 intervals, as determined. On the other hand, the JF factor reaches its maximum values at W1 intervals of increasing average length. Increases of up to 25% occur in the relevant intervals. When the effect of the α1 angle is considered, as depicted in Fig.13a, the change from each W1 value tends to have roughly similar characteristics. It is observed that the α1 angle decreases between 75° and 105°. As stated previously, the pressure penalty in this angle range is significantly greater than the heat transfer enhancement, causing this characteristic. The highest enhancement is typically seen between 45° and 135° of α1. In Fig.13a, the effect of row spacing is taken into account. Examining the figure closely reveals that the average W1 and SVG values provide the greatest enhancements. In general, according to the results of the GARS method, the JF factor varies between 0.78 - 1.38. The maximum value of JF factor is obtained for 5 rows of vortex pairs whose LVG = 0.48 mm, W1 = 0.35 mm, α1 = 72° and α2 = 125° and Re = 974. The JF factor reaches its minimum value of 0.78 under conditions of Re = 2282, NVG = 7, LVG = 0.50 mm, W1 = 0.32 mm, α1 = 97°, and α2=95°. Similarly, the enhancement in heat transfer compared to the empty channel ranges from 6% to 230%. The maximum increase in Colburn - j factor occurs for Re = 969 and microchannel properties of NVG = 7, LVG=0.45 mm, W1=0.14 mm, α1=92°, α2=120°. The corresponding increase in the friction factor f is 950%. It is observed that the pressure loss changes in

terms of friction factor f in the range of 5% to 1466% compared to the empty channel. In the results obtained by the GARS method, when α1 and α2 values are 0°, the values of j and f are dominantly the minimum regardless of the dimensions of the other parameters. The relevant results are also in excellent agreement with the numerical study results. According to the results of the GARS method, local sensitivity analysis has been also applied within the scope of the study. Local sensitivity study concentrates on the sensitivity around a set of parameters and examines the local effect of variation in input factors on model response. Such sensitivity is frequently evaluated at these factor values using gradients or partial derivatives of the output functions, which means that when evaluating the local sensitivity of one input factor, the values of the other input factors are held constant. In the context of the research, sensitivity analysis has been performed to investigate the impact of five different geometrical parameters (LVG, NVG, W1, α1, α2), in addition to incoming velocity, on j, f, and JF (Fig. 13b). In general, it is comprehended that each factor has a substantial impact on the relevant outputs. However, it appears that the effects of W1 and NVG on the Colburn-j factor are minimal. The sensitivity analysis for f yields comparable results. All three output values are determined to decrease as the velocity increases. LVG is a crucial input variable and parameter for both pressure drop and heat transfer. It is evident that the difference between the smallest and largest values influences more than 30% of the three outputs. Compared to other parameters, the number of rows has a relatively minor impact on the outputs. The effect of NVG on j is approximately 15%, while its effect on f and JF is approximately 10%. Observably, the angle of attack has a significant effect on j and f. However, its effect on JF decreases. Taking into account the smallest and largest values of the effects of W1 value on JF, a result change of up to 20% is recognized.

Conclusion

In the current study, we have aimed to increase the heat transfer capacity of a microchannel equipped with vortex generators (VGs). For this, we have decorated the microchannel with rectangular VGs of different sizes, numbers, and angles of attack for heat transfer enhancement and used CFD to predict the flow structure and heat transfer. While evaluating the heat transfer and pressure loss characteristics as Colburn-j factor, friction factor f and JF factor, we have used streamline, velocity, temperature, and TKE contours to understand the effects of VGs on flow and heat transfer structure. We have performed the Genetic Aggregation Response Surface (GARS) methodology to find the optimum design for VGs by considering heat transfer and pressure loss, and to define the effect of design parameters on the results. The most essential findings of the study are as follows:

• Heat transfer is significantly enhanced by the fluid mixing produced by swirling flow of vortices. A rise in the fluid mixing is ensured by secondary flow structures, which transfer the hot fluid on the hot surface to the main flow region and the cold fluid from the main flow region to the cold surfaces. • As VG attack angles and lengths increase, the vortex formation lengths increases, and the heat transfer and pressure losses are highly dependent on the vortex formation length. Colburn-j factor increases with the thinning of the thermal boundary layer. • Colburn-j factor increases considerably, especially at the 90° angle of attack. However, since the pressure losses also increase in this case, the JF factor, performance evaluation criterion, takes maximum values at the attack angles range of 30°-45° and 135°-150° of depending on the number of VG rows. • The compatibility of less than 5% is discovered between the numerical analysis results and the GARS Methodology. • According to the GARS technique, the highest value of the JF factor is around 1.38 in the channel with Re = 974, NVG = 5, LVG = 0.48 mm, W1 = 0.35 mm, α1 = 72°, α2 = 125°. • The highest augmentation in j factor is around 270% in the case of Re = 2287 and microchannel specifications NVG = 7, LVG=0.45 mm, W1=0.14 mm, α1=92°, α2=120°, while the corresponding increase in f friction factor is 950% when compared to the empty channel. • It has been determined that both j and f values generally present minimum results for α1 and α2 = 0°. • The GARS technique is employed for sensitivity analysis, and it is comprehended that all input parameters considered have a significant impact on j, f, and JF. Furthermore, the sensitivity analysis reveals that LVG has the greatest impact on j and f values, whereas Vf is the most influential parameter on JF. We believe our research will shed light on efforts to improve heat transfer in microchannels. Despite the fact that we have carefully considered many parameters, it is always possible to design higher-performance microchannels. It may be possible to manufacture microchannels with much higher heat transfer capacity by using different types of VG geometries, changing the microchannel structure, and combining more than one flow control method (active or passive).

Nomenclature

Area, m2 Channel cross-section area, m2 Heated surface area (base area), m2 Specific heat coefficient of fluid, J.kg-2.K-1 Hydraulic diameter, m

f H h k L Nu N j JF p Pw Pr . Q Re S Th Ti To u u´ Vf W W1 x ΔTm ΔP

Friction factor [-] Channel height [m] Heat convection coefficient, W.m-2.K-1 Turbulent kinetic energy, J.kg-1 Channel length, m Average Nusselt number [-] Number of VG pairs Colburn-j factor [-] Thermal & hydraulic performance criteria [-] Pressure, Pa Channel wetted perimeter, m Prandtl number [-] Heat transfer rate, W Reynolds number [-] The longitudinal pitch between VG pairs, mm Heated surface temperature, °C Fluid inlet temperature, °C Fluid outlet temperature, °C Velocity component, m s-1 Fluctuating velocity component, m s-1 Incoming fluid velocity, m.s-1 Channel width, m The traverse pitch between VG pairs, mm Coordinate Logarithmic mean temperature difference, °C Pressure drop, Pa

Greek symbols α Angle of attack, ° Δ Difference δij Knocker delta λ Thermal conductivity, W m-1 °C-1 ω The specific dissipation, 1.s-1 μ Dynamic Viscosity, Pa.s ν Kinematic viscosity, m2 s-1 , = μ ρ-1 ρ Density, kg.m-3 Subscripts f Fluid eff Effective h Heated surface i Inlet i, j, k Direction of coordinate o Outlet 0 Empty channel (without VGs) exp Experimental studies num Numerical studies GARS Data from GARS studies s Solid t Turbulent VG Vortex generator

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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