Characteristics of MEMS Heat Sink Using Serpentine Microchannel for Thermal Management of Concentrated Photovoltaic Cells

This work explains the simulation-based study to understand the thermohydraulic characteristics (thermal resistance and pumping power) of a MEMS heat sink using serpentine microchannels employed for thermal management of concentrated photovoltaic cells; the planar dimensions of both are 1 cm by 1 cm. In this study, water is the coolant and the MEMS heat sink is constructed in silicon. Over the Reynolds number varying from 50 to 1000 and for concentration ratio of 20, the thermal resistance and pumping power of a MEMS heat sink using serpentine microchannel, in comparison with that using straight microchannel, is lower and higher, respectively; the benefit of switching microchannels outweigh the cost as up to 64% reduction in thermal resistance is achieved with just 126% rise in pumping power. Studies are done for understanding the contribution of geometry of the serpentine microchannel on the characteristics of the MEMS heat sink. Irrespective of the geometry, rise in Reynolds number leads to the rise and decrease in the pumping power and thermal resistance, respectively. For a particular Reynolds number, decrease in hydraulic diameter, rise in offset width, and decrease in offset length led to rise in pumping power and decrease in thermal resistance. The contribution of concentration ratio on characteristics of MEMS heat sink using serpentine microchannel is investigated and found to be independent of concentration ratio. Nusselt and Poiseuille numbers of serpentine microchannel are provided for the benefit of heat sink designers; these parameters are higher for serpentine microchannel in comparison with a similar straight microchannel.


I. INTRODUCTION
Solar energy is a heavily relied renewable energy source for meeting the energy demands of the world [1]. Photovoltaic (PV) cells are used for converting solar irradiance to electrical work and concentrated photovoltaic (CPV) cells are a type of PV cells capable of handling increased solar irradiance. Generally, solar irradiance on the surface of the earth is 1000 W/m 2 and regular PV cells are subjected to this solar irradiance; however, CPV cells are subjected to solar The associate editor coordinating the review of this manuscript and approving it for publication was Ravi Mahajan.
irradiance that is several orders of magnitude greater than this. Figure 1 provides the schematic of the operation of CPV cells; the natural solar irradiance is concentrated using optics which then is applied on the CPV cell to generate electrical power. As CPV cells have conversion efficiency lower than 100%, only a certain percentage of the power applied on it is converted into electrical power and the remaining power is primarily absorbed by it. The absorbed power if not removed will lead to increase of its temperature which can negatively affect the conversion efficiency of CPV cells and as well, can permanently damage CPV cells. Thus, it is important to continuously remove this unconverted power VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ from CPV cells in order for them to keep generating electrical power [2], [3], [4]. Therefore, thermal management plays a significant role in the proper functioning of the CPV cells. Liquid based cooling that involves microchannel heat sinks is an effective approach for cooling heat dissipating devices including CPV cells [2], [3], [4]. Straight microchannels are the most commonly used channel configuration of MEMS heat sinks; however, there is great interest in modifying the microchannels for enhancing the performance of MEMS heat sinks and for this, various designs have been proposed [5], [6]. Nevertheless, there is continuing interest in realizing newer designs of microchannels for enhancing the performance of MEMS heat sinks and this work proposes a microchannel design for use in MEMS heat sinks for thermal management of CPV cells.
In this work a MEMS heat sink using serpentine microchannels is proposed for thermal management of CPV cells. Figure 2.a provides the schematic of this MEMS heat sink. The MEMS heat sink consists of multiple parallel serpentine microchannels realized in a substrate that is sealed to prevent leakage. In this study, silicon substrate that is sealed with glass is used along with water as the coolant. In this study the planar area of the CPV cell and the region occupied by the serpentine microchannels is the same, at 1 cm by 1 cm. Flow initially enters the manifold and is subsequently split equally among the multiple serpentine microchannels. The arrows show the direction of flow. As the coolant moves through the serpentine microchannels, it absorbs the heat dissipated from the CPV cell thereby leading to increase in the temperature of the coolant as well as that of the substrate. Heat is transferred through the substrate via conduction and across the interface, between the substrate and microchannel, via convection. The maximum temperature of the coolant and the substrate exists in the exit plane of the serpentine microchannel. The coolant exiting each serpentine microchannel is collected in the outlet manifold and then transferred out of the heat sink as shown in Fig. 2.a. MEMS heat sinks using serpentine microchannels have not yet been considered for thermal management of CPV cells and this work is the first to do so. Additionally, the foot print of the region occupied by the microchannels of the proposed heat sink is same as that of the CPV cells unlike that commonly proposed; matching the foot print minimizes the pumping power which is crucial in making liquid-based cooling an economically viable option for CPV cells. Radwan and Ahmed [7] studied different MEMS heat sinks using straight microchannel for liquid based cooling of CPV cells that are subjected to low concentration ratio (CR = 5 -20). They considered MEMS heat sinks with single and multiple microchannels, single and two layers of multiple microchannels, and parallel and opposing flow in neighboring microchannels and layers. Based on the findings of their study, the lowest thermal resistance (R th ) is exhibited by the MEMS heat sink using two layers of multiple microchannels with parallel flow irrespective of the mass flow rates. At low mass flow rates the second lowest R th is exhibited by the MEMS heat sink using a single layer of multiple microchannels with parallel flow and at the higher mass flow rates, the MEMS heat sink using a single layer of multiple microchannels with opposing flow exhibits the second lowest R th . They identified that the maximum temperature of the CPV cell as well as its temperature variation decreased with rise in mass flow rate for all MEMS heat sinks. The MEMS heat sinks using single layer of multiple microchannels, irrespective of flow direction, exhibited the highest pumping power (PP c ) while the MEMS heat sinks using two layers of multiple microchannels exhibited the least PP c . They also identified that the thermal efficiency 1 and electrical efficiency 2 of all designs of the MEMS heat sinks increased and decreased with rise in CR, respectively; both efficiencies of the MEMS heat sinks with opposing flow are below that of other MEMS heat sinks at low mass flow rates unlike at high mass flow rates where the efficiencies are almost same for all MEMS heat sinks. Di Capua H et al. [8] studied the characteristics of a MEMS heat sink using straight microchannels with right-triangular sidewall ribs for thermal management of a 3 mm by 3 mm CPV cell subjected to CR of 1000. They considered the effect of both in-line and staggered arrangements of the ribs on the characteristics of the MEMS heat sink and characterized its performance in terms of R th , PP c , Nusselt number (Nu), and Poiseuille number (Po). Di Capua H et al. [8] carried out studies for Reynolds number (Re) ranging from 100 to 400 and over this range, the MEMS heat sink with in-line arrangement of ribs exhibited lower R th and maximum temperature of CPV cell in comparison with the MEMS heat sink with staggered arrangement. Also, PP c of the MEMS heat sink with in-line arrangement is identified to be greater than that of the MEMS heat sink with staggered arrangement. Also, Di Capua H et al. [8] calculated the ratio of electrical power from the CPV cell to PP c and found this ratio to rise with rise in Re for both the MEMS heat sinks. Soliman and Hassan [9] evaluated the influence of heat spreader dimensions, flow configuration, and nanofluids on the characteristics of a MEMS heat sink employed for purposes of thermal management of CPV cells. They considered Re up to 100 in their investigation. Three flow configurations on the basis of direction of flow in the MEMS heat sink, having a single layer of multiple microchannels, are considered by Soliman and Hassan [9]; the direction of flow through all microchannels of the MEMS heat sink is same in the first configuration, the direction of flow through the microchannels of the first half of the MEMS heat sink is opposite that through the remaining microchannels in the second configuration, and the direction of flow through the microchannels of the first and third quarters of the MEMS heat sink is opposite that through microchannels of the second and fourth quarters of the MEMS heat sink in the third configuration. For all configurations employing water, increase in planar area of the heat spreader in the low Re range decreased both the maximum temperature and the maximum temperature difference of the CPV cell while increasing the efficiency of CPV cell; however, with increase in the planar dimensions of the heat spreader, in the high Re range, the maximum temperature and maximum temperature difference of the CPV cell as well as efficiency of CPV cell diminished for all configurations. Similar trends are observed even with nanofluids, except that the performance metrics are better in the case of nanofluids (water based SiC nanofluid). The net electrical power for a particular Re, irrespective of the configurations and planar dimensions of the heat spreader, increased with change in coolant from water to nanofluid. Abo-Zahhad et al. [10] proposed MEMS heat sinks with straight microchannels for thermal management of CPV cells subjected to CR of 1000. The unique feature of these microchannels is that they have 1 to 2 stepwise changes in width along the length of the microchannel. The purpose of these changes in width is to increase the thermal performance of the MEMS heat sink. They studied five MEMS heat sinks; the first used straight microchannels without changes in width, the second used 1 step reduction in width downstream from the inlet and this resulted in splitting each of the original straight microchannels into two straight microchannels, the third used 2 step reductions in width such that the 1 st step reduction happened downstream from the inlet and the 2 nd step reduction happened downstream from the 1st step change and this resulted in splitting each of the original straight microchannels initially into two straight microchannels with each subsequently split into two straight microchannels, and the fourth used 1 step reduction in width downstream from the inlet and this resulted in splitting each of the original straight microchannel into four straight microchannels, and the fifth used straight microchannel without changes in width except that this width is smaller than that of the first. For all MEMS heat sinks, the temperature of the CPV cell decreased with rise in mass flow rate. Among the MEMS heat sinks, the lowest temperature of the CPV cell, lowest R th , and highest thermal and electrical efficiencies, for a particular mass flow rate, are achieved for both fourth and fifth MEMS heat sinks; however, the fourth exhibited better temperature uniformity than the fifth MEMS heat sink. The pressure drop ( P c ) associated with the fourth and fifth MEMS heat sinks is the highest among all in the low mass flow rate range; however, P c associated with the fifth MEMS heat sink is the highest among all in the high mass flow rate range. Ahmed et al. [11] did simulations to study the influence of employing water-based Al 2 O 3 nanofluid and waterbased SiO 2 nanofluid in MEMS heat sinks using straight microchannels for purposes of thermal management of a 1 cm by 1 cm CPV cell subjected to CR ranging from 500 to 2000. They identified that the use of water-based nanofluid as coolant lowered the R th of the MEMS heat sink below that employing water as the coolant and the reduction increased with increase in the concentration of nanoparticles; however, nanofluids exhibited higher P c than water and the P c increased with concentration of nanoparticles. They also identified that at low Re, the maximum temperature of the CPV cell with water-based SiO 2 nanofluid is lower than that of water-based Al 2 O 3 nanofluid; however, in the high Re range, the temperature is observed to be independent of the type of nanofluid. AlFalah et al. [12] used simulations to understand the characteristics of a MEMS heat sink using microscale pin-fins for thermal management of a 1 cm by VOLUME 11, 2023 1 cm CPV cell subjected to CR between 500 and 2500. They considered cylindrical pin-fins and conical pin-fins in both in-line and staggered configurations for the MEMS heat sink. They identified that rise in the Re leads to drop in the maximum temperature of the CPV cell while raising P c associated with the flow. Over the CR range considered in the study, the maximum temperature of the CPV cell is the least when employing the MEMS heat sink using in-line arrangement of cylindrical pin-fins while the highest is observed for the MEMS heat sink using staggered arrangement of conical pinfins. They identified that the thermal efficiency of the MEMS heat sinks using in-line arrangement of pin-fins rose with rise in mass flow rate while that of using staggered arrangement initially rose and then decreased after peaking with rise in mass flow rate. Regarding the electrical efficiency, it rose with rise in mass flow rate for all heat sinks; nevertheless, the heat sink using in-line arrangement of cylindrical pin-fins showed the best electrical efficiency.
One of the performance metrics of heat sinks is R th and the other performance metric is PP c . R th , and PP c are mathematically stated in Eq. (1) and Eq. (2), respectively [13]. R th consists of three components including caloric thermal resistance (R th,cal ), convection thermal resistance (R th,conv ), and conduction thermal resistance (R th,cond ). R th,cal is stated in Eq. (3) while the R th,conv is stated in Eq. (4) [13]. Regarding R th,cond , it is best determined from R th and the sum of the R th,cal and R th,conv as stated in Eq. (5). It can be concluded from Eq. (3) that R th,cal depends on mass flow rate of the coolant and specific heat capacity of the coolant. Regarding R th,conv stated in Eq. (4), it is dependent on the heat transfer coefficient (h) and the heat transfer surface area between the coolant and the substrate (A w ). Thus, R th,conv can be reduced by raising either h or A w . Raising A w can lead to rise in the size of the heat sink. With augmentation of h, it is possible to reduce R th,conv without raising the size of the heat sink [13]. With reduction in the hydraulic diameter (D hy ), h rises due to the reduction in the thickness of the boundary layer and this has been the primary motivation behind the conceptualization of microchannels for employment in heat transfer devices [13]. In recent years there has been great interest in augmenting h in microchannels and this has led to the use of non-straight microchannels in heat transfer devices [5], [6]. In non-straight microchannels there is repeated disturbance of boundary layer as well as existence of secondary flows and these lead to augmentation of h. The microchannel proposed in this work is a non-straight microchannel as presented in Fig. 2. With the rise in h there is reduction in the coolant temperature adjacent to the walls of the microchannel which in turn reduces the substrate temperature. R th,cond is dependent on the thermal conductivity of the substrate and conduction shape factor; however, it is determined based on R th , R th,cal , and R th,conv as shown in Eq. (5). Employment of microchannel, including serpentine microchannel, allows for reducing the thickness of the heat sink which leads to minimization of R th,cond . where is the specific heat capacity, q ′′ (W/m 2 ) is the heat flux applied at the bottom surface of the heat sink; c represents coolant, in represents inlet of the microchannel, b represents the base of the heat sink, w represents the interface between the coolant and heat sink, out represents the outlet of the microchannel, cal represent caloric thermal resistance, conv represents convection thermal resistance, cond represents conduction thermal resistance. As mentioned earlier, this is the first work to analyze MEMS heat sinks using serpentine microchannels for thermal management of CPV cells. The contribution of the geometric features associated with the MEMS heat sink on its characteristics is investigated in this work. These investigations are done for Re ranging from 50 to 1000 for CR of 20. In addition, the influence of CR on the characteristics of the MEMS heat sink using serpentine microchannels is investigated as well; for this, the CR is varied from 20 to 100 for Re = 50, 250, 500, and 1000. The characteristics of the MEMS heat sinks in all these simulations are evaluated in terms of R th and PP c and the characteristics of the serpentine microchannel are evaluated in terms of Nu and Po. The information on Nu and Po provided here would be particularly useful for designers who prefer to use analytical equations for designing the proposed MEMS heat sink; Mathew and Weiss [13] have provided the analytical equations for designing any MEMS heat sink. Nu and Po can be determined as stated in Eq. (6) and Eq. (7), respectively [14], [15].
where Nu (−) is the Nusselt number, D hy (m or µm) is the hydraulic diameter, k (W/m.K or W/m. • C) is the thermal conductivity, LMTD (K or • C) is the Log Mean Temperature Difference, Po (−) is the Poiseuille number, f (−) is the friction factor, Re (−) is the Reynolds number, L (m or cm) is the length of the microchannel, ρ (kg/m 3 ) is the density, and U (m/s) is the average velocity of the coolant in the microchannel.

II. MATHEMATICAL MODELING
The mathematical model of the MEMS heat sink using serpentine microchannels is detailed in this section. It is made up of the continuity equation, Navier-Stokes equations, and energy equation of the microchannel of the MEMS heat sink as well as the energy equation of the substrate of the MEMS heat sink. Equations (8) and (9) state the continuity equation and Navier-Stokes equations of the MEMS heat sink's microchannel, respectively [16]; Eq. (10) states the energy equation of the microchannel of the MEMS heat sink [16]. Equation (11) states the energy equation of the substrate of the MEMS heat sink [16]. As shown in Fig. 1 the MEMS heat sink is in direct contact with the CPV cell and thus, the temperature of the bottom surface of the MEMS heat sink is taken to be the temperature of the CPV cell.
where V (m/s) represents the velocity vector, g (m/s 2 ) represents the acceleration due to gravity vector, and µ (Pa.s) represents the viscosity. The computational domain is made up of one serpentine microchannel and the corresponding substrate as depicted in Fig. 2.b and it is commonly referred to as the repeating unit; the MEMS heat sink shown in Fig. 2.a is composed of multiple repeating units. This allows for determining the field variables in the entire device with the computational resource requirement being only a fraction of that required for simulating the entire device. The boundary conditions associated with the computational domain are detailed here. At the inlet of the microchannel, the three components of the velocity of the coolant and its temperature are known. The velocity at the inlet in the axial direction is stated in Eq. (12) while the other velocities at the inlet are zero; the temperature at the inlet is stated in Eq. (13). All three components of velocity on the walls of the serpentine microchannel are zero due to the existence of no-slip condition on the same. The thermal condition on the walls of the serpentine microchannel include continuity of temperature and heat flux and these two conditions are stated in Eq. (14.a) and Eq. (14.b), respectively. It is assumed that the coolant exits the serpentine microchannel into the atmosphere and thus, the gage pressure at the exit is assumed to be zero. Beyond the exit of the serpentine microchannel, there is no change in coolant temperature and this is a boundary condition as well.
where u (m/s) is the axial component of the velocity vector V and n represents the direction normal to the surface of interest. The wall of the substrate of the MEMS heat sink accepting the power rejected from the CPV cell is the bottom wall of the MEMS heat sink as can be identified from Fig. 2; power in the form heat is rejected from the CPV cell to the heat sink. No CPV cell is considered in this study and thus, whole of the solar irradiance on the CVP cell is assumed to be rejected from it in the form of heat. The characteristics of the MEMS heat sink as evaluated would be same as that evaluated with the actual heat flux; this is because the performance metrics, i.e. R th and PP c , of the MEMS heat sink are independent of the heat flux. Though the inclusion of CPV cells would represent the actual scenario, the fact remains that the CPV cell acts as the heat source for the heat sink and thus, it can be represented by a heat flux condition on the surface of the MEMS heat sink. Similar approaches are the standard while designing heat sinks for microelectronic chips. Several studies have considered the presence of the CPV cell while evaluating the characteristics of the heat sink and for this, they assume typical values for the efficiencies, temperature of the ambient, h between the ambient and the surface of the CPV cell, thermal coefficients, and emissivity of the surfaces [7], [8], [9], [10], [11], [12], [17], [18], [19]; these typical values are not universal as they depend on the CPV cell and thus, all findings of such an approach are only of limited value. Several researchers have carried out studies of heat sink for CPV cells without the inclusion of CPV cells and in these cases, heat is assumed to be directly applied on the heat sink [20], [21]. The heat flux applied on the bottom surface of the heat sink, i.e. surface in contact with the CPV cell, of this study is stated in Eq. (14.b). The other walls of the repeating unit of the heat sink are assumed to be either insulted or adiabatic.
Temperature uniformity of CPV cells is an expectation from the thermal management of the same. The temperature uniformity of the bottom surface of the MEMS heat sink is an indication of the temperature uniformity of the CPV cell. Temperature uniformity has in the past been evaluated as the difference between the maximum and minimum temperature of the CPV cells [7], [10]. However, temperature uniformity of CPV cell would be best evaluated as the standard deviation of the temperature of its surface which is same as VOLUME 11, 2023 the bottom surface of the MEMS heat sink. This is stated in Eq. (16). The parameter N represents the number of locations of the bottom surface of the MEMS heat sink at which temperature is measured and as mentioned previously, the temperature of the bottom wall of the MEMS heat sink is same as the temperature of the CPV cell, i.e. T s | z=0 = T CPV .
Augmentation of h is usually achieved at the expense of increased PP c . Thus, it is important to be able to evaluate the benefit (reduction in R th ) with regards to the cost (increase in PP c ). John et al. [22] presented a Figure of Merit (FOM) for this and it is used in this study as well. The FOM is stated in Eq. (17) and it considers two terms; the first is R th,non (nondimensional R th ) and the second is PP c,non (non-dimensional PP c ). R th,non represents the R th of the MEMS heat sink using serpentine microchannels relative to that of the MEMS heat sink using straight microchannels. PP c,non represents the PP c of MEMS heat sink using serpentine microchannels relative to that of the MEMS heat sink using straight microchannels. Thus, higher FOM indicates better combination of design and operating conditions of the MEMS heat sink using serpentine microchannels. By shifting from straight microchannel to serpentine microchannel, R th,non is expected to be less than unity to in turn raise FOM and for this, R th,non and FOM should have an inverse relationship in Eq. (16). On the other hand, PP c,non is expected to be greater than unity, because of the shift from straight microchannel to serpentine microchannel; to reduce FOM and therefore, PP c,non and FOM should have an inverse relationship in Eq. (16). Equation (17) includes two weightages, i.e. w R th,non and w PP c,non , and these account for the contribution of R th,non and PP c,non to FOM. As with weightages, the sum of the weightage of R th,non and weightage of PP c,non is unity and for equal contribution of R th,non and PP c,non to FOM, both weightages are equal to 0.5.
R th,non = R th,serpentine R th,straight PP c,non = PP c,serpentine PP c,straight w R th,non + w PP c,non = 1 Fluent module of Ansys Workbench is used in this study for obtaining the solution of the governing equations subjected to the mentioned boundary conditions. The first step of using Ansys Workbench starts with creating a CAD (computer aided drawing) of the computational domain. In this study, the CAD is created in SolidWorks (a commercially  available CAD package) and afterwards, the CAD is imported into Fluent. The next step involves meshing the computational domain. The details of the mesh setting are provided in the next section of this paper. Once the computational domain is meshed successfully, the process for obtaining the solution of the governing equations is initiated. For this, the thermophysical properties of the entities of the computational domain, boundary conditions, solution algorithm, and convergence criteria are inputted into Fluent. In this study, water and silicon are used as the coolant and substrate, respectively. As mentioned earlier, the heat flux in this study is taken to be 20,000 W/m 2 . SIMPLE solution algorithm is selected for this work and the relative convergence criteria is set equal to 10 −5 . Once the field variables are determined, the required temperatures, inlet pressure, and mass flow rates are determined to subsequently calculate PP c , R th , R th,conv , R th,cal , Nu, and Po using the equations already provided. The temperature of the coolant used in these equations is the mass weighted average while the pressure of the coolant used in these equations is the area weighted average. The wall temperature VOLUME 11, 2023

III. GRID DEPENDENCY
The independency of the grid setting in determining the field variables is checked and detailed in this section of the article. The computational domain is populated with nodes in three different instances with the total number of nodes in each instance being greater than the previous instance. The gird setting with the least number of nodes is referred to as coarse grid setting. The grid setting with the second highest number of nodes is referred to as fine grid setting while the grid setting with the third highest number of nodes is referred to as finer grid setting. Figure 3 provides the different views of the computational domain meshed with the third grid setting. The number of nodes in the finer grid setting is at least 30% more than that of the fine grid setting and similarly, the number of nodes in the fine grid setting is at least 30% more than that of the coarse grid setting; 30% increase in number of nodes between grid settings is considered as standard in literature [23], [24], [25], [26]. Simulations are carried out for Re = 50, 250, 500, and 1000 using all the three grid settings. Two temperatures (T CPV ,max and T c,out ) and the pressure drop ( P c ) are compared for all the three grid settings as shown in Table 1. It can be noticed that the variation of T CPV ,max , T c,out , and P c between the second and third grid settings is less than 1% and for this reason, the finer gird setting is used for all simulations conducted as part of this study. As can be noticed from the next section, the finer mesh setting provides results that match very well with the experimental data. Also, the goal of this study is to obtain the average Nu and Po in the microchannel and the finer mesh setting is sufficient to meet this goal; refinement of mesh near the boundaries is not necessarily required to meet this goal.

IV. MODEL VALIDATION
Data available from literature is used for validating the model developed in this work. Fedorov and Viskanta [23] provides data on R th and Po related to a MEMS heat sink using straight microchannels and this data is used for validating the current model. The dimensions of the computational domain VOLUME 11, 2023 (substrate + microchannel) are 1 cm (length) by 100 µm (width) by 900 µm (height) while that of the microchannel are 1 cm (length) by 57 µm (width) by 180 µm (height). The heat flux applied on one of the surfaces of the computation domain is 90 W/cm 2 . Fedorov and Viskanta [27] provided experimental data for Re between 50 and 400. Figure 4 compares the experimental and simulation data of R th and Po. It can be noticed from Fig. 4 that the difference between the simulation data and experimental data are within the uncertainty of experimental data thereby validating the model. Figure 5 depicts the changes of R th , T CPV ,max , and PP c with respect to Re for MEMS heat sinks using serpentine and straight microchannels. Both R th and T CPV ,max are represented in the same plot, Fig. 5, since the former can be considered as a scaled version of the latter. It is identifiable that both T CPV ,max and R th of the MEMS heat sink using serpentine microchannels are lower than that of the MEMS heat sink using straight microchannels for all Re between 50 and 1000. This is because the R th,conv associated with serpentine microchannel is less than that associated with straight microchannel; the difference in the R th,conv , of serpentine and straight microchannels, for a particular Re, can be identified from Fig. 5.b. R th,conv is reduced in serpentine microchannels compared with straight microchannels because the h in the former is greater than in the latter for all Re. The h is higher in a serpentine microchannel compared with a straight microchannel because in the former, there is disturbance of boundary layer and existence of secondary flows. Figure 6 compares the velocity contour plot of the region of interest (ROI) of the horizontal plane passing through the middle of the serpentine microchannel for Re = 50, 500, and 1000; Fig. 6 also provides enlarged view of the velocity profile (in-plane vector plot overlaid on contour plot) in the offset width. On reviewing Fig. 6, it is identifiable that there is repeated disturbance of boundary layer in the serpentine microchannel. It is also identifiable from Fig. 6 of the existence of circulation near to the walls in a serpentine microchannel. It is stressed here that the arrows provide qualitative information rather than quantitative information and thus, comparison of the length of the arrows between figures should be avoided though comparison of length of the arrows within each figure is acceptable. Figure 7 provides the velocity profile (in-plane vector plot overlaid on contour plot), of the cross-section, at several locations along the length of serpentine microchannel. Existence of secondary flows is identifiable in these plots; once again it is stressed here that the arrows are provided for qualitative rather than quantitative purposes. Figure 5.b indicates that R th,cal is same for both serpentine and straight microchannels, for a particular Re, and this is because it depends only on the mass flow rate which in same for both microchannels. Figure 8 compares the contour plot of T c,out of the serpentine and straight microchannels for Re = 50, 500, and 1000. It can be viewed from Fig. 8 that T c,out of serpentine microchannel, for a particular Re, exhibits better uniformity than the T c,out of straight microchannel; improved temperature uniformity is indictive of improved mixing. From the simulation results, it is calculated that the standard deviation of T c,out in straight microchannel is 1.2 times of that associated with serpentine microchannel for Re = 50 and it becomes 1.4 and 1.8 for Re = 500 and 1000, respectively. Figure 5.a indicates that the PP c of the MEMS heat sink using serpentine microchannels is higher than that of the MEMS heat sink using straight microchannels. The reason for this is the increased P c that is associated with the MEMS heat sink using serpentine microchannels in comparison with the MEMS heat sink using straight microchannels. P c is higher in a serpentine microchannel due to the disturbance of boundary and existence of secondary flows unlike in the case of a straight microchannel. By switching from straight to serpentine microchannels it is possible to reduce the R th to 94% at Re = 50 and 64% at Re = 1000 while raising PP c to 110% at Re = 50 and 126% at Re = 1000. Thus, the benefits (reduction in R th ) of switching from straight to serpentine microchannels outweigh the costs (increase in PP c ) associated with such a transition over most of the Re range of 50 to 1000. Figure 5.b also provides the variation of SD T CPV with respect to Re for both serpentine and straight microchannels. It can be noticed that SD T CPV of both serpentine and straight microchannels decrease with increase in Re. Also, it can be noticed that SD T CPV of serpentine microchannel is lower than that of straight microchannel for the same Re. Thus, MEMS heat sinks using serpentine microchannels provides better temperature uniformity of CPV cell, as desired, in comparison with MEMS heat sink using straight microchannels. Figure 5.c provides the comparison between the Nu and Po of serpentine and straight microchannels; it can be noticed that the Nu and Po of a serpentine microchannel is higher than that of a straight microchannel for a particular Re. Higher Nu is observed in the case of a serpentine microchannel because of the higher h associated with it. Similarly, higher Po is observed in the case of a serpentine microchannel because of the higher P c associated with it. Also, both Nu and Po VOLUME 11, 2023 of serpentine and straight microchannels rise with rise in Re because of the associated increase in h and P c . Figure 5.c represents the variation of FOM with respect to Re for the MEMS heat sink using serpentine microchannels; the weightages are kept equal at 0.5. It can be noticed that for equal contributions of R th,non and PP c,non , the use of serpentine microchannels in MEMS heat sinks is beneficial over using straight microchannels in MEMS heat sinks. Moreover, it is identifiable from Fig. 5.c that FOM rises with rise in Re. This is attributed to the fact the rise in R th,non with Re is greater than the PP c,non with Re as can be concluded from Fig. 5.a.

V. RESULTS AND DISCUSSIONS
As already mentioned, the contribution of geometric features of the MEMS heat sink using serpentine microchannels on the characteristics of the same are investigated in this section. The geometric features considered in the study include D hy , L e , and w e . The influence of L e ( = 0.25, 0.5, and 1 mm) on the variation of T CPV ,max , R th and PP c with Re of the MEMS heat sink using serpentine microchannels is presented in Fig. 9.a. It can be identified that with rise in L e , both T CPV ,max and R th rises while PP c reduces. The rise in R th with rise in L e is because of the associated rise in R th,conv . Figure 9.b shows the variation of R th,conv and R th,cal with respect to Re and it can be noticed that R th,conv rises with rise in L e while R th,cal remains independent of L e . The rise in R th,conv with L e , for a particular Re, is due to the reduction in h which in turn is due to the reduction in the frequency of boundary layer disturbance and initiating secondary flows. Figure 13 compares the contour plots of T c,out for the three different serpentine microchannels under consideration and it can be noticed that the uniformity of T c,out improved with decrease in L e which is indicative of improved mixing; at Re = 1000, the standard deviation of T c,out of serpentine microchannel with L e = 0.5 mm and L e = 1 mm are calculated to be higher than that of the serpentine microchannel with L e = 0.25 mm by 1.23 and 1.42 times, respectively. PP c decreases with rise in L e as can be observed from Fig. 9.a. and this is due to the associated reduction in P c which in turn is due to the reduction in the frequency of boundary layer disturbance and initiating secondary flows. It can be seen from Fig. 9.c that rise in L e , for a particular Re, reduces Nu and Po associated with the serpentine microchannel. The decrease in Nu and Po with rise in L e , for a particular Re, is due to the associated decrease in h and P c , respectively. Similarly, it can be identified from Fig. 9.c. that both Nu and Po rises with rise in Re for a particular L e . This is due to the rise in h and P c with Re for a particular L e . For the 400% change (0.25 mm to 1 mm) in L e , R th increased by ∼25% at the highest Re; the change in R th at the lowest Re is negligible. VOLUME 11, 2023 Regarding the PP c , it decreased by ∼7% at the lowest Re and by ∼16% at the highest Re.
The influence of w e ( = 5, 10, 15, 20, 25, and 30 µm) on the relationship between Re and T CPV ,max , R th , and PP c of the serpentine microchannel is analyzed using Fig. 10.a; it can be noticed that rise in w e reduces both T CPV ,max and R th while raising PP c for a particular Re. This is because of the fact that rise in w e , for a particular Re, result in reduction of R th,conv as can be noticed from Fig. 10.b. It can also be noticed from Fig. 10.b that R th,cal is independent of w e and thus it does not contribute to the reduction in R th with w e . The reduction in R th,conv with rise in w e , for a particular Re, is due to the associated increase in h. Regarding the rise in PP c with rise in w e , it is due to the associated rise in P c . With rise in w e there is rise in the extend of the disturbance of boundary layer along with the rise in the intensity of secondary flows and these lead to the observed rise in h and P c . Figure 14 compares the contour plots of T c,out for different w e at Re = 50, 500, and 1000 and it is identifiable that rise in w e improves the uniformity of T c,out . The standard deviation of T c,out decreased with rise in w e for a particular Re and this is indicative of the improvement in mixing with rise in w e . Figure 10.c provides the variation of Nu and Po with Re for different w e and it can be noticed that rise in w e leads to rise in both Nu and Po. The reason for the observed relationship between Re and Nu as well as that between Re and Po reflects the relationship between Re and h as well as between Re and P c , respectively. It can be identified from Fig. 10.a that there exists a threshold for w e beyond which no decrease in R th happens for further rise in w e . The reason for this is the saturation of the effect of the disturbance of boundary layer and secondary flows on the augmentation of h as indicated through Nu as shown in Fig. 10.c. Corresponding to the 600% increase in w e , R th decreased by ∼26% at the highest Re while at the lowest Re, the change in R th is negligible. For the same increase in w e , PP c increased from ∼13%, at the lowest Re, and ∼34%, at the highest Re. Figure 11.a compares the T CPV ,max , R th and PP c for different D hy (= 125, 150, 175, and 200 µm) over the Re range of 50 to 1000. It is identifiable that rise in D hy raises both T CPV ,max and R th while decreasing PP c . Figure 11.b indicates that R th,cal is less than R th,conv , for a particular D hy , over most of the Re range considered in the study. Also, it can be identified from Figure 11.b that the variation of R th,conv and R th,cal with Re and D hy is not similar; for a particular Re, the variation in R th,cal with rise in D hy is much less than the variation in R th,conv with rise in D hy . These two facts coupled leads to R th , which is primarily the sum of R th,cal and R th,conv , rising with rise in D hy as observed in Fig. 11.a. It can also be noticed from Fig. 11.a that rise in D hy leads to decrease in PP c . With rise in D hy there is reduction in the P c which in turn is the reason for the observed reduction in PP c with rise in D hy . Figure 11.c represents the variation of Nu and Po with respect to D hy and Re. As expected, the Nu and Po, for a particular Re, increased with decreased in D hy ; this behavior is because decrease in D hy leads to increase in h and P c , for a particular Re. Also, it can be noticed from Fig. 11.c that rise in Re leads to rise in both Nu and Po for a particular D hy and the reason for this is the increase in h and P c with rise in Re. For the 160% increase in D hy considered in this study, R th increased from ∼6%, at the lowest Re, to ∼30%, at the highest Re. Correspondingly, PP c decreased by ∼68% to ∼75%. Figure 12 shows the influence of CR on the R th and PP c of a MEMS heat sink. It can be identified that R th and PP c are both independent of CR. By definition, R th is the temperature difference per unit heat flux and thus, R th of a specific MEMS heat sink while operating at a particular Re should be independent of heat flux which in Fig. 12 is equivalent to CR. On the other hand, PP c is observed to be independent of CR as the former depends on parameters such on the velocity of flow and dimensions of the microchannel which are independent of CR.
This study is dedicated to understanding the influence of different parameters on the characteristics of the MEMS heat sink; the purpose of this study is not the optimization of the proposed MEMS heat sink. Optimization will lead to identifying the set of dimensions that will achieve the objective function while satisfying the constraints. However, objective function and constraints of any MEMS heat sink are specific to the application and thus, the set of dimensions that result from optimization will be application specific. Nevertheless, the process detailed in this work can be used for carrying out optimization once the objective function and constraints are set. Few examples of objective function of a MEMS heat sink are lowest pumping power, lowest operating cost, and lowest thermal resistance and few examples of the constraints of a MEMS heat sink are maximum temperature of the substrate, limits of the dimensions of the microchannel, and maximum pressure drop.

VI. CONCLUSION
This work presents a MEMS heat sink for liquid based cooling of concentrated photovoltaic cells and as well studies the working of the same, over the Reynolds number ranging from 50 to 1000, using Fluent of Ansys Workbench. The MEMS heat sink uses serpentine microchannels; structurally the MEMS heat sink comprises of a silicon substrate in which the serpentine microchannels are created and it is sealed with glass. Using serpentine microchannels, it is possible to reduce the thermal resistance of the MEMS heat sink and this allows for reducing the temperature of the concentrated photovoltaic cell below that possible with a MEMS heat sink with straight microchannels. This reduction in thermal resistance is achieved at the expense of additional pumping power. In addition, simulations are carried out as part of this study to understand the influence of several geometric features of the serpentine microchannel.
Reduction in the offset length, for a particular Reynolds number, reduces and raises the thermal resistance and pumping power, respectively. This is because with reduction in offset length there is rise in heat transfer coefficient and pressure drop and this is related to the rise in the frequency of disturbing the boundary layer and initiating the secondary flows. Rise in Reynolds number, for a particular offset length, reduces and raises the thermal resistance and pumping power, respectively; this is due to augmentation of heat transfer coefficient and pressure drop which in turn is due to the enhancement in the disturbance of boundary layer and secondary flows. The Nusselt and Poiseuille numbers of the serpentine microchannel rose with rise in Reynolds number, for a particular offset length, and this is due to the enhancement in the disturbance of boundary layer and secondary flows. Also, the Nusselt and Poiseuille number rises with reduction in offset length, for a particular Reynolds number, due to the associated increase in heat transfer coefficient and pressure drop.
Regarding the offset width, this study has revealed that rise in the offset width, for a particular Reynolds number, raises the pumping power while decreasing the thermal resistance. This is attributed to the increase in pressure drop as well as heat transfer coefficient with rise in offset width. The increase in pressure drop and heat transfer coefficient with respect to rise in offset width is attributed to the enhancement in the disturbance of boundary layer and the secondary flows. It is also revealed that rise in Reynolds number, for a particular offset width, raised the pressure drop and reduced the thermal resistance of the MEMS heat sink and this again is attributed to the associated rise in pressure drop and heat transfer coefficient. With rise in Reynolds number for a particular offset width, both Poiseuille number and Nusselt number increased due to the associated rise in heat transfer coefficient and pressure drop. It is also observed that rise in the offset width, for a particular Reynolds number, raised the Nusselt and Poiseuille numbers of the serpentine microchannel and the associated rise in heat transfer coefficient and pressure drop are the reasons for this as well.
The influence of hydraulic diameter of the serpentine microchannel on the thermal resistance and the pumping power of the MEMS heat sink has been studied in this work. It is identified that reduction in the hydraulic diameter, for a particular Reynolds number, leads to reduction and rise in the thermal resistance and pumping power, respectively. The reason for this is the rise in heat transfer coefficient as well as pressure drop which in turn are due to the enhancement in the disturbance of boundary layer as well as secondary flows. Also, it is identified from this work that rise in the Reynolds number, for a particular hydraulic diameter, decreases and raises the thermal resistance and pumping power, respectively. This trend is attributed to the increase in heat transfer coefficient and pressure drop with Reynolds number. Regarding the Nusselt number and Poiseuille number, they are observed to rise with reduction in hydraulic diameter, for a particular Reynolds number, as well as with rise in Reynolds number, for a particular hydraulic diameter, due to the associated rise in heat transfer coefficient and pressure drop.