In present work, the entropy generation minimiza tion technique (EGM) is applied to study the performance of a microchannel heat sink combined with a new proposed parameter called irreversibility index and energy harvesting concept. Three different cases ha ve been investigated using geometry of a microchanel heat sink selected from experimental work in the literature. The constraints considered in this study, are fixed channel height and maximu m pressure drop. It has been observed that with fixed chan nel height constraint, while the as pect ratio changes from 1 to 10, the optimum operating condition fa ll in the range of Reynolds number equal to 2000 and aspect ratio of 2.25. Moreover, the extra constrain on ma ximum pressure drop imposes a limitation on applicable aspect ratio range. The maximum aspect ratio of the channel for stable flow field in this case cannot be higher than 5 impos ed by criteria of laminar flow regime. The obtained optimum values are Reynolds number of 1850 and aspect ratio of 2. Us ing a combined new defined irreversibility index and Energy Harvesting Concep t (EHC), it has been shown that the optimum design values for industrial applications are not necessary ones obtained from EGM method and may shift to a new operating point based on th e method considered for energy harvesting. Keywords: Entropy generation, Energy harvesting, irreve rsibility index, Microchannel heat sink, design constraints Introduction As the electronic devices get smaller, the heat diss ipation rate gets larger and special treatments are needed to handle heat management. One of the solutions proposed by Tuckerman & Pease [1] is using microchannel heat sink. The advantages of microchannels ar e their smaller size, minimum coolant amount, high surface to volume ratio and lo w cost of maintenance and manufacturing. On the other hand, the most negative drawback of microchannel heat sinks that restricts their application is, the high pressure drop penalty paid for enhancing heat transfer rate. Since the idea of microscale heat sink wa s proposed, several experi mental and numerical investigations have been conducted to inspect fluid flow and heat transfer within such small scale devices. Early researchers attempted to answer the question whether flow fiel d and heat transfer in microchannels follow the conventi onal fluid flow theories. Desp ite some early discrepancies reported around 1990s [2-7], later and most recent e xperimental and numerical investigations, have proven that for incompressible fluid flow within the microchannels and tubes, Navier-Stokes equation can be applied both for sing le and multiphase flows with high level of confidence [8-12]. Optimization of thermal device and heat exchangers is always an interesti ng topic in engineering field. Enhancing heat transfer rate while the pr essure drop is fixed is the main goal of most optimization studies. Shortly, after microchannel heat sink idea was emerged, analytical and numerical studies have been performed to en hance the device performance through geometry optimization. Kim [13] compared three different m odels used to decrease the thermal resistance of microchannel heat sink within a fixed pressure dr op. Garimella and Singhal [14] and Jang and Kim [15] analyzed experimentally th e pumping requirements of microchannel heat sinks and optimized the size of the microchannels for minimum pumpi ng requirements. Jang and Kim [15] showed that the cooling performance of an optimized microc hannel heat sink subject to an impinging jet is enhanced by about 21% compared to that of th e optimized microchannel heat sink with parallel flow under the fixed-pumping-power condition. Besides, the conventional method of enhancing thermal performance in a fixed pumping power, the second law of thermodynamics has been also applied to microchannel through entropy minimization technique. Cruz et al . [16] found that the best configuration wa s a copper aluminum heat sink with ammonia gas under considering thr ee materials( silicon, aluminum, copper) and two working fluids( air, ammonia gas). Khan and Cu lham [17] used the EGM method to optimize the heat sink by considering thermal and hydraulic re sistances simultaneously. They obtained the best design but very high penalty for pressure drop was paid. From the literature, it can be seen that most studi es that utilized the second law of thermodynamics on optimization of micro-channel heat sinks geomet ry, tried to decrease the heat resistance while high pumping power and pressure drop were re quired. The high pumping power means high flow rate that force the microchannels to work in turbul ent flow regime which is not practical. From the reported results available in the literature, even in a reasonable fixed pr essure drop, the required optimum volume flow rate restricts the application of such devices in real life if liquids such as water are considered as working fluid. On the othe r hand, most previous studies investigated micro- scale heat sinks with gases and there is little wo rk done on EGM optimization of heat sinks dealing with liquids such as water as working fluid. Moreover, the analysis based on the second law of thermodynamics ignores the fact to tie the obtained optimum results to real world applicatio n of microchannel heat si nks. The optimum results obtained from EGM are not applicable in real life , unless an efficient energy harvesting process is introduced. Without combining the EGM results w ith efficient energy harvesting techniques, the obtained optimum geometries can be interprete d only as hypostasize ideal values without any significant industrial use. In this study, the second law of thermodynamics is applied to a microchannel heat sink with water as working fluid and an analytic al expression is derived for en tropy generation rate. The entropy generation rate consists of two competing term s, entropy generation due to pressure drop and viscous friction as well as entr opy generation due to heat transfer through finite temperature difference. In order to optimize the device geometr y, the rate of entropy gene ration is considered as the objective function and then it is minimized to get the best solution based on the imposed constraints. Effort has been added to tie the results obtained from EG M method to engineering application through a new proposed irreversibility index and ener gy harvesting concept. With combined approach, EGM and Energy Harvesting c oncept, it has been obser ved that the optimum design values for heat sinks may shift to a ne w operating condition more suitable for real word industrial application. 1. Geometry and Physical model Figure 1, shows the heat sink geometry used in this study. Due to symmetry, only one channel has been analyzed and then the results are generalize d to whole heat sink geometry by considering the total number of parallel channels in heat sink, “N”. The top surface is assumed as perfectly insulated. Fig.1) Schematic of the microchannel heat sink geometry used in this study The height of channel is H c, the width is Wc, the lateral solid thickness is depicted by t. The thickness of base solid part is considered to be tb. The total length of the ch annel is considered as L, and the total width of the heat sink is Wtotal. From the geometrical constraint, it can be shown that the number of channels in the heat sink, N, is: total cWNWt=+ (1) The first and the second law of thermodynamics as well as fundamental thermodynamic relation, for a differential element of length “ dx” along the channe l axis, leads to: "*( )cdhqWtmdx+= (2) "*( ) gen c gen bdS qWt dsSmdx dx T+′== −  (3) 1 fdh ds dPTdx dx dx ρ=+ (4) Where, q" is heat flux, m is the mass flow rate for a single channel (Totalm N), P is the pressure, ρ is the fluid bulk density, “h” is the enthalpy of the working fl uid (water is this study), T b is the temperature of the base solid ch anging with x along the channel. T f , is the bulk temperature of the fluid at any location, x. T W is the average temperature of macr ochannel walls which is considered constant along the perimeter of the cha nnel at each longitudinal location, x. Combining equations (2), (3) and (4), the foll owing relation between en tropy generation, heat flux and pressure drop is obtained for the heat sink. "*( )gen b f gen c bf fdPmdS T TdxSq W tdx T T T ρ−′== + −  (5) The side walls of the channel can be treated as thin fins with adiabatic tip. It is a reasonable approximation as the thickness of side walls are mu ch smaller than the channel length, (t << L in Fig. 1) and they behave like fins. Figure 2 also de picts the equivalent thermal resistance network for the differential element of lengt h “dx” along the heat sink length. Equations 6 and 7 present the equivalent thermal resistance and local he at flux transferred to working fluid. () () () () ()__ __ 1 __ _ _1 1 2 11 1 2b base sc bottom wall conv ave c side walls conv ave c f b total base bottom wall conv side walls conv s c ave c c ftRkW t d x RhW d x R hHd x tRd xRRR k W t hW Hη η−=+ = = ⎛⎞ =+ + = +⎜⎟⎜⎟ + +⎝⎠ (6) () () ()1 2bf bf c b total sc ave c c fTT TTqW tt R kW t hW H η−−+= = ++ + (7) Where, h ave , is the convective heat coefficien t at any location x along the channel. ηf is the fin efficiency (side walls of the channel th at are considered as thin fins). k s , is the thermal conductivity of heat sink base material that cons idered to be copper in this study. Fig. 2) Equivalent thermal resistance network for a differential element of length “dx” along the heat sink length investigated in this study (see Fig. 1) Substituting Eq. (7) in (5), the correlation for en tropy generation rate is changed as Eq. (8). ()."2 ' "*( ) genc total gen f total c f fdPmqWt R dS dxSdx T qR W t T T ρ+== −⎡⎤ ++⎣⎦  (8) Assuming uniform heat flux over the heat sink length and using the Darcy friction factor, Temperature of the fluid at a ny location x as well as pressure drop are obtained, Eq. (9): " , 2 2 2*( )() 1 22c ff i n p c hh cqWtTx T xmc dP dP f mfV m A Vdx D dx D Aρρρ+=+ =⇒ = ⇒ =  (9) The final form of entropy generation rate for th e differential element of length dx is obtained by combining equations (9) and (8). (). "2 3 ' 22 " " ,*( ) 2 *( )()gen c total gen hc f total c f f c ff i n pdS qWt R fmSdx D A T qR W t T T qWtTx T xmcρ+== −⎡⎤ ++⎣⎦ +=+  (10) Total entropy generation due to fr iction and heat transfer for whol e microscale heat sink, is obtained by taking the integral from both sides in Eq. (10) microchannel. ' 0L gen genSS d x=∫ (11) In order to solve this integral and calculate the entropy generati on rate, the following correlations have been used in this study to evaluate Nusselt nu mber and friction factor in microscale heat sinks. Friction factor [17]: () () ()2 2 ,0 0.57 2 3453.2Re Re Re Re 96 1 1.3553 1.9467 1.7012 0.9564 0.2537c ave L fd c h fdff L D f α αααα−⎡⎤ ⎢⎥⎢⎥=+ ⎢⎥⎛⎞⎜⎟⎢⎥⎝⎠⎣⎦ =− + − + − (12) Where, α is the channel aspect ratio, c cH Wα= . Nusselt Number [18]: ** 44 *2 13 33 22 5 1 1 2 42 3 2 3 4 231 1< <10, () 2.757*10 3.274*10 7.464*10 4.476 6.391*10 1.604*10 2.622*10 2.568*10 13.11 15.19 6.0947.301th CNu C x xCz C Where C CC Cα ααα αα ααα−−− − −− −=+ <+ =− + + + = =− + =−+− ()* *6 6 5 5 4 4 3 3 22 2 non-dimensional thermal longitudinal positionRe Pr 1.275*10 4.709*10 6.902*10 5.014*10 1.769*10 1.845*10 0.05691 h thx xD x αααα αα−−−− −−= =− + − + −+ + non-dimens ional Thermal developing length The average Nusselt number for whole ch annel is evaluated using Eq (13). 3 4 4*d ave dNuNu NuNu⎛⎞=⎜⎟ ⎝⎠ (13) () ()234 5 3 23 4 5 48.235 1 1.883 3.767 5.814 5.361 2 8.235 1 2.0421 3.0853 2.4765 1.0578 0.1861d dNu Nuαα α α α α αααα=− + −+− = − +−+− Figures 3 and 4 illustrate the co mparison between correlations (1 2)-(13) and some experimental data form literature. It can be seen that the used correlations in this study is accurate and follow the physical trend by a maximum deviation of 10 percent. 00.20.40.60.8 0 500 1000 1500Present Work, Eq. (12) Experiment, Qu and Mudawar (2002) Reynolds NumberPressure Drop (bar) Fig. 3) Comparison between experimental data [18] and Eq. (12) for pressure drop 0246810 0 500 1000 1500Present Work, Eq. (13) Experiment, Kim (2016) Reynolds NumberNusselt Number, Nu Fig. 4) Comparison between experimental data [19] and Eq. (13) for Nusselt number 2. Optimization case studies The main assumption for all case studies in this work, are: a: Water as working fluid, b: Rectangular cross section for mi crochannels in heat sink with aspect ratio range of 1 to 10. c: Laminar flow regime with maximum Reynolds number of 2000. d: Incompressible fluid flow with temperature dependent properties of () , () ,()f ff TT K Tρμ . e: Insulated top surface for heat sink (see Fig. 1) with no heat loss with surrounding environment 2.1) Case I To apply the EGM technique, we first considered the geometry of microchannel heat sink used in the experimental work by Qu and Mudawar [ 18]. They used a recta ngular cross-sectional micro scale heat sink with aspect ratio of 3 and Reynolds number range of 130-1700. The channels were made of copper and deionized wa ter was used as the working fluid. Table 1 summarizes the geometry of micro heat sink us ed in experimental study by Qu and Mudawar [18]. Table 1) The channel geometry, Reynolds number range and heat flux used in Case I, [18] Hc ( mμ) Wc ( mμ) N (#Channels) Re q"(2w cm) 713 231 21 130-1700 100 The entropy generation rate is calculated by ta king the integral in Eq . (11) simultaneously considering equations (12) and (13). The aim is to find the optimum mass flow rate and corresponding Reynolds number in laminar flow regime that minimizes the entropy generation rate in heat sink. This will be the best operati ng point for the micro-channel heat sink used in the experiment based on the s econd law approach (EGM). Figure 5, illustrates the total entropy generation rate as a function of Reynolds number for heat sink in this section. It can be seen that the minimum entropy ge neration rate occurs at Reynolds number of 1250. This point is the best operating condition for the device accord ing to second law of thermodynamics (EGM technique). 10-42x10-45x10-410-32x10-3 200 500 1000 2000 Reynolds NumberEntropy Generation Rate (W/k) Figure 5) Entropy generation rate as a function of Reynolds number for case I In order to investigate further the obtained re sults, the definition of Be jan number as well as a new parameter defined by author , "Irreversibility index", ar e considered in Eq. (14). (Due to heat transfer) (Total entropy generation) work power + lost work heat removed heat applied from bottom to heat sink power *gen genSBeS total PumpingIrreversibility indexTotal Total Pumping Pψ= == = =Δ ./ work *amb genm Lost T Sρ =(14) The Irreversibility index, ψ, is an indicator of total work whic h measures the work done by pump to overcome the friction force plus the energy th at can be harvested as useful work. Figures 6 to 8, present the Beja n number, total work and irrevers ibility index as a function of Reynolds number for case I. It can be noticed that, Bejan number is high at low Reynolds number range and by increasing the Reynolds number, it gets smaller. At low Reynolds number, The entropy generation within the heat sink is dominated by heat transfer mechanism, but at Reynolds number larger than 1250, the irre versibility due to fluid friction becomes more significant than the entropy generation due to heat tran sfer. From figures 5 and 6, it can be noticed that the best operating point obtained from second law approach (EGM) is almost coinci des with the transition point (Bejan number=0.5), where entropy generation rate changes from heat dominated to pressure dominated. 00.20.40.60.81.0 0 500 1000 1500 2000 Reynolds NumberBejan Number Figure 6) Bejan number as a function of Reynolds number for case I 01234 0 500 1000 1500 2000 Reynolds NumberTotal Work (W) Figure 7) Total work defined in Eq. (14) as a function of Reynolds number Based on the second law of themodynamics, the obtained value for Reynolds number from EGM technique in this section, corre sponds to the best thermal desi gn point (optimum Reynolds number or mass flow rate) that minimi ze entropy generation rate in heat sink for removing constant amount of heat. To further investigate this conclusion, author defined ψ, irreversibility i ndex in Eq. (14). The entropy generation times ambient temperature, gi ves the total work that have been lost during the process of removing heat in heat sink. The EGM technique and second law of thermodynamics inherently rely on Energy harvesting concept. The obtained Reynolds numb er from EGM technique in this section, RE ~ 1250, is the best operating conditi on if efficient energy harvesting techniques are designed to convert the lost work to useful power and comp ensate some portion of required pumping power. In the absence of such efficien t energy harvesting techniques in current academic and industrial environment, the auth or suggests that it is more prac tical to consider combination of lost work and pumping power as objective function to obtain the optimum design point. It can be interpreted as a bridge to fill the gap between ideal optimum condition obtained from EGM technique and optimum thermal design point for real industrial application. For example, in ideal case with EGM technique it can be assumed that some energy can be harvested using an ideal thermoelectric device working between heat sink bottom wall and bulk fluid temperatures. Using this concept, some ener gy can be harvested and used to run the pump and therefore reduce the total pumping power. The op timum operating point therefore would be one obtained from EGM i.e. Re = 1250. However, in th e microchannel heat sink in this case, no energy has been harvested; therefore, the irreversibility index has been proposed to address the total penalty paid to remove fixed amount of heat th rough heat sink. The best operating point based on irreversibility index theref ore, is the condition that has lowest total work or correspondingly lowest irreversibility index, see Fig. 8. 00.010.020.030.040.05 0 500 1000 1500 2000 Reynolds NumberIrreversibility Index Figure 8) Irreversibility index defined in equation 14 as a function of Reynolds number As it can be seen from figure 8, the optimum desi gn point shifts from Reynolds number of 1250 to 450 if total lost work is c onsidered as objective function and no energy is harvested. It can be noticed that that the obtained results from EG M technique (Rey ~ 1250) does not necessarily coincide with the optimum values from i rreversibility concept. In a nut shell, the energy harvesting concept and defined irreversibility i ndex indicate that the optimum design values emerged from EGM technique is reliable if there are efficien t methods available for energy harvesting. Otherwise, the optimum values can be interpreted as the ideal optimum parameters that cannot be used in industrial environment. Without an efficient energy harvesting method, the author suggests using the irreversibility in dex as objective function, since it minimizes th e total work in the process, (pumping power and lost work). 2.2) Case 2, Optimization of channel Aspect Ratio 2.2.1) Case A, Fixed Channel Height In the section 2.1 (Case I), the en tropy generation rate for heat sink used by Qu and Mudawar [18] in deferent Reynolds numbers has been inves tigated and optimum working condition has been obtained using EGM technique. In this section, the effect of channel geometry on its performance is studied. For this purpose, first, the height of channel is kept constant and the aspe ct ratio changes from 1.25 to 9.75. The main goal is to determine in what aspect ratio, the minimu m entropy generation rate o ccurs and then at that optimum aspect ratio, the optimum mass flow ra te and corresponding Reynolds number is evaluated by EGM technique. The minimum entropy generation rate within the channels with different aspect ratio and constant height is presen ted in figure 9. It can be seen th at the aspect ratio of 2.25 is the best design point with constrai nt of constant channel height and objective function of lowest entropy generation. 00.00040.00080.0012 1 4 7 10 Aspect RatioEntropy Generation Rate (W/k) Figure 9) Entropy generation rate as a function of channel aspect ratio, fixed channel height Constraint In order to further investigate th e channel with optimum aspect rati o of 2.25, the entropy generation in different Reynolds numbers has been calculated and shown in figure 10. Figure 10 illustrates that the optimum Reynolds number in a channel with opt imum aspect ratio of 2.25 is 2000 if the flow regime considered laminar. It is consistent with the result in figure 11. As it can be seen, the Bejan number is still higher than 0.5 which means that for all Reynolds number in this aspect ratio the entropy generation due to heat transfer is domin ant and therefore increa sing the Reynolds number leads to transferring heat in a lo wer temperature difference. It mean s that the irreversibility reduces in the heat sink. 00.0020.004 0 500 1000 1500 2000 Reynolds NumberEntropy Generation Rate (W/k) Figure 10) Entropy generation rate as a function of Reynol ds number for channel with optimum aspect ratio of 2.25 0.50.60.70.80.91.0 500 1000 1500 2000 Reynolds NumberBejan Number Figure 11) Bejan number as a function of Reynolds number for case with optimum aspect ratio of 2.25 In order to see if the optimum working point obtained fr om EGM method coincides with optimum design point based on the irreversibility index, the lost work and irreversibility index have been shown in figures 12 and 13, respectively. It can be seen that the best operating condition based on the irreversibility index is ar ound Reynolds number of 700. It indicates that with no energy harvesting the optimum design point fall around Reynolds number of 700 instead of 2000. This fact results in a large pumping power difference as mass flow rate will decrease significantly within heat sink. 00.51.01.5 0 500 1000 1500 2000 Reynolds NumberLost Work (W) Figure 12) Total work defined in equation 14 as a function of Reynolds number for case with optimum aspect ratio of 2.25 00.0050.0100.0150.0200.025 0 500 1000 1500 2000 Reynolds NumberIrreversibility Index Figure 13) irreversibility index defined in Eq. (14) as a fu nction of Reynolds number for case with optimum aspect ratio of 2.25 2.2.2) Case B, Maximum pressu re drop constraint at Fixed Channel Height In industrial environment and real application of microcannel heat sinks, pressure drop is a main constraint limiting the performan ce of micro scale heat sinks. Th e overall pressure drop is high as the ratio of surface area to volume is very high in microscale devi ces. Therefore, it is assumed that the maximum pressure drop should not exceed 0.5 bar in this section. This constrain is applied to instigate the optimum aspect ratio and working condition for microscale heat sink presented in Fig. 1. From figure 14, it can be seen that if Reynol ds number is allowed to increase up to 2000 and flow field is considered laminar, the pressure drop constraint of 0.5 ba r, reduces the highest practical channel aspect ratio to 5 instead of 10 . For higher aspect ratios, the pressure drop within heat sink exceeds the threshold of 0.5 bar before optimum Reynolds number is reached. It can be inferred that for aspect ratio higher than 5, the maximum pu mping power associated with maximum pressure drop threshold (0.5 bar) is not sufficient enough to keep stable flow filed within microchannel heat si nk before minimum entropy generatio n rate appears. It is also clear that by applying the extra constraint on pres sure drop, the optimum as pect ratio shifts from 2.25 to 2.0. Entropy generation and Bejan number within a ch annel with aspect ratio of 2 and maximum pressure drop constraint of 0.5 ba r, have been presented in figures 14 and 15. It can be seen that the extra constraint on pressure drop change s the optimum working condition from Reynolds number of 2000 to almost 1850. 00.00050.00100.0015 1 4 7 10Maximum Pressure Drop Constraint at constant channel height Constant Channel Height Constraint Aspect RatioEntropy Generation Rate (W/k) Figure 14) Effect of aspect ratio on entropy generation rate at constant channel height with and without maximum pressure drop constraint 00.0020.0040.006 0 500 1000 1500 2000Maximum Pressure Drop Constraint at constant channel height Constant Channel Height Constraint Reynolds NumberEntropy Generation Rate (W/k) Figure 15) Entropy generation rate as a function of Reynol ds number for optimum aspect ratio at constant channel height with and without maximum pressure drop constraint 0.60.70.80.91.0 0 500 1000 1500 2000Constant Channel Height Constraint Maximum Pressure Drop Constraint at constant channel height Reynolds NumberBejan Number Figure 16) Bejan number as a function of Reynolds number for optimum aspect ratio at constant channel height with and without maximum pressure drop constraint The lost work and irreversibility index have b een shown in figures 17 and 18 as well. it can be inferred from the figures that the best design point from irreversibility inde x approach shifts from Reynolds 700 to Reynolds 900 due to the extra constrain of maximum pressure drop. 00.51.01.52.0 0 500 1000 1500 2000Maximum Pressure Drop Constraint at constant channel height Constant Channel Height Constraint Reynolds NumberLost Work (W) Figure 17) Total work defined in Eq. (14) as a function of Reynolds number for optimum aspect ratio at constant channel height with and without maximum pressure drop constraint 00.010.020.03 0 500 1000 1500 2000Maximum Pressure Drop Constraint at constant channel height Constant Channel Height Constraint Reynolds NumberIrreversibility Index Figure 18) Irreversibility index defined in Eq. (14) as a function of Reynolds number for optimum aspect ratio at constant channel height with and without maximum pressure drop constraint Conclusion The performance of a microcanel heat sink has b een investigated using EG M approach. First, one experimental set-up used by QU a nd Mudawar [18] investigated to obtain the optimum working condition for mass flow rate without any constrai ns. The results show that the optimum Reynolds number is around 1250 proposed by EGM technique relying on second law of thermodynamics. Moreover, in the second case, the constraint was a pplied to channel height (F ixed channel height of 713 micrometers) while the aspect ra tio changes from 1 to 10. In th is case the optimum aspect ratio and corresponding Reynolds number have been determined as 2.25 and 2000, respectively. In the Third case, an extra constrain is applied to maximum pressure drop along the channel while the channel height was kept constant. Applying th is extra condition, the a pplicable aspect ratio range reduces to 5 instead of 10 and moreover the optimum aspect ratio and Reynolds number becomes 2 and 1850 respectively. The irreversibility index has been pr oposed to magnify the effect of energy harvesting on optimum values derived from EGM technique. The irreversibility index indicates that without energy harvesting, the optimum values obtained from EGM method can only be interpreted as ideal valu es with a little i ndustrial application. In other words, Results indicate that if an efficient thermodynamic machine is designed to harvest lost work, the best operating condition is one derived from EGM technique. The author suggestion is to combine thermoelectric devices with microscale heat sinks to harvest some portion of lost work and run the pump with less external power source.