International Journal of Heat and Mass Transfer 117 (2018) 787–798 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Optimization of thermal performance of multi-nozzle trapezoidal microchannel heat sinks by using nanofluids of Al2O3 and TiO2 Ngoctan Tran a,⇑, Yaw-Jen Chang b, Chi-Chuan Wang a,⇑ a b Department of Mechanical Engineering, National Chiao Tung University, 1001 University Road, Hsinchu 300, Taiwan Department of Mechanical Engineering, Chung Yuan Christian University, Chung-Li City, Taiwan a r t i c l e i n f o Article history: Received 28 June 2017 Received in revised form 2 October 2017 Accepted 12 October 2017 Keywords: Nanofluids Trapezoidal microchannel heat sink Substrate materials High heat flux Hydraulic diameters Novel equation Inlet-coolant temperatures a b s t r a c t In this study, a new multi-nozzle trapezoidal microchannel heat sink (MNT-MCHS) was proposed. Five substrate materials, two nanofluids with nanoparticle volume fractions, 0.1% u 1%, and channel hydraulic diameters, 157.7 mm Dh 248.2 mm, were numerically examined in detail. In addition, heat fluxes in the range of 100–1450 W/cm2 subject to inlet coolant temperature from 15 °C to 75 °C were examined in detail. A locally optimal MNT-MCHS was defined, and a novel equation was proposed for predicting the maximum temperature on the locally optimal MNT-MCHS depending on the heat flux, coolant inlet temperature, and the Reynolds number. It was found that at a Reynolds number of 900, the overall thermal resistance of a MNT-MCHS using copper as a substrate material is improved up to 76% as compared to that using stainless steel 304. The locally optimal MNT-MCHS, using TiO2-water nanofluid with u ¼ 1%, could dissipate a heat flux up to 1450 W/cm2 at a Re of 900. A minimum thermal resistance in the present study is improved up to 11.6% and 36.6% in association with those of a multinozzle MCHS and a double-layer MCHS, respectively. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction In parallel with the development of scientific technology, electronic products not only have continued to shrink their sizes, but also have added more functions and operated at even higher processing speeds. As a consequence, a huge amount of heat is generated in a rather small volume that makes traditional air-cooling heat sinks in a dilemma. Hence, liquid-cooling heat sinks had been implemented to resolve the tough thermal management in many high-power applications. A great breakthrough in the implementation of liquid-cooled heat sinks was created by Tuckerman and Pease  in 1981. They proposed microchannel heat sinks (MCHS) using water as the working fluid. One of the MCHSs could dissipate a heat flux up to 790 W/cm2 with an overall thermal resistance of 0.09 °C/W and kept a maximum junction temperature under 94 °C. Thereafter, many researchers were involved in this field due to its promisingly heat-dissipation capability. Investigations in MCHSs can be classified into three main groups. Most research efforts devoted to the first group by examining the effects of geometric parameters on the heat transfer and ⇑ Corresponding authors. E-mail addresses: firstname.lastname@example.org, email@example.com (N. Tran), firstname.lastname@example.org (Y.-J. Chang), email@example.com, ccwang@ hotmail.com (C.-C. Wang). https://doi.org/10.1016/j.ijheatmasstransfer.2017.10.051 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved. fluid flow characteristics of the MCHSs, including influences of channel shapes, channel dimensions, manifold shapes, manifold dimensions, channel layers, channel distributions, substrate dimensions and the like. The second group investigated the effects of the thermophysical parameters of the coolants on the heat transfer and fluid flow characteristics of the MCHSs, and the third group, with the fewest reports in this field, studied the effects of the thermophysical properties of the substrate’s materials on the heat transfer of the MCHSs. In the first group, a great number of the studies had concentrated on the performance of single-layer parallel microchannel heat sinks (SL-P-MCHS). However, the SL-P-MCHSs had not been widely applied due to concerns of high-pressure drop and non-uniform temperature distribution. To tailor the non-uniform temperature distribution, Vafai and Zhu  first proposed a double-layered microchannel heat sink (DL-MCHS) in 1999. The inlets of the upper layer of a DL-MCHS were located on top of the outlets of the lower layer, thereby lowering the temperature near the outlet region, and a more uniform temperature prevails on the bottom surface. However, the DL-MCHS design showed no appreciable improvements in overall thermal resistance and pressure drop as compared to those of the SL-MCHSs. To ease pressure drop penalty of the MCHSs, Boteler et al.  were the first to propose a manifold microchannel heat sink (M-MCHS). The coolant paths of the M-MCHS were shortened 788 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 Nomenclature a Abt,s b Bth Cp Dh Dc Hs Lc Lm Ln Lt MCHS Q qw Re RT Tbt,max length of the top edge of the trapezoidal channel (m) heat transfer area of the bottom wall of the heat sink (m2) length of the bottom edge of the trapezoidal channel (m) bottom thickness (m) specific heat at constant pressure (J/kg K) channel hydraulic diameter (m) channel depth (m) substrate height (m) channel length (m) model length (m) inlet-nozzle length (m) top cover length (m) microchannel heat sink heat transfer rate (W) heat flux applied on the bottom wall (W/cm2) Reynolds number thermal resistance (°C/W) maximum temperature on the bottom wall (°C) considerably when compared to those of the SL-MCHSs and DLMCHSs. In this regard, the M-MCHS remarkably reduced the pressure drop and considerably increased the temperature uniformity as compared to the SL-MCHSs and the DL-MCHSs. Although the pressure drop and temperature uniformity of the M-MCHS were conspicuously improved, it still suffers from comparatively high overall thermal resistance as compared to that of the SL-MCHSs or DL-MCHSs. For further improvements to balance the pressure drop penalty, temperature uniformity, and thermal resistance into a microchannel heat sink, Tran et al.  recently proposed a multinozzle microchannel heat sink (MN-MCHS). The key features of the MN-MCHS are those channel lengths are much shorter as compared to those of the SL-MCHS and DL-MCHS, and channel heat transfer areas are larger than that of the M-MCHS subject to the same channel shape and hydraulic diameter. They showed that the overall thermal resistance, temperature uniformity and the pressure drop of the MN-MCHS were significantly improved. Normally, water was utilized as a working fluid in MCHSs for its superior thermal-physical properties. For further augmentation of the heat-transfer capability of the coolants, Choi and Eastman  first proposed a nanofluid concept. Eastman et al.  reported that by adding 5% CuO nanoparticles, it could improve thermal conductivity by approximate 60% in comparison with that of pure water. In recent years, the number of investigations on nanofluids in the MCHSs increased drastically. For example, Hung et al. [7,8] used Al2O3-water nanofluids as coolants in a SL-MCHS and a DLMCHS, respectively. They concluded that the thermal resistance of SL-MCHS and DL-MCHS could be minimized by properly adjusting the nanoparticle volume fraction. Kuppusamy et al.  created various combinations amid nanoparticles (Al2O3, CuO, SiO2, and ZnO) and base fluids (water, ethylene glycol, and engine oil), and conducted tests in a triangular grooved microchannel heat sink (TG-MCHS). They reported that the thermal performance of the TG-MCHS with Al2O3-water nanofluid having a volume fraction of 0.04% (nanoparticle diameter of 25 nm) outperformed the straight MCHS using pure water. A MCHS using CuO-water nanofluids as working fluids was experimentally investigated by Rimbault et al. . Their results showed that the thermal performance of the MCHS using CuO-water nanofluids with u 6 1:03% was slightly increased when compared to that using water. Sakanova et al. [11,12] reported results from a DL-MCHS and a double-side MCHS (DS-MCHS) using Al2O3-water nanofluid , Ti DT Wm Ws inlet-coolant temperature (°C) temperature difference between inlet and outlet coolant (°C) model width (m) substrate width (m) Greek symbols l dynamic viscosity of the coolant (kg/ms) q density of the coolant (kg/m3) u nanoparticle volume fraction (%) Subscripts bf base fluid bt,max bottom maximum f fluid i,o inlet_outlet nf nanofluid p nanoparticle with a wavy MCHS using three different nanofluids . They showed that the thermal performances of the heat sinks using the Al2O3-water nanofluid with u ¼ 1% and u ¼ 5% were substantially better than those using water. The convective heat transfer and friction factor of Cu-water nanofluids in a cylindrical microchannel heat sink were experimentally studied by Azizi et al. [13,14]. For the same Reynolds number, the corresponding Nusselt numbers of the heat sink with u being 0.05%, 0.1% and 0.3% were enhanced approximately 17%, 19%, and 23%, compared to that using pure water, respectively. A ribbed MCHS using Al2O3-water nanofluids was numerically investigated by Ghale et al. . They revealed that the Nusselt number and friction factor of nanofluids in the ribbed microchannel were higher than those of a straight microchannel without rids inside. Nebbati and Kadja  presented a numerical study on a microchannel heat sink using Al2O3-water nanofluid. They reported that the heat transfer coefficient of a heat sink using nanofluid was significantly increased in comparison with pure water. Parametthanuwat et al.  presented an experimental study on thermal properties of silver nanofluids. They revealed that a silver nanofluid with u ¼ 1% at a temperature of 80 °C yielded the highest heat transfer enhancement. PCM slurry and nanofluid coolants were numerically investigated in a DL-MCHS by Rajabifar . The author revealed that the pumping power of both PCM slurry and nanofluid was significantly raised for their higher viscosity. A M-MCHS using nanofluids was numerically investigated by Yue et al. . They concluded that the increase in the nanoparticle diameter led to the decrease in the Nusselt number, pumping power, and performance index. Recent investigations using nanofluids in microchannel heat sinks were reported by Kim et al. , Radwan et al. , Xia et al. , Yang et al. , and Zhao et al. . The results in  showed that the thermal resistance of the flat-plate heat pipes with Al2O3-Acetone nanofluids containing sphere-, brick- and cylinder-shaped nanoparticles were decreased by 33%, 29%, and 16%, respectively, as compared to that using pure acetone. Radwan et al.  concluded that the use of nanofluids in MCHSs was quite effective. Xia et al.  revealed that the thermal conductivity and dynamic viscosity of Al2O3 and TiO2 nanofluids were both increased with the increase in u. Yang et al.  reported that the nanoparticle volume fraction and the fins aspect ratio played a very important role in the enhancement in the heat transfer of the MCHSs. Zhao et al.  reported that the pressure drop could N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 be improved by increasing the nanoparticle volume fraction and decreasing particle size. Very recently, investigations on nanofluids in MCHSs were reported by Abdollahi et al. , Anbumeenakshi and Thansekhar , Duangthongsuk and Wongwises , Sarafraz et al. , and Tafarroj et al. . In these reports, the thermal performances of the nanofluids were experimentally examined, and nanofluids with high thermal conductivities were recommended in MCHSs. Based on the foregoing reviews, it is found that the multi-nozzle microchannel heat sinks could significantly improve the overall thermal resistance, remarkably decrease the pressure drop, and considerably enhance the temperature uniformity in association with those of the SL-MCHSs, DL-MCHSs, and M-MCHSs. However, the MN-MCHSs had been only examined by using water as a working fluid with rectangular and square microchannel shapes . Recently, Tran et al.  reported performance for five different channel shapes. They found that the thermal performances of trapezoidal channel shapes with 0:33 6 c 6 1 outperformed that of a square channel shape at the same Reynolds number and hydraulic diameter. However, in their study, the height of the trapezoidal channels was fixed at 200 mm, and the trapezoidal channel is not yet implemented in MN-MCHSs. In addition, the investigated MN-MCHSs only utilized silicon and copper as substrate’s materials in two individual studies. Therefore, more studies pertaining to the MN-MCHSs should be carried out for elaboration of related details. For fulfilling and bridging the gap of the aforementioned lacking datum in the MN-MCHS field, and a strong desire for finding out a heat sink that can dissipate a higher heat transfer rate in a smaller size with a more uniform temperature and a lower pressure drop is the objective of this study. Effects of the substrate materials, nanofluids, and channel hydraulic diameters on the thermosfluids characteristics of a multi-nozzle trapezoidal microchannel MNT-MCHS in association with some optimal layouts will be reported in this study. 789 2.2. Geometric design of a new multi-nozzle trapezoidal microchannel heat sink Fig. 1(a) presents a structure of a multi-nozzle microchannel heat sink with trapezoidal channel shapes (MNT-MCHS). Gray parts on the top of the Fig. 1(a) are the top cover of the heat sink. A yellow part is a substrate. Blue and red parts are inlet and outlet channel nozzles, respectively. Fig. 1(b) presents a unit cell of the MNT-MCHS, which includes parts of the MNT-MCHS such as the substrate, channel, channel inlet, channel outlet and top cover. In this study, the unit cell was used as a computational model for analysis. The detailed dimensions of the substrate, computational model, and channels are shown in Fig. 1(c) and (d). All dimensions, which were presented by specific numbers in the Fig. 1(c) and (d), are constant in this study, while the dimensions, presented by symbolic texts, will be altered in Section 3.2. The function of the top cover is not only to cover the channel, but also to separate the inlet and outlet nozzle of the channel. Therefore, the top cover length must always exist, and its length, Lt, can be calculated by Lt ¼ Lc 2Ln : ð1Þ The substrate height, Hs, is fixed, but the channel depth, Dc, is variable in the investigative process leading to the values of the bottom thickness, Bth , to be changed depending on the channel depth, and can be calculated by: Bth ¼ Hs Dc : ð2Þ With this structure of the MNT-MCHS, the channels can be arrayed on the substrate following both row and column directions. The number of channels can be calculated for a row, Nr, a column, Nc, or a total number of the channels on the heat sink, Nt, respectively, as follows: Nr ¼ W s =W m ; ð3Þ Nc ¼ Ls =Lm ; ð4Þ 2. Methodology Nt ¼ Nr Nc : ð5Þ 2.1. Work description where Ws, Ls, Wm, and Lm are the substrate width, substrate length, model width, and model length, respectively. In the present study, the effects of substrate materials, channel hydraulic diameters, nanofluids, heat fluxes, and inlet-coolant temperatures on thermodynamic properties of multi-nozzle trapezoidal microchannel heat sinks were individually investigated in four sections. In Section 3.1, five substrate’s materials, including copper, aluminum, silicon, iron, and stainless steel 304, were examined in detail based on the following given parameters: a constant heat flux, qw = 250 W/cm2, Reynolds numbers from 100 to 900, a top-channel edge, a = 100 mm, a bottom-channel edge, b = 300 mm, a channel depth, Dc = 200 mm, a constant inletcoolant temperature, Ti = 24 °C, and three coolants, including two nanofluids Al2O3-water and TiO2-water and pure water. It is noted that the nanoparticle volume fraction was fixed at u = 0.5% in Sections 3.1 and 3.2. In Section 3.2, the channel hydraulic diameters, from 157.7 mm to 248.2 mm, were examined in detail by using the above-given parameters and altering the channel depth (Dc), from 150 mm to 350 mm. Based on the results obtained in Sections 3.1 and 3.2, a substrate material and a channel hydraulic diameter, which yielded the best thermal performance in these sections, would be selected as a locally optimal model for analysis in subsequent sections. In Section 3.3, the nanofluids with various nanoparticle volume fractions, u, of 0.1%, 0.5% and 1.0%, were examined in detail. Finally, heat fluxes in the range from 100 W/cm2 to 1450 W/cm2 and inlet-coolant temperature ranging from 15 °C to 75 °C were investigated in detail in Section 3.4. 2.3. Mathematic model For simplifying in analyzing processes, assumptions of the computational model were analogous to some previous studies [4,19,30,31]: (1) thermophysical properties of the fluids and the solid were assumed to be temperature independent; (2) the flow was assumed to be three-dimensional, single phase, steady state, incompressible and laminar; (3) the gravitational force was neglected, and there was no internal heat generation within the model; (4) the walls of the channel had a no-slip condition for velocity and temperature. According to the above assumptions and some literatures [4,7,8,12,30,32–35], the governing equation for computing thermodynamic processes in this study could be written as follows: Continuity equation @u @ v @w þ ¼ 0; þ @x @y @z ð6Þ where u, v and w are the velocity components of the coolant in x-, y-, and z-directions, respectively. 790 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 Fig. 1. A multi-nozzle trapezoidal microchannel heat sink. Momentum equations ! @u @u @u @p @2u @2u @2u ¼ þl q u þv þw þ þ ; @x @y @z @x @x2 @y2 @z2 q u @v @v @v þv þw @x @y @z ¼ ! @p @2v @2v @2v ; þl þ þ @y @x2 @y2 @z2 ð7aÞ ð7bÞ ! @w @w @w @p @2w @2w @2w þw ¼ þl q u þv þ 2 þ 2 ; @x @y @z @z @x2 @y @z ð7cÞ where l, q and p are the dynamic viscosity, density, and pressure drop of the coolants, respectively. The boundary conditions of the inlet flow are u ¼ 0; v ¼ 0 and w ¼ w0 . The boundary condition of the outlet flow is p = p0, where p0 is the pressure of the ambient environment (atmosphere pressure). 791 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 Energy equation for the coolants u kf @T @T @T þv þw ¼ @x @y @z qf cp;f 2.5. Thermophysical properties of nanofluids ! @2T @2T @2T þ þ ; @x2 @y2 @z2 ð8aÞ where T, cp;f , qf , and kf are temperature, specific heat at a constant pressure, density, and thermal conductivity of the fluid (coolants), respectively. The boundary condition for the inlet is T = Ti, where Ti is the inlet temperature of the coolant. Energy equation for the solid ks @2T @2T @2T þ þ @x2 @y2 @z2 ! ¼ 0; ð8bÞ where ks is the thermal conductivity of the solid. Boundary conditions for the solid are given as follows: the bottom wall of the substrate is a constant heat flux and the four side walls of the computational mode are symmetric while the other walls are adiabatic. For no-slip condition, uwall = vwall = wwall = 0. For a steady state condition and an incompressible flow, @@tq ¼ 0; @V ¼ 0; @T ¼0 @t @t and@p ¼ 0. @t C p;nf ¼ In the present study, three-dimensional, fluid-solid, conjugate modules of the CFD-ACE+  software package were employed to analyze the computational models. A high accurate meshing method, which was proposed in , was employed for the meshing process, and one of the meshed computational models is presented in Fig. 2(a). According to the assumptions of the computational model as presented in Section 2.3, all outer walls except the bottom wall of the computational model are symmetrical or thermally insulated, and there is no heat generation within the model. Therefore, Eq. (9) is utilized to verify all the simulated results with maximum percentage errors being less than 1.5%. ð9Þ where qw is the heat flux applied on the bottom wall of the computational model, M is the mass flow rate of the coolant passed through the channel, C p is the specific heat of the coolant, T o is the outlet-coolant temperature, and T i is the inlet-coolant temperature. Lm and W m are the length and width of the computational model, respectively. In addition, experimental results of a cylindrical microchannel heat sink, which was reported by Azizi et al. , were utilized to verify the present simulated results. The C-MCHS included 86 channels with a channel width of 0.526 mm, a channel depth of 0.6 mm, a channel length of 50 mm, and a rib width of 0.5 mm. The channels were equilaterally circularly arrayed outside of a cylinder with an outside diameter of 30 mm, as presented in Fig. 2(b). A constant heat flux was applied along the centered axis of the C-MCHS by using a single cartridge heater with an outside diameter of 10 mm and a length of 50 mm. A unit cell as presented in Fig. 2(c), including parts of the cylindrical heat sink such as the channel, substrate, rib, inlet and outlet, was utilized as a verifying model. All boundary conditions for the verifying model were applied by the experimental conditions reported in . Fig. 2(d) shows relevant dimensions of the unit cell and Fig. 2(e) presents a comparison between simulated results and experimental results by Azizi et al. . As depicted in the figure, good agreements between the simulation and measurements were achieved with a maximum deviation being less than 3%. Fig. 2(f) presents a temperature profile of the verifying model at a Reynolds number of 400. The temperature distribution of the temperature profile was similar to that reported in . ð1 uÞqbf C p;bf þ uqp C p;p qnf ¼ ð1 uÞqbf C p;bf þ uqp C p;p ; ð1 uÞqbf þ uqp ð10Þ where C p , u, and q are the specific heat at a constant pressure, nanoparticle volume fraction, and density, respectively. The subscripts, nf, bf, and p, stand for nanofluids, base fluid, and nanoparticle, respectively. It is noted that the subscripts for nanofluids mentioned here are used for all sections in this study. The density of the nanofluids was calculated by the following equation [9,19]: qnf ¼ qp u þ qbf ð1 uÞ; ð11Þ where u is the nanoparticle volume fraction. The thermal conductivity of nanofluids with spherical particles, which was proposed by Yu and Choi  as follows [2,9,14]: knf ¼ 2.4. Computational model validation qw ¼ MC p ðT o T i Þ=ðLm W m Þ; For calculations of the performance of nanofluids in microchannels, the thermophysical properties of nanofluids are needed and can be obtained from some published literatures. The specific heat of the nanofluids was calculated by using Eq. (10) [8,14,18]: kp þ 2kbf þ 2uðkp kbf Þ kbf ; kp þ 2kbf þ uðkp kbf Þ ð12Þ where k is the thermal conductivity, u is the nanoparticle volume fraction. The dynamic viscosity of nanofluids was calculated as follows [14,38,39]: lnf ¼ lbf ð1 þ 2:5u þ 6:2u2 þ . . .Þ; ð13Þ where lnf and lbf are the dynamic viscosity of nanofluids and the base fluid, respectively. u is the nanoparticle volume fraction. Specifications of nanoparticles, which were used in this study, are listed in Table 1 . Thermophysical properties of nanofluids based on water, which were examined in this study, are listed in Table 2. 3. Results and discussion 3.1. Substrate material examination To examine the effects of substrate materials on the MNTMCHS, five substrate materials, including copper, aluminum, silicon, iron, and stainless steel 304, were examined in detail. The unit cell in Section 2.2 (Fig. 1(b)) was utilized as a computational model. The given parameters mentioned in Section 2.1 were applied in analytical processes. A channel depth of 200 mm, an inlet-nozzle length of 400 mm, and a top cover length of 800 mm were considered. Three coolants, including Al2O3-water and TiO2-water with a nanoparticle volume fraction of 0.5%, and pure water, were used in the examinations. The thermophysical properties of the substrate and top cover’s materials are listed in Table 3 with Reynolds number ranging from 100 to 900: Re ¼ w q Dh l ; ð14Þ where l, w, and q are the dynamic viscosity, inlet velocity, and density of the coolant, respectively. Dh is the channel hydraulic diameter, and calculated by the following equation: Dh ¼ 4A ¼ P 2ða þ bÞDc qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 2 ða þ bÞ þ ðb aÞ þ 4D2c ð15Þ where a, b and Dc are the channel-top edge, channel-bottom edge, and channel depth, respectively. The overall thermal resistance, RT , was calculated by the following equation [1,4]: 792 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 Fig. 2. A comparison between Num. and Exp. results, which presented by Azizi et al. . Table 1 Specifications of nanoparticles. Nanoparticles Diameter (nm) Density (kg/m3) Specific heat (J/kg K) Purity (%) Al2O3 TiO2 5 5 3680 4230 79.77 69.09 99 99 793 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 Table 2 Thermophysical properties of nanofluids based on deionized water. Nanofluids u (%) k (W/mK) Cp (J/kgK) q (kg/m3) m (kg/ms) Al2O3 0.1 0.5 1.0 0.621 0.633 0.642 4163 4116 4056 1000 1010 1024 0.000990 0.000995 0.001010 TiO2 0.1 0.5 1.0 0.631 0.638 0.651 4163 4104 4034 1000 1013 1030 0.000993 0.001016 0.001019 DI water 0 0.611 4178 996 0.000859 Table 3 Thermophysical properties of substrates and top cover’s materials. Materials k (W/mK) Cp (J/kgK) q (kg/m3) Copper Aluminum Silicon Iron Stainless steel 304 PMMA 386 220 148 71.8 15.5 0.19 380 896 702 452 510 1960 8954 2707 2329 7897 7955 1420 RT ¼ T max T min T bt;max T i ¼ ; Q Q substrate with lower thermal conductivity depicts a larger temperature difference between the bottom wall of the substrate and the bottom wall of the channel. A maximum temperature of the model using stainless steel as a substrate material was much higher than that using copper as seen in Fig. 3(d). In this regard, the effective thermal resistance of stainless steel significantly exceeds other materials as shown in Fig. 3(a). A further comparison of Fig. 3(c) and (d), the temperatures on the bottom of the MNT-MCHSs were very uniform for each single substrate material. This is one of the improvements of the MNT-MCHS in comparison with those of SL-MCHS, and DL-MCHS. ð16Þ 3.2. Influence of channel hydraulic diameter where T max is a maximum temperature of the heat sink, T min is a minimum temperature of the heat sink, and Q is the heat transfer rate of the heat sink. Fig. 3(a) presents an overall thermal resistance, RT, of the MNTMCHS, using TiO2-water with u ¼ 0:5% as a working fluid, subject to the influence of substrate materials and Reynolds number (Re). As expected, RT declined consecutively with the rise of Reynolds number and the trend is in line with previous efforts . When the Re is increased from 100 to 900, the RT of the MNT-MCHS using copper as a substrate material and TiO2-water nanofluid decreased from 0.289 °C/W to 0.089 °C/W. It was found that with the same coolant and Reynolds number, the overall thermal resistance of a MNT-MCHS using a substrate material with higher thermal conductivity was smaller than those with smaller thermal conductivities. At a Reynolds number of 900 and a heat flux of 250 W/cm2, the overall thermal resistance of a MNT-MCHS using copper as a substrate material is improved up to 15.3%, 21.6%, 41.1%, and 76% as compared to those using aluminum, silicon, iron, and stainless steel 304, respectively. It is noted that the thermal conductivity of copper is higher than those of aluminum, silicon, iron, and stainless steel being 1.75 times, 2.6 times, 5.3 times, and 24.9 times, respectively. Fig. 3(b) presents the overall thermal resistance versus Reynolds number for MNT-MCHSs subject to various nanofluids. For the present nanofluids, it was found that the overall thermal resistance of a MNT-MCHS using a coolant with higher thermal conductivity was smaller than those with smaller thermal conductivities . Fig. 3(c) presents temperature profiles of the computational model using copper as a substrate material with three different coolants at a Reynolds number of 450. In this case, all boundary conditions and the substrate material are the same except the difference of the coolants. The profiles show that the bottom temperature of the model using TiO2-water as a coolant was smaller than those using Al2O3-water and pure water. This reflects the distribution of the overall thermal resistance associated with the different coolants in Fig. 3(b). Fig. 3(d) presents surface temperature distributions using TiO2-water as a working fluid with five different substrate’s materials at a Reynolds number of 450. With the same wall thickness, a To examine the effects of channel hydraulic diameter on the performance of the MNT-MCHS, the channel depth was altered from 150 mm to 350 mm that corresponds with the hydraulic diameter from 157.7 mm to 248.2 mm. To maintain a constant mean velocity of the coolant along the channel from the inlet to the outlet, the nozzle length must be satisfied with the following equation. Ln ¼ Dc aþb ¼ 2Dc ; 2a ð17Þ where Dc is the channel depth, a is the top edge of the channel, and b is the bottom edge of the channel. In this case, the inlet-nozzle length was altered from 300 mm to 700 mm. Copper and TiO2water nanofluid were selected as the substrate material and the working fluid for the simulation. Fig. 4(a) presents the overall thermal resistance versus the Reynolds number subject to the influence of channel depths. It was found that the RT of a model with deeper channel depth was smaller than those with shallower channel depths. At a Reynolds number of 900, the overall thermal resistances of the MNT-MCHS, with channel depths of 150 mm, 200 mm, 250 mm, 300 mm, and 350 mm, were 0.114 °C/W, 0.089 °C/W, 0.088 °C/W, 0.083 °C/W, and 0.076 °C/W, respectively. Fig. 4(b) shows that the pressure drop also rose with the increase in Reynolds number [14,40] and the decrease in channel depth (shallower channels) [4,41]. Fig. 4(c) presents temperature profiles of the computational model with five different channel depths at a Reynolds number of 450. Since the effective surface area rises with the steeper channel, a smaller thermal resistance is therefore expected, and the incurred maximum temperature of a deeper channel depth was smaller than those with shallower channel depth as presented in Fig. 4(c). Fig. 4(d) presents the velocity profiles of the computational model with different channel depths at a Reynolds number of 450. At the same Reynolds number, the coolant velocity in a channel with a deeper channel depth or a greater hydraulic diameter was smaller than those with shallower channel depths or smaller hydraulic diameters (Eq. (14)). The significant rise in the pressure drop, in Fig. 4(b), is associated with the considerable rise of the coolant velocity as presented in Fig. 4(d). 794 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 Fig. 3. RT vs. Re with different coolants and substrate’s materials. 3.3. Different nanofluids with different nanoparticle volume fractions examination Fig. 5(a) presents the overall thermal resistance of the MNTMCHS with different nanofluids, nanoparticle volume fractions, and Reynolds numbers. It was found that the overall thermal resistance of the MNT-MCHS using a nanofluid with a greater thermal conductivity and a greater nanoparticle volume fraction was smaller than those with smaller thermal conductivities and smaller nanoparticle volume fractions [23,24]. On the other hand, the negligible difference in the overall thermal resistances between Al2O3-water and TiO2-water nanofluids is encountered. However, a minimum overall thermal resistance of the MNT-MCHS using TiO2-water nanofluid at a nanoparticle volume fraction of 1% was improved up to 6.7% relative to that using pure water [9,11,12,16]. In fact, at a Reynolds number of 900, the minimum overall thermal resistances of the MNT-MCHS, using TiO2-water nanofluid at a nanoparticle volume fraction of 1% and using water, were 0.0761 °C/W and 0.0818 °C/W, respectively. Fig. 5(b) presents temperature profiles of the MNT-MCHS with different nanofluids and different nanoparticle volume fractions at a Reynolds number of 450. The profiles show that the maximum temperature on the bottom wall of the MNT-MCHS using a nanofluid with a higher thermal conductivity and a higher nanoparticle volume fraction was smaller than those using nanofluids with smaller thermal conductivities and smaller nanoparticle volume fractions or pure N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 795 Fig. 4. RT and pressure drop vs. Re with different channel depths (different hydraulic diameters). water, corresponding to the RT distribution presented in Fig. 5(a) [22,24]. Fig. 5(c) presents velocity profiles of the MNT-MCHS using different nanofluids with different nanoparticle volume fractions at a Reynolds number of 450. The velocity distributions show that the velocities of the MNT-MCHS using nanofluids with different nanoparticle volume fractions were insignificantly different. The velocity of the MNT-MCHS using water was slightly smaller than those using nanofluids. The velocity distributions indicated that the pressure drops among the MNT-MCHSs using nanofluids were insignificantly different [25,27], but were slightly higher than that using water. For all cases in Sections 3.1–3.3, it was found that a minimum overall thermal resistance of 0.0761 °C/W or the best thermal performance was achieved by a MNT-MCHS with a channel depth of 350 mm using TiO2-water nanofluid with a nanoparticle volume fraction of 1% as a working fluid and copper as a substrate’s material. Therefore, this model was selected as a locally optimal model and was utilized for analysis in the rest of this study. 3.4. Heat flux and inlet-coolant temperature examination To examine the effects of the heat flux and the inlet-coolant temperature on the thermodynamic characteristics of the MNTMCHS, the locally optimal model was employed for the analysis. Heat fluxes, from 100 W/cm2 to 1450 W/cm2, were examined in detail using the boundary conditions mentioned in Section 3.3 with an inlet-coolant temperature of 24 °C. Inlet-coolant temperatures, from 15 °C to 75 °C, were also examined in detail with a fixed heat flux of 250 W/cm2. 796 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 Fig. 5. RT vs. Re with different coolants and different nanoparticle volume fractions. Fig. 6(a) presents overall thermal resistances and the maximum temperatures of the locally optimal MNT-MCHS versus heat fluxes. The results show that the overall thermal resistances were independent of the heat flux ; however, the maximum temperature linearly increased with the increase in the heat flux. With the increase in the heat flux on the same heat transfer area, the heat transfer rate would be increased that leads to the increase in the temperature difference between the bottom wall of the heat sink and the bottom wall of the channels. In addition, with the increase in heat transfer rate at the same mass flow rate, the temperature difference between coolant inlet and outlet would also be increased. It was found that the locally optimal MNT-MCHS, using the TiO2-water nanofluid with a nanoparticle volume fraction of 1%, could dissipate a heat flux up to 1450 W/cm2, and kept the temperature rise above the inlet-coolant under 97.3 °C at a Reynolds number of 900. At the same Reynolds number of 900 with copper as the substrate material, the minimum overall thermal resistance of the MNT-MCHS in the present study is improved up to 11.6% and 36.6% as compared to those of a MN-MCHS with a square channel shape which was reported by Tran et al. , and an optimal DL-MCHS was also reported by Hung et al. , respectively. Fig. 6(b) presents maximum temperatures on the bottom wall of the heat sink and the temperature differences between inlet and outlet versus inlet coolant temperatures. The results show that the maximum temperature of the bottom wall of the heat sink increased with the increase in the inlet coolant temperature. It was found that the overall thermal resistance and the temperature difference between coolant inlet and outlet were independent on the inlet-coolant temperature. In this case, the heat flux, the N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 797 Fig. 6. Tmax, RT, and DT vs. heat flux and Ti. bottom wall area of the substrate, the thermophysical properties of the coolant, and the overall thermal resistance were constant; therefore, when the inlet coolant temperature increased, the heat sink changed to another thermal equilibrium, and the maximum temperature would be increased exactly equal to the increase in the inlet temperature (Eq. (16)). Based on the results obtained in this study, a novel equation for predicting the maximum temperature on the bottom wall of the optimal MNT-MCHS depending on the inlet coolant temperature, heat flux, and Reynolds number was proposed as follows: The maximum temperatures of the optimal MNT-MCHS with different heat fluxes at a Re = 900 depending on the inlet-coolant temperature, which were obtained by the proposed equation, are presented in Fig. 6(c). Maximum temperatures of the optimal MNT-MCHS with different Reynolds numbers at a heat flux of 250 W/cm2, which were also obtained by the proposed equation, are presented in Fig. 6(d). T bt;max ¼ T i þ P qw R Abt;s ; In this study, a new MNT-MCHS was proposed. A plate of 9 9 0.5 mm was considered as a fixed substrate. Effects of substrate materials, channel hydraulic diameters, nanofluids, heat fluxes, and inlet-coolant temperatures were reported in detail. Based on the present analyzed results, conclusions can be summarized as follows: ð18Þ where T bt;max is the maximum temperature on the bottom wall of the heat sink, T i is the inlet-coolant temperature, qw is the heat flux applied on the bottom wall of the heat sink, R is a thermal resistance, Abt;s is the heat transfer area of the bottom wall of the heat sink, and II is a novel dimensionless number depending on Reynolds number. For the optimal MNT-MCHS using TiO2-water nanofluid with u ¼ 1% as a working fluid, the dimensionless number, II, was proposed as follows: II ¼ 8:909 Re0:818 þ 0:042 where Re is Reynolds number. ð19Þ 4. Conclusions 1. The thermal performance of a MNT-MCHS using a substrate material with higher thermal conductivity was better than those with smaller thermal conductivities. 2. An overall thermal resistance of a MNT-MCHS with a deeper channel depth was smaller than those with shallower channel depths. 798 N. Tran et al. / International Journal of Heat and Mass Transfer 117 (2018) 787–798 3. The overall thermal resistance of the MNT-MCHS, using TiO2water nanofluid with a nanoparticle volume fraction of 1%, could improve up to 6.7% as compared to that using pure water. 4. 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