Effect of porous substrates on thermohydraulic performance enhancement of double layer microchannel heat sinks

International Journal of Heat and Mass Transfer 131 (2019) 52–63

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effect of porous substrates on thermohydraulic performance enhancement of double layer microchannel heat sinks Ali Ghahremannezhad a, Huijin Xu b, Mohammad Alhuyi Nazari c, Mohammad Hossein Ahmadi d, Kambiz Vafai a,⇑ a

Department of Mechanical Engineering, University of California, Riverside, United States China-UK Low Carbon College, Shanghai Jiao Tong University, China Renewable Energy and Environmental Engineering Department, University of Tehran, Tehran, Iran d Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran b c

a r t i c l e

i n f o

Article history: Received 31 August 2018 Received in revised form 6 November 2018 Accepted 7 November 2018

Keywords: Microchannel heat sink Double layer microchannel heat sink Porous substrate Thermal resistance Pumping power Heat transfer optimization

a b s t r a c t Effect of utilizing porous substrates on thermal and hydraulic performance of double-layered microchannel heat sinks (MCHSs) is comprehensively analyzed in this work. Thermal resistance and pumping power of the porous double layer MCHSs are evaluated to find optimized designs which improve heat transfer while requiring lower pumping power compared to conventional MCHSs. Conjugate heat transfer is numerically simulated by developing three dimensional models of porous MCHSs with different solid and porous fin thicknesses at the top and bottom channels. For design optimization, various performance parameters are evaluated and compared to conventional microchannels by changing the porous substrate and solid fin thickness. The results show that for every combination set of geometrical parameters in double-layered MCHS, an optimized porous double-layered MCHS can be found which can enhance thermal and hydraulic performance. Studying the heat transfer effectiveness and pumping power effectiveness of the new porous double-layered MCHSs simultaneously indicate this. The enhancement is shown in all scenarios where the top and bottom channel can have different solid-porous thickness. The superior performance of porous double-layered microchannels is verified for different range of Reynolds number, porosity of substrates, and heat sink material. Ó 2018 Published by Elsevier Ltd.

1. Introduction Controlling high temperatures and heat fluxes generated in nowadays’ miniature-scale and high performing circuits are the primary challenge in the thermal management of microelectronics [1–6]. Microchannel heat sinks (MCHS) introduced in 1981 by Tuckerman and Pease [7,8] have been shown to have an excellent potential as cooling devices because of their convenient size, reliability, and cooling efficiency. Therefore, many works have studied the heat transfer capability of this promising cooling technology [9–18]. In addition, several studies have addressed their heat transfer and pressure drop limitations, and proposed new methods to optimize their thermal and hydraulic performances [19–25]. Vafai and Zhu [20] proposed a double layer microchannel heat sink with counter current flow arrangement to improve the heat transfer capability of the conventional single layer MCHSs. The ⇑ Corresponding author. E-mail address: [email protected] (K. Vafai). https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.040 0017-9310/Ó 2018 Published by Elsevier Ltd.

double-layered design was shown to have a significant reduction in the streamwise temperature rise on the base surface and provided more uniform cooling compared to the one-layered design. It was also found that the required pressure drop is lower than that of the single-layered MCHSs. Wei et al. [22] studied the thermal performance of a stacked microchannel heat sink by performing experimental measurements and numerical simulations. Different flow arrangements were examined which showed counterflow arrangement provided more uniform temperature profile. Chong et al. [21] performed an optimization study for the single layer and double layer MCHSs to find the optimal thermal resistance under a set of design constraints. Xie et al. [26] numerically studied the heat transfer and pressure drop in a double-layer microchannel heat sink under different inlet/outlet flow conditions and performed optimizations for the height of the upper and lower branch of the MCHS. Hung et al. [23] numerically analyzed the characteristics of a double layer MCHS for different materials, coolants, and geometric parameters. Lu et al. [27] performed experimental studies on a copper-based single- and double-layer microchannel heat exchangers and indicated that double-layer microchannels require

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

53

Nomenclature A C cf cs Dh f FOM hm ke kf ks Kp lf Lx Ly Lz N Nu P Pout qw Q_ Re RT T in Tw T w T w;max tB tT tM

Heat sink’s bottom surface area (m2) Forchheimer’s constant Heat capacity of the coolant (J kg
1 K
1) Heat capacity of the solid (J kg
1 K
1) Hydraulic diameter of the channel (m) Friction factor, Eq. (11) Figure of merit, Eq. (18) Average heat transfer coefficient (W m
2 K
1) Effective thermal conductivity of the porous medium (W m
1 K
1) Thermal conductivity of the coolant (W m
1 K
1) Thermal conductivity of the solid (W m
1 K
1) Permeability of the porous medium (m2) Height of channel at inlet (m) Length of the heat sink (m) Height of the heat sink (m) Width of the heat sink (m) Number of channels Nusselt number, Eq. (14) Pressure (Pa) Outlet pressure (Pa) Heat flux on the base surface of the heat sink (W m
2) Volume flow rate of the channel (m3 s
1) Reynolds number, Eq. (10) Thermal resistance (K W
1) Inlet temperature (K) Wall temperature at the centerline of the base surface (K) Average temperature along the base surface (K) Maximum wall temperature of the heat sink Thickness of the solid at bottom part of the channel (m) Thickness of the solid at top part of the channel (m) Thickness of the solid at middle part of the channel (m)

much lower pressure drop for the same heat removal as compared to the of that of single-layer microchannels. Wu et al. [28] numerically studied the effect of geometric and flow parameters on the performance of double-layer MCHS and discussed the optimal aspect ratio and flow rates to improve the performance. Leng et al. [29,30] proposed a double-layer microchannel heat sink with truncated top channel and showed that it has more uniform bottom temperature and lower thermal resistance. Besides the multi-layering technique, several works have utilized capabilities of a porous medium to enhance the heat dissipation in microchannel heat sinks. Fully developed force convection in rectangular microchannels filled with a porous medium is studied by Hooman [31]. Hung et al. [32] compared different porous MCHS configuration designs and performed numerical simulations to find the preferred configuration considering the thermal resistance along with the pressure drop. Porous MCHSs with sandwich configuration are numerically studied by Hung et al. [33] and Hung and Yan [34], and their optimum performance is obtained through an optimization method with various initial guesses of input parameters. Effect of using porous fins instead of solid fins in single layer MCHSs has been numerically studied by Chuan et al. [35]. They indicated a velocity slip of the coolant at the walls which will result in a 43% to 47.9% decrease in the pressure drop at different coolant flow rates compared to the conventional heat sinks with only a 5% increase in thermal resistance. Shen et al. [36,37] used and internal Y-shaped porous bifurcation made of metal foams inside microchannels and numerically investigated their advanta-

tS tP t ST t SB t PT t PB uin ! V wO wT wB x y z

Thickness of the solid fins when identical channels are assumed (m) Thickness of the porous treatment when identical channels are assumed (m) Thickness of the top channel’s solid fins (m) Thickness of the bottom channel’s solid fins (m) Thickness of the top channel’s porous substrates (m) Thickness of the bottom channel’s porous substrates (m) Inlet velocity of the coolant (m/s) Velocity vector (m/s) Total width of a single channel (m) Width of the top channel’s coolant pathway (m) Width of the bottom channel’s coolant pathway (m) x coordinate y coordinate z coordinate

Greek

e

DP

lf qs qf X

eh ep

Porosity Pressure drop from the inlet to the outlet (Pa) Dynamic viscosity (kg m
1 s
1) solid density (kg m
3) Coolant density (kg m
3) Pumping power of a channel (W) Average heat transfer coefficient ratio Pumping power effectiveness

Subscripts e Effective property of the porous medium f Fluid phase m Mean value p Porous medium s Solid phase

geous performance compared to the conventional MCHSs. Other configurations for utilizing a porous medium to enhance the performance of single layer MCHSs can be found in literature [38,39]. Most of the works on utilizing porous medium in MCHSs have studied single layer microchannels and studies on porous multilayer MCHSs are relatively rare. In a recent study, Wang et al. [40] have used porous fins in a double-layered MCHS and numerically investigated the thermal resistance and pumping power for a wide range of Reynolds numbers. They have discussed the slip effect of the coolant and friction reduction caused by the porous fins. The porous double-layered designs indicated a 45.3– 48.5% reduction in pumping power, while there was only a 14.8– 16.2% increase in the thermal resistance. In most of the previous research works, improving the thermal performance of the MCHSs resulted in increasing the pumping power which is not favored. In a previous study [41], the possibility of simultaneous increase in thermal and hydraulic performance of single layer microchannel heat sinks by utilizing porous substrates is discussed. In this paper, a comprehensive optimization study is performed on double-layered microchannel heat sinks to investigate their thermal and hydraulic performance at various poroussolid fin thicknesses. Two scenarios are considered; a case in which both channels have the same geometrical properties; and a case that the top and bottom channels can have different solid and porous fin thicknesses. Sensitivity analysis is performed for the optimal designs and the effect of flow rate, porosity, and heat sink material is discussed in detail.

54

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

2. Analysis

ing equations which are continuity, momentum, and energy will be as follows [46]:

To study the effect of adding porous substrates on doublelayered microchannel heat sinks, three dimensional models of the double-layered porous MCHSs are developed. Fig. 1 shows the porous treatment configuration on the bottom and top channel walls. Fig. 2(a) shows a schematic of the periodic unit which is used to simulate the double layer microchannel heat sinks. Lx is the length, Ly is the height, and Lz is the width of the MCHS. The geometrical parameters for a porous double layer channel are shown in Fig. 2(b). Each double layer microchannel has a total height of Ly and a width of wo , half vertical solid fins with thicknesses of tSB and tST for bottom and top branches respectively, bottom horizontal solid fin with thickness of tB , middle horizontal solid fin with thickness of tM , and top horizontal solid fin with thickness of tT . Two vertical porous structures with thicknesses of t PB for the bottom branch and tBT for the top branch are placed on the vertical solid fins. The coolant is assumed to be pure water, and the solid parts of the heat sink as well as the porous domains are considered to be stainless steel. Opposite flow directions are assumed at the bottom and the top branch for the counter-flow double layer MCHS and the heat is applied at the bottom surface as in most heat sink applications.

r: eqf V

3. Numerical procedure Three dimensional models of the double-layered MCHSs are developed and conjugate heat transfer is numerically simulated using ANSYS Fluent 18.2. The sintered porous domains are assumed to be homogeneous, isotropic, and fully saturated with the coolant. The fluid flow is considered to be incompressible, laminar, and steady. For the working temperature ranges, constant thermo physical properties are used and the gravitational and radiation effects are neglected. The heat sink is insulated from the outside environment. The solid and the coolant are assumed to be in local thermal equilibrium throughout the porous domains [42–47]. Forchheimer-Brinkman-Darcy equation based on a volume-averaging method is used to model the fluid flow through porous domains [45,46]. Based on these assumptions, the govern-




!

¼ 0;

ð1Þ !

lf qf C ! ! lf 2 ! qf
! ! V :r V ¼ rP
þ pffiffiffiffiffiffi  V  V þ r V 2 e Kp e Kp
!  ðqcÞe V :rT ¼ ke r2 T

ð2Þ ð3Þ

where e is the porosity of the porous domain; qf density of the fluid; ! V velocity vector of the fluid; P pressure; K p permeability; C the Forchheimer’s constant; lf viscosity of the fluid; T temperature; ke effective thermal conductivity and ðqcÞe is the effective heat capacity of the porous domain. The effective thermal conductivity, ke is obtained from ke ¼ ekf þ ð1
eÞks and ðqcÞe is ðqcÞe ¼ eqf cf þ ð1
eÞqs cs in which f and s are indices for fluid and solid respectively [44,46]. The energy equation for the solid domain is as follows:

ks r2 T s ¼ 0

ð4Þ

where ks is the thermal conductivity of the solid domain. The boundary conditions are listed as follows: Inlets for bottom and top channels

u ¼ uin ;

v ¼ 0;

w ¼ 0;

T ¼ T in

at

x ¼ 0 ðbottom channelÞ and x ¼ Lx

ðtop channelÞ

ð5Þ

Outlet for bottom and top channels

P ¼ Pout at x ¼ Lx ðbottom channelÞ and x ¼ 0 ðtop channelÞ

ð6Þ

Solid-fluid interface

! V ¼ 0;

Tf ¼ Ts;


kf rT f n ¼
ks rT s jn ;

ð7Þ

where n is the normal unit vector. Bottom wall

qw ¼
ks rT s jn ;

ð8Þ

At symmetric interfaces and insulated walls


ks rT s jn ¼ 0

ð9Þ

Darcy-Brinkman-Forchheimer effects are considered at the porous computational cells, and porous and fluid domains are solved simultaneously. The inlet velocity is considered in a range from 0.2 to 2 m/s and a uniform temperature of 300 K is assumed at the inlet of the channels. Constant pressure is assumed at outlet and no slip condition is considered at the solid walls. The top surface is insulated and a heat flux of qw ¼ 100 W/cm2 is applied at the bottom surface of the MCHSs. A generic channel is considered as the simulation domain and left and right surfaces are assumed to be symmetric. Effect of assuming a non-uniform distribution at the inlet was investigated for a number of cases and the difference in the results was found to be less than 0.5% on the average. Therefore, a uniform distribution is assumed at the inlet for simplicity considering the significant number of cases to be evaluated and the efficiency in the computational time. Other parameters used to evaluate the performance of MCHSs are described as follows [19,20,24,25,32]: Reynolds number Fig. 1. Schematic of a conventional double layer MCHS and a double layer porous MCHS.

Re ¼

qf uin Dh ; lf

ð10Þ

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

y

55

x

z

(a)

(b)

(c) Fig. 2. (a) Schematic of the proposed double Layer MCHS; A generic channel is used as the simulation domain; (b) Geometric parameters of the generic double layer porous channel; (c) Heat flux is applied at the bottom surface of the channel.

compare the heat transfer capability between two designs, an average heat transfer effectiveness is defined:

Friction factor

f ¼

1 Dh DP ; 2 Lx qf u2in

ð11Þ

eh ¼

Total pumping power





 þ Q_ DP Bottom Top     ¼ uin lf wf DP Bottom þ uin lf wf DP Top ;

X ¼ Q_ DP



ð12Þ

q hm ¼   w  ; T w
T in

ð13Þ

Nusselt number

Num ¼

hm D h ; kf

RT ¼

T w;max
T in ; qw A

Xbase ; Xnew

 FOM ¼

ð15Þ

where Dh is the hydraulic diameter for each of the bottom and top     channels Dh ¼ 2 lf wf = lf þ wf ; Q_ volume flow rate for each channel; T in inlet temperature; T w average wall temperature along the centerline of the bottom surface; T w;max maximum bottom wall temperature of the MCHS; and A is the surface area of the bottom surface. Some comparative parameters are defined in order to compare the performance of a new designed MCHS to a basic MCHS [32]. To

ð17Þ

Figure of merit (FOM) is defined to compare two different designs in order to determine the improvement of one design both in terms of heat transfer and pumping power at the same time [38,32]:

ð14Þ

Thermal resistance

ð16Þ

In a previous work for single-layered microchannels [41], we have defined a ‘‘pumping power effectiveness” parameter to compare hydraulic performance of two MCHSs designs:

ep ¼

Average heat transfer coefficient of the MCHS

hm;new ; hm;base

hm;new =hm;base



ðXnew =Xbase Þ1=3

;

ð18Þ

To perform a complete comprehensive parametric optimization with multiple parameters, about 1500 double-layered MCHSs with different geometrical and material properties are simulated. The finite volume approach is employed to numerically solve the governing equations with a SIMPLE (Semi-Implicit Method for Pressure Linked Equations) scheme [48]. Grid independency tests are performed for each case. Heat transfer coefficient and friction factor for three different grids are shown in Table 3 for a candidate double-layered MCHS. Based on the results, the grid scheme of 120  130  50 is selected for reasonable accuracy and computational time. Criterion for solution convergence of the equations is

56

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

set at 10
6 . Experimental works of Hetsroni et al. [49] and Jiang et al. [50] on porous channel heat sinks are used as a reference to validate the numerical procedure. A comparison of the current numerical results for identical porous-MCHSs with various porous samples used in experiments is shown in Fig. 3. An excellent agreement with the empirical data is observed and therefore current numerical method is considered to be reliable.

Table 1 Thermal properties of coolant, solid section, and porous structure. Materials Coolant: Pure water Substrate: Steel Porous medium: Steel

q

(kg m
3)

Cp (J kg
1 K
1)

k (W m
1 K
1)

l (Pa s)

998.2 8030 8030

4182 8030 8030

0.6 16.27 16.27

0.000855

4. Results and discussion To analyze the thermal and hydraulic characteristics of doublelayered porous MCHSs shown in Fig. 2, effect of varying solid and porous fin thicknesses of the bottom and top channels is investigated. To perform a parametric optimization with multiple involved parameters and the high number of cases, two scenarios are defined in order to categorize the results. In the first scenario, the bottom and top channels are identical, and the solid fin and the porous fin thicknesses of the two channels are considered to be equal. In the second scenario, the solid and porous fin thicknesses of the bottom and top channels can be different. For this case, various porous fins are tested on double-layered MCHSs with candidate solid fin thickness for the bottom and top channel. Detailed descriptions of the two categories are provided in the following sections. A basic conventional double-layered microchannel heat sink with no porous substrate is selected as the reference MCHS to compare the performance of the new porousMCHS designs. For the reference double-layered MCHS, the vertical solid fin of the bottom and top channels are t SB ¼ t ST ¼ 0:1 mm, and the thickness of the horizontal bottom, middle, and top solid fins are tB ¼ 0:1 mm, t M ¼ 0:1 mm, and tT ¼ 0:1 mm, respectively. Improvement in the performance parameters of the new designs are evaluated by comparing them to the reference doublelayered MCHS. Thermo-physical properties of the coolant, the solid domains, and the porous substrates are listed in Table 1. Assumed values for the geometrical parameters of the double-layered MCHSs are listed in Table 2. Depending on the inlet velocity and the hydraulic diameter of the bottom and top channels, the Reynolds number changes between 20 and 170. 4.1. Heat transfer and pumping optimization 4.1.1. First scenario Thermal and hydraulic performance of double-layered MCHSs is evaluated at various solid and porous substrate thicknesses for the

0.5

Present results for 20 μm pore size MCHS Present results for 60 μm pore size MCHS Hetsroni et al. 20 μm pore size MCHS Hetsroni et al. 60 μm pore size MCHS Present results for a porosity of ε = 0.402 Present results for a porosity of ε = 0.444 Jiang et al. porosity of ε = 0.402 Jiang et al. porosity of ε = 0.444

0.45 0.4

ΔP (MPa)

0.35 0.3 0.25 0.2 0.15 0.1

0.05 0 0

1000

2000

3000

4000

5000

Re Fig. 3. Comparison of the present results with the experimental data of Hetsroni et al. [49] and Jiang et al. [50].

Table 2 Geometrical parameters used for the doublelayered porous MCHSs. Parameter

Value

Lx (mm) Ly (mm) Lz (mm) wo (mm) wB (mm) wT (mm) tT (mm) tB (mm) tM (mm) tPB (mm) tPT (mm) tSB (mm) tST (mm)

10 3 12.2 0.8 0.05–0.7 0.05–0.7 0.1 0.1 0.1 0.005–0.35 0.005–0.35 0.025–0.35 0.025–0.35

bottom and top channels. In the first category, the bottom and top channels are considered to be identical which means they have the same solid fin and porous fin thickness (tSB ¼ tST ; t PB ¼ t PT ). The vertical solid fin thicknesses of the bottom and top channels, t SB and tST , have been increased from 0.025 mm to 0.3 mm with a step size of 0.025 mm, while for each solid thickness, the bottom and top porous substrate thicknesses, tPB and t PT , have been changed from 0.025 mm to 0.275 mm with the same step size 0.025 mm. For each case, the fluid pathway for the bottom and top channels, wB and wT , are changed based on the solid and porous fin thicknesses and they are kept at least at 0.05 mm. For all considered MCHSs, temperature increases from inlet to the exit and the maximum temperature occurs at the bottom surface of the heat sink where the coolant fluid is exiting the channel. The average heat transfer coefficient of the double-layered MCHSs is shown in Fig. 4 for different solid fin thicknesses versus porous substrate thickness. For almost all the solid fin thicknesses, increasing the thickness of the porous substrate will increase the average heat transfer coefficient until an optimum point is reached. After this point, adding porous substrate results in a decrease of the heat transfer coefficient. Furthermore, at a specific porous substrate thickness, higher heat transfer coefficients occur at higher solid fin thicknesses. Average heat transfer coefficient of the reference double-layered MCHS with a solid fin thickness of tSB ¼ tST ¼ 0:1 mm is also shown in Fig. 4. For all double-layered porous MCHSs, there is a range of the porous fin thicknesses which results in a higher heat transfer than the basic MCHS. The highest thermal performance is observed for high solid fin thickness of tSB ¼ tST ¼ 0:3 mm and low porous fin thickness of tPB ¼ tPT ¼ 0:025 mm which has the highest effective thermal conductivity. The same thermal behavior at an optimal solid-fin thickness has been previously reported for single layer porous MCHS [41]. Although better thermal performance of the double-layered porous MCHSs is observed at high solid-fin thicknesses, higher pressure drops occur at a high fin thickness as shown in Fig. 5. Pressure drop is plotted for double-layered porous MCHSs with different solid fin thicknesses versus porous substrate thicknesses.

57

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63 Table 3 Grid independence study.

tS = 0.05 mm tS = 0.075 mm tS = 0.1 mm tS = 0.125 mm tS = 0.15 mm tS = 0.175 mm tS = 0.2 mm tS = 0.225 mm tS = 0.25 mm tS = 0.275 mm tS = 0.025 mm tS = 0.3 mm Basic MCHS, tS = 0.1 mm (with no porous substrate)

1.7

Number of nodes

hm (kW m
1 K
1)

f

100  100  40 120  130  50 140  180  60

400,000 780,000 1,512,000

8.0819 8.0801 8.0796

0.12137 0.12130 0.12128

1.5 1.3

FOM

Grid

1.1

14 0.9

12 0.7

hm (kW/m2.K)

10

0.5

8

0

6

2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Porous Substrate Thickness (mm) Fig. 4. Heat transfer coefficient of double-layered porous MCHSs with different bottom and top solid fin thicknesses (t S ¼ t SB ¼ t ST ) versus bottom and top porous substrate thicknesses (tP ¼ tPB ¼ tPT ).

tS = 0.05 mm tS = 0.075 mm tS = 0.1 mm tS = 0.125 mm tS = 0.15 mm tS = 0.175 mm tS = 0.2 mm tS = 0.225 mm tS = 0.25 mm tS = 0.275 mm tS = 0.025 mm tS = 0.3 mm Basic MCHS, tS = 0.1 mm (with no porous substrate)

16 14

ΔP (kPa)

12 10 8 6

4 2 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.15

0.2

0.25

0.3

0.35

Fig. 6. FOM of double-layered porous microchannel heat sinks versus the bottom and top porous substrate thicknesses (t P ¼ tPB ¼ tPT ) for different bottom and top solid thicknesses (tS ¼ tSB ¼ t ST ).

0 0

0.1

Porous Substrate Thickness (mm)

tS = 0.075 mm tS = 0.05 mm tS = 0.1 mm tS = 0.125 mm tS = 0.15 mm tS = 0.175 mm tS = 0.2 mm tS = 0.225 mm tS = 0.25 mm tS = 0.275 mm tS = 0.025 mm tS = 0.3 mm Basic MCHS, tS = 0.1 mm (with no porous substrate)

4

0.05

0.35

Porous Substrate Thickness (mm) Fig. 5. Pressure drop of double-layered porous microchannel heat sinks versus the bottom and top porous substrate thicknesses (tP ¼ tPB ¼ tPT ) for different bottom and top solid thicknesses (t S ¼ t SB ¼ tST ).

Pressure drop of the basic double-layered MCHS is also shown as reference. For all MCHSs, as the thickness of the porous substrate increases, the pressure drop increases while an exponential rise is observed at a high solid-porous fin thickness. Moreover, for a specific porous substrate thickness, higher solid fin thickness will increase the pressure drop, and therefore, more pumping power is required to run the MCHSs. As can be seen in Fig. 5, pressure drop at a high solid-fin thickness—where minimum thermal resistance occurs—is considerably higher than the pressure drop of the basic double-layered MCHS. Therefore, higher pumping power is the cost for lower thermal resistance [1,4,20,38,32,35,39,41]. Figure of Merit (FOM) is used

to evaluate the extent of improvement in both thermal and hydraulic aspects when comparing a new design to a basic design [38,32,41]. In Fig. 6, FOM is plotted for double-layered MCHSs with different solid fin thicknesses versus their porous substrate thickness. At low solid fin thicknesses (0.025 mm to 0.225 mm), FOM increases by adding more porous substrate until an optimum point and then it decreases. For higher solid fin thicknesses (0.225 mm to 0.3 mm) close to the optimal heat transfer point, adding a porous substrate will result in a decrease of FOM. This is due to higher impact of pressure drop on FOM compared to the heat transfer aspect at high solid-porous fin thicknesses. The same behavior can be observed when considering the effect of varying solid fin thickness at a constant porous substrate thickness. FOM increases by increasing tSB and t ST until an optimum point and then starts to decrease. Highest FOM occurs for the double-layered MCHS with a tSB and tST close to (but smaller than) the optimal heat transfer point and small t PB and tPT . This double-layered MCHS has only 5% lower heat transfer coefficient compared to the doublelayered MCHS with the highest hm , while having 58% lower pressure drop. This suggests that FOM can be a reliable and useful performance parameter when choosing a candidate MCHS considering both thermal and hydraulic aspects. 4.1.2. Second scenario For this category, thermal and hydraulic performance of doublelayered MCHSs are evaluated where the bottom and the top channel can have different solid and porous fin thicknesses. The vertical solid fin thickness of the bottom channel, tST , is changed from 0.1 mm to 0.3 mm with a step size of 0.1 mm, while for each t ST , the vertical solid fin thickness of the top channel is also changed from 0.1 mm to 0.3 mm with a step size of 0.1 mm. For each combination of tSB and t ST , the bottom and top porous substrates, t PB and tPT , are changed from 0.025 mm to 0.275 mm with a step size of 0.025 mm. Fig. 7 show variation of average heat transfer coefficient of double-layered porous microchannel heat sinks for different bottom and top vertical solid fin thicknesses versus bottom and top porous substrates. hm is plotted for three different tSB in Fig. 7(a), (b), and (c) for t ST ¼ 0:1 mm, tST ¼ 0:2 mm, and t ST ¼ 0:3 mm respectively. As can be seen in Fig. 7(a), (b), and (c), for all double-layered MCHSs where t SB and tST are either 0.1 or 0.2 mm, increasing tPB or t PT will result in an increase in the heat transfer coefficient until a maximum point is reached, and afterwards,

58

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

tSB = 0.3 mm tSB = 0.2 mm tSB = 0.1 mm

(a)

tSB = 0.3 mm tSB = 0.2 mm tSB = 0.1 mm

(b)

tSB = 0.3 mm tSB = 0.2 mm tSB = 0.1 mm

(c) hm (kW/m2.K) Fig. 7. Average heat transfer coefficient of double-layered porous MCHSs with different t SB versus bottom and top porous substrate thicknesses for (a) tST ¼ 0:1 mm, (b) t ST ¼ 0:2 mm, and (c) tST ¼ 0:3 mm.

causes a decrease. For cases where tSB ¼ 0:3 mm (or tST ¼ 0:3 mm), adding porous substrate to the bottom (or top) channel results in a reduction in the thermal performance since the porous-fin thickness increases beyond the optimal heat transfer point of each channel. Moreover, it can be noticed from Fig. 7 that thermal performance of the double layered porous MCHS is more sensitive to the solid-porous fin thickness of the bottom channel compared to the top channel because it is closer to the heat source. As shown in Fig. 7, at a constant t SB , tST , and t PT , variations in t PB can result in

about 28% change in hm while at a constant t SB , tST , and tPB , variations in tPT cause about 8% maximum change in hm . The same tendency can be seen when changing the thickness of the solid part of the fin. At a constant t ST , by increasing tSB the heat transfer coefficient jumps to a higher level while at a constant t SB , increasing t ST results in a slight increase in level of heat transfer compared to the rise caused by a change in t SB . Higher sensitivity of heat transfer to the top channel solid-porous fin thickness occurs when the bottom channel has higher solid-porous fin thickness. Furthermore, allowing different values for the bottom and top channels’ solid and porous fin thicknesses reveals that at the optimal thermal performance point in double-layered MCHSs the geometrical parameters for the bottom and top channels are different. Fig. 7 shows that the highest heat transfer occurs when the bottom channel has a solid-porous fin thickness of about 0.325 mm while the top channel has a lower solid-porous fin thickness of about 0.275 mm. Therefore, porous and solid fin thicknesses at the optimal thermal point are different for each channel which should be considered in the design of double-layered porous MCHSs. Fig. 8 shows the pressure drop variation of double-layered MCHSs with different bottom and top solid fin thicknesses versus the bottom and top porous substrate thickness. Pressure drop increases by increasing the bottom or top channels’ solid or porous fin thicknesses. The rise in pressure drop is linear at small solidporous fin thicknesses while at higher thicknesses it increases exponentially. Because of the same inlet conditions for the bottom and top channels, changing t SB or t ST and tPB or t PT has an identical effect on the rise in the pressure drop. To reach a high thermal performance in double-layered porous MCHSs, the porous-solid fin thickness of the bottom channel needs to be relatively high which is in the region where there is also considerable pressure drop. However, as discussed before, the porous-fin thickness of the top channel does not need to be as high as the bottom channel, and at the optimal thermal point of the MCHS, the fin thickness of the top channel is smaller than the bottom channel. This can be a benefit compared to single-layered porous MCHSs [41] and a potential area to improve the thermal and hydraulic performance of microchannels. Here, FOM can be a useful parameter to evaluate both performance parameters at the same time. Fig. 9 shows the variation of FOM for double-layered porous MCHSs with different bottom and top solid fin thicknesses versus the bottom and top porous substrate thickness. The doublelayered porous MCHSs generally have higher FOM at higher tSB . At a constant t ST and t PT , for tSB ¼ 0:1 mm and tSB ¼ 0:2 mm, adding porous substrate to the bottom channel increases the FOM until a maximum point is reached after which it starts to decrease. However, for all double-layered porous MCHSs, increasing t PT results in a reduction of FOM. The same trend is seen for the t ST . Since the sensitivity of heat transfer to the top channel’s solid-porous fin thickness is relatively low while its thickness has considerable effect on the pressure drop, higher FOM occur at lowest t ST and tPT . Maximum FOM happens at t SB ¼ 0:3 mm, low tPB of 0.025 mm, and lowest t ST of 0.1 mm and tPT of 0.025 mm. For this design, the thermal resistance is only 7% higher than that of the optimal thermal performance design while its pressure drop is 30% lower. Therefore, the solid and porous fin thicknesses of the top channel in double-layered porous MCHSs can be designed to achieve lower pressure drops while keeping the thermal performance at the same level.

4.2. Sensitivity analysis Earlier the performance of double-layered porous microchannel heat sinks were analyzed and compared to the conventional double-layered MCHSs. In this part, a sensitivity analysis is

59

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

tSB = 0.3 mm

tSB = 0.3 mm

tSB = 0.2 mm

tSB = 0.2 mm

tSB = 0.1 mm

tSB = 0.1 mm

(a) (a)

tSB = 0.3 mm

tSB = 0.3 mm tSB = 0.2 mm

tSB = 0.2 mm

tSB = 0.1 mm

tSB = 0.1 mm

(b) (b)

tSB = 0.3 mm

tSB = 0.2 mm

tSB = 0.3 mm

tSB = 0.1 mm

tSB = 0.2 mm tSB = 0.1 mm

(c) (c)

FOM ΔP (kPa)

Fig. 8. Pressure drop of double-layered porous MCHSs with different tSB versus bottom and top porous substrate thicknesses for (a) t ST ¼ 0:1 mm, (b) tST ¼ 0:2 mm, and (c) tST ¼ 0:3 mm.

performed on variations of the porous substrate thickness for porous-MCHSs which have a solid fin thickness very close to the basic conventional MCHSs. The reason to choose very close solid fin thicknesses is to have the same range of fin effective thermal conductivity while using the hydraulic advantage of the porous substrates. For this purpose, the heat transfer effectiveness, eh , and the pumping power effectiveness, ep , of different porous double-layered MCHSs are compared to the basic conventional

Fig. 9. Variation of FOM for double-layered porous MCHSs with different tSB versus bottom and top porous substrate thicknesses for (a) tST ¼ 0:1 mm, (b) t ST ¼ 0:2 mm, and (c) tST ¼ 0:3 mm.

ones. For a specific porous MCHS, an eh and/or eP larger than 1, indicates better heat transfer and/or lower pumping power. Variations of eh and eP for different double-layered porous MCHSs with bottom and top solid thickness close to tS ¼ t SB ¼ t ST ¼ 0:1 mm is shown in Fig. 10(a). The thickness of the bottom and top porous substrates are assumed to be the same. The solid line represents the basic conventional double-layered MCHS with identical channels of tSB ¼ tST ¼ 0:1 mm. MCHSs that fall above this line have a larger eh and ep than 1 which means a

60

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

better thermal and hydraulic performance compared to the basic MCHS. As shown in Fig. 10(a), for all double-layered porous MCHSs with bottom and top solid thicknesses close to—but smaller than—the basic MCHS, there is a range of bottom and top porous substrate thicknesses which result in a larger eh and ep than 1. The dashed areas represent these regions for each porous MCHS. Optimized range of bottom and top porous substrate thicknesses is shown with t p;opt . Therefore, a porous double-layered MCHS can have a higher solid-porous fin thermal conductivity while having a lower required pumping power. Fig. 10(b) shows variations of eh and eP for different double-layered porous MCHSs with bottom and top solid thickness close to tS ¼ tSB ¼ tST ¼ 0:275 mm. Higher FOM was achieved around this thickness as discussed in the previous section. The thickness of the bottom and top channels’ porous substrates are kept the same for simplicity. As can be seen for all double-layered MCHSs with bottom and top solid thickness between 0.25 mm and 0.27 mm, there is a range of porous substrate thicknesses that will increase the heat transfer coefficient and decrease the pumping power at the same time compared to the conventional MCHS with tS ¼ 0:275 mm. These areas are shown by the dashed lines and the optimized porous range for each MCHS is shown with t p;opt . The solid line represents the basic conventional double-layered MCHS with bottom and top channel solid fin thickness of 0.275 mm. For both cases, the dashed area

1.1

tP,opt

(a)

tP,opt

1.1

tP,opt tP,opt

1.05

1

1 tP,opt εh : tS = 0.075 mm εh : tS = 0.085 mm εh : tS = 0.095 mm εp : tS = 0.08 mm εp : tS = 0.09 mm Basic MCHS, tS = 0.1 mm

0.95

εh : tS = εh : tS = εp : tS = εp : tS = εp : tS =

0.08 mm 0.09 mm 0.075 mm 0.085 mm 0.095 mm

εp = Ω0/Ω

εh = RT0/RT

1.05

0.95

0.9

0.9 0

0.02

0.04

0.06

0.08

0.1

Porous Substrate Thickness (mm) 1.1

becomes larger when the solid thickness gets closer to the basic MCHS thickness. At solid thicknesses close to the basic MCHS, effective conductivity of the fin is less affected and smaller porous substrates are required to compensate for the lower thermal conductivity. This allows a larger range of porous substrates which can be added to the basic MCHSs to have higher eh and ep . For each case, the best performing double layered porous MCHS is the one where lines of eh and eP intersect. Among all the peaks for different solid thicknesses, the highest one falls within the region where the solid thickness is closest to the basic conventional MCHS. This is due to the higher solid-porous fin effective conductivity and larger range of porous substrate thicknesses where eh and ep are greater than 1. Fig. 11 shows effect of porous substrate thickness on primary thermofluid parameters, eh and eP , for different bottom and top solid fin thicknesses around 0.325 mm. As can be seen, at this solid thickness due to exponential rise in pressure drop with a slight increase in the porous substrate thickness, the ep lines are steeper. Also for all considered double-layered MCHSs, since the solid thicknesses are close to the optimal thermal point, eh lines rise to a maximum point and decrease afterwards. As shown in Fig. 11, at lower solid thicknesses such as 0.3 mm and 0.305 mm, eh lines reach their optimum point before hitting the basic MCHS line and eh stays smaller than 1. However, for solid thicknesses closer to the basic thickness of 0.325 mm, optimized regions are achieved by adding porous substrates. This is due to higher porous-solid fin effective conductivity which allows the optimized double layered porous MCHSs to simultaneously have higher heat transfer and lower pressure drop than the basic double-layered MCHS. The optimized areas are shown with the dashed lines. The concept of optimizing double-layered MCHSs utilizing porous substrates to improve their thermal and hydraulic performance simultaneously is shown in Fig. 12. A solid reduction in either top or bottom channel and replacing it with a porous substrate with higher optimized thickness can increase the solidporous fin’s effective conductivity while requiring lower pumping power to run the double-layered MCHS. The simultaneous improvement in thermal and hydraulic performance occurs at a specific range of porous substrate thickness. The procedure to achieve this optimized region is shown for some candidate double-layered MCHSs. The same concept can be applied for double-layered MCHSs with any geometrical configuration. This procedure has also been shown for single-layered MCHSs [41].

1.1

(b) 1.02

tP,opt

0.95

0.9 0

0.01

0.02

εh : t S = 0.255 mm εh : t S = 0.265 mm εp : t S = 0.25 mm εp : tS = 0.26 mm εp : t S = 0.27 mm

0.03

0.04

εh = RT0/RT

1

1 tP,opt εh : tS = 0.25 mm εh : tS = 0.26 mm εh : tS = 0.27 mm εp : tS = 0.255 mm εp : tS = 0.265 mm Basic MCHS, t S =0.275 mm

tP,opt tP,opt tP,opt

1.01

εp = Ω0/Ω

εh = RT0/RT

1.05

tP,opt

tP,opt

1.01

1

1

0.99

0.99

0.95

0.9 0.05

Porous Substrate Thickness (mm) Fig. 10. Effect of bottom and top porous substrate thickness on primary thermofluid parameters, eh and eP , for different bottom and top solid fin thicknesses around (a) t S ¼ t SB ¼ t ST ¼ 0:1 mm and (b) t S ¼ t SB ¼ tST ¼ 0:275 mm. (Dash lines represent eP and solid lines represent eh ).

εp = Ω0/Ω

tP,opt

1.05

1.02

εh : tS = 0.3 mm εh : tS = 0.305 mm εh : tS = 0.31 mm εh : tS = 0.315 mm εh : tS = 0.32 mm Basic MCHS, t S =0.325 mm

0.98

0.98

0.97 0

0.01

0.02

0.03

0.04

0.97 0.05

Porous Substrate Thickness (mm) Fig. 11. Effect of porous substrate thickness on primary thermofluid parameters, eh and eP , for different bottom and top solid fin thicknesses around tS ¼ tSB ¼ tST ¼ 0:325 mm. (Dash lines represent eP and solid lines represent eh ).

61

Porous Double-layered MCHS

Conventional Double-layered MCHS

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

Fig. 12. Porous substrate setup for double-layered microchannel heat sinks to improve their hydraulic and thermal performance simultaneously.

50

50

1

RT, u = 0.2 m/s RT, u = 1 m/s RT, u = 2 m/s FOM, u = 0.2 m/s FOM, u = 1 m/s FOM, u = 2 m/s

35 30 25 20

0.8 0.6 0.4

15 10

0.2

5

Thermal Resistance (K/W)

40

1.2

45

FOM

Thermal Resistance (K/W)

45

40

1

35

RT, Steel

30

RT, Copper

0.8

FOM, Steel

25

FOM, Copper

0.6

FOM

1.2

20 0.4

15 10

0.2

5

0

0 0

0.1

0.2

0.3

0.4

0

0 0

0.1

0.2

0.3

0.4

Solid Thickness (mm)

Solid Thickness (mm)

Fig. 13. Effect of bottom and top solid thickness tS ¼ tSB ¼ tST and inlet velocity on the thermal resistance and Figure of Merit for t PB ¼ t PT ¼ 0:025 mm. (Dash lines represent FOM and solid lines represent thermal resistance).

Fig. 14. Comparison of thermal resistance and Figure of Merit for steel and copper double-layered porous microchannel heat sinks. (Dash lines represent FOM and solid lines represent thermal resistance).

4.3. Effect of velocity, material, and porosity

effective thermal conductivity. Higher flow heat transfer coefficient and higher fin effective conductivity results in higher total heat transfer of the double layered porous MCHS. Because the heat transfer is still high for thicker solid-porous fins at higher velocities, the reduction in FOM is smaller. The effect of heat sink material on the performance of doublelayered porous microchannel heat sinks is shown in Fig. 14. The same porous substrates are placed for the bottom and top channels with a thickness of t PB ¼ t PT ¼ 0:025 mm, and the effect of increasing the bottom and top solid thicknesses on the thermal resistance and FOM are plotted for copper and steel MCHSs. The same trend for steel double-layered MCHSs is also seen for copper doublelayered MCHSs. The thermal resistance of the copper porous MCHSs are lower than the steel ones since copper has higher thermal conductivity. The FOM lines show combined thermal and hydraulic performances of the double-layered MCHSs. At higher solid fin thicknesses, FOM for steel double-layered MCHSs is higher than copper double-layered MCHSs. It can be seen that utilizing porous substrates in steel MCHSs is slightly more effective than

Fig. 13 shows the effect of bottom and top solid thickness and inlet velocity on the thermal resistance and Figure of Merit for bottom and top porous substrates thickness of tPB ¼ t PT ¼ 0:025 mm. To investigate the effect of velocity on the performance of the double-layered porous MCHSs, the inlet velocity value is changed from 0.2 m/s to 2 m/s. Bottom and top channels are assumed to be identical and their solid thickness is changed for the same porous substrate thickness of 0.025 mm where the highest FOM and lowest thermal resistance where achieved as shown earlier. To impose a higher inlet velocity, more pumping power is required. However, as shown in Fig. 13, lower thermal resistances can be achieved at higher inlet velocities. FOM for all velocities increases until a maximum point and starts to decrease afterwards. At higher velocities the optimal heat transfer point occurs at higher bottom and top solid-porous fin thicknesses. Increasing the inlet velocity will increase the coolant’s heat transfer coefficient and as a result the thickness of the solid-porous fin can be increased for a higher

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63

40

1.4

35

1.2 RT, ε = 0.32, tp RT, ε = 0.44, tp RT, ε = 0.32, tp RT, ε = 0.44, tp FOM, ε = 0.32, FOM, ε = 0.44,

30 25

= 0.025 mm = 0.025 mm = 0.15 mm = 0.15 mm tp = 0.025 mm tp = 0.025 mm

1 0.8 0.6

20

FOM

Thermal Resistance (K/W)

62

0.4

15

0.2

10

0 0

0.1

0.2

0.3

0.4

Solid Thickness (mm) Fig. 15. Effect of porosity on thermal resistance and Figure of Merit of doublelayered porous MCHSs for different bottom and top solid fin and porous substrate thicknesses. (Dash lines represent FOM and solid lines represent thermal resistance).

copper MCHSs. Moreover, the optimal thermal and FOM performances for copper MCHSs occur at slightly lower solid-porous fin thicknesses. Effect of porosity on thermal resistance and Figure of Merit of double-layered porous MCHSs for different bottom and top solid fin and porous substrate thicknesses is shown in Fig. 15. The same porous substrate is placed on the bottom and top channels with thickness of 0.025 mm and 0.15 mm. Thermal resistance and FOM are plotted versus varying bottom and top solid fin thicknesses for porosities of 0.32 and 0.44. The same optimal thermal point behavior is noticed for a porosity of 0.44. As can be seen, at porous substrate thicknesses of 0.025 mm and 0.15 mm, the thermal resistance of the double layered porous MCHSs with porosity of 0.44 is higher than the MCHSs with porosity of 0.32. The change in thermal resistance is more noticeable at higher porous substrate thickness of 0.15 mm. At higher porosity of 0.44 the effective conductivity of the solidporous fin decreases and results in an increase in the thermal resistance. However, this change in thermal resistance is small specially for lower porous substrate thickness of 0.025 mm. On the other hand, the pumping power at higher porosity of 0.44 is reduced. To compare the changes in the thermal resistance and pumping power, FOM for two porosities is shown. As can be seen, FOM for a porosity of 0.44 is higher than for porosity of 0.32. This indicates that although the heat transfer of the double-layered MCHS is lower at a higher porosity, the reduction in the pumping power is more effective resulting in a higher FOM. 5. Conclusions Thermal and hydraulic performance of the double-layered porous microchannel heat sinks is comprehensively analyzed in this work. Effect of utilizing porous substrates is investigated and superior designs based on thermal resistance and pumping power are discussed. The conclusions are as follows: (1) Porous substrates can be utilized in double-layered microchannel heat sinks to increase their thermal and hydraulic performance. (2) By a careful design based on an optimization study, thermal resistance and pumping power of the double-layered microchannel heat sinks can be reduced simultaneously.

(3) To compare the changes in heat transfer and pressure drop of new MCHS designs at the same time, Figure of Merit can be used. Porous double-layered MCHSs have higher FOM compared to the conventional MCHSs. (4) For all double-layered MCHSs, there is an optimum solidporous fin thickness for the bottom and the top channel for which the highest heat transfer coefficient is achieved. Also, FOM is maximized at an optimum thickness. Optimal FOM point occurs at a slightly lower thickness compared to optimal thermal point. (5) Thermal performance of double-layered porous MCHSs is more sensitive to the solid-porous fin thickness of the bottom channel than the top channel. Effect of the bottom and top fin thickness on the hydraulic performance is the same. (6) At the optimal thermal point, the solid-porous fin thickness of the bottom channel is larger than the thickness of the top channel. (7) Since the sensitivity of the thermal resistance to the top solid-porous fin thickness is relatively low, highest FOM of the double-layered porous MCHS occurs at larger bottom fin thickness and lower top fin thickness. (8) Lower thermal resistance and higher pressure drop is observed at higher flow rates. The optimal thermal and FOM points for higher flow rates occur at higher poroussolid fin thicknesses. (9) Using a higher conductive material for fins decreases the thermal resistance for the same pumping power. At higher thermal conductivity, optimal thermal and FOM point occur at lower solid-porous fin thickness. (10) Using a porous substrate with higher porosity results in a rise in thermal resistance and a reduction in pumping power. However, the reduction in pumping power is more significant than the rise in thermal resistance. Effect of changing porosity is more noticeable at higher porous substrate thicknesses.

Conflict of interest There is no conflict of interest. This manuscript has not been submitted to anywhere else.

References [1] A. Adham, N. Mohd-Ghazali, R. Ahmad, Thermal and hydrodynamic analysis of microchannel heat sinks: A review, Renew. Sustain. Energy Rev. 21 (2013) 614–622. [2] B. Agostini, M. Fabbri, J. Park, L. Wojtan, J. Thome, B. Michel, State of the art of high heat flux cooling technologies, Heat Transfer Eng. 28 (2007) 258–281. [3] C. Zhao, T. Lu, Analysis of microchannel heat sinks for electronics cooling, Int. J. Heat Mass Transf. 45 (2002) 4857–4869. [4] I. Hassan, P. Phutthavong, M. Abdelgawad, Micro-channel heat sinks, an overview of the state-of-the-art, Microscale Thermophys. Eng. 8 (2004) 183– 205. [5] I. Mudawar, Assessment of high-heat-flux thermal management schemes, Trans. Compon. Packag. Technol. 24 (2001) 122–141. [6] P. Smakulski, S. Pietrowicz, A review of the capabilities of high heat flux removal by porous materials, microchannels and spray cooling techniques, Appl. Therm. Eng. 104 (2016) 636–646. [7] D.B. Tuckerman, Heat-Transfer Microstructures for Integrated Circuits, Stanford University, 1984. [8] D.B. Tuckerman, F.R. Pease, Microcapillary thermal interface technology for VLSI packaging, in: Digest of Technical Papers—Symposium on VLSI Technology, 1983, pp. 60–61. [9] A. Weisberg, H. Bau, J. Zemel, Analysis of microchannels for integrated cooling, Int. J. Heat Mass Transf 35 (1992) 2465–2474. [10] T. Harms, M. Kazmierczak, F. Gerner, Developing convective heat transfer in deep rectangular microchannels, Int. J. Heat Fluid Flow 20 (1999) 149–157. [11] A. Fedorov, R. Viskanta, Three-dimensional conjugate heat transfer in the microchannel heat sink for electronic packaging, Int. J. Heat Mass Transf. 43 (2000) 399–415.

A. Ghahremannezhad et al. / International Journal of Heat and Mass Transfer 131 (2019) 52–63 [12] J. Ryu, D. Choi, S. Kim, Three-dimensional numerical optimization of a manifold microchannel heat sink, Int. J. Heat Mass Transf. 46 (2003) 1553– 1562. [13] R. Chein, J. Chen, Numerical study of the inlet/outlet arrangement effect on microchannel heat sink performance, Int. J. Therm. Sci. 48 (2009) 1627–1638. [14] S. Kim, D. Kim, Forced convection in microstructures for electronic equipment cooling, J. Heat Transf. 3 (1999) 639–645. [15] W. Qu, I. Mudawar, Analysis of three-dimensional heat transfer in microchannel heat sinks, Int. J. Heat Mass Transf. 45 (2002) 3973–3985. [16] J. Ryu, D. Choi, S. Kim, Numerical optimization of the thermal performance of a microchannel heat sink, Int. J. Heat Mass Transf. 45 (2002) 2823–2827. [17] Z.Y. Guo, Z. Li, Size effect on single-phase channel flow and heat transfer at microscale, Int. J. Heat Fluid Flow 24 (2003) 284–297. [18] E. Sparrow, J. Abraham, P. Chevalier, A DOS-enhanced numerical simulation of heat transfer and fluid flow through an array of offset fins with conjugate heating in the bounding solid, J. Heat Transfer 127 (2005) 27–33. [19] S. Lu, K. Vafai, A comparative analysis of innovative microchannel heat sinks for electronic cooling, Int. Commun. Heat Mass Transfer 76 (2016) 271–284. [20] K. Vafai, L. Zhu, Analysis of two-layered micro-channel heat sink concept in electronic cooling, Int. J. Heat Mass Transf. 42 (1999) 2287–2297. [21] S. Chong, K. Ooi, T. Wong, Optimisation of single and double layer counter flow microchannel heat sinks, Appl. Therm. Eng. 22 (2002) 1569–1585. [22] X. Wei, Y. Joshi, M. Patterson, Experimental and numerical study of a stacked microchannel heat sink for liquid cooling of microelectronic devices, J. Heat Transf. 129 (2007) 1432–1444. [23] T. Hung, W.-M. Yan, W.-P. Li, Analysis of heat transfer characteristics of double-layered microchannel heat sink, Int. J. Heat Mass Transf. 55 (2012) 3090–3099. [24] A. Khaled, K. Vafai, Cooling augmentation using microchannels with rotatable separating plates, Int. J. Heat Mass Transf. 54 (2011) 3732–3739. [25] K. Vafai, A. Khaled, Analysis of flexible micro-channel heat sink systems, Int. J. Heat Mass Transf. 48 (2005) 1739–1746. [26] G. Xie, Y. Liu, B. Sunden, W. Zhang, Computational study and optimization of laminar heat transfer and pressure loss of double-layer microchannels for chip liquid cooling, J. Thermal Sci. Eng. Appl 5 (1) (2013), 011004. [27] B. Lu, W.J. Meng, F. Mei, Experimental investigation of Cu-based, doublelayered, microchannel heat exchangers, J. Micromech. Microeng. 23 (2013). [28] J. Wu, J. Zhao, K. Tsenga, Parametric study on the performance of doublelayered microchannels heat sink, Energy Convers. Manage. 80 (2014) 550–560. [29] C. Leng, X.-D. Wang, T.-H. Wang, An improved design of double-layered microchannel heat sink with truncated top channels, Appl. Therm. Eng. 79 (2015) 54–62. [30] C. Leng, X.-D. Wang, T.-H. Wang, W.-M. Yan, Multi-parameter optimization of flow and heat transfer for a novel double-layered microchannel heat sink, Int. J. Heat Mass Transf. 84 (2015) 359–369. [31] K. Hooman, Heat and fluid flow in a rectangular microchannel filled with a porous medium, Int. J. Heat Mass Transf. 51 (2008) 5804–5810. [32] T. Hung, Y. Huang, W. Yan, Thermal performance analysis of porousmicrochannel heat sinks with different configuration designs, Int. J. Heat Mass Transf. 66 (2013) 235–243.

63

[33] T. Hung, Y. Huang, T. Sheu, W. Yan, Numerical optimization of the thermal performance of a porous-microchannel heat sink, Numer. Heat Transfer, Part A: Appl. 65 (2014) 419–434. [34] T. Hung, W. Yan, Optimization of design parameters for a sandwichdistribution porous-microchannel heat sink, Numer. Heat Transfer, Part A: Appl. 66 (2014) 229–251. [35] L. Chuan, X.-D. Wang, T.-H. Wang, W.-M. Yan, Fluid flow and heat transfer in microchannel heat sink based on porous fins, Int. Commun. Heat Mass Transfer 65 (2015) 52–57. [36] H. Shen, X. Liu, H. Yan, G. Xie, B. Sunden, Enhanced thermal performance of internal y-shaped bifurcation microchannel heat sinks with metal foams, ASME J. Thermal Sci. Eng. Appl. 10 (2017), pp. 011001–011001-8. [37] B. Shen, H. Yan, B. Sunden, H. Xue, G. Xie, Forced convection and heat transfer of water-cooled microchannel heat sinks with various structured metal foams, Int. J. Heat Mass Transf. 113 (2017) 1043–1053. [38] B. Wang, Y. Hong, L. Wang, X. Fang, P. Wang, Z. Xu, Development and numerical investigation of novel gradient-porous heat sinks, Energy Convers. Manage. 106 (2015) 1370–1378. [39] G. Lu, J. Zhao, L. Lin, X.-D. Wang, W.-M. Yan, A new scheme for reducing pressure drop and thermal resistance simultaneously in microchannel heat sinks with wavy porous fins, Int. J. Heat Mass Transf. 111 (2017) 1071–1078. [40] S.-L. Wang, X.-Y. Li, X.-D. Wang, G. Lu, Flow and heat transfer characteristics in double-layered microchannel heat sinks with porous fins, Int. Commun. Heat Mass Transfer 93 (2018) 41–47. [41] A. Ghahremannezhad, K. Vafai, Thermal and hydraulic performance enhancement of microchannel heat sinks utilizing porous substrates, Int. J. Heat Mass Transf. 122 (2018) 1313–1326. [42] B. Alazmi, K. Vafai, Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer, Int. J. Heat Mass Transf. 44 (2001) 1735–1749. [43] A. Amiri, K. Vafai, T.M. Kuzay, Effect of boundary conditions on non-darcian heat transfer through porous media and experimental comparisons, Numer. Heat Transfer J. Part A 27 (1995) 651–664. [44] K. Vafai, R. Thiyagaraja, Analysis of flow and heat transfer at the interface region of a porous medium, Int. J. Heat Mass Transf. 30 (1987) 1391–1405. [45] K. Vafai, Convective flow and heat transfer in variable porosity media, J. Fluid Mech. 25 (1984) 233–259. [46] K. Vafai, C. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Transfer 24 (1981) 195–203. [47] C. Hsu, P. Cheng, Thermal dispersion in a porous medium, Int. J. Heat Mass Transf. 33 (1990) 1587–1597. [48] S. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere/McGrawHill, 1980. [49] G. Hetsroni, M. Gurevich, R. Rozenblit, Sintered porous medium heat sink for cooling of high-power mini-devices, Int. J. Heat Fluid Flow 27 (2006) 259–266. [50] P.-X. Jiang, M. Li, T.-J. Lu, L. Yu, Z.-P. Ren, Experimental research on convection heat transfer in sintered porous plate channels, Int. J. Heat Mass Transf. 47 (2004) 2085–2096.