Applied Thermal Engineering 182 (2021) 116131

Available online 2 October 20201359-4311/© 2020 Elsevier Ltd. All rights reserved.

Conjugate heat transfer enhancement in the mini-channel heat sink by realizing the optimized flow pattern

Hui Xiao , Zhichun Liu , Wei Liu *

School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China

A R T I C L E I N F O

Keywords: Mini-channel heat sinks Inclined parallelepiped ribs Thermal hydraulic performance Longitudinal swirls flow Thermal management

A B S T R A C T

Mini-channel heat sinks are popular in cooling high heat flux devices due to the high convective heat transfer performance associated with moderate pressure drop penalty. It is of great significance to further improve the thermal hydraulic performance in the mini-channel heat sink so as to adapt it to higher heat flux conditions. In this paper, a new method of realizing the optimized flow field to enhance the thermal hydraulic performance was carried out successfully in the mini-channel heat sink. The chosen mini-channel heat sink was 30 mm × 30 mm in substrate size with Reynolds number ranging from 100 to 1100. Through conjugate heat transfer optimization based on exergy destruction minimization principle, the optimized flow pattern was characterized by three pairs of longitudinal swirls flow. Subsequently, the inclined parallelepiped ribs were proposed to realize the optimized flow pattern in the mini-channel heat sink. The heat transfer enhancement mechanism was investigated by analyzing velocity distributions, temperature distributions, and heat convection intensity. Besides, the total thermal resistance was decreased and the exergy destruction minimization principle was verified. As a result, the variation ranges of Nu/Nu0, f/f0, efficiency evaluation criterion (EEC), and overall performance criterion (R3) were 1.35–5.92, 1.27–8.75, 0.68–1.12, and 1.31–4.22, respectively. The maximum average heat flux could achieve 3.2 × 106 W/m2 within a temperature difference of 60 K between substrate and fluid. The configured parameters pitch ratio (PR) and width ratio (WR) were recommended as 1 and 0.2, respectively. This work is conducive to the structural design of mini-channel heat sinks.

1. Introduction

Cooling plays a significant role in many different engineering ap-plications, such as electronic industry, nuclear industry, power battery, solar cell, furnace engineering, biomedical engineering, and chemical vapor deposition instruments, etc. [1]. Especially in electronic tech-nology, the power density of the electronic components is rising with the miniaturization of the volume. In 2010, the average heat flux of the chip rose to 50 W/cm2 [2]. After 2020, the heat dissipation capability de-mand of the chip will reach 100 W/cm2 [3]. Electronic devices are usually recommended to work below the temperature of 85 ◦C to ensure the reliability and life span. In order to decrease the working tempera-ture of these high-power density devices, an efficient cooling solution should be configured. Mini/micro channel heat sink, which was firstly introduced in 1981 by Tuckerman and Pease [4], has been usually applied to cope with the challenge of massive heat dissipation in mini-ature devices in the past few decades [3].

Generally, the primary impediment of channel cooling solution is the

high thermal resistance between the coolant and the substrate. The decrease of channel width always results in the increase of convective heat transfer coefficient. Besides, to an extent, the heat transfer surface area increases with the use of high aspect ratio channel. Thus, the microscopic channel is desirable. However, the accompanying pressure drop is also significantly increased as the channel hydraulic diameter decreases. Another disadvantage of the present single layer mini/micro channel heat sink is the non-uniform temperature distribution along the flow direction. An applicable solution to decrease the temperature non- uniformity is increasing the mass flow rate of working fluid by using a large hydraulic diameter. Therefore, in order to balance the gains in heat transfer performance and the two disadvantages of high pressure drop and temperature non-uniformity, the mini-channel is favored by engi-neers and researchers. Here, by the way, the mini-channel is used to refer to the cooling channel whose hydraulic diameter is within 0.2–3 mm [5]. As for mini-channel heat sinks, the heat transfer enhancement measure is necessary. Hence, this paper aims to conceive an efficient heat transfer enhancement structure in the mini-channel and investigate the corresponding conjugate heat transfer mechanism as well as flow

* Corresponding author. E-mail address: w_liu@hust.edu.cn (W. Liu).

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Applied Thermal Engineering

journal homepage: www.elsevier.com/locate/apthermeng

https://doi.org/10.1016/j.applthermaleng.2020.116131 Received 2 June 2020; Received in revised form 22 September 2020; Accepted 27 September 2020

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Applied Thermal Engineering 182 (2021) 116131

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characteristics. In order to reduce the thermal resistance of micro/mini channel heat

sinks, various measures have been taken into consideration in the lit-eratures, such as inlet and outlet arrangement, porous metal, special coolant, various channel shape, and fluid disruption [1–3].

A suitable design of inlet and outlet arrangement is conducive to heat transfer enhancement and temperature uniformity. Vajravel et al. [6] carried out an experimental investigation on the heat transfer perfor-mance in a new miter bended mini-channel heat sink with arranging four inlets and outlets. The results indicated that the averaged reduction in thermal resistance and substrate temperature gradient were 43% and 60% respectively, with Reynolds number ranging from 175.02 to 553.4. Liu et al. [7] proposed a novel T type heat sink containing several inlet and outlet ports. Numerical and experimental investigations showed that eight inlet ports led to the best temperature uniformity. Lu and Vafai [8] comprehensively summarized the works about multiple layer micro-channel heat sinks and characterized the respective diagrams, thermal resistance, pumping power, advantages and disadvantages. On the basis of multiple layer micro-channel, Osanloo et al. [9] numerically investigated the heat transfer performance of a double layer micro- channel heat sink with tapered channels. The results showed that ther-mal performance was improved and that the suggested convergence angle was 4◦. Arabpour et al. [10] numerically investigated the thermal hydraulic performance of nanofluid in the double-layer micro-channel with truncated top layer. The maximum performance evaluation crite-rion was achieved at a truncated factor of 1/3.

The application of porous metal in micro/mini channel heat sinks usually contributes to large heat transfer area and low flow resistance. Hung et al. [11] numerically investigated the performance of a porous micro-channel heat sink with enlarged channel outlet. The overall thermal resistance was first larger and later smaller than that of the conventional micro-channel heat sink, as the given pump power increased. Dehghan et al. [12] analytically investigated the heat transfer performance for micro-channels with porous inserts. The porous inserts were suggested to be implemented within flow passages especially when the flow was in the slip-flow regime. In allusion to the lotus-type porous

copper heat sink, Liu et al. [13] optimized the pore structure by applying the straight fin model. With experimental test of the optimal pore structure, a heat transfer coefficient of 7.86 W/(cm2∙K) was achieved with a pressure drop of 100 kPa.

The high thermal conductivity of special coolant is conducive to the enhancement of convective heat transfer coefficient. Ho and Chen [14] carried out an experimental investigation on the thermal performance of a mini-channel heat sink cooled by Al2O3/water nanofluid. The average heat transfer coefficient could be increased more than 72%, and the particle fraction of 6 wt% was suggested for greater merit parameters. Muhammad et al. [15] numerically analyzed the laminar flow and heat transfer of a mini-channel heat sink cooled by liquid metal. The dissi-pated heat flux of 158.3 W/cm2 was achieved with the pressure drop of 467.7 Pa under the temperature difference constraint of 43 K.

More works for heat transfer enhancement of mini/micro channel heat sinks lay in the aspects of increasing heat transfer area and dis-turbing fluid flow. Fluid disruption is usually implemented by equipping structures such as pins, fins, grooves and etc. Wang et al. [16] experi-mentally and numerically studied the convective heat transfer perfor-mance in a trapezoidal micro-channel whose hydraulic diameter was 155 μm. The fully developed Nusselt number was 4, and the numerical results obtained by solving classical Navier–Stokes and energy equations were in good agreement with the experimental results. Ghani et al. [17] numerically studied the effect of the combination of sinusoidal cavities and rectangular ribs on the hydrothermal performance of a micro- channel heat sink. Due to large flow area and Dean vortices, the over-all performance evaluation criteria could achieve 1.85 at the Reynolds number of 800. Chamanroy et al. [18] proposed a wavy miniature heat sink equipped with pin-fins. The numerical analysis showed that the heat transfer coefficient and pressure drop were augmented 0.17–1.95 times and 5.1–13.6 times, respectively, compared with the smooth wavy miniature heat sink. Datta et al. [19] numerically analyzed the entropy generation and thermal performance of a micro-channel heat sink with cavities and ribs. The highest thermal performance factor was 2.31 for copper heat sink and 1.96 for silicon heat sink, respectively. Gorasiya and Saha [20] numerically investigated the effect of cylindrical and

Nomenclature

A,B,C0 Lagrange multipliers Ab substrate area, m2

Ac conjugate interface area, m2

cp specific heat capacity, J/(kg∙K) Dh hydraulic diameter, m e rib thickness, m EEC efficiency evaluation criterion Exd exergy destruction, W f friction factor h heat transfer coefficient, W/(m2∙K) H heat sink height, m Hc mini-channel height, m L heat sink length, m L1 outlet length in simulations, m Lu local heat convection number Nu Nusselt number p pressure, Pa pi inlet rib pitch, m pl longitudinal rib pitch, m P power consumption, W Pr Prandtl number PR longitudinal pitch ratio q substrate heat flux, W/m2

R thermal resistance, K/W

R3 overall performance criterion Re Reynolds number T temperature, K T0 ambient temperature, K Tb substrate temperature, K Tin inlet fluid temperature, K Tm mean fluid temperature, K Tw conjugate interface temperature, K um inlet mean velocity, m/s u,v,w velocity components, m/s Vh heat convection velocity, m/s W heat sink width, m Wb simplified unit width, m Wc mini-channel width, m WR width ratio x,y,z Cartesian coordinates, m

Greek symbols α inclined angle of ribs,◦β synergy angle,◦

κf fluid thermal conductivity, W/(m∙K) κs solid thermal conductivity, W/(m∙K) μ fluid dynamic viscosity, kg/(m∙s) ρ density, kg/m3

Φ viscous heat dissipation, W/m3

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parallelepiped fins on the heat transfer and flow characteristics. The enhancements in heat transfer were achieved 32% and 28% for cylin-drical and parallelepiped fins respectively, while the corresponding in-crements in pressure drop were 3% and 10% respectively. Alfellag et al. [21] proposed a combined structure of inclined slotted plate-fins and triangular pins in the mini-channel. With Reynolds number ranging from 100 to 1600, the numerical results showed that the Nusselt number and overall performance could be enhanced 1.84 times and 1.54 times, respectively. Bi et al. [22] numerically studied the effect of dimples, cylindrical grooves and low fins on the convective heat transfer in mini- channel heat sinks with Reynolds number ranging from 2700 to 6100. The order of the overall performance achieved by the three enhance-ment measures from high to low was: dimple surface, cylindrical groove surface, and low fin surface. Liu et al. [23] numerically investigated turbulent heat transfer in rectangular channels with novel cylindrical shaped grooves. The design with rounded transitional grooves was su-perior to conventional cylindrical grooves and square ribs.

Recent years, attentions have been attracted to longitudinal vortex generators in order to disturb fluid efficiently. Ahmed et al. [24] numerically enhanced the heat transfer performance of backward-facing step channels by using four different types of longitudinal vortex gen-erators in laminar flow. The best performance was observed at Reynolds number of 180 in the case of rectangular wings with an attack angle of 60◦. Zhang et al. [25] optimized the parameters of longitudinal vortex generators in the micro-channel with Taguchi method and response surface analysis. The longitudinal separation distance of longitudinal vortex generator pairs was the main factor for thermal hydraulic per-formance. Hosseinirada et al. [26] investigated the effect of rectangular non-uniform height vortex-generators on thermal hydraulic perfor-mance in the mini-channel. Due to the increased heat transfer area, chaotic advection, and boundary layer interruption, the overall perfor-mance was enhanced with relatively uniform temperature distribution. Lu and Zhai [27] proposed a combined structure of dimples and vortex generators in the micro-channel and numerically investigated the ther-mal hydraulic performance with Reynolds number ranging from 167 to 834. The optimal comprehensive performance value could reach 1.28.

These fruitful works indicate that this research topic is significant and hot at present. As for the present paper, the attention is attracted to enhancing the convective heat transfer in the mini-channel heat sinks by disturbing fluid flow. However, the structure design for heat transfer enhancement relies on the experience of researchers and the results are random. In this way, the enhancement structures may result in excessive pressure drop with very little heat transfer enhancement, which limits the design efficiency. In order to achieve high heat transfer coefficient with moderate pressure drop, the enhancement structure should lead to a well-organized flow field. Thus, three questions arise: what kind of flow field is a well-organized flow field in the mini-channel? How to achieve this optimized flow field by real structures? What is the thermal hydraulic performance of the enhancement structure designed in such way?

Recent years, researchers have proposed a concept called heat transfer optimization. By this way, the optimal flow field can be ob-tained by minimizing the irreversibility in second law of thermody-namics with variation method. Particularly, Liu et al. [28] carried out a heat transfer optimization with exergy destruction minimization prin-ciple in a circular tube and obtained multiple longitudinal swirls in laminar flow. Xiao et al. [29] implemented a turbulent heat transfer optimization in a rectangular duct with variation method based on exergy destruction minimization principle. The results indicated that the multiple longitudinal swirls flow was also an optimized flow field in the rectangular duct. Furthermore, He and Tao [30] found that the syner-getic effect of temperature field and velocity field was improved so that convective heat transfer was enhanced when the longitudinal swirls flow was formed. Xiao et al. [31] mathematically explored the mechanism of convective heat transfer enhancement and showed that the longitudinal swirls flow could significantly intensify the local heat convection. They

also realized this flow pattern by inserting inclined wedge-shaped plates in the circular tube. Liu et al. [32] employed slant square rods to form the longitudinal swirls flow so as to improve multiple physical quantities synergy. In addition, Zheng et al. [33] carried out a comprehensive re-view on heat transfer enhancement structures which formed multiple longitudinal swirls flow. The multiple longitudinal swirls flow always achieved high thermal hydraulic performance. This work can be used as a structure library for forming longitudinal swirls flow.

The multi-longitudinal swirls flow is a kind of well-organized flow field. However, little research has been carried out on forming the flow pattern of multiple longitudinal swirls flow in mini-channel heat sinks in the past few decades. Inspired by the aforementioned works, this paper will find the optimized locations of longitudinal vortex cores in the present mini-channel by using heat transfer optimization method firstly. Subsequently, the inclined parallelepiped ribs will be proposed to form the optimized flow pattern. Furthermore, the conjugate heat transfer and flow characteristics will be investigated to obtain the thermal hy-draulic performance and maximum heat flux that can be transferred. This work will promote the development of heat transfer enhancement in mini-channel heat sinks.

2. Mini-channel heat sinks

As depicted in Fig. 1, the selected size of present conventional mini- channel heat sink is 30 mm× 30 mm, which is similar to that in the literature [5]. The substrate is made of copper and the working fluid is water. The heat flux is imposed on the substrate and then taken away by the working fluid. In order to investigate the thermal performance of the mini-channel heat sink, this paper chooses a single channel with sym-metry boundary on the left and right sides. The research model consists of a test section (L = 30 mm) and an outlet section (L1 = 5 mm). The height (H) and width (Wb) of the research model are 4 mm and 2 mm, respectively. The mini-channel height (Hc) and width (Wc) are 3 mm and 1 mm, respectively.

In the following, this paper will obtain the optimized flow pattern in the single channel research model firstly. Subsequently, the enhanced mini-channel will be proposed to realize the optimized flow pattern.

3. Research methods

3.1. Governing equations

As for the present three-dimensional steady laminar conjugate heat transfer problem, the following assumptions are given as: (a) the working fluid is continuum and incompressible; (b) the gravity and ra-diation are neglected; (c) the viscous dissipation is negligible in the energy equation; (d) the physical properties are constant except that the dynamic viscosity is dependent on temperature.

Thus, the governing equation in the solid domain is written as:

∂∂xi

(

κs∂T∂xi

)

= 0 (1)

where κs = 387.6W/(m∙K). The governing equations in the fluid domain are expressed as: Continuum equation:

∂(ρui)∂xi

= 0 (2)

Momentum equation:

∂∂xj

(ρuiuj

)= −

∂p∂xi

+∂

∂xj

[

μ(

∂ui∂xj

+∂uj∂xi

)]

(3)

Energy equation:

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Applied Thermal Engineering 182 (2021) 116131

4

∂∂xi

(

ρuicpT − κf∂T∂xi

)

= Φ (4)

where ρ = 998.2kg/m3, cp = 4182J/(kg∙K), κf = 0.6W/(m∙K), μ sat-isfies the piecewise linear function of temperature as shown in Table 1 [34].

3.2. Boundary conditions

At the inlet of fluid domain, uniform velocity is specified associated with a uniform temperature distribution of 293.15 K. At the outlet of the fluid domain, the pressure outlet is specified. At the bottom of the solid substrate, two kinds of thermal boundary conditions of constant heat flux and constant wall temperate are specified separately for thorough investigations. At the top of the research model, the adiabatic thermal condition is specified for all domains while the no slip velocity boundary is specified for the fluid domain. The symmetry boundary condition is specified on the left and right sides of the research model. As for the present conjugate heat transfer problem, the boundary condition of solid–fluid interface is specified as coupled thermal boundary condition and no slip velocity condition.

3.3. Parameter definitions

In this paper, some parameters are introduced for data reduction. The hydraulic diameter (Dh), Reynolds number (Re), and friction factor (f) are respectively defined as:

Dh =2WcHc

Wc + Hc(5)

Re =ρumDh

μ (6)

f =Δp

(L/Dh)ρu2m/2(7)

where um is the inlet mean velocity and Δp is the pressure difference in

the test section. The equivalent convective heat transfer coefficient (h) is used to

evaluate the Nusselt number (Nu) in the mini-channel. The definitions are expressed as:

h =qAb

Ac(Tw − Tm)(8)

Nu =hDhκf

(9)

where q is the heat flux at the bottom of the substrate, Tw is the average temperature of the conjugate fluid–solid interface, Tm is the mean temperature of the working fluid, and Ab and Ac are the area of the substrate and fluid–solid interface in the conventional mini-channel, respectively.

The thermal resistance can be defined as:

Rtot =Tb,max − Tin

qAb, Rsol =

Tb,max − TwqAb

, Rflu =Tw − Tin

qAb(10)

where Rtot, Rsol, and Rflu are total thermal resistance, substrate thermal resistance, and fluid thermal resistance, respectively.

As for the performance evaluation, the quantitative criterion R3 is adopted to evaluate the overall thermal hydraulic performance. The R3 criterion is the heat transfer performance ratio of enhanced channel to plain channel at identical pump power consumption. It is expressed as:

R3 = Nu/Nuc (11)

where Nuc is the equivalent Nusselt number in the conventional mini- channel at an identical pump power consumption. The increase of R3 means the increase of thermal hydraulic performance.

The criterion EEC is an auxiliary evaluation criterion which is used for evaluating the efficiency performance at identical mass flow rate. It is given as:

EEC =Nu/Nu0

f/f0(12)

where Nu0 and f0 are the Nusselt number and friction factor in the conventional mini-channel at an identical mass flow rate. A large EEC means that the increased friction factor is moderate compared with the enhanced Nusselt number.

The present study is in a state of simultaneously hydrodynamic and thermal developing laminar flow, the Nu0 can be evaluated with Sie-der–Tate correlation as [35]:

Nu0 = 1.86(RePrDh/L)1/3,Re < 2300 (13)

Fig. 1. The selected mini-channel heat sink and simplified research model.

Table 1 The relationship between dynamic viscosity and temperature.

T, K μ, 10− 6kg/(m∙s)

T, K μ, 10− 6kg/(m∙s)

T, K μ, 10− 6kg/(m∙s)

283.15 1305.9 313.15 652.7 343.15 403.6 293.15 1001.6 323.15 546.5 353.15 354.1 303.15 797.2 333.15 466.0 363.15 314.2

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Besides, for the transitional and fully developed turbulent flow, the Nu0 can be evaluated with Gnielinski correlation [36].

In addition, in the conventional mini-channel heat sink, the laminar friction factor in the entrance region satisfies the following correlation as [5]:

fx =4

Re⋅

a1 + a3(x + )0.5 + a5x +

1 + a2(x + )0.5 + a4x + + a6(x + )1.5(14)

where x+ = x/(DhRe), and the corresponding coefficients taken from Ref. [5] are listed in Table 2 for mini-channels with different aspect ratios. The numbers of 0.5 and 0.2 in the symbols of fx,0.5 and fx,0.2 stand for aspect ratios (Wc/Hc) of different mini-channels in Table 2.

As for the present conventional mini-channel with an aspect ratio (Wc/Hc) of 0.333, the friction factor fx,0.333 can be obtained with linear interpolation of fx,0.2 and fx,0.5. The relationship is expressed as:

f0 = fx,0.333 = 0.557fx,0.2 + 0.443fx,0.5 (15)

In this way, Nu0 and f0 can be obtained. In addition, in accounting for property variations due to temperature changes, the correction term M, which is defined as (μb/μw)

c, is applied to correct Nu0 and f0. As for the present heating condition, exponent c is − 0.58 and 0.14 for friction factor and Nusselt number, respectively.

3.4. Computational methods

The whole computational domain of the research model is covered by a structured mesh which is generated by the commercial software ICEM 16.0. The numerical results are obtained based on the platform of commercial software FLUENT 16.0. The finite volume method is adop-ted to discretize the governing equations with applying second order scheme for pressure and second order upwind scheme for other trans-port quantities. Besides, the puzzle of velocity and pressure coupling is solved by SIMPLE algorithm [37–39]. When the relative residuals are decreased to 10− 4 for continuity equation, 10− 9 for energy equation, and 10− 6 for other equations, or all quantities change little, the calculation can be considered converged. Besides, the parameters of inlet and outlet temperature, inlet and outlet pressure, conjugate wall temperature, and conjugate wall heat flux in the test section are monitored to ensure that the parameter changes are less than 1% after 1000 iterations.

4. Conjugate heat transfer optimization

4.1. Exergy destruction minimization

The exergy destruction minimization principle was introduced by Liu et al. [28] to the area of convective heat transfer enhancement. With specifying constant wall heat flux for the duct flow, the decrease of exergy destruction may lead to smaller temperature difference in heat transfer process, thereby enhancing the heat transfer performance. In other words, given a constant power consumption, the optimized flow pattern for heat transfer enhancement can be obtained by minimizing the exergy destruction in the condition of constant wall heat flux.

As for the present conjugate heat transfer problem, the total exergy destruction of heat transfer can be expressed as:

Exd =∫∫∫

Ω

T0κ(∇T)2

T2dV (16)

where the ambient temperature T0 is equal to 293.15 K and κ represents κs and κf in the solid and fluid domain, respectively.

The total pump power consumption is equivalent to the sum of ki-netic energy loss and viscous dissipation in the fluid domain [40]. It is expressed as:

P =∫∫∫

Ω

[ρU⋅(U⋅∇)U + Φ ]dV (17)

where the viscous dissipation Φ is given as:

Φ =μ2(∇U +∇UT

)2 (18)

Thus, given a constant pump power consumption with the constraint of continuity and energy equations, the Lagrange function of exergy destruction in the whole domain can be constructed as:

J =∫∫∫

Ω

⎧⎪⎨

⎪⎩

T0κ(∇T)2

T2+ C0[ρU⋅(U⋅∇)U + Φ ]+

A∇⋅(ρU) + B[∇⋅(κ∇T) − ρcpU⋅∇T

]

⎫⎪⎬

⎪⎭dV (19)

where A and B are functions of space, C0 is constant, κ represents κs and κf in the solid and fluid domain, respectively, and the velocity is equal to zero in the solid domain.

With appropriate treatments to the multiplier A and boundary con-ditions, the variation of functional J can be simplified as:

δJ =∫∫∫

Ω

C0((

U⋅∇)(ρU) − ∇⋅(μ(∇U +∇UT

) )− ρcp

BC0

∇T +∇p)

δUdV

+

∫∫∫

Ω

∇⋅(ρU)δAdV +∫∫∫

Ω

(∇⋅(κ∇T) − ρcpU⋅∇T

)δBdV

+

∫∫∫

Ω

(

∇⋅(κ∇B) + ρcpU⋅∇B +2T0T2

(κT(∇T)2 − ∇⋅(κ∇T)

))

δTdV

+

∫∫

©

Γ

[(

2T0κ∇TT2

− κ∇B − ρcpBU)

δT + Bδ(κ∇T)]

dS

(20)

Letting the δJ equal zero, the governing equations for the optimized flow field can be obtained.

In the solid domain, in addition to Eq. (1), the scalar equation B should be also satisfied. It is expressed as:

∇⋅(

κscp

∇B)

+2T0cpT2

(κsT(∇T)2 − ∇⋅(κs∇T)

)= 0 (21)

In the fluid domain, in addition to Eqs. (2) and (4), the governing equations include the momentum equation with a virtual force and the scalar equation B, which are expressed as:

ρU⋅∇U = − ∇p+∇⋅(μ(∇U +∇UT

) )+ ρcpB∇T/C0 (22)

− ρU⋅∇B = ∇⋅(

κfcp

∇B)

+2T0cpT2

(κfT(∇T)2 − ∇⋅

(κf∇T

) )(23)

In Eqs. (21) and (23) of scalar equation B, the coupled boundary is specified at the solid–fluid interface. At other boundaries, the boundary conditions are dependent on the thermal boundary condition. The relationship can be expressed as: (

2T0κ∇TT2

− κ∇B − ρcpBU)

δT + Bδ(κ∇T) = 0 (24)

Thus, B = 0 can be specified at constant wall temperature boundary while ∇B = 2T0∇T/T2 can be specified at constant wall heat flux boundary.

Through solving the above scalar B equations and modified

Table 2 Coefficients for f correlation in conventional mini-channels.

a1 a2 a3 a4 a5 a6

fx,0.5 142.05 − 5.4166 1481 1067.8 13,177 − 108.52 fx,0.2 142.1 − 7.3374 376.69 800.92 14,010 − 33.894

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momentum equations combined with continuity and energy equations in the whole domain, the optimized flow pattern can be obtained. In addition, the source term in Eq. (22) can be added by using UDF, while the scalar equation B can be added by using UDS.

4.2. Optimized flow pattern

The heat transfer optimization is carried out at Re = 300 with specifying a constant heat flux of 105 W/m2 at the substrate bottom of the research model. With giving a multiplier C0 as 7× 106, the optimized results are obtained. Fig. 2 shows the temperature distribution, tangential velocity distribution, and the corresponding simplified flow pattern at the transverse section of x/L = 0.5 in the fluid domain. As depicted in Fig. 2(a), the temperature isopleths are wavy at the trans-verse section. It is due to the disturbance caused by the optimized flow field. In order to characterize the optimized flow pattern, the tangential velocity distribution at the transverse section is shown in Fig. 2(b). Multi-vortices are found at the transverse section and the corresponding simplified flow pattern is displayed in Fig. 2(c). Thus, the main char-acteristic of the optimized flow pattern is the longitudinal swirls flow. In addition, three pairs of longitudinal vortices are expected for the present research model.

5. Optimized flow pattern realization

In this part, the inclined parallelepiped ribs will be proposed as heat transfer enhancement structures to realize the optimized flow pattern in the present mini-channel heat sink. Besides, the flow characteristics and heat transfer mechanism will be revealed in the enhanced mini-channel heat sink. Furthermore, the thermal hydraulic performance will be investigated to verify the effectiveness of this heat transfer enhancement method.

5.1. Heat transfer enhancement structure

The research model of the proposed enhanced mini-channel heat sink is shown in Fig. 3. The inclined parallelepiped ribs are fitted in the mini- channel so as to generate multi-longitudinal swirls flow. The size of the mini-channel is the same as the conventional one. The thickness and the inclined angle of the ribs are 0.2 mm and 45◦, respectively. The inlet distance (pi) between the inlet and the first rib is 1 mm. The width ratio (WR = 3Wr/Hc) is 0.1, 0.2, 0.3, respectively. The longitudinal pitch ratio (PR = pl/Dh) is 1, 2, 3, 4, respectively. The Reynolds number (Re) ranges from 100 to 1100.

5.2. Results verifications

In order to ensure reliable results, the grid verification and results validation are carried out in this part.

Fig. 4(a) shows the variations of Nu and f with the increase of grid number in the condition of WR = 0.3, PR = 2, Re = 700, and q = 105 W/ m2. Six grid systems of Mesh1, Mesh2, Mesh3, Mesh4, Mesh5, and Mesh6 are 1.07 million, 1.69 million, 2.67 million, 3.79 million, 4.75 million, and 7.07 million in grid number, respectively. The grid convergence index (GCI) [41] is a measure of the deviation percentage in the present numerical value relative to the asymptotic numerical value. The GCI method is applied to report the grid independent veri-fication results with Mesh1, Mesh3, and Mesh6. The GCI values of Mesh3 are 0.23% and 0.95% for Nu and f, respectively. Besides, as the grid number increases from 2.67 million to 7.07 million, the deviations in Nu and f are 0.46% and 0.15%, which are little. Hence, the mesh system of Mesh 3 is dense enough for present research. Thus, Mesh 3 is selected for research in this paper. Fig. 4(b) shows the mesh details of Mesh 3. As depicted, the mesh quality is pretty good as well as the mesh is densified further near the sold-fluid interface. Thus, Mesh 3 is capable of capturing the complex flow in the mini-channel heat sinks.

As depicted in Fig. 5, the present results are verified with empirical correlations and experimental results in the literature. The results of present Nu0 and f0 are taken from the conventional mini-channel in the present paper with constant substrate heat flux. The corresponding empirical results for Nu0 and f0 are taken from the works of Lee et al. [35] and Kandlikar et al. [5], respectively. Besides, the experimental results of Nu1 and Nu2 are Nusselt numbers taken from G3 channel (without LVGs) and G6 channel (with LVGs) in the work of Liu et al. [42], respectively. The constant wall temperature condition is specified. It is similar for f1 and f2. As shown in Fig. 5, the deviations between present results and literature results are quite small. The maximum deviations in Nu0, Nu1, and Nu2 are approximately 4.2%, 9.7%, and 8.5%, respectively, compared with the literature results. The maximum deviations in f0, f1, and f2 are approximately 5.6%, 11.6%, and 8.4%, respectively. It indicates that the present results are reliable whether constant wall temperature or constant wall heat flux is specified at the substrate bottom. Therefore, the present research method is right and the obtained results in the following are accurate.

5.3. Heat transfer enhancement mechanism

5.3.1. Flow characteristics and temperature distributions The streamlines and tangential velocity distributions are displayed in

Fig. 6. The streamlines are taken from the enhanced mini-channel heat

Fig. 2. The optimized results at x/L = 0.5 in the fluid domain: (a) temperature distribution; (b) tangential velocity distribution; and (c) simplified flow pattern.

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sink in the condition of WR = 0.1, PR = 2, Re = 300, and q = 105 W/m2. As depicted in Fig. 6, the streamlines are bent periodically due to the disturbance caused by parallelepiped ribs. Besides, backflow seldom

appears. As the fluid flows through the enhanced min-channel, the fluid is deflected from the front edge to the rear edge of the inclined paral-lelepiped rib. Thus, a pair of vortices are generated near the rib. At cross

Fig. 3. The mini-channel heat sink with inclined parallelepiped ribs.

Fig. 4. (a) Grid independence verification and (b) selected mesh system details.

Fig. 5. Verifications with results in the literature: (a) Nu; (b) f.

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sections, three pairs of vortices are generated. In this way, the multi- longitudinal swirls flow is formed in the fluid domain. It indicates that the optimized flow pattern is realized through using the inclined parallelepiped ribs.

Fig. 7 shows the comparisons of temperature distributions at cross sections between the enhanced mini-channel heat sink and the con-ventional one in the condition of WR = 0.1, PR = 2, Re = 300, and q =105 W/m2. As a whole, the maximum temperature of the enhanced mini- channel heat sink is lower obviously. In the fluid domain of the con-ventional mini-channel heat sink, the thermal boundary layer becomes thicker and thicker along the flow direction, which deteriorates the convective heat transfer performance. On the contrast, in the enhanced mini-channel heat sink, the fluids are mixed fully so that the thermal boundary layer is thinner than that in the conventional mini-channel, which improves the convective heat transfer coefficient. Hence,

according to the Newton cooling formula, the maximum temperature is decreased in the enhanced mini-channel heat sink with specifying con-stant substrate heat flux. Particularly, due to the effect of longitudinal swirls flow in the enhanced mini-channel, an alternate temperature distribution of warm and cold fluids is formed in the core flow region of the cross section. It is similar to the temperature distribution obtained by heat transfer optimization. It indicates that the optimized temperature distribution is also realized through using the inclined parallelepiped ribs.

The local Nu and f are plotted along the flow direction in Fig. 8. The local Nu and f are obtained at Re = 300 with specifying constant sub-strate heat flux. As depicted in Fig. 8, the local Nu and f are both decreased in the conventional mini-channel as the flow and thermal develop gradually in the channel. In the enhanced mini-channel, the local Nu and f are enhanced periodically compared with the

Fig. 6. Streamlines and tangential velocity distributions at cross sections.

Fig. 7. Comparisons of temperature distributions at cross sections: (a) conventional mini-channel heat sink; (b) enhanced mini-channel heat sink.

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conventional one. At the front edge of inclined parallelepiped ribs, the local f is increased sharply due to the rib obstruction. As the fluid flows away from the rear edge of inclined parallelepiped ribs, the fluid kinetic energy is partly converted into static pressure potential energy, which could reduce the pressure drop, thereby resulting in smaller f compared with the conventional mini-channel. Besides, the local Nu in the enhanced mini-channel is almost always larger, which demonstrates the effectiveness of longitudinal swirls flow for heat transfer enhancement.

5.3.2. Heat convection velocity analysis Heat convection velocity [31,43] is the component of fluid velocity

in temperature gradient direction. Its magnitude can be evaluated as

Vh =U⋅∇T|∇T|

= |U|cosβ (25)

where β is the included angle between velocity and temperature gradient.

In the whole fluid domain, the mean heat convection velocity is defined as

Vh,m =

∫∫∫

Ω|U||∇T|cosβdV∫∫∫

Ω|∇T|dV

(26)

Heat convection velocity analysis is a new method for investigating the heat transfer enhancement mechanism. It includes three aspects: (a) visualizing the local heat convection performance by using local heat convection velocity analysis; (b) investigating the contributions of ve-locity distribution variation and temperature distribution variation to convective heat transfer enhancement, so as to obtain the principle of heat transfer enhancement in the view of heat convection analysis; (c) evaluating the convective heat transfer degree and the effectiveness of exerted disturbance.

Firstly, the local heat convection performance can be visualized through heat convection velocity analysis. In order to describe the local heat convection performance effectively, the Lu number is introduced as

Lu =ρcp|Vh|Dh

κf(27)

The increase of Lu usually leads to a larger U⋅∇T, which results in more intense heat convection. In this way, the local heat convection performance can be visualized by displaying the contour of Lu.

Secondly, the relationship between Nu and the mean heat convection velocity can be constructed to study the convective heat transfer enhancement mechanism in the perspective of heat convection analysis.

In Ref. [31], their relationships were constructed through analyzing three typical cases: boundary layer flow, duct flow, and closed cavity flow. In the following, this paper will review their relationship in duct flow.

As for the present channel flow in the heat sinks, integrating Eq. (4) in the fluid domain, it yields ∫∫∫

Ω

ρcpU⋅∇TdV =∫∫

in+out

n→⋅κf∇TdS+∫∫

interface

n→⋅κf∇TdS+∫∫∫

Ω

ΦdV (28)

Neglecting the sum of inlet and outlet heat conduction as well as the viscous heat dissipation, it becomes ∫∫∫

Ω

ρcpU⋅∇TdV =∫∫

interface

n→⋅κf∇TdS = Q (29)

where Q = qAb. With substituting Eqs. (26) and (29) into Eq. (8), the equivalent

convective heat transfer coefficient can be also expressed as

h =qAb

Ac(Tw − Tm)=

ρcpVh,mAc(Tw − Tm)

∫∫∫

Ω

|∇T|dV (30)

Thus, the Nusselt number can be expressed as

Nu = ρcpVh,m2

κfAc

∫∫∫

Ω

|∇T|dV (31)

where the dimensionless temperature gradient is defined as

∇T =∇T

(Tw − Tm)/(Dh/2)(32)

With using ΔTm to denote the integration of dimensionless temper-ature gradient magnitude in Eq. (31), the expression of Nusselt number becomes

Nu = ρcpVh,m2

κfAΔTm (33)

Thus, the enhanced ratio of Nusselt number can be expressed as

NuNu0

=Vh,mVh,m,0

ΔTmΔTm,0

(34)

As known from Eq. (34), there are four suggestions for heat transfer enhancement [43]: (a) increasing the synergetic effect between velocity field and temperature field; (b) deflecting the working fluid from the low

Fig. 8. Comparisons of local Nu and f along the flow direction.

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temperature gradient region to the high temperature gradient region; (c) decreasing the mean temperature difference between solid wall and working fluid; (d) forming a temperature distribution with alternate warm fluid and cold fluid.

Thirdly, the convective heat transfer degree can be evaluated with heat convection intensity factor (Hf). The Hf is defined as the ratio of the mean heat convection intensity to the heat capacity of the fluid. It is expressed as

Hf =ρcpVh,mρcpum

=Vh,mum

(35)

The increase of heat convection intensity factor always accompanies with heat transfer enhancement, which indicates that the imposed disturbance is more efficient. Besides, a larger heat convection intensity factor also indicates a higher convective heat transfer degree.

In the following text of this part, this paper will reveal the heat transfer enhancement mechanism in the perspective of heat convection analysis.

In the test section of the enhanced mini-channel with WR = 0.1, PR = 2, Re = 300, and q = 105 W/m2, the average Nu/Nu0, Vh,m/Vh,m,0, and ΔTm/ΔTm,0 are 1.88, 1.38, and 1.36, respectively, which verifies the relationship in Eq. (34). Fig. 9 displays the contributions of local Vh,m/ Vh,m,0 and local ΔTm/ΔTm,0 to local Nu/Nu0 along the flow direction in this enhanced mini-channel. These three ratios vary periodically along the flow direction. The variation range of ΔTm/ΔTm,0 is small and stable, and the value is always larger than 1. It means that the temperature distribution in the enhanced mini-channel is conducive to heat transfer enhancement. The variation range of Vh,m/Vh,m,0 is large, which in-dicates that the heat convection intensity can be enhanced significantly. In the enhanced mini-channel, the value of Vh,m is increased sharply near the inclined parallelepiped ribs due to the increase of heat convection intensity. In the conventional mini-channel, the value of Vh,m,0 is decreased due to the decrease of inlet effect. Thus, the corresponding peak values of Vh,m/Vh,m,0 are increased gradually along the flow di-rection, as depicted in Fig. 9. Besides, the Vh,m/Vh,m,0 is the dominated contribution to Nu/Nu0 near the inclined parallelepiped ribs, while the ΔTm/ΔTm,0 is the dominated contribution to Nu/Nu0 between two in-clined parallelepiped ribs in the flow direction. Furthermore, the vari-ation regularity of Nu/Nu0 is similar to that of Vh,m/Vh,m,0. It indicates that enhancing the heat convection intensity is a promising method for improving the convective heat transfer performance.

As depicted in Fig. 10, in order to investigate the local heat con-vection intensity, this paper displays the comparisons of Lu distributions

at several cross sections near the fifth ribs along the flow direction. The Lu distributions change greatly. In the conventional mini-channel, the Lu is relatively larger in the core flow region. In the enhanced mini- channel, the region with larger Lu is closer to the conjugate interface, where the temperature gradient is larger. It is conducive to convective heat transfer. Fig. 11 shows the variations of heat convection intensity factor. As the Reynolds number increases, the heat convection intensity factor decreases. It indicates that the heat convection degree is reduced. As the WR increases at an identical Reynolds number, the heat con-vection intensity factor is increased. It indicates that the effective disturbance exerted on the fluid is increased.

5.3.3. Thermal resistance and exergy destruction Fig. 12 displays the comparisons of thermal resistance between the

conventional mini-channel and the enhanced mini-channel with PR = 2 and WR = 0.1. At a given Reynolds number, the total thermal resistance and fluid thermal resistance are both decreased significantly whether specifying constant substrate heat flux or specifying constant substrate temperature. It indicates that the temperature difference is reduced, thereby improving the temperature uniformity performance. Besides, the variation of solid thermal resistance relies on the thermal boundary condition. With specifying constant substrate temperature, the solid thermal resistance changes little. On the other hand, with specifying constant substrate heat flux, the solid thermal resistance is decreased due to the more uniform substrate temperature in the enhanced mini- channel. It means that the adopted inclined parallelepiped rib not only enhances the convective heat transfer but also improves the heat con-duction performance in the solid domain.

The relationships between Nusselt number and exergy destruction are shown in Fig. 13. With specifying constant heat flux, the decrease of exergy destruction accompanies with the decrease of temperature dif-ference, thereby enhancing the convective heat transfer. Hence, the Nusselt number is increased as the exergy destruction is decreased at a given Reynolds number, as depicted in Fig. 13(a). It proves the cor-rectness of the exergy destruction minimization principle. On the other hand, with specifying constant substrate temperature, the transferred heat is increased due to the convective heat transfer enhancement. Hence, the Nusselt number may be increased with the increase of exergy destruction at a given Reynolds number, as depicted in Fig. 13(b).

5.4. Performance evaluation

5.4.1. Thermal hydraulic performance Fig. 14 shows the variations of Nu/Nu0, f/f0, EEC, and R3 with Re

Fig. 9. Contributions of Vh,m/Vh,m,0 and ΔTm/ΔTm,0 to Nu/Nu0.

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ranging from 100 to 1100. As depicted in Fig. 14(a) and (b), Nu/Nu0 and f/f0 both increase as Re and WR increase, and increase with the decrease of PR. The variation ranges of Nu/Nu0 and f/f0 are 1.35–5.92 and 1.27–8.75, respectively. The EEC is used to evaluate the enhanced per-formance at identical mass flux rate. The variation range of EEC is 0.68–1.12. It indicates that the increased friction factor is moderate compared with the enhanced heat transfer performance. The R3 is used to evaluate the overall thermal hydraulic performance at identical pump power consumption. The R3 values are always beyond 1 and the vari-ation range of R3 is 1.31–4.22. It indicates that the enhanced mini-

channel is always effective in the range of this research. According to the R3 evaluation criterion, the configured parameter of PR = 1 and WR = 0.3 is the best based on the results shown in Fig. 14(d). However, the configured parameter of PR = 1 and WR = 0.3 generates the smallest EEC value as shown in Fig. 14(c). It means that the friction factor increased much with PR = 1 and WR = 0.3. Considering the demand of moderate flow resistance, the configured parameter of PR = 1 and WR =0.2 is recommended as a compromise in this paper.

Fig. 10. Comparisons of Lu distributions at cross sections: (a) conventional mini-channel; (b) enhanced mini-channel.

Fig. 11. The variations of heat convection intensity factor: (a) constant substrate heat flux; (b) constant substrate temperature.

Fig. 12. Comparisons of thermal resistance: (a) constant substrate heat flux; (b) constant substrate temperature.

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5.4.2. Maximum average heat flux With specifying constant substrate temperature of 353.15 K, the

average substrate heat flux and the corresponding pressure drop are obtained and shown in Fig. 15. The obtained average substrate heat flux is the maximum average heat flux that can be dissipated within a tem-perature difference of 60 K between substrate and fluid. As shown in Fig. 15(a), the maximum average heat flux in the enhanced mini- channel is much larger than that in the conventional mini-channel. The maximum average heat flux can achieve 3.2 × 106 W/m2 at Re =1100. The corresponding pressure gradient is 148.9 × 103 Pa/m, as shown in Fig. 15(b). In addition, with applying the recommended pa-rameters of PR = 1 and WR = 0.2, the variation ranges of the maximum average heat flux and pressure gradient are 0.66 × 106–2.81 × 106 W/

m2 and 1.7 × 103–80.9 × 103 Pa/m, respectively, as the Reynolds number ranges from 100 to 1100.

5.4.3. Performance comparisons Fig. 16 shows the comparisons with results in the literature. The

chosen heat transfer enhancement works are cavities and ribs [19], pin- fins [21], dimples and vortex generators [27], and wavy channel [44]. The enhanced mini-channel heat sink with PR = 1 and WR = 0.2 in this work is chosen for comparison. As shown in Fig. 16(a), the Nu/Nu0 in the present work is relatively high. Thus, the convective heat transfer can be improved more significantly by using inclined parallelepiped ribs. Be-sides, the corresponding f/f0 is also moderate in the present work. Therefore, the performance of this work is excellent.

Fig. 13. Relationships between exergy destruction and Nusselt number: (a) constant substrate heat flux; (b) constant substrate temperature.

Fig. 14. Variations of (a) Nu/Nu0, (b) f/f0, (c) EEC, and (d) R3.

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6. Conclusions

In this paper, the conjugate heat transfer optimization was carried out based on exergy destruction minimization principle to find the optimized flow pattern in the mini-channel heat sink. Subsequently, the inclined parallelepiped ribs were applied to realize the optimized flow pattern so as to enhance the conjugate heat transfer performance in the mini-channel heat sink. Some conclusions were drawn as follows:

(a) The main characteristic of the optimized flow pattern was three pairs of longitudinal swirls flow.

(b) The inclined parallelepiped ribs were applied to realize the optimized flow pattern successfully. Besides, the Vh,m/Vh,m,0 was the dominated contribution to Nu/Nu0 near the inclined paral-lelepiped ribs, while the ΔTm/ΔTm,0 was the dominated contri-bution to Nu/Nu0 between two inclined parallelepiped ribs in the flow direction. Enhancing the heat convection intensity was a promising method for improving the convective heat transfer performance. In addition, the exergy destruction minimization principle was verified with specifying constant substrate heat flux.

(c) The variation ranges of Nu/Nu0, f/f0, efficiency evaluation cri-terion, and overall performance criterion were 1.35–5.92, 1.27–8.75, 0.68–1.12, and 1.31–4.22, respectively. Within a temperature difference of 60 K between the substrate and work-ing fluid, the maximum average heat flux that could be dissipated achieved 3.2 × 106 W/m2 with a corresponding pressure gradient of 148.9 × 103 Pa/m at Re = 1100. In addition, the recommended pitch ratio and width ratio were 1 and 0.2, respectively.

CRediT authorship contribution statement

Hui Xiao: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing. Zhichun Liu: Resources, Supervision, Project administration. Wei Liu: Resources, Supervision, Project administra-tion, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant No. 51736004). Particular acknowledgements to Ms. Qiao Tang for her help in data curation.

References

[1] N.A.C. Sidik, M.N.A.W. Muhamad, W.M.A.A. Japar, Z.A. Rasid, An overview of passive techniques for heat transfer augmentation in microchannel heat sink, Int. Commun. Heat Mass Transfer 88 (2017) 74–83.

[2] H.E. Ahmed, B.H. Salman, A.S. Kherbeet, M.I. Ahmed, Optimization of thermal design of heat sinks: a review, Int. J. Heat Mass Transf. 118 (2018) 129–153.

[3] N.H. Naqiuddin, L.H. Saw, M.C. Yew, F. Yusof, T.C. Ng, M.K. Yew, Overview of micro-channel design for high heat flux application, Renew. Sustain. Energy Rev. 82 (2018) 901–914.

Fig. 15. Variations of (a) maximum average heat flux and (b) pressure drop.

Fig. 16. Comparisons with results in the literature: (a) Nu/Nu0; (b) f/f0.

H. Xiao et al.

http://refhub.elsevier.com/S1359-4311(20)33611-5/h0005http://refhub.elsevier.com/S1359-4311(20)33611-5/h0005http://refhub.elsevier.com/S1359-4311(20)33611-5/h0005http://refhub.elsevier.com/S1359-4311(20)33611-5/h0010http://refhub.elsevier.com/S1359-4311(20)33611-5/h0010http://refhub.elsevier.com/S1359-4311(20)33611-5/h0015http://refhub.elsevier.com/S1359-4311(20)33611-5/h0015http://refhub.elsevier.com/S1359-4311(20)33611-5/h0015

Applied Thermal Engineering 182 (2021) 116131

14

[4] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE Electron Device Lett. 2 (5) (1981) 126–129.

[5] S. Kandlikar, S. Garimella, D. Li, S. Colin, M.R. King, Heat Transfer and Fluid Flow In Minichannels and Microchannels, second ed., Elsevier, 2014.

[6] L.V. Vajravel, S. Kuppusamy Swaminathan, S. Baskaran, R. Kalpoondi Sekar, Experimental investigations on heat transfer in a new minichannel heat sink, Int. J. Therm. Sci. 140 (2019) 144–153.

[7] H.-L. Liu, Y.-Q. Shao, Z.-T. Chen, Z.-L. Xie, Heat transfer and flow performance of a novel T type heat sink with GaInSn coolant, Int. J. Therm. Sci. 144 (2019) 129–146.

[8] S. Lu, K. Vafai, A comparative analysis of innovative microchannel heat sinks for electronic cooling, Int. Commun. Heat Mass Transfer 76 (2016) 271–284.

[9] B. Osanloo, A. Mohammadi-Ahmar, A. Solati, M. Baghani, Performance enhancement of the double-layered micro-channel heat sink by use of tapered channels, Appl. Therm. Eng. 102 (2016) 1345–1354.

[10] A. Arabpour, A. Karimipour, D. Toghraie, O.A. Akbari, Investigation into the effects of slip boundary condition on nanofluid flow in a double-layer microchannel, J. Therm. Anal. Calorim. 131 (3) (2018) 2975–2991.

[11] T.-C. Hung, Y.-X. Huang, W.-M. Yan, Thermal performance of porous microchannel heat sink: effects of enlarging channel outlet, Int. Commun. Heat Mass Transfer 48 (2013) 86–92.

[12] M. Dehghan, M.S. Valipour, S. Saedodin, Microchannels enhanced by porous materials: heat transfer enhancement or pressure drop increment? Energy Convers. Manage. 110 (2016) 22–32.

[13] X. Liu, Y. Li, H. Zhang, Y. Liu, X. Chen, Effect of pore structure on heat transfer performance of lotus-type porous copper heat sink, Int. J. Heat Mass Transf. 144 (2019), 118641.

[14] C.J. Ho, W.C. Chen, An experimental study on thermal performance of Al2O3/ water nanofluid in a minichannel heat sink, Appl. Therm. Eng. 50 (1) (2013) 516–522.

[15] A. Muhammad, D. Selvakumar, J. Wu, Numerical investigation of laminar flow and heat transfer in a liquid metal cooled mini-channel heat sink, Int. J. Heat Mass Transf. 150 (2020), 119265.

[16] G. Wang, L. Hao, P. Cheng, An experimental and numerical study of forced convection in a microchannel with negligible axial heat conduction, Int. J. Heat Mass Transf. 52 (3) (2009) 1070–1074.

[17] I.A. Ghani, N.A.C. Sidik, N. Kamaruzaman, Hydrothermal performance of microchannel heat sink: The effect of channel design, Int. J. Heat Mass Transf. 107 (2017) 21–44.

[18] Z. Chamanroy, M. Khoshvaght-Aliabadi, Analysis of straight and wavy miniature heat sinks equipped with straight and wavy pin-fins, Int. J. Therm. Sci. 146 (2019), 106071.

[19] A. Datta, V. Sharma, D. Sanyal, P. Das, A conjugate heat transfer analysis of performance for rectangular microchannel with trapezoidal cavities and ribs, Int. J. Therm. Sci. 138 (2019) 425–446.

[20] A. Gorasiya, S.K. Saha, Single phase laminar fluid flow and heat transfer in microchannel with cylindrical and parallelepiped micro-fins, Heat Mass Transf. 55 (3) (2019) 613–626.

[21] M.A. Alfellag, H.E. Ahmed, A.S. Kherbeet, Numerical simulation of hydrothermal performance of minichannel heat sink using inclined slotted plate-fins and triangular pins, Appl. Therm. Eng. 164 (2020), 114509.

[22] C. Bi, G.H. Tang, W.Q. Tao, Heat transfer enhancement in mini-channel heat sinks with dimples and cylindrical grooves, Appl. Therm. Eng. 55 (1) (2013) 121–132.

[23] J. Liu, G. Xie, T.W. Simon, Turbulent flow and heat transfer enhancement in rectangular channels with novel cylindrical grooves, Int. J. Heat Mass Transf. 81 (2015) 563–577.

[24] H.E. Ahmed, A.S. Kherbeet, M.I. Ahmed, B.H. Salman, Heat transfer enhancement of micro-scale backward-facing step channel by using turbulators, Int. J. Heat Mass Transf. 126 (2018) 963–973.

[25] J.-F. Zhang, L. Jia, W.-W. Yang, J. Taler, P. Oclon, Numerical analysis and parametric optimization on flow and heat transfer of a microchannel with longitudinal vortex generators, Int. J. Therm. Sci. 141 (2019) 211–221.

[26] E. Hosseinirad, M. Khoshvaght-Aliabadi, F. Hormozi, Evaluation of heat transfer and pressure drop in a mini-channel using transverse rectangular vortex-generators with various non-uniform heights, Appl. Therm. Eng. 161 (2019), 114196.

[27] G. Lu, X. Zhai, Analysis on heat transfer and pressure drop of a microchannel heat sink with dimples and vortex generators, Int. J. Therm. Sci. 145 (2019), 105986.

[28] W. Liu, P. Liu, J.B. Wang, N.B. Zheng, Z.C. Liu, Exergy destruction minimization: a principle to convective heat transfer enhancement, Int. J. Heat Mass Transf. 122 (2018) 11–21.

[29] H. Xiao, J. Wang, Z. Liu, W. Liu, Turbulent heat transfer optimization for solar air heater with variation method based on exergy destruction minimization principle, Int. J. Heat Mass Transf. 136 (2019) 1096–1105.

[30] Y.-L. He, W.-Q. Tao, Convective heat transfer enhancement: mechanisms, techniques, and performance evaluation, in, Adv. Heat Transfer, Elsevier (2014) 87–186.

[31] H. Xiao, Z. Dong, R. Long, K. Yang, F. Yuan, A study on the mechanism of convective heat transfer enhancement based on heat convection velocity analysis, Energies 12 (21) (2019) 4175.

[32] W. Liu, P. Liu, Z.M. Dong, K. Yang, Z.C. Liu, A study on the multi-field synergy principle of convective heat and mass transfer enhancement, Int. J. Heat Mass Transf. 134 (2019) 722–734.

[33] N. Zheng, F. Yan, K. Zhang, T. Zhou, Z. Sun, A review on single-phase convective heat transfer enhancement based on multi-longitudinal vortices in heat exchanger tubes, Appl. Therm. Eng. 164 (2020), 114475.

[34] IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam, in: International Steam Tables: Properties of Water and Steam Based on the Industrial Formulation IAPWS-IF97, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008, pp. 7–150.

[35] P.-S. Lee, S.V. Garimella, D. Liu, Investigation of heat transfer in rectangular microchannels, Int. J. Heat Mass Transf. 48 (9) (2005) 1688–1704.

[36] V. Gnielinski, New equations for heat and mass-transfer in turbulent pipe and channel flow, Int. Chem. Eng. 16 (2) (1976) 359–368.

[37] H. Xiao, J. Wang, Z. Liu, W. Liu, A consistent SIMPLE algorithm with extra explicit prediction — SIMPLEPC, Int. J. Heat Mass Transf. 120 (2018) 1255–1265.

[38] F. Moukalled, L. Mangani, M. Darwish, The Finite Volume Method in Computational Fluid Dynamics, Springer, New York, 2016.

[39] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington D.C., 1980.

[40] W. Liu, Z. Liu, L. Ma, Application of a multi-field synergy principle in the performance evaluation of convective heat transfer enhancement in a tube, Chin. Sci. Bull. 57 (13) (2012) 1600–1607.

[41] P.J. Roache, Perspective: a method for uniform reporting of grid refinement studies, J. Fluids Eng. 116 (3) (1994) 405–413.

[42] C. Liu, J.-T. Teng, J.-C. Chu, Y.-L. Chiu, S. Huang, S. Jin, T. Dang, R. Greif, H.- H. Pan, Experimental investigations on liquid flow and heat transfer in rectangular microchannel with longitudinal vortex generators, Int. J. Heat Mass Transf. 54 (13) (2011) 3069–3080.

[43] H. Xiao, Z. Dong, Z. Liu, W. Liu, Heat transfer performance and flow characteristics of solar air heaters with inclined trapezoidal vortex generators, Appl. Therm. Eng. 115484 (2020).

[44] M. Khoshvaght-Aliabadi, M. Sahamiyan, M. Hesampour, O. Sartipzadeh, Experimental study on cooling performance of sinusoidal–wavy minichannel heat sink, Appl. Therm. Eng. 92 (2016) 50–61.

H. Xiao et al.

http://refhub.elsevier.com/S1359-4311(20)33611-5/h0020http://refhub.elsevier.com/S1359-4311(20)33611-5/h0020http://refhub.elsevier.com/S1359-4311(20)33611-5/h0025http://refhub.elsevier.com/S1359-4311(20)33611-5/h0025http://refhub.elsevier.com/S1359-4311(20)33611-5/h0030http://refhub.elsevier.com/S1359-4311(20)33611-5/h0030http://refhub.elsevier.com/S1359-4311(20)33611-5/h0030http://refhub.elsevier.com/S1359-4311(20)33611-5/h0035http://refhub.elsevier.com/S1359-4311(20)33611-5/h0035http://refhub.elsevier.com/S1359-4311(20)33611-5/h0035http://refhub.elsevier.com/S1359-4311(20)33611-5/h0040http://refhub.elsevier.com/S1359-4311(20)33611-5/h0040http://refhub.elsevier.com/S1359-4311(20)33611-5/h0045http://refhub.elsevier.com/S1359-4311(20)33611-5/h0045http://refhub.elsevier.com/S1359-4311(20)33611-5/h0045http://refhub.elsevier.com/S1359-4311(20)33611-5/h0050http://refhub.elsevier.com/S1359-4311(20)33611-5/h0050http://refhub.elsevier.com/S1359-4311(20)33611-5/h0050http://refhub.elsevier.com/S1359-4311(20)33611-5/h0055http://refhub.elsevier.com/S1359-4311(20)33611-5/h0055http://refhub.elsevier.com/S1359-4311(20)33611-5/h0055http://refhub.elsevier.com/S1359-4311(20)33611-5/h0060http://refhub.elsevier.com/S1359-4311(20)33611-5/h0060http://refhub.elsevier.com/S1359-4311(20)33611-5/h0060http://refhub.elsevier.com/S1359-4311(20)33611-5/h0065http://refhub.elsevier.com/S1359-4311(20)33611-5/h0065http://refhub.elsevier.com/S1359-4311(20)33611-5/h0065http://refhub.elsevier.com/S1359-4311(20)33611-5/h0070http://refhub.elsevier.com/S1359-4311(20)33611-5/h0070http://refhub.elsevier.com/S1359-4311(20)33611-5/h0070http://refhub.elsevier.com/S1359-4311(20)33611-5/h0075http://refhub.elsevier.com/S1359-4311(20)33611-5/h0075http://refhub.elsevier.com/S1359-4311(20)33611-5/h0075http://refhub.elsevier.com/S1359-4311(20)33611-5/h0080http://refhub.elsevier.com/S1359-4311(20)33611-5/h0080http://refhub.elsevier.com/S1359-4311(20)33611-5/h0080http://refhub.elsevier.com/S1359-4311(20)33611-5/h0085http://refhub.elsevier.com/S1359-4311(20)33611-5/h0085http://refhub.elsevier.com/S1359-4311(20)33611-5/h0085http://refhub.elsevier.com/S1359-4311(20)33611-5/h0090http://refhub.elsevier.com/S1359-4311(20)33611-5/h0090http://refhub.elsevier.com/S1359-4311(20)33611-5/h0090http://refhub.elsevier.com/S1359-4311(20)33611-5/h0095http://refhub.elsevier.com/S1359-4311(20)33611-5/h0095http://refhub.elsevier.com/S1359-4311(20)33611-5/h0095http://refhub.elsevier.com/S1359-4311(20)33611-5/h0100http://refhub.elsevier.com/S1359-4311(20)33611-5/h0100http://refhub.elsevier.com/S1359-4311(20)33611-5/h0100http://refhub.elsevier.com/S1359-4311(20)33611-5/h0105http://refhub.elsevier.com/S1359-4311(20)33611-5/h0105http://refhub.elsevier.com/S1359-4311(20)33611-5/h0105http://refhub.elsevier.com/S1359-4311(20)33611-5/h0110http://refhub.elsevier.com/S1359-4311(20)33611-5/h0110http://refhub.elsevier.com/S1359-4311(20)33611-5/h0115http://refhub.elsevier.com/S1359-4311(20)33611-5/h0115http://refhub.elsevier.com/S1359-4311(20)33611-5/h0115http://refhub.elsevier.com/S1359-4311(20)33611-5/h0120http://refhub.elsevier.com/S1359-4311(20)33611-5/h0120http://refhub.elsevier.com/S1359-4311(20)33611-5/h0120http://refhub.elsevier.com/S1359-4311(20)33611-5/h0125http://refhub.elsevier.com/S1359-4311(20)33611-5/h0125http://refhub.elsevier.com/S1359-4311(20)33611-5/h0125http://refhub.elsevier.com/S1359-4311(20)33611-5/h0130http://refhub.elsevier.com/S1359-4311(20)33611-5/h0130http://refhub.elsevier.com/S1359-4311(20)33611-5/h0130http://refhub.elsevier.com/S1359-4311(20)33611-5/h0135http://refhub.elsevier.com/S1359-4311(20)33611-5/h0135http://refhub.elsevier.com/S1359-4311(20)33611-5/h0140http://refhub.elsevier.com/S1359-4311(20)33611-5/h0140http://refhub.elsevier.com/S1359-4311(20)33611-5/h0140http://refhub.elsevier.com/S1359-4311(20)33611-5/h0145http://refhub.elsevier.com/S1359-4311(20)33611-5/h0145http://refhub.elsevier.com/S1359-4311(20)33611-5/h0145http://refhub.elsevier.com/S1359-4311(20)33611-5/h0150http://refhub.elsevier.com/S1359-4311(20)33611-5/h0150http://refhub.elsevier.com/S1359-4311(20)33611-5/h0150http://refhub.elsevier.com/S1359-4311(20)33611-5/h0155http://refhub.elsevier.com/S1359-4311(20)33611-5/h0155http://refhub.elsevier.com/S1359-4311(20)33611-5/h0155http://refhub.elsevier.com/S1359-4311(20)33611-5/h0160http://refhub.elsevier.com/S1359-4311(20)33611-5/h0160http://refhub.elsevier.com/S1359-4311(20)33611-5/h0160http://refhub.elsevier.com/S1359-4311(20)33611-5/h0165http://refhub.elsevier.com/S1359-4311(20)33611-5/h0165http://refhub.elsevier.com/S1359-4311(20)33611-5/h0165http://refhub.elsevier.com/S1359-4311(20)33611-5/h0175http://refhub.elsevier.com/S1359-4311(20)33611-5/h0175http://refhub.elsevier.com/S1359-4311(20)33611-5/h0180http://refhub.elsevier.com/S1359-4311(20)33611-5/h0180http://refhub.elsevier.com/S1359-4311(20)33611-5/h0185http://refhub.elsevier.com/S1359-4311(20)33611-5/h0185http://refhub.elsevier.com/S1359-4311(20)33611-5/h0190http://refhub.elsevier.com/S1359-4311(20)33611-5/h0190http://refhub.elsevier.com/S1359-4311(20)33611-5/h0195http://refhub.elsevier.com/S1359-4311(20)33611-5/h0195http://refhub.elsevier.com/S1359-4311(20)33611-5/h0200http://refhub.elsevier.com/S1359-4311(20)33611-5/h0200http://refhub.elsevier.com/S1359-4311(20)33611-5/h0200http://refhub.elsevier.com/S1359-4311(20)33611-5/h0205http://refhub.elsevier.com/S1359-4311(20)33611-5/h0205http://refhub.elsevier.com/S1359-4311(20)33611-5/h0210http://refhub.elsevier.com/S1359-4311(20)33611-5/h0210http://refhub.elsevier.com/S1359-4311(20)33611-5/h0210http://refhub.elsevier.com/S1359-4311(20)33611-5/h0210http://refhub.elsevier.com/S1359-4311(20)33611-5/h0215http://refhub.elsevier.com/S1359-4311(20)33611-5/h0215http://refhub.elsevier.com/S1359-4311(20)33611-5/h0215http://refhub.elsevier.com/S1359-4311(20)33611-5/h0220http://refhub.elsevier.com/S1359-4311(20)33611-5/h0220http://refhub.elsevier.com/S1359-4311(20)33611-5/h0220

Conjugate heat transfer enhancement in the mini-channel heat sink by realizing the optimized flow pattern1 Introduction2 Mini-channel heat sinks3 Research methods3.1 Governing equations3.2 Boundary conditions3.3 Parameter definitions3.4 Computational methods

4 Conjugate heat transfer optimization4.1 Exergy destruction minimization4.2 Optimized flow pattern

5 Optimized flow pattern realization5.1 Heat transfer enhancement structure5.2 Results verifications5.3 Heat transfer enhancement mechanism5.3.1 Flow characteristics and temperature distributions5.3.2 Heat convection velocity analysis5.3.3 Thermal resistance and exergy destruction

5.4 Performance evaluation5.4.1 Thermal hydraulic performance5.4.2 Maximum average heat flux5.4.3 Performance comparisons

6 ConclusionsCRediT authorship contribution statementDeclaration of Competing InterestAcknowledgementReferences