Numerical investigation of the fluid flow and heat transfer characteristics of tree-shaped microchannel heat sink with variable cross-section

Journal Pre-proof Numerical investigation of the fluid flow and heat transfer characteristics of tree-shaped microchannel heat sink with variable cross-section Pingnan Huang, Guanping Dong, Xineng Zhong, Minqiang Pan

PII:

S0255-2701(19)31036-0

DOI:

https://doi.org/10.1016/j.cep.2019.107769

Reference:

CEP 107769

To appear in:

Chemical Engineering and Processing - Process Intensification

Received Date:

21 August 2019

Revised Date:

18 October 2019

Accepted Date:

27 November 2019

Please cite this article as: Huang P, Dong G, Zhong X, Pan M, Numerical investigation of the fluid flow and heat transfer characteristics of tree-shaped microchannel heat sink with variable cross-section, Chemical Engineering and Processing - Process Intensification (2019), doi: https://doi.org/10.1016/j.cep.2019.107769

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Numerical investigation of the fluid flow and heat transfer characteristics of treeshaped microchannel heat sink with variable cross-section Pingnan Huang

Guanping Dong Xineng Zhong

Minqiang Pan*

School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China

*Corresponding author. Tel.: +86 20 8711 4634; fax: +86 20 8711 4634 E-mail address: [email protected] (M. Pan)

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Graphical abstract

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Highlights A tree-shape microchannel heat sink with a variable cross-section was firstly designed.



The coupling effects of bonic fractal and variable cross-section on heat transfer was studied.



Cavities in the TMHS can enhance branching effects, while ribs can suppress branching effects.



The TMHS-C exhibits a highest comprehensive performance.



Higher Reynolds number enhances heat transfer and the coupling effects.

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

Abstract: The bionic fractal microchannel heat sink (MHS), which is characterized by a uniform velocity distribution and higher heat transfer efficiency, is an ideal heat sink candidate, while the structure of the

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variable cross-section has a great impact on the fluid flow and heat transfer performance of the MHS.

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Considering the coupling effects of the bionic fractal and variable cross-section, a tree-shaped MHS (TMHS) with a variable cross-section is firstly proposed. The flow characteristics, heat transfer characteristics,

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pressure drop characteristics, and comprehensive performance of TMHSs with cavities and ribs were

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analyzed numerically using the commercial software Fluent, and compared with the traditional smooth TMHS. The results indicate that the structure of varibale cross-section can further enhance heat transfer

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performance of the TMHS. The TMHS with ribs (TMHS-R) exhibited the highest heat transfer performance, but also had the highest pressure drop. Finally, the heat transfer index was put forward to consider both heat

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transfer and pressure drop characteristics, and the results found that the TMHS with cavities (TMHS-C) had the highest comprehensive performance at the range of the calculating Reynolds number. Keywords: Microchannel heat sink; Tree-shaped; Variable cross-section; Heat transfer characteristics; Pressure drop characteristics

Symbols used 2

Nomenclature Inner wall/fluid contact area based on the TMHS-S (m2)

Cp

Specific heat capacity (J∙kg-1∙K)

D

Hydraulic diameter (m)

H

Height of the microchannel (m)

h

Heat transfer coefficient (W∙m-2)

L

Length of the microchannel (m)

𝑚̇

Mass flow rate (kg∙s-1)

n

Number of branching levels (-)

Nu

Nusselt number (-)

P

Pressure (Pa)

∆P

Pressure drop (Pa)

q

Heat flux (W∙m-2)

Q

Quantity of heat (W)

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R

Radius of disk-shape TMHS (m) Reynolds number (-)

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Re

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S T

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A

Spacing between cavities or ribs (m) Temperature (K)

TMHS-C

Tree-shape microchannel heat sink with cavities

TMHS-R

Tree-shape microchannel heat sink with ribs

TMHS-S

Smooth tree-shape microchannel heat sink

𝑢

Inlet velocity (m∙s-1) 3

U

Volume flow rate (m3∙s-1)

𝒗

Velocity (m∙s-1)

W

Width of the microchannel (m)

Greek letters Length ratio (-)

𝑤

Width difference (mm)

𝜇

Kinetic viscosity of fluid medium (Pa∙s)

𝜌

Density of liquid water (kg∙m-3)

𝜆

Thermal conductive (W∙m-1K-1)

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Subscripts cavity

f

fluid

i

Index of the fractal level

in

inlet

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na outlet

sm w

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s

Puming power

ur

pum r

re

c

out

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𝛽

rib solid smooth wall

4

1. Introduction With the development of technology, electronic chips are moving toward miniaturization as well as increasingly larger heat flux. Heat dissipation has become a very challenging task in the application of electronic chips. Microchannel heat sink (MHS) with hydraulic diameters in the range of 10-1000 μm is an ideal choice for heat dissipation in electronic chips due to its compactness and high heat transfer performance. After the landmark work by Tuckerman and Pease [1], much effort [2-7] has been exerted to

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improve the heat transfer performance of the MHS. Among them, bionic fractal MHS, with higher heat transfer performance and lower pressure drop, is an effective and potential option to improve the performance of MHS.

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Inspired by the natural transport systems, such as the root and leaf vein system of plants, or mammalian

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circulatory and respiratory systems, the bionic fractal microchannel was firstly put forward by professor Adrian Bejan [8]. He then described the great potential of bionic fractal microchannel by combining

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constructal theory and entropy production principles [9, 10]. Because of its bright future, there are a large

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number of studies focus on the bionic fractal MHS in recent decades. Chen and Cheng [11] designed a fractal branching MHS and established the relationships of heat transfer and pressure drop to the specific

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fractal MHS and the traditional parallel MHS. The results showed that the fractal MHS can enhance heat transfer. Soon after, they carried out experimental research on the thermal efficiency of a fractal tree-like

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MHS. The results further confirmed that when the fractal level is less than 4, the thermal efficiency of a fractal tree-like MHS is much higher than that of a traditional parallel MHS [12]. Senn and Poulikeakos [13] designed a tree-like MHS and numerically investigated the heat transfer and pressure drop characteristics, and then compared the results to a serpentine MHS. The results demonstrated that the tree net with six branching levels generates only about half the pressure drop of a serpentine pattern, and 5

provides a larger heat transfer capability. Wang et al. [14] designed a leaf-like MHS and compared the fluid flow and heat transfer performance between the symmetrical and asymmetrical architecture. The results found that asymmetrical architecture can significantly reduce the pressure drop while maintaining a maximum temperature difference. Xu et al. [15] designed four different MHSs and analyzed them numerically. The results showed that a tree-like MHS can remove the most heat at the same inlet flow rate. Xia et al. [16] designed a novel fractal tree-like MHS in the cooling sleeve of the spindle. They also found

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that the tree-like MHS exhibited a lower pressure drop and a more uniform temperature distribution. The coefficient of performance of the tree-like MHS was almost twice that of the traditional helical MHS. Peng et al. [17-19] designed a vein-like fractal-architecture vapor chamber; their results also confirmed that the

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leaf-vein-like fractal architecture exhibits better performance than does a parallel structure. As illustrated

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by the above literature review, it is evident that the bionic fractal MHS has a higher thermal performance, and it is more suitable for MHS design. However, most studies of the bionic fractal MHS have focused on

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structural parameter optimization or the introduction of a new structure. Research on further strengthening

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design based on the existing bionic fractal MHS is seldom reported. Studies [20-22] have found that the structure of variable cross-section in the MHS can enhance the

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forced convective heat transfer. Therefore, the structure of variable cross-section is an effective way to further enhance heat transfer performance of the MHS. Recently, a large number of studies are focusing on

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the microchannels with variable cross-section for further enhancing heat transfer. Xu et al. [23] designed a microchannel with dimples at the bottom, and numerically investigated the characteristics of flow and heat transfer in microchannel. The results showed that in comparison to straight channels, dimpled surface reduced the local flow resistance and also improved thermal performance of the MHS. Chai et al. [24-26] designed microchannels with aligned and offset fan-shaped ribs, and numerically studied the effects of fan6

shaped ribs on the fluid flow, heat transfer, pressure drop and comprehensive performance. The results found that the geometry of fan-shaped ribs can increase the flow velocity, destroyed the fluid and thermal boundary layer, thereby effectively enhance heat transfer with a higher pressure drop. Zhai et al. [27] designed four complex microchannels with different variable cross-section and investigated the physical mechanism of heat transfer enhancement in the microchannels based on the field synergy principle and entropy generation analysis. The results found that the introduction of variable cross-section can cause the

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good synergy between the velocity vector and the temperature gradient, and the destruction of the thermal boundary, finally leading to the irreversibility reduction and heat transfer enhancement. Pan et al. [28] designed and constructed a microchannel heat exchanger with fan-shaped cavities (FSCs), and

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experimentally studied the effects of structural parameters on the heat transfer performance of the

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microchannel heat exchanger. The results found that the structure of FSCs can not only introduce the spurt and throttling effects, but also cause the interrupted and redeveloped thermal and hydraulic boundary layers

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heat exchanger without FSCs.

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along microchannel, thereby effectively enhance heat transfer performance compared to the microchannel

However, unlike the MHS with variable cross-section, the structure of variable cross-section in bionic

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fractal MHS will cause the coupling effects. To the author’s knowledge, the coupling effects of bionic fractal and variable cross-section on heat transfer are seldom reported. Whether the coupling effects of

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variable cross-section and bionic fractal structures in the MHS can further improve the heat transfer performance, and what kind of cross-sections has more effect on heat transfer performance of the TMHS are still unknown. To bridge this information gap, tree-shaped MHSs with fan-shaped cavities and ribs on the sidewall were proposed in this study, and the fluid flow, heat transfer, and pressure drop characteristics, as well as the comprehensive performance, were numerically analyzed and compared with the traditional 7

smooth TMHS.

2. Numerical model 2.1 Physical model The fractal-like TMHS has been studied extensively. In this study, a typical disk-shaped TMHS based on the model presented by Peng et al. [17] is established, as depicted in Fig. 1. The physical model consists of three parts: the cover plate at the top, the TMHS substrate, and the heat source attached to the TMHS

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substrate. To accelerate the convergence speed and improve the computational efficiency, one-twelfth of the total structure of a simplified TMHS is considered for numerical simulation. It consists of one inlet,

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four outlets, and tree-shaped microchannel networks connecting them.

Fig. 1. Structure of the TMHS.

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During heat sink operations, a quantity of heat is transferred into the TMHS from the bottom surface, and is then removed by the working fluid that emerges from the inlet and is expelled from the four outlets. Therefore, the structure of the fluid channel has an important role in the performance of the TMHS. In this study, the setting of the parameters of the structure is described in detail. Fig. 2 shows the two-dimensional view and parameter settings of one-twelfth of the TMHS. The radius of the disk-shaped TMHS is set as R, 8

while the radius of the inlet and outlet are set as Rin and Rout, respectively. The fractal angle is set as 60°. In addition, based on the self-similarity of the fractal structure, the sizes of the microchannels can be determined by the sizes of the level 0 microchannel according to the following scaling laws: 𝑑𝑖+1 𝑑𝑖 𝑙𝑖+1 𝑙𝑖

= 𝛽,

(1)

= 𝛾,

(2)

where 𝑑𝑖 and 𝑙𝑖 represent the hydraulic diameter and length of the channel segment in the ith level,

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respectively, and 𝛽 and 𝛾 represent the diameter ratio and length ratio, respectively. To minimize the pumping power, most of the existing research has set 𝛽 and 𝛾 to be 2-1/3 and 2-1/2, respectively [29, 30]. However, Pence and Enfield [31] found that there is a major problem in considering high branching levels

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when using the conventional hydraulic diameter to define the branching ratio to characterize rectangular

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channels for a fixed channel height. Therefore, in this study, the length ratio is kept at 0.7, while the diameter ratio is represented by the width difference, that is, 𝑤 = 𝑊𝑖 − 𝑊𝑖+1 , which is the same as the parameters

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used in Peng et al. [32]. The TMHS with a variable cross-section is formed by adding 12 semicircle cavities

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or ribs at regular intervals to each microchannel of the smooth TMHS, as shown in Figs. 2b and 2c. The diameters of the cavities and ribs are set as half of the width of the corresponding microchannel. The

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locations of the first cavity and rib, and the intervals between different levels, obey the scaling law. The

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detailed parameter settings are listed in Table 1.

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(a) L2

0.005

W1

Ro

W2 ¦È

l1 ve Le

W0

Rin

0

ut

L0

el3 Lev

Level 0

W

3

L1

Level 2

(b)

Rc

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A 2:1

0.005

0 Sc

A

re

Lc

L3

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-0.005

(c)

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-0.005

Rr B 2:1

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0.005

ur

0

Sr

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Lr

B

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Fig. 2. Two-dimensional view of one-twelfth of the TMHS. (a) TMHS-S; (b) TMHS-C; (c) TMHS-R. Table 1. Structural parameter settings. 10

Parameter

Variable

Parameter

L0 (mm)

10

Rc0, Rr0 (mm)

0.15

W0 (mm)

0.6

Lc0, Lr0 (mm)

1.5

H (mm)

0.5

Sc0, Sr0 (mm)

1.5

Rin (mm)

2.5

𝜃

60°

Rout (mm)

0.15

𝛽

0.7

R (mm)

27.5

𝑤

0.1

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Variable

2.2 Model assumptions & Governing equations

To make the mathematical models simpler and more reasonable, several assumptions are made:

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(1) Steady and laminar flow;

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(2) Incompressible, Newtonian, and viscous fluid;

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(3) Non-slip velocity and temperature at a rough surface;

(4) Temperature and heat flux continuity at all solid boundaries.

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The steady-state conservation equations for mass, momentum, and energy in fluids are as follows. Mass conservation equation:

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∇ ∙ 𝒗 = 0.

(3)

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Momentum conservation equation: 𝜌(𝒗 ∙ ∇)𝒗 = −∇𝑃 + 𝜇∇2 𝒗.

(4)

𝜌𝐶𝑝,𝑓 (𝒗 ∙ ∇𝑇) = 𝜆𝑓 ∇2 𝑇.

(5)

Energy conservation equation:

For solids, the energy conservation equation is given by:

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𝜆𝑠 ∇2 𝑇 = 0.

(6)

2.3 Numerical method In this study, the computational fluid dynamics (CFD) software Fluent was adopted to simulate the fluid flow and heat transfer in the TMHS. Copper, which has a thermal conductivity of 387.6 W/(m∙K), was chosen as the material for the TMHSs, and liquid water with temperature dependent thermophysical properties was used as the working fluid [26, 33]. The inlet is set as the velocity inlet, and the temperature

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is kept at 298.15 K. Both outlets are set as pressure outlets, and the relative pressure is 0. In addition, a heat source with a constant heat flux q=105W/m2 is applied to the bottom of the heat sink, as shown in Fig. 1, and the other walls were assumed to be insulation. The convection terms in the momentum equation and

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the energy conservation equation are discredited by a second-order upwind scheme. The SIMPLE method

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is adopted to deal with the relationship between velocity and pressure. The computations are considered to be converged when the normalized residuals are less than 10-4 for the flow equations and 10-7 for the energy

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equations.

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2.4 Grid independece test and rationality verification of the model To reduce the calculation error and improve the simulation precision, a detailed grid independence

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study was carried out, in which three element sizes (0.1, 0.05, and 0.03 mm) were compared based on the TMHS-C model at an inlet velocity of 0.1 m/s. The pressure drop and the maximum temperature rise over

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the bottom surfaces of the heat sources are compared. The results are exhibited in Table 2. The deviations in the pressure drop and temperature rise between the 0.05 and 0.03 mm element sizes are 0.179% and 0.180%, respectively. This indicates the calculation with element size of 0.05 mm is accurate enough. Hence, all the computations were carried out using the mesh with 0.05 mm element size to reduce the computational load. The final mesh of the TMHS-C is shown in Fig. 3. 12

Table 2. Sensitivity analysis of the TMHS-C. Element size Mesh number Pressure drop (Pa) Difference Temperature rise (K) Difference 3141649

1532.1691

0.948%

37.1231

1.676%

0.05

7499528

1534.1802

0.818%

37.7504

0.014%

0.02

17778859

1546.8341

0

37.7558

0

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0.1

Fig. 3. Computational domain mesh.

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In order to verify the rationality of the results, the outlet temperature of numerical results is compared

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with that of theoretical calculation results. According to 𝑄 = 𝐶𝑝 𝑚̇∆𝑇, the outlet temperature can be calculated [34, 35]. Fig. 4 shows the verification of the numerical and theoretic results for the outlet temperature versus different Reynolds numbers. It can be seen that the numerical and theoretic results are in good agreement with each other, the maximum relative error is less than 1%. Therefore, the rationality and validity of the numerical results are verified. 13

Numerical results of TMHS-S Numerical results of TMHS-C Numerical results of TMHS-R Theoretic results

350

340

330

320

310 200

300

400

500

600

700

900

1000

1100

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Re

800

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Outlet Temperature (K)

360

3. Performance evaluation parameters

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Fig. 4. Verification of numerical and theoretic results for outlet temperature.

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To evaluate the flow characteristics, the dimensionless number, i.e., the entrance Reynolds number, is defined as:

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𝑅𝑒 =

𝜌𝑢𝐷 𝜇

,

(7)

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where 𝜌 and 𝜇 are the density and dynamic viscosity of liquid water, respectively, while 𝑢 and D are the inlet velocity and diameter, respectively.

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To analyze the heat transfer performances, the average heat transfer coefficient and Nusselt number are given by:

𝑄

ℎ = 𝐴∆𝑇 = 𝑁𝑢 =

ℎ𝐷 𝜆𝑓

1/12𝜋𝑅 2 𝑞 𝐴(𝑇𝑠 −𝑇𝑓 )

,

,

(8) (9)

where 𝑅 is the radius of the disk-shaped heat sink, A is the inner wall/fluid contact area based on the 14

TMHS-S, 𝑇𝑠 and 𝑇𝑓 are the volume-average temperatures in the solid and fluid zones, respectively, and 𝜆𝑓 is the mass-average thermal conductivity of water. The average hydraulic diameter 𝐷 is defined as: 𝐷=

𝐷0 𝐿0 +2𝐷1 𝐿1 +2𝐷2 𝐿2 +4𝐷3 𝐿3 𝐿0 +2𝐿1 +2𝐿2 +4𝐿3

,

(10)

where 𝐷𝑖 represents the hydraulic diameter of the microchannel at the ith level, which can be calculated as: 2𝑊 𝐻

𝑖 𝐷𝑖 = (𝑊 +𝐻) .

(11)

𝑖

In addition, pressure drop is another significant parameter by which to evaluate the performance of

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the TMHS. For a heat sink system, the larger the pressure drop, the greater the required pumping power. In this study, the pressure drop and pumping power are defined as Eqs. (11) and (12). ∆𝑃 = 𝑃𝑖𝑛 − 𝑃𝑜𝑢𝑡 ,

(12)

(13)

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𝑄𝑝𝑢𝑚 = ∆𝑃 ∙ 𝑈

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where 𝑃𝑖𝑛 and 𝑃𝑜𝑢𝑡 are the pressure of the inlet and outlet, respectively. U is the volume flow rate.

index 𝜂 is defined as [36]:

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Finally, to analyze both the heat transfer and pressure drop characteristics, the thermal performance

𝑁𝑢/𝑁𝑢𝑠𝑚

𝜂 = (∆𝑃/∆𝑃

𝑠𝑚 )

1/3

,

(14)

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where 𝑁𝑢𝑠𝑚 and ∆𝑃𝑠𝑚 are the average Nusselt number and pressure drop of the TMHS-S, respectively.

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4. Results and Discussion

4.1 Comparative analysis of fluid flow characteristics

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Fluid flow characteristics have a great influence on heat transfer and pressure drop characteristics [37]. In this study, the fluid flow characteristics in the TMHS were investigated. Fig. 5 shows the velocity contours and streamlines on the middle planes of the channels with X in the range of 0.011 to 0.0142 m at the Reynolds number of 498. For the TMHS-S, it can be seen that the maximum velocity value occurs in the channel center, and the minimum velocity value occurs near the sidewall for the main channel. When 15

the fluid flows into the branching channel, the maximum velocity shifts inward and stagnant regions appears near the outer sidewall, which will leads to a higher heat transfer performance near the inner sidewall and a lower heat transfer performance near the outer sidewall (i.e. the branching effect). According to the analysis of the Introduction, it is obvious that the branching effect will enhance heat transfer performance. Therefore, the intensity of branching effect is an evaluation criterion of heat transfer performance. As for the TMHS-C, it can be seen that the introduction of cavities will cause the continuous

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destruction and regeneration of the fluid boundary layer, and the generation of secondary flow, which not only continuously bring the boundary heat into the main flow, but also changes the fluid-solid flow resistance to fluid-fluid flow resistance. Hence, the introduction of cavities can effectively enhance heat

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transfer and decrease pressure drop performance in the main channel. When the fluid flows into branching

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channels, it can be seen that the secondary flow in the inner side cavities has been intensified compared to that in the main channel. Additionally, it is note that the cavities in the outer side cause the expansion of the

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stagnant regions. That is to say, cavities will enhance the branching effects (i.e. further strengthen the heat

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transfer near the inner sidewall and weaken the heat transfer near the outer sidewall). Fig. 5c shows the velocity contours and streamlines of the TMHS-R. It can be seen that the maximum velocity value increases

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and appears just behind the ribs. This is because the introduction of ribs causes the constriction of the fluid channel, leading to the increase in flow velocity. In addition, it can be seen that the introduction of ribs

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causes the interrupt and regeneration of streamlines, and the generation of the recirculation before and behind the ribs. As for the coupling effects of ribs and branch, it can be seen that ribs not only increase the velocity value, but also change the flow direction, which might speed up the process of fluid flow to fully develop. It also can be seen that after introducing ribs, the inner boundary layer thickness increases, while the outer boundary layer thickness decreases. That is to say, the introduction of ribs can inhibits the inner 16

side heat transfer, while enhance the outer side heat transfer (i.e. the coupling effects of ribs and branch can

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na

lP

re

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suppress the branching effects).

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Fig. 5. Velocity contours and streamlines on the middle planes of the channels at Re = 498. (a) TMHS-S; (b) TMHS-C; (c) TMHS-R. 4.2 Comparative analysis of the heat transfer characteristics Fig. 6 shows the temperature distribution on the middle planes of the channels with X ranging from 0.011 to 0.0142 mm at the Reynolds number of 498. It can be seen for the main flow channel in the TMHS 17

that the temperature in the channel center is lower, while it is higher near the sidewalls. After flowing into the branching channels, the lower temperature value and higher temperature gradient approaches the inner sidewall due to the offset of the main flow. Instead, a smaller temperature gradient approaches the outer sidewall due to the increase of the stagnant region. After the introduction of the cavities to the channel sidewalls, the heat boundary layer redevelops near the cavities, and a higher temperature gradient is attained. In addition, the generation of a secondary flow could continuously bring the boundary heat into the main

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flow; thus, the outlet temperature of the main flow channel is slightly higher than that in the smooth TMHS. As for the branching channels, it can be seen that the inner side temperature gradient increases, while the outer side temperature gradient decreases, compared to the smooth one. This is because the introduction of

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cavities intensify the branching effects. Therefore, the coupling effects between cavities and branch can

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enhance heat transfer in the microchannel.

Fig. 6c shows the temperature distribution of the TMHS-R. It can be seen that the temperature

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difference between the channel center and sidewall decreases. This is because the construction and

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expansion effect leads to the increase of the velocity and the generation of recirculation, which then contribute to a thinner boundary layer and lower heat transfer resistance, and ultimately results in the

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increase of the heat transfer efficiency. Compared to the TMHS-S and TMHS-C, the maximum temperature value decreases significantly. In other words, the TMHS-R has the greatest effect on heat transfer. As for

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the branching channels, the temperature changes with the flow velocity, and the high-temperature colored tape near the outer sidewall decreases, while the temperature distribution after the introduction of ribs is much more uniform. These findings indicate that the introduction of ribs can inhibit the disadvantageous effects of branching and improve the total heat transfer by increasing the flow velocity and the uniformity of temperature distribution. 18

The maximum temperature value and temperature uniformity of the heat source are important indicators for evaluating the performance of the heat sink. Fig. 7 shows the temperature contours over the bottom surface of the heat source at Re = 498. It can be seen that after the introduction of the variable crosssection, the maximum temperature values decrease, and the temperature distribution uniformity increases. Combined with the Nu number distribution versus Re (Fig. 8), it is clear that the ranking of heat transfer performance is as follows: TMHS-R > TMHS-C > TMHS-S. In addition, as the Reynolds number increases,

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the difference becomes larger, which indicates that a larger Reynolds number is more favorable for

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ur

na

lP

re

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enhancing heat transfer at a variable cross-section.

19

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Fig. 6. Temperature distribution on the middle planes of the channels at Re = 498. (a) TMHS-S; (b) TMHS-

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C; (c) TMHS-R.

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Fig. 7. Temperature contours over the bottom surfaces of the heat sources at Re = 498. (a) TMHS-S; (b)

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TMHS-C; (c) TMHS-R.

21

18

TMHS-S TMHS-C TMHS-R

16

Nu

14 12

8 6 200

300

400

500

600

700

1000

re

Fig. 8. Nu versus Re. 4.3 Comparative analysis of the pressure drop

900

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Re

800

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10

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The pressure distribution on the middle planes of the channels at Re = 498 is displayed in Fig. 9. The

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pressure continuously decreases due to the frictional losses along the channels. After flowing into the branching channels, the pressure drop consists of not only the along flow resistance term, but also the terms

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resulting from the changing flow direction and expanding flow channel. It can be seen from Fig. 9a that there is an increase of pressure in the branching node in the inner sidewall, which was caused by the turning

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effects and the sudden decrease in velocity. After introducing cavities, there is a jet-shrinkage effect in the main flow channel, which will lead to an inverse pressure gradient and the generation of secondary flows in the cavities. Moreover, the generation of secondary flow will change the fluid-solid friction to fluid-fluid friction in the cavities, leading to the decrease of pressure drop, which can be seen from the shift forward of the color band. As for the branching channels, the main flow offsets to the inner sidewall and the cavities 22

intensify the branching effect, which leads to a much more uneven pressure distribution in the microchannel, as showed in Fig. 9b. In other words, a larger pressure appears near the inner sidewall, and a smaller pressure appears near the outer sidewall. The uneven pressure distribution will cause a little increase in the total pressure drop, while the cavities in the main flow channel will causes a little decrease in pressure drop, finally leading to small difference of pressure drop between TMHS-S and TMHS-C. As for the pressure distribution in the TMHS-R, it can be seen from Fig. 9c that the pressure rapidly decreases with the

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constriction of flow channel, which also indicates that ribs have a greater influence on pressure than do cavities. After flowing into the branching channels, the ribs can suppress the branching effects and obtain a much more uniform pressure distribution in the channel. Therefore, the coupling effects of branching and

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ribs can decrease the pressure drop, which can alleviate the pressure drop caused by ribs.

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In order to more intuitively understand the effects of variable cross-section on pressure drop characteristics, the total pressure drop distribution of different TMHS with different Reynolds numbers is

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shown in Fig. 10. There is almost no difference between the TMHS with cavities and the smooth TMHS,

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which is the same as our previous analysis. This is because the cavities in the branching channels will increase the branching effect and cause a much more uneven pressure distribution, though the cavities in

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the main flow channels will decrease the pressure drop. As for the TMHS-R, it can be seen that the pressure drop is larger than in the other heat sinks, and as the Reynolds number increases, the differences between

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the pressure drops also increase due to the increasing effect of ribs.

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(c) TMHS-R.

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Fig. 9. Pressure distribution on the middle planes of the channels at Re = 498. (a) TMHS-S; (b) TMHS-C;

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16000

TMHS-S TMHS-C TMHS-R

14000

10000 8000 6000 4000 2000 0 200

300

400

500

700

600

900

1000

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Re

800

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Pressure drop (Pa)

12000

re

Fig. 10. Pressure drop versus Re.

4.4 Comparative analysis of comprehensive performances

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From the above analysis, it is evident that although the heat transfer efficiency of the TMHS-R is the

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best, the pressure drop is also the largest, which may be confusing when attempting to determine the TMHS with the best comprehensive performance. To comprehensively analyze the heat transfer and pressure drop

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characteristics, the thermal performance index 𝜂 was examined, as showed in Fig. 11. All the values of 𝜂 increase with the increase of the Reynolds number, which indicates that a higher Reynolds number is

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beneficial for the improvement of the comprehensive performance of heat sinks with a variable crosssection. This is because the larger Reynolds number means a larger flow velocity value and stronger secondary flow intensity, thereby increasing the influence of heat transfer and decreasing the influence of the viscous force. It is also notable that when the Reynolds number is less than 750, the 𝜂 value of the TMHS-R is less than 1; this means that although it has a very high heat transfer efficiency, the 25

comprehensive performance of the TMHS-R is worse than the smooth TMHS. In the Reynolds number range of simulation calculation, the variable cross-section with cavities demonstrates a better comprehensive performance than the variable cross-section with ribs. Combining with Fig. 11b, it can be more intuitively understand the effects of cavities and ribs on the performance of TMHS. Fig.11b shows the relationship between pumping power and Nusselt number. It can be seen that as the pumping power increases, heat transfer performance increases and the enhancement of ribs and cavities on heat transfer

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increases. In particular, the Nusselt number in the TMHS-C is the largest, when pumping power is lower than 0.01 W, follow by that in the TMHS-R, and the TMHS-S is the least. However, when pumping power is larger than 0.01 W, the Nusselt number of TMHS-R is the largest, the TMHS-C is the second, followed

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by the TMHS-S. Therefore, the TMHS-C is a better choice for heat sink design at lower pumping power

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(Qpum < 0.01 W), when the TMHS-R is a better choice for heat sink design at higher pumping power (Qpum > 0.01 W).

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(a) 1.10

na

1.05

200

300

400

500

600

700

14

TMHS-S TMHS-C TMHS-R

10 8

ur

Jo

0.90

16

12

TMHS-C TMHS-R

0.95

(b)

Nu



1.00

18

6 4 800

900

0.00

1000

0.01

Re

0.02

0.03

0.04

0.05

0.06

Pumping power (W)

Fig. 11. Performance comparison. (a) 𝜂 versus Re; (b) Nu versus Pumping power.

5. Conclusions In this study, disk-shaped TMHSs with a variable cross-section were designed. The flow 26

characteristics, heat transfer characteristics, pressure drop characteristics, and comprehensive performance of TMHSs with cavities and ribs were analyzed numerically using the commercial software Fluent, and compared with the traditional smooth TMHS. The following conclusions were drawn: 1) Branching channels will cause a branching effect on the TMHS, i.e., the maximum velocity shifts inward and stagnant regions appears near the outer sidewall, leading to a higher heat transfer performance near the inner sidewall and a lower heat transfer performance near the outer sidewall;

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2) The variable cross-section with cavities promotes the generation of a secondary flow, while ribs promote a higher flow velocity and the generation of recirculation, leading to the continuous destruction and regeneration of the flow and heat boundary layer, which effectively improves the heat transfer

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performance;

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3) Cavities can enhance the branching effect, while ribs can suppress the disadvantageous influence of branching and obtain a much more uniform temperature and pressure distribution;

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4) The TMHS-R has the highest heat transfer efficiency and a more uniform temperature distribution,

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but also exhibits a higher pressure drop. The TMHS-C has a higher heat transfer efficiency and slightly lower pressure drop than the TMHS-S. Considering both the heat transfer and pressure drop characteristics,

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the TMHS-C exhibits a higher comprehensive performance. 5) A higher Reynolds number is beneficial for the improvement of the comprehensive performance

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of heat sinks with a variable cross-section, and as the Reynold number/pumping power increases, the enhancement of variable cross-section on heat transfer increases.

Declarations of interest: none Declaration of Competing Interest The authors declare no competing financial 27

Acknowledgment This research was supported by the Science and Technology Planning Project of Guangdong Province, China, Project No. 2016A050503019, and National Natural Science Foundation of Guangdong, China, No.1714050000297, and Science and Technology Program of Guangzhou, China, Project Nos. 201804010137 and 201807010074, and the Fundamental Research Funds for the Central Universities, Project No.2018ZD28.

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References

[1] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE Electron device letters, 2(5) (1981) 126-129.

[2] M.A. Arie, A.H. Shooshtari, S.V. Dessiatoun, E. Al-Hajri, M.M. Ohadi, Numerical modeling and

re

Journal of Heat and Mass Transfer, 81 (2015) 478-489.

-p

thermal optimization of a single-phase flow manifold-microchannel plate heat exchanger, International

[3] V. Leela Vinodhan, K.S. Rajan, Computational analysis of new microchannel heat sink configurations,

lP

Energy Conversion and Management, 86 (2014) 595-604.

[4] D. Mei, L. Liang, M. Qian, X. Lou, Modeling and analysis of flow distribution in an A-type microchannel reactor, International Journal of Hydrogen Energy, 38(35) (2013) 15488-15499.

na

[5] M. Pan, Y. Zhong, Y. Xu, Numerical investigation of fluid flow and heat transfer in a plate microchannel heat exchanger with isosceles trapezoid-shaped reentrant cavities in the sidewall, Chemical Engineering

ur

and Processing-Process Intensification, 131 (2018) 178-189. [6] M. Pan, P. Huang, K. Vafai, Investigation of the Blockage Conditions in a Laminated-Sheet

Jo

Microchannel Reactor, Chemical Engineering & Technology, 40(12) (2017) 2283-2294. [7] M.-O. Coppens, A nature-inspired approach to reactor and catalysis engineering, Current Opinion in Chemical Engineering, 1(3) (2012) 281-289. [8] A. Bejan, M.R. Errera, Deterministic tree networks for fluid flow: geometry for minimal flow resistance between a volume and one point, Fractals, 5(04) (1997) 685-695. [9] A. Bejan, From Heat Transfer Principles to Shape and Structure in Nature: Constructal Theory, Journal of Heat Transfer, 122(3) (2000) 430-449. 28

[10] A. Bejan, S. Lorente, Constructal theory of generation of configuration in nature and engineering, Journal of applied physics, 100(4) (2006) 5. [11] Y. Chen, P. Cheng, Heat transfer and pressure drop in fractal tree-like microchannel nets, International Journal of Heat & Mass Transfer, 45(13) (2002) 2643-2648. [12] Y. Chen, P. Cheng, An experimental investigation on the thermal efficiency of fractal tree-like microchannel nets ☆, International Communications in Heat & Mass Transfer, 32(7) (2005) 931-938. [13] SENN, M. S., POULIKAKOS, Laminar mixing, heat transfer and pressure drop in tree-like microchannel nets and their application for thermal management in polymer electrolyte fuel cells, Journal

ro of

of Power Sources, 130(1) (2004) 178-191.

[14] X.-Q. Wang, P. Xu, A.S. Mujumdar, C. Yap, Flow and thermal characteristics of offset branching network, International Journal of Thermal Sciences, 49(2) (2010) 272-280.

[15] S. Xu, J. Qin, Y. Yang, A numerica1 study of fluid flow and heat transfer in different microchannel

-p

heat sinks for electronic chip cooling, Journal of Mechanical Science & Technology, 26(4) (2012) 1257-

re

1263.

[16] C. Xia, J. Fu, J. Lai, X. Yao, Z. Chen, Conjugate heat transfer in fractal tree-like channels network heat

lP

sink for high-speed motorized spindle cooling, Applied Thermal Engineering, 90 (2015) 1032-1042. [17] Y. Peng, W. Liu, W. Chen, N. Wang, A conceptual structure for heat transfer imitating the transporting principle of plant leaf, International Journal of Heat and Mass Transfer, 71 (2014) 79-90.

na

[18] Y. Peng, W. Liu, N. Wang, Y. Tian, X. Chen, A novel wick structure of vapor chamber based on the fractal architecture of leaf vein, International Journal of Heat and Mass Transfer, 63 (2013) 120-133.

ur

[19] Y. Peng, X. Zhu, B. Cao, Y. Luo, W. Zhou, Heat transfer and permeability of the tree-like branching networks, International Journal of Heat and Mass Transfer, 129 (2019) 801-811.

Jo

[20] L. Chai, G. Xia, M. Zhou, J. Li, Numerical simulation of fluid flow and heat transfer in a microchannel heat sink with offset fan-shaped reentrant cavities in sidewall, International Communications in Heat and Mass Transfer, 38(5) (2011) 577-584. [21] G. Xia, L. Chai, M. Zhou, H. Wang, Effects of structural parameters on fluid flow and heat transfer in a microchannel with aligned fan-shaped reentrant cavities, International Journal of Thermal Sciences, 50(3) (2011) 411-419. [22] L. Chai, G. Xia, L. Wang, M. Zhou, Z. Cui, Heat transfer enhancement in microchannel heat sinks 29

with periodic expansion–constriction cross-sections, International Journal of Heat and Mass Transfer, 62 (2013) 741-751. [23] M. Xu, H. Lu, L. Gong, J.C. Chai, X. Duan, Parametric numerical study of the flow and heat transfer in microchannel with dimples, International Communications in Heat and Mass Transfer, 76 (2016) 348357. [24] L. Chai, G.D. Xia, H.S. Wang, Parametric study on thermal and hydraulic characteristics of laminar flow in microchannel heat sink with fan-shaped ribs on sidewalls – Part 2: Pressure drop, International Journal of Heat and Mass Transfer, 97 (2016) 1081-1090.

ro of

[25] L. Chai, G.D. Xia, H.S. Wang, Parametric study on thermal and hydraulic characteristics of laminar flow in microchannel heat sink with fan-shaped ribs on sidewalls – Part 3: Performance evaluation, International Journal of Heat and Mass Transfer, 97 (2016) 1091-1101.

[26] L. Chai, G.D. Xia, H.S. Wang, Parametric study on thermal and hydraulic characteristics of laminar

Journal of Heat and Mass Transfer, 97 (2016) 1069-1080.

-p

flow in microchannel heat sink with fan-shaped ribs on sidewalls – Part 1: Heat transfer, International

re

[27] Y.L. Zhai, G.D. Xia, X.F. Liu, Y.F. Li, Heat transfer in the microchannels with fan-shaped reentrant cavities and different ribs based on field synergy principle and entropy generation analysis, International

lP

Journal of Heat and Mass Transfer, 68 (2014) 224-233.

[28] M. Pan, H. Wang, Y. Zhong, M. Hu, X. Zhou, G. Dong, P. Huang, Experimental investigation of the

na

heat transfer performance of microchannel heat exchangers with fan-shaped cavities, International Journal of Heat and Mass Transfer, 134 (2019) 1199-1208. [29] D. Pence, Improved thermal efficiency and temperature uniformity using fractal-like branching Proc. International Conference on Heat Transfer and Transport Phenomena in

ur

channel networks, in:

Microscale, 2000, pp. 142-148.

Jo

[30] G.B. West, J.H. Brown, B.J. Enquist, A general model for the origin of allometric scaling laws in biology, Science, 276(5309) (1997) 122-126. [31] D. Pence, K. Enfield, Inherent benefits in microscale fractal-like devices for enhanced transport phenomena, WIT Transactions on Ecology and the Environment, 73 (2004). [32] Y. Peng, W. Liu, B. Liu, J. Liu, K. Huang, L. Wang, W. Chen, The performance of the novel vapor chamber based on the leaf vein system, International Journal of Heat and Mass Transfer, 86 (2015) 65630

666. [33] F.P. Incropera, Liquid cooling of electronic devices by single-phase convection, Wiley-Interscience, 1999. [34] J.M. Wu, J.Y. Zhao, K.J. Tseng, Parametric study on the performance of double-layered microchannels heat sink, Energy Conversion and Management, 80 (2014) 550-560. [35] D.D. Ma, G.D. Xia, J. Wang, Y.C. Yang, Y.T. Jia, L.X. Zong, An experimental study on hydrothermal performance of microchannel heat sinks with 4-ports and offset zigzag channels, Energy Conversion and Management, 152 (2017) 157-165.

ro of

[36] Y. Luo, W. Liu, L. Wang, W. Xie, Heat and mass transfer characteristics of leaf-vein-inspired microchannels with wall thickening patterns, International Journal of Heat and Mass Transfer, 101 (2016) 1273-1282.

[37] Y.-C. Tsai, F.-B. Liu, P.-T. Shen, Investigations of the pressure drop and flow distribution in a chevron-

Jo

ur

na

lP

re

-p

type plate heat exchanger, International communications in heat and mass transfer, 36(6) (2009) 574-578.

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