# Flow and Heat Transfer Performances of Liquid Metal Based Microchannel Heat Sinks under High Temperature Conditions

^{*}

## Abstract

**:**

## 1. Introduction

^{2}. Sohel et al. [29,30] showed analytically that 0.5–4.0 vol% CuO nanofluid flow in a copper microchannel heat sink having circular channels provides far better heat transfer performance and lower friction factor than Al

_{2}O

_{3}and TiO

_{2}nanofluids. Sivakumar et al. [31] also showed that CuO nanofluid provides better heat transfer coefficient enhancement than Al

_{2}O

_{3}nanofluid. Salman et al. [32,33] showed numerically that dispersing SiO

_{2}nanoparticles in ethylene glycol (base liquid) provides the highest Nu, followed by ZnO, CuO and Al

_{2}O

_{3}, and Nu increases with decreasing nanoparticles size. Kumar et al. [34] conducted thermofluidic analysis of Al

_{2}O

_{3}-water nanofluid cooled, branched wavy heat sink microchannels using a numerical method. The results showed that apart from disruption of the boundary layer and its reinitialization, vortices were formed near the secondary channel, which improved thermal performance. The heat transfer coefficient increased with increasing nanofluids concentrations for any investigated Reynolds number. Wang et al. [35] numerically investigated the forced convection in microchannel heat sinks using multi-wall carbon nanotube-Fe

_{3}O

_{4}hybrid nanofluid as coolant working fluid. According to the results, the heat sinks, which consist of metallic foam, have better cooling performance and are able to decrease the surface temperature. However, the performance improvements brought by nanofluids are still limited due to the base fluids used. Particularly, the mixed solution may easily be subject to additional troubles such as susceptibility to fouling, particle deposition or conglomeration, degeneration of solution quality and flow jamming over the channels [36,37]. Besides, the low evaporation point of these liquids may imply potential dangers in preventing the device from burning out, since the liquids may easily escape to the ambient air.

## 2. Modeling and Numerical Methods

#### 2.1. Physical Model

_{c}× W

_{c}× H

_{c}= 13 mm × 0.4 mm × 1 mm. Besides rectangular microchannel cross-section shape, three cross-section shapes (circle, trapezoid and parallelogram) are also studied in this paper. Different microchannel cross-section shapes are shown in Figure 2. The four different cross-section shapes have the same hydraulic diameter (D

_{h}= 0.57 mm).

^{3}, 800 J/(kg·K) and 80 W/(m·K) respectively.

^{3}, J/(kg·K), W/(m·K) and Pa·s.

#### 2.2. Mathematical Model

#### 2.2.1. Governing Equations and Boundary Conditions

- (1)
- Both the fluid flow and heat transfer are steady.
- (2)
- The fluid flow is incompressible and single phase.
- (3)
- There is no slip between fluid and wall.
- (4)
- Radiation heat transfer and viscous dissipation effect are neglected.

^{3}is density and η in Pa·s is viscosity.

_{f}in W/(m·K) is thermal conductivity and Cp in J/(kg·K) is specific heat capacity.

_{m}[49],

_{w}is the heat transfer area between the fluid and the walls; T

_{f}and T

_{w}are the average temperature of the fluid and the walls respectively; Q is the heat exchange capacity which could be calculated as,

_{b}are the heat flux and area of the heat sink bottom surface.

#### 2.2.2. Mesh Independence and Model Validation

_{H}is the hydraulic resistance, ΔP is the pressure drop, V

_{fr}is the volume flow rate.

_{max}is the maximum temperature of the whole heat sink, T

_{in}is the inlet temperature of the liquid metal, Q is the heat exchange capacity.

## 3. Results and Discussions

#### 3.1. Effects of Working Fluid

^{2}, inlet temperature 600 K and inlet velocity 3 m/s as typical condition, Figure 4 shows the temperature distributions of the heat sink with different working fluids (temperature contours have the same scale). It could be seen that taking Li as the working fluid has the best cooling effect while taking K as the working fluid has the worst. For the heat sinks with Na-K alloy as the working fluid, their cooling performances are intermediate between the Na-based and K-based heat sinks. Moreover, the bigger the weight fraction of Na, the better the cooling effect.

_{m}, is compared under the same pressure drop. Figure 9 shows the changing trend of h

_{m}with different pressure drop at rectangular microchannel cross-section, inlet temperature 600 K, heat flux 200 W/cm

^{2}. Obviously, since using liquid Li could get the biggest heat transfer coefficient, it is the optimum working fluid among all the investigated alkalis.

#### 3.2. Effects of Microchannel Cross-Section

^{2}, inlet temperature is 600 K and inlet velocity is 3 m/s. Obviously, the temperature distribution is hardly influenced by the change of microchannel cross-section shape.

_{m}= Nu·k/D

_{h}, the variation of the mean heat transfer coefficient is accordance with Nusselt number. So, as shown in Figure 11a, using a circular cross-section obtains the highest heat transfer coefficient. Compared to the other three cross-section shapes, using circular cross-section could increase the mean heat transfer coefficient by about 14,000 W/(m

^{2}·K), which indicates the heat sink with circle microchannel cross-section has the best flow performance.

_{0}and f

_{0}are the benchmark Nusselt number and mean flow resistance coefficient. In this paper, Nu and f of the heat sink with rectangular microchannel cross-section under heat flux 100 W/cm

^{2}are appointed as Nu

_{0}and f

_{0}.

#### 3.3. Effects of Inlet Velocity

^{2}, inlet temperature 600 K and Li as the working fluid. It could be seen that the inlet velocity has a significant effect on the temperature distribution. With the increase of inlet velocity, the temperature of the heat sink becomes obviously lower. Under the inlet velocity of 1 m/s, the high temperature region covers most of the heat sink, indicating the heat sink’s poor cooling effect. When the inlet velocity increases to 3 and 5 m/s, the high temperature region becomes much smaller. When the inlet velocity reaches 7 and 9 m/s, the entire heat sink is basically at a relatively low temperature level.

^{2}) (according to Equation (26)).

## 4. Conclusions

- (1)
- Among all the seven investigated alkalis, lithium is the best option for working fluid because the lithium-based microchannel heat sink has the best cooling ability and the lowest pressure drop.
- (2)
- For the four considered microchannel cross-section types (rectangle, circle, trapezoid and parallelogram), utilizing a circular microchannel cross-section obtains a higher mean heat transfer coefficient, while using a parallelogram obtains the lowest pressure drop. Considering flow and heat transfer performances comprehensively, the circle is the optimum choice for microchannel cross-section shape because using a circular microchannel has the highest PEC value.
- (3)
- Inlet velocity has a significant influence on the heat sink’s flow and heat transfer performances. When the inlet velocity rises from 1 m/s to 9 m/s, the heat transfer coefficient enhances 74.35% at most, while the pressure drop increases up to 65 times. In order to obtain a favorable overall performance, the inlet velocity should be selected carefully.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | heat transfer area (m^{2}) |

at | mole fraction |

Cp | specific heat capacity (J/(kg·K)) |

D_{h} | hydraulic diameter (mm) |

f | flow resistance coefficient |

H | height (mm) |

h_{m} | mean heat transfer coefficient (W/(m^{2}·K)) |

k | thermal conductivity (W/(m·K)) |

L | length (mm) |

Nu | Nusselt number |

PEC | performance evaluation criteria |

Pr | Prandtl number |

p | pressure (Pa) |

Q | heat exchange capacity (W) |

q | heat flux (W/m^{2}) |

Re | Reynold number |

R_{H} | hydraulic resistance (Pa·s/m^{3}) |

R_{Thm} | thermal resistance (K/W) |

r | radius of inlet and outlet passages (mm) |

T | Kelvin temperature (K) |

t_{f} | Fahrenheit temperature (°F) |

u | velocity in x direction (m/s) |

V_{fr} | volume flow rate (m^{3}/s) |

v | velocity in y direction (m/s) |

W | width (mm) |

w | velocity in z direction (m/s) |

wt | weight fraction |

x | Cartesian coordinate (m) |

y | Cartesian coordinate (m) |

z | Cartesian coordinate (m) |

Greek symbols | |

ΔP | pressure drop (Pa) |

η | dynamic viscosity (Pa·s) |

λ_{f} | thermal conductivity (W/(m·K)) |

ρ | density (kg/m^{3}) |

Subscripts | |

b | heat sink bottom |

c | microchannel |

f | fluid |

K | potassium |

m | manifold |

Na | sodium |

w | wall |

## References

- Gangtao, L.; Issam, M. Review of single-phase and two-phase nanofluid heat transfer in macro-channels and micro-channels. Int. J. Heat Mass Transf.
**2019**, 136, 324–354. [Google Scholar] - Tuckerman, D.B.; Pease, R. High-performance heat sinking for VLSI. IEEE Electr. Device Lett.
**1981**, 1, 126–129. [Google Scholar] [CrossRef] - Wu, P.Y.; Little, W.A. Measurement of friction factor for the flow of gases in very fine channels used for micro miniature Joule-Thomson refrigerators. Cryogrnics
**1983**, 23, 273–277. [Google Scholar] - Xu, B.; Ooi, K.T.; Wong, N.T.; Choi, W.K. Experimental investigation of flow friction for liquid flow in micro channels. Int. Comm. Mass Transf.
**2000**, 27, 1165–1176. [Google Scholar] [CrossRef] - Judy, J.; Maynes, D.; Webb, B.W. Characterization of frictional pressure drop for liquid flows through micro channels. Int. J. Heat Mass Transf.
**2002**, 45, 3477–3489. [Google Scholar] [CrossRef] - Qu, W.; Mudawar, I. Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink. Int. J. Heat Mass Transf.
**2002**, 45, 2549–2565. [Google Scholar] [CrossRef] - Lee, P.S.; Garimella, S.V.; Liu, D. Investigation of heat transfer in rectangular micro channels. Int. J. Heat Mass Transf.
**2005**, 48, 1688–1704. [Google Scholar] [CrossRef][Green Version] - Liu, D.; Garimella, S.V. Investigation of liquid flow in micro-channels. J. Thermophys. Heat Transf.
**2004**, 18, 65–72. [Google Scholar] [CrossRef][Green Version] - Kandlikar, S.G.; Garimella, S.; Liu, D. Heat Transfer and Fluid Flow in Mini Channels and Micro Channels, 2nd ed.; Elsevier Science: Kidlington, UK, 2013. [Google Scholar]
- TAdams, M.; Abdel-Khalik, S.I.; Jeter, S.M.; Qureshi, Z.H. An experimental investigation of single-phase forced convection in micro channels. Int. J. Heat Mass Transf.
**1998**, 41, 851–857. [Google Scholar] [CrossRef] - Adams, T.M.; Dowling, M.F.; Abdel-Khalik, S.I.; Jeter, S.M. Applicability of traditional turbulent single-phase forced convection correlations to non-circular micro-channels. Int. J. Heat Mass Transf.
**1999**, 42, 4411–4415. [Google Scholar] [CrossRef] - Yousef, A.; Mohammad, Z.T.; Mohammad, M.H.; Nima, G. Effect of a micro heat sink geometric design on thermo-hydraulic performance: A review. Appl. Therm. Eng.
**2020**, 170, 114974. [Google Scholar] - Dalei, J.; Lei, H. Numerical studies on the hydraulic and thermal performances of microchannels with different cross-sectional shapes. Int. J. Heat Mass Transf.
**2019**, 143, 118604. [Google Scholar] - Gunnasegaran, P.; Mohammed, H.A.; Shuaib, N.H.; Saidur, R. The effect of geometrical parameters on heat transfer characteristics of microchannels heat sink with different shapes. Int. Commun. Heat Mass Transf.
**2010**, 37, 1078–1086. [Google Scholar] [CrossRef] - Xia, G.D.; Jiang, J.; Wang, J.; Zhai, Y.L.; Ma, D.D. Effects of different geometric structures on fluid flow and heat transfer performance in microchannel heat sinks. Int. J. Heat Mass Transf.
**2015**, 80, 439–447. [Google Scholar] [CrossRef] - Kumar, R.; Singh, G.; Mikielewicz, D. A new approach for the mitigating of flow maldistribution in parallel microchannel heat sink. J. Heat Transf.
**2018**, 140, 072401. [Google Scholar] [CrossRef] - Kumar, R.; Singh, G.; Mikielewicz, D. Numerical study on mitigation of flow maldistribution on parallel microchannel heat sink: Channels variable width versus variable height approach. J. Electron. Packag.
**2019**, 141, 021009. [Google Scholar] [CrossRef] - Ahmed, H.E.; Ahmed, M.I. Optimum thermal design of triangular, trapezoidal and rectangular grooved microchannel heat sinks. Int. Commun. Heat Mass Transf.
**2015**, 66, 47–57. [Google Scholar] [CrossRef] - Zhu, J.-F.; Li, X.-Y.; Wang, S.-L.; Yang, Y.-R.; Wang, X.-D. Performance comparison of wavy microchannel heat sinks with wavy bottom rib and side rib designs. Int. J. Therm. Sci.
**2019**, 146, 106068. [Google Scholar] [CrossRef] - Wang, S.-L.; Chen, L.-Y.; Zhang, B.-X.; Yang, Y.-R.; Wang, X.-D. A new design of double-layered microchannel heat sinks with wavy microchannels and porous-ribs. J. Therm. Anal. Calorim.
**2020**, 141, 547–558. [Google Scholar] [CrossRef] - Ermagan, H.; Rafee, R. Numerical investigation into the thermo-fluid performance of wavy microchannels with superhydrophobic walls. Int. J. Therm. Sci.
**2018**, 132, 578–588. [Google Scholar] [CrossRef] - Gong, L.; Li, Y.; Bai, Z.; Xu, M. Thermal performance of micro-channel heat sink with metallic porous/solid compound fin design. Appl. Therm. Eng.
**2018**, 137, 288–295. [Google Scholar] [CrossRef] - Ghahremannezhad, A.; Xu, H.; Nazari, M.A.; Ahmadi, M.H.; Vafai, K. Effect of porous substrates on thermohydraulic performance enhancement of double layer microchannel heat sinks. Int. J. Heat Mass Transf.
**2019**, 131, 52–63. [Google Scholar] [CrossRef] - Li, X.Y.; Wang, S.L.; Wang, X.D.; Wang, T.-H. Selected porous-ribs design for performance improvement in double-layered microchannel heat sinks. Int. J. Therm. Sci.
**2019**, 137, 616–626. [Google Scholar] [CrossRef] - Hussien, A.A.; Abdullah, M.Z.; Mohda, A.N. Single-phased heat transfer enhancement in micro/minichannels using nanofluids: Theory and applications. Appl. Energy
**2012**, 89, 150–155. [Google Scholar] - Chen, Z.; Qian, P.; Huang, Z.; Luo, C.; Liu, M. Study on flow and heat transfer of liquid metal in a new top-slotted microchannel heat sink. IOP Conf. Ser. Earth Environ. Sci.
**2021**, 624, 012054. [Google Scholar] [CrossRef] - Jang, S.P.; Choi, S.U.S. Cooling performance of a microchannel heat sink with nanofluids. Appl. Therm. Eng.
**2006**, 26, 2457–2463. [Google Scholar] [CrossRef] - Farsad, E.; Abbasi, S.P.; Zabihi, M.S.; Sabbaghzadeh, J. Numerical simulation of heat transfer in a micro channel heat sink using nanofluids. Heat Mass Transf.
**2011**, 47, 479–490. [Google Scholar] [CrossRef] - Sohel, M.R.; Saidur, R.; Sabri, M.F.M.; Kamalisarvestani, M.; Elias, M.M.; Ijam, A. Investigating the heat transfer performance and thermophysical properties of nanofluids in a circular micro-channel. Int. Commun. Heat Mass Transf.
**2013**, 42, 75–81. [Google Scholar] [CrossRef] - Sohel, M.R.; Khaleduzzaman, S.S.; Saidur, R.; Hepbasli, A.; Sabri, M.F.M.; Mahhubul, I.M. An experimental investigation of heat transfer enhancement of a minichannel heat sink using Al
_{2}O_{3}-H_{2}O nanofluid. Int. J. Heat Mass Transf.**2014**, 74, 164–172. [Google Scholar] [CrossRef] - Sivakumar, A.; Alagumurthi, N.; Senthilvelan, T. Experimental investigation of forced convective heat transfer performance in nanofluids of Al
_{2}O_{3}/water and CuO/water in a serpentine shaped micro channel heat sink. Heat Mass Transf.**2016**, 52, 1265–1274. [Google Scholar] [CrossRef] - Salman, B.H.; Mohammed, H.A.; Kherbeet, A.S. Heat transfer enhancement of nanofluids flow in microtube with constant heat flux. Int. Commun. Heat Mass Transf.
**2012**, 39, 1195–1204. [Google Scholar] [CrossRef] - Salman, B.H.; Mohammed, H.A.; Munisamy, K.M.; Kherbeet, A.S. Three-dimensional numerical investigation of nanofluids flow in microtube with different values of heat flux. Heat Transf. Asian Res.
**2015**, 44, 599–619. [Google Scholar] [CrossRef] - Kumar, R.; Tiwary, B.; Singh, P.K. Thermofluidic analysis of Al
_{2}O_{3}-water nanofluid cooled branched wavy heat sink. Appl. Therm. Eng.**2022**, 201, 117787. [Google Scholar] [CrossRef] - Wang, J.; Xu, Y.-P.; Qahiti, R.; Jafaryar, M.; Alazwari, M.A.; Abu-Hamdeh, N.H.; Issakhov, A.; Selim, M.M. Simulation of hybrid nanofluid flow within a microchannel heat sink considering porous media analyzing CPU stability. J. Petro. Sci. Eng.
**2022**, 208, 109734. [Google Scholar] [CrossRef] - Miner, A.; Ghoshal, U. Cooling of high-power-density microdevices using liquid metal coolants. Appl. Phys. Lett.
**2004**, 85, 506–508. [Google Scholar] [CrossRef] - Buongiorno, J. Convective transport in nanofluids. ASME J. Heat Transf.
**2006**, 128, 240–250. [Google Scholar] [CrossRef] - Deng, Y.; Jiang, Y.; Liu, J. Liquid metal technology in solar power generation—Basics and applications. Sol. Energy Mater. Sol. Cells
**2021**, 222, 110925. [Google Scholar] [CrossRef] - Deng, Y.; Liu, J. A liquid metal cooling system for the thermal management of high power LEDs. Int. Commun. Heat Mass Transf.
**2010**, 37, 788–791. [Google Scholar] [CrossRef] - Ma, K.; Liu, J. Liquid metal cooling in thermal management of computer chips. Front. Energy Power Eng. China
**2007**, 1, 384–402. [Google Scholar] [CrossRef] - Liu, J.; Zhou, Y.X. A Computer Chip Cooling Method Which Uses Low Melting Point Metal and Its Alloys as the Cooling Fluid. China Patent 02131419.5, 10 October 2002. [Google Scholar]
- Ghoshal, U.; Grimm, D.; Ibani, S.; Johnston, C.; Miner, A. High performance liquid metal cooling loops. In Proceedings of the IEEE 21st Annual Semiconductor Thermal Measurement and Management Symposium, San Jose, CA, USA, 15–17 March 2005; pp. 16–19. [Google Scholar]
- Hodes, M.; Zhang, R.; Lam, L.S.; Wilcoxon, R.; Lower, N. On the potential of Galinstan-based minichannel and minigap cooling. IEEE Trans. Compon. Packag. Technol.
**2014**, 4, 46–56. [Google Scholar] [CrossRef] - Zhang, R.; Hodes, M.; Lower, N.; Wilcoxon, R. Water-based microchannel and Galinstan-based minichannel cooling beyond 1 kW/cm
^{2}heat flux. IEEE Trans. Compon. Packag. Technol.**2015**, 5, 762–770. [Google Scholar] - Jarger, W. Heat transfer to liquid metals with empirical models for turbulent forced convection in various geometries. Nucl. Eng. Des.
**2017**, 319, 12–27. [Google Scholar] - IAEA. Thermophysical Properties of Materials for Nuclear Engineering: A Tutorial and Collection of Data; International Atomic Energy Agency: Vienna, Austria, 2002. [Google Scholar]
- He, Z.; Yan, Y.; Zhang, Z. Thermal management and temperature uniformity enhancement of electronic devices by micro heat sinks: A review. Energy
**2021**, 216, 119223. [Google Scholar] [CrossRef] - Wu, T. Numerical Heat Transfer; Xi’an Jiaotong University Press: Xi’an, China, 2001. [Google Scholar]
- Holman, J.P. Heat Transfer, 9th ed.; McGraw-Hill: New York, NY, USA, 2002. [Google Scholar]
- Kandikar, S.G.; Garimella, S.; Li, D.Q.; Colin, S.; King, M.R. Heat Transfer and Fluid Flow in Minichannels and Microchannels; Elsevier: Oxford, UK, 2006. [Google Scholar]
- Mortensen, N.A.; Okkels, F.; Bruus, H. Reexamination of Hagen-Poiseuille flow: Shape dependence of the hydraulic resistance in microchannels. Phys. Rev.
**2005**, 71, 057301. [Google Scholar] [CrossRef][Green Version] - Muhammad, A.; Selvakumar, D.; Wu, J. Numerical investigation of laminar flow and heat transfer in a liquid metal cooled mini-channel heat sink. Int. J. Heat Mass
**2019**, 150, 119265. [Google Scholar] [CrossRef] - Ambreen, T.; Saleem, A.; Park, C.W. Numerical analysis of the heat transfer and fluid flow characteristics of a nanofluid-cooled micropin-fin heat sink using the Eulerian-Lagrangian approach. Powder Technol.
**2019**, 345, 509–520. [Google Scholar] [CrossRef] - Ambreen, T.; Saleem, A.; Park, C.W. Pin-fin shape-dependent heat transfer and fluid flow characteristics of water- and nanofluid-cooled micropin-fin heat sinks: Square, circular and triangular fin cross-sections. Appl. Therm. Eng.
**2019**, 158, 113781. [Google Scholar] [CrossRef] - Ambreen, T.; Saleem, A.; Ali, H.M.; Shehzad, S.A.; Park, C.W. Performance analysis of hybrid nanofluid in a heat sink equipped with sharp and streamlined micro pin-fins. Powder Technol.
**2019**, 355, 552–563. [Google Scholar] [CrossRef] - Ambreen, T.; Saleem, A.; Park, C.W. Analysis of hydro-thermal and entropy generation characteristics of nanofluid in an aluminium foam heat sink by employing Darcy-Forchheimer-Brinkman model coupled with multiphases Eulerian model. Appl. Therm. Eng.
**2020**, 173, 115231. [Google Scholar] [CrossRef] - Chai, L.; Xia, G.D.; Wang, H.S. Numerical study of laminar flow and heat transfer in microchannel heat sink with offset ribs on sidewalls. Appl. Therm. Eng.
**2016**, 92, 32–41. [Google Scholar] [CrossRef]

**Figure 2.**Diagram of different microchannel cross-section types. (

**a**) Rectangle; (

**b**) circle; (

**c**) parallelogram; (

**d**) trapezoid.

**Figure 4.**Comparison between theoretical and numerical results of dimensionless hydraulic resistance.

**Figure 5.**Temperature distributions of the heat sink with different working fluids. (

**a**) Na; (

**b**) K; (

**c**) Li; (

**d**) Na22-K78; (

**e**) Na44-K56; (

**f**) Na52-K48; (

**g**) Na56-K44.

**Figure 6.**Temperature and velocity distributions at the center section of microchannels. (

**a**) Temperature distribution; (

**b**) velocity distribution.

**Figure 7.**Mean heat transfer coefficients and Nusselt numbers with different working fluids. (

**a**) Mean heat transfer coefficient; (

**b**) Nusselt number; (

**c**) Mean heat transfer coefficient and Nusselt number under heat flux 200 W/cm

^{2}.

**Figure 10.**Temperature distributions of the heat sink with different microchannel cross-section types. (

**a**) Rectangle; (

**b**) circle; (

**c**) trapezoid; (

**d**) parallelogram.

**Figure 11.**Mean heat transfer coefficients and Nusselt numbers with different micro channel cross-section types. (

**a**) Mean heat transfer coefficient; (

**b**) Nusselt number.

**Figure 12.**Pressure drops and flow resistance coefficients with different microchannel cross-section types. (

**a**) Pressure drop; (

**b**) flow resistance coefficient.

**Figure 13.**Velocity vectors at the center section of the heat sink with different microchannel cross-section types. (

**a**) Rectangle; (

**b**) circle; (

**c**) trapezoid; (

**d**) parallelogram.

**Figure 15.**Temperature distributions of the heat sink with different inlet velocities. (

**a**) 1 m/s; (

**b**) 3 m/s; (

**c**) 5 m/s; (

**d**) 7 m/s; (

**e**) 9 m/s.

**Figure 16.**Mean heat transfer coefficients and Nusselt numbers with different inlet velocities. (

**a**) Mean heat transfer coefficient; (

**b**) Nusselt number.

**Figure 17.**Pressure drops and flow resistance coefficients with different inlet velocities. (

**a**) Pressure drop; (

**b**) flow resistance coefficient.

Cell Number | Calculated Pressure Drop | Relative Difference |
---|---|---|

1,627,410 | 9709.92 Pa | / |

3,480,055 | 10,320.50 Pa | 5.9% |

5,741,015 | 10,414.30 Pa | 0.9% |

Microchannel Height | R_{Thm} (K/W) Present | R_{Thm} (K/W) Ref. [52] | Relative Difference |
---|---|---|---|

3 | 0.1174 | 0.1184 | 0.84% |

4 | 0.09944 | 0.1002 | 0.76% |

5 | 0.09076 | 0.09156 | 0.87% |

6 | 0.08648 | 0.08718 | 0.80% |

7 | 0.08429 | 0.08478 | 0.58% |

9 | 0.08282 | 0.08346 | 0.77% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, T.; Wang, L.; Tang, Y.; Yin, C.; Li, X.
Flow and Heat Transfer Performances of Liquid Metal Based Microchannel Heat Sinks under High Temperature Conditions. *Micromachines* **2022**, *13*, 95.
https://doi.org/10.3390/mi13010095

**AMA Style**

Wu T, Wang L, Tang Y, Yin C, Li X.
Flow and Heat Transfer Performances of Liquid Metal Based Microchannel Heat Sinks under High Temperature Conditions. *Micromachines*. 2022; 13(1):95.
https://doi.org/10.3390/mi13010095

**Chicago/Turabian Style**

Wu, Tao, Lizhi Wang, Yicun Tang, Chao Yin, and Xiankai Li.
2022. "Flow and Heat Transfer Performances of Liquid Metal Based Microchannel Heat Sinks under High Temperature Conditions" *Micromachines* 13, no. 1: 95.
https://doi.org/10.3390/mi13010095