## Abstract

Liquid metal owns the highest thermal conductivity among all the currently available fluid materials. This property enables it to be a powerful coolant for the thermal management of large power device or high flux chip. In this paper, a high-efficiency heat dissipation system based on the electromagnetic-driven rotational flow of liquid metal was demonstrated. The velocity distribution of the liquid metal was theoretically analyzed and numerically simulated. The results showed that the velocity was distributed unevenly along longitudinal section and the maximum velocity appears near the anode. On the temperature distribution profile of the heat dissipation system, the temperature on the electric heater side was much higher than the other regions and the role of the rotated liquid metal was to homogenize the temperature of the system. To analyze the heat dissipation of the system performance, a second-order R-C network thermal resistance model of the experimental device was established with the parameters determined. The total thermal resistance of the dissipation system presented an increasing tendency with the increase of the heating power and gradually stabilized to about 4.42 °C/W. Besides, the relationship between the temperature of the electric heater and the heating power was experimentally determined. And it exhibits linear characteristic with the slope value of about 1.033 ^{o}C/W. With such corresponding relations, the heating power could be conveniently determined once the maximum control temperature was given. The heat dissipation method introduced in the paper provides a novel way for fabricating compact chip cooling system.

## 1 Introduction

With the rapid advancement of electronic information technology, the integration degree of the semiconductor integrated circuit is growing up with its function strengthening. And the ever-increasing power density of the integrated circuit device will definitely lead to large temperature rise which reduces the stability and the lifetime of the device [1,2]. Empirical data show that the failure rate of microelectronic system doubles with every 10 °C increase in temperature [3]. Therefore, efficient heat dissipation is receiving more and more attention. Up to now, there are several main heat dissipation methods including air cooling [4], liquid cooling [5], heat pipe cooling [6], and semiconductor cooling [7]. Air cooling is the simplest heat dissipation method and is not suitable for high heat flow density chip cooling condition due to the large-size fins and the high thermal resistance [8,9]. Heat pipe cooling is a passive heat transfer method and contains liquid evaporation and gas condensation processes. There are various factors influencing the heat pipe performance including capillary limit, boiling limit, viscous limit, entrainment limit, vapor limit, and so on [10,11]. Semiconductor cooling method takes advantage of the Seeback, Peltier, and Thomson effects to implement active cooling. Water vapor condensation is easy to occur on the cold side of the semiconductor radiator and could lead to short-circuit fault. And the size of the semiconductor radiator becomes larger with the increase of the radiating power [12]. Liquid cooling is one of the most widely used heat dissipation strategies [13–15]. And the working medium can be various liquid including water [16], water-based nanofluids [17], liquid metal [18,19], and so on. Water is the most commonly used liquid cooling medium. Although it has a high heat capacity of 4.2 J/(g · °C), the low boiling point (100 °C at 1 atm) hinders its application in high temperature cooling environment. Compared with water or other liquid-phase working medium, liquid metal has the highest thermal conductivity in all the fluid materials. For example, Ga is a kind of room temperature liquid metal with melting point of 29.8 °C. And the thermal conductivity of Ga is about 50 times larger than that of water. In addition, the boiling point of Ga is 2403 °C and the liquid-phase range is 2373.2 °C [20]. The high thermal conductivity and the high boiling point of liquid metal enable it to be an ideal liquid cooling medium in various heat dissipation working conditions including chip cooling [21,22], laser diode array cooling [23], nuclear cooling [24,25], concentrated solar cooling [26,27], and so on. The application of liquid metal in chip cooling research has received more and more attention in recent years and many related heat dissipation coolants and techniques have been developed. Ma and Liu [28] proposed a concept of nano liquid metal coolant fabricated by addition of high thermally conductive nano particles into liquid metal. Deng and Liu [29] demonstrated a hybrid liquid metal–water cooling system combining the merits of both liquid metal and water. Yang and Liu [30] fabricated an electrically driven chip cooling device by using liquid metal and aqueous solution as the coolants. Yang et al. [31] demonstrated the flow and thermal model of liquid metal-based mini-channel heat sink. However, existing heat dissipation structures with hybrid coolants combining liquid metal and other fluids are relatively complex and are not conductive to effective integration. This paper is dedicated to introduce a new type of compact liquid metal cooling system with the rotational flow of liquid metal activated by electromagnetic field.

## 2 Methodology

### 2.1 Materials.

Ga_{68.5}In_{21.5}Sn_{10} (melting point ∼11 °C, nontoxic) is selected as the liquid metal coolant and its weight is 17.9 g. The preparation process of Ga_{68.5}In_{21.5}Sn_{10} is as follows: Gallium, indium and tin with purity above 99.99% purchased from Zhuzhou Keneng New Material Co., Ltd. were weighed according to the mass ratio of Ga: In: Sn = 68.5: 21.5: 10. And then they were put in a beaker. The beaker was placed under vacuum conditions and in 90 °C constant temperature water bath heated for about 2 h, followed by stirring the metals continuously and cooling to room temperature after uniform mixing.

A piece of annulus-shaped rubidium-iron-boron magnet is adopted to produce a uniform magnetic field. The size of the magnet is 40 mm inner diameter and 50 mm outer diameter and 5 mm length. The magnetic induction strength of the magnet is measured by a digital TESLA meter (YJT-100, Xi'an Qingcheng Electromechanical Equipment Co., Ltd.), and the measurement result is presented in Fig. 1(a). It can be seen that there is a uniform distribution of magnetic induction strength in the hollow part of the magnetic ring. And this guarantees a uniform Ampere force distribution subjected by liquid metal.

A micro high temperature ceramic electric heater is used for the heat source which is presented as the inset of Fig. 1(b). The size of the electric heater is 10 mm × 10 mm × 1 mm. Figure 1(b) shows its temperature–time, heating power–time, and heating voltage–time relationships. The temperature of the electric heater was measured by a T-type thermocouple attached to its surface with the temperature detection error within 0.1 °C. The measurement was performed two times under room temperature of 22.1 °C. And the standard deviation of the two measurements is 10.02 °C with the data acquisition points of 187. It can be seen from Fig. 1(b) that there is a better consistency between the two measurements as the time is less than about 21 s and the temperature is less than about 131 °C. As the temperature continues to rise, the measurement error intermittent increases while the two measurement results turn out to be quite consistent at the points indicated by the red arrows. Such law of change may be due to the heat transfer characteristics of the electric heater itself. In addition, it can also be seen that the heating power and the temperature gradually tend to be stabilized at constant applied voltage, and the time constant of the heating power is approximately the same as the stable time of the temperature. Generally speaking, there is a corresponding relationship between heating power and temperature of the electric heater, and the heating power of 4.2 W corresponds to the temperature of about 230 °C.

Graphite is selected as the electrode material. It is a kind of nonmetal conductor with electrical conductivity of 3 × 10^{3} S/m. The anode is a cylinder with diameter of 5 mm and length of 8 mm, while the cathode is a ring-shaped object with 30 mm internal diameter, 40 mm external diameter, and 10 mm height, respectively. As liquid metal has a poor wettability on the surface of graphite electrode, the low motion resistance enables its smoothly moving under electromagnetic actuation force [32].

### 2.2 Experimental Set Up.

The schematic diagram and the external appearance of the experimental device are presented in Figs. 2(a) and 2(b), respectively. The electric heater is attached to the bottom surface of a piece of square shaped AlN ceramic plate by using thermal silicone grease. And on the upper surface of the ceramic plate, there is a pair of dual ring-shaped graphite electrodes. Liquid metal is located between the cylindrical-shaped anode and the annulus-shaped cathode. Magnetic field is provided by an annulus-shaped permanent magnet located on the top of the cathode, and its direction is perpendicular to the top surface of liquid metal. Liquid metal can be driven by Ampere force to rotate around its central axis when the electrodes are electrified. In order to conduct the heat to air environment timely, a piece of copper fin is attached to the ceramic plate. To detect real-time temperature of the heat dissipation system, seven temperature monitoring points have been set up on the experimental device which are indicated as numbers 1–7 presented in Fig. 1. The points 1–5 represent different positions on the surface of liquid metal. And the points 6 and 7 represent the temperature of the electric heater and the radiator, respectively. In order to fix the junction of the thermocouple firmly at point 6, a piece of copper plate with a slot was set between the electric heater and the ceramic plate, and the junction was put in the slot. Then the surface temperature of the electric heater will thus be measured. Seven T-type thermocouples were placed in the graphite electrodes or the liquid metal to perform the temperature monitoring in real time, and the temperature measurement error is within ±0.1 °C. These thermocouples were fabricated with copper and constantane wires which were purchased from Shanghai Sanguang Electric Alloy Co., Ltd. In addition, an Agilent 34970A is used to monitor and collect the real-time information of temperature and voltage at a sample rate of 2 samples/s.

### 2.3 Numerical Analysis.

**is velocity vector. Assume the electric field is constant during the system power-up, the electromagnetic equations in the International System of Units can be expressed as**

*v***H**is electric field intensity vector,

*c*is speed of light,

*σ*is electric conductivity,

*μ*is magnetic permeability, and its value is about 1 as to Ga

_{68.5}In

_{21.5}Sn

_{10}liquid metal.

*is the electrical current density vector of liquid metal,*

**J****is the electrical filed intensity vector which is in the same direction as that of**

*E***, and**

*J***is the magnetic induction intensity vector of the magnetic field in liquid metal and is provided by the magnet.**

*B**ρ*is the density,

*ν*is the kinetic viscosity, and

*P*is the pressure. The last term of the Eq. (5) is due to the action of electromagnetic field on liquid metal.

*ɛ*is internal energy per unit mass of liquid metal,

**is energy flow density and can be expressed as**

*q**w*is the enthalpy per unit mass of liquid metal,

*σ′*is the viscous stress tensor, and

*k*is the thermal conductivity of liquid metal. The second and the third terms on the right-hand side of Eq. (7) are the Poynting vector which represents the flow of electromagnetic power density, while the fourth and the fifth terms are the heat flow densities related to viscosity and heat conductivity, respectively.

*T*is the temperature,

*t*is the time,

*k*′ is the thermal conductivity of AlN ceramic plate,

*ρ*is the density, and

*c*is the heat capacity at constant pressure.

_{P}*η*is dynamic viscosity,

*i*,

*j*= 1, 2, 3. The left and right sides of Eq. (9) are the formation heat and dissipation heat of liquid metal per unit volume and time, respectively. And the three terms on the right of the equal sign are the dissipation energies due to the heat conduction, viscous dissipation, and Joule heating of liquid metal in the electromagnetic field per unit volume, respectively. As liquid metal is still and electrodes are not electrified, the viscous dissipation and Joule heating terms can be neglected and Eq. (9) can be simplified to Eq. (8).

*q*is the heat flux, and

*h*is the local convective heat transfer coefficient which can be expressed as [35]

*k*is the thermal conductivity,

*L*is the size of the fin,

*U*is the wind velocity,

*ν*is the kinetic viscosity, and Pr is the Prandtl number of the air.

The initial and boundary conditions of the numerical simulation are shown in Table 1.

Initial condition | Temperature of the system | 23.3 °C (296.45 K) |
---|---|---|

Velocity of the liquid metal | 0 m/s | |

Boundary condition | Pressure on the surface of liquid metal | 1 atm |

Electric potential difference between the two electrodes | 1.5 V | |

Residual magnetic flux density (y direction) | 60 mT | |

Generalized inward heat flux (on the electric heater) | 12.276 W/cm^{2} | |

Convective heat flux (from the system to the atmosphere) | 10 W/(m^{2} · °C) | |

Wall boundary condition | Free slip walls |

Initial condition | Temperature of the system | 23.3 °C (296.45 K) |
---|---|---|

Velocity of the liquid metal | 0 m/s | |

Boundary condition | Pressure on the surface of liquid metal | 1 atm |

Electric potential difference between the two electrodes | 1.5 V | |

Residual magnetic flux density (y direction) | 60 mT | |

Generalized inward heat flux (on the electric heater) | 12.276 W/cm^{2} | |

Convective heat flux (from the system to the atmosphere) | 10 W/(m^{2} · °C) | |

Wall boundary condition | Free slip walls |

A simulation model is developed to present the velocity distribution of the liquid metal after electrifying the electrodes. The numerical model was implemented in COMSOL Multiphysics 5.4 which is a powerful software platform for investigating the multi-physical coupled fields.

### 2.4 The Thermal Resistance Model.

*R*

_{total}is the total thermal resistance of the system,

*Q*

_{total}is the total radiating power of the heat transfer system including the heating power of the heater as well as the electric power of the liquid metal.

*T*

_{heater}and

*T*

_{amb}are the temperature of heater and ambient, respectively.

*R*

_{heater-lm},

*R*

_{lm-rad}, and

*R*

_{rad-amb}are the thermal resistance between heater and liquid metal, liquid metal and fan radiator, and fan radiator and ambient, respectively.

*R*

_{heater-lm},

*R*

_{lm-rad}, and

*R*

_{rad-amb}can further be expressed as

*R*

_{heater-AlN plate}is the contact thermal resistance between heater and AlN plate,

*R*

_{AlN plate-lm}is the convective thermal resistance between AlN plate and liquid metal at the heating end.

*R*

_{lm-AlN plate}is the convective thermal resistance between AlN plate and liquid metal at the cooling end.

*R*

_{AlN plate-rad}is the contact thermal resistance between AlN plate and fan radiator.

*T*

_{lm}is the temperature of liquid metal, and it can be considered as the average of the temperatures at the monitoring points 1, 2, 3, 4, 5.

*T*

_{rad}is the temperature of fan radiator. The schematic diagram of the three parts thermal resistances of the cooling system is shown as Fig. 3(a).

*R*and

_{i}*C*(

_{i}*i*= 1, 2) are thermal resistances and heat capacity of the heat transfer model, respectively. Transient thermal resistance

*R*(

*t*) is expressed as

## 3 Results and Discussion

### 3.1 Numerical Model.

According to practical sizes of the constituent parts of the experimental setup, a 3D finite element method model based on self-adaption grid division was established to simulate the multi-physical fields coupled together including electromagnetic field, thermal field and flow field. Since the heat transfer process and the fluid flow process are all unsteady, a transient solver was used during the calculation. And segregated algorithm and conjugate gradient iterative procedure were carried out to solve the conservation equations. A grid independence test was performed and is presented in Fig. 4. The temperature of the electric heater was monitored when the grid number is 20169, 35712, 67387, 171213, 276106, and 661177, respectively. It can be seen from Figs. 4(a) and 4(b) that the temperature of the electric heater gradually decreased to a nearly stable value with the increase of grid number. And there is an approximately linear relationship between computational time and grid number. Considering the factors of calculating accuracy and computational time, grid number of 661177 was finally selected for simulation. The computational domain and the grid structure of the simulation model of the experimental setup are shown in Fig. 4(c) and 4(d), respectively. And the statistical information of the model grid is listed in Table 2.

Number | Type | Value |
---|---|---|

1 | Number of elements | 661177 |

2 | Number of degrees of freedom | 167993 |

3 | Shape of element | Tetrahedron |

4 | Minimum element quality | 0.1845 |

5 | Average element quality | 0.6623 |

6 | Element volume ratio | 3.109E-4 |

7 | Grid volume | 0.001 m^{3} |

Number | Type | Value |
---|---|---|

1 | Number of elements | 661177 |

2 | Number of degrees of freedom | 167993 |

3 | Shape of element | Tetrahedron |

4 | Minimum element quality | 0.1845 |

5 | Average element quality | 0.6623 |

6 | Element volume ratio | 3.109E-4 |

7 | Grid volume | 0.001 m^{3} |

To validate the numerical results, a comparison between numerical and experimental steady temperature results of the monitor points 6 (the heater) and 4 (the liquid metal) is presented in Fig. 5. The heat flow density of the heater is 12.276 W/cm^{2} and the values are acquired at 300 s. As to the monitor points 6 and 4, the percentage errors between the numerical and experimental results are, respectively, 1.62% and 1.64%, which shows a good consistency. And the numerical results can be considered reliable.

### 3.2 The Velocity and Temperature Distribution of the Experimental Device.

Let us analyze qualitatively the velocity and temperature distribution of the experimental device. It can be seen from Fig. 6(a) that the velocity is generally centrosymmetric distributed around the central cylinder anode. And the maximum velocity appears near the anode. The velocity decreases with the increase of the distance from the anode and gradually increases near the cathode. The reason can be elucidated that the current density has the maximum value in the region near the anode, and the magnetic induction intensity in the region near the cathode is higher than that in other regions.

As can be seen from Fig. 6(b), the minimum velocity appears in the regions near the bottom of the electrodes. It is due to the high friction force between the liquid metal and the graphite electrodes. In the middle area between the two electrodes, the velocity in the upper region is large than that in the lower region. It can be attributed to the surface oxidation layer which increases the moving resistance of the liquid metal.

The temperature distribution of the heat dissipation device is shown in Figs. 6(c) and 6(d). It can be seen that the temperature decreases gradually from the electric heater to the surrounding areas. And the temperature in the area above the top edge of the electric heater is much higher than that in the area below the bottom edge of the electric heater. Therefore, the role of the rotated liquid metal is to distribute the temperature evenly throughout the heat dissipation system.

### 3.3 The Thermal Resistance Analysis of the Heat Dissipation System.

The transient thermal resistance–time changing curve is characteristic of nonlinearity in the heating process of the heater. Figure 7(a) are the total radiating power–time, *R*_{total}-time, *R*_{heater-lm}-time, *R*_{lm-rad}-time, and *R*_{rad-amb}-time curves when the heating power is set as 7.1 W and the electric power of the liquid metal is set as 0.87 W. It can be seen that the heating power increases sharply at the beginning and then keeps steady with the time. *R*_{total}(*t*), *R*_{heater-lm}(*t*), *R*_{lm-rad}(*t*), and *R*_{rad-amb}(*t*) are in exponential relationship with the time.

By fitting the experimental data with the analytic model, *R _{i}* and

*C*for

_{i}*R*

_{total}(

*t*),

*R*

_{heater-lm}(

*t*),

*R*

_{lm-rad}(

*t*), and

*R*

_{rad-amb}(

*t*) are presented in Table 3.

R (°C/W)_{i} | C (J/°C)_{i} | Fitting residual | Standard deviation | |||
---|---|---|---|---|---|---|

1 | 2 | 1 | 2 | |||

R_{total}(t) | 2.2768 | 2.1226 | 2.2409 | 56.5494 | 0.1954 | 0.0183 |

R_{heater-lm}(t) | 0.6606 | 1.4698 | 6.0641 | 2.6608 | 1.9874 | 0.0584 |

R_{lm-rad}(t) | 0.1233 | 0.0329 | 5.4888 | 30.5797 | 0.2587 | 0.0211 |

R_{rad-amb}(t) | 0.0000 | −15968 | 2.0000 | 54.0000 | 0.1650 | 0.0168 |

R (°C/W)_{i} | C (J/°C)_{i} | Fitting residual | Standard deviation | |||
---|---|---|---|---|---|---|

1 | 2 | 1 | 2 | |||

R_{total}(t) | 2.2768 | 2.1226 | 2.2409 | 56.5494 | 0.1954 | 0.0183 |

R_{heater-lm}(t) | 0.6606 | 1.4698 | 6.0641 | 2.6608 | 1.9874 | 0.0584 |

R_{lm-rad}(t) | 0.1233 | 0.0329 | 5.4888 | 30.5797 | 0.2587 | 0.0211 |

R_{rad-amb}(t) | 0.0000 | −15968 | 2.0000 | 54.0000 | 0.1650 | 0.0168 |

*R*

_{total},

*R*

_{heater-lm},

*R*

_{lm-rad}, and R

_{rad-amb}with the increase of the heating power. It indicates that

*R*

_{total}presents an increasing tendency and gradually stabilizes to about 4.42 °C/W. As to

*R*

_{heater-lm}, it increases gradually with the increase of the heating power, which is different from that of

*R*

_{lm-rad}and

*R*

_{rad-amb}. The relationship between

*R*

_{heater-lm},

*R*

_{lm-rad}, and

*R*

_{rad-amb}can be expressed as follows:

*Q*

_{heater}is the heating power of the electric heater.

*R*

_{heater-lm},

*R*

_{lm-rad},

*R*

_{rad-amb,}and

*R*

_{total}as shown in Fig. 7(b) are fitted with exponential functions. And the fitting expressions and the R-squared values are as follows:

Figure 8 shows the changing relationship among *R*_{heater-lm}-time, *R*_{lm-rad}-time, and *R*_{rad-amb}-time curves when the heating power/voltage are set as typical values including 0.083 W/0.904 V, 0.331 W/1.809 V, 0.738 W/2.716 V, 3.755 W/6.386 V, 5.913 W/8.252 V, and 13.641 W/13.949 V, respectively. It can be seen that when the heating power is 0.083 W, *R*_{heater-lm} is a negative value which is about –0.37 °C/W (Fig. 8(a)), and as the heating power increases to 0.331 W, *R*_{heater-lm} changes from a negative to a positive value which is about 0.23 °C/W (Fig. 8(b)). As the heating power is set as 0.738 W, *R*_{heater-lm} is larger than *R*_{lm-rad} but lower than *R*_{rad-amb} (Fig. 8(c)). The stable value of *R*_{heater-lm} becomes larger and larger with the increase of the heating power (Fig. 8(d)). It equals to that of *R*_{rad-amb} when the heating power is 5.913 W (Fig. 8(e)), and it becomes the maximum value of the three parts thermal resistances as the heating power continues to increase (Fig. 8(f)).

*T*

_{heater},

*T*

_{p}_{1},

*T*

_{p}_{2},

*T*

_{p}_{3},

*T*

_{p}_{4},

*T*

_{p}_{5},

*T*

_{rad}, Ratio, respectively. Here,

*T*

_{heater}and

*T*

_{rad}represent the temperature of the electric heater and the radiator, respectively.

*T*

_{p}_{1},

*T*

_{p}_{2},

*T*

_{p}_{3},

*T*

_{p}_{4}, and

*T*

_{p}_{5}represent the temperature of the five system monitoring points which are 1, 2, 3, 4, and 5, respectively.

*m*represents the ratio of the heating power to the liquid metal electrifying power, and its specific meanings are as follows:

*Q*

_{lm}is the liquid metal electrifying power.

It can be seen from Fig. 9 that all the measured curves are approximately linear with different slope values. As to *T*_{heater}, the slope value is about 1.033 °C/W. When 0 < *m <* 1, *T*_{heater} can be lower than the average temperature of liquid metal, and *R*_{heater-lm} can be a negative value which is mentioned above. The demarcation point between *R*_{heater-lm} > 0 and *R*_{heater-lm} < 0 is *m =* 1 which corresponds to *Q*_{heater} = 0.92 W. And if the temperature of the electric heater is to be controlled within 80 °C which corresponds to *Q*_{heater} = 12.32 W, then the heating power of the electric heater should be less than 12.32 W. The system presented above exhibits excellent heat dissipation performance compared with liquid metal-based heat dissipation systems introduced in related references as shown in Table 4. As the heat flow density of the heating source are, respectively, set as 3.15 W/cm^{2} and 3.5 W/cm^{2}, the temperature of the heating source can be controlled within 38.6 °C and 39.8 °C which is a great improvement compared with previous works [5,36].

## 4 Conclusion

In summary, we have presented an electromagnetic driven liquid metal-based heat dissipation system. Liquid metal is situated between the dual ring-shaped electrodes and rotates around the anode by an electromagnetic field. The velocity profile throughout the liquid metal is demonstrated and the results show that the velocity is distributed unevenly along longitudinal section. The temperature profile of the experimental device shows that high temperature region is located on the electric heater side and liquid metal plays an important role in the flatting of temperature distribution. The thermal resistance model of the heat dissipation system is established. The total thermal resistance *R*_{total} of the system contains three parts and the size relationship between each part is analyzed. And *R*_{total} tends to be a stable value of 4.42 °C/W. As to the heat dissipation system, there is an approximately linear relationship between the temperature of the electric heater and the heating power. And the heating power should be less than 12.32 W when the temperature of the electric heater is to be controlled within 80 °C. Liquid metal-based thermal management introduced in the paper serves as an effective solution for the electronic heating systems with limited space.

## Acknowledgment

This work was partially supported by the National Science Foundation for Young Scientists of China (Grant No. 51706234) and CAS Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry (CRYOQN201909).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*c*=speed of light (m/s)

*h*=convective heat transfer coefficient (W/(m

^{2}· °C))*i*=subscript index

*j*=subscript index

*m*=ratio

*q*=heat flux (W)

*t*=time (s)

*v*=velocity (m/s)

*w*=enthalpy per unit mass (J/kg)

*L*=size (m)

*P*=pressure (Pa)

*Q*=the radiating power of the heat transfer system (W)

*R*=transient thermal resistance (°C/W)

*T*=temperature (°C)

*U*=wind velocity (m/s)

=*q*energy flow density (W/m

^{2})=*v*velocity vector (m/s)

=*B*magnetic induction intensity vector (mT)

=*E*electrical filed intensity vector (V/m)

=*F*Lorentz force (N)

=*H*magnetic field intensity vector (A/m)

=*J*electrical current density vector (A/m

^{2})*c*=_{P}heat capacity at constant pressure (J/(kg · °C))

*C*=_{i}heat capacity of the heat transfer model (J/(kg · °C))

*Q*_{heating power}=the total radiating power of the heat transfer system (W)

*Q*_{total power}=the total radiating power of the heat transfer system (W)

*R*_{AlN plate-lm}=the convective thermal resistance between AlN plate and liquid metal (°C/W)

*R*_{AlN plate-rad}=the contact thermal resistance between AlN plate and fan radiator (°C/W)

*R*_{heater-AlN plate}=the contact thermal resistance between heater and AlN plate (°C/W)

*R*_{heater-lm}=the thermal resistance between heater and liquid metal (°C/W)

*R*=_{i}thermal resistances of the heat transfer model (°C/W)

*R*_{lm-AlN plate}=the convective thermal resistance between AlN plate and liquid metal (°C/W)

*R*_{lm-rad}=the thermal resistance between liquid metal and fan radiator (°C/W)

*R*_{rad-amb}=the thermal resistance between fan radiator and ambient (°C/W)

*R*_{total}=total thermal resistance (°C/W)

*T*_{amb}=the temperature of ambient (°C)

*T*_{heater}=the temperature of the electric heater (°C)

*T*_{lm}=the temperature of liquid metal (°C)

*T*_{p1},*T*_{p2},*T*_{p3},*T*_{p4},*T*_{p5}=the temperature of the five system monitoring points (°C)

*T*_{rad}=the temperature of fan radiator (°C)

- Pr =
Prandtl number