A diamagnetic-airflow hybrid levitation structure

Abstract

In this paper, a diamagnetic-airflow hybrid levitation structure is reported, with a floating magnet rotor stably levitated above a diamagnetic disc just using a lifting magnet and airflow. Compared with the typical diamagnetically stabilized structure, the rotation speed of the rotor is increased from 9570 rpm to 16666 rpm, and the levitation gap is increased from 0.3 mm to 0.7 mm when the airflow rate exceeds 2198 sccm under standard temperature and pressure (STP). At the same time, the rotor can be stably levitated at any height from 0.2 mm to 0.8 mm by adjusting the vertical position of the nozzles. With the hybrid levitation structure, it’s possible to overcome the structural limitation of typical diamagnetically stabilized levitation structure and enhance the working performance of the rotor. This levitation structure is expected to be applied to sensing, energy harvesting and air bearing under actuation of airflow.

Keywords

Diamagnetic airflow stabilized levitation simulation energy harvesting

1. Introduction

Levitation has attracted significant research attention in recent years. Basically, magnetic levitation [1–3], electrostatic levitation [4,5], ultrasonic levitation [6,7], optical levitation [8,9] and airflow levitation [10,11] are used widely. Diamagnetism [12] was firstly discovered in 1778 by observing that bismuth was repelled by a magnetic field. In 1842, the British physicist Earnshaw proposed the famous Earnshaw theory [13], which was applied to the static equilibrium of objects. According to Earnshaw theory, it is impossible to obtain a stable magnetic levitation between permanent magnets or between permanent magnets and ferromagnetic materials. Diamagnetism was verified by Braunbek with the levitation of small pieces of graphite and bismuth in a non-uniform strong electromagnetic field in 1939 [14]. Braunbek pointed out that magnetic material with relative magnetic permeability less than 1 could remain stable in a static magnetic field.

Magnetic material with a relative magnetic permeability less than 1 is usually a diamagnetic material. At present, pyrolytic carbon is of the largest diamagnetism and is with the magnetic susceptibility−4.7e-4 at room temperature. Andre Geim, winner of the 2010 Nobel Prize in physics, levitated a frog firstly with a 16 T magnetic field in 1997 [15]. Metal bismuth, water and most of the organic matter are diamagnetic. Diamagnetically stabilized levitation has attracted gradually increasing attraction because of low friction [16,17] and small energy consumption [18]. Due to short range of diamagnetic force, the movement gap is usually limited in diamagnetic levitation structure. With joint action of magnetic force or other mechanism, diamagnetic levitation found many applications. For example, it is used in the micro-nano rotating machinery [19], vibration energy harvesting [20,21], high sensitivity sensing [22,23]. However, the structure of the typical diamagnetically stabilized levitation inhibits its own potential, because the gap between the two diamagnetic discs is too small.

In this paper, a diamagnetic-airflow hybrid levitation structure is proposed, with the magnet rotor driven by two centrosymmetric nozzles. The levitation structure not only eliminates the gap limitation of the two diamagnetic discs but also increases the rotation speed of the floating magnet rotor.

2. Structural model

Figure 1 shows the schematic diagram of the levitation structure. The lifting magnet, the upper diamagnetic disc, the magnet rotor, and the lower diamagnetic disc constitute a typical diamagnetic stabilized levitation system. The gravity of the magnet rotor is balanced by the attracting force between the magnet rotor and the lifting magnet, and the upper and lower diamagnetic discs help to keep the magnet rotor in equilibrium by exerting repelling force. The balance of the floating magnet rotor would be broken and moved upward if the upper diamagnetic disc is removed directly.

Fig. 1.

Schematic diagram of the levitation structure.

In this study, it was found that the lifting magnet, the centrosymmetric nozzles, the magnet rotor, and the lower diamagnetic disc made up a diamagnetic-airflow hybrid levitation system. In which the magnet rotor could be stably levitated above the lower diamagnetic disc by the airflow without the upper diamagnetic disc.

Two nozzles are arranged symmetrically about the axis of the magnet rotor, and the two centrosymmetric nozzles are placed at the same horizontal plane which is also the central plane of the floating magnet rotor. Two streams of airflow acting on the magnet rotor to form a force couple. The magnet rotor can be driven to rotate above the lower diamagnetic discs by the airflow. The diamagnetic-airflow hybrid levitation is realized by combining the action of magnetic attraction, diamagnetic force, and airflow driving force.

3. Theoretical analysis of the diamagnetic-airflow hybrid levitation

Since the forces exerted on the magnet rotor are balanced in the vertical direction, the magnet rotor can be stably levitated and rotate above the disc plate. The airflow not only drives the rotor to rotate but also exerts a downward force on the rotor to keep the rotor stable in the vertical direction. Therefore, rotor rotation and airflow driving force are two necessary conditions for stable hybrid levitation.

3.1. Levitation analysis

The mechanical analysis of the hybrid levitation structure is shown in Fig. 2. Since the two magnets are arranged the same magnetization direction, the magnetic attraction F _L can offset the gravity of the magnet rotor. Two downward force F _n from the nozzles are acting on the magnet rotor to overcome the upward diamagnetic force F _D from the lower diamagnetic disc.

Fig. 2.

Mechanical analysis of the hybrid levitation structure.

In order to calculate the magnetic and diamagnetic forces, a 3-D model is established with finite element software COMSOL Multiphysics 5.3. A vertical section along Z axis is made for showing magnetic field lines clearly. The numerical simulation of the magnetic field lines is carried out, and the results are shown in Fig. 3.

Fig. 3.

Schematic diagram of magnetic field lines.

Because the meshing is not completely symmetric and there are residuals during the FEA simulation, the results are a little bit asymmetric in Fig. 3. However, it can be seen as symmetric along the z axis in the allowable range. Therefore, the axis of the rotor is aligned with the center axis of the lifting magnet. At the same time, the magnet rotor can rotate freely about the central axis of the lifting magnet.

The diamagnetic force is often neglected in daily life because it is so weak. In fact, the effect of the diamagnetism is obvious when the diamagnetism and magnetic field are strong enough. In magnetic field H, the repelling force of the diamagnet can be calculated using the following equation. $\begin{eqnarray}\displaystyle \vec{F}={\mu}_{0}{\chi}_{m}\int _{V}{\nabla}\vec{H}\cdot H\vec{d}V & & \displaystyle\end{eqnarray}$ (1) where 𝜇₀ is the relative magnetic permeability, 𝜒_m denotes the magnetic susceptibility, V is the volume of the diamagnet.

The simulation results obtained by COMSOL Multiphysics 5.3 are shown in Fig. 4 and Fig. 5. Since the levitation gap of the diamagnetic levitation is very small, take the data with a range of 3 mm near the equilibrium point of magnetic force and gravity. Similarly, the diamagnetic data with a range of 3 mm is taken. The lifting magnet and the diamagnetic disc are fixed. As the rotor moves in the vertical direction, the distance L decreases with the increase of distance D. Ensure that the sum of L and D is constant (L + D = 52 mm) during the simulation.

Fig. 4.

Trend of the magnetic force versus the distance L.

Fig. 5.

Trend of the diamagnetic force versus the distance D.

Figure 4 shows the relationship between the magnetic force and the distance L, the magnetic force decreases as the distance L increases. According to Fig. 4, the static levitation gap is decided when the magnetic force is equal to the gravity of the rotor, 47740 μN. Figure 5 indicates the trend of the diamagnetic force versus the distance D. It can be seen that the diamagnetic force of the rotor increases rapidly as the distance D decreases.

When the flow velocity increases, the pressure on the interface between the rotor and the gas will decrease [24]. Because of the Bernoulli effect, the rotor is subjected to a force in the vertical direction. $\begin{eqnarray}\displaystyle & \displaystyle p+1/2{\rho}{\nu}^{2}+{\rho}gh=\text{constant} & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle & \displaystyle F_{n}=pS & \displaystyle\end{eqnarray}$ (3) where p is the pressure, 𝜌 is the fluid density, v is the fluid velocity, g is the gravitational acceleration, and h is the height of the point.

A set of analysis is carried out to obtain the force from the airflow acting on the rotor by COMSOL Multiphysics 5.3 by adjusting the distance of the nozzle from the center plane of the rotor. In the analysis, the symmetrical plane of the rotor is set as the zero plane, the upward direction is selected as positive and the inlet flow rate was set at 2198 sccm under STP. The analysis results are shown in Fig. 6.

Fig. 6.

Trend of airflow force versus the offset distance.

In Fig. 6, the pressure difference is the difference between the pressures on both sides of the rotor. The offset distance is the distance between the nozzle and the zero plane in the vertical direction. It can be seen from Fig. 6 that the force acting on the rotor is related to the position of the nozzle. When the vertical position of the nozzle is about −0.7 mm, the rotor receives the maximum force and downward force can reach up to −857 μN.

In the vertical direction, the rotor is subjected to magnetic force, diamagnetic force, its gravity and actuation of the airflow. $\begin{eqnarray}\displaystyle F=F_{L}+F_{D}-F_{n\text{max}}-G & & \displaystyle\end{eqnarray}$ (4) where F is the resultant force of the rotor, F _L is the upward attractive force from the lifting magnet, F _D is the upward diamagnetic force from the lower disc plate, F _nmax is the maximal downward force from the airflow in the vertical direction.

Figure 7 shows the relationship between the resultant force of the rotor and the distance D. It can be seen from the figure that when the range of the distance D is from 0.11 to 0.78 mm, the resultant force of the rotor is not more than 0. When the resultant F is 0 N and the offset distance of the nozzle is −0.7 mm, the maximum downward force from airflow is to keep the rotor stable in the vertical direction. As the resultant force is less than 0 N, the downward airflow force does not need to be maximal to keep the rotor stable. Therefore, the range of levitation gap for the rotor is from 0.11 to 0.79 mm by adjusting the offset distance of the nozzle in the vertical position.

Fig. 7.

The vertical resultant force exerted on the rotor versus the distance L.

3.2. Analysis of driving force and air resistance

The airflow volume rate Q is converted into airflow velocity as following. $\begin{eqnarray}\displaystyle v_{0}=\frac{Q}{{\pi}r_{0}^{2}} & & \displaystyle\end{eqnarray}$ (5) where r ₀ is the radius of the nozzle, v ₀ is the outlet velocity of the airflow.

The contact area between the airflow and the circular arc rotor blade is taken as the control body. As shown in the Fig. 8(a), the control body satisfies the momentum equation.

Fig. 8.

Schematic diagram of the interaction between the air flow and the magnet rotor. (a) Control body of the airflow, (b) air resistance exerted on the magnet rotor.

Based on the momentum equation, the following equations are obtained. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle -\oint _{S}p\cos (n,x)dS+\iiint _{{\tau}}{\rho}f_{x}d{\tau}-F_{x}=\frac{\partial }{\partial t}\iiint _{{\tau}}{\rho}ud{\tau}+∯_{S}(u\cdot n)u{\rho}dS\\[12.0pt] \displaystyle -\oint _{S}p\cos (n,y)dS+\iiint _{{\tau}}{\rho}f_{y}d{\tau}-F_{y}=\frac{\partial }{\partial t}\iiint _{{\tau}}{\rho}vd{\tau}+∯_{S}(v\cdot n)v{\rho}dS\\[12.0pt] \displaystyle -\oint _{S}p\cos (n,z)dS+\iiint _{{\tau}}{\rho}f_{z}d{\tau}-F_{z}=\frac{\partial }{\partial t}\iiint _{{\tau}}{\rho}wd{\tau}+∯_{S}(w\cdot n)w{\rho}dS\end{array}\right. & & \displaystyle\end{eqnarray}$ (6) where S is the area of the control surface, n is the unit vector of the outer normal, τ is the volume of control body, 𝜌 is the density of the fluid, F is an external force of acting on the control body, f is the mass force, uvw is the flow rate of the fluid.

The force exerted on the control body in the Y direction can be calculated as following. $\begin{eqnarray}\displaystyle F=\iint _{S}(v\cdot n)v{\rho}dS. & & \displaystyle\end{eqnarray}$ (7)

The torque produced by the airflow from the two centrosymmetric nozzles is calculated as following. $\begin{eqnarray}\displaystyle T=2\cdot FL=\int _{R1}^{R}\frac{2{\rho}Q^{2}r\sin {\theta}}{{\pi}r_{0}^{2}}dr=\frac{{\rho}\sin {\theta}}{{\pi}r_{0}^{2}}(R^{2}-R_{1}^{2})Q^{2} & & \displaystyle\end{eqnarray}$ (8) where L is the vertical distance from the nozzle to the axis of rotation, θ is the angle between the airflow and the action surface, R is the maximum radius of the rotor, R ₁ is the radius from the root of the blade to the center of the rotor. Air resistance is mainly composed of viscous resistance and windward resistance. The air resistance force on the rotor can be calculated in two parts as shown in the Fig. 8(b).

Due to the existence of the lower diamagnetic disc, the viscous resistance of the lower surface is much larger than the viscous resistance of the upper surface, so the resistance of the upper surface can be ignored in the calculation. Viscosity resistance of the lower surface can be calculated by the following equation. $\begin{eqnarray}\displaystyle & \displaystyle f_{1i}=-{\mu}A\frac{v}{2h}=-{\mu}A\frac{{\omega}r_{i}}{2h} & \displaystyle\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle & \displaystyle M_{1}=-2\int _{R_{0}}^{R}{\mu}A\frac{{\omega}r^{2}}{2h}dr=-\frac{{\mu}A{\omega}}{3h}(R^{3}-R_{0}^{3}) & \displaystyle\end{eqnarray}$ (10) where A is on the surface area, h depends on the levitation gap, R ₀ is the minimum radius of the rotor.

Windward resistance of the rotor side walls can be calculated by the following equation. $\begin{eqnarray}\displaystyle & \displaystyle f_{2i}=-bv=-b{\omega}r_{i} & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle & \displaystyle M_{2}=-\int _{R_{1}}^{R}3b{\omega}r^{2}dr=-b{\omega}(R^{3}-R_{1}^{3}) & \displaystyle\end{eqnarray}$ (12) where b depends on the performance of the fluid.

Fig. 9.

Fluidic field simulation analysis. (a) Streamline of airflow acting on the magnet rotor, (b) driving torque versus flowrate.

The rotor is in equilibrium when the driving force and resistance are equal in magnitude. $\begin{eqnarray}\displaystyle & \displaystyle T+M=T+M_{1}+M_{2}=0 & \displaystyle\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle & \displaystyle {\omega}=\frac{3{\rho}h(R^{2}-R_{1}^{2})\sin {\theta}}{{\pi}r_{0}^{2}[uA(R^{3}-R_{0}^{3})+3hb(R^{3}-R_{1}^{3})]}Q^{2}. & \displaystyle\end{eqnarray}$ (14)

Fluidic field of the airflow was simulated and analyzed with COMSOL Multiphysics 5.3. Figure 9(a) shows the streamline diagram of the airflow. The airflow is mainly divided into two streams when it is acting on the blade of the magnet rotor. One stream of airflow moves clockwise along the arc of the blades and the other one flows counter clockwise along the arc of the blades, which is consistent with the physical model of the torque analysis performed. Parametric scanning of flow rate is used to obtain the torque of the rotor under different airflow rates. As shown in Fig. 9(b), the relationship between torque T and airflow rate Q is a quadratic curve, which is consistent with the Eq. (8). Based on the theoretical analysis and simulation, the relationship between the airflow rate and the rotation speed of the rotor is to be expected, and then verified by experiment.

4. Experimental configuration and discussions

In the experiment, a continuous nitrogen flow was directed to the floating magnet rotor through two centrosymmetric nozzles (𝛷 = 0.6 mm), which were positioned in proximity to the magnet rotor and connected to a compressed nitrogen source. Two mass flow controllers (AITOLY MFC300) were used to regulate the flowrate of nitrogen flow with the computer software. An oscilloscope (Tektronix TDS2012B) was used as a data processor to measure the frequency of the voltage signal of the photoelectric rotation measuring apparatus (KEYENCE LV-NH35, 80 μs response time). The rotation speed of the floating magnet rotor could be calculated by means of calculating the frequency of the voltage signal. The elevating platform (LWZ40-L200, 0.02 mm accuracy) was used to adjust the vertical position of the upper diamagnetic disc.

The necessary components and parameters for levitation are detailed in Table 1. Pyrolytic graphite was chosen as the diamagnetic disc (𝜒_PG = −45 × 10⁻⁵). In order to achieve the highest levitation gap, the strong NdFeB magnets (B _r = 1.45 T) were selected. The typical levitation gap between the floating magnet rotor and the two diamagnetic discs both are 0.3 mm. The gravity of the magnet rotor is 0.04774 N.

Table 1
Necessary components for levitation list

Component Material Dimension (mm)

Lifting magnet NdFeB-52 𝛷25 × 10

Upper diamagnetic disc Pyrolytic graphite 𝛷25 × 2

Magnet rotor NdFeB-52 𝛷18 × 3

Lower diamagnetic disc Pyrolytic graphite 𝛷25 × 2

Nozzle Plastic 𝛷0.6 (inner diameter)

Component	Material	Dimension (mm)
Lifting magnet	NdFeB-52	𝛷25 × 10
Upper diamagnetic disc	Pyrolytic graphite	𝛷25 × 2
Magnet rotor	NdFeB-52	𝛷18 × 3
Lower diamagnetic disc	Pyrolytic graphite	𝛷25 × 2
Nozzle	Plastic	𝛷0.6 (inner diameter)

Figure 10 shows the experimental apparatus of the typical levitation structure. The levitation state of the floating magnet rotor can be seen from the figure. Firstly, the magnet rotor can be stably levitated between the upper and lower diamagnetic discs by adjusting the elevating platform. Next, the mass flow controllers are used to control the flowrate of the airflow. Then, the rotor rotates under the driving of airflow, and range of the flow rate is from 0 to 2748 sccm under STP. Finally, the rotation speed of the rotor is measured by the photoelectric rotation measuring apparatus and recorded with the oscilloscope.

Fig. 10.

Experimental apparatus. Levitation state and zoom view of levitating components in the typical levitation structure.

Figure 11 shows the realistic configuration for the diamagnetic-airflow hybrid levitation. The levitation state of the floating magnet rotor can be observed from the figure. When the air flowrate reaches 2198 sccm, the upper diamagnetic disc is removed, and the rotor can still maintain stable levitation above the lower diamagnetic disc. Then, the rotor rotates under the driving of airflow, and the range of the flow rate is from 2198 to 2748 sccm. At the same time, the rotation speed of the rotor is measured by the photoelectric rotation measuring apparatus and read out with the oscilloscope.

Fig. 11.

Realistic configuration for diamagnetic-airflow hybrid levitation. Levitation state and zoom view of levitating components.

After the upper diamagnetic disc was removed, the gap between the lower diamagnetic disc and the floating magnet rotor rises from 0.3 mm to 0.7 mm. The joint influence by the high-speed airflow and the atmosphere pressure forms air films around the magnet rotor, which acts as an air bearing to maintain the stable levitation of the magnet rotor. It is possible to achieve a stable rotational levitation of the floating magnet rotor only when the two nozzles are correctly adjusted symmetrically about the axis of rotation.

Figure 12 shows the relationship between the rotation speed of the magnet rotor and the airflow rate. The red line represents the hybrid levitation structure. While the black line represents the typical levitation structure. The rotation speed increases approximately quadratic curve with the increase of airflow rate, which is consistent with the Eq. (14). When the airflow rate is no less than 2198 sccm, the hybrid levitation structure can be formed by removing the upper diamagnetic disc. The rotation speed in the hybrid levitation structure is much bigger than that in the typical levitation structure. The rotation speed of the floating magnet rotor is increased from 9570 rpm to 16666 rpm when the airflow rate is 2198 sccm.

Fig. 12.

The relationship between the rotation speed of the magnet rotor and the airflow rate.

Figure 13 shows the relationship between the rotation speed of the magnet rotor and the displacement distance of the upper diamagnetic disc when the airflow rate is 2198 sccm. In the process of moving up the upper diamagnetic disc until it is removed, the rotation speed increases gradually. When the displacement of the upper diamagnetic disc exceeds 3.6 mm, the speed of the rotor remains unchanged.

Fig. 13.

The variation trend of the rotor speed during the upper diamagnetic disc removal process.

Due to the viscous action, the airflow adheres to the surface of the magnet rotor and generates damping resistance and energy dissipation. Viscosity depends on the density of the air flow. When the upper diamagnetic disc moves up until it is removed, the levitation gap increases and the air density between the upper diamagnetic disc and the magnet rotor decreases. Correspondingly, the rotation speed increases as the damping decreases.

Next, the levitation gap of the rotor in the hybrid levitation structure is studied, so as to explore the changing range of the gap between the rotor and the lower magnetic disc. The levitation gap is changed from 0.2 mm to 0.8 mm by adjusting the offset distance of the nozzle along the vertical direction when the rotor is levitated at a flow rate of 2198 sccm. The experimental results are shown in Fig. 14. It means that the rotor can be levitated at any height within this range by adjusting the nozzles. The results are consistent with the simulation results, where the range of levitation gap is between 0.11 mm and 0.79 mm.

Fig. 14.

The range of the levitation gap under the airflow rate of 2198 sccm under STP.

In the typical diamagnetic stabilized levitation system, the levitation gap between the floating magnet rotor and the two diamagnetic discs both are 0.3 mm. By removing the upper diamagnetic disc, the levitation gap reaches 0.7 mm. At the same time, the rotation speed of the rotor is increased greatly. Then, by adjusting the vertical position of the nozzles, the rotor can be levitated at any height from 0.2 mm to 0.8 mm.

5. Conclusions

In this paper, a diamagnetic-airflow hybrid levitation structure is presented. The hybrid levitation structure overcomes the structural constraints of typical diamagnetically stabilized levitation, in which the rotation speed and levitation gap of the magnet rotor are both limited. In the proposed diamagnetic-airflow hybrid levitation structure, it was first discovered that the limitation of the upper diamagnetic disc was eliminated, so as to improve the dynamic characteristic of the levitated rotor. The probable application of the levitation structure is in air bearing due to its advantages of frictionless, high rotation speed and large levitation gap. It also could be used in sensors and energy harvesters because of the good linearity of its rotational characteristics. In the future, the output characteristics will be studied by introducing inductive coils to the structure.

Footnotes

Acknowledgements

This study is supported by the National Natural Science Foundation of China under grant no. 51475436, and Henan Province University Science and Technology Innovation Team under grant no. 18IRTSTHN016l.

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