- Email: [email protected]
Magnetically suspended virtual divergent channel
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 289 (2005) 389–391 www.elsevier.com/locate/jmmm
Magnetically suspended virtual divergent channel Ryuichiro Yamanea,, Shuzo Oshiamab, Myeong-Kwan Parkc a Kokushikan University, 4-28-1 Setagaya, Setagaya-ku, Tokyo 154-8515, Japan Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan c Pusan National University, 30 Changjeon-dong, Kumjeong-ku, Pusan 609-735, Republic of Korea b
Available online 30 November 2004
Abstract Two permanent magnets are set face-to-face and inclined with each other to produce the long cuspidal magnetic field. The diamagnetic liquid is levitated and flows through it without contact with the solid walls as if it is in the virtual divergent channel. Analysis is made on the shape of the virtual channel, and the results are compared with the experimental ones. The divergence angle increases with the increase in the inclination of the magnets. r 2004 Elsevier B.V. All rights reserved. PACS: 47.65; 75.50.Mm; 83.80.Gv; 85.70.Rp Keywords: Magnetic fluid; Levitation; Diamagnetic liquid; Virtual channel
1. Introduction The contamination of highly pure molten metals and chemically active materials is drastically reduced, resulting in high quality of the products, if the flow and the shape are controlled without contact with the surrounding walls. The diamagnetic material can be levitated by the strong magnetic field. By placing two long magnets (one of the long surfaces is N-pole and the opposite surface S-pole unlike a conventional bar magnet) face-to-face with each other, the long cuspidal magnetic field is produced, and the diamagnetic liquid column can be levitated stably at the center of the magnetic field. If the liquid is supplied at one end and extracted from the other, it can flow in the air or in the vacuum without contact with any solid walls just as if it is in a virtual channel. If the magnets are parallel, the channel is a circular pipe, and if the magnets are inclined with each other, the channel is conical and divergent. Corresponding author. Tel./fax: +81 3 5481 3332.
E-mail address: [email protected] (R. Yamane).
In order to form the channel in the air or in the vacuum and to suspend the liquid in it, the very strong magnets such as the superconducting magnets are required. But with the help of the magnetic fluid as a surrounding fluid, the magnetic force is increased, and it is possible to simulate the phenomena even with the conventional magnets. In the previous research [1–6], this principle was confirmed to be successful in levitating the non-magnetic droplet and cylinder. In the present research analysis is made on the shape of the virtual channel formed by the magnetic field, and is compared with the experimental results.
2. Analysis As shown in Fig. 1 the north poles of the two long magnets are placed face-to-face with each other in the same vertical plane to produce the long cuspidal magnetic field. The magnets are slightly inclined with each other to form a divergent channel. The diamagnetic
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.110
ARTICLE IN PRESS R. Yamane et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 389–391
390
Magnet
S
Using the saturation magnetization Ms and the initial susceptibility wi as the limit at H is infinity and zero, respectively, Eq. (4) can be written as 3wi M ¼ MSL H (5) MS
S
N
N
r z N
Diamagnetic liquid flow N
and the local susceptibility w is 3 2 MS 3wi 3wi dM M 5: S þ ¼ M S4 w¼ 2 2 3wi dH H sin h H
S S
Magnet
(6)
MS
Fig. 1. Analytical model.
The shape of the diamagnetic liquid is obtained numerically step by step from Eqs. (1), (2) and (6) as
liquid is levitated at the center of the cuspidal magnetic field and flows from left to right. The flow of the diamagnetic liquid is assumed to be very slow. Then, the configuration of the liquid is symmetrical, and the induced flow of the surrounding fluid is very slow. The dynamic pressure is neglected, and the pressure is constant inside the diamagnetic liquid flow. Then, neglecting the effect of the surface tension, the magnetic pressure on the diamagnetic liquid surface is assumed to be constant, that is,
1 m wH 2 C; (8) 2 0 where C is a constant determined by the boundary conditions. FðrÞ ¼
;
The experimental results are reported in the previous paper [6]. The magnets are the rare earth permanent magnets of 200 20 20 mm3 in length, height and width, respectively, and are inclined by an angle of 0–2.81. The magnetic flux density at the pole surface is 60 50
where K and a are constants to be determined by the strength and the setting of the magnets. In the vacuum or in the air w is constant, and from Eqs. (1) and (2) the shape of the diamagnetic liquid flow is represented as
40
1 az e : K
(3)
The radius r increases in the z-direction, which shows the formation of the virtual divergent channel. In the simulation using the magnetic fluid as the surrounding fluid, w is not constant, depending on the strength of the magnetic field. The magnetization of the magnetic fluid is assumed with the Langevin function as
M kA/m
(2)
r¼
(7)
(1)
where m0 is the magnetic permeability, w the susceptibility and H the strength of the magnetic field. In the cuspidal magnetic field the magnetic flux density B is assumed proportional to the radius r and to decrease exponentially in the axial direction. Then B is represented in the form B ¼ Kre
Fðrn1 Þ
; qF=qr rðn1Þ
3. Experiment
1 m wH 2 ¼ const; 2 0
az
rn ¼ rn1
30 20 10 0
0
200
400 H kA/m
600
Fig. 2. Magnetization of HC-50.
M ¼ NmLðxÞ; LðxÞ ¼ coth x
1 x
x¼
m0 mH ; kT
ð4Þ
where M is the magnetization, N the number density of the magnetic particles, m the magnetic moment, k the Boltzmann constant and T the temperature.
Fig. 3. Diverging water flow (1.81, 1.5%).
800
ARTICLE IN PRESS R. Yamane et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 389–391
Cal. 1.0deg Cal. 1.8deg Cal. 2.4deg Cal. 2.8deg
Exp.1.0deg Exp.1.8deg Exp. 2.4deg Exp. 2.8deg
6
6
5 4
4
r mm
r mm
5
3
3 2
2
1
1 0
Exp.1.0deg Exp.1.8deg Exp. 2.4deg Exp. 2.8deg
Cal. 1.0deg Cal. 1.8deg Cal. 2.4deg Cal. 2.8deg
7
391
0 0 0
50
100
50
100
150
z mm
150
z mm
Fig. 6. Radius of water flow (1.5%).
Fig. 4. Radius of water flow (0.75%).
Cal. 1.0deg Cal. 1.8deg Cal. 2.4deg Cal. 2.8deg
respectively. The radius of the water increases with distance, suggesting that a divergent channel is formed. The divergence angle increases with the increase in the inclination of the magnets. The calculated results are in good accordance with the experimental ones.
Exp.1.0deg Exp.1.8deg Exp. 2.4deg Exp. 2.8deg
6 5
5. Conclusions
r mm
4 3 2 1 0
0
50
100
150
z mm
(1) The virtual divergent channel can be formed with two magnets set face-to-face and inclined with each other. (2) The shape of the channel can be estimated analytically. (3) The divergence angle increases with the increase in the inclination of the magnets.
Fig. 5. Radius of water flow (1.0%).
0.4 T. The magnetic fluid is the hydrocarbon-based one, HC-50, and the magnetization is shown in Fig. 2. It is diluted to the concentration of 0.5–1.5%. The diamagnetic liquid is water. Fig. 3 shows an infrared photograph of the diverging flow of the water in the magnetic fluid. The flow is from left to right. The inclination of the magnets is 1.81, and the concentration of the magnetic fluid is 1.5%. The flow is vertically elongated to the axial direction.
4. Results The analytical calculation is made using the parameters of the experimental conditions. Figs. 4–6 show the analytical and experimental results for the concentration of the magnetic fluid, 0.75, 1.0 and 1.5%,
Acknowledgements Authors are deeply indebted to Mr. Y. Ishikawa and Mr. C. Basandash for their help in the experiment.
References [1] N. Fujisaki, et al., Trans. Japan Soc. Mech. Eng. B 63 (606) (1997) 417. [2] R. Yamane, et al., Proc. ASME/JSME Fluids Eng. Conf. (1999). [3] J. Mai, et al., Fluid Dyn. Res. (24) (1999) 147. [4] J. Mai, et al., Eur. J. Mech/Fluids (2) (2002) 237. [5] R. Yamane, et al., J. Magn. Magn. Mater. 252 (2002) 268. [6] R. Yamane, et al., Int. J. Appl. Electromagn. Mech. 19 (2004) 557.
Journal of Magnetism and Magnetic Materials 289 (2005) 389–391 www.elsevier.com/locate/jmmm
Magnetically suspended virtual divergent channel Ryuichiro Yamanea,, Shuzo Oshiamab, Myeong-Kwan Parkc a Kokushikan University, 4-28-1 Setagaya, Setagaya-ku, Tokyo 154-8515, Japan Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan c Pusan National University, 30 Changjeon-dong, Kumjeong-ku, Pusan 609-735, Republic of Korea b
Available online 30 November 2004
Abstract Two permanent magnets are set face-to-face and inclined with each other to produce the long cuspidal magnetic field. The diamagnetic liquid is levitated and flows through it without contact with the solid walls as if it is in the virtual divergent channel. Analysis is made on the shape of the virtual channel, and the results are compared with the experimental ones. The divergence angle increases with the increase in the inclination of the magnets. r 2004 Elsevier B.V. All rights reserved. PACS: 47.65; 75.50.Mm; 83.80.Gv; 85.70.Rp Keywords: Magnetic fluid; Levitation; Diamagnetic liquid; Virtual channel
1. Introduction The contamination of highly pure molten metals and chemically active materials is drastically reduced, resulting in high quality of the products, if the flow and the shape are controlled without contact with the surrounding walls. The diamagnetic material can be levitated by the strong magnetic field. By placing two long magnets (one of the long surfaces is N-pole and the opposite surface S-pole unlike a conventional bar magnet) face-to-face with each other, the long cuspidal magnetic field is produced, and the diamagnetic liquid column can be levitated stably at the center of the magnetic field. If the liquid is supplied at one end and extracted from the other, it can flow in the air or in the vacuum without contact with any solid walls just as if it is in a virtual channel. If the magnets are parallel, the channel is a circular pipe, and if the magnets are inclined with each other, the channel is conical and divergent. Corresponding author. Tel./fax: +81 3 5481 3332.
E-mail address: [email protected] (R. Yamane).
In order to form the channel in the air or in the vacuum and to suspend the liquid in it, the very strong magnets such as the superconducting magnets are required. But with the help of the magnetic fluid as a surrounding fluid, the magnetic force is increased, and it is possible to simulate the phenomena even with the conventional magnets. In the previous research [1–6], this principle was confirmed to be successful in levitating the non-magnetic droplet and cylinder. In the present research analysis is made on the shape of the virtual channel formed by the magnetic field, and is compared with the experimental results.
2. Analysis As shown in Fig. 1 the north poles of the two long magnets are placed face-to-face with each other in the same vertical plane to produce the long cuspidal magnetic field. The magnets are slightly inclined with each other to form a divergent channel. The diamagnetic
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.110
ARTICLE IN PRESS R. Yamane et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 389–391
390
Magnet
S
Using the saturation magnetization Ms and the initial susceptibility wi as the limit at H is infinity and zero, respectively, Eq. (4) can be written as 3wi M ¼ MSL H (5) MS
S
N
N
r z N
Diamagnetic liquid flow N
and the local susceptibility w is 3 2 MS 3wi 3wi dM M 5: S þ ¼ M S4 w¼ 2 2 3wi dH H sin h H
S S
Magnet
(6)
MS
Fig. 1. Analytical model.
The shape of the diamagnetic liquid is obtained numerically step by step from Eqs. (1), (2) and (6) as
liquid is levitated at the center of the cuspidal magnetic field and flows from left to right. The flow of the diamagnetic liquid is assumed to be very slow. Then, the configuration of the liquid is symmetrical, and the induced flow of the surrounding fluid is very slow. The dynamic pressure is neglected, and the pressure is constant inside the diamagnetic liquid flow. Then, neglecting the effect of the surface tension, the magnetic pressure on the diamagnetic liquid surface is assumed to be constant, that is,
1 m wH 2 C; (8) 2 0 where C is a constant determined by the boundary conditions. FðrÞ ¼
;
The experimental results are reported in the previous paper [6]. The magnets are the rare earth permanent magnets of 200 20 20 mm3 in length, height and width, respectively, and are inclined by an angle of 0–2.81. The magnetic flux density at the pole surface is 60 50
where K and a are constants to be determined by the strength and the setting of the magnets. In the vacuum or in the air w is constant, and from Eqs. (1) and (2) the shape of the diamagnetic liquid flow is represented as
40
1 az e : K
(3)
The radius r increases in the z-direction, which shows the formation of the virtual divergent channel. In the simulation using the magnetic fluid as the surrounding fluid, w is not constant, depending on the strength of the magnetic field. The magnetization of the magnetic fluid is assumed with the Langevin function as
M kA/m
(2)
r¼
(7)
(1)
where m0 is the magnetic permeability, w the susceptibility and H the strength of the magnetic field. In the cuspidal magnetic field the magnetic flux density B is assumed proportional to the radius r and to decrease exponentially in the axial direction. Then B is represented in the form B ¼ Kre
Fðrn1 Þ
; qF=qr rðn1Þ
3. Experiment
1 m wH 2 ¼ const; 2 0
az
rn ¼ rn1
30 20 10 0
0
200
400 H kA/m
600
Fig. 2. Magnetization of HC-50.
M ¼ NmLðxÞ; LðxÞ ¼ coth x
1 x
x¼
m0 mH ; kT
ð4Þ
where M is the magnetization, N the number density of the magnetic particles, m the magnetic moment, k the Boltzmann constant and T the temperature.
Fig. 3. Diverging water flow (1.81, 1.5%).
800
ARTICLE IN PRESS R. Yamane et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 389–391
Cal. 1.0deg Cal. 1.8deg Cal. 2.4deg Cal. 2.8deg
Exp.1.0deg Exp.1.8deg Exp. 2.4deg Exp. 2.8deg
6
6
5 4
4
r mm
r mm
5
3
3 2
2
1
1 0
Exp.1.0deg Exp.1.8deg Exp. 2.4deg Exp. 2.8deg
Cal. 1.0deg Cal. 1.8deg Cal. 2.4deg Cal. 2.8deg
7
391
0 0 0
50
100
50
100
150
z mm
150
z mm
Fig. 6. Radius of water flow (1.5%).
Fig. 4. Radius of water flow (0.75%).
Cal. 1.0deg Cal. 1.8deg Cal. 2.4deg Cal. 2.8deg
respectively. The radius of the water increases with distance, suggesting that a divergent channel is formed. The divergence angle increases with the increase in the inclination of the magnets. The calculated results are in good accordance with the experimental ones.
Exp.1.0deg Exp.1.8deg Exp. 2.4deg Exp. 2.8deg
6 5
5. Conclusions
r mm
4 3 2 1 0
0
50
100
150
z mm
(1) The virtual divergent channel can be formed with two magnets set face-to-face and inclined with each other. (2) The shape of the channel can be estimated analytically. (3) The divergence angle increases with the increase in the inclination of the magnets.
Fig. 5. Radius of water flow (1.0%).
0.4 T. The magnetic fluid is the hydrocarbon-based one, HC-50, and the magnetization is shown in Fig. 2. It is diluted to the concentration of 0.5–1.5%. The diamagnetic liquid is water. Fig. 3 shows an infrared photograph of the diverging flow of the water in the magnetic fluid. The flow is from left to right. The inclination of the magnets is 1.81, and the concentration of the magnetic fluid is 1.5%. The flow is vertically elongated to the axial direction.
4. Results The analytical calculation is made using the parameters of the experimental conditions. Figs. 4–6 show the analytical and experimental results for the concentration of the magnetic fluid, 0.75, 1.0 and 1.5%,
Acknowledgements Authors are deeply indebted to Mr. Y. Ishikawa and Mr. C. Basandash for their help in the experiment.
References [1] N. Fujisaki, et al., Trans. Japan Soc. Mech. Eng. B 63 (606) (1997) 417. [2] R. Yamane, et al., Proc. ASME/JSME Fluids Eng. Conf. (1999). [3] J. Mai, et al., Fluid Dyn. Res. (24) (1999) 147. [4] J. Mai, et al., Eur. J. Mech/Fluids (2) (2002) 237. [5] R. Yamane, et al., J. Magn. Magn. Mater. 252 (2002) 268. [6] R. Yamane, et al., Int. J. Appl. Electromagn. Mech. 19 (2004) 557.