Transient characteristics of convection and diffusion of oxygen gas in an open vertical cylinder under magnetizing and gravitational forces

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Abstract

A simplified model was derived for magnetizing force convection of oxygen gas and solved for transient characteristics of oxygen gas in a vertical open pipe. Oxygen gas in the cylindrical pipe initially flows out downward since oxygen gas is heavier than air. However, the magnetizing force works to attract the oxygen gas in the lower half of the vertical pipe, and the oxygen gas rises back to the central part of the pipe where the magnetic field is strongest. After a long time, all the oxygen gas in the pipe is replaced with air due to diffusion. This model represented moderately well the transient concentration of oxygen gas measured experimentally in a similar system.

Introduction

The recent development of a superconducting magnet appears to have marked the start of a new era of magnet applications in industrial processes. Typical developments in this area were reported at the 3rd International Symposium on Electro-magnetic Processing of Materials. In the plenary lecture of this meeting, Kitazawa, Hirota, Ikezoe, and Uetake (2000) described various phenomena of a steep gradient of magnetic field, i.e., magnetizing force effects, which the group had discovered (Ikezoe, Hirota, Nakagawa, & Kitazawa, 1998; Ikezoe et al., 1998a; Uetake, Nakagawa, Hirota, & Kitazawa, 1999; Nakagawa, Hirota, Kitazawa, & Shoda, 1999). Magnetizing force becomes dominant in a steep gradient of magnetic field expressed by the square of magnetic induction for a fluid with a high magnetic susceptibility. This force induces various interesting phenomena, such as lifting (or lowering) of a part of a liquid surface (Moses effect), microgravity effects, levitation of a liquid droplet in a gravity field, etc. Wakayama and coworkers (Wakayama 1991, Wakayama 1991a, Wakayama 1993; Wakayama, Ito, Kuroda, Fujita, & Ito, 1996; Bai, Yabe, Qi, & Wakayama, 1999) have also been actively experimenting with magnetizing force fields. Their findings include a jet-like nitrogen gas flow in a decreasing magnetic gradient in air, the so-called Wakayama jet, and the magnetic promotion of combustion in diffusion flames. Braithwaite, Beaugnon, and Tournier (1991) reported the enhancing effects of natural convection of paramagnetic fluid on Rayleigh–Benard convection. They reported enhanced or decreased heat transfer rates of natural convection. Ohgaki and Matsumura (1995) measured the effect of magnetic gradient on the diffusion and mixing of oxygen-containing gas.

All the above reports concern the magnetizing force induced by a steep gradient of a strong magnetic field. This force acts selectively on materials, such as oxygen gas, with a high magnetic susceptibility. This phenomenon was discovered by Faraday (1847), who reported that a bubble of oxygen gas is attracted to the center of a strong magnetic field. Pauling, Wood, and Sturdivant (1946) developed a paramagnetic oxygen analyzer based on this phenomenon. Knowledge of this magnetizing force thus has a long history, but its application has been rather neglected until the recent development of the superconducting magnet. A strong magnetic field allows diamagnetic material to be floated in a gravity field, and this is expected to be employed for processing new materials.

The present report aims to develop a model equation for the transient flow and diffusion characteristics of oxygen gas in a vertical open cylinder under a steep gradient of a magnetic field, i.e., magnetizing force. The transient decrease in the oxygen concentration in a pipe was computed and compared with the measured concentration in a similar experimental set-up.

Section snippets

The experimental results

To study the flow and diffusion characteristics of oxygen gas under a magnetizing force, the following simple system was adopted. A vertical pipe was filled with oxygen gas. At time zero the top and bottom caps are removed, and at intervals thereafter the transient concentration of oxygen gas was measured. Fig. 1 shows the schematics of the experimental apparatus. A glass pipe of 50mm inner diameter×600mm length was placed in the bore of a superconducting magnet (12T100mmφ), and a small amount

Derivation of model equation

The above system was subsequently studied numerically. There appear to be no simple model equations for such a system, and we therefore developed the following model equations for a magnetizing force field in a similar way to those for natural convection. Wakayama, Ito, Kuroda, Fujita, and Ito (1996) derived an equation for the magnetizing force of oxygen gas, which can be rewritten in terms of magnetic induction b asfm=μm2χO2(T)YO2ρ∇H2χO2ρ·YO2mb2,where the relationship b≒μmH was

Computed result

The model equations in Section 3 were numerically solved by a standard finite difference method (Hirt, Nichols, & Pomeco, 1975) for the system described in Fig. 3. The system considered for the computation is rather small in comparison with the experimental one of Fig. 1, since the experimental domain is extremely large and would require enormous amounts of time for computation. The system was assumed to be axially symmetrical. The system has a radius rp and an axial length ℓp, and ℓp/rp=10.

Conclusion

Based on the model for magnetizing force proposed by the Wakayama group, a magnetizing force term was included in a Navier–Stokes equation. The model equation was derived and numerically solved for the transient convection and diffusion of pure oxygen gas in a vertical pipe outward through the top and bottom openings. The transient characteristics of the oxygen concentration at the middle level agreed quite well with those of experimental measurements in a similar system. These results suggest

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