ISTP-16, 2005, PRAGUE 16TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA

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MAGNETIC EFFECT ON THE WATER MIST FLOW AND NUMERICAL SIMULATION FOR BROWNIAN MOTION

X. Wang*, H. Hirano**, T. Tagawa*, H. Ozoe*

* Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Japan ** Okayama University of Science, Japan

Corresponding author: ozoe@cm.kyushu-u.ac.jp, Tel: 81-92-5837834, Fax: 81-92-5837838

Keywords: water mist, magnetic field, numerical simulation, Brownian motion

Abstract In the present report, water mist flow in a

super-conducting magnet was studied both experimentally and numerically. The water mist was produced by ultrasonic atomizers and fed to the magnet through a Plexiglas pipe placed in the magnet. The magnet was inclined for φ = 0º (horizontal), 30º and 90º (vertical). For the cases of 0º and 30º, the water mist was stopped in the magnetic field at b = 10 T. For the vertical case 90º, the water mist fell down with a sandglass shape above the magnetic coil at b = 10 T. Numerical simulation was carried out for 1000 water droplets of 3μm. The Brownian motion was considered and Langevin equation was used. The numerical results show that for the cases of φ = 0º and 30º, most of the droplets make sedimentation on the pipe wall near the inlet of the pipe. At φ = 30º, only a few number of droplets go through the magnetic coil and their trajectories are like an arch due to the magnetic force at bc = 10.75 T. For the vertical case φ = 90º, the sandglass shape was computed at bc = 10.75 T.

1 Introduction

The strong magnetic field has been employed to seek for various new phenomena after a recent development of a super-conducting magnet. Most of the new findings are related to the fluid flow, levitation of materials, combustion and large particles [1-4]. There appears to be almost no previous works on the

effect of magnetic field on the behavior of micron-scaled particles. However we could expect extensive application of the magnetic field for treatment of powders and particles since they are widely employed in the practical industries. In the present work, the effect of a strong magnetic field on the water mist flow is studied. 2 Experiment Apparatus

Figure 1 shows the experimental schematics. The experimental apparatus consists of (1) a super-conducting magnet (maximum 10 Tesla), (2) a Plexiglas cylindrical pipe, and (3) ultrasonic atomizers to produce the water mist. The pipe is 1000 mm long and 90 mm in inner-diameter. According to Lang [5] and original to Rayleigh [6], the equation for particle size is given as follows:

2 1/3 0.34 0.34( 8 / )med cd fλ πσ ρ= = (1)

Where dmed is the mediate diameter of the particles, λc is the wavelength of the capillary waves, σ is the surface tension (72 mN/m (25 °C)), f is the applied ultrasonic frequency (1700 kHz used for calculation). Then,

2 .9 1m e dd mµ≅ (2)

The room temperature for the current experiment was about 25 Co .

X. Wang, H. Hirano, T. Tagawa, H. Ozoe

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3 Experimental Results

Figure 2 shows the experimental pictures at φ = 30º for (i) b = 0 T and (ii) b = 10 T. From left to right, they are (a) an end view, (b) a side view at the outlet and (c) at the inlet. Atφ = 30º, when there was no magnetic field, the water mist was fed from the inlet ((i) (c)) and flew out from the outlet ((i) (a) (b)). When a strong magnetic field was applied, most of the mist was stopped in the magnet and accumulated at the inlet of the pipe ((ii) (c)). However, as seen

in pictures (ii) (a) and (b), a little mist passed down and made sedimentation on the pipe bottom wall. The pipe wall was wet ((ii) (a)) due to the sedimentation of the mist.

4 Numerical Method

In the computation, due to the small size of water droplets, the Brownian motion was considered. In 1943, Chandrasekhar developed a mathematical model for Brownian motion [7], that is Langevin model. Then in 1980, Ermak and Buckholtz formulated this model for the numerical analysis using Monte Carlo method [8].

Figure 3 shows the geometrical configuration of the present model system with coordinates. In the present computation, the water mist was simulated with 1000 water droplets and they were scattered randomly in the upper half of the cylindrical pipe at the beginning. The sizes of the water droplets were assumed to be 3 mµ according to eqs. (1) and (2). The sample magnitude of the cylindrical pipe is 0.45m in length (l) and 0.045m in radius (r0). An electric coil with rc=0.09m in radius was set coaxially at the middle of the pipe to produce the magnetic field in stead of the practical multi-layer coils of the real magnet. The air was assumed to be static at 1 atm and not to be affected by the water droplet motion due to the dilute number of water droplets. Table 1 shows the physical properties of water and air. In this report, the parameters with subscript f stand for surrounding fluid, and for a water droplet, there is no subscript.

Fig. 1. Schematic view of the experimental apparatus.

Fig. 3. Computational model system.

Fig. 2. Experimental pictures at φ = 30º for (i) b= 0 T and (ii) b= 10 T. They are (a) an end view, (b) a side view at the outlet and (c) at the inlet.

(a) (b) (c)

(i)

(ii)

MAGNETIC EFFECT ON THE WATER MIST FLOW AND NUMERICAL SIMULATION FOR BROWNIAN MOTION

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Water Air

3 [ / ]m kgχ 99.05 10−− × 73.0 10−× 3 [ / ]kg mρ 29.97 10× 1.185

[ /( )]f kg m sµ ⋅ - 51.835 10−× 2 [ / ]f m sα - 52.218 10−×

The nondimensional Langevin equation for the motion of Brownian particles is as follows [8]: The velocity components of a water droplet in the R, ϕ and Z directions:

0 exp 1 expi i i uiU U StF LSt Stτ τ∆ ∆

= − + − − +

(3)

The location of a water droplet displacement in the R and Z directions:

0 0 1 exp

1 exp

i i i

i xi

X X StUSt

StF St LSt

τ

ττ

∆= + − −

∆+ ∆ − − − +

(4)

The location of a water droplet in the circumferential ϕ direction:

0

00

0

1 exp

1 exp

R

x

R

StUStX X

X

StF St LSt

X

ϕ

ϕ ϕ

ϕ ϕ

τ

ττ

∆− −

= + +

∆∆ − − − +

(5)

uiL and xiL are the random variable vectors to be satisfied with the following relations [8]:

( )00expui iL A d

Stτ τ τ

τ τ τ∆ ′∆ − ′ ′= − +

∫ (6)

( )001 expxi iL St A d

Stτ τ τ

τ τ τ∆ ′∆ − ′ ′= − − +

∫ (7)

0ui xiL L< >=< >= (8)

2

23 1 expaui xi

a

L L StStmu

κθ τ∆< ⋅ >= − −

(9)

22

23 1 expa

ui

a

LStmu

κθ τ∆< >= − −

(10)

2 223

22 3 4exp exp

axi

a

L Stmu

St St St

κθ

τ τ τ

< >=

∆ ∆ ∆− + − − −

(11)

Here, subscript i stands for three components in the R, ϕ and Z direction. The forces in the R-, ϕ - and Z- directions are as follows,

( )2

1 cos sin

12

R f

ff

f

F G X

G BR

ϕρ φ

γρ

χ

= −

∂+ −

∂

(12)

( )1 cos cosfF G Xϕ ϕρ φ= − (13)

( )21

1 sin2f

Z f ff

G BF G

Zγ

ρ φ ρχ

∂= − + −

∂

(14)

3

14

R dSB

Rπ×

= − ∫rr

r (15)

5 Numerical Results Figure 4 shows various vector fields for the vertical case ( φ = 90º) on the R-Z plane. (a) Magnetic field B

r, (b) 2B∇ , (c) magnetic force

vectors on a water droplet ( ) 21/ / 2f f fG Bγ χ ρ− ∇ and (d) gravitational +

magnetic force vectors on a water droplet ( ) ( ) 21 1/ / 2f f f fG Z G Bρ γ χ ρ− ∇ + − ∇ at

1000fγ = , 33.15fχ = − , 31.19 10fρ −= × and 61.8 10G = × . The dimensional magnetic

induction at the center of the coil is bc=10.75 Tesla for 1000fγ = . The magnetic force is effective only in a small region about Z= (-1.5, 1.5) as shown in Fig. 4 (c). The total force of magnetic and gravitational forces has the

Table 1. Physical properties (1 atm and 300 K) of water droplet and air.

X. Wang, H. Hirano, T. Tagawa, H. Ozoe

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smallest value at about Z= (0.8, 1.2) as shown in Fig. 4 (d). That means in this region, the magnetic and gravitational forces balance each other mostly. Figure 5 (a) shows the distribution of 3 mµ water droplets for the horizontal caseφ = 0º. (a) is for no magnetic field at 0fγ = and

15τ = (t=1372.5s), and (b) shows a series of transient distribution of 3 mµ water droplets during 15τ = at 1000fγ = . The droplets make sedimentation on the right-hand side of the pipe wall at 0fγ = . At (b) 1000fγ = , although the droplets fall down, some of them near the magnetic coil are repelled away from the magnetic coil with time proceeding. Figures 6 and 7 show the corresponding results for the inclined angle φ = 30º and vertical caseφ = 90º. At 30φ = o , all the droplets make sedimentation when there is no magnetic fields (Fig.6 (a)). At

1000fγ = (Fig.6 (b)), most of the droplets make sedimentation on the right-hand side of the pipe wall and a small number of droplets goes

through the coil to make sedimentation on the left-hand side of the pipe wall. The trajectories of these droplets are deformed due to the combined forces of gravity and magnetic field. For the vertical case, as shown in Fig. 7, almost all the droplets fall down out of the bottom opening of the pipe at 15τ = when there is no magnetic field (Fig.7 (a)). But at (b) 1000fγ = , the water droplets fall down slowly above the magnetic coil and the bulk area of the droplets becomes as a sandglass shape due to the strong radial repelling magnetic force toward the center axis as shown in Fig.4 (c) and (d).

g

(a) (b) (c) (d) -5

-4

-3

-2

-1

0

1

2

3

4

5

-5

-4

-3

-2

-1

0

1

2

3

4

5

Z

R

Fig. 4. Computed vector fields at φ = 90º on R-Z plane. (a) B

r, (b) 2B∇ , (c)

( ) 21/ / 2f f fG Bγ χ ρ− ∇ and

(d) ( ) ( ) 21 1/ / 2f f f fG Z G Bρ γ χ ρ− ∇ + − ∇ at

1000fγ = , 33.15fχ = − , 31.19 10fρ −= ×

and 61.8 10G = × . 0τ =

1.5τ =

3τ =

15τ =

0,fγ = 15τ =

Fig. 5. The distribution of 3 mµ droplets at 0φ = o . (a) No magnetic field at 0fγ = and 15τ = , and (b) the transient distribution during 15τ = at 1000fγ = .

(a)

(b)

MAGNETIC EFFECT ON THE WATER MIST FLOW AND NUMERICAL SIMULATION FOR BROWNIAN MOTION

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6 Conclusions

The water mist flow in a super-conducting magnet was studied in the present work. The repulsive magnetic force on the water mist was observed clearly with inclined angle φ = 0º, 30º and 90º. The water mist was stopped in the middle part of the strong magnetic field (10 T) at φ = 30º. In the numerical simulation, the trajectories of 3 mµ -sized water droplets were computed. The trajectories of droplets are deformed when they pass through the magnetic coil. The sandglass shape of the water droplets is computed for the vertical case.

Nomenclature Ar

non-dimensional random force br

magnetic field, T ba /m c ai xµ= , T d diameter of water droplet, m F nondimensional force on a water droplet G Galilei number, 3 2/ fax g α= l The length of the cylindrical pipe, m L nondimensional random variable r0 the radius of the cylindrical pipe, m St Stokes number, 22 /(18 )f f ad xρα µ= U nondimensional velocity of a water droplet X nondimensional position of a water droplet xa reference value = r0 , m

fα thermal diffusivity of fluid, m2/s

fγ magnetic strength, 2 /( )f a m ab gxχ µ

fµ viscosity of air, Pa·s

mµ permeability of air, H/m ρ density of a water droplet, kg/m3

0τ =

1.5τ =

3τ =

15τ =

0,fγ = 15τ =

(a)

(b)

Fig. 6. The distribution of 3 mµ droplets at 30φ = o . (a) No magnetic field at

0fγ = and 15τ = , and (b) the transient distribution during 15τ = at 1000fγ = .

0τ = 1.5τ = 3τ = 15τ = 0,fγ = 15τ =

(a) (b)

Fig. 7. The distribution of 3 mµ droplets at 90φ = o . (a) No magnetic field at

0fγ = and 15τ = , and (b) the transient distribution during 15τ = at 1000fγ = .

X. Wang, H. Hirano, T. Tagawa, H. Ozoe

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fρ density of air, kg/m3

fρ /fρ ρ χ magnetic susceptibility of water, m3/kg

fχ magnetic susceptibility of air, m3/kg

fχ /fχ χ τ nondimensional time

τ∆ nondimensional time step φ inclined angle of the magnetic field Subscripts a reference value c related to the center of the coil f surrounding fluid i,j components of vector R a component in the radial direction u related to the velocity of a water droplet x related to the displacement of a water droplet Z a component in the axial direction ϕ a component in the circumferential direction 0 initial value References [1] Beaugnon S E and Tournier R. Levitation

of organic materials. Nature, Vol. 349, pp 470-470, 1991.

[2] Berry M V and Geim A K. Of flying frogs and levitrons. Eur. J. Phys., Vol. 18, pp 307-313, 1997.

[3] Shigemitsu R, Tagawa T and Ozoe H. Numerical computation for natural convection of air in a cubic enclosure under combination of magnetizing and gravitational forces. Numerical Heat Transfer, Part A, Vol. 43, pp 449-463, 2003.

[4] Wakayama N I. Behavior of flow under gradient magnetic fields. J. Appl. Phys., Vol. 69, No. 4, pp 2734-2736, 1991.

[5] Lang R J. Ultrasonic atomization of liquids. J. Acoust. Soc. Am., Vol. 34, pp 6-8, 1962.

[6] Rayleigh L. Theory of Sound. Dover press, Chap.20, 1898.

[7] Chandrasekhar S. Stochastic problems in physics and astronomy. Rev. Mod. Phys., Vol. 15, pp 1-89, 1943.

[8] Ermak D L and Buckholtz H. Numerical integration of the Langevin Equation: Monte Carlo simulation. J. Comput. Phys., Vol. 35, pp 169-182, 1980.