Metal Flow in a Packed Bed under Magnetic Field Imposition

Abstract

A liquid metal flow suppression function of a static magnetic field has been theoretically and experimentally investigated, and this function has been utilized as an electromagnetic brake in a continuous casting process of steel. Because macro segregation is induced by macro scale flow, this function must be a powerful tool for the suppression of the macro segregation. However, effect of the static magnetic field on the flow in a liquid-solid coexisting region has not been clarified until now. In this study, for clarification of the magnetic field effect on the suppression of the liquid metal flow in the liquid-solid coexisting region, a model experiment using a packed bed composed of a liquid tin and solid particles made of copper or alumina has been done. The Reynolds number decreased and the friction factor increased by imposing the static magnetic field in the experiment. This was remarkable when the solid particles in the packed bed were the copper rather than the alumina. Furthermore, an equation which can predict the relation between the friction factor and the Reynolds number in the liquid-solid coexisting region under the static magnetic field imposition has been derived based on the Ergun equation and the Hartmann flow analysis under the condition that the solid particles were perfectly conducting material or insulating material.

1. Introduction

Suppression of macro segregation is essential for high quality steel production because chemical composition deviation from the designed chemical composition of an alloy decreases corrosion resistance, mechanical properties and so on. For instance, center segregation of a continuously cast slab¹⁾ induces hydrogen induced cracking,²⁾ internal cracking. Thus, a lot of investigations on the macro segregation have been done until now. The macro segregation is classified into V-segregation, inverse V-segregation, freckle, center segregation and so on. These macro segregations form in liquid-solid coexisting region, and it has been said that the flow induced by shrinkage, gravity contributes the macro segregation formation.³⁾ The flow caused by the liquid density change during solidification induces the V-segregation, the inverse V-segregation and the freckle.⁴⁾ Suzuki et al. artificially formed the inverse V-segregation in a steel ingot, and they found that the inverse V-segregation formed when the product of solidification speed to the power of 1.1 and cooling rate was lower than a certain value.⁵⁾ Yamada and his co-workers have done a lot of investigations on the macro segregation using the steel ingots. They found that size and region of the defect formed in the ingot center had a correlation with the ratio of the height of the ingot to its width.⁶⁾ They also found that the starting point of the inverse V-segregation depended not on the density of the solute concentrated liquid but on the secondary dendrite arm spacing, and the segregation started when the secondary dendrite arm spacing became a critical value.⁷⁾ Furthermore, they showed that the inverse V-segregation formed in a 12Cr steel decreased by Ta, Nb addition because of the refine structure.⁸⁾ Beckermann et al. clarified that the freckle formed in a single-crystal nickel-base superalloy if the Rayleigh number which is an index whether a natural convection occurs or not was over a critical value.⁹⁾

Relating to the continuously cast slab, effect of solidification shrinkage and/or bulging on the center segregation have been investigated.^10,11,12) Miyazawa et.al. numerically simulated the center segregation of the cast slab under consideration of the solidification shrinkage and the bulging.¹⁰⁾ Kajitani el.al. numerically found out that the solute distribution observed in the industrial cast slab agreed with the numerically obtained one only when both the solidification shrinkage and the bulging were considered in the numerical simulation.¹¹⁾ Numerical investigation by Murao et. al. focused on the relation between the center segregation and the bridging.¹²⁾ That is, negative segregation was observed above the bridging region while positive segregation was observed below the bridging region when the bridging was artificially formed in their numerical calculation. This agreed with the segregation distribution in the cast slab. And their estimated mechanism of the solute distribution was that solute concentrated liquid flows into the negative pressure region which is formed below the bridging region through its side where solid fraction is relatively low. A lot of methods to suppress the center segregation have been proposed until now. Soft reduction technique in which decrease of the cast slab volume due to the phase change is compensated by mechanical compression of the cast slab is one of the solutions to reduce the center segregation.^13,14) Kajitani el.al. pointed out from their numerical results¹¹⁾ that the optimum condition of the soft reduction might depend on the amount of the bulging. Furthermore, shorten of roll pitch,¹⁵⁾ non-periodical pitch of the rolls,¹⁶⁾ intensification of secondary cooling of the cast slab, maintenance of the rolls and so on have been proposed.

For prediction of the macro-segregation, evaluation of permeability in the liquid-solid coexisting region is required, and thus many investigations have been done until now.^{17,18,19,20,21,22,23,24)} For example, Takahashi et al. measured the permeability of a steel¹⁷⁾ and Natsume et al. numerically evaluated the permeability.^18,19) Based on these results, numerical prediction of the macro-segregation for the continuously cast steel has been done.²⁵⁾ However, a new method for the liquid metal flow suppression in the liquid-solid coexisting region has not been proposed though it is essential for decrease of the macro segregation.

Applications of a static magnetic field to a high temperature process have been investigated until now.^{26,27,28,29,30,31,32)} One of their advantages is that it can suppress the liquid metal motion surrounded by solid phase under non-contacting operation. Thus, this function has been industrially realized as an electromagnetic brake in a continuous casting process of the steel. Moderate flow in a mold by imposing the static magnetic field prevents breakout, entrapment of inclusions and so on. The braking function of the static magnetic field for a laminar flow of an electrically conductive liquid in a pipe is called the Hartmann flow.³³⁾ This flow has been theoretically analyzed for an electrically insulative pipe and for a perfectly conductive pipe. Since the braking function must be valid for the liquid metal flow in a liquid-solid coexisting region during the solidification, it is a promising candidate for preventing the macro segregation. However, the braking function for the flow in the liquid-solid coexisting region has not been investigated until now. A flow in a packed bed is similar with that in the liquid-solid coexisting region when equi-axed solids are dispersed in the liquid. If an alloy is adopted as a model material for the experimental simulation of the flow in the liquid-solid coexisting region, its temperature in the experimental apparatus should be precisely controlled to prevent fluctuations such as the solid fraction change and non-uniform solid shapes. The packed bed is attractive from this viewpoint. The flow in the packed bed is often expressed by the Kozeny-Carman equation, the Ergun equation and so on.^34,35,36) These equations give the relation between friction factor and liquid velocity (or Reynolds number) in the packed bed as a function of its configuration and physical properties of the liquid. However, the effect of the magnetic field on their relation has not been clarified until now. Thus, for clarification of the magnetic field effect on the velocity decrease in the liquid-solid coexisting region and on the relation between the friction factor and the liquid velocity, experimental work using the packed bed and theoretical work based on the Ergun equation and the Hartmann flow have been done in this study.

2. Experiments

2.1. Experimental Apparatus and Procedure

The flow phenomena during the solidification of an alloy under the magnetic field imposition must exist between the flow phenomena with a perfectly conductive solid phase and the flow phenomena with an insulative solid phase because of electrical conductivity difference between the liquid phase and the solid phase. Thus, copper and alumina (Al₂O₃) were adopted as the solid phase in the packed bed, and tin was chosen as a model liquid metal because of its low melting point and easiness of treatment. The electrical conductivity of the solid copper is completely larger than that of the liquid tin while the solid alumina is considered as an electrically insulating material as shown in Table 1. Spherical particles with a diameter, D_p of 2 mm, made of the alumina or the copper were chosen as solid particles in the packed bed. A ceramic tube of a 100 mm length, a 13 mm inner diameter was filled with the solid particles, and its both ends were covered by non-magnetic stainless steel wire meshes with mesh size of 20 mesh. This was used as the packed bed. Porosity in the packed bed, ε was calculated using the following equation.   

$$\epsilon =1-{\displaystyle \frac{V_p}{{\displaystyle \frac{\pi }{4}} D_b^2 L_b}}$$(1)

Here D_b is inner diameter of the packed bed, L_b is length of the packed bed and V_p is total volume of the solid particles in the packed bed, respectively.

Table 1. Electrical conductivity of materials used in experiment at 573 K.

	Sn	Cu	Al2O3
Electrical conductivity [S/m]	2.04 × 106	2.78 × 107	3 × 10−14

The porosities under no magnetic field condition in the cases of the alumina particles and the copper particles were 0.422 and 0.419, respectively.

The experimental apparatus is shown in Fig. 1. A metal reservoir with an inner diameter of 90 mm had an overflow port at its upper side to keep the liquid metal height constant during each run. The lower part of the metal reservoir was connected with inlet of the packed bed through an upper tube of a 220 mm length. The outlet of the packed bed was connected with a 40 mm length lower tube. The inner diameter of the upper and lower tubes was 9 mm. A metal receiver with an inner diameter of 50 mm was placed below the outlet of the lower tube to receive a flowing liquid metal. A scale of 10 mm interval was set on its inner wall for measuring the liquid metal height by a video camera.

Fig. 1.

Experimental apparatus. (Online version in color.)

The experimental procedure was as follows. The metal reservoir, the packed bed and the upper and lower tubes were heated and kept at 573 K under argon atmosphere to prevent the liquid metal from oxidition and solidifcation. The outlet of the lower tube was closed by a stopper. Then the liquid metal was poured into the metal reservoir. After the metal height in it reached a set value, the stopper was released. And the metal height in the receiver was recorded using the video camera. During each run, the metal height in the reservoir was kept constant by continuous pouring of the liquid metal, and a certain magnitude of a transeverse magnmetic field was imposed on the flowing liquid metal in the packed bed.

Distribution of the magnetic field along the metal flow direction is shown in Fig. 2. Vertical axis in this figure indicates the non-dimensional magnetic field magnitude nolmarized by the average in the packed bed region. The non-dimensional magnetic field magnitude at the center of the packed bed was 1.03 while that at the inlet and the outlet of the packed bed was 0.92. Thus, the magnetic field is roughly considered to be uniform in the packed bed.

Fig. 2.

Distribution of non-dimensional magnetic field.

2.2. Velocity and Friction Factor without Magnetic Field

The rising velocities of the metal height in the receiver without the magnetic field in the cases of the alumina particles and the copper particles are shown in Fig. 3. The rising velocities in the both cases fluctuated until about 30 seconds from the stopper release and then they became almost constant. Thus, the average rising velocity after 30 seconds from the stopper release was used for calculation of the superficial velocity in the packed bed, u_s.

Fig. 3.

Rising velocity of metal height in receiver.

The friction factor for the packed bed, f and the Reynolds number for the packed bed, Re. are defined in this paper as   

$$f=\frac{\mathrm{\Delta }{P}_b}{\rho u_s^2}\frac{D_p}{L_b}\frac{{ϵ}^3}{(1-ϵ)}$$(2)

  

$$Re=\frac{\rho u_s{D}_p}{\eta (1-ϵ)}$$(3)

where ΔP_b is pressure drop in the packed bed, η is viscosity and ρ is density, respectively.

The Ergun equation³⁴⁾ is composed of pressure drop, inertial and viscous terms. It relates the friction factor for the packed bed with the Reynolds number for the packed bed.   

$$f=\frac{150}{Re}+1.75$$(4)

The friction factor for the packed bed was calculated using two methods. In the first one, the friction factor was evaluated using Eqs. (3) and (4). In the second one, the assumption that the gravitational energy of the metal in the reservoir was equivalent with the pressure drop in the packed bed was adopted.   

$$\mathrm{\Delta }{P}_b=\rho g\mathrm{\Delta }z$$(5)

Here g is gravitational accerelation and Δz is the relative height between the free surface of the metal in the reservoir and the outlet of the packed bed, respectively.

The friction factor for the packed bed was calculated using Eqs. (2), (3) and (5). The friction factor for the packed bed obtained by the two methods are shown in Table 2. The values calculated by the two methods agree each other. Thus, the assumption adopted in the second method was valid in this experiment.

Table 2. Friction factor for packed bed evaluated (a) using Ergun equation and (b) using assumption that gravitational energy is equivalent with pressure drop.

Solid particles in packed bed	Friction factor for packed bed [−]
	(a)	(b)
Al2O3	2.02	2.10
Cu	1.99	1.97

The ratio of the momentum energy to the gravitational energy at the outlet of the packed bed, r defined by Eq. (6) was evaluated to confirm the validity of the assumption under the magnetic field impositoin.   

$$r={\displaystyle \frac{{\displaystyle \frac{1}{2}}\rho u_r^2}{\rho g\mathrm{\Delta }z}}$$(6)

  

$$u_r=\frac{1}{ϵ}\frac{L_r}{L_b}{u}_b=\frac{1}{ϵ}C{u}_s$$(7)

  

$$C=\frac{L_r}{L_b}=1.58$$(8)

where C is the ratio of the real flow path length in the packed bed, L_r to the packed bed length, and u_r is the average of the real velocity in the packed bed, respectively.

The ratio, C was set to 1.58. This value is usually adopted in the Kozeny-Carman equation.³⁷⁾ The ratio, r was less than 0.0097 in the both cases of the alumina particles and the copper particles. That is, the momentum energy at the outlet of the packed bed was less than 1% of the gravitational energy and almost of the residual energy loosed as thermal energy through the viscous dissipation in the packed bed. Imposition of the magnetic field must decrease the ratio, r because it causes the Joule dissipation. Therefore, the assumption that the gravitational energy of the metal in the reservoir is equivalent with the pressure drop in the packed bed is valid in all the experimental conditions in this paper.

2.3. Velocity and Friction Factor with Magnetic Field

Non-dimensional electromagnetic force magnitude, Hartmann number is square root of the ratio of an electromagnetic force to a viscous force acting on an electrically conductive liquid flowing in the magnetic flux density, B.³¹⁾ Its square for the packed bed is defined by the following equation in this paper.   

$$H{a}^2=\frac{\sigma B^2}{\eta }\frac{D_p^2{ϵ}^2}{{(1-ϵ)}^2}$$(9)

Here, Ha is the Hartmann number for the packed bed and σ is electrical conductivity of the liquid, respectively.

The Hartmann number for the packed bed and the porosity in the magnetic field imposition experiment are summarized in Table 3. The Hartmann number was ranged between 0 and 15.6. The porosity was adjusted to be 0.42 under all the experimental conditions. The friction factor for the packed bed was evaluated using the assumption mentioned in the previous section (Eq. (5)) and the superficial velocity calculated from the rising velocity of the metal height in the receiver, and the result is plotted as a function of the Reynolds number for the packed bed as shown in Fig. 4. The Ergun equation (Eq. (4)) which expresses the relation between the friction factor and the Reynolds number for the packed bed, is indicated by the solid curve in this figure. The relation between the friction factor and the Reynolds number for the packed bed under the imposition of the magnetic field deviated from the Ergun equation. As increase in the Hartmann number, the friction factor increased while the Reynolds number decreased. This tendency was independent of the solid material in the packed bed though degree of the deviation depended on the solid material.

Table 3. Hartmann number and porosity.

solid material	porosity/−	Hartmann number
Al2O3	0.422	0.0
	0.422	5.2
	0.422	10.4
	0.422	15.6
Cu	0.419	0.0
	0.419	5.1
	0.421	10.3
	0.421	15.5

Fig. 4.

Friction factor for the packed bed as a function of Reynolds number.

The ratio of the superficial velocity in the packed bed with the magnetic field, u_sB to that without the magnetic field, u_s0 is shown in Fig. 5 as a function of square of the Hartmann number. The ratio was smaller than unity and it decreased with increase in the Hartmann number. This is the braking effect of the magnetic field. The braking effect in the case of the copper particles was larger than that in the case of the alumina particles under the same Hartmann number condition. In the case of the copper particles, the eddy current induced in the liquid tin must return through the solid copper particles under an ideal condition because of their large electrical conductivity. Thus, the electromagnetic force acting on the liquid tin is mainly suppression force as shown in Fig. 6(a). On the other hand, in the case of the alumina particles, the eddy current circulates in the tin because of their electrically insulative nature as shown in Fig. 6(b). Therefore, the electromagnetic force acts as a suppression force in some area while it acts as an acceleration force in other area. This is the reason why the braking effect in the case of the alumina particles was smaller than that of copper particles under the same Hartmann number condition.

Fig. 5.

Ratio of superficial velocities as a function of square of Hartmann number.

Fig. 6.

Eddy current circuit in the cases of copper particles and alumina particles.

3. Theoretical Evaluation of Velocity with Magnetic Field

3.1. Pressure Drop in Packed Bed

The pressure drop is balanced with the electromagnetic force induced by the magnetic field imposition in ideal Hartmann flow. For the ideal Hartmann flow of an incompressible fluid flowing in a circular pipe of an inner radius, a and with an insulating wall or with a perfectly conducting wall, the relation between the average velocity, U and the pressure gradient, dP/dx was theoretically analyzed under the assumptions of uniform transverse magnetic field, steady state, laminar flow, uniformity in the flow direction, constant physical properties and large Hartmann number.³⁸⁾ The approximate solutions in the cases of the insulating wall and the perfectly conducting wall are expressed by Eqs. (10) and (11), respectively.   

$$\frac{dP}{dx}\approx -1.2247\frac{{\eta }^2}{\rho a^3}{H}_{p}R{e}_p$$(10)

  

$$\frac{dP}{dx}\approx -\frac{{\eta }^2}{ \rho a^3}{H}_p^{2}R{e}_p$$(11)

  

$$H_p=aB\sqrt{\frac{\sigma }{\eta }}\gg 1$$(12)

  

$$R{e}_p=\frac{\rho Ua}{\eta }$$(13)

Here H_p is the Hartmann number for the pipe, Re_p is the Reynolds number for the pipe flow, respectively.

These equations were modified for evaluation of the pressure drop in the packed bed with the magnetic field under the assumption that the packed bed was a bundle of thin flow paths which was adopted for derivation of Kozeny-Carman equation. And braking efficiency of the magnetic field, b was introduced as a parameter, because parallel component of the flow to the magnetic field does not contribute to the braking effect, and because magnitude and path of the eddy current in the packed bed must depend on the distribution and material of the solid particles. As a result, the pressure drop in the packed bed caused by the magnetic field imposition was derived as   

$$\frac{\mathrm{\Delta }{P}_b}{L_b}=\gamma ERe$$(14)

  

$$E=\frac{{\eta }^2{(1-ϵ)}^3}{\rho D_p^3{ϵ}^3}$$(15)

  

$$\gamma =\{\begin{array}{l}{\gamma }_i,\mathrm{\hspace{0.17em} }\text{for}\mathrm{\hspace{0.17em} }\text{insulating}\mathrm{\hspace{0.17em} }\text{particles}\\ {\gamma }_c,\mathrm{\hspace{0.17em} }\text{for}\mathrm{\hspace{0.17em} }\text{conducting}\mathrm{\hspace{0.17em} }\text{particles}\end{array}$$(16)

  

$${\gamma }_c=b{C}^{2}H{a}^2$$(17)

  

$${\gamma }_i=\frac{3}{0.8165}b{C}^{2}Ha$$(18)

where E is constant defined by Eq. (15), γ, γ_c, γ_i are parameters defined by Eqs. (16), (17) and (18), respectively.

The pressure drop is proportional to the Reynolds number for the packed bed. In the case of the insulating particles, it is proportional to the Hartmann number while it is proportional to the square of the Hartmann number in the case of the perfectly conducting particles.

3.2. Relation between Velocity and Friction Factor

The pressure drop in the packed bed is expressed as a function of the Reynolds number for the packed bed without the magnetic field, Re₀ using Eqs. (2), (3) and (4).   

$$\frac{1}{E}\frac{\mathrm{\Delta }{P}_b}{L_b}=1.75R{e}_0^2+150R{e}_0$$(19)

For evaluation of the velocity decrease in the packed bed by imposing the magnetic field, the electromagnetic force term expressed by Eq. (14) was added to Eq. (19), and it was rewritten as a function of the Reynolds number for the packed bed with the magnetic field, Re_B.   

$$\frac{1}{E}\frac{\mathrm{\Delta }{P}_b}{L_b}=1.75R{e}_B^2+150R{e}_B+\gamma R{e}_B$$(20)

This is the modified Ergun equation under the magnetic field imposition. Ratio of the Reynolds number with the magnetic field to that without the magnetic field, X was easily derived under the assumption that the pressure drops with and without the magnetic filed imposition were same as mentioned in section 2.2.   

$$X\equiv \frac{R{e}_B}{R{e}_0}=\frac{u_{sB}}{u_{s0}}=\frac{1}{2}\left[-(p+q)+\sqrt{{(p+q)}^2+4(1+p)}\right]$$(21)

  

$$p=\frac{150}{1.75R{e}_0}$$(22)

  

$$q=\frac{\gamma }{1.75R{e}_0}$$(23)

Here, the non-dimensional number, p indicates the ratio of the viscous force to the inertial force while the non-dimensional number, q indicates the ratio of the electromagnetic force to the inertial force, respectively.

From Eq. (21), the velocity decrease by imposing the magnetic field can be estimated. Then, the friction factor under the magnetic field imposition can be estimated from Eq. (2) because the pressure drop is easily calculated using Eq. (19). That is, the relation between the friction factor and the superficial velocity (the Reynolds number for the packed bed) can be obtained under the magnetic field imposition.

3.3. Comparison of Theoretical and Experimental Results

The ratio of the Reynolds numbers, X evaluated by the theoretical method mentioned in the previous section and that evaluated by using the experimental data are shown as a function of square of the Hartmann number for the packed bed in Fig. 7, in which the braking efficiency, b was set to 1.2 to fit the theoretically calculated values with the experimental ones. The calculated values agree with the experimental ones except in the case of the copper particles with the Hartmann number of 5.1. One of the possibilities of this discrepancy is that the Hartmann number is not so large to satisfy the assumption mentioned in the section 3.1. The magnetic field gradient in the flowing direction in the experiment might also induce the error in the theoretical estimation. However, the reason of the discrepancy is not clear. Clarification of the effect of the magnetic field gradient and determination of the braking efficiency are future work.

Fig. 7.

Comparison of Reynolds numbers ratios in experiment and in theory.

3.4. Theoretical Prediction of Braking Effect

The ratio of the Reynolds number with the magnetic field to that without the magnetic field was calculated as a function of the Hartmann number under the condition that the Reynolds number for the packed bed without the magnetic field was 1 and 100. In the both cases the braking efficiency, b was set to 1.0. The result is shown in Fig. 8. The ratio of the Reynolds numbers decreases as increase in the Hartmann number. In the large Hartmann number region, the Reynolds number ratio for the conducting particles is smaller than that for the insulating particles. This is a reasonable result because the braking effect of the Hartmann flow with the conducting wall is larger than that with the insulating wall. However, the Reynolds number ratio in the case of the conducting particles is larger than that in the case of the insulating particles in the region where the Hartmann number is less than 3.7. This might be caused by inaccuracy of Eqs. (10) and (11) in the relatively small Hartmann number region. The braking effect in the case that Reynolds number is 1 is effective in comparison with that in the case that Reynolds number is 100 under the same Hartmann number condition. Therefore, the braking effect is remarkable for a slow flow with the conducting wall. The actual phenomena must exist between the curve for the conductive particles and that for the insulating particles. The Hartmann number for the steel is calculated at 20.6 under the condition that the porosity is 0.4, the spherical solid particles diameter is 3 mm and the magnetic field magnitude is 0.3 T. Under this Hartmann number condition, the relative velocity for the insulating particles is 0.45 and that for the conducting particles is 0.13, when the Reynolds number is 1. Thus, the velocity suppression by the magnetic field imposition is expected in some cases while this function depends on the size of the grains and the magnetic field magnitude. This figure is useful for the macro segregation decrease prediction when the relation between the velocity decrease and the macro segregation is clarified.

Fig. 8.

Ratio of Reynolds numbers with to without magnetic field as a function of Hartmann number.

4. Conclusions

In this study, the relation between the superficial velocity and the friction factor in the packed bed under the imposition of the magnetic field has been experimentally and theoretically evaluated. And the experimental results were compared with the theoretically derived ones.

The mainly obtained results are as follows.

• Friction factor for the packed bed increased and the Reynolds number (the superficial velocity) decreased by imposing the magnetic field.

• Decrease of the superficial velocity by imposing the magnetic field was remarkable when the solid particles in the packed bed were conducting rather than insulating.

• The equation for prediction of the relation between the Reynolds number (the superficial velocity) and the friction factor in the packed bed under the imposition of the magnetic field has been proposed.

Acknowledgements

A part of this research was financially supported by Grant-in-Aid for Scientific Research (B) 18H01758, and “3D/4D analysis of segregation and defects in solidification process” research group” in the Iron and Steel Institute of Japan.

Nomenclatures

a [m]: inner radius of pipe

B [T]: magnetic flux density

b [−]: braking efficiency by magnetic field

C [−]: length ratio defined by Eq. (8)

D_b [m]: inner diameter of packed bed

D_p [m]: diameter of solid particle

E [−]: constant defined by Eq. (15)

f [−]: friction factor for packed bed

g [m/s²]: gravitational acceleration

Ha [−]: Hartmann number for packed bed

H_p [−]: Hartmann number for pipe

L_b [m]: length of packed bed

L_r [m]: real flow path length in packed bed

P [Pa]: pressure

p [−]: non-dimensional number defined by Eq. (22)

q [−]: non-dimensional number defined by Eq. (23)

r [−]: ratio of momentum energy to gravitational energy

Re [−]: Reynolds number for packed bed

Re₀ [−]: Reynolds number for packed bed without magnetic field

Re_B [−]: Reynolds number for packed bed with magnetic field

Re_p [−]: Reynolds number for pipe

U [m/s]: average velocity in pipe

u_r [m/s]: average of real velocity in packed bed

u_s [m/s]: superficial velocity in packed bed

u_s0 [m/s]: superficial velocity in packed bed without magnetic field

u_sB [m/s]: superficial velocity in packed bed with magnetic field

V_p [m³]: total volume of solid particles in packed bed

X [−]: ratio of the Reynolds numbers with to without magnetic field

x [m]: axial direction of pipe

γ [−]: parameter defined by Eq. (16)

γ_c [−]: parameter defined by Eq. (17)

γ_i [−]: parameter defined by Eq. (18)

ΔP_b [Pa]: pressure drop in packed bed

Δz [m]: relative height between metal free surface in reservoir and outlet of packed bed

ε [−]: porosity in packed bed

η [Pa.s]: viscosity of liquid metal

ρ [kg/m³]: density of liquid metal

σ [S/m]: electrical conductivity of liquid

References

• 1)   T.  Kawawa,  H.  Sato,  S.  Miyahara,  T.  Koyano and  H.  Nemoto: Tetsu-to-Hagané, 60 (1974), 486 (in Japanese).
• 2)   K.  Murakami,  Y.  Tsuchida,  Y.  Kobayashi,  M.  Nakada and  S.  Endo: Tetsu-to-Hagané, 85 (1999), 301 (in Japanese).
• 3)   M. C.  Flemings: ISIJ Int., 40 (2000), 833.
• 4)  Solidification of Steel, ed. by Research Group of Solidification Process, The 19th Committee (Steelmaking) of the Japan Society for the Promotion of Science, Shokadoh Book Sellers, Kyoto, (2015), 37 (in Japanese).
• 5)   K.  Suzuki and  T.  Miyamoto: Tetsu-to-Hagané, 63 (1977), 53 (in Japanese).
• 6)   H.  Yamada,  T.  Sakurai,  T.  Takenouchi and  Y.  Iwanami: Tetsu-to-Hagané, 73 (1987), 1706 (in Japanese).
• 7)   H.  Yamada,  T.  Sakurai and  T.  Takenouchi: Tetsu-to-Hagané, 75 (1989), 97 (in Japanese).
• 8)   H.  Yamada,  T.  Sakurai,  T.  Takenouchi and  Y.  Iwanami: Tetsu-to-Hagané, 73 (1987), 107 (in Japanese).
• 9)   C.  Beckermann,  J. P.  Gu and  W. J.  Boettinger: Metall. Mater. Trans. A, 31 (2000), 2545.
• 10)   K.  Miyazawa and  K.  Schwerdtfeger: Arch. Eisenhüttenwes., 52 (1981), 415.
• 11)   T.  Kajitani,  J. M.  Drezet and  M.  Rappaz: Metall. Mater. Trans. A, 32 (2001), 1479.
• 12)   T.  Murao,  T.  Kajitani,  H.  Yamamura,  K.  Anzai,  K.  Oikawa and  T.  Sawada: Tetsu-to-Hagané, 99 (2013), 94 (in Japanese).
• 13)   K.  Isobe,  H.  Maede,  K.  Syukuri,  S.  Satou,  T.  Horie,  M.  Nikaidou and  I.  Suzuki: Tetsu-to-Hagané, 80 (1994), 42 (in Japanese).
• 14)   M.  Zeze,  H.  Misumi,  S.  Nagata,  S.  Mizoguchi,  T.  Shirai and  A.  Tsuneoka: Tetsu-to-Hagané, 87 (2001), 77 (in Japanese).
• 15)   S.  Ueyama,  M.  Niizuma and  K.  Yonezawa: Shinnittetsu Sumikin Giho, 394 (2012), 103 (in Japanese).
• 16)   H.  Yamamoto,  K.  Matsumura,  A.  Yamagami,  N.  Tsuru,  T.  Sera and  A.  Yoshii: Tetsu-to-Hagané, 73 (1987), S913 (in Japanese).
• 17)   T.  Takahashi,  M.  Kudoh and  S.-i.  Nagai: Tetsu-to-Hagané, 68 (1982), 623 (in Japanese).
• 18)   Y.  Natsume,  D.  Takahashi,  K.  Kawashima,  E.  Tanigawa and  K.  Ohsasa: Tetsu-to-Hagané, 99 (2013), 117 (in Japanese).
• 19)   R.  Sugawara,  T.  Itoh,  Y.  Natsume and  K.  Ohsasa: Tetsu-to-Hagané, 99 (2013), 126 (in Japanese).
• 20)   D.  Apelian,  M. C.  Flemings and  R.  Mehrabian: Metall. Trans., 5 (1974), 2533.
• 21)   N.  Streat and  F.  Weinberg: Metall. Trans. B, 7 (1976), 417.
• 22)   T.  Takahashi,  M.  Kudoh and  K.  Yodoshi: J. Jpn. Inst. Met., 43 (1979), 1086 (in Japanese).
• 23)   K.  Murakami,  T.  Shiraishi and  T.  Okamoto: Acta Metall., 31 (1983), 1417.
• 24)   K.  Murakami,  T.  Shiraishi and  T.  Okamoto: Acta Metall., 32 (1984), 1423.
• 25)   K.  Oikawa,  N.  Hirata and  K.  Anzai: Tetsu-to-Hagané, 103 (2017), 747 (in Japanese).
• 26)   K.  Takatani: Tetsu-to-Hagané, 90 (2004), 751 (in Japanese).
• 27)   T.  Toh: Tetsu-to-Hagané, 90 (2004), 983 (in Japanese).
• 28)   K.  Okazawa,  I.  Sawada,  H.  Harada,  T.  Toh and  E.  Takeuchi: Tetsu-to-Hagané, 84 (1998), 490 (in Japanese).
• 29)   N.  Kubo,  T.  Ishii,  J.  Kubota and  T.  Ikagawa: ISIJ Int., 44 (2004), 556.
• 30)   Y.  Miki,  H.  Ohno,  Y.  Kishimoto and  S.  Tanaka: Tetsu-to-Hagané, 97 (2011), 423 (in Japanese).
• 31)   S.  Asai: Electromagnetic Processing of Materials, Springer, Dordrecht, Heidelberg, London, New York, (2012), 78.
• 32)   Y.  Yuan,  K.  Sassa,  K.  Iwai,  Q.  Wang,  J.  He and  S.  Asai: ISIJ Int., 48 (2008), 901.
• 33)   R.  Moreau: Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, Boston, London, (1990), 110.
• 34)   R.  Byron Bird,  W. E.  Stewart and  E. N.  Lightfoot: Transport Phenomena, John Wiley & Sons, New York, (1960), 196.
• 35)  Society for Chemical Engineers, Japan: Kagaku Kougaku Binran, 5th ed., Maruzen, Tokyo, (1988), 242 (in Japanese).
• 36)   S.  Ergun: Chem. Eng. Prog., 48 (1952), 89.
• 37)   R.  Aoki: J. Soc. Instrum. Control Eng. (Keisoku to Seigyo), 11 (1972), 648 (in Japanese).
• 38)   F.  Sakao: Bull. Inst. Space Aeronaut. Sci., Univ. Tokyo, 2 (1961), 456 (in Japanese).