EMC 공정에서의 자기장과 유동장 분석을 통한 자유표면 형상 계산

진호영, 강인석, 박재일*

포항공과 대학교 화학공학과,

포스코 개발 주식회사 미니밀 엔지니어링팀*

Computation of meniscus shape through magnetic and flow fields analyses

in EMC (ElectroMagnetic Casting) process

Ho Young Chin, In Seok Kang, Jae Il Park*

Department of Chemical Engineering, POSTECH.,

Mini Mil Engineering Team, POSEC*.

Introduction

The periodic scratches called oscillation marks are produced at the surface of solidified metal slab in continuous casting process. The oscillation marks due to mold reciprocation affect much the surface quality because they can become the sites of crack formation and impurity penetration. EMC (ElectroMagnetic Casting) process is the advanced casting process that improve the surface quality by changing the free surface shape of molten metal by applying magnetic field around the continuous casting mold. Alternating magnetic field produced by an alternating electric current generates magnetic pressure at slab surface and induces flow within molten metal. Meniscus shape of molten metal is formed through equilibrium of magnetic pressure, dynamic pressure, and static pressure.

In this work, we compute the meniscus shape in EMC process by improving our earlier version[1] and show the results. We determine magnetic field distribution and the magnetic pressure at the free surface by using the boundary integral equation method (BIEM). In flow field analysis, we use the streamfunction-vorticity formulation. Orthogonal grid system, finite difference method and alternating direction implicit (ADI) method are employed for flow field computation. Meniscus shape is determined through normal stress balance at free surface.

Problem Statement

We assume all materials to be linear isotropic homogeneous materials. The generalized ohm's law is used for current density. In molten metals, the magnetic Reynolds number (Rm) cannot exceed [eqn: 10^-2] on the laboratory scale. Therefore we can neglect the term related with Rm. Then we can get B-field governing equation[2]. We introduce the vector potential A and take curl to the B-field equation. Then we can write A-field governing equation by using vector identity. In 2-dimensional B field problem, A has only the third component. If we consider AC magnetic field, [eqn: {A}_{z}=( {psi}_{B}+i {phi}_{B}){e}^{i omega t}]. To get the dimensionless governing equation for □ and [eqn: phi], we define the characteristic values and the dimensionless values as:

[eqn: {bold J}_{c}] = total ampere turn/(a□b), [eqn: {psi}_{Bc}=mu{{l}_{c}}^{2}{bold J}_{c}] , [eqn: {phi}_{Bc}=mu{{l}_{c}}^{2}{bold J}_{c}]

[eqn: {bold B}_{c}][eqn: =mu{{l}_{c}}{bold J}_{c}] , [eqn: {M}_{2}= mu {sigma }_{2}{omega}{{l}_{c}}^{2}], [eqn: {M}_{3}= mu {sigma }_{3}{omega}{{l}_{c}}^{2}] , [eqn: l_c =R]

where a is the width of coil, b is the height of coil, and R is a half width of slab. Then the dimensionless governing equations of magnetic field are as following:

region [eqn: { OMEGA }_{1}]: [eqn: {del}^{2}{psi}_{B}=- {bold J}_{0}] , [eqn: {del}^{2}{phi}_{B}=0]

region [eqn: {OMEGA}_{2}]: [eqn: {del}^{2}{psi}_{B}=-{M}_{2}{phi}_{B}] ,[eqn: {del}^{2}{phi}_{B}=-{M}_{2}{psi}_{B}]

region [eqn: {OMEGA}_{3}]: [eqn: {del}^{2}{psi}_{B}=-{M}_{3}{phi}_{B}] ,[eqn: {del}^{2}{phi}_{B}=-{M}_{3}{psi}_{B}] .

The regions are shown in Fig.2. The boundary conditions are that[eqn: {psi}_{B}],[eqn: {phi}_{B}],[eqn: {{Partial psi}_{B}} / {partial n}], and [eqn: {{Partial phi}_{B}} / {partial n}] are continuous on boundary.

The governing equations of flow field are the continuity equation and Navier-Stokes equation. We assume steady state, laminar flow condition. If the frequency of electric alternating current increase, magnetic field cannot penetrate the molten metal deeply. Then skin layer thickness([eqn: delta] ) is almost zero and Lorenz force is not considered as body force. The Navier-Stokes equation can be expressed in terms of [eqn: bold omega] and we eliminate the pressure term taking curl. To get the governing equation of dimensionless form we define the characteristic values as:

[eqn: l_c = R] , [eqn: u_c = u_{i n}] , [eqn: {bold del}_c = 1/l_c]

Then the dimensionless governing equation of motion is the followings.

[eqn: Re [del Times (u Times omega)] + {del}^{2} { omega} =0]

One of the net effects of the alternating magnetic field is to replace the no-slip condition at a solid mold interface by an induced tangential velocity. The free-shear stress condition at a free surface is replaced by an induced tangential stress. For [eqn: xi > > delta], the conditions are the followings.

[eqn: {u}_{s}={M{delta}}over {4 {l_c} }del({{{B}_{s}}^{2}} over {2})]at solid surface,[eqn: {partial {u}_{s}}over{partial xi }=-{M}over{2}del({{{B}_{s}}^{2}}over{2})]at free surface

where [eqn: M={delta{B_{c}}^{2}}/ {2 mu rho nu u_c}]. For incompressible 2-dimensional flows, the Navier-Stokes equation can be simplified by introducing the streamfunction,[eqn: PSI].

The dimensionless form of time averaged governing equation of normal stress at free surface under AC magnetic field is written as:

[eqn: ({M{u_c}^2}/ {Re {delta} g } )({{ LEFT | B_s (x) RIGHT | }^2}/ {2}) + h+({{u_c}^2}/ {g l_c})({u^2}/ {2})= const.]

Numerical Analysis

(1)magnetic field analysis

When the skin layer thickness is zero, [eqn: M_{3}] becomes infinite according to definition of [eqn: delta]. Then [eqn: psi_{B} = phi_{B} = 0] in [eqn: OMEGA _{3}]. If we assume mold is an insulator, [eqn: OMEGA_{1}] and [eqn: OMEGA_{2}] have the same properties and they are combined to form a region [eqn: OMEGA_{m}] where [eqn: M_{2}] is zero. Because [eqn: phi_{B} =0] in all boundaries, [eqn: phi_{B} == 0] in all regions. Then the governing equation of magnetic field in [eqn: OMEGA_{m}] is as following:

[eqn: {del}^{2}{psi}_{B}=- 1] at coil part;  [eqn: 0] at other part on [eqn: OMEGA_{m}]

In order to solve the above equation, we use the boundary integral equation method. Because B-field has the derivative of [eqn: psi_{B}], we can get the derivative of [eqn: psi_{B}] directly from the solution by BIEM.

(2)Flow field analysis

We intend to rearrange the streamfunction equation and vorticity equation in the following general form.

[eqn: q_0 ({partial A}over{partial t})_x ={f}^{2}{{partial}^{2}A}over{partial {xi}^{2}} +{{partial }^{2}A}over{partial {eta}^{2}} + {q}_{1}{partial A}over{partial xi} +{q}_{2}{ partial A}over{partial eta} +{q}_{3}A +{q}_{4}]

where f is distortion function([eqn: f={{h}_{eta}}/ {{h}_{xi}}]). We use the ADI(alternating direction implicit) method in orthogonal grid system.

(3)Meniscus shape determination

First, we assume the initial meniscus shape. Then we modify the meniscus shape in the direction to minimize the error in the normal stress balance which the molten metal volume is conserved.

Results and Discussion

We calculated magnetic field distribution, flow field distribution and meniscus shape. We set the nozzle angle to be 10□ and modified the intensity of the current([eqn: M]). First, to see the effects of [eqn: M], we varied [eqn: M] from 0 to 1000, while Re is set to be 10 and 100. The results for the case ([eqn: M]=0, Re=10) and ([eqn: M]=1000, Re=10) are shown in Fig.3, 4. We can see a flow circulating in the clockwise direction induced by the magnetic field in the region near the coil in Fig.4.

At solid surface, the effective flow runs in the direction of positive derivative of magnetic field. That is, we can expect the flow direction according to the density of magnetic stream lines. Therefore we can catch that the flow runs from the lower solid surface parts to the upper solid surface parts. At free surface, the effective shear stress is induced by the magnetic field. The induced flow is expected to rotate in the clockwise direction because the flow direction is from the place of lower magnetic field strength to the place of higher magnetic field strength. When we see Fig.4, circulation flow of molten metal near coil part means that the flow induced by the effective shear stress at the free surface dominates the flow induced near the solid surface. Currently we are analysing the flow fields based on k-[eqn: epsilon] turbulence model. The new results will be presented in the near future.

Acknowledgment

The authors would like to acknowledge the financial support from KOSEF through ARC at POSTECH and the financial support from POSCO.

Reference

[1]Park, J. I., M.S. thesis, POSTECH (1997)

[2]Chandrasekhar, S., "Hydrodynamic and hydromagnetic stability", Dover (1961)

Fig.3 M=0, Re=10        Fig.4 M=1000, Re=10