ELECTROMAGNETIC STIRRING
2
NUMERICAL MODELLING OF THE CONTINUOUS CASTING OF STEEL WITH ELECTROMAGNETIC STIRRING
Report Prepared by: M Hughes
Report Approved by: M R Malin
Signature:
Abstract
This report describes work performed under CHAM contract 05/022/04 for Centro Sviluppo Materiali S.p.A. The work is concerned with the development and application of a PHOENICS-based computer model for simulating a continuous casting process whereby liquid steel enters a mould cavity, and is then electromagnetically stirred before the partially-solidified strand is withdrawn from the bottom of the mould at a uniform casting speed. The strand forms a curved bar of solidifying steel after leaving the model, and so a body-fitted coordinate system is employed for representing the process geometry. The solidification model [1] already implemented in PHOENICS V3.2 for CSM is upgraded for use in PHOENICS V3.5.1 in this study. This report does not present any results from the simulations, but rather it focuses on the mathematical formulation and brief instructions on how to use the model.
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2. THE PROBLEM CONSIDERED..................................................................................... 4 2.1 Introduction................................................................................................................ 4 2.2 Main Features of the Model ....................................................................................... 4 2.3 The Solution Domain ................................................................................................. 5 2.4 Electromagnetic Properties and Assumptions ........................................................... 5 2.5 Flow & Thermal Properties ........................................................................................ 7
3. MATHEMATICAL FORMULATION ................................................................................ 7 3.1 Electromagnetic Considerations ................................................................................ 7 3.2 Flow and Thermal Considerations ........................................................................... 10
4. BOUNDARY CONDITIONS.......................................................................................... 14 4.1 Electromagnetics ..................................................................................................... 14 4.2 Flow and Thermal Conditions ................................................................................. 17
APPENDIX B – Electromagnetic Q1 Input File....................Error! Bookmark not defined.
APPENDIX C – Electromagentics GROUND file .................Error! Bookmark not defined.
APPENDIX D - Flow Analysis and Solidification Q1 Input File ..........Error! Bookmark not defined.
APPENDIX E - Flow Analysis and Solidification GROUND File ........Error! Bookmark not defined.
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1. INTRODUCTION
The purpose of this report is to provide a brief description of the work performed for Centro Sviluppo Materiali S.p.A (CSM) on the modelling of a continuous casting process with electromagnetic stirring. The worked was carried out under CHAM contract No. US/05/022/04.
The objective of the project is to provide a PHOENICS-based computer model which can be used by CSM to gain more insight into the physical behaviour of the continuous casting process. The version of PHOENICS used was PHOENICS V3.5.1.
This report comprises six main sections of which this introductory section is the first. Following this section, a description of the problem considered is provided in Section 2. Next, the mathematical model is described in Section 3, and then the boundary conditions are discussed in Section 4. Explanations on how to apply the model to the CSM casting system are provided in Section 5. The report closes with concluding remarks in Section 6, and supplementary material is provided via the References, Nomenclature and Appendices.
2. THE PROBLEM CONSIDERED
2.1 Introduction
Electromagnetic (EM) stirring of steel during continuous casting of billets and slabs is a well-established technique that improves the quality of the cast products by stirring the molten pool. Mixing of the liquid metal positively influences the solidification process at the microscopic crystal level. The purpose of the work undertaken during the course of this contract is to provide a means to study the large-scale electromagnetically-influenced flow and thermal behaviour in the sub-mould region of continuous casting. The solidification model which was has been employed takes no account of microscopic metallurgical effects.
It should be noted that modelling coupled flow and electromagnetic equations in three- dimensional eddy-current problems is a current and open area of research. This is largely due to the free boundary nature of Maxwell’s equations, and the subsequent difficulty of restricting the solution to the region of interest.
2.2 Main Features of the Model
The main features of the model can be summarised as follows:
Three-dimensional model. Split simulation, the electromagnetic and flow simulations are uncoupled. Electric field is solved. Coil Implementation is external to the solution domain. Alternating current low-frequency conditions. Integral-differential approach to the EM solution which limits the solution domain to
the CFD domain. Steady-state time averaged solution of EM equations.
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2.3 The Solution Domain
The casting geometry considered in this study is illustrated in Figures 1 and 2. Heated liquid steel flows into the domain through a nozzle of area 1.257e-3mm2 at a temperature of 1530° C. Heat losses specified by CSM are applied at different rates over progressive cooling sections of the domain. The steel solidifies below a temperature of 1450° C. Upon solidification, the velocity is fixed in the axial direction to 41.5mm/s representing the effect of roller motion.
Electric currents are induced in the molten metal over the stirrer region; these are known to produce circumferential stirring of the metal in the plane that is orthogonal to the axial direction. Because of the non-symmetrical coil positioning, the solution profile is not symmetric and so the full 360° mesh is required for the simulations.
2.4 Electromagnetic Properties and Assumptions
These properties can be readily changed from the PHOENICS Q1 input file.
Electric conductivity 7.7e5 mho/m Frequency 10 Hertz Coil current 80 Amp.
The following assumptions are made:
The mould material and position does not influence the electromagnetic field. The electric conductivity is considered constant with temperature. Ferromagnetic effects from the steel are insignificant because of the high
temperatures.
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2.5 Flow & Thermal Properties
The density of the liquid and solid material is taken as constant at 7300 kgm-3
The laminar kinematic viscosity is constant .9e-6 m2s-1
The turbulent kinematic viscosity is derived from the k- turbulence model. The mean specific heat over the temperature range varies with temperature as:
Cp = 0.5*0.136*T-37.15+523. Jkg-1K-1
Thermal conductivity varies as a function of temperature, given by
1117011.31
Latent heat of solidification is 2.64e+5 J/kg.
The solid fraction, fs, is calculated from the local temperature using the following formulae:
0.0f;TT
TT






Ts and TL are solidus and liquidus temperatures and m is a material-dependent index taken as 1.0 in the present model.
3. MATHEMATICAL FORMULATION
3.1 Electromagnetic Considerations
The Electromagnetic Equations Maxwell’s equations in differential form describe the local relationship between the variables of the electromagnetic field. In the case of non-magnetic materials, (non ferrous metals or steel above Curie temperature), the magnetic permeability may be assumed constant throughout the domain of interest. Additionally, in materials that are sufficiently conducting, such as molten metals, there are no localised electric charges and Maxwell’s equations can be expressed as:
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)4(0
)3(

where B is the magnetic flux density, E is the electric field intensity, t is time and J is the current density. These equations together with Ohm’s law complete the electromagnetic description. For isotropic electrical conductivity , Ohm’s law is expressed as
)5(EJ
Assuming that B is sufficiently continuous so that it’s temporal and spatial derivatives can be interchanged, taking the curl of (2) and substituting for B gives an equation solely in terms of E
)6( t
EE
The electrical conductivity depends on temperature which in turn depends on time. However, on the scale of the electromagnetic processes may be assumed constant in time, and the equation above may be expressed as
)7( t
EE
The frequency of the AC field used for the induction stirring results in a time scale for the electromagnetic phenomena being at least 50 times smaller then that of the flow. Consequently, modelling of the fluid flow in the process ideally requires the pseudo steady-state solutions of the electromagnetic fields, as they are time harmonic.
Assuming that a periodic solution exists with a circular frequency : E=ER cos(t)+EI
sin(t), with ER the real component and EI the imaginary. Substitution into equation [7] results in
)9(μσω
)8(ωμ


Physically, ER represents the solution of the electric field at time t0 and EI the solution at time period offset by t=, this is implicit to the complex plane. The solution of these equations provides sufficient information to calculate time-averaged forces for the flow equations.
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This above system consists of six scalar equations for six unknowns. The transient terms have disappeared by nature of the periodic solution and have been replaced by source terms. Mathematically this is possible because the fields are expressed as phasors of the form A=A0e
jwt. Note that the solution will be in terms of the field amplitude A0. All of the electromagnetic variables are expressed in this way.
The magnitude of the RHS of equations (8) and (9) will be substantial for frequencies in the kHz range and the equations would need to be linearised to achieve convergence. This is possible despite the sources not being in terms of the solved variables. However, with the low operational frequency of the CSM stirrer this is not required as the PHOENICS solvers are sufficiently robust to provide convergence.
The magnetic field is split into applied and induced contributions, from the coils and electromagnetic induction respectively. Both sets of contributions are calculated from the Biot-Savart integrals shown below. The source current for the applied field is known from the induction coils, and the induced current is calculated from the solution of the electric field.
dV ||
(10)
In the above equations Icoil is coil current, dlcoil is an elemental length of coil descretisation, J is current density and dV is a control volume.
The Lorentz Forces
The time-averaged Lorentz forces Fem can be obtained by integration of the cross product of complex current density and magnetic field over the time period t=0,2/, [see Appendix A1].
As a momentum source term S this is expressed as
S = )Re( 2
1 BJ
The units of this force are Nm-3 and hence it is integrated volumetrically within the governing equations to obtain the electromagnetic force.
Vol
Vol
10
The symbol * denotes the complex conjugate, and the units of the Joule heating source term is in Watts.
3.2 Flow and Thermal Considerations
Darcy Solidification Sources
The additional source terms in the momentum equation arising from the solidification of the molten metal are modelled using the Darcy-law solidification model of Prakash and Voller [4]. The terms are dependent on the local solid fractions fs
a)f1(
s S
where is the velocity component of the momentum equation; ‘A’ is constant and dependent on the morphology of the media and ‘a’ is a small number that is included to prevent division by zero, when completely solid (fs=1). The motion of the solid is caused by the actions of small rollers that pull in the axial (z) direction at a given velocity; motion in the r-θ plane is fixed to zero. The force is expressed within the governing equations as :
)( a)f1(
;
where 1215.4 mse for w(axial) velocity, and =0 for u,v velocities
Latent Heat Sources
The phase change of liquid metal to a solid state releases the latent heat of solidification into the energy equation. This source being calculated from a conservation equation for the convection of heat
0SΔH).(ρ L U
where is density, U the velocity, H is the local value of latent heat and SL the source of latent heat. The latent heat content of the metal H is a linear function of the solid fraction, fs and hence
L)f(1H s ;
where L is the latent heat of solidification.
With reference to Figure 3, the latent heat source can be expressed from a balance formula as
HHpLLpnnpssspepwwwL ΔHAρUΔHAρUΔHAρUΔHAρUΔHAρUΔHAρUS
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The upwinding principal is applied for the calculation of the latent heat at the cell faces, such that for example in the IX-direction
Eep
pep
pww
Www
ΔHΔH0U
ΔHΔH0U
ΔHΔH0U
ΔHΔH0U



If variable density were employed, a similar upwinding would be applied to the density to calculate mass fluxes.
Figure 3: Nomenclature for latent heat convection



MINMAX
MINMAX
MINMAX
Transport of latent heat out of a cell is given by:
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where om is the total mass outflow from the cell.
The latent heat can be expressed in terms of temperature, and hence the heat source can be written as
L) TT
TT (1mS




In the case of m=1 as used in this study, the above source can be represented in PHOENICS COVAL form as
)T(T TT
4.1 Electromagnetics
Coil Configuration
There are four coils which are unevenly distributed at approximately 90° spacing around the casting. The distances from the centre of the coils to the centre of the billet have been provided, from which the exact coil positions can be deduced.
Figure 4 is a schematic illustrating coil position and Figure 5 shows the current configuration. Two coil configurations are available; the first has four coils at 90 degrees and symmetrically spaced around the billet. This configuration was used for developing and testing the code as it generates a symmetrical magnetic field and hence symmetrical results.
The second configuration is that defined by CSM, (see Figures below). Additionally, the coils are split into vertical and horizontal components. The vertical component is the strongest influence towards generating the Lorentz forces as it is longest. Both these current contributions can be activated or deactivated through the Q1 input file.
Electromagnetic
At the boundary of the domain the following two conditions apply
0
t
The former is Faraday’s law which states that a time changing magnetic field induces an electric field; the second is Kirchoff’s law which states that the normal component of current at the boundary is zero.


Figure 4: CSM coil arrangement
Figure 5: Electric current Configuration
The subscripts zr ,, are the coordinate directions and R,I represent the complex real and imaginary planes.
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The above equations can be represented in terms of fluxes at the boundaries. If the vertical components of the coil loops are much longer then the horizontal components, the induced current can be approximated as being one-dimensional and hence only the vertical component of the Electric field Ez need be solved. In this case the boundary conditions for Ez can be written as
θR zI



Additionally, the fields will need to be fixed to a reference value at some point in the domain, setting Ez to zero at the far end from the coils is a sensible condition.
If the horizontal components of the coil loops are to be included in the model the full set of coordinate equations for the Electric field should be solved.
In this case the boundary conditions on the side of the billets are:
0.Eand0E θ
E B
0Eand0E θ











These conditions are applied together with a condition setting Eθ to a reference value of zero at a suitable point in the domain far from the magnetic field.
On the side of the domain Er is only zero at the surface boundary. Because this surface boundary is approximated by a body-fitted (BFC) mesh as a flat surface its value will be non-zero and hence its derivatives with respect to θ and z will need to be estimated. These derivatives are approximated in the following way:
Let ra be the curvature radius of the surface in the plane (r,θ). In the vicinity of the observation point on the surface, the intersection of plane (r,θ) with the surface swept by ra
is a circular arc., [see Figure 6].
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Let α be the central angle sweeping that arc. Then the projection onto axes r of the tangential vector of magnitude Eθ is En = - Eθ sinα. Differentiating this with respect to θ
then gives a
Similarly, b

, where rb is the radius of curvature in the (r,z) plane.
Figure 6: Boundary condition approximisation
4.2 Flow and Thermal Conditions
Inlet steel temperature 1530°C Liquidus temperature 1500°C Soild temperature 1450°C
The thermal boundary conditions are listed below, and the regions over which they apply are illustrated in Figure 7.
From the start of the casting to the end of the mould & stirrer region a heat flux loss is applied through the supplied formula:
Q = 1.404e+6*(hlength/Zps);
where hlength is the length of mould, and Zps is given by:
ZWLAST*(ZFRAC(II)+ZFRAC(:II-1:))/2
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beyond this section, from regions 3 to 9 as shown in Figure 7, the following heat loss conditions are applied:
Ring : htc=800 W/mK, T=400K I zone: htc=500 W/mK, T=300K II zone: htc=350 W/mK, T=300K Below: htc=200 W/mK, T=300K
The steel inlet velocity through the nozzle is 0.97 m/s and the casting speed is 0.0416 m/s
Figure 7: Thermal boundary conditions
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5. APPLICATION OF THE MODEL
The electromagnetic calculations are uncoupled from the flow equations, this is possible as the motion of the billet does not affect the magnetic field. It is therefore convenient to model the problem in two stages and separate Q1 and ground files have been developed.
The electromagnetic field is solved by application of the first run. The resulting output, ‘phi’, file from this simulation is then used to read the Lorentz forces into the second (flow) simulation.
5.1 Running the model
The following steps are a guide to running the model.
1. Copy the file named em.q1 to q1. This file can be modified in terms of the available parameters as listed below. The user could modify the parameters interactively through running the satellite program, in which case the Q1 will need to be edited to set TALK=T. Otherwise the changes should be hand made in an editor of choice.
2. Run the Satellite program, this creates a file called EARDAT, which contains the input required for the simulation run. An XYZ file is also produced, this contains co- ordinate geometry for the body-fitted mesh.
3. Copy the file em.grd to ground.for and run the command bldear to produce an earth executable file earexe.
4. Run the earth executable earexe, when this has successfully finished the electromagnetic simulation has completed and an output file named phi, will have been produced. Rename this file as phie.
5. To run the flow simulation, copy the file flow.q1 to Q1, make any desired parameter changes or set Talk=T in the Q1 and make the changes interactively. Ensure that the working directory contains the above mentioned phie file. Run the Satellite program.
6. Copy the file FnT.grd to ground.for, compile and link by invoking the bldear script to produce an earexe executable.
7. Run the earth program earexe, after successful completion the final phi file will be produced.
5.2 Default settings and variables
Run 1- The electromagnetic Q1
The ratio of the electromagnetic stirrer length to billet length is <2%, additionally the curvature of the billet in the stirrer region is quite gentle and hence the solution of the
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electromagnetic field over a straight cylinder is a reasonable approximation to this part of the model.
However in an attempt to include the asymmetry in the fields caused by the curvature of the billet, the model has been further developed to provide a solution attempted over the exact CSM geometry. This solution includes more approximations than the counterpart solution on a straight cylinder, for example approximations to the axial direction electric field differentials and may therefore be less accurate.
This curvature in the mesh is activated through the Boolean variable curved=T.
The stirrer coils are represented as line sources of current. The longest and hence dominant contributors to the Lorentz forces are the vertical components. These are activated through the PIL logical LG(3)=T; the horizontal components are secondary and activated through the logical LG(1)=T.
The PIL logical LG(4) controls the coil arrangement. The CSM coil arrangement is activated by LG(4)=F, otherwise the testing and development coil arrangement is used. It should be noted that the latter should only be used with CURVED=F.
If only the vertical current sources are included, then the electric current induced inside the billet can be approximated in 1-D through the solution of the axial electric field Ez,
otherwise the all three components of the electric field should be solved. The Boolean variable FULLMODL=T provides a switch to these different solutions.
The induced current in the billet produces a secondary magnetic field. In the specified frequency range this magnetic field is found to be two orders of magnitude less then the applied field and hence can be ignored. The integration required to calculate this field is very expensive and of order n2, it is by default deactivated through the PIL logical LG(2)=F. If this contribution is activated, the mesh density spacing would need to be changed, (even spacing was found to work well, and the variables solved in a wholefield manner.
Table 1 Summary of electromagnetic variables
Variable PIL Default Value
Frequency RG(3) Hertz
Coil-Top & Bottom LG(1) True
Coil- Sides LG(3) True
Run 2- The Flow and Thermal equations
The forces from the electromagnetic model run are read into the fluid simulation from a restart. These forces are placed in the storage locations FLX,FLY,FLZ, and subsequently converted to polar coordinates, (FLXP,FLYP,FLZP) before being integrated into the governing flow equations as volumetric body forces. The forces generated by the 80 amp current sources as specified for this study have been found to be too small to significantly stir the molten steel, this is entirely in line with the literature{1,2}, where for purposes of the model simulation, current sources are taken to be of order 0(1000A).
The reality of the stirrer is probably that each section of the coil contains many 80A windings along with an iron yoke (pole piece) that would bolster the effect of the magnetic field considerably. Consideration of the physics indicates that application of a scaling factor is a valid approach, in fact considering the lack of detail regarding the coils it is necessary. Attempts have been made to obtain detailed coil information but ABB the manufacturers, have not been forthcoming.
Essentially, the magnetic field generated from the coils is linearly proportional to the effective coil current. Additionally, an increase in the coil current will generate a linearly proportional increase in the magnetic field strength. In turn the Lorentz force is proportional to the square of the magnetic field. This means that if an effective coil current can be estimated; taking into account that the B-field may be bolstered several times by an iron yoke; then a scaling factor for the forces can be guessed as (effective_current ÷ 80)2.
Specific heat and conductivity can be set to constant values or to vary with temperature via the Boolean variables concp and contk.
To simplify the code development, the fluid model was initially developed on a straight cylindrical geometry, the Boolean variable curved switches from this domain to the actual curved CSM billet geometry. Note that it is not necessary to run the electromagnetic and flow runs with the same geometry, however the grid size and distribution should be the same in both models as an interpolation procedure for the forces has not been developed.
The Boolean logical solmod is a legacy switch from development, it activates or deactivates the solidification sources, however it should be noted that deactivation of these sources will remove latent heat energy and hence invalidate the heat loss boundary conditions.
Additionally, the model assumes that there is complete solidification of the fluid before reaching the outlet was a fixed mass-flux loss boundary condition is set. This outlet condition would be inappropriate if complete solidification has not occurred and hence an initial condition is set for temperature and fs in this region. Additionally, a fixed pressure boundary is also applied at the inlet area as a means to absorb up any imbalance in mass continuity, which should be tiny at convergence.
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Variable PIL Value
Limit Temp Incs LG(3) LG(3)=T
Liquid density RHOL 7300kgm-3
Temp solid RG(8) 1723K
Temp liquid RG(9) 1773K
Latent heat - solidification RG(11) 2.64e+5Jkg-1K-1
Large const in Darcy force RG(14) 5e+6
Small const in Darcy force RG(15) 5e-10
Lower limit for solidification RG(25) 0
Specific heat constants ACP
RG(27) 0.136
6. CONCLUDING REMARKS
6.1 Final Comments
The work undertaken in this study provides a framework for the modelling of the CSM continuous casting system. The solution of the electromagnetic equations under these conditions is an open area of research, and as such it should be realised that the solution of the full 3-D electric field on the curved geometry is a novel approach. Consideration could be given as to whether the 1-D simulation gives a better approximation of the electromagnetic forces. Although they are circumferential, the forces generated from an 80 amp line current source will be far too small to produce stirring and a scaling factor is required.
6.2 Future Work
Information regarding coil configuration was not very detailed, and hence they could only be represented as line current loops. Additionally a coil arrangement with phase shift of 180º as specified by CSM does not produce circumferential forces. Attempts to procure information directly from the coil manufacturers were unsuccessful but would be useful, this would enable the magnitude of the forces can be better estimated. Alternatively one of the coil containers might be disassembled so that the internal configuration can be accessed and a separate FE analysis performed. Alternatively, an experimental calibration could be performed.
The thermal boundary conditions have been taken from a previous continuous-slab casting Consultancy Contract carried out by CHAM for CSM [5]. These boundary conditions might be examined further as they may be extracting too much heat from the current curved-strand system.
Temperature-dependent properties for density and viscosity could be implemented.
7. REFERENCES
1. Hamill, I.S and Jureidini, R.H, Hydrodynamic and Thermal Analysis of a Continuous Casting Machine, CHAM Technical Report TR 311/02, Vol30,pp 1709-1720, (1991).
2. T.T. Natarajan & Nagy El-Kaddah, Finite element analysis of electromagnetic and fluid flow phenomena in rotary electromagnetic stirring of steel, Applied Mathematical Modelling 28, 47-61, (2004).
3. N. El-Kaddah & T.T.Natarajan, Electromagnetic stirring of steel: Effect of stirrer design on mixing in horizontal electromagnetic stirring of steel slabs, Second Internation Conference on CFD in the Minerals and Process Industries, CSIRO, Melbourne, Australia, 6-8 December, (1999).
4. Voller, V.R and Prakash C, A Fixed-Grid Numerical Modelling Methodology for Convection-Diffusion Mush region Phase-Change problems, Int.J.Heat Mass Transfer, Vol30, 1709-1720, (1987).
5. Heritage, J., Lopez, E. and Malin, M.R, Modelling Electromagnetic Braking in Continuous-Slab Casting, CHAM Consultancy Contract File C/04362, (1998).
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Symbol Meaning Units
B Magnetic flux T or kg/(m s) E Electric Field V/m or N/C H Magnetic field A/m or C/(m s) J Current density A/m2
Conductivity 1/( m) or s C2/(kg m3)
o Permeability H/m or kg m/C2
25
t
ttttt
Code can be provided on application