Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
Numerical Simulation of Induction Stirred Ladle M Pal*1, S Kholmatov and P Jönsson 1
Civil and Computational Engineering Centre-University of Wales, Swansea, Division of metallurgy, Material Science and Engrineering, KTH SE-10044 Stockholm, Sweden *Corresponding author:
[email protected]
Abstract: Induction ladle plays an important role in a broad range of metal-processing operations. The treatment of steel in the ladle is as old as the use of ladles in steelmaking. The main purpose for ladle treatment of hot metal and liquid steel include desulphurization, deoxidation, alloying, and inclusion shape control. Over last few years efforts are made to develop simulation models of induction ladle [1][8] , in order to study heat transfer and fluid flow in gas and induction stirred ladles. These models provide more information about the industrial processes used in ladle treatment of steel. In this paper a simulation model of a laboratory scaled induction ladle is presented. The simulation model so developed will make it feasible to have information about the fluid flow phenomenon and thermal heat transfer. In order to perform the numerical simulation of the furnace, physical processes involved are expressed as a coupled-nonlinear system of partial differential equations arising from a thermal-magneto-hydrodynamic problem. The simulation model is formulated in a twodimensional domain. The equations of electromagnetic model to describe magnetic diffusion inside the ladle through magnetic stirrer are expressed by well known system of Maxwell’s equations. Moreover, the heat equations governing the induction heating are provided. The hydrodynamic model for fluid flow in the molten metal is described by wellknown incompressible Navier-Stokes equation. Numerical simulation are performed by solving the coupled system of equations using the commercial software COMSOL Multiphysics® application mode by combining Electromagnetics, Fluid Dynamics and Heat transfer modules. Keywords: Induction ladle, induction heating, electromagnetic forces, Maxwell’s equations, Navier-Stokes equations.
1. Introduction Electromagnetic stirring is widely used in ladle metallurgy and continuous casting. The application of the electromagnetic stirring improves strand quality, increases metallic yield and continuous caster productivity, and increase production flexibility. Typically, such systems consist of a mould, made out of nonmagnetic material, which is surrounded by induction coils. Upon passing an alternating current through these coils, a fluctuating magnetic field is generated, and a current is induced in the melt contained in the crucible. The net result is essentially two-fold: 1) the induced current gives rise to Joule heating; 2) the interaction between the magnetic field and the current gives rise to electromagnetic force field, Lorentz force, which is responsible for stirring. Induction furnace systems are modeled by: • Maxwell’s equations for calculating induced current, and electromagnetic force field; • Navier-Stokes equations to calculate velocity field; • Differential thermal balance equation to calculate the temperature field. The study of these flows is called magnetohydrodynamics (MHD), which is concerned with the mutual interaction of fluid flow and magnetic fields. The mutual interaction of a magnetic field, B, and a velocity field, U, arises partially as a consequence of Faraday, and Ampère, and partially because of the Lorentz force experienced by a current-carrying body. It is convenient, although somewhat artificial, to split the process into three parts1 • The relative movement of a conducting fluid and a magnetic field causes as e.m.f. (of order
U x B ) to develop in accordance to
Faraday’s law of induction.
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
•
•
These induced currents must, according to Ampère’s law, give rise to a second, induced magnetic field. This adds to the original magnetic field, and change is usually such that the fluid appears to ‘drag’ the magnetic field lines along it. The combined magnetic field (imposed plus induced) interacts with the induced current density, J, to give rise to Lorentz force (per unit volume), J x B. This acts on the conductor and is generally directed so as to inhibit the relative movement of the magnetic field and the fluid.
Previously a model of induction stirring using straight stirrer (to explore the fluid flow phenomenon inside the ladle) was presented in [9]-[10] . In this paper cylindrical stirrer configuration is explored for fluid flow and induction heating. In a cylindrical stirrer combination of high and low frequency current can be used for both induction heating and stirring of the molten steel. This paper is organized as follows: Section 2 gives a brief description of formulation of the problem to be solved. In section 3 we summarize the governing equations of the induction stirring and fluid flow in the ladle. Next, in section 4, details of the heat transfer process involved in induction stirring and heating of the ladle are presented. Section 5 presents the problem formulation for simulation purpose with the details of boundary conditions, mesh and solver settings. Results and discussion are presented in section 6. Conclusions follow in section 7. 2.
Formulation of the Problem
In this research, a CFD model of the induction stirred ladle is developed. At first, electromagnetic Lorentz forces are calculated. Secondly, the resulting flow field caused by the Lorentz forces is calculated. The physical orientation of the induction stirrer and the ladle is shown in Figure 1. The induction coils inside the stirrer produce a high frequency magnetic field. This magnetic field induces currents on the surface of the conductor whose distribution is such as to shield the interior of the conductor from the imposed field.
3.
Governing Equations
3.1. Governing equations for Electromagnetic Forces The Ampère-Maxwell’s simplified equation for magnetic field:
∇×B=J
(1)
Faradays law for induces electric field:
∂B ∂t
∇ × E=− B=µ H
(2) (3)
Ohms law and Lorentz forces due to interactions of the electromagnetic field are given by: J = σ (E + U × B ) (4)
J = σ (E + U × B ) F=J×B
(5)
As a practical matter, in case of axi-symmetric problem it is convenient to work in terms of the vector potential A, because it has single component, given as:
B =∇ × A and
E=
− ∂A , ∂t
(6)
(7)
with
∇⋅A = 0
(8)
It may be shown, moreover, that upon combining the above definition with the Maxwell equations, we obtain:
∇ 2 A = µ mσ e ∂A ∂t ,
(9)
This equation is of the same form as the diffusion equation. For Cylindrical symmetry, which is appropriate to most furnace applications and to the induction stirring of cylindrical billets, Eq. (9) may be written in cylindrical coordinate form as follows:
1 ∂ ∂A θ r r ∂r ∂r
2 ∂A θ ∂ A θ Aθ + − = µ σ m 2 ∂t r2 ∂z
(10)
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
With A z = A r = 0
ρ (V ⋅ ∇ )V = −∇P + ∇ ⋅ τ + FB
(16)
Upon using Eq. (6) we have:
1 ∂ (rA θ ) Bz = r ∂r ∂A Bz = − θ ∂t
(11) (12)
∂A θ ∂t
(13)
Now using the above Eq. (11)-(13), the two components of the body force field i.e., the nonzero terms of the ( J × B) product are then given as:
Fγ = −J θ B z
(15)
Induction Coils
(14)
Axis of Symetry
Fz = − J θ B r
LADLE
∂u r ∂u + uz r = ∂r ∂z
ρ ur
Moreover, upon using Ohm’s law Eq. (4) together with Eq. (7), we obtain:
J θ = −σ
For parameters and designations, please see the Nomenclature. However, for cylindrical symmetry the equation of motion can be written in the following form:
∂ µe ∂ (ru z ) ∂P ∂r r ∂r − + ∂r ∂ ∂u r + µ e ∂z ∂z + J θ Bz and,
∂u z ∂u + uz z = ∂r ∂z 1 ∂ ∂u z r ∂r µ e r ∂r ∂P − + ∂z ∂ ∂u z + µ e ∂z ∂z − J θ Br
ρ ur
Upon introducing the vorticity,
ξ=
(18)
ξ
∂u r ∂u z − , ∂z ∂r
(19)
and the stream function, ψ
Fig. 1 Axi-Symmetric Ladle Geometry used in
1 ∂ψ r ∂z 1 ∂ψ uz = − r ∂r
Simulation Model
The relation between
ur =
3.2. Governing Equations of Fluid Flow In general, steady fluid flow in a turbulent, molten-metal system which is induction stirred, may be represented by the following timesmoothed, vectorial form of the Navier-Stokes Eq. (4):
(17)
(20) (21)
ξ ,ψ
is given through
∂ 1 ∂ψ ∂ 1 ∂ψ + − ξ = 0 (22) ∂z r ∂z ∂r r ∂r Using Eq. (19)-(21) the equation of motion then can be written as:
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
∂ ξ ∂ψ ∂ ξ ∂ψ − ∂z r ∂r ∂r r ∂z
+
+
1
ρ 1
ρ
∂ µe ∂ ( rξ ) (23) ∂r r ∂r + ∂ µ e ∂ ( rξ ) ∂z r ∂z ∂ ∂ ( J θ B z ) + ( J θ Br ) = 0 ∂ z ∂ r
Where, J θ B z and J θ B r are the only remaining terms of J x B for cylindrical symmetry. The effective viscosity is defined as: (24) µe = µ + µt , Where µ t is the turbulent viscosity, which may be evaluated by solving conservation equations for turbulent kinetic energy and the frequency of turbulent fluctuations. Further details about the formulation can be found in the literature [9]-[14].
flow Q, through a slab is proportional to the gradient of temperature difference:
∆Q ∆T = − kA ∆t ∆x
(25)
Here, k is known as the coefficient of thermal conductivity. The negative is required to yield a positive number for the heat flow. Thermal radiation, natural phenomena known as electromagnetic radiation, i.e. the broadcasting of energy by subatomic transport processes, is the process of heat transfer by which equilibrium is achieved between two bodies without a mutual contact. This can be excited by the passage of an electric current-chemically, by electron bombardment and thermally, as a simple consequence of the temperature level of a body. For a black body the total rate of radiant emission per unit surface is given by
Wb = σ T 4 ,
(26)
in which σ = 56.7 ∗ 10 kW m K . The transfer of heat through the motion induced by the natural volume or density changes associated with temperature differences in a fluid is known as natural or free convection. The rate of heat transfer per unit area of the heated body, q, is proportional to the difference in temperature between the hot body, T, and the ambient flow, Ta. If the constant of proportionality is h, then we have: −12
4.
Heat Transfer
Heat transfer is an important component of all ladle metallurgy and injection systems. When molten steel is held in a ladle, heat losses will occur to the surroundings and in particular to the ladle walls. In addition to a modest rate of heat loss, this will also result in stratification of the melt. When molten steel is being agitated in a ladle by gas injection or induction stirring, the rate of heat loss will be increased, particularly because of radiative heat transfer from the free surface. In many metallurgy operations, thermal energy is added to the melt either by induction or, more frequently through the electric arcs [8][10] . In a induction ladle high frequency magnetic field is used to agitate the melt and a low frequency magnetic field is deployed for induction heat thus induction heating and stirring are can be carried out at the same time in a induction stirred ladle. Although in many practical situations all modes of heat transfer occur together, we will consider conduction, radiation, and convection separately. 4.1. Equations Governing the Heat Transfer The law of heat conduction, also known as Fourier’s law, states that the time rate of heat
2
4
•
Q = qA = hA(T − Ta )
(27)
Where h is heat transfer coefficient. 5.
Modeling in the solver
The simulation model of the above described metallurgical process is developed. To incorporate induction stirring and heating in the simulation model Electromagnetic, Fluid Mechanics and Heat Transfer modules of the solver are coupled [15]. The simulation is performed using two-dimensional axi-symmetric model as shown in Figure 1. The simulation domain consists of a conducting Ladle domain surrounded by induction coils. The coil and ladle domain is enclosed in another domain where the
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
boundary conditions were set as magnetic insulation at the boundary of enclosing domain. The details of the geometry used for modeling and simulation in COMSOL Multiphysics® are based on [2]. Physical properties of steel, induction coil and medium used in simulation can be found in the literature [9], [13], and [14]. WTW
LTW
The initialized mesh consists of 2315 mesh elements and 4684 number of degrees of freedom. Then a mesh refinement was applied on the ladle in order to get better resolution of the forces and the velocity inside the ladle. The Figure 3 shows the mesh generated on the whole domain. From this figure it can be seen that mesh is made denser close to the ladle and the stirrer, and is kept coarser at the boundary.
C O
I
WRW
LADLE
LRW
WLW
L
S
CBW
5.2. Mesh setting for the 2D Axial-symmetric model
LBW
WBW Fig. 3 Initialized Mesh over the 2D domain.
Fig. 2 Wall BC used in Simulation Model
5.1. Assumptions and Boundary Conditions 5.3. Solver settings The following assumptions are made in the statement of the mathematical model of induction-stirred ladle: • The calculations are performed using transient solution mode and one time step. • No thermal radiation is considered for heat transfer simulation Accordingly, following wall boundary conditions were used for the model (Figure 2): 1. The electromagnetic module: • Ladle domain is inactive; • WLW – symmetry axis; • LTW – no shear: velocity gradient is zero; • LRW, LBW – velocity component is zero 2. Induction heating: • WLW – symmetry axis; • WBW, WTW, WRW temperature T0;
–
specified
First the solution was obtained for the electromagnetic module with the use of timeharmonic, direct linear solver. In this way the force fields were calculated and these force fields were used as source terms to solve incompressible Navier-stokes equation where time-harmonic, non-linear iterative solver was used. For heat transfer by conduction timedependent direct linear solver was used. 6.
Results and Discussions
In this section we will present the results obtained from the simulation model of the induction stirred ladle. The results are presented for 2D axial symmetry configuration of the ladle. We will first present the results for the distribution of the magnetic flux distribution inside the ladle because of the induction coils. Figure 4 shows the contours of the magnetic flux density distribution.
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
Fig. 4 Magnetic Flux Density Contours
Induced current density inside the coil is shown in Figure 5 along with the streamlines of the flux inside the ladle. It can be clearly seen from the figure that the induced current density is higher at the surface of the ladle closer to the induction coils. Figure 7 shows the streamlines for the flow of molten steel inside the ladle due to induction stirring. The results are in close agreement with the results presented in [2]. The temperature distribution inside the ladle is shown in Figure 8. The temperature is higher at the surface closes to the inductor due to induction heating and almost zero at the enclosing domain boundary
Fig. 7 Close Streamlines Plot of
Flow inside the Ladle
Fig.5 Magnetic Flux norm Plot
In Figure 5 surface plot of the magnetic flux density is shown. It can be seen from the figure that the flux density is maximum near the coil and almost zero at the boundary of the enclosed domain.
Fig. 7 Surface Plot of Temperature Distribution inside the Ladle
7.
Fig.6 Induced Current density and Flux Streamlines
Conclusions
Two-dimensional axial-symmetric simulation model of induction stirring is presented. The model enables to study the fluid flow pattern along with induction heating phenomenon in
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
case of cylindrical stirrer. It was observed that as the ladle is brought closer to the stirrer the induced current density inside the melt increases thereby increasing the induced force field and hence the velocity of the melt increases. These results obtained from modeling of the ladle refining process with the help of the solver confirm the physics involved in this phenomenon. The results also match with the velocity profile of the melt calculated in [2]. This also validates the solver model.
4.
5.
6. Nomenclature
µ ρ σ k
Magnetic permeability Density of the liquid melt Electrical conductivity Thermal conductivity Heat transfer coefficient Magnetic flux density Electrical potential Volumetric Lorentz force Magnetic flux density Pressure, Pa Heat flow
h B E F H P Q
7. 8.
9.
10.
•
Q Si U Xi U,V,W
Heat flow rate Source term in Xi coordinate Velocity of the conductor in the magnetic field i-component of the coordinate system Axial, radial, and tangential velocity components
11. 12. 13.
References 1.
2.
3.
J Szekely and K Nakanish: Stirring and its Effect on Aluminum De-oxidation in the ASEA-SKF Furnace, Part II, Mathematical Representations of the Turbulent Flow Field and of Tracer Dispersion. Metallurgical Transaction, 1975, 6B, 245 J Szekely and C W Chang: Turbulent electromagnetically driven flow in metals processing: Part 2 practical applications, Ironmaking and Steelmaking, 1977, No3, 196-204 J L Meyer, J Szekely and N EL-Kaddah: A New Method for Computing Electromagnetic Force Fields in Induction
14.
15.
Furnaces Trans, IEEE Transaction on Magnetics, 1987, 23, 1806-1810 W-S Kim and J-K Yoon: Numerical Prediction of Electromagnetically Driven Flow in ASEA-SKF Ladle Refining Straight Induction Stirrer, Ironmaking and Steel 1991, 6, 446-453 N S Saluja: PhD Thesis 1991, Electromagnetic stirring of Metallic Melts: Theory and experiments, Department of Material Science and Engineering, MIT-USA L Jonsson: PhD Thesis, 1998, Department of Material Science and Engineering, KTH Stockholm. Y Sundberg: Lecture notes on Induction Stirring. KTH-Library. 1971 J Alexis: PhD Thesis, 2000, Modeling of the Ladle Furnace with Emphasis on Electromagnetic Phenomenon, Department of Material Science and Engineering, KTHStockholm M Pal: MSc Thesis, 2004, Mathematical Modeling of Induction Stirred Ladle, Department of Material Science and Engineering, KTH-Stockholm M Pal, R Eriksson, and P Jönsson: A CFD model of a 20kg Induction stirred laboratory scaled ladle, In Proceeding of FEMLAB® Conference Oct 2006-Stockholm, Sweden J Szekely: Fluid Flow Phenomena in Metals Processing, Academic Press Inc-US, 1979 J Szekely, G Carlsson, L Helle. Ladle Metallurgy, Springer-Verlag NewYork Inc., 1989 J J Moore: The application of Electromagnetic Stirring in Continuous Casting of Steel, Continuous Casting, Volume 3 ISSAIME, 1984 P A Davidson: An introduction to magnetohydrodynamics. Cambridge University Press, 2001 COMSOL® 3.2 User guide Chemical Engineering, Electromagnetic Module and Heat transfer module.
