Modelling of Turbulent Flow and Multi-Phase Heat Transfer Under Electromagnetic Force

DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS–SERIES B Volume 8, Number 3, October 2007 pp. 695–706

MODELLING OF TURBULENT FLOW AND MULTI-PHASE HEAT TRANSFER UNDER ELECTROMAGNETIC FORCE

Yong Hong Wu Department of Mathematics & Statistics Curtin University of Technology, GPO Box U1987 Perth WA 6845 Australia B. Wiwatanapataphee Department of Mathematics, Faculty of Science Mahidol University, 272 Rama 6 Road Bangkok, ZIP 10400, THAILAND

Abstract. In this paper, we develop a mathematical model and a numerical technique to study the coupled turbulent flow and heat transfer process in continuous steel casting under an electromagnetic force. The complete set of field equations are established and solved numerically. The influences of the electromagnetic field on a flow pattern of molten steel and the temperature field as well as steel solidification are presented in the paper.

1. Introduction. Continuous steel casting is a heat extraction process for casting steel products from molten steel. In this process, molten steel is poured continuously from a tundish through a submerged entry nozzle into a water-cooled mould where intensive cooling results in a thin solidified steel shell to form around the edge of the casting, as shown in figure 1. The solidified steel shell with a liquid pool in the centre is then continuously withdrawn from the bottom of the mould at a constant speed. To control the fluid flow pattern and the steel solidification process, an electromagnetic field, generated from the source current through the coil, is imposed to the system. The magnetic field induces electric currents in molten steel and consequently generates a body force, namely the electromagnetic force or Lorentz force. This body force acts on the molten steel and consequently influences the flow of the molten steel and the steel solidification process. Over the last few decades, extensive studies have been carried out worldwide to model various aspects of the continuous casting process, in particular the heat transfer and steel solidification process [9, 23], the electromagnetic stirring [11, 13, 14, 17], the flow phenomena [18, 21] and the formation of oscillation marks [10]. However, as analyzed by Thomas [19], the continuous casting process involves a staggering complexity of at least eighteen interacting phenomena at the mechanistic level. Due to this complexity, in the past, research was focussed mainly on the modelling of each individual phenomenon or interaction of two or three phenomena only. Hence, it is a worthwhile undertaking to develop a sophisticated model capable of dealing with the staggering complexity of the interacting phenomena including turbulence, convection heat transfer, phase change and electromagnetic stirring.

2000 Mathematics Subject Classification. Primary: 65M60, 76D05; Secondary: 35Q60. Key words and phrases. Turbulent flow, heat transfer,electromagnetic force, modelling. The first author is affiliated with Curtin University of Technology, Perth, Western Australia. 695 696 Y.H. WU AND B. WIWATANAPATAPHEE

Figure 1. The continuous steel casting process.

In this work, we further develop our coupled heat transfer - turbulent flow model [21, 24] by incorporating the effect of the electromagnetic field. The rest of the paper is organized as follows. In section two, a complete set of field equations are presented. In section three, a brief description of the solution method is given. In section four, a numerical study is presented to demonstrate the influence of the electromagnetic field on the flow of molten steel in the central liquid pool and the distribution of temperature as well as solidification of steel.

2. Mathematical model. The continuous casting process involves many complex phenomena including turbulent flow, heat transfer with phase change and electromagnetic stirring. These phenomena interact with one another and thus the modelling of the continuous casting process constitutes one of the most outstanding mathematical modelling problems. The key element in simulating the heat transfer, the steel solidification process, is the tracking of the phase change interface. At present two major types of methods, diffuse and sharp interface methods [2, 5, 12, 20, 25], are available for tracking the phase change interfaces. Diffuse interface methods transmit the effects of the interface to the temperature and species field equation solver through source terms in the transport partial differential equations, while sharp interface methods construct the spatial differential operators at points that adjoin the interface in such a way that the interfacial jump conditions are incorporated. In comparison with sharp interface methods, diffuse interface methods are much easier to implement and are particularly effective for problems in which the interface is not a sharp surface but spreads out over a certain thickness. In continuous steel casting, phase change occurs over a temperature range and hence the melt/solid interface can be described as a mushy zone. Typically, the thickness of the mushy zone is about 20-25 mm at the exit of the casting mould of width 200 mm. A single domain enthalpy method, as a diffuse interface method, has thus been developed to simulate the heat transfer and steel solidification process. In this method, based on the principle of energy conservation, the equation for the heat transfer in the region undergoing a phase change is MULTI-PHASE HEAT TRANSFER & TURBULENT FLOW 697

∂Ht ρ + uj Ht,j = (k T,j )j (1) ∂t 0 where ( ),j denotes differentiation with respect to xj ,uj represents the velocity · component of fluid in the xj direction, ρ and k0 are respectively the density and the thermal conductivity of steel. The enthalpy Ht is defined as the sum of sensible heat h = cT and Latent heat H = Lf as follows

Ht = cT + Lf (2) where c and L are the speciﬁc heat and Latent heat of liquid steel, and f is the liquid fraction which could be approximated linearly by

0 if T TS T −TS ≤ f(T ) =  TL−TS if TS

∂T ∂H ρc + uj T,j = (k T,j ),j ρ + ujH,j (4) ∂t 0 − ∂t It should be remarked here that the last term on the right hand side of equation (4) is equal to zero everywhere except in the region where phase change occurs. Hence, equation (4) can be applied to all the regions including the solid region, the mushy region and the molten steel region. To model the flow of molten steel in the central liquid pool, the molten steel is assumed as an incompressible Newtonian fluid. The flow in the mushy region is modeled by Darcy’s law for porous media. Thus, the unified field equations governing the fluid flow for all the regions with or without phase change are as follows ui,i = 0, (5)

∂ui ρ + uj ui,j + p,i (µf (ui,j + uj,i),j = Fi(ui, xi,t)+ ρgi + Femi, (6) ∂t − where Fi is determined, according to [1], by 2 µf [1 f(T )] Fi(ui, xi,t) = C − (ui (Ucast)i) (7) ρf(T )3 − The influence of the electromagnetic field on the transport of momentum is modeled by the addition of the electromagnetic force in the momentum conservative equation (6). Based on on our previous work in [3], the electromagnetic force can be determined by Fem = J ( A) (8) × ∇× where A is the magnetic vector potential which is governed by the following equation 1 ( A) = J (9) µ∇× ∇× ∂A where J = Js σ σ φ, µ and σ denote the magnetic permeability and elec- − ∂t − ∇ troconductivity, Js is the source current density, φ is a scalar potential function. It should be addressed that in deriving equation (9) from the Maxwell’s equations, we have neglected the influence of the displacement current and flow induced current on the magnetic field generated by the source current. 698 Y.H. WU AND B. WIWATANAPATAPHEE

The influence of turbulence on the transport of momentum and energy is modeled by the addition of the turbulent viscosity µt to the laminar viscosity µ0 and the cµt turbulent conductivity kt = σt to the molecular conductivity k0, yielding the effective viscosity µf and the effective thermal conductivity kf given by

µf = µ0 + µt, kf = k0 + kt, (10) where σt is the turbulent Prandtl number [1]. Thus, the turbulent effect can be easily incorporated into the model by replacing the laminar viscosity and molecular conductivity in equations (5)-(6) respectively by the effective viscosity and effective conductivity. Various models, such as the mixing-length type model, the one-equation model and the two-equation (K ε) model, have been proposed for − calculating µt. As the flow in the continuous casting process is complex with circu- lation and with a phase change boundary, the low-Reynold number K ε model is used in this work, namely the turbulent kinetic energy K and the dissipation− rate ε are determined by

∂K µt µt ρ + uj K,j ((µ0 + )K,j ),j = βgj T,j + µtG ρε, (11) ∂t − σK − σt − 2 ∂ε µt εµt ε ε ρ + uj ε,j ((µ0 + )ε,j ),j = C1(1 C3) βgj T,j + C1 µtG ρC2fε , ∂t − σε − Kσt K − K (12) where G = 2ǫij ǫij with ǫij = (ui,j + uj,i) /2. The constants involved in the above two equations are empirical constants and are taken to have the following values [7, 21]: σt = 0.9, σk = 1, σε = 1.25, Cµ = 0.09, C1 = 1.44, C2 = 1.92, C3 = 0.8. Based on the work in [1, 15, 22], the turbulent viscosity is determined by 0.09f ρK2 µ = µ , f = f(T )exp( 3.4/(1 + R /50)2), (13) t ε µ t p − 2 where Rt = ρK /(µε) denotes the local turbulent Reynold number. To ensure that all the terms in eqs(11)-(12) are deﬁned as K approaches zero in the near-wall region, the last term on the right hand side of equation (12) is multiplied by a 2 −Rt damping function fε =1 Aεe , where Aε is a constant and is chosen as one if −4 − K < 10 or otherwise Aε =0.3 [6, 22].

3. Method of solution. As shown in section two, the electromagnetic field problem can be uncoupled from the fluid flow - heat transfer problem. Thus, the electromagnetic field problem is solved first to yield the electromagnetic force for sub- sequent fluid flow and heat transfer analysis. For the electromagnetic field, in this work, we are concerned with two-dimensional problems with A, J and φ taking the following forms in the coordinate system as shown in Figure 2,

A = (0, A2(x,z,t), 0), J = (0, J2(x,z,t), 0), φ = constant. (14) Substituting equation (14) into (9), we have

∂A2 A ,jj = µσ µJs . (15) 2 ∂t − 2 iωt For the case of sinusoidal source current, i.e. Js2 = jse , the above equation admits a solution of the following form iωt A2 = a(x, z)e (16) MULTI-PHASE HEAT TRANSFER & TURBULENT FLOW 699

Figure 2. Computational domain (a = 0.1 m.) and equation (15) becomes 2 a,jj β a = µjs, (17) − − where β2 = σµωi and i = √ 1. To solve equation (17) numerically,− we firstly develop the following associated variational boundary value problem: Find a H1(Ω) such that w H1(Ω) ∈ 0 ∀ ∈ 0 2 (a,j, w,j )+ β (a, w) = µ(js, w) (18) where w is the so called weight function or test function, ( , ) denotes the inner 2 · · 1 product on the square-integrable function space L (Ω) and H0 (Ω) is defined as follows ∂v ∂v H1(Ω) = v v, , L2(Ω) and v =0 on ∂Ω . (19) 0 | ∂x ∂z ∈ The Galerkin finite element method is then used to discretize the problem in space to yield the following system of equations N 2 (ψi,j , ψk,j )+ β (ψi, ψk) ai = µ(js, ψk) (k =1, 2, ..., N). (20) Xi=1 Once the solution of the above system is obtained, we can determine the magnetic vector potential A by A = eiωt[0, a(x, z), 0] (21) and consequently we can calculate J and Fem. For the coupled fluid flow - heat transfer problem, various numerical methods can be used for the solution, such as the projection method [4] and the penalty function method [8]. In this study, we use the penalty function method to weaken the continuity requirement by uj,j = δp (22) − where the penalty parameter δ is a small positive number. The effect of penalization is to relax the incompressibility condition (5). For more details on the mathematical aspect of the method, the reader is referred to [8, 16]. Thus, by substituting (22) into (6), the pressure variable can be eliminated from the momentum equations. 700 Y.H. WU AND B. WIWATANAPATAPHEE

(a) (b)

Figure 3. Influence of the penalty parameter δ on the downward velocity profile on a typical horizontal cross-section of the casting at 250 mm below the meniscus: (a) downward velocity distribution across the cross-section for different δ values; (b) variation of the downward velocity with δ at a typical point 20 mm from the mould wall on the cross-section ( for -log10δ=0, 1, 2, 3 and 4, δ=1, 0.1, 0.01, 0.001, 0.0001, respectively).

Figure 4. Model predictions of the downward velocity at the location 250 mm below the meniscus and 25 mm from the mould wall, obtained by using seven diﬀerent ﬁnite element meshes respectively with 1115, 2278, 3064, 5839, 7949, 9306 and 10649 elements.

Consequently, for two dimensional cases, we have a closed system of five partial differential equations, (4), (6) and (11)-(12), in terms of five coordinate and time- dependent unknown functions (u1,u2,T,K and ε). To find the numerical solution, the governing partial differential equations are discretized in space by the Galerkin finite element method to yield the following system of nonlinear ordinary differential equations M ˙q + Kq = f(q), (23) N where q = (u1i, u2i,Ti,Ki, εi) i=1 represent the values of u1, u2,T,K and ε on the finite element{ nodes (i =1, 2,} ..., N). The matrix M corresponds to the transient terms in the governing partial differential equations, the matrix K corresponds to the advection and diffusion terms, and the vector f depends nonlinearly on ui,T,K MULTI-PHASE HEAT TRANSFER & TURBULENT FLOW 701

(a) (b)

Figure 5. Comparison of temperature profiles and downward velocity distributions on a horizontal cross-section at 400 mm below the meniscus, obtained by models with turbulence effect (solid line) and with no turbulence effect (dotted line): (a) temperature profile; (b) downward velocity and ε. To keep details of the paper to minimum, the specific form of each of the matrices and vectors are omitted here. The numerical solutions to the nonlinear discretization system with appropriate boundary conditions are then obtained by using an iterative scheme [21] developed based on the backward Euler differentiation scheme. The following convergence condition was used in the simulation Rm+1 Rm k i − i k tol, (24) Rm ≤ k i k where the superscript m + 1 and m denote iterative computation steps, Ri denotes the solution vector of the ith variable on the finite element nodes, is the Euclidean norm and tol is a small positive constant. k·k

4. Numerical investigation and discussion. The influence of the electromagnetic field on the coupled turbulent flow and steel solidification is investigated in the present study. The example under investigation is a rectangular caster which has a width of 0.1 m and a depth of 0.4 m in the x z plane. The computation region is as shown in Figure 2. The finite element mesh,− used in this study, consists of 7949 triangular elements with a total of 101,443 degrees of freedom. The system parameters are as listed in Table 1. To determine a proper value for the penalty parameter δ, a sensitivity study is carried out to investigate how the numerical results are affected by the parameter. In the investigation, the parameter δ is varied from 0.5 to 0.2, 0.1, 0.05, 0.005 and 0.0005, while all other parameters are kept at the same values as in Table 1. The numerical results show that at large value, δ has very significant influence on the velocity field while the influence decreases as δ decreases. Figure 3 shows the profiles of the downward velocity component on a typical horizontal cross-section of the casting at 0.25 m below the meniscus obtained using different δ values. Our numerical results show that when δ < 0.005, the velocity field in molten steel almost remains unchanged for any further decrease in the δ value. Hence, the use 702 Y.H. WU AND B. WIWATANAPATAPHEE

e 2 (1) Js = 10, 000A/m

e 2 (2) Js = 50, 000A/m

e 2 (3) Js = 100, 000A/m

Figure 6. Influence of external current density on (a) the magnetic flux density B; (b) the magnetic potential Az; (c) The electromagnetic force Fem. of δ=0.0005 will ensure that the numerical results obtained are independent of the penalty parameter δ. As numerical error decreases as element size decreases, a grid resolution study is carried out to investigate the effect of finite element mesh on numerical results. Seven different finite element meshes, respectively having 1115, 2278, 3064, 5839, 7949, 9306 and 10649 triangular elements, are used in this investigation. Figure 4 shows the model predictions of the downward velocity, at a typical location 250 mm below the meniscus and 10 mm from the mould wall, obtained by using the seven MULTI-PHASE HEAT TRANSFER & TURBULENT FLOW 703

Figure 7. Influence of source current density on the magnitude of electromagnetic force at the horizontal section 55 mm below the meniscus. different meshes. The results show that the velocity starts to converge when the number of elements reaches 7949. Hence, in this study, we use the finite element mesh with 7949 elements for the analysis and the results obtained can be considered to be independent of the finite element mesh used. To demonstrate the effect of turbulence, a numerical study is carried out using a model with turbulence and a model with no turbulence. Figure 5 shows comparison of the temperature profiles and the downward velocity distributions, on a horizontal cross-section at 400 mm below the meniscus, obtained respectively by the two different models. The results indicate that turbulence has considerable effect on the velocity and temperature fields.

Table 1. Parameters used in numerical simulation

Parameter Symbol Value Unit o Pouring temperature Tm 1530 C o Molten temperature TL 1525 C o Solidiﬁcation temperature TS 1465 C Density ρ 7850 kg/m3 Viscosity µ0 0.001 pa s Speciﬁc heat c 465 J/kg·oC o Thermal conductivity of steel k0 35 W/m C Latent heat L 2.72 105 J/kg Morphology constant C 1.8 ×106 m−2 × Casting speed Ucast 0.028 m/s Magnetic permeability of vacuum µ 4π 10−7 Henry/m Electric conductivity σ × Ω−1m−1 - steel 4.032 106 - coil 1.163×107 Electric permittivity of vacuum ǫ 8.8540 ×10−12 Farad/m Penalty parameter δ 0.0005× Tolerance tol 1.0 10−6 time step ∆t 0.0001× sec. 704 Y.H. WU AND B. WIWATANAPATAPHEE

e 2 (1) Js = 10, 000A/m

e 2 (2) Js = 50, 000A/m

e 2 (3) Js = 100, 000A/m

Figure 8. Influence of external current density on the fluid flow and heat transfer (a) velocity field of molten steel; (b) Temperature profiles.

Figure 6 shows the magnetic ﬂux density vector B (i.e. A), the contour ∇× plot of a(z,t) and the electromagnetic force Fem corresponding to diﬀerent external source current density js. The results show that the electromagnetic force acts on the molten steel basically in the horizontal direction toward the central line. This force will contribute to preventing molten steel from sticking to the mould wall and MULTI-PHASE HEAT TRANSFER & TURBULENT FLOW 705

Figure 9. Influence of source current density on the temperature profile at the horizontal section 400 mm below the meniscus. smoothing the steel casting surface. The results have also shown that the magnitude of the force can be controlled by controlling the imposed source current density. The magnitude of the force increases as the current density increases as shown in Figure 7. Figure 8 shows the influences of source current density on velocity and temperature fields in molten steel. The electromagnetic field applied to the system suppresses the melt flow and results in reduction of velocity in the mould region and leads to more uniform melt flow below the mould. The suppression of the jet melt flow, by the imposed electromagnetic field, causes the reduction of ad- vective heat transfer to the casting surface. Therefore, superheat is not removed sufficiently on the casting surface, resulting in the increase of energy level in the overall liquid region and the increase in temperature gradients near the solidified shell. The increased temperature gradient near the solidifying shell increases the diffusion heat flux to the shell surface, resulting in a thicker solidified shell. The temperature profiles on a horizontal section 400 mm below the meniscus (i.e. at the exit of the mould) for various different current densities are shown in Figure 9. With the increase of current density, the thickness of the solidified steel shell increases significantly.

Acknowledgements. The authors gratefully acknowledge the Australian Research Council for the support of the project through an ARC Discovery Project Grant. The authors are also grateful to the referees for their valuable comments and sug- gestions.

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