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 The solidified steel shell with a liquid pool in the centre is then continuously withdrawn from the bottom of the mould at a constant speed.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 electro- magnetic stirring. These phenomena interact with one another and thus the mod- elling 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 inter- face 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 oc- curs 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 <TL (3) 1 if T TL ≥ Substituting eq (2) into eq (1), we have ∂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 mod- eled 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 molec- ular 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.