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Solution of Macrosegregation in Continuously Cast Billets by a Meshless Method

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2012 IOP Conf. Ser.: Mater. Sci. Eng. 27 012058 (http://iopscience.iop.org/1757-899X/27/1/012058) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 50.2.15.218 This content was downloaded on 05/05/2017 at 12:36 Please note that terms and conditions apply.

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The 3rd International Conference on Advances in Solidification Processes IOP Publishing IOP Conf. Series: Materials Science and Engineering 27 (2011) 012058 doi:10.1088/1757-899X/27/1/012058

Solution of Macrosegregation in Continuously Cast Billets by a Meshless Method R Vertnik1,2, B Šarler1 and B Senčič2 1 University of Nova Gorica, Vipavska 13, 5000 Nova Gorica, Slovenia 2 Štore Steel, d.o.o., Železarska c. 3, 3220 Štore, Slovenia E-mail: [email protected], [email protected], [email protected] Abstract. The main aim of this paper is to demonstrate the applicability and the advantages of a novel meshless method for simulation of macrosegregation in steel billets. The physical model is established on a set of macroscopic equations for mass, energy, momentum, species, turbulent kinetic energy, and dissipation rate in two dimensions. The mixture continuum model is used to treat the solidification system. The mushy zone is modelled as a Darcy porous media with Kozeny-Karman permeability relation, where the morphology of the porous media is modelled by a constant value. The incompressible turbulent flow of the molten steel is described by the Low-Reynolds-Number (LRN) k-epsilon turbulence model, closed by the Launder and Sharma closure coefficients and damping functions. The microsegregation equations rely on lever rule. The numerical method is established on explicit timestepping, and collocation with multiquadrics radial basis functions on non-uniform five-noded influence domains, and adaptive upwinding technique. The velocity-pressure coupling of the incompressible flow is resolved by the explicit Chorin’s fractional step method, with the intermediate velocity field, calculated without the pressure term. A recently proposed standard continuous casting configuration with Fe-C system has been used for verification of the model. The advantages of the method are its simplicity and efficiency, since no polygonisation is involved, easy adaptation of the nodal points in areas with high gradients, almost the same formulation in two and three dimensions, high accuracy and low numerical diffusion.

1. Introduction Continuous casting [1] is the most common process for production of steel. Nowadays, there is an increasing demand on improving the process and casting machines to achieve higher casting speed, higher quality, lower production cost and defect free casting. Macrosegregation is one of the major strand defects, and represents the non-uniformity of chemical composition in the cast section. The cause of segregation is the difference in solubility of solute in the solid and liquid phase. Usually, for most of the alloys, the solubility in the liquid phase is larger than in the solid phase. Therefore, during the solidification, the alloying elements are rejected from the solid-liquid interface into the liquid. In the liquid region, these solutes are further re-distributed by the convection (forced and natural) and diffusion, and this leads to macrosegregation. It is almost impossible to get a suitable measurements of the macrosegregation inside the mould during the casting process, due to the very high temperatures of liquid steel pouring into the mould. Our knowledge and understanding of these phenomena mostly relay on the physical models [2], solved by various numerical models [3,4,5]. In this paper, recently developed meshless Local Radial Basis Function Collocation Method (LRBFCM) [6] is used to solve the mass, momentum, energy and species conservation equations

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The 3rd International Conference on Advances in Solidification Processes IOP Publishing IOP Conf. Series: Materials Science and Engineering 27 (2011) 012058 doi:10.1088/1757-899X/27/1/012058

which govern the solidification, fluid flow, heat transfer and solute transport model of the continuous casting process. The method was first developed for diffusive problems [6], than for convectivediffusive problems with phase-change [7], direct-chill casting problems of aluminium alloys with material moving boundaries [8], and coupled turbulent fluid flow and heat transfer with solidification of the continuous casting process [9]. The method was recently used in solving the macrosegregation test case [10]. The velocity-pressure coupling is solved by the fractional step method [11], where the pressure equation is solved directly through the sparse matrix [12]. The low-Reynolds number two-equation turbulence model [13,14] is used to incorporate the turbulence effects. This numerical procedure was previously tested for solving various incompressible turbulent flows such as 2D channel and a backward facing step [15] flow. 2. Governing equations The incompressible turbulent flow of the continuous casting of carbon steel can be reasonably described by the following Reynolds time-averaged transport equations for mass, energy, momentum and species conservation ∇⋅u = 0 , ρ

ρ

K 0 (1 − f L2 ) ∂u 2 + ρ∇ ⋅ ( uu ) = −∇P + ∇ ⋅  2 ( µ + µt ) S  − ρ∇k − µ L (u − uS ) + f (1),(2) ∂t 3 f L3

  µ ∂h + ρ∇ ⋅ ( uh ) = ∇ ⋅ ( λ∇T ) + ρ∇ ⋅  f S ( hL − hS )( u − u S ) + ∇ ⋅  f L t ∇hL  and ∂t  σT 

(3)

∂C + ρ∇ ⋅ ( uC ) = ∇ ⋅ ( f S DS ∇CS + f L DL ∇C L ) + ρ∇ ⋅ ( u − u S )( C L − C ) ∂t ,   f L ⋅ µt + ∇⋅ ∇C L   σC 

(4)

ρ

with u , P , h , T and C standing for velocity, pressure, enthalpy, temperature and solute concentration, respectively, and ρ , µ , µ t , f L , λ , Ds and Dl are standing for density, molecular dynamic viscosity, turbulent dynamic viscosity, liquid fraction, thermal conductivity, diffusivity of solute in solid phase and diffusivity of solute in liquid phase, respectively. The third term in equation (2) represents the Darcy term, where K 0 is the morphology constant of the porous medium and u S is velocity of the solid phase. S stands for the Newtonian strain-rate tensor and f for the vector of thermal and solutal buoyancy forces. The turbulent dynamic viscosity is defined as

µ t = ρ cµ f µ

k2

ε

(5)

where k and ε are kinetic energy and dissipation rate, respectively. They are calculated by the following transport equations K 0 (1 − f L2 )  ∂k µt   k, ρ + ρ∇ ⋅ ( uk ) = ∇ ⋅  µ L +  ∇k  + Pk − ρε + ρ D + µ L σk   ∂t f L3 

(6)

K 0 (1 − f L2 )  ∂ε µt   ε + ρ∇ ⋅ ( uε ) = ∇ ⋅  µ L + ρ ε.  ∇ε  + ρ ( c1ε f1 − c2 ε f 2ε ) + ρ E + µ L σε   ∂t k f L3 

(7)

In (3)-(7) cµ , f µ , c1ε , f1 , c2ε , f 2 , σ T , σ C , σ k and σ ε are the closure coefficients, and D and

E are the extra source terms of the low-Reynolds turbulent model, defined in [13], except σ C = 1.0 . 2

The 3rd International Conference on Advances in Solidification Processes IOP Publishing IOP Conf. Series: Materials Science and Engineering 27 (2011) 012058 doi:10.1088/1757-899X/27/1/012058

In order to calculate heat and fluid flow on a fixed domain Ω with boundary Γ , the system of equations (1)-(4), (6) and (7) have to be solved. The liquid fraction f L , concentration of solute in liquid phase C L , partition ratio k p , and liquidus temperature TL are determined based on the level rule microsegregation model, i.e. fL = 1−

1 T − TL C , TL = Tm + (Te − Tm ) , 1 − k p T − Tm Ce

CL =

(8),(9)

C C and k p = S , CL 1 + f S ( k p − 1)

(10),(11)

where Tm , Te , C e and k p are melting temperature, eutectic temperature, solute concentration in eutectic alloy and partition coefficient, respectively. They are obtained from the equilibrium phase diagram for the iron-carbon system (i.e. Tm = 1539 °C, Te = 1147 °C, C e = 4.3 % and k p = 0.48 ).

3. Solution procedure We seek the solution of the velocity field, pressure field, temperature field, solute concentration field and k and ε fields at time t + t0 by assuming known fields u , P , T , C , k and ε at time t0 and known boundary conditions at time t > t0 . The coupled set of mass conservation (1) and momentum conservation (2) are solved by the fractional step method [11], where the continuity of the mass (1) is considered by constructing the pressure Poisson’s equation. The governing equations are discretized by using the explicit time discretization. This leads to the following algorithm, described step-by-step 1) The intermediate velocity components are calculated without the pressure gradient  K 0 (1 − f L2 ) 2 ∆t  − + u = u +  − ρ∇ ⋅ ( uu ) + ∇ ⋅  2 ( µ + µ t ) S  − ρ∇k − µ L u u f  ( ) S ρ  3 f L3  *

n

n

(12)

2) The pressure Poisson equation (13) is treated by solving the related sparse matrix [12]. The velocity components are corrected by the pressure gradient, i.e. (14).

∇2 p n+1 =

ρ ∆t

∇ ⋅ u* , un+1 = u∗ −

∆t

ρ

⋅ ∇P n+1

(13),(14)

3) After the solution of the velocity field, the transport equations for energy and solute concentration are solved

h

n+1

  µ ∆t  = h +  − ρ∇ ⋅ ( uh ) + ∇ ⋅ ( λ∇T ) + ρ∇ ⋅  f S ( hL − hS )( u − u S ) + ∇ ⋅  f L t ∇hL   ρ   σT  n

C n+1 = C n +

n

(15)

∆t

 − ρ∇ ⋅ ( uC ) + ∇ ⋅ ( f S DS ∇CS + f L DL ∇CL ) + ρ∇ ⋅ ( u − uS )(CL − C ) + ρ    f ⋅µ + ∇ ⋅  L t ∇CL   σC 

(16)

n

4) The kinetic energy k n+1 and the dissipation rate ε n+1 of the turbulence model are solved in the same way as equation (15) for the enthalpy or equation (16) for the solute concentration.

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The 3rd International Conference on Advances in Solidification Processes IOP Publishing IOP Conf. Series: Materials Science and Engineering 27 (2011) 012058 doi:10.1088/1757-899X/27/1/012058

5) The turbulent viscosity (equation (5)), solute concentration in liquid (equation (10)) and solid phase (equation (11)), and liquidus temperature (equation (9)) are updated for each node. The solution is set ready for the next time step. All derivatives of the transport equations are calculated by the LRBFCM, where the collocation is made locally on overlapping sub-domains. On each sub-domain, the scalar function Φ (standing for temperature, velocity component, pressure, solute concentration, k and ε ) is represented over a set of (in general) non-equally spaced nodes p n ; n = 1,2,..., l N in the following way K

Φ ( p ) ≈ ∑ψ k ( p )α k , ψ k ( p ) =  rk2 + c 2  , 1/2

(17),(18)

k =1

where ψ k stands for the multi-quadric radial basis shape functions [16], α k for the coefficients of the shape functions, and K represents the number of the shape functions. In (18) c represents the shape parameter and rk the radial distance between two points in the sub-domain. The detailed procedure of calculating the derivatives with the LRBFCM can be found in [6,9]. 4. Numerical example The solution of the simplified model of the continuous casting process in 2D is presented. The steadystate solution is shown, approached by a false transient calculation using a fixed time-step of 5.0·10-4 s. The geometry of the simplified casting machine is shown in figure 1. The length of the mould is 0.8 m, and the length of the whole computational domain is 1.8 m in order to account for the spray cooling of the billet surface bellow the mould. The cross-section of the billet is a square. The simplified material properties of the carbon steel are used (i.e. ρ =7200 kg/m3, cp,S=cp,L=700 Jkg/K, λS=λL=30 W/(m⋅K), K0=1.67⋅1010 m-2, DS=1.6⋅10-11 m2/s and DL=1.0⋅10-10 m2/s). These simplifications are used mainly for straightforward verification of the developed numerical model, and in order to reduce the CPU time of the simulations. Although, due to the simple inclusion of physics in the developed meshless numerical method and dimensional flexibility, a very few changes are needed to extend the numerical algorithm to perform the calculations in curved geometry and three dimensions with more sophisticated physics. 4.1.1. Boundary conditions The boundary conditions for the velocity and temperature are set as follows: SEN outlet: The velocity component in the casting direction u y , k and ε are pre-calculated with the numerical model for the 2D turbulent channel flow. Temperature and solute concentration are constant, equal to the pouring temperature and initial concentration C0 = 0.8 wt%, respectively. Billet end: The pressure outlet is prescribed, where the following boundary conditions are used P = 0;

∂u y ∂ux ∂k ∂ε ∂T ∂C =0; =0; =0; = 0; =0; =0. ∂y ∂y ∂y ∂y ∂y ∂y

(19)

Top surface (meniscus): At the meniscus, the symmetry boundary conditions are used (free surface flow). The vertical velocity u y is set to zero and the normal derivatives of all other variables are set to zero. Moving walls: The walls with the solidified steel are moved with the casting velocity along the casting direction. At the walls, where the liquid phase exists, the no-slip boundary conditions for the velocity are set. In the mold, the Robin boundary condition is used, with the surface heat transfer coefficient 2000 W/(m2K). Bellow the mould, at the secondary cooling system, the heat transfer coefficient equals to 800 W/(m2K). The normal derivative of the solute concentration is set to zero.

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The 3rd International Conference on Advances in Solidification Processes IOP Publishing IOP Conf. Series: Materials Science and Engineering 27 (2011) 012058 doi:10.1088/1757-899X/27/1/012058

4.1.2. Numerical results The velocity field, temperature field, and solute concentration fields, obtained at casting superheat ∆T = 30 K, are shown in figure 1. Due to the very large value of the morphology constant, appearing in the Darcy’s term, very small amount of the carbon are rejected from the mushy zone into the liquid region. The effect of the casting superheat ∆T on the carbon segregation is presented in figure 2, where the segregation ratio profiles at two different vertical positions are shown. As expected, the segregation ratio increases with higher casting superheat.

Figure 1. From left to right: Casting geometry with typical dimensions (d2=0.035 m, ab=0.14 m). Node arrangement details. Absolute velocity field. Temperature field (black lines represents liquidus and solidus isotherms). Carbon concentration field.

Figure 2. Segregation ratio profile of the carbon at different vertical positions as a function of superheat. Left: at 0.8 m, right: at 1.8 m. 5. Conclusions This paper represents the solution of the macrosegregation simulation of the continuous casting of steel by a meshless method. The turbulent fluid flow is modelled by the low-Re number turbulence model with the closure coefficients proposed by Launder and Sharma. The continuum mixture formulation, introduced by Bennon and Incropera, is used to govern the macroscopic physical model. The results of the simplified 2D numerical model are shown. The effect of the casting superheat on the

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The 3rd International Conference on Advances in Solidification Processes IOP Publishing IOP Conf. Series: Materials Science and Engineering 27 (2011) 012058 doi:10.1088/1757-899X/27/1/012058

carbon segregation is analysed, where a higher segregation was obtained at a higher casting superheat. However, more realistic results are expected with the full 3D numerical model which development is underway. The meshless numerical model is based on the local collocation with the radial basis functions for spatial disretization and first order explicit finite difference method for time discretization. Due to its locality and explicit time stepping, the method is very appropriate for parallelization. Non-uniform node arrangement is easily generated, since it does not rely on polygons. The partial differential equations are solved in their strong form, hence no integration is needed. The transition from twodimensional to three-dimensional cases is quite straightforward. The main advantages of the present numerical approach represent simple coding and lack of any polygonisation. Acknowledgement Financial support from the Slovenian Research Agency in the framework of the Research Programme P2-0379 and project L2-3651 is kindly acknowledged. References [1] Irwing W R 1993 Continuous Casting of Steel (London: The Institute of Materials) [2] Bennon W D and Incropera F P 1987 A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems - I. Model formulation Int. J. Heat Mass Tran. 30 2161-2170 [3] Kang K G, Ryou H S and Hur N K 2005 Coupled turbulent flow, heat, and solute transport in continuous casting processes with an electromagnetic brake Numer. Heat Tr. A-Appl. 48 461-481 [4] Yang H, Zhao L, Zhang X, Deng K, Li W and Gan Y 1998 Mathematical simulation on coupled flow, heat, and solute transport in slab continuous casting process Metall. Mater. Trans. B 29 1345-1356 [5] Reza Aboutalebi M, Hasan M and Guthrie R I L 1995 Coupled turbulent flow, heat, and solute transport in continuous casting processes Metall. Mater. Trans. B 26 731-744 [6] Šarler B and Vertnik R 2006 Meshfree explicit radial basis function collocation method for diffusion problems Comput. Math. Appl. 51 1269-1282 [7] Vertnik R and Šarler B 2006 Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems Int. J. Numer. Method. H. 16 617640 [8] Vertnik R, Založnik M and Šarler B 2006 Solution of transient direct-chill aluminium billet casting problem with simultaneous material and interphase moving boundaries by a meshless method Eng. Anal. Bound. Elem. 30 847-855 [9] Vertnik R 2010 Heat and fluid flow simulation of the continuous casting of steel by a meshless method : dissertation (Nova Gorica: University of Nova Gorica) (ECCOMAS awarded) [10] Kosec G, Založnik M, Šarler B and Combeau H. 2011 A meshless approach towards solution of macrosegregation phenomena CMC-Comput. Mater. Con. 22 169-195 [11] Chorin A J 1967 A numerical method for solving incompressible viscous flow problems J. Comput. Phys. 2 12-26 [12] Lee C K, Liu X and Fan S C 2003 Local muliquadric approximation for solving boundary value problems Comput. Mech. 30 395-409 [13] Launder B E and Sharma B I 1974 Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc Lett. Heat Mass Trans. 1 131–138 [14] Wilcox D C Turbulence modeling for CFD (California: DCW Industries, Inc.) p 477 [15] Vertnik R and Šarler B 2009 Solution of incompressible turbulent flow by a mesh-free method CMES-Comp. Model. Eng. 44 66-95 [16] Buhmann M D 2003 Radial Basis Function: Theory and Implementations (Cambridge: Cambridge University Press) p 258

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