Simulation of continuous casting of steel by a meshless technique R. Vertnik*1 and B. Sˇarler2 A recently developed local radial basis function collocation method is used for the solution of the transient convective–diffusive heat transport in continuous casting of steel. The solution of the thermal field with moving boundaries due to phase-change and the growing computational domain is based on the mixture continuum formulation. The growth of the domain and the movement of the starting block are described by activation of additional nodes and by the movement of the boundary nodes through the computational domain, respectively. Time-stepping is performed in an explicit way by a simple characteristic procedure. A two-dimensional transient test case solution is shown at different times and its accuracy is verified by comparison with the reference finite volume method results. The method is very attractive in the present context due to its trivial implementation of curved geometry for two and three dimensions, accuracy and stability of the results. Keywords: Steel, Continuous casting, Simulation, Meshless, Radial basis functions, Simple characteristic procedure
Introduction Continuous casting1 is currently the most common process for production of steel. During casting, the process parameters are changing with time. The fluctuations of the process parameters, i.e. casting temperature or temperature of the cooling water, can significantly influence the quality of the strand. Surface and subsurface defects usually appear at places where transient phenomena occur. To keep a uniform quality of the strand and to prevent defects during the casting process, an accurate online control must be obtained. The accuracy and flexibility of the online control can be achieved by appropriate preliminary studies on the basis of off-line computer models. Off-line computer models can greatly improve the understanding of transients. On the basis of off-line models, simplified on-line models are developed, which are used to control and automate the casting process by compensating for the transient changes. Many different numerical methods have been used in the past to solve the related continuous casting models. In this paper, the recently developed local radial basis function collocation method (LRBFCM) is used to calculate transient temperature fields in the curved strand. The method was already developed for diffusive problems,2 convective–diffusive problems with phasechange3 and direct-chill (DC) casting problems for aluminium alloys with growing domains.4 The method was extended to cope with the curved geometry of a casting machine and with highly convection dominated 1ˇ 2
Store Steel d.o.o., Zˇelezarska cesta 3, SI-3220 Sˇtore, Slovenia University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia
*Corresponding author, email
[email protected]
ß 2009 W. S. Maney & Son Ltd. Received 17 June 2008; accepted 12 September 2008 DOI 10.1179/136404609X368064
problems,5 specific for continuous casting of steel. The adaptive upwind technique (AUT) was used to stabilise the numerical approximation due to the convection dominated situation. Here, a simple characteristic procedure (SCP)6 is proposed as an alternative to the AUT, because of its trivial numerical implementation.
Governing equations Consider a connected growing domain V with boundary c occupied by a phase change material described with the density rb, specific heat at constant pressure cb of the phase b, effective thermal conductivity k, and the specific latent heat of the solid–liquid phase change hm. The mixture continuum formulation of the enthalpy conservation for the assumed system is L ðrhÞz+:ðr~ vhÞ~+:ðk+T Þ Lt
(1)
The mixture density is defined as r~fSV rS zfLV rL , the mixture velocity is defined as r~ v~fSV rS~ vS zfLV rL~ vL , and the mixture enthalpy is defined as h~fSV hS zfLV hL , with subscripts S and L denoting the solid and the liquid phase, respectively. The constitutive mixture temperature – mixture enthalpy relationships are ðT ðT cS dT, hL ~hS (T)z (cL {cS )dTzhm (2) hS ~ Tref
TS
with Tref and TS standing for the reference temperature and solidus temperature, respectively. All material properties can arbitrarily depend on the temperature. The liquid volume fraction fLV is assumed to linearly vary from 0 to 1 between solidus TS and liquidus temperature TL. By assuming the initial temperature, velocity field, and boundary conditions at time t0, the
International Journal of Cast Metals Research
2009
VOL
22
NO
1–4
311
Vertnik and Sˇ arler
Simulation of continuous casting of steel by a meshless technique
1 Geometry of billet caster with typical node arrangement and 5 point influence domain schematics
+:ðk0 +T0 Þ(x{Dx,y{Dy) &+:(k0 +T0 )h0
mixture temperature at the time t0zDt can be found, where t0 is the starting time of the casting process and Dt is the time step.
{Dx
The governing equation is discretised in 2D real curved geometry (see Fig. 1) with Cartesian coordinates px, py with base vectors ix, iy by using the SCP in its explicit form. In this approach, the equation (1) is written in Lagrange formulation as:
W(p)&
(4)
where v- x and v- y are average velocity values of vx and vy along the characteristic Lv v- x ~v0x 1{Dt LvLx0x , v- y ~v0y 1{Dt Ly0y , (5) where v0x and v0y are known velocity values at the initial time. The characteristic is a line that connects the material point at the beginning of the time step with the same material point at the end of the time step. In equation (3) the convective term disappears and diffusive and source terms are averaged quantities along the characteristic. The explicit time discretisation of equation (3) along the characteristic is 1 h{h0 ðx{Dx,y{DyÞ ~+:ðk0 +T0 Þðx{Dx,y{DyÞ (6) Dt
Numerical examples The transient simulation of the Sˇtore–Steel billet caster9 with the simplified boundary conditions and material properties presented. The boundary conditions are: (i) the inner and outer surface – Robin boundary condition with the following heat transfer coefficients: mould 2000 W m22 K21, radiation and sprays 600 W m22 K21, and rolls
It is known that the solution of the above equation in moving coordinates leads to mesh updating and presents difficulties. An alternative way is to expand the terms into a Taylor expansion Lh0 Lh0 h0 (x{Dx,y{Dy) &h0 {Dx (7) (x,y) {Dy (x,y) , Lx Ly
VOL
22
(9)
where yn stands for the shape functions, an for the coefficients of the shape functions, rn for the radial distance between two collocation points in a subdomain, and c for the shape parameter, respectively. The coefficients an are calculated by collocation with the multiquadric radial basis function7,8 as the shape function. The growth of the domain is performed by moving the bottom boundary nodes according to the casting velocity and time step length in casting direction Dcl5v0l?Dt. The unknown values of the moving boundary nodes G(t0zDt) are set from equation (9). When the distance between the moving boundary nodes and the fixed domain nodes exceeds two times the typical grid distance of the node arrangement, new inner nodes are inserted between the moving boundary nodes and the fixed inner nodes. Their values are obtained by equation (9).
(3)
Dx~v- x Dt, Dy~v- y Dt:
2009
yn (p)an ; yn (p)~ r2n (p)zc2 ;
r2n (p)~(p{pn ):(p{pn ),
where Dx and Dy are the distances travelled in the x- and y-direction described as:
International Journal of Cast Metals Research
N X n~1
L(rh) (p(t),t)~+:(k0 +T0 ); p(t) Lt ~(px (t0 )zDx):ix z(py (t0 )zDy):iy
312
(8)
In the LRBFCM the domain and the boundary are covered by overlapping influence domains. Each of the influence domains includes N neighboring nodes. The arbitrary function W is represented over a set of N nodes on each of the influence domains in the following way
Solution procedure
r0
L : L ½+ (k0 +T0 )(x,y) {Dy ½+:(k0 +T0 )(x,y) Lx Ly
NO
1–4
Vertnik and Sˇ arler
Simulation of continuous casting of steel by a meshless technique
new inner nodes. A minimal discretisation man-effort was put into the transition of the discretisation from the flat to the real curved geometry. The extension of the method to cope with higher dimensions is straightforward. The stability problem of the dominated convection is for the first time solved in the LRBFCM context by using the explicit SCP. The accuracy of the new method is tested by comparison of the results with the Fluent software package. A very good agreement between both methods is observed. The method was chosen as the core in the simulation system of the Sˇtore Steel billet caster.9,10 Our ongoing research is focused on the extension of the method to calculate the turbulent fluid flow of the continuous casting processes and to couple the macroscopic with the microscopic simulations. 2 Calculated temperature on centreline (upper curves) and at outer side (lower curves). t5340 s
1200 W m22 K21, with the reference temperature set to 293?15 K (ii) the top surface – Dirichlet boundary condition with the temperature 1763?5 K and (iii) the bottom surface – Neumann boundary condition with zero heat flux with the following simplified material properties: rS ~rL ~7800 kg m{3 , kS ~35 W m{1 K{1 ,
Acknowledgements The first author would like to acknowledge the Public Agency for Technology of the Republic of Slovenia for support in the framework of the PhD programme. The second author would like to thank the Slovenian Research Agency for support in the framework of the project L2–7204 Modelling and Optimisation of Continuous Casting, which was part-financed by the European Union, European Social Fund.
kL ~52 W m{1 K{1 , cS ~700 J kg1 K{1 ,
References
cL ~800 J kg1 K{1 and hm ~227 kJ kg{1
1. W.R. Irwing: ‘Continuous Casting of Steel’, 1993, London, The Institute of Materials. 2. B. Sˇarler and R. Vertnik: Comp, Math.Applic., 2006, 51, 1269– 1282. 3. R. Vertnik and B. Sˇarler: Int.J.Numer.Methods Heat Fluid Flow, 2006, 16, 617–640. 4. R. Vertnik, M. Zalozˇnik and B. Sˇarler: Eng.Anal., 2006, 30, 847– 855. 5. R. Vertnik, B. Sˇarler, Z. Bulinski and G. Manojlovic´: Proc. 2nd Int. Conf. of ‘Simulation and Modelling of Metallurgical Processes in Steelmaking’, Graz, Austria, September 2007, ASMET. 6. O.C. Zienkiewicz and R.L. Taylor: ‘The Finite Element Method, Fitfh edition, Volume 3: Fluid dynamics’, 2000, London, Butterworth–Heinemann. 7. G.R. Liu: ‘Mesh Free Methods’, 2003, Boca Raton, CRC Press. 8. M.D. Buhmann: ‘Radial Basis Function’, 2003, Cambridge, Cambridge University Press. 9. B. Sˇarler, R. Vertnik, H. Gjerkesˇ, A. Lorbiecka, G. Manojlovic´, J. Cesar, B. Marcˇicˇ and M. Sabolicˇ Mijovicˇ: WSEAS transactions on systems and control, 2006, 2, 294–299. 10. B. Sˇarler, R. Vertnik, S. Sˇaletic´, G. Manojlovic´ and J. Cesar: Bergund Hu¨ttenma¨nnische Monatshefte, 2005, 9, 300–306. 11. A. Fic, A.J. Nowak and R. Białecki: Eng.Anal. Boundary Elements, 2000, 24, 215–223. 12. B. Sˇarler, R. Vertnik and J. Perko: Comp.Mat.Cont., 2005, 2, 77– 83. 13. Y.T. Gu and G.R. Liu: Comp.Mech., 2006, 38, 171–182.
The results are compared with the results obtained from the fluid dynamics software Fluent, which is based on the finite volume method (FVM). In Fluent, the growth of the domain was realised by the dynamic mesh mode. Figure 1 represents the geometry of the caster with the node arrangement during the simulation. Due to the high temperature gradients at the boundary, the node arrangement is denser near the boundary. Figure 2 shows the calculated temperature along the billet on the outer side and the centreline obtained with the LRBFCM and the FVM.
Conclusions This paper represents a solution of the transient temperature field in the two-dimensional curved geometry of the steel billet caster by a meshless technique. The governing equation is solved in its strong form on a moving domain, where both the material and the inter-phase boundary are simultaneously moving. The growing of the computational domain is performed by moving the bottom boundary nodes and by inserting
International Journal of Cast Metals Research
2009
VOL
22
NO
1–4
313
