199 mathematical modeling of mold-filling and solidification - Acta ...

More documents

Recommendations

Info

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 206 where T l and T s are the liquidus and the solidus temperatures (K), respectively. Another widely used expression is the Scheil equation, which assumes uniform solute concentration in the liquid but no diffusion in the solid [23]: −1/1−k ⎛ T o m − T ⎞ f s = 1 − ⎜ ⎟ Tm T ⎝ − l ⎠ (12) where T m is the melting point of the solvent metal (K),and k o is the equilibrium partition coefficient of the alloy. Eq. (10) defines the heat flux [3] which is released during liquid cooling and solidification. In classical models, the heat generated after solidification is assumed to be zero, i.e., for T< T s , Q & = 0. However, experimental investigations showed that lattice defects and vacancies are condensed in the already solidified part of the crystal and the enthalpy of the solid increases and thus the latent heat will decrease. Based on this fact, another way to represent the change of the solid fraction during solidification can be written as [3]: 2 ⎡ π ( T − T ) ⎤ ( T − T ) + ( Ts − T )(1 − cos l l l ⎢ ⎥ π ⎣ 2( Ts − Tl ) f = ⎦ s ( Tl − Ts )(( 1− 2 / π ) (13) Combining Eqs. (7), (9) and (10), the new unsteady state heat conduction equation can be expressed in the form [4; 23]: ∂ ( ρ c´ T ) = ∇.( k∇ T) ∂ t (14) At the range of temperatures where solidification occurs for binary metallic alloys, the physical properties will be evaluated taking into account the amount of liquid and solid that coexists in equilibrium at each temperature, as shown by Eqs. (9), (10) and (14). 3. Conclusions A comprehensive theoretical basis regarding equations of fluid mechanics and heat flow has been shown. Like other CFD problems, the momentum equation, the continuity equation, and the energy equation are the partial differential equations that are essential in the numerical analysis of the casting process. The moving boundary liquid metal/air is modeled using the VOF method to trace the free surface in mold filling processes. The latent heat release during the unsteady solidification phenomenon is taken into account as a source term in the energy equation. The implementation of such equations is aimed to furnish important information such as filling dynamics and the thermal field along the casting process. Acknowledgements The authors acknowledge financial support provided by CNPq (The Brazilian Research Council), FAPESP (The Scientific Research Foundation of the State of São Paulo, Brazil) and CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior.
Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 207 Literature [1] Guo Z., Saunders N., Miodownik A.P., Schill J.-Ph.: Materials Science and Engineering, vol. A 413–414, 2005, pp.465–469. [2] Shepel S.V., Paolucci S.: Applied Thermal Engineering, vol.22, 2002, pp-229-2418. [3] Radovic Z., Lalovic M.: Journal of Materials Processing Technology, vol.160, 2005, pp.156–159. [4] Katzarov I.H.: International Journal of Heat and Mass Transfer, vol.46, 2003, pp.1545- 1552. [5] Zaidi K., Abbes B., Teodosiu C.: Computer Methods in Applied Mechanics and Engineering, vol.144, 1997, pp.227-233. [6] Son G., Dhi V.K.: International Journal of Heat and Mass Transfer, vol.51, 2008, pp.1156– 1167. [7] Handbook Ansys. 2008 Inc. Canonsburg, PA. [8] Aboutalebi M.R., Guthrie R.I.L., Seyedein S.H.: Applied Mathematical Modelling, vol.31, 2007, pp.1671–1689. [9] Sharma A.K., Velusamy K. Balaji C.: Annals of Nuclear Energy. vol.35, 2008, pp.1502- 1514. [10] Cheng T.S., Yang W.J.: Computers & Fluids, vol.37, 2008, pp.194–206. [11] Ha M.Y., Lee H.G., Seong S.H.: Journal of Materials Processing Technology, vol.133, 2003, pp.322-339. [12] Pericleous K., Bojarevics V., Djambazov G., Harding R.A., Wickins M.: Applied Mathematical Modelling, vol.30, 2006, pp.1262–1280. [13] Im I.-T., Kim W.-S., Lee K.-S.: International Journal of Heat and Mass Transfer, vol.44, 2001, pp.1507-1515. [14] McBride D., Croft T.N., Cross M.: Computers & Fluids, vol. 37, 2008, pp.170–180. [15] Hirt C.W., Nichols B.D.: Journal of Computational Physics, vol. 39, 1981, pp. 201-225. [16] Kermanpur A., Mahmoudi S., Hajipour A.: Journal of Materials Processing Technology. vol.206, 2008, pp.62-68. [17] Kim K.D., Yang D.Y., Jeong J.H.: Computer Methods in Applied Mechanics and Engineering, vol.195, 2006, pp.6799–6821. [18] Babaei R., Abdollahi J., Homayonifar P., Varahram N., Davami P.: Computer Methods in Applied Mechanics and Engineering, vol.195, 2006, pp.775–795. [19] Miao Y.-C., Zhang X.-M., Di H.-S., Wang G.D.: Journal of Materials Processing Technology, vol.174, 2006, pp.7–13. [20] Janik M., Dyja H.: Journal of Materials Processing Technology, vol.157–158, 2004, pp.177-182. [21] Ferreira I.L., Spinelli J.E., Pires J.C., Garcia A.: Materials Science and Engineering, vol. A 408, 2005, pp.317–325. [22] Santos C.A., Fortaleza E.L., Ferreira C.R.F., Spim J.A., Garcia A.: Modelling and Simulation in Materials Science and Engineering, vol.13, 2005, p.1071–1087. [23] Shi Z., Guo Z.X.: Materials Science and Engineering, vol.A365, 2004, pp. 311–317.
Page 1 and 2: Acta Metallurgica Slovaca, 15, 2009
Page 7: Acta Metallurgica Slovaca, 15, 2009

199 mathematical modeling of mold-filling and solidification - Acta ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?