Como o leitor pode ter observado, considera¸c˜oes importantes precisam ser acrescen- tadas na modelagem. Este trabalho utilizou o M´etodo de Volumes Finitos baseado em Elementos (EbFVM) afim de resolver numericamente a equa¸c˜ao energia para obter campo de temperatura e utilizou a equa¸c˜ao da conserva¸c˜ao do momento linear para resolver o campo deslocamentos. Assim, a expectativa para os pr´oximos trabalhos ´e a utiliza¸c˜ao do EbFVM na simula¸c˜ao do processo de lingamento cont´ınuo tridimensional, e o tratamento, com o mesmo m´etodo, dos fenˆomenos f´ısicos acoplados, como for¸cas de contato com os rolos e o calor gerado no processo de fric¸c˜ao, e o efeito advectivo do a¸co l´ıquido dentro do molde.
Apresenta-se, abaixo, uma lista de poss´ıveis trabalhos que podem ser desenvolvidos:
I - Implementa¸c˜ao do comportamento Elasto-pl´astico bidimensional/tridimensional; II - Estudos direcionados ao comportamento viscoel´astico e/ou viscopl´astico do mate-
rial;
III - Implementa¸c˜ao do acoplamento forte termomecˆanico em dom´ınio bidimensional/tridimensional;
IV - Efeito advectivo produzido pelo a¸co l´ıquido dentro do tarugo, com efeito fer- rost´atico e for¸ca de corpo;
V - O efeito das trocas de calor dentro do molde levando em conta a forma¸c˜ao de “gaps” entre as paredes do molde e o lingote.
REFERˆENCIAS
[Anjos 2013]. T. P. D. Anjos. Metodologia num´erico-experimental para determina¸c˜ao
da qualidade de a¸cos obtidos por lingotamento cont´ınuo. Disserta¸c˜ao de Mestrado, Universidade Federal do Cear´a, Programa de p´os-gradua¸c˜ao em Engenharia e Ciˆencia de Materiais, Fortaleza-CE (2013)
[Bailey e Cross 1995]. C. Bailey e M. Cross. A finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh. International
journal for numerical methods in engineering, 38(10), 1757–1776 (1995)
[Baliga e Patankar 1983]. B. R. Baliga e S. V. Patankar. A control volume finite- element method for two-dimensional fluid flow and heat transfer. Numerical Heat Trans-
fer, 6(3), 245–261 (1983)
[Bellet et al. 2004]. M. Bellet, A. Heinrich e others. A two-dimensional finite element thermomechanical approach to a global stress-strain analysis of steel continuous casting. ISIJ international, 44(10) (2004)
[Bortoleto 2010]. E. M. Bortoleto. Modelamento num´erico-computacional das trans-
forma¸c˜oes de fase nos tratamentos t´ermicos de a¸cos. Tese de Doutorado, Universidade de S˜ao Paulo (2010)
[Brimacombe et al. 1984]. J. K. Brimacombe, I. V. Samarasekera e J. E. Lait.
Continuous Casting: Heat Flow, Solidification and Crack Formation. Vol. 2. Iron & Steel Society (1984)
[Chandio et al. 2004]. M. S. Chandio, K. S. Sujatha e M. F. Webster. Consis- tent hybrid finite volume/element formulations: model and complex viscoelastic flows.
[Chen et al. 2009]. W. Chen, Y. Z. Zhang, C. J. Zhang, L. G. Zhu, W. G. Lu, B. X. Wang e J. H. Ma. Thermo-mechanical simulation and parameters optimization for beam blank continuous casting. Materials Science and Engineering: A, 499(1), 58– 63 (2009)
[Cordazzo e others 2006]. J. Cordazzo e others. Simula¸c˜ao de reservat´orios de petr´oleo utilizando o m´etodo EbFVM e multigrid alg´ebrico (2006)
[Crisfield 1997]. M. A. Crisfield. Non-linear finite element analysis of solids and struc-
tures: Advanced topics. John Wiley & Sons, Inc. (1997)
[Das 1999]. S. K Das. Thermal modelling of DC continuous casting including submould boiling heat transfer. Applied thermal engineering, 19(8), 897–916 (1999)
[Fengming et al. 2014]. D. Fengming, W. Xudong, Y. Man e Z. Xiaobing. Analysis of the non-uniform thermal behavior in slab continuous casting mold based on the inverse finite-element model. Journal of Materials Processing Technology (2014) [Ferziger e Peri´c 2002]. J. H. Ferziger e M. Peri´c. Computational methods for fluid
dynamics. Vol. 3. Springer Berlin (2002)
[Filippini 2011]. G. Filippini. O M´etodo de volumes finitos baseado em elementos apli-
cado a problemas de elasticidade. Tese de Doutorado, Universidade Federal de Santa Catarina, centro de tecnol´ogico, Programa de p´os-gradua¸c˜ao em Engenharia Mecˆanica, Florian´opolis, SC (2011)
[Frink e Pirzadeh 1998]. N. T. Frink e S. Z. Pirzadeh. Tetrahedral finite-volume so-
lutions to the Navier-Stokes equations on complex configurations. National Aeronautics and Space Administration, Langley Research Center (1998)
[Huespe et al. 2000]. A. E. Huespe, A. Cardona e V. Fachinotti. Thermomecha- nical model of a continuous casting process. Computer methods in applied mechanics
Mecˆanica Industrial (2014). http://www.mecanicaindustrial.com.br/conteudo/ 282-lingotamento-continuo-na-industria. Accessed: 2014-11-14
[Janik e Dyja 2004]. M. Janik e H. Dyja. Modelling of three-dimensional temperature field inside the mould during continuous casting of steel. Journal of Materials Processing
Technology, 157, 177–182 (2004)
[Janik et al. 2004]. M. Janik, H. Dyja, S. Berski e G. Banaszek. Two-dimensional thermomechanical analysis of continuous casting process. Journal of Materials Proces-
sing Technology, 153, 578–582 (2004)
[Li e Thomas 2004]. C. Li e B. G. Thomas. Thermomechanical finite-element model of shell behavior in continuous casting of steel. Metallurgical and Materials transactions
B, 35(6), 1151–1172 (2004)
[Limache e Idelsohn 2007]. A. Limache e S. Idelsohn. On the development of finite volume methods for computational solid mechanics. Mec´anica Computacional, 26(11), 827–843 (2007)
[Maliska 2004]. C. R. Maliska. Transferˆencia de calor e mecˆanica dos fluidos compu-
tacional 2a
Ed. Rio de Janeiro -RJ: LTC - Livros T´ecnicos e Cient´ıficos Editora S.A. (2004)
F. Marcondes e K. Sepehrnoori (2007). Unstructured grids and an element based conservative approach for compositional reservoir simulation. Pages 5–9 of: The 19th
International Congress of Mechanical Engineering, November
[Marcondes e Sepehrnoori 2010]. F. Marcondes e K. Sepehrnoori. An element- based finite-volume method approach for heterogeneous and anisotropic compositional reservoir simulation. Journal of Petroleum Science and Engineering, 73(1), 99–106 (2010)
[Schneider e Zedan 1983]. G. E. Schneider e M. Zedan. Control-Volume-Based Finite Element Formulation of the Heat Conduction Equation, in Spacecraft Thermal Control, Design, and Operation. Prog. Astronaut, 86, 305–327 (1983)
[Slone et al. 2003]. A. K. Slone, C. Bailey e M. Cross. Dynamic solid mechanics using finite volume methods. Applied mathematical modelling, 27(2), 69–87 (2003) [Taylor et al. 1999]. G. A. Taylor, C. Bailey e M. Cross. Computational solid
mechanics using a vertex-based finite volume method. Finite Volumes for Complex
Applications II: Problems and Perspectives, 507–515 (1999)
[Vertnik e ˇSarler 2014]. R. Vertnik e B. ˇSarler. Solution of a continuous casting of steel benchmark test by a meshless method. Engineering Analysis with Boundary
Elements, 45, 45–61 (2014)
[Voller 2009]. V. R. Voller. Basic control volume finite element methods for fluids and
solids. Vol. 1. World Scientific Publishing (2009)
[Wheel 1999]. M. A. Wheel. A mixed finite volume formulation for determining the small strain deformation of incompressible materials. International journal for nume-
A. APˆENDICE
Este apˆendice tem por finalidade expor as tabelas utilizadas na simula¸c˜ao.
A.1 Tabelas Utilizadas na Simula¸c˜ao
Tabela A.1: Parˆametros da equa¸c˜ao do coeficiente global de transferˆencia de calor na interface molde/metal
A B C
Raio Externo 1652,54982 0,076 1012,3554
Raio Interno 1382,75331 0,11003 892,5904
Face Lateral 1323,34964 0,10339 908,72076
Tabela A.2: Vaz˜ao dos sprays (ls−1)
Zonas dos Sprays
1a Zona 2a Zona 3a Zona
3, 07 3, 02 2, 67
Tabela A.3: Coeficientes de transferˆencia de calor nas regi˜oes dos sprays
Zonas dos Sprays Radia¸c˜ao/Convec¸c˜ao natural
1a Zona 2a Zona 3a Zona 154(W m−2K−1)
741(W m−2K−1) 734(W m−2K−1) 679(W m−2K−1)
Fonte: Adaptado de Anjos [2013].
Tabela A.4: Propriedades mecˆanicas para o a¸co 0,3% carbono
T (K) E(GP a) ν α(K−1) σ Y(M P a) 1173 32,378 0,33 25, 0 × 10−6 14,0 1273 20,0 0,33 11,0 1373 14,542 0,33 8,0 1473 12,896 0,33 29, 092 × 10−6 5,5 1573 11,954 0,33 4,0 1673 8,608 0,36 41, 653 × 10−6 3,5 1723 5,062 0,40 1728 65, 649 × 10−6 1,9 1763 0,097 0,41 0 0,5 1770 0,002 0,41 0 0.5