• No results found

2. ANALYSE AV DYPFORVITRING

2.2 Geologisk bakgrunn

Individualmente, cada modelo apresentou resultados consistentes, e comparativamente foram semelhantes entre si, sendo as pequenas diferenças de resultados devidamente compreendidas e justificadas. Apesar das várias simplificações adotadas, os resultados forem satisfatórios. Cada forma de abordagem, seja experimental, analítica ou numérica, complementa as lacunas deixadas pelas outras, dessa forma, podemos dizer que os 3 modelos se completam na busca de uma melhor compreensão do fluxo em meios granulares.

Conclui-se por fim, que para o material estudado, o gnaisse britado, podemos definir 3 padrões distintos de comportamento, em função de sua razão D/d. Para valores de razão D/d menores que 4, dificilmente ocorrerá fluxo granular, sendo portanto dispensáveis ensaios nesta faixa.

Para valores da razão D/d entre 4 e 5, entra-se numa zona de transição, na qual a ocorrência de fluxo é incerta. Nesta faixa, recomenda-se o uso da simulação numérica, através do método dos elementos discretos, pois este permite analisar os possíveis problemas de obstrução que podem ocorrer nesta faixa.

Por último, para razões D/d maiores que 5, a ocorrência de fluxo granular é praticamente certa, e desta forma, o modelo cinemático se torna a melhor opção, pois permite resultados rápidos, robustos, como campos vetoriais e perfis de velocidade, e é de fácil implementação. Todavia, em todos os casos, é necessário o conhecimento de parâmetros constitutivos do material, e estes devem ser medidos com critério para que sejam representativos.

Em trabalhos futuros, poderá ser abordado o tratamento tridimensional destes modelos, ou também a modelagem de materiais compostos por mais de uma faixa granulométrica, ou seja, materiais polidispersos. Pode-se também testar outras formas de obtenção de parâmetros constitutivos, como a densidade e ângulo de repouso. O ângulo de repouso, por exemplo, poderia ser estimado a partir de um ensaio de inclinação, também conhecido como tilt test. Por fim, como última sugestão, pode-se variar a geometria do silo, desde o ângulo de suas paredes à própria escala do modelo.

REFERÊNCIAS BIBLIOGRÁFICAS

ANAND, A. et al. Predicting discharge dynamics from a rectangular hopper using the discrete element method (DEM). Chemical Engineering Science, v. 63, n. 24, p. 5821-5830, 2008. BALEVIČIUS, R. et al. Discrete-particle investigation of friction effect in filling and unsteady/steady discharge in three-dimensional wedge-shaped hopper. Powder Technology, v. 187, n. 2, p. 159-174, 2008.

BALEVIČIUS, R. et al. Analysis and DEM simulation of granular material flow patterns in hopper models of different shapes. Advanced Powder Technology, v. 22, n. 2, p. 226-235, 2011.

BAZANT, M. Z. A Theory of Cooperative Diffusion in Dense Granular Flows. p. 26, 2003.

BELHEINE, N. et al. Numerical simulation of drained triaxial test using 3D discrete element modeling. Computers and Geotechnics, v. 36, n. 1-2, p. 320-331, 2009.

BEVERLOO, W.A., LENIGER, H. A., VAN DE VELDE, J. The flow of granular solids through orifices. Chemical Engineering Science, September 1961, Vol.15 (3-4), pp.260-269. BRADY, B. H. G., BROWN, E. T. Rock Mechanics for underground mining. 3rd ed. 2004, XVIII, 626 p.

CAMONES, L. A. M. et al. Application of the discrete element method for modeling of rock crack propagation and coalescence in the step-path failure mechanism. Engineering Geology, v. 153, p. 80-94, 2013.

CAPRIZ, G. et al. Mathematical Models of Granular Matter. Springer. 2008. 212 p.

CHEN G. Stochastic modeling of rock fragment flow under gravity. International Journal

of Rock Mechanics and Mining Sciences. v. 34. Issue 2. p. 323-331 1997.

CHIALVO, S.; SUN, J.; SUNDARESAN, S. Bridging the rheology of granular flows in three regimes. Phys Rev E Stat Nonlin Soft Matter Phys, v. 85, n. 2 Pt 1, p. 021305, Feb 2012.

CHOI, J.; KUDROLLI, A.; BAZANT, M. Z. Velocity profile of granular flows inside silos and hoppers. Journal of Physics-Condensed Matter, v. 17, n. 24, p. S2533-S2548, Jun 22 2005.

CLEARY, P. W.; SAWLEY, M. L. DEM modelling of industrial granular flows: 3D case studies and the effect of particle shape on hopper discharge. Applied Mathematical

Modelling, v. 26, n. 2, p. 89-111, Feb 2002.

COETZEE, C. J.; ELS, D. N. J. The numerical modelling of excavator bucket filling using DEM. Journal of Terramechanics, v. 46, n. 5, p. 217-227, 2009.

COETZEE, C. J.; ELS, D. N. J. Calibration of discrete element parameters and the modelling of silo discharge and bucket filling. Computers and Electronics in Agriculture, v. 65, n. 2, p. 198-212, 2009.

COETZEE, C. J.; ELS, D. N. J.; DYMOND, G. F. Discrete element parameter calibration and the modelling of dragline bucket filling. Journal of Terramechanics, v. 47, n. 1, p. 33-44, 2010.

CRISPIM, P. H. S. Parâmetros de influência na lavra por abatimento em Sublevel

Caving. Trabalho de conclusão de curso. Universidade Federal de Ouro Preto. 2010

GODA, T. J.; EBERT, F. Three-dimensional discrete element simulations in hoppers and silos. Powder Technology, v. 158, n. 1-3, p. 58-68, 2005.

GÖNCÜ, Fatih. Mechanics of Granular Materials: Constitutive Behavior and Pattern Transformation. PhD Thesis. Delft University of Technology – Delft Center for Computational Science and Engineering – DCSE. 2012.

GONZÁLEZ-MONTELLANO, C. et al. Validation and experimental calibration of 3D discrete element models for the simulation of the discharge flow in silos. Chemical

Engineering Science, v. 66, n. 21, p. 5116-5126, 2011.

HAMBLEY, D. F. Design of ore pass systems for underground mines. CIM Bulletin, 80, pp. 25-30, 1987.

HILL, J. M.; SELVADURAI, A. P. S. Mathematics and Mechanics of Granular Materials. Springer. 2005. 324 p.

ITASCA CONSULTING GROUP. PFC2D – Particle Flow Code in 2D dimensions. Version 4.0 – 32 bit. 2008.

JAEHYUK, C.; KUDROLLI, A.; BAZAN, M. Z. Velocity profile of granular flows inside

silos and hoppers. J. Phys. Condens. Matter, special issue on granular media, p. 1-18. 2005.

JANELID I.; KVAPIL R. Sublevel caving. International Journal of Rock Mechanics and

Mining Sciences & Geomechanics Abstracts. v. 3. Issue 2. p. 129-132. 1966.

JENIKE, A. W. Storage and Flow of Solids. University of Utah, 1964. 198 p.

JUST, G. D.; FREE, G. D.; BISHOP, G. A. Optimization of ring burden in sub-level caving.

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. v. 10. p. 119-131. 1973.

KETTERHAGEN, W. R. et al. Granular segregation in discharging cylindrical hoppers: A discrete element and experimental study. Chemical Engineering Science, v. 62, n. 22, p. 6423-6439, 2007.

KETTERHAGEN, W. R.; HANCOCK, B. C. Optimizing the design of eccentric feed hoppers for tablet presses using DEM. Computers & Chemical Engineering, v. 34, n. 7, p. 1072- 1081, 2010.

KVAPIL R. Gravity flow of granular materials in hoppers and bins. International Journal of

Rock Mechanics and Mining Sciences & Geomechanics Abstracts. v. 2. Issue 1. p. 25-34.

1965.

KVAPIL R. Gravity flow of granular materials in hoppers and bins II. International Journal

of Rock Mechanics and Mining Sciences & Geomechanics Abstracts. v. 2. Issue 3. p. 277-

292. 1965.

LANGSTON, P. A. et al. Distinct element modelling of non-spherical frictionless particle flow. Chemical Engineering Science, v. 59, n. 2, p. 425-435, 2004.

LIMA, E. L. Geometria analítica e álgebra linear. 2ª ed. Rio de Janeiro: Instituto de Matemática Pura e Aplicada – IMPA, 2006. 324 p.

LI, Y.; XU, Y.; THORNTON, C. A comparison of discrete element simulations and experiments for ‘sandpiles’ composed of spherical particles. Powder Technology, v. 160, n. 3, p. 219-228, 2005.

LI, W. C. et al. Discrete element modeling of a rainfall-induced flowslide. Engineering

Geology, v. 149-150, p. 22-34, 2012.

LITWINISZYN, J. The model of a random walk of particles adopted to researches on problems of mechanics of granular. Bulletin of the Polish Academy Science. v11 p. 9-61, 1963

MASSON, S.; MARTINEZ, J. Effect of particle mechanical properties on silo flow and stresses from distinct element simulations. Powder Technology, v. 109, n. 1-3, p. 164-178, Apr 3 2000.

MENDONÇA, G. M. Manual de normalização para apresentação de trabalhos

acadêmicos. Salvador: UNIFACS. 2009. 83 p.

MEHTA, A. Granular Physics. Cambridge University Press. 2007. 305 p. MINITAB INC. Minitab 16. Version 16.1.1. 2010.

MISHRA, B. K.; THORNTON, C. An improved contact model for ball mill simulation by the discrete element method. Advanced Powder Technology, v. 13, n. 1, p. 25-41, 2002.

MULLINS, W.W. Experimental evidence for the stochastic theory of particle flow under gravity. Powder Technology. v. 9, p. 29-37. 1974.

NEDDERMAN, R. M. Statics and Kinematics of Granular Materials. Cambridge University Press. 1992. 372 p.

NEDDERMAN, R. M. et al. The flow of granular materials I - Discharge rates from hoppers,

Chemical Engineering Science. v. 37. Issue 11. p. 1597-1609. 1982.

NEDDERMAN, R. M. The measurement of the velocity profile in a granular material discharging from a conical hopper. Chemical Engineering Science. v. 43. Issue 7. p. 1507- 1516. 1988.

NEDDERMAN, R. M. The use of the kinematic model to predict the development of the stagnant zone boundary in the batch discharge of a bunker. Chemical Engineering Science. v. 50. Issue 6. p. 959-965. 1995.

NEVES, C. E. V. Comportamento de materiais granulares usando o método dos

elementos discretos. 2009. 166 p. Dissertação (Mestrado em Geotecnia) – Universidade de Brasília – UnB, Brasília, 2009.

PLASSIARD, J. P.; BELHEINE, N.; DONZE, F. V. A spherical discrete element model: calibration procedure and incremental response. Granular Matter, v. 11, n. 5, p. 293-306, Oct 2009.

PUCKETT, J. G. et al. Trajectory entanglement in dense granular materials. Journal of

Statistical Mechanics-Theory and Experiment, Jun 2012.

RAJI, A. O.; FAVIER, J. F. Model for the deformation in agricultural and food particulate materials under bulk compressive loading using discrete element method. I: Theory, model development and validation. Journal of Food Engineering, v. 64, n. 3, p. 359-371, 2004. RAO K. K., NOTT P. R. An Introduction to Granular Flow. 1ª ed. Cambridge University Press. 2008. 513 p.

RICHEFEU, V.; COMBE, G.; VIGGIANI, G. An experimental assessment of displacement fluctuations in a 2D granular material subjected to shear. Geotechnique Letters, v. 2, p. 113- 118, Jul-Sep 2012.

RYCROFT, C. H.; ORPE, A. V.; KUDROLLI, A. Physical test of a particle simulation model in a sheared granular system. Phys Rev E Stat Nonlin Soft Matter Phys, v. 80, n. 3 Pt 1, p. 031305, Sep 2009.

SAHNI, E.; YAU, R.; CHAUDHURI, B. Understanding granular mixing to enhance coating performance in a pan coater: Experiments and simulations. Powder Technology, v. 205, n. 1- 3, p. 231-241, 2011.

SILVA, J. M. Estudo do fluxo de material fragmentado na mineração subterrânea, com

uso de modelos físicos. Tese de doutorado. Universidade Federal de Minas Gerais – UFMG. 2005.

SILVA, J. M.; GRIPP, M. F. A.. Fluxo de material fragmentado em passagem de minério nas minas subterrâneas: a prática corrente. Rem: Rev. Esc. Minas, Ouro Preto, v. 59,n. 3,Sep 2006.

SILVA FILHO, A. B.; SILVA, J. M.; CURI, A. Efeito da rugosidade recortada polida no escoamento em modelo de passagem de minério. Rem: Rev. Esc. Minas, Ouro Preto, v. 63,n. 4,Dec. 2010.

STUART, J. Calculus: Early Transcendentals, McMaster University. 6ª ed. Thomsom Brooks/Cole. 2008. 1308 p.

TAO, H. et al. Discrete element method modeling of non-spherical granular flow in rectangular hopper. Chemical Engineering and Processing: Process Intensification, v. 49, n. 2, p. 151-158, 2010.

TIJSKENS, E.; RAMON, H.; BAERDEMAEKER, J. D. Discrete element modelling for process simulation in agriculture. Journal of Sound and Vibration, v. 266, n. 3, p. 493-514, 2003.

TÜZÜN, U.; NEDDERMAN, R. M. Experimental evidence supporting kinematic modelling of the flow of granular media in the absence of air drag. Powder Technology. v. 24, Issue 2, November – December 1979, Pages 257-266.

TÜZÜN, U et al. The flow of granular materials II - Velocity distributions in slow flow,

Chemical Engineering Science. v. 37. Issue 12. p. 1691-1709. 1982.

ULRICH, S.; ZIPPELIUS, A. Stability of freely falling granular streams. Phys Rev Lett, v. 109, n. 16, p. 166001, Oct 19 2012.

VAN LIEDEKERKE, P. et al. A discrete element model for simulation of a spinning disc fertilizer spreader I. Single particle simulations. Powder Technology, v. 170, n. 2, p. 71-85, 2006.

VAN LIEDEKERKE, P. et al. DEM simulations of the particle flow on a centrifugal fertilizer spreader. Powder Technology, v. 190, n. 3, p. 348-360, 2009.

VAN ZEEBROECK, M. et al. The discrete element method (DEM) to simulate fruit impact damage during transport and handling: Case study of vibration damage during apple bulk transport. Postharvest Biology and Technology, v. 41, n. 1, p. 92-100, 2006.

WAMBAUGH, J. F.; HARTLEY, R. R.; BEHRINGER, R. P. Force networks and elasticity in granular silos. Eur Phys J E Soft Matter, v. 32, n. 2, p. 135-45, Jun 2010.

WHITTLES, D. N. et al. An investigation into the parameters affecting mass flow rate of ore material through a microwave continuous feed system. Advanced Powder Technology, v. 16, n. 6, p. 585-609, 2005.

WONG, C. X.; DANIEL, M. C.; RONGONG, J. A. Energy dissipation prediction of particle dampers. Journal of Sound and Vibration, v. 319, n. 1-2, p. 91-118, 2009.

YANG, S. C.; HSIAU, S. S. The simulation and experimental study of granular materials discharged from a silo with the placement of inserts. Powder Technology, v. 120, n. 3, p. 244-255, Oct 22 2001.

WIKIPEDIA, Granular material, <http://en.wikipedia.org/wiki/Granular_material>. Feb 10 2014.

ZHU, H. P. et al. Discrete particle simulation of particulate systems: Theoretical developments. Chemical Engineering Science, v. 62, n. 13, p. 3378-3396, 2007.

A. TABELA DE DADOS DOS ENSAIOS DE DENSIDADE