282654 Analysis of Melt Flow of Bridgman Process Using Lagrangian Coherent Structures

Wednesday, October 31, 2012: 12:55 PM
Frick (Omni )
Mojtaba Izadi, Chemical & Materials Engineering, University of Alberta, Edmonton, AB, Canada, Youssef Belhamadia, University of Alberta, Edmonton, AB, Canada and Stevan Dubljevic, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada

Bridgman process is one of the commonly used methods for the production of melt-grown crystals. In the inverted Bridgman method, the lower zone of the furnace with temperatures above the melting point of the material is separated from the upper zone with a temperature below melting point by an adiabatic baffle. The crucible containing the solid material is lowered into the hot zone and after temperature stabilization, the crucible is raised slowly into the upper zone, so that the crystal grows from the melt.

In this process, thermal convection plays an important role by affecting heat and mass transfer. It has been demonstrated that during the formation of a crystal from its melt, mixing properties of the melt flow affect transport phenomena. If complete mixing in the melt does not occur, a gradient of impurities concentration exists in the melt near the crystal interface.

In the framework of dynamical systems theory, it is well-known that there exist special flow patterns in a fluid domain which are the skeletons of observed tracer patterns and govern transport structure and mechanics of the flow. These patterns are referred to as coherent structures and when captured in terms of quantities derived from particle trajectories, they are called Lagrangian coherent structures (LCS). Recently, effective computational algorithms have been developed for the identification of these structures based on the eigenvalues and eigenvectors of the finite-time deformation tensor.

In this work, we study the kinematics of mixing of the melt flow in Bridgman process by computing the LCS in the melt domain. To this end, velocity field is obtained based on the mixed finite element resolution of the coupled governing equations including Navier-Stokes, continuity and energy equations. Having velocity data, fluid element trajectories are obtained numerically and used for the computation of deformation tensor at each point in the melt domain, which finally reveals the LCS. Providing a novel insight into the transport of the flow, LCS can be considered as stable and unstable manifolds in the phase space of the dynamical systems. Because all of these manifolds are invariant, meaning that fluid particles do not cross them, the fluid that is trapped in the intersection of these manifolds (lobes) is confined to remain in them as time evolves. Therefore, the motion of lobes in terms of entrainment and detrainment helps explain the transport and mixing processes.

References:

1. G. Haller, Distinguished Material Surfaces and Coherent Structures in Three-Dimensional Fluid Flows, Physica D 149, 2001, 248--277.

2. S. C. Shadden, F. Lekien, J. E. Marsden, Definition and Properties of Lagrangian Coherent Structures from Finite-Time Lyapunov Exponents in Two-Dimensional Aperiodic Flows, Physica D 212, 2005, 271--304.

3. S. C. Shadden, J. O. Dabiri, J. E. Marsden, Lagrangian Analysis of Fluid Transport in Empirical Vortex Ring flows, Phys. Fluids 18, 2006, 047105.

4. G. Haller, A Variational Theory of Hyperbolic Lagrangian Coherent Structures, Physica D 240 (7), 2011, 574--598.


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See more of this Session: The Use of CFD In Simulation of Mixing Processes
See more of this Group/Topical: North American Mixing Forum