Thermal Transport During Seeding and Shouldering in the Czochralski Crystal Growth

2000 ◽  
Author(s):  
A. K. Saha ◽  
H. Zhang ◽  
V. Prasad
Keyword(s):  
Free Surface ◽  
Thermal Transport ◽  
Grown Crystal ◽  
Crystal Surface ◽  
Silicon Single Crystal ◽  
Interface Shape ◽  
Governing Equations ◽  
Finite Volume Discretization ◽  
Adaptive Grid Generation ◽  
Cylindrical Crystal

Abstract Czochralski (Cz) growth during seeding and shouldering of silicon single crystal has been studied. During seeding, the diameter increases from a small value to the desired value of the fully-grown crystal. The convection in the melt, conduction in the crystal and radiation from the melt surface and crystal, surface tension at the free surface of the melt and crystal rotation have been considered to investigate the effect of seeding on thermal transport. The rotational Reynolds number, the radiation from the crystal and Marangoni number are varied to investigate their effect on the interface shape. A few selected cases have been studied using a cylindrical crystal to understand the effect of geometry on the heat transfer rate and subsequently on the interface shape. The governing equations are solved using the curvilinear finite volume discretization scheme and the grids are redistributed using the multizone adaptive grid generation after each iteration. The movement of free surface is taken care by employing suitable equations for energy and stress balance. Results show the dependency of both the radiation from the crystal and rotation of the crystal on the interface shape.

A New Computational Method for Free Surface Problems

2005 ◽  
Author(s):  
Jeremy Rice ◽  
Amir Faghri
Keyword(s):  
Free Surface ◽  
Capillary Tube ◽  
Accurate Method ◽  
Computational Method ◽  
Surface Velocity ◽  
Governing Equations ◽  
Two Fluids ◽  
Correction Technique ◽  
Instability Growth ◽  
A New Technique

A new technique, called the surface velocity correction technique (SVC), is developed to track a free surface such as a liquid-vapor interface. SVC is a computationally inexpensive, and accurate method to capture interfacial fluid phenomena. This method uses a finite volume technique to discretize the governing equations, and a semi-Legrangian mesh to locate the interface between two fluids. The effectiveness of this technique is demonstrated through several classical examples and the results are also compared to both analytical and VOF solutions. The examples include: the shape of a meniscus in a capillary tube in mechanical equilibrium, the rise of a meniscus in a capillary tube, and the instability growth of a free flowing cylindrical column of fluid.

Unstructured Grid Based Reynolds-Averaged Navier-Stokes Method for Liquid Tank Sloshing

2005 ◽  
Vol 127 (3) ◽  
pp. 572-582 ◽  
Author(s):  
Shin Hyung Rhee
Keyword(s):  
Free Surface ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Navier Stokes ◽  
Governing Equations ◽  
Reconstruction Scheme ◽  
Geometric Reconstruction ◽  
Membrane Type ◽  
Tank Sloshing ◽  
Good Agreement

The present study is concerned with liquid tank sloshing at low filling level conditions. The volume of fluid method implemented in a Navier–Stokes computational fluid dynamics code is employed to handle the free-surface flow of liquid sloshing. The geometric reconstruction scheme for the interface representation is employed to ensure sharpness at the free surface. The governing equations are discretized by second order accurate schemes on unstructured grids. Several different computational approaches are verified and numerical uncertainties are assessed. The computational results are validated against existing experimental data, showing good agreement. The capability is demonstrated for a generic membrane-type liquefied natural gas carrier tank with a simplified pump tower inside. The validation results suggest that the present computational approach is both easy to apply and accurate enough for more realistic problems.

scholarly journals Propagation of Plane Waves in a Thermally Conducting Mixture

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Baljeet Singh
Keyword(s):  
Free Surface ◽  
Longitudinal Wave ◽  
Plane Waves ◽  
Generalized Thermoelasticity ◽  
Elastic Solid ◽  
Reflection Coefficients ◽  
Governing Equations ◽  
Transverse Waves ◽  
Longitudinal Waves ◽  
Thermally Conducting

The governing equations for generalized thermoelasticity of a mixture of an elastic solid and a Newtonian fluid are formulated in the context of Lord-Shulman and Green-Lindsay theories of generalized thermoelasticity. These equations are solved to show the existence of three coupled longitudinal waves and two coupled transverse waves, which are dispersive in nature. Reflection from a thermally insulated stress-free surface is considered for incidence of coupled longitudinal wave. The speeds and reflection coefficients of plane waves are computed numerically for a particular model.

Effects of Interface Wedge Angle in Metal Side on Bonding Strength in Ceramic to Metal Joint

2011 ◽  
Author(s):  
Masayoshi Tateno ◽  
Eiichirou Yokoi
Keyword(s):  
Free Surface ◽  
High Temperature ◽  
Bonding Strength ◽  
Wedge Angle ◽  
Fracture Pattern ◽  
Interface Conditions ◽  
Interface Shape ◽  
Electric Discharge Machining ◽  
Metal Joint ◽  
Wire Electric Discharge Machining

This study was performed to clarify dependences of bonding strength on the interface wedge angle in the metal side of ceramic-to-metal joint. Each plate Si3N4 and Ni used for this experiment is produced by wire electric discharge machining. The geometric interface shape at the edge of the interface is characterized by wedge angle on both side of the ceramic and metal defined as a configuration angle between the free surface of each material and the interface. As the wedge angle of Si3N4 is a right angle, the wedge angle of Ni is set from 30° to 180°. Joint specimens were bonded at high temperature using braze metal of 0.05mm thickness under vacuum and cooled slowly. The tensile bonding strength of the ceramic-to-metal joint was evaluated to determine the optimum interface shape. The highest bonding strength appeared under identical interface conditions where the fracture pattern changed. This study provided a useful geometric interface shape to improve the tensile bonding strength of ceramic-to-metal joint.

Axis symmetry of silicon molten zone interface shape under a mirror-shifting-type infrared convergent-heating floating-zone method

CrystEngComm ◽  
2015 ◽  
Vol 17 (48) ◽  
pp. 9452-9458 ◽  
Author(s):  
Md. Mukter Hossain ◽  
Satoshi Watauchi ◽  
Masanori Nagao ◽  
Isao Tanaka
Keyword(s):  
Grown Crystal ◽  
Molten Zone ◽  
Interface Shape ◽  
Floating Zone ◽  
Zone Method ◽  
Floating Zone Method ◽  
L System

Axis symmetry of the silicon molten zone shape, which is dependent on the position of the mirror-lamp (M-L) system, is related to the shape of the grown crystal.

Numerical Simulation of 2D Hydraulic Jump Flow in a Vertical Plane

2012 ◽  
Vol 482-484 ◽  
pp. 16-20
Author(s):  
Wen Li Wei ◽  
X.J Zhao ◽  
Y. L Liu
Keyword(s):  
Free Surface ◽  
Hydraulic Jump ◽  
Flow Model ◽  
Vertical Plane ◽  
Surface Profile ◽  
Governing Equations ◽  
Proposed Model ◽  
2D Flow ◽  
Time Changes ◽  
Volume Method

This paper was concerned with a vertical two-dimensional (2D) flow model with free surface. The water governing equations were discretized with finite difference method. The function of volume method was employed to track the moving free surface. The model was used to predict the characteristics of hydraulic jump flow in a 2D vertical plane. The surface profile and time averaged velocity were calculated, which shows the proposed model can be capable of capturing sharp water and gas interface configuration as time changes.

Thermal Transport of an Electrically Conducting Fluid over a Stretching Sheet in Manufacturing Processes

2010 ◽  
Vol 148-149 ◽  
pp. 406-409
Author(s):  
Chien Hsin Chen ◽  
Che Na Chen ◽  
Yu Rung Chen
Keyword(s):  
Thermal Transport ◽  
Stretching Sheet ◽  
Mhd Flow ◽  
Conducting Fluid ◽  
Material Surface ◽  
Manufacturing Processes ◽  
Governing Equations ◽  
Coupled Flow ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid

The properties of the final product greatly depend on the cooling rate from the material surface in many manufacturing processes, such as stretching sheets or filaments that are extruded continuously from a die. Therefore, the thermal transport behavior may play an important role in such manufacturing processes. In this paper we present an analysis of momentum and thermal transport for the magnetohydrodynamic (MHD) flow of an electrically conducting fluid over a stretching sheet with prescribed surface temperature. The effects of free convection, Joule heating and viscous dissipation are taken into consideration. The transformed governing equations are solved numerically for this non-similar coupled flow problem. To reveal the tendency of the solutions, typical results for velocity profiles, temperature distributions and the local Nusselt number are presented for different values of governing parameters.

Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading

2000 ◽  
Vol 417 ◽  
pp. 183-209 ◽  
Author(s):  
WEN-LING HONG ◽  
DAVID T. WALKER
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Reynolds Stress ◽  
Surface Current ◽  
Free Surface Flows ◽  
Governing Equations ◽  
Stress Anisotropy ◽  
Averaged Equations ◽  
Surface Fluctuations ◽  
High Froude Number

The goals of this study were to develop a set of Reynolds-averaged governing equations for turbulent free-surface flow, and to use the resulting equations to determine the origin of the surface current in high-Froude-number jet flows. To develop the Reynolds-averaged equations, free-surface turbulent flow is treated as a two-fluid flow separated by an interface. It is shown that the general Navier–Stokes equations written for variable property flow embody the field equations applicable to each fluid, as well as the boundary conditions for the interface and, therefore, can be applied across the entire fluid domain, including the interface. With this as a starting point, a formulation of the Reynolds-averaged governing equations for turbulent free-surface flows can be developed rigorously. The resulting Reynolds-averaged equations are written in terms of density-weighted averages, their derivatives, and the probability density function for the free-surface position. These equations are similar to the conventional Reynolds-averaged equations, but include additional terms which represent the average effect of the forces acting instantaneously on the free surface, forces normally associated with the boundary conditions. These averaged equations are applied to the interaction of a turbulent jet with the free surface in order to establish, for arbitrary-Froude-number flows, the origin of the surface current, the large outward velocity which occurs in a thin layer adjacent to the surface. It is shown via an order-of-magnitude analysis that the outward acceleration associated with the surface current results from a combination of the Reynolds-stress anisotropy and the free-surface fluctuations. For low Froude number, the surface current is mainly driven by the Reynolds stress anisotropy, consistent with the results of Walker (1997); when the Froude number is large, the Reynolds-stress anisotropy is smaller and the free-surface fluctuations make a significant contribution.

Numerical Study of Liquid Dynamics in Partially Filled Circular Tanks

2005 ◽  
Author(s):  
Liang Xu ◽  
Liming Dai
Keyword(s):  
Free Surface ◽  
Natural Frequencies ◽  
Numerical Study ◽  
Governing Equations ◽  
Liquid Motion ◽  
Liquid Pressure ◽  
Circular Area ◽  
Lateral Direction ◽  
Generalized Eigenvalue ◽  
Liquid Dynamics

The liquid dynamics in partially filled circular tanks has been studied by numerically solving the natural frequencies and transient liquid motion. The governing equations for liquid in tanks are based on the potential flow theory. Instead of direct discretization in the 2D circular area, the governing equations are rearranged in such a way that the discretization is performed in a fixed square area by continuous coordinate mappings to overcome the difficulties in dealing with the boundary conditions on the circular edge and the free surface. The natural frequencies of liquid sloshing in partially filled circular tanks are determined by solving generalized eigenvalue problem of liquid under different fill levels. Transient liquid motion is simulated when the tank is subjected to motion in the lateral direction, which is represented by different prescribed lateral accelerations. The forces caused by the change of liquid pressure on the tank walls are calculated.

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