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Article
Publication date: 2 May 2017

Mirosław Seredyński, Sara Battaglioli, Robin P. Mooney, Anthony J. Robinson, Jerzy Banaszek and Shaun McFadden

Numerical models of manufacturing processes are useful and provide insight for the practitioner; however, model verification and validation are a prerequisite for…

Abstract

Purpose

Numerical models of manufacturing processes are useful and provide insight for the practitioner; however, model verification and validation are a prerequisite for expedient application. This paper aims to detail the code-to-code verification of a thermal numerical model for the Bridgman solidification process of alloys in a two-dimensional axisymmetric domain, against an established commercial code (ANSYS Fluent); the work is considered a confidence building step in model development.

Design/methodology/approach

A grid sensitivity analysis is carried out to establish grid independence, and this is followed by simulations of two transient solidification scenarios: pulling rate step change and ramp input; the results of which are compared and discussed.

Findings

Good conformity of results is achieved; hence, the non-commercial model is code-to-code verified; in addition, the ability of the non-commercial model to deal with radial heat flow is demonstrated.

Originality/value

The ability of the home made model for Bridgman furnace solidification to deal with cases where significant radial heat transfer occurs in the sample was demonstrated. The introduction of front tracking to model the macroscopic growth of dendritic mush and the region of undercooled liquid is identified as the next step in model development.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 February 1996

C.W. Lan and C.C. Ting

A pseudo steady‐state model is developed to study heat transfer, fluidflow, and the interface shape in the liquid encapsulated vertical Bridgmancrystal growth. The model…

Abstract

A pseudo steady‐state model is developed to study heat transfer, fluid flow, and the interface shape in the liquid encapsulated vertical Bridgman crystal growth. The model, which is governed by momentum, heat, and overall mass balances in the system, is solved by a finite‐volume/Newton method. Flow and temperature fields, as well as unknown melt/crystal and melt/encapsulant interfaces, are calculated simultaneously. Sample calculations are mainly conducted for the GaAs/B2O3/PBN system. Calculated results for the Germanium/graphite system are compared with finite element calculations by Adornato and Brown, and they are in good agreement. The effects of some process parameters, including the growth speed, ambient temperature profile and heat transfer conditions, on flow patterns, temperature fields and the interface shape are illustrated through calculated results. Interface inversion from concave to convex, by modifying the ambient temperature profile, is also demonstrated through computer simulation. Particularly, through an inverse problem approach, a flat interface can be easily obtained for various operation conditions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 6 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 April 1992

S. BRANDON and J.J. DERBY

A finite element method for the analysis of combined radiative and conductive heat transport in a finite axisymmetric configuration is presented. The appropriate…

Abstract

A finite element method for the analysis of combined radiative and conductive heat transport in a finite axisymmetric configuration is presented. The appropriate integro‐differential governing equations for a grey and non‐scattering medium with grey and diffuse walls are developed and solved for several model problems. We consider axisymmetric, cylindrical geometries with top and bottom boundaries of arbitrary convex shape. The method is accurate for media of any optical thickness and is capable of handling a wide array of axisymmetric geometries and boundary conditions. Several techniques are presented to reduce computational overhead, such as employing a Swartz‐Wendroff approximation and cut‐off criteria for evaluating radiation integrals. The method is successfully tested against several cases from the literature and is applied to some additional example problems to demonstrate its versatility. Solution of a free‐boundary, combined‐mode heat transfer problem representing the solidification of a semitransparent material, the Bridgman growth of an yttrium aluminium garnet (YAG) crystal, demonstrates the utility of this method for analysis of a complex materials processing system. The method is suitable for application to other research areas, such as the study of glass processing and the design of combustion furnace systems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 2 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 17 May 2011

Diego Celentano, Marcela Cruchaga, Jorge Romero and Mohammed El Ganaoui

The purpose of this paper is to present a 2D numerical simulation of natural convection and phase‐change of succinonitrile in a horizontal Bridgman apparatus. Three…

Abstract

Purpose

The purpose of this paper is to present a 2D numerical simulation of natural convection and phase‐change of succinonitrile in a horizontal Bridgman apparatus. Three different heat transfer mechanisms are specifically studied: no growth, solidification and melting.

Design/methodology/approach

The analysis is carried out with a preexisting thermally coupled fixed‐mesh finite element formulation for generalized phase‐change problems.

Findings

In the three cases analyzed, the predicted steady‐state liquid‐solid interfaces are found to be highly curved due to the development of a primary shallow cell driven by the imposed furnace temperature gradient. In the no growth case, the heating and cooling jackets remain fixed and, therefore, a stagnant liquid‐solid interface is obtained. On the other hand, the phase transformation in the solidification and melting cases is, respectively, controlled by the forward and backward movement of the jackets. In these last two growth conditions, the permanent regime is characterized by a moving liquid‐solid interface that continuously shifts with the same velocity of the jackets. The numerical results satisfactorily approach the experimental measurements available in the literature.

Originality/value

The numerical simulation of the no growth, solidification and melting cases in a horizontal Bridgman apparatus using a finite element based formulation is the main contribution of this work. This investigation does not only provide consistent results with those previously computed via different numerical techniques for the no growth and solidification conditions but also reports on original numerical predictions for the melting problem. Moreover, all the obtained solid‐liquid interfaces are validated with experimental measurements existing in the literature.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 December 1998

Yasunori Okano, Susumu Sakai, Takahiro Morita and Jun Shimizu

A finite difference simulation for the bulk single crystal growth of indium phosphide by the liquid encapsulated vertical gradient freezing (LE‐VGF) method with a flat…

Abstract

A finite difference simulation for the bulk single crystal growth of indium phosphide by the liquid encapsulated vertical gradient freezing (LE‐VGF) method with a flat bottom crucible is presented. In order to treat a curvature interface, the boundary fixing method is applied. The transient behavior of the flow and temperature fields, the melt/crystal interface shape and the growth rate during growth are studied numerically. The crystal growth rate is not constant although the temperature lowering rate is constant. The effect of crucible thickness, thermal conductivity of the crucible and temperature of the growth furnace wall on the crystal growth behavior are discussed.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 8 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 8 July 2019

Mirosław Seredyński and Jerzy Banaszek

Proper selection of the stability parameter determines the accuracy of dendrite tip kinetics at a single crystal scale. Recently developed sophisticated phase field…

Abstract

Purpose

Proper selection of the stability parameter determines the accuracy of dendrite tip kinetics at a single crystal scale. Recently developed sophisticated phase field modelling of a single grain evolution provides evidence that this parameter is not constant during the process. Nevertheless, in the commonly used micro-macroscopic simulations of alloy solidification, it is a common practice to use a constant value of the stability parameter, resulting from the marginal stability theory. This paper aims to address the issue of how this inaccuracy in modelling crystal growth kinetics can influence numerically predicted zones of columnar and equiaxed dendrites and the macro-segregation formation.

Design/methodology/approach

Using the original authors’ micro-macroscopic computer simulation model of binary alloy solidification, the calculations have been performed for the Kurz-Giovanola-Trivedi (KGT) crystal growth kinetics with two different values of the stability parameter, and for two different compositions of Al-Cu alloys. The computational model is based on single domain-based formulation of transport equations, which are discretized on control-volume mesh. To identify zones of different grain structures, developing within the two-phase liquid-solid region, an envelope of columnar dendrite tips is tracked on a fixed non-orthogonal, triangular control volume grid. The models of porous and slurry media are used, along with the concept of the switching function, to account for diverse flow resistances in the columnar and equiaxed crystal zones. The numerical predictions are carefully studied to address the question of how the chosen stability parameter influences macroscopic structures of a cast, the most important issue from the engineering point of view.

Findings

The carried-out comprehensive numerical analysis shows that the value of the stability parameter of the KGT-constrained dendrite growth model does not have a direct significant impact on the macrosegregation formation. It, however, visibly influences the undercooling along the front, separating different dendritic structures and the size of the undercooled melt region where the equiaxed grains can develop. It also affects the amount of eutectic phase created.

Originality/value

To the best of the authors’ knowledge, this is the first attempt at estimating the influence of some inaccuracies, caused by possible ambiguities in choosing the stability constant of the KGT law, on numerically predicted macroscopic fields of solute concentration, the developing zones of columnar and equiaxed crystals and the macrosegregation patterns.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 February 1994

Yiqiang Zhang, J.I.D. Alexander and J. Ouazzani

Free and moving boundary problems require the simultaneous solution ofunknown field variables and the boundaries of the domains on which thesevariables are defined. There…

Abstract

Free and moving boundary problems require the simultaneous solution of unknown field variables and the boundaries of the domains on which these variables are defined. There are many technologically important processes that lead to moving boundary problems associated with fluid surfaces and solid‐fluid boundaries. These include crystal growth, metal alloy and glass solidification, melting and flame propagation. The directional solidification of semi‐conductor crystals by the Bridgman—Stockbarger method1,2 is a typical example of such a complex process. A numerical model of this growth method must solve the appropriate heat, mass and momentum transfer equations and determine the location of the melt—solid interface. In this work, a Chebyshev pseudospectral collocation method is adapted to the problem of directional solidification. Implementation involves a solution algorithm that combines domain decomposition, a finite‐difference preconditioned conjugate minimum residual method and a Picard type iterative scheme.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 4 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 7 March 2016

Igor Vušanović and Vaughan R Voller

When a multi component alloy solidifies the redistribution of solute components leads to the formation of macrosegregation patterns. Blending ideas from a number of recent…

Abstract

Purpose

When a multi component alloy solidifies the redistribution of solute components leads to the formation of macrosegregation patterns. Blending ideas from a number of recent publications the purpose of this paper is to provide a “best practice” on how grid convergence of a given macrosegregation simulation can be measured and determined.

Design/methodology/approach

The best practice is arrived at by considering a benchmark problem consisting of a 2D-casting simulation of an idealized Al-4.5%Cu alloy in a side cooled square (76×76 mm) cavity. The model for this simulation is based on a mixture treatment of the relevant heat and mass transfer equations. Simulations are made using three increasingly refined grid sizes.

Findings

The best practice to determine grid resolution involves two steps: first, a visual evaluation of predicted segregation images leading to the evaluation of solute profiles along selected transects; and second, the construction of a cumulative distribution function (CDF) of the predicted segregation field. On application to the benchmark problem, it is concluded that current computer resources are insufficient to grid resolve macrosegregation patterns but that the CDF provides a useful signal of the nature of macrosegregation in a given system.

Research limitations/implications

The benchmark is chosen to be representative. Exact convergence behavior, however, may depend on the system chosen.

Originality/value

In addition to establishing a best practice for measuring grid resolution of macrosegregation simulations the work also highlights, even in the absence of complete grid convergence, how the recently proposed CDF treatment can inform solidification modeling and process understanding.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Content available
Article
Publication date: 4 January 2013

158

Abstract

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 1
Type: Research Article
ISSN: 0961-5539

Content available
Article
Publication date: 2 May 2017

Andrzej J. Nowak

244

Abstract

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 5
Type: Research Article
ISSN: 0961-5539

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