Thermal Buoyancy Convection in Vertical Zone Melting Configurations

Abstract

A numerical method has been performed to explain and interpret oscillating convection phenomena in the vertical zone melting technique. The governing equations are solved in a cylindrical coordinate system using the finite volume method. The thermal boundary conditions were taken from experimental data. The aspect ratio ar was varied within the range of 1 5 ar 5 3. Two-dimensional calculations with application of parabolic boundary temperature show always steady axisymmetric behaviour of the convection. Only the three-dimensional calculation yields the transition to nonaxisymmetric, time-dependent (unsteady) flow. Furthermore the necessity of using thermal boundary conditions by a heat flux density q instead of a fixed parabolic temperature profile was demonstrated. The torus-like flow changes at ar = 1.45. . . 1.5 to a single roll flow pattern. This single roll shows an unsteady behaviour with increasing aspect ratios and Rayleigh numbers, respectively. The comparison of the numerical results with experimental data shows a good qualitative correlation. MSC (1991): 76R05

Optimization of semiconductor melts

“…3. Finally we note that the present work extends the numerical investigations of zone melting presented by the first author, König and Seifert in [2].…”

Boundary control of a Boussinesq equation for a crystal growth problem: aspects of numerical solution

“…The used geometrical and material parameters for the crystal (Bi 0.25 Sb 0.75 ) 2 Te 2 , a composition of bismuth point fifty antimony one point fifty tellurium two, are summarized in table 1 (see also [1]). Figure 1 shows the physical domain of the melt zone.…”

“…For example we construct an optimization system for the case of a boundary temperature profile as a control variable and a cost functional (1). For the calculus of optimization and the derivation of an optimality system we use the above described dimensionless mathematical model (3)-(5) with the boundary conditions (6).…”

“…This leads to tracking type optimization problems (i) with functionals like (1) and problems with optimization functionals of the form…”

“…A good overview of the crystal growth problems which will be discussed in this paper as a real-world application problem is given in [1].…”