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A numerical algorithm is presented which allows three-dimensional time-dependent simulations of Czochralski and other crystal growth processes. Besides the computation of turbulent flow and heat transfer in the melt, it includes the movement of the crystallization interface between melt and crystal, the dynamics of the melt free surface with the meniscus and the variation of the three-phase boundary. Thus, there is the potential that crystal diameter variations, which occur during growth of the start and the end cone can be predicted. This is illustrated by results of sample computations.
Delta-winglet vortex generators (VGs) are known to enhance the heat transfer between the energy-carrying fluid and the heat transfer surfaces in plate-fin-and-tube banks. In this study optimal angles of attack of the delta-winglets are investigated based on the Pareto optimal strategy. The optimization process combines a CFD analysis, genetic algorithms and the response surface methodology. The angle of attack of a pair a delta-winglet-type VGs mounted behind each tube is varied between β = −90° and +90°. Three circular tube rows with inline and staggered tube arrangements are investigated for Reynolds numbers from 200 to 1200 (based on the inlet height and inlet velocity). The flow structure and heat transfer behavior is analyzed in detail for certain cases and the staggered and the inline tube arrangements are compared. Finally, for each of these arrangements the optimal sets of angles of attack for different Reynolds numbers are presented.
In the literature, numerical computations of the Czochralski process for crystal growth are conducted using a quasi-steady state assumption for the crystallization neglecting time-dependent effects. In the present work, an algorithm is developed, which allows to calculate the transient behavior of the crystallization interface including the movement of the three-phase boundary and the free surface of the melt. Thus, in conjunction with the computation of the turbulent melt flow and heat transfer, more realistic predictions of the crystal growth can be achieved. Test cases show that with the algorithm, realistic phenomena like crystal diameter increase and decrease during the growth process could be reproduced.
A DNS of the turbulent flow and heat transfer in an idealized Czochralski configuration was conducted using a very fine grid. The results show the irregular forming of buoyant thermal plumes as well as the occurrence of a large, stable vortical structure in the bulk of the melt. In the averaged flow field a B´enard-cell-like pattern can be recognized. The DNS data served also as a reference for the validation of the LES method. The LES computations were conducted using different grid sizes, SGS models, and discretization methods. For relatively fine grids and central differences, the results agree very well with the DNS. Using an upwind discretization introduces numerical errors. In combination with coarser grids, this leads to large deviations and even qualitative differences. However, overall the computational effort could be reduced significantly by LES. Thus a compromise between accuracy achieved and effort required has to be found.
For numerical simulations of the Czochralski crystal growth process, in the literature the quasi-steady state approach is used, i.e. the diameter of the crystal is kept constant. However, this leads to errors in the computation of the phase interface. In the present work, for the first time in 3D, the three-phase junction between melt, crystal and atmo-sphere is allowed to move in horizontal direction. Thus, diameter changes during the crystal growth process could be simulated accordingly. In the future, the vertical movement will be allowed as well, considering the free surface stresses. This will enable realistic predictions of the transient behavior of the crystallization front.