Numerical simulation heat transfer by natural convection in liquid metal with a sinusoidal temperature (original) (raw)
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Heat and Mass Transfer, 2006
Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.
On the numerical modelling of heat transfer during solidification processes
International Journal for Numerical Methods in Engineering, 1988
An extensive amount of work has been published on the treatment of heat transfer associated with phase change. A few recent advances are discussed in this paper with some emphasis on the phase change of metals. The apparent capacity, effective capacity, enthalpy, post iterative, source based and semi-analytical methods are discussed and relative advantages and disadvantages of each are analysed. Recent developments in modelling the flow during pouring and natural convection with applications of two widely used convective diffusive codes, the Los Alamos MAC and the Imperial College TEACH, are presented. An alternative stream function-vorticity approach is also discussed. Applications of these methods to turbulent convection during mould filling and continuous casting are presented. Areas of interest for further research work are identified as modelling of turbulence in liquid metals, flow through mushy regions and improvement of the performance of weak methods in multidimensional problems when the ratio of latent heat to sensible heat is large.
Journal of Thermophysics and Heat Transfer, 2005
An experimental and numerical study of natural convection and solidification in a two-dimensional cavity driven by constant and oscillating temperature gradients is presented. Finite element models are developed to predict the flowfield, the temperature distribution, and the solid-liquid interphase shapes during solidification. Both the fixedgrid and moving-grid methods are applied in the numerical simulations, using the former to illustrate the oscillating thermal gradient conditions and using the latter to illustrate the constant temperature gradient conditions. An experimental system is set up where succinonitrile is used as a working fluid. The flow pattern, the velocity field, and the solidification interface shape are measured using the laser-based particle-image-velocimetry system. Numerical simulations and experiments are conducted for various configurations and different thermal gradients. In most cases, convection is dominated by one recirculating loop. With an inverted temperature gradient, however, multiple convection loops are observed. Both the convective flow pattern and the velocity strongly affect the solid-liquid interface shapes during solidification. In a majority of the cases studied, the model predictions are in good agreement with the experimental measurements.
Oscillatory Regimes for the Horizontal Thermal Convection in Solidification Problems
1998
A numerical study of the buoyancy driven natural convection of liquid materials is presented. There is experimental evidence that at low values of the Grashof number (Gr) the ow is laminar and steady while is periodic and generally time dependent as the value of Gr increases. This behaviour a ects the products in many applications, for instance the quality of semi-conductor crystals grown by the Czochralski and Bridgman techniques. The bifurcation pattern of this ow has been almost completely identi ed in the two dimensional case whereas in three dimensions we cannot mention any published result due to the inadequacy of the computational resources. In this work we shall present the ow con gurations occuring in the case of a shallow (4 1 1) box where the temperature gradient is applied at the smaller vertical faces. The uid considered has Prandtl number equal to 0.015. The vorticity-velocity formulation of the Navier-Stokes equations has been integrated by means of a fully implicit nite di erence scheme. By starting from a steady ow con guration as Gr increases we shall describe the transition to a periodic ow, and then from this one the transition to a non-periodic timedependent ow.
Freezing/solidification process involves continuous transfer of heat in the material medium to change from liquid to solid. The problems that describe these processes satisfy certain initialand boundary conditions along with the Stefan condition at the interface. These problems referred as Stefan problems or Moving boundary problems or Phase change problemsin which there exists continuously changing moving fronts.At any time, tracking the position of themoving fronts and the temperature plays an important role to solve these non-linear problems. Phase change problems has applications in the field of energy conservation techniques, Material Science,industrial units and many more.In this paper, solidification of water is studied to understand the mechanics of heat transfer problems. A convective upper boundary condition is considered as sinusoidal function of time, to study the changes in the temperature fields, which in general treated as constant. Understanding the effect of heat transfer coefficient on the movement of moving fronts and the temperature distributionhelp to design ice storage units in energy conservations technologies.
Modelling of convection during solidification of metal and alloys
Sadhana, 2001
The role of convection during solidification is studied with the help of a mathematical model. The effect of various mush models on convection and consequent macrosegregation is examined with the help of numerical simulations. The predicted macrosegregation profiles are compared with published experimental data. Subsequently, the importance of proper auxiliary relationship for thermo-solutal coupling in the mushy region is highlighted through some careful numerical simulations. Finally, the role of material parameters on double-diffusive convection is illustrated through comparative study of solidification of aqueous ammonium chloride, ironcarbon and lead-tin binary systems. Important results of these studies are presented and discussed.
Convection in melts and crystal growth
Advances in Space Research, 1983
Convection plays a significant role in determining the electronic properties of semiconductor crystals grown from the melt. Inhomogeneities (doping striations) in these crystals can be caused by unsteady natural convection. Therefore, the origins of unsteady natural convection, which means fluctuating temperatures in a fluid, are investigated in this paper. Several effects have been found at increasing Rayleigh numbers Ra which cause fluctuating temperatures. But it was also observed, that unsteady convection became again steady if Ra was increased to higher values. This effect, which is called relaminarization, is very interesting for crystal growth and is discussed in this paper.
Mixed convection with liquid metals: Review of experiments and model development
2018
In this paper, a review of experiments related to liquid metal heat transfer under mixed convection is performed. This study is relevant because heat transfer during start-up and shutdown procedures, and operational transients is influenced by natural convection, resulting in mixed convection, which differs considerably from forced convection. Up to now, simulation tools like TRACE, RELAP, etc. apply only forced convection models for liquid metal heat transfer. The influence of mixed convection on the heat transfer during the above mentioned transients is completely ignored. Hence, it is not possible to simulate mixed convection with best-estimate system codes like TRACE or RELAP. In order to perform realistic simulations of plants and experimental facilities mixed convection must be addressed and considered. Therefore, the literature is reviewed for experimental data with liquid metal heat transfer under mixed convection and generally applicable statements and models will be provided. A clear distinction in the heat transfer behavior for low and high Péclet number flows can be identified. Thereby, a Péclet number dependency is visible for higher Péclet numbers (Pe > 100). Furthermore, the heat transfer (Nusselt number) cannot be presented as a function of one dimensionless parameter. To identify underlying phenomena, especially when comparing different experimental scenarios several dimensionless numbers are needed (Gr*, B, Z, etc.). Based on this study, it is possible to derive a model for the heat transfer under mixed convection. Nevertheless, these findings and the sparse number of experiments also indicate the need for new and comprehensive experiments.
Analysis of Steady Convective Heating for Molten Materials Processing within Trapezoidal Enclosures
Industrial & Engineering Chemistry Research, 2008
Material processing involving natural convection within a trapezoidal enclosure for uniformly and nonuniformly heated bottom wall, insulated top wall, and isothermal sidewalls with inclination angle have been investigated. The penalty finite element method is used to obtain isotherm and streamline profiles for model liquids e.g. molten metal, salt water and olive oil. Parametric study for the wide range of Rayleigh number (Ra), 10 3 e Ra e 10 5 and Prandtl number (Pr) for model fluids with various tilt angles ) 45°, 30°, and 0°have been obtained. Secondary circulations were observed during molten metal processing. Streamlines show that the strength of convection is larger for ) 45°and flow intensities are also found to be larger for olive oil compared to molten metal and salt water. Heat transfer rates are shown via local and average Nusselt number plots. Local heat transfer rates are found to be relatively more for ) 0°than those with ) 45°and ) 30°. Average Nusselt number plots show higher heat transfer rates for ) 0°except for the nonuniform heating of the bottom wall with Pr ) 0.015 (molten metal). Overall, less heat transfer rates are observed for molten metal processing.
WIT transactions on engineering sciences, 1970
The influence of latent heat and natural convection in the melt and the shape of the melt-crystal interface are analyzed for a vertical Bridgman crystal growth system by direct numerical simulation. The temperature and the velocity field in the melt and in the crystal are computed using an homogenization technique. Initially the ampoule contains poly-crystal material, these container is translated inside a furnace which includes two heating zones characterized by two temperatures which are respectively greater and lower than the melted temperature of the material. The two heating zones in the furnace are separated by an adiabatic zone when the solid-liquid phase change occur. The control of the solidification conditions (phase change front velocity, gravity ...) permits to produce with these technique very high quality single crystals. The results presented in this paper show the effects of various gravitational conditions upon the flow in the melted part of the material inside the ampoule. We have also analyzed the effect of the latent heat upon the shape of the melt-crystal interface and the convective cells in the melted pool.