Study of melt convection and interface shape during sapphire crystal growth by Czochralski method

Global simulation is performed to predict electromagnetic field, heat transfer, melt flow, and interface shape during the different growth stages of sapphire crystal by RF-heated Czochralski (Cz) process. Melt flow in the Cz-sapphire growth system includes Marangoni convection due to the variation of surface tension at the free surface, natural convection due to the complex thermal boundary conditions on the crucible walls, and forced convection due to the crystal rotation. As the crystal grow longer, the effects of the convections on the melt flow, temperature distribution and interface shape varies due to the drop of melt, which can be characterized by Grashof number (Gr) with the melt height as the characteristic length. It is found that as melt height reduces, natural convection gets weaker, and the interface becomes flatter. To examine systematically the effects of the convections on the crystal growth process, three dimensionless parameters, N_σ, Gr/Re² and N_σ2, are further introduced. N_σ denotes the influence of surface tension in the boundary layer as compared with buoyancy in the melt, Gr/Re² is the ratio of convective forces to the normalizing convective velocities with viscosity, and N_σ2 represents the relative strength of Marangoni flow to forced convection. The effects of the parameters on the melt flow and heat transfer are examined, and their relationships to the interface convexities are obtained. The understanding of melt convection and interface shape is not only important for the optimization of a stable sapphire crystal growth, but also critical for an accurate modeling of the growth process.     

Numerical investigation of the interfacial characteristics during Bridgman growth of compound crystals

The effects of the Bridgman process parameters and the double diffusive convection in the melt on the melt/solid interface shape and the solute distribution were numerically investigated for the II–VI semiconductor material HgCdTe. The results show that the melt/solid interface is concave at the center and the solute concentration at the interface is not uniform. A clockwise eddy forms near the interface in the melt when double diffusive convection is considered. With this eddy flow, the solute near the interface is well mixed so the solute distribution at the interface is very different from the case of no flow, but there are no significant changes in the interface shape or the thermal field. However, the growth parameters such as Bi, St, U and A significantly affect the melt/solid interface shape, position and the solute distribution at the interface.     

Time evolution of double-diffusive convection in a vertical cylinder with radial temperature and axial solutal gradients

The characteristics of transient double-diffusive convection in a vertical cylinder are numerically simulated using a finite element method. Initially the fluid in the cavity is at uniform temperature and solute concentration, then constant temperature and solute concentration, which are lower than their initial values, are imposed along the sidewall and bottom wall, respectively. The time evolution of the double-diffusive convection is investigated for specific parameters, which are the Prandtl number, Pr = 7, the Lewis number, Le = 5, the thermal Grashof number, Gr_T = 10⁷, and the aspect ratio, A = 2, of the enclosure. The objective of the work is to identify the effect of the buoyancy ratio (the ratio of solutal Grashof to thermal Grashof numbers: N = Gr_S/Gr_T) on the evolution of the flow field, temperature and solute field in the cavity. It is found that initially the fluid near the bottom wall is squeezed by the cold flow from the sidewall, a crest of the solute field forms and then pushed to the symmetry line. In the case of N > 0, a domain with higher temperature and weak flow (dead region) forms on the bottom wall near the symmetry line, and the area of dead region increases when N varies from 0.5 to 1.5. More crests of the solute field are formed and the flow near the bottom wall fluctuates continuously for N < 0. The frequency of the fluctuation increases when N varies from −0.5 to −1.5. Corresponding to the variety of the thermal and solutal boundary layers, the average rates of heat transfer (Nu) at the sidewall remain almost unchanged while the average rates of mass transfer (Sh) at the bottom wall change much in the cases of N = 1, 0, −1.     

Laminar mixed convection boundary layers induced by a linearly stretching permeable surface

The boundary layer flow on a linearly moving permeable vertical surface is studied when the buoyancy force assists or opposes the flow. Similarity and local similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, injection parameter d, and the mixed convection parameter λ=Gr_x/Re_x², which determine the velocity and temperature distributions, the heat transfer coefficient, and the shear stress at the surface are studied. The heat transfer coefficient increases as λ assisting the flow for all d for uniformly or linearly heated surface and as Pr increases it becomes almost independent of λ. However, as the temperature inversely proportional to the distance up the surface, the buoyancy has no effects on the heat transfer coefficient. Critical buoyancy parameter values are obtained for vanished shear stress and for predominate natural convection. Critical values are also presented for predominate buoyancy shear stress at the surface for assisting or opposing flow. A closed form analytical solution is also presented as a special case of the energy equation.     

Analysis of heat and mass transfer in asymmetric system

This study aims to investigate theoretically the effect of water evaporation from a wetted channel wall, on natural convection heat transfer and the effect of channel width. Major nondimensional groups identified are Gr_T, Gr_M, Pr, Sc and φ. Results are presented for an air–water system under various heating conditions. The influence of heated wall temperatures, wetted wall temperatures, channel width and the relative humidity of the moist air in the ambient on the heat and mass transfer are examined in great detail.     

Effect of an External Magnetic Field on the 3-D Oscillatory Natural Convection of Molten Gallium During Phase Change

In this article, a numerical study of the effect of an external magnetic field on three-dimensional (3-D) oscillatory natural convection during phase change is carried out. A parallelepiped enclosure filled with a molten gallium and subjected to an external magnetic field applied in X-, Y-, and Z-directions, separately, is considered. The finite-volume method with enthalpy formulation is used to solve the mathematical model in the solid and liquid phases. The Hartmann number is fixed to Ha = 20.The computer program developed in this study was validated with the experimental data founded in the literature. The critical Grashof numbers Gr _Cr and critical corresponding frequencies F r _Cr are determined with and without magnetic field. The results show that the oscillatory natural convection during phase change are characterized by low-frequency oscillations in the presence and absence of the magnetic field. The pattern flow shows a spiral development of the flow in Z- direction. A strong dependence between the direction of the magnetic field and the critical Grashof number and their corresponding frequency is determined. A strong stabilization of the flow field is shown when the magnetic field is oriented horizontally.     

Numerical visualization of mass and heat transport for mixed convective heat transfer by streamline and heatline

Based on the method we suggested in a previous paper [Int. J. Heat Mass Transfer 45 (2002) pp. 2373-2385], the present work is to investigate the mixed convection problem. A two-dimensional, steady, laminar displacement ventilation model is adopted here for the interaction between the buoyancy driven natural convection and the external forced convection is important to achieve the goal of ventilation effectiveness. The solution is determined by the non-dimensional parameters Gr and Gr/Re², the influences of which on the resulting heat and fluid flow are discussed. To optimize the ventilation system, different outlet locations are investigated. Results and comparisons show that the displacement ventilation guarantees a high indoor air quality (IAQ) and is therefore a desired air-conditioning system.     

Combined forced and natural convection heat transfer from a horizontal cylinder embedded in a porous medium

This paper reports the results of an experimental study of heat transfer by combined forced and natural convection from a horizontal cylinder embedded in a porous medium composed of randomly packed glass spheres saturated with water. The direction of the flow of water was horizontal and normal to the longitudinal axis of the cylinder. The diameter of the cylinder, D, was 11.45mm and the equivalent diameter of the glass spheres was 3.072mm. It is shown that the condition Gr_k/Re²_D ⩽ 0.5 represents a conservative criterion for segregating heat transfer data that are predominantly governed by forced convection from those in which natural convection effects are significant. A correlation hypothesis for convection heat transfer which is based upon four assumptions, primary among which is that the flow can be (conceptually) regarded as being composed of ‘coarse’ and ‘fine’ components, is presented. This hypothesis is shown to provide a basis for successfully correlating a set of experimental heat transfer data that extends from the Darcy regime into the turbulent regime and spans the intervening Forchheimer and transition regimes. It is suggested that the correlation procedure adopted here may yield useful results if applied to other geometries such as, for example, forced convection heat transfer in ducts packed with porous media.     

Developing laminar mixed convection with heat and mass transfer in horizontal and vertical tubes

The effects of the solutal and thermal Grashof numbers on the flow, temperature and concentration fields in tubes with uniform heat flux and concentration at the fluid-solid interface have been investigated numerically using a three-dimensional axially parabolic model. Results show a complex development of the flow field which is strongly influenced by the values of the two Grashof numbers and by the tube inclination. For vertical tubes the flow field is also influenced by the relative direction of the flow and the buoyancy forces. In general, very close to the tube inlet forced convection boundary layer development dominates. Further downstream, the effects of solutal buoyancy predominate while those of thermal origin determine the flow field far downstream and, in particular, the fully developed conditions. The axial evolution of the wall shear stress τ_z, the Nusselt number Nu_z and the Sherwood number Sh_z in both horizontal and vertical tubes are presented for different combinations of the two Grashof numbers. For horizontal tubes and vertical tubes with upward flow these three variables are greater than the corresponding ones for forced convection. The opposite is true for downward flow in vertical tubes.     

Onset of oscillatory double-diffusive buoyancy instability in an inclined rectangular cavity

Double-diffusive buoyancy convection in an inclined rectangular closed cavity with imposed temperatures and concentrations along two opposite sidewalls is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude (buoyancy ratio R_ρ = ?1). In this case a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Gr_c. For both infinite and finite layers, linear stability analysis shows that, when the cavity inclination α with respect to gravity decreases from 0° to ?90°, Gr_c for the onset of stationary instability increases exponentially while that for the onset of oscillatory instability decreases exponentially. Below a critical α_c, the first onset of instability is oscillatory, rather than stationary. For the infinite layer, the influences of α on the critical wave number and frequency of the oscillatory mode are shown and the corresponding flow structure of the eigenfunction consists of counterrotating vortices travelling from one end to the other. For a bounded layer, the neutral stability curves of the first two oscillatory modes, centrosymmetric and anticentrosymmetric, cross each other successively at a series of double Hopf bifurcation points as the aspect ratio increases. These two curves are not smooth, but each contains several abrupt changes, after every one of which a pair of counterrotating vortices is added to the flow field and thus the parity of the mode remains unchanged. The neutral curves showing the influences of Pr and Le are also obtained. The present work expands the work of Bergeon et al. (1999) [8] in which the same physical problem was studied and yet no oscillatory onset of instability was considered.     

Effects of flow inertia on vertical,natural convection in saturated,porous media

An analytical study is made for the effect of flow inertia on vertical, natural convection in saturated, porous media. Within the framework of boundary-layer approximations, Forchheimer's model was transformed into a set of non-similar equations. Effects of flow inertia are measured and examined in terms of the dimensionless inertia parameter ξ = Gr_xFo_x where Gr_x is the local Grashof number of determined by the bulk properties of saturated porous media, and Fo_x is a new dimensionless parameter governed by the microstructure of porous matrix. The non-similar solutions are presented and discussed for two types of flow: (1) the uniform heat flux surface; and (2) plane plume flows. Results show that thermal boundary layer in the non-Darcy regime is thicker than the corresponding pure-Darcy flow. In addition, the local wall heat fluxes for the first case and the maximum temperature gradient for the second case decrease with increasing ξ.     

Thermodiffusion in porous media: Multi-domain constitutant separation

A numerical study is carried out on double-diffusive, natural convection within a vertical annular, porous cylinder. The flow is driven by buoyancy force due to externally applied constant temperature difference on the vertical cylinder while the horizontal surfaces are impermeable and adiabatic. The effects of cross phenomena “Soret effect” were considered in the analysis. It is demonstrated that the cylindrical annulus permits a higher thermal gradient. It is established that such system has an optimal separation effect for a given thermal Rayleigh number, Ra_T. To overcome such limitation, two sub-domains (buffer) allowing filtration separation was proposed and investigated. The flow field, temperature and concentration distributions are obtained in terms of the governing parameters. The effect of the sub-domains scales, the Darcy number, Da, and the curvature, R, on flow, temperature and species separation ability is found to be significant.     

A correlation for mixed convection heat transfer from converging,parallel and diverging channels with uniform volumetric heat generating plates

A numerical study of steady, laminar, mixed convection heat transfer from volumetrically heat generating converging, parallel and diverging channels is carried out. Air is considered as the working fluid. The maximum angle of deviation (ϕ) of the channel plates from the vertical position for the converging and diverging channels is 2° and − 2°, respectively. A parametric study has been carried out for a wide range of Re_S, Gr_S^⁎, γ and ϕ to investigate their effect on the fluid flow and heat transfer characteristics. A universal correlation, to estimate the non-dimensional maximum temperature occurring in the converging, parallel or diverging channels is presented.     

Effects of mass transfer and free convection on the unsteady mhd flow past a vertical porous plate with constant suction

Effects of free convection currents and mass transfer on the unsteady flow of an electrically conducting and viscous incompressible fluid past an infinite vertical porous plate subjected to uniform suction, in the presence of transverse magnetic field, have been studied taking into account that the external flow velocity varies periodically with time in magnitude but not in direction. The effect of the induced magnetic field has been neglected. Approximate solutions to the transient flow, the amplitude and the phase of the skin-friction and the rate of heat transfer have been derived. During the course of the discussion, the effects of the Grashoff number Gr, the modified Grashoff number Gc (depending on the concentration difference), the Schmidt number Sc, the Eckert number Ec, the magnetic field parameter M, and the frequency ω have been discussed.     

MHD non-Darcian free convection from a vertical wavy surface embedded in porous media in the presence of Soret and Dufour effect

The objective of this paper is to examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection along a heated vertical wavy surface embedded in an electrically conducting fluid saturated porous medium, subject to the diffusion-thermo (Dufour), thermo-diffusion (Soret) and magnetic field effects. Diffusion-thermo implies that the heat transfer is induced by concentration gradient, and thermo-diffusion implies that the mass diffusion is induced by thermal gradient. The boundary-layer regime is considered where the Darcy–Rayleigh number is very large. A suitable coordinate transformation was considered to reduce the governing boundary-layer equations into non-similar form. The resulting nonlinear, coupled differential equations were solved numerically employing the Runge–Kutta algorithm with shooting iteration technique. Dimensionless velocity, temperature, concentration distributions, as well as local Nusselt number and Sherwood number are presented graphically for various values of Dufour number D_u, Soret number S_r, Buoyancy ratio N, amplitude of the wavy surface a, Lewis number Le, Grashof number Gr, and magnetic field effect M_g.     

Combined free and forced convection flow past a vertial porous plate

Local similarity solutions of the forced and free convection flow past a semi-infinite vertical porous plate have been obtained numerically. Velocity and temperature profiles are shown for air (Pr = 0.72) and for different values of Gr (Gr > 0, cooling of the plate and Gr<0, heating of the plate) (Grashof number), E (Eckert number) and f_w (f_w>0, suction, f_w <0, injection). the effects of these parameters are discussed.     

Mixed convection in a lid driven square cavity with an isothermally heated square blockage inside

Laminar mixed convection characteristics in a square cavity with an isothermally heated square blockage inside have been investigated numerically using the finite volume method of the ANSYS FLUENT commercial CFD code. Various different blockage sizes and concentric and eccentric placement of the blockage inside the cavity have been considered. The blockage is maintained at a hot temperature, T_h, and four surfaces of the cavity (including the lid) are maintained at a cold temperature, T_c, under all circumstances. The physical problem is represented mathematically by sets of governing conservation equations of mass, momentum, and energy. The geometrical and flow parameters for the problem are the blockage ratio (B), the blockage placement eccentricities (?_x and ?_y), the Reynolds number (Re), the Grashof number (Gr), and the Richardson number (Ri). The flow and heat transfer behavior in the cavity for a range of Richardson number (0.01–100) at a fixed Reynolds number (100) and Prandtl number (0.71) is examined comprehensively. The variations of the average and local Nusselt number at the blockage surface at various Richardson numbers for different blockage sizes and placement eccentricities are presented. From the analysis of the mixed convection process, it is found that for any size of the blockage placed anywhere in the cavity, the average Nusselt number does not change significantly with increasing Richardson number until it approaches the value of the order of 1 beyond which the average Nusselt number increases rapidly with the Richardson number. For the central placement of the blockage at any fixed Richardson number, the average Nusselt number decreases with increasing blockage ratio and reaches a minimum at around a blockage ratio of slightly larger than 1/2. For further increase of the blockage ratio, the average Nusselt number increases again and becomes independent of the Richardson number. The most preferable heat transfer (based on the average Nusselt number) is obtained when the blockage is placed around the top left and the bottom right corners of the cavity.     

OSCILLATORY CONVECTION IN SOLIDIFYING PURE METALS

A two-dimensional numerical study that models solidification of a pure metal in a horizontal crucible by using a finite-volume-enthalpy method has been carried out. The primary goals of the investigation are to assess the influence of phase change on the known transition to time-periodic convection and to predict the critical parameters associated with this phenomenon. Numerical simulations reveal the influence of the dimensionless solidification temperature T_s, on the critical value of the Grashof number Gr_crit. The influence is found to be directly related to an effective aspect ratio of the liquid part of the cavity. When T_s is increased, it is found that oscillatory convection takes place at a higher Grashof number and with slightly higher frequency. However, varying the Stefan number Ste has small influence on the critical parameters. The critical value of Gr for the onset of the oscillation is determined by carrying out a series of time-dependant calculations.     

Correlations for combined heat and mass transfer from an open cavity in a horizontal channel

Combined heat and mass transfer from a horizontal channel with an open cavity heated from below is numerically examined in this paper. Air is the fluid considered (Pr = 0.7). The main focus of the study is mass-transfer driven flows (|N| > 1). The governing parameters considered are the buoyancy ratio N, Lewis number Le, Reynolds number Re, and Grashof number Gr. Based on the scale analysis, correlations for the entire convection regime, from natural, mixed, to forced convection, were proposed.