Buoyancy surface tension

Numerical simulations that combine the details of the thermal-capillary models described previously with the calculation of convection in the melt should be able to predict heat transfer in the CZ system. Sackinger et al. (175) have added the calculation of steady-state, axisymmetric convection in the melt to the thermal-capillary model for quasi steady-state growth of a long cylindrical crystal. The calculations include melt motion driven by buoyancy, surface tension, and crucible and crystal rotation. Figure 24 shows sample calculations for growth of a 3-in. (7.6-cm)-diameter silicon crystal as a function of the depth of the melt in the crucible. 

External fields such as electric, magnetic, and acoustic fields can be employed for assembly in fluidic environment, while fluid properties such as buoyancy, surface tension, electrical insulation, viscosity, and interfaces can assist. Various tasks can be accomplished using the advantages of each specific external field, and complex assemblies can be formed through feedback control. 

The origin of several important physical concepts including those of inertia, buoyancy, surface tension of liquids, the density of air, and its resistance to motion (drag). 

Inertial Viscous Buoyancy Surface tension Pressure... 

Viscous Buoyancy Surface tension Pressure gL Ap pU Ca= Y Y LAp pU Ap gLAp LAp Y... 

We expect more insight from simulations in the future, particularly in situations where these multicomponent systems show effects of coupling between the different degrees of freedom, surface tensions depending on temperature and concentration, hydrodynamic flow induced by concentration gradients in addition to thermal buoyancy. 

Bubble size at departure. At departure from a heated surface, the bubble size may theoretically be obtained from a dynamic force balance on the bubble. This should include allowance for surface forces, buoyancy, liquid inertia due to bubble growth, viscous forces, and forces due to the liquid convection around the bubble. For a horizontally heated surface, the maximum static bubble size can be determined analytically as a function of contact angle, surface tension, and... 

W-3 CHF correlation. The insight into CHF mechanism obtained from visual observations and from macroscopic analyses of the individual effect of p, G, and X revealed that the local p-G-X effects are coupled in affecting the flow pattern and thence the CHF. The system pressure determines the saturation temperature and its associated thermal properties. Coupled with local enthalpy, it provides the local subcooling for bubble condensation or the latent heat (Hfg) for bubble formation. The saturation properties (viscosity and surface tension) affect the bubble size, bubble buoyancy, and the local void fraction distribution in a flow pattern. The local enthalpy couples with mass flux at a certain pressure determines the void slip ratio and coolant mixing. They, in turn, affect the bubble-layer thickness in a low-enthalpy bubbly flow or the liquid droplet entrainment in a high-enthalpy annular flow. 

Saturation (v) is the volume fraction of the total void volume occupied by a specific fluid at a point. Saturation values can vary from zero to 1 with the saturation of all fluids equal to 1. Residual saturation (Sr) is the saturation at which the NAPL becomes discontinuous and immobile due to capillary forces. Residual saturation is dependent upon many factors, including pore size distribution, wettability, fluid viscosity and density ratios, interfacial surface tension, gravity and buoyancy forces, and hydraulic gradients. 

Tate s law makes use of bubble formation in the measurement of the surface tension of a liquid, and hence the fact that surface tension influences the bubble volume is obvious. At flow rates tending to zero, the bubble volume is such that the upward force due to buoyancy is balanced by the downward force of surface tension. So, an increase in surface tension should... 

When the density of a liquid is increased, the buoyancy force corresponding to a specific size of the bubble increases, whereas the surface-tension force may remain constant. Thus, for a definite amount of surface-tension force, the bubble volume obtained is smaller. 

Considering the case of vanishingly small flows (Q a 0), both the first and the second terms on the right-hand side of Eq. (55) which contain Q, become zero, and the force balance bubble volume is obtained directly by equating the surface tension force with the buoyancy force. Even in the detachment stage,... 

Referring to Eq. (55), it is easily seen that if both n and Q are small, that of surface tension and that of buoyancy are the only important forces, and an... 

These authors have assumed the bubble to be expanding at the orifice, and have used the force balance equation at the time of detachment. The various forces considered by these authors are buoyancy, force due to the addition of mass (P2), excess pressure force, surface tension force, drag force, and force due to the inertia of the liquid. 

The procedure is the same as for constant flow conditions. The only forces to be considered are the inertia force, the surface-tension force, and the buoyancy force. The viscous effects are neglected. Making a force balance, we obtain ... 

The first stage is assumed to end when the vertical components of the expansion drag and the surface-tension force together become equal to the buoyancy. 

Krishnamurthy et ah (K13) have confirmed the above conclusion and have developed an expression for evaluating the bubble volume under conditions of flow when the corresponding volume for static conditions is given. These authors used capillaries of different diameters ground at the tip as nozzles. The capillaries were arranged horizontally, and the fluid travelled vertically so as to add to the buoyancy. The liquid viscosity was varied from 1 to 30 cp and the surface tension from 62 to 70 dyn/cm. 

The surface force will act along the perimeter of the plate (i.e., length [Lp] + width [Wp]). The plate is often very thin (less than 0.1 mm) and made of platinum, but even plates made of glass, quartz, mica, and filter paper can be used. The forces acting on the plate consist of the gravity and surface tension downward, and buoyancy due... 

Thus, by using very thin plates with thickness 0.1 to 0.002 mm, one can measure surface tension with very high sensitivity. In practice, by using very thin platinum plates of well-known dimension (length = 1.00 or 2.00 cm), one can calibrate the apparatus with pure liquids, such as water and ethanol. The buoyancy correction is made very small (and negligible) by using a very thin plate and dipping the plate as little as possible. 

Convection in Melt Growth. Convection in the melt is pervasive in all terrestrial melt growth systems. Sources for flows include buoyancy-driven convection caused by the solute and temperature dependence of the density surface tension gradients along melt-fluid menisci forced convection introduced by the motion of solid surfaces, such as crucible and crystal rotation in the CZ and FZ systems and the motion of the melt induced by the solidification of material. These flows are important causes of the convection of heat and species and can have a dominant influence on the temperature field in the system and on solute incorporation into the crystal. Moreover, flow transitions from steady laminar, to time-periodic, chaotic, and turbulent motions cause temporal nonuniformities at the growth interface. These fluctuations in temperature and concentration can cause the melt-crystal interface to melt and resolidify and can lead to solute striations (25) and to the formation of microdefects, which will be described later. 

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