Buoyancy-driven convection

Because of the large thermal and concentration gradients, polymerization fronts are highly susceptible to buoyancy-induced convection. Garbey et al. performed the Hnear stability analysis for the Hquid/liquid and liquid/soHd cases [77-79]. The bifurcation parameter was a frontal Rayleigh number  

Let us first consider the liquid/solid case. Neglecting heat loss, the descending front is always stable because it corresponds to heating a fluid from above. The front is always flat. If the front is ascending, convection may occur depending on the parameters of the system. 

If the reactor is not vertical, there is no longer the question of stability-there is always convection. Bazile et al. studied descending fronts of acrylamide/bis-acrylamide polymerization in dimethyl sulfoxide (DMSO) as a function of tube orientation [81], The fronts remained nearly perpendicular to the vertical but the velocity projected along the axis of the tube increased with the inverse of the cosine of the angle. 

McCaughey et al. tested the analysis of Garbey et al. and found the same bifurcation sequence of antisymmetric to axisymmetric convection in ascending fronts [83] as seen with the liquid/solid case. 

Garbey et al. also predicted that, for a descending Hquid/hquid front, instability 

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Figure 3.3. Various features of diffusion and convection associated with crystal growth in solution (a) in a beaker and (b) around a crystal. The crystal is denoted by the shaded area. Shown are the diffusion boundary layer (db) the bulk diffusion (D) the convection due to thermal or gravity difference (T) Marangoni convection (M) buoyancy-driven convection (B) laminar flow, turbulent flow (F) Berg effect (be) smooth interface (S) rough interface (R) growth unit (g). The attachment and detachment of the solute (solid line) and the solvent (open line) are illustrated in (b).

Figure 3.6. Schlieren photographs showing the changes in thickness of the diffusion boundary layer and the behavior of buoyancy-driven convection shown in relation to bulk supersaturation [1], [2]. The figure shows the (111) faceofaBa(N03)2 crystal from an aqueous solution. In region I, only the thickness of the diffusion boundary layer increases in region II, we see unstable lateral convection (HA) and intermittently rising plumes (IIB) and in region III we see steady buoyancy-driven convection.

K. Onuma, K. Tsukamoto, and I. Sunagawa, Effect of buoyancy driven convection upon the surface microtopographs of BalNOjjj and Cdl crystalsJ. Crystal Growth, 98,1989, 384-90... 

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. 

Convection in the crystal growth systems discussed earlier cannot be characterized by analysis with either perfectly aligned vertical temperature gradients or slender cavities, because these systems have spatially varying temperature fields and nearly unit aspect ratios. Even when only one driving force is present, such as buoyancy-driven convection, the flow structure can be quite complex, and little insight into the nonlinear structure of the flow has been gained by asymptotic analysis. 

Accident scenarios that assume no loss of hydrogen due to buoyancy-driven convection can considerably overestimate the risk incurred in hydrogen escapes. In reality, hydrogen s very low density prevents this accident scenario from occurring. 

Figure 12-4. The Rayleigh-Benard configuration d for buoyancy-driven convection in a horizontal fluid...

This is essentially just the neutral stability condition for direct-mode buoyancy-driven convection. In the Ra, Ra plane, this is a straight line that defines a possible stability boundary. Note, however, that we do not get conditions for validity of the assumption aImag = 0. For this, we must examine the other case, crjmag / 0. 

Problem 12-13. Raleigh-Benard Convection - Nonisothermal Boundaries. The assumption of isothermal boundaries in the buoyancy-driven convection instability problem is generally an oversimplification. A more realistic picture is that the upper and lower surfaces are in contact with reservoirs that are maintained at a constant temperature so that the thermal boundary conditions are better approximated as... 

Figure 4-13 represents a history of reactor development. The most well-known type of reactor is shown in Figure 4-13a, which is the horizontal reactor. Horizontal reactors are well studied and understood. These reactors produce good materials and devices. Such reactors have been scaled to hold several 50 mm wafers. These reactors require sufficient gas flows to counter buoyancy driven convection (hot gases rise) and to counter reactor depletion along the flow path. In-position rotation of the wafer minimizes depletion effects. Dramatic increases in wafer numbers have come about by spreading the linear horizontal tube into a circular symmetric device as described below. 

The analysis with disturbance quantities of the form of Eqs. (10.6.15) indicates a periodic structure in the x, z plane but the shape of the cells associated with the solution is not specified and higher order nonlinear theory is required to define a particular cellular structure. Palm (1960) has shown that in the parallel Rayleigh problem for steady buoyancy driven convection of a liquid film heated from below, the cells approach a hexagonal form as a consequence of the variation of the kinematic viscosity with temperature. 

A new type of discrepancy was uncovered by Jeffreys (J5) who showed in 1951 that, in contrast to the observations of Benard, buoyancy-driven convection must lead to a free liquid surface that is convex over centers of ascending warm liquid. 

Recently, Sen and Davis (20) studied capillary flow in bounded cavities in which d/ is small, end effects are present, and the flow is very slow and the cavity is heated from the side. Cowley and Davis (21) studied the high Marangoni number Thermocapillary analogue of a buoyancy driven convection problem solved by Roberts (22). Later, we shall make some comparisons between our results for the deflection of the surface and those of Sen and Davis (20). 

Contrary to our expectations, die propagating front could not be generated near the wall of the test tube. A buoyancy-driven convection prevented the formation of the front. Experiments were carried out changing the size of the test tube and the temperature. In test tubes having inner diameters of 23,13, or 8 mm, AIBN was dissolved in 15.0, 5.0, or 1.7 mL of MMA. The depth of MMA liquid in each test tube and the concentration of AIBN (2.0 wt%) was kept constant. Hie tubes were then placed in a 50°C oil bath to start the polymerization. In all cases, the front was not generated near the wall of the test tubes. 

Furthermore, we neglect forced or natural (buoyancy-driven) convection, surface oscillations, and spreading phenomena, not to mention more sophisticated aspects such as Leidenfrost effects, i.e., retardadmi of evaporation on a hot substrate due to droplet levitation by the vapor, or aspects of thin film evaporation in the microregion, where the material properties deviate considerably from the bulk properties. 

As an application example, in the field of magnetohydrodynamics, the profile sensor measured the temporal evolution of the velocity field near the electrode at a copper electrolysis experiment under the influence of a magnetic field in order to study the interaction between Lorentz force and buoyancy-driven convection [6]. 

Surface tension is affected both by chemical concentration and by temperature. Figure 9.10 shows how a hot spot can cause convection by locally lowering the surface tension. The cooler fluid has a higher surface tension and draws the warm fluid towards itself. If the temperature gradient is perpendicular to the interface, both buoyancy-driven convection and Marangoni (surface-tension-driven) convection are possible (Antar and Nuotio-Antar, 1993). 

Pojman discusses thermal frontal polymerization in Chapter 4. He focuses on thermal frontal polymerization in which a locahzed reaction zone propagates through the coupling of thermal diffusion and the Arrhenius dependence of the kinetics of an exothermic polymerization. Frontal polymerization is close to commercial apphcation for cure-on-demand appHcations and is also showing value as a way to make some materials that are superior to those prepared by traditional methods. It also manifests many types of instabihties, including buoyancy-driven convection, surface-tension-driven convection, and spin modes. 

If there is a free interface between fluids, gradients in concentration and/or temperature parallel to the interface cause gradients in the surface (interfacial) tension, which cause convection [85]. This convection, also known as Marangoni convection, is especially noticeable in thin layers (or weightlessness) in which buoyancy-driven convection is greatly reduced. 

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