Gas Bubble Growth

For isothermal gas bubble growth at atmospheric pressure, N/ is usually small (Section II,B,1), so that P3 s This result is identical with that 

This equation is solved exactly by Epstein and Plesset, and is also extended to include surface tension effects. Generally, for gas bubble growth, 1, 

SjR 1 it is seen from Eq. (101) that, for this limiting case also, 

However, eventually 5jR 1 for a dissolving bubble, so that the steady-state solution prevails. For gas bubbles dissolving in water at atmospheric pressure with i (0) 0.2 mm, Houghton, Ritchie, and Thomson (H8) find that the transient term becomes negligible after about 100 sec. After this time the result reduces to the quasi-steady solution, governed by the Laplace equation, whose solutions have been extensively studied. For a bubble attached to a wall with zero contact angle, Liebermann (L6) thus shows that the rate of solution is reduced by a factor of In 2, leading to the result 

Manley (M3) extends this result to allow for nonzero contact angle. 

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Bruijn (B24) employs the quasi-stationary approximation, as discussed in Bankoif (B8), to the growth of vapor bubbles in superheated binary liquid mixtures. As noted previously, this neglect of the convective term in the diffusion equation is justified only when Aj< l, which is usually not the case in atmospheric boiling. On the other hand, this technique would be applicable to isothermal gas bubble growth in three-component systems, where two of the components are dissolved gases. 

For isothermal gas bubble growth in liquids of arbitrary initial concentration distribution, where the boundary layer volume is large compared to the bubble volume, it is recommended (SI2) that a straightforward modification of the Epstein and Plesset (E3) quasi-stationary technique be employed. 

Figures 2 and 3 illustrate the evolution of the microstructure for the samples comprising 1 wt.% of calcium carbonate heat treated at 700-900 °C. The foams sintered under 850°C present well-dispersed, ovoid shaped, small pores (0.2-0.4 mm in diameter). For higher sintering temperature, due to a lowering of the liquid glass phase viscosity, gas bubbles growth and contacts lead to larger cavities (0.5-3 mm in diameter). As the mechanical and the insulation properties are influenced by the process temperature in opposite directions, a compromise around 800°C has been found.

The boiling mechanism can conveniently be divided into macroscopic and microscopic mechanisms. The macroscopic mechanism is associated with the heat transfer affected by the bulk movement of the vapor and Hquid. The microscopic mechanism is that involved in the nucleation, growth, and departure of gas bubbles from the vaporization site. Both of these mechanistic steps are affected by mass transfer. 

For group B and D particles, nearly all the excess gas velocity (U — U,nj) flows as bubbles tnrough the bed. The flow of bubbles controls particle mixing, attrition, and elutriation. Therefore, ehitriation and attrition rates are proportional to excess gas velocity. Readers should refer to Sec. 17 for important information and correlations on Gel-dart s powder classification, minimum fluidization velocity, bubble growth and bed expansion, and elutriation. 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

M. Amon and C. D. Denson [33-34] attempted a theoretical and experimental examination of molding a thin plate from foamed thermoplastic. In the first part of the series [33] the authors examined bubble growth, and in the second [34] — used the obtained data to describe how the thin plate could be molded with reference to the complex situation characterized in our third note. Here, we are primarily interested in the model of bubble growth per se, and, of course, the appropriate simplification proposals [33]. Besides the conditions usual for such situations ideal gets, adherence to Henry s law, negligible mass of gas as compared to mass of liquid, absence of inertia, small Reynolds numbers, incompressibility of liquid, the authors postulated [33] several things that require discussion ... 

Internal bubble growth occurs because flaws at an elastomer-substrate interface act as nucleation sites where gas which is coming out of solution in the mbber can collect, following a reduction in pressure. If there are flaw sites, based on energy considerations, gas will take the easiest option— which is to collect at an already preexisting bubble and cause it to grow. 

Louisnard O, Gomez F (2003) Growth by rectified diffusion of strongly acoustically forced gas bubbles in nearly saturated liquids. Phys Rev E 67 036610... 

In the third study, Miyazaki and Henry (1978) carried out vapor bubble growth experiments with water drops in hot silicone oil under various pressures of argon gas. As conducted, the oil temperature was set so that the interface temperature was below the homogeneous nucleation temperature of water. When bubbles did appear, their growth was followed by... 

The physical transport of mass is essential to many kinetic and d3mamic processes. For example, bubble growth in magma or beer requires mass transfer to bring the gas components to the bubbles radiogenic Ar in a mineral can be lost due to diffusion pollutants in rivers are transported by river flow and diluted by eddy diffusion. Although fluid flow is also important or more important in mass transfer, in this book, we will not deal with fluid flow much because it is the realm of fluid dynamics, not of kinetics. We will focus on diffusive mass transfer, and discuss fluid flow only in relation to diffusion. 

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