Growth of bubbles

As indicated in Sec. IIB, ordinary nucleate boiling is a two-step process. First, nuclei must appear. Second, the nuclei must grow into bubbles large enough to move away from the nucleation sites. The rate of heat absorption by the liquid may be controlled by either one or both of these two processes. The growth of a nucleus (tiny bubble) into ordinary bubbles has received attention recently. The theoretical attack of Forster and Zuber was discussed in Sec. IIB2. Inasmuch as the theory of Zwick and Plesset (P3, P4, Zl, Z2) represents another attempt to obtain exact expressions for bubble growth, and since the theory fits well with the few data for steam bubbles in superheated water, their theoretical method is summarized below. 

For a bubble to grow, vapor must pass from the superheated liquid into the bubble. Thus latent heat of vaporization is removed from the surrounding liquid, and the liquid cools. The drop in liquid temperature near the bubble means a decrease in the driving force between liquid and bubble. This temperature drop strongly affects the bubble rate of growth. The rate can be shown to approach asymptotically a condition whereby the radius increases according to the square root of time. 

The growth involves not only heat transfer but also mass transfer. Because of this, and also because growth is a dynamic process, the mathematical problem is quite complex. In order to describe the process in mathematical terms, a few reasonable assumptions can be used. The sole resistance to heat transfer is assumed to lie in a thin liquid film surrounding the bubble. The vapor in the bubble is assumed to be 

It is not necessary to assume the liquid film to be completely stagnant. Radial motion can be allowed for, but with some difficulty. It was noted in Sec. IIB2 that Forster and Zuber state that conduction is the chief mode of heat transfer (compared with convection due to radial motion). Eddies or motions of the liquid tangent to the bubble are neglected. The Zwick-Plesset theory likewise excludes eddies. The derivation is lengthy therefore the final typical equations are presented here without proof. 

The integrated expression relating radius to time and temperature is a nonlinear, integrodifferential equation. It can be put into tractable forms by imagining four overlapping growth periods. 

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Part of the gas can escape from the solution at a specific concentration and a fixed temperature, as the pressure level falls to under P < Pg. This takes place in two phases appearance of nuclei, and growth of bubbles of the free gas phase. Thermodynamic conditions for stable nucleation are formulated in [1], They are analogous to the conditions for starting the boiling of low-molecular liquids. The following changes take... 

When heat is transferred to a pure liquid near its boiling point, vaporization begins with the formation of microscopic bubbles on the tube wall in a process identified as nucleation. The analysis of heat transfer in nucleate boiling can be separated into two parts, the formation and growth of bubbles on a surface, and the subsequent growth of these bubbles after they leave the surface. Both processes are complex and in an attempt to better understand the basic mechanisms they are most often studied in nonflowing batch systems. 

The bubble formed in stable cavitation contains gas (and very small amount of vapor) at ultrasonic intensity in the range of 1-3 W/cm2. Stable cavitation involves formation of smaller bubbles with non linear oscillations over many acoustic cycles. The typical bubble dynamics profile for the case of stable cavitation has been shown in Fig. 2.3. The phenomenon of growth of bubbles in stable cavitation is due to rectified diffusion [4] where, influx of gas during the rarefaction is higher than the flux of gas going out during compression. The temperature and pressure generated in this type of cavitation is lower as compared to transient cavitation and can be estimated as ... 

Growth of bubble by the mass transfer of volatiles from the magma to the bubble in a sphere coordinate is given by (Proussevitch and Sahagian,... 

The growth of bubbles in a volume-heated liquid has been determined by Dergarabedian (Dl) using 1000 frames/sec. The bursting of bubbles... 

Verification of the Forster-Zuber formulas has been of two kinds, checks of Eq. (15) and of Eq. (21). Ellion (El) conducted a photographic study to determine the rate of growth of bubbles of steam and of carbon... 

The microstructure of the multiphase media is often the product of phase transitions, e.g. (i) capillary condensation in the porous media, (ii) phase separation in polymer/polymer and polymer/solvent systems, (iii) nucleation and growth of bubbles in the porous media, (iv) solidification of the melt with a temporal three-phase microstructure (solid, melt, gas), and (v) dissolution, crystallization or precipitation. The subject of our interest is not only the topology of the resulting microstructured media, but also the dynamics of its evolution involving the formation and/or growth of new phases. 

From studies on the behaviour of fluidized beds it is already known that bubbles are of great importance if one seeks to describe these systems. Mori and Wen( )have shown an influence of the ratio dp/D on the growth of bubbles, and it is well known that bubbles grow when they rise through the bed. Clearly,it... 

The growth of bubbles is controlled by the rates at which volatiles in the melt can diffuse towards the bubbles, and the opposing viscous forces. Near a bubble, volatiles are depleted such that melt viscosity increases dramatically, and diffusivities drop, making it harder for volatiles to diffuse through and grow the bubble. These opposing factors are described by the nondimensional Peclet number (Pe), which is the ratio of the characteristic timescales of volatile diffusion (T(1 = r lD, where r is the bubble radius and D the diffusion coefficient of the volatile in the melt) and of viscous relaxation (t = 17/AP where 17 is the melt dynamic viscosity and AP the oversaturation pressure, i.e., Pe = Dingwell... 

The correlation of Werther (W9) gives nearly the same results as Eq. (2-5). Both correlations predict growth of bubble diameter and decrease of bubble frequency due to coalescence. However, experimental data in fluid beds have been excluded from the above correlations. 

The factors affecting the stability and growth of bubbles in aqueous foams have been reviewed in depth by deVries (3). In order to disperse a given volume of gas in a imit volume of liquid, one must increase the free energy of the system by an amount of energy AF as follows ... 

Models that directly relate the gas bubble flux to the current density implicitly assume that the nucleation and growth of bubbles is very fast. This assumption is not needed in the present modelling approach as gas evolution is related to supersaturation instead of current density. 

One of the most interesting problems of ultrasonic cavitation is the existence of the diffusive growth of bubbles in a sound field. As, without any field, a gas bubble should slowly dissolute due to gas diffusion from the bubble to a liquid, directional gas diffusion from liquid to the bubble arises under conditions of the bubble surface... 

This physical step follows a cycle. The bubble growth starts at defects (such as cavities) on the electrode surface. The nucleation and growth of bubbles is possible only if the electrolyte in the vicinity of the electrode is supersaturated with dissolved gas. During their growth, the bubbles are fed from the highly supersaturated surrounding electrolyte [12,113,114]. Two forces act on the bubbles the buoyancy force FB and the capillary force Fc (Fig. 3.7) ... 

The local, phase-volume-averaged treatment of flow and heat and mass transfer has been addressed by Voller et al. [154], Prakash [155], Beckermann and Ni [156], Prescott et al. [157], and Wang and Beckermann [158], and a review is given by Beckermann and Viskanta [143]. Here we will not review the conservation equations and thermodynamic reactions. They can be found, along with the models for the growth of bubble-nucleated crystals, in Ref 158. 

The explanation for the existence of boiling at much lower superheats than predicted by homogeneous nucleation theory is that bubbles are initiated from cavities on the heat transfer surface. Gas or vapor is trapped in these cavities as shown in Fig. IS.lb-d. Once boiling is initiated, these cavities may remain vapor-filled and continue to be active sources for the initiation and growth of bubbles from the surface. The growth process from a conical cavity whose mouth radius is rc is illustrated in Fig. 15.8. 

R velocity of bubble interface, rate of growth of bubble... 

A variety of polymers are extrusion-foamed, including polyolefins, polystyrene, and polyesters if the polymer is subjected to high-pressure gas, and the pressure is suddenly decreased or the temperature rapidly increased, the gas will try to escape from the polymer, causing antiplasticization. This rapid escape of gas can cause the nucleation and growth of bubbles within the polymer. Once a significant amount of gas escapes, the of the polymer drops, and thus the structure formed is frozen . 

Suppose that together with growth of bubbles brought in from the outside, there occurs nucleation of the new bubbles of radius R , at the rate N . It is easy to see that a bubble nucleated at the depth Yo, will, at the end of the layer, have the radius Ri = R (l +XiF// tga). In order for this to happen, the bubble must travel the distance Xi = (Ri/R — l)/ tga/F. Thus, in order for the bubble of initial radius R to reach the surface of the layer, it should be nucleated at a distance no less than Xi from the end of the layer. Out of all the bubbles, only those will reach the exit, for whom the inequality Yo < Yon- is satisfied. 

Fig. 23.13 The radius distribution of bubbles as the mixture leaves the layer for different lengths of the layer L, m 0- initial distribution 7 -10 2 - 50 3 - 100 I, II - with and without due consideration for diffusion growth of bubbles.

THERMAL GROWTH OF BUBBLES IN SUPERHEATED SOLUTIONS OF POLYMERS... 

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