Bubble asymptotic

In the stirred tank, the final mean size of particles was reduced by the increase of stirring rate, being consistent with increased fluid shear induced particle disruption relative to aggregation. Use of three different gas velocities in the bubble column, however, results in no significant difference in agglomerate size but since the size is relatively small, it may simply reflect an asymptotic value. 

The life of a single bubble may be summarized as occurring in the following phases nucleation, initial growth, intermediate growth, asymptotic growth, possible col-... 

At a later stage of bubble growth, heat diffusion effects are controlling (as point c in Fig. 2.9), and the solution to the coupled momentum and heat transfer equations leads to the asymptotic solutions and is closely approximated by the leading term of the Plesset-Zwick (1954) solution,... 

The rise velocity, Uo, in general depends on the bubble size, or the bubble Reynolds number but as bubble size increases, as in two-phase upflow, Ua approaches an asymptotic value that is independent of Reynolds number. The following expressions have been accepted for a single bubble rising in an infinite medium, and for one rising in a swarm of surrounding bubbles, respectively (Duckler and Taitel, 1991b) ... 

II provides a transition between the two asymptotic limits. Viscous stresses now scale by the local thickness of the film, h, and the bubble shape varies from the constant thickness film to the spherical segment. Here the surfactant distribution along the interface may be important. Fortunately, for small capillary numbers, dh/dx < 1 and the lubrication approximation may be used throughout. Region II is quantified below. 

Werner (1980) has studied devolatilization in corotating twin-screw extruders when the volatile component was stripped from the polymeric solution by applying a vacuum to the system. Rough estimates of the equilibrium partial pressure of the volatile component in the feedstream for each of the systems studied by Werner indicate that this pressure was less than the applied pressure, which means that bubbles could have been formed. Figure 17 shows the influence of the externally applied pressure on the exit concentration for a methyl methacrylate-poly(methyl methacrylate) system of fixed concentration. Note that the exit concentration decreases as the pressure is decreased, but seems to approach an asymptotic value at the lowest pressures studied. Werner also reported that at a fixed flow rate and feed concentration the exit concentration did not vary with screw speed (over the range 150-300 min" ), which also suggests that ky alay, is independent of screw speed. Figure 18 is a plot of data obtained by Werner on an ethylene-low-density poly(ethylene) system and also shows that decreases in the applied pressure result in decreases in the exit concentration, but here a lower asymptote is not observed. 

In reality, Eqs. (13) and (14) should be solved simultaneously with Eqs. (8) and (9), but no analytical solution is available. However, we can examine the asymptotic solutions to Eqs. (13) and (14) to determine the bubble growth rate when heat transfer limits the growth, i.e., when P r) — Pq and r Tb so no inertial effects are present. For this extreme,... 

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. 

Fig. 38. Asymptotic growth of a steam bubble in water at 220.1 F., 1 atm. The line is a graph of Zwick s theoretical prediction (Zl).

If the bubble distribution analysis does not take into account a certain fraction, for example R < Rn, then the linear character of the distribution curves in the logarithmic probability system is sharply disturbed close to the point corresponding to radius R the curves acquire a vertical asymptotic character. 

Lisseter and Fowler (1992) have derived a simple set of equations for bubbly flow through a vertical tube. They have shown that under steady flow conditions, the void fraction will relax from its inlet value to an asymptotic value within only a short distance from the inlet. They have obtained a relationship between the inlet void fraction and the imposed pressure drop and derived a simple expression for the equihbrium void fraction. They have also considered the wall friction in their analysis of bubbly flows. 

The formulation of a proper e - equation for the case of bubbly flow was found to be more severe. As a first approach they adopted the above equation developed from the single phase transport equations (5.3). However, analyzing the two physical situations mentioned above, they found that this model formulation fails to produce both the asymptotic value and the time constant of homogeneous decay of grid generated bubbly flow turbulence. That is, the modified single-phase model did not break down, but it gave rise to unphysical solutions for such cases. 

To parameterize the new quantities occurring in these equations a few semi-empirical relations from the literature were adopted. The asymptotic value of bubble induced turbulent kinetic energy, fesia, is estimated based on the work of [3]. By use of the so-called cell model assumed valid for dilute dispersions, an average relation for the pseudo-turbulent stresses around a group of spheres in potential flow has been formulated. Prom this relation an expression for the turbulent normal stresses determining the asymptotic value for bubble Induced turbulent energy was derived ... 

Lopez de Bertodano [92] stated that this simple modification has a big effect on the dynamic and the asymptotic behavior of the model. At a later stage, [93] also stated that the bubble induced time constant, which is proportional to the residence time of a bubble, is usually very short compared to the time constant of the shear induced turbulence. They concluded that for most practical cases the transport equation for bubble induced turbulence (5.13) can be reduced to fcei = fceia-... 

In this section, we consider these problems in some detail, although with the major simplifications of assuming that the processes are isothermal and that the liquid is incompressible. As we shall see, the governing equations for even this simplified ID problem are nonlinear, and thus most features can be exposed only by either numerical or asymptotic techniques. In fact, the problem of single-bubble motion in a time-dependent pressure field turns out to be not only practically important, but also an ideal vehicle for illustrating a number of different asymptotic techniques, as well as introducing some concepts of stability theory. It is for this reason that the problem appears in this chapter. 

Now, the dynamics of changes in bubble radius with time, starting from some initial radius that differs slightly from an equilibrium value, is a problem that is ideally suited to solution by means of a regular asymptotic approximation. Of course, the governing equation is still the Rayleigh Plesset equation. Before beginning our analysis, we follow... 

Another important characteristic of the gas bubble is its response to a periodic oscillation of the ambient pressure / ,. For large-amplitude oscillations of the pressure, or for an initial condition that is not near a stable equilibrium state for the bubble, the response can be very complicated, including the possibility of chaotic variations in the bubble radius.22 However, such features are outside the realm of simple, analytical solutions of the governing equations, and we focus our attention here on the bubble response to asymptotically small oscillations of the ambient pressure, namely,... 

The basis of our analysis is, again, a very simple variation of the regular perturbation technique in which we assume that the bubble radius can be expressed in terms of a regular asymptotic expansion of the form... 

Thus, in the absence ofviscous damping, we find that the bubble radius oscillates periodically with an amplitude of 0(e) in response to the oscillating pressure field, provided only that co / co0, as we have assumed. A plot is given in Fig. 4-12 showing the time dependence of gi /p l for several different values of (co/coq). A key point to note about the solution, (4-233), however, is that the magnitude of gi becomes unbounded in the limit <0 —> >o-Indeed, in the limit co = co0, no bounded solution of the asymptotic form, (4-227), exists. This is a consequence of the resonant interaction that occurs when the forcing frequency co is equal to the natural oscillation frequency of the bubble, co0. 

The resonant case co = coQ in the limit of weak damping is itself quite interesting and also amenable to asymptotic analysis. We have seen, in this case, that an approximation of the form (4-227) is not possible for either Re, = 0 or Re x / 0(sp) - that is, a response of O(e) does not occur in spite of the fact that the pressure variation is 0(e) and e <obvious question is whether the amplitude of changes in bubble radius actually remains bounded in the resonant case o> = o> ] when Re, <following pages for the undamped case Re l =0. The problem for very small, but nonzero, damping is similar, but we shall not consider it here. 

In the present case, we consider a bubble whose interface is described in terms of a spherical coordinate system in the asymptotic form... 

Here, e is a small parameter that will form the basis of an asymptotic approximation for the dynamics of the bubble surface. The question here is whether a bubble with a nonspherical initial shape of small amplitude 0(e) will return to a sphere - that is, fn(9,[Pg.271]

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