Bubble analytical solutions

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

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,... 

One of the most important analytic solutions in the study of bubbles, drops, and particles was derived independently by Hadamard (HI) and Rybczynski (R5). A fluid sphere is considered, with its interface assumed to be completely free from surface-active contaminants, so that the interfacial tension is constant. It is assumed that both Re and Rep are small so that Eq. (1-36) can be applied to both fluids, i.e.,... 

In the following sections, the solutions of the models as well as examples will be presented for the case of trickle-bed reactors and packed bubble bed reactors. Plug flow and fust-order reaction will be assumed in order to present analytical solutions. Furthermore, the expansion factor is considered to be zero unless otherwise stated. Some solutions for other kinetics will be also given. The reactant A is gas and the B is liquid unless otherwise stated. 

In this section, some analytical solutions of fluidized-bed models are presented. Specifically, model solutions will be given for the case of a gas-phase reactant and a single solid-catalyzed reaction of the form A —> products and bubbling fluidized bed (Type B fluidization). The same analysis holds for a reaction of the form A + B —> products, if the reaction depends only on the concentration of A. Some solutions for the cases of a single reversible reaction, for two reactions in parallel, and two reactions in series will be given as well. 

Die Orcutt model is very simple, offering analytical solutions, and thus is a useful tool for a rough estimation of the effect of various parameters on the operation of fluidized beds (Grace, 1984). However, it should be used only for qualitative comparisons, since its predictions have often been inaccurate compared to the experimental values obtained. The sources of those failures are the predicted uniform concentration of gas in the dense phase, which is not the case in experiments, and the assumption of the absence of solids in the bubble phase, which results in underestimating the conversion in the case of fast reactions. 

Here b is the radius of curvature at the particle apex, where the two principal curvatures are equal (e.g., the bottom of the bubble in Figure 5.7a). Unfortunately, Equation 5.99, along with Equation 5.106, has no closed analytical solution. The meniscus shape can be exactly determined by numerical integration of Equation 5.102. Alternatively, various approximate expressions are available. Eor example, if the meniscus slope is small, z 1, Equation 5.99 reduces to... 

As a consequence of this nonlinearity, it is impossible to obtain analytic solutions of the Rayleigh-Plesset equation for most problems of interest, in which po j (t) is specified and the bubble radius R(t) is to be calculated. Indeed, most comprehensive studies of (4-208) have been carried out numerically. These show a richness of dynamic behavior that lies beyond the capabilities of analytic approximation. For example, a typical case might have Poo(t) first decrease below p,Xl(()) and then recover its initial value, as illustrated in Fig. 4-10. The bubble radius R(t) first grows up to a maximum (which typically occurs after the minimum... 

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,... 

This completes the solution to 0(Re l/2). It should be noted that the first two terms in (10-246) are, in fact, nothing but the first two terms in the inviscid solution, evaluated in the inner region, namely, (10-213). Thus, to 0(Re x/2), we see that the solution in the complete domain consists of the inviscid solution (10-155) and (10-156), with an 0(Re l/2) viscous correction in the inner boundary-layer region to satisfy the zero-shear-stress boundary condition at the bubble surface. Because the viscous correction in the inner region is only C)( Re l/2), the governing equation for it is linear. Hence, unlike the no-slip boundary layers considered earlier in this chapter, it is possible to obtain an analytic solution for the leading-order departure from the inviscid flow solution. 

The aim of this chapter is to present the fundamentals of adsorption at liquid interfaces and a selection of techniques, for their experimental investigation. The chapter will summarise the theoretical models that describe the dynamics of adsorption of surfactants, surfactant mixtures, polymers and polymer/surfactant mixtures. Besides analytical solutions, which are in part very complex and difficult to apply, approximate and asymptotic solutions are given and their range of application is demonstrated. For methods like the dynamic drop volume method, the maximum bubble pressure method, and harmonic or transient relaxation methods, specific initial and boundary conditions have to be considered in the theories. The chapter will end with the description of the background of several experimental technique and the discussion of data obtained with different methods. 

The theoretical description of a diffusion process of a surfactant to, or from, the surface of a floating bubble is impossible without information on the floating velocity and the hydrodynamic field around the bubble. The first of these quantities can be found comparatively easily experimentally, whereas the Navier-Stokes equation is used to define the hydrodynamic field around the floating bubble. A solution of the equation must satisfy all boundary conditions at the bubble surface. It should be stated that a general analytical solution of this... 

Saville (1973) solved the convective diffusion equation numerically and gave the same value of retardation coefficient as obtained by Dukhin (1965, 1981). Listovnichii (1985) has succeeded in obtaining simple approximation formulas for the concentration distribution not only along a bubble surface but also across the diffusion layer, based on numerical solution of Eq. (8.85). He has also shown that the analytical solutions Eqs (8.69) and (8.79) deviate from the exact solution less than 1 %, at m > 10 and m < 0.1. 

The coupling of the transport of momentum with the mass transport practically excludes any analytical solution in the field of physico-chemical hydrodynamics of bubbles and drops. However, a large number of effective approximate analytical methods have been developed which make solutions possible. Most important is the fact, that the calculus of these methods allows to characterise different states of dynamic adsorption layers quantitatively weak retardation of the motion of bubble surfaces, almost complete retardation of bubble surface motion, transient state at a bubble surface between an almost completely retarded and an almost completely free bubble area. 

For the emulsion film case, independent of the type of the function r(c) no analytical solution is available and numerical methods have to be apphed. To obtain a link to the experiment, an additional relationship, equivalent to the adsorption isoflierm, is required, relating the surface concentration r with the measured capillary pressure P = 2y/r in the bubble or film pressure of the curved foam film, which in turn is proportional to the surface pressure n. 

Dhaouadi, H., Poncin, S., Homut, J.M., andMidoux, N. (2008), Gas-liquid mass transfer in bubble column reactor - Analytical solution and experimental confirmation, Chemical Engineering and Processing Process Intensification, 47(4) 548-556. 

In Table 2.3, empirical correlations for the prediction of the bubble velocity during gas-liquid Taylor flow, as well as the analytical solution of Bretherton (1961) are summarised. 

For higher Reynolds numbers, analytical solutions do not exist, so the numerical solutions must be considered. When k->-oo, this problem corresponds to the viscous flow around a rigid particle and was studied by several authors [10—15]. When k = 0, this problem corresponds to the viscous flow around a spherical bubble and was also studied by several authors [15-18]. The significant phenomena are very well explained in the books of Clift et al. [1] and Sadhal et al. [2]. Values of drag coefficients from numerical solutions for bubbles and rigid spheres are presented in Table 5.2, which shows a good agreement between the different studies. 

Analytical solution of the mole balance equations is only likely to be possible when a number of simplifying assumptions can be made such as those adopted previously where we assumed a single irreversible first-order reaction, no change in molar flow due to reaction, isothermal reactor, negligible variation in pressure, plug flow of gas in the bubble phase, and either perfect mixing or plug flow in the dense phase (see Ref. [46]). Assumptions must also be made with respect to the respective... 

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