Bubble numerical solutions

An attempt has been made by Johnson and co-workers to relate such theoretical results with experimental data for the absorption of a single carbon dioxide bubble into aqueous solutions of monoethanolamine, determined under forced convection conditions over a Reynolds number range from 30 to 220. The numerical results were found to be much higher than the measured values for noncirculating bubbles. The numerical solutions indicate that the mass-transfer rate should be independent of Peclet number, whereas the experimentally measured rates increase gradually with increasing Peclet number. The discrepancy is attributed to the experimental technique, where-... 

Fig. 10. Numerical solutions of the forced-convection mass-transfer equation for the case of irreversible first-order chemical reaction [after Johnson et al. (J4)] (Solid lines— rigid spheres dashed lines—circulating gas bubbles).

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

At larger Re and for more marked deformation, theoretical approaches have had limited success. There have been no numerical solutions to the full Navier-Stokes equation for steady flow problems in which the shape, as well as the flow, has been an unknown. Savic (S3) suggested a procedure whereby the shape of a drop is determined by a balance of normal stresses at the interface. This approach has been extended by Pruppacher and Pitter (P6) for water drops falling through air and by Wairegi (Wl) for drops and bubbles in liquids. The drop or bubble adopts a shape where surface tension pressure increments, hydrostatic pressures, and hydrodynamic pressures are in balance at every point. Thus... 

Attempts to obtain theoretical solutions for deformed bubbles and drops are limited, while no numerical solutions have been reported. A simplifying assumption adopted is that the bubble or drop is perfectly spheroidal. SalTman (SI) considered flow at the front of a spheroidal bubble in spiral or zig-zag motion. Results are in fair agreement with experiment. Harper (H4) tabulated energy dissipation values for potential flow past a true spheroid. Moore (Mil) applied a boundary layer approach to a spheroidal bubble analogous to that for spherical bubbles described in Chapter 5. The interface is again assumed to be completely free of contaminants. The drag is given by... 

Pritchett et al. (1978) were the first to report numerical solutions of the nonlinear equations of change for fluidized suspensions. With their computer code, for the first time, bubbles issuing from a jet with continuous gas through-flow, could be calculated theoretically. Figure 14 shows as an illustration the computed mo-... 

Ryskin, G., and Leal, L. G., Numerical solution of ffee-boundary problems in fluid mechanics. Part II, Buoyancy-driven motion of a gas bubble through a quiescent liquid. J. Fluid Mech. 148, 19 (1984b). 

To simulate the effects of reaction kinetics, mass transfer, and flow pattern on homogeneously catalyzed gas-liquid reactions, a bubble column model is described [29, 30], Numerical solutions for the description of mass transfer accompanied by single or parallel reversible chemical reactions are known [31]. Engineering aspects of dispersion, mass transfer, and chemical reaction in multiphase contactors [32], and detailed analyses of the reaction kinetics of some new homogeneously catalyzed reactions have been recently presented, for instance, for polybutadiene functionalization by hydroformylation in the liquid phase [33], car-bonylation of 1,4-butanediol diacetate [34] and hydrogenation of cw-1,4-polybutadiene and acrylonitrile-butadiene copolymers, respectively [10], which can be used to develop design equations for different reactors. 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

G. Ryskin and L.G. Leal, Numerical Solution of Free-Boundary Problems in Fluid Mechanics. Part 2. Bouyancy-Driven Motion of Gas Bubble Through a Quiescent Liquid, J. Fluid Mech, 148 (1984). 

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 theory of weak retardation of surface motion allows relations to be obtained to give an estimate of the minimum surfactant concentration for the appearance of a stagnant cap which exerts and effects the buoyant velocity. The theory of strong retardation yields a maximum surfactant concentration which separates the transient state from a complete retardation of the bubble surface. Thereby, the transition between the theory of limiting states of the dynamic adsorption layer and the theory of the transient state is obtained, which is important for two reasons. First of all, the theories were developed by different teams of scientists independently. Secondly, it allows to conclude the appropriateness of the approximate methods employed which gives a complete picture of different states of the dynamic adsorption layer. This is possible without huge efforts necessary for numerical solutions. 

A numerical solution would be needed to allow for the change in bubble size, area, oxygen concentration and mass transfer coefficient as the bubble rises in a quiescent liquid. In a stirred tank, small bubbles could be carried downward in some regions and have more time to dissolve. 

Knowledge of the secondary current distribution around bubbles on electrodes would be of some interest, for example in electroplating where gas pits in the plated metal are a problem. Calculating the secondary current distribution around bubbles on an electrode, however, requires a numerical solution. Nevertheless some conclusions can be drawn from dimensional analysis of the problem. Newman81 has shown that two parameters characterize nonuniform secondary current distributions ... 

Some results of the numerical solution of Eqs. (22.60) are shown in Figs. 22.5 to 22.7. Dependences on parameters f and n that characterize the influence of the surfactant, and also on the parameter K that determines the influence of pressure, are of greatest interest. At small values of r, the effect of the surfactant causes a reduction of surface tension, and the bubble grows faster than it would grow in the absence of the surfactant. At n = 0, the surfactant layer on the surface does not impede the transition of dissolved substance into the bubble, and the influence of the surfactant is manifest only in the reduction of surface tension. With an increase of n, the rate of bubble growth slows down considerably. An increase of the parameter K (the fall of pressure poo) results in enhancement of the bubble growth rate. Fig. 22.6 shows the change of surface concentration of surfactant at K = 4.45 n = 0 xioo = 1, and = 1 for various values of fi. An in-... 

For higher values of Ca (Ca > 5 x 10 ), numerical studies are required to obtain the shape of the bubble and the thickness of the film. A number of numerical solutions to the problem have been suggested, which also validate Bretherton s approach at very low values of Ca. As the front profile and the thickness of the film of a long Taylor bubble with sufficient separation between the bubbles can be approximated by that of an infinitely long bubble, many early studies considered semi-infinite bubbles. 

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. 

The numerical solution of this equation defines the temporal evolution of the radius of an isolated spherical bubble. Knowing the radius at any time enables the evolution of the temperature (thermodynamic value) and pressure inside the bubble (e.g., by means of Eqs. 23 and 24) to be deduced. Tgas represents the temperature inside the bubble for any value of R (derived by numerical integration of Eq. 22) and Rq the initial radius. To is the equilibrium temperature within the bubble, Pgas the gas pressure in the bubble at any time, Pq the equilibrium pressure in the bubble, and a the radius of the van der Waals hard core ... 

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