Bubble number density

Their G//OST -codc essentially consisted of a mass balance for the gas, a transport equation for the bubble number density nb, and a force balance for a single bubble, respectively, which run as... 

Interaction parameters, a(x,t), calculated from the bubble population balance itself, e.g., the total bubble density in flowing and stationary foam, the higher moments of the bubble number density distribution, etc. 

The following balance equation can be derived for the bubble number density in this space (19,20)... 

The net rate of bubble generation, H, describes redistribution of mass in bubble-bubble interactions. Thus, H is a nonlinear functional of F(x,m,t) and Equations (2) and (3) are a pair of coupled, nonlinear, integro-differential equations in the bubble number density, similar to Boltzmann s equation in the kinetic theory of gases (26,27) or to Payatakes et al (22) equations of oil ganglia dynamics. 

The transport equation for the bubble number density distribution can be derived by use of the population balance method ... 

A meaningful simplification of the bubble population balance can be achieved by expanding the bubble number density into the moments of bubble mass. 

The purpose of this Appendix is to volume-average the population balance of bubble number density... 

The intrinsic domain average , i=f,t, of the bubble number density is defined as... 

Fig. 8.13. The calculated bubble number density at axial level 2 (m) above the column inlet. ( ) is experimental data [61]. = 8 cm/s). Reprinted with permission...

It is noted that all the terms in the continuum population balance equation have common units, (l/m s[m]). By discretizing the continuous bubble number density in groups or classes, a PBE formulation equivalent to the discrete macroscopic framework is obtained. 

Priedlander SK (2000) Smoke, Dust, and Haze Fundamentals of Aerosol Dynamics, Second Edition, Oxford University Press, New York Guido Lavalle G, Carrica PM, Clausse A, Qazi MK (1994) A bubble number density constitutive equation. Nucl Engng Des 152 213-224 Hagesaether L, Jakobsen HA, Svendsen HF (1999) Theoretical Analysis of Fluid Particle Collisions in Turbulent Flow. Chem Eng Sci 54(21) 4749-4755 Hagesaether L, Jakobsen HA, Hjarbo K, Svendsen HF (2000) A Coalescence and Breakup Module for Implementation in CFD-codes. Computer-Aided Chemical Engineering 8 367-372... 

In equation 4, the subscripts f and t refer to flowing and trapped foam, respectively, and ni is the foam texture or bubble number density. Thus, nf and t are, respectively, the number of foam bubbles per unit volume of flowing and stationary gas. The total gas saturation is given by Sg = 1 — Sw = S + St, and Qb is a source—sink term for foam bubbles in units of number per unit volume per unit time. The first term of the time derivative is the rate at which flowing foam texture becomes finer or coarser per unit rock volume, and the second is the net rate at which foam bubbles trap. The spatial term tracks the convection of foam bubbles. The usefulness of a foam bubble population-balance, in large part, revolves around the convection of gas and aqueous phases. 

Bubble size is required to calculate, for example, the drag force imparted on a bubble. Most Eulerian-Eulerian CFD codes assume a single (average) bubble size, which is justified if one is modeling systems in which the bubble number density is small (e.g., bubbly flow in bubble columns). In this case, the bubble-bubble interactions are weak and bubble size tends to be narrowly distributed. However, most industrially relevant flows have a very large bubble number density where bubble-bubble interactions are significant and result in a wide bubble size distribution that may be substantially different from the average bubble size assumption. In these cases, a bubble population balance equation (BPBE) model may be implemented to describe the bubble size distribution (Chen et al., 2fX)5). 

The development of the model for the wave phenomena is based on a hierarchy of length scales [61 On the microscale this is the bubble radius R, on the mezoscale it is the cell radius A, and on the macroscale we deal with a characteristic wave length L. We make the assumption that the bubble distribution is monodisperse, that bubbles keep their spherical shape and do not fragment. The void fraction (p is then related to the bubble number density n and the radius R by... 

To circumvent this problem. Lane et al. [48] used a simplified approach. Instead of computing the size distribution in the fermenter, they modeled and computed the mean bubble diameter at each point in the tank. They computed the bubble number density, defined as =, , where now indicates the mean bubble... 

The source terms are assumed to be functions of bubble size birth rate for bubbles of size di, Bc,u can be obtained by summing all coalescence events that form a bubble of size di. This gives ... 

Guido LavaUe G, Carrica PM, Clausse A, Qazi MK (1994) A bubble number density constitutive equation. Nucl Eng Des 152 213-224... 

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