Coalescence source term

Furthermore, due to the limited internal coordinate resolution, the calculation of the coalescence and breakage terms requires some trick to enable an accurate determination of the mass of each group. The numerical implementation of the coalescence source terms is rather complex and requires a... 

Filbet and Laurengot [58] developed a particular finite volume method (FVM) scheme for dicretizing the Smoluchowski equation for purely coalescing systems. For the application of the FVM they established a continuous flux form of the PBE coalescence source terms. The FVM thus ensures that the poly-disperse particle fluxes between the individual sections are conserved in the system. This approach deviates from the conventional sectional methods which are applied to the standard discrete form of the PBE source terms. Kumar et al. [113] adapted the FVM scheme for solving the transformed coalescence source terms to pure breakage and simultaneous breakage and coalescence systems. 

Lee et al [66] and Prince and Blanch [92] adopted the basic ideas of Coulaloglou and Tavlarides [16] formulating the population balance source terms directly on the averaging scales performing analysis of bubble breakage and coalescence in turbulent gas-liquid dispersions. The source term closures were completely integrated parts of the discrete numerical scheme adopted. The number densities of the bubbles were thus defined as the number of bubbles per unit mixture volume and not as a probability density in accordance with the kinetic theory of gases. 

In this section the macroscopic population balance formulation of Prince and Blanch [92], Luo [73] and Luo and Svendsen [74] is outlined. In the work of Luo [73] no growth terms were considered, the balance equation thus contains a transient term, a convection term and four source terms due to binary bubble coalescence and breakage. 

A fairly general framework has been formulated for the source terms considering particle breakage, fluid particle coalescence, solid particle agglomeration/aggregation and similar processes (e.g., [109, 80, 81, 37, 114, 43, 25, 94]). Detailed discussions of the particle breakage and coalescence modeling and the mathematical properties of the constitutive equations can be found in the papers by Barrow [4], Laurencot and Mischler [64, 65]. 

As for the integral source terms considered hitherto, the integral in physical space is not appropriate as it is based on the assumption that two particles that coalesce at a particular location in space can produce a larger particle elsewhere. Following the same procedure as described above, the sink term due to coalescence can be re-defined as ... 

T source term in generalized Boltzmann type of equation representing the effects of particle coalescence, breakage and collisions J c)) collision term in the Boltzmann equation... 

The uniformly distributed redispersion of a drop (formed by coalescence at any instant) of size y into two other drops implies that the probability density for the size of either of the newly formed pair is l/y. Thus, the source term for drops of volume x is given by... 

The source terms for the bubble numbers are due to breakage and coalescence of bubbles, and mass transfer induced size change. Other sources (such as formation of small bubbles through nucleation mechanisms) were neglected in this study. The discretized population balance equation can then be written in the following form... 

The discrete source term will be written in a rather compact form by introducing the idea of the interaction matrices for droplet breakage and coalescence which decouples the working variables vector, (p, from the grid structure. Consequently, the interaction matrices are generated only once a time even for time dependent fi uencies. Now the discretized PBE (3) is projected onto the droplet diameter coordinate using the identity ... 

In a first modeling approach, a macroscopic population balance is formulated directly on the averaging scales in terms of number density functions [80, 102], A corresponding set of macroscopic source term closures are presented as well. Reviews of numerous fluid particle breakage and coalescence kernels on macroseopie scales can be found elsewhere [60,73,74, 122], This modeling framework resembles the mixture model concept. 

The source terms are assumed to be functions of bubble size bubble number density m and time t. The birth of bubbles of size di due to coalescence stems from the coalescence between aU bubbles of size smaller than di. Hence, the 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 ... 

In accordance with the work of Coulaloglou and Tavlarides [17] and Prince and Blanch [102], Luo [79] assumed that all the macroscopic source terms determining the death and birth rates could be defined as the product of a collision density and a probability. Thus modeling of bubble coalescence means modeling of a... 

The corresponding source terms for breakage and coalescence are defined by (9.87)-(9.90) ... 

The discrete pivot method provides information on the particle size distribution that is needed in the multi-fluid framework and can also be used for proper validation of the source term closures. However, in order to cover a broad range in the particle size distribution, a large number of sections are often needed making these algorithms rather time consuming. To ensure mass conservation, the smallest particles are generally not allowed to break and the largest particles are normally not involved in the coalescence process. 

The integer corresponds to the index of the cell such that (+1/2 - Cfc e k- Kumar et al. [ 113] transformed the conventional pure breakage PBE with discrete source terms into a flux form by adapting the same modeling framework as proposed by Filbet and Laurengot [58] for purely coalescing systems. The breakage equation was thus expressed as ... 

Overall source term due to breakage and coalescence Upper triangular matrix... 

The kernel functions of bubble breakup and coalescence are required to supply the source term in PBE to predict the bubble size distribution. These kernel functions are generally some phenomenological models together with some derivation using statistical analysis and classical theory of isotropic turbulence. PBE has been coupled with CFD in Hterature and the predicted bubble size agrees weU with the experiments at low superficial gas velocity less than 0.01 m/s or small gas volume fraction. The bubble size is usually overpredicted at relatively higher superficial gas velocity or gas volume fraction because the coalescence rate is always overpredicted. Hence, correction factors are used by some studies, either as a constant or as a function of gas holdup or Stokes number. However, these correction factors are empirical and only work weU for limited operating conditions or specified kernel functions. 

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