Coalescence kernel bubble

However, in other cases the model predictions deviate much more from each other and were in poor agreement the experimental data considering the measurable quantities like phase velocities, gas volume fractions and bubble size distributions. An obvious reason for this discrepancy is that the breakage and coalescence kernels rely on ad-hoc empiricism determining the particle-particle and particle-turbulence interaction phenomena. The existing param-eterizations developed for turbulent flows are high order functions of the local... 

If the continuous phase is a liquid, the main obstacle to coalescence is the drainage of the film of liquid in the small space in between the two particles. The efficiency is in these cases usually quantitied as a function (generally a negative exponential function) of the ratio of the characteristic time for droplet contact and film drainage. For example, in the case of small bubbles coalescing due to turbulent velocity fluctuations the coalescence kernel assumes the form (Buffo et al, 2012 Laakkonen et al, 2006 Petitti et al, 2010)... 

Just like in the context of simulating solids suspension, one may wonder whether much may be expected from just sticking to the two-fluid approach combined with population balances. A better way ahead might rather be to combine population balances with LES, while proper relations for the various kernels used for describing coalescence and break-up processes could be determined from DNS of periodic boxes comprising a certain number of bubbles (or drops). The latter simulations would serve to study the detailed response of bubbles or drops to the ambient turbulent flow. 

Venneker et al [118] made an off-line simulation of the underlying flow and the local gas fractions and bubble size distributions for turbulent gas dispersions in a stirred vessel. The transport of bubbles throughout the vessel was estimated from a single-phase steady-state flow fleld, whereas literature kernels for coalescence and breakage were adopted to close the population balance equation predicting the gas fractions and bubble size distributions. 

It has been shown in [147] that aerated model emulsions of palm kernel oil, coconut oil, as well as butter, which are used in making ice-cream can be obtained at an optimum surfactant/protein ratio. A combination of Tween 60 and skimmed milk powder provided a higher stability than using mono- and diglycerides. In particular, this can be due to the fact that Tween 60, at the same time, effectively inhibited the coalescence of gas bubbles. 

A simphfied way is to decouple the two approaches first, the stabihty condition serves as the close law for the simphfied algebraic conservation equations, as described by the EMMS model for gas—hquid and gas—solid systems. The nonlinear optimization problem can therefore be solved to obtain the global or local structure parameters which are then be used to derive the closure law or correlations for the drag, bubble-induced turbulence and even the correction factors for the kernel functions of bubble coalescence and breakup for PBEs. 

There are three types of closure models in CFD simulation of gas—hquid flow in bubble columns, i.e., drag force, bubble-induced turbulence, and kernel functions of bubble breakup and coalescence. We will show how we utilize the EMMS approach to derive new models and integrate them into CFD simulation. 

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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