Bubble Coalescence Modeling in the Population Balance Framework

In the churn-turbulent bubbly flow regime with highly nonuniform bubble size distributions, bubble breakage and coalescence are important processes because they govern the bubble size distribution and consequently directly affect the interfacial mass, momentum, and heat transfer fluxes through the renewal bubble surfaces. At present, accurate prediction of bubble size distributions of dispersed gas–liquid flows by use of the population balance (PB) equation is a difficult task. The modeling of bubble breakup and coalescence rates is very complex and is based on the knowledge of collision and breakup frequencies, breakage daughter size distributions, and probability of coalescence. In this work, we focus on the coalescence phenomenon. The coalescence models are still on an empirical level and the mechanisms are not fully understood. This motivates the analysis of the suitability of the coalescence closures for the prediction of experimental data obtained from coalescence dominated gas–liquid flows. For this task, a cross-sectional averaged combined multifluid-PB model is adopted. Based on different theories for the coalescence efficiency, the simulation results show a similar trend in the prediction of the experimental data. Good prediction of the Sauter mean diameter is achieved although the shape of the bubble size distribution is not completely reproduced. The second aim of this work is to review the PB framework. Here, focus is placed on the coalescence term and the combined multifluid-PB model based on kinetic theory approach.     

A Survey of Multicomponent Mass Diffusion Flux Closures for Porous Pellets: Mass and Molar Forms

Heterogeneous catalysis is of paramount importance in many areas of gas conversion and processing in chemical engineering industries. In porous pellets, the catalytic reactions may be affected by diffusional limitations such that the global rate can be different from the intrinsic reaction rate. In the literature, a number of multicomponent diffusion flux closures have been applied to characterize the diffusion process within different units in chemical process plants. The main purpose of this paper is to outline the derivation of the different diffusion flux models: the rigorous Maxwell–Stefan and dusty gas models, and the simpler Wilke and Wilke–Bosanquet models. Usually the diffusion fluxes are derived and presented with respect to the molar average velocity definition. In this study, also the diffusion flux closures with respect to the mass average velocity definition is outlined. Thus, if the temperature equation and the momentum equation are used in the pellet model, a consistently closed set of pellet equations is obtained on mass basis holding only the mass average velocity. On the other hand, for the closed set of pellet equations on molar basis, the component balances hold the molar averaged velocity whereas the temperature and momentum equations hold the mass average velocity due to the physical laws applied deriving these fundamental balances. Nevertheless, the Maxwell–Stefan and dusty gas models are manipulated and put on the convenient Fickian form. The second purpose of this article is the evaluation of the diffusion flux closures derived. For this purpose, a transient model is developed to describe the evolution of the species composition, pressure, velocity, temperature, total concentration, and fluxes within a spherical pellet. The catalyst problem has been simulated for the methanol dehydration process producing dimethyl ether (DME), with computed efficiency factor values in the range 0.06–0.6 for pellet pore diameters of 0.1–100 nm. Identical results are expected for the mole and mass based pellet equations. However, deviations are obtained in the component fractions comparing the mass and mole based pellet model formulations where the mass fluxes were described according to the Wilke and Wilke–Bosanquet models. On the other hand, the rigorous Maxwell–Stefan and dusty gas models gave identical results.     

A bubble breakage model for finite Reynolds number flows

The breakage frequency of bubbles in turbulent liquid flows is modeled as the inverse of the breakage time by Martinez-Bazan et al. [J. Fluid Mech. 401: 157–182; 1999]. In this definition of the breakage frequency, it is assumed that the breakage probability is unity and hence all bubbles will break. This assumption is reasonable in turbulent flows at extremely high Reynolds numbers in which the turbulence energy dissipation is very high. For systems characterized by finite Reynolds numbers the energy dissipation rate decreases rapidly and the breakage probability is reduced significantly. In the present study, the breakage frequency model by Martinez-Bazan et al. has been extended to include the effect that only a fraction of the bubbles breaks at finite Reynolds numbers. For this model extension, an adjusted version of the breakage probability formula proposed by Coulaloglou and Tavlarides [Chem. Eng. Sci. 32: 1289–1297; 1977] was employed. The extended breakage frequency model for finite Reynolds number flows has been evaluated by comparison to recent experimental single bubble breakage data. It can be concluded that extensive experimental analyses are required to gather sufficient experimental data for improved understanding of the physical phenomena and for model validation. In particular, the bubble breakage analysis must be performed simultaneously with the characterization of the local turbulence properties in the flow.     

Numerical Solution of the Drop Population Balance Equation Using Weighted Residual and Finite Volume Methods

This article presents a comparison of numerical results obtained by two different approximations of population balances—the spectral orthogonal collocation and finite volume methods. In particular, the population balance equation for a homogeneous dispersed liquid–liquid system in a batch reactor was considered in the present numerical study. The focus was placed on the accuracy of the numerical approximation of the particle property density distribution. An advantage of the finite volume method is the easy of distributing the points in a nonuniform discretization. It is supposed that the spectral-element orthogonal collocation method may benefit by dividing the computational domain into elements of various polynomial orders. For the present problems studied, the orthogonal collocation in the spectral framework does not perform as well as the finite volume method.