Coalescence redispersion models

Figure 1.6. Four micromixing models that have appeared in the literature. From top to bottom maximum-mixedness model minimum-mixedness model coalescence-redispersion model three-environment model.

Harada and co-workers (HI) developed two coalescence-redispersion models to describe micromixing in a continuous-flow reactor. In the first model, the incoming dispersed-phase fluid is assumed to consist of uniformly sized droplets. These droplets undergo 0 to n coalescences and redispersions with surrounding droplets of a constant average concentra-... 

A generalization of these population balance methods to reactions with arbitrary RTD was given by Rattan and Adler [126]. They expanded the phase space of the distribution functions to include the life expectation as well as concentration of the individual fluid elements i/ (C, A, 0- The population balance then reduces to all of the previous developments for the various special cases of segregated or micromixed flow, the perfect macromixing coalescence-redispersion model, and can be solved as continuous functions or by discrete Monte Carlo techniques. Goto and Matsubara [127] have combined the coalescence and two-environment models into a general, but very complex, approach that incorporates much of the earlier work. 

Pojman, J. A. Epstein, I. R. Kami, Y. Bar-Ziv, E. 1991a. Stochastic Coalescence-Redispersion Model for Molecular Diffusion and Chemical Reactions. 2. Chemical Waves, J. Phys. Chem. 95, 3017-3021. 

Describe the coalescence/redispersion model in less than three sentences. 

Bajpai, R. K., D. Ramkrishna, and A. Prokop (1976). Coalescence redispersion model for drop-size distributions in an agitated vessel, Chem. Eng. ScL, 31(10), 913-920. 

The coalescence-redispersion (CRD) model was originally proposed by Curl (1963). It is based on imagining a chemical reactor as a number population of droplets that behave as individual batch reactors. These droplets coalesce (mix) in pairs at random, homogenize their concentration and redisperse. The mixing parameter in this model is the average number of collisions that a droplet undergoes. 

Coalescence and redispersion models applied to these reaction systems include population balance equations, Monte Carlo simulation techniques, and a combination of macromixing and micromixing concepts with Monte Carlo simulations. Most of the last two types of models were developed to... 

Luss and Amundson (LI 3) employed this model to analyze reactor stability and control for segregated two-phase systems. The Monte Carlo simulation was employed to model the age distribution of segregated drops in the vessel. Conditions of operation under which heat-transfer effects may control the design of the reactor were given. It was shown that some steady states may be obtained in which the temperature of some drops greatly exceeds the average dispersed-phase temperature. The coalescence-redispersion problem was not considered here because of unreasonable computation times. 

The drops behave as segregated entities between flow and coalescence-redispersion simulation. The coalescence and breakage frequencies can be varied with vessel position. The computational time was related to coalescence frequency data available in the literature. Figure 15 shows the steady-state dimensionless droplet number size distribution as a function of rotational speed for continuous-flow operation. As expected the model predicts smaller droplet sizes and less variation of the size distribution with increase in rotational speed. Figure 16 is a comparison of the droplet number size distribution with drop size data of Schindler and Treybal (Sll). 

The model can be employed to predict the effects of droplet size distribution and droplet coalescence-redispersion on conversion and selectivity for reacting dispersions. The reactions can occur in either phase simultaneously with interphase heat and mass transfer. 

Villermaux, J. and Devillon, J.C., 1975. Representation de la coalescence et de la redispersion des domains de segregation dans un fluide par un modele d interaction phenomenologique. In Proceedings of the second international conference of chemical reaction engineering. Amsterdam, pp. Bl-13. 

In this model the segregation is perfect isotropic microscale segregation. All drops are assumed to have the same size. The chance of coalescing with a neighboring drop is for each drop the same, and independent of time and of the concentration of the drop. Redispersion occurs immediately after coalescing and exchange of concentration. Coalescing and redispersion is assumed to have no effect on mass transfer. 

The model assumes that drop coalescence is followed by immediate redispersion into two drops sized according to a uniform distribution. By assuming that the coalescence frequency is independent of drop size, the solution of the resulting form of Eq. (107) is exponential for the equilibrium drop volume distribution. Comparison of the distribution to experimental data is favorable. The analysis is useful in that a functional form for the distribution is obtained. The attendant simplifications necessary for solution, however, do not permit more rational forms of the interaction frequency of droplet pairs in order to account for the physical processes which lead to droplet coalescence and breakage as discussed in Section III. A similar work was presented by Inone et al. (II). 

A simplified homogeneous dispersed-phase mixing model was proposed by Curl (C16). Uniform drops are assumed, coalescence occurs at random and redispersion occurs immediately to yield equal-size drops of the same concentration, and the dispersion is assumed to be homogeneous. Irreversible reaction of general order s was assumed to occur in the drops. The population balance equations of total number over species concentration in the drop were derived for the discrete and continuous cases for a continuous-fiow well-mixed vessel. The population balance equation could be obtained from Eq. (102) by taking the internal coordinate to be drop concentration and writing the population balance equation in terms of number to yield... 

Erickson et al. (E3) developed a model for batch growth in fermentations with two liquid phases present in which the growth-limiting substrate is dissolved in the dispersed phase. The model accounts for drop size distribution and considers the effect of droplet coalescence and redispersion by an interaction model similar to that of Eq. (110). Droplet interactions were shown to be important if drop size distributions have large variance. 

Multiatom migration/ coalescence/ emission Crystallites of all sizes move, contacting crystallites or adatoms coalesce, crystallites and adatoms emitted diffusion or interfacial control. Reduce to KR or NMC models show effects of loading, solubility, emission, and diffusion good f( sintering or redispersion. Numerical computation elaborate multiatom emission very slow. 

The macroscopic problem is more intricate. The type of model utilized depends upon the ratio of the diffusion and reaction rates and thus upon the importance of micro- and macro-mixing. In a pipe reactor the values of the axial dispersion coefficients for both phases are required. For modeling, micro-mixing models are used, which describe the mutual interlinking of coalescence and redispersion processes. 

Fluid element or particle models where the fluid is broken up into small elements and mass transfer occurrs by coalescence and redispersion or diffusion. 

The main advantage of the Eulerian-Lagrangian formulation comes from the fact that each individual bubble is modeled, allowing consideration of additional effects related to bubble-bubble and bubble-liquid interactions. Mass transfer with and without chemical reaction, bubble coalescence, and redispersion, in principle, can be added directly to an Eulerian-Lagrangian hydrodynamic model. The main disadvantage of the Eulerian-Lagrangian approach is that only a limited number of particles (bubbles) can be tracked, such as when the superficial gas velocity is low (Chen et al., 2005), due to computer limitations. 

---