Monte coalescence

Tavlarides presents a sophisticated model for representing coalescence and breakage of droplets in liquid-liquid dispersions. The model relies on the population balance equation and still requires the adjustment of 6 parameters. The solution of such equations is difficult and requires the use of Monte-Carlo methods... 

Spielman, L. A., and Levenspiel, O., A Monte-Carlo treatment for reacting and coalescing dispersed phase systems. Chem. Eng. Sci. 20, 247 (1965). 

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. 

These coalescence-redispeision models can also be formulated as a direct numerical Monte Carlo computation, as shown by Spielman and Levenspiel [123]. Similar results are obtained, but, of course, each situation must be individually simulated. 

The Monte Carlo approach was extended to reactors with a plug flow macro-mixing RTD by Rattan and Adler [124]. Here, the coalescing fluid elements are moved through the reactor at a speed corresponding to the constant mean fluid velocity. Rattan and Adler [124] were able to simulate experimental results of Vassilatos and Toor [125]. The coalescence frequency was found from data for extremely rapid reactions, where the observed rate is essentially completely controlled by the micromixing in this situation with a flat velocity profile. Then, these coalescence rates were used to predict the experimental results for rapid and slow reactions taking place in the same equipment. 

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. 

Martin, M., Garcia, J.M., Montes, F.J., and Galan, M.A. (2008a), On the effect of the orifice configuration on the coalescence of growing bubbles, Chemical Engineering and Processing Process Intensification, 47(9-10) 1799-1809. 

More complex computational models using Monte Carlo methods have attempted to predict bubble size distributions for a combination of breakup and coalescence. These models typically treat bubble coalescence by analogy with the kinetic theory where bubbles are assumed to act as solid particles [18,19]. They use a binary collision rate (probability) and a collision efficiency factor to account for collisions that do not lead to coalescence. Since collision is assumed to be a random process in these models, turbulence of the same scale as the bubbles or smaller would increase collisions and, therefore, also increase the coalescence rate. 

Spielman and Levenspiel (1965) appear to have been the earliest to propose a Monte Carlo technique, which comes under the purview of this section, for the simulation of a population balance model. They simulated the model due to Curl on the effect of drop mixing on chemical reaction conversion in a liquid-liquid dispersion that is discussed in Section 3.3.6. The drops, all of identical size and distributed with respect to reactant concentration, coalesce in pairs and instantly redisperse into the original pairs (after mixing of their contents) within the domain of a perfectly stirred continuous reactor. Feed droplets enter the reactor at a constant rate and concentration density, while the resident drops wash out at the same constant rate. Reaction occurs in individual droplets in accord with nth-order kinetics. 

Artificial realization of the system behavior (35). This method is commonly applied to complex particulate processes, which are described in some detail here. In the artificial realization, the direct evaluation of integral and differential functions is replaced by the simulation of the stochastic behavior modeled by using a randomness generator to vary the behavior of the system (20). The important probabilistic functions in the original model equations, such as coalescence kernels for granulation processes, are still essential in Monte Carlo simulations and are shown later. 

Monte Carlo methods for the artificial realization of the system behavior can be divided into time-driven and event-driven Monte Carlo simulations. In the former approach, the time interval At is chosen, and the realization of events within this time interval is determined stochastically. Whereas in the latter, the time interval between two events is determined based on the rates of processes. In general, the coalescence rates in granulation processes can be extracted from the coalescence kernel models. The event-driven Monte Carlo can be further divided into constant volume methods... 

Key equations needed in Monte Carlo simulations include the interevent time Atg representing the time spent from q—l to q Monte Carlo steps, coalescence kernel Kij, normalized probability ptj for a successful collision between particles i and j, and a number of intermediate variables. The coalescence kernel can be divided into particle property independent part Kc and dependent part ky (X X ) as follows ... 

The simulation procedure for the constant number Monte Carlo method applied to coalescence processes consists of the following key steps ... 

It can be seen that the Monte Carlo methods are applicable to both one-dimensional and multidimensional coalescence processes without any theoretical and algorithmic hurdles. However, most reported results with good agreement with experimental data are limited to one-dimensional systems except that reported by Wauters (52). This is mainly because of the lack of reliable multidimensional kernel models rather than the applicability of Monte Carlo methods. 

Spielman LA, Levenspiel O. A Monte Carlo treatment for reaction and coalescing dispersed systems. Chem Eng Sci 1965 20 247. 

Tavlarides (1981) 1981 Monte Carlo methods Direct Used phenomenological models describing coalescence and dispersion to predict DSD, with results compared with experiments. A review. 

Bapat et al. (1983) 1983 Interval of quiescence Monte Carlo method Direct Used breakage and coalescence functions to predict spatially varying drop size distribution and mass transfer rates. 

Table 13-12 summarizes the main simnlation methods that have been or are in use. In the discussion that follows, Enlerian methods based on time-averaged (or Reynolds-averaged) balance equations for the component concentrations and segregation will be emphasized, but the Lagrangian-oriented engulfment model and Monte Carlo coalescence-dispersion models are also presented. 

Monte Carlo Coalescence-Dispersion Simulation of Mixing... 

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