Coalescence-dispersion model

Here, (jf n) indicates (p value on the nth particle. In the coalescence/dispersion model proposed by Curl (1%3), the state of/at f + Zif is calculated from the collision frequency co. The compositions of a pair of particles, which are selected at random (denoted by wi and W2), change as ... 

Another model for the micromixing term can be derived from a mechanistic model, the coalescence-dispersion model of Curl [11] it... 

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. 

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

An attempt has been made by Tsouris and Tavlarides[5611 to improve previous models for breakup and coalescence of droplets in turbulent dispersions based on existing frameworks and recent advances. In both the breakup and coalescence models, two-step mecha-nisms were considered. A droplet breakup function was introduced as a product of droplet-eddy collision frequency and breakup efficiency that reflect the energetics of turbulent liquid-liquid dispersions. Similarly, a coalescencefunction was defined as a product of droplet-droplet collision frequency and coalescence efficiency. The existing coalescence efficiency model was modified to account for the effects of film drainage on droplets with partially mobile interfaces. A probability density function for secondary droplets was also proposed on the basis of the energy requirements for the formation of secondary droplets. These models eliminated several inconsistencies in previous studies, and are applicable to dense dispersions. 

Delichatsios and Probstein (D4-7) have analyzed the processes of drop breakup and coagulation/coalescence in isotropic turbulent dispersions. Models were developed for breakup and coalescence rates based on turbulence theory as discussed in Section III and were formulated in terms of Eq. (107). They applied these results in an attempt to show that the increase of drop sizes with holdup fraction in agitated dispersions cannot be attributed entirely to turbulence dampening caused by the dispersed phase. These conclusions are determined after an approximate analysis of the population balance equation, assuming the size distribution is approximately Gaussian. 

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-... 

The next term reflects the existence of fluctuations in the system (a is the fluctuation strength, hj (x) are elements of the Nxn diffusion matrix). The third term uses a coalescence-dispersion mechanism (for details, see [1,4]) to model turbulent mixing (23 is the inverse characteristic mixing time). The last term describes the flow of reactants into and out of the CSTR (a is the inverse residence time p (2c,t) is the probability density for input reactant stream concentrations). 

Models reported in the literature have tended to focus on one of two parts of the problem. Simple reaction models (e.g reacting slabs, random coalescence-dispersion, or multienvironment models) are designed to fit overall reaction data. More complex theoretical approaches require a model of the turbulence (see Section 2-5). The problem here is the adequacy of the turbulence model over a wide range of flow conditions. There still is no theory that takes into account structural aspects of turbulence with or without superimposed chemical reaction, although steady progress is being made in this direction [some cnrrent approaches are discussed by Fox (1998)]. 

The industrial wastewater used in the experiment is considered as having non-coalescing air electrolyte dispersion. Thus the equations discussed above would be used as a theoretical model for the estimation of oxygen transfer rate in the liquid phase, and compared with the experimental data obtained. 

There have been several attempts at models incorporating breakup and coalescence. Two concepts underlie many of these models binary breakup and a flow subdivision into weak and strong flows. These ideas were first used by Manas-Zloczower, Nir, and Tadmor (1982,1984) in modeling the dispersion of carbon black in an elastomer in a Banbury internal mixer. A similar approach was taken by Janssen and Meijer (1995) to model blending of two polymers in an extruder. In this case the extruder was divided into two types of zones, strong and weak. The strong zones correspond to regions... 

Chesters, A. K., The modeling of coalescence processes in fluid-liquid dispersions A review of current understanding. Trans. Inst. Engrs. 69, 259-270 (1991). 

---