Eddy breakup model

Spalding developed a successful eddy-breakup model for describing rates of flame spread in high-intensity flows of this type [143]-[146]. There are a number of difYerent versions of the model one employs... 

The mean species source term, It, requires further examination. Because this term is usually highly nonlinear as illustrated in Equation 4.8 and its value directly controls the reaction progress, a proper closure model is necessary. Indeed, various methods are available from the literature.21-22 For the purpose of discussion, we consider the eddy breakup model of Magnussen and Hjertager.23 Note that this model is chosen for its simplicity rather than accuracy, much as the A - e model was selected for turbulence closure. In this model, a turbulent reaction rate is computed which is then compared with the kinetic rate. The smaller of the two is used as the reaction rate because it limits the reaction progress. The kinetic rate is simply the laminar reaction rate evaluated at the mean temperature, pressure, and concentrations ... 

An analytical NOX chemical kinetic model has been developed by the DLR to investigate the influence of various parameters on the formation of pollutants. It has been coupled with an eddy breakup model for the combustion process [133]. 

There are some alternative models to describe the micromixing eddy breakup model [41], flamelet model [42], and engulfment model [43]. 

If the chemical reactions are very fast compared to the mixing rate, it may be assumed that any mixed reactants are immediately reacted. No rate expression is therefore necessary. The simplest model to represent such cases is called the eddy break up (EBU) model (Spalding, 1970 Magnussen and Hjertager, 1976). In the EBU model, the effective rate of chemical reactions is equated to the smaller of rate calculated based on kinetic model and that based on the eddy break-up rate. The eddy breakup rate is defined as the inverse of a characteristic time scale kle. Therefore, for fast reactions, the rate of consumption or formation is proportional to the product of density, mass fraction and the eddy break-up rate elk). The model is useful for the prediction of premixed and partially premixed fast reactive flows. EBU, however, was originally developed for single-step chemical reactions. Its extension to multiple step reactive systems should be made with caution. 

It is important to note that the last equation describes the final step of viseous molecular diOusion however, other homogenization proeesses such as gulf entraimnent or engulfinent and eddy breakup are not only rate eontiolUng but also decisive for the importanee of main and secondaiy reactions. A big variety of models for the prediction of time eonstants ean be found in the hterature. All these equations have in eommon that the times are always proportional to the root JvTe. The time constant r for the engulfinent step is... 

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. 

S. V. Apte, M. Gorokhovski, and P. Moin. Large-eddy simulation of atomizing spray with stochastic modeling of secondary breakup. Int. J. Multiphase Flow, 29 1503-1522, 2003. 

Narsimhan et al, on the other hand, consider breakup by bombardment of the drop by eddies (smaller than the drop), with at least, as much energy required to create the minimum amount of new interface. The breakage frequency is calculated as the ratio of the probability that an eddy of the appropriate amount of energy is incident upon the drop surface, to the average arrival time of the eddies. Thus, in this model the temporal element lies in the waiting period for the appropriate eddy to arrive, but upon its arrival breakup occurs instantly. 

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