TAB model

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

The maj or limitation of the TAB model i s that it can only keep track of one oscillation mode, while in reality there are many oscillation modes. Thus, more accurately, the Taylor analogy should be between an oscillating droplet and a sequence of spring-mass systems, one for each mode of oscillations. The TAB model keeps track only of the fundamental mode corresponding to the lowest order spherical zonal harmonic 5541 whose axi s i s aligned with the relative velocity vector between the droplet and gas. Thi s is the longest-lived and therefore the most important mode of oscillations. Nevertheless, for large Weber numbers, other modes are certainly excited and contribute to droplet breakup. Despite this... 

A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et aU556l to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski. 311 ... 

As described previously, in the atomization sub-model, 232 droplet parcels are injected with a size equal to the nozzle exit diameter. The subsequent breakups of the parcels and the resultant droplets are calculated with a breakup model that assumes that droplet breakup times and sizes are proportional to wave growth rates and wavelengths obtained from the liquid jet stability analysis. Other breakup mechanisms considered in the sub-model include the Kelvin-Helmholtz instability, Rayleigh-Taylor instability, 206 and boundary layer stripping mechanisms. The TAB model 310 is also included for modeling liquid breakup. 

Figure 8.4. Main window of Gepasi. The main window of Gepasi consists of menus (File, Options, and Help), icons, and four tabs (Model definition, Tasks, Scan, and Time course). Activation of any of the tab opens an indexed page. At the start of Gepasi, the Model definition page is opened. Enter name of the metabolic pathway to the Title box. Click Reactions button to define enzymatic reactions (e.g., E + A+B = EAB for R1, EAB = EPQ for R2, and EPQ = E + P + Q for R3 shows 3 reactions and 7 metabolites), and then click Kinetics button to select kinetic type. Activate Tasks tab to assign Time course (end time, points, simufile.dyn), Steady state (simufile.ss) and Report request. Activate Scan tab to select scan parameters. Activate Time course tab to select data to be recorded and then initiate the time course run.

Odell and Keller [97] developed a thermally activated barrier to scission (TABS) model. They incorporated the expression of F ((11), bead-rod model) into (17) and accounted for the nonuniform distribution of tension along the contours of the... 

Regardless of breakup morphology, [17] demonstrated that early drop motion obeys a constant acceleration model. Therefore, (6.6) and (6.8) can be applied directly to the calculation of the initial drop trajectory. However, (6.7) requires modification for the case of non-Newtonian liquids. Unfortunately, experimental deformation data is currently unavailable. Analytical models, such as the TAB model or its derivatives, discussed in Chap. 7, could be modified to include purely viscous or viscoelastic non-Newtonian effects. However, this has yet to be done and as a result the accuracy of such a modification is unknown. 

Abstract In an effort to characterize fuel sprays using Computational Fluid Dynamics (CFD) codes, a number of spray breakup models have been developed. The primary atomization of liquid jets and sheets is modeled considering growing wave instabilities on the liquid/gaseous interface or a combination of turbulence perturbations and instability theories. The most popular approaches for the secondary atomization are the Taylor Analogy Breakup (TAB) model, the Enhanced-TAB (E-TAB) model, and the WAVE model. Variations and improvements of these models have also been proposed by other researchers. In this chapter, an overview of the most representative models used nowadays is provided. 

The Enhanced-TAB Model (E-TAB) has been developed by Taimer in 1997 [7] and reflects a cascade of droplet breakups, in which the breakup condition is determined by the Taylor droplet oscillator dynamics (this method is further described in the next section). The droplet size is reduced in a continuous manner, until the product droplets reach a stable condition. The model maintains the droplet deformation dynamics of the TAB model [5]. According to this approach, the droplet distortion is described by a forced damped harmonic oscillator, in which the forcing term corresponds to the aerodynamic droplet-gas interaction, the restoring force is due to surface tension, while damping is attributed to the liquid s viscosity. 

In the CAB model the breakup condition is determined by means of the drop deformation dynamics of the standard Taylor analogy breakup model [5] (cf. TAB model above). In this approach, the drop distortion is described by a forced, damped, harmonic oscillator in which the forcing term is given by the aerodynamic droplet-gas interaction, the damping is due to the liquid viscosity and the restoring force is supplied by the surface tension. More specifically, the drop distortion is described by the deformation parameter, y = Ixjr, where x denotes the largest radial distortion from the spherical equilibrium surface, and r is the drop radius. The deformation equation in terms of the normalized distortion parameter, y, as provided in Eq. 9.29 is... 

Drop breakup enters the spray equation via the source term/bu in (19.45). There are various ways of accounting for drop breakup, most of which are also used for a rudimentary description of the atomization process. Some of these approaches are discussed in more detail in Chap. 9, and include the TAB model of O Rourke and Amsden [37], the Wave Breakup model of Reitz and coworkers [46, 40], the Unified Spray Breakup model of Chryssakis and Assanis [10], and the Cascade Atomization and Drop Breakup model of Tanner [54]. 

Based on an analogy between the oscillations of a two-dimensional (2D) droplet and a mass spring system (similar to the Taylor analogy breakup (TAB) model), we assume that the deformation of our 2D liquid droplet is dependent on the viscous (Fv), surface tension (Fj), and inertial (Fa) forces. So, performing a force balance in the X2-direction for the half element (shaded) in Fig. 29.2c, we can write... 

Speed of sound [m/s] Coefficients in TAB model [-] Liquid discharge coefficient [—] Drag coefficient [-]... 

The Enhanced-TAB (ETAB) model developed in Tanner [22] maintains the droplet deformation dynamics of the TAB model, that is droplet breakup occurs when the normalized droplet distortion y exceeds the critical value of 1. However, for each breakup event the ETAB model assumes that the rate of product droplet generation is proportional to the number of the product droplets. From this, the rate of droplet creation, in conjunction of with the mass cmiservation principle, leads to the basic ETAB law... 

Both MMD and distribution span have been extracted from numerical simulations to compare with experimental data as shown in Fig. 18.24. AU the breakup models predict a decrease in MMD with increasing atomization pressure as observed in the experiment. However, these models perform very differently in describing the span of the droplet-size distribution. From the comparisOTi with the experimental data, it can be concluded that the TAB model, which assumes that the child-droplet size... 

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