Model drop breakup

F.X. Tanner, G. Weisser, Simulation of liquid jet atomization for fuel sprays by means of a cascade drop breakup model, SAE Technical Paper Series 980808, 1998. 

F.X. Tanner. Development and validation of a cascade atomization and drop breakup model for high-velocity dense sprays. Atomization and Sprays, 14(3) 211-242, 2004. 

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]. 

Tanner FX (2004) Development and Validation of a Cascade Atomization and Drop Breakup Model fa- High-Velocity Dense Sprays. Atomization Sprays 14(3) 211-242... 

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

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 ... 

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. 

Equation (35) for drop breakup caused by acceleration and Eqs. (37) to (39) to model drop breakup by turbulent stresses can be used to interpret drop behavior for the twin-fluid nozzle shown in Figure 7. The predicted size of the ethanol drops dispersed in supercritical carbon dioxide is compared with measured values in Figure 13. One can see that the model predicts well the... 

The main limitation of the TAB model is that it can only keep track of one oscillation mode, while in reality more than one mode exists. The model keeps track only of the fundamental mode, corresponding to the lowest order harmonic whose axis is aligned with the relative velocity vector between droplet and gas. This is the most important oscillation mode, but for large Weber numbers other modes are also contributing significantly to drop breakup. Despite this limitation, a rather good agreement is achieved between the numerical and experimental results for low Weber numbers. 

The Rayleigh-Taylor (RT) component has been added to the breakup model by Patterson et al. [11] to improve predictions of the secondary breakup of the droplets. The RT model predicts instabilities on the surface of the drop that grow until a certain characteristic breakup time is reached, when the drop finally breaks up. The RT waves are only allowed to form on droplets with diameters larger than the wavelength of the fastest growing disturbance. When the disturbances exceed the elapsed breakup time, the droplet is split into smaller droplets, with diameters proportional to the wavelength of the disturbances. 

A fragmented liquid core is simulated by injecting large drops which break up into smaller and smaller product droplets, until the latter reach a stable condition. The primary breakup, that is, the first drop breakup after injection, is modeled by delaying the initial drop breakup in accordance with experimental correlations. The drop distortion and the breakup criterion are obtained from Taylor s drop oscillator. The properties of the product droplets are derived from principles of population dynamics and are modeled after experimentally observed droplet breakup mechanisms. 

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... 

The primary breakup of these highly unstable initial drops is modeled by artificially prolonging their lifetime such that they agree with experimentally observed breakup lengths. More precisely, the value for the breakup time, is obtained from the experimental jet breakup length correlation of Levich [23]... 

The jet breakup modeling is illustrated in Fig. 9.3 for a non-evaporating spray. This figure illustrates the drop breakup cascade, which results in a fragmented liquid core at the nozzle exit. 

Keywords Atomization Chemical reactions Craiservation equations Constitutive equations Drop breakup Drop deformation Drop collisions Evaporation LES Newtonian fluids RANS Spray modeling Spray PDF Stochastic discrete particle method Source terms Turbulence... 

The remainder of the chapter focuses on the actual spray modeling. The exposition is primarily done for the RANS method, but with the indicated modifications, the methodology also applies to LES. The liquid phase is described by means of a probability density function (PDF). The various submodels needed to determine this PDF are derived from drop-drop and drop-gas interactions. These submodels include drop collisions, drop deformation, and drop breakup, as well as drop drag, drop evaporation, and chemical reactions. Also, the interaction between gas phase, liquid phase, turbulence, and chemistry is examined in some detail. Further, a discussion of the boundary conditions is given, in particular, a description of the wall functions used for the simulations of the boundary layers and the heat transfer between the gas and its confining walls. 

Note that the drop distortion parameters can influence the drop drag via its change of the cross-section and the drag coefficient because of the change of shape. Such investigations have been reported by Hwang et al. [23]. More importantly, the drop distortion parameters play a fundamental role in the determination of drop breakup and in the modeling of the atomization process. 

M. A. Gorokhovski The stochastic Lagrangian model of drops breakup in the computation of liquid sprays. Atomization and Sprays, 11, 505-520 (2001). 

Drop size distributions are typically described using raie of four methods empirical, maximum entropy formalism (MEF), discrete probability function (DPF) method, or stochastic. The empirical method was most popular before about the year 2000, when drop size distributions were usually determined by fitting spray data to predetermined mathematical functions. Problems arose when extrapolating to regimes outside the range of experimental data. Two analytical approaches were proposed to surmount this, MEF and DPF, as well as one numerical approach, the stochastic breakup model. 

The stochastic breakup model assumes that the spray fragments following a cascade for which the probability of forming a daughter drop via breakup is independent of its parent drop size. 

The ligament breakup model predicts drop diameter given a set of initial conditions. A distribution of drop sizes results because the initial cmiditions fluctuate, due to vibration of the atomizer, variations in liquid delivery rate, unsteady exit velocity, inhomogeneous liquid physical properties, cavitation-induced pulsations, turbulent flow fields, etc. PDFs are required for all fluctuating quantities. 

When modeling high Weber number secondary atomization, the probability of breakup is first applied to a parent drop of the size of the nozzle diameter. Once the first daughter drop(s) is(are) formed, time is reinitialized and the daughter drop becomes a parent drop (with the probability for second daughter drop breakup independent of the original parent drop size). This breakup cascade occurs until the drop critical radius (a function of the local Weber number) is reached. 

Vinckier a id. [278] used the third coalescence model for the analysis of experimental coalescence data in the PIB/PDMS system without drop breakup. The point of departure was that the probability function for coalescence was assumed... 

The dynamic behavior of polymer blends under low strain has been theoretically treated from the perspective of microrheology. Table 2.3 lists a summary of this approach. These models well describe the experimental data within the range of stresses and concentrations where neither drop-breakup nor coalescence takes place. The two latter models yield similar predictions as that of Palierne. The last entry in the Table 2.3 is an empirical modification of Palieme s model by replacement of the volume fraction of dispersed phase by its efiective quantity (Eq. (2.24)), which extends the applicability of the relation up to 0 < 0.449. However, at these high concentrations the drop-drop interactions absent in the Palierne model must complicate the flow and coalescence is expected. The practical solution to the latter problem is compatibilization, but the presence of the third component in blends has not been treated theoretically. 

The idea of a dynamic fragmentation model, which calculates the characteristic melt diameter as a function of instantaneous hydrodynamic conditions, was first proposed by Camp in Ref, 30. A model using this idea was later incorporated into a version of the Thermal Explosion Analysis System (TEXAS)one-dimensional FCI code by Chu and Corradini, using an empirical correlation derived from data obtained in the FITS experiments. The fragmentation model in IFCI is a version of a dynamic fragmentation model developed by Pilch based on Rayleigh-Taylor instability theory and the existing body of gas-liquid and liquid-liquid drop breakup data. 

This formulation was developed from the observation that, in high Weber number drop breakup experiments, the drop experiences primary breakup into 3-5 primary fragments in a dimensionless time T+ of 1-1.25. While primary breakup is occurring, smaller fingers continuously develop and break off, forming a cloud of droplets this effect is included via a surface entrainment model, given as... 

The foregoing example is interesting because it shows population balance models can account for the occurrence of physicochemical processes in dispersed phase systems simultaneously with the dispersion process itself. Shah and Ramkrishna (1973) also show how the predicted mass transfer rates vary significantly from those obtained by neglecting the dynamics of drop breakage. The model s deficiencies (such as equal binary breakage) are deliberate simplifications because its purpose had been to demonstrate the importance of the dynamics of dispersion processes in the calculation of mass transfer rates rather than to be precise about the details of drop breakup. 

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