Weber number drop breakup

Dealing with the breakup of fluids resulting in the formation of bnbbles or drops the Weber number is the decisive criterion The stability of fluid particles can be described by a certain value of this number as will be shown later. 

Keywords Bag breakup Breakup mode Breakup time Catastrophic breakup Fragments Fragment size distribution Initiation time Multimode breakup Newtonian drops Non-Newtonian drops Ohnesorge number (Oh) Secondary atomization Secondary breakup Sheet-thinning breakup Total breakup time Vibrational breakup Weber number (We)... 

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]. 

Isolated Droplet Breakup—in a Velocity Field Much effort has focused on defining the conditions under which an isolated drop will break in a velocity field. The criterion for the largest stable drop size is the ratio of aerodynamic forces to surface-tension forces defined by the Weber number, N (dimensionless). 

Most correlations show that di2 is proportional to the Weber number raised to the power of -0.6, which is consistent with the theory of drop breakup by turbulent shear forces. Strictly, these correlations should be applied only where the drop size is in the inertial subrange of turbulence, i.e.,... 

Where W is the Weber number and a.- is the shear stress, the same on both sides of the Interface. The critical condition for drop breakup in Newtonian liquid mixtures is given by ... 

FIGURE 11.8 The effect of the viscosity ratio, drop over continuous phase inn/tlc), on the critical Weber number for drop breakup in various types of laminar flow. The parameter a is a measure of the amount of elongation occurring in the flow for a — 0, the flow is simple shear for a — 1, it is purely (plane) hyperbolic. 

Also if t]c breakup is difficult. At t]D/t]c = 10 4, which is about the magnitude in most foams, Wecr would be as large as 30. At such a small viscosity ratio, the bubble or drop is deformed into a long thread before breaking. However, for some protein surfactants, the surface layer of the drop can be stagnant (Section 10.8.3) and then the drop can presumably break at a smaller Weber number. 

At lower gas speed and higher drops surface tension, formation of bag structures and breakup into smaller droplets have been observed following the initial drop deformation into a disk [5], In the current experiment, these smaller drops are not observed, primarily due to the large Weber number. A mist with scales smaller than the camera resolution, which Wcis 1.2 /mi/pixel, was visible both... 

Then, in view of the correction for internal viscosity, the critical Weber number that corresponds to the drop s state prior to breakup is equal to... 

The two most important forces governing the breakup of drops are the disruptive aerodynamic force and the restorative surface tension force. Their ratio results in the nondimensional Weber number ... 

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. 

Fig. 9.1 Schematic illustration of a drop breakup caused by Kelvin-Helmholtz (KH) or Rayleigh-Taylor (R-T) instabilities. The breakup mechanisms are ciassified with respect to the (increasing) Weber number as bag, stripping (shear) and catastrophic breakup...

As discussed in the original TAB article, a necessary condition for drop breakup has been established from shock wave experiments such that the drop Weber number (based on relative velocity) satisfies We > WCcrit = 12, where d denotes the drop diameter. 

The breakup parameter, Kbu, in the bag breakup and the stripping breakup regime is proportional to the characteristic breakup frequencies suggested by [5]. The characteristic breakup frequency for the catastrophic breakup regime is derived from the study of the RT instability by Bellman and Pennington [21] as reported by Patterson and Reitz [11]. The constant k = 0.05 has been determined such that the drop radii match the phase Doppler measurements of Schneider [22], whereas the values for the constants ki and k are chosen such that Kbu is continuous at the regime-dividing Weber numbers, Web,s and Wcg c-... 

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. 

Figure 35.4 shows the variation of ellipticity with respect to the Weber, Reynolds, and capillary numbers at various axial locations. As observed fi-om these figures, the droplets are big close to the injector and the Ohnesorge numbers are small. The Weber number is much larger than the critical Weber number ( 6) and the drops undergo breakup. The deformation predicted by the above correlation... 

In addition to the critical Weber number for a drop-on-demand breakup, the criterion of We > 8 for a Rayleigh breakup has also been reported [16]. This limit can also be motivated by the lower limit of jet formation in the case of dripping out of a vertical capillary with diameter Dnozzis under the action of gravity. The static pressure pstat inside a hanging droplet is... 

However, this classification originally was introduced for jet breakups and not for drop-on-demand breakups with short pressure pulses. Since the Weber number can be written using the Reynolds number and the Ohnesorge number... 

The previous discussion focused on the breakup of liquid thread suspended in a quiescent Newtonian fluid. In real mixing operations quiescent conditions will usually not occur, except perhaps for short periods of time. The more important issue, therefore, is how the breakup occurs when the system is subjected to flow. Good reviews on the breakup of liquid threads are available from Acrivos [304], Rallison [305], and Stone [306]. Probably the most extensive experimental study on drop breakup was performed by Grace [286] data was obtained over an enormous range of viscosity ratios 10- to 10 Grace determined the critical Weber (Capillary) number for breakup both in simple shear and in 2-D elongation the results are represented in Fig. 7.152. 

A series of articles was published by Utracki et al. [318-322] on the modeling of mixing of immiscible fluids in a twin screw extruder. The fourth paper in the series [321] incorporates several refinements of the earlier model, one of the most important refinements being the incorporation of the effect of coalescence. The model considers two breakup mechanisms, both based on the micro-rheology. One breakup mechanism is the drop fibrillation and disintegration into fine droplets when the Weber number is greater than four times the critical Weber number. The second mechanism is drop splitting that occurs when the Weber number is below four times the critical Weber number. 

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... 

Fig. 22.3 Influence of the cross-wind flow on the wave number and the growth rate of waves at surface tension-driven thread breakup. Larger wave number corresponds to shorter critical wavelengths resulting in smaller drops. On the other hand, the growth rate increases with increasing gas-Weber number. This leads to shorter breakup time and breakup length [11]...

Fig. 22.8 Dimensionless mean drop size depending on the relative velocity of cross-wind flow. The drop size is compared for two flow rates and viscosities. Increasing gas-Weber number leads to a slight increase in drop size. The breakup diameter at breakup is larger for shorter threads, as the cross-wind flow initiates shorter breakup length [11]...

Fig. 22.10 Plot of the mean drop size for different process conditions depending on the gas-Weber number. The drop size up to a critical value increases slowly with increasing gas velocity. Above this critical gas-Weber number, the drop size rises suddenly. This can be explained by the transition of the breakup regime from axisymmetric breakup to wind-induced breakup. The oscillating thread is more sensitive to this transition, as the critical gas-Webta-number is reached earlier for similar process conditions...

Above the critical gas-Weber number, the increase of span value is more distinct (not visible in Fig. 22.12). The stochastic breakup with multiple small drops from the liquid bridge between two main drops (compare Fig. 22.11) is the reason for this behavior. The higher viscous damping and intensive stretching of the bridge generate satellites drops. In addition, coalescence of drops is promoted by the irregular breakup. 

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