Droplet velocity distribution

Let us imagine that droplets of three sorts are present in the EPR droplet ensemble. We call them figuratively as big , small , and medium droplets with dimensionless parameters given in the Table below. Let they be present in the ensemble in the proportions (probabilities) p1 = 1/3, p3 = 1/2, and p2 = 1/6 correspondingly, px + p2+ /L = 1 Figure 3.24,A presents the droplet velocity distributions within an EPR with the relative height 6 = 0.5. 

The results of calculations of the monodisperse flow with the parameters determined by (3.121) has been presented by the dotted curves. The droplet velocity distribution of the equivalent uniquely sized droplet ensemble is seen in Fig. 3.24,A. By the way, the shape of this curve is strictly given by... 

For the given parameters (3.138), the model predicts all the aerothermal profiles in the main flow region, i.e. the air velocity profile U(z) and the vapour concentration E(z), droplet velocity distribution u(z) and the vapour concentration near their surfaces e(z) which is in an empirical relation (3.94) with the droplet temperature t. The approximate formulas were found as... 

Figure Bl.14.13. Derivation of the droplet size distribution in a cream layer of a decane/water emulsion from PGSE data. The inset shows the signal attenuation as a fiinction of the gradient strength for diflfiision weighting recorded at each position (top trace = bottom of cream). A Stokes-based velocity model (solid lines) was fitted to the experimental data (solid circles). The curious horizontal trace in the centre of the plot is due to partial volume filling at the water/cream interface. The droplet size distribution of the emulsion was calculated as a fiinction of height from these NMR data. The most intense narrowest distribution occurs at the base of the cream and the curves proceed logically up tlirough the cream in steps of 0.041 cm. It is concluded from these data that the biggest droplets are found at the top and the smallest at the bottom of tlie cream.

Although NMRI is a very well-suited experimental technique for quantifying emulsion properties such as velocity profiles, droplet concentration distributions and microstructural information, several alternative techniques can provide similar or complementary information to that obtained by NMRI. Two such techniques, ultrasonic spectroscopy and diffusing wave spectroscopy, can be employed in the characterization of concentrated emulsions in situ and without dilution [45],... 

Spatially-resolved measurement of the droplet size distribution can be accomplished by the implementation of velocity compensated pulse sequences, such as the double PGSTE [81] in a spatially resolved imaging sequence. Accurate measurements of spatially resolved droplet size distributions during flow and mixing of emulsions would provide truly unique information regarding flow effects on the spatial distribution of droplets. 

To compute the motion of two immiscible and incompressible fluids such as a gas liquid bubble column and gas-droplets flow, the fluid-velocity distributions outside and inside the interface can be obtained by solving the incompressible Navier-Stokes equation using level-set methods as given by Sussman et al. (1994) ... 

Weiss and Worsham 259 indicated that the most important factor governing mean droplet size in a spray is the relative velocity between air and liquid, and droplet size distribution depends on the range of excitable wavelengths on the surface of a liquid sheet. The shorter wavelength limit is due to viscous damping, whereas the longer wavelengths are limited by inertia effects. 

Wu, Ruff and Faethl249 made an extensive review of previous theories and correlations for droplet size after primary breakup, and performed an experimental study of primary breakup in the nearnozzle region for various relative velocities and various liquid properties. Their experimental measurements revealed that the droplet size distribution after primary breakup and prior to any secondary breakup satisfies Simmons universal root-normal distribution 264]. In this distribution, a straight line can be generated by plotting (Z)/MMD)°5 vs. cumulative volume of droplets on a normal-probability scale, where MMD is the mass median diameter of droplets. The slope of the straight line is specified by the ratio... 

In many atomization processes, physical phenomena involved have not yet been understood to such an extent that mean droplet size could be expressed with equations derived directly from first principles, although some attempts have been made to predict droplet size and velocity distributions in sprays through maximum entropy principle.I252 432] Therefore, the correlations proposed by numerous studies on droplet size distributions are mainly empirical in nature. However, the empirical correlations prove to be a practical way to determine droplet sizes from process parameters and relevant physical properties of liquid and gas involved. In addition, these previous studies have provided insightful information about the effects of process parameters and material properties on droplet sizes. 

The variations of the mean droplet size and the droplet size distribution with axial distance in a spray generated by pressure swirl atomizers have been shown to be a function of ambient air pressure and velocity, liquid injection pressure, and initial mean droplet size and distribution 460]... 

Some quantitative studies1498115011 on droplet size distribution in water atomization of melts showed that the mean droplet size increases with metal flow rate and reduces with water flow rate, water velocity, or water pressure. From detailed experimental studies on the water atomization of steel, Grandzol and Tallmadge15011 observed that water velocity is a fundamental variable influencing the mean droplet size, and further, it is the velocity component normal to the molten metal stream Uw sin , rather than parallel to the metal stream, that governs the mean droplet size. This may be attributed to the hypothesis that water atomization is an impact and shattering process, while gas atomization is predominantly an aerodynamic shear process. 

In the second method, i.e., th particle method 546H5471 a spray is discretized into computational particles that follow droplet characteristic paths. Each particle represents a number of droplets of identical size, velocity, and temperature. Trajectories of individual droplets are calculated assuming that the droplets have no influence on surrounding gas. A later method, 5481 that is restricted to steady-state sprays, includes complete coupling between droplets and gas. This method also discretizes the assumed droplet probability distribution function at the upstream boundary, which is determined by the atomization process, by subdividing the domain of coordinates into computational cells. Then, one parcel is injected for each cell. 

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

For agiven system of metal/alloy and atomization gas, the 2-D velocity distributions of the gas and droplets in the spray can be then calculated using the above-described models, once the initial droplet sizes and velocities are known from the modeling of the atomization stage, as described in the previous subsection. With the uncoupled solution of the gas velocity field in the spray, the simplified Thomas 2-D nonlinear differential equations for droplet trajectories may be solved simultaneously using a 4th-orderRunge-Kutta algorithm, as detailed in Refs. 154 and 156. 

---