Droplet size cumulative distribution

Fig. 23. (a) Distribution of drop sizes for mother droplets and satellite droplets (solid lines) produced during the breakup of a filament (average size = 2 x 10 5 m) in a chaotic flow. The total distribution is also shown (dashed line). A log-normal distribution of stretching with a mean stretch of 10 4 was used, (b) The cumulative distribution of mother droplets and satellite droplets (solid line) approaches a log-normal distribution (dashed line). 

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... 

As AD is made smaller, a histogram becomes a frequency distribution curve (Fig. 4.1) that may be used to characterize droplet size distribution if samples are sufficiently large. In addition to the frequency plot, a cumulative distribution plot has also been used to represent droplet size distribution. In this graphical representation (Fig. 4.2), a percentage of the total number, total surface area, total volume, or total mass of droplets below a given size is plotted vs. droplet size. Therefore, it is essentially a plot of the integral of the frequency curve. 

Figure 4.2. Cumulative droplet size distribution curve based on mass.

Table I lists some of the basic mathematical expressions of importance in droplet statistics. The expressions are given in terms of an arbitrary ptb-weighted size distribution. The specific forms are obtained for various integral values of p. For example, the substitution of p = 2 into the equations of Table I yields the cumulative distribution, arithmetic mean, variance, geometric mean, and harmonic mean of the surface-weighted size distribution. Analogous expressions valid for frequencies or mass distributions are obtained by setting p equal to 0 or 3, respectively.

Unfortunately, most emulsions do not have a single droplet size. There are small, medium and large droplets present, and it is important to be able to characterise the emulsion for this. This is done by counting the number of particles that is smaller than a specific size, for many different sizes. The resulting data can then be plotted on a curve, the cumulative distribution curve. Alternatively, one can count all particles that have a size within an interval of sizes (e.g., 1-2 pm), and do this for all intervals. Plotting all the numbers obtained for all intervals, then results in a frequency distribution. The two distributions are closely related the derivative of the cumulative curve to the particle size, will give a (continuous) curve that is similar to the discrete frequency distribution obtained earlier, and the smaller the intervals are chosen, the closer the derivative will follow the frequency distribution (see Figure 15.4). 

Droplet size distributions are, as are all particle size distributions, either represented as volume density distributions 3(dp) or as cumulative volume distributions Qi (dp) ... 

Finally the studies [314, 392] should be mentioned, which also deal with the effect of mixing time on the droplet size distribution. Logarithmic cumulative volume distributions were presented as a function of droplet volume for different mixing times, was made so small, that the mixing time only caused parallel shifts of the distribution curves (without changing the slope) this confirmed, that the disintegration process did not change with time. 

From Fig. 4.18, which showed log normal as weU as cumulative internal droplet size distribution, it can be observed that emulsion prepared using 4000 rpm impeUer speed resulted in a narrow size distribution with internal droplets mean diameter 0.024 pm. In contrast, emulsion prepared using 8000 rpm impeUer speed yields broader droplet size distribution with lower mean diameter of 0.015 pm. 

Figure 10.6 Droplet size distribution for zinc phosphate-based anticorrosive paint at 163 m/s a) frequency distribution (histogram), b) cumulative distribution. (From Lyulin, 1988.)...

Lastly, droplet distribution curves and droplet size characteristic measurements were made for each of the sprays. Figure 15.13 shows the droplet size characteristics for the pressure swirl injector and Fig. 15.14 shows the droplet size characteristics for the liquid jet injector. The droplet size distribution is shown in both a cumulative and normalized sense. The cumulative volume fraction is used to determine DvO.l, DvO.5, and DvO.9 measurements. The calculated values for DIO, D31, and SMD are also shown on each plot. In general, the pressure swirl injector analyzed had a tighter distribution of injected droplets. The tighter distribution means that there are fewer larger droplets measured, which create smaller values for all of the key droplet characteristics, measured (D31, DIO, DvO.l, Dv05, DvO.9, and D32). 

Most distribution functions contain an average size and a variance parameter typicaUy based on the cumulative droplet number or volume distributions. For example, the Rosin-Rammler function uses the cumulative Hquid volume as a means of expressing the distribution. It can be expressed as... 

Figure D3.4.7 Change in cumulative particle size distribution of a 20% (w/v) oil-in-water emulsion stabilized by 2% (w/v) Tween 20 at the lower port (A) and upper port (B). (C) Change in mean droplet diameter and volume fraction of the emulsions as a function of time.

Figure 5 shows the respective distributions of the two techniques plotted as cumulative percent by number against size. The two curves appear quite similar. The number median of the MgO data is 60 fxm. This result represents a less than 4% difference, which is a very favorable comparison. The MgO analysis indicates a broader distribution. This may be attributed to the limited range of the visibifity technique as previously noted. Owing to this limitation, any droplet larger than 94... 

Microemulsions were characterized by Dynamic Light Scattering (DLS) to determine the size distribution of the water pools. Measurements were performed on a Malvern Zetasizer. Interactions existing between aqueous droplets were neglected which is a prerequisite to use the hard sphere model. The linear mode of the apparatus was chosen this corresponds to a cumulative analysis. 

Fig. 23.3 The cumulative drop size distributions of emulsion SE-la and SE-lb, plotted as a function of emulsion droplet diameter for different mean drop sizes of 2, 4, and 10 pm...

In the bidisperse case. Figure 4.4(b), fi ctionation does occur. The large droplets cream faster than the small ones and two sharp boundaries form at the base and rise to die top at two discrete rates. The two creaming rates allow two hydrodynamic sizes to be inferred fiom eqn. (4.1). The rates at which die boundary rises at two volume fiacdons (ordinates yi and 2) are sufficient to define completely the cumulative size distribution of a bidisperse dispersion. Polydisperse dispersions are treated as an extension of the bidisperse case, the number of ordinates examined being increased as required until die size distribution is sufficiendy well defined. However, this simplistic analysis is only applicable to very dilute emulsions, where Stokes law is valid (i.e at infinite dilution in an infinite medium). In closed concentrated emulsions, droplets will interfere with one another and the effect of back-flow by the continuous phase becomes significant. 

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