Estimating Drop Size Distribution

The preceding section provides us with a technique for estimating the volume-averaged drop size of a collection of droplets flowing within a pipe under mist-annular flow conditions. And while this is often all that one may wish to know about the droplet distribution, it is sometimes of interest to know, or at least estimate, the entire drop size distribution. Such would be the case if one wished to perform a cyclone simulation study which required, as input, an estimate of the inlet drop size distribution which may exist within the upstream pipe feeding the cyclone. 

Fortunately, it turns out (AIChE, 1978) that the width of the drop size distribution is strongly dependent upon the volume or mass average droplet size, Xmedi as previously computed. Furthermore, if the drop size distribution [z.e., F x)] is normalized by dividing each x by Xmed then, as a rough approximation, all droplet distributions are identical and can be represented as shown in Table 13.4.1 and 13.4.2  

For this distribution the mean size x) is almost equal to the median size 

In Appendix 13.A, an example calculation showing how to use this standard droplet distribution is given. 

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Fig. 13.A.1. Estimated drop size distribution for the example problem on basis of the AlChE design technique...

Computer programs accounted for the presence of oil drops below- the detection limit of the Coulter Counter. The data processing procedure, which assumed that the oil-drop size distribution was lognormal, yielded accurate estimates of the true mean and standard deviation describing the emulsion drop size distribution. The data-analysis procedure did not affect the actual measured drop populations which were used in the kinetic studies. The computer programs are described in detail by Bycscda.8... 

The correlations presented above allow only a rough estimation of the mass transfer coefficients, since many relevant parameters are not accoimted for, as, for instance, shape of column internals, intensity of energy input, neighboring drops, and drop size distribution (BlaB et al. 1985). 

Regardless of which constraints are chosen, it is necessary to estimate the parameters that appear there. When this is dmie, there is similarity between predicted and measured drop size distributions, and the predicted drop velocity distribution is consistent with measured profiles. However, agreement with experimental data is achieved by adjusting the source term magnitudes. Hence, the MEF is similar to curve fitting. 

The industrial complexity of liquid—liquid dispersions and emulsions is complicated by the fact that small amounts of impurities make marked differences in the bubble size produced. There is a proliferation of data on bubble sizes and droplet sizes in industrial systems and also in academic smdies where the presence of trace chemicals can often be eliminated. However, there is a multitude of equations giving the bubble size for various kinds of two-phase weU-purified systems, and no equation has been set up to give bubble-size distributions more than on an estimated or predictive basis, particularly where there is a statistical distribution of drop sizes. Many of the papers in the literature are very useful in ratio form. It means that we expoimaitally determine a drop-size distribution and then use the effect of other liquid properties, the effect of geometry, tank size, and the effect of baffles, etc. in a relative sense to predict what would happen to bubble size when we scale up or scale down. 

Aj can be obtained from the drop size distribution. However, direct measurement or calculation of would seem to be problematic. It is, of course, as we have seen, determined by many factors most of which cannot be independently estimated. These factors include those that are intrinsic to a given antifoam dispersion, as well as those that are determined by the properties of both the surfactant solution, the structure of the foam, and the method of foam generation. That both the phenomenological approach and this statistical approach are mutually consistent and also consistent... 

In addition, Chandavimol et al. (1991a,b) have estimated the kinetic rate at which the bubbles go from initial size to the maximum equilibrium size as a function of energy dissipation. The rate of dispersion was found to be approximately proportional to energy dissipation rate. [See Figure 7-24 for a comparison of bubble breakup rate between vortex (HEV) and spiral (KMS type) static mixers.] In general, the equilibrium drop size is reached in a few pipe diameters. However, the drop size distribution is narrowed as the simultaneous processes of drop breakup and coalescence are continued, depending on the mixer design and fluid properties. See also Hesketh et al. (1987, 1991). 

Mean Drop Size and Drop Size Distribution. Estimates of drop size that are achievable by various mixing devices for dispersing immiscible liquid-liquid systems are also shown in Table 8-1. In the table, the specific... 

Grosso M, Maffettone PL (2007) A new methodology for the estimation of drop size distributions of dilute polymer blends based on LAOS flows. J Non-Newton Ruid 143 48-58... 

Since the drop size distribution developed by the SP Pack can be conservatively estimated as a straight line,... 

This data collected for determination of the Knox parameters can also be used to establish the linearity of the pressure versus linear velocity curve to evaluate compression of the bed. Lastly, these data can be used to estimate the effective particle size from the pressure drop. The pressure drop data are aseful to assess the effective particle size with the vendors nominal particle size and particle size distribution data. Calculation of the effective particle size is given by Eq. (7.9), where dp is the particle size in cm, u is the linear velocity in cm/s, p is the viscosity in cP, L is the bed length in cm, k[) is the column permeability (e.g. 1 x 10 for irregular particles and 1.2 x 10 for spherical), and AP is the pressure in psi. 

Average particle sizes were determined with TEM. For that purpose a drop of the colloidal solution was placed on a carbon covered copper grid (Balzers) and analyzed with a high resolution transmission electron microscope (model JEOL 200 CX). Particle size distributions were determined by optical inspection of the photographs. From this data, metal areas of the catalysts were estimated assuming spherical particle shape and a rhodium surface density of 1.66 10 mol Rh/m [10]. As a reference material for characterization and testing, a commercial rhodium on carbon catalyst (5w% Rh, Aldrich) was used. 

The two most important factura influencing the pressure drop over a bed of reain are the size distribution of the beads and the voidaga of the bed. Modem reains are spherical in shape due to their method of mansfacture, but older reains were sometimes granular. Many manufacturers publish data on pressure drop as a function of flow rate for (heir reains at stated water temperatures. These data can also be obtained readily in laboratory experiments or estimated from the punicle size distribution of resin and the appropriate solution properties. 

The scavenging coefficient for this simplified scenario is shown in Figure 20.7 for Po = 1 mmh 1 for drops of diameters Dp = 0.2 and 2 mm. This figure indicates the sensitivity of the scavenging coefficient to sizes of both aerosols and raindrops, suggesting the need for realistic size distributions for both aerosol and drops in order to obtain useful estimates for ambient scavenging rates. 

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