Cumulative volume fraction

Keywords Characteristic drop diameter Cumulative volume fraction Discrete probability function (DPF) Drop size distribution Empirical drop size distribution Log-hyperbolic distribution Log-normal distribution Maximum entropy formalism (MEF) Nukiyama-Tanasawa distribution Number distribution function Probability density function (pdf) Representative diameter Root-normal distribution Rosin-Rammler distribution Upper limit distribution Volume distribution... 

The Rosin-Rammler is a cumulative volume fraction distribution ... 

A cumulative volume fraction that represents the fraction of particles with volume at most v, denoted F(v, r, t), is given by... 

We begin with examining Equation (4.6.19) and recognize that the cumulative volume fraction F x, t) of particles of mass at most x is a cumulative distribution function that can be attributed to a random process X (t), which may be interpreted as the mass of a single particle at time t. We imagine that this particle, whose initial size is a random variable with... 

It is again expedient to use the equation in terms of the cumulative volume fraction presented in Chapter 3, viz., Eq. (3.3.9). Thus we have... 

The mathematical statement of the inverse problem is as follows Given measurements of F x, t), the cumulative volume (or mass) fraction of particles of volume ( x) at various times, determines, b x), the breakage frequency of particles of volume x, and G x x ), the cumulative volume fraction of fragments with volume ( x) from the breakage of a parent particle of volume x. Obviously, the experimental data on F x, t) would be discrete in nature. We assume that G x x ) is of the form (5.2.9) and rely on the development in Section 5.2.1.1 using the similarity variable z = b x)t. Self-similarity is expressed by the equation F x, t) = 0(z), which, when substituted into (6.1.1), yields the equation... 

The breakage process was simulated by Sathyagal et al (1995) using the single particle technique of Section 4.6.4 to obtain the cumulative volume fraction of drops of various sizes x at different times t. Their test of similarity, made through a plot of In t versus In x for 14 different values of the cumulative fraction is represented in Fig. 6.1.1 below. 

That conformation to self-similarity is excellent is evidenced by a plot of the arc length versus particle size curve shown in Figure 6.1.2. This plot was obtained by assembling different segments of the curve for different cumulative volume fractions by fitting a smooth quadratic curve to In t versus In x from all of the data at hand. 

FIGURE 6.1.1 Similarity test of simulated data on the t-x plane at fixed cumulative volume fractions (from Sathyagal et al 1995) (Reprinted with permission from Elsevier Science.)... 

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). 

This quantity, which is often referred to as magma density or soHds concentration (mass of crystals per unit system volume), is often an important process variable. A cumulative mass fraction of crystals having a size less than U can also be defined as... 

This is the mean abscissa of a graph of cumulative area fraction versus size. Otherwise known as the Surface Volume Mean (or Sauter Mean)... 

Figure 1 Is a flow sheet showing some significant aspects of the Iterative analysis. The first step In the program Is to Input data for about 50 physical, chemical and kinetic properties of the reactants. Each loop of this analysis Is conducted at a specified solution temperature T K. Some of the variables computed In each loop are the monomer conversion, polymer concentration, monomer and polymer volume fractions, effective polymer molecular weight, cumulative number average molecular weight, cumulative weight average molecular weight, solution viscosity, polymerization rate, ratio of polymerization rates between the current and previous steps, the total pressure and the partial pressures of the monomer, the solvent, and the nitrogen.

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.

Pc> Pb> Pf cumulative number fraction of grid squares that exhibit craze formation, craze fibril breakdown, and catastrophic fracture, respectively probability that a given entangled strand survives craze fibril formation disentanglement time of i strands in a fibril that survive fibril formation craze interface velocity volume fraction of polymer within craze... 

Low-level wastes high in sodium content have been collecting at the ICPP for years. Though these wastes represent but a small fraction of the total wastes produced, their cumulative volume is considerable (106 gal). The typical composition of high sodium concentration waste is shown in Table IV. 

The effectiveness of this approach was demonstrated by measuring the cumulative fluorescence of 1-micron-size blue fluorescent microspheres versus cumulative volume withdrawn from our WEP collector after capture from an aerosol suspension of about 140 particles/ml drawn from a test chamber over a 5-minute period at a sampling rate of 500 liters/minute. As the liquid was being withdrawn from the WEP collector, it was filtered at a rate of 1 ml/second and the filter back-flushed with small portions of filtrate at 1-minute intervals. The relative cumulative concentration of captured particles in the first back-flushed 1-ml fraction was around 56 as compared with a value of 1.4 in the first 60-ml filtered fraction, which constitutes a liquid-to-liquid concentration enhancement by a factor of 40 and an air-to-liquid concentration factor olYl.25 x 10. ... 

Fraction No. Cumulative volumes (ml) of methanol added to 4,465 gm of Copolymer in 70 mlofTHF Polymer recovered (%) Inherent viscosity of fraction (at 30 C) Allyl ester content (%)... 

It should be pointed out that the determination of vapor fractions can begin at any convenient time other than time zero. The volume fraction liquid is merely the slope of the cumulative liquid residence time time curve. Hence, summing the liquid residence times after slug flow becomes well developed will avoid the experimental errors inherent in this technique for the mist flow regime. 

The mathematical relations expressing the different amounts of time that fluid elements spend in a given reactor may be expressed in a variety of forms [see, e.g., Leven-spiel (1-3) and Himmelblau and Bischoff (4)]. In this book we utilize the cumulative residence-time distribution curve [F(0], as defined by Danckwerts (5) for this purpose. For a continuous flow system. Fit) is the volume fraction of... 

Figure 12 Water droplets in model oils with asphaltene and resin fractions extracted from a crade oil normalized cumulative volume showing the effect of asphaltene and resin content. Key emulsion 1 - high asphaltene, no resin 2 -lowasph., no resin 3 - high asph., high resin 4 - high asph., low resin.

Figure 3.10. Relative and cumulative frequencies as a function of normalized ID (a) Type I, mono-sized particles/voids of 5.35, and a minimum inter-particle/void distance of 1.5 (b) Type II mono-sized particles/voids of 5.35, and a minimum inter-particle/void distance of zero (c) Type III, log-normally sized particles/voids with a mean of 5.57 and a standard deviation of 1.13 (d) Type IV, log-normally sized particles/voids with a mean of 5.91 and a standard deviation of 2.46 and (e) Type V, log-normally sized bi-modal particles/voids with a similar mean particle/void size of 5.74 but different standard deviations of 1.11 and 2.47 and respectively volume fractions of 0.096 and 0.054. The curves represent cumulative Gaussian distribution functions [36]...

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