Cumulative mass fraction

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... 

Sequentially adding up each individual segment on the histogram gives the Cumulative mass fraction, m d)... 

This is the mean abscissa of a graph of cumulative mass fraction versus size. 

Note that by definition XK — 1 and thus Xp is the cumulative mass fraction of the first P species. The inverse transformation corresponding to Eq. (93) is... 

The correction terms can be computed using either Yx or Xa. However, it is clear that the correction terms will depend on which choice is used since the moments controlled by DQMOM are different (i.e., (Y ) vs. (X(")). Thus, for example, with fixed N one set of moments may lead to singular correction terms, but not the other. In general, we can continue to solve for the mass fractions in the CFD model, but with the correction terms computed using the cumulative mass fractions as follows. 

Given the mass fractions in each environment Yan, use Eq. (93) to compute the cumulative mass fractions Xm. 

Sieve No. Mesh Size (mm) Average Particle Size Mass (kg) Mass Fraction Cumulative Mass Fraction Oversize Cumulative Mass Fraction Undersize... 

FIGURE 15.24 Cumulative mass fraction undersize vs. screen aperture. 

The cumulative mass fraction plot is another method of representing the data. The cumulative mass fraction oversize is the relative mass of particles retained on each sieve. Similarly, the cumulative mass fraction undersize is the relative mass of particles passing through each sieve. Figures 15.24 and 15.25 show these cumulative plots. The cumulative mass fraction oversize is calculated by adding the mass... 

Each mass fraction is that which is retained on the sieve. For example, the cumulative mass fraction oversize for sieve 3 would be 0.49 ... 

Cumulative Mass Fraction Undersize = 1 - Cumulative Mass Fraction Oversize... 

Determine the screen size, screen openings, mass fraction, and cumulative mass fraction through each screen for all runs. 

C. Cumulative mass fraction (or weight percent) (undersize) through each screen vs. average screen opening... 

Given the following data for a screen analysis, calculate the mass fraction, cumulative mass fraction, and relative frequency. 

Such calculations give the mass of crystals below size L, and the results are shown in column 4 of Table II. Column 5 gives the cumulative mass fraction distribution ... 

Min particle diameter Max particle diameter Mean particle diameter Mass fraction % Mass fraction in sieve fractionj mean diameter Cumulative mass fraction... 

Mean particle diameter at 16% 84% of cumulative mass fraction... 

Cumulative mass fraction finer than mesh size after noted grinding time... 

FIGURE 13.2 Cumulative mass fraction of ground material as a function of grinding time (Example 13.2). 

Figure 27.15a relates the cumulative mass fraction to the dimensionless length z. For given values of G and t, z may be converted by Eq. (27.28) to the crystal size L, and Fig. 27.15a then becomes a plot of the cumulative screen analysis. This procedure is illustrated in Example 27.6. 

Figure 1 Release of ferritin (500-kDa protein) from a matrix of EVAc. (a) The cumulative fraction of mass released from matrices containing 35% ( ) or 50% ( ) ferritin by mass is plotted versus time, (b) The same cumulative mass fraction released from the 50% loaded matrices is plotted versus the square root of time. The dashed line represents the fit to the linear model of desorption from a slab, Eq. (5). Data points represent the mean cumulative fraction of mass of ferritin released from four EVAc matrices incubated in buffered saline at 37 °C. The error bars represent 1 SD of the mean. Some error bars are smaller than the symbols.

In what follows we seek a characterization of the preceding breakage process in terms of the cumulative mass fraction of particles with mass in the range [0, x] defined by... 

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