CUMULATIVE . VOLUMES

Fig. 3. Cumulative volume distilled as a function of boiling point from A, light B, intermediate and C, heavy cmde oils (not including condensates).

Volume median mass median) D,m- This has no fundamental meaning but is easy to determine since it is at the midpoint of a cumulative-volume plot. 

There are a variety of ways to describe the droplet population. Figures 14-88 and 14-90 illustrate one of the most common methods, the plot of cumulative volume against droplet size on log-normal graph paper. This satisfies the restraint of not extrapolating to a negative drop size. Its other advantages are that it is easy to plot, the results are easy to visualize, and it yields a nearly straight line at lower drop sizes. 

Cumulative volume over the range of 1 to 50 percent can also be shown to vary approximately as D. This is equiv ent to finding that the number of droplets of a given size is inversely proportional to the droplet area or the surface energy of the droplet. 

Gal-Or (G4) has recently reported bubble-size distribution data in air-water dispersions. The equipment used to evaluate the bubble-size distribution is a new type of multistage gas-liquid contactor without pressure drop in each stage, in which the gas is drawn in from the bottom of the vessel. Typical bubble-size—cumulative-volume data are given in Fig. 2.f The data show that for 99% of the bubbles, 0.1 < 1.4 mm. The surface mean radius a32... 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

Organic Syntheses, published by Wiley, New York is a collection of procedures for the preparation of specific compounds. The thin annual volumes have appeared each year since 1921. For the first 59 volumes, the procedures for each 10- (or 9-) year period are collected in cumulative volumes. Beginning with vol. 60, the cumulative volumes cover five-year periods. The cumulative volumes published so far are... 

Corrected Temp, C Cumulative Volume Distilled % of Sample Corrected Temp, C Cumulative Volume Distilled % of Sample... 

We consider a 100-m length of an aquifer with a porosity of 30% and a nominal dispersivity of 10 cm the dispersivity reflected in the calculation results will be somewhat larger than this value, due to the effects of numerical dispersion. The domain is divided into 100 nodal blocks, each 1 m long. We assume local equilibrium, so time enters into the calculation only as a measure of the cumulative volume of fluid that has passed through the aquifer. Specifying the aquifer s pore volume be replaced 30 times over the course of the simulation, and setting the time span to 30 years, each year in the simulation corresponds to a single replacement of the aquifer s pore fluid. 

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... 

Figure 16 Variability (time dependency) of differential GI flow rates (DFR) in the small intestine of Labradors. VR represents the cumulative volume of chyme collected at midgut following oral administration of 200 mL glucose solution 20% (I) and 200 mL NaCl 0.9% (J). Source From Ref. 10.

The plot of residuals versus some measure of the time at which experiments were run can also be informative. If the number of hours on stream or the cumulative volume of feed passed through the reactor is used, nonrandom residuals could indicate improper treatment of catalyst-activity decay. In the same fashion that residuals can indicate variables not taken into account in predicting reaction rates, variables not taken into account as affecting activity decay can thus be ascertained. 

The volume pore size distribution, which is defined as the pore volume per unit interval of the pore radius can be determined by building the first derivation of the cumulative volume by the pore radius. 

The total pore volume can directly be determined by the raw data, as it is equal to the cumulative volume at the highest pressure applied. 

The specific surface area is calculated as the area of the intrusion curve that results by plotting the cumulative volume versus the pore radius [117]. 

Index of Reviews in Organic Chemistry, Second Cumulative Volume 1976 , compiled by D. A. Lewis and P. Charnock, The Chemical Society, London, 1977. 

A plot of the intruded (or extruded) volume of mercury versus pressure is sometimes called a porogram. The authors will use the terms intrusion curve to denote the volume change with increasing pressure and extrusion curve to indicate the volume change with decreasing pressure. Figure 11.1 shows a typical porosimetry curve of cumulative volume plotted versus both pressure (bottom abscissa) and radius (top abscissa). The same data plotted on semilog paper is illustrated in Fig. 11.2. 

Cumulative volume curves generated by intruding mercury into porous samples are not followed as the pressure is lowered and mercury extrudes out of the pores. In all cases the depressurization curve lies above the pressurization curve and the hysteresis loop does not close even when the pressure is returned to zero, indicating that some mercury is entrapped in the pores. Usually after the sample has been subjected to a first pressurization-depressurization cycle, no additional entrapment occurs during subsequent cycles. In some cases, however, a third or even fourth cycle is required before entrapment ceases. 

The effect of UPF is verified by observing the similar pore size distributions of the wet granulation and the direct compression tablets in Frogerais, after 25% of the cumulative volume. 

In an apparatus based on Figure 6.16b, the volume of mercury forced into the pores of the solid can be measured as a function of the applied pressure. Equation (86) shows that higher pressures are required for smaller pores. Therefore incremental increases in p will result in the filling of pores of progressively smaller radii. The volume V that has intruded a porous solid at a pressure p gives the cumulative volume of all pores larger than the size associated with p. Since plots of V versus p give information about the cumulative pore distribution, it is the derivative of such data that measures the increment in pore volume associated with an increment in Rc. Written as a formula, dV/dp oc dV/(—dRc) since Fincreases as R( decreases. 

The data for significant runs were treated by plotting concentration vs. cumulative volume of effluent (Figures 8 to 14). Demineralization capacities were obtained by graphical integration. In most cases the run was not carried to the point where the concentration of the effluent equaled that of the influent. Consequently, the demineralization capacity is somewhat uncertain. Estimates of the demineralization capacity represented by the uncompleted portion of the curve are always on the conservative side. 

---