Integral size-distribution curves

Figure VI.24 shows (in probability—logarithmic coordinates) the integral size-distribution curves for particles adhering to surfaces painted with polyurethane enamel and made oily with Avtol, placed at various angles to the air-flow axis, for various velocities of the air flow.

Discrete data point, extracted irom the log file, can be viewed. The data can also be viewed in tabular form and as a size distribution curve. Data can also be integrated over any selected range. A Statistical Process Control (SPC) option enables the file data to be viewed in standard control chart format either as an X or R chart. 

Malvern (Insitec) ECPS2 is designed to monitor and control particle size distributions from 0.5 to 1,500 pm, at concentrations up to 10,000 ppm, directly in pneumatic powder flow streams. Up to one thousand size distribution measurements per second are carried out at flow velocities from static to ultrasonic. Discrete data point, extracted from the log file, can be viewed. The data can also be viewed in tabular form and as a size distribution curve. Data can also be integrated over any selected range. A Statistical Process Control (SPC) option enables the file data to be viewed in standard control chart format either as an X or R chart. Various interface arrangements have been described, [203] ... 

To follow the coagulation process, samples of the smoke were taken from the flask at intervals over a period of 4 min and were passed into a centrifugal aerosol collector and classifier. Size distribution curves were measured, together with values for the total number of panicles per unit volume, obtained by the graphical integration of the size distribution curves. The volume fraction of aerosol material was 0 = 1.11 x 10. Theory and experiment are compared in Figs. 7,9 and 7.10, In Fig. 7.9, the experimental points for... 

Fig. V-33. Integral and differential particle size distribution curves...

Inpolydisperse systems the blurriness of sedimentation boundary is related to both the diffusion and the differences in the sedimentation rates of particles having different sizes. In cases when diffusion is negligible, the c(R) dependence represents the shape of integral particle size distribution curve at any moment of time. 

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. 

The values of the effective permeabilities vary over orders of magnitude, and this corresponds to the different results of the models. Furthermore, as discussed in various papers,the effective permeability of Natarajan and Nguyen (curve e) varies significantly over a very small pressure range, although they state that their capillary-pressure equation mimics data well. With respect to the various equations, the models that use the Leverett J-function all have a similar shape except for that of Berning and Djilali (curve a), who used a linear variation in the permeability with respect to the saturation. The differences in the other curves are due mainly to different values of porosity and saturated permeability. As mentioned above, only the models of Weber and Newman (curve d) and Nam and Kaviany (curve f) have hydrophobic pores, which is why they increase for positive capillary pressures. For the case of Weber and Newman, the curve has a stepped shape due to the integration of both a hydrophilic and a hydrophobic pore-size distribution. 

Typical pore size distribution data are shown in Fig. 5, where the integral penetration of mercury into the pores is plotted as a function of applied pressure. The calculated pore diameters in angstroms are shown across the top. The integral curve clearly shows a bimodal pore distribution with mean pore diameters at 20,000 and 50 A. The latter is at the lower limit of the technique. A nitrogen desorption isotherm is required to obtain an accurate measure in the region below 100 A. 

Figure 45b (upper part) shows the residual stress contribution of the strained filler clusters for the different pre-strains, obtained by subtracting the polymer contributions (solid lines) from the experimental stress-strain data (symbols) of Fig. 45a. The resulting data (symbols) are fitted to the second addend of Eq. (47) (solid lines), whereby the size distribution of filler clusters Eq. (37), shown in the lower part of Fig. 45b, has been used. The size distribution (x ) is determined by the adapted mean cluster size =< ifd>=26 and the pre-chosen distribution width Q=-0.5, which allows for an analytical solution of the integral in Eq. (47). The tensile strength of filler-filler bonds is found as Q b/d3=24 MPa. The different fit lines result from the different stress-strain curves ctR1( ) that enter the upper boundary of the integral in Eq. (47). Note that this integral, representing the contribution of the strained filler clusters to the total stress, becomes zero at = max for every pre-strain.

Fig. 46 a Stress contributions of the strained filler clusters for the different pre-strains (upper part), obtained as in Fig. 45b. The solid lines are adapted with the integral term of Eq. (47) and the log-normal cluster size distribution Eq. (55), shown in die lower part. The obtained parameters of the filler clusters are Qe /d 3=26 MPa, =25, and b=0.8. b Uniaxial stress-strain data (symbols) as in Fig. 45c. The insert shows a magnification for the smaller strains, which also includes equi-biaxial data for the first stretching cycle. The lines are simulation curves with the log-normal cluster size distribution Eq. (55) and material parameters as specified in the insert of Fig. 45a and Table 4, sample type C40...

The distribution function is presented graphically both as integral and differential distribution curves. In the integral distribution F(R) curve the abscissa axis depicts the size and the ordinate axis the fraction or percentage content of the total bubble number or the total volume of those bubbles whose size is bigger or smaller than R. In the differential distribution F(R) curve the abscissa axis depicts again the size but the ordinate axis the fraction content, i.e. number of bubbles entering a definite radius interval. The latter is more often employed. 

Figure 5. Light scattering per unit volume of aerosol material (G) as a function of particle diameter (dp), integrated over the wavelengths 360-680 nm for a refractive index of 1.5. The curve is independent of the particle size distribution. From Friedlander (2000) by Oxford University Press, Inc. Used by permission.

When a raw coal is wet ball-milled for a sufficient time to produce a slurry with a particle diameter mode in the range of 4 pm there results two forms of mineral matter That fractured away from the coal and that which is still enveloped in the coal particles. Figure 3 illustrates a typical particle size distribution for the separated product coal as compared to the separated free mineral matter (90 weight percent ash) from one milling test. The separated mineral matter is clearly smaller in diameter than the coal which is probably due to its more brittle properties. Note that in Figures 3 and 4 an integration of the curves will yield 100% of the mineral matter (or ash) under consideration rather than the ash content of the coal as was the case in Figures 1 and 2. 

Figure 5.8 Light scaltering per unii volume of aerosol material us a function of parlidc size, integrated over all wavelengths for a relVaetive index, in = 1.5. The incident radiation is as.sumcd to have the standiird distribution of soiar radiation at. sea level (Bolz and Tuve 1970). The limits of integration on wavelength were 0.36 to 0.680 w. The limits of visible light are approximately 0.350 to 0.700 m. The curve is independent of the particle size distribution.

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