classes. We then haveNiparticles in size classiand
Ni¼Zxiþ (^12) Dx
xi^12 Dx
fðxÞdðxÞð 9 : 3 Þ
whereDxis the class width andxiis the value ofxcharacterizing classi
(midpoint of class). The approximated frequency is now given byNi=Dx, the
cumulative distribution bySNi.
Figure 9.9 illustrates this. At the left, the (presumed) measured values
are indicated; for convenience, the points are connected by straight lines.
The right-hand graph shows the same data in the form of a histogram.
Notice that the class width is not the same for every class. A smoothed curve
is drawn through the histogram to show the continuous frequency
distribution; since only limited information is available (8 points), this
curve is to some extent conjectural.
Besides the number distribution, distributions ofmass, volume, surface
areaor other characteristics of the particles can be made. In such a case, the
number frequency should be multiplied by mass, volume, etc. of the
corresponding particles. For instance, if it concerns spheres, wherexequals
the diameter, the volume frequency distribution is given bypx^3 fðxÞ=6. In
other words, one can choose both for the abscissa and for the ordinate the
kind of variable that is most suitable for the presentation of the results,
depending on the problem studied. Plotting volume frequency versus
particle diameter is the most common presentation.
FIGURE9.9 Example of a particle size distribution, given as a cumulative and as a
frequency distribution. The points denote the measured values. A continuous
frequency distribution is also shown.