distribution. This type of average should be inserted into Eq. (9.1a) for a
polydisperse system. The most commonly used average may bed 32 , which is
generally not far removed from the modal volume diameter. It gives the
relation between volume fraction and specific surface area of the particlesA
(dimension [L^1 ]), according to
A¼pS 2 ¼6 j
d 32ð 9 : 7 Þtaking into account thatj¼pS 3 =6.d 32 also is the appropriate average in
Eq. (9.1b) for a polydisperse system, and in relations for pore size or
permeability [e.g., Eq. (5.25)].
Width. If the size distribution is very narrow, i.e., almost
monodisperse, an average suffices to characterize it, and the various types
of average differ only slightly from each other. Otherwise, thedistribution
widthis important. It can best be expressed as a relative width, according to
Eq. (9.6). Takingn¼2, the resulting parameter isc 2 , i.e., the standard
deviation of the size distribution weighted with the particles’ surface area,
divided byd 32 (¼the mean of the distribution weighted with the surface
area). Values ofc 2 are rarely below 0.1 (very narrow) or over 1.3 (very wide);
in most fabricated foods the range is between 0.5 and 1.
An absolute width, e.g., expressed as a standard deviation in
micrometers, tells very little. Figure 9.10 shows a quite wide distribution
ðc 2 & 1 : 07 Þ; its absolute width would be about 0.7mm, assuming it to represent
a homogenized emulsion. A collection of glass marblesðd&10 mmÞtends to
be very monodisperse (e.g.,c 2 ¼0.02), but the standard deviation of the
diameters would then be 200mm, i.e., about 300 times as high.
Shape. Ifc 2 is not small, say>0.4, theshapeof the size distribution
may become of importance. Several equations exist for frequency
distributions (among which the log-normal one depicted in Figure 9.10
represents several dispersions rather well), but for a small width they all give
nearly the same curve (assumingd 32 to be the same). For larger width,
differences in shape become important. For instance, distributions can be
more or less skewed, or truncated, or even bimodal.
We will not treat mathematical equations for frequency distributions.
It may be added, however, that the often used normal distribution is rarely
suitable for particle size distributions. For instance, it allows the presence of
particles of negative size; in other words,fðxÞ>0 forx 4 0. A log-normal
distribution (in which logxhas a normal distribution) givesfðxÞ¼0 for
x¼0.