Eq 3

The only other parameter that appears to meet the three basic requirements outlined above has been derived by Hilliard (Ref 40) from a geometric theorem relating the total surface and the total volume of three-dimensional objects to the first and the fourth moment of the intercept length distribution--Mj(1) and M4(1) (see Ref 34). After normalizing to 1 for spheres again, this yields the shape parameter:


Nl and VV are defined above. The main limitation of FH is that it fulfills Eq 3 in principle only; in practice, M4(1) is subject to large experimental errors due to the fact that the longest intercepts, which usually are present with a low numerical frequency, contribute most to M4(1).

Numerous other shape parameters have been proposed by Underwood (Ref 35, 36), Ministr (Ref 41), and others. However, most equations are in pronounced contradiction to one or more of the basic requirements for true shape factors. Due to the problems outlined in this section, two-dimensional shape characteristics provide practical alternatives and as such are used almost exclusively in practical work.

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