Yahya Ghasemi Print III pdf

3.3.2.2. Equivalent polyhedron shape 
In order to calculate the specific surface area of spherical particles, the mean diameter of the 
particle sizes d
i
and d
i+1 
of fraction 
i
, as the characteristic particle size, is required. The mean 
diameter d
i
can be calculated using either arithmetic mean or geometric mean using Eq. (3.28) 
and (3.29) respectively.
݀
௜
,
௔௥௜௧௛
=
݀
௜
+
݀
௜ାଵ
2
(3.28)
݀
௜
,
௚௘௢
=
ට݀
௜
ଶ
+
݀
௜ାଵ
ଶ
(3.29)
When replacing the spheres with other polyhedron shape, the length of the sides of 
polyhedrons should be calculated based on the mean diameter of the spheres. The length of 
the sides of polyhedrons can be computed based on the assumptions of how the shapes are 
defined in respect to geometric properties using the concepts of circumshpere and midsphere,
see Figure 3.9. In geometry, a circumscribed sphere or circumsphere of a polyhedron is 
a sphere that contains the polyhedron and touches each of the polyhedron's vertices. 
Midsphere is defined as a sphere that touches all of the polyhedron edges. The midsphere 
does not necessarily pass through the midpoints of the edges, but is rather only tangent to the 
edges at same point along their lengths (Cundy and Rollett, 1961). 
It is also possible to calculate the side length of the polyhedrons with the assumption that the 
polyhedrons have the same volume as the sphere it replaced (volumetric equivalency).

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28
Mix design approaches
Circumsphere -Cube
Midsphere - Cube
Equivalent volume
Figure 3.9. Circumsphere, Midsphere and volume equivalency of a cube, edge length of a. 
The geometric equivalent edge lengths of the polyhedrons, a (a
c
for circumsphered and a
m
for 
midsphere edge length), can be calculated by equations listed in Table 3.2. Median radius of 
equivalent spheres, r, can be calculated by either Eq. (3.28) or (3.29).
Table 3.2: Edge lengths (Circumsohered a
c
and Midsphered a
m
) of polyhedron.