Yahya Ghasemi Print III pdf

particles are embedded in a matrix of small particles (De Larrard, 1999).
This study deals with the following particle packing models: Toufar (Toufar et al, 1976),
CPM (De Larrard, 1999), and 4C models (Pade et al, 2009; 4C user’s manual, 2009). 
3.2.1. Toufar Model
Toufar model is developed for calculating the packing of binary mixes where the diameter 
ratio of the class falls in the range of 0.22 < d
1
/d
2
< 1.0 (Toufar et al, 1976). According to the 
model, for diameter ratio larger than 0.22, the smaller particles (d
1
) will be too large to fit in 
the voids between larger particles (d
2
). As a result, factor k
d
was introduced which relates the 
packing density to the diameter ratio of the two particle classes. In addition, Toufar model 
assumes that each of the fine particles is placed between exactly four of the coarse particles 
which is utilized in factor k
s 
(Fennis, 2011). The packing density,
ߙ
௧
, is described by 
following equations (Goltermann et al. 1997):
ߙ
௧
=
1
ݕ
ଵ
ߙ
ଵ
+
ݕ
ଶ
ߙ
ଶ
െ ݕ
ଶ
ቀ
1
ߙ
ଶ
െ
1
ቁ ݇
ௗ
݇
௦
(3.7) 
݇
ௗ
=
݀
ଶ
െ
݀
ଵ
݀
ଵ
+
݀
ଶ
(3.8) 
݇
௦
= 1
െ
1 + 4
ݔ
(1 +
ݔ
)
ସ
(3.9) 
ݔ
=
ݕ
ଵ
ߙ
ଶ
ݕ
ଶ
ߙ
ଵ
(1
െ ߙ
ଶ
)
(3.10) 
w
KHUHĮ
t
is the calculated packing, y
i
LVWKHFODVVLVKDUHRIYROXPHDQGĮ
i
is the packing of 
class i. 

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18
Mix design approaches
The Toufar model was later modified by (Golterman et al., 1997) since comparisons with test 
results showed that the packing density of a sample of coarse particles does not increase when 
a small amount of fine particles is added to the mixture as a result of the unrealistic 
assumption that each fine particles is placed in between four coarse particles. Here coarse 
particles are considered relative to the fines size and not a certain range e.g. for a binary 
mixture the constitute with larger diameter is considered as coarse. The behaviour was 
corrected by modifying the k
s
factor:
݇
௦
=
0.3881
ݔ
0.4753
For x < 0.4753 
(3.11) 
݇
௦
= 1
െ
1 + 4
ݔ
(1 +
ݔ
)
ସ
For x
൒
0.4753
(3.12) 
where x is calculated based on Eq. (3.10). 
Toufar and modified Toufar models are fairly easy to use. However, estimation of packing 
density by these models will lose accuracy in multicomponent systems and tends to 
underestimate the packing density.