Biomedical Engineering Reference
In-Depth Information
11.10 Schematic illustration of a single aggregate consisting of the
backbone (black circles) and dead ends (gray circles). The aggregate is
decomposed into dead ends with the fluid and the backbone (Prasher
et al., 2006c).
where n(r) is the radius distribution function, which represents the fractal
characteristics of the space distribution of clusters.
Prasher et al. (2006c) analyzed the effects of aggregation and its kinetics
to predict the effective thermal conductivity of nanofluids. They suggested
that a fractal cluster is embedded within a sphere of radius equal to r and is
composed of a few approximately linear chains called the backbone of the
cluster, with other particles called dead ends (Fig. 11.10). The volume
fraction of particles belonging to dead ends was calculated as
f
nc
= f
(r)
d
l
3
is the volume fraction backbone particles.
The thermal conductivity of the aggregate due to dead end particles is
calculated from the Bruggeman (1935) equation as:
f
c
in which
f
c
¼ð
r
=
a
Þ
k
f
k
nc
k
p
k
nc
ð
1
f
nc
Þ
2k
nc
þ
f
nc
2k
nc
¼
0
½
11
:
13
k
f
þ
k
p
Assuming the backbone particles to form randomly oriented cylindrical
chains, the model of Nan et al. (2004) can be used to predict the thermal
conductivity of an aggregate sphere with both chains and dead ends as:
3
þ
f
c
½
2
b
11
ð
1
L
11
Þþ
b
33
ð
1
L
33
Þ
k
a
¼
k
nc
½
11
:
14
3
f
c
ð
2
b
11
L
11
þ
b
33
L
33
Þ
