Biomedical Engineering Reference
In-Depth Information
to considerable development in formulating a single correlation for any shape and
orientation. One correlation is given by Haider and Levenspiel (1989) which provides
accurate representation for certain irregular shapes. The correlation defines the
C
D
as a function of the particle Reynolds number and a shape factor which is given as
Re
p
1
a
Re
p
b
+
24
c
Re
p
C
D
=
+
(8.13)
d
+
Re
p
where the coefficients are
exp
2
.
3288
2
.
4486
φ
2
a
=
+
6
.
4581
φ
+
b
=
0
.
0964
+
0
.
5565
φ
exp
4
.
905
10
.
2599
φ
3
18
.
4222
φ
2
c
=
−
13
.
8944
φ
+
−
exp
1
.
4681
15
.
8855
φ
3
20
.
7322
φ
2
d
=
+
12
.
2584
φ
−
+
The shape factor
φ
is defined as
A
s
A
p
φ
=
(8.14)
where
A
s
is the equivalent surface area of a sphere having the same volume as the
non-spherical shaped particle, and
A
p
is the actual surface area of the particle. An
advantage of the method by Haider and Levenspiel (1989) is that it provides a simple
correlation to fit all types of shapes; however, its generality makes it inaccurate and
unsuitable for shapes that are very non-spherical.
A later development for non-spherical particles is the technique presented by
Tran-Cong et al. (2004) which uses agglomerates of spheres to represent different
particle shapes. Two equivalent diameters, surface-equivalent-sphere
d
A
and the
nominal diameter
d
n
, and a shape factor called the particle circularity (Wadell 1933)
are used for the drag correlation. The surface equivalent sphere diameter is defined as
4
A
p
π
d
A
=
(8.15)
where
A
p
is the projected area of the sphere. The volume equivalent sphere diameter,
also known as the nominal diameter, is defined as
3
6
V
π
d
n
=
(8.16)
where
V
is the particle volume. The shape factor used is based on the surface
sphericity and is defined as
c
=
π
d
A
P
P
(8.17)
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