Chemistry Reference
In-Depth Information
Alternatively, the maximum flow velocity can be used (Kowalewski,
1980):
v
max
1
r
R
n
1
n
v
=
−
+
(3.18)
Yet another approach to determine
n
is to use the wall shear rate ˙
γ
w
,
which can be expressed using the flow rate
Q
as:
3
n
4
Q
π
+
1
γ
=
˙
(3.19)
w
4
n
R
3.6.3
Herschel-Bulkley fluid model
The analysis in the previous section can be extended to other rheological
models such as Herschel-Bulkley with the yield stress
τ
0
:
n
τ
=
τ
0
+
K
˙
γ
(3.20)
The corresponding equation for the velocity is given by:
n
1
n
R
1
+
r
R
−
1
+
1
1
n
1
nR
P
2
LK
R
R
R
R
v
=
−
−
(3.21)
+
1
n
in which
r
P
is the radius of the plug, which
can also be determined directly from the flow profile (Dogan
et al.
,
2003b). The local shear rate and viscosity are respectively given by
>
R
∗
and
R
∗
=
2
L
τ
0
/
1
n
P
2
LK
R
∗
)
n
γ
=
˙
(
r
−
(3.22)
K
1
−
1
n
τ
˙
P
2
LK
r
η
=
=
(3.23)
γ
R
∗
)
n
(
r
−
3.6.4
Other models
Power law and Herschel-Bulkley are the two most often applied models,
but literature also describes the use of Ellis (Wunderlich and Brunn,
1999), Casson (Dogan
et al.
, 2003b), Eyring (Dogan
et al.
, 2005b) and
Sisko (Birkhofer, 2007; Wiklund, 2007) models.
