Chemistry Reference
In-Depth Information
The advantage of the gradient method is that it does not require an a
priori flow model. The resulting usable shear rate range is rather limited
as the low shear rate region towards the centre of the pipe is given by
the velocity resolution of the measurement system, and due to averaging
effects it is also difficult to have an accurate measurement towards the
pipe wall, which limits the maximum shear rate.
3.6.2
Power law fluid model
The simple power law model
n
τ
=
K
˙
γ
(3.12)
respectively
n
−
1
η
=
K
˙
γ
(3.13)
=
<
represents Newtonian (
n
1), shear-thinning (
n
1) or shear-thickening
(
n
>
1) non-Newtonian fluids, where
K
is the consistency index and
n
is the flow exponent. Combination with Equation 3.9 and integration
results in an equation for the radial velocity profile:
R
1
/
n
1
1
+
1
/
n
r
R
R
P
2
LK
v
=
−
(3.14)
(1
+
1
/
n
)
Having measured the velocity profile and the pressure drop, Equation
3.14 can be used to fit
n
and
K
, usually by minimising the sum of the
squares of the differences of the measured and the calculated velocity
profile.
The local shear rate and viscosity are respectively given by:
1
/
n
Pr
2
LK
γ
=
˙
(3.15)
K
1
−
1
/
n
Pr
2
LK
η
=
(3.16)
It would also be possible to derive on
ly
n
when the volume flow rate
is
Q
, and thus the mean flow velocity
v
is known using (Wilkinson,
1960)
v
3
n
1
r
R
+
1
n
+
1
n
=
−
v
(3.17)
n
+
1
