Chemistry Reference
In-Depth Information
representative shear rate ranges for cone-and-plate and capillary rheometers. The
last viscometer type, which bears a superficial resemblance to the orifice in an
extruder or injection molder, is the most widely used and will be the only type
considered in this nonspecialized text.
Equation (4-90)
[cf. Eq. (3-87)] gives the relation between flow rate and vis-
cosity for a fluid under pressure
P
in a tube with radius
r
and length
l
. In such a
device the apparent shear stress,
τ
a
5
Pr/
2
l
; and the apparent shear
rate,
r
3
, where
Q
, the volumetric flow rate, is simply the
Q
/
t
term of Eq. (3-
87). That is,
γ
a
5
4
Q
=π
Pr
4
t
8
Ql
5
η5
π
Pr
=
2
l
r
3
5
τ
a
(4-90)
4
Q
=π
γ
a
The shear stress and shear rate here are termed apparent, as distinguished
from the respective true values at the capillary wall,
γ
w
. The Bagley cor-
rection to the shear stress allows for pressure losses incurred primarily by acceler-
ating the polymer from the wider rheometer barrel into the narrower capillary
entrance
[24]
. It is measured by using a minimum of two dies, with identical radii
and different lengths. The pressure drop, at a given apparent shear rate, is plotted
against the
l/r
ratio of the dies, as shown in
Fig. 4.31
. The absolute values of the
negative intercepts on the
l/r
axis are the Bagley end-corrections,
e
. The true
shear stress at each shear rate is given by
τ
w
and
P
2
r
1
τ
w
5
(4-91)
e
τ
w
can be measured directly by using a single long capillary with
l/r
about 40. The velocity gradient in Fig. 3
Alternatively,
6 is assumed to be parabolic, but this
is true strictly only for Newtonian fluids. The Rabinowitsch equation
[25]
corrects
for this discrepancy in non-Newtonian flow, such as that of most polymer melts:
4
Q
π
3
n
1
1
γ
w
5
(4-92)
4
n
r
3
.
Pressure
l
L/R
Bagley correction
FIGURE 4.31
Bagley end correction plot.
