Chemistry Reference
In-Depth Information
the edge of the cone and plate if a constant shear rate is required. Unfortu-
nately, the DIN standard bob is poor in this respect.
The double-gap geometry provides a high sensitivity with a light construc-
tion. However, the engineering is subtle in that the inner gap must be slightly
smaller than the outer one if a uniform shear rate is to be achieved throughout
the sample. In addition, it is less satisfactory in terms of the onset of flow
instabilities. Taylor
2
showed that the Reynolds number for the onset of such
instabilities was lower if the inner cylinder is made to rotate and, with the
double-gap arrangement, this occurs in the inner section. This will significantly
lower the upper range of shear rate available, although not to the level in a cone
and plate with a large included angle.
So far we have restricted our discussion to Newtonian liquids, but
the analysis will change somewhat if the systems are non-Newtonian. A
useful illustration of the problems that arise is the case of a Bingham plastic.
This gives us a linear response, as does a Newtonian liquid, but in this case
there is an intercept or yield stress. The constitutive equation for a Bingham
plastic is
s
¼
Z
ð
N
Þ
g
þ
s
B
ð
3
:
21
Þ
Here, the yield stress is the Bingham yield value and the value of Z(N) is the
linear value reached at high shear, often referred to as the plastic viscosity.
The calculation of the material behaviour follows the same route as with the
Newtonian case so
moment
¼
sum of moments from all the elements
i.e.
M
cc
¼
sum of
½
2pLr
2
ð
stress from each element
Þ
ð
3
:
22
Þ
We must use the constitutive equation for the Bingham plastic for the stress.
This then gives the angular velocity of the outer cylinder from:
Z
do
¼
Z
dr
O
R
o
M
cc
2pLZ
ð
N
Þ
r
3
s
B
Z
ð
N
Þ
r
ð
3
:
23
Þ
R
i
0
giving the Riener-Riwlin equation for plastic flow in a Couette
M
cc
4pLZ
ð
N
Þ
1
R
i
1
R
o
s
B
Z
ð
N
Þ
ln
R
o
R
i
O
¼
ð
3
:
24
Þ
Clearly the Riener-Riwlin equation reduces to the Margules equation when the
Bingham yield value is zero. There is an important consequence in that it is
assumed that all the material is flowing, i.e. the shear stress at the wall of the
outer cylinder must be
M
cc
2pLR
o
s
B
ð
3
:
25
Þ
