Chemistry Reference
In-Depth Information
Now, the extension rate e in the geometry is given by the rate of change of
velocity with distance:
¼
Q
pk
e
¼
dV
ð
l
Þ
dl
¼
d
dl
Ql
pk
ð
6
:
111
Þ
This expression indicates that for a constant flow rate there is a constant
extension rate for this shape of die in a total-slip condition. So, you might
suppose that all that is required is to have a lubricated die, a pump and a
method of measuring the stress in flow and the extensional viscosity can be
obtained from the stress divided by the rate. However, a practical geometry has
to have a finite length and the radius has to be large enough to allow our
material out of the end! So, the fluid can only spend a certain amount of time in
the flow - this is termed the residence time t
r
. It depends on the size of the die
and the flow rate. If we divide the total volume of the die V, by the volume
flowing per second Q we get the time that the material is resident in the
geometry. We can use a volume of rotation integral to determine V between
lengths L
1
and L
2
:
Q
p
Z
L
2
Z
L
2
t
r
¼
V
Q
¼
1
Þ
2
dl
¼
pk
Q
l
1
dl
ð
R
ðÞ
ð
6
:
112
Þ
L
1
L
1
Evaluating the integral and using the extension rate we obtain the following
expression:
¼
w
L
2
L
1
t
r
e
¼
ln
ð
6
:
113
Þ
You will notice that w depends only upon the geometry and is invariant with
rate. The residence time is the amount of time the material spends at that
extension rate; it is our experimental measurement time. It depends on the
length of the geometry for a given design constant k. This has the same form as
the Hencky strain, where a rod is grabbed at each end and subjected to an
exponential increase in length. This gives a uniaxial extension. The importance
of this is shown when one considers what would happen to a material measured
as a function of extension rate. Suppose we have a material that in extension
behaves as a Maxwell model. The extensional viscosity as a function of time
and rate as measured in our geometry is given by
¼
Z
e
ðÞ
1
exp
t
r
t
e
w
et
e
Z
e
e
; ðÞ¼
Z
e
ðÞ
1
exp
ð
6
:
114
Þ
So, the Hencky strain determines the apparent viscosity we measure. Each rate
defines a different point in time that is not necessarily the steady-state value. If
w/ ect
e
the material will appear extension thinning and if w/ e
{
t
e
the
viscosity will be constant. If the material displays a significant stress overshoot
