Capillary Rheometer
The capillary rheometer (or viscometer) is the most
common device for measuring viscosity. Gravity, compressed gas or a piston is used to
generate pressure on the test fluid in a reservoir. A capillary tube of radius R and length L is connected to the bottom of the reservoir. Pressure drop and flow rate through this tube
Capillary Rheometer Analysis
The flow situation inside the capillary rheometer die is essentially
identical to the problem of pressure driven flow inside a tube (Poiseuille flow).
We can record force on
piston, F (or the pressure
drop ∆ P), and volumetric
flow rate, Q
Capillary Rheometer Analysis
Hagen-Poiseuille law for pressure driven flow of Newtonian fluids inside a tube:
Q 8
−µ
=
∆
Shear stress profile inside the tube:
∆
=
τ L
P 2
r
At the wall (r=R):
∆
=
τ L
P 2
R
W
Velocity profile inside the tube:
− µ
= ∆
2 2
R 1 r
L 4
R ) P
r ( u
Recall from Fluid Mechanics
(1)
Capillary Rheometer Analysis
apparent
R
3Q R 4
L 2
P dr
du = γ
= π µ
= ∆
=
γ ɺ ɺ
Shear rate:
True shear rate for Newtonian fluids but
Apparent shear rate (
app) for non-Newtonian fluids
∴ For non-Newtonian fluids if we use the apparent shear rate then we can only calculate an Apparent Viscosity:
app w
app
γ
= τ
η ɺ
γ ɺ
Capillary Rheometer Analysis
For non-Newtonian fluids the Rabinowitch analysis is followed - From the definition of the volumetric flow rate through a tube:
∫
∫ → = π − π
π
=
R0 R 2
0 2 parts
by g integratin R
0
dr
dr r du u
r Q
dr u(r) r
2 Q
- Applying the “no-slip” boundary condition and eliminating r with the aid of eq. (1)
τ
τ π =
τ ∫τ d
dr du R
Q
W0
2 3
3 W
Capillary Rheometer Analysis
After several manipulations we obtain the Rabinowitch equation
+ τ γ
=
+ τ
= π γ
W app
W W 3
ln d
lnQ d
4 1 4
3 ln
d
lnQ d
4 1 4
3 R
Q
4 ɺ
ɺ
∴ The Real Viscosity of the polymer melt is:
W w
γ
= τ η ɺ
(2)
Capillary Rheometer Analysis
To obtain the “true” shear rate we must plot Q vs τw on logarithmic
coordinates to evaluate the derivative dlnQ/dlnτw for each point of the curve
For power-law fluids, it turns out that the slope is:
n 1 ln
d
lnQ d
W
τ =
∴The Rabinowitch equation becomes:
n 4
1 n
3 R
Q 4
W 3
+
= π
γ ɺ
Entrance Pressure Drop
In the previous analysis we have assumed that the measured ∆P by the instrument corresponds to the pressure drop inside the capillary die, ∆Pcap
∆ P
cap∆ P
res~0
∆ P
e=
Entrance Pressure DropR 2 L
P
W
= ∆ τ
Therefore from eq. (1) we had:
Entrance Pressure Drop
Newtonian Fluids and some melts such as HDPE and PP
Fluids with pronounced non- Newtonian behaviour
Bagley Correction for ∆∆∆∆ P e
Unless a very long capillary is used (L/D>100), entrance pressure drop may considerably affect the accuracy of the measurements.
The Bagley correction is used to correct for this, by assuming that we can represent this extra entrance pressure drop by an equivalent length of die, e:
− Three or four capillaries are used and results are plotted as DP vs L/R:
+
= ∆ τ
R e 2 L
P
W
The true shear stress is:
Summary of Corrections
Calculate the apparent shear rate: app 3
R Q 4
= π γ ɺ
Correct the shear rate by using the Rabinowitch correction:
+ τ γ
= γ
W app
W
d ln
lnQ d
4 1 4
ɺ 3 ɺ
Obtain true shear stress by using Bagley correction:
+
= ∆ τ
R e 2 L
P
W
Calculate true viscosity: w
