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
are used to determine
viscosity
Fluidodinamica
15/10/2007
FDS
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
Fluidodinamica
15/10/2007
FDS
Capillary Rheometer Analysis
Recall from Fluid Mechanics
Shear stress profile inside the tube:
τ =
r  ∆P 


2 L 
At the wall (r=R):
τW =
R  ∆P 


2  L 
(1)
Velocity profile inside the tube:
2

∆P R
 r  
u(r) =
1 −   
4 µ L 
 R  
2
Hagen-Poiseuille law for pressure driven flow of Newtonian
fluids inside a tube:
8Q −4
∆P = µL
R
π
Fluidodinamica
15/10/2007
FDS
Capillary Rheometer Analysis
Shear rate:
du ∆P
4Q
R=
= γɺ apparent
γɺ =
=
3
dr 2µL
πR
True shear rate for
Newtonian fluids but
Apparent shear rate ( γɺ app)
for non-Newtonian fluids
∴ For non-Newtonian fluids if we use the apparent shear rate then we
can only calculate an Apparent Viscosity:
η app
Fluidodinamica
τw
=
γɺ app
15/10/2007
FDS
Capillary Rheometer Analysis
For non-Newtonian fluids the Rabinowitch analysis is followed
- From the definition of the volumetric flow rate through a tube:
R
Q = 2 π ∫ r u(r) dr
0
   
→ Q = πr u 0 − ∫
integrating by parts
2
R
R
0
 du 
πr  dr
 dr 
2
- Applying the “no-slip” boundary condition and eliminating r with the aid
of eq. (1)
τ 3W Q
=
3
πR
Fluidodinamica
∫
τW
0
 du 
τ 
 dτ
 dr 
2
15/10/2007
FDS
Capillary Rheometer Analysis
After several manipulations we obtain the Rabinowitch equation
 3 1 d lnQ 
4Q  3 1 d lnQ 
 +
 = γɺ app  +

γɺ W =
3 
πR  4 4 d lnτ W 
 4 4 d lnτ W 
(2)
∴ The Real Viscosity
of the polymer melt is:
τw
η=
γɺ W
Fluidodinamica
15/10/2007
FDS
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:
d lnQ 1
=
d lnτ W n
∴The Rabinowitch equation becomes:
4Q 3n + 1
γɺ W =
πR 3 4n
Fluidodinamica
15/10/2007
FDS
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
Therefore from eq. (1) we had: τ W
∆P
=
L
2
R
∆Pres~0
∆Pe= Entrance Pressure Drop
∆Pcap
In reality
∆Ptotal = ∆Pres + ∆Pe + ∆Pcap
Fluidodinamica
15/10/2007
FDS
Entrance Pressure Drop
Newtonian Fluids and some
melts such as HDPE and PP
Fluidodinamica
Fluids with pronounced nonNewtonian behaviour
15/10/2007
FDS
Bagley Correction for ∆Pe
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:
The true shear stress is:
τW =
Fluidodinamica
15/10/2007
∆P
L

2 + e 
R

FDS
Summary of Corrections
γɺ app
Calculate the apparent shear rate:
4Q
=
πR 3
Correct the shear rate by using the Rabinowitch correction:
 3 1 d lnQ 

γɺ W = γɺ app  +
 4 4 d lnτ W 
Obtain true shear stress by using Bagley correction:
τW =
Calculate true viscosity:
Fluidodinamica
η=
τw
γɺ w
15/10/2007
∆P
L

2 + e 
R

FDS