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August 25, 2003 Particles 2003 Herschel Bulkley Flow
The flow of Herschel-Bulkleyfluids
Ian Morrison, Cabot Corporation
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Thin polymer-filled layers coated from dispersions
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Typical shear thinning rheogramPhotoconductive Pigment in Nonaqueous Polymer Solution
Shear rate (s-1)
0.1 1 10 100 1000 10000
Visc
osity
(mPa
s)
0.001
0.01
0.1
1
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Data is reasonably fitted by a power law
10
nη η γ −=
n is the power law index and is one for a Newtonian fluid.
Photoconductive Pigment in Nonaqueous Polymer Solution
Shear rate (s-1)
0.1 1 10 100 1000 10000
Visc
osity
(mPa
s)
0.001
0.01
0.1
1
0.750.580.25n
η γ −==
August 25, 2003 Particles 2003 Herschel Bulkley Flow
The flow of a shear thinningliquid through a pipe
The reduced flow rate of a shear thinning liquid is:
1 11 3 11
nn rvn R
+ + = − +
Macosko, Rheology, pp. 88-89, 1994.
n= 0.25
0 0.5 10
0.5
1
1.5
2
2.5
Radial postion from center
Rel
ativ
e ve
loci
ty
2.5
0
PLj
10 rj
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Same data replotedShear stress versus shear rate
Shear rate (s-1)
0 200 400 600 800 1000 1200 1400 1600
Shea
r stre
ss (P
a)
0
1
2
3
4
5
6
7
August 25, 2003 Particles 2003 Herschel Bulkley Flow
How yield points approximate power law flowShear stress versus shear rate
Shear rate (s-1)
0 200 400 600 800 1000 1200 1400 1600
Shea
r stre
ss (P
a)
0
1
2
3
4
5
6
7
The slopeis the viscosity.
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Herschel-Bulkley rheology
0nmτ τ γ= +
Model the flow as shear thinning above some yield point:
Shear stress versus shear rate
Shear rate (s-1)
0 200 400 600 800 1000 1200 1400 1600
Shea
r stre
ss (P
a)
0
1
2
3
4
5
6
7
0.710.64 0.31τ γ= +
Note the value of “n” is now higher, apparently less shear thinning.
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Effect of a yield point on flow
If a fluid has a yield point, τ0, and is under pressure, p, a column of radius r and length L in the center moves as a plug. The plug radius can be calculated.
02
0 0
(2 ) or
2 2
rLprLrp P
τ ππ
τ τ
=
= =
P is the pressure drop.
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Flow of an Herschel-Bulkley fluid through a pipe
( )
0
112 20 0 0
2 3 3 3 2 2 2 3 3 20 0 0
20
40 0
2
3 1 (2 1) (( ) (Pr ) )2 2 2
2 1
4 113 3
nn n
rP
P n n R PR PRvelocity
n R P n RP R P n R P nelse
RPvelocity
RP RP
τ
τ τ ττ τ τ
τ
τ τ
+−
>
− + + − − − +=
− + + −
− =
− +
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Flow of Herschel Bulkley fluid
0 0.002 0.004 0.006 0.0080
0.5
1
1.5
2
2.5
Herschel-BulkleyFluid velocity
Rad
ial p
ositi
on fr
om c
ente
r
HBj
rj
Note: The radius of the plug is:
The fitted Herschel-Bulkley parameters:n = 0.75tau = 0.5 Pa
with a reasonable pressure dropP = 106 Pa/0.1m
5
2 0.5 Pa 110 Pa/0.1xr m
mµ= =
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Comparison of predicted flows:Power law and Herschel-Bulkley
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Some Problems Require More Study
August 25, 2003 Particles 2003 Herschel Bulkley Flow
August 25, 2003 Particles 2003 Herschel Bulkley Flow
Surface Energy Characterization ofFibers, Fillers and Paper
D. Steven Keller, Empire State Paper Research Institute,State University of New York, College of Environmental
Science and Forestry, 1 Forestry Drive, Syracuse, NY13210
E-mail: dskeller@syr.edu
Ian D. Morrison, Cabot Corporation, 157 Concord Road,Billerica, MA 01821
E-mail: Ian_Morrison@Cabot-Corp.com
Philadelphia, August 22-26, 2004228th ACS National Meeting
Pennsylvania Convention Center,Philadelphia
PROGRAM CHAIR R. Nagarajan, Department ofChemical Engineering, Penn State University, UniversityPark, PA 16802 E-mail: rxn@psu.edu
Submit abstracts via Online Abstract Submittal System(OASYS) at www.acs.org/meetingsDeadline for submission of 150 words abstract is April 15,2004.
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