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Predict Friction
Loss in
Slurry Pipes
ChemicalEngineering;
Sep 1992; 99, 9; ABI/I-NFORM Global
pg. 116
T
o select pumps and design piping systems to transport
slurries, the engineer must predict the pressure drop due
to pipe friction. This is inherently a far-more-complicated
matter than for flow of liquids alone. However, the procedure
and equations presented here make it possible to do so by
scaling up laboratory rheological data on the slurry. The
approach is valid for either laminar or turbulent flow.
It is also suitable for a wide range of concentrated slurries.
Most of the highly loaded suspensions encountered in engi-
neering practice are pseudohomogeneous (i.e., essentially
homogenous) non-Newtonian fluids whose properties can in.
fact be measured in the laboratory, via laminar-tube-flow or
viscometer studies.
Such suspensions generally consist of particles that are
relatively small (i.e., below 100 Fro) or of low density or both.
These materials frequently exhibit a measurable yield stress,
and can be pumped in laminar or turbulent flow through
pipes. Among the many examples are slurries of clay, lime-
stone, iron ore, bauxite or coal.
The starting point for the procedure consists of using
either or both of two relatively simple equations’ that are
widely accepted as rheological models for homogeneous sus-
pensions. One is the Bingham-plastic model
and the other is the power-law model
O)
"q = m~
"-1
(2)
In a nutsheli, the overall sequence consists of fitting either
or both of these equations* to laboratory data so as to get
NOTATION
a
fr
He
K’
K
L
I11
n
Reo
Re
V
"re
Tw
parameter given by Eq. (20)
parameter given by Eq. (21)
parameter given by Eq. (11)
parameter given by Eq. (11)
pipe diameter, m
Fanning friction factor, Eq. (5)
laminar friction factor
turbulent friction factor
transition friction factor
Hedstrom number
parameter in Eqs. (12), Pa s
n
parameter in HerscheI-Bulkey model, Pa s
n
pipe length, m
power-law consistency coefficient, Pa s
n
power-law flow index
pressure drop in pipe, Pa
Bingham Reynolds number
critical power-law Reynolds number for
transition to turbulence
power-law Reynolds number
Reynolds number
velocity [m/s]
parameter in Eqn (13), given by Eqn (16)
shear rate, s
-1
non-Newtonian viscosity, Pa s
Bingham-plastic limiting viscosity, Pa s
density, kg/m
3
yield stress, Pa
wall stress, Pa
Ron Darby
Texas A&M University
Robert Mun and Dovid V. Boger
University of Melbourne
estimates of To and
F~o
or of m and n (or of all four), then
employing those estimated values in other equations set out
below, as explained later. In general, either the Bingham or
the power-law leads to good results. However, because the
]~ingham model includes a realistic high-shear (or "plastic")
limiting viscosity
F
¢o, it is the more reliable if it is necessary
to extrapolate beyond the range of laboratory conditions.
Accordingly, its edge over the power-law model is especially
notable for turbulent flow.
Tube-flow
equation
s
For laminar flow in a tube, the theoretical solution for a fluid
modeled by the power law is well established, It can be
expressed in te~s of the relation between the wall stress
(-APD/4L)
and the effective or apparent shear rate (8
V/D)
as
8V/D
= [4n/(3n +
1)]m-V’~[-APD/4L]
u"
(8)
or in dimensionless form as
fL = 16/T~ep
(4)
where
fL = -APD/2Lp~
(5)
Re~ = 2’~-’D~V~’~p/m[(3n +
1)/n]
’~
(6)
For fluids modeled as Bingham plastics, the corresponding
solution is the Buckingham-Reiner equation, which can be
expressed as:
(’0
or in dimensionless form as
f,~ = [~6/zes] [~ + (~/6)(ge/~e~) -
O/~)(g~/f~es~)] (~)
where
Re~ = DVp/F~, He = D~p’ro/p,~
~
(9)
*Another common theological model is the Herschel-Bu]kley model’.
.~ = .ty/.] + K,,in’-I
The Bingham and power-|aw models are obviously special eases of the Herschel-Bulkley
model. ,Qthough Hersehel.Bulkley is the most versatile of the three mod.e]s (beouse its
three parameters provide greater flexibility in fitting data)
i
the range, quality, ann nature
of viscosity measurements are not always adequate to justify this more-complex model, and
may not enable accurate or reliable determination u~ all three parameters. Even with
good
theological data. it is unusual to obtain a range of shear rates greater than about two
ordure of magnitude, in which case either of the two siinpler models should be quite
adequate. Furthermore Herschel-Bulkley does not predict ahigh-shear limiting viscosity,
so
it may be unrel able for extrapo sting data.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Parr ~
FRICTION LOSS
SLURRY
PIPES
A new approach to solving this difficult problem gives good results.
Laboratory measurement of slurry viscosity is the starting point
He
is known as the Hedstrom number.
For turbulent flow, a number of empirical or semi-empiri-
cal expressions have be~n proposed in the literature [11, 13].
One of the most popular and widely used expressions is that
of Dodge and Metzner [6]:
(10)
where
An = 4.0/n
’°’75,
Cn = -0.4/n
’1"~-
(11)
~’,~ = K’ [SV/D]
n’
(12a)
n’ = dlog
r,Jd(SV/D)
(12b)
and
Re’ = D
n’In-~’p/ K ’8
"’4 (12c)
If the fluid obeys the power-law model, then
n’=n,
Re’=Rep,
and/!’=m
(3n+l)/4n.
If it doesn’t, then
n’
is
evaluated as the point slope of a log-log plot of ~,v vs
8V/D,
at a value of rw =
-APD/4L.
This requires an iterative
solution for AP, which assumes that viscosity data are
available for a range of shear stresses encompassing those
encountered in turbulent flow (often not the case).
Hanks and Ricks [8] have also presented an analysis of
turbulent flow of a power-law fluid. Their results are ex-
pressed in graphical form.
Darby [3] has presented a set of empirical curve-fit equa-
tions that represent the results both of Dodge and Metzner
and of Hanks and Ricks reasonably well for power-law
fluids. These equations, which encompass the entire range of
laminar, transition and turbulent flow, are
f = fL(1-a)
+
a/(fw
"~
+ f~:R-~)
(13)
wherefL is given by Equation (4) and
fT = O.0682n-WRep
1/(1"~ +
(1,1)
f~R
=
1.79
X 10~
X
exp(-5.24n)Rep
(0.414+0,757n)
(15)
a = 1/1(1 + 4-~), ~ =
Re~-Re¢
(16)
Re~
= 2100 + 875(1- n)
(17)
Re~
is the Reynolds number corresponding to the laminar-
turbulent transition. (Note that a is either zero or one,
except in the near vicinity of
Re = Re¢.)
For Bingham plastics, one of the works referenced most
often is that of Hanks and Dadia [9]. This is a computer
solution based upon a semi-empirical model, the results of
which are given in graphical form with the friction factor as
a function of Reynolds (Re)
and Hedstrom
(He)
numbers.
Darby and Melson [4] have also presented an empirical
curve-fit equation that accurately represents the Hanks and
Dadia results from laminar to full turbulence:
d= (fp + fp)~/~
(18)
Here, f~ is the solution to Equation (8), and
f~ = 10
~ Re~
~.~a
(19)
a =-1.4711 + 0.146exp(-2.9 × 10-~He)]
(20)
and b = 1.7 +
40,O00/Ree (21)
Because the Hanks method frequently overpredicts the value
of f in turbulent flow [14], Equation (20) has been slightly
modified from that given by Darby and Melson [4]. Specifical-
ly, the value 1.47 replaces the original value of 1.38.
Putting the equations to work
The above expressions may be used directly to scale up
laboratory tube-flow data, or data from concentric-cylinder
(cup-and-bob) viscometers, to predict the friction loss in pipe
flow under any flow conditions for a given suspension. If a
concentric-cylinder viscometer is used, plot its data for vis-
cosity vs. shear rate by Equations (1) and (2) to determine the
values of m, n, ro and ~o. It is best to do this on log-log
coordinates for both models, because of the wide range of
values normally encountered. It is important that the proper
correction factors be applied to the measured angular veloci-
ty values to provide values of the true shear rate; for details,
see Reference [5].
If instead laminar tube-flow data for pressure drop versus
flow rate are available, fit those data directly by Equations
(3) and (7) to determine the values of m and n, and ~o and
~ ~. This is most easily done by trial and error, with the help
of a spreadsheet.
Once these rheological parameters have thus been deter-
mined, the pressure loss for that slurry in any pipe at any
flow rate can readily be determined. Proceed as follows:
CHEMICAL ENGINEERING/SEPTEMBER 1992
11’/
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Couette Figure 1
~2,4% Fine
Uminite
and tube flo,w viscos~ty data
1
t"18
CHEMICAL ENGINEERING[SEPTEMBER1992
MEGAN REID[
FEATURe: R~:PORT
FIGURES 1,2. Laboratory data (inset) on a limlnite suspension
lead to flow predictions that match observed behavior well
FIGURES 3.5, Unlike the studies described In Figures 1 and 2,
those shown here (made with an iron oxide suspension) employ
observations that have been made with two different pipe sizes
(not one) to test and demonstrate the prediction methodology
If you are using the power4aw model, calculate the Reyn-
olds number from Equation (6), then insert that value into
(14) and (15) to findfw andJ~,R. The value of
Rec
is calculated
from Equation (17), which determines a via (16). Finally, use
(13) to find f and insert that value into (5) to determine the
pressure loss.
With the Bingham model, use the theo!ogical parameters
to calculate the Reynolds and Hedstrom numbers from
Equations (9). Then employ (20) and (21) to find a and b. Next,
apply iteration to Equation (8) to obtain fL, and use (19) to
give fT. The friction factor is then determined from (18),
which is fed into (5) to yield the pressure loss.
Some examples
Figure 1 shows viscosity datafrom Mun [11] for a suspension
containing 52.4 wt.% fine liminite (hydrated iron oxide) solids.
These data were obtained from both a concentric-cylinder
viscometer and a laminar-tube-flow apparatus. The curves
show the
"best
fit" of both the power-law and Bingham
plastics models. It is evident that the Bingham model gives a
better fit, although the power-law fit is reasonable.
Figure 2, based on measurements with this slurry from
laminar through transition flow in a 0.0103-m-dia. tube, demon-
strates the validity of the overall procedure by comparing the
observed pressure drop with that predicted via the power-law
model and Equation (13), as well as with that predicted via the
Bingham plastic model and Equation (18). The pressure drops
are expressed in terms of APD/4L, the flowrates as 8V/D.
Figure 3 shows viscosity dat~ from Cheng and Whittaker [2]
for an 18% iron oxide suspension, obtained with a concentric
cylinder viscometer. Also shown are the "best fit" power-law
and Bingham plastic curves, with again the better fit being
provided by the Bingham plastic.
Figm’e 4 shows the flow data for this suspension in a 0.079-
m-dia, tube, from laminar through turbulent flow, as well as
the predictions from the power-law and Bingham plastic
models. As seen, both models predict essentially the same
pressure drop in turbulent flow, in reasonable agreement with
the data.
Figure 5 shows the data for this same suspension in a 0.053-
m tube, and the model predictions. The prediction based upon
the Bingham model agrees with the data, whereas the power-
law-model prediction is about 20% low. This may well be
because the Bingham model provides a better fit of the
viscosity data and is more reliable for extrapolation to higher
shear rates, and because the data in Figure 5 cover a shear and
Reynolds number
(Rep)
range up to three times greater than
that in Figure 4.
Yet another case (not shown here in diagrams), involving
-21.4% bauxite tailings [1], should ~be noted because data points
for this slurry in turbulent flow are available from commercial
pipelines, of 0.335-m and 0.303-m diameters. In the larger pipe,
the measured pressure gradient at a mass flow rate of 6,000
dry tons/d is 270 Pa/m, while the value predicted by Eq. (18) for
the Bingh~:m plastic model is 252 Pa/m. For the smaller pipe at
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Pad ~
the same flow rate, the measured gradient is 380 Pa/m, while
the power-law and Bingham-plastic predictions are 349 and 377
Pa/m, respectively.
The table lists actual (measured) properties of the three
slurries discussed in this article, plus two others. It also shows
the range of flow conditions (e.g., Reynolds number) for the
pipe-flow data from the literature that were used to test the
equations.
Sunnning
up
The method outlined here has further been evaluated by
comparison with literature data for other suspensions. The
studies cover a variety of different materials, with a substan-
tial range of properties, in a wide range of pipe sizes and flow
rates.
With the power-law model, the turbulent-flow predictions
are at worst within about 20%, and in most cases much better.
And the predictions using the Bingham-plastic-based equation
generally lie within the range of scatter of the data.
The authors wish to thank Mr. Andrew Mims and the
Comalco Co. of Weipa, Australia for providing the turbulent-
pipeline-flow data for the bauxite-tailings suspension. .
Edited by Nicholas P. Chopey
References for Part
1
1. Alien, T. "Particle Size Measurement," 3rd ed., Chapman and Hall,
London, 1981.
2. Brown N. P., An Instrument to Measure Chord-averaged Concentration
Profi es in Two-phase Pipeline Flows.
JPipelines,
7, pp. 177-89 (1988).
3. Cheng, D. C-H., A Review of On-line Rheological Measurement,
Food
Sci. Technol. Today, ~
(4}, pp. 242-9 (1990).
4. Cheng, D. C-H., others, "Status Report on Process Control Viscometers’.
Current Applications and Future Needs," Warren Spring Laboratory,
Stevenage, UK, 1985.
5. Dealv, J. M. (1984), Viscometers for On-line Measurement and Control,
Che~. E~tg.,
pp. 62-70, 1 Oct., 1984.
6. Johnsen, H. K., others Development and Field Testing of a High-
accuracy Full-bore Return Flow Meter, presented at Pro IADC/SPE
Driling Conference, Texas, Feb. 28 - blar. 2, 1988.
7. Heywood N. I, and Mehta, K., A Survey of Non-invasive Flowmeters for
Pipeline Flow of High Concentration Non-settling Slurries, "Proc. of
Hydrotransport lr’,-Stratford, UK. Paper C2, pp. 131-56 (1988).
8. Kao, D. T., and Kazanskij, I. (1979), On Slurry Flow Velocity and Solid
Concentration MeasuringTechniques, "Proc. 4th Slurry Transport Assn.
Conference," pp. 102-20 (1979).
9. Shook, C. A., Flow Of Stratified Slurries Through a Horizontal Venturi
Meter, Can. J. Chem. Eng.,
60, pp. 342-5 (1982).
10. Shook, C. A. and Masliyah, J. H., Flows of a Slurry Through a Ventari
Meter, Ca~. J. Chem. Eng.,
52, pp. 228-33 (1974).
11.
~V, yatt,. D. G., Electromagnetic Flowmeter Sensitivity with Two-phase
rtow,
~nl. J. MultiphaseFlow,
12, pp. 1009.17 (1986).
References for Part
2
1. Boger, D.V. and Nguyen, Q.D., "Flow Properties of Weipa #3 and #4
Plant Tailings" report prepared for Camalco Aluminum Limited, Weipa,
Austra in, May 1987.
2. Cheng, D. G.H. and Whittaker, W., "Applications of the Warren Spring
Laboratory Pipeline Design Method to Settling Suspensions", Hydro-
transport 2, C 3-21, BHRA, Bedford, England, 1972.
3. Darby, R., Hydrodynamics of Slurries and Suspensions, "Encyclopedia of
Fluid Mechanics," Vol. 5, N.P. Cheremisinoff (Ed.), Ch. 2, pp. 49-91, 1986.
4. Darby, R., and Melson, J.M., A Friction Factor Equation for Bingham
P|astic Sluries and Suspensions for All Flow Regimes,
Chem. Eng.,
pp.
59.61, 28 December, 1981.
5. Darby, R., Couette Viscometer Data Analysis for Fluids With a Yield
Stress, J.
Rheology, 29,
pp. 369-378, 1985.
6. Dodge, D.W. and Metzner, A.B., Turbulent Flow of Non-Newtenian
Systems,
AICHE Journal,
5 (2), pp. 189-204, 1959.
7. Gorier, G.W. and Aziz, K., ’"rhe Flow of Complex Mixtures in Pipes",
Van Nostrand Reinhold, 1972.
8. Hanks, R.W. and Ricks, B.L. Transition and Turbulent Pipe Flow of
Pseudoplast c Fluids , J.
Hydronautics,
9 (39), 1975.
9. Hanks, R.W. and Dadia, B.H., Theoretical Analysis of the Turbulent
Flow of Non-Newtonian Slurries in Pipes,
AIChEJournal,
17 (3), pp. 554-
557, 1971.
10. Metzner, A.B. and Reed, J.C. Flow of Non-Newtonian Fluids - Correla-
tion of the Laminar, Trans tonal and Turbulent Flow Regions,
AIChE
Journal,
1 (4), pp. 434-440, 1955~
11. Mun, R., "The Pipeline Transportation of Suspensions with a Yield
S~css", Thesis, Master of Engineering Science, The University of Mel-
bourne, December 1988.
12. Ngnyen, Q.D. and Boger, D.V., Yield Stress Measurement for Concen-
trated Suspensions, J.
Rheology, 27,
pp. 321-349, 1983.
13. Skelland, A.H.P., "Non-Newtenian Flow and Heat Transfer", Wiley,
New York, 1967.
14. Thomas, A.D., Slurry Pipeline Rheology, "Proc. Second National Conf.
on Rheology," M. Keentek (Ed.), Brit,sh Soc. of Rheology, Australian
Branch, The University of Sydney, 1981. []
The author
Ron Darby is a professor in the chemical engineering department of Texas
A&M U. (College Station, TX 77843-3122; Tel. 409 845-3361), which he joined in
1965 after three yearn as a senior scientist at LTV Research Center, Dallas.
His research has been mainly in transport phenomena, fluid mechanics and
theology of non-Newtonian fluids. He has ~ublished more than 50 papers, as
well as a book on viscoelastic fluids, and ~s the recipient of several honors
related to writing and teaching. A registered engineer in Texas, he belongs to
AIChE, the Soc. of Plastics Engineers, the Soc. of Rheology and the American
Soc. for Engineering Education. He holds baccalaureate degrees as well as a
Ph.D., all in chemical engineering from Rice U., and was a postdoctoral fellow
at Cambridge U.
Robert
Mun is a postgraduate s~udent at Australia’s U. of Melbourne
studying non-Newtonian fluid mechanics with current emphasis on droplet
formation and jet breakup of concentrated colloidal susp~_ nsions. He holds a
B.E. (Chem.) and a M. Eng. Sci. from Melbourne, and has worked as a
research fellow in developing coal-water fuels.
David V. Boger is professor of chemical engineering at the U. of Melbourne,
and deputy director of an Advanced Mineral Products Centre funded by the
Australian Research Council. His research is mainly in non-Newtenian fluid
mechanics. He has been an invited plenary speaker at many international
conferences, but is perhaps best known for the so-called Boger fluids, elastic
fluids of constant viscosity. Last year, he received the Esso Award for
Excellence in Research, at the 19th Australasian Chemical Engineering
Conference. He received a B.S. in chemical engineering from Buckne]t U., and
a Ph.D. in chemical engineering from the U. of Illinois. He is a member of the
Australian Academy of Technological Sciences and Engineering.
TABLE, Parameters here are for measurements that were made in studies related to the models discussed In the text
SUSPENSIONS EVALUATED
Source
Material
P
m
n
Rep
ReB
Reo
He
(kg/m
~)
(Pa)
Mun (1988}
52.4%
<50
2435
8,7
0.27
3O
0.016
0,48 to
29 to
2740!
1,78 X !0’
Fine Llminite
6200 6900
Cheng and
t8%
<50
1t70
0.16
0.48 0.78
0.0045
32to
675 to
2560
1.3 to 2.8
Whiflaker[1972
Iron Oxide
32,000
48,500
XIO
s
Thomas
7,5%
Colloidal
tt03
0,71
0,425
7.5
0,005
12 to
14,3. to
2600
t.7 X t0’ to
(1981)
Kaolin Clay
13,500 85,000
3.6 X t0
=
Cheng and
58%
< 160
1530
0,62
0.42
2.5
0.015
226t0.
1300to
2600
4,4X I0
~
Whlflaker
Limestone
9300
t3.000
Beget and
2t.4%
<200
1163 0.43
0,49
8.5
0.004t
34,500
3,4 X I0
~
2546
6.6 X 107
Nguyen [1987]
Bauxite
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