Comparison of Friction Factor Equations For Non-Newtonian Fluids in A Pipe Flow
Comparison of Friction Factor Equations For Non-Newtonian Fluids in A Pipe Flow
Comparison of Friction Factor Equations For Non-Newtonian Fluids in A Pipe Flow
ABSTRACT
INTRODUCTION
APF -
--- 2fLvZ
P D
The friction factor for laminar flow of Newtonian, power law, Bingham
plastic, and Herschel-Bulkley fluids can be obtained from a single rela-
tionship. However, numerous correlations exist to estimate the friction
factor of these fluids under turbulent flow conditions. Unlike the equa-
tions for Newtonian fluids, the predictions for non-Newtonian fluids may
differ greatly depending on the relationship used (Heywood and Cheng
1981). The purpose of this article is to summarize and compare friction
factor equations for non-Newtonian fluids flowing in pipes.
REVIEW OF LITERATURE
Fluid Models
The flow behavior of many fluid foods and other materials may be
described by the Herschel-Bulkley (H-B) model (Herschel and Bulkley
1926):
r = Kin (3)
7 = pi (5)
Laminar Flow
The friction factor for the laminar flow of Herschel-Bulkley fluids was
presented by Hanks (1978):
COMPARISON OF FRICTION FACTOR EQUATIONS 95
and
+
R e = 2 H e (1 A
3n ) 2
(i)
where
For power law and Newtonian fluids, the friction factor can be estimated
directly from Eq. (6) since f o = 0 and rG. = 1 when ro = 0. For Bingham
plastic and H-Bfluids, however, f o must be calculated through iteration
of Eq. (10). The range of f o for laminar flow is 1 > f o > f O c where
f o c is the laminar-turbulent transition value of to.Once f o is known, the
friction factor is calculated from Eq. (6),(7)and (8). tOc will be discussed
in more detail later in the paper.
96 E. J. GARCIA and J. F. STEFFE
f=- 0.0791
n5ReP
where
and
COMPARISON OF FRICTION FACTOR EQUATIONS 97
Equation (17) was verified with starch pastes and lime slurries with flow
behavior indexes ranging from 0.178 to 0.952 and RepT from 1.5 X lo3
t o 3 x 104.
Thomas (1960a) suggested different coefficients for the Dodge and Metz-
ner relationship as
Based on the integrated mean velocity of the turbulent core, Clapp (1961)
developed a friction factor relationship and confirmed the equation with
experimental data of Carbopol solutions having flow behavior indexes rang-
ing from 0.698 to 0.8 13 over a Reynolds number range of 5480 to 42,800.
His equation can be considered a special case of the relationship developed
by Torrance (1963) for H-Bfluids which will be presented later. Clapp
(1961) compared his results with the relationship of Dodge and Metzner
(1959) and found that his equation gave a beter representation of experimen-
tal data than Eq. (12). Results for Reynolds numbers below 15000 were
similar; however, Eq. (12) predicts higher friction factors than Clapp’s
equation for higher Reynolds numbers. In addition, Clapp’s relationship
is confirmed for higher Reynolds numbers but Dodge and Metzner (1959)
had few data points above 20,000.
Kemblowski and Kolodziejski (1973) derived an empirical equation based
on experimental data of aqueous suspensions of keolin with flow behavior
indexes ranging from 0.14 to 0.83 and generalized Reynolds number from
2680 to 98,600. These authors found that the relationship of Dodge and
Metzner (1959) under predicted their experimental results. They observed
that the friction factor curves for non-Newtonian fluids were below the
Newtonian curves for Reynolds numbers greater than Re tC = 3000 but
approached the Newtonian values as the Reynolds numbers increased. They
defined this as the transitional flow region with the upper Reynolds number
given by
x 104
Ret = 3.16n0.43S (19)
98 E. J . GARCIA and J. F. STEFFE
For Rec < Re < Ra, their relationship for the friction factor is
where
A = exp n4'2)
1
E = 8.9 X 10-3exp(3.57n2) (22)
and
For Re > Re, they used a Blasius equation for Newtonian fluids:
Hanks and Ricks (1975) derived a friction factor relationship for the
flow of power law fluids under transitional and turbulent flow conditions.
Their model correlated well with the experimental data from Dodge and
Metzner (1959), Shaver and Merrill (1959) and Clapp (1961). The rela-
tionship is a special case of the model of Hanks (1978) for H-B fluids which
will be presented later.
Szilas et al. (1981) analytically determined a friction factor relation-
ship which included the effect of relative roughness of the pipe. They
reported their relationship to have the best fit compared to the relation-
ship of Dodge and Metzner (1959), Shaver and Merrill (1959), Tomita
(1959) and Clapp (1961) for experimental results with crude oil flowing
with a generalized Reynolds number ranging from 104 to lo5, and flow
behavior indexes of 0.6 and 0.9. Their relationship predicts increased fric-
tion factors as the relative roughness ratio increases. For smooth walls,
their friction factor relationship is
COMPARISON OF FRICTION FACTOR EQUATIONS 99
and
lo in Eq. (26) and (27) is defined by Eq. (9) and is estimated as if the
flow was laminar, i.e., from Eq. (10) with n = 1. Equation (28) was con-
firmed with slurries over a ReBT range of 2 X lo3 to 1 X lo5 and a He
range [as defined by Eq. (ll)] of 1.46 x 104 to 5.51 X lo5.
Dodge and Metzner (1959) suggested that their relationship for power
law fluids [Eq. (12)] could be used as an approximation to predict friction
factors for Bingham plastic fluids if n and K are placed by n ' and K ', respec-
tively. The latter parameters are obtained from a logarithmic plot of T~
versus 8v/D where n ' is the slope of the line and K ' is the intercept on
the ordinate at the point where 8v/D equal unity. n' and K' should be
evaluated at the wall shear stress for the prevailing turbulent flow condi-
tions. Dodge and Metzner (1959) successfully correlated turbulent flow
data for clay suspensions by this technique. However, the validity of this
approach has been questioned by several authors (Hanks and Ricks 1975;
100 E. J. GARCIA and J . F. STEFFE
Govier and Axis 1972). Heywood and Cheng (1981) presented an assess-
ment of this procedure as well as other methods to estimate friction factors.
Thomas (1960b, 1962, 1963) developed a Blasius type equation based
on the turbulent flow of Bingham plastics: flocculated suspensionsof kaolin,
titanium dioxide and thorium oxide in water. His experimental data covers
a He number range of 2.9 X lo3 to 8.3 x lo5 and results showed the
friction factors for these suspensionsltobe below those for Newtonian fluids.
Friction factors approached the Newtonian values as the Reynolds number
increased for yield stresses below 24.0 Pa but tended to diverge from the
Newtonian values for yield stress values greater than 24.0 Pa. The rela-
tionship of Thomas cannot be written in terms of the He number for the
purpose of comparison; hence, it will not be considered further.
A semi-theoretical analysis for Bingham plastic fluids was presented by
Hanks and Dadia (1971). Their relationship can be considered a special
case of the analysis of Hanks (1978). Kenchington (1974) found that the
relationship of Hanks and Dadia (1971) predicted his experimental results
better than the correlations of Dodge and Metzner (1959), Tomita (1959)
and Kemblowski and Kolodziejski (1973). Darby and Melson (1981)
developed an expression to approximate the friction factor relationship
of Hanks and Dadia (197 1).
-$- = 0.45 -
Then Eq. (29) and (30) give f as a function of n, Re and He. Notice
that Eq. (30) is only a manipulation of dimensionless numbers and is valid
COMPARISON OF FRICTION FACTOR EQUATIONS 101
for any flow conditions. If f is substituted into Eq. (30) using Eq. (6),
Eq. (10) for laminar flow is obtained.
Equation (29) reduces to the relationship of Clapp (1961) for power law
fluids for to = 0. Notice also that this equation reduces to the Bingham
plastic model for n = 1. Equation (29) has not been verified experimen-
tally for Bingham plastic or H-B fluids. Torrance (1963) also developed
an equation to account for turbulent flow in rough pipes and further infor-
mation can be found in Torrance (1963) and Govier and Aziz (1972).
The Dodge and Metzner (1959) approach for Bingham plastic fluids
described in the previous section can also be used to approximate friction
factor predictions for H-B fluids in turbulent flow. Heywood (1980) sug-
gested that n’ could be approximated as n’ = d ln(TO + K+G)/d In(+,)
when only rotational viscometric data was available.
Probably the most comprehensive analysis for H-B fluids in turbulent
flow has been presented by Hanks (1978). Unlike the relationships presented
up to now [with the exception of Hanks and Ricks (1975) and Hanks and
Dadia (197l)], Hanks’ analysis deals with transitional flow and includes
the laminar-turbulent transition criterion developed by Hanks and Ricks
(1974). As stated before, this model reduces to the models of Hanks and
Ricks (1975) for power law fluids (to=0) and Hanks and Dadia (1971)
(with some improved modifications) for Bingham plastic fluids. Even
though the model of Hanks (1978) has not been experimental verified,
it has been successfully confirmed for the special cases of Bingham plastic,
power law and Newtonian fluids.
The laminar-turbulent transition criterion developed by Hanks and Ricks
(1974) is
where Re, is the critical Reynolds number, the point where laminar flow
ends. to, is the laminar-turbulent transition value of to and is given im-
plicity by
where 4,is determined from Eq. (7)by replacing to with to,. For Re
< Re,, the flow is laminar and f is estimated from Eq. (6) through (1 1)
with 1 > to > to,.For Re = Re,, the flow is at the critical point and
102 E. J. GARCIA and J . F. STEFFE
the critical friction factor (f,) is estimated from Eq. (6) with 4 = 4 and
Re = Re,. For transitional and turbulent flow, Re > Re,, the friction
factor is obtained from Hanks (1978) relationships which can be expressed
in terms of the generalized Reynolds number:
R-R, (37)
4=JSB
B is an empirical parameter given for H-B fluids as
22 O.OO352He
(38)
B = - [ nI + (1 + 0.W504He)2)
COMPARISON OF FRICTION FACTOR EQUATIONS I03
0.1
0.01
0.031
0.m1
10 10' 10' 10'
Re
0.1
0.01
0.WI
o.Ooo1
10' 10. 10' 10'
Re
0.1 11111 I 1 1 1 1 1 1 1 I 1 1 . 1 .
0.01 r
0.001 :
1
o.oo01
10' 10' 10. 10' 10'
Re
0.01 :
0.1 , 1 1 1 , 1 1 1 ,
0.01 :
\
r
0.001 --
'\
0.WoI
10' 10'
1 . .. 1 ...a
10'
1
10'
Re
,
0
n
0
c
-9 ?
0 0 0
L
110 E. J. GARCIA and J. F. STEFFE
and Dadia (1971) is the only one that deals with the transition from laminar
to turbulent flow. This figure shows the variation of Re, with He (dashed
line) obtained from Eq. (31) and (32). The curves to the left of Re, cor-
respond to the laminar flow region and are obtained from Eq. (6) through
( 1 1). For Fig. (9) and (10) the turbulent curves for the different He values
are extended until they meet the corresponding laminar flow curve. The
interception of these curves could be considered as the laminar-turbulent
transition point. As seen from Fig. (9), the friction factors for Bingham
plastic fluids predicted by the Tomita (1959) relationship are below the
Newtonian values (He = 0). These predictions approach the Newtonian
values as Re increases. Conversely, the Torrance (1963) relationship [Fig.
(lo)] predicts values above the Newtonian numbers. As in the Tomita (1959)
curves, the Bingham plastic curves of Torrance (1963) correlation merges
with the Newtonian curve as Re increases. The relationship of Hanks and
Dadia (1971) [Fig. (13)], on the other hand, predicts friction factors below
the Newtonian values for He numbers below 5 X lo5,similar values for
a He range of 5 x lo5 to 1 x lo6, and higher values for He numbers
greater than 1 X lo6. For He > lo5, the Bingham plastic curves even-
tually merge with the Newtonian curve (black dots) as Re increases. For
lower He values, however, the Bingham plastic friction factor predictions
stay below the Newtonian values following a curve similar to the New-
tonian curve.
Figures (1 1) and (12) show the Torrance (1963) relationship for H-B
fluids with n = 0.5 and n = 0.2, respectively. The laminar flow curves,
obtained from Eq. (6) through (1 l), are also shown and are extended up
to the intersection of the Torrance (1%3) curves for the various He numbers
considered. The Torrance (1963) equation predicts higher friction factors
for H-B fluids than the corresponding power law fluid (He = 0, n # 1).
The H-B curves also merge with the power law curves as Re increases.
Figures 14 and 15 show the friction factor prediction for H-B fluids ac-
cording to the analysis of Hanks (1978) for n = 0.5 and n = 0.2, respec-
tively. The corresponding power law curves (He = 0) are indicated with
black dots. In Fig. (13), the H-B curves fall below their corresponding
power law curves for lower He numbers. As n decreases, however, these
curves start falling above the power law curves. For n = 0.2, for exam-
ple, most of the friction factor predictions for H-B fluids are above the
corresponding power law predictions for the Re range shown. Notice also
that for smaller n and high He values there is an abrupt laminar-turbulent
transition. This also occurs at higher n values but low He numbers. For
high n and He numbers, very extended transitional regimes are observed.
Also the critical Reynolds number increases with increasing He numbers
and has a smaller range as n decreases.
R
v)
COMPARISON OF FRICTION FACTOR EQUATIONS
FIG. 1 I . RELATIONSHIP OF TORRANCE (1963) FOR H-B FLUIDS WITH n = 0.5, EQ. (29) AND (30)
111
I12 E. J. GARCIA and J . F. STEFFE
FIG. 12. RELATIONSHIP OF TORRANCE (1963) FOR H-B FLUIDS WITH n = 0.2,
EQ. (29) AND (30)
Re
FIG. 13. RELATIONSHIP OF HANKS AND DADIA (1971) FOR BINGHAM PLASTIC FLUIDS I
WITH IMPROVED MODIFICATION OF HANKS (1978), EQ. (6) THROUGH ( 1 1 ) AND (31) THROUGH (39)WITH n = 1 t;
1 I4
y1
B
E. J. GARCIA and J. F. STEFFE
t
Y
n
0
0
1"
0)
E
is necessary to select the best relationship. (4). The use of power law or
Newtonian relationships to predict friction factors for fluids having a yield
stress may lead to significant errors.
LIST OF SYMBOLS
A = parameter in the relationship at Kemblowski and Kolod-
ziejski (1973), Eq. (21)
B = empirical wall effect parameter in mixing length theory,
Eq. (38)
D = pipe inside diameter, m
E = parameter in the relationship of Kemblowski and Kolod-
ziejski (1973), Eq. (22)
f = Fanning friction factor, Eq. (1)
fBT = Tomita (1959) friction factor for Bingham plastic fluids,
Eq. (26)
fc = laminar - turbulent transition value of f
fPT = Tomita (1959) friction factor for power law fluids, Eq. (15)
He = generalized Hedstrom number, Eq. (1 1)
K = consistency coefficient, Pa s"
L = pipe length, m
m = parameter in the relationship of Kemblowski and Kolod-
ziejski (1973), Eq. (23)
n = flow behavior index, dimensionless
P = parameter in the relationship of Shaver and Merrill (1959),
Eq. (14)
rw = radius at inside wall of pipe (rw = D/2), m
R = turbulent parameter in the relationship of Hanks (1978),
Eq. (34)
= laminar-turbulent transition value of R
= generalized Reynolds number, Eq. (7)
= Tomita (1959) Reynolds number for Bingham plastic fluids,
Eq. (27)
1 I8 E. J. GARCIA and J. F. STEFFE
Greek Symbols
= rate of shear, s-l
= pressure drop due to friction, Pa
= dimensionless rate of shear
= plastic viscoisty, Pa s
= dimensionless mixing length, Eq. (36)
= Newtonian viscosity, Pa s
= dimensionless radial coordinate, 7 / ~ ~
= dimensionless unsheared plug radius, 70/rw
= laminar-turbulent transition value of to
= fluid density, kg/m3
= yield stress, Pa
= shear stress, Pa
= value of 7 at rw
= parameter in mixing length, Eq. (37)
= laminar flow function, Eq. (8)
= laminar-turbulent transition value of $
COMPARISON OF FRICTION FACTOR EQUATIONS 119
REFERENCES
CHENG, D. C. -H. 1970. A design procedure for pipe line flow of non-
Newtonian dispersed systems. In Proceedings of the First International
Conference on the Hydraulic Transport of Solids in Pipes
(Hydrotransport 1). Sept. 14. Paper J5. p. 52-25 Cranfield, Bedford,
England.
CHENG, D. C. -H. 1975. Pipeline design for non-Newtonian fluids. The
Chem. Eng. (London) 301, 525-528, 532; 302, 587-588, 595.
CLAPP, R. M. 1961. Turbulent heat transfer in pseudoplastic non-
Newtonian fluids. International Development in Heat Transfer. A. S.
M. E., Part 111, Sec. A., p. 652-661.
DARBY, R. and MELSON, J . 1981. How to predict the friction factor
for flow of Bingham plastics. Chem. Eng. 88(26), 59-61.
DODGE, D. W. and METZNER, A. B. 1959. Turbulent flow of non-
Newtonian systems. A. I. Ch. E. J . 5(7), 189-204.
GARCIA, E. J. 1985. Optimum economic tube diameter for pumping
Herschel-Bukley fluids. M. S. Thesis Dept. Food Sci. and Human Nutri-
tion. Michigan State University, East Lansing, MI.
GARCIA, E. J . and STEFFE, J. F. 1986. Optimum economic pipe
diameter for pumping Herschel-Bulkley Fluids in laminar flow. J. Food
Process Engr. 8, 1 17-136.
GOVIER, G. W. and AZIZ, K. 1972. The Flow of Complex Mixtures in
Pipes. Van Nostrand Reinhold Co., New York.
HANKS, R. W. 1978. Low Reynolds number turbulent pipeline flow of
pseudohomogeneous slurries. In Proceedings of the Fifth International
Conference on the Hydraulic Transport of Solids in Pipes
(Hydrotransport 5). May 8- 1 1. Paper C2. p. C2-23 to C2-34. Hannover,
Federal Repuplic of Germany. BHRA Fluid Engineering, Cranfield,
Bedford, England.
HANKS, R. W. and DADIA, B. H. 1971. Theoretical analysis of the tur-
bulent flow of non-Newtonian slurries in pipes. A. I. Ch. E. J . 17(3),
554-557.
HANKS, R. W. and RICKS, B. L. 1974. Laminar-turbulent transition
in flow of pseudoplastic fluids with yield stresses. J. Hydronautics. 8(4),
163-166.
HANKS, R. W. and RICKS, B. L. 1975. Transitional and turbulent
pipeflow of pseudoplastic fluids. J. Hydronautics. 9(l), 39-44.
HERSCHEL, W. H. and BULKLEY, R. 1926. Konsistenzmessungen von
gummi-benzollosungen. Kolloid-Zeitschr. 39, 29 1-300.
I20 E. J. GARCIA and J . F. STEFFE