Non Newtonian Fluid Flow in Ducts

2nd Mercosur Congress on Chemical Engineering

4th Mercosur Congress on Process Systems Engineering

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NON-NEWTONIAN FLUID FLOW IN DUCTS: FRICTION FACTOR

AND LOSS COEFICIENTS Adelson Belizário Leal, Luis Américo Calçada, Cláudia Mirim Scheid*

Departamento de Engenharia Química - Universidade Federal Rural do Rio de Janeiro Abstract. The knowledge of the head loss in the flow of non-Newtonian fluids is very important for the execution of pipelines and pumping systems designs, common in plants of almost all kinds of industries. The determination of the total head loss involves the establishment of the friction factor corresponding to pressure drop in the straight section and the loss coefficients caused by each fittings and valves existent in piping systems. In this context, the head loss in globe valve, sudden contraction, 90 degree standard elbow and straight smooth circular pipe was studied. The fluids used in the tests were aqueous solutions of carboxymethylcellulose (CMC) and xanthan gum, in different concentrations, flowing in fully-developed turbulent flow. The rheology was determined by a capillary rheometer, in the same temperature condition of the head loss experiments. All solutions employed in this work showed non-Newtonian and pseudoplastic behavior. The rheological data were adjusted for the Ostwald-de Waele model and the correlation coefficients were above 0.99. The experimental data of friction factors were compared with predictive correlations proved in literature. On the other hand, the experimental data of loss coefficients were compared to that obtained to water flow and non-Newtonian flow those data available in literature.

Keywords: Fittings, pressure drop and valve.

1. Introduction The non-Newtonian fluids, unlike the Newtonian, are defined as materials which do not conform to a direct

proportionality between shear stress and shear rate. Among different models to represent this relationship, the

Ostwald-of Waele model, though empirical, represents a great many non-Newtonian fluids given by,

nkγ=τ , (1)

where τ e γ are shear stress and shear rate respectively. While n is flow behavior index and k is fluid consistency

index, both rheological parameters.

Due to the necessity of the knowledge of the head loss in the execution designs of piping systems and

pumping, the study of the flow of fluids non-Newtonian in ducts is very important for engineering. These fluids

are present in many processes of the plants of almost all of the types of industries.

The total head loss, ht, caused by some piping system consists of the sum of losses regarding to the straight

pipe section, hd, and the losses regarding to the several fittings and valves present in this system, hs. In the case

of a steady and isothermal flow of the incompressible fluid, without pump and no work involved, the

* To whom all correspondence should be addressed. Address: LSP/DEQ/IT, UFRRJ, BR-465, Km 07, 23890-000 Seropédica – Rio de Janeiro–Brazil E-mail: [email protected]



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total head loss can be given by the macroscopic mechanical energy balance applied to two points any of the

piping (Bird et al, 1960), is given by,

21

22

2121

t zzg2

vvgpp

h −+−

+ρ−

= , (2)

where p1 and p2, v1 and v2, z1 and z2, are pressures, average velocities, the heights at points 1 and 2 respectively.

ρ is the fluid density and g is the gravitational acceleration.

1.1. Friction factor

The turbulent flow in straight cylindrical duct of length L and diameter D, lying in a horizontal plane so that

z1=z2 and v1=v2, the eq. (2) can be simplified to the equation given by

gP

gpphh 21

dt ρ∆

=ρ

== − , (3)

where ∆P is the static pressure difference between 1 and 2. The dimensional analysis of the fully turbulent flow

in straight cylindrical tube leads to the definition of the Fanning friction factor, f, given by,

2v2D

LPf

ρ

∆= . (4)

It is common to find in the literature the definition of Darcy friction factor; relationship by fD=4f.

An important review about friction factor was made by Kemblowski and Kolodziejski (1973) and Coelho

(1982), in which it is shown the three trends followed by the researchers in the attempt of correlating the factor

friction in the turbulent flow of non-Newtonian fluids. The main correlations mentioned by these authors are

presented in the table 1, with their respective definitions of Reynolds number and range of behavior index. All

the equations, except the eq. (8), use the Fanning friction factor definition given by the eq. (4).

The first trend, which maybe the oldest, uses the equations developed for the turbulent flow of Newtonian

fluids to correlate the flow fluids of non-Newtonian, using several viscosity definitions in the expression for the

Reynolds number. The second correlates the experimental data of friction factor and Reynolds number to the

equations of the type Blasius, given by,

bReaf = , (9)

where a and b are functions of the rheological properties fluids. For Newtonian fluids, a is 0.079 and b is -0.25.

The correlation proposed by Shaver and Merrill (1959), eq. (5), is an exemple of this trend and was developed



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with base in the experimental data obtained in the turbulent flow of aqueous solutions of CMC, carbopol and

polyisobutylene in cyclohexeno.

The last of those trends makes use of the "Prandtl law" to predict the friction factor. Equations of this type

have the general form given by,

( ) CfRelogAf1 B += , (10)

where the parameters A, B and C are functions of the rheological properties of the fluid. The correlations given

by the equations 6, 7, 8 are considered the most important for this case.

Dodge and Metzner (1959), through a theoretical and experimental study using solutions of CMC, carbopol

and clay, presented a correlation in which the hypothesis of the Newtonian turbulent flow was extended to non-

Newtonian turbulent fluid flow. In the special case of fluids that follow the model of Ostwald-of Waele, this

correlation is given by the eq. (6), where the Reynolds number used was defined by Metzner and Reed (1955)

with base in the laminar flow. The deviation observed is this case were smaller than ±2,5%, except for CMC

solutions that present viscoelastic behavior.

Clapp (1961), working with the flow of pseudoplastic fluids, presented, as well as Dodge and Metzner

(1959), a correlation where the parameters A, B and C are functions of the rheological properties of the fluid.

The correlation of Clapp is given by the eq. (7) with a maximum deviation of ± 4% in the range worked

(Skelland, 1967).

Tomita (1959) proposed a correlation for the turbulent friction factor based on the similarity criteria and

Prandtl’s mixing length theories (Skelland, 1967). Unlike Dodge and Metzner (1959) and Clapp (1961), the

correlation proposed by Tomita presents the parameters A, B and C independent of the rheology of the fluid and

another definition for friction factor. This definition for friction factor consists of an extension of the Fanning

friction factor. In the case of fluids that obeyed the Ostwald-of Waele model, Tomita correlation is given by the

eq. (8) and Tomita friction factor corresponding is defined by

⎟⎠⎞

⎜⎝⎛

++

ρ∆

=1n31n2

vL3PD2f 2T . (11)

It is valid to remind that the correlations of Dodge and Metzner, eq. (7), and Tonita, eq. (9) assume, when n

equal to the unit (Newtonian fluids), the well-known expression of Karman-Nikuradse given by,

( ) 4,0fRelog0,4f1

−= . (12)



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Table 1. Correlations for turbulent flow of non-Newtonian fluids

Author n Reynolds number Correlation Shaver and

Merril (1959)

0.5–1.0 n

1n

n2n

SM 1n3n4

k8vDRe ⎟

⎠⎞

⎜⎝⎛

+ρ

= −

−

( ) n5.1063.2

SM5 Ren

079,0f = (5)

Dodge and Metzner (1959)

0.4–1.0 n

1n

n2n

MR 1n3n4

k8vDRe ⎟

⎠⎞

⎜⎝⎛

+ρ

= −

−

2.12

n2

MR75.0 n4.0fRelog

n0.4

f1

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

− (6)

Clapp (1961) 0.7–0.8

k8vDRe 1n

n2n

Cl −

− ρ=

n75.2n45.0fRelog

n53.4

f1 2

n2

Cl−

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

− (7)

Tomita (1959) 0.2–0.9

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ +

ρ=

−

−

n1n22

n1n36

kvDRe

n

n1

n2n

T ( ) 4.0fRelog0.4

f1

TTT

−= (8)

2.2. Loss Coefficient

The head loss in fittings and valves, hs, is resulted of the wall friction, of changes in the cross-section and/or

direction of the flow and is expressed in function of the loss coefficients K, given by,

g2vKh

2

s = . (13)

The loss coefficient, K, is characteristic of each fitting and valve type while, hs, it can be obtained through

the eq. (2). However, it should be taken into account the particularities of each fitting and valve during the

simplifications of this equation, for example: the variation in the cross-section flow and the heights. It is valid to

observe that in the case of the fittings, as a sudden contraction, in which there is variation in the average velocity

before and after the fitting, the loss coefficient is related with the velocity regarding to the smallest diameter.

The Crane Company (1976) provided a very extensive tabulation of loss coefficients for turbulent flow

thorough various types and sizes of fitting and valves for Newtonian fluids. In the case of non-Newtonian fluids,

Turian et al. (1998) studied the pressure drop in different types and sizes of fittings, valves and venturi meters

for the flow of non-Newtonian slurries. These authors obtained loss coefficients similar to the flow of water for

majority of the fittings and valves. Already Etmad (2004), accomplishing similar study with aqueous solutions

of the CMC, obtained loss coefficients dependent of the rheology of the non-Newtonian fluid.

In this context, this work seeks to test the correlations of friction factor and to compare the loss coefficients

existent in the literature to that obtained experimentally, all in turbulent regime. For such an objective, polymeric

solutions of xanthan gum and CMC were used in different concentrations, flowing through smooth cylindrical

tube and two basic types of fittings (90° elbow and 1-3/4 sudden contraction ) and one globe valve.



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2. Materials and Methods

The experimental apparatus used for the tests is showed in figure 1. It consists of a reservoir tank, with 500

liters of capacity, a centrifugal pump of 1.5 HP (model CAM W-6C made by Dancor, Brazil) and 6 m of pipe

system in a horizontal plane working in loop. The friction factor was studied in a section of length 387cm with

2.65 cm of internal diameter. The friction loss coefficient was determined for globe valve and 90 degree

standard, elbow with 1” diameter both and a sudden contraction 1” to ¾”. All the parts, except valve were made

of PVC. The pressure drop in the fittings and the straight section was obtained by tube in U manometer in

turbulent flow without “end effects”. The pressure drops in the fittings and the straight section were measured

by means of U-tube manometers containing mercury and tetrachloromethane. The flow rate was calculated using

a measuring cylinder and stop watch. For the tests, aqueous solutions of carboxymethylcellulose (CMC) and

xanthan gum in different concentrations were used.

The rheological behaviors of the solutions were obtained by the capillary rheometer in the average

temperature and in the same range of shear rate of each experiment. All the fluids presented non-Newtonian and

pseudoplastic behavior allowing the adjustment for the model of Ostwald-of Waele with coefficients of

correlation over than 0.99 for all worked concentrations. The results of the rheology are presented in table 2.

Table 2. Concentration, temperature, rheological parameters and density of the polymeric solutions

n: Flow behavior index; k: Fluid consistency index; γ: Shear rate; T: Temperature and ρ: Fluid density.

Solutions % (p/v)

T (°C) n k

(dyna sn /cm2) γ

(s-1) ρ

(g/cm3) 26.0 0.965 0.0144 100 - 2044 1.01

0.010 28.0 0.979 0.0121 110 - 2028 1.01 26.0 0.907 0.0219 85 - 1739 1.01

0.012 28.5 0.906 0.0215 88 - 2052 1.01

26.9 0.786 0.0590 45 - 2012 1.01 0.020

28.5 0.799 0.0520 52 - 1720 1.01

0040 25.4 0.579 0.3185 27 - 1063 1.01 24.5 0.399 1.8490 2 - 496 1.01

XG

0.100 26.5 0.397 1.8680 2 - 485 1.01 28.0 0.921 0.0227 81 - 1527 1.00

0.100 30.0 0.933 0.0206 85 - 1570 1.00

0.150 31.0 0.895 00314 64 - 1544 1.00 0.200 26.0 0.858 0.0521 43 - 1198 1.00 0.250 27.5 0.840 0.0645 36 - 1100 1.00 0.400 29.0 0.832 0.0869 25 - 818 1.00 0.500 29.5 0.817 0.1109 20 - 682 1.00 0.900 30.0 0.830 0.1964 6 - 251 1.00 1.000 32.7 0.825 0.2404 5 - 209 1.00

CMC

1.350 32.9 0.816 0.3160 3 - 158 1.00



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Fig. 1. Schematic of the experimental apparatus.

3. Results and Discussions

In order to check the validity of the friction factor and loss coefficients data obtained in the unit of flow tests

were done using water as standard fluid. The friction factors and the loss coefficients were evaluated throught

the eq. (4) and eq. (13) and the data of ∆P e v obtained experimentally. The deviations for the friction factor

were always smaller than 5% in module for every range of Reynolds studied, indicating that the unit of test can

be used in the determination of the friction factors for other fluids. In case of loss coefficients, the table 3 shows

the values provided by the references and obtained in the experimental unit, with their respective standard

deviations and correlation coefficients, for all the fittings and valve studied. The deviations differences observed

can be explained by the different geometric configurations of the fittings employed in the determination of the

loss coefficients while the low standard deviations and the correlation coefficients near the unit, confirm the

capacity of the experimental unit to supply data for other fluids.

Table 3. Comparison of Newtonian experimental data with available results in the literature

Experimental Crane (1976)

Turian (1998)

Fittings and

Valve K σ R2 K K

Globe valve 10.40 0.10 0.99 7.82 10.00±0,75

90º Elbow 1.33 0.05 0.99 0.69 1.11±0,163 1-3/4 Sudden

contraction 0.41 0.03 0.99 0.20 -

K: Average loss coefficients, R2: Correlation coefficients and σ: Standard deviation.

90º Elbow

50

Manovacuumeter

50 50

387

50

Manometer

V=500L

Centrifugal Pump

50 50

Globe Valve

12080

*Distances are in cm.

Straight duct

Sudden contraction



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The friction factors obtained experimentally through the eq. (4) were compared with the friction factors

predicted by the eq. (5), eq. (6), eq. (7). and eq. (8), with the purpose of evaluating the performance of each

correlation in turbulent flow, ReMR > 4000.

The figure 2a shows that the deviations between the experimental and predicted friction factors by the

correlation of Shaver and Merrill (1959), eq. (5), are less than 15% in module for the solutions of CMC. In the

case of xanthan gum solutions, the deviations vary among -15% for solutions with n=0.965 to value larger than

+90% for solutions with n=0.399. It is pertinent to remind that Shaver and Merrill (1959) observed deviations

among +33% to -15% for solutions with 0.53≤n≤1.0 and it is not indicated the use of their correlation to

solutions with n <0.40.

The performance of correlation of Dodge and Metzner (1959), eq. (6), is presented in the figure 2b. It can be

observed in this figure that the predicted friction factors are larger than obtained experimentally, for all of the

solutions of CMC and xanthan gum studied. In the case of the xanthan gum, the deviations were between -35%

and -15% for the solutions with n=0.399 and n=0.965 respectively. Taking into account the case of CMC flow

with 0.81<n<0.92, the deviations between the experimental and predicted by the eq. (6) were smaller than -10%.

The figure 2c shows the deviations between the experimental friction factors and predicted by correlation of

Clapp (1961), given by the eq. (7). It is pertinent to point out that this correlation is valid for 0.698≤n≤0.813, but

it was used to predict frictions factors of xanthan gum and CMC solutions with 0.399≤n≤0.965 showing

deviations which are among +5% and-20%.

The figure 2d shows the deviations between the experimental and predicted friction factors by the correlation

of Tomita (1959), eq. (8). This equation overestimates the frictions factors for all the solutions. In the case of

CMC solutions, the deviations are among -10% to -20% while for the xanthan gum solutions the deviations were

between -20% and -70%.

The table 4 shows the results of the statistical treatment applied to the experimental friction factors

(totalizing 151 points) and calculated by the equations 5, 6, 7 and 8. It can be observed in this table that the

correlation of Clapp (1961), eq. (7) is the best one while the correlation of Shaver and Merrill (1959), eq. (5),

supplies the worst results for the friction factor. In this table, y is average, σ is standard deviation and DMA is

average absolute deviation.

Table 4. Statistical treatment for several correlations

Statistical Shaver and Merrill

Dodge and Metzner Clapp Tomita Equation

y 1.49 0.89 0.93 0.79 ∑−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

1i i.pred

.exp

ff

n1y

σ 1.66 0.09 0.06 0.14

5.0n

1i i.pred

.exp yff

1n1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−=σ ∑

=

DMA (%) 15.98 13.66 8.66 33.26

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −= ∑

=

n

1i .exp

.pred.exp

fff

n100DMA



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0.000 0.002 0.004 0.006 0.008 0.010 0.0120.000

0.002

0.004

0.006

0.008

0.010

0.012+10%

+20%+40%+90%

-10%

f Exp

erim

enta

l

fPredicted-Shaver and Merrill

n=0.816 -CMC n=0.399 -XG n=0.817 -CMC n=0.579 -XG n=0.825 -CMC n=0.773 -XG n=0.830 -CMC n=0.907 -XG n=0.832 -CMC n=0.965 -XG n=0.840 -CMC n=0.858 -CMC n=0.895 -CMC n=0.921 -CMC

(a)

0.002 0.004 0.006 0.008 0.010

0.002

0.004

0.006

0.008

0.010

-10%

-20%

-30%

+10%

f Exp

erim

enta

l

fPredicted - Dodge and Metzner

n=0.816 - CMC n=0.399 - XG n=0.817 - CMC n=0.579 - XG n=0.825 - CMC n=0.773 - XG n=0.830 - CMC n=0.907 - XG n=0.832 - CMC n=0.965 - XG n=0.840 - CMC n=0.858 - CMC n=0.895 - CMC n=0.921 - CMC

(b)

0.002 0.004 0.006 0.008 0.010 0.012

0.002

0.004

0.006

0.008

0.010

0.012

-20%

-10%

+10%

f Exp

erim

enta

l

fPredicted-Clapp

n=0.816 - CMC n=0.399 - XG n=0.817 - CMC n=0.579 - XG n=0.825 - CMC n=0.773 - XG n=0.830 - CMC n=0.907 - XG n=0.832 - CMC n=0.965 - XG n=0.840 - CMC n=0.858 - CMC n=0.895 - CMC n=0.921 - CMC

(c)

0.002 0.004 0.006 0.008 0.010 0.012

0.002

0.004

0.006

0.008

0.010

0.012 n=0.816 - CMC n=0.399 - XG n=0.817 - CMC n=0.579 - XG n=0.825 - CMC n=0.773 - XG n=0.830 - CMC n=0.907 - XG n=0.832 - CMC n=0.965 - XG n=0.840 - CMC n=0.858 - CMC n=0.895 - CMC n=0.921 - CMC

-20%

-30%

-50%

-70%

-10%

f Exp

erim

enta

l

fPredicted-Tomita

(d)

Fig. 2. Comparison of experimental friction factor and the factor predicted by eq. (5), (6), (7) e (8).

The curves of hs versus v2/2g for elbow 90°, globe valve and sudden contraction are presented in the figures

3a, 3b and 3c respectively, for all the solutions. In these plotters the slope represents the loss coefficients of each

fitting and it can be observed that the rheological behavior of the fluid is not important in the loss coefficient

results. For the globe valve and the elbow 90°, the dispersion of the points was smaller than in the sudden

contraction presented deviations smaller than 5%. For sudden contraction this deviations were in maximum

15%. The table 5 presents a summary of the medium loss coefficients for non-Newtonian fluids and water with

their respective deviations.



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0 10 20 30 40 50 60 70 800

20

40

60

80

100

120 n=0.816 - CMC n=0.397 - XG n=0.817 - CMC n=0.579 - XG n=0.825 - CMC n=0.786 - XG n=0.830 - CMC n=0.906 - XG n=0.832 - CMC n=0.979 - XG n=0.840 - CMC n=0.858 - CMC n=0.895 - CMC n=0.933 - CMC

h s (cm

)

v2/2g (cm)

(a)

0 10 20 30 40 50 60 700

100

200

300

400

500

600

700

800 n=0.817 - CMC n=0.397 - XG n=0.825 - CMC n=0.579 - XG n=0.830 - CMC n=0.799 - XG n=0.832 - CMC n=0.906 - XG n=0.840 - CMC n=0.979 - XG n=0.858 - CMC n=0.895 - CMC n=0.933 - CMC

h s (cm

)

v2/2g (cm)

(b)

0 20 40 60 80 100 120 140 160 180 200 2200

20

40

60

80

100

120 n=0.816 - CMC n=0.397 - XG n=0.817 - CMC n=0.579 - XG n=0.825 - CMC n=0.786 - XG n=0.830 - CMC n=0.906 - XG n=0.832 - CMC n=0.979 - XG n=0.840 - CMC n=0.858 - CMC n=0.895 - CMC n=0.933 - CMC

h s(cm

)

v2/2g(cm)

(c)

Fig. 3. Loss coefficients for: a: 90° elbow, b: globe valve and c: 1-3/4” sudden contraction.

Table 5. Mean losses coefficients for turbulent flow of xanthan gum and CMC

XG CMC XG - CMC Water Fittings and

Valve K σ K σ K σ K σ

Globe valve 10.95 0.40 10.53 0.420 10.67 0.457 10.40 0.10

90º Elbow 1.42 0.071 1.42 0.067 1.42 0.067 1.33 0.05

1-3/4 Sudden

contraction 0.49 0.062 0.43 0.051 0.45 0.062 0.41 0.03

K: Average loss coefficients and σ: Standard deviation.



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4. Conclusions

For the studied non-Newtonian solutions, it can be concluded that the correlations of Clapp (1961), Dodge

and Metzner (1959) and Tomita (1959), in the average, overestimate the friction factor. While the opposite

occurs with the correlation proposed for Shaver and Merrill (1959). It is important to point out that the

correlation of Clapp (1961) was the one that presented better performance and Shaver and Merrill (1959) is the

worst. In relation to the loss coefficients, it can be concluded that these do not depend on the rheological

behavior of the fluid, what it indicates that the defined loss coefficients for Newtonian fluids can be used in the

calculation of pressure drop for the non-Newtonian fluid flow.

References

Bird, R. B., Stewart, W. E., Lightfoot, E. N. (1960). Transport Phenomena, John Wiley & Sons, NY.

Clapp, R. M. (1961). International Developments in Heat Transfer, Part III, 652-61; D-159: D-211-5. A.S.M.E., NY. In

Skelland, A. h. P. (1967). Non-Newtonian Flow and Heat Transfer. John Wiley & Sons, NY.

Coelho, G. L. V., (1982). Reologia e Escoamento Turbulento de Suspensões de Minério de Ferro, Thesis de M. Sc.,

Campinas, Brasil.

Crane Co. (1976). Flow of Fluids Through Valves, Fittings and Pipe. Tech. Paper No. 410, 16th printing. Crane Co., 300 Park

Avenue, NY.

Dodge, D. W., Metzner, A. B. (1959). Turbulent Flow of non-Newtonian Systems. A.I.Ch.E. Journal, 5, 191.

Etmad, S. Gh. (2004). Turbulent Flow Friction Loss Coefficients of fittings for Purely Viscous non-Newtonian Fluids. Int.

Comm. Heat Mass Transfer, 31, 763.

Kemblowski, Z., Kolodziejski, J. (1973). Flow Resistances of non-Newtonian Fluids in Transitional and Turbulent Flow. Int.

Chem. Eng., 13, 265.

Metzner, A. B., Reed, J. C. (1955). Flow of non-Newtonian Fluids-Correlation of the Laminar, Transition and Turbulent-

Flow Regions. A.I.Ch.E. Journal, 1, 434.

Shaver, R. G., Merrill, E. W. (1959). Turbulent Flow of Pseudoplastic Polymer Solutions in Straight Cylindrical Tubes.

A.I.Ch.E. Journal, 5, 181.

Skelland, A. h. P. (1967). Non-Newtonian Flow and Heat Transfer. John Wiley & Sons, NY.

Tomita, Y. (1959). A study on non-Newtonian Flow in Pipe Lines. Bulletin of J.S.M.E, 2, 10.

Turian, R. M., Ma, T. W., Hsu, F. L. G., Sung, M. D. J., Plackmann, G. W. (1998). Flow of Concentrated non-Newtonian

Slurries: 2. Friction Losses in Bends, Fittings, Valves and Venturi Meters. Int. J. Multiphase Flow, 24, 243.

Acknowledgments The authors acknowledge the financial support of the CAPES.

Documents

Non Newtonian Fluid Flow in Ducts