Turbulent pipe flow of power-law fluids

P e r g a m o n

Int. Comm. Heat Mass Transfer, Vol. 24, No. 7, pp. 977-988, 1997 Copyright © 1997 Elsevier Science Ltd Printed in the USA. All rights reserved

0735-1933/97 $17.00 + .00

P I I S0735-1933(97)00083-3

TURBULENT PIPE FLOW OF P O W E R - L A W FLUIDS

M.R. Malin C H A M Limited, Wimbledon

London SW19 5AU UK

(Communicated by J.P. Hartnett and W.J. Minkowycz)

A B S T R A C T This paper reports on the numerical computation of the turbulent flow of power-law fluids in smooth circular tubes. The turbulence is represented by means of a modified version of an existing two-equation turbulence model. Numerical results are presented for the fully- developed friction factor and velocity profile, and compared with experimental data. The model is shown to produce fairly good agreement with experiment over a wide range of values for the power-law index and generalised Reynolds number. O 1997 Elsevier Science Ltd

Introduction

Flows of non-Newtonian fluids through pipes are relevant in many engineering applications,

especially within the manufacturing, process and wastewater industries. The turbulent flow of these

fluids is less common than laminar flow, but turbulence may be encountered in some situations, e.g.

drilling hydraulics, sewage transport, and applications involving relatively high heat transfer rates. The

present paper will deal with the pipeline flow of time-independent viscous fluids described by the power-

law rheological model, which relates the shear stress to the strain rate via the consistency index K and

the power-law index n. For values of n<i, the fluid is pseudoplastic (shear thinning), and for values of

n>l, it is dilitant (shear thickening). If n= 1 the fluid is Newtonian.

Extensive experimental and theoretical studies of turbulent non-Newtonian pipe flows were carried

out by Metzner and co-workers during the 1950s [I-3]. These workers indicated how the friction factor f

varies with generalised Reynolds number Re [l] in the laminar, transitional and turbulent regimes. For

the fully-developed turbulent flow of power-law fluids, Dodge and Metzner (DM) [3] developed a semi-

977

978 M.R. Malin Vol. 24, No. 7

theoretical expression for the mean velocity profile together with a correlation for f versus Re. The study

revealed that a decreasing n slightly delays transition to higher Re, and that friction factors reduce with

decreasing values of n. Subsequently, several workers [4-7] derived alternative expressions for the mean

velocity profile, and the relative merits and demerits of these have been discussed elsewhere [8-10].

More recently, Hartnett and Kostic [I 1] examined the available turbulent friction-factor correlations, and

found that the DM correlation [3] produced the best agreement with the available measurements. The

transition regime has been considered by Reed and Pilehvari [12] who developed a procedure to calculate

transitional friction factors by combining the laminar and turbulent friction factors of DM [3].

Mohammed et al [13] developed a method for calculating the fully-developed friction factor and mean

velocity profile by numerical integration of the mean-momentum equation. The Reynolds stresses were

modelled by analogy with the power-law model, and by use of Prandtl's hypothesis [14]. Their

calculations showed fairly good agreement with the measurements of Bogue [15] over a wide range of Re

and n, although the model coefficients were evaluated by reference to this data. The planar turbulent flow

of a power-law fluid through a film bearing was calculated numerically by Pierre and Boudet [16], but no

comparisons were made with experimental data. These workers also closed the Reynolds stresses by

analogy with the power-law model, but the eddy viscosity was determined by use of the low-Reynolds-

number two-equation k-e turbulence model of Lam and Bremhorst (LB) [ 17].

The present study also employs the LB low-Re k-e model [17], but the more conventional practice of

determining the eddy viscosity from the linear Boussinesq stress-strain relationship is adopted. Earlier

work [18] has demonstrated that the LB model performs quite well for the calculation of Bingham plastic

fluids in pipelines. This earlier work is extended here to power-law fluids, so as to calculate the frictional

resistance, along with the velocity profile, by solving numerically the basic flow equations for fully-

developed laminar and turbulent flow in smooth-walled tubes. The performance of the model is assessed

by comparing predictions with the available experimental information. It will be shown that the original

LB k-e model exhibits significant quantitative deficiencies with regard to the friction factor at low values

of the power-law index. Therefore, a modification to the viscous damping is proposed which is inactive

for Newtonian fluids, but is shown to improve the predictions for non-Newtonian fluids.

Mathematical Model

For compactness, the Reynolds-averaged mean-flow equations for a steady turbulent incompressible

flow are written as follows:

Vol. 24, No. 7 TURBULENT PIPE FLOW OF POWER-LAW FLUIDS 979

V.(U)=O ; V.(pU®U)=V.(p{v+vdVU)-Vp +V.(p{v+v,l(VU) ~) (1)

where p is the fluid density, p the pressure, U the velocity vector, vt is the turbulent eddy viscosity arising

from closure of the Reynolds stresses through use of the Boussinesq stress-strain relationship, and v is

the apparent kinematic viscosity of the power-law fluid which is given by:

/" 1 .~(n-I)/2 v=(KIp)['~(A:A) I ; A = 0 . 5 [ V U + (VU) T ] (2)

where K is the consistency index, n is the power-law index, A is the deformation tensor, and the

superscript T denotes that the transpose of the dyadic VU is taken. For pseudoplastic fluids, v decreases

with increasing deformation rate, whereas for dilitant fluids it increases with increasing deformation rate.

The eddy viscosity v, is determined from the LB k-e model [17], which uses transport equations for

the turbulent kinetic energy k and its rate of dissipation e. This model calculates vt from:

v, = c,,f , ,k ~ / e (3)

where Ca--0.09 and f~ is a damping function, to be defined below. The turbulence parameters k and e are

calculated from the following transport equations:

V . ( p U k ) = V . (p(v+vdoi) V k ) + P(Pk - e) (4)

V . ( p U E ) = V . (p(v + vJo¢ ) V £ ) + p E (Clt flPk - CU t"2 e)/k (5)

where Pk is the volumetric production rate of turbulent kinetic energy

Pk = 2v , A :V U (6)

The model coefficients are given by ak=l.0, ot=1.314, C~t=1.44, Cu=1.92; and the damping functions f~,

fl and t"2 are determined from:

f/~ =[I -exp(-0.0165Ren/nl/4)]2(! +20.51Ret) ; f~ = 1 + (0.05 / f~)3 ; f2 = l+exp(-Ret 2) (7)

The damping function f~ differs from that proposed by Lam and Bremhorst [17] only in that it

includes the empirical parameter n TM, which will be shown to improve the accuracy of the predictions

when dealing with strongly non-Newtonian fluids. The Reynolds numbers Ren and Ret are defined by


Re, = .qr~-y, / v and Re r = k 2 / (EV) , where y, is the normal distance to the wall. This distance is easy

to compute for the present geometry, but difficult for an arbitrary boundary topography in multi-

dimensional problems, ttere, it is estimated from a general, economical method proposed by Spalding

[ 19], which involves the solution for a scalar variable, ~ , which obeys the differential equation V2~ =-1

within the fluid, and which equals zero within solid materials and at no-slip surfaces. The wall distance is

deduced from the solution for O from consideration of a simple geometry, namely that between two

parallel walls, and then to presume that the resulting relationship:

y~ =[ (VO) 2 + 2 ~ ] 1 n _Vqb (8)

has general validity.

Solution Method

The flow is one-dimensional, axisymmetric and fully-developed, and boundary conditions are needed

only at the flow axis and wall boundary. At the flow axis a zero-flux condition is employed for all

variables, while at the wall k=0, ~)f/3y=0, and the axial velocity w=0. The model equations are solved

numerically with the finite-volume solution procedure embodied in the general-purpose PttOENICS

computer code [19]. The solution is obtained iteratively by means of a solver option which determines

the axial pressure gradient from overall continuity for a specified mass flow rate. Typically, the

calculations utilise 120 radial grid cells, with the grid spacing increasing in geometric progression away

from the wall. The progression ratio is between 1.02 and 1.05, and the near-wall grid node is located at y÷

{=pw.~2-")y"/K}-_-0.5, where w.{=('t:ctp) I'~} is the friction velocity and x w is the wall shear stress.

Results and Discussion

Numerical computations are performed leading to the friction factor for fully-developed laminar and

turbulent flow. The parameters are the Fanning friction factor f=2xw/(p wb 2 ), the power-law index n and

the generalised Reynolds number of Metzner and Reed [l]:

2 - n n

Re = OWb D (9) K(0.75 + 0.25 / n)"8 n-~

wherein wb is the bulk velocity and D is the pipe diameter. Figures 1 and 2 present computed and

measured friction factors against generalised Reynolds number for various values of the power-law

index. Fig. 1 presents results obtained with the original LB k-e model, while Fig. 2 displays results


produced by the modified LB model. The experimental friction factors in the laminar and turbulent

regimes are represented, respectively, by the analytical [ I ] and Dodge-Metzner [3] correlations:

1 A l l 0.4 . . ~ . u . . ~ # . ( 2 _ , ) / 2 . L f = 16 / Re ; x~- = n-~-75 l°gl°(Kes s n'2 (10)

In the limiting case of a Newtonian fluid, n=l and the generalised Reynolds number Re reduces to the

conventional Reynolds number, and the Dodge-Metzner correlation reduces to the Karman-Nikuradse

correlation for Newtonian fluids.

1 . 8 E - 8 1

f

1 . 8 E - 8 2

×

O D O t

1.8E-83 1.8E+82

I I

P r e d i c t i o n s ~ ~ o o o

1 . 8 E + 8 3

n

1 .2

1 . 8

8 . 8

8 . 6

8 . 4

i i , i i i ,

1 . B E ' 8 4 R e 1 . 8 E + 8 5

FIG. 1

Frictional resistance: original Lam-Bremhorst model

In purely laminar flow, Figs. 1 and 2 show that there is perfect agreement between the numerical

predictions and the analytical correlation. As noted in earlier work [18], the LB k-e model tends to

overestimate the friction factor for turbulent Newtonian fluids (n=l) at Re > 2.104, whereas at lower

Reynolds numbers the model shows fairly close agreement with the measurements.


1 .BE-81

1.BE-82

×

o I"1oi-

1 . 8 E - B 3

1.8E+62

I I

- - P r e d i c t i o n s

L a m i n a r co~Pe la t i on Turbulent c o r r o l a t i o n

1.2

1.8

8.8

8.6

8.4

1 i i 1 i i 1 1 i t | i i i l , I , i i i i t L

1.6E +B3 I. 8E +64 Re I. BE "-85

FIG. 2

Frictional resistance: modified Lam-Bremhorst model

In the turbulent regime, it is evident from Fig. 1 that the unmodified turbulence model significantly

underestimates the frictional resistance for highly non-Newtonian pseudoplastics (n=0.4), whereas for

dilitant fluids (n=l.2) the friction factor is progressively overestimated with increasing Re. These

discrepancies arise because the original fo damping function ( which depends on n via the apparent

viscosity in the Reynolds numbers Re, and Ret ) produces too low vt values in the former case, and too

high values in the case of dilitant fluids. Fig. 2 reveals that the modified fg damping function significantly

reduces this oversensitivity of the original model, as fairly good agreement is achieved for all values of n

over the entire Reynolds-number range.

It is interesting to comment on the ability of the model to predict the critical Reynolds number Rec at

which the flow undergoes transition to turbulence. In the calculations this is defined by the point where f

achieves its minimum value, but it should be mentioned that no attempt has been made to determine the

precise values of Rec. It is seen from Fig.2 that the calculated transition occurs at essentially the same

Reynolds number for 0.6<n<l.0, whereas experiments [3] suggest that Rec increases slowly with

decreasing values of n. In the calculations, this feature is observed for strongly non-Newtonian fluids


(n--0.4) where transition occurs at Rec=3,000, which agrees quite well with the value reported by Dodge

and Metzner [3]. However, an adequate prediction of transition cannot really be expected unless the

model is extended to account for the intermittent nature of the transitional flow.

2 , S

un~

2 . 0

1 . 5

1 . 0

0 . 5

.=2,.o , , , 1 Re = 509

P r o d i c t ions - a ~ 0 o Amt lgtica I %

e . o I I I I O.O 9.Z 0.4 0.6 6.8 r/R 1.0

FIG. 3

Laminar axial velocity profiles

The implementation of the power law model in the numerical solution procedure is further verified by

comparing the predictions obtained for the fully-developed velocity profile in laminar flow at Re=500

with those obtained by the exact analytical solution [9,10]:

R) n+l)ln .~w = ~ 3 n + 1 [1 - ] (11) w b n + 1

where R is the pipe radius. The analytical and numerical results are shown in Fig. 3 for different values

of n at Re=500. It can be seen that the predictions are in excellent agreement with the analytical values,

and the figure illustrates the well-known effect of n on the velocity profile; i.e. for pseudoplastics the

profile becomes progressively flatter; and for dilitant fluids the profile becomes progressively linear. Of

course, for Newtonian fluids the velocity profile is parabolic.


Figures 4 and 5 compare the predicted mean-velocity profiles with those measured by Bogue [15] for

fully-developed turbulent flow. Fig. 4 shows results obtained at Re=197,000 for a Newtonian fluid, and

at Re=107,000 for a slightly non-Newtonian fluid (n=0.825). Fig. 5 presents results for a moderately non-

Newtonian fluid (n=0.615) at Re=23,240, and for a strongly non-Newtonian fluid (n=0.465) at

Re=12,900. In general there is good agreement between the calculated and measured profiles, although

the lack of any data beyond r/R=0.9 precludes validation of the model closer to the wall.

1 . 5

W/Idb

1 . 8

8 .S

1 .5

t,l,,"lJb

1 . 8

8 . S

i I I I

[ ]

- - P red lc t ion~ D a ~ a [ 1 5 ] Re = 1 .97E5

n = 1 . 8

P ~ d l c t l o n s D a t a [ 1 S ]

Ra = 1 . 8 7 E 5 n = 6 . 8 2 5

8 . B I I I I 8 . 8 8 . Z 8 . 4 8 . 6 8 . 8 r / R 1 . 8

FIG .4

Turbulent mean velocity profiles for Newtonian and slightly non-Newtonian fluids

For fully turbulent flow at Re------~O,000, velocity profiles in w+-y + coordinates are plotted in Figure 6

for various values of n. The figure compares numerical predictions of the mean velocity with the

following semi-theoretical 'two-layer' velocity distribution:

• + + 0 . 4 x / 2 ( 1 2 ) w + = mm(w I ,w, ) ; w~ =fy+),/n ; w + =2.46n°25tln(y+)lln +A(n)+B(~,n)]- nL 2

where w+=w/w., w~ is the laminar sub-layer velocity [3], w + is the velocity in the turbulent region [7],

and


A(n) = 1.3676 + in2(2+n)/2 n ; B(~,n) 10.1944 0.1313 0.3876 0.0109 ex [ -n2 - 08 2 = - ~ + n ~ --~ } p ~ - - ~ ( ¢ - . ) ] (13)

n n

wherein ~=y/R and y is the distance from the wail. The velocity predictions of Fig. 6 are also plotted in

Figure 7, but normalised in terms of wb and R. The corresponding calculated profiles of turbulent kinetic

energy are presented in Figure 8.

1.5

W/IJb

1.8

8.5

1.5

W/Wb

1.8

8 .5

I I I I

P r e d i c t i o n s

D a t a [ 1 5 1 Be = 1 . 2 9 E 4

n = 8 . 4 6 5

- - P r e d i c t i o n s

D a t a [ 1 5 )

B e = 2 . 3 2 4 E 4 n = I ] . 6 1 5

8 . 8 I I I I

8 . 8 8 . 2 8 . 4 8 . 8 8 . 8 r / D 1 , 8

FIG. 5.

Turbulent mean velocity profiles for moderate and strongly non-Newtonian fluids

Figure 6 shows that the thickness of the viscous sublayer reduces with decreasing n, a trend which

might have been anticipated from the laminar velocity profiles shown earlier in Fig. 3. The influence of

turbulence is to transport higher-momentum fluid towards the wall, thereby leading to broader velocity

profiles, as is evident from the profiles shown in Fig. 7. The velocity predictions shown in Fig. 6 agree

fairly well with the logarithmic distributions, except in the transition zone which is not correctly

described by the two-layer approximation. It is interesting to note that the limits of the transition zone

indicated by the present model agree reasonably well with those proposed by Clapp [4]: 5 n < y+ < y~-,

where y~- is the boundary between the transition and turbulent zones y~- = exp[(3.8 + 3.05n) / 2.22].

986 M.R, Malin Vol. 24, No. 7

30

W+

20

10

0 1.0E-01

I I I I

n=O .4 i i 1

Re = 4E4 / n=0.6

/ / ~ ] n=O .B

o Q

Pved i ct ions o Non-Hewtonian log-law [71

+ Newtonlan log-law [14]

i , , , , I H I I I * * , * , , I I I ' ' l ' l ' l I [ I l l l l l

1.0E+00 1.0E+01 1,0E+02 1.8E+03 y+ 1.0E+04

FIG. 6

Effect of n on turbulent mean velocity profiles plotted in wall coordinates

I I I I I I

0 . 8 ~

0.6 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 r / R 1 . 8

FIG. 7

Effect of n on turbulent mean velocity profiles


Finally, Figure 8 demonstrates the effect of the power-law index on the turbulence energy profiles at

Re=40,000. It is seen that in addition to the narrowing of the viscous sub-layer, there is also a

pronounced attenuation of the turbulence with decreasing power-law index. From Figs. 6 and 7, it is

evident that the velocity profiles become progressively flatter owing to the increase in apparent viscosity,

and hence turbulence production diminishes over the entire pipe cross section.

8 . 8 1 5

2

R/W b

8 . 8 1 e

8 . 8 0 5

I I I I

R e = d E 4 _ n

- 1 . 2 "

8 . G

I I I I 8 . B88 8 . 8 8 . 2 8 . 4 8 . 6 8 . 8 t - /B 1 . 8

FIG. 8

Effect of n on turbulence-energy profiles

C o n c l u s i o n s

A series of numerical computations has been performed to calculate fully-developed laminar and

turbulent flow of power-law fluids in smooth tubes. A modified version of the Lam-Bremhorst k-e

model has been tested against experimental data on the friction factor and mean velocity profile for

various generalised Reynolds numbers, with different values of the power-law index n. Generally, the

model produced fairly good agreement with the measured data, but the delay of transition to turbulence

was predicted only for strongly non-Newtonian fluids, i.e. n<0.6.


References

1. A.B.Metzner and J.C.Reed, AIChE J 1, 4,434, (1955).

2. A.B.Metzner, ind.Eng.Chem. 49, 9, 1429, (1957).

3. D.W.Dodge and A.B.Metzner, AIChE J 5, 189, (1959).

4. R.M.Clapp, Int. Dev. in Heat Transfer, Part III, 652, D-159, D-211-5, ASME, New York, (1961).

5. D.C.Bogue and A.B.Metzner, Ind. Eng. Chem. Fundam., Vol.2, 143, (1963).

6. W.B.Krantz and D.T.Wasan, AIChE J, 17, 1360, (1971).

7. A.V.Shenoy and D.R.Saini, Can.J.Chem.Eng. 60, 694, (1982).

8. A.V.Shenoy, In Encyclopedia of Fluid Mechanics, Ed.N.P.Cheremisinoff, Vol.1, Chap.31, 1034, Gulf, Houston, Texas, USA, (1981).

9. A.H.P.Skelland, Non-Newtonian Flow and Heat Transfer, Chap.6, 180, John Wiley, (1967).

10.G.W.Govier and K.Aziz, Tile Flow of Complex Mixtures in Pipes, Chap.5, 182, Krieger, (1977).

1 ! .J.P.Hamett and M. Kostic, Int. Comm. Heat Mass Transfer 17, 59, (1990).

12.T.D.Reed and A.A.Pilehvari, SPE 25456, Prod. Operations Syrup., 39, Oklahoma City, USA, (1993).

13.A.LY.Mohammed, N.N.Gunaji and P.R.Smith, ASCE J.Hyd. Div., HY7, 885, (1975).

14.H.Schlichting, Boundary Layer Theory, McGraw Hill, 6th Edition, (1968).

15.D.C.Bogue, PhD Thesis, University of Delaware, Newark, USA, (1960).

16.J.F.Pierre and R.Boudet, 4th int.Conf Num.Meth.Laminar and Turbulent Flow, 235, Swansea, Pineridge Press (1985).

17.C.K.G.Lam and K.A.Bremhorst, ASMEJ.Fluids Engng 103, 456, (1981).

18.M.R.Malin, to appear in ha.Carom.Heat Mass Transfer, (1997).

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Turbulent pipe flow of power-law fluids