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Journal of Hydraulic Research

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A general correlation for turbulent velocity profilesof dilute polymer solutions

K. C. Tam , C. Tiu & R. J. Keller

To cite this article: K. C. Tam , C. Tiu & R. J. Keller (1992) A general correlation for turbulentvelocity profiles of dilute polymer solutions, Journal of Hydraulic Research, 30:1, 117-142, DOI:10.1080/00221689209498951

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A general correlation for turbulent velocity profiles of dilute polymer solutions Une relation générale pour les profils de vitesse turbulente dans des solutions de polymère dilué

K. C. TAM Department of Chemical Engineering, Monash University, Clayton, Victoria, Australia

C. TIU Department of Chemical Engineering,

Monash University, Clayton, Victoria, Australia ^ | | ^ ? %^̂

Received August 17, 1990. Open for discussion till August 31, 1992.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30, 1992, NO. 1 117

R. J. KELLER Department of Civil Engineering, Monash University, Clayton, Victoria, Australia

Keywords: Turbulence, velocity profiles, polymer solutions, drag reduction.

SUMMARY Dilute polymer solutions of sufficiently high molecular weight generally exhibit the phenomenon of drag reduction in turbulent flows. This has been usually studied by conducting either pressure drop/flow rate measurements or axial velocity profile measurements. A general correlation, based on Prandtl's mixing length model of the form,

_,+ 2.43, 4 (6.5-p r\

is developed herein to correlate the velocity data obtained in a 9 mm glass tube using an LDA system. This new correlation can be used to describe the velocity data of dilute solutions of polyacrylamide, xanthan gum and polyethylene oxide. The parameters a, /? and T are unique functions of u*l(u~) and are independent of polymer type, concentration, mechanical degradation and Reynolds number. Good agreement between two independent data sets and predictions of the velocity profile using the above correlation was observed.

RESUME Des solutions diluées de polymère de poids moleculaire élevé présentent généralement la propriété de dimi-nuer la trainee dans les écoulements turbulents. Cela a été étudié soit par des mesures de perte de charge en fonction du débit, soit par des mesures de profils de vitesse. Une correlation générale a été développée, basée sur la notion de longueur de mélange de Prandtl,

, ;+ 2,43, + fc,S-/? r\

afin de décrire les mesures de vitesse par vélocimétrie laser dans un tube de verre de 9 mm. Cette nouvelle relation peut être utilisée pour décrire les mesures de vitesse de solutions diluées de polyacrylamide, gomme

de xanthan et oxyde de polyethylene. Les paramètres a, /? et r sont des functions uniques de u*j{ü) et sont indépendantes du type de polymère, de la concentration, de la degradation mécanique et du nombre de Reynolds. Un bon accord a été constaté entre deux families de données indépen-dantes et les predictions de profils de vitesse a partir de la relation ci-dessus.

1 Introduction

In recent years, interest in the flow of dilute polymer solutions under turbulent conditions has increased dramatically. The incentive for such study is partly a result of favourable economic consequences in areas such as long distance transport of crude oil (Ram et al., 1967; Brod et al., 1971; Lescarbourna et al., 1971; Bulina, 1979; Burger et al., 1980) and mineral suspensions (Herod et al., 1974; Poreh et al., 1970; Gust, 1976; Fujimoto and Tagori, 1974), sewers and open channels (Sellin, 1977, 1978; Sellin and Ollis, 1980; Overfield et al., 1969; Sellin and Barnard, 1970; Peter­son et al., 1973), hydraulic machinery (Latto and Czaban, 1974; Bilgen and Vasseur, 1977), central heating systems (Fitzgerald et al., 1967; El'perin et al., 1971) and marine applications (Dove, 1966; Emerson, 1965, Canham et al., 1971; Thurston and Jones, 1965). The introduction of small quantities of linear high molecular weight polymers in the flow systems results in a phenomenon which has been universally termed Drag Reduction. This phenomenon is defined as any modification to a turbulent fluid flow system which results in a decrease in the rate of energy loss. It was first demonstrated by Toms (1948) and Mysels (1949). An extensive research effort has been devoted to elucidate possible mechanisms resulting in numerous publications. A number of reviews have been published. Hoyt (1972), a pioneer re­searcher in drag reduction, compiled a comprehensive review and discussion of the subject. Lumley (1969,1973) and Landahl (1973) considered the subject from a fluid mechanics viewpoint proposing various mechanisms and discussing the role of rheology. An extensive review of the role of linear random-coiling molecules in turbulent flow was described by Virk (1975). He presented comprehensive experimental evidence which graphically established various impor­tant features relating to the physical mechanism of drag reduction. Berman (1978) discussed the dynamics of drag reduction in the light of new evidence. The latest review of Sellin, Hoyt and Scrivener (1982a, b) re-emphasized some of the basic aspects of drag reduction and discussed the existing industrial applications and possible future utility of this phenomenon. Most of the experimental studies have been conducted in smooth pipes yielding three possible types of data, namely: 1. Pressure drop versus flow rate. 2. Time-smoothed axial velocity profiles. 3. Dynamic data in the form of spectra.

The first type is the most common and is reported in almost all drag reduction studies. It is easy to measure but lacks the details needed to describe the complex behaviour in turbulent flows. The time-smoothed velocity profiles provide information that sheds light on the nature of the boundary layer near the wall of the pipe. Many researchers have correlated a change in the boundary layer thickness with the reduction in drag at the wall of the pipe. The most detailed and comprehensive study of drag reduction can be made by examining the dynamic response of the velocity fluctuations at various radial positions within the duct. Statistical analysis of these data yields useful information on the effect of macromolecules on the turbulent structure. The results of many years of research on this subject is reflected in the many models, correlations and mechanisms proposed to describe this unique phenomenon. From an engineering viewpoint,

118 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30, 1992, NO. 1

it would be desirable to develop a general correlation that can adequately describe the experi­mental results and is simple to use. In this study, time-smoothed axial velocity profiles are measured for four different types of poly­meric systems. A general correlation, based on the Prandtl mixing length model is developed to represent the experimental results.

2 Review of previous work on time-smoothed velocity profiles

Table 1 summarises the published work on time-smoothed velocity profiles of drag-reducing polymer solutions. A comprehensive review was documented by Shenoy (1989) in the Encyclope­dia of Fluid Mechanics. Most of the work was done in the mid 60's and early 70's, the earliest reported data being those attributed to Elata et al., (1966). They measured the axial velocity in a 50.8 mm pipe, using a relatively simple Pitot tube (O.D. 3 mm), in solutions of 50 to 10,000 ppm of Guar Gum. By modifying the conventional Newtonian law-of-the-wall expression, they proposed a simple correlation to describe the velocity distribution across the pipe of the form:

where

£/+ = 2.5 1nj>+ + 5.5 + a'lnDe (1)

U+ = the normalized local velocity at normalized distance y+ from the boundary a' = an empirical coefficient De = the Deborah number, defined by

lu*2

De= (la)

where

X = the polymer molecular relaxation time u* = the shear velocity v = the kinematic viscosity

Meyer (1966) analysed the velocity data of Ernst (1966) and proposed a modified form of the universal law of the wall to represent the velocity profile,

U+ = 2.5\ny+ + a*\n — ) (2) Writ/

where

a* = an empirical coefficient and u*ril is the shear velocity at the onset of drag reduction

A more general equation was proposed by Seyer and Metzner (1969) by replacing equation (1) with a general function B{De) which could be empirically determined. The equation has the form:

U+ = A\ny+ + B (3)

where

^ =2.46 B = 5.6+].55De forQ<De<\0

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30, 1992, NO. 1 119

Table 1. Summary of published works on turbulent velocity profile measurements on polymer solutions

researchers

Elata

Meyer, Ernst

Virk

Goren

Patterson

Seyer

Rudd

Rollin

Arunachalam

Chung

year

1966

1966

1967

1967

1969

1969

1969

1972

1972

1972

polymertype

Guar Gum

CMC

polyethylene oxide

polyethylene oxide

PIB

polyacrylamide

polyacrylamide

polyacrylamide

polyethylene oxide

polyacrylamide

x l ( T 6

M.W. range

1.7

0.7

0.7

5.0

2.3

3.0

3.0

3.0

5.0

3.0

(ppm) cone. range

50-10,000

500

1,000

2.5-10

2,000

1,000

100

100 1,000

0.55-33

100 200

pipe I D . (mm)

50.8

0.65 1.43

25.4

50.8

25.4

25.4

12.7*

25.4 69.8

13.6

11.2 11.9

measurements techniques

Pitot tubes

Pitot tubes

Pitot tubes

Pitot tubes

hot wire anemometry

bubble streak photography

Laser-Doppler anemometry

bubble streak photography

photochronic dye photography

Laser-Doppler anemometry

max Rex 10~3

457.2

-

178.1

149.5

55.4

49.1

50.2

103

37.6

5.3

* Square duct

Further development in the velocity profile correlation was advanced by the extensive studies of Virk et al., (1967). An elastic sub-layer model was proposed based on the physical observation that polymer molecules subjected to turbulent shear flow create a zone called the elastic sub­layer which is characteristic of the drag reduction phenomenon. This sub-layer originates at onset and thickens with increasing drag reduction ultimately occupying the entire pipe cross-section at maximum drag reduction. A more extensive description of the model can be found in Virk (1975). According to the model, the velocity profile can be described by two equations in each of the defined regions:

a. Newtonian plug region

t/+ = 2.5 1ny+ + 5.5 + 9.2 1n I ^ M for yt<y+<R (4) \ysLi

where >>E and y$L are defined in Fig. 1 R = the pipe radius

b. Elastic sub-layer region (Virk asymptote)

U+=UJ\ny+~n.O for ytL<y+<y+E (5)

The above two regions are illustrated in Fig. 1.

120 JOURNAL DE RECHERCHES HYDRAUL1QUES, VOL. 30, 1992, NO. 1

U = 2 . 5 l n y + + 5 . 5 + AB

Drag Reducing Flow

'iscous sublayer Turbulent Core

Fig. 1. Dimensionless velocity profiles according to Virk's three layer model (Sellin et al., 1986). Profils de vitesse adimensionnels selon Ie modèle a trois couches de Virk (Sellin et al., 1986).

A more complex expression was proposed by Shenoy and Talathi (1985) to correct for the dis­parity of previous correlations which fail to predict a zero velocity gradient at the centre of the pipe. Following the procedure suggested by Bogue and Metzner (1963) and Stein et al., (1980), a new expression of a form similar to that of Seyer and Metzner (1969) was developed as follows:

U+ = A (In y+ + C({, De)) + B(De)

where

C(Z,De) = <j](De) exp 0.8

a2(De)

(6)

(6a)

and oi(De), oi(De) are adjustable parameters which have been empirically determined to satisfy the boundary condition of zero velocity gradient at the centreline. They have the form,

ffi(De) = 0.4398 + 0.123De + 0.0135Z)e2

a2(De) = 0.254(1 + 0.2De)

(6b)

(6c)

For De = 0, the values of CT,(0) and <r2(0) reduce to those obtained by Stein et al., (1980) for Newtonian fluids. Integrating equation (6) over the entire cross-section of the turbulent core yields an implicit expression for friction factor. The parameters A and B(De) were determined by comparing the solution with the result of Seyer and Metzner (1969). The final result of the equa­tion has the form:

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30. 1992, NO. 1 121

U+ = 2.46{ln y+ + (0.4398 + 0.123Z)e + 0.0135/V) exp (£) + 1.3676(1 - 0.09De - O.OlDe2) + 5.6 + \.55De - G\ (7)

(7a)

and G is an adjustable parameter ranging from 3 to 4 for all values of De (from 1 to 10) over a Reynolds number range of 104 to 106. The value of £ can be obtained from,

where

Re = the Reynolds number ƒ = the Fanning friction factor

Equation (7) is fairly complex and a number of empirical and adjustable parameters have to be introduced in order to correct for any observable deviations. It is apparent that although equation (7) overcomes the disparity discussed earlier, it is debatable whether the complexity and empiri­cisms justify its use over the correlation of Seyer and Metzner (1969). All of the above correlations are derived on the basis that the slope of the velocity profiles is similar to that of Newtonian fluid with the exception of the Virk drag reduction asymptote. Recent results of Lodesova and Lodes (1989) and of the extensive investigation reported herein suggest the contrary. For some polymeric systems, the transition to the Virk maximum drag reduction asymptote is gradual and is dependent on the polymer type, concentration, Reynolds number, degradation and ionic characters in the case of aqueous polymer solutions. Thus there is a need to consider such behaviour in formulating a general correlation. This is the main subject addressed in this paper.

3 The correlation equation

The development of the general correlation equation for time-averaged axial velocity profiles was based on Prandtl's mixing length theory. According to this theory, the turbulent shear stress for axial flow in a tube may be expressed by:

where

y — R — r is the radial distance from the pipe wall L = the Prandtl mixing length, given by

L = kxy (10)

where

k\ = a universal constant.

For turbulent flows of polymer solutions, the experimental evidence of previous investigations

122 JOURNAL DE RECHERCHES HYDRAULIQUES. VOL. 30, 1992, NO. 1

where,

(£-0.8)2

[ 0.129(1 +0.2De)2

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30. 1992, NO. 1 123

(Vick, 1975; Lodes and Macho, 1989; Kozlov, 1989) leads to the deduction that there is an apparent thickening of the laminar sub-layer. For such flows it is appropriate to rewrite equation (10) as:

Lv = ak\y (11)

where

a = the ratio of the Prandtl's mixing length for the polymer solution to that for water

Equation (9) may then be rewritten for a polymer solution as:

ï? = ea\f(^j (12)

An alternative expression for fr(zl) may be obtained from the time-averaged equation of motion in cyclindrical co-ordinates, yielding:

rl!) = T w / l - ^ ) (13)

where

rw = the wall shear stress

Following an approach analogous to that adopted by Prandtl for Newtonian fluids, the right hand sides of equations (12) and (13) were equated and the resulting expression integrated to give

U; = ^\ny++U;0 (14)

where

*P = (1 /«*I ) (15)

£/p+0 = defined in Fig. 2

For Newtonian fluids equation (14) reduces to

K = <pp\ny++U^0 (16)

where

0N = 1 / ^ = 2 . 5

t/fJo = defined in Fig. 2 and has the value of 5.5

At this point, two additional parameters are introduced, namely

r = up+

0 - (/N+

0 ( 1 7 )

and

/? = (l-a)c/N+

0 (18)

Furthermore, from equations (14) and (16),

«=4^4 (19) " N — ^No

Fig. 2. Schematic diagram representing the equations and parameters of the modified Prandtl mixing length model. Diagramme schématique représentant les equations et les paramètres du modèle de longueur de mélange de Prandtl modifié.

The various parameters used in equations (14) to (19) are shown schematically in Fig. 2. It is evident that the parameter a, defined by equation (19), represents the ratio of the slope of the two velocity profiles. Combining equations (17), (18) and (19) and rearranging yields:

t/^ = a ( t / P+ - r ) + /? (20)

It is evident from equation (20) that the velocity profiles from polymer solutions may be shifted onto the Newtonian profile by correcting Up

+ data with parameters a, r and /?. It is postulated that each of these parameters is a unique function of an independent variable which defines the extent of drag reduction in turbulent flow. An appropriate independent variable is the friction factor,/, which may be expressed as a function of u*j{ü). Ifthispostulation is correct and the unique func­tions can be determined, equation (20) yields a basis for predicting the mean velocity distribution in ducts carrying dilute polymer solutions from measurements of pressure drop and flow rate.

4 Experimental procedure

a. Test fluids and preparation Four different polymers were investigated, each of which is known to exhibit drag reduction characteristics. Table 2 provides a brief description of each of the polymer types. Each of these polymers has previously been commercially tested in many drag reduction studies.

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Table 2. Summary of polymer types used in turbulent flow studies

polymer type

polyacrylamide

polyethylene oxide

xanthan gum

manufacturer

Dow Chemicals

Union Carbide

Kelco Inc.

trade name

Separan AP30 Separan AP302

Polyox Coagulant

Keltrol

molecular weight (xirr6) 4 8

5

3.7

For each polymer, a stock solution of 1% was prepared by dispersing the polymer powder in 5 litres of distilled water placed on a mechanical shaker. The polymer solution was left on the shaker for 24 hours until homogeneity was achieved. Normally the stock solution was prepared a day before the pipe flow tests were carried out. A required concentration of polymer was prepared by mixing a known amount of 1% stock solution in about 180 litres of tap water in a premixing tank. The polymer solution was homogenised using a pneumatic mixer for 15 minutes before it was introduced into the test loop. The solution viscosity was measured using the Contraves Low Shear 30 and the Rheomat 30 instruments over the shear rate range of 10~2 to 103 s~'.

b. Test loop A test loop was constructed for this study and a schematic layout is shown in Fig. 3. It consisted of the following major items: a. Two 200 litre tanks for premixing and storage of the polymer solutions. b. A Lightnin pneumatic mixer for homogenising the polymer solution before it was introduced

into the loop. c. A centrifugal pump for delivering the fluid through the test loop. d. A glass test section of 9 mm ID and 3.0 m in length. The maximum flowrate which the pump could deliver was approximately 8 x 10~5 nrVs, corre­sponding to an average velocity of about 1.5 to 2 m/s and a Reynolds number of between 10,000 and 15,000, depending on the polymer concentration. Special care was taken in the design and construction of the test section to ensure that the glass tube sections butted smoothly into the brass collars which contained the pressure tapping. The inside surfaces of the brass collars was polished to ensure surface characteristics as close as possible to those of glass. The pressure tappings were only 1/8 the size of the test section, thereby ensuring that no secondary flows were generated at their entrance. The test section was supported by a simple stand which could be adjusted to permit leveling of the duct.

c. Instrumentation Velocity characteristics were measured using a Laser-Doppler Anemometer (LDA). A commer­cial system was utilized, designated as Type 9100-3, and manufactured by TSI Inc. (TSI, 1981). The LDA incorporated a 35 mW Helium-Neon Laser as the light source and was operated in the forward scatter mode. For a detailed description of the physics of an LDA system, reference should be made to one of the reviews of Watenanewicz and Rudd (1976), Durrani and Greated (1973) and Durst et al. (1981). The whole laser optical unit rests on a stable mounting base of a 3-axes milling table. A linear magnetic scale (Sony Magnescale) is attached to each axis on the traversing table and a 3-axes digital display (Sonly LH10) is used to locate the position of the measuring volume in the pipe.

JOURNAL OF HYDRAULIC RESEARCH. VOL. 30, 1992, NO. 1 125

This unit has an accuracy of 0.01 mm at 20 °C, easily meeting the accuracy requirements of the present study. No seeding of particles was necessary as there were sufficient dust particles in the tap water to give good signals. For all the runs, a sampling rate of 20 samples/sec was set over a time duration of 60 sec. This gives a sample volume of 1200, the maximum practically possible for reasons of data storage space. Only minimal differences in the average velocity were observed when the sample rate was increased to 60 samples/sec and higher. Thus, a sample size of 1200 was adequate. A more detailed description of LDA system and its data acquisition technique is given elsewhere (Tarn, 1990). The mean local velocities at various positions in the pipe, from close to the wall to the centre of the pipe were measured for water and for each of the four different polymers.

Fig. 3. Flow loop used for the study of turbulent flow of dilute polymer solutions in a 9 mm glass tube. Boucle d'essais utilisée pour l'étude de l'écoulement turbulent de solutions de polymères diluées dans un tube de verre de 9 mm.

5 Results and discussion

a. Local instantaneous velocity The instantaneous velocity may be represented by the sum of the time-smoothed (average) velo­city ü and the velocity fluctuation «':

ul= u + u (21)

It is an irregular oscillation function and the magnitude of the fluctuating component u', decrea­ses from the pipe wall to the center. Figs. 4a to 4c shows such a trend at various positions in the pipe. These fluctuations are generally represented in terms of the turbulence intensity, which decreases with increasing distance from the pipe wall.

126 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30, 1992, NO, 1

b. Accuracy of velocity data The bulk average velocity (w) is related to the local mean velocity by,

2 R

<")="^2 f " ' d/" A n

(22)

The above integration was carried out using Simpsons rule and the resulting (it) was compared with the bulk mean velocity determined from flowrate measurements. An accuracy of ± 5% was observed for all of the experimental runs.

c. Velocity profile for Newtonian fluid (water) The turbulent velocity profile for water obtained in the flow system at Re= 11,000 is plotted in dimensionless form üjüm.M vs rjR in Fig. 5. The solid curve represents the best fit of the experi­mental data obtained in 2 runs. The accuracy of experimental results was established not only from the reproducibility of the data points, but also from the flowrate measurements as discussed in (b).

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30, 1992. NO. 1 127

Fig. 4. Turbulent velocity fluctuations of water at radial position (a) (/•//?) = 0.01, (b) (/•//?) = 0.45 and (c) (/■/ƒ}) = 1.0.

Fluctuations de vitesse turbulente en position radiale (a) (/■//?) = 0,01, (b) (/■//?) = 0,45 et (c)(/-/iR)=l,0.

r/R

Fig. 5. Dimensionless time smoothed axial velocity profile of water at Re= 10,000 to 11,000. Profil de vitesse axiale de Peau, adimensionnel et lissé dans le temps pour Re de 10000 a 11000.

Traditionally, turbulent velocity profiles are represented in terms of £/+ vs y+. This is demon­strated for four water runs over a narrow range of Re between 10,000 and 11,000 in Fig. 6. The linearity of the velocity distributions on a semi-log plot concurs with the semi-theoretical profile as predicted from the Prandtl mixing length theory. The expression representing the straight line shown in Fig 6 has the form

£# = 2.43 ln.y+ + 6.5 (23)

Fig. 6. Reduced turbulent velocity profile of water at Reynolds number of 10,000 to 11,000. Profil de vitesse turbulente réduite pour l'eau a des nombres de Reynolds entre 10000 et 11000.

128 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30, 1992, NO. 1

The two numerical coefficients, representing the slope and the intercept of the straight line, agree well with previously published results. For example, the classical data from Deissler (1955) and from Laufer (1953) yielded a slope of 2.77 and an intercept between 3.8 and 10.0.

d. Velocity of polymer solutions

i. Effect of polymer concent ra t ions Figs. 7 to 10 show the effect of polymer concentrations on four different types of drag reducing polymers at Reynolds number ranging from 10,000 to 15,000. Separan AP30 and AP302 exhibit a sharp deviation from the Newtonian velocity profile as the polymer concentration is increased. They approach the Virk assymptote at polymer concentrations of about 100 and 50 ppm for Separan AP30 and AP302 respectively (Figs. 7 and 8). This is not evident for solutions of xanthan gum (Keltrol) and polyethylene oxide (Polyox coagulant), whose velocity profiles at different polymer concentrations have similar slope (Figs. 9 and 10). In the present study, the drag reduc­tion capability of polyethylene oxide is low compared to some previously published results. This is due to the difference in the pre-sheared history of the polymer. Polyethylene oxide is known to degrade rapidly when exposed to the mechanical shearing of the action of the impellers in a centrifugal pump. The effects of this degradation prompt caution in comparing the drag reduc­tion capabilities of various polymers.

ii. Effect of polymer types The velocity profiles of 50 ppm of 4 different types of drag reducers are shown in Fig. 11. Their behaviour is markedly different; Separan AP302 showing the largest deviation (largest drag reduction) from the Newtonian profile, followed by Separan AP30, Keltrol and Polyox coagulant. There is a strong correlation between the effectiveness of drag reduction and the molecular weight of the polymer. In this case, Separan AP302 has the largest molecular weight (MW ~ 8 x 10"), followed by Separan AP30 (MW ~ 4 x 106) and then Keltrol (MW ~ 3.7 x 106). These molecular weights are given in the manufacturers specification and they are given here to show qualitatively the difference between each of them. It must be stressed that other factors that do influence the size of polymer coils are the ionic content of polymers, molecular weight distri­bution and the solvent environment. The anomaly lies with the polyethylene oxide which has a molecular weight of about 5 x 106. It is seen that the velocity profile for the 50 ppm Polyox coa­gulant is only marginally above the water line. As stated earlier, polyethylene oxide is highly sus­ceptible to shear-degradation. Much of its drag reduction capability would have been lost due to scission of the chains under the mechanical action of the pump impeller. As a matter of fact, fresh Polyox coagulant (un-degraded) has been found to achieve the maximum reduction at a much lower polymer concentration than Separan AP30, i.e., about 30 ppm compared to about 100-150 ppm for Separan AP30 (Kenis, 1971).

iii. Effect of mechanical degradat ion The evidence of mechanical degradation is usually reflected in the loss of drag-reduction effec­tiveness. This has been observed in both simple shear (Barnard and Sellin, 1972; Yu et al., 1979) and elongational straining flows (James et al., 1987). There is no doubt that the most rapid degra­dation of a polymer takes place under intense shearing conditions encountered inside the im­peller of a centrifugal pump. This has been demonstrated by the low drag reduction effectiveness

JOURNAL OF HYDRAULIC RESEARCH. VOL. 30, 1992. NO. 1 129

7. Dimensionless turbulent velocity profile of various concentrations of Separan AP30 at Re = 10,000 to 15,000. Profil de vitesse turbulente adimensionnelle pour diverses concentrations de Separan AP30 avec Re entre 10000 et 15000.

Fig. 8. Dimensionless turbulent velocity profile of various concentrations of Separan AP302 at Re= 10,000 to 15,000. Profil de vitesse turbulente adimensionnelle pour diverses concentrations de Separan AP302 avec Re entre 10000 et 15000.

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30, 1992, NO. 1 130

Fig. 9. Dimensionless turbulent velocity profile of various concentrations of xanthan gum at Re = 10,000 to 15,000. Profil de vitesse turbulente adimensionnelle pour diverses concentrations de gomme de xanthan avec Re entre 10000 et 15000.

Fig. 10. Dimensionless turbulent velocity profile of various concentrations of Polyox Coagulant at Re= 10,000 to 15,000. Profil de vitesse turbulente adimensionnelle pour diverses concentrations de Polyox Coagulant avec Re entre 10000 et 15000.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30, 1992, NO. 1 131

Fig. 11. Dimensionless turbulent velocity profile of 50 ppm of different types of drag reducers at Re= 10,000 to 15,000. Profil de vitesse turbulente adimensionnelle avec 50 ppm de différents types de réducteurs de trainee avec Re entre 10000 et 15000.

of polyethylene oxide which generally exhibit good drag reduction characteristics in many experiments where the centrifugal pump is not used. The effect of mechanical degradation of Separan AP30 on the velocity profiles is shown in Fig. 12.

Fig. 12. Dimensionless turbulent velocity profiles of 100 ppm Separan AP30 under different shear condi­tions at Re of about 11,000. Profil de vitesse turbulente adimensionnelle avec 100 ppm de Separan AP30 sous diverses condi­tions de cisaillement avec Re aux environs de 11000.

132 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30, 1992, NO. 1

The polymer solutions were recycled, each time passing through the centrifugal pump. Reduc­tion in the slope of the profiles is evident for each subsequent pass through the pump and the flow loop.

iv. Effect of Reynolds number The degree of drag reduction is highly dependent on the Reynolds number. This is generally evident from the friction factor-Reynolds number plot. The effect of Re can also be shown from the velocity profile plots as seen in Fig. 13. Such dependence is consistent with the observation of Ollis (1981) who demonstrated that the percentage drag reduction for 10 ppm of Polyox WSR301 increases with u* and reaches a plateau at u* of 0.1 to 0.4 m/s, depending on the pipe diameter. It is evident that for the 50 ppm Separan AP30 solution, the level of drag reduction is higher at Re= 11,000 as compared to Re = 6,000.

Fig. 13. Dimensionless turbulent velocity profiles of 50 ppm Separan AP30 at two different flowrates. Profil de vitesse turbulente adimensionnelle avec 50 ppm de Separan AP30 a deux debits différents.

e. General correlations and master plots

Following the derivation of the velocity profiles based on the modified Prandtl mixing length model, it is apparent from equation (20) that master curves of the velocity data for all the runs could be obtained by plotting the parameter [a(t/p

+ — /")+/?] versus In y+. The parameter r, /? and a were obtained from the velocity data as prescribed by equations (17), (18) and (19) respect­ively. Figs. 14 to 17 show the master plots of the velocity profiles of Separan AP30, AP302, Keltrol and Polyox coagulant respectively. The number of data points vary from 80 for Polyox coagulant to 170 for Separan AP302. These data represent all the three conditions mentioned earlier, namely varying polymer concentration, degradation and Reynolds number. It is apparent from these figures that all the polymer data collapse onto the water line represented by,

[a(Up+-r)+IJ] = 2A3\ny+ + 6.5 (24)

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30, 1992, NO. 1 133

[a(u

;-r)+

p]

Fig. 14. Master plot of dimensionless turbulent velocity profile of Separan AP30. Droite d'ajustement du profil de vitesse turbulente pour du Separan AP30.

[a(u

;-r)

+ p]

Fig. 15. Master plot of dimensionless turbulent velocity profile of Separan AP302. Droite d'ajustement du profil de vitesse turbulente pour du Separan AP302.

134 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30, 1992, NO. 1

[a(u

;-r)

+ p]

Fig. 16. Master plot of dimensionless turbulent velocity profile of xanthan gum. Droite d'ajustement du profil de vitesse turbulente pour de la gomme de xanthan.

[o(u

p*-r

j + p

]

Fig. 17. Master plot of dimensionless turbulent velocity profile of Polyox Coagulant. Droite d'ajustement du profil de vitesse turbulente pour du Polyox Coagulant.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30, 1992. NO. 1 135

This is a universal correlation which is applicable for the conditions tested with Reynolds number ranging from 4,000 to 15,000. At the limiting condition of Newtonian fluid, a becomes unity, T and /? are each zero, and equa­tion (24) reduces to equation (23). The practical application of the above correlation, requires the development of relationships for the three parameters or, r and /?. Based on the dynamics of the drag reduction phenomenon in turbulent flow, it is reasonable to postulate that these parameters are related to the ratio of friction velocity and the average bulk velocity, (w*/(«)-represented by e) as shown below:

a r, p =fn(e) (25)

The velocity ratio u*/(ü), (e) is equivalent to if ft. The above relationship was tested by plotting all of the available data for each of the polymer solutions as shown in Figs. 18 to 20. As can be observed, a unique relationship exists between each of these parameters and the ratio of the fric­tion velocity to the average velocity or the friction factor which is independent of polymer type, concentration, mechanical degradation or Reynolds number. The relatively minor data scatter evident in the figures is thought to be due to experimental error, largely a result of interpolating the velocity data to obtain a, /? and F. The parameters can be fitted by second order polynomials using the method of least square fitting. The empirical equations for each of the parameters are as follows:

The extent of data scatter is reflected in the values of/?2 which suggests that each of the param­eters can be adequately represented by the mathematical expressions as given by equations (26) to (28).

Fig. 18. Dependence of a on the reduced friction velocity of various polymeric systems. Variation de a en fonction de la vitesse de frottement réduite pour divers systèmes de polymères.

136 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30, 1992, NO. 1

a = - 2 . 8 6 4 + 105.5e-654.4fi2 R2 = 0.96

P =24.98 -679.9fi + 4140fi2 R2 = 0.96

r = -396.1 + 1.629 x 104£- 1.665 x 10 V for e < 0.05 I ^2 = fJ 9 4

= 2.2 for £>0.05)

(26)

(27)

(28)

Fig. 19. Dependence of/? on the reduced friction velocity of various polymetric systems. Variation de /? en fonction de la vitesse de frottement réduite pour divers systèmes de polymères.

f. Comparison of prediction of velocity using new correlation The usefulness of the proposed general correlation rests on the basic premise that the turbulent velocity profile of any polymer solution can be predicted from measurements of pressure drop and volumetric flowrate which yield the two parameters, namely the friction velocity, "*(=v(Tw/e)) and the bulk mean velocity, (w) or the friction factor ƒ Knowing u* and (u) enables a, /? and F to be determined from equations (26) to (28). The velocity profile can then be

Fig. 20. Dependence of F on the reduction friction velocity of various polymetic systems. Variation de F en fonction de la vitesse de frottement réduite pour divers systèmes de polymères.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 30, 1992, NO. 1 137

Fig. 21. Comparison of prediction using equation (24) and experimental results of two sets of independent data. Comparaison entre les predictions faites a partir oe l'équation (24) et les résultats expérimentaux issus de deux séries d'essais indépendantes.

evaluated from equation (24). Comparisons were made for two independent sets of velocity data and the results are shown in Fig. 21. Good agreement is evident, suggesting that the proposed correlation (equation 24) provides a simple means for predicting the turbulent velocity profiles of any polymer solutions, at least within the Reynolds number range of 4,000 to 15,000. It is thought that this correlation can be applied to flows with higher Reynolds number, although this has yet to be specifically tested.

6 Conclusions

On the basis of the Prandtl well-known mixing length model, a simple equation has been develop­ed to predict mean velocity distributions in flows of dilute polymer solutions. Three parameters, a, /? and r, require separate determination before the correlation equation can be utilized. On the basis of the well-known study reported herein, the following conclusions are drawn: 1. a,/? andT are unique functions of u*j{ü) and hence, may be explicitly determined for a given

flow system. 2. The developed correlation, of the form;

a(t/p+ -r) + j3 = 2A3 Iny+ + 6.5

provides a simple and accurate means of predicting the mean velocity distribution for any polymer solutions.

3. Testing of the correlation is, to date, restricted to four types of polymers with concentrations between 10 and 200 ppm and Reynolds number between 4,000 and 15,000.

138 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 30. 1992. NO. 1

Acknowledgement

One of the authors (K. C. Tam) acknowledges the financial support in the form of a post-graduate scholarship from Monash University. The authors also wish to acknowledge the assistance of Dr. W. T. Wong in setting up the LDA system and Mr. T. N. Fang in conduct ing some of the experiments.

Notations

A,B constants in the velocity distribution defined in equat ion (3) ƒ Fanning friction factor fD Doppler frequency shift g gravitational acceleration G empirical constant as defined in equat ion (7) H length of pipe ^i universal constant as defined in equat ion (10) L Prandtl mixing length p pressure P p — Qgz, total po tent ia l r radial posi t ion R pipe radius u velocity component in z-direction ü t ime-smoothed (time-average) axial velocity u' axial velocity fluctuation ( M ) bulk average velocity U+ u/u*, dimensionless local velocity (7 p

+0 in tercept of d imens ion l e s s veloci ty profile of po lymer at y + = 1.0

C/NO intercept of d imens ion le s s velocity profile of N e w t o n i a n fluid at y + = 1.0 u* friction velocity, / ( T W / < ? ) ;7max t ime smoothed max imum velocity «Jrit critical friction velocity at onset of drag reduction v velocity component in /--direction w velocity component in 0-direction y R — r, distance from pipe wall y+ dimensionless distance, {yu*o)jrj y* dimensionless distance, yjR De Deborah number Re Reynolds number a ratio of Prandtl 's mixing length of Polymer to water (equat ion (11)) a* coefficient appearing in equat ion (2) /? constant as defined in equat ion (18) e ratio of u*\{ü) r constant as defined in equat ion (17) t] viscosity 9 half angle of intersecting beams A wavelength of laser source; relaxation t ime in Deborah number

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f dimensionless distance from the wall defined by equation (8) Q fluid density aua2 function of De defined by equations (6b) and (6c) respectively f/z' laminar shaer stress or laminar momentum flux f£' Reynolds stress or turbulent momentum flux T W wall shear stress 0 slope of the velocity profile

Subscript P polymer N Newtonian (water) O Boundary condition at z = 0 H Boundary condition at z = H SL laminar sub-layer E outer edge of the elastic sublayer as defined by Virk (1975)

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