Flow of dilute polymer solutions in rough pipes

(1)

IN ROUGH PIPES

by

Michael Poreh

Sponsored by Office of Naval Research Contract Nonr-1611( 03)

'

liotheek van d

Afdeling Scheeps

r'-e ,

DOCUMENTATIE

56 -12,

IIHR Report No. 126

Iowa Institute of Hydraulic Research

The University of Iowa

Iowa City, Iowa

April 1970

This document has been approved for public release and sale; its distribution is unlimited.

(2)

Flow of Dilute Polymer Solutions in Rough Pipes

' INTRODUCTION

The use of polymers to reduce friction losses has _{been intensively}

investigated during the last decade, but most of the investigators have

been concerned with flows past smooth boundaries. _{The question of drag}

reduction in the case of rough boundaries is of practical importance since most commercial pipes and surfaces of marine vehicles usually have some

degree of roughness. _{Experiments with rough pipes [1, 2, 3] have shown that}

polymers are less effective in this case and that drag reduction tends to

zero as the roughness is increased. _{No theory has been offered to explain}

these observations.

A simplified analytical mode], describing the _{combined effect of} uniform roughness and PolYmer solutions on the floV-it propoded herein

The effect of nonuniform roughness,

with

_{And without polymers, is then}

re-lated to the case- of uniforM roughness.

Flow-of dilute polymer solutions in smooth pipes. _{Although the}

exact mechanism of drag reduction is not fully understood, _{it is apparent from}

velocity-distribution measurements that drag reduction is associated with an

increased thickness of the viscous sublayer and that the _{structure of the}

flow outside the wall region is hardly changed in dilute _{polymer solutions.}

It is also evident that drag reductioh occurs only when the shear stress near

the wall exceeds a critical value. _{Measurements in smooth pipes have led}

Meyer

[4]

_{to suggest that the mean longitudinal velocity u in dilute polyMer}

solutions is described by the logarithmic equation

u/V* A log (tV*/ ) + B + Aut

(1)

au+

= a log (V*/V*crit)

₍₂₎

where z is the distance from the wall, a is a concentration dependent

parameter, V*

= VT7T7

is the shear velocity, V* is the shear velocity crit

(3)

.Equation (1)

is

assumed to be valid for V* > V* whereas for

crit +

V* < V* Au: 77: 0, Equation (1) may also be written as

crit

u/V* = A log.[(zV*/v )(Vs/V:rit)a/A].

+' ( 3)

Integrating (1) over the pipe cross section, an approximate

ex-pressiOn for the Darcy-Weisbach friction coefficient, is obtained:

1/V77= A

log Re

IF+

+ At+/V

.

(4)

The integration gives A. 2.03 and B hoWeVer,

T = 240

and = -048 give better agreement with the data and are usually assumed.

According to (2) the value of Au+ depends on two characteristic

parameters of the polymer solution: a, and V4L.it. Data collected by

different investigators, with Supposedly the same polymer solutions, however,

do not always yield identical values of a or One reason for the

large scatter in the measured values of a

and Vrit

is undoubtedly the

different characteristics of polymers which have the same brand name. In

addition, polymer solutions can be degraded during the preparation of the solution or the experiment itself. The success of models used to predict

the values of a and V* theoretically have so far been limited, and

crit

it is thus required to determine them experimentally [5,

6, 7, 8, 9].

Flow of Newtonian fluids in rough pipes. Like

roughness leaves the structure of the flow away from the

the velocity-defect law remains valid in both cases [10] possible to express the velocity profiles of a Newtonian

boundaries by the equation

polymers, surface wall unaltered and

, It is 'therefore

fluid near rough

Where F

is

the downward shift of the logarithmic profile due to surface

roughness. The dimensionless parameter F is found to be a function Of the

(4)

'Reynolds

number.

Of the roughness, .kV/\, where k is the Characteristic size of the roughness. Its exact form depends, however, _{on the shape of} the roughness.

Figure 1, taken frot Schliehting [10], _{describes the measurements}

of Niktradse for pipes with uniform sand

_roughness..

_{The parateter} _B' _{_ in}

.thit figure is related to

F by

3' =5.75 log (kV*/v) ± 5.5

-(6)

The broken line in Fig. 1 describes the following approximation of F:

F(7,t) = 0

w < 3,35 j (7.1)

F(w) = 0.26 (w - 3.35) , 0,0026 (w - < < 20 , _(7.2)

F(w) = 5,75 log .(N.T - 2.0

_17.4./V77-)

-

3.0 ; 20 < w , _(7.3)

where w = kV*/v.

The asymptotic form of F for very large values of kV*/v -is

Fas. = A log (kV*/v) t C ₍₈₎

where C = -3.0. _{Thus, the velocity profile for large}

kV*/v is given

by

U/V = A log (z/k) + B- C

(9)

and is independent of the viscosity.

In the region where Fat. > _{(5) may be written in the form}

V* = A log (zV*/y) _{= Fas.} _R(kV*/v) ₍₁₀₎

where

R(kV*/ )

- F(kV*/v)/Fas. (11)

The Reynolds httber kV*/v _May _{be expressed in terms of the ratio Of k}

(5)

thickness-can be defined by the intersection of(1) and the equation -u/V* = tV*/v. Accordingly, for Newtonian fluids without polymers,

(SV*/v = A log (dB*/vd) + B ₍₁₂₎

which gives, for A =5.75 and B = 5.5,

avif/v = 11.6 ₍₁₃₎

kV*A) = (6V*/v) (k/6) = 11.6 k d . (14)

Equation (6) may thus be written as

u/V* = A log (tV*/v) + B -F p(k/o) (15)

Where-, p(k/6) = 0 ; k/d < 0.29 p(k/d) = {0.26 [11.6 (k/6) - 3.35] - 0.0026 [11.6 (k/6) - 3.35] log (11.6 k/d) - 3.0] ; 0.29 < 3/c3 < 1.72 P(k/6) = {5.75 log [11.6 k/d - 2.0 - 17.4/(11.6 k/6)] - 3.0}/ /[5.75 log (11.6 k/6) - 3.0] ; 1.72 < k/d _(16.3)

A MODEL FOR FLOWS OF POLYMER SOLUTIONS IN PIPES WITH UNIFORM ROUGHNESS Since neither rough boundaries nor polymer additives affect the structure of the turbulent flow away from the wall, it is reasonable to assume that the combined effect of both will not alter the velocity defect law either. _{It is therefore possible to express the velocity distribution,} see Fig. 2, as

u/V* = A log (tV*/ ) + + B.

(6)

or

-5-Gas. = A log [(kV*/V)(V*/V* .t ) gri

As already pointed out by Spangler [1], there is no drag reduction in very

.rough pipes and.the velocity

profile

is therefore. described by (6). This

in-dicates the .asymptotic value of G

in

(18) is

- 3.0 . (19)

-Thus the function G

in

(18) Varies from zero at small kid and approaches

G as kid becomes large. Since (5) and.(8).can be considered the special

-

-Case of

(18)

And (19) when

a

= 0; the assumption will be made that, similar

to (15),

u/V* = A log [( V ./

_{)v*./V:t,it)c'}

] Gas: P(X/6) (20)

where

Gas. is given by (19), p is given by (16) and determined

by

dV*/v = A log (61/*/v) + B + a (log 1/*/Vrit) (21)

This satisfies the condition that

Eqs.

(20) and (21) reduce to (15) and

:(12) When a = 0. The paraMeter AB in equation (17),WhiCh describes the

botbined effect Of the polymers and the roughness on the velocity profile (see Fig. 2), is thus given by

AB = log (V* - Gas.' p(k/d) = Au+

_-Gas

_P(k/6) (22)

The value of dV*/v for polymer solutions is determined by a

and V*/V* and thus AB is a function of the dimensionless parameters:

crit

a, kV*/v

andkV*rit

c /v. Figure 3 described the variation of LB according

to (22) as a function of kV*/v for kV*it /v

= 6

and a = 17.2. Since

cr

k, V* and a are constants for a given pipe and a particular polymer

crit

solution, an increase of kV*/v in this figure corresponds to an increase

in the velocity. When kV*/v < kV*rit /v the curve of AB for the polymer

c

(7)

and negative thereafter. At

kV*/v >

kV*/vcrit the value of AB increases. due to the effect of the

polymers. _{In this case it is positive over a Certain range of kV*/v, which} indicates that the friction there is smaller than in the case of a-Newtonian fluid in a smooth pipe. At large kV*iv _{the Curve approaches the}

asympto-tic curve for a completely rough flow. _{The broken line} _{a log. (V* "*crit)}

in

this figure designates the value of AB for the flow of this polymer

solution in a smooth pipe.

Similar curves for various values of kV*/v and a = 17,2

crit

are given in Fig. 4. _{It is seen from this figure that for} _{IcIrrit/v > 15}

drag reduction is very small and is limited to _{a narrow range of shear}

stresses. For kV*/v larger than 100, the drag reduction achieved is

small even for efficient polymers with mall _V*

crit'

CONFARISON WITH EXPERIMENTAL RESULTS

Unfortunately, it is not possible to determine the values of a

and V* for all the published data. _{Spangler [1] has reported a large} crit

number of measurements with a solution of Polyhall 295 _{in a smooth pipe}

and in three rough pipes. Pclyhall 295, a product of Stein-Hall _{and CO}

is an efficient drag reducing polymer with _{a molecular weight of 5-6}

millions.

The roughness was produced by threading the inside _{of brass tubes}

with specially-made taps. On the basis of his measurements he _{concluded that}

for the particular concentration tested, c = 31 ppm, a = 17.2 and

V* = 0.0968 ft/sec (2.95 cm/sec). _{The roughness in these pipes which}

crit

we shall designate by 1, 2, and 3, were designed to give _{R/k = 18, 36, and}

72 which correspond to kV4L.it/v a 16, 8 and 4, respectively.

Figure 5 sumiarizes Spangler's measurements of the _{friction factor}

f _{without polymers and compares them with results from Nikuradse's}

measure-ments shown by broken lines. _{The measurements with pipe "2" are quite}

close to Nikuradse's line for P/k = 36 except for a deviation of

approxi-mately 20% at low Reynolds numbers. The data from pipe "1" deviates slightly

from Nikuradse's,line for R/k = 18. A better fit was obtained using R/k = 16

(8)

-7-considerably from the corresponding curve for R/k = 72 and instead

follow the curve for R/k

= 47.

_{These measurements demonstrate clearly the}

fundamental difficulty encountered in predicting friction _{losses in rough}

pipes. _{The value of k}

for a particular type of roughness cannot be pre-dicted with great accuracy. _{Even when an equivalent value of k} _is

deter-mined experimentally by measurements in the region kV*/v > 200, deviations

of more than 20% in _{f may be obtained at the lower Velocity} _range.

Spangleres measurements with Polyhall 295 _{are summarized in Figs.}

6 and

7.

Figure _{6 was presented by Spangler who used the estimated}

values

of R/k:

18, 36,

and 47. The shape of the experimental _{curves in this}

figure is similar to that of the theoretical _{curves presented in Fig. 2.}

The data from pipe "1" shows only small drag reduction near kV*/v = 50

as suggested by the proposed model for this _{range of kV!rit/v.}

The values

of kV*. /v

_crt

corresponding to R/k = 18 and

₁₆

_{are approximately}

₁₆

and 18.

The data from pipe "2" (kV4L.it/v

= 8)

_{follow more or less the}

shape of the corresponding theoretical line. _{The theoretical values of}

AB

are found, however, to be slightly smaller than the _{experimental values near}

kV*/v = 20 _{and larger than the experimental values near}

kV*/v

= 80.

The measurements in pipe "3" appear to be close to the theoretical

curve for kV*it /v

= 5

_{which correspond to} _{R/k = 56.}

cr

.The values of f _{calculated using the proposed model are compared}

with the measurements of f in Fig. 5. _{The difference between the}

theore-tical values which are described by broken lines, and the experimental _{ones is}

only slightly larger than the discrepancies _{found for flows without polymers}

(see Fig. 5).

EXTENTION OF THE MODEL TO NONUNIFORM ROUGHNESS

The derivation of the function p _{in the proposed model for}

uniform roughness is based on the _{assumption that} _F(kV*/v) _in

_{(6) is}

smaller than

Fas. and that the ratio F /Fas. is finite for every value

of kV*/v. _{This assumption is satisfied by Nikuradse's data as well as by}

the measurements of Hama [11] in boundary _{layers on uniformly roughened}

flat plates. _{However, it has been shown by Colebrook [12]}

(9)

in commercial pipes with nonuniform roughness is not necessarily _{zero in}

. the region kV*/u < 3.35,

although its asymptotic value is the same as for

uniform sand roughness. _{A simplified model which explains the effect of}

roughness nonunifornity on F is proposed in this section. Using this

model it is possible to extend to nonuniform roughness the previously proposed method for estimating drag reduction.

Consider a surface with 'xi sizes of roughness elements ki,,kn.

The velocity profile in this case can be described by

u/V* F A log V*/)+ B - P(kV*/u) . (23)

where_ R is the equivalent size of the nonuniform roughness. _{it is proposed}

to estitate the function P as follows:

54(kV*/v) =

I

ai F(kiV /v) (24)

i=1

where F is given by (22), and ai

is

the relative effective influence

of the corresponding size, The coeffiCients ai Are positive and

ai = 1 . (25)

The ,equivalent roughness 17 is determined at large values of EV*iv.

where

P

is equaa to F. Addordingly

F.

(Ey AO

a F ( as. i=1 as' or log (V/ ) - 3.0 = _i=

I

a. [(log (k.V*/v) - 3.0)] It follows that a k ₍₂₈₎

(10)

Consider for example a blend of two roughness elements, _ki

k, with _{al.= a2 = 0.5,} and

k2 = Z.Ic1' Where- Z is a parameter

describing. the roughness honUniformity.. It follows. from (28) that

E _{= Zk1 = k2/} ₍₂₉₎

When ki and a. are known, the function -11(17.V*/v) can be calculated

from (24), Figure 8 describes the shape of P for Z = 2 and 3. When Z > 1 the effect of roughness is recognized for KV*/v > 3.35. Physically,

this demonstrates the effect of the larger elements. On the other hand,

the effect of the nonuniformity at large values of iV*/v is rather small.

The curve for Z = 3 is quite similar to those obtained in commercial pipes

Let us examine now the effect of drag reducing polymers in rough

pipes of nonuniform roughness. Using this model with Z = 2, the values

of AB have been calculated and plotted in Fig. 9 for

EV/v = 4,

8,

and 02. Again, the effect of the roughness becomes dpparent at values of EV*/v < 3.35. It is seen that near kV*/v =20, _{AB is amdller when} Z = 2 than it is when Z = 1; while near kV*/v = 80, the mixed roughness

gives more drag reduction than uniform roughness with the same

E.

As point,

ed out earlier the deviation of the experimental data from the theoretical

curves for uniform roughness (Z = 1) are in the same direction. The

deviation of Spangler's measurements with water at low Reynolds numbers also suggests that the threads produced an effect which is better described

with the mixed roughness model. The values of f for pipe "2" with R/k = 36

and Z = 2 have been calculated and are shown in Fig. 7 by a solid line. This line is a better approximation of the experimental data than the broken

line calculated with Z = 1.

The function P can also be calculated for a continuous

distri-bution of roughness. In this case

CO

P(EV*/v)= J

(k) F(kV*/v) dk , ₍₃₀₎

0

(11)

The equivalent roughness k is determined in this cae by the equation

log E

= a(k) log k.dk ₍₃₂₎

o

Conclusions

The lack of detailed theories for drag reduction and for turbulent flaws in rough pipes makes it impossible to derive a rigorous theory to

des-cribe the effects of polymers on flow in rough pipes, The sitplified model

proposed in this work is based on the assumption that the effect of the

relative roughness size is.similar for flows with and without polymers. The

model appears to be successfUl in describing, at least qualitatively, the

experimental tesults. The deviation of the experimental results from the theoretical calculations with the model is of the same order of magnitude

as the one obtained in flows without polymers.

'Since the efficiency of the polymers depends on the i-elative

roughhess Size, the effedt of nonuniform roughness can be estimated by

cal-culating the effect of each roughness size separately. The effect of

non-uniform roughness calculated in this manner is in better agreement with the experimental results both with and without polymers.

AOknowledgements

The author wishes to thank J.G..Spangler for sending him the

original data pUblished

in [1],

and to thank

L.

Landwebor for reviewing the

manuscript and for his helpful Suggestions.

(k) dk = ₍₃₁₎

(12)

,References

[1] _{Spangler, J.G "Studies of Viscous Drag Reduction with Polymers.}

Including Turbulence Measurements and Roughness Effects", Viscous

Drag Reduction, edited by C.S. Wells, Plenum Press,

_1969.

Brandt, H., MCDOnald, A.T and Boyle, F.W., "Turbulent Skin _Friction

of Dilute Polymer Solutions in :Rough Pipes", Viscous Drag Reduction,

edited by C.S. Wells, Plenum Press,

1969-White, H,, "Some Observations on. the Flow of Characteristics of Certain Dilute Macromolecular Solutions", Viscous Drag Reduction,

edited by C.S. Wells, Plenum Press,

_1969.

Meyer, W.A., "A Correlation of the Frictional Characteristics for Turbulent Flow of Dilute Non-Newtonian Fluids in Pipes", A.I.Ch.E.

Journal, 12, 3, 1966.

Elata, C, Lehrer, J., and Kahanovitz, A., "Turbulent Shear Flaw of

Polymer Solutions", Israel

Journal

of Technology,

4, 87, 1966.

Poreh, M., Rubin, H., and Elata, C., "Studies in Rheology and Hydro-olynamics of Dilute Polymer Solutions", Publication No. 126, Civil

Engineering Department, Technion, Haifa, Israel,

_1969.

Virk, P.S., "The Tons Phenomenon - Turbulent Pipe Flow of Dilute Polymer Solutions", Sc.D. Thesis, Massachusetts Institute of Tech-nology,

1966

(see also Virk, P.S., and Merril, G.W., Viscous Drag

Reduction, edited by C.S. Wells, Plenum Press,

_1969).

Paterson, R.W., "Turbulent Flow Drag Reduction and Degradation with

Dilute Polymer Solutions", Technical Report, Contract

N00014-67-A-0298-0002,

Harvard University,

1969.

Whitsitt, N.F., Harrington, and Crawford, Viscous Drag

Reduction, edited by C.S. Wells, Plenum Press,

_1968.

Schlichting, H., Boundary Layer Theory, McGraw-Hill,

_1959.

Hama, P.R., "Boundary Layer Characteristics for Smooth and Rough Surfaces", Transactions, The Society of Naval Architects and Marine Engineers,

62, 1954.

Colebrook, C.F., "Turbulent Flow in Pipes with Particular Reference to the Transition Region between Smooth and Rough Pipe Laws", Journal

of the Institution of Civil Engineers,

1939

(see also Engineering

(13)

8

transition 5.0

02 0+

0.8

as

10 1.2 14 1.8 1.8 2.0

22

2.4

24

2,8

10

Fig. J. nkuradsets measurements in

pipes

with uniform sand roughness

(according to Schlichting [10]r)(see Eqs.

6 and

7):

. rough completely 10

a,

9

(14)

Fig.

.

Shift of the Idgatithmioi=file due to roughness and polymer additives.

Curve. (I)

flow

in

smooth pipes without polymers..

Curve (2)

-Fiow in smooth pipes with-polymers..

("AUt_7 0)

Curve. (3)

Flow in rough pipes. without polymers.

(F 5. 0)

'Curve

(4)

Flow

in rough pipes with polymers..

(G

.0)

Curve (5)

Flow

in

very rough pipes with and without pblymers.

=

Au4'

(15)

(16)

20

18

16

14 AB 12 10 8

6

4 2 2

4

6

K Vs:fa /9 co

₂₀

₁₅ ₁₀ 10 1 1 1 1 1 1

100 KV*A)

Fig.

4.

Variation of

a

for a = 17.2.

1 1 1 1 1 1

1000

(17)

0.10 _0.05

_0.02

0,01, _

LAMINAR \

I II REFERENCE

PIPE

9 PIPE 2

0 PIPE 3

9

SMOOTH

!III

:I ILI e e 4111 11) seam. a_e 9 ee 9

eeeeee ee

. )

.005

103 Fig.

Friction coefficients without polymers

(experimental data of Spangler [1]).

1 1 1 1 1 1 1 1 1 ,1 I 1 111'11 1 :1, 1 1 I 1

(18)

Fig. 6.

Variation of AB according to Spangler [1].

I0

100

(19)

0.10

_0.05

_0,02

0:01

_.005

103 'I I

11111

I

tIIIII

-Is.

4..1: 16

-

--... _

--11,0 .42....a

Z

-.... ---...., 6.--_ 2* CD.,

CD""--REFERENCE

`-...

."4...'

PIPE

PIPE 2

PIPE 3

SMOOTH

+ft. cD m

0.

t 1 1

11111

e e 00 ° 9 04

Re

Fig. 7.

Friction coefficients with 31 ppm Pplyhall 295 (experimental data of Spangler [1]).

(20)

Fig,

8.

The effect of roughness

honUnifOrmity on

(21)

-20

18 16 14 12 10

8 -6

4 -2

2

6

I I I

111111

I I

1 IIIj

I I I 1 =Mb Fas I I I I

IIIII

UNIFORM ROUGHNESS NONUNIFORM ROUGHNESS Z

2 10 K I

II 1

11111

I I I

III II

100 K V79

Fig.

9. The effect of roughness_nonunifortity

(22)

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1 4 73

S/N 0101-807-6811

(PAGE 1)

Unclassified DOCUMENT CONTROL DATA - R & D

(Seeority class:fie:blob of title, hody of obstmet boa inclexine mmotntion most be entered i%' hen the overall report Is ebb:silted)

1. 04-091r4 A TING AC Ii yiTy (cOlporo(e author)

IOWA INSTITUTE OF HYDRAULIC RESEARCH

O. REPORT SECURITY cLAssTFICATION

Unclassified

Th. GROUP

3. REPORT TITLE

Flow of Dilute Polymer Solutions in Rough

Pipes

4. DESCRIPTIVE NOTES (Type of report apd.inclusive dates) 5. AUTHOR(5) (First name, middle initial; /fist-name)

Michael Poreh

6. REPORT DATE

_April 1970

76. TOTAL NO. OF PAGES. 7b. NO. OF REFS

Sn. CONTRACT OR GRANT NO.

- .Nonr-1611(03) 6. PROJECT NO. Task 062-217 c. ' d.

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IIHR Report No. 126'

gb. OTHER REPORT NO(S) (Any other numbers that may be assigned

this itt-pit)

10. DISTRIBUTION STATEMENT

This document has been approved for pUblic release and

Sale;

its distribution is unlitited..

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY AC.TIVITY

Office of Naval Research

13."ABSTRACT,

. A simplified model is developed to describe the effects

of

boundary roughness on dtag reduction adhieved by polymer additives. The

model is suitable for both uniform and nonuniform roughness. Predictions

of friction coefficients by means of the model are in reasonable agreement

with experimental results.

. ..

(32)

--DD,T,"1.514 73 ./N 0101-60.7. 6 t

Unclassified

(BACK)

_{lJncle.ssified}

Security Classification A-31409 Security Classification

14. _{KEY WORDS} LINKA LINK B LINK C

ROLE WT ROLE WT ROLE WT

Drag reduction.

Polymer

solutiOng

Rough pipes