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1/12

SOCIETYOF PETROLEUMENGINEERSOF AIME

6200

NcvthCentralExpressway

Dallas,Texas 75206

~ R

SPE 4885

TRIS IS A PREPRINT

--- SUBJECTTO CORRECTION

Determination of Laminar, Turbulent, and

Transitional Foam Flow Losses in Pipes

By

R.il.Blauer,B.J, Mitchell,and C.A. Kohlhaas,MembersSPE-AIME,

. .-

~o~~do Schoolof Mines

@Copyright 1974

Ameriwan Institute of Mining Metallurgical. and Petroleum Engineers. inc.

Thispaperwas preparedfor the llthAnnualCaliforniaRegior:aleetingof the Societyof

PetroleumErxgineersf AIME, to be held in SanFrancisco,Calif.,April

4-5 1974.

Permissionto

copy is restrictedto an abstractof not morethan 300 words.

Illustrationsmay not be copied.

. .

The abstractshouldcontainconspicuousacknowledgmentof whereand by whom the paper is present

Publicationelsewhereaftermlblicationin the JOURNALOF PETROLEUMTECHNOLOGYor the SOCIETYOF

PETROLEUMENGINEERSJOURNALis usuallygrantedupon requestto the Editcwof the appropriate

journalprovidedasreementto givepropercreditis made.

Discussionof thispaperis invited.

Three copiesof any discussionshouldbe sentto the

Societyof PetroleumEkgineersoffice,

Such discussionmaybe presentedat the abovemeetingan

with tinepaper,may be ccnsi.deredarpublicationin one of the two SPE magazines.

ABSTRACT

,,

foam in oilwell treatments were the

In arriving at a mekhod for pre-

calculation of friction loss, wellhead

dieting friction losses iillaminar,

pressure, and the resulting density and

carrying capacities during treatment.

transitional, and turbulent. flow re-

gimes for flowing foam, it was found

Lockhart and Martinelli,l devel-

Reynolds numbers and Fanning friction

factors could be calculated with ef-

oped the most popular emperical corre-

lation for predicting friction losses

fective foam viscosity, actual foam

density, average velocity, and true

of flowing two-phase fluids wi~hin

horizontal pipes. This method was

pipe diameter.

Further, it was found

the relationship between Reynolds

sufficiently accurate only for pressure

losses within pipes smaller than 2-

number and Fanning friction factor for

inches in diameter.2

Other emperical

foam was identical to that of single-

3 o5 based on mechanical

orrelations I

phase fluids.

The test data were taken

energy or momentum losses proved in-

under controlled flow in capillary tubes

sufficient also.

and l-1/4-inch and 2-3/8-inch oilfield

seamless tubing.

Friction losses of

foam within 2-7/8-inch tubing and

Bertuzzi, Tek, and Poettmanns6

4-1/2-inch and 5-1/2-inch casing dur-

energy dissipation function which has

ing fracture treatments have been

extensive oilfield use fox predicting

pressure losses within horizontal pipes

accurately predicted using the proposed

for non-uniformly mixed, two-phase

method.

.

fluids was found not reliable for foam.

INTRODUCTION

Failures of these emperical two-

Foam has been used for several

phase correlations for the prediction

of preswme losses for foam flow may be

years as a wellborec lean-out and drill-

lodged in the theoretical developments

ing fluid and to a very limited extent

by Einst,ein7

and HatschekB* .

They

as a fluid loss, diverting, and frac-

turingfluid.

Applications have been

point o~t foam should be treated as a

limited because the behavior of the

single-phase fluid with viscosities

foam could not be confidently pre-

significantly greater than either phase

Mitchelll used these theories to ex-

dicted.

Major problems when using

perimentally find viscosities of foam.

References and illustrations at end.of.

paper.


2/12

DETERMINATION OF IJUKINAR, TURBULENT, AND TRANSITIONAL

-

~nn~ n~nv.~

nnrnm rnn7 r

nccwe

TM DT17i~CS

SPE 4885

L

z u

-L ~, J-lWwv L- ~ ~. K . I. J =* -U

Hel

also showed foam approximately

bubbles are no longer spheres and re-

exhibits Bingham plastic behavior and

form to parallelepipeds during flow.

presented plastic viscosities and yield

strength data.

Einsteins theory is valid for

foam qua+ities less than 0.52. The

Krug and Mitchells12 modifica-

derivation of his two-phase viscosity

tions in existing single-phase fluid-

is based on an energy balance and the

flow equations account for both the following assumptions;

compressibility of the gas within the

foam and the resulting changes in the 1.

Solid spherical particles

viscosity. The resultant equations

are suspended in a homo-

were applicable to the circulation of

foam within a wellbore; however, these

geneous fluid.

equations were limited to laminar flow

2.

All particles are homoge-

and lacked emperical justification.

neous, weightless, and

have identical volumes

The foregoing theories and empir-

and diameters.

icism and additional field and labora-

tory work have been combined to devise

3.

Spacing of the particles

a method for predicting the type of

is uniform and the parti-

regime in which the foam is flowing

cles do not touch.

and a method for calculating the pres-

sure losses of foam flowing within 4. There is no slip at the

horizontal o: vertical pipes.

surface of the particles.

EFFECTIVE VISCOSITY OF FOAM

Einsteins equation for the vis-

cosity of foam is

Foam in this study is a homogene-

ous mixture of air or nitrogen, fresh

water; and a surface-active agent.

llf

The gas phase exists as microscopic

=pl l.orE2.5rT p).. . . 2

?

gas-bubbles suspended in the water and

surfactant solution.

In practice these

Hatscheks theories explain foam

bubbles may occupy between 10 and 95

viscosity for the bubole-interference

percent of the total foam volume.

and bubble-deformation quality ranges.

The interference between the spherical

Foam quality is the ratio of gas

bubbles in foams with qualities between

volume to the total foam volume.

0.52 and 0.74 requires additional work

be applied to initiate and maintain

flow.

This additional work accounts fo

J . . . . . . . . .

T,P Vf

(1)

h+gh foam apparent viscosity. Hatscheks

viscosity for bubble-interference foam

is

Since the gas is compressible, tempera-

ture and pressure of the foam must be

lJf

specified.

= P1 1.0 + 4.5 rT,p) . . . . (

Figure 1 was developed from the

His next argument was the deforma-

theories of Einstein and Hatschek and

tion of the bubbles within foams which

laboratory measurements by Mitchell.

have,qualities above 0.74 causes the

Three separate theories, each with a

geometric shape of the bubbles to pro-

physical model dependent on quality,

teed from spheres to dodecahedra and

were required to fully determine and finally to parallelepipeds, and this

understand foam viscosity.

bubble configuration is the only one

which can flow in laminae.

The vis-

The spherical gas-bubbles in foams

cosity of the foam caused by the shear

with qualities between 0.00 and 0.52

of the fluid between the parallelepipeds

are uniformly dispersed in the liquid

gas bubbles is

and do not contact other bubbles. Flow

is Newtonian.

At 0.52 quality the

spherical bubbles are packed loosely

in a cubic arrangement and begin to

Pf =

Ml+ . . . . . . . (

interfere by contacting each other

T,P

during flow.

Above 0.74 quality the


3/12

,

)?4885 R. E. BLAUER. B. J.

MITI

Mitchell showed that foam behaves

approximately as a Bingham plastic

fluid in fully developed laminar flow.

His shear stress-shear rate relation-

ship for

foams

with shear rates above

20,000 see-l is linear for any quality.

The relationship for foam flowing with

a shear rate below 20,000 see-l can be

linearized by subtracting the apparent

yield strength.

His shear stress-shear

rate equation for Bingham Plastic Foam

is

(T-ry)=llp@ . . . . . . . . ..(5)

Figure 1 shows Mitchells plastic

viscosity and Figure 2 shows the ap-

parent yield strength. He showed

experimentally determined plastic

viscosities agree well with Einsteins

and Hatscheks theories.

Additional work showed ar.effec-

tive viscosity which combines plastic

viscosity and yield point is signifi-

cantly more reliable than plastic

viscosity alone in the determination

of Fanning friction factors. The use

of effective viscosity of Bingham

plastic fluids is technically precise;

however, it is interesting to note

that most Bingham plastic fluids en-

countered in the oilfield have small

yield strengths compared with their

plastic viscosities. Thus, the ef-

fective viscosity and the Bingham

plastic viscosity are of similar magni-

tude and are commonly interchanged.

However, foams high yield strength

requires the use of effective viscosity.

The effective viscosity in this

study is developed with the Hagen-

Poiseuille law and the B.ckingham-

Reiner equation.

The Buckingham-Reiner equation

for laminar flow of

fluids within pipes

Q=

n AP D 9

128 Bp LG

Bingham plastic

is

[1-:>

+*(>)Kl

. . . . .

O.**

(6]

The Hagen-Poiseuille law for

Newtonian

laminar

flow within pipes is

n AP Dq gc

Q

. {7;

~28peL .

IELL, C. A. KOHLHAAS

For the Newtonian turbulent flow

relationships to be valid for a Bingha

plastic fluid, the flow rate in equa-

tion 6 must equal the flow rate in

equation 7. A practical solution of

this equality shows the effective vis-

cosity of the Bingham plastic foam is

gc~D

Ye

= ~p + ~vy . . . . . .

.

Figure 3 illustrates the effectiv

viscosity of foam based on equation 8.

EXPERIMENTAL APPARATUS AND PROCEDURE

Osborne Reynolds experiments in

which streams of dye were injected int

water flowing within pipes showed the

frictional pressure losses were propor

tional to average fluid velocity when

the dye remained as a stream. This fl

regime was called laminar or stream-

line flow.

When mixing of the dye and

water occurred the pressure losses wer

approximately proportional to the squa

of average fluid velocity. Due to the

appearance of the dye mixing with wate

the flow regime was called turbulent.

Therefore, the flow regimes of a fluid

within pipes can be determined by

measuring pressure losses and flow rat

and comparing their functional relatio

ships.

To generat~ these functional re-

lationships for foam two horizontal

capillary tubes and two horizontal

strings of tubing were used in flow

tests.

The large diameter difference

of the tubes and tubing assured any

friction loss correlation developed

would have wide application.

The tube

and tubing also

served to

produce full

developedlaminar and turbulent flow.

regimes, respectively.

Pressures at each end of either

the tubes or tubing, flowing tempera-

ture, gas volume, and liquid weight

were measured for each flow test. A

mass balance, the real gas law, and

derived equations were used to calcu-

1 late foam-volumetric flow rate, qualit

foam average velocity, shear rate, sh

stressl and effective viscosity.

The capillary tubes used were ea

24-inches long with internal diameters

of 0.0483-inch and 0.0924-inch.

The

foam for these tests consisted of an

aqueous solution of 1 Adofoarn BF-1


4/12

FOAM FLOW FRICTIO

9 commercial anionj.c surface active

agent, and nitrogen.

Two calibrated

temperature-compensated gauges recorded

flowing pressures. The foam was flowed

into a plastiic separator which was con-

nected to a wet-test meter.

The separa-

tor was weighed to determine the mass

of liquid which was flowed during a

test.

The effect of quality changes on

&he foam properties were kept to a mini-

num by keeping the tubes discharge

pressure above 500 psi.

Calculated

velocities were less than 20 feet per

second. Figure 4 is a diagram of the

capillary tube apparatus.

The large diameter tubes were made

of eight joints of new J-55 tubing. The

l-1/4-inch 2.40-pound tubing was 194.75

feet long with an internal diameter of

1.380-inch.

The 2-3/8-inch, 4.7-pound

tubing was 181.32 feet long with an

internal diameter of 1.995-inch. An

aqueous solution of 1 Dowell F58B,

an anionic surface-actice agent, was

pumped through the tubing strings at

rates as high as 7 BPM.

Nitrogen was

combined with the water at rates to

3000 SCFM.

This produced foam volu-

metric rates to 34.5 BPM.

A water

storage tank was gauged before and

after each test to determine water mass

and volume rates.

Pump displacements

of liquified nitrogen were used to

determine nitrogen gas volumetric rate.

Pressure losses at the extreme ends of

the tubing during the laminar flow tests

were measured by a single water mano-

meter inclined at 85 from vertical.

Turbulent flow pressure losses were

recorded by six differential pressure

transducers which were connected across

each joint of tubing.

Pressures at eacl

end of the tubing strings Were measured

by two calibrated laboratory gauges.

Test pressures to 3000 psi were used.

Figures 5 and 6 show the tubing appar-

atus.

EXPERIMENTAL CORRELATIONS

Mitchell showed that average foam

flowing velocity and quality can be ac-

curately determined using a mass bal-

ance and the real gas law.

The methods

used to calculate flowing properties in

tubing were only slightly different

from the capillary tube. Equations for

the capillary tube are presented in the

following discussion.

Since the flow tube is in a state

of dynamic equilibrium, the mass enter-

ing the tube as gas or liquid must leav

the tube,

DETERMINATION OF LAMINAR, TURBULENT, AND TRANSITIONAL

LOSSES IN PIPES

SPE 48

M~n=Mout . . . . . . . . . . (9

in which

M~n = Mil + M

+ M

lg gg

+M;lo . (1

M

out

= lC ld

+ lg + Wt + g

gd

. (1

The mass of liquid in the separator

is

s Wlc. , . . .

* (

lC = gc

Applying the real gas law and Dalto

law of partial pressures, the mass of

gas and water vapor flowing through th

wet-test meter is

M

Wt

=M

w

+ lm

(Pm -

Pv)

Vmm

+

v 1 m

RT

9 9

Z1 R T1

The mass of water vapor and

placed from the separator is

(

gas dis

(Pa -

Pv) v~c fiq

M

Vs

gd= Z9RT9

+

v lC 1

. (

lRTl

Total measured flowing mass of liq

and gas is

lt =

gt

thus ,

L+

Pv Vmnl

1 gc zl R T1

,-

(Pa

- Pv) Vmm

RT

9 9

(Pa -

Pv) Vlc Mg

R Tg

.*

9

(

(


5/12

FM *002

m. a.

Pun

UQJS m. u .

A.LL&b-uu.

b.

n.

-U

A.L.IA-u

The volume of gas entering into

Average shear rate and shear rate

solution is,negligible.

Equations 15

at the wall of the tube for a Bingham

and 16 are valid for large diameter

plastic fluid is

tubing if the mass and volume are re-

placed by mass rate and volumetric rate.

@a=@w= * (Zz

Volumes of saturated gas flowing

at the tube pressure and temperature

are determined by the perfect gas law.

Shear stress at the wall is

The average pressure of the flow tube

can be used only if the differential

pressure is small in comparison. Data

.~-

T

4L

. (

points with .high differential pressure

w

Y

and low average pressure were deleted

for the final analysis.

The Reynolds number and Fanning

friction factor are

The volume of saturated gas in the

tube is

vf D of

R=~

v

=V

Mgt R t

Sg

gt

vlv=-

Mg (Pt-pv) Zg

f

AP D

= L ~f Vf2

.*9**

*

tpv

+~

(Pt-Pv) (17)

Plots of the shear stress-shear

rate relationships are given in Figure

7 through 9.

Foams of constant qualit

The volume of liquid in the tube

show two distinct flow regimes.

Below

is

critical shear rates for each pipe

diameter and foam quality the slope of

. lt

the shear stress-shear rate function

It pl

Iv

(18

) 1.0, indicating larninarBingham plasti

flow.

Above this critical shear rate

the slope is approximately 2.0 and the

foam is in turbulent flow. Calculated

True foam quality is

slopes of the turbulent shear stress-

shear rate relationship are between 1

Vs

and 2.01.

T,P= V +V;

(19)

Sg

l?hecritical shear rate for foam

corresponds with a critical Reynolds

number between 2000 and 2500.

Figure

True foam density is

10 is a conventional single-phase flu

Moody diagram with the foam data suPe

lt + C3t

imposed.

The algebraic standard devia

Pf=v

,.

(20) tion of friction factors calculated w

gt Vlt

this data compared with the Colebrook

method normally used for single-phase

fluids is 0.05 for the laminar flow

Average foam-flowing velocity in

regime and 0.188 for the turbulent fl

the flow tube is

regime.

vfv .***.***

APPLICATION OF METHOD FOR PREDICTION

(21) OF FRICTIONAL LOSSES IN PIPES

EXAMPLE CALCULATIONS

The effective foam viscosity, tru

e

density and pipe diameter, and average

1.

Laminar Flow

foam velocity is used to calculate

shear stress at the wall of the pipel

Determine. the pressure-loss

average shear rate, Reynolds

number ?

gradient of 0.80 quality foam

and Fanning friction factor.

flowing within a l.,00-inch

diameter pipe at 1.0 BPM.

)


6/12

DETERMINATION OF LAMINAR, TURBULENT, AND TRANSITIONAL

6

FOAM FLOW FRICTION LOSSES IN PIPES

SPE 4885

.

1. a

Average foam velocity,

f

= 17.16 fps

b.

From equation 8, effec-

tive viscosity, ue =

29 CPS

c. Foam density may be ap-

proximated by neglecting

gas in solution and the

vapor pressure of water.

Approximate foam density,

Pf

= Pl(l-r)

= 1.67 PP9

d. From equation 24, Reynolds

=917 NR < 2000,

umber, NR

therefore, flow is laminar.

e. From equation 6, laminar

pressure-loss gradient,

AP

= 0.590 psi/ft.

AL

2. Turbulent Flow

Determine pressure-loss gradient

for the laminar flow example with

a flow rate of 5.0 BPM.

a.

Average foam velocity,

f

= 85.8 fJX3

b.

Effective foam viscosity,

Pe

= 10.9 Cps

c. Approximate foam density,

Pf = 1.67 ppg

d.

Reynolds number, NR =

1.21 x 104

NR < 2000

Therefore flow

is turbulent

Other wells have been fractured

through 4-1/2-inch and 5-1/2-inch casin

and the calculated wellhead pressures

have been within 2 of the actual pres

sures during treatment.

CONCLUSIONS

1. Foam behaves as a single-phase

Bingham plastic fluid.

2. Effective viscosity of foam

must be used for calculating

pressure losses for flowing

foam.

3. Friction losses for foam may

be determined as for a single-

phase fluid using conventional

Reynolds number and Moody

diagram.

4. Friction losses in oilfield

pipes can be accurately pre-

dicted for any combination of

foam flow rate and pipe size.

NOMENCLATURE

D

=

inside diameter of tube, in

f

= Fanning friction factor

9C

= gravitational constant, 32.2

L=

tube length, ft

M= mass at atmospheric conditions

Lbm

M =

mass at flowing conditions, Lb

E= molecular weight

R

= Reynolds number

e. From Figure 12 Fanning

friction factor, f = .0075

P = pressure, psia

f.

From equation 25, turbulent

AP =

pressure drop between ends of

pressure-loss gradient,

tube, psid

Q=

foam quality

P

= 3.57 psi/ft

AL

FIELD AiPLICATION I R= universal gas constantl

Wellhead pressure during a fracture

10.732

(PSIA)(FT5)

treatment using foam flowing at 14 BPM

(Lb-mole) (OR)

in 2-7/8-inch tubing at 7000 feet was

predicted using the proposed method.

T

= temperature, R

~he total calculated-wellhead pressure

was 5250 psi and measured pressure was

5350 psi.

t

= time, sec

v=

volume, ft3

v

=

velocity, ft/sec


7/12

-. - . ---

M,uAaw xi*\, . u . L,AAALAAUALU* L. n.

nw44.uA-L2

t

w weight, Lbf

GREEK SYMBOLS

z

=

compressibility factor r

T,P

= foam quality at specified

temperature and pressure

SUBSCRIPTS

4

shear rate, sec

-1

=

a

=

average tube

pf =

true foam density, Lbm/ft3

e = effective

T

= shear stress, Lbf/ft2

f = foam

Y

= yield strength, Lbf/ft2

9

= gaseous phase

B

=

viscosity, CP

gd

= gas displaced from separator

ACKNOWLEDGEMENTS

99

= gas-phase in gaseous state

The authors wish to thank Dowell

gl

= gas-phase absorbed into the

and NOWSCO for their assistance in the

liquid phase tubing tests.

gm = saturated gas measured by

REFERENCES

wet-test meter

1.

Lockhart, R. W. and Martinelli, R. C

gt = gas flowing in tube Proposed Correlations of Data for

Isothermal Two-Phase, Two Component

in = entering tube Flow in Pipes, Chem. Eng. Procir.,

1949

v.

45

p.

39

1 = liquid

2. Baker, O.: Experiences with Two-

lC =. liquid in the separator Phase Pipelines, paper presented

before the joint meeting of Cana-

ld =

liquid-phase vapor displaced

dian Natural Gas Processing Assn.

from separator and.the Natural Gasoline Assn. of

America, Calgary, Alberta, Sept. 15

lg =

liquid-phase in g..:~us state

1960.

11 = , liquid-phase in liquid state 3* Dukler, A. E., et al: Frictional

Pressure Drop in Two-Phase Flow:

lm =

liquid-phase vapor measured by

B-An Approach Through Similarity

wet-test meter

Analysis, AICHE J., 1964, v. 10,

p. 44-51.

lt = liquid at flowing conditions

4.

Hughmark, G. A. :

Hold-Up In Gas-

m= wet-test meter

Liquid Flow, Chem. Eng. Progr.?

1962, V. 58, p. 62-65.

out = existing tube

5. Eaton, Ben A., et al:

The Pre-

P = Bingham plastic diction of Flow Patterns, Liquid

Hold-Up and Pressure Losses Occur-

Sg = saturated gas

ring During Continuous Two-Phase

Flow in Horizontal Pipelines, g.

t = tube Petrol. Technol. , June 1967, p. 315-

328.

v

=

vapor

6.

Bertuzzi, A. F., Tek, M. R., and

Vs = saturated gas displaced from Poettmann, F. H.:

Simultaneous

separator

Flow of Liquid and Gas Through

Horizontal Pipe, Trans. AIME,

w=

wall 1956, V.

207 p. 17-24.


8/12

DETERMINATION OF IAMINAR, TURBULENT, AND TRANSITIONAL

8

7.

Einstein, Albert:

Eine Neve

Bestimmung Der Molekuldimensionenl;

Annalen Der Physik, 1906, v. 19?

SFR 4, p. 289.

8.

Hatschek, Emil:

Die Viskositat

Der Di.spersoide. 1. Suspensoide~

Kolloid 2., 1910, v. 7, p. 301-

304.

9.

:

Die Viskositat Der

Dispersoide.

11 Die Emulsionen

Uno Emulsoide,

Kolloid Z.,

1910

V

8 p.

34-39.

IL

o

.--- ---

}

f

5

0

L

0.0

TN PIPlw

SPE 4885

10.

11.

12.

Mitchell, B. J.: Viscosity of Foam

Ph.D. thesis, Univ. Okla., 1969.

.

Test Data Fill

Theory Gap & Using Foam as a

Drilling Fluid,

Oil and Gas J.,

Sept. 6, 1971, p. 96-100.

Krug, J., Mitchell, .B. J.: Charts

Help Find Volume, Pressure Needed

for Foam Drilling, Oil and Gas J.,

Feb. 7, 1972, p. 61-64.

. -

~ e... ..- -

-~

R //

i

-r --~ .

.-

.

I

.

---- ..._j____--.-- - -- -

I

I

I

DISPERSION

llNTERFEf{E~CE

I

I

1/

1

-. -

i /

I

--1

I

I

I

, 3

lDEFOW\TION-

1

I

I

I

L

>

0.4

0.6

OB

I.0

QUALITY


9/12

Fig.2

- Yield stressoffoam.

1oo-

90

In 80

- .- -

8

: 70

G

o

v

U)

60

-. .

s

z 50

u

g

40 ~

LLJ

>

g 30

.

w

u

20 :

-. -.-

10

- . -.

0

--i___-. .,..

0.6

0.2

0.4

QU

--. ..

.

--

-. .-

,

.-.

1

:

~H~AR

RAf~

s~~-1

1

.--..-.. .

.-+

c).6

79

ALITY

Fig. 3 -

Effective viscosity of foan.


10/12

Fig. 4

- Capillary tube apparatus.

100bbl

AQUEOUS

SOLUTION

NITROGEN

TANK

O-3QG0 ~CF

CHECK VALVE

--.

m

CEMENT SLURRY

CHECK

PUMP, O-78PM

5000psi PRESSURE

TRANSDUCER

@

::::Epsi

TO RECORDER

OILF~~L~_Iu.fj_ ~

ADJUSTABLE

.. ---

- ,- ~-

CHOKE

5000 psi,500psi-

TO RECORDER

DIFFERENTIAL

PRESSURE

TRANSDUCERS

Fig. 5 -

Turbulent flow regime tubing apparatus.


11/12

~

CHOKES

-

FOAM

NITROGEN

GENERATOR

55 gal

AQUEOUS SOLUTION

SUPPLY

&-

1,

8 J OI NTS O~LFIELD TUBIN

I

I

J

INCLINED

WATER-MANOMETER

Fig.

6 -

Larninar flow regime tubing ap~.aratus.


12/12

IO J.

;1,0

ntn

L-

1

t

t

t 1

I

I I

I

I

I

-

102

[Os

SHEAR RATE,

L-J

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Fig. 10 -

Floody diagram for foam.

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