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1965

Construction of a capillary viscometer and the study of non-Construction of a capillary viscometer and the study of non-

Newtonian liquids Newtonian liquids

Hsun Kuang Yang

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Recommended Citation Recommended Citation Yang, Hsun Kuang, "Construction of a capillary viscometer and the study of non-Newtonian liquids" (1965). Masters Theses. 7238. https://scholarsmine.mst.edu/masters_theses/7238

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... ··f •

CONSTRUCTION OF A CAPILLARY VISCOMETER "' -' '. ·~ i :

AND

THE STUDY OF NON-NEWTONIAN LIQUIDS

BY

HSUN KUANG YANG I fl ;/() 1\?: f

A

THESIS

submitted to the faculty of the

UNIVERSITY OF MISSOURI AT ROLLA

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE IN CHEMICAL ENGINEERING

Rolla, Missouri

1965

Approved by

(advisor)

ii

ABSTRACT

A capillary tube viscometer was built for the purpose of

investigating the fluid properties of non-Newtonian aqueous CMC

and Carbopol solutions. The viscometer was tested with Newtonian

liquids (glycerine, water and two different oils) having known

viscosities to insure that the viscometer was operating correctly.

The shear stress- shear rate data obtained for different concen­

trations of CMC and Carbopol solutions were correlated using

the simple power law model. The power law constants were only

slightly affected by saturating the solution with iodine and carbon

tetrachloride. Aging had very little effect on the viscosities of

the CMC solutions but had a considerable effect on the Carbopol

solutions.

iii

TABLE OF CONTENTS Page

ABSTRACT ii

LIST OF TABLES vi

LIST OF FIGURES viii

I. INTRODUCTION 1

II. LITERATURE REVIEW 3

A. Classification of Non-Newtonian Fluids 3

1. The Bingham Plastic Model 7

2. The Ostwald - de Waele Model 7

3. The Eyring Model 8

4. The Ellis Model 10

5. The Sisko Model 10

B. Viscometers 11

1. Capillary Viscometer 11

2. Rotational Viscometer 12

3. Other Types of Viscometer 12

c. Treatment of Data from Capillary Viscometer Using the Power Law Model 12

1. Newtonian Fluids 13

2. Non- Newtonian Fluids 14

D. Reynolds Number and Friction Factor 17

E. Effect of Turbulence 19

F. Err.or in Capillary Viscometry 19

III.

IV.

EXPERIMENTAL

A. Object of Investigation

B. Materials

1. Non- Newtonian Liquids

2. Newtonian Liquids

C. Apparatus

1. Liquid Reservoir

2. Capillary Tubes

3. Pressure Gages

4. Piping, Valves and Fittings

5. Pressure Regulator

6. Constant Temperature Bath

7. Weight and Time Measurements

iv

22

22

22

22

25

25

26

29

32

33

33

33

34

D. Operation of the Viscometer 35

E. Inside Diameter of "Thermometer" Capillary 39

F. Test of Viscometer System 39

G. Analysis of Data for Non-Newtonian Liquids 43

H. Correction or Elimination of Data Points 52

DISCUSSION 56

A. Effect of Aging the Solution 65

B. Effect of Concentration of the Polymer 65

C. Effect of Solution on Non-Newtonian Behavior 66

D. Recommendations 67

v

1. Constant Temperature Around the Capillary 67

2. Gaskets

3. Fluid Head Correction

v. CONCLUSIONS

IV. APPENDICES

A. Capillary Data Tables

B. Figures for Aged Non-Newtonian Liquids

C. Notation

VII. BIBLIOGRAPHY

VIII. ACKNOWLEDGEMENTS

IX. VITA

68

68

69

71

101

107

110

112

113

vi

LIST OF TABLES

Table Page

1 Description of Capillary Tubes 30

2 Calculation of Inside Diameter of "Thermometer" Capillary Tube 40

3 Results Using Calibration Liquids 42

4 Constants of Power Law Model 64

5 Capillary Data for 0. 05% Carbopol at 22. S°C 72

6 Capillary Data for 0. 1% Carbopol at 22. 7°C 74

7 Capillary Data for 0. 2% Carbopol at 22. S°C 76

Capillary Data for 0 s 0. 1% CMC at 23. S C 7S

9 Capillary Data for 0. 2% CMC at 23. 5°C so

10 Capillary Data for 0~2% Carbopol (without solute) at 22. S°C Sl

0 11 Capillary Data for 0.2% CMC at 23. 5 C S3

12 Capillary Data for 0. 05% Carbo pol (aged) at 22. S°C S4

13 Capillary Data for 0. 1% Carbopol (aged) at 22. 7°C S6

14 Capillary Data for 0.2% Carbopol {aged) at 22. S°C ss

0 15 Capillary Data for 0. 1% CMC {aged) at 23. S C 90

16 Capillary Data for 0. 2% CMC {aged) at 23. 5°C 92

A-1 Capillary Data for Oil Number 243 at 25°C 94

A-2 Capillary Data for Oil Number 67S at 25°C 95

Table

A-3

A-4

Capillary Data for glycerine at 25°C

Capillary Data for glycerine at 20°C

vii

Page

97

99

Figure

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

LIST OF FIGURES

Basic Shear Diagram

Flow Chart of Pseudoplastic Liquid

Appearance of Turbulence

General Description of Capillary Viscometer

Pressure Vessel and Temperature Bath

Capillary Tube

Flow Chart for Glycerine at 25°C

Flow Chart for Oil Number 243 at 25°C

Flow Chart for Oil Number 678 at 25°C

Friction Factor - Reynolds Number for Oil 0

Number 243 at 25 C

Friction Factor - Reynolds Number for Oil Number 678 at 25°C

Flow Chart for 0. 1 o/o Carbopol at 22. 7°C Showing the Effects of Kinetic Energy and Fluid Head

0 Flow Chart for 0. 05% Carbopol at 22. 8 C

Flow Chart for 0. 1 o/o Carbopol at 22. 7°C

Flow Chart for 0. 2% Carbopol at 22. 8°C

Flow Chart for 0. 1% CMC at 23. 8°C

Flow Chart for 0. 2% CMC at 23. 5°C

Flow Chart for 0. 2% Carbopol (Without Solute) at 22. 8°C

Flow Chart for 0. 2% CMC {Without Solute) at 23. 5°C

viii

Page

4

9

20

27

28

31

44

45

46

47

48

53

57

58

59

60

61

62

63

Figure

20

21

Flow Chart for 0. 05% Carbopol (saturated with iodine and carbon tetrachloride and aged for one year) at 22. 8°C

Flow Chart for 0. 1% Carbopol (saturated with iodine and carbon tetrachloride and aged for one year) at 22. 7°C

22 Flow Chart for 0. 2% Carbopol {saturated with iodine and carbon tetrachloride and aged

ix

Page

102

103

for one year) at 22. 8°C 104

23 Flow Chart for 0. 1% CMC (saturated with iodine and carbon tetrachloride and aged for one year} at 23. 8°C

24 Flow Chart for 0. 2% CMC (saturated with iodine and carbon tetrachloride and aged for one year) at 23. 5°C

105

106

1

I. INTRODUCTION

Real fluids have been classified into the two categories of

Newtonian or non-Newtonian according to their behavior under

imposed shearing forces. In Newtonian fluids the shear stress

is linearly related to the shear rate; whereas, in non-Newtonian

fluids this relationship is not, in general, linear. There exist

quite a number of shear stress - shear rate functional relation­

ships describing non-Newtonian fluids; most of these relations

are semi-empirical. Many investigations of the rheology of non­

Newtonian fluids are under way.

In recent years, the problems of describing heat and mass

transfer processes in non-Newtonian fluids have begun to attract

the attention of research workers. In this department, recent

studies of mass transfer from liquid droplets falling in non­

Newtonian liquids have indicated a considerable difference between

the mass transfer mechanism in Newtonian and non-Newtonian

systems. (1 ). In order to help explain these mass transfer dif­

ferences, it was felt that the fluid dynamic characteristics of

the non-Newtonian liquids used in these mass transfer studies

should be investigated.

Therefore, the purpose of this project was to construct

a capillary tube viscometer and study the rheological properties

2

of the non- Newtonian fluids studied in reference ( 1 ). Although

the aqueous non-Newtonian systems used are rather common, the

effect of the added solute iodine was unknown and had to be investi­

gated in this project. Two different concentrations of carboxymethy­

cellulose (CMC) from the Hercules Powder Company in distilled

water and three different concentrations of carboxypolymethylene

(Carbopol) from the Goodrich Chemical Company in distilled

water were used in this project. The concentrations and tempera­

tures were the same as in the mass transfer experiments (1).

Before the non-Newtonian fluids were examined, it was necessary

to test the viscometer with Newtonian liquids of known viscosities.

3

II. LITERATURE REVIEW

The literature review will be divided into the following

sections: (1) a general discussion and classification of non-

Newtonian fluids; (2) a brief discussion of types of viscometers;

(3) a detailed analysis of the relations necessary to describe a

"power-law" non-Newtonian fluid flowing through a capillary

viscomter; {4)Reynolds numbers andfrictionfactors; (5) effect

of turbulence; and (6) errors in capillary viscometry.

A. Classification of Non-Newtonian Fluids

The discussion presented in this section will be very

brief. For more details, the reader is referred to other sources

(2-13).

Any discussion on non-Newtonian fluids should start with

a description of a Newtonian fluid. A plot of the shear stress

Trx (force per unit area) versus shear rate dux/dr for a New-

tonian fluid should give a straight line through the origin. See

figure 1, page 4. Mathematically this is expressed as

(2. 1)

where T rx = shear stress, force per unit area

= flux of x-momentum in the r -direction (4)

4

Bingham Plastic

Newtonian Dilatant

To Pseudoplastic

SHEAR RATE, du/dr

Figure 1 • Basic shear diagram

5

= fluid velocity in the x-direction

= rate of shear

IJ.N = the "viscosity", the slope of the straight line

For pipe flow with the notation used above, x refers to the

axial direction in a pipe and r refers to the radial direction. The

subscripts on T and u will be dropped later for convenience. The

viscosity of a given Newtonian fluid is a constant for a given tern-

perature and pressure, and values for the viscosity of many

Newtonian fluids may be found in the literature.

Equation (2. 1) may be used to derive the following equation

for laminar flow of an incompressible fluid in uniform, circular

ducts:

where Q = volumetric flow rate of fluid, volume per unit time

L = length of pipe

R = radius of pipe

{2. 2)

~p = pressure drop of the fluid over the distance

Lin the pipe

For a given R, !JoN and L, Q is linearly related to AP.

For such fluids as colloidal solutions, polymer melts and

solutions, clay and paper pulp slurries, dispersions and certain

lubricating oils, the simple linear relation as given by eq_ua:tion

6

(2. 1) is no longer applicable. These fluids are defined ther f • e ore,

as non-Newtonian. At the present time there is no simple re _

lation between shear stress and shear rate which is applicable to

all non-Newtonian fluids.

Most non-Newtonian fluids may be described by the re-

lation

t::: ""~a(du/dr) (2. 3)

where f.La may be a complex function

~a ::: f1 (r. du/dr, t) (2. 4)

::: apparent viscosity

In this review, only time independent non- Newtonian fluids Will

be considered, where the following rna y hold:

f.La = f 2 (t", du/ dr) (2. 5)

In the regions in which f.La decreases with increasing rate of shear

(-du/dr), the behavior is termed "pseudoplastic"; 1n regions in

which f.La increases with increasing rate of shear, the behavior

is termed "dilatent" {4 ). See figure (1) for a qualitative description.

There are a number of relations which have been proposed

for the function f 2 . Each relation contains empirical constants.

The choice of the proper relation can not apparently be predicted

in advance of experimentation. Chemical structure of the con-

stituent molecules plays an important role in the relation between

7

the shear stress and shear rate. Examples of some of these re-

lations are given as follows:

( 1) The Bingham Plastic Model. Fluids obeying this

model have a threshold value of the shear stress, 7:'0 , which

must be exceeded before flow can occur. See figure 1. This

is expressed mathematically as follows:

l = r 0 - f.Lo (du/dr) (2. 6a)

du/dr = 0 (2.6b)

For this model !J.a, the apparent viscosity, can be shown to be

f.La =To - f.Lo du dr

-(du/dr) (2. 7)

Thus, the apparent viscosity of a Bingham plastic decreases

with increase in rate o£ shear. Examples of fluids which have

been stated to approximate this model are drilling muds, sus-

pensions of chalk and sewage sludge ( 14 ).

(2) The Ostwald - de Waele Model

' = - K] du/ dr 1 n-l (du/ dr) (2. 8)

This: two parameter equation is also known as the "power

law" model (4 ). For n equal to unity, this model reduces to

equation (2. 1) for Newtonian fluids, where f.LN = K. Thus the

deviation from unity indicates the degree of deviation from

Newtonian behavior. For values of n less than unity, the behavior

8

1s pseudoplastic, whereas for n greater than unit, the behavior

is dilatant (4 ). See figure 1.

The fluids studied in this project were pseudoplastic over

the range of shear rates studied, and the power law with n < 1 was

employed. It should be noted that frequently n is a function of

shear rate. Ram and Tamir (5) have described this behavior

(although not restricted to the power law model) which is shown

on figure 2, page 9 for a typical pseudoplastic fluid. Note that

a Newtonian region (f.L=constant} appears to exist at very low shear

rates and another appears to exist at higher shear rates. At

medium shear rates, a pseudoplastic structure region appears.

When attempting to extend the flow curve by increasing the shear

rate at high shear rates, a sharp increase of apparent viscosity

appearswhich indicates turbulence. However, "apparent" vis-

cosity is restricted by definition to the ratio of shear stress to

rate of shear in the laminar region only (5 ).

(3) The Eyring Model.

r = ao arc sinh (-.!. du_) a1 dr

(2. 9)

This model predicts pseudoplastic behavior at finite values

of L but reduces asymptotically to Newtonian behavior, with

1-LN = a 0 /a1, as T' approaches zero. The constants a 0 and al' are

in practice empirically determined (4 ).

/

/ /

/

Turbulence

7

/ /

SHEAR RATE, du/dr

Figure 2. A typical complete flow curve for a

pseudoplastic liquid

9

10

(4) The Ellis Model.

(2. 10)

In this implicit function for ?', the constants <f> 0 , <1>1, and

a are empirically determined. When a is greater than unity, the

model approaches Newtonian behavior for small T; if a is less than

unity, Newtonian behavior is approached for large 't . Notice that

the model reduces to equation {2. 1 )(if <P1 = 0) and equation (2. 8)

(if <}> 0 = 0) as special cases {4).

(5) Sisko Model. ( 15)

b2 = b 0 (du/ dr )+ b 1 (du/ dr} (2. 11)

This model is useful because it predicts a limiting apparent vis-

cosity at large shear rates, a phenomenon which is often experi-

mentally observed.

The Bingham plastic, Ostwald - de Wade, and Sisko model

would appear to be special forms of an even more complex

empirical equation {25 ). c3 -1

~ = c 0 + (c1 + c 2 ] du/ dr l } {du/ dr) (2. 12}

Of course, with four constants this relation could be made to fit

almost any flow curve.

(6) Reiner .. Philippoff Mode1.(17)

- du =t 1 dr IJ. oo + IJ.o - IJ.oo

1 + {f/rs>2

(2.13)

11

where !J. 00 , IJ.o and Ls are adjustable parameters.

B. Viscometers

There are many types of viscometers available; most are

reviewed in the book by Van Wazer, Lyons, Kim, and Colwell

(17). The types most commonly used are the capillary and rota­

tional viscometer s.

( 1) Capillary Viscometer. There are two basic types

of capillary viscometers. In the first, the fluid flows through

the capillary due to the fluid "head" of the liquid itself. This

type of viscometer is not very useful for non- Newtonian fluids

because only one point on the shear stress - shear rate curve

is obtained for a given capillary tube. In the second, the fluid

is forced through the capillary by means of different applied

pres sure s (usually by means of gases). This type of viscometer

is often called a "rheometer". With a set of interchangeable

capillary tubes of different diameters and a means of adjusting

the applied pressure, the operating rang~ of the rheometer is

almost unlimited. This type of viscometer is widely used in the

study of non-Newtonian liquids, and a viscometer of this type

was constructed and used for this research project. The data

obtained from this viscometer are the pressure drop across a

measured length of tubing of known diameter and the weight of

fluid flowing through the tube in a known time.

12

(2) Rotational Viscometers. There are many types of

rotational viscometers, e. g., cone-plate, coaxial cylinder,

rotating disks, etc. The viscosity is determined by measuring

the torque required to rotate a cone, disk or cylinder (depending

on the type) in the viscous medium at a definite angular velocity.

These viscometers may be used for non-Newtonian fluids; however,

they are difficult to construct and usually have a more limited

range of shear rates than the capillary viscometers. There are

many commercial varieties available (17).

(3) Other Types of Viscometers. The rate of movement

of an object through the viscous medium, such as in the Falling-

Ball or Rolling-Ball viscometers, can often be related to the

viscosity of the liquid. Another type of viscometer measures

the resistance to flow by the damping of a rapidly vibrating reed.

The reader is referred to Karam (22) and Van Wazer, et al., (17)

for information concerning other types of viscometers.

C. Treatment of Data from Capillary

Viscameters Using the Power Law Model

The power law model will be used in this work to approxi-

mate the shear stress ... shear rate relation, and the constants

in this "law" will. be determined from capillary tube data. In ,,

this section, expressions for r w (shear stress at the tube wall)

13

and (du/ dr >w (shear rate at the tube wall) will be related to the

constants in the power law model and capillary tube data.

Newtonian fluids will be considered first, followed by a detailed

consideration of non-Newtonian fluids.

(1) Newtonian Fluids. The momentum flux distribution

in a circular tube was first derived by G. G. Stokes in 1851 and

is expressed as follows:

'r= AP r 2L

For a Newtonian fluid, 't is given by equation (2. 1 ).

- 1-LN (du/dr) = (AP/2L)r

- du/dr = (AP/2LiJ.N)r

(2. 14)

Thus

(2.15)

(2. 16)

For a given value of AP, L and 1-LN we see that 7: and du/ dr are

functions of r. It will be convenient later in the analysis of non-

Newtonian fluids to consider just the shear stress and shear rate

at the wall and determine these values in terms of AP, L, Rand

U where U = Q/-rr R 2 = average mixing-cup velocity of the liquid.

The shear stress at the wall is

lw = RAP/2L (2. 17)

The shear rate at the wall (- du/ dr) is obtained as follows: Equation

(2. 15) is integrated, assuming no slip at the wall, to give

u = APR2 [ 1 - (r/R)2]

4LtLN

(2. 18)

Noti<:e that since this is a Newtonian liquid 1-LN is constant across

14

the tube cross-section. The average mixing-cup velocity of the

fluid, U, rna y be determined from the following relation:

U -- loR J< u21Trdr

-f"oRn-----j< 21T rdr

(2. 19)

Substituting equation {2. 18) into equation (2. 19) and integrating,

one obtains

{2. 20)

at the wall

T w = - jJ. N (du/dr>w (2. 21)

From equation (2. 17) and (2. 21)

- (du/dr>w = RAP/2L!J.N (2. 22)

Rearranging equation {2. 20) gives

4 U/R= RAP/2L!J. N (2. 23)

From equation {2. 22) and (2. 23 ):

- (du/dr>w = 4 U/R (2. 24)

Thus measuring AP, L, R, and U one could plot Tw versus

- (du/ dr >w using equations (2. 17) and (2. 24 ).

(2) Non-Newtonian Fluids. For non-Newtonian fluids,

expres.sions will be developed for tw and - (du/ dr >w· Equation

(2. 17) is equally valid for non-Newtonian and Newtonian fluids;

howeve'!l', equation {2. 24) must be modified for non-Newtonian

fluids. Metzner and Reed (20) modified equation (2. 24) using

15

the Rabinowitsch-Mooney (18, 19) relation; this relation is given as

follows:

- (du/dr) w = 3 (QhrR 3 ) +(RAP) d(QhrR 3) 2L d(RAP/2L)

Substituting Q = UrrR2 into equation (2. 25)

-(du/dr)w= ~ ( 4U) + _!_(4U) 4 R 4 R

or

d(4U/R) 4U/R

d(R~P/2L)

RAP/2L

- (du/dr)w = 3 ( 4U) + I ( 4U) d[ ln (4U/R}] 4 R 4 R d( ln (RAP/2U]

or

- (du/dr>w =[43 +

41 d[ ln {4U/R)] } ( 4 u)

d[ ln (RAP/2L)] R

Let

n' = d( ln (RA.P/2L)] d [ ln ( 4 U I R )]

where n' is called the "flow behavior index"

(2. 25)

(2. 26)

(2.27)

(2. 28}

Equation (2. 28) is applicable to Newtonian fluids and such non-

Newtonian fluids as Bingham Plastics or Power Law Fluids. In

order to use equation (2. 28 ), one must determine n', which is

the slope of the curve ln (RAP/2L} versus ln(4U/R). I£ n' is a

constant, equation (2. 27} may be integrated to give

16

I

(R.6.P/2L) = K' (4U/R)n (2. 29)

K' the integration constant, is called the "consistency index".

From equations (2. 17) and (2. 29 ), the following is obtained:

L w = K' (4U/R)n' (2. 3 0)

The relation between equation (2. 3 0) and the Power Law

Model [ equation (2. B)]

L = - K (du/dr)n (2. 31)

is of interest and will now be derived (2 ). Basically a method

for determining K and n from K' and n' will be developed.

Consider equation (2. 31) to apply at the tube wall.

L w = -K (du/dr~

and therefore

ln !w = lnK +!-n ln (-du/dr)w

or

d(lnt"w) = nd [ln(-du/dr)w]

or

n = d (1n t'w) _ d [ ln (-duL d:r:-~]

From equation (2. 28)

ln (-du/dr)w = ln ( 3n' + 1 )·· + In (4U/R) 4n'

Equation (2~35) is· divided by d On'Twl to give:

(2. 32)

(2. 33)

(2. 34)

(2. 35)

17

d[ ln(-du/dr)w] d(lnTw>

= d pn ( 3~~,+ 1)) + dln (4U/R) (2. 36)

d(lnTw>

Comparing equation {2. 33) and (2. 36) gives:

n = d {ln 'Z'"w) d ( ln ( 3 n' + 1 ) J + d ( ln { 4 U / R )J

4n'

= 1 d [ ln (3n1 + l)J - d (ln4n') + 1

d(ln Tw) n'

= n' {2. 37)

1 - ( 1 ) f dn' ] . 3n' + 1 d (ln'tw)

From equation {2. 3 7 ), it is seen that when n' is constant with

shear stress, n is equal to n'. Also when n' is a constant, one

may find by comparing

T w = K' (4U/R)n' {2. 3 0)

with

{2.38)

that

K' * . , n'

K(3n'+l) = 4n'

{2. 39)

D. Reynolds Nu,mper and Friction Factor.

Data for non-Newtonian fluid flow are often correlated in

teriQS o£ modified Rey;1;1olds nUlllbers and friction factors {3, 6-9,

20,,, Z;3 ). :'f:h~ 4:is,cussi,.n, in tl)is s~ction is limited to fluids

approximately described by the power law model.

The friction factor is the same as the usual Fanning

friction factor

where D = pipe diameter

P = fluid density

gc = gravitational constant

If equation (2. 29) is incorporated into equation (2. 40), the

following is obtained:

or

if

f = K' (8U /D)n'

euZ/2gc

f = 16gcK' sn'-l

n' 2-n' 0 D u r

'{ = gc K' gn-1

f = 16 '{

nn' u2-n'r

18

(2.40)

(2.41)

{2. 42)

(2. 43)

The relation between the Fanning fraction factor and the

Reynolds number for Newtonian fluids in laminar flow is

f = 16/NRe (2. 44)

A Reynolds number for n~~-Newtonian fluids ~ay be obtained

from equationS {i. 43) and {Z. 44)

19

(2. 45)

Brodkey gives other Reynolds numbers (23 ).

Capillary data for systems approximated by the power

law model may be plotted in the form off versus NR.e· This

plot should overlap the straight line f = 16/N' in the laminar Re

region. This type of plot provides a critical test of the accuracy

of the data and calibration of the viscometer (24 ).

E. Effect of Turbulence.

The reader is referred to references (8) and (9) for in-

formation on turbulent flow in non-Newtonian fluids. In this

work, the rheological parameters are obtained in all cases for

conditions of laminar flow in the capillary tube viscometer. It

is possible to detect the onset of turbulence by observing the

nature of the shear stress-shear rate curves. For example,

in figure 3, page 20, at a shear rate of about 1. 5 x 104 , the

data for the largest tube suddenly departs from linearity and

moves upward. While departure from linearity is possible for

fluids not obeying the power law, the sudden change is generally

thought of as indicating the onset of turbulence.

F. Error in Capillary Viscometry

Bowen (11) and Van Wazer, et al. (17) discuss many of

E-< <X;

({) (/)

~ E-< ({)

20

0 0.00525" I.D. tube 4~-

0.00525" I.D. tube-}~

I ---l--

0 0.00126 11 I.D. tube

1 0.00063" I.D. tube

I

10-1 ~--~~~------~----~--~--~~----~ 4 6 2 4 6 8 105 2

FLOW FUNCTION, 8U/D, sec-1

Figure 3. Flow chart for 0.05% Carbopol (saturated with

iodine and carbon tetrachloride and aged for one

year) at 22.8°C showing the appearance of

turbulence in the largest tube

it- N;e> 2100

Note: Kinetic energy correction have been made.

m=2.16 in m u2/gc was found in this work to correlate

the data.

21

the possible errors in capillary viscometry and suggest possible

corrections.

End effects may lead to erroneous results when the ratio

of length to inside diameter of capillary tubing is low. It has

been suggested that when the ratio is greater than about sixty­

five, entrance effects due to the sudden constriction of the fluid

streamlines are negligible. At the entrance to the capillary there

are kinetic energy changes which lead to a pressure drop. The

loss in kinetic energy is usually expressed as mfu2 I gc in

lbf/sq. ft. The constant m varies from 0. 5 to 1. 55; however,

it has been suggested that a value of 1. 0 be used (11 ). Viscous

end effects, thermal effects, and wall effects will not be discus sed

here.

22

III. EXPERIMENTAL

A. Object of Investigation

The object of this investigation was to study the rheological

properties of aqueous non-Newtonian liquids. It was necessary

to design and construct a suitable capillary viscometer. This

viscometer was tested with Newtonian liquids before the non­

Newtonian liquids were analyzed. The non-Newtonian aqueous

CMC and Carbopol solutions were, in almost all cases, saturated

with iodine and carbon tetrachloride to simulate the conditions of

the mass transfer studies. In two cases, the solutions were not

saturated with iodine and carbon tetrachloride in order to study

the effect of these solutes on the fluid flow process.

B. Materials

All materials described in this section refer to the liquid

systems used in this study. Materials used in the construction

of the viscometer will be discus sed in the next section.

1. Non-Newtonian Liquids.

CMC Solutions. Two different concentrations of

sodium carboxymethylcellulose {CMC) in distilled water were

prepared: a 0. I weight percent and a 0. 2 weight percent solution.

The CMC, which was in the £orm of a powder, was slowly added

to the distilled water in an agitated tarik in order to prevent the

23

formation of lumps of the polymer. The stirring process con­

tinued for about eight hours. Prior to the addition of the polymer

the distilled water was saturated with iodine and carbon tetrachloride,

a process which took several days. After the polymer was added

to the water, additional iodine and carbon tetrachloride was added

to insure saturation. The solutions were placed in an air tight

glass container until they were used (usually within two days) in

order to prevent the sublimation of the iodine.

In order to study the effect of the iodine and carbon

tetrachloride, a 0. 2 percent solution of CMC in pure distilled

water was also prepared.

In addition to these solutions, the original solutions used

in the mass transfer studies (during the previous year) were

available. The rheological properties of these "aged" solutions

were studied in a peripheral set of experiments which were

actually unrelated to the original objective of the investigation

but which was of interest to the investigator.

The polymer used .in this investigation and in the original

mass transfer studie.s was from the same batch:

So<;lium carboxymethylcel~ulose (CMC-7HP)

f#gh ,yisco.sity p~~mium grade. Lot number 44077

Hercules Powder Company

24

Carbopol Solutions. Three different concentrations of

carboxypolymethylene (Carbopol) in distilled water were studied:

0. 05 weight percent, 0. 1 weight percent and a 0. 2 weight percent

Carbopol. The Carbopol powder was slowly added to the water in

an agitated tank. According to a letter of instructions from the

B. F. Goodrich Chemical Company, it was necessary to add 0. 42

parts of sodium hydroxide per part Carbopol (by weight). The

sodium hydroxide was added in the form of a 1 Oo/o aqueous solution.

The above solutions were saturated with iodine and carbon tetra­

chloride. A 0. 2 percent solution of Carbopol in pure distilled

water was also prepared to study the effect of the absence of

iodine and carbon tetrachloride. One-year "aged" solutions of the

above three Carbopol concentrations were also studied.

Carboxypolymethylene (Carbopol 934)

Commercial grade. Lot number 125

B. F. Goodrich Company

Additives in the Non-Newtonian Liquid-s.

Iodine.

Baker Analyzed Reagent

A. C. S. Specifications

J. T. Baker Chemical Co.

Carbon Tetrachloride.

Comhlercial grade

25

2. Newtonian Liquids. The test runs on the viscometer

were made with thr,ee different liquids:

Glycerine.

Fisher Scientific Company

Fisher Certified Reagent

Cat. No. G-33

Hydraulic Oil.

Socony-Mobil Oil Company, Stock Number 243

Solvent Refined Naphthenic 500 Second Oil.

Socony-Mobil Oil Company, Stock Number 678

C. Apparatus

In order to obtain the values of the parameters in the

power law model using the capillary viscometer for the previously

described non-Newtonian fluids, the following data must be

obtained:

W = weight of fluid pas sing through the capillary

T = time for the flow of W

ap :::: pressure drop through the capillary

R, L = the radius and length of the capillary, respectively

p = mass density of the liquid flowing through the

capillary

T~e. analysi.~. Q(~his ~ta ..,vjll be outline~ ~ater.

26

The major components of the viscometer system are the

1' • d ' ( "b II) 1qu1 reservou or omb , the capillaries, the pressure gages,

the high pressure gas source and piping to the bomb, the water

bath, and the pressure regulator and valves. These are described

in the following sections. Figure 4, page 27 shows a schematic

diagram of the apparatus.

1. Liquid Reservoir. The 1. 5 liter pressure vessel was

constructed from a 12 inch length of extra heavy 3 inch stainless

steel pipe. The outside of both ends of the pipe was threaded to

fit extra heavy three inch pipe flanges. These flanges have a

workable pressure of 250 psig. and they were mounted on each

end of the pipe. Refer to figure 5, page 28. An extra heavy

blind flange was bolted to the top flange. The blind flange had

two inlets; one inlet was fitted with a connector and led to the

high pressure gas source. The other inlet was fitted with a hex

head plug and was used for introducing the fluids into the pressure

vessel. An extra heavy three inch pipe flange was bolted to the

bottom flange. A ten inch square steel plate with a three inch

diameter opening was placed between the bottom two flanges and

was used to mount the pressure vessel to the supporting table.

An extra heavy, three inch solid plug was placed in the bottom

flange. A 1/4 inch hole was drilled through this plug, and a

connector was mounted in this hole. All capillary tubes were

Precision Scientific Water Bath

Capilla~; Tube

Thermometer

Circulating Pump

\'later Bath

--.... ~,Pressure vessel

Valve V

PRESSURE GAGES

30 in Hg ¥.ano- 0-30 0-300 meter psig psig

~---~ (') r.:-----" '' \ I '

\ "/ \ / • I

'\.. ~/ . ....__ .. · "- .. /

II(A) ICX)

L------L----~-- ___ _]

IV III

Figure 4. General description of capillc>.ry viscometer

Nitrogen Cylinder

~ -..J

28

--------------- ------------ P. -----3

li ie-~---··- ----;:

L

M

G

H

I

A i" CONNECTOR TO NITROGEN ,___________________ J

PIPELINE

B HEX HEAD PLUG

l FLUID FEEDING·

c 3 11 BLIND PIPE FLANGE K

D WATER BATH

E 3 11 EXTRA HEAVY PIPE FLANGE

F STEEL PLATE, 1011X10 11 J CAPILLARY TUBE

G TABLE K RUBBER STOPPER

H 3" SOLID PLUG L 311 STAINLESS STEEL PIPE

I .2." CONNECTOR M RUBBER GASKET 8

Figure 5. Pressure vessel

29

attached to the pressure vessel at this connector.

All gaskets were made of rubber. The inner surfaces of

the bomb were painted with an epoxy enamel. The versatility

of the liquid reservoir may be improved by using teflon gaskets

instead of the rubber gaskets.

2. Capillary Tubes. Three stainless steel capillary tubes

and two glass capillary tubes were used in the experiments. The

capillary dimensions are listed in Table 1, page 3 0. The 3 04

stainless steel capillary tubes are products of the "Small Parts

Company", Miami, Florida. Tube IV is a glass capillary obtained

from "Fischer and Porter Company", Warminster, Pennsylvania.

During the course of the experimental study, it was found necessary

to have a capillary with an even smaller inside tube radius than

the above capillaries in order to reduce the possibility of turbulence

in the "thin" liquids. A very satisfactory capillary was made of

a broken glass thermometer and is listed as tube V.

Since the capillaries had to be interchanged very often it

was advantageous to devise a simple method of attaching the capil­

laries to the one connector at the bottom of the liquid reservoir.

The stainless steel tubes were silver soldered in a 3/8" diameter

stainless steel rod which was 2" long. The dimensions are indi­

cated on figure 6, page 31. The connector was designed to clamp

TABLE 1. DESCRIPTION OF CAPILLARY TUBES

Tube Number

I

II

III

IV

v

Tube Type

HTX':~-7

HTX*-11

HTX':~-14

F&P Glass Tube

Glass Thermometer

Tube

R ft

0. 00625 (±0. 000093 )*~~

0. 003917(±0. 000093}**

0.002625{±0.000093}**

0.0006297(±0.0000019)**

o. 0003166~'~0:~

* HTX refers to stainless steel capillaries.

~~~:~ deviations are those given by manufacturer.

L

ft

2.00156

2.00156

2.00156

1. 319

I. 156

*':~':c calculated from calibration procedure (See page 34. ).

30

L/2R

160

256

381

1048

1825

I silver- ~

solder I II I I

I 3 ,, '

-a1! o I I

! I

I

i I II ; I

I! JA

-x---1

I '' 2. I _.:i_

r-

Figure 6. Capillary tube

31

32

the 3/8" rod very tightly. The two glass capillaries were forced

through a central hole in 3/8" by 2" plastic rods and the capillaries

were mounted in the same manner as the stainless steel capillaries.

In all cases, the capillaries extended up through the plug and into

the pressure vessel in order to minimize entrance effects caused

by the plug.

The inside diameter of the capillary tubes was checked

with Newtonian fluids of known viscosities. In the case of the

capillary constructed from the thermometer, the radius was

calculated using published viscosity values for water. The cali­

bration results will be discussed in a later section.

3. Pressure Gages. As is indicated on figure 4, page 27,

three pressure gages were employed. Two Helicoid test gages

were used; one covered the range 0-30 psig in increments of 0. 2

psi and the other covered the range 0-300 psig in increments of

2 psi. These gages are "type 410 RTD, bronze". Although the

gages were supposedly calibrated at the factory they did not agree

with one another at about 25 psig and hence they were recalibrated

in the laboratory. A 30 inch, open-end mercury manometer was

used to measure the applied gas pressure in the low pressure

ranges. The manometer could be read to ±0. 025 inches of

mercury.

33

4. Piping,- Valves and Fittings. All piping was of 3/8"

stainless steel tubing, and brass self-aligning fittings from the

Weatherhead Company were employed. Five Hoke needle valves

(Model Number R380M, brass, 3000 psig service) were used as

indicated on figure 4, page 2 7.

5. Pressure Regulator. A Matheson 2-stage regulator

(Model 90580; 5-250 psig) was used for the adjustment of the gas

pressure in the range of 0-150 psig. In the low pressure range

(less than 10 psig), it was necessary to use needle valve I for

finer control.

Although almost any gas may be used to apply pressure on

the liquid in the pressure vessel, nitrogen was used because of

its inert nature. High pressure nitrogen was obtained from the

Matheson Company in a standard pressure cylinder.

6. Constant Temperature Bath. The purpose of this in­

vestigation was to measure the rheological properties of the same

non-Newtonian fluids as used in previous mass transfer studies.

Therefore, for any useful results, the viscosity measurements

must be at the same temperature as the mass transfer experi­

ments as visc·osity is usually strongly dependent on temperature.

Toward this end, the liquid reservoir was immersed in a .tempera­

ture controlled water lSath.-"; · The bath was constructed from a

34

steel drum which is 15. 5 inches high and 14 inches in diameter.

The inner surface of the water container and outer surface of the

pressure vessel were painted with an epoxy enamel. There were

rubber gaskets between the bottom of the water bath and the steel

support plate to prevent leakage of water. The bath was bolted

to the table as shown in figure 5, page 28. Water was circulated

between the bath around the pressure vessel and a Model M-3

Precision Scientific Company water bath. The temperature relay

in the Precision Scientific water bath was cap:ible of holding the

temperature in that bath to within ±0. 05 °C. However, the flow

rate of the water from the Precision Scientific bath was rather

low so that it is questionable whether this temperature control

was extended to the water around the pressure vessel. The water

in the constant temperature bath was agitated by using a small

centrifugal pump to remove the water and reintroduce it in a jet.

The temperature of the water around the pressure vessel was

measured with a conventional mercury in glass thermometer and

found to be within ::1:0. I °C. It was possible to hold the temperature

in the bath to within ±0. 1 °C of the desired temperature.

7. Weight and Time Measurements. The weight of the

liquid passing through the capillary tube was measured using an

Ohaus triple beam balance. The balance could be read to within

::1:0. 05 grams. In over 90% of the runs, the total weight of the

35

fluid collected exceeded 20 grams. The time required to collect

the above mass of liquid was recorded using a hand actuated stop

watch accurate to ±0. 1 seconds. In over 95o/o of the runs, the total

time was greater than 60 seconds.

D. Operation of the Viscometer

The first step was to prepare the fluids. In the case of

the non-Newtonian fluids, the polymers and solute were mixed

with distilled water as indicated earlier. It is important to note

that all of the ''fresh" non-Newtonian fluids were studied in the

capillary viscometer within one to two days after they were pre­

pared. The "aged" solutions had already been stored one year,

after being used in the mass transfer experiments.

A capillary tube was chosen and attached at the bottom of

the pressure vessel. A rubber stopper was used to close the

bottom of the capillary tube. Prior to each run, the room tempera­

ture was adjusted to within ±2 °C of the desired run temperature.

The fluid to be studied was introduced to the pressure vessel,

and the temperature of the bath surrounding the vessel was adjusted

to the desired run temperature. A thermometer was lowered into

the fluid to be studied {in the pressure vessel) through the liquid

feed port in order to determine when the desired temperature was

attained. This usually took from one-half an hour to one hour. The

36

thermometer was then removed from the bomb and placed in the

water bath. The temperatures of the fluid and the water bath

were always identical, within the accuracy of the thermometer.

The operation of tre capillary tube viscometer is outlined

as follows:

l. Close all valves to pressure gages.

2. Open the nitrogen cylinder valve (A).

3. Using the regulator, adjust the system pressure to the approxi-

mate value of the desired gage pressure (as indicated on the regu-

lator pressure gage).

4. Open the valve to either the high pressure gage, low pressure

gage, or manometer depending on the operating pressure.

5. Remove the rubber stopper at the exit of the capillary tube in

order that flow through the capillary may begin.

6. The gas pressure above the liquid is adjusted once again using

either the regulator valve or valve I to obtain the desired pressure

under flow conditions.

1. When the pressure is constant, a previously weighed, empty

container is placed in position to receive the fluid leaving the

capillary tube. After a reasonable amount of liquid has been

~ollected, the container is removed from the stream of liquid.

The time to collect the fluid is determined by means of a stop

'·\' ·,

watch.'

37

8. The flow through the capillary is stopped by reinserting the

rubber stopper on the end of the capillary.

9. The applied gas pressure, weight of the fluid collected, and

time to collect the fluid is recorded.

10. If there is enough liquid in the pressure vessel for a run at

anothe-r pressure, the process is repeated beginning with step

3. If there is not enough fluid in the pressure vessel, it is neces-

sary to refill the pressure vessel. If the previously used pressure

was less than 14. 7 psig {that is, the manometer was being used},

valve V was closed cind the feed port in the pressure vessel was

slowly opened in order to introduce the liquid. This procedure

was necessary in order to prevent mercury from the manometer

being blown toward the pressure vessel. However, if the mano-

meter had not been used in the previous run, the manometer was

removed from the piping network, and valve IV was used to vent

the system. Valve I was closed first and then valve IV was slowly

opened allowing the pressure in the vessel to reach atmospheric

pressure. The feed port was opened and the pressure vessel was

refilled.

Approximately six to ten different pressures were used

for each capillary tube. ·Progressively. higher pressures were

used in the series of tuns.

11. Whep. aJJ,of .th~J~esi:r~c:i applied gas pressures were used for - ·~' . . . ..

38

one capillary, the regulator valve was closed and vent valve IV

was opened. The capillary tube was removed and replaced with

another tube, and the entire procedure was repeated.

12. After use, the capillary tubes were cleaned with either water

or benzene, depending upon whether aqueous non- Newtonian liquids

or oils were studied.

13. When the entire series of capillary tubes was used (usually

three of the five), the nitrogen cylinder valve was closed, the

regulator valve was opened, the vert valve was opened, and the

pressure gage valves were opened. The constant temperature

bath system and agitator were turned off. The inside of the pres­

sure vessel was cleaned with either water or benzene depending

on the system studied. An air aspirator drew air through the

pressure vessel in order to vaporize any remaining cleaning fluid.

In conclusion, for each fluid a series of capillary tubes

with known (or to be determined) values of R and L were used.

For each tube, six to ten applied pressures were used, in each

case measuring the time required for a measured mass of fluid

to pass through the capillary. From a knowledge of the cross­

sectional area of the capillary and the density of the liquid at the

operating temperature, the average fluid velocity through the

capillary could be determined.

Twelve different non-Newtonian liquid systems were

39

studied. Five Newtonian liquid systems were studied for both

calibration and test purposes.

E. Inside Diameter of "Thermometer" Capillary

The capillary with the smallest diameter was constructed

from a broken thermometer. Attempts were made to calculate

the inside diameter by filling the tube with mercury, but the re-

sults were inaccurate. The diameter was so small that the weight

of mercury was too small to be weighed accurately. The capillary

diameter was ultimately determined by calibration with the flow

of distilled water at 25 °C through the capillary. Equation (2. 2)

was used to calculate the radius R, using the value of the viscosity

of water given in reference (27). Kinetic energy and fluid head

corrections were negligibly small (< 1 o/o) for the calibration runs.

The data and results are summarized in Table 2, page 40.

F. Test of Viscometer System

The viscosities of three different Newtonian liquids were

determined using the capillary viscometer constructed for this

investigation, and these values were compared with values of the

· · · h s The liquids were pure v1scos1ty obta1ned from ot er source •

glycerine, oil number 243, and oil number 678*, all studied at

25 oc. . t d" d at 20°C but it is suspected that it Glycer1ne was s u 1e '

... . d 1· r in this chapter •

.,c The oils are descr1be ear 1e

TABLE 2. CALCULATION OF INSIDE DIAMETER OF

"THERMOMETER" CAPILLARY TUBE

Run Ap w T D

psig gram sec ft

1 61.5 30.70 677.3 0. 0006319

2 80.0 35.84 599.8 0.0006340

3 100. 0 45.24 600.2 0. 0006354

4 113.0 49.80 600.0 0. 0006313

Dave = 0. 0006331 ± 0. 0000013 ft.

40

41

was contaminated with absorbed water vapor. The raw data are

presented in Tables A-1-A-4 ' and the calculated results are pre-

s ented in Table 3, page 42.

The value of n in the power law model was calculated for

each of the fluids to be sure that the fluids were actually Newtonian.

As may be seen from Table 3, glycerine arrl oil number 678 are

probably Newtonian; the 95o/o confidence limits on n include the

value of n equal to unity. Oil number 243 apparently is slightly

non-Newtonian. For comparison purposes, all liquids were assum­

ed to be Newtonian (n= I. 00), and the viscosity, J.lN, was calculated.

The average value of calculated viscosity is listed in Table 3 with

the 95o/o confidence limits on J.lN· These values are compared with

other sources as indicated on Table 3. When one considers the

9 5o/o confidence limits, it is seen that agreement between the vis­

cosity calculated using this viscometer and other sources is very

good.

The difference in the case of glycerine at 20. 0°C rna y be

explained by the fact that glycerine is very hygroscopic and any

adsorbed water would lower the viscosity of the glycerine. The

25. 0 °c run was made first and the same sample of glycerine was

reused for the 20. 0 oc run. There were ample opportunities for

adsorption of water prior to the 20. 0 °C run. As may be seen,

42

TABLE 3. RESULTS USING CALIBRATION LIQUIDS

Liquid n++ JJ.N

Temp centiEoise oc dimensionless this other

vis co- sources meter

Glycerine + 20. 0 1. 01 7±0. 03 1400. 0*''±25 1499*~:c

Glycerine 25.0 1.002 ±0.018 953~C ±21 945 ~:c~:c

Oil No. 243 25. 0 0. 978 ±0. 0065 4.17*±0.32 4.25t

Oil No. 678 25.0 o. 982 ±0. 0263 211. 6):c::7. 65 216.8t

+ Possibly contaminated by H 20 absorbed from the atmosphere.

++ Twice the standard deviation of n is indicated, which gives

the 95% confidence limits on n.

* Calculated assuming n = 1. 000.

):o:c Reference (27 ).

t Measured in Cannon- Ubbelohde viscometer (26 ).

43

the viscosit yr of the 20. 0°C run is 6. 7% lower than the literature

value.

The .:flow diagrams for glycerine at 25. 0 °C, oil number

243, and o i 1 number 678 are presented in figure 7, 8, and 9.

Plots of the :friction factor versus the Reynolds number for the

two oils ar ~ presented in figures 10 and 11. In general the data

points are "'i.T ~ ry close to the theoretical curve, except for the

largest capi 1lary for oil number 678.

The deviations in the case of the largest capillary are

probably d~~ to the fluid head and kinetic energy losses which

become sigr:1...:ificant, especially for the high liquid flow rates.

~,,

Also, it is E:'! :::x:pected that for this capillary some error may be

attributed tc:> the short timing periods. See Table A-4 for the

data for thi ~ capillary.

G. Analysis of Data for Non-Newtonian Liquids

The .shear stress- shear rate data was analyzed using

the power lc:L ~ model

{= K (-du/dr)n (2. 31}

and the obj €! c:t of the analysis was to determine the best values

of K and n £ <> r each non-NewtGnian: liquid:. • The determination of

the constants K and n, is dependent upon the determination of K'

and n' in th~ :following relation:

4r-----r-----r-~--~~----

0 0.00525" I.D. tube

0.007SJ" I.D. tube

2 - 0 0.01250" I.D. tube

1 0 1 ..------ ---·--·· , ___ ··-··

8 ----------· ·····--· ........ ~- - ····-·-·····--· ···-· .. I

6 ....

4-

. ···-· --· -t' -·------!----- ! -···· . ' I

I

I I ·------·· t ...... j J

: !

I

-1 FLOW FUNCTION, SU/D, sec

Figure 7. Flow chart. for glycerine at 25°C

45

6r---r----r------.-----~---

o 0.00126" I.D. tube

4t-----r ---~·--- ---------

... H ~ 2 --- -- -- . 0.. ~ u

0 10 -····------ ...

I

.... ·------·--· -1-------·-····-------····-

I

t I I I : ! i

-------... --.. ~. -4-·······-··---~ -· I I I

i

4~----._------~--------~----------LI ____ _j 6

4 6 2

FLOW FUNCTION, 8U/D, -1

sec

Figure 8. Flow chart for Oil Number 243 at 25°C

o· (f)

~ .o r--l

2

0

A

7 t----

2 --···-----

0.00525" I.D. tube

0.00783" I.D. tube

---------- -------..----1

I ···-·-- ·---l " - . -····-- ------

i - -- --------··---. _______ J __ _ 7

4 ------~----~--~--L-----~--__J 102 2 4 2 4

FLOW FUNCTION, 8U/D, sec-1

Figure 9. Flow chart for Oil Number 678 at 25°C

47

5

5 - --t---t--~~~--+---r---·:--

1 I 2 ---+---r·

i l

f=16/NRe

i ,: i l I

: I i i

'

' ! :

5 j I

... -----·-t ---Oj

I :

i I 2 --·-··-+·----t-----1----1----+----1

I '

I I I

10~------~'----~------~----~----~------------_J 5 2 5

Dn 'u2-n' p REYNOLDS NUMBER, --1-----

2

Figure 10. Friction factor - Reynolds number correlation for

Oil Number 243 at 25°C

5

48

I I' I 5 -------------- ------ ---j-----r- . -------

_______ L_ :~---l-----~---0 a: ! I I

0 0.01250" I.D. tube I I I' I l 0 ---- ---r--'--- -I

2

0.00783" I.D. tube

0.00525" I.D. tube 0

! '

5 ~------~~----~----~~------~~------~----_J 2 5 2 5 101

2

n 1 2-n' REYNOLDS NUMBER, D U-{ _ (l

Figure 11. Friction factor- Reynolds number correlation

for Oil number 678 at 25°C

49

(RA.P/2L) = K 1 (4U/R)n' {2. 29)

The analysis is outlined as follows for any particular non-Newtonian

liquid:

1. Calculate U, the average velocity in the capillary

u = w

2. For each data point, calculate RAP/2L and 4U/R.

3. Plot log R..::l?/2L versus log 4U/R to visually check for

the effects of turbulence, fluid head, and kinetic energy errors.

In this work, data points requiring any of the above three corrections

were ignored in the further analysis. These data points comprised

a small fraction of the total data points. These corrections will be

discus sed in the next section.

4. Equation (2. 29) is linearized by taking logarithms of

both sides to give:

log (R.AP/2L) = log K' + n' log (4U/R)

i. e. ,

y = log K' + n'x

The logarithms of R.AP/2L and 4U/R for each data point were fed

• • I to a least -squares regression analys1s program to determ1ne n

and the 95% confidence limits of n'.

According to equation (2. 3 7) when n' is constant with

Yw, n' is equal to n. This would be true if the above relation was

50

truly linear. The difference between a linear relation. · , 1. e.,

y ;;; log K 1 + n'x

and a 2nd degree polynomial relation

in terms of absolute percentage deviation was found in all cases

to be insignificant. Therefore, n 1 was considered to be a constant.

Visual observation of the data also supported this conclusion.

5. I

If RAP/ZL is plotted versus (4U/R)n, a linear relation

is obtained with K' as the slope. K' was determined in this

I

manner by feeding RAP/ZL and (4U /R)n into a least squares

regression analysis program. The 95o/o confidence limits of K'

were also obtained.

The author also determined the best value of K' from the

procedure in step 4 which gave the best value of log K' (and the

95% confidence limits of log K' ).

The value of K 1 calculated by the above two methods in

all twelve cases agreed to within 2% of each other; and in eight

of these twelve cases they agreed to within 1% of each other.

Because of the logarithmic transformation, the 95% confidence

limits of K' calculated from the log K' are skewed. For example,

· ·· · f th two methods for 0. 2% CMC without cons1der a compar1son o e

any solute:

I

;. ' I '·

51

K' obtained from log K'

{0. 003162-0.000763}<K'<(O. 003162+0.001008)

or 0. 00249< K'< 0. 00417

K' obtained from RAP/2L versus (4U/R)n'

(0. 0003116-0. 000133}<K'<(O. 003116+0. 000133)

or 0.00298<K'<0.00325

The value of K' determined by the method involving log K' has

lar.ger confidence limits than the second method. It is the opinion

of the author that the slope technique is more accurate than the

intercept technique. The value of K' used in subsequent analysis

was that obtained from the curve fitting of RAP/2.L versus

I

(4U/R)n.

6. The following relation was derived in the literature

review and applies if n' is a constant (therefore, n' = n):

n K'=K(~)

4n

(2. 39)

K was determined from the previously determined values of K'

and n using equation (Z.. 39 ). The 95o/o confidence limits of K could

be determined from the following relation

.AK = AK'

( 3::1) by a simplification of a more complex equation (2.5). This

simple relation essentially assumes that the error of n has a

52

negligible effect on the error of K T his was found to be true.

H. Correction or Elimination of Data Points

The gage pressure of the gas above the liquid in the pres­

sure vessel is only an approximation to the actual liquid pressure

at the entrance to the capillary tube. One rather obvious correc-

tion is that due to the liquid head above the capillary entrance. In

other words, the liquid pressure at the capillary entrance is

greater than that read on the gas pressure gage, by an amount

equal to:

(AP~ = h pl (g/gc}

where ~1 = the density of the liquid

g = the acceleration of gravity

gc = the gravitational constant

· h = the average height of the liquid above the capillary entrance during a run

The fluid head correction becomes a problem when the

applied gas pressure is low. For example, consider figure 12,

page 53. The three data points to the extreme left indicate the

effect of neglecting the fluid head correction. Unfortunately,

with the present design of the pressure vessel it is nearly impos-

sible to estimate the liquid level in the pressure vessel directly.

The maximum liquid height above the capillary entrance is 12

inches, with an average value of six inches. For six inches of a

0' (f)

~ .0 rl

101

8_ I 0

6 e

4 11>,

0

A

2 -·-- -

! I

10° I J 8 I

6 I ----~

I I I

4 ---·-- j I

I I

I 2 !

I

«> I

1o-1 I 6 8

10 2

0.00525" I.D. tube -+---+-·-- -·· .

0.00525" I.D. tube* -4--i----- ....

0.00525" I.D. tube+ --+----- ----···.

0.00126 11 I. D. tube 0

0.00063" I. D. tube

2 4 6 8 103

FLOW FUNCTION, 8U/D, sec-1

2 4

Figure 12. Flow chart for o. 1% Ca.rbopol {saturated with iodine

and carbon tetrachloride) at 22.7°C showing the effect" of

kinetic energy and fluid head

53

6

* After kinetic energy correction using mr0.0 in K.E.=mpu2/g {11) c

+ Need some fluid head correction

54

typical liquid, the pressure correction is equal to approximately

0. 26 psi. Since it was impossible to accurately determine the

liquid height, all data exhibiting the dip below the straight line

at low shear stress were ignored in the data analysis. At large

applied pressures, this effect is negligible.

Another undesirable, but unavoidable problem, is that

due to the kinetic energy effect at the tube entrance. As the liquid

accelerates from an essentially zero velocity to the velocity in

the tube, part of the pressure head is used in accelerating the

fluid. The effective head to be used for estimating wall shear

stress is less than that obtained by adding the fluid head correc-

tion to the applied gas pressure. This pressure correction is

estimated by the following expression:

(AP)K. E. = mpl (U2-U~)/gc

where U = average velocity in the capillary

= the liquid velocity in the pressure vessel,

essentially zero

m = a correction constant

The difficulty in applying this correction is that m is not a unique

constant. Although values around 1. 0 to 1. 5 have been mentioned

in the literature for non-Newtonian fluids (11 ), it was found in

this work that the value of m ranged from I. 0 to 3. 0 or even

Capillary diameter and type of non- Newtonian greater depending on the

55

liquids. These values of m were found by trial and error. The

high shear stress data points were forced to be on the straight line

which visually appeared to pass through the data points at moderate

shear stresses. For example, refer to figure 12, page 53, once

again. Consider the 0. 005 25" I. D. capillary; at a shear stress

greater than 1. 0 lbf/ sq £t the data points diverge. The average

velocity in the 0. 00525 11 I. D. capillary is much greater than in

the other two capillaries (refer to Table A-4. ), and the kinetic

energy correction is important. Using a value of m equal to 3. 0

lowers the data points and improves th.e fit considerably. This

process was so tedious, for the small number of data points in­

volved, that those data points were ignored. It must be emphasized

that m varied from capillary to capillary and was a function of the

type of non-Newtonian liquid.

56

IV. DISCUSSION

The object of this investigation was to study the rheological

properties of the aqueous non-Newtonian liquids used in previous

mass transfer studies. The solutions actually used in the mass

transfer runs (aged one year) were studied along with freshly­

prepared solutions. Two solutions, 0. 2o/o CMC and 0. Zo/o Carbopol,

were prepared without saturating these solutions with iodine and

carbon tetrachloride in order to study the effect of the solute.

The flow diagrams for the freshly-prepared non-Newtonian

fluids are presented on figure 13 to 19, pages 57 to 63. The

flow diagrams for the aged solutions appear in Appendix B. Only

those data points used in the analyses for the constants in the

power law model are shown on the diagrams. However, all of

the data are given in the Appendices, including those data not used

in the analyses.

The constants in the power law model and the 95% confidence

limits of these constants are presented in Table 4, page 64. The

average absolute percentage deviation of the observed shear stress

from that calculated using the power law relation is also given in

this table for each of the fluids. The largest percentage deviation

was 6. 3o/o for the 0. z% Carbopol solution. Most of the liquids

were described by the power law model with an average absolute

o• tf)

~ ,0 M

101

8 ----·--- """"

6 ---0.00126 11 I.D. tube 0

4 - 6 0.00063" I.D. tube

lo0

2

8 r------'------ --

6 ----- ------· --

4-------

2

2

FLOW FUNCTION, 8U/D,

I

- ------- ---.1----- ----- -- -

4

-1 sec

I

6 8 105

Figure 13. Flow chart for' ·.05% Garbopol (saturated with

iodine and carbon tetrachloride) at 22.8°C

57

2

··-·- ····· ~-4.-

a . -- -·--· -- --· __ .__ __ - 0 0.00126" I.D. tube

6

5 -~- ~r ~-~=-=-~-==~---=-~· =- .. -' I 4 ---·-· . -*~ • ~- •¥--···--· ... ···~-···----··

3 --· ··- -

FLOI'i FUNCTION, SU/D,

-~---1·- .. -··, ~--·

-1 sec

4 5 6

Figure 14. Flow chart for 0.1% Carbopol' (saturated

with iodine and carbon tetrachloride) at 22.7°C

t! cr

~ ..0 r-1

.s ~ ~ .. ...:l ~ < ~

~ tJ'.)

~ E-o (Jj

0:: <t t::

til

101

o O.OJ525" I.D. tube

5 A 0.00783" I.D. tube

0 0.0125011 I.D. tube

2 -----~~

l l i l I i ! o ___ _J_ _ ___!_• ----t----i -r--T-1 I I

10°

\ . I I . ill. ! ! ---1 I ~-~--,.. · . I l \

5

I 2 __ l __ _

-1 10 1

10 2 5

I I·

\

---------t·-----------1-- I

102

I I

l I t ' I i j .

2 5

-1 FLUd FUNCTION, 8U/D, sec

103 2 5 104

0 Figure 15. Flow chart for 0.2% Carbopol (saturated ~~th iodine anc carbon tetractloride) at 22.8 C Vl

'-()

t! 0" CD

~ ,Q l""i .. ~

~ Q .. ~ ...:I

::: -" I

4 '

3 ' 0

6

I I 2

I

I I I

I ,---+ . I

0.00126 11 I.D. tube I I I I i -I l ! '

0.00063" I.D. tube I I I I I I I !

; 100 I I v- I I I

I I <---~ I I

I l . t ~----···- ---~

-:6'-' -+-----+--·---· -.-- ·-+1 ---- j --+-t---4----- ;-----------1 I I I ' I .

I j I 1 i

4

I I . ' . . ' . I ; , • l : 1 l 3 . . . . ' 2 3 4 5 7 104

2

FLG.-i FUFCTION, 8U/D,

3

_, sec

4 5 7 105 2

Fib~re 16. Flow chart for 0.1% C!:C (saturated with iod~ne and carbon tetrachloride) at 23.8°C C> 0

c ..;

0 0.00126" I.D. tube

t! .3 t. 0.00063" I.D. tube 1 ! --1--g- j_ l I I ........._ : i

~~ I .~~~----~------+-----r----. ,...j 2 I ' : ! ; I' .. I I i i ~ ' I I

-:t I' I ! I i .......... ' I '

~ \ '. I I I i .. I I I ! I

~0 I I- . +----+ l f :3:

10 ' I ! 1 : I ! ~ l I I : i I I

"' i I 11 ; ; ! J \JJ i l i _;... • .

~ 7 1---------.----·r- ! ; · 1 ; 1

E- : i I I i I I

tf.) : I I I l I I 0::: . ' ·-+-----1 ~ 5 --:~--- l j. I ::I: I ' . I

tf.) I I : I 'I f I ' I I . ' I I : 3L_------~--~--------~--~------~--~----~--~---

1o3 2 3 5 7 104 2 3 5 7

105

FLO-., L":P'C'~'"'C;"'' 8U/D - 1 .. .r c" Ll. ,, , , sec

Fib~re 17. Flow char~ for 0.2% CEC (saturated -wi~h iodine and carbon tetrachloride) at 23.5°C

0'-

101

6

~ 4 ' cr

4D

~ ,Q r-i 2 .. ...:I

.::!. ~

0 .. 10 j ~

~ 6 tf)

~ 4 E-< 1:1')

~

~ :X: U) 2

10_,

I 1

o 0 .. 00525" I.D. tube -··-~-t------~------...,.,.c;---------t

l

~ 0.00783" I.D. tube

o 0-.01250" I.D. tube

' ~ ----t--------~·---------~·----~------~------~

2 4 6 102 2 4 6

I i

I

103

-1 Fill.\' rJ~;CTIO~, BU/D, see

0 Fib~re 18. Flow chart for 0.2% Carbopol at 22.8 C

L --

2 4 6 104 2

0"· 7'0

t! 0' Ul

~ ..0 r-i

-~

~ -;!'

~ Cl

~

:j < ::?: E-4 < Cll

~ i-< Cll

c:: ~ :c Cll

5 I o 0.00126 11 I.D. tube

3 t:l 0.00063" I.D. tube

2

10°

7.

i ! I

I I

i I I I I I I

_L

I i I I

~ I I I I I . 1 - __ j

/

I I ---- --~-~----.-

! I I I 5

3 I I

ta3 · · I I 2 J

Figure 19.

104

FLCJ.i FUNCTim;, BU/D,

5 7

-1 sec

2

0 Flow chart fo:- 0.2% CHC at 23.5 C

J 5 7 105

0' w

64

TABLE IV. CONSTANTS IN POWER LAW MODEL

Liquid Temp nit** KX1o4*'H~ % D11J +

(lbf seen' of

oc Dimensionless '"( sq ft )

0.05% Carbopol* 22.8 0.967+0.015 0.447:!:.0.008 3.09

0.05% Carbopol 22.8 0.917±0.039 0.935±o.052 6.29

0.1% Carbo pol* 22.7 0.864±{).022 . 2.453.±0. 105 3.13

0.1% Carbo pol 22.7 0.767j_D.004 12.42 ±o.070 0.18

0.2% Carbopol* 22.8 o. 799.!0.014 14.39 ±o.35 2.03

.0.2% Ca.rbopol 22.8 0.514.i{).011 557.6 ±12.5 4.65

0.2% Carbopol** 22.8 0.504±0.014 639.7 ±18.3 6.30

0.1% 9.32 :!:0.16 CHC* 23.8 0.720±0.012 2.32

0.1% 9.53 :!:0.15 2.25 CHC 23.8 o. 724±o.011

0.2% 27.34 .±o.52 1. 52 CMCif- 23.5 ·0.647!Q.011

0.2% 30.59 ±.o.76 2.01 CHC 23.5 o.643~.o14 .

0.2% 28.71 ±1.22 2.38 CMC-lHf- 23.5 0.651::0.028

* Aged one year · ** Not saturated with iodine and carbon tetra~hl~r1~ed which gives **t~ Twi · t" f nor K is 1nd1ca e , ce the standard dev1a 10n o

the 95% confidence limits on n or K % 100~ fTobs -Teale I

DEV =-w- 'lobs l:!

65

percentage deviation of less than about 3%.

A. Effect of Aging the Solutions

For the two concentrations of CMC studied, aging had

essentially no effect on the power law index, n. The effect of

age on the constant K was a very slight decrease inK with age.

Aging the polymer solutions one year had a more pronounced

effect for the Carbopol solutions than for the CMC solutions. In

all cases, n increases with aging. The percentage change* inn

for the 0. 05%, 0. 1 o/o, and 0. 2% Carbopol solution is, respectively,

5. 4%, 11. 2%, and 35. 7%. In all cases, K decreases with aging.

The percentage decrease in K for the 0. 05o/o, 0. 1%, and 0. 2% Car-

bopol solutions is 1 09o/o, 417o/o and 3900o/o, respectively*. The

apparent viscosity also decreases with aging.

B. Effect of Concentration of the Polymer

The object of this investigation was not to systematically

study the effect of polymer concentration on rheological properties

over a complete range of polymer concentrations. However, as

expected, it was observed that over the range of concentrations

studied n decreased with increase in polymer concentration. That ' .

is, the fluid became more non-Newtonian as the percentage of

d The rate Of Change Of n between polymer in the water increase •

* difference in the value times 100 divided by_ a~ed co~dition

66

0. 1 and 0. 2 weight percent polymer was much greater for the

freshly-prepared CMC solutions. Also as expected, the value of

K increased with increase in polymer concentration. The increase

was very rapid for the Carbopol solutions.

C. Effect of Solute on Non-Newtonian Behavior

Because of a lack of time, only one concentration for each

type of aqueous polymer solution was studied in the absence of

both iodine and carbon tetrachloride. The results are indicated

in Table 4 for the 0. 2% CMC and 0. 2% Carbopol solutions.

For the 0. 2% aqueous Carbopol solution, saturation with

iodine and carbon tetrachloride increased the constant n from

0. 5 04 (with no solute) to 0. 514. Therefore, the solute causes

the fluid to behave slightly more Newtonian. The increase is very

small, however, as may be seen from 95% confidence limits, and

one might infer that the difference is not statistically significant

at the 95% confidence level. The value of K decreased from 639

(no solute) to 557 (when saturated with the solute). Therefore,

1 · t b "th · r" the addition of solute causes the Carbopol so utlon o e lnne

or less viscous.

For the 0. 2 CMC solutions, the effect of the solute is

difficult to state. I£ one neglects the 95% confidence limits, it

appears that the addition of iodine and carbon tetrachloride is just

67

the opposite of that observed for the 0. 2% Garbopol solution.

Whereas for the Carbopol solution, n increased with the addition

of iodine and carbon tetrachloride, in the case of CMC, n decreased

with the addition of the two compounds. However, when one con-

siders the 95% confidence limits, it is possible to state with 95%

confidence that the solute had no effect on the constant n in the

CMC solutions. Whereas for the Carbopol solutions, K decreased

with the addition of iodine and carbon tetrachloride~ in the case of

the CMC solution, K increased slightly with the addition of the

solute.

D. Recommendations

Several modifications in the design of the capillary visco-

meter are recommended.

1. It is suggested that a constant temperature bath be

placed around the capillary tube during a run. The currently

designed viscometer exposes the capillary to atmospheric tempera-

ture fluctuations. D · the room temperature was adjusted ur1ng a run

. h d . ed liquid temperature, but control as close as poss1ble tot e eslT

S . 1 the pressure vessel (liquid reservoir) varied by ±2°C. 1nce on Y

tant temperature bath, all we is currently surrounded by the cons

that the ll.qUl' d entering the capillary tube is can be sure of is

within ±0. 1 °C of the desired temperature. Furthermore at high

68

shear stress, the fluid may be heated by viscous dissipation in the

tubes. This :m:> dification would aid in approaching an isothermal

flow process.

2. The gaskets currently used in the viscometer are made

of rubber. If the viscometer is to be used to study organic solvents,

these gaskets should be replaced, preferably with teflon gaskets.

3. It would have been desirable if the fluid head correction

could have been easily and accurately applied. In order to make

this correction, either a device indicating the level of the liquid

in the pressure vessel, or a method of measuring the liquid pres­

sure just before it enters the capillary is necessary. A simpler,

but more tedious, tethnique would be to measure the volume of

liquid added to the pressure vessel accurately and to record the

volume of the fluid discharged both during the start-up period and

the actual run. These volumes could be related to the height of

liquid inthe pressure vessel.

69

V. CONCLUSIONS

A capillary viscometer was constructed, tested and used

to study non-Newtonian liquids. Two major types of non-Newtonian

liquids were studied: (1) aqueous solutions of carboxymethycellulose

(CMC) and; (2) aqueous solutions of carboxypolymethylene (Carbo-

pol). The effect of saturating these solutions with iodine and carbon

tetrachloride was studied. The following conclusions may be drawn:

1. In the range of shear rates studied, the "power law"

model adequately described the shear stress - shear rate behavior

of all solutions used in this investigation.

2. The addition of iodine and carbon tetrachloride (saturated)

to the aqueous non- Newtonian solutions had a small effect on the

constants in the "power law" model. These effects, though small,

were directionally different for the CMC and Carbopol solutions.

3. Aging the solutions one year had very little effect for

the CMC solutions; whereas for the Carbopol solutions a large

change in the power law constants was observed. The Carbopol

b N Wtonl·an and less viscous as they aged. solutions ecame more e

4. Correction is very tedious to apply The kinetic energy

t It must be found by because the parameter m is not constan •

different for each capillary and each trial and error and is

70

non-Newtonian liquid. Values ranging from one to greater than

three were found. One way to avoid these errors is to work with

low average velocities in the capillaries. This was accomplished

without sacrificing the range of shear stress - shear rate data by

using smaller capillary tube diameters.

5. The correction for fluid head is difficult to make because

of the design of the viscometer used in this investigation.

71

APPENDIX A

'";f

Capil­lary Number

Table V.

AP

lbr <sq in)

0 Capillary data for 0.05% Carbopol* (saturated •r.ltp r 2 and CC14) at 22,8 C

w

(gram)

T

(sec)

u

f't <sec>

40/R __,

(sec-1)

.RAP/21

lb! <sq ft)

(4U/R)n'

(see-n 1 )

(du/dr)w

(sec-1}

__ liL_ ______ l_,Q8 158.70 127.6 2.03 3097.61 .102 1584.94 3168.04

I \ \ I

1.74 196.70 125.5 2.56 3903,57 .164 1959.18 3992.32 2.28 229.20 126.8 2.95 4501.91 .215 2232.78 4604.26 ?.88 276.30 123.8 3.64 5558.55 .272 2708.81 5684.92 3.46 320.10 127.6 4.10 6247.93 .327 3015.23 6389.98 3.97 339.30 124.2 4.46 6803.99 .375 3260.33 6958.68 4.69 195.80 62.0 5.16 7865,43 .443 3723.68 804~.25 5.35 227.10 65.0 5.71 8701.72 .505 4085.05 8899.56 5.43 222.30 61.4 5.91 9017.21 .513 4220.61 9222.22 5.94 241.80 61.8 6.39 9744.71 .561 4531.72 9966.26 \ .

\ i IV 5.10 12.70 602.6 .59 3812.06 .175 1917.04 3898.73 I 10.40 21.30 503.4 1.20 7653.35 .357 3631.54 7827.35 ,------fs--: 1 o 3 2 ~ 3 o 5o 1 • o 1 • 8 3 11 6 6 1 • 3 8 • 53 9 s 3 4 2 . 4 8 11 9 2 6 • s 1 ' 20.50 43.30 50?..2 2.45 15595.40 .704 6973.74 15949.97 1-- 3 o • o o ? 1 • 2 o '~ o 1 • o 3 • 6 3 2 3 c 9 4 • 6 3 1 • o 3 o 9 9 9 4 • s_ 1 2 3 5 i. s • 6 9

40.00 70.10 402.4 4.95 31509.78 1.373 13287.78 32226.17 50.50 90.10 401.4 6.38 40600.64 1.734 16763. 3 41523.71

_____________ _ }o. ?0 ____ 9~-.! a :) ___ )_oo_. 2. __ ~_0_5 59!:?_Q_._f Q 2_!.!!-_2_1 _____ 2J_7 7 3_._~ 9__ Q.Q: ~-t· 3_3_ 90,50 98.30 225.4 12.47 792E~.53 3.108 30959.15 31:37,09

110.00 99,30 Sl.~ 15.57 99014.18 3.777 37953.Sl 1012~5.30 131 00 67 3 ~ ~l ' 1= C~ 1~~?=7 ~= L LCC L~~·= :o 1?~,?? l• ___ ...,.-...--.--<- _ -* • .. __ .-•~-~-- - • - -- • . "- • L ·-- ~ -~-~-~·-- · • · " - - J J..L._e_ .:_ • to..._ . --~-·-~-

* Li'l_·dd dc:1:: .:_ V',j = 0, 99~ :::;-1; <:•; cr.:

~ ..

-...1 1\)

Capil- 6P w T lary

lbr Number {gram) (sec) <sq in)

v 40.00 13.30 917.4 50.00 16.60 903.2 60.00 20.30 906.0 80.00 27.17 905.6

100.00 35.03 903.0 120.00 42.8 5 903.2 140.00 51.00 901.0

-~---·---~-- ------· -. --

Table V -- Continued

u 4U/R ~P/21

:f't {See) {sec-1} lbr

(sq ft)

1.62 20586.05 .788 2.06 26097.82 .985 2.51 31816.17 1.182 3.37 42602.33 1.577 4.35 55084.90 1.971 5.33 67366.97 2.365 6.36 80375.84 2.759

(4U/R)n'

(see-n')

8994.79 11179.79 13406.17 17519.48 22172.60 26665. 20_ 31349.57

{du/dr)w

(sec-1}

21054.08 26691.16 32539.52 43570.91 56337.27 68898.58 82203.22

-...] w

Capil­lary Number

Table VI. Capillary data i'or 0.1% Carbopol* (saturated with r 2 and CC14) at 22~7°C

AP

lbr (sqin)

w

(gram)

T

(sec)

u

rt <sec>

4U/R

(sec-1)

RAP/21

lbr <sq !t)

(4U/R)n'

(sec-n 1 )

(du/dr)w

(sec-1)

! III 3.64 190,40 308,4 1.00 1537,29 ,344 277.37 1654,28 I 4.49 224.20 299.8 1.22 1862.12 .424 321.28 2003.83 4.98 105.30 122.0 1.41 2149.18 .470 358,61 2312.74

l 5.52 115.90 120,8 1.56 2389.03 .521 388.91 2510.84 6.05 128.50 122.4 1.71 2614.13 .571 416.71 2813.06 6.70 151.50 125.0 1.98 3017.92 ,633 465.21 3247.59 \ 7 • 4 0 15 8 • 3 0 1 2 2 • 0 2 • 1 2 3 2 3 0 • 9 2 • 6 9 9 4 9 0 • 18 3_426_..._8_Q_

l 8.47 169,80 113.8 2.43 3715.36 .800 545.60 3998.10 ' 9.49 167.40 97.6 2.80 4270.81 .896 607.11 4595,83 --------~1~1~.32 143~?.0 68.0 3.4~ 5243.72 1.069 710,55 ~642.77 j 13.92 156.80 58.8 4.35 6640.09 1.315 851.53 7145.41

18.60 147.80 40.8 5.91 9020.27 1.756 1076.96 9706.73 ;__ ____ l_l_,85 160.20 37.0 7.07 10781.18 2.063 1234,73 11601.64

26.80 207.60 38.6 8.78 13392.00 2.530 1458.05 1~411.15 36.00 217.00 30.6 11.58 17658.09 3.399 1802.37 19001.90

* Liquid density = 0,998 gm/cu em

-..J ~

Capil- AP w T lary

lbf Number (gram) (sec) <sq in)

IV 37.00 39.14 903.6 50.00 58.08 902.6 70.50 67.35 665.8 81.00 50.80 421.6 90.50 59.40 425.6

101.00 48.60 303.0 120.00 54.30 268.8 141.00 62.30 251.2

v 100.00 11.55 967.2 110.00 13.55 979.2 122.00 15.32 983.2 ].40.00 18. 50 974.0 160.00 21.30 976.6

. -- ----- -- .

Table VI. - Continued

u 4U/R RAP/21

rt (SeC) {sec-1)

lbr (sq ft)

1.23 7833.10 1.270 1.83 11636.45 1.717 2.87 18292.93 2.421 3.42 21789.77 2.781 3.96 25239.12 3.108 4.56 29005.68 3.468 5.74 36530.87 4. 121 7.05 44849.53 4.842

l. 34 16953.13 1.971 1.55 19645.00 2. 168 1.75 22120.81 2.404 2. 13 26964.79 2.759 2.50 31690.13 3.154

(4U/R)n'

(see-n')

966.53 1309.15 1851.84 2117.60 2370.13 2636.84 3146!90 3682.89

17 46_~_9 5 1955.90 2142."'?. 2l. a ) , ,, . " . ~ -- - ..

22-21.98

(du/dr)w

{sec-1)

84_2 9. 21 12522.00 lS->5.05 23448.00 27159.86 31213.06 ~9310!!~ 48262.64

1824~. ~ 21140.01 23=304.24 25016 ,_S!t_ 34101.79

-J VI

Capil-lary Number

I

I I

*

Table VII. Capillary data for 0.2% Carbopol* (saturated with r 2 and CC14) at 22.8°C

--AP w T u 4U/R IW'/21 (4U/R)n' (du/dr)w

lbr (sq in) (gram) (sec) !t C-;ec) {sec-1) lb!

(sq ft) (see-n 1 ) (sec-1)

1.00 28.30 303.4 .02 17.19 .225 4.31 21.25 2.45 73.10 182.8 .11 73.71 .552 9.12 91.13 3.93 112.00 121.2 .26 170.34 .883 14.03 210.60 4.83 104.70 79.6 .37 242.46 1.088 16.82 299.76 6.28 155.50 76.0 .58 377.17 1.413 21.11 466.29 7.56 186.50 67.6 .79 508.57 1.701 24.62 628.74

2S. S 0 , ,..... - ~

~:J.,v

1£..7 •. ~0 267.3:·

34.6 ,~.C'

;. • 1 3 J - • '- . "; "'- 5 C·54. 81

~- -~- _ _ _ L 3 • ':' :. ___ ----~ 2 ;,_~ • _; -~~ __ ____ 3_:_ -~---::_ ____ ~ ·-~ L _____ _ S __ l.23__._ ~ ~~~---

T~r••,;rr d·"·,., .. :a .. ~' Q Qq•l -7!"/C'' ;.::~ ..l..t...l ·-! .......~,.._.._ \..i. - l .. ~- J. <J J • / _, I :_; HI ....... ~. ~

-.J

"'

Table VII. -- Continued

Capil- ~p W' T u 4U/R JWl/21 (4U/R)n' (du/d.r)w lary Number lbr

(gram) (sec) ft (sec-1) lbr (see-n') {sec-1) (sqin) csec> <sq ft)

i I I I 5.01 7.12 358.6 .03 49.40 ,473 7,42 61.07 .

I

6.73 13.26 361.6 .o 5 91.24 .635 10.17 112.80 8.21 20.34 362.2 .09 139,72 .776 12.67 17 2. 74 9.85 30.58 367.8 .13 206.87 .930 15.50 255.75

11.49 41.40 360.8 • 18 285.50 1.085 18.29 352.96 13.15 55.00 359.8 .24 380.34 1.241 21.20 47J.21 14.81 71.20 361.0 • 32 490.73 1.398 24. 17 6("-:.69

. ------17.60 40,90 145.2 .45 700.85 1.661 29.03 865.46 19.70 50.70 144.4 .57 873.60 1.860 32.51 1080.03 21.70 61.40 143.8 .69 1062.39 2.049 35.95 1313.42 23.55 62.20 144.6 • 70 1070.27 2.223 36.09 1323.17 26.65 67.60 114.6 .96 1467.69 2.516 42.45 1314.50 27.75 61.60 90.2 1. 11 1699.21 2.620 45 •. ~-~ 2100.73 30.00 68.50 89.2 1.25 1910.73 2.832 48.62 2362.23 36.50 100.60 89.0 1.84 2812.43 3.446 59.31 3476.99 42.00 124.00 89.0 2.27 3466.61 3.965 66.04 4285,76 49.50 149. 50 89.8 2.71 4142.27 4.674 72.37 5121.07 60.00 145.90 53.4 4.46 6798.11 5.665 93.36 8404.47 70.00 136.40 40.6 5.48 8359.16 6.609 103.83 10~~4!!~9 82.50 189.90 L-0. 2 7.71 11753.66 7.790 123.71 14531.00 90.00 218 ;30 40.0 8.91 13579.01 8.498 133.25 16787.67

100.00 193.50 30.2 10.46 15942.20 9.442 lL-4.70 19709.27

------- -·-- ---- ----------·--- -------------------. ----~-~-------~--- - -- -------- .. - .. - - -- -- ··- -- -

-.J -.J

Tab~e VIII. Capillary data for 0.1% CM0~ (saturated with I 2 and CC14 ) at 23.8°C

Capil- AP w T u 4U/R BAP/21 (4U/R)n' {du/dr)w lary Number lbr

(gram) (sec) ft (sec-1) lbf (see-n') (sec-1) (sq in) (See) (sq ft)

~III 2.48 80.10 151.4 .86 1317.78 .234 181.37 1443.40 I 4.91 91.07 60.2 2.47 3768.03 .463 388.05 4127.25 i 7.86 123.08 44.8 4.49 6842.98 .742 597.70 7495.35 j 10.99 183.90 45.0 6.67 10179.01 1.038 796.77 11149.40 i 14.59 166.30 29.2 9.30 14185.53 1.377 1013.17 15537.88 ; 20.20 230.30 30.0 12.54 19120.93 1.907 1257.62 20943.78 I 24.80 291.10 29.4 16.18 24662.16 2.341 1512,04 27013.28

IV 8.32 6.70 506.4 .37 2393.32 .286 279.37 2621.48 12.15 11.38 501.8 .64 4102.34 .417 412.68 4493,43

~ 15.32 16.10 502.4 .91 5796.91 .526 530.06 6349.54 1 21.03 25.45 500.4 1.44 9200.06 .722 740.53 10077.13

Z? __ .33 33.20 502.2 1.88 11958.63 .869 895_,3__5 13098~-29.70 41.30 500.0 2.35 14941.71 1.020 1051.99 16366.14 40.00 61.80 501.2 3.50 22304.76 1.373 1405.96 24431.14 50.00 60.00 361.6 4.72 30015.33 1.717 1743.JO 32876,78 60.50 54.30 250.6 6.16 39195.75 2.077 2114.58 42932.39 70.00 50.15 190.2 7.50 47595.86 2.404 2437.44 52242.84 80.50 46.00 145.3 8.97 57071.67 2.764 2775.61 62512.47

~-------cf[.-(S0 ___ 45.00 121.0 10.58 67274.02 3.125 3126.56 i3-68-i~44-lOO.OO 33.30 80.4 11.96 76046.80 3.434 3416.69 33296.56

I 120.00 72.60 135.6 15.23 S68~9.46 4.121 ~070.3~ 106082.39 I 141.50 78.20 120.0 18.54.117881.53 4.859 4692:65 ___ f"Ti"Il9.50 I . -* llcuid density = o. 998 en/ cu em -.J

en

Capil- 4P w T lacy

lbr Number <sq in) (gram) ·(sec)

·- -·

Table VIII, Continued

u 40/R

tt (;) (sec-1)

RAP/21

lbt <sq ft)

(4U/R)n'

(see-n')

(du/dr)..,

(sec-1)

t v 50 e 0 0 8 I 0 2 9 2 3 I 6 t 9 7 12 3 3 1 t 2 0 I 9 8 5 9 15 I 4 5 1 : ::•:.·s._._n ) 60.00 10.64 904.0 1.32 16714130 1.182 1140.93 l~j~?l72 I 70.25 13.30 900.6 1~65 20971.75 1~384 1344.62 22971.04 . 80.50 16133 905.2 2.02 25618.67 1.586 1554.27 28060.97

91.50 19.42 905.6 2.41 30452.84 1.803 1761.46 33356.00 101.00 22.62 902.4 2.81 35596.60 1.991 1972.16 38990.12 120.00 29.00 902.4 3.61 4563~6_.66 2.365 236_D_,80 49987133 141.50 35.90 90114 4.47 56557.72 2.789 2757.49 61949.52

---·---·-···- .

......., -.()

Table IX. Capillary data for 0.2% CHC-h- (saturated with r 2 and CC14 ) at 23.5°C

Capil- ~p W' T u 4U/R Rt.P/21 (4U/R)n' (du/dr\,. lary Number lbr

(gram) (sec) tt . · (sec-1) lbt (see-n 1 ) (sec-1) (sq in) (See) (sq ft)

III 5.89 53.40 81.0 1.07 1641.44 .556 116.72 1869.35 9.82 19 3. 30 89.8 3.51 5359.52 .927 249.77 6103.67

14.00 15 2. 50 48.6 5.12 7812.75 1.322 318.26 8897.53 18.10 196.20 41.2 7.78 11856.93 1.709 416.16 13503.23 23.70 238.00 35.6 10.92 16645.53 2.237 517.58 18956.70 29.00 319.50 36.0 14.50 22097.29 2.738 621.00 25165.43 29.00 298.00 36.0 13.52 20610.31 2.738 593.80 23471.93

IV 10.50 7.28 909.6 .22 1447.22 .360 107.64 1648.16 14.40 14.25 10 6 7. 2 .37 2414.48 .494 149.59 2749.72 20.20 26.45 1152.0 .65 4151.71 .693 211.96 4728.17 30.00 38.90 941.4 1.17 7471.88 1.030 309.26 8509.32 40.00 59.60 902.4 1.87 11942.68 1.373 418.09 13600.83 50.00 85.50 906.4 2.68 17056.93 1. 717 525.77 19425.22 60.00 75.90 601.2 3.59 22828.50 2.060 6 34. L3 25998.16

~-. 71.00 58.20 362.2 4.57 29055.56 2.438 740.50 330!39.83 ao.oo 81.80 421.6 5.51 35083.36 2.747 835.93 39955.13 91.00 71.00 301.2 6. 70 42624.37 3.125 947.39 49542.63

100.00 54.60 201.4 7.71 4'?021.64 3!!l..3G. l026t5l 553Z9~.LL 120.00 54.00 152.3 10.05 63903.56 4. 121 1229.13 72776.35

,, 60.00 5. 30 922.S • S5 gt..62.49 ! . 1 ~ ~ 335 .·13 9S3 7. G..]__ ao.oo 9.60 907.4 1.06 13G.53.92 1.577 451.3~ 15321.95

100.00 12.20 90 7.4 l. 31 l91Jqs.~o 1.971 3-:,3.17 21735.~') ___ l2Q.QO _______ t_.'l .9_0 __ ___1_2?~~-- 2.0 S 7 -:, lJ3 • (' _2__ __ 'Lr_3 ~ 5 ?C?_L._; 7 707~~ ~ l

~---...:.---~ lG.l.5t) 2~.2') '101.G. 2. I)G. 333--3S.25 2.719 3•')?.-s9 33'}2!..32 ~ * Uquid density== 0.'?98 g>;</cu em

. Table X. Capillar,y data for 0.2% Carbopol* at 22.8°C

J

l

Capil­lary Number

I

\;; 1

I I

\ l

AP

lbr (__ .;_)

2.45 3.93 5.45 7.61

2.55 5.25 7.66 9.58

12.18 14.98 20.00 25.20 30.00

--------- -- -- - - - - ,.-

w T

(gram) (sec)

134.40 360.2 77.90 91.0

129.00 84.6 162.60 60.6

11.70 361.6 44.00 306.0 96.50 300.2 11.30 150.2

125.00 150.2 124.00 100 .o. 10 5. 30 50.0 124.00 40.2 144.10 34.2

-- .. ---- -

4 Liquid density = 0.999 gm/cu em

u

f't (SeC)

.10

.24 .43 .77

.02

.10

.23

.37

.61

.90 1.54 2.26 3.08

. - --

4U/R

(sec-1)

68.78 157.80 281.08 494.61

24.23 107.67 240.72 385.39 623.22 928.58

1577.10 2309.92 3155.29

RAP/21

lbt <sq ft)

.552

.883 1.226 1.712

.359

.740 1.079 1.349 1.716 2.111 2.818 3.550 4.227

··- ------ - -- -- ··-- --- . - -- - -

.

(4U/R)n'

(see-n')

8.43 12.81 17.14 22.79

4.98 10 ~57 15.85 20.10 25.61 31.31 40.89 49.57 58.01

(du/dr)11

(sec-1)

85.70 196.62 350.24 616.31

30.19 134.17 299.94 480.22 776.55

1157.05 1965.12 Z878.2~ 3931.59

---- - -. - -·- - --- - . - --- ... - - - . ~ ... -- -

<» -

Capil­lary' Number

-- ~-

III

I I

4P

lbt c._.;_)

5.20 7.71

10.56 12.77 15.37 20.00 25.00 30.00 40.00 50.00

w T

(gram} (sec)

5.30 305.1 12.40 • 301.4 29 .8o 302.8 41.40 301.2 66.40 302.2 66.80 183.2 98.60 1a1. 0

137 .ao 1a2.2 149.00 120 .a 194.90 107 .o

Table X. -- Continued

u

.t't (See)

.02

.06"

.16

.22

.35

.59

.sa 1.23 2.01 2.97

4U/R

(sec-1)

43.22 102.36 244.86 341.99 546.69 907.24

1355.41 18a1.ao 3068.97 4532.13

RAP/2L

lbt ( __ 4"~}

.491

.728

.997 1.206 1.452 1.888 2 60 2.a32 3.777 4.721

(4U/R)n'

(see-n')

S.67 10.30 15.99 18.92 23.97 30.95

7 89 44.70 57.20 69,62

(du/dr)11

(sec-1)

53.85 127.55 305.11 426.13 681.20

1130.45

2344.79 3a24.04 564 71! 18

~()) l\)

Table XI. Capillary data :for 0.2% CHC~- at 23.5°C

i

\ l

\ I

Capil­lary Number

III

IV

I v

' I

oP

lbr <sq in)

12.10 17.70 22.80 28.90

16.80 28.00 40.00 49.50 69.50 89.50

110.00

80.00 121.00 140.00

-·--·- -----~ -· < ---·-- -· • - ..

w T

(gram) (sec)

123.20 39 .o 178.20 35.4 310.50 45.2 310. 10 36.0

15.60 900.0 36.85 10 20.0 47.88 780.2 59.60 701.0 56.35 377.4 65.90 301.0 74.40 250.2

11.20 1267.4 18.20 923.0 20.80 919.4

* Liquid density = 0.998 gm/cu em

u

rt (SeC)

5.16 8.22

11.22 14.07

.49 1.0 2 1.74 2.41 4.24 6.22 8.45

.99 2.21 2.54

4U/R

(sec-1)

7865.32 12533.58 17.103.85 21447.17

3134.27 6532.68

11096.93 15373.86 26998.93 39588.91 53770.03

12544.51 27991.04 32115.02

R.6P/2L

lbr (sq ft)

1tl42 1.671 2.152 2.728

.576 • 961

1.373 1.700 2.386 3.073 3.777

1.577 2.385 2.759

(4U/R)n'

(see-n')

:24:2.11 464.74 568.97 659.26

188.55 304. 11 429.34 530.82 765.82 982.46

1199.10

465.01 784.02 857.38

(du/dr)w

(see-1)

8220.18 14214.52 19397.73 24323.55

3554.62 7408.82

12585.19 17435.72 30619.90 44898.38 60981.40

14226.91 31745.06 36422.12

I I '

!

e

~

Capil- AP w T lary

lbr Number (gram) (sec) (sqin)

IV 5.18 27.50 909.8 7.12 25.50 630.8 9.50 32.40 606.6

11.88 28.10 422.0 14.37 34.50 427.4 18.00 36.70 364.2 22.45 38!50 ;204.4 26.70 36.70 241.8 33.00 34.60 18 2. 6 41.00 30.90 132.0 61.00 36.40 106.0

l ao.oo 49.80 101.8 I 100.00 65.10 104.0

\ 122.00 7 5. 40 100.2

l

v 40.00 14.60 904.6 60.00 22.66 900.4 80.00 31.12 902.6

100.00 39.7 5 901.7 120 .oo 49.07 90 2. 8 140.80 58.35 902.5

Table XII. -- Continued

0 4U/R RAP/2L

rt <s-ee) {sec-1) lbr <sq ft)

.85 5466.69 tl78 1.14 7311.16 .244 1.51 9660.08 .326 1.89 12042.93 .408 2.29 14598.99 .493 2.86 18224.87 .618 3.59 22874.Q~ .111 4.31 27450.36 • 916 5.39 34269.96 1.133 6.65 42337.27 1.408 9.76 62106.04 2.095

13.91 88474.86 2.747 17.80 113210.31 3.434 21.40 136094.93 4.190

1.81 22915.50 .788 2.82 35732.01 1.182 3.87 48952.77 1.577 4.95 62590.46 1.971 6.10 77171.61 2.365 7.26 91796.62 2.775

(4U/R)n'

{see-n')

4096,~4 5425.58 7102.06 8788.72

10585.56 13116.74 16338.30 19487.03 24147.95 29621.83 42898.66 60391.70 76639.81 91564.96

16366.50 25142.95 34084.15 43222.04 52918.22 62581.75

(du/dr)w

(sec-1)

7374.57 9743.85

l·2 I4 7 14725.59 18382.91 23Q13.Q_l_ 27688.41 345.67.15 42704.41 62644.62 89242.10

114192.06 137275.13

23114.22 36041.87 49377.28 63133.24 77340.84 92592.67

())

""

Ta.b.le XIII. Ca.pillar.y data for 0.1% Carbopol* (saturated with I 2 and cc14 and aged for one year) at 22.7°C

Capil- _op w T" u 40/R RAP/21 (4U/R)n' (du/dr)v lary

lbt Number {gram) (sec) tt (sec-1) lbt (see-n') (sec-1) <sq in) <sec> (sq tt)

III 1.92 121.27 122.7 1.61 2460.59 .181 849.96 2557 2.30 142.68 121.9 1.91 2914.00 .218 983.68 3028.80 2.73 166.00 131.6 2.06 3140.38 .258 ° 1049.35 3264.10 3.25 186.25 131.0 2.32 3539.61 .307 1163.64 3679.06

i,. 3.47 191.85 125.7 2.49 3799.76 .328 1237.17 3949.46 3.78 212.95 133.9 2.59 3959.38 .357 1281.94 4115.37

I 4.16 214.so 122.4 2.s6 4369.01 .393 1395.73 4541.13 . 4.48 221.40 120~9 2,99 4559.12 .423 1448.04 4738.74

I l I t

' 1

5.10 266,00 128.6 3.37 5149.57 ,482 1608.69 5352.45 6.11 170.00 67.8 4.09 6242.37 ,577 1899,64 648 7.36 146.30 48.8 4,89 7463.71 .695 2216.73 7757.76 9.77 206.20 54.2 6.21 9471.52 .923 2723.28 9844.67

12.24 264.40 55.4 7.79 11881.80 1.156 3312.47 12349 14.77 262.50 46.4 9.24 14084,51 1.395 3836.69 14639.40 20.00 282.60 38.0 12.15 18514.80 1.888 4859.19 19244.23 29.75 509.80 52.1 15.98 24360.86 2.809 6159.05 ~5320.60 20.00 431.60 60.2 11.71 17849,07 1.888 4707,88 18552.26

* Liquid density = 0.999 gm/cu em

'

C» ~

Capil­lary Number

IV

v

AP

lbf <... ... .; -)

9.65 12.82 16.08 20.45 31.00 40.05 61.00 80.50

100.80 122.00

104.00 123.50 140.50 142.50 162.00

w T

(gram) (sec)

20.80 908.4 28.20 899.0 35.80 900.8 58.30 1115.0 29.70 362.8 41.10 366.8 54.20 302.0 74.50 299.0 82.80 254.4

104.20 254.6

15.70 710.4 21.50 713.0 23.55 7 11.5 25.20 712.9 29.50 710 .o

Table XIII. Continued

u

tt <sec>

.65

.89 1.13 1. 48 2.32 3.18 5.10 7.08 9.25

11.63

2.48 3.38 3.71 3.97 4. 66

4U/R

(sec-1)

4140.00 5671.57 7185.70 9453.83

14801.43 20259.42 32449.41 45050.50 58847.46 73998.66

31369.38 42801.42 46981.33 50174.29 58975.68

RttP/21

lbt <sq ft)

.331

.440

.552

.702 1.064 1. 3 75 2.095 2.764 3.461 4. 190

2.050 2.434 2.769 2.809 3.193--

(4U/R)n'

(see-n')

1332.30 1748.62 2145.22 2718.89 4004.82 5252.28 7890.01

10475.44 13194.87 16082.61

7662.63 10022.10 10862.17 11497.03 13219.7Q

(du/dr)..,

(sec-1)

4303,10 5895.02 7468.79 9826.29

15384.55 21057.57 33727.81 46825.34 61165.86 76913.96

32605.23 44487.66 48832.24 52150.99 6122_9. 13

~

Ts.b~e XI.V. Cs.pi~ar,y data for 0.2%·carbopo~* (saturated with I2 and CCl4 and aged for one year) at 22,8°C

Capil- · AP w T u 40/R RAP/21 (40/R)n' (du/dr)11

lary lbr Number (gram) (sec) .f't (sec-1) lb.f' (see-n 1 ) (sec-1)

<sq in) (See) (sq !t)

l . . I I 1.63 195.50 75.7 .74 475.95 .366 137.41 505.97 t 2.63 437.40 104.8 1.20 769.18 .592 201.60 817.70

5.45 358.20 45.4 2.27 1454,05 1.226 335.22 1545.78 a.oo soo.3o 46.6 3.09 l978.ss ____ J.ao_o ______ 428.70 2103.40 9,97 611.30 47.6 3.69 2366.78 2.242 494.63 2516.09

13.10 606.00 38.7 4.50 2885.84 2.947 579,49 3067,89

co co

Capil­lary Number

~p

lb.r <sq in)

w

{gram)

2. 26 25.60

T

(sec)

Table.XVI. -- Continued

·u

!t <sec>

40/R

(sec-1)

&6P/2L

lbr <sq !t)

{4U/R)n'

(see-n')

{du/dr)w

{sec-1)

2.93 33.87 122.6 .45 687.20 .276 184.25 730.55 3.89 46.73 120.8 .63 962.25 .367 241.08 1022.96 4.79 60.65 122.2 .81 1234.58 .45? ,q~-16~---- 1312.47 5.77 77.00 122.6 1.02 1562.29 .545 •355.00 1660.85 7.05 100.58 124.8 1.31 2004.74 .665 433.22 2131.21 7.60 _106.95 120.4 1.45 2209.61 _.717 ___ 468.22u_ - 2349.01 8.29 120.50 122.6 1.60 2444.89 .783 507.62 2599.12 8.83 142.40 127.2 1.82 2784.74 .834 563.22 2960.42

10.21 165.00 122.4 2.20 3353.24 .964 653.28 3564.78 l 11.10 182.42 121.6 2.44 3731.65 1.048 711.51 3967.06 1 12.25 210.90 124.6 2.76 4210.38 1.157 783.50 4475.99 \ 13.70 236.10 122.3 3.15 4802.11 1.294 870.25 5105.05 ·, 16.35 247.60 100.8 4.00 6110.16 1.543 1054.83 6495.62 : 20.50 322.00 101.6 5.17 7883.61 1.935 1292.87 8380.94 I 25.00 264.30 69.4 6.21 9473.28 2.360 1497.12 10070.90

\ \ <

30.00 337.78 66.0 8.35 12730.72 2.832 1895.60 13533.83 40.05 416.60 60.0 11.33 17271.53 3.781 2418.42 18361.11 so.oo 478.80 53~6 14.58 22220.42 4.721 2957.36 23622.19 60.00 385.90 36.4 17.30 26371.59 5.665 3390.79 28035.24 70.00 267.60 21.0 20.80 31697.85 6.609 3927.32 33697.50

.·

co "'

Table XYI. Capillary data for 0.2% CHC~- (saturated with r2 and cc14 and aged for one year) at 23.8°C

Capil- oP w T u 4.0/R RAP/21 (4U/R)n' {du/dr).., lacy

lbr lb! Number (gram) (sec) !t (sec-1 ) (see-n') (sec-1) (sq in) <sec> <sq rt>

III 2. 16 50.50 132.4 .62 950.09 .204 139.13 - ---1042.54 4.17 75.10 76.4 1.60 2448.55 .394 275.03 2686.82 6.87 153.80 65.6 3.83 5840.02 .649 514.18 6408.33 9.13 151.30 50.0 4.94 7537.57 .862 617.85 8271.06 [ - - -

11.27 201.60 49.0 6.72 10248.42 1.064 770.77 11245.71 I ' 13.26 208.90 42.0 8.13 12389.44 1.252 883.56 13595.07 l 16. 10 164.50 28.6 9.40 14327.23 1.520 980.99 15721.44 l "21.85 230.00 27.2 13.82 21063.05 2.063 1294.59 23112.74 I IV 6.01 555.4 6.80 .30 1957.55 .233 234. 11 2148.05 ' 10.34 10.60 541.8 .55 3539.26 .355 358.55 3883.67 I J . 13.36 17.40 606.8 .81 5187.40 .458 472.14 5692.20 I 16.90 21.80 544.2 1.13 7246.76 .580 600.60 79 51_. 96

I 20.85 30.07 548.0 1.56 9926.57 .716 753.27 10892.54

1 24.65 38.12 542.4 1.99 12713.92 .846 900.15 13951.14 28.65 52 .o 5 603.0 2.45 15615.28 .983 1043.70 17134.83 40.00 66.10 480.8 3.91 24870.44 1.373 1459.06 27Z90.62 51.00 68.12 360.6 5.37 34173.96 1.751 1834.08 37499.49 60.00 57.82 2 41. 0 6.80 43293.99 2.060 2174.5~ 473)7,00 70.50 53.37 18 1. 2 8.313 53282.60 2.421 2525.01 58~67.62

80.00 45.10 130.0 9. 87 62759.52 2.747 2540.78 62~66.76 90.00 53.30 l32.C 11. ~ s 73nG.6.55 3.C·90 31SR.75 Q015C..83

100.00 60.00 130.4 13.09 83237.70 3.l..34 3l..81.10 91337.71 120.00 76.10 130.8 16.55 105250.30 4.121 4121.62 115492.38

"' --~--- ---- ~-- . -------------------- 0

* Liquid density = 0.998 gm/cu em

Table XV. Continued

Capil- .t.P w T u 4U/R R.6P/21 (4U/R)n 1 (du/dr).., lar;y

lb.r f't lbr {see-n 1 ) {sec-1) Number <sq in) (gram) (sec) (See) {sec-1) (sq ft)

... - ···--· . ··- --- ------- -- -~ .... . . - - . - ~ -- -- . - . . -· -;

v 51.00 8.95 905.8 1. 11 14032.39 1.005 966.41 15397.91 I 60.50 11.69 903,8 l. 45 18368,91 1.192 1173.13 20156.42

l 70,00 14.55 903.8 1.80 ·22862.92 1.379 1373.30 25087,76 80,00 17.53 904.4 2.17 27527.23 1.577 1569.65 30205.95 91.00 21.05 903.0 2.62 33105,91 1.793 1792.63 36327.50

100.00 24.24 906.4 3.00 37979,90 1.971 1978.92 41675.80 120 .oo 31.40 902.0 3.91 49438.38 2.365 2 39 2! 51 54249,32 141.00 38.86 900,8 4.84 61265.44 2.779 2791,94 67227.29

I '--··-- .. -·- -- ·- --- . -- - -- - ~ - - -- - ---

...0 .....

Capil-lary Number

I I I

Table XVI.

.t.p

lbr <sq in)

.53 2.26 3.66 4.96 6.63 8.32 9.82

11.79 14.22 16.00 19.50

Capillary data for 0.2% CNC~- (saturated with 12 and cc14 and aged for . one year) at 23.5°C.

w T u 4U/R RAP/21 (4U/R)n' {du/dr).w

(gram) (sec) rt

(SeC) (sec-1) lbr

(sq ft} (see-n 1 ) (sec-1)

55.70 300.6 .30 461.38 .050_ 52.86 524.37 I

118.60 243.2 .79 1214.28 .213 98.85 1380.04 I 145.50 183.6 1.29 1973.28 .345 135.32 2242.65.\ 135,10 119,4 1.84 2817.40 .468 170.37 3202.01 151.70 94.4 2.62 4oo1.4o .626 213.77 4547.63 I 183,00 85.6 3.49 5323.23 .786 257.12 6049.91 ' 155.70 59.0 4.31 6571.05 .927 294,64 746~~ 203.00 60.8 5.45 8313.63 1.113 343.06 9448.53 200.00 47.4 6.89 10506.31 1.343 399.14 11940.52 215.00 44.2 7.94 12111.97 1.510 437.60 13765,37 __ _ 246.40 40.6 9.91 15111.70 1.841 504.93 17174.60

* Liquid density = 0,998 gm/cu em

..0 I\)

Capil­lary Number

AP

lbr <sq in)

W T

(gram) . (see)

Table XVI.

u

!t (See)

Continued

4U/R

{see-1)

RAP/21

. lb:r (__ N)

(4U/R)n'

{see-n 1 )

(du/dr )w

(see-1)

f~~~,_-:.::.:::.:.::~~--=-~ -:_-. -----------: .. -:--. -:-::". __ :--_ -=-=----~~---~-----:-:-~ .. -::-:::::-.::::::: __ ================= IV 10.10 8.00 902.8 .25 1602.43 .346 118.27 182~~

20.40 26.40 1015.2 .73 4702.56 .700 237.30 5344.51 30.00 42.60 921.6 1.31 8358.91 1.030 344.27 9499.98 i

1 40.00 65.70 908.8 2.05 13073.13 1.373 459.75 1,.857.74 i I 5o.oo 55.6o 526.4 3.oo 191oo.35 1.111 587.53 211o1.14 :

I 60.00 66.50 476.8 3.96 25221.32 2.060 703.27 28664.29 --------~70.00 70.70 401.0 5.01 31882.88 2.404 818.39 36~L22__ 1 8o.5o 12.20 331.o 6.2o 39444.98 2.764 939.18 44829.62 l 91.00 65.00 246.4 7.50 47704.04 3.125 1062.06 54216.13 ' 100.00 73.00 240.4 8.63 54912.46 3.434 1163.27 62408.57 I 122.00 79,90 199.8 11.37 72315.90 4.190 1390,01 82187.76 \ t __ v ---~-Q..!..QQ_ ~ • ~- . ~ - • ~ •. ~ •.•.• ~ • /..._?() Q()t;_A _7{; 0717 ()O 1.182 379.48 11043.58 I ao.oo -- ·- · · -- ·-·-· -I 100.00 I 12o.oo

'-J,'-JU '-J!U.b 1e£t:. !:>4j4.~~ 1.577 511.88 17541.15 14.30 904.4 1.77 22446.71 1.971 652.20 25510.91 19. 10 906.6 2.36 29908.52 2.365 785.24 33991.33

' l 141.50 25.90 956.8 3.04 38428.71 2.789 923.45 43674.62

I

'-0 \,.)

Table A-1. Capillar.y data.for Oil Number 243* at 25°C

Capil­lary Number

III

IV

v

,_

.6P

lbf <sq in)

_2.48 5.15 8. 27

10 I 12

13.85 19.40 25.25 30.00 40.00 50.00 61.00 80.00 99.50

121.00

31.00 39.50

-------~-9-~ 00 60.00 80.00

w T

(gram) (sec)

130,00 150.6 93.00 60.6

117.20 48.6 l6816Q ~5.8

15.00 615.8 20.80 601.2 27.60 600.6 32.40 601.2 39.00 542.0 43.70 480.0 40.20 360.4 39.80 271.2 37.00 200.8 40.50 179 .a

3.60 968.2 4.50 907.6 5.60 903.2 6.80 903.0 9. 10 903.2

100.00 11.50 902.6 --------------··-· 120.00 13.9 3 899.8 141.00 16. 28 902.6

----- .... -- ----·- ·--~---·----------~----------------

* liquid density = 0.875 gra/cu em

u

.f't <sec>

l! 64 2.92 4.58 11QQ

.80 1.14 1.52 1.78 2.38 3.01 3.69 4. 86 6.10 7.46

.48 .64 .81 .98

1.31 1.66 2.02 2.35

4U/R

(sec-1)

2502.78 4449.53 6991.91

lQ613.25

5129.08 7285.05 9676.37

11347.88 15151.44 19170.30 23487.13 30901.67 38799.55 47430.10

6146.40 8195.99

1 02_49 .14 12448.14 16654.85 21061.33 25591.07 29815.52

~P/21

lbr L ... N}

.23~

.487

.781

.255

.475

.666

.867 1.030 1.373 1.717 2.095 2.747 3.417 4.155

• 611 .778 .985

1.182 1.577 1.971 2.365 2.779

.

(4U/R)n'

(see-n')

2102.51 3690.33 5740.83 8681.30

4240.49 5976!06 7887.67 9217.43

12227.94 15390.50 18771.10 24'546.38 30664.19 37317.81

5061.11 6705.69 R343.87

10090.31 l3ld 2. 93 16873 .}Q 20ld3.55 23702.50

(du/dr)..,

-1) (sec

2511.03 4474.87 7031.72 1013~.03

5158.29 7326.54 9731.47

11412.50 15237.72 19279.46 23620.87 31077.6Lt 39020.49 47700.18

6181.40 8242.66

t0)_2_7_. 5Q_ 12.d9.02 lt.4~.69

2 : .l.:':...~ . . ~ 9 __ 25736.80 29985.30

i i

"' .J:-

Table A--2. Capillary. data for Oil Number 678-:< at 25°C

n' Capil- uP W T U 4U/R RAP/21 (4U/R) (du/dr)w lary Number lbf ft 1 lbf r n I 1

( . ) (gram) {sec) (-) (sec- } ( q ft) {sec- ) (sec- ) sq m sec s

I

____ I 2.80 92.00 120.6 .24 157.62 .629 143.53 158.36 5.13 113.40 90.0 .40 260.34 1.154 234.88 261.57 8.10 152.70 80.2 .61 393.41 1.822 352.24 395.26

11.59 147.10 62.0 .76 490.23 2.606 437.14 492,54 15.40 165.70 50.4 1.06 679.32 3.462 602.11 682.52 24.00 166.20 34.8 1.54 986.81 5.395 868.63 991.46 40.00 211.80 35.2 1.94 1243.27 8.992 1089.71 1249.13 59.00 134.00 12.4 3.48 2232.89 13.264 1936.00 2243.41

II 4.83 22.80 131.6 .14 145.42 ,681 132.62 146.11 7.36 30.10 120.2 .20 210.19 1.038 190.39 211.18

10.31 43.00 120.4 .29 299.78 1.453 269.76 301.19 12.77 53.50 120.4 .36 372.98 1.799 334.27 374.74 15.23 66.00 120.4 .45 460.13 2.146 410.78 462.29 23.00 54.00 64.6 .68 701.65 3.240 621.53 704.96 30.00 49.80 45.6 .89 916.70 4.227 80?_.0_1 921.02

1 4o.oo 65.oo 45.o 1.18 1212,4s 5.636 1063.19 121s.16 1 50.50 67.80 40.0 1.39 1422.76 7.115 1243.92 1429.47 i--·--------~0_.00 74.60 34.8 1.76 1799.38 8_-_lt2.i._ ___ 1_2_96!.J8 1JtQ]_'!..!~_6_ l 80.00 131.00 45.2 2.38 2432.74 11.272 2105.94 2444.21

100.00 167.50 45.4 3.03 3096.86 14.090 2668.91 3111.46 -- ---· -·~-·- - ----------- ----------·~----------- -------

* Liquid density = 0.891 gm/cu em

'-() Vt

I l l r I \

'

Capil­lary Number

I I I

AP

lbr (sq in)

_1_4. 4 0 18.00 22.00 26.00 29.70 40.00 51.00 60.00 80.00

100.00 122.00

w T

(gram) (sec)

22.80 200.6 21.60 151.2 20.70 121.2 20.40 10 2. 2 21.30 92.8 2l~. 60 81.4 24.60 62.2 29.30 62.0 38.30 60.6 l~8. 40 60.2 59.20 60.2

Table A-2. -- Continued

u

ft (;eo)

.20

.26 .31 .36 .42 .55 .72 • 86

1.15 1.47 1.79

4U/R

(sec-1)

316.98 398.41 476.32 556.69 640.13 842.84

1103.01 1317.99 1762.63 22lr2 .25 2742.59

RAP/21

lb:f' (sq ft)

1.359 1.699 2.077 2.455 2.804 3.777 4.815 5.665 7.554 9.442

11.520

(4U/R)n'

(see-n')

284_,94 356.63 424.96 495.24 567.99 74l~. 07-968.91 ---

1153.95 15 3'~. 9 7 l9lr3.97 2368.91

(du/dr)w

(sec-1)

318,48 400.29 478.57 559.31 643.14 8lt6.81

11_08 ._2__1_ 1324.?0 1770.94 2252.82 2755.52

-.o 0"-

Capil­lary Number

I

t.p

lbr (.._ ~~>

Table A-3. Capillary data for glycerine* at 25°C

w T

(gram) (sec)

u

ft (SeC)

4U/R

(sec-1}

RAP/21

lbf (__ N)

(4U/R)n'

(see-n')

(du/dr)w

{sec-1)

12.38 169.65 183.4 .21 135.39 2.783 135.39 135.39 j 16.40 237.90 184.2 .29 189.04 3.687 189.04 189.04 I

l 16.60 122.00 94.0 .29 189.96 3.732 189.96 189.96 -~~- 20.69 227.20 140.1 .37 237.36 4.651 237.36 237.36 I 25.o5 1s3.os 89.4 .46 299.69 5.631 299.69 299.69 1

1 29.1o 2s6.1s 103.9 .s6 361.69 6.542 361.69 361.69 ~--------40.30 333.00 102.6 .74 475.06 9.060 475.06 475.06

51.50 405.77 101.8 .91 583.42 11.578 583.42 583.42 61.00 500.98 101.6 1.12 721.73 13.714 721.73 721.73

- 61.00 77.10 1~.0 1.10 705.32 13.714 705,32 705.32 70.00 391.08 71.0 1.25 806.23 15.737 806.23 806.23 78.00 98 .• 75 16.2 1.39 892.22 17.536 892.22 892.22

____ ._.._.:81_~0 __ 40 4. 2 5 62.0 1. 49 954.3 5 18.32 3 95l~. 35 9 5lt_,_l2_ 90.80 373.80 53.6 1.59 1020.76 20.414 1020.76 1020.76 99.00 433.40 56.7 1.74 1118.81 22.257 1118.81 1118.81 99.00 1?7.90 16.6 1.76 1127.75 22.257 1127.75 112~~.75

119.50 168,00 18.5 2.07 1329.19 26.866 1329.19 1329.19 136.50 295.50 27.8 2.43 1555.84 31.138 1555.84 1555.84

----·rc-------6-4.o5 s1.1o 63.4 .46 479.25 9.024 479.25 479.25 82.20 54.77 52.9 .60 615.62 11.582 615.62 615.62

102.00 52.43 41.0 .74 761.09 14.372 761.09 76[.09 ---·----li2.00 82.90 53.6 .90 919.64 17.190 919.64 -919.64 __ _

141.00 108.05 59.9 1.05 1072.57 19.867 1072.57 1072.57

--* Iir;uid den-~ity-~ 1 ~260 g.;;~~ em '-0 -..J

Capil- AP w T lary Number lbr

(gram) (sec) <sq in) -----

I I I 10.50 46.34 261.8 ' 86.10 55.59 260.0 I.

I 87.10 55.65 258.3 I

. 102.50 61.37 240.7 . 114.00 71.45 247.7 l

' ------ --------

Table A-3. -- Continued

u 4U/R ~P/21

ft (-;ec) {sec-1}

lbf (sq ft)

.22 349.69 6,657

.27 422.40 8.130

.27 425.64 8.224 .33 503,71 9,678 .37 569.87 10.764

---- + ·-----~---- -- •• -·--

(4U/R)n'

{see-n')

3_49 ._69 422.40 425.64 2Q3.71 569.87

- - - .. - - - -- - -- -.

(du/dr)w

{sec-1)

342.62 422.40 425.64 50.3...._ll_ 569.87

- --- ---· ---------

'-0 co

Table A-4. . Capillary_ data for glycerine-r, at 20°C

-~~····-'"

Capil- AP w T u lary

lbr ft Number <sq in) (gram) (sec) (SeC)

I 11.40 51.64 76.4 • 15 l4 .60 50.07 60. 1 .19 19.30 70.08 64.1 .24 2't. 7 2 83.47 60.4 .31

\ 29.06 102.23 66.4 .35 40.00 140 ·'~3 59.6 .53

l------~~00 __ 141._87 60.4 • 53 I so.oo 166.95 61.9 .61

' 50.00 158.69 61.6 .58 70.00 221.76 61.0 .83

I 90.00 30 5. '~1 62.5 1. 11 I 100.00 360.05 64.8 1.26 . ________ 1J_O_._o o 38 2. 7 5 61.8 1. lc 1

120.50 416.81 61.6 1.54 1'~0. 90 49 3. 60 62.5 1.80

..

I I 61.20 5 l. 1(, 10 1. 8 .29 81.00 6 7. u~ 101. 1 .38

____ 99_._ 3_0 _______ 8'':_• __ 5Q __ 1_9_3_. 8 ·'t 7 120.50 10 3. 40 102.4 .58 1'~0. 00 118.80 10 2. 8 .67

--- -----~---- ---··--·· . - --------· ---------------·-

* Liquid cirn:·.:L·. : 1.26 [;;-'/cu Cl:t

4U/R

{sec-1}

98.84 121.83 159.88 202.10 225.16 344.58 343.50 394.43 376.74 531.66 714.63 812.58 9 05.75 989.55

1154.98

298.56 394.54 483 ._QJ 599.90 686.56

MP/21

lbf c... ... N)

2.563 3.282 4.339 5.557 6.533 8.992 8.992

11 • 2 '~ 1 11.241 15.737 2 0. 2 3 1~

22.482 24_._ 7}Q_ ___ 27.091 31.677

8.623 11.413 1_2_.~9.1 16.978 19.726

(4U/R)n'

{see-n 1 )

98.84 121.83 159.88 202.10 225.16 344.58 3'} 3. 50 394.43 376.74 531.66 71Lt.63 812.58 905.75 989.55

1154.98

298.56 394.5 't

{du/dr )w

-1) (sec

98.84 121.83 159.88 202.10 225.16 344.58 3'~3. 50 394.43 376.74 I

531_.66 714.63 812.58 9Q5 __ J..L 989.55

1154.98

298.56 39'~. 54

lt_~_}_._f:_J ______ !_t 3-~-. _Q 3_ 599.90 59~.90

626.56 686.56 ---·- -------.

-.() -.()

Table A-4.

Capil- ~p w T 0 lary

lbr Number (gram) (sec) rt

<sq in) <sec)

\ I I I 80.00 41.7 2 301.5 .17

I 90.20 46.98 29 7. 6 .20 100.00 52.60 304.1 .22 110.80 58.52 307.3 .24

I 121.50 63.38 304.0 .27 I 131.00 69.40 307.7 .29 I 141.00 75.11 305.0 .31 I

Continued

4U/R R~P/21

{sec-1) lbr

<sq ft)

2 73. 14 7.554 311.61 8. 517 341.43 9.442 375.90 10.462 411.54 11.472 445.21 12.369 486.10 13.3l't

(4U/R)n'

(see-n')

273.14 311.61 341.43 3_?5. 90 411.54 4Lr5. 21 486.10

(du/dr)w

(sec-1)

273.14 311.61 341.43 375.90 411.54 445.21 ~ 8 6_._l__Q__

...... 0 0

101

APPENDIX B

0

2

6 ---- ---

4 ·-

2

0.00126" I.D. tube

0.00063" I.D. tube

FLOW FUNCTION, 8U/D, -1 sec

102

Figure 20. Flow chart for 0.05% Carbopol (saturated with

iodine and carbon tetrachloride and aged for one

6

0

4 A

4

3 4 6

0.00126" I D t • • ube

0.00063" I.D. tube

----------·----1------l

···-- ..• -···-· . ··-·--· - __ ......... -

·--·-··-·······-- _____ .. _______ ---· -·-·· ... i l

a

I ----· ....... -- --·-. t

2 3

I -1 FLOW FUNCTION, 8U D, sec

I

I 4 6

Figure 21. Flow chart for 0.1% Carbopol (saturated with iodine 0

and carbon tetrachloride and aged for one 7ear) at 22.7 C

1 ".' l.;

E_, <X!

U)

g~ (}. . [-f ())

I -I -

I 0 0.00525" I.D. tube 6..-----,----.-- --r----+------l---- --- -

4 -------- ------~-- -- ----- ·~·------· ·---~···

I I 2 ----· -- .. I ............. .

I

-~i- t--~~ --~=-. ___________ J___ l

I 1 o0 ___ .. -.J . ·----- .. ---j·-- - --·-·-····--·· ...

8 - ------- - ··--,---1-- ,---- ---------· 6 -----+-----·· _j_- ---- I :

I I I I

. --·· --! --'" --- i -· -· ..... -· 4 ·-··---· ----

I

10- 1 ~----~--~~~------~----~--~~~~----~ 2 4 6 2 4 6 2

FLOW FUNCTION, 8U/D, -1 sec

Figure 22. Flow chart for 0.2% Carbopol (saturated with iodine

and carbon tetrachloride and aged for one year) at 22.8°C

~ 0'

~ .0 rl .. ~ ...;:t.

~ Q ..

5

o 0.00126" I.D. tube

3 A 0.00063" I.D. tube

l I

2 1 ' I I . - I I ~I ; I I i 1 j , l ' i I

:j ~ 10°

I I i I I ~-----·---- ; . !

I \ ; I E-< < U)

~ 7 I l ------~-- --- ~ I -·-'·· -- ~- ---.

I E-< C/)

~ < r=:l ::X:: U)

51 -+·----

3~--------------------------------------------------------------~ 5 7 4 2 3 5 7 5 2

10 10 2 3

Figure 23. flow chart fo~ 0.1%

one year) at ;j.8°C

r'"lf("\ \...-.!'.:.\....

FLO., FUJ\C':'ICJ;, 8U/D, sec_,

( ... ---.,r-•ea· , . ...:t'­~c.. •. d .... 0.0 n..l .~ iocine anc cc.rbcr. · ... e:rac:-.1o~ide anc aged for c \J1

~ ~

c:r rt)

......... ~

,Q ~ .. ....::1 -.:t ~ s .. ~ < ~

e-. < (f.)

~ til 0::

~ :I: (f.)

c "' I I

0 0.00126" I.D. tube

3 I ~ 0.00063" I.D. tube

2

i I i

I I

\ I 10° I I IT I ~---

! •7 ' ! I '

7~~----------- I

I I

I l I I

5 --------·-·----· --!---·--- ---.

; i

3~----------------------------------------------------------------~ -~ 2 3 5 7 4 2 3 5 7 5

1CJ 10 10

FLO.l F'Ul~CTION, 8U/D, sec-1

F'i.g-1.1re 24. Flow chart for 0.2% CEc· (saturated with iodine anc cc..rbon tetrachloride and aged

for one year) at 23.5°C 0 ry...

107

APPENDIX C

108

NOTATION

D = inside diameter of tube

f = friction factor

= the gravitational constant

K, n = power law constants defined by Equation (2. 32)

K' = consistency index defined by Equation (2. 3 0)

L = length of pipe

= kinetic energy correction constant defined by 2

m mpu /g c

NRe = Reynolds number nup

= ~N N' = Reynolds number for non-Newtonian fluid defined by

Re Equation (2. 45)

n' = flow behavior index defined by Equation (2. 27)

tt,.p ::I pressure drop

Q = volumetric flow rate

R = radius

T = time

u = average velocity in the capillary

w = weight of fluid passing through the capillary

Greek Letters

'( = defined by Equation (2. 42)

"C = shear stress

= shear stress at wall

'robs -= shear stress observed from experimental

'rcalc

j.la

=

=

=

shear stress calculated from Equation {2. 32)

apparent viscosity

viscosity of non-Newtonian liquid

~ = mass density of liquid

109

1.

2.

3.

4.

5.

6.

7.

8.

9.

1 o.

11.

12.

13.

14.

15.

110

VII. BIBLIOGRAPHY

Wu, H. C., "Mass Transfer to Droplets: Effect of Non­Newtonian Continuous Phase"; Masters Thesis, Missouri School of Mines and Metallurgy, 1964.

Wilkinson, W. L., "Non-Newtonian Fluids", 1 ed., Chapters 1 & 2, Pergamon Press, New York (1960).

Metzner, A. B., "Non-Newtonian Technology, Fluid Mechanics, Mixing, and Heat Transfer" in B. D. Thomas and J. W. Hoopes Jr., "Advances in Chern. Eng.", vol. 1, pp. 78-150, Adademic Press, New York (1956).

Bird, R. B. , W. E. Stewart, and E. N. Lightfoot, "Transport Phenomena~', 1 ed., pp. 10-15, John Wiley, New York (1963).

Ram, A., and A. Tamir, "A Capillary Viscometer for Non­Newtonian Liquids", Ind. Eng. Chern., 56, 47 (1964).

Bowen, Jr., R. L., Chemical Engineering, pp. 243-248 (June 12, 1961).

Ibid. , PP· 12 7- 13 0 (June 2 6, 19 61).

Ibid. , PP· 147-150 (July 10, 1961).

Ibid. , PP· 143-150 (July 24, 19 61 ). --Ibid. , PP· 12 9 - 13 2 (August 7, 19 61 ).

Ibid. , PP· 119-122 (August 21, 19 61 ). -Ibid. , PP· 131-146 (September 4, 19 61 ). -

"ll E w "Non-Newtonianism in Thin Liquids: Molecular Merr1 , . · , Ch E "

and Physical Aspects", in A. Acrivos, "Modern .em. ng. , I 141 -195 Reinhold Publishing Corporatlon, New

vol. , PP· ' York ( 1963 ).

A B Chern. Eng. Prog., 50, 27 (1954). Metzner, · · ,

Sisko, A. W., Ind. Eng. Chern. , SO, 1789 (1958).

111

16. Philippoff, W., Kolloidni Z., 71, 1 (1935).

17. Van Wazer, Lyons, Kim, and Colwell, "Viscosity and Flow Measurements", Inter science, New York 1963.

18. Rabinowitsch, B., Z. Physik. Chern., A145, 1 (1929).

19. Mooney, M., J. Rheology,~· 210 (1931).

20. Metzner, A. B. , and J. C. Reed, AIChE Journal, .!_, 434 (1955 ).

21. Oka, S. , "Principles of Rheometry" in F. R. Eirich, "Rheology", vol. III, pp. 22-25, Academic Press, New York (1960).

22. Karam, H. J. , Ind. Eng. Chern. , ~~ 38 ( 1963 ).

23. Brodkey, R. S., Ind. Eng. Chern., 54, 44 (1962).

24. Hershey, H., Personal Communication to the Author, February,

1965.

25. Wellek, R. M., Personal Communication to the Author, May,

1965.

26. Chen, Y. C., Personal Communication to the Author, March,

1965.

27. Lange, N. A., Handbook of Chemistry, 8 ed., pp. 1708, 1709, Handbook Publishers, Sandusky, Ohio (1952).

112

VIII. ACKNOWLEDGEMENTS

The author is deeply indebted to Dr. Robert M. Wellek,

Assistant Professor in Chemical Engineering, who suggested this

investigation and served as research advisor. His help, guidance

and encouragement is sincerely appreciated.

113

IX. VITA

The author was born on February 2, 1938, in Shanghai, China.

He attended high school in Taipei, Taiwan, graduating in 1956. After

high school, the author attended Tunghai University, Taichung, Taiwan,

graduating in 1961 with a degree of Bachelor of Science in Chemical

Engineering. After graduation, the author performed his national

service in the Chinese Navy from October 1961 to October 1962.

In September 1963 the author entered graduate school at the

University of Missouri at Rolla.