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Masters Theses Student Theses and Dissertations
1965
Construction of a capillary viscometer and the study of non- Newtonian liquids
Hsun Kuang Yang
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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
1 I. INTRODUCTION 3 II. LITERATURE REVIEW 3 A. Classification of Non-Newtonian Fluids 7 1. The Bingham Plastic Model 7 2. The Ostwald - de Waele Model 8 3. The Eyring Model 10 4. The Ellis Model 10 5. The Sisko Model 11 B. Viscometers 11 1. Capillary Viscometer 12 2. Rotational Viscometer 12 3. Other Types of Viscometer
Treatment of Data from Capillary Viscometer c. 12 Using the Power Law Model 13 1. Newtonian Fluids 14 2. Non-Newtonian Fluids 17 D. Reynolds Number and Friction Factor 19 E. Effect of Turbulence 19 F. Err.or in Capillary Viscometry iv
III. EXPERIMENTAL 22
A. Object of Investigation 22
B. Materials 22
1. Non-Newtonian Liquids 22
2. Newtonian Liquids 25
C. Apparatus 25
1. Liquid Reservoir 26
2. Capillary Tubes 29
3. Pressure Gages 32
4. Piping, Valves and Fittings 33
5. Pressure Regulator 33
6. Constant Temperature Bath 33
7. Weight and Time Measurements 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
IV. 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 68
3. Fluid Head Correction 68
v. CONCLUSIONS 69
IV. APPENDICES
A. Capillary Data Tables 71
B. Figures for Aged Non-Newtonian Liquids 101
107 C. Notation 110 VII. BIBLIOGRAPHY 112 VIII. ACKNOWLEDGEMENTS 113 IX. VITA 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
0 s Capillary Data for 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 vii
Table Page
A-3 Capillary Data for glycerine at 25°C 97
A-4 Capillary Data for glycerine at 20°C 99 viii
LIST OF FIGURES
Figure Page
1 Basic Shear Diagram 4
2 Flow Chart of Pseudoplastic Liquid 9
3 Appearance of Turbulence 20
4 General Description of Capillary Viscometer 27
5 Pressure Vessel and Temperature Bath 28
6 Capillary Tube 31
7 Flow Chart for Glycerine at 25°C 44
8 Flow Chart for Oil Number 243 at 25°C 45
9 Flow Chart for Oil Number 678 at 25°C 46
10 Friction Factor - Reynolds Number for Oil 0 Number 243 at 25 C 47
11 Friction Factor - Reynolds Number for Oil Number 678 at 25°C 48
12 Flow Chart for 0. 1 o/o Carbopol at 22. 7°C Showing the Effects of Kinetic Energy and Fluid Head 53
0 13 Flow Chart for 0. 05% Carbopol at 22. 8 C 57
14 Flow Chart for 0. 1 o/o Carbopol at 22. 7°C 58
15 Flow Chart for 0. 2% Carbopol at 22. 8°C 59
16 Flow Chart for 0. 1% CMC at 23. 8°C 60
17 Flow Chart for 0. 2% CMC at 23. 5°C 61
18 Flow Chart for 0. 2% Carbopol (Without Solute) at 22. 8°C 62
19 Flow Chart for 0. 2% CMC {Without Solute) at 23. 5°C 63 ix
Figure Page
20 Flow Chart for 0. 05% Carbopol (saturated with iodine and carbon tetrachloride and aged for one year) at 22. 8°C 102
21 Flow Chart for 0. 1% Carbopol (saturated with iodine and carbon tetrachloride and aged for one year) at 22. 7°C 103
22 Flow Chart for 0. 2% Carbopol {saturated with iodine and carbon tetrachloride and aged 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 105
24 Flow Chart for 0. 2% CMC (saturated with iodine and carbon tetrachloride and aged for one year) at 23. 5°C 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: {2. 2)
where Q = volumetric flow rate of fluid, volume per unit time
L = length of pipe
R = radius of pipe
~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 (t", du/ dr) = 2 (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 (2. 7) -(du/dr)
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 1n-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_) (2. 9) a1 dr
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 ). 9
Turbulence
7
/ / / / / /
SHEAR RATE, du/dr
Figure 2. A typical complete flow curve for a
pseudoplastic liquid 10
(4) The Ellis Model.
(2. 10)
In this implicit function for ?', the constants (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 (2. 12} ~ = c 0 + (c1 + c 2 ] du/ dr l } {du/ dr) Of course, with four constants this relation could be made to fit almost any flow curve. (6) Reiner .. Philippoff Mode1.(17) - du 1 (2.13) dr =t IJ. oo + IJ.o - IJ.oo 1 + {f/rs>2 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 (2. 14) 2L For a Newtonian fluid, 't is given by equation (2. 1 ). Thus - 1-LN (du/dr) = (AP/2L)r (2.15) - du/dr = (AP/2LiJ.N)r (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] (2. 18) 4LtLN 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 -- J 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) 3 w = 3 (QhrR ) +(RAP) d(QhrR3) (2. 25) 2L d(RAP/2L) Substituting Q = UrrR2 into equation (2. 25) d(4U/R) -(du/dr)w= ~ ( 4U) + _!_(4U) 4U/R 4 R 4 R d(R~P/2L) RAP/2L or - (du/dr)w = 3 ( 4U) + I ( 4U) d[ ln (4U/R}] 4 R 4 R d( ln (RAP/2U] or - (du/dr>w 3 1 d[ ln {4U/R)] } ( (2. 26) =[ 4 + 4 4 u) d[ ln (RAP/2L)] R Let n' = d( ln (RA.P/2L)] (2.27) d [ ln (4 U I R )] where n' is called the "flow behavior index" (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~ (2. 32) 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) _ (2. 33) d [ ln (-duL d:r:-~] From equation (2. 28) ln (-du/dr)w = ln ( 3n' + 1 )·· + In (4U/R) (2. 34) 4n' (2. 35) Equation (2~35) is· divided by d On'Twl to give: 17 d[ ln(-du/dr)w] = d pn ( 3~~,+ 1)) + dln (4U/R) (2. 36) d(lnTw> 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 . , n' {2. 39) K' * K(3n'+l) = 4n' 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 18 approximately described by the power law model. The friction factor is the same as the usual Fanning friction factor (2.40) where D = pipe diameter P = fluid density gc = gravitational constant If equation (2. 29) is incorporated into equation (2. 40), the following is obtained: f = K' (8U /D)n' euZ/2gc or f = 16gcK' sn'-l (2.41) n' 2-n' 0 D u r if {2. 42) '{ = gc K' gn-1 f = 16 '{ (2. 43) nn' u2-n'r The relation between the Fanning fraction factor and the Reynolds number for Newtonian fluids in laminar flow is (2. 44) f = 16/NRe A Reynolds number for n~~-Newtonian fluids ~ay be obtained {i. 43) and {Z. 44) from equationS 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 20 E-< 0 0.00525" I.D. tube 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 mfu 2 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 1'1qu1 • d reservou' ( or "b omb II) , 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 Thermometer Precision Scientific Circulating PRESSURE GAGES Water Bath Pump 30 in Hg \'later ¥.ano- 0-30 0-300 Bath meter psig psig --.... ~,Pressure ~---~ r.:-----" vessel I ' \ / • I (')'\.. ~/ .....__ .. · \ "-"/.. / '' \ Nitrogen Capilla~; Cylinder Tube Valve V IV III II(A) ICX) L------L----~-- ____] ~ Figure 4. General description of capillc>.ry viscometer -..J 28 ------P. -----3 li ie-~---··-;: ---- L M G H I A i" CONNECTOR TO NITROGEN ,______J PIPELINE B HEX HEAD PLUG FLUID FEEDING· K c 3 11 BLIND PIPE FLANGE l D WATER BATH E 3 11 EXTRA HEAVY PIPE FLANGE F STEEL PLATE, 1011 X10 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 30 TABLE 1. DESCRIPTION OF CAPILLARY TUBES Tube Tube Type R L L/2R Number ft ft I HTX':~-7 0. 00625 (±0. 000093 )*~~ 2.00156 160 II HTX*-11 0. 003917(±0. 000093}** 2.00156 256 III HTX':~-14 0.002625{±0.000093}** 2.00156 381 F&P IV Glass 0.0006297(±0.0000019)** 1. 319 1048 Tube Glass v Thermometer o. 0003166~'~0:~ I. 156 1825 Tube * HTX refers to stainless steel capillaries. ~~~:~ deviations are those given by manufacturer. *':~':c calculated from calibration procedure (See page 34. ). 31 I silver- ~ -x--- solder I 1 II 2.I '' I I I 3 ,, ' _.:i_ -a1! o I r- I ! I I iIII ; I JAI! Figure 6. Capillary tube 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 (< 1o/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 • all studied at glycerine, oil number 243, and oil number 678*, oc. . t d" d at 20°C but it is suspected that it 25 Glycer1ne was s u 1e ' ... . d 1· r in this chapter • .,c The oils are descr1be ear 1e 40 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. 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 JJ.N Liquid Temp n++ 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 10 1 ..------·--·· , ___ ··-·· 8 ------· ·····--· ...... ~- - ····-·-·····--· ···-· .. I 6 .... .···-· --· -t' -·------!----- ! -···· . ' I I I j J 4- ·------·· t ...... : ! 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~ I 100 -····------... t I I I : I ! i ------... --.. ~. -4-·······-··---~ -· .... ·------·--· -1------·-····------····- I I I i 4~----._------~------~------LI ____ _j 2 6 4 6 -1 FLOW FUNCTION, 8U/D, sec Figure 8. Flow chart for Oil Number 243 at 25°C 2 0 0.00525" I.D. tube A o· 0.00783" I.D. tube (f) ~ 7 t------..----1 .o r--l 2 --···----- ···-·-- ·---lI " - . -····------··---.i ______J__ _ 7 4 ------~----~--~--L-----~--__J2 4 102 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---·:-- f=16/NRe 1 I 2 ---+---r·i l i ,: i l I : I i i ' '! : ... -----·-t --- 5 j Oj I I I: ' i 2 --·-··-+·----t-----1----1----+----I 1 I I I 10~------~'----~------~----~----~------_J 2 5 5 2 5 Dn 'u2-n' p REYNOLDS NUMBER, --1----- Figure 10. Friction factor - Reynolds number correlation for Oil Number 243 at 25°C 48 I I 5 ------j-----r-I' .------ 2 ______L_ a: :~---l-----~---0! I I 0 0.01250" I.D. tube I I I' I l 0 0.00783" I.D. tube -I 0 0.00525" I.D. tube ------r--'--- ! ' 5 ~------~~----~----~~------~~------~----_J2 5 2 5 2 101 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 This would be true if the above relation was Yw, n' is equal to n. 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. I 5. 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' I were also obtained. ;. ' I '· 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: 51 K' obtained from log K' {0. 003162-0.000763} K' obtained from RAP/2L versus (4U/R)n' (0. 0003116-0. 000133} or 0.00298 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 (2. 39) K'=K(~) 4n 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 = (3::1) AK' 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 53 101 8_I 0 0.00525" I.D. tube -+---+-·-- -·· . 6 e 0.00525" I.D. tube* -4--i-----.... 11>, 0.00525" I.D. tube+ 0' 4 --+------···. (f) 0 ~ 0 0.00126 11 I. D. tube .0 rl A 0.00063" I. D. tube 2 -·-- - ! I 10° I 8 J I I 6 ----~ I I ---·-- j 4 I I I I 2 ! I «> I 1o-1 I 6 8 2 2 4 6 8 2 4 6 10 103 FLOW FUNCTION, 8U/D, sec-1 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 2 * After kinetic energy correction using mr0.0 in K.E.=mpu /g c {11) + 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 U = average velocity in the capillary where = 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 57 101 8 ----·--- """" 6 --- 0 0.0012611 I.D. tube 4 - 6 0.00063" I.D. tube o• tf) ~ ,0 M 2 lo0 I 8 r------'------.1------I 6 ------· -- 4------ 2 2 4 6 8 105 2 -1 FLOW FUNCTION, 8U/D, sec Figure 13. Flow chart for' ·.05% Garbopol (saturated with iodine and carbon tetrachloride) at 22.8°C ··-·- ····· ~-4.- a .-- -·--· -- --· __ .__ __ - 0 0.00126" I.D. tube 6 5 -~- ~r ~-~=-=-~-==~---=-~· =- .. - 4 ---·-· ' . -*~ • ~- •¥--···--· ... ···~-···----·· I 3 --· ··- - -~---1·- .. -··, ~--· 4 5 6 -1 FLOI'i FUNCTION, SU/D, sec Figure 14. Flow chart for 0.1% Carbopol' (saturated with iodine and carbon tetrachloride) at 22.7°C 101 o O.OJ525" I.D. tube 5 A 0.00783" I.D. tube t! 0 11 cr 0.01250 I.D. tube ~..0 r-1 2 .s -----~~ ~ l l i l ~ .. 10° I i ! o ____J_ _ ___!_•----t----i ...:l ~ < I ~ -r--T-1 I ~ \ \ill. .! I! I ---1 I. tJ'.) 5 ~-~--,.. · . I l ~ E-o (Jj \ 0:: I 0 Vl Figure 15. Flow chart for 0.2% Carbopol (saturated ~~th iodine anc carbon tetractloride) at 22.8 C '-() ::: -" I I 4 ' I I I I ,---+. I 0 0.00126 11 I.D. tube I t! I i - I I! I' 0" 3 ' CD 6 0.00063" I.D. tube l I I ~ I ,Q I I I I I ! I l""i.. 2 ~ ~ Q .. ~ ...:I 100 ; I I I I I v- I I <---~ I I I. l t ~----···- ---~ 4 -:6'-' -+-----+--·---· -.-- ·-+1 ---- j --+-t---4----- ;------1 I ' I . I I I j I 1 i I I .' '. .I ; . , 3 .• . l :. . 1 l' 2 3 4 5 7 2 3 4 5 7 2 104 105 _, FLG.-i FUFCTION, 8U/D, sec 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: Ii ~~ 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 ~010 I I- . +----+ l f :3: ' I ! 1 : I ! ~ l I I : i I I 1 ; ; "'\JJ ii Il 1 i _;... !• J . ~ 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 .: 3L_------~--~------~--~------~--~----~--~--- 2 3 5 7 2 3 5 7 1o3 104 105 1 FLO-.,.. .rL":P'C'~'"'C;"'' c" Ll. ,, , 8U/D , sec - Fib~re 17. Flow char~ for 0.2% CEC (saturated -wi~h iodine and carbon tetrachloride) at 23.5°C 0'- 101 I 1 6 -··-~-t------~------...,.,.c;------t o 0 .. 00525" I.D. tube l ~ 0.00783" I.D. tube ~ 4 cr' 4D o 0-.01250" I.D. tube ~ ,Q r-i 2 ...:I.. .::!. ~ 0 .. 10 j ~ ~ 6 tf) ~ 4 E-< ' ~ ----t------~·------~·----~------~------~ 1:1') I ~ i ~ :X: U) 2 I L -- 10_, 2 4 6 2 4 6 2 4 6 2 102 103 104 -1 Fill.\' rJ~;CTIO~, BU/D, see 0 Fib~re 18. Flow chart for 0.2% Carbopol at 22.8 C 0"· 7'0 5 I / o 0.00126 11 I.D. tube t! 3 t:l i 0.00063" I.D. tube I 0' I Ul I I I ~ I ..0 r-i 2 _L -~ ~ -;!' i ! I ~Cl ~ :j I I < i I I ::?: 10° E-4 I I < I~ I. I Cll 1 - __ j 7. I I ------~-~----.- ~i-< Cll c:: ! I I I ~ 5 :c Cll 3 I I ta3 · · I 2 J 5 7 2 5 7 104 J 105 -1 FLCJ.i FUNCTim;, BU/D, sec 0 Figure 19. Flow chart fo:- 0.2% CHC at 23.5 C w0' 64 TABLE IV. CONSTANTS IN POWER LAW MODEL + Liquid Temp nit** KX1o4*'H~ % D11J of oc (lbf seen' 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% 2.32 CHC* 23.8 0.720±0.012 9.32 :!:0.16 0.1% 2.25 CHC 23.8 o. 724±o.011 9.53 :!:0.15 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 Twi · t" f nor K is 1nd1ca e , ****t~ 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 The rate Of Change Of n between polymer in the water increase d • * 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, the addition of solute causes the Carbopol so1 utlon · t o b e "thlnne · r" 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- · the room temperature was adjusted ture fluctuations. D ur1ng a run . h d . ed liquid temperature, but control as close as poss1ble tot e eslT . 1 the pressure vessel (liquid reservoir) varied by ±2°C. S1nce on Y tant temperature bath, all we is currently surrounded by the cons the ll.qUl' d entering the capillary tube is can be sure of is that Furthermore at high within ±0. 1 °C of the desired temperature. 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 ·an and less viscous as they aged. solutions b ecame more N e Wtonl rrection is very tedious to apply 4. The kinetic energy Co t It must be found by because the parameter m is not constan • for each capillary and each trial and error and is different 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 0 Table V. Capillary data for 0.05% Carbopol* (saturated •r.ltp r 2 and CC14) at 22,8 C Capil AP w T u 40/R .RAP/21 (4U/R)n' (du/dr)w lary __, Number lbr f't lb! (sec-1) (see-n 1 ) (sec-1} __liL______l_,Q8 158.70 127.6 2.03 3097.61 .102 1584.94 3168.04 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 I 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 I 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 c9 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\) ~ .. '";f Table V -- Continued Capil- 6P w T u 4U/R ~P/21 (4U/R)n' {du/dr)w lary Number lbr :f't 1 lbr 1 v 40.00 13.30 917.4 1.62 20586.05 .788 8994.79 21054.08 50.00 16.60 903.2 2.06 26097.82 .985 11179.79 26691.16 60.00 20.30 906.0 2.51 31816.17 1.182 13406.17 32539.52 80.00 27.17 905.6 3.37 42602.33 1.577 17519.48 43570.91 100.00 35.03 903.0 4.35 55084.90 1.971 22172.60 56337.27 120.00 42.8 5 903.2 5.33 67366.97 2.365 26665. 20_ 68898.58 140.00 51.00 901.0 6.36 80375.84 2.759 31349.57 82203.22 -~---·---~------· -. -- -...] w Table VI. Capillary data i'or 0.1% Carbopol* (saturated with r 2 and CC14) at 22~7°C Capil AP w T u 4U/R RAP/21 (4U/R)n' (du/dr)w lary lbr lbr Number rt 1 1 1 (sqin) (gram) (sec) ! III 3.64 190,40 308,4 1.00 1537,29 ,344 277.37 1654,28 4.49 224.20 299.8 1.22 1862.12 .424 321.28 2003.83 I 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 ~ Table VI. - Continued Capil- AP w T u 4U/R RAP/21 (4U/R)n' (du/dr)w lary Number lbf rt 1 lbr 1 IV 37.00 39.14 903.6 1.23 7833.10 1.270 966.53 84_2 9. 21 50.00 58.08 902.6 1.83 11636.45 1.717 1309.15 12522.00 70.50 67.35 665.8 2.87 18292.93 2.421 1851.84 lS->5.05 81.00 50.80 421.6 3.42 21789.77 2.781 2117.60 23448.00 90.50 59.40 425.6 3.96 25239.12 3.108 2370.13 27159.86 101.00 48.60 303.0 4.56 29005.68 3.468 2636.84 31213.06 120.00 54.30 268.8 5.74 36530.87 4. 121 3146!90 141.00 62.30 ~9310!!~ 251.2 7.05 44849.53 4.842 3682.89 48262.64 v 100.00 11.55 967.2 l. 34 16953.13 1.971 17 46_~_9 5 1824~. ~ 110.00 13.55 979.2 1.55 19645.00 2. 168 1955.90 122.00 15.32 21140.01 983.2 1.75 22120.81 2.404 2142."'?. 23=304.24 ].40.00 18. 50 974.0 2. 13 26964.79 2l. a ) , ,, 2.759 . " . - ~ 160.00 21.30 -- .. 25016 ,_S!t_ 976.6 2.50 31690.13 3.154 22-21.98 34101.79 . ------. -J VI Table VII. Capillary data for 0.2% Carbopol* (saturated with r 2 and CC14) at 22.8°C -- Capil- AP w T u 4U/R IW'/21 (4U/R)n' (du/dr)w lary lbr !t lb! Number 1 (see-n 1 ) (sec-1) (sq in) (gram) (sec) C-;ec) {sec- ) (sq ft) I 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 I I 2S. S 0 1£..7 •.~0 34.6 ;. • 1 3 , ,..... - ~ J - • ~:J.,v 267.3:· ,~.C' '- . "; "'- 5 C·54. 81 ~- -~- _ _ _ L 3 • ':' :. ___ ----~ 2 ;,_~ • _; -~~ ______3_:_ -~---::_ ____ ~ ·-~ L______S __ l.23__._ ~ ~~~--- -.J T~r••,;rr d·"·,., .. :a .. ~' Q Qq•l -7!"/C'' ;.::~ "' * ..l..t...l ·-! ...... ~,.._.._ \..i. - l .. ~- J. Capil- ~p W' T u 4U/R JWl/21 (4U/R)n' (du/d.r)w lary Number lbr ft lbr (gram) 1 1 (sqin) (sec) csec> (sec- ) .. - .. ------·------·------. ----~-~------~------··- -- - -.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 lbr Number ft 1 lbf 1 (sq in) (gram) (sec) (See) (sec- ) (sq ft) (see-n') (sec- ) ~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 Table VIII, Continued Capil- 4P w T u 40/R (4U/R)n' lacy RAP/21 (du/dr).., Number lbr tt lbt (gram) ·(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 tt lbt . · (sec-1) 1 ) 1 (sq in) (gram) (sec) (See) (sq ft) (see-n (sec- ) 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>; Capil AP w T u 4U/R RAP/21 (4U/R)n' (du/dr)11 lary lbr Number f't 1 lbt 1 (__ .;_) (gram) (sec) (SeC) (sec- ) I 2.45 134.40 360.2 .10 68.78 .552 8.43 85.70 J 3.93 77.90 91.0 .24 157.80 .883 12.81 196.62 5.45 129.00 84.6 .43 281.08 1.226 17.14 350.24 l 7.61 162.60 60.6 .77 494.61 1.712 22.79 616.31 \;; 2.55 11.70 361.6 .02 24.23 .359 4.98 30.19 1 5.25 44.00 306.0 .10 107.67 .740 10 ~57 134.17 7.66 96.50 300.2 .23 240.72 1.079 15.85 299.94 9.58 11.30 150.2 .37 385.39 1.349 20.10 480.22 12.18 125.00 150.2 .61 623.22 1.716 25.61 776.55 II 14.98 124.00 100 .o. .90 928.58 2.111 31.31 1157.05 \ 20.00 10 5. 30 50.0 1.54 1577.10 2.818 . 40.89 1965.12 l 25.20 124.00 40.2 2.26 2309.92 3.550 49.57 Z878.2~ 30.00 144.10 34.2 3.08 3155.29 4.227 58.01 3931.59 ,.- .. - . ------··------··-- --- . ------. - -·------. - --- ... - - - . ~ ... -- - 4 Liquid density = 0.999 gm/cu em -<» Table X. -- Continued Capil 4P w T u 4U/R RAP/2L (4U/R)n' (du/dr)11 lary' Number lbt .t't lbt c._.;_) (gram} (sec) (See) (sec-1) ( __ 4"~} (see-n') (sec-1) -- ~- III 5.20 5.30 305.1 .02 43.22 .491 S.67 53.85 7.71 12.40 • 301.4 .06" 102.36 .728 10.30 127.55 I 10.56 29 .8o 302.8 .16 244.86 .997 15.99 305.11 I 12.77 41.40 301.2 .22 341.99 1.206 18.92 426.13 15.37 66.40 302.2 .35 546.69 1.452 23.97 681.20 20.00 66.80 183.2 .59 907.24 1.888 30.95 1130.45 25.00 98.60 1a1. 0 .sa 1355.41 2 60 7 89 30.00 137 .ao 1a2.2 1.23 18a1.ao 2.a32 44.70 2344.79 40.00 149.00 120 .a 2.01 3068.97 3.777 57.20 3a24.04 50.00 194.90 107 .o 2.97 4532.13 4.721 69,62 564 71! 18 ~()) l\) Table XI. Capillary data :for 0.2% CHC~- at 23.5°C Capil oP w T u 4U/R R.6P/2L (4U/R)n' (du/dr)w lary lbr Number lbr rt 1 (see-n') (see-1) III 12.10 123.20 39 .o 5.16 7865.32 1tl42 :24:2.11 8220.18 17.70 178.20 35.4 8.22 12533.58 1.671 464.74 14214.52 22.80 310.50 45.2 11.22 17.103.85 2.152 568.97 19397.73 i I 28.90 310. 10 36.0 14.07 21447.17 2.728 659.26 24323.55 I IV 16.80 15.60 900.0 .49 3134.27 .576 188.55 3554.62 ' \ 28.00 36.85 10 20.0 1.0 2 6532.68 l • 961 304. 11 7408.82 40.00 47.88 780.2 1.74 11096.93 1.373 429.34 12585.19 49.50 59.60 701.0 2.41 15373.86 \ 1.700 530.82 17435.72 ! I 69.50 56.35 377.4 4.24 26998.93 2.386 765.82 30619.90 89.50 65.90 301.0 6.22 39588.91 3.073 982.46 44898.38 110.00 74.40 250.2 8.45 53770.03 3.777 1199.10 60981.40 I v 80.00 11.20 1267.4 .99 12544.51 1.577 465.01 14226.91 121.00 18.20 923.0 2.21 27991.04 2.385 784.02 31745.06 I 140.00 20.80 919.4 2.54 32115.02 2.759 857.38 36422.12 ' -·--·------~ -· < ---·-- -· • - .. * Liquid density = 0.998 gm/cu em e ~ Table XII. -- Continued Capil- AP w T 0 4U/R RAP/2L (4U/R)n' (du/dr)w lary Number lbr rt 1 lbr 1 (sqin) (gram) (sec) IV 5.18 27.50 909.8 .85 5466.69 tl78 4096,~4 7.12 25.50 630.8 1.14 7311.16 .244 5425.58 7374.57 9.50 32.40 606.6 1.51 9660.08 .326 7102.06 9743.85 11.88 28.10 422.0 1.89 12042.93 .408 8788.72 l·2 I4 7 14.37 34.50 427.4 2.29 14598.99 .493 10585.56 14725.59 18.00 36.70 364.2 2.86 18224.87 .618 13116.74 18382.91 22.45 38!50 ;204.4 3.59 22874.Q~ .111 16338.30 23Q13.Q_l_ 26.70 36.70 241.8 4.31 27450.36 • 916 19487.03 27688.41 33.00 34.60 18 2. 6 5.39 34269.96 1.133 24147.95 345.67.15 41.00 30.90 132.0 6.65 42337.27 1.408 29621.83 42704.41 61.00 36.40 106.0 9.76 62106.04 2.095 42898.66 62644.62 l ao.oo 49.80 101.8 13.91 88474.86 2.747 60391.70 89242.10 I 100.00 65.10 104.0 17.80 113210.31 3.434 76639.81 114192.06 \ 122.00 7 5. 40 100.2 21.40 136094.93 4.190 91564.96 137275.13 l v 40.00 14.60 904.6 1.81 22915.50 .788 16366.50 23114.22 60.00 22.66 900.4 2.82 35732.01 1.182 25142.95 36041.87 80.00 31.12 902.6 3.87 48952.77 1.577 34084.15 49377.28 100.00 39.7 5 901.7 4.95 62590.46 1.971 43222.04 63133.24 120 .oo 49.07 90 2. 8 6.10 77171.61 2.365 52918.22 77340.84 140.80 58.35 902.5 7.26 91796.62 2.775 62581.75 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 Number lbt tt 1 lbt 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 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 I l 14.77 262.50 46.4 9.24 14084,51 1.395 3836.69 14639.40 I t 20.00 282.60 38.0 12.15 18514.80 1.888 4859.19 19244.23 ' 1 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» ~ ' Table XIII. Continued Capil AP w T u 4U/R RttP/21 (4U/R)n' (du/dr).., lary lbf tt lbt Number (sec-1) (see-n') (sec-1) <...... ; -) (gram) (sec) IV 9.65 20.80 908.4 .65 4140.00 .331 1332.30 4303,10 12.82 28.20 899.0 .89 5671.57 .440 1748.62 5895.02 16.08 35.80 900.8 1.13 7185.70 .552 2145.22 7468.79 20.45 58.30 1115.0 1. 48 9453.83 .702 2718.89 9826.29 31.00 29.70 362.8 2.32 14801.43 1.064 4004.82 15384.55 40.05 41.10 366.8 3.18 20259.42 1. 3 75 5252.28 21057.57 61.00 54.20 302.0 5.10 32449.41 2.095 7890.01 33727.81 80.50 74.50 299.0 7.08 45050.50 2.764 10475.44 46825.34 100.80 82.80 254.4 9.25 58847.46 3.461 13194.87 61165.86 122.00 104.20 254.6 11.63 73998.66 4. 190 16082.61 76913.96 v 104.00 15.70 710.4 2.48 31369.38 2.050 7662.63 32605.23 123.50 21.50 713.0 3.38 42801.42 2.434 10022.10 44487.66 140.50 23.55 7 11.5 3.71 46981.33 2.769 10862.17 48832.24 142.50 25.20 712.9 3.97 50174.29 2.809 11497.03 52150.99 162.00 29.50 710 .o 4. 66 58975.68 3.193-- 13219.7Q 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 .f't lb.f' 1 Number 1 (see-n ) (sec-1) co co Table.XVI. -- Continued Capil ~p w T ·u 40/R &6P/2L {4U/R)n' {du/dr)w lary lb.r .· Number !t 1 lbr (see-n') 1 2. 26 25.60 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 Number !t 1 lb! 1 (sq in) (gram) (sec) 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 I 11.27 201.60 49.0 6.72 10248.42 1.064 770.77 11245.71 ' 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 6.80 6.01 555.4 .30 1957.55 .233 234. 11 2148.05 I' 10.34 10.60 541.8 .55 3539.26 .355 358.55 3883.67 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 24.65 38.12 542.4 1.99 12713.92 .846 900.15 13951.14 1 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 Number f't 1 lbr 1 1 ... - ···--· . ··------~ ...... - - . - ~ -- -- . - . . -· - ; 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..... Table XVI. Capillary data for 0.2% CNC~- (saturated with 12 and cc14 and aged for . one year) at 23.5°C. Capil- .t.p w T u 4U/R RAP/21 (4U/R)n' {du/dr).w lary lbr Number rt 1 lbr 1 1 I I I I .53 55.70 300.6 .30 461.38 .050_ 52.86 524.37 2.26 118.60 243.2 .79 1214.28 .213 98.85 1380.04 I 3.66 145.50 183.6 1.29 1973.28 .345 135.32 2242.65.\ 4.96 135,10 119,4 1.84 2817.40 .468 170.37 3202.01 6.63 151.70 94.4 2.62 4oo1.4o .626 213.77 4547.63 I 8.32 183,00 85.6 3.49 5323.23 .786 257.12 6049.91 ' 9.82 155.70 59.0 4.31 6571.05 .927 294,64 746~~ 11.79 203.00 60.8 5.45 8313.63 1.113 343.06 9448.53 14.22 200.00 47.4 6.89 10506.31 1.343 399.14 11940.52 16.00 215.00 44.2 7.94 12111.97 1.510 437.60 13765,37__ _ 19.50 246.40 40.6 9.91 15111.70 1.841 504.93 17174.60 * Liquid density = 0,998 gm/cu em ..0 I\) Table XVI. Continued (4U/R)n' Capil AP W T u 4U/R RAP/21 (du/dr )w lary . lb:r lbr !t 1 Number {see-1) (__ N) {see-n ) (see-1) 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 '-J,'-JU-- '-J!U.b·- · 1e£t:.· -- ·-·-·!:>4j4.~~ - 1.577 511.88 17541.15 I 100.00 14.30 904.4 1.77 22446.71 1.971 652.20 25510.91 I 12o.oo 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 Capil .6P w T u 4U/R ~P/21 (4U/R)n' (du/dr).., lary lbf Number .f't 1 lbr -1) III _2.48 130,00 150.6 l! 64 2502.78 .23~ 2102.51 2511.03 5.15 93.00 60.6 2.92 4449.53 .487 3690.33 4474.87 8. 27 117.20 48.6 4.58 6991.91 .781 5740.83 7031.72 10 I 12 l6816Q ~5.8 11QQ lQ613.25 .255 8681.30 1013~.03 i IV 13.85 15.00 615.8 .80 5129.08 .475 4240.49 5158.29 i 19.40 20.80 601.2 1.14 7285.05 .666 5976!06 7326.54 25.25 27.60 600.6 1.52 9676.37 .867 7887.67 9731.47 30.00 32.40 601.2 1.78 11347.88 1.030 9217.43 11412.50 40.00 39.00 542.0 2.38 15151.44 1.373 12227.94 15237.72 50.00 43.70 480.0 3.01 19170.30 1.717 15390.50 19279.46 61.00 40.20 360.4 3.69 23487.13 2.095 . 18771.10 23620.87 80.00 39.80 271.2 4. 86 30901.67 2.747 24'546.38 31077.6Lt 99.50 37.00 200.8 6.10 38799.55 3.417 30664.19 39020.49 121.00 40.50 179 .a 7.46 47430.10 4.155 37317.81 47700.18 v 31.00 3.60 968.2 .48 6146.40 • 611 5061.11 6181.40 39.50 4.50 907.6 .64 8195.99 .778 6705.69 8242.66 ,_ ------~-9-~ 00 5.60 903.2 .81 1 02_49 .14 .985 R343.87 t0)_2_7_. 5Q_ 60.00 6.80 903.0 .98 12448.14 1.182 10090.31 12.d9.02 80.00 9. 10 903.2 1.31 16654.85 1.577 l3ld 2. 93 lt.4~.69 1.971 16873 .}Q 2 : .l.:':...~ .. ~ 9__ ------··-·100.00 11.50 902.6 1.66 21061.33 120.00 13.9 3 899.8 2.02 25591.07 2.365 20ld3.55 25736.80 141.00 16. 28 902.6 2.35 29815.52 2.779 23702.50 29985.30 ----- .... ------·- ·--~---·------~------"'.J:- * liquid density = 0.875 gra/cu em 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 ( sq m . ) (gram) {sec) (-)sec (sec- } ( s q ft) {sec- ) (sec- ) 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 Table A-2. -- Continued Capil AP w T u 4U/R RAP/21 (4U/R)n' (du/dr)w lary lbr Number ft 1 lb:f' 1 (sq in) (gram) (sec) (;eo) (sec- ) (sq ft) (see-n') (sec- ) I I I _1_4. 4 0 22.80 200.6 .20 316.98 1.359 284_,94 318,48 18.00 21.60 151.2 .26 398.41 1.699 356.63 400.29 22.00 20.70 121.2 .31 476.32 2.077 424.96 478.57 26.00 20.40 10 2. 2 .36 556.69 2.455 495.24 559.31 29.70 21.30 92.8 .42 640.13 2.804 567.99 643.14 40.00 2l~. 60 81.4 .55 842.84 3.777 74l~. 07- 8lt6.81 I 51.00 24.60 62.2 .72 1103.01 4.815 968.91--- 11_08 ._2__1_ l 60.00 29.30 62.0 • 86 1317.99 5.665 1153.95 1324.?0 l r 80.00 38.30 60.6 1.15 1762.63 7.554 15 3'~. 9 7 I 1770.94 \ 100.00 l~8. 40 60.2 1.47 22lr2 .25 9.442 l9lr3.97 2252.82 ' 122.00 59.20 60.2 1.79 2742.59 11.520 2368.91 2755.52 -.o 0"- Table A-3. Capillary data for glycerine* at 25°C Capil t.p w T u 4U/R RAP/21 (4U/R)n' (du/dr)w lary Number lbr ft lbf (.._ ~~> (gram) (sec) (SeC) (sec-1} (__ N) (see-n') {sec-1) I 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 16.60 122.00 94.0 .29 189.96 3.732 189.96 189.96 l-~~- 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 '-0 --* Iir;uid den-~ity-~ 1 ~260 g.;;~~ em -..J Table A-3. -- Continued Capil- AP w T u 4U/R ~P/21 (4U/R)n' (du/dr)w lary lbr Number ft 1 lbf 1 ----- I I I 10.50 46.34 261.8 .22 349.69 6,657 3_49 ._69 342.62 I.' 86.10 55.59 260.0 .27 422.40 8.130 422.40 422.40 87.10 55.65 258.3 .27 425.64 8.224 425.64 425.64 I . .I 102.50 61.37 240.7 .33 503,71 9,678 2Q3.71 50.3...._ll_ l 114.00 71.45 247.7 .37 569.87 10.764 569.87 569.87 - .. ' ------+ ·-----~------•• -·------. ------· ------ '-0 co Table A-4. . Capillary_ data for glycerine-r, at 20°C -~~····-'" Capil- AP w T u 4U/R MP/21 (4U/R)n' {du/dr )w lary lbr lbf Number ft 1 -1) I 11.40 51.64 76.4 • 15 98.84 2.563 98.84 98.84 l4 .60 50.07 60. 1 .19 121.83 3.282 121.83 121.83 19.30 70.08 64.1 .24 159.88 4.339 159.88 159.88 2't. 7 2 83.47 60.4 .31 202.10 5.557 202.10 202.10 29.06 102.23 66.4 .35 225.16 6.533 225.16 225.16 \ 40.00 140 ·'~3 59.6 .53 344.58 8.992 344.58 344.58 l------~~00 __141._87 60.4 • 53 343.50 8.992 3'} 3. 50 3'~3. 50 I so.oo 166.95 61.9 .61 394.43 11 • 2 '~ 1 394.43 394.43 ' 50.00 158.69 61.6 .58 376.74 11.241 376.74 376.74 I 70.00 221.76 61.0 .83 531.66 15.737 531.66 531_.66 I 90.00 30 5. '~1 62.5 1. 11 714.63 2 0. 2 3 1~ 71Lt.63 714.63 .I 100.00 360.05 64.8 1.26 812.58 22.482 812.58 812.58 ______1J_O_._o o 38 2. 7 5 61.8 1. lc 1 9 05.75 24_._ 7}Q_ ___ 905.75 9Q5__ J..L 120.50 416.81 61.6 1.54 989.55 27.091 989.55 989.55 1'~0. 90 49 3. 60 62.5 1.80 1154.98 31.677 1154.98 1154.98 .. I I 61.20 5 l. 1(, 10 1. 8 .29 298.56 8.623 298.56 298.56 81.00 6 7. u~ 101. 1 .38 394.54 11.413 394.5 't 39'~. 54 ____99_._ 3_0______8'':_•__ 5Q __1_9_3_. 8 ·'t 7 483 ._QJ 1_2_.~9.1 lt_~_}_._f:_J______!_t 3-~-. _Q 3_ 120.50 10 3. 40 102.4 .58 599.90 16.978 599.90 59~.90 1'~0. 00 118.80 10 2. 8 .67 686.56 19.726 626.56 686.56 ------~------··--·· . ------· ------·- ---·------. Liquid cirn:·.:L·. : 1.26 [;;-'/cu Cl:t * -.() -.() Table A-4. Continued Capil- ~p w T 0 4U/R R~P/21 (4U/R)n' (du/dr)w lary lbr lbr Number rt 1 (see-n') 1 \ I I I 80.00 41.7 2 301.5 .17 2 73. 14 7.554 273.14 273.14 90.20 46.98 29 7. 6 .20 311.61 8. 517 311.61 311.61 100.00 52.60 304.1 .22 341.43 9.442 341.43 341.43 I 110.80 58.52 307.3 .24 375.90 10.462 3_?5. 90 375.90 I 121.50 63.38 304.0 .27 411.54 11.472 411.54 411.54 I 131.00 69.40 307.7 .29 445.21 12.369 4Lr5. 21 445.21 I 141.00 75.11 305.0 .31 486.10 13.3l't 486.10 ~ 8 6_._l__Q__ I ...... 0 0 101 APPENDIX B 102 0 0.00126" I.D. tube 0.00063" I.D. tube 2 6 ------ 4 ·- 2 -1 FLOW FUNCTION, 8U/D, sec Figure 20. Flow chart for 0.05% Carbopol (saturated with iodine and carbon tetrachloride and aged for one 1l.; ".' 6 0 0.00126" I • D• t ube 4 A 0.00063" I.D. tube ------·----1------l ···-- ..• -···-· . ··-·--· - __ ...... - ·--·-··-·······-- _____.. ______---· -·-·· ... li I ----· ...... ----·-. t 4 I I 3 3 4 6 4 6 a 2 -1 FLOW FUNCTION, 8U I D, sec Figure 21. Flow chart for 0.1% Carbopol (saturated with iodine 0 and carbon tetrachloride and aged for one 7ear) at 22.7 C I -I - 0 0.00525" I.D. tube I 6..-----,----.----r----+------l------ 4 ------~------·~·------· ·---~··· I 2 ----· -- .. I ...... I I 1o 0 ___ .. -.J .·----- .. ---j·-- - --·-·-····--·· ... 8 ------··--,---1-- ,------· -~i- t--~~ --~=- E_, .______J___ l 10- 1 ~----~--~~~------~----~--~~~~----~ 2 4 6 2 4 6 2 -1 FLOW FUNCTION, 8U/D, sec Figure 22. Flow chart for 0.2% Carbopol (saturated with iodine and carbon tetrachloride and aged for one year) at 22.8°C 5 o 0.00126" I.D. tube A 0.00063" I.D. tube ~ 3 0' l ~ .0 I rl 1 ' I1 . - I I ~I ; I I .. 2 i I j , l ' i I ~ ...;:t. ~ Q .. :j I I i I ~ 10° I ~-----·---- ; . ! E-< < I \ ; I U) ~ 7 I -·-'··l -- ~- ---. ------~-- --- ~ E-< C/) ~ I < r=:l 51 -+·---- ::X:: U) 3~------~ 2 3 5 7 4 2 3 5 7 5 2 10 10 FLO., FUJ\C':'ICJ;, 8U/D, sec_, r'"lf("\ ... ---.,r-•ea· ,. ...:t' Figure 23. flow chart fo~ 0.1% \...-.!'.:.\.... ( ~c.. •. d .... 0.0 n..l .~ iocine anc cc.rbcr. ·... e:rac:-.1o~ide anc aged for c one year) at ;j.8°C \J1 c "' I I 0 0.00126" I.D. tube ~ ~ 0.00063" I.D. tube ~ 3 I c:r ...... rt) ~ ,Q ~.. 2 ....::1 -.:t i ~ i I s I .. I ~ < \ ~ I I IT 10° I ~--- e-. ! I •7 ' ! I < (f.) ~ 7~~------I I til I 0:: I l ~ I I :I: 5 ------·-·----· --!---·------. (f.) ; 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 2 mpu /g m = kinetic energy correction constant defined by c nup = Reynolds number NRe = ~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 -= shear stress observed from experimental 'robs 109 = 'rcalc shear stress calculated from Equation {2. 32) j.la = apparent viscosity = viscosity of non-Newtonian liquid ~ = mass density of liquid 110 VII. BIBLIOGRAPHY 1. Wu, H. C., "Mass Transfer to Droplets: Effect of Non Newtonian Continuous Phase"; Masters Thesis, Missouri School of Mines and Metallurgy, 1964. 2. Wilkinson, W. L., "Non-Newtonian Fluids", 1 ed., Chapters 1 & 2, Pergamon Press, New York (1960). 3. 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). 4. Bird, R. B. , W. E. Stewart, and E. N. Lightfoot, "Transport Phenomena~', 1 ed., pp. 10-15, John Wiley, New York (1963). 5. Ram, A., and A. Tamir, "A Capillary Viscometer for Non Newtonian Liquids", Ind. Eng. Chern., 56, 47 (1964). 6. Bowen, Jr., R. L., Chemical Engineering, pp. 243-248 (June 12, 1961). 7. Ibid. , PP· 12 7- 13 0 (June 2 6, 19 61). 8. Ibid. , PP· 147-150 (July 10, 1961). 9. --Ibid. , PP· 143-150 (July 24, 19 61 ). 19 61 ). 1 o. Ibid. , PP· 12 9 - 13 2 (August 7, 19 61 ). 11. -Ibid. , PP· 119-122 (August 21, 19 61 ). 12. -Ibid. , PP· 131-146 (September 4, "ll E w "Non-Newtonianism in Thin Liquids: Molecular 13. Merr1 , . · , Ch E " and Physical Aspects", in A. Acrivos, "Modern .em. ng. , -195 Reinhold Publishing Corporatlon, New vol. I , PP· 141 ' York ( 1963 ). Chern. Eng. Prog., 50, 27 (1954). 14. Metzner, A · B · , Sisko, A. W., Ind. Eng. Chern. , SO, 1789 (1958). 15. 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.