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A Study Of The Behavior Of Thixotropic Fluids

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A Study Of The Behavior Of Thixotropic Fluids

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Content 70-16,892 TAO, Fan-sh.eng, 1936- A STUDY OF THE BEHAVIOR OF THIXOTROPIC FLUIDS. University of Southern California, Ph.D., 1970 Engineering, chemical University Microfilms, Inc., Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED A STUDY OP THE BEHAVIOR OP THIXOTROPIC FLUIDS by Fan-sheng Tao A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy (Chemical Engineering) January 1970 UNIVERSITY O F SO U TH ER N CALIFORNIA TH E GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 9 0 0 0 7 T his dissertation, w ritten by u nder the direction of h. . i s . . D issertation C o m m ittee, an d a p p ro ve d by all its m em bers, has been presented to and accepted by T h e G ra d u ate School, in p artial fu lfillm en t of require ments of the degree of Fan-_s henjS. . Ta o D O C T O R O F P H I L O S O P H Y Dean D a te..Januarx. I .9. 7. Q . DISSERTATION COMMITTEE » / / „ Chairman ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. C. J. Rebert for his invaluable advice and guidance during this investigation. Appreciation is also extended to his wife, Rosa, for her constant interest and encouragement. The author is grateful to the continuous financial support of this work by Texaco, Inc., Archimedes Circle, and the Department of Chemical Engineering, University of Southern California. il TABLE OF CONTENTS Page ACKNOWLEDGMENTS ........................................ ii LIST OF TABLES ......................................... vi LIST OF FIGURES ........................................ vii SUMMARY ................................. ............... X CHAPTER I. THE NATURE OF THIXOTROPY ................. 1 1. Definition of Thixotropy ................. 1 2. Perfect and Imperfect Thixotropic Fluids . 3 3. Formation of Thixotropy .................. 8 CHAPTER II. HOW TO CHARACTERIZE THIXOTROPY......... 12 1. Previous Theoretical and Experimental Approaches ................................. 12 2. Theoretical Model for the Present Study .. 24 3. A Frequency Domain Presentation ......... 43 CHAPTER III. EXPERIMENTAL WORK ...................... 46 1. Purpose .................................... 46 2. Instrument ................................. 47 3. Calibrations ............................... 48 4. Preparation and Testing Procedures ...... 50 CHAPTER IV. EXPERIMENTAL RESULTS .................... 55 1. Shear and Recovery Curves ................ 55 iii Page 2. The Basic Shear Diagram................... 57 3. Graphical Method for Determination of Constants .................................. 61 4. Statistical Regression Analysis for Determination of Constants ............... 71 5. A Comparison of Calculated versus Experimental Data ......................... 73 6. Comparison of the Proposed Mathematical Model with Others ......................... 82 7. A Frequency Domain Presentation .......... 83 CHAPTER V. CONCLUSIONS ............................... 87 CHAPTER VI. RECOMMENDATIONS ......................... 89 NOMENCLATURE ........................................... 91 REFERENCES .............................................. 95 APPENDICES .............................................. 100 APPENDIX A. TABULATED EXPERIMENTAL DATA .... 101 APPENDIX B. DETERMINATION FOR THE CONSTANTS OF THEORETICAL EQUATION (22) ___ 115 APPENDIX C. CALIBRATION OF ROTATIONAL SPEEDS AND CHART DRIVE SPEEDS .......... 126 APPENDIX D. COMPUTER PROGRAM AND RESULTS FOR THE FREQUENCY DOMAIN PRESENTATION 130 APPENDIX E. COMPUTER PROGRAM FOR REGRESSION ANALYSIS ......................... 137 iv APPENDIX F Page COMPUTER PROGRAM FOR COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL VALUES .............. 146 LIST OF TABLES Table Page 1 Classifications of Thixotropic Fluids ........ 6 2 Constants for Equation (22) ................... 31 3 Dimensions of Bob and Cup for Rotovisco Viscometer ...................................... 48 4 Torsion Spring Constants (For SV=I Type Rotor of Rotovisco Viscometer) ...................... 49 5 Constants Determined by Regression Analysis .. 72 6 Linear Interpolation Constants for k (or Xq) in Recovery Equations ......................... 73 7 A Comparison of T(calculated) vs. T(experimental) ................................ 81 8 Arbitrarily Selected Equations ................ 82 9 Comparing Equations by Popovics' Method for Hand and Body Lotion (Number of observations: 144) ............................................ 84 vi LIST OF FIGURES Figure Page 1 The Hysteresis Loop ............................. 12 2 Weltmann's Downcurves Diagram .................. 13 3 Experimental vs. Calculated Values ............. 41 4 Shear Curves of the PVC Plastisol at 25°C...... 56 5 Recovery Curves of the PVC Plastisol at 25°C. . 56 6 Recovery Curves of 8% Wyoming Aquagel at 25°C. 58 7 The Basic Shear Diagram ......................... 59 8 The Basic Shear Diagram, PVC Plastisol at 25°C.............................................. 59 9 The Basic Shear Diagram, Hand and Body Lotion at 25°C........................................... 60 10 The Basic Shear Diagram, 8% Wyoming Aquagel at 25°C........................................... 60 11 Shear Stress Recovery Plot of PVC Plastisol (Pre-sheared for 10 minutes) ................... 62 12 Shear Stress Recovery Plot of PVC Plastisol (Pre-sheared for 1 hour) ....................... 62 13 Shear Stress Recovery Plot of Hand and Body Lotion (Pre-sheared for 10 m9nutes) ........... 63 14 Shear Stress Recovery Plot of Hand and Body Lotion (Pre-sheared for 1 hour) ............... 63 15 Shear Stress Recovery Plot of 8% Wyoming Aquagel (Pre-sheared for 10 minutes) .......... 64 vii Figure Page 16 Shear Stress Recovery Plot of 8% Wyoming Aquagel (Pre-sheared for 1 hour) .............. 64 17 A Log-log Plot of the Basic Shear Diagram, PVC Plastisol at 25°C................................ 66 18 A Log-log Plot of the Shear Stress Recovery Curves, PVC Plastisol at 25°C. (Pre-sheared for 10 minutes) ................................. 66 19 Reduced Log-log Plot for Shearing of PVC Plastisol at 25°C. (n=0.69) ..................... 67 20 Reduced Log-log Plot for Recovery of PVC Plastisol at 25°C. (n=0.59) ..................... 68 21 A Comparison of Calculated versus Experimental Values for Shearing of PVC Plastisol at 25°C. . 69 22 A Comparison of Calculated versus Experimental Values for Shearing of Hand and Body Lotion at 25°C............................................... 70 23 A Comparison of Calculated versus Experimental Values for Shearing of 8% Wyoming Aquagel at S = 105.7 sec"1 (25°C.) ........................ 74 24 A Comparison of Calculated versus Experimental Values for Shearing of 8% Wyoming Aquagel at S = 35.2 sec-1 (25°C.) .......................... 75 25 A Comparison of Calculated versus Experimental Values for Shearing of 8% Wyoming Aquagel at S = 11.75 sec-1 (25°C.) ........................ 76 26 Calculated vs. Experimental Values for Recovery of PVC Plastisol at 25°C........................ 77 27 Calculated vs. Experimental Values for Recovery of Hand and Body Lotion at 25°C................. 78 viii Figure Page 28 Calculated vs. Experimental Values for Re covery of 8% Wyoming Aquagel at 25°C. (Pre-sheared for 10 minutes) .................. 79 29 Calculated vs. Experimental Values for Re covery of 8% Wyoming Aquagel at 25°C. (Pre-sheared for 1 hour) ...................... 80 30 An Amplitude-Frequency Plot of PVC Plastisol at 25°C.......................................... 86 31 Log-log Plot of X vs. k2(l + t) ('y^’ )ne. t for PVC Plastisol at 25°C...................... 118 32 Log-log Plot of X vs. k0(l + t) \ 3 t r / for Hand and Body Lotion at 25°C.............. 121 33 Log-log Plot of T vs. k2(l + * t for Q% Wyoming Aquagel at 25°C................. 124 ix SUMMARY Macroscopic studies of thixotropic fluids deal with the examination of the time-dependent rheological proper ties of certain fluids as influenced by their microscopic structural breakdown and reformation. Specifically, the quantitative relationship among the variables, shear stress, X , shear rate, S, and time is sought. Theoretical models of thixotropic behavior have been proposed by many investigators. To date, none of these models has gained wide acceptance or use because of mathematical complexity or lack of generality. The present theory assumes that when a thixotropic fluid is subjected to a continuous shearing, the structur al breakdown and recovery occur simultaneously. The ob served rate of change of the shear stress is the differ ence between the rate of change due to structural break down and the rate of change due to structural recovery. The resulting non-linear differential equation is solved by further assuming separable functions of shear rate and time. By equating the function of shear rate to the wide ly accepted power-law relationship with a yield value, the x model takes the form: T= ( To + K(S)n ] (1 + t)m (1) where X0 = yield value k,n,m = constants S = shear rate t = time of shearing (or resting). The above equation can be used to describe many types of fluids which include both time-dependent Newtonian or non-Newtonian, and time-independent Newtonian or non- Newtonian fluids. The validity of the several assumptions has been verified experimentally by observing the breakdown and re covery properties of three thixotropic fluids in a series of stress-time experiments. In one set of experiments, one observes at a constant shear rate the net time depend ence of the shear stress. In a second set of experiments the recovery of shear strength is observed. This experi mental method is considered to give more information about the flow properties of a thixotropic fluid than the pre viously accepted hysteresis loop method. The experimental results can be plotted on a basic shear diagram with different time levels. This treatment provides a useful way to classify the thixotropic fluid in a conventional manner but with a time parameter, demonstra ting that the functions of shear rate and time are separ able. Previous shear history has been found to have great influence to the recovery of a thixotropic fluid. This history can be classified into two types, namely, the in tensity (shear rate) and period (pre-sheared time) for the fluid subjected to the shearing action. A complete re covery could be reached sooner for a fluid pre-sheared at a lower shear rate than a higher shear rate. The pre sheared time could influence the starting shear stress for the recovery of a fluid; however, this phenomenon can be adjusted by the constant, k, in equation (1). Excellent agreement has been found on comparing the values predicted by equation (1) with the experimental data for three thixotropic fluids, namely, PVC plastisol, hand and body lotion, and 8% Wyoming aquagel in water. xii CHAPTER I THE NATURE OF THIXOTROPY 1. Definition of Thixotropy The term thixotropy (or thixotropie, in German) was first used by Freundlich (13) early in 1926 to specify a sol-gel transformation (14) of a fluid. He defined a thixotropic gel to be a gel capable of an isothermal, re versible, sol-gel transformation. A thixotropic gel will liquefy on shaking and then spontaneously gel on standing, showing no change of volume (15). After Freundlich's dis covery, studies were conducted on materials such as water solutions of iron oxide, gelatine, and agar. Green and Weltmann (22) in 1943 extended the studies to printing inks and oils. They interpreted the word "thixotropy" in a broader sense, that is, the characteris tic of a fluid whose structure breaks down continuously upon shearing and builds up upon resting. From this point of view, observers have reported thixotropy in diverse fluid mixtures including polymer solutions and suspensions of finely divided solids, as well as gel forming mixtures. Hahn, Ree, and Eyring (25,26) defined the thixo- 2 tropic phenomenon as having the following characteristics: a) It accompanies an isothermal structural change brought about by applying mechanical disturbance to a fluid system; b) When the mechanical disturbance is removed, the system recovers its original structure; c) The flow curve (shear rate vs. stress) of the system shows a hysteresis loop. I The hysteresis loop, first observed by Green (20), I has been used extensively for characterizing the thixo tropic fluids; however, this method is questionable because the loop consists of consecutive but different actions in finite time steps and the time required to make a loop is j I entirely dependent on the individual who operates the vis cometer. Therefore, it is difficult to convert the loops i ; i into useful data or to compare two loops made by different j i operators. Further discussion on hysteresis loop will be presented in Section 1, Chapter III. The term "reversible," which appeared in the defini tion of thixotropy by Freundlich, is discussed in the next section. i i 3 2. Perfect and Imperfect Thixotropic Fluids When a mechanical disturbance on a thixotropic fluid is removed, the fluid will not necessarily recover fully | ! its original structure. Therefore, it is necessary to de fine a fluid which is fully recoverable to its original structure as a "perfect" thixotropic fluid, and a fluid which is only partially recoverable as an "imperfect" thixotropic fluid. The reasons for the "imperfection" are !considered to be: i I i a) Incomplete Dispersions: The incomplete disper- I j sions come from either improper mixing or insufficient mix- ' i ! j |ing of the fluid mixture components. Sometimes these con- j i ditions exist for practical reasons. When an incompletely j dispersed fluid mixture is subjected to shearing action |in order to observe the resulting shear stress, the change in shear stress is due simply to further mixing. This j change is considered permanent (12,16). If this additional j ^dispersion is accompanied by thixotropic breakdown, it will ; confuse the observations on the recovery of such a fluid, and hence, the fluid will be classed as having imperfect j thixotropic properties. ! b) Structural Breakage: Linear polymers or fiber | 4 structures, such as glass fibers (19) or attapulgite (16) in water, can be broken down into shorter pieces upon shearing. This structural breakage is also permanent. If this phenomenon is accompanied with thixotropy, as for the case of attapulgite in water, it will make the thixotropy imperfect. c) Irreversible Structures: Sometimes a structural I frame is formed immediately upon preparation, but the structure is irreversible, i.e., once it is disturbed it i will rarely reform again. The following cases belong to this category: i) Green (21) found that pigments like the | zinc oxide suspension in linssed oil with a suitable| acid concentration, will form a firm structural soap. If such a structure is broken, it will not reform unless there is sufficient acid remaining to ! | form more soap; ! ii) Dispersion of a thixotropic agent in a medium at an elevated "incorporation" temperature may form an irreversible gel structure upon cooling ; to room temperature. An example is a dispersion ! 5 of Thixcin R in a low viscosity epoxy resin. d) Size Reduction Due to Solvation: In a lyophil- ic colloid system swelling or solvation may occur, when a part of the colloid particle is softened by the surrounding liquid. When the system is subjected to shearing, the softened part is gradually dispersed into the surrounding liquid, and hence, the size of the particle is gradually reduced. The size reduction is continuous but permanent, so that an imperfect thixotropic fluid results. e) Other Factors: Aging, precipitation, chemical changes, and temperature rise due to shearing (35) may confuse the measurements, but they are generally control lable by improving the experimental techniques. Twenty fluid mixtures suspected of showing thixo tropic behavior were prepared and are classified in Table *Thixcln R is a thixotropic agent produced by Baker Castor Oil Company. A 4% Thixcin R dispersed in Epon 812 (Shell Chemical Co.) was prepared at 130°F. incorporation temperature (1). 6 Table 1. Classifd cations of Thixotropic Fluids Material Thixotropic Classification Possible Reasons for Imperfection PVC Plastisol Perfect Hand and Body Lotion (Silk'n Satin) Perfect Hand Lotion (Alyce La Mont Product) Perfect Bentonite in Water (R. Gesell, Inc.) Perfect Wyoming Aquagel in Water Perfect Aviloids (FMC) in Water Perfect Mayonnaise Perfect 5% Methocel MC (Dow) in Water Perfect 2% Methocel 65 HG in Water Perfect 10% Attagel 30 in Water Imperfect Imperfect dispersion and structural breakage Separum AP-30 (Dow) in Water Imperfect Same as above 4% Thixcin R (Baker Castor Oil) in Low Viscosity Epoxy Resin, Mineral Oil, or Propylene Glycol Imperfect Irreversible Structures Aviloids in Glycerol, Dioctyl Sebacate, or Epoxy Resin Imperfect Precipitations Heavy Mineral Oils Not thixotropic Change of shear stress with time due to temperature rise of fluid Polyester Resins Not thixotropic Same as above 7 Table 1 (con’t.) Material Thixotropic Classification Possible Reasons for Imperfection Epoxy Resins Not thixotropic Same as above Silicone Oils: Dow Corning 199, 200, 510 Union Carbide L-520 Not thixotropic No thixotropic response Silica in Water (Cab-o-sil, Cabot) Not thixotropic No thixotropic response Starch (laundry) in Water Not thixotropic No thixotropic response Silene D, EF in Water (Harwick Standard Chemical Co.) Not thixotropic No thixotropic response Pure Mineral Oils (350 cps) (drug use) Not thixotropic No thixotropic response Vegetable Oils Not thixotropic No thixotropic response Corn Syrup Not thixotropic No thixotropic response 8 3. Formation of Thixotropy A. Green's Postulates One theory of the cause of thixotropic formation as outlined by Green (21) is based on the following assump tions: 1) Assume that the thixotropic phenomenon arises from structure; I 2) The structure possesses rigidity. He also stated that in order to produce flow, the structures must be broken, and as long as flow continues, i j the broken structures will not reform even in part, unless j j the rate of shear is lessened. Furthermore, he postulates,: i without proof, that the thixotropic structure is an orient-; i ed one and that each particle acts like a magnet possessing i i | a north and a south pole. Thixotropic structures will form | only when unlike poles come in contact. Brownian motion or some other disturbances may be the source to bring the | unlike poles in contact; therefore, this procedure requires; I jtime. This model may explain why recovery requires time. I I A careful study of the structural formation of the bentonite-in-water system gives substantial support to ! Green's model. Microscopic studies indicate that the structure of the Bentonie is very thin platelets j 9 (41,44,45). Also, these platelets are proved to have nega tive charges on the face, but positive charges at the edge (8,44). The charge distribution is assumed to be caused by the atomic structural differences within the platelet particles (45). When a suspension of platelet Bentonite particles flocculate, edge-to-face particle association occurs be cause of the opposite charges of the edge and face (46). The edge-to-edge, and face-to-face associations are not likely to happen because of the repulsion force existing between the two similar charges. Since the association by electrical charges of the particles are net strong as com pared with the chemical bonds, it can be broken by mechani cal disturbance, and the edge-to-face association forms again after the disturbances are removed. Brownian motion is assumed to help in bringing the particles together (48). This may explain the time lag in ■the recovery process. In order to form the above so-called "card house" edge-to-face structure, a minimum of 2% Bento nite is required to fill the available volume (47). From these considerations, it is obvious that the following conditions have to be satisfied in order to form 10 a thixotropic fluid that conforms to Green's model: 1) Opposite charges are required on the particles in order to form the edge-to-face or other associations so that a continuous thixotropic structure may be built up. Face-to-face association is not preferred because it will cause aggregation of the system. i 2) Because the electrical charge always exists, | [ once the structure is broken by shearing (or disturbance), j | ! ieven if the disturbance continues, the recovery process al- j ways proceeds simultaneously. 3) A certain concentration is required to form a continuous structure. The concentration required is re lated to the size of the particle. j | | ; j B. Molecular Entanglement-Disentanglement Theory | Hahn, Ree, and Eyring (25) stated that in a thixo- I ’ tropic system, there are two kinds of molecules, entangled land disentangled. Entangled molecules make a three-dimen- |sional network, in which disentangled and solvent molecules are enclosed. The entangled molecules behave as non-New- |tonian units, and the disentangled molecules behave as New tonian units. The transition from entanglement to disen- j tanglement, which is brought about by stress, destroys the ! 11 network, changing a solid-like substance to a liquid-like substance. If the stress is relieved, the transformed molecule returns to its original state only after marked delay. This theory explains the thixotropic behavior of the solvent systems where electrical charges may not exist among particles. However, the reason why the transformed molecule returns to its original state is unexplained. CHAPTER II HOW TO CHARACTERIZE THIXOTROPY 1. Previous Theoretical and Experimental Approaches A. The Hysteresis Loop Plotting the hysteresis loop was one of the first widely used methods to characterize the time-dependent I fluids. The experimental procedure for deriving the hyster-j I esis loop is to observe the peak torques on the same fluid J I ! sample by means of a rotational viscometer. An upcurve is ! ! derived by starting at the lowest rotational speed and pro- j i ! ceeding step by step to the highest speed where each step requires a finite time period. When the highest speed j i is reached, the procedure is reversed yielding a downcurve. j The upcurve and downcurve, when plotted on a basic shear diagram, have a shape as shown in Figure 1. The differences between the upcurve and down curve indicate the degree of the thixotropic breakdown. In order to characterize 12 Shear Rate, S Figure 1. The Hysteresis Loop I 13 the thixotropic breakdown more specifically by means of mathematical equations, Weltmann (51) obtained the downcurve for a pigment suspension in a slightly different manner to make i the set of downcurves shown in Figure 2, start ing from downcurve 1 to E. The downcurve, E, is supposed to be an equilibrium curve. The coefficient, B, was used to characterize the thixo tropic breakdown with time, t, as follows: where B = coefficient of thixotropic break down with time; U = plastic viscosity. Green and Weltmann (23) have used another coeffi cient, M, to characterize the thixotropic breakdown with increase of rate of shear: < D S o* u o H RPM Figure 2. Weltmann's Downcurves Diagram 14 where M = loss in shearing force per unit area per unit increase of rate of shear; k = constant for the upcurve of the loop; ca) = top angular velocity. Dahlgren (9) in 1955 studied the above effects and concluded that eight constants are necessary to character ize a thixotropic material fully. B. Application of the Molecular Entanglement-Disen tanglement Theory to the Hysteresis Loop Hahn, Ree, and Eyring (25) modified their theoreti cal flow equation, which contains Newtonian and non-New tonian parts: r = S f l s * 5 sinh-1 p , s dl (2) where T = shear stress S = shear rate °i = (x ^ 2 K 3)i/2kT' 1 = 1' 2 ft = l/-r- 2K’ , i = 1, 2 i K ^ 2/ A 3 are parameters in Eyring*s theory of flow k = Boltzmann's constant k' = specific rate of flow when there is no stress 15 x^ = fraction of the area of the shear surface occupied by the itk kind of flow units, i = 1, 2 T = absolute temperature. They considered the net rate of the transition from en tanglement to disentanglement to be: 2 dx9 M' CS _ (l-zncs2 - ■^7“ = x2kfe kT - x-|kbe kT (3) where kb = specific rate for the reverse reaction at zero stress kf = specific rate for the forward reac tion at zero stress c = constant yu* = reaction constant The hysteresis loop is used to evaluate the above equation. For the upcurve, the rate of shear, S, is changed with time according to the relation: S = f t (4) where $ is a constant to be determined by the experimental conditions. Equations (3) and (4) yield: -kfrl x2 = (x2^0 0 16 ry 2 where I = ey dy J0 y - t/r = 7 3 % 2 = ®/rP (x2^0 = fraction of the area occupied by the entangled molecules at t = 0. Substituting equation (5) in equation (2) yields: T = [l - (x2)0e"kfrIj|i- S + ^ ( x 2>0e~kfrI sinh"1^ (6) Equation (6) is applicable to the upcurve of a hysteresis loop. It is time-dependent because the shear rate, S, is dependent on the time for making the upcurve. For the downcurve, the following equation applies: T= [l - (x2)0e"kfrIm)^ s + ^(x 2)0e"kfrIm sinh"V2S (7) where Im is the value of I at the apex. Although the above model characterizes the hystere sis loop, the following drawbacks limit the use of the equations: 1) The theory does not explain why a thixotropic i fluid recovers, or if the transformation from disentangle- j I ment to entanglement is possible. i 17 2) No characterization of the recovery of the thixotropic fluids by the equations has been attempted. 3) The equations are not applicable for the fluid systems with yield stresses. 4) Eight constants are to be evaluated for shear equation alone, which is too complicated to handle. C. Modeling Thixotropy by an Analogy to Reaction Kinetics Many investigators (4,11,18,37,39) have used an analogy to reaction kinetics to represent the thixotropic breakdown and recovery. j I Goodeve and Whitefield (18) have described a thixotropic system as consisting of "cells'1 having solid ; i properties. The cells per a specific volume is denoted by ; j x, and the maximum possible value of x is Xjjj. The break- j down process, then, can be represented by: - § -rf* where S = shear rate k = proportional constant for forward reaction. The building up process can be represented by: 18 g = l,'-(xm -x)a where a = order of building up unlinked state k'' = the backward reaction proportional constant. Finally, they stated that the apparent viscosity is proportional to x: n - n 0 = where = apparent viscosity at equilibrium Under the condition of steady shear, high shear j i rate or weak internal structure, the equilibrium value of x j will be small compared with xm, so that x = k'' xa/kS, and, n-= V • i = ! < 8 > i where Q = coefficient of thixotropy Equation (8) gives the relationship of the steady j i state behavior of the thixotropic fluid. Moore (37) has represented the thixotropic behavi or by linkage formation and breakdown. The linkage is indicated by X , which is the total number of links present ; at the state of interest divided by the number of links j present in the completely built-up state. His equation i i 19 has the form: ^ = a (1 - X) + bXs (9) dt where a, b = constants Equation (9) together with the following constitu tive equation: where n0 = viscosity of completely broken down substance c = material constant. This equation does not include yield stress for the materials. Denny and Brodkey (11) have developed a theory based upon reaction kinetics to describe the structural changes of the thixotropic materials and the following T = (n0 + c X)s (10) I equation is obtained: d(unbroken) dt • P / 3 = k1Sir (unbroken) - k2 (broken) where (unbroken) = concentration of material unbroken (broken) = concentration of material breakdown k^ = forward reaction rate constant k2 = backward reaction rate constant 20 c* = order of forward reaction p = order of reverse reaction p = material constant. The unbroken and broken concentration are defined as follows: unbroken = JAq - JJ-oo broken = >^e ~ JUq — where J*0 = maximum point viscosity = minimum point viscosity /Jq = the point viscosity at any moment The result of above assumptions provides the fol lowing equation: / o “A V dt 7 kl /fo - A 2 A - A Integration of equation (11) at a constant shear rate will give the variation of point viscosity with time. The point viscosity is dependent upon the shear i rate, so that when the shear rate changes, the maximum point viscosity, yM0, and the minimum point viscosity, , change accordingly. Hence, the use of this theory is 21 limited. Pinder (39) assumed that the apparent viscosities of the system follows the kinetic rule as: *1 j M l v- J *2 k2 The paired second and zero order reactions are found for the system of tetrahydrofuran-hydrogen sulphide gas hydrate slurry as follows: d/S i = -*2 The drawback is that different sets of constants k^ and k2 are required for different shear rates. Casey (4), in modifying Moore's theory (37), re tained the linkage equation (eq. 9) but used a different constitutive equation as follows: T = n (S) S + T 0 + \T-j_ j where TQ = value of critical shear stress ! measured in sustained flow experi-i ments T x = additional shear stress above the steady value which is necessary to initiate flow n(S) = c/(l + dS), shear rate dependent viscosity 1 22 c,d = material constants. The resulting breakdown equation has the form: T - + <a+b5)] d 2 > For buildup, the equation has the form: r*= t t t s + T° + T i (1 - e_atr) (13) i | where tr = rest time after shearing. | i Six material parameters, a, b, c, d, an^ T q I must be evaluated for the above equations. ! From the point of view of reaction kinetics, the j breakdown and buildup can be represented in terms of link- i age units. Ideally, these units can be broken down upon shearing and hence, equilibrium is reached. In practice, | i Jhowever, this equilibrium is very hard to reach during continuous shear. The time required to reach equilibrium j | usually has a magnitude of hours. Therefore, it is quite : hard to evaluate the equilibrium conditions or parameters I suggested by these investigators. Further, there are at least six constants to be determined for most of the break down equations alone, which are mentioned in this section. These facts limit the use of the above theories. 23 D. Concept of Viscoelasticity as Applied on Thixotropy The idea of viscoelasticity has been applied to thixotropic fluids by many investigators (32,33). The vis coelastic theory was originally made to characterize the viscoelastic fluids. The theory usually assumes a model to be made up of a combination of springs and dash-pots (52) to represent the elastic and viscous properties of the fluid. A parallel combination of spring and dash-pot gives a model called the Voigt Body and the series combination j gives a model called the Maxwell Body. j Lewis (33) has applied the Maxwellian stress re tardation model to describe transient and equilibrium shear i stress at a constant rate of shear. The equation describing this thixotropy is: i - = k2KTVP+P2 <CFt > " k2TTp2 [<1 " Ft )2 q2) (14> I i where FT = fraction of thixotropic structure unchanged I T t = thixotropic shear stress C = concentration ]<2 ,P2 = constants ; K,P = equilibrium constants j 24 The relationship of measured shear stress, X , and the elastic retardation at constant shear rate is provided by the following equation: T _ T m -*etx t — * • rp (1 — e ) where ke is an elastic constant dependent upon shear rate. 47% polymethylmethacrylate in diethylphthalate at 40°C has been used for fitting equation (14), and the vari ation of k with shear rate is tabulated. | I A total of six constants are involved in this i ! theory; the parameter, k, is dependent upon the shear rate, ! j which makes the equation more complicated to use. j | j 2. Theoretical Model for the Present Study A. Mathematical Considerations I When a perfect thixotropic fluid, maintained at | constant temperature, is subjected to a shearing (or re- ! i covery) operation, the response of shear stress, T , is :assumed to be a function of shear rate, S, and the time, t,| for the fluid subjected to the shearing (or recovery). It j can be expressed as follows: j T = F(S, t) 25 The shear rate, S, and time, t, are independent variables. It is assumed that these two variables possess independent, or separable functions which have the form: T = f(S) g(t) (15) where f(S) = a function of shear rate, S, only g(t) = a function of time, t, only. In reviewing the time-independent fluid studies, |the power-law model, I T = k(s)n (16) I I | where k and n are constants, j 'has been widely used (6,7,31,38,43) for characterizing the I ; i !time-independent fluids. The power-law model with yield j .value (2,5) has the form: j T = T 0 + k(s)n d7) : i where To = yield value. j I I Equation (16) or (17) can be used to substitute j f(S) in equation (15) if the forms are uneffected at differ- I I lent time levels. The resulting equation has the general | I I 'form: r -i I T = |_T +k(S)n Jg(t) (18) ! B. Theoretical Considerations It is assumed that when a perfect thixotropic fluid is subjected to a continuous shearing at constant shear rate, S, the structure of the fluid breaks down; however, a jpart of the broken structure recovers simultaneously. As a result, a net change of structure is observed which may be reflected by the measured shear stress. The relationship may be written as follows; \'bt/s,net \ ,breakdown \^ t/S,recovery Equation (19) indicates that the net rate of change j I of shear stress, -( ). , at a constant shear rate, S, y d t /S, net is the result of the difference between rate of breakdown j ( , , and rate of recovery f-4-x). . The V3t/s,breakdown \ 3 tys,recovery minus sign for the first term indicates a decrease of shear : stress. It also means that a thixotropic fluid is dealt i | with, instead of a rheopectic fluid. 1 I The rate of breakdown is assumed proportional to j I the shear stress, T , with th order at that moment; ( ■ t t )- (20> 0 XJS,breakdown The rate of recovery is assumed inversely 27 proportional to the effective time, T, to the (3 th power, where the effective time, T = t + 1, and t is the total elapsed time. ( m c < H r (21) S,recovery T* Substituting equations (20) and (21) into equation !(19), a non-linear differential equation, which describes i ! I both breakdown and recovery for a perfect thixotropic fluid I 'results in: S ,net 1 2 (t + 1)P (22) where k^ = a constant for structural break- | down j k2 = a constant for recovery. i The boundary condition for equation (22) is | "T = at t = 0. i The solution of equation (22) for shear stress, X , in terms of time, t, at constant shear rate, S, will give the form of g(t) required in equation (18). I Considered at constant shear rate, S, with a sub- jstitution of T = t + 1, equation (22) can be written as an jordinary differential equation as follows: 28 dT T c T * k2 ki c " ^ r Let = X + 1, and X changed to the following form: K P (23) r j - L , equation (23) can be _2 (24) j Equation (24) is Riccati's equation which can be solved by making the following substitutions (36): Let U = e dl/ CT Xkx) xdT , then, dT X k xxu (25) d2U dT2 ( ^ i f + k2\ *2>u (26) Substituting equation (26) into (24), yields: d2n -P - X.k,k0T U = 0 (27) dT 1 2 Equation (27) is a second-order linear equation which has a form of the following Bessel equation: _ 41 _|2 + k -2 T-P u = 0 dT^ T dT 1 (28) The solution of equation (28) by setting o^' = 0 is: 29 U = TP [ AJ v (kTq) + BJ_ y (kTq)J where p=-j(ot'+l)=-^ (29) k = 2 - p 2 - P 2k i 2 - p The solution then can be expressed in terras of the modified Bessel functions, 1^ (z) = e-^ j ^ (2 i) ; U = T3 * [ AXI p (i>Tq) + C f Tq) J = T% (30) where = 2 J ^1^2 (<* - 1) 2 - p Af, B^ = constants From equation (25), X = U' (31) Substituting equation (30) into (31) , yields: X = ^k-jT^ qj J£L Xkx ] 2T = 1 _1 + IqT^1 [a x i' v (iTq) + B^jy (j?Tq)] 2T qj y 30 - 1 + i \kxk2 Tq_1[l^(iTq) + cil„ (JpTq)J^ Xki 2T I ^ (JfTq) + CI^WTS) where C = b1/A1 = the constant of integration. The final form of equation (32) is -0'” 1 = a 1— ' ± J(o<- 1 ) ^ 2 T ^(i^JlT1 ) + CI^UT T )] (o<- l)kx 2T 1- P 1- I„( T + Cl T 2) (33) Equation (33) provides the shear stress, X , in jterms of T, where constants ^ , k^ and k2 are deter- ! | i jmined by equation (22) and C can be determined by the bound- j >ary condition. However, the terms of the modified Bessel : I functions are too tedious to evaluate. Further simplifica tion must be done in order to change equation (33) into a i i i tractable form. : i One reasonable simplification results from an exa- | ; i_ i ! |mination of the order of magnitude of JT 2. Since J) = j j2 k1k2(a ~ I --- , k.^ k2, oi and (3 are evaluated. Table 2 summarizes these constants for equation (22) 31 which are evaluated from the experimental data of PVC plas- tiso, hand and body lotion, and 8% Wyoming aquagel. De tails are provided in Appendix B. Table 2. Constants for Equation (22) S, sec-i oi P kl k2 ll. 75 15.4 0.95 1.2 X 10~33 6.04 PVC Plastisol 35.2 15.4 0.95 2.6 X 10~3S 10"43 11.54 105.7 15.4 0.95 3.0 X 22.07 11.75 7.9 0.90 7.4 X 10"17 5.41 Hand and Body Lotion 35.2 7.9 0.90 1.8 X 10"17 10"18 6.89 105.7 7.9 0.90 3.9 X 8.78 11.75 8.3 0.86 9.0 X 10" 2 2 23.5 8% Wyoming Aquagel 35.2 8.3 0.86 6.1 X 10-22 21.7 105.7 8.3 0.86 3.6 X 10"22 19.7 From Table 2, the range of Jl is approximately — 7 —IQ n ~2 19.6 x 10 to 1.8 x 10 . Therefore, the value of X T *is usually very small, at least valid up to 10 hours of shearing. Even for 10 hours duration, T = 36,000 sec.r n 1" "5 -4 the maximum value of IT 4 is 1.44 x 10 Based upon the above argument, the modified Bessel ! i- 1 jfunctions I^Cj^T 2) in equation (33) can be further simplified. Since the definition of I„(x) has the form: 32 .y Z_i m! P ^+2m Ii»(x) m! n^+m+l) (34) m=0 So, for a positive value of t 3 , Ip(x) approaches a finite value between 0 and 1 when x approaches 0 with 0 ^^ ^1; but Ij, (x) approaches infinity when x approaches zero with ^ being a negative value (34). Rearranging equation (33), note that 1^ (x) = j ^I„_i(x) + Iv/+i(x)J : 1 1 2T W-Dki _ P r 1- — 1 - — 1 1 - — P J^-Dit^T 2) + i ^ U t 2)J+ ci^( t 2)| ! , P !_ / 3 I„(J?T ^) + CI_„ (JT ^) ; (35) | I ! Since the values of Iy+1(x) and Ij,(x) are small as compared with Iv_i(x) and I_,>(x) , respectively, they are j ! I | I dropped out from equation (35), yielding: j ' T rf_ 1 = I 1 r 1-1 > 1- £i 1 ___________ - i l r l--£ • 1 - ^ 7 1 J(c(-1)^ 2 T 2 2(I » - 1 ^ T 2) + CI.^UT 2)J > (oi-Dk, I 2T , 1 - 4 x 1 I ci_p ( tx ?) y (36) Each of the above modified Bessel functions con verges very rapidly (50) so that only the first terms are 33 retained as an approximation; thus equation (36) becomes: o< - 1 1-4 M ( oC-DTc- l “4 1 ( M z l 2? 1- £ P(.^T V " ^ ) 2T C(%J|T 1- r(^+D _£ = W ^ | - T t + J**-« w “ + W T ) " 4 cF(i>) P i - p ,1_ 2\-l W ^ ( m + t' 2 I ■ ■ ^ iTe" 2 ) ~ X + (JT1" 2 J-1] I + — •( 2-p \ [— i-( _______ (2-p)C'2 2- *1*2 ( • ‘-D. Tl- | ) -1 P 2-p (c*-l)k -X + f(o(-l)k,k~ T O m J X Z P 2T 1 4Cjk1k2U-l) £ -i T'2 - _1 2T 4(o(-l)k1C T -1 (37) T = U w - W *'1 <38) Substituting T = t + 1 into equation (38) yields: 1 T - l (t + l) -1 (39) From the boundary condition, T = at t = 0, the 34 integration constant, C, can be evaluated as follows: = .. c = ^W-DkiCp-i 1______ crt-l i 4(o(-l)k1 T. Substituting into equation (39) yields: ___ T = T ±(t + i) -i (40) Equation (40) is an approximate solution of the pro posed differential equation (eq. 22), where is the iinitial condition or the peak value obtained at each con- I I t • stant shear rate, S, at time, t = 0. Compare equation (40) i iwith equation (18) and we have: j : • i . j Ti = [Tq + ^'(s)n Jg(0) = T Q + h'(s)n (41) | where Tq, h', and n' designate the constants | for the shear operation. i ! i j Substituting of equation (41) into (40) yields: j ! T = JTq + k' (S)n ' J (1 + t)ra' (42) ■ 1 i where X q = yield value oL - 1 Equation (42) is a general shear equation for the 35 thixotropic fluids. As far as recovery of a thixotropic fluid, the / ar \ breakdown term ("TT / * , , in equation (19) is V d t ' S,breakdown ^ dropped, so that equation (22) is reduced to: ( ^ t)s, net ^ (1 + t)p Integration directly yields: -r 1-/3 T = ( 1 + + C' ( 4 3 ) *2 For a boundary condition T = = —--— at t = 0, C' = 0 Therefore, T = T^l + t)1"^ (44) This recovery equation is similar to equation (40), so that Xi can be substituted with a power-law model with a yield value as follows: T = [-To + k"(S)n" ] ( l + t)"’ (45) where m" = 1 - (3 I I 3 c", and n" are constants for re covery of a thixotropic fluid. Equations (42) and (45) can be written as the 36 following generalized form: 'X = [Tq + k(S)n ] (1 + t)m (46) where Tq = yield value k, n, m are constants dependent upon the nature of the fluids and type of operations. Equation (46) is a generalized model which charac terizes both shear and recovery behavior of the thixotropic fluids. When T q = 0/ the fluids behave as time-dependent power-law models; when Tq = 0 and n = 1, the fluids behave ! as time-dependent Newtonian models; when k = yM and n = 1, the fluids behave as time-dependent Bingham plastic models; i land when m = 0, the fluids behave as time-independent j |models. i I j C. Statistical Considerations I i 1) To obtain constants for the theoretical model: The constants T q » an<^ n tn equation (46) i I I should be, theoretically, determined by the initial i conditions or the peak values at time, t = 0 of the experiments. However, it is generally better to have all constants evaluated statistically from the whole set of experimental data instead of 37 only using initial values. a) Power-law fluid without yield value: From a basic shear diagram (a plot of shear stress, X / versus shear rate, S, at fixed time levels), it can be determined if the fluid has a yield value, T q . For a time-dependent power-law fluid without yield value, equation (46) is reduced to: X = k(S)n (1 + t)m (47) Taking the logarithm of both sides, the equation becomes, In T = In k + n In S+mln(l+t) Corresponding to: yi = a0 + alxl + a2x2 ao = In k al = n a2 = m X1 = In • s x2 = In (1 The constants can be evaluated by ap plying a regression analysis on the above 38 equation (54). b) Power-law fluid with yield value: For a time-dependent power-law fluid with yield value: After taking the logarithm of both sides, the equation appears non-linear and thus difficult to solve by previous techniques; however, if the yield value can be evalu ated by a plot of initial peaks on a basic shear diagram, the following procedure is suggested: i) The first step is to evaluate m in equation (49). At a fixed shear rate, S, the shear stress, T , is a function of time, t, only: T = ( T 0 + kSn) (1 + t)m (49) T = c1(l + t)m T = c2(i + t)m at S at s 1 2 After taking logarithm: 39 In *T = In + m ln(l + t) In T = In C2 + m ln(l + t) at s ± at s2 The above linear equations have a com mon slope, m, that is, they are paral lel lines. These parallel lines may be lumped together into a single line having a slope, m, by applying regres sion analysis on the equation: y = aQ + ai*i (5°) where a0 = intercept of the re- = In(1 + t) ii) After m is known, equation (49) can be rearranged to: After taking the logarithm, it be comes : In [— ^ - T0 ] = In k + n In S l(l+t)m J gression line obtained from the whole set of data a^ = m (1+t) T n = k(S)n (51) 40 which can be written as y = aQ + a1x1 (53) where y = In (l+t)m - T °] a0 = In k al = n 0 X1 = In s Equation (53) can be solved by ap plying a standard regression analysis method. 2) To compare theoretical results with experiment al data: Theoretical results can be calculated by means of time-dependent power-law equation, equation 46. The question arises as to how good these theoreti cal results are as compared with the experimental data. Many statistical methods such as non-para- metric sign test and rank sum test, as well as parametric student-t test, difference of two means method (29) and analysis of variance (27) can be applied. All of the tests may provide a qualita tive answer of "yes” or "no," based upon certain 41 levels of confidence limits about the compatibility of theoretical results with the experimental data. However, the method suggested by Popovics (40) may give more information about it. Popovics considered each pair of related ex perimental and calculated values to be a coordinate of a point (x0,y0), where x0 is experimental value and yQ is theoretical or calculated value, as shown in Figure 3. The deviations of these points from the line of equality, y = x, are used to measure the goodness of fit of the experimental values to the line, y = x. The variance of the points about the line, y = x is defined to be: n rH t a o •H 4J 0 ) H O < U x, Experimental Figure 3. Experimental vs. Calculated Values Sp/x n 2 (y° " X°> The square root of the S p/x (54) can be called the standard error of fit (SEF) which represents an average deviation for the set of points from the 42 line of equality, that is, between the computed and experimental values. SEF/y provides a relative measure of the standard error of fit (RSEF), where y is the mean value of y. The fit coefficient, f, is defined as follows: f2 = 1 - sp/x/(sl + sy> <55> 2 where SL = variance of the least- squares estimated value of y about the line of equality = n 2 ( y ' - y)2 n 1 ' s T —1 2 = n J> (a - bxQ - xQ) 2 Sy = variance of the yQ values about the mean y = n 2 <yo - ?) 2 The fit coefficient is similar to the correla tion coefficient. One-sidedness of the deviations can be measured by defining a deviation factor, DF, as: DF = 2 > o " ]|>o (56) The larger the absolute value of DF compared to 43 the value of SEF, the more one-sided are the devia tions. DF/^n is a relative measure of one sided ness of data per each observation. From the above considerations, it is possible to compare the theoretical results with the experi mental data quantitatively by means of SEF, RSEF, f, DF, and DF/yn, so that a conclusion of how well the theoretical equation (equation 46) fits the data can be made. 3. A Frequency Domain Presentation An alternate way to present the thixotropic fluid i trum can be generated and examined. With this standpoint, is to express the change of shear stress, X , at a specific jshear rate, S, in terms of a Fourier integral, then a spec- equation (18) may be written as: (57) in which, f(t) e iwt dt (58) where g(w) eiwsds o ^ f (B) eiwsds (59) 44 g(w) is a function of frequency, w. If g(w) is known, f(t) can be determined through equation (58). Thus, we have two different representations of the function of our discus sions: f(t) in the time domain, and g(w) in the frequency domain. f(t) can also be written in a discrete sinusoidal form as follows: The Fourier coefficients, a(wn) and b(wn) indicate that these coefficients are frequency dependent, where the f(t) = a(0) (60) term, wn, represents the frequency of the n1 -*1 harmonic in Sin the expansion, and where 2p = period. The Fourier coefficients, a(wn) and b(wn) can be |determined by the following relationships: (61) Equation (61) is parallel to equation (59). 45 The amplitude corresponding to the frequency, wn of the nth harmonic in the expansion of equation (60) is defined as: A(wn) = J [ a<wn)]2 + [t> (wn) J2 (62) and A(0) = a(0). Usually, a spectrum of f(t), i.e., a plot of A(wn) versus frequency, wn, is presented in order to examine the the characteristics of the function, f(t). CHAPTER III EXPERIMENTAL WORK 1. Purpose The purpose of the experiment is to verify the pro posed theory by observing the following variables for dif ferent thixotropic fluids: a) shear stress,T ; b) shear rate, S; c) elapsed time, t, for shear or recovery. Other variables which could be involved in the ex periment, such as temperature and composition of the (fluids, are kept constant. j | An experimental technique, different from that for hysteresis loop, is used and generally is as follows: a) The net structural breakdown of the thixotropic I fluid is achieved by applying various constant shear rates | each to a fresh, unsheared sample, so that a continuous | curve for shear stress, “ X , versus time of shearing, t, may be obtained. b) The structural buildup immediately follows for the same pre-sheared sample by allowing it to rest for a 46 period of time, t. Then, the sample is sheared at the same shear rate as before, in order to get a peak value of shear stress, Xp* After meesuring a series of samples, a plot of peak shear stress, Xp# versus period of resting, t, can | be made to represent the recovery of the thixotropic fluid. I I 2. Instrument The experimental instrument consists of a combina tion of rotational viscometer and automatic recorder. A rotational viscometer called the Haake Rotovisco (manufactured by Gebruder-Haake in Berlin, West Germany) is utilized in making viscometric measurements. The vis- j i i i cometer consists mainly of a bob rotating within a station- j ! ary cup. A very small gap is allowed between the bob and j i 'cup for holding samples. Resistance to the bob rotation j i :caused by the sample material is transmitted by a torsion i spring and measured as angular deflection. There are two i |torsion springs and ten different rotational speeds avail able, capable of covering a wide range of viscosities from 130 to 400,000 centipoises. The dimensions of the bob and cup are shown in Table 3. The measuring accuracy is reported to be + 2% (24). 48 This instrument is described in detail by Van Wazer et al (49) . Table 3. Dimensions of Bob and Cup For Rotovisco Viscometer Dimensions, mm Diameter Height Bob 20.2 (outside) 60 Cup 23.1 (inside) 70 An automatic recorder of the Sargent Model SR is connected to output signals corresponding to angular de- ! flection from the viscometer, resulting in a continuous | ! plot of time versus angular deflection. Accuracy of the i recorder is reported to be %%. i , I I 3. Calibrations | A. Torsion Spring Constants ! The torsion spring constants of the viscometer were ■ determined by calibrating against a standard fluid (from I | | i Cannon Instrument Company, State College, Pa.) with a vis- | 1 cosity of 1296 centipoises at 25°C. The results are indi cated in Table 4. Table 4. Torsion Spring Constants (For SV-I Type Rotor of Rotovisco Viscometer) A Measuring Head Spring Constant C, dynes/cm^ 50 10.8 500 115.0 Shear stress, T , at a specific rotation speed can be obtained from the following relationship: T = CR (63) j where X = shear stress, dynes/cm2 j j R = scale reading on the recorder chart | C = spring constant provided by | Table 4, dynes/cm2. ! ! | j B. Rotational Speeds | i ! The rotational speeds of the viscometer were cali- ! | ! |brated against a stop watch. The results are indicated in j Appendix C-l. Statistical inference is applied to these | 'results. Based upon 99% confidence limit, the observed i I I | rotational speeds for the viscometer are in agreement with j ' | the manufacturer's reported speeds. Speeds tested are *Measuring Head 50, and 500, correspond to two different torsion springs used by the viscometer. 50 10.8 RPM, 32.4 RPM, and 97.2 RPM. The shear rate, S, can be calculated from the rota tional speeds by the following equation: S = A N (65) where A = constant dependent on the dimen sions of the bob and cup N = rotational speed, RPM. The constant, A, for the SV-I type rotor and-cup is specified as 1.088. C. Chart Drive Speeds The chart drive speeds of the Sargent automatic re corder were also calibrated by stop watch. Ten observa tions were made for each speed. The results are indicated in Appendix C-2. Based upon 99% confidence limit, the ob served chart drive speeds are in agreement with the manu facturer's reported speeds. 4. Preparation and Testing Procedures A. Sample Preparation Three different thixotropic fluids were prepared and tested, which were: a) Polyvinylchloride (PVC) plastisol b) Hand and body lotion 51 c) 8% Wyoming Aquagel in water. PVC Plastisol was a manufacturer's sample, prepared by the Plastikam Corporation, Burbank, California, which was formulated as follows: PVC 54.03% by weight Diisodecyl phthalate 21.38% Isobutyrate 21.38% Epoxy difatty acid 1.61% Barium soap 1.61% Color trace Total 100.00% The sample was prepared by mixing the above ingre- j dients in a high speed blender, then degassed in vacuum j | and settled for approximately two months before testing. | Hand and body lotion is a commercial sample pro- I I duced by Chas. Pfizer & Co., Inc., New York, N. Y., with I the brand name, "Silk'n Satin." I 8% Wyoming Aquagel in water was prepared by mixing j ; | 8% Wyoming Aquagel powder and distilled water in a Waring j i i blender at high speed for about 30 minutes, then immediate ly placed in 7-dram plastic bottles. Stability of the samples were tested and it was discovered that the sample 52 slowly flocculates, but is nearly stable four days after preparation. Thus, all shearing tests were performed after the fluid stabilized. B. Sampling Techniques Individual samples were filled into 7-dram wide- mouth plastic bottles after preparation. Each plastic bottle contained only enough material for one test. The following advantages were gained by this procedure: The amount of shearing action experienced by the I individual fluid sample following preparation until actual j | use was minimized. If a large container with sufficient material for numerous tests had been used, the fluid could I i | stratify. Thus, samples from the top would differ from ; those at the bottom. Furthermore, the bulk of the fluid would be disturbed with each sampling. Finally, random , sampling could easily be achieved merely by randomizing | | the plastic bottles and taking a random samples from these ! bottles. C. Design of Experiments i i The response (or dependent) variable for the ex- j periment is shear stress, T , the independent variables are j 53 shear rate, S, time of shearing, T, and time of resting, t'. Shear rate, S, has been chosen at three levels: 51 = 11.75 sec""1 52 = 35.2 sec"1 53 = 105.7 sec” 1 Time of shear, t, has been chosen at two levels: t^ = 10 minutes t2 = 1 hour Time of recovery, t', has been chosen at four levels: t£ = 10 minutes ±2 = 1 bour t^ = 3 hours t^ = 16 hours Altogether, 24 different combinations are required to complete a cycle of the experiments for a specific fluid. With r replications performed in order to improve accuracy, there are 24r different combinations for each I fluid. Randomizations are made among samples, shear rate, time of shear, and time of recovery. The procedure of testing is as follows: 54 1) Select a sample randomly from the 7-dram bottles; 2) Select a shear rate, S, for the test; 3) Select a time of shear, t, and shear the sample at the chosen shear rate, record shear stress, T, as a function of time; 4) Select a time of recovery, t1, and let the pre-sheared sample rest for that period; 5) Obtain a peak of shear stress by shearing at the same shear rate as selected in (2), after it has been rested; 6) Repeat procedure (1) through (5) until all combinations are tested. CHAPTER IV EXPERIMENTAL RESULTS 1. Shear and Recovery Curves When a thixotropic fluid is subjected to a constant and continuous shear rate, the typical observable results are as shown in Figure 4. These shear stress-time curves are for PVC plastisol. Hand and body lotion and 8% Wyo ming aquagel in water gave similar plots. For recovery of the thixotropic fluid after shear- I ing, typical recovery curves also represented by PVC plas tisol are shown in Figure 5. The recovery plots consist of two families of curves, one with the fluid pre-sheared i for 10 minutes, and another pre-sheared for 1 hour. The fluid pre-sheared for 10 minutes always has a higher re- j covery peak shear stress than that for 1 hour when rested j for the same period in both cases, but the general shape j of the two families of curves are similar. ! I i In some cases, such as 8% Wyoming aquagel and 4% j • f f Aviloid in water, the rate of recovery when pre-sheared *Aviloid CT-E-581 is produced by FMC Corp., Marcus Hook, Pa.; 4% water solution appears white and gellied. 56 S = 105.7 sec 1 S = 3 5 .2 sec S = 11.75 sec"1 800 600 200 ~ 0 20 4.0 60 80 100 120 1000 2000 Shear Time, t, sec. Figure 4.. Shear Curves of the PVC Plastisol at 25°C Pre-shearedj 1 hr. 10 min S= 105.7 sec!1 — a— S= 35.2 sec" 1 - S = 11.75 sec *_ S 600 400 20 0 2 4. 6 8 10 12 14. 16 1 Recovery Time, t, hours Figure 5. Recovery Curves of the PVC Plastisol at 25°0 57 at a lower shear rate appeared faster than pre-sheared at a higher shear rate, as seen in Figure 6. 2. The Basic Shear Diagram The basic shear diagram shown in Figure 7, has been generally accepted as a basis for characterizing fluids with time-independent fluid properties. These fluids are named by the shape of the curve in Figure 7, as follows: Curve A: Newtonian fluid Curve B: Pseudoplastic fluid Curve C: Dilatant fluid Curve D: Bingham plastic fluid Curve E: Pseudoplastic fluid with a yield value. The basic shear diagram can be extended to describe thixotropic fluids by plotting the shear stress versus shear rate at different levels of time for shear or re covery. Figures 8, 9, and 10 provide the basic shear dia grams of PVC plastisol, hand and body lotion, and 8% Wyoming aquagel, respectively. From the curves, it can be seen that PVC plastisol and hand and body lotion are 58 800 CM a S' 600 & T3 ^ 400 ( 0 t o © J h +3 CO & 200 Jl CO 16 6 8 12 10 U 2 0 Recovery Time, t, hours O Figure 6. Recovery Curves of 8# Wyoming Aquagel at 25 C 59 « % W C O © u CO & ( 1 ) S3 t o » Shear Rate, S Figure 7. The Basic Shear Diagram 800 2600 4.00 §! 200 100 20 • ^*1 Shear Rate, S, sec Figure 8. The Basic Shear Diagram, FVC Plastisol at 25°C 60 Figure 9t The Basic Shear Diagram, Hand and Body Lotion at 25°C CM ( 3 1 0 d > c o m ( D CO d j§ CO 800 600 400 200 O SEC. 6 SEC. 1 - 0 MIN. I HOUB 20 4.0 60 80 Shear Rate, S, sec 100 Figure 10» The Basic Shear Diagram, 8$ Wyoming Aquagel at 25°C 61 time-dependent pseudoplastic fluids, and Q% Wyoming aquagel is a time-dependent Bingham plastic fluid. Figures 11 through 16 give the recovery curves of the same three materials. A recovery curve is a plot of the observed peak shear stress versus shear rate, where each point has a fixed and known history. For example, point A, Figure 11, is the observed peak shear stress for PVC plastisol at a shear rate of 35.2 sec"'*’. The sample has previously been sheared for 10 minutes at 35.2 sec-1 i i ! I ! and allowed to rest without disturbance for 1 hour. Each i j figure indicates that the complete recovery to their ori- ! ginal peaks is quite slow if the fluid has been pre-sheared ; at a higher shear rate, but comparatively faster if the fluid has been pre-sheared at a lower shear rate. It has | i i been proved by experiments for PVC plastisol that point i i I B (Figure 11), pre-sheared for 10 minutes at a shear rate j of 11.75 sec”1, can be reached by 16 hours of resting; i point C, pre-sheared at 35.2 sec-1, can be reached by 150 i hours of resting; but, point D, pre-sheared at 105.2 j -i ! ' sec , can be reached only after 72 days of resting. j 3. Graphical Method for Determination of Constants Although the constants in the proposed mathematical K, 600 62 Shear Rate, S, sec" Figure 11, Shear Stress Recovery Plot of PVC Plastisol (Pre-sheared for 10 minutes) - 4.00 ® 200 4.0 60 8 Shear Rate, S, sec 100 Figure 12. Shear Stress Recovery Plot of PVC Plastisol (Pre-sheared for 1 hour) 63 300 200 100 O' 0 60 0 20 40 80 100 Shear Rate, S, sec Figure 13, Shear Stress Recovery Plot of Hand and Body Lotion (Pre-sheared for 10 minutes) c \ i 300 200 HR- & i°° 20 100 • —1 Shear Rate, S, sec A Figure 14 Shear Stress Recovery Plot of Hand and Body Lotion (Pre-sheared for 1 hour) 800 CM a o \ m , ® 600 s, 'd 03 03 ( 1 3 -P CO 4 . 00 c 3 200 Jl CO 0 ultimate _ ^6 HOURS 3 HOURS HOUR 20 4-0 To" 80 Shear Rate, S, sec"’’ * " 100 Figure 15- Shear Stress Recovery Plot of 8$ Wyoming Aquagel (pre-sheared for 10 minutes) 800 ULTIMATE HOURS 0 4-00 ( 0 m 1 4 +3 co 1 200 CO 10 MIN. O MIN 100 20 -1 Shear Rate, S, sec Figure 16. Shear Stress Recovery Plot of 8$ Wyoming Aquagel (pre-sheared for 1 hour) 65 model (equation 46) have been evaluated by means of statis tical analysis, a graphical analysis is enlightening and reassuring. For time-dependent fluids, PVC plastisol and hand j and body lotion, TTq = 0 in equation (46) as shown in Fig ures 8 and 9. A log-log plot of the basic shear diagram i (Figure 17), and recovery curve (Figure 18) can be made first. Both plots give parallel lines. This result indi- I cates that the functions, f(S) and g(t) in equation (15) | are separable functions. The constant, n, in equation (46)j I for shear and recovery can be obtained by taking the aver- | age slope of the graphs, respectively. i ! After the constant, n, is determined, a log-log i plot of T / ( S ) n versus (1 + t) for the case of PVC plasti- ; i ; sol (also applicable for hand and body lotion) can be ob- j ; i i ! tained as shown in Figures 19 and 20 for shear and recov- ! ' ery, respectively. The constant, m, in equation (46) for j ;shear and recovery can be calculated by taking the slopes j iof the graphs in Figures 19 and 20, respectively. The in- i ; i i itercepts give the constant, k, for equation (46). j I For a fluid with a yield value such as 8% Wyoming aquagel, the constant, m, can be first obtained from the 66 800 600 b 400 « 200 p J § 100 co 10 4.0 60 80 100 20 200 • Shear Rate, S, sec Figure 17> A Log-log Plot of the Basic Shear Diagram, PVC Plastisol at 25°C 800 % 600 w Q ) $ 400 •s i 200 a ) P co 5 3 j § 100 co i O ao ioo 200 20 • —* ] Shear Rate, S, sec Figure 18, A Log-log Plot of Recovery Curves, PVC Plastisol at 25°C (Pre-sheared for 10 minutes) 100 — -1 = 105.7 sec = 35.2 sec -1 = 11.75 sec 20 10 100 1000 10000 Figure 19. Reduced Log-log Plot for Shearing of PVC Plastisol at 25°C • (n=0.69) 100 — 10 min. 1 hr. 80 — 0 X S = 105.7 -1 sec 60 — A + S = 35.2 -1 sec □ V S = 11.75 -1 sec Pre-sheared: -fr 10 1 10 100 1,000 10,000 100,000 I + t sec o Figure 20. Reduced Log-log Plot for Recovery of PVC Plastisol at 25°C. (n=0.59) 03 69 average slope of T versus 1 + t plot* the yield value, T 0 can be obtained from extrapolating at S = Os Then, a plot X °f (i+t)m 0 versus s can be obtained, from which n can be calculated as a slope and k can be calculated as an intercept on the plot. 4. Statistical Regression Analysis for Determination of Constants The constants k, n, m, and Xo in equation (46) have been calculated by means of a digital computer, ac cording to the statistical regression analysis method pre viously described. Details are shown in Appendix E, and the results are tabulated here in Table 5. The constants, k (also T0 / in the case of 8% Wyo ming aquagel) in the recovery equation is dependent upon j i pre-sheared time, t; since the values are changing slowly, j I I linear interpolation formula can be used as an approxima- | tion: k = kQ + a t (66) (or Tq = X q + a t, for Wyoming aquagel) where k (or Tq'^ = value of K (or Xq) extrapolated at zero pre-shear t ime, i.e. at t = 0 70 a = constant, dynes, secn"m_1/cm^ t =» pre-sheared time, seconds. Table 5. Constants Determined by Regression Analysis Fluids Shear or Recovery k n m To Hand and Body Lotion Shear 99.06 0.59 -0.14 0 Recovery Pre-sheared 10 minutes 1 hour 31.31 30.35 0.22 0.22 0.10 0.10 0 0 PVC Plastisol Shear 30.88 0.69 -0.07 0 Recovery Pre-sheared 10 minutes 1 hour 27.68 24.88 0.59 0.59 0.05 0.05 0 0 Shear 0.57 1.00 -0.14 692.0 i 8% Wyoming Aquagel Recovery Pre-sheared 10 minutes 1 hour -99.48 -99.48 0.09 0.09 0.14 0.14 j 289.8 279.9 The k0 and a for equation (66) are tabulated in i Table 6. 71 Table 6. Linear Interpolation Constants for k (or T 0) In Recovery Equations ko a PVS Plastisol Hand and Body Lotion 8% Wyoming Aquagel 28.24 31.50 291.78 er') o -0.00093 -0.00032 -0.0033 5. A Comparison of Calculated versus Experimental Data The values of calculated shear stress, ~C , obtained by equation (46) with constants evaluted in the previous i section are compared graphically in Figures 21 through 29. j | Comparisons can also be made according to the method of j | I jPopovics, which has been described previously. The calcu- j l j llations can be facilitated by using computer programs as j ; j shown in Appendix F. The results are summarized in Table 7. The column of RSEF gives an average deviation be- | | tween the calculated value and experimental data. In sum- imary, the average deviations for PVC plastisol, hand and body lotion, and 8% Wyoming aquagel are in a range of ap- i proximately 3.5% to 6.7%. The column of DF/yN gives a relative measure of 900 Experimental Calculated 700 600 S = 105.7 SFr: 500 4.00 300 s= 35.2 SEC'1 200 S = 11.75 SEC'1 100 20 1000 3000 80 100 Shear Time, t, seconds 120 2000 4000 Figure 21, A Comparison of Calculated versus Experimental Values for Shearing of PVC Plastisol at 25°C. Shear Stress, T , dynes/cm' 02 300 200 100 0 20 40 * or 9 Experimental Calculated 60 J. _L 80 100 120 1000 Shear Time, t, seconds 2000 3000 4000 Figure 22. A Comparison of Calculated versus Experimental Values for Shearing of Hand and Body Lotion at 25 °C GJ Shear Stress 900 800 Experimental Calculated 700 600 500 400 300 200 100 0 20 40 60 80 100 120 1000 2000 3000 4000 Shear Time, t, seconds Figure 23. A Comparison of Calculated versus Experimental Values for Shearing of 8$ Wyoming Aquagel at S =105.7 sec“^ (25°C). 900 800 Experimental Calculated 700; cv 2 600 500 M 4.00 300 200 100 120 2000 3000 4.000 0 20 4-0 60 80 100 Shear Time, t, seconds Figure 24.. A Comparison of Calculated versus Experimental Values for Shearing of % Wyoming Aquagel at S = 35.2 sec-- * - (25°C). 1000 U1 Shear Stress, T , dynes/cm' 900 800 Experimental Calculated 700.- 600 500 4.00 300 200 100 3000 4000 120 1000 2000 100 20 Shear Time, t, seconds Figure 25. A Comparison of Calculated versus Experimental Values for Shearing of 8$ Wyoming Aquagel at S = 11.75 sec-1 (25’C). ^ O' Experimental: Calculated: Pre-sheared 1 hr 10 min Pre-sheared 10 min, Pre-sheared 1 hr. S = 105.7 sec’1 S — 35.2 sec--- S = 11.75 sec’1 800 600 S o C O © & 'd ^ \ 4-00 C Q C O 0 U -P CO £ } 200 ® si co v_ Time, hours Figure 26. Calculated vs. Experimental Values for Recovery of PVC Plastisol at 25°C -j <i Shear Stress, T , dynes/cm' Experimental: Calculated: Pre-sheared 1 hr. 10 min Pre-sheared 10 min Pre-sheared 1 hr. S = 105.7 sec"1 S = 35.2 sec-1 S = 11.75 sec"1 300 200 100 12 ; i6 is Time, hours 22 26 Figure 27. Calculated vs. Experimental Values for Recovery of Hand and Body Lotion at 25°C -o 00 800 ^ 600 C O < D & T3 ^ 400 01 0 1 © I h +3 CO a I 200 = 105.7 sec"?- = 35.2 sec“| = 11.75 sec"1 Calculated Values 18 6 8 16 10 12 0 2 4 Time, hours Figure 28. Calculated vs. Experimental Values for Recovery of 8$ Wyoming Aquagel at 25°C (Pre-sheared for 10 minutes) ^3 VO 800 600 4.00 S = 105.7 sec S = 35.2 sec“^ S = 11.75 sec 200 - Calculated Values 8 10 Time, hours. Figure 29. Calculated vs. Experimental Values for Recovery of 8$ Wyoming Aquagel at 25°C (Pre-sheared for 1 hour.) 81 Table 7. A Comparison of T(calculated) vs.% (experimental) Standard Error of Estimate Corre lation Coeff. Fit Coeff. SEF RSEF DF/yN No. of Obser vations PVC Plastisol Shear 16.6555 0.9990 0.9956 18.28 0.0567 -0.0028 216 Recovery pre sheared 10 min. 11.8354 0.9996 0.9983 11.86 0.0349 0.0058 12 1 hr. 10.4587 0.9996 0.9983 10.63 0.0349 0.0018 12 Hand and Body Lotion Shear 7.5549 0.9981 0.9871 8.44 0.0668 -0.0118 144 Recovery pre sheared 10 min. 9.3706 0.9986 0.9894 9.50 0.0564 0.0000 14 1 hr. 9.4547 0.9987 0.9899 9.47 0.0558 0.0040 15 8% Wyoming Aquagel Shear 21.9494 0.9987 0.9846 26.13 0.0634 -0.0343 108 Recovery pre sheared 10 min. 25.3503 0.9987 0.9713 27.63 0.0531 0.0017 11 i 1 hr. 26.8937 0.9983 0.9626 29.52 0.0607 0.0024 11 the one-sidedness of the fit. The negative values indicate | that the majority of the predicted values are greater than ! the observed values; and the positive values indicate that j the predicted values are less than the observed values. ; Therefore, the shear equation of 8% Wyoming aquagel pre- | j i | diets a result approximately 3.43% higher than observed. j The hand and body lotion result is approximately 1.18% higher than observed values. The one-sidedness of the 82 other cases are relatively less significant. 6. Comparison of the Proposed Mathematical Model with Others Other mathematical functions, g(t), have been se lected arbitrarily as possible alternate forms to the pro posed model. Some of the resulting equations are listed in jTable 8. Table 8. Arbitrarily Selected Equations Exponential Forms T = k(S)n eat T = k(S)n e(a+bt) - e-t T = k(S)n e W t T - To» = k(S)n eat Logarithmic Forms k(S)n In t/c T= k(§)n In t Hyperbolic Forms X = k(S)n sinh t T = k(S)n csch t/C + B X = k(S)n coth t/C + B Polynomials T = (S)n (aQ + a^t + a^t2) T = (S)n ------ i ^ a + a^t + a2t Other Forms 1 T = k(S)n a - btS. t - k(§)a * : b a T = k(§)n t™ 83 Using the experimental data for hand and body lo tion, the statistical parameters of Popovics were computed for each expression. The results are indicated in Table 9 for three forms with higher correlation coefficients. The polynomial g(t) is obviously poor in every i | other respect. The exponential g(t) does not fit the data as well as the proposed model as reflected by the fit co efficient and the relative standard error of fit. Further more, the exponential forms are considerably more one- | sided. Finally, the relative precision, defined as the j ratio of the variance of the proposed equation to the vari-j i ance of one prediction equation expressed as a percentage, | | was calculated. The theoretical equation, which has a re- j ; lative precision of 100% is considered to be a base for j ' t j comparison. The exponential forms of the equation only j give relative precisions of 4.6% and 21.5% as compared I i with the theoretical equation, and the quadratic form of i the equation only gives 0.47% relative precision. | 7. A Frequency Domain Presentation i ’ ; j Usually, the observed data are used to determine the Fourier coefficients of equation (61). However, an Table 9. Comparing Equations by Popovics' Method for Hand and Body Lotion (Number of observations: 144) Equations Constants Correlation Coefficient Fit Coefficient SEF RSEF DF/yN Relative Precision k(S)n (l+t)m k = 99.06 n = 0.22 m = -0.14 0.9981 0.9879 8.4434 0.0668 -0.0029 100.00% k(S) eat k = 61.69 n = 0.22 a = -0.00023 0.9599 0.547 39.1740 0.3244 0.0415 4.60% e-t k(S)n ea 9+t k = 51.15 n = 0.22 6 = 69.00 a = 0.47 0.9928 0.9317 18.1997 0.1522 0.0509 21.50% k(S)n (ag+a-^+a^2) a0 o 64.93 a± = -0.012 a2 = -0.000012 0.6489 0.4806 123.716 1.5449 0.3633 0.4% oo 85 an approximate result may be obtained by generating data from equation (57). A shear equation for PVC plastisol with the form -r^r = M l + t)m (67) (S)n where m = -0.07 k = 30.88 is used for generating data, where the shear time of 1 hour is divided into 1024 intervals. The Fourier coeffi- | cients, a(wn) and b(wn), as well as the amplitudes, A(wn), j are determined according to the computer program indicated i i j in Appendix D. A spectrum of the amplitude-frequency plot i | is presented in Figure 30. ! From Figure 30, no predominant frequency is ob- i served for the case of shearing of PVC plastisol. Amplitude Frequency, rad./sec. Figure 30. An Amplitude-Frequeney Plot for PVC Plastisol at 25° CHAPTER V CONCLUSIONS The shear and recovery properties of a thixotropic fluid can be measured by a stress-time experiment at a constant shear rate. The theoretical predictions of PVC plastisol, hand and body lotion, and 8% Wyoming aquagel agree very well I with the experimental results, both for shear and re covery, with an average deviation less than 6.7%. A plot of a basic shear diagram with different time i levels provides a useful way to classify the thixo tropic fluid in a conventional manner, but with a time parameter, demonstrating that f(S) and g(t) are sep- i arable. j The proposed mathematical model requires only four constants, n, k, T 0 / and m to characterize the behavi-j | or of a thixotropic fluid, under strain. The first three constants have a high degree of acceptance for characterizing certain classes of time-independent 88 fluids. Although another set of constants, n, k, T 0» and m, are required to characterize the recovery of the peak shear stress of a thixotropic fluid, the same model equation applies. There is no precedent for character izing this phenomenon. In the case of recovery, the constant k, in a time- dependent power-law equation (or T 0 in a power-law equation with a yield value) is dependent upon the previous shear history. Although at the present time, the constant is obtained by a linear model, k = kQ + j | at, it could be alternately obtained from the final j I i conditions of a shear equation with less accuracy, provided more observations are made. A complete recovery could be reached sooner for a fluid pre-sheared at a lower shear rate than a higher shear rate. The frequency domain presentation for a thixotropic fluid does not appear superior than a time domain pre sentation for the present investigation. CHAPTER VI RECOMMENDATIONS Further research is recommended in the following phases: 1. A wide survey of thixotropic fluids is recommended in order to test the generality of the present theory. 2. The effect of varying concentrations, temperatures, or other factors on the shear and recovery behavior of thixotropic fluids could be valuable. ! 3. An alternate way to perform the recovery experiment which may be suggested is to vary the shear rates for i getting the peaks after a fluid has been pre-sheared ! | at a fixed shear rate and rested for a fixed period. 4. A study could be made to evaluate the recovery constant k, directly from the final condition of the shear equa- j tion, instead of using a linear interpolation formula. ! This study could be achieved by taking more samples than the present investigation. 89 90 Rheopectic fluids could be applicable to the present time-dependent power-law equation. This effect has to be verified by experiments. NOMENCLATURE conversion factor, revolution""*" Order of building up unlinked state coefficient of thixotropic breakdown with time material constant concentrat ion material constant material constant deviation factor fit coefficient fraction of thixotropic structure unchanged fy v2 defined to be J ef dy o value of I at the apex constant for the power-law model Boltzmann constant, or constant for the for ward reaction, or equilibrium constant, or constant for the upcurve of the loop in Green and Weltmann's equation. constant for shearing operation, or proportion al constant, or specific rate of flow when there is no stress. constant for recovery, or reverse reaction proportional constant 92 forward (breakdown) reaction rate constant reverse (recovery) reaction rate constant specific rate for the reverse reaction at zero stress elastic constant dependent upon shear rate specific rate for the forward reaction at zero stress loss in shearing force per unit area per unit increase of rate of shear constant for time-dependent power-law model number of observations rotational speed, RPM constant for power-law model viscosity of complete breakdown substance shear rate dependent viscosity equilibrium constant constant relative standard error of fit shear rate, sec"1 variance of the least-squares estimated value of y about the line of equality variance of the fit variance of the yQ values about the mean y 93 standard deviation for the differences of ex perimental and theoretical values standard error of fit time of shear rest time after shearing time of recovery (or resting) absolute temperature; also means T = t + 1 plastic viscosity fraction of the area of the shear surface occupied by the ith kind of flow units experimental values j I fraction of the area occupied by the en- j tangled molecules at t = 0 | theoretical or calculated values j i i average of the differences of experimental and | calculated values | | j order of forward reaction order of reverse reaction linkage state of Noor's equation parameters in Eyring's theory of flow j maximum point viscosity i minimum point viscosity the point viscosity at any moment 94 reaction constant shear stress, dynes/cm yield value, dynes/cm^ additional shear stress above the steady value which is necessary to initiate flow 2 thixotropic shear stress, dynes/cm apparent viscosity apparent viscosity at equilibrium constant in equation (4) coefficient of thixotropy REFERENCES 1. 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L., "Time Dependent Rheology of the Tetra- hydrofuran-Hydrogen Sulphide Gas Hydrate Slurry," Canadian J. of Chem. Eng., 42, 3, 132-138 (1964). I ! ! I ! 40. Popovics, S., "A Method for Evaluating How Well Ob- | served Data Fit the Line Y x," Materials Research and Standard, _7, 5, 195-202, May 1967. ! 41. Rogers, W. F., Composition and Properties of Oil Well j Drilling Fluids, p. 247, 3rd ed., Gulf Publishing Co., Houston, Texas (1963). : 42. Sargent, E. H. & Co., Instruction Manual, Sargent j ! Laboratory Recorder, Model SR. j ; 43. Tao, F. S., "Heat Transfer from Non-Newtonian Fluids,"! j M.S. Thesis, Department of Chemical Engineering, ! j University of Utah, Salt Lake City, Utah (1964). ! 44. Thiessen, P. A., "Wechselseitige Adsorption von | Kolloiden," Z. Elektiochem., 48, 675-681 (1942). i ; I : 45. Van Olphen, An Introduction to Clay Colloid Chemis- | try. Inerscience Publishers, p. 90, 1963. I 46. Ibid. , p. 99. 47. Ibid., p. 103. • CD Ibid. , p. 137. 99 49. Van Wazer, J. R., J. W. Lyons, K. Y. Kim and R. E. Colwell, Viscosity and Flow Measurement. Interscience Publishers (1963). 50. Watson, G. N., A Treatise on the Theory of Bessel Functions, 202-207, 2nd ed., Cambridge Univ. Press (1944). 51. Weltmann, R. N., "Breakdown of Thixotropic Structure as Function of Time," J. of Applied Physics, 14, 343-350, July 1943. 52. Wilkinson, W. L., Non-Newtonian Fluids, pp. 11-12, Pergamon Press (1960). 53. Wylie, C. R., Jr., Advanced Engineering Mathematics, j p. 453, 3rd ed., McGraw-Hill Book Co. (1966). 54. Yule, G. U., and A. Kendall, An Introduction to the Theory of Statistics, pp. 343-364, 4th Impression, Charles Griffin & Co., Ltd., London (1965). I APPENDICES 100 I APPENDIX A TABULATED EXPERIMENTAL DATA 101 1. PVC Plastiso1 - Shear Data S = 105.7 sec-1 Curve F-■72 F-•66 F--59 F-■57 F-33 F-31 Time of shear sec. * S.R. - 7- dynes 2 cm'1 S.R. dynes cm^ S.R. ^djmes cm^ S.R. -^djmes cm^ S.R. ~j'j dynes cm^ S.R. *j'} dynes cm^ 0 77.65 838.6 80.50 869.4 81.00 874.8 81.05 875.3 74.05 799.7 74.78 807.6 3 67.25 726.3 70.25 758.7 69.85 754.4 68.60 740.9 63.50 685.8 62.00 669.6 6 63.60 686.9 65.05 702.5 63.50 685.8 63.25 683.1 59.85 646.4 59.20 639.4 12 59.25 639.9 60.55 653.9 58.70 634.0 59.45 642.1 58.05 626.9 56.80 613.4 30 55.15 595.6 56.25 607.5 53.80 581.0 55.10 595.1 54.25 585.9 53.40 576.7 60 52.25 564.3 53.35 576.2 51.75 558.9 52.70 569.2 52.00 561.6 51.65 557.8 120 50.15 541.6 51.05 551.3 49.70 536.8 50.70 547.6 50.25 542.7 49.80 537.8 300 47.30 510.8 48.50 523.8 47.00 507.6 48.10 519.5 48.20 520.6 47.70 515.2 600 45.50 491.4 46.30 500.0 45.50 491.4 46.00 496.8 46.55 502.7 46.25 499.5 1200 43.50 469.8 44.55 481.1 44.05 475.7 44.25 477.9 45.20 488.2 44.75 483.3 2400 41.50 448.2 42.55 459.5 42.35 457.4 42.35 457.4 43.30 467.6 43.50 469.8 3600 40.85 441.2 41.23 445.3 41.23 445.3 41.25 445.5 42.55 459.5 42.80 462.2 • f g Note: S.R. = scale reading of the automatic recorder chart. H O to PVC Plastisol § - Shear = 35.2 Data (con't.) sec-1 Curve F-■68 F--54 F--50 F-46 F-39 F-•30 Time of shear sec. S.R. -r dynes * ■ ’ 2 cm^ S.R. T)d2nes cm S.R. ^djmes cm^ S.R. T> dynes cm^ S.R. r dynes cm^ S.R. 'T dynes ’ 2 0 35.50 383.4 32.40 349.9 33.35 360.2 32.75 353.7 35.55 383.9 34.80 375.8 3 32.22 348.0 30.30 327.2 31.30 338.0 30.75 332.1 31.35 338.6 32.70 353.2 6 30.65 331.0 29.20 315.4 29.55 319.1 29.05 313.7 29.85 322.4 31.00 334.8 12 28.55 308.3 27.30 294.8 27.95 301.9 27.70 299.2 28.50 307.8 29.35 317.0 30 26.25 283.5 25.75 278.1 25.80 278.6 25.85 279.2 26.75 288.9 27.25 294.3 60 24.45 264.1 24.20 261.4 24.35 263.0 24.45 264.1 25.15 271.6 25.85 279.2 120 23.05 248.9 23.30 251.6 23.25 251.1 23.25 251.1 23.85 257.6 24.80 267.8 300 21.60 233.3 21.80 235.4 21.80 235.4 21.78 235.2 22.55 243.5 23.95 258.7 600 20.65 223.0 20.85 225.2 20.80 224.6 20.95 226.3 21.70 234.4 22.95 247.9 1200 19.55 211.1 19.80 213.8 19.80 213.8 19.80 213.8 20.90 225.7 22.05 238.1 2400 18.68 201.7 18.65 201.4 18.70 202.0 18.72 202.2 19.75 213.3 21.00 226.8 3600 18.40 198.7 18.30 197.6 18.50 199.8 18.50 199.8 19.05 205.7 20.30 219.2 103 PVC Plastisol S - Shear = 11.75 Data (con't.) sec-' * ' Curve F-•70 F-■52 F-■49 F-44 F-•38 F-•36 Time of shear sec. S.R. t dynes 2 cirr S.R. Tj dynes cm^ S.R. T. dynes cm^ S.R. 'f dynes ’ 2 cnr S.R. • £ - dynes cm^ S.R. ~t- dynes 2 cm 0 15.70 169.6 15.35 165.8 15.25 164.7 15.80 170.6 14.20 153.4 15.18 163.9 3 14.75 159.3 14.25 153.9 14.55 157.1 14.50 156.6 13.50 145.8 13.55 146.3 6 14.50 156.6 13.70 148.0 14.20 153.4 13.85 149.6 13.35 144.2 12.80 138.2 12 13.75 148.5 13.15 142.0 13.55 146.3 13.35 144.2 12.75 137.7 12.00 129.6 30 12.85 138.8 12.25 132.3 12.75 137.7 12.75 137.7 11.85 128.0 11.45 123.7 60 12.40 133.9 11.75 126.9 12.20 131.8 12.20 131.8 11.45 123.7 10.75 116.1 120 11.65 125.8 11.20 121.0 11.50 124.2 11.70 126.4 11.10 119.9 10.25 110.7 300 10.80 116.6 10.55 113.9 10.85 117.2 10.95 118.3 10.60 114.5 9.78 105.6 600 10.25 110.7 10.25 110.7 10.30 111.2 10.70 115.6 9.80 105.8 9.70 104.8 1200 9.85 106.4 9.95 107.5 9.80 105.8 10.25 110.7 9.35 101.0 9.25 99.9 2400 9.35 100.0 9.70 104.8 9.65 104.2 9.75 105.3 9.05 97.7 9.05 97.7 3600 9.20 99.4 9.35 101.0 9.35 101.0 9.60 103.7 8.50 91.8 8.55 92.3 H O PVC Plastisol - Recovery Data Presheared Time 10 minutes 1 hour Time of Recovery Shear rate -1 S, sec Shear stress dynes ^ J cm2 Shear rate • S, sec Shear stress dynes ^ ' cm2 0 105.7 497.0 105.7 449.0 10 min. 105.7 556.2 105.7 494.6 1 hr. 105.7 592.4 105.7 519.5 3 hrs. 105.7 642.6 105.7 586.4 16 hrs. 105.7 651.2 105.7 594.0 0 35.2 230.2 35.2 203.5 10 min. 35.2 293.2 35.2 256.0 1 hr. 35.2 334.8 35.2 287.8 3 hrs. 35.2 345.1 35.2 307.8 16 hrs. 35.2 360.2 35.2 323.5 0 11.75 109.8 11.75 98.2 10 min. 11.75 140.4 11.75 133.4 1 hr. 11.75 158.8 11.75 153.4 3 hrs. 11.75 167.9 11.75 159.8 16 hrs. 11.75 172.8 11.75 169.6 105 2. Hand and Body Lotion - S = 105.7 sec” Shear Data 1 Curve F-117 F--115 F-113 F-■90 Time of shear sec. CO • • _ dynes L' 2 cm* S.R. _ dynes T' 2 cm S.R. ~ dynes * • ' 2 cm* S.R. _ dynes T' 2 cm* 0 27.65 298.6 28.00 302.4 27.35 295.4 27.15 293.2 3 21.00 226.8 21.30 230.0 21.00 226.8 21.25 205.2 6 19.00 205.2 19.00 205.2 19.25 207.9 19.00 205.2 12 17.75 191.7 16.50 178.2 17.20 185.8 17.35 187.4 30 15.35 165.8 14.35 155.0 14.70 158.8 15.05 162.5 60 13.65 147.4 12.80 138.2 13.15 142.0 13.75 148.5 120 13.30 143.6 11.80 127.4 12.05 130.1 12.78 138.0 300 11.75 126.9 11.00 118.8 10.80 116.6 11.35 122.6 600 10.78 116.4 10.30 111.2 10.30 111.2 10.80 116.6 1200 10.22 110.4 9.70 104.8 9.72 105.0 10.00 108.0 2400 9.25 99.9 8.75 94.5 8.72 94.2 9.10 98.3 3600 8.40 90.7 8.25 89.1 8.48 91.6 8.40 90.7 106 Hand and Body Lotion - Shear Data (con't.) S = 35.2 sec Curve F->108 F-107 F->105 F-109 Time of shear sec. S.R. -n - dynes '•t 2 cm^ S.R. T>d*n|s cm^ S.R. T,dynes cm^ S.R. nes cnr 0 21.32 230.3 21.23 229.3 24.15 260.8 22.90 247.3 3 17.50 189.0 16.75 180.9 18.65 201.4 17.55 189.5 6 15.20 164 .2 14.80 159.8 16.20 175.0 15.60 168.5 12 13.40 144.7 12.85 138.8 14.00 151.2 13.78 148.8 30 11.50 124.2 11.30 122.0 11.80 127.4 11.95 129.1 60 10.23 110.5 10.05 108.5 10.30 111.2 10.35 111.8 120 9.45 102.1 9.15 98.8 9.55 103.1 9.65 104.2 300 8.38 90.5 8.40 90.7 8.40 90.7 8.65 93.4 600 7.85 84.8 7.85 84.8 7.95 85.9 7.98 86.2 1200 7.30 78.8 7.30 78.8 7.22 78.0 7.45 80.5 2400 6.85 74.0 6,80 73.4 6.80 73.4 6.95 75.1 3600 6.48 70.0 6.60 71.3 6.60 71.3 6.70 72.4 H O Hand and Body Lotion - Shear Data (con't.) • S = 11.75 sec” 1 Curve F-■98 F-■96 F--95 F-•93 Time of shear sec. S.R. T.diags S.R. T ,3G2SS cm^ S.R. -j- dynes cm^ S.R. r dynes cm^ 0 15.78 170.4 15.70 169.6 16.50 178.2 15.20 164.2 3 13.45 145.3 13.95 150.7 13.50 145.8 13.40 144.7 6 12.50 135.0 12.70 137.2 12.30 132.8 12.00 129.6 12 11.15 120.4 11.10 119.9 11.05 119.3 11.00 118.8 30 9.68 104.5 9.65 104.2 9.70 104.8 9.65 104.2 60 8.75 94.5 8.45 91.3 8.55 92.3 8.55 92.3 120 7.85 84.8 7.80 84.2 7.95 85.9 7.70 83.2 300 6.90 74.5 6.70 72.4 6.78 73.2 6.65 71.8 600 6.35 68.6 6.20 67.0 6.15 66.4 6.20 67.0 1200 5.78 62.4 5.75 62.1 5.80 62.6 5.75 62.1 2400 5.27 56.9 5.15 55.6 5.20 56.2 5.30 57.2 3600 5.25 56.7 5.10 55.1 4.95 53.5 5.10 55.1 108 Hand and Body Lotion - Recovery Data Pre-sheared Time 10 minutes 1 hour Time of Recovery Shear rate • —1 S, sec x Shear stress dynes ^ cm2 Shear rate * —i S, sec x Shear stress dynes L cm2 0 105.7 110.4 105.7 92.3 10 min. 105.7 190.6 105.7 154.1 1 hr. 105.7 206.8 105.7 193.9 2 hrs. - - 105.7 201.4 3 hrs. 105.7 219.8 105.7 207.4 16 hrs. - - 105.7 238.1 17 hrs. 105.7 255.4 - - 24 hrs. - - 105.7 264.6 26 hrs. 105.7 259.2 - - 0 35.2 85.1 35.2 71.6 10 min. 35.2 137.9 35.2 137.7 1 hr. 35.2 159.6 35.2 150.1 3 hrs. 35.2 166.9 35.2 174.4 16% hrs. - — 35.2 223.2 17 hrs. 35.2 203.0 - - 19 hrs. - - 35.2 207.4 21 hrs. 35.2 210.6 — — 109 Hand and Body Lotion - Recovery Data (con't.) Presheared Time 10 minutes 1 hour Time of Recovery Shear rate -1 S, sec Shear stress o- dynes L' 2 cm'* Shear rate • S, sec Shear stress _ dynes T' 2 cnr 0 11.75 67.2 11.75 55.1 10 min. 11.75 105.3 11.75 86.4 1 hr. 11.75 110.2 11.75 112.3 4 hrs. 11.75 143.4 11.75 136.9 18% hrs. 11.75 171.7 — - 24 hrs. 11.75 171.7 H1 H o 3. 8% Wyoming Aquagel - shear Data (Aged for 11 days) • S = 105.7 sec"1 Curve E-30 E-25 E-•20 Time of shear dynes cm2 dynes dynes ^ • cm2 i sec. S.R. S.R. T, „™2 • cm S..R. 0 68.00 734.4 65.85 711.2 69.25 747.9 3 56.80 613.4 51.60 557.3 60.25 650.7 6 51.80 559.4 46.75 504.9 54.55 589.1 12 46.20 499.0 42.35 457.4 49.20 531.4 30 40.70 439.6 37.55 405.5 43.20 466.6 60 37.35 403.4 34.70 374.8 39.60 427.7 120 34.65 374.2 31.80 343.4 36.30 392.0 300 31.45 339.7 28.70 310.0 32.50 351.0 600 28.80 311.0 26.50 286.2 29.85 322.4 1200 27.25 294.3 24.80 267.8 28.20 304.6 2400 26.25 283.5 23.25 251.1 26.70 288.4 3600 25.25 272.7 22.75 245.7 25.80 278.6 111 8% Wyoming Aquagel - Shear Data (con't.) S = 35.2 sec-- 1 - Curve E-28 E-24 E-18 Time of shear sec. S.R. dynes cm2 S.R. dynes ' * • > cm2 S.R. dynes ^ • cm2 0 66.00 712.8 67.95 733.9 64.00 691.2 3 53.75 580.5 55.25 596.7 51.50 556.2 6 48.50 523.8 50.50 545.4 47.35 511.4 12 42.25 456.3 44.90 484.9 42.85 462.8 30 36.20 391.0 38.00 410.4 37.35 403.4 60 32.50 351.0 33.55 362.3 33.50 361.8 120 29.30 316.4 31.70 342.4 30.45 328.9 300 26.25 283.5 27.30 294.8 26.75 288.9 600 24.00 259.2 25.65 277 .0 24.80 267.8 1200 22.50 243.0 23.90 258.1 23.30 251.6 2400 21.15 228.4 22.75 245.7 22.00 237.6 3600 20.75 224.1 22.30 240.8 21.35 230.6 H H to 8% Wyoming Aquagel - Shear Data (con't.) S = 11.75 sec" -1 Curve E-27 E--22 E-16 Time of shear sec. S.R. dynes T’ cm2 S.R. rv dvnes t ' 2 cm^ S.R. dvnes V 2 cm 0 64.30 694.4 65.25 704.7 62.60 676.1 3 52.20 563.8 52.95 571.9 49.85 538.4 6 48.90 528.1 48.10 519.5 46.00 496.8 12 43.25 467.1 43.70 472.0 41.85 452.0 30 37.50 405.0 38.70 418.0 37.15 401.2 60 33.75 364.5 35.35 381.8 33.25 359.1 120 29.75 321.3 32.20 347.8 29.85 322.4 300 25.45 274.9 26.90 290.5 25.80 278.6 600 23.30 251.6 25.00 270.0 23.65 255.4 1200 21.75 234.9 23.30 251.6 21.95 237.1 2400 20.75 224.1 22.30 240.8 20.75 224.1 3600 20.25 218.7 22.00 237.6 20.25 218.7 H H OJ • 1 8% Wyoming Aquagel - Recovery Data Time of Recovery 10 Shear rate -1 S, sec Pre minutes Shear stress _ dvnes 2 cm^ sheared Time 1 Shear rate -1 S, sec hour Shear stress dvnes T' 2 cm 0 105.7 306.5 105.7 265.7 10 min. 105.7 349.4 105.7 321.8 1 hr. 105.7 447.7 105.7 435.8 3 hrs. 105.7 541.6 105.7 470.3 16 hrs. 105.7 669.6 105.7 653.4 0 35.2 268.0 35.2 231.8 10 min. 35.2 372.6 35.2 346.1 1 hr. 35.2 467.6 35.2 437.4 3 hrs. 35.2 550.8 35.2 526.0 16 hrs. 35.2 737.6 35.2 657.2 0 11.75 259.0 11.75 225.0 10 min. 11.75 426.6 11.75 386.6 1 hr. 11.75 486.0 11.75 472.5 3 hrs. 11.75 689.0 11.75 658.8 16 hrs. 11.75 703.1 11.75 739.8 H H APPENDIX B DETERMINATION FOR THE CONSTANTS OF THEORETICAL EQUATION (22) The constants for equation (22) are determined ac cording to the method provided in this section. Equation (22) has the following form: where the constants , (5 , k^, and k^ are to be determined. Constants k2 and ( 3 can be determined from the recovery data by equation (21) which has the form: Equation (21) can be obtained by differentiating the following form of equation: determined by collecting the recovery data and performing a regression analysis on the data. (22) (21) T = k^ (t + l)r The constants, k^, and r in equation (68) can be (68) The constants, oi and k^ can be determined from a 115 116 plot of shear stress, T , versus ko (1 + t) ^ ) . * vo t/s,net on log-log paper, where the terra can be deter- \ ^ t /S,net mined by taking the slopes from the plots of shear stress, T, versus time of shear, t, as shown in Figure 4 or Fig ures 22 through 26. The following results were obtained for PVC plasti- sol, hand and body lotion, and 8% Wyoming aquagel: 1) PVC plastisol: Shear rate, S *2 r *2 ft 11.75 sec-1 120.8 0.05 6.04 0.95 35.2 t l 230.7 0.05 11.54 0.95 105.7 I I 441. 3 0.05 22.07 0.95 • s = 11.75 sec-1 Time of shear, dynes t/ 0 -(■&) k, (t+iyp-f ) t, sec. cm 't/net V at/net 0 164.7 3.4167 9.4567 3 153.2 2.6083 4.2267 6 148.3 1.5340 2.4850 12 141.4 0.7784 1.3066 30 133.0 0.5142 0.7455 60 127.4 0.0982 0.2198 120 121.3 0.0400 0.1034 300 114.4 0.0380 0.0647 600 109.8 0.0106 0.0244 1200 105.2 0.0042 0.0114 2400 101.8 0.0020 0.0057 3600 98.2 0.0010 0.0035 117 -1 S = 35.2 sec Time of shear f.ilBJSS. _ fVTj k (t+j/-(VT) t, sec. cm^ vT^/net ^ V ^^/net 0 367.8 11.8387 23.3787 3 339.5 6.6544 9.7466 6 322.7 4.5548 6.3719 12 304.8 2.3571 3.663 30 283.8 0.7273 1.1693 60 267.2 0.4366 0.6690 120 254.7 0.1267 0.2479 300 240.3 0.0664 0.1174 600 230.2 0.0249 0.0513 1200 219.4 0.0117 0.0254 2400 207.9 0.0045 0.0116 3600 203.5 0.0029 0.0077 -1 S = 105.7 sec Time of shear T dynes -(XL) k? (t+lf1 *-( X L ) t, sec. cm vdt/net \ ^ t/net 0 844.2 74.7826 96.8526 3 722.6 17.6591 23.5728 6 674.0 9.3418 12.8169 12 635.0 4.6892 6.6192 30 590.3 1.5260 2.3713 60 564.7 0.6419 1.0863 120 543.0 0.1250 0.3568 300 516.3 0.1208 0.2183 600 497.0 0.0479 0.0985 1200 479.3 0.0158 0.0420 2400 460.0 0.0115 0.0251 3600 449.8 0.0077 0.0169 A plot of shear stress, T , versus k2(l + t) ^Y^-^net is Presented in Figure 31. The constant, oi is determined to be 15.4, and k-^ is determined to be 10,001 sec * sec H«75 sec'i 100 0.001 0.01 . ________________I 0.1 1 10 k2(l + dynes/cm^ sec. . k2(l + for PVC Plastisol at 25°C 100 Figure 31. Log~i0g_ Plot Qf ^ ^ 118 119 —43 * -*X * “38 • 3.01 x 10 (at S = 105.7 sec ); 2.63 x 10 (at S = 35.2 sec"1); and 1.17 x 10"33 (at S = 11.75 sec"1). 2) Hand and body lotion: Shear rate, S *2 r * to ft 11.75 sec' -1 54.1 0.10 5.41 0.90 35.2 I I 68.9 0.10 6.89 0.90 105.7 • 1 87.8 0.10 8.78 0.90 S = 11.75 sec"1 Time of shear - { & ) k,(t+lf t, sec. cm2 V3t/net 2 VaVnet 0 170.6 20.5000 25.9100 3 146.6 6.1000 7.6546 6 133.7 3.8293 4.7685 12 119.6 1.8310 2.3693 30 104.4 0.4756 0.7216 60 92.6 0.2627 0.3965 120 84.5 0.0453 0.1175 300 73.0 0.3239 0.0641 600 67.3 0.0163 0.0334 1200 62.3 0.0038 0.0130 2400 56.5 0.0026 0.0075 3600 55.1 0.0018 0.0052 120 S = 35.2 sec"1 Time of shear T-^yneB _ /_|£\ k,(t+1) “P- t, sec. cm^ \3t/net 2 V ®r /net 0 241.9 72.5000 79.3900 3 190.2 13.5000 15.4799 6 166.9 5.8000 6.9961 12 145.9 2.1770 2.8626 30 125.7 0.8054 1.1187 60 110.5 0.3189 0.4893 120 102.1 0.0458 0.1378 300 91.3 0.0320 0.0725 600 85.4 0.0163 0.0380 1200 79.0 0.0079 0.0196 2400 74.0 0.0031 0.0093 3600 71.3 0.0029 0.0072 S = 105.7 sec"1 Time of shear r dynes k~(t+l) ^ ) t, sec. cm2 i/net W^'net 0 297.4 84.2800 93.0600 3 228.3 16.5300 19.0530 6 205.9 5.8100 7.3343 12 185.8 2.5440 3.4176 30 160.5 0.8792 1.2785 60 144.0 0.3809 0.5980 120 134.8 0.1019 0.2191 300 121.2 0.0508 0.1024 600 113.9 0.0206 0.0483 1200 107.1 0.0029 0.0178 2400 96.7 0.0025 0.0105 3600 90.5 0.0020 0.0075 -(3 A plot of shear stress, X / versus 3^2(1 + t) ^ net 1S Presente(^ - * - n Figure 32, and the constant, o(. is S = 105.7 sec"?- S = 35.2 sec"j S = 11.75 sec 1,000 S o 01 < D a >> ■d 100 •» 01 0 1 < D J h -P CO U c d ® X! CO 100 10 0.01 0.1 0.001 Figure 32. Log-log Plot of T vs. K0(l + tj^ for Hand and Body Lotion at 25°C. | _ » * rVfJet to 122 determined to be 7.93 from the graph, is determined as 3.86 x 10"18 (at S = 105.7 sec*"1) ; 1.77 x 10"17 (at S = 35.2 sec"1); and 7.44 x 10"17 (at S = 11.75 sec"1). 3) 8% Wyoming aquagel in water: Shear rate, S k2 r k2 ■ 0 11.75 sec"1 167.7 0.14 23.5 0.86 35.2 154.8 0.14 21.7 0.86 105.7 140.6 0.14 19.7 0.86 S = 11.75 sec-1 Time of shear T<t e - (4t) k2 (t+lf^- f e ) t, sec. cm^ '®’ c/net '®t'net 0 691.9 65.0000 88.5000 3 560.0 32.3000 39.4429 6 514.8 10.9430 15.3520 12 462.3 5.3979 7.9866 30 407.2 2.2844 3.5105 60 369.2 1.0330 1.7181 120 330.5 0.4884 0.8685 300 281.3 0.1549 0.3285 600 259.0 0.0603 0.0958 1200 241.2 0.0123 0.0528 2400 229.7 0.0250 0.0540 3600 225.0 0.0025 0.0230 123 S = 35.2 sec -1 Time of shear t, sec. T dynes " 2 ~ B£)net net 0 712.6 57.0000 78.0000 3 594.4 31.4000 37.9957 6 529.0 13.5320 17.6033 12 470.2 8.3840 10.7744 30 400.2 2.2614 3.3936 60 359.8 1.0868 1.7195 120 326.9 0.3256 0.6765 300 289.1 0.1438 0.3041 600 268.0 0.0559 0.1443 1200 250.9 0.0200 0.0487 2400 237.2 0.0071 0.0340 3600 231.8 0.0042 0.0232 S = 105.7 sec"1 Time of shear dynes _/ 1>T ^ k~ (t+1)"^- t, sec. cm^ \ ^ Vnet W^/net 0 731.2 50.0000 69.7000 3 623.7 28.2000 34.1878 6 556.2 18.5650 22.2611 12 498.1 6.8520 9.0220 30 437.6 2.1300 3.1578 60 402.8 0.9516 1.5259 120 370.6 0.2913 0.6098 300 333.6 0.1192 0.2647 600 306.5 0.0619 0.1422 1200 288.9 0.0226 0.0668 2400 274.3 0.0071 0.0315 3600 265.7 0.0063 0.0235 T r , versus k2(l + t) ■^rj t is presented in Figure 33, and the constant , o£ 10,000 S = 105.7 sec"?- S = 35.2 sec-} S = 11.75 sec'-1 - 100 10 100 1,000 0.01 0.1 , 2 cm sec Figure 33. Log-log Plot of 'T vs. K2 (1 + t) - ( Tf) for 8$ Wyoming Aquagel at 25°C. 'a * ' 'Net is determined as 8.29 from the mmO 0 * 3.61 x 10 (at S = 105.7 sec • 1 «,oo i sec ); and 9.00 x 10 (at S 125 graph, is determined as 1); 6.13 x 10"22 (at S = = 11.75 sec"1). APPENDIX C CALIBRATION OF ROTATIONAL SPEEDS AND CHART DRIVE SPEEDS 1. Rotational Speeds of the Rotovisco Viscometer: Calibration of the rotational speeds of the Rotovis co viscometer is performed by counting the revolutions in dicated by the watch window on the measuring head of the viscometer. A standard stop watch of the type A-8 ser. no. AF-44-2030, manufacturer's part no. 130 (Waltham), has been used for recording the time. The results are indicated as j follows: RPM Report ed Time Revolutions Difference of reported and experi mented Z *"058 . t99% tabu lated Count Calcu lated Result 10.80 1123"5 1'23"3 1123"3 1'23"2 15 15 15 15 10.78 10.80 10.80 10.81 0.02 0.00 0.00 -0.01 0.0025=Z 0.40 4.54 Accept Ho 32.40 46" 3 46 "3 46"3 46"2 25 25 25 25 32.39 32.39 32.39 32.47 0.01 0.01 0.01 -0.17 -0.035=Z -0.22 -4.54 Accept H° 97.20 1'01"8 1'01"8 1'01"6 1' 01" 7 100 100 100 100 97.09 97.09 97.39 97.24 0.11 0.11 -0.19 -0.04 -0.0025=Z -0.05 -4.54 Accept Ho 126 127 At 10.8 RPM: N 2 'Ziz - z)2 _ z2 ,.-2 0-0005 sz “ N N ^ ' 4 “ t _ (z -xpJlPI = 0.0025 x 1.73 _ obs sz 0.0001188 = 0.40 At 32.4 RPM: S2 = °-v 9- - (-0.035)2 = 0.0715 z 4 t = -0-035 x 1.73 _ -o.22 obs 0.0715 At 97.2 RPM: Sz = °‘461^ “ (-0.0025)2 = 0.009469 -0.0025 X 1.73 _ n nr- tobs --------------------0.05 0.009469 0025)2 = 0.0001188 0.0025 x 1.73 0.0109 2. Chart Drive Speeds of the Automatic Recorder: Time scale reported, t1 Time calibrated by a stop watch Difference Z = t-t' z2 ^obs. t99% Result 1 min./inch 1 inch 60.0 sec. 60.5 sec. 59.8 sec. 60.0 sec. 60.3 sec. 60.0 sec. 60.4 sec. 60.0 sec. 59.7 sec. 60.0 sec. 0.0 0.5 -0.2 0.0 0.3 0.0 0.4 0.0 -0.3 0.0 Z=0.07 0.00 0.25 0.04 0.00 0.09 0.00 0.16 0.00 0.09 0.00 Jz2=0.63 0.871 2.821 Accept Ho 5 min./inch 5 in. 300.5 sec. 301.1 sec. 299.4 sec. 299.5 sec. 300.1 sec. 299.9 sec. 300.9 sec. 300.2 sec. 299.6 sec. 298.9 sec. N| II 1 1 1 II O H O O O O O O O H O • • • « • • • • • • • O H i M O V D H H U i m H U l H" 0.25 1.21 0.36 0.25 0.01 0.01 0.81 0.04 0.16 1.21 £ z ^ = 4 . 3 1 0.045 2.821 Accept Ho 128 129 For 1 min./inch: s i - ^ z '- 2)2 = % £ - / = -ajga- - (0.07,2 = 0.0581 * < * » = = - ^ r S i i r x J 1 3 ^ = ° - 8 7 1 For 5 min./inch: Sz 2 = - (0.01) 2 = 0.4309 t . = , —?*01 x 3 = 0.045 obs J0.4309 APPENDIX D COMPUTER PROGRAM AND RESULTS FOR THE FREQUENCY DOMAIN PRESENTATION 130 The computer program for calculating the Fourier coefficients and the amplitudes are presented as follows: C PVC PLAST I SO L SHEARING EQUATION 0001 DIMENSION F (1024),S(1024) ,S INTAB(300),U(1024),T( 0002 READ 10,XK,XM 0003 READ 20,N,DELTA 0004 T IME=0. 0005 K= 0 0006 1 K=K+1 0007 IF (K-N) 2,2,3 0008 2 F (K)=XK*{(l.+TIME)**XM) 0009 TIME=TIME+DELTA 0010 GO TO 1 0011 3 YK=0. 0012 PRINT 49 0013 PRINT 50,F 0014 K=0 0015 XN =N 0016 6 YK=YK+1. 0017 K=K+1 0018 IF (YK-XN/4.) 4,4,5 0019 4 SINTAB(K)=SIN(2.*3.14159*YK/XN) 0020 GO TO 6 0021 5 N= XN 0022 CALL SINCOS(F,S,SINTAB,N) 0023 PRINT 38 00 24 WRITE (6,30)(S (L ),L=1,1024) 0025 M= 1 0026 N l=N/2 0027 T(1 ) =S(1) 0028 7 M=M + 1 131 j 0029 NN =M+512 ! 0030 IF (M-N1 ) 9,9,8 1 0031 9 T(M )= SQRT( S ( M )*S(M) + S (NN)*S(NN)) ! 0032 GO TO 7 1 0033 8 PRINT 70 | 0034 WRITE (6,30)(TIM),M=1,512) 0035 70 FORMA 1 (1H , 9HAMPLITUTE) ! 0036 DO 60 1=1,1024 0037 S(I)=S(I)/1024• 0038 60 CONTINUE ■ 0039 CALL SWING!S»N) 0040 PRINT 25 ! 0041 PRINT 30,S 0042 CALL SINCOStS,U,SINTAB,N) 0043 CALL SWING(U»N) 0044 PRINT 39 0045 PRINT 40,U 0046 10 FORMAT (2F10.2) 0047 20 FORMAT (110,F10.2) 0048 25 FORMAT (1H0,20HSWINGED COEFFICIENTS) 0049 30 FORMAT (8F10.4) 0050 38 FORMAT (1H0,33HC0EFFICIENTS OF FOURIER TRANSFORM) 0051 39 FORMAT (1H0,30HINVERSI0N OF FOURIER TRANSFORM) 0052 40 FORMAT (8F10.2) 0053 49 FORMAT (1H0,49HDATA GENERATED BY THE MODIFIED POWER LAW EQUATION) 0054 50 FORMAT (8F10.2) ' 0055 STOP ! 0056 END 132 133 Subroutine SINCOS (F,S,SINTAB,N): Sincos is a Fortran callable subroutine which com putes the sine cosine series for a tabulated function on 2N points. F is the input array in real, floating point form; S is the output array as follows: N S (I) = 2 F (I) 1=1 For k = 2,....... N/2+1: N S(k) = 2 F(I) cos (27T-I* (k-l)/N) 1=1 For k = N/2 + 2,....... ,N: N S(k) = 2 F(I) sin (2 T •I(k-N/2-l)/N) 1=1 SINTAB is a table of sines which must be generated by the calling program and supplied to the subroutine for k = 1,...N/4-1. SINTAB (k) = SIN (2 7T • k/N) where N is the number of points in the F array and must be a power of 2. 0001 SUBROUTINE SWING(F,N 0002 DIMENSION F (2) 0003 ND2=N/2 0004 ND2P2=ND2+2 0005 ID=ND2*3+2 0006 IM AX=ID/2 0007 DO 100 1=2,ND2 0008 11 = I+ND2 0009 A =F(I)-F( I I) 0010 F(I)=F(I)+F( 11) 0011 100 F (I I ) = A 0012 DO 101 I=ND2P2»IMAX 0013 I 1=ID-1 0014 A= F ( I ) 0015 F (I)=F( I I ) 0016 101 F ( 11)=A 0017 RETURN 0018 END THE AMPLITUDE OF THE NTH HARMONIC 9171.8984 141.6943 83.0940 60.3163 48.2619 40.4812 35.0671 31 .0654 27.9801 25.5227 23.5179 21.8496. 623.6880 129.5302 79.2612 58.6070 47.1065 39.7009 34.5024 30.6368 27.6426 25.2504 23.2930 21.6603 391.0125 119.4941 75.8045 56.8285 46.0116 38.9545 33.9582 30 .2219 27.3147 24.9844 23.0730 21.4752 293.3035 111.0573 72.6698 55.1669 44.9727 38 .2390 33.4337 29.8194 26.9960 24.7252 22.8577 21 .2937 237.9373 103.8541 69.8128 53.6113 43.9854 37.5529 32.9272 29.4295 26.6856 24.4725 22.6474 21.1157 = 1,2,.. ..,512 201.7780 176.0893 156.7882 97.6246 92.1780 87.3711 67.1965 64.7913 62.5716 52.1515 50.7784 49.4840 43.0460 42.1509 41.2968 36.8940 36.2612 35.6524 32.4380 31.9651 31.5079 29.0508 28.6833 28.3264 26.3833 26.0890 25.8023 24.2252 23.9841 23.7484 22.4415 22.2402 22.0428 20.9414 20.7706 20,6028 134 20.4391 20.2773 19.2292 19.0900 18.1813 18.0596 17.2639 17.1571 16.4546 16.3599 15.7354 15.6512 15.0920 15.0166 14.5139 14.4460 13.9913 13.9297 13.5169 13.4610 13.0852 13.0336 12.6899 12.6426 12.3274 12.2842 11.9939 11.9540 11.6866 11.6495 11.4022 11 .3684 11.1392 11.1076 10.8947 10.8655 10.6680 10.6406 10.4568 10.4312 10.2587 10.2364 10.0763 10.0543 9.9055 9.8844 9.7447 9.7254 9.5951 9.5769 9.4551 9.4380 9.3248 9.3082 9.2012 9.1867 9.0872 9.0729 8.9792 8.9662 8.8784 8.8671 8.7854 8.7741 8.6974 8.6873 8.6163 8 .6065 20.1192 19.9639 18.9534 18 .8188 17.9402 17.8231 17.0522 16.9487 16.2669 16.1753 15 .5683 15.4859 14.9419 14.8685 14.3785 14.3120 13.8685 13.8082 13 .4055 13.3504 12.9827 12.9326 12 .5962 12 .5502 12.2412 12.1991 11.9145 11 .8755 11.6134 11.5774 11 .3346 11 .3010 11.0761 11.0454 10.8365 10.8079 10.6138 10.5868 10.4062 10 .3814 10.2130 10.1895 10 .0323 10 .0106 9.8637 9.8434 9 .7064 9.6874 9.5592 9.5415 9.4214 9.4048 9.2924 9.2769 9.1718 9.1575 9.0591 9.0455 8 .9538 8.9409 8.8550 8.8432 8 .7629 8.7517 8.6769 8 . 6664 8 .5968 8.5872 19.8113 19.6617 18.6870 18.5573 17.7076 17.5940 16.8470 16.7467 16.0847 15.9954 15.4050 15.3252 14.7957 14.7239 14.2462 14.1815 13.7488 13.6897 13.2961 13.2425 12.8832 12.8339 12.5047 12.4596 12.1572 12.1158 11.8372 11.7986 11.5416 11.5063 11.2681 11.2355 11.0148 10.9844 10.7794 10.7510 10.5603 10.5343 10 .3566 10.3320 10.1663 10.1437 9.9893 9.9677 9.8235 9.8035 9.6686 9.6501 9.5240 9.5064 9.3883 9.3720 9.2615 9.2464 9.1430 9. 1288 9.0321 9.0189 8.9283 8.9159 8.8311 8.8198 8.7408 8.7300 8.6564 8.6462 8.5777 8.5680 19.5147 19.3707 18.4298 18.3044 17.4819 17.3721 16.6479 16.5505 15.9075 15.8210 15.2464 15.1689 14.6532 14.5832 14.1172 14.0539 13.6314 13.5739 13.1894 13.1367 12.7853 12.7373 12.4150 12.3707 12.0747 12.0342 11.7609 11.72 34 11.4713 11.4366 11.2030 11.1707 10.9544 10.9244 10.7231 10.6954 10.5082 10.4822 10.3077 10.2836 10.1209 . 10.0987 9.9465 9.9255 9.7838 9.7641 9.6314 9.6133 9.4892 9.4719 9.3558 9.3398 9.2312 9.2162 9.1146 9.1005 9.0054 8.9925 8.9035 8.8913 8.8081 8.7966 8.7192 8.7084 8.6361 8.6262 8.5588 8.5495 8.5404 8.5311 8.4698 8.4613 8.4042 8.3963 8.3436 8.3366 8.2883 8.2812 8.2367 8.2306 8.1898 8. 1842 8.1470 8.1419 8.1087 8. 1039 8.0739 8 .0698 8.0432 8. 0397 8.0164 8.0133 7.9934 7.9905 7.9737 7.9716 7.9576 7.9562 7.9457 7.9443 7.9367 7.9358 7.9319 7.9311 8.5220 8.5132 8 .4529 8.4444 8.3890 8.3810 8 .3294 8.3224 8.2746 8.2681 8.2245 8 .2186 8.1785 8.1731 8 .1368 8.1322 8.0995 8.0950 8 .0659 8.0620 8.0360 8.0327 8 .0104 8.0074 7.9881 7.9854 7 .9695 7.9675 7.9546 7.9528 7 .9431 7.9420 7.9352 7.9345 7.9308 7.9305 8.5043 8.4955 8.4363 8.4281 8.3733 8.3658 8.3152 8.3085 8.2616 8.2556 8.2127 8.2069 8.1679 8.1626 8.1273 8. 1225 8.0906 8.0866 8.0581 8.0543 8.0292 8.0261 8.0044 8.0013 7.9831 7.9807 7.9654 7.9633 7.9512 7.9497 7.9407 7.9398 7.9338 7.9331 7.9305 7.9302 4869 8.4783 4201 8.4124 3586 8.3511 3015 8.2949 2490 8.242 8 2012 8.1954 1572 8.1522 1177 8.1132 0821 8.0781 0505 8.0466 0226 8.0195 9985 7.9959 9782 7.9762 9615 7.9598 9483 7.9469 9387 7.9377 9326 7.9320 9302 7.9301 H 00 cn 8 8 8 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 APPENDIX E COMPUTER PROGRAM FOR REGRESSION ANALYSIS 137 1. Time-dependent Power-law Model without Yield Value: X = K(S)n (1 + t)m Let YY(K) = shear stress; XXI(K) = time; XX(K) = shear rate C PVC PLASTISDL (PK 5851LV) SHEARING EQUATION 0001 DOUBLE PRECISION YY(500),XX1(500) ,XX2(500),V (500),XI(500),X2(500) 0002 DOUBLE PRECISION XL(500) 0003 K= 0 0004 PRINT 14 0005 PRINT 15 0006 1 K =K+1 0007 READ 10,YY(K),XX1(K),XX2(K) 0008 IF (YY(K ) ) 5,5,1 0009 5 N= 0 0010 K=0 0011 Z=0. 0012 SUMY=0 • 0013 SOY=0• 0014 SUMX1=0. 0015 S0X1=0. 0016 SUM X2 = 0 . 0017 SO X2 = 0 . 0018 XXI2=0 . 0019 YX1 = 0 • 0020 Y X2=0• 0021 2 K=K+1 0022 IF (YY(K ) ) 3,3,22 0023 22 Y(K)=DL0G(YY(K )) 0024 XL(K)=1.+XX1(K) 0025 XI(K)=DL0G(XL(K )) 00 26_________________X 2 ( K ) = DLQG ( XX2 ( K ) ) . 138 j 002 7 PRINT 30,YY(K),Y(K),XX1(K),X1(K),XX2(K ),X2(K ) ! 0028 GO TO 2 | 0029 3 N=N+1 ! 0030 IF ( YY (N.) ) 4,4,6 I 0031 6 Z=Z+1. ! 0032 SUMY=SU'MY + Y ( N ) 0033 SUMX1=SUMX1+X1(N) 0034 SUMX2=SUMX2+X2(N) 0035 SQY=SQY+Y(N)*Y(N> 0036 S0X1=SQX1+X1(N)*X1(N) 0037 S0X2=SQX2+X2<N)*X2(N) 0038 XX12=XX12+X1(N)*X2(N) j 0039 YX1=YX1+Y(N)*X1(N) 0040 Y X2=YX2+Y(N)*X2(N ) 0.041 GO TO 3 0042 4 PRINT 40»SUMY,SUMX1,SUMX2 0043 PRINT 41,S0Y,S0X1,S0X2,YX1 0044 PRINT 42,YX2,XX12,Z 0045 A=YXl*XX12-YX2*SQXl-(SUMY*SQX1-SUMX1*YX1)/(SUMX2*SQX1-SUMX1*XX12)* 1(XX12*XX12-SQX1*SQX2) 0046 B=SUMX1*XX12-SUMX2*SQXI- (Z*SQXl-SUMXl*SUMXl)/( SUMX2*SQX1-SUMX1*XX1 12)*(XX12*XX12-SQX1*SQX2) i 0047 AO=A/B 0048 A2=(SUMY*S0X1-SUMX1*YX1-A0*(Z*SQX1-SUMX1*SUMX1))/(SUMX2*SQX1-SUMX1 I 1*XX12) ; 0049 A1=(SUMY-A0*Z-A2*SUMX2)/SUMX1 0050 PRINT 50,AO,A1,A2 0051 XK=2.71828**AO j 00 52 PRINT 100,XK 0053 U=SQY—A0*SIJMY-A1*YX1-A2*YX2 0054 PRINT 80,U 139 0055 10 FORMAT (2F10.1,F10 . 2 ) 0056 14 FORMAT (64H0Y = L0G(SHEAR STRESS ) X (1) = LOG(1+AT ) X(2)=L0G(SH 141EAR RATE)) 0057 15 FORMAT (60H0SHEAR STRESS Y TIME T X (1) SHEAR RATE X 151(2) ) 0058 30 FORMAT (F10.1,F10.4,F10.1,F10.4,F10.2,F10.4) 0059 40 FORMAT (1H0,10HSUM OF Y =,F10.4,18H SUM OF X (1) =,F10.4,18H 401 SUM OF X (2) =,F10.4) 0060 41 FORMAT (1H ,5HSQY = ,F10.4,UH SQX1 =,F10.4,11H SQX2 =,F10. 4114 * 19H SUM OF YX(1) =,F10.4) 0061 42 FORMAT (1H ,14HSUM OF YX(2 ) =,F10.4,22H SUM OF X (1 )X (2) = *F10 421.4,21H NO. OF SAMPLES =,F10.4) 0062 50 FORMAT (1H ,3HY =,F10.5,2H +,F10.5,7H X (1) +,F10.5,5H X (2)) 0063 60 FORMAT (2F10.2) 0064 80 FORMAT (1H0,9HRESI DUE =,F10.5) 0065 100 FORMAT (5H0K = ,F10.4) 0066 11 STOP 0067 END 2. Time-dependent Bingham Plastic Model: T = [T0 +/( S J (1 + t)m 0001 DOUBLE PRECISION YY(300),XXI(300),XX2(300),Y (300) ,X1(300 ) ,X2(300) 0002 DOUBLE PRECISION XL(300),XM(300)*R(300) 0003 K =0 0004 READ 60*AA* AINCRE 0005 X=10.*AINCRE+AA 0006 1 K=K+1 0007 READ 10*YY(K)*XX1(K)*XX2(K) 0008 IF (YY(K )) 5,5,1 0009 5 AA=AA+AINCRE 0010 ..... .. N= 0 140 0011 K=0 0012 Z=0. 0013 SUM Y =0. 0014 SO Y=0 • 0015 SUMX1 = 0. 0016 S0X1 =0. 0017 SUMX2 = 0. 0018 S0X2 =0. 0019 X X12 =0. 0020 YX 1 = 0 . 0021 Y X2 = 0 • 0022 2 K=K+1 0023 IF ( YY(K )) 3,3,22 0024 22 Y ( K ) =OLOG(YY(K )) 0025 X L ( K )=1.+XX1(K ) 0026 XI (K) =DLOG(XL(K )) 0027 XM(K) = 692 .0+AA-XX2 (K) 0028 X2 (K )=DLOG(XM(K)) 0029 GO TO 2 0030 3 N=N+1 0031 IF (YY(N ) ) 4,4,6 0032 6 Z=Z + 1• 0033 SUMY =SUMY+YIN) 0034 SUMX1 =SUMX1+X1(N ) 0035 SUMX2=SUMX2+X2(N ) 0036 SOY=SQY+Y(N)*Y(N) 0037 S0X1 =SQX1+X1(N)*X1 (N) 0038 SO X2 = SQX2+X2(N)*X2 (N) 0039 X XI2 =XX12+X1(N )*X2 (N) 0040 YX 1 = YXl+Y(N )*X1(N ) 0041 YX2=YX2+Y(N)*X2(N) 0042 GO TO 3 0043 4 PRINT 40,SUMY,SUMX1,SUMX2 0044 PR INT 41,SOY,S0X1,S0X2,YX1 0045 PRINT 42,YX2,XX12,Z 0046 A1=(YX1-XX12)/SQX1 0047 PRINT 200,A1 0048 K= 0 0049 U =0 . 0050 7 K=K+1 0051 IF (YY(K )) 9,9,8 0052 8 R(K)=Y(K)-A1*X1(K)-X2(K) 0053 U=U+R(K)*R(K) 0054 GO TO 7 0055 9 PRINT 80,AA,U 0056 IF (X-AA) 11,11,5 0057 10 FORMAT (2F10.1,F10.2) 0058 14 FORMAT (64H0Y=L0G(SHEAR STRESS) X (1)=LOG(1+AT) X(2)=LGG(SH 141EAR RATE)) 0059 15 FORMAT (60H0SHEAR STRESS Y TIME T X(1) SHEAR RATE X 151(2) ) 0060 30 FORMAT <F10.1,F10.4,F10.1,F10.4,F10.2,F10.4) 0061 40 FORMAT (1 HO,10HSUM OF Y =,F10.4,18H SUM OF X 11) =,F10.4,18H 401 SUM OF X (2) =,F10.4) 0062 41 FORMAT (1H ,5HS0Y =,F10.4,11H SQX1 =,F10.4,11H SQX2 =,F10. 4114,19H SUM OF YX(1) =,F10 .4) 0063 42 FORMAT (1H ,14HSUM OF YX(2 ) =,F10.4,22H SUM OF X (1)X ( 2 ) = , F10 421 .4,21H NO. OF SAMPLES =,F10.4) 0064 50 FORMAT (1H , 3HY =,F10.5,2H +,F10.5,7H X (1) +,F10.5, 5H X (2 ) ) 0065 60 FORMAT (2F10.2) 0066 80 FORMAT (1H , 18HVALUE OF A ASSUMED,F10.2,14H RESIDUE =,F10.5) 0067 200 FORMAT (5H0A1 =,F10.4) 0068 1 1 STOP 0069 END 142 3. Time=dependent Power-law Fluid with a Yield Value: 1 = [T0 + K(S)n ) (1 + t)m First, obtain the constant, m, according to the procedure described in Appendix E-l; also, obtain the yield value, 0 . by a basic shear diagram. Then, apply the fol lowing program to obtain the constants, n, and k: 0001 DOUBLE PRECISION YY ( 300 ) ,XX1(300),XX2(300),Y(300) ,X1(300 ) ,X2(300) 0002 DOUBLE PRECISION XM(300),R(300),YYY(300) 0003 K=0 0004 READ 60,AA, AINCRE 0005 X=10.*AINCRE+AA 0006 1 K=K+1 0007 READ 10fYY(K),XX1(K),XX2(K) 0008 IF (YY(K)J 5,5,1 0009 5 X =X-AINCRE 0010 AA=X 0011 N=0 0012 K=0 ! 0013 Z=0. 0014 SUMY = 0. 0015 S0Y=0. 0016 SU MX 1 =0 * 0017 S0X1=0. 0018 SUMX2=0. 0019 S 0X2=0 * I 0020 XX12 = 0 . j 0021 Y XI = 0. £ i ,_002 2 YX 2=0 • w 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054 2 K=K+1 IF (YY(K ) ) 3,3,22 22 X1(K) = DL0G(XX2(K) ) XM(K)=1.+XX1(K) X2(K)=DL0G(XM(K) ) YYY(K)=(AA*((XM(K ))**0.143)-YY(K )) Y (K ) =L)L0G ( YYY ( K ) ) GO TO 2 3 N=N + 1 IF (YY(N )) 4,4,6 6 Z=Z+1. SU MY = SUMY+Y(N) SUMX1=SUMX1+X1(N) SUMX2=SUMX2+X2(N) SOY=SQY+Y(N)*Y(N) S0X1=SQX1+X1(N)*X1(N) S0X2=SQX2+X2(N)*X2(N> XX12=XX12+X1(N)*X2(N) YX1=YX1+Y(N)*X1(N) YX 2=YX2+Y(N )*X2(N ) GO TO 3 4 PRINT 40»SUMY,SUMX1,SUMX2 PRINT 41,SOY,SQX1,S0X2,YX1 PRINT 42,YX2,XX12»Z A 1=((YX1-0.143*XX12)/SUMXl+(0.143*SUMX2-SUMY)/Z)/(SQX1/SUMX1-SUMX1 1/Z ) A0=(SUMY-Al^SUMX1-0 .143*SUMX2)/Z PRINT 50 , AO , A1 U =0 . K=0 7 K=K+1 IF ( YY(K )) 9,9,8 8 XK=2.71828**A0 144 0055 R(K)=YY(K)-(AA-XK=MXX2(K)**A1) ) * ( XM ( K ) **0 . 14-3 ) . 0056 U=U+R(K)*R(K) ! 0057 GO TO 7 j 0058 9 PRINT 80,AA,U i 0059 IF (AA) 12,12,5 i 0060 10 FORMAT (2F10.1,F10.2) ! 0061 14 FORMAT (64H0Y = L0G(SHEAR STRESS) X (1)=LOG(1+AT ) X(2)=L0G(SH ! 141EAR RATE)) ! 0062 15 FORMAT (60H0SHEAR STRESS Y TIME T X(1) SHEAR RATE X 1 151(2) ) : 0063 30 FORMAT (F10.1,F10.4,F10.1,F10.4,F10.2,F10.4) { 0064 40 FORMAT (11H0SUM OF Y =,F10.4,21H SUM OF X (1) =,F10.4,18H ! 401 SUM OF X (2) =,F10 .4) I 0065 41 FORMAT (1H ,5HSQY =,F10.4,11H SQX1 =,F10.4,11H SQX2 =,F10. ! 4114,19H SUM OF YX(1) =,F10.4) ! 0066 42 FORMAT (1H ,14HSUM OF YX(2) =,F10.4,22H SUM OF X (1)X (2) =,F10 j 421.4,21H NO. OF SAMPLES =,F10.4) ■ 0067 50 FORMAT (1H ,3HY =,F10.5,2H +,F10.5,18H X (1) + 0.143 X {2)) i 0068 60 FORMAT (2F10.2) j 0069 80 FORMAT (19H VALUE OF A ASSUMED,F10.5,14H RESIDUE =,F10.1) \ 0070 12 STOP i 0071 END ! 145 APPENDIX F COMPUTER PROGRAM FOR COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL VALUES 146 1. Time-dependent Power-law Model without Yield Value: T = K(S)n (1 + t)m ' 0001 DIMENSION Y0(500 ) ,XI(500),X2(500),X0(500) i 0002 Z=0. 0003 M=0 0004 N= 0 0005 SUMY0=0. 0006 SUMX0=0. 0007 S QY0 = U • 0008 S0X0=0. 0009 Y0X0=0 • 0010 READ 20,XK,A1,A2 0011 PRINT 13 0012 PRINT 14 0013 PRINT 15 0014 1 M=M+1 0015 READ 10 » XO (M ) , X1 ( M ) , X2 ( M ) 0016 IF (X 0(M ) ) 5,5,1 0017 5 N=N+1 0018 IF (XO(N )) 4,4,6 0019 6 Z=Z+1.0000 0020 YO(N)=XK*(X2(N)**A2)*((1.0000+X1(N))**A1) 0021 SUMY0=SUMY0+Y0(N ) 0022 SlJMX0 = SUMX0+X0(N) 0023 S0X0=SQX0+X0(N)*X0(N) 0024 S0Y0=SQY0+Y0(N)*Y0(N) 0025 XOYO=XOYO+XO(N)-YO(N) 0026 PRINT 21,YO(N),X0(N),XI(N),X2(N ) 0027 GO TO 5 0028 4 VARY0=SQY0/Z-SUMY0*SUMY0/(Z*Z) 0029 VARX0=SQX0/Z-SUMX0*SUMX0/(Z*Z) 003 0_________________B= ( Z*X0YQ—SUMXO^SUMYO ) / ( Z-SQXO-SUMXO-SUMXO ) 147 0031 A=SUMY0/Z-SUMX0*B/Z 0032 SOR=XOYO*XOYO/(SQX0*SQY0) 0033 R=SQRT(SQR ) 0034 AAA=(SQY0-A*SUMY0-B*X0Y0)/Z 0035 S YX = SURT(AAA ) 0036 PRINT 24,SUMY0,SUMX0,SQY0,SQX0 0037 S L=((1.00-B)-(1.00 —B)*SQX0-2.0 0*A*(1.00-B)*SUMX0+Z*A*A)/Z 003 8 SQSPX=(SQY0+SQX0-2.0000*X0Y0)/Z 0039 F =SQRT(1.0000-SQSPX/(SL+VARYO) ) 0040 DF=SUMXO-SUMYO 0041 SPX=SQRT(SOSPX) 0042 RSEF=SPX*Z/SUMYO 0043 D FN = DF/Z 0044 PRINT 27» XOYO 0045 PRINT 25,VARX0,VARY0,SYX 0046 PRINT 26,A,B 0047 PRINT 30,R,F 0048 PRINT 40,SPX,RSEF,DF,DFN 0049 10 FORMAT ( 2F 10 . 1, F10 .2 ) ' 0050 13 FORMAT (31H0 Y(0) X (0) ) 0051 14 FORMAT (54H SHEAR STRESS SHEAR STRESS TIME T SHEAR RATE) 0052 15 FORMAT (33H CALCULATED OBSERVED ) 0053 21 FORMAT (F10.1,6H ,F10.1,6H ,F10.1,3H ,F10.2) . 0054 20 FORMAT (3F10.2) JD055 24 FORMAT (13H SUM OF Y ( 0 ) = , F10 .1 ,18H SUM OF X ( 0 ) =,F10.1,13H 241 SOY(0) =,F10.0,13H SQX(O) =,F10.0) ... 0056 26 FORMAT (7H Y(0) =,F10.4,2H +,F10.4,5H X (0) ) 0057 27 FORMAT (18H SUM OF X(0)Y(0) =,F10.0) 0058 25 FORMAT (11H VAR X(0) =,F10.3,15H VAR Y (0) =,F10.3,33H STAN 251DARD ERROR OF ESTIMATE =,F10.4) 0059 30 FORMAT (26H CORRELATION COEFFICIENT =,F10.4,22H FIT COEFFICIEN 301T =,F10.4) 0060 40 FORMAT (6H SEF =,F10.4,11H RSEF =,F10.4,9H DF =,F10.4,11H 401 DF/N =,F10.4) 0061 END 148 2. Time-dependent Bingham Plastic Model: T = + /* S ] (1 + t)m 0001 DIMENSION Y0(500),XI(500),X2(500),X0(500) 0002 Z=0. 0003 M =0 0004 N=0 0005 SUMY0=0 • 0006 SUMX0=0. 0007 SQY0=0 * 0008 S0X0=0. 0009 Y 0X0=0 > 0010 READ 20,XK,A1 0011 PRINT 13 0012 PRINT 14 0013 PRINT 15 0014 1 M=M+1 0015 READ 10,X0(M),X1(M),X2(M) 0016 IF (XO(M )) 5,5,1 0017 5 N=N + 1 0018 IF (X0(N)) 4,4,6 0019 6 Z=Z+1.0000 0020 YO (N ) = (692.00+XK*X2(N) )*( ( 1 .0000 + X1 (N) )**A1 0021 SUMY0=SUMY0+Y0(N) 0022 SUMX0=SUMX0+X0(N ) 0023 S0X0=SQX0+X0(N)-XO(N) 0024 S0Y0=SQY0+Y0(N)*Y0(N) 0025 X0Y0=X0Y0+X0(N)*Y0(N) 0026 PRINT 21,YO(N),X0(N),XI(N),X2(N) 0027 GO TO 5 0028 4 VARY0=SQY0/Z-SUMY0*SUMY0/(Z*Z) 0029 VARXO=SQXO/Z—SUMXO^SUMXO/(Z*Z) 0030 B=(Z*X0Y0-SUMX0*SUMY0)/(Z*SQXO-SUMXO*SUMXO) 0031 A=SUMYO/Z-SUMXO*B/Z 0032 SQR=XOYO*XOYO/(SQXO*SQYO) 0033 R=SQRT(SOR) | 0034 AAA=(SQY0-A*SUMY0-B*X0Y0)/Z ! 0035 SYX=SQRT(AAA) I 0036 PRINT 24,SUMY0,SUMX0,SQY0rSQX0 ! 0037 . . . SL=((1.00-B)*(1.00-B)*SQX0-2.00*A*(1.00-B)*SUMX0+Z*A*A)/Z j 0038 SQSPX=(SQY0+SUX0-2.0000*X0Y0)/Z I 0039 F=SQRT(1 .OOOO-SQSPX/(SL+VARYO)) I 0040 DF=SUMXO—SUMYO j 0041 S PX=SQRT(SOSPX) | 0042 RSEF=SPX*Z/SUMYO 0043 DFN=DF/Z 0044 PRINT 27 ? XO YO 0045 PRINT 25tVARXO,VARY0,SYX 0046 PRINT 26» A» B 0047 PRINT 30 * R »F ' 0048 PRINT 40,SPX,RSEF,DF,DFN j 0049 10 FORMAT ( 2F 10 .1, F10 .2 ) 0050 13 FORMAT (31H0 Y (0) X (0) ) 0051 14 FORMAT (54H SHEAR STRESS SHEAR STRESS TIME T SHEAR RATE) : 0052 15 FORMAT (33H CALCULATED OBSERVED ) | 0053 21 FORMAT (F10.1,6H ,F1Q.1,6H ,F10.1,3H ,F10.2) i 0054 20 FORMAT (2F10.2) i 0055_ 24 FORMAT^ (13H SUM OF Y ( 0 ) = , F10 .1,18H SUM OF X(0) =,F10.1»13H 241 SQY(O) =,F10.0,13H SQX(O) =,F10.0) 0056 26 FORMAT (7H Y(0) =,F10.4,2H +,F10.4,5H X (0)) ; 0057 27 FORMAT (18H SUM OF X(0) Y (0) =,F10.0) j 0058 25 FORMAT (11H VAR X(0) =>F10.3,15H VAR Y (0) =,F10.3,33H STAN ! 251DARD ERROR OF ESTIMATE =,F10.4) j 0059 30 FORMAT (26H CORRELATION COEFFICIENT =,F10.4,22H FIT COEFFICIEN i 150 0060 0061 3. Time-dependent Power-law Model with a Yield Value: T = [ X 0 + K(S)n ) (1 + t)m c 8% WYUMING AQUAGEL RECOVERY EQUATION PRE-SHEARED FOR 1 HR. 0001 DOUBLE PRECISION YO(500),XI(500),X2{500),XO(500),XX(500),XL(500) 0002 ss=o. 0003 R R=0 • 0004 Z=0. 0005 SYX=0. 0006 sx=o. 0007 K =0 0008 L= 0 0009 M =0 0010 N= 0 0011 SUMY0=0. 0012 SUMX0=0. 0013 S QY 0=0. 0014 soxo=o. 0015 Y 0X0=0 . 0016 READ 100,XM,XN,XK 0017 XK=2.71828--XK 0018 i M = M+1 0019 READ 10,X0(M),X1(M),X2(M) 0020 IF (XO(M )) 103rl03»l 0021 103 Z=Z+1.0000 0022 L= L+1 30 IT =»F10 .4) _ a. i i u 40 FORMAT (6H SEF =,F10.4,11H RSEF =,F10.4,9H DF -,F10.4,11H 401 DF/N = * F10.4) END 151 0023 IF (XO(L )) 102,102,7 0024 7 XL(L)=1.+X1(L) 0025 R R=RR+XO(L )/(XL(L)**XM) 0026 SS=SS+X2(L)**XN 0027 GO TO 103 0028 102 Z=Z-1.0000 0029 T=(RR+XK*SS)/Z 0030 PRINT 500,T 0031 PRINT 13 0032 PRINT 14 0033 PRINT 15 0034 5 N=N+1 0035 IF (XO(N )) 4,4,8 0036 8 YO(N)=(T-XK*(X2(N)**XN))*(XL(N)**XM) 0037 SUMYO=SUMYO+YO(N) 0038 SUMXO=SUMXO+XO(N ) 0039 SOXO=SQXO+XO(N)*XO(N) 0040 SOYO=SQYO+YO(N)*YO(N) 0041 XOYO=XOYO+XO(N)*YO(N) 0042 PRINT 21,YO(N),X0(N),XI(N)»X2(N) 0043 GO TO 5 0044 4 VARYO=SQYO/Z-SUMYO*SUMYO/(Z*Z) 0045 VARXO=SQXO/Z-SUMXO-SUMXO/(Z*Z) 0046 B=(Z^XOYO-SUMXO^SUMYO)/(Z*SQXO-SUMXO*SUMXO) 0047 A=SUMYO/Z-SUMXO*B/Z 0048 SOR=XOYO*XOYO/(SQXO-SQYO) 0049 R =SQRT(SOR} 0050 AAA= { .^OYO-A-TL• • I MY0-B*XOYO }/Z 0051 5YX = i.0f (mAA) 005? SL-"( Q.OO-B)*( 1.00-B)*SQX0-2.00*A*( 1.00-B)*SUMXO+Z*A*A)/Z 0655 SOSPX=(SQY0+SQX0-2.0000*X0Y0)/Z 0054 F= SOR‘ f i 1.OOOO-SQSPX/ ( SL + VARYO ) ) H Ul to 0055 DF=SUMXO—SUMYO 0056 SPX=SQRT(SOSPX) 0057 RSEF=SPX*Z/SUMYO 0058 DFN=DF/Z 0059 PRINT 24,SUMYO,SUMXO,SQYO,SQXO 0060 PRINT 27,XOYO 0061 PRINT 25,VARXO,VARYO,SYX 0062 PRINT 26,A,B 0063 PRINT 30 , R, F 0064 PRINT 40,SPX,RSEF,DF,DFN 0065 10 FORMA! (2F10 • 1,F10.2) 0066 13 FORMAT (31 HO Y (0) X(0) ) 0067 14 FORMA! (57H SHEAR STRESS SHEAR STRESS TIME T SHEAR RAT 141E ) 0068 15 FORMA! (33H CALCULATED OBSERVED ) 0069 21 FORMAT (F10.1,6H ,F10.1,6H ,F10.1,3H ,F10.2) 0070 20 FORMA! (2F10.2) 0071 24 FORMAT (13 H SUM OF Y (0) = ,F10 .1,18H SUM OF X(0) =,F10.1»13H 241 SQY(O) =,F10.0,13H SQX(O) =,F10.0) 0072 26 FORMAT (7H Y(0) =,F10.4,2H +,F10.4,5H X 10) ) 0073 27 FORMA! (18H SUM OF X(0)Y (0 ) =,F10.0) 0074 25 FORMAT (11H VAR X(0) =,F10.3,15H VAR Y(0) =,F10.3,33H STAN 251DARD ERROR OF ESTIMATE =,F10.4) 0075 30 FORMAT (26H CORRELATION COEFFICIENT =,F10.4,22H FIT COEFFICIEN 30 1T =, F 10 .4 ) 0076 40 FORMAT (6H SEF =,F10.4,11H RSEF =,F10.4,9H DF =,F10.4,11H 401 DF/N =,F10.4) 0077 94 FORMAT (4H0K =,F10.4) 0078 100 FORMA! (3F10.3) 0079 101 FORMAT (2F10.2) 0080 105 FORMA! (1 HO,14HT(0) ASSUMED =,F10.2,14H RESIDUE =,F10.5) 0081 500 FORMAT ( 1H , 18HYIELD VALUE T(0) =,F10.2) H* 0082 END $

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Asset Metadata

Creator Tao, Fan-Sheng (author)

Core Title A Study Of The Behavior Of Thixotropic Fluids

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Rebert, Charles J. (committee chair), Kaplan, Richard E. (committee member), Lenoir, John M. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-407812

Unique identifier UC11362297

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Legacy Identifier 7016892.pdf

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Document Type Dissertation

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

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