LIBRARY Michigan State University ‘— PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES Mum on or before date due. DATE DUE DATE DUE DATE DUE A ~ ~ «7 E ITI— j MSU I. An Affinnativo Action/Equal Opportunity Institutio CMMM1 Q ‘gfi FLOW MODEL FOR A ROTATING DIE PRE-PREGGER by Nancy Stoneking Losure A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1991 ABSTRACT A FLOW MODEL FOR A ROTATING DIE RRE-PREGGER by Nancy Stoneking Losure In this research, a rotating die impregnation device, which pre-shears a thermoplastic melt prior to the fiber/melt contact zone, is evaluated with a shear thinning viscoelastic fluid (polyacrylamide solution) and a constant viscosity viscoelastic fluid (polyisobutylene solution). The operating conditions necessary to attain impregnation were explored experimentally. In the absence ofia fiber tow, a mathematical model for the rotating die pne-pregger assisted in the identification of design conditions for maximum flow rates. The theory, which employs the CEF model for viscoelastic behavior, predicts the existence of an optimum flow rate for certain designs and fluid characteristics. Although only qualitative agreement between the model calculations and the flow capacity data.i11 the prototype impregnation die was obtained, the theory nevertheless suggests that practical impregnation rates of' 20 cm/sec for a 3,000 fiber tow may be attained using this approach. To my husband, Ronald J. Losure, who took on extra chores, rearranged his schedule, and hugged me when I was grouchy. ii ACKNOWLEDGMENTS I wish to thank the State of Michigan for supporting this work through the Research Excellence Funding (REFO. II am grateful for support through the Dow Chemical Company Foundation Fellowship Fund, and for support through the Amoco Doctoral Fellowship. I also want to thank the Composite Materials and Structures center at Michigan State University for the use of their space and equipment, and the people at Hycar for generously supplying resins. Most of all, I wish to thank my advisors, Dr. Charles Petty and Dr. K. Jayaraman for their guidance and patience. iii TABLE OF CONTENTS List of Tables List of Figures List of Nomenclature Chapter Chapter Chapter Chapter #bhbbh Chapter UIUIUIU'IUIUI-b-bb Chapter 6 Chapter 7 Appendix Appendix Appendix Appendix Appendix Appendix U'I-hUNi-J U'IwaI-t ”MOON? mQO‘ Introduction Objectives Background The Rotating Die Pre-pregger The Centripetal Pump viscoelastic Fluids Selection Criteria for Experimental Fluids Mathematical Model Introduction Pre-pregger Flow Geometry Kinematics The Rheological Model The Macroscopic Mechanical Energy Balance Dimensional Analysis Flow Capacity for a Boger Fluid Parametric Study Experiments Introduction Rheological Parameters Experimental Apparatus and Procedure Experimental Results Experimental Discussion Conclusions Recommendations Kinematic Tensors Stress . sors The Mechanical Energy Balance Compute: Program Listing Rheolog cal Data Pre-pregger Date List of References iv vi viii 10 10 15 19 23 26 26 27 28 31 34 41 46 49 6O 60 63 74 87 99‘ 164 106 118 120 128 134 142 159 LIST OF TABLES Table 5.1 Experimental Fluids; Constituents and Suppliers Table 5.2 Rheological Constants for Experimental Fluids ‘9 61 71 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Fiugre Figure Figure Figure LIST OF FIGURES A Device for Impregnating Continuous Fibers with a Viscoelastic Fluid Schematic of the Centripetal Pump Rod-climbing Behavior in Viscoelastic Fluids Flow Structure in Disk Region The Effect of the Viscosity Exponent on the Velocity Ratio. The Effect of Elasticity on the Velocity Ratio. The Effect of Land Length on the Velocity Ratio. The Effect of the Disk Radius on the Velocity Ratio. The Effect of the Viscosity Exponent on the Velocity Ratio. The Effect of the Normal Stress Coefficient Exponent on the Velocity Ratio. The Cone and Plate Rheometer Shear Stress Data from RMS-800 for Low Viscosity Fluids. Shear Stress Data from RMS-800 for High Viscosity Fluids. Normal Stress Data from RMS-800. Relationship between b and n for Various Fluids. The Rotating Die Pre-Pregger. Schematic of the Rotating Die Pre-pregger. Typical Pre-pregger Data. Flow rate data for 0.3 wt% PIB. Flow rate data for Fresh Separan AP-30. Flow rate data for Aged Separan AP-30. Model compared with experiment for 0.3 wt% PIB. Value of Me for experimental fluids, from model calculations. Model compared with experiment for Fresh Separan AP-30. Model compared with experiment for Aged Separan AP-30. vi 14 24 32 50 53 54 55 57 59 64 67 69 70 75 77 78 82 85 86 88 90 92 93 95 Figure 5.16 Model compared with data from Good et a1. 97 (Fig. 6, 1974). Figure A.1 The Cone and Plate Rheometer Coordinates 115 Figure F.1 Spreadsheet Diagram 142 vii LI ST OF NOMENCLATURE a fluid elasticity coefficient [dyne secb / cm’] b fluid elasticity exponent [dimensionless] (a, c2,|<=—>| . I - .| I I. . ///z//////7///J/ gfi/fl/fl/fl/fl . ' / '/////////// Figure 3.1: Schematic of the Centripetal Pump.' 15 3.2 The Centripetal Pump The centripetal pump was first described by Maxwell and Scalora in 1960, and a patent was issued to Bryce Maxwell in 1962. In its simplest form, a centripetal pump consists of two disks facing each other across a small separation gap. One disk is held motionless, and the other is rotated about their common axis (see Figure 3.1). Because of the elasticity of the fluid, a flow field is established wherein fluid entering at the periphery of the disks is transported as an inward spiral toward the center, whence it exits through the die tube. Maxwell (1959, 1962, 1970, 1973) has published several papers on ways the pump may be used in processing polymers, and on the calculations necessary to scale up the centripetal pump from laboratory size to full production sizes. He has studied the effects of the radius, R, and rotation rate, 0, of the rotating disk and the width, H, of the gap on the rate of output, Q, of a polymer melt. Maxwell noted that when the exit die is the major restriction to flow in the pump, then the flow rate is insensitive to the gap width and depends strongly on the rotation rate and the diameter of the disk. This was later confirmed by studies by D'Amato (1975). However, when the exit die is not restrictive to flow, then the flow rate is very sensitive to the gap width. Thus for each rotation rate, there is a gap width which would produce a maximum 16 volumetric flow rate. The existence of a critical gap width was also confirmed by Good, et al. (1974). Maxwell (1970), Goppel (1969), and D'Amato (1975) claim that the centripetal pump is capable of mixing action superior to the screw extruder, with shorter residence times and with a large part of the heat necessary to melt the polymer being provided by mechanical work rather than thermal input. Also claimed is better versatility with regards to the form of the feed, the lack of pulsation in the output stream, and the ability of the pump to handle fiberous additives without damaging them. Maxwell (1970) also demonstrated that the centripetal pump can be scaled for output rates equivalent to the output rates of commercial screw extruders. However, Maxwell did not study the effect of fluid rheology on output rates, except to state that fluids with low viscosity and high elasticity were pumped at higher rates than fluids with high viscosity and low elasticity. Starita (1972) blended two thermodynamically incompatible polymers in a centripetal pump and studied the micro- structure of the resulting blends. He found that the rheological characteristics of the two polymers were factors in the mixing process, though he did not mention whether they significantly affected the flow rates achieved. Kocherov and Lukach (1973) used a pump with a glass stator to photograph flow patterns in the pump as it was filled 17 with polymer and as dyed particles were mixed into the polymer. They found that secondary flows developed in the pump and that these contributed significantly to the mixing action. Kataoka, et al. (1976) also studied the mixing action of the pump. Kocherov (1973) and D'Amato (1975) both found that an efficient feed mechanism is an important factor in the operation of the centripetal pump. They noted that inefficient delivery of the feed decreased the flow rate and caused "instabilities" in the operation. Both these authors were feeding solids to the pump, but Good et al. (1974) noted that their fluids tended to climb over the rotating plate of their pump when rotation rates were too high. They did not conclude that a more efficient feed mechanism was necessary, but they did limit their investigation to rotation rates where the fluid remained within the pump. Other studies have focused on the velocity and pressure profiles of the flow fields in the pump. Blyler (1966) used a pump with a glass stator and developed a photographic technique to study the velocity profiles both in the radial and gapwise directions. He found that the tangential velocity of the fluid varies in a linear manner across the gap, except for regions very near the stator where the deviation from linearity can be attributed to the thickness of the boundary layer which adheres to the stator. It was also observed that the tangential velocity does not depend 18 upon the radial position of the fluid for a given position in the gap. Blyler also noted that the profile of the radial velocity was parabolic, but that the profile was unsymmetric about the center of the gap. Also, plug flow profiles of shear thinning fluids were observed near the center of the gap, as opposed to a strictly parabolic profile for a fluid which has constant viscosity. Blyler did not attempt to quantify the effect that variation in rheological properties of the fluid might have on the velocity profiles or on the output rate of the pump. Remnev and Tyabin (1971) derived a linear profile for the tangential velocity and a parabolic profile for the radial velocity from the equation of motion for power law fluids. However, the predicted parabolic profile, unlike the experimental observations of Blyler, was symmetrical across the gap. Tomita and Kato (1967) derived pressure profiles in the radial direction of the pump, and then performed confirmatory experiments. Good, et al. (1974) also solved the equation of motion for the velocity profiles in a centripetal pump and derived a flow equation which incorporates the fluid characteris- tics. The shear stress was modeled as a power law, and the first normal stress difference coefficient was modeled as a polynomial function of the strain rate. The results of the calculation show that there is a gap width for any given fluid and rotation rate that yields a maximum flow rate. 19 This is confirmed by their experiments, and confirms the previous work of Maxwell (1973) and D'Amato (1975). The rotating die pre-pregger of the present study was designed so that physical dimensions of the disk, gap, and die tube would be similar to the device employed by Good et a1. (1974). Thus, the present work complements this earlier study. 3.3 Viscoelastic Fluids The stress necessary to deform a Newtonian fluid is proportional to the strain rate. When the applied stress goes to zero, the deformation of the fluid will instantaneously cease. An elastic or Hookean solid is also deformed by stress, with the extent of deformation being proportional to the stress. The solid is called elastic because it recovers its original shape when the applied stress is removed. A viscoelastic fluid has a combination of viscous (dissipation) and elastic (storage) behavior. The ratio of the amount of energy that is dissipated to that which is stored depends on the characteristics of the specific fluid, and on the time span over which the stress is applied and the flow behavior is observed. The matter of observation time is crucial. Glaciers will be perceived to flow if observations occur over a span of years, and the time scale of a "belly flop" dive is short enough that water may be 20 perceived as a solid. The ratio of a time scale characteristic to the fluid to a time scale characteristic of the flow is often called the Deborah number, De. Ordinary viscous flow occurs for De << 1, and solid behavior occurs for De >> 1. Viscoelastic behavior is characterized by De 5 1. Viscoelastic behavior is sometimes modeled as a superposition of viscous and elastic effects. However, in practice, it is very difficult to separate fluid behavior due to elasticity from that due to viscosity, except for a limited class of flows, and a select class of fluids. The following constitutive model will be employed in this study to relate the stresses in the fluid to the strain rate field: 65 3 = 2n [§ - Ag] ”-3) § 2% W? + 079)") (3-4) as as ., ésfi+ufl§ -(Vl.1)'§ " §°V9 (3'5) In Eq. (3.3), the parameters n and A represent the shear viscosity and a characteristic time associated with the 21 fluid. n and A will be positive functions of the invariants of the strain rate dyadic S. Eq. (3.3) is a special case of the Criminale-Erickson-Fibley (CEF) equation for which the second order term is zero (Bird et al., p. 503, 1987). Appendix A shows that Eq. (3.3) gives a zero second normal stress coefficient for simple shear flows and that 2nk = w the primary normal stress coefficient. 1: This equation was chosen to explore the behavior of a rotating impregnation die because it provides a good approximation for simple shear flows of viscoelastic fluids subjected to large deformations for which it, 5 0 (Tanner, see p. 126 and p. 222, 1988). Furthermore, Eq. (3.3) gives an explicit equation for the stress dyadic, I, once the flow field has been specified. The viscosity coefficient, n, and the primary normal stress coefficient, i1, can be represented by the following empirical expressions over a limited range of flow conditions £1 n = k (2 §:§)2 (3.6) i_:-_2_ 211k 5 {'1 = a (2 §:§) 2 (3.7) The parameters a, b, k, and n are intrinsic properties of the fluid and must be determined experimentally. For a = 0 and n f 1, Eq. (3.3) reduces to a model for 22 a purely viscous fluid, whose viscosity is described by a power law expression such as Eq. (3.6). When n < 1, the fluid is shear-thinning: when n > 1 the fluid is shear- thickening. For the case of a = 0 and n = 1, Eq. (3.3) reduces to a Newtonian model. The validity of the above model has been discussed by Tanner (p. 222, 1988) for viscometric flows. Its utility for this study is that it provides an unambiguous, albeit approximate, link between experimentally obtained fluid characteristics and process flow conditions. This will provide a means to interpret the complex flow behavior of viscoelastic fluids through the rotating die. When n = 1 and b = 2, the shear stress and the first normal stress difference are constants independent of the strain rate, and the fluid is a special case of a "Boger" fluid. "Boger" fluids are often employed to study the shear-thinning effects and elastic effects in non-viscometric flows (Choplin, 1983). Tanner (1973) has observed that b 5 2n for a wide range of polymer solutions. According to Eqs. (3.6) and (3.7), for fluids with b = 2n and n < 1, the primary normal stress coefficient decreases significantly as the strain rate increases, inasmuch as ¢1‘-tf. Paradoxically, the shear thinning nature of thermoplastic melts, which may provide a means to improve the intrinsic impregnation rates of fiber tows, may simultaneously hinder the transport of 23 fluid to the tow interface by centripetal pumping (see Figure 1.1). One phenomenon associated with elastic fluids is rod climbing (see Figure 3.2). When a rod is rotated in a beaker of fluid with no elasticity, the momentum transfer from the rod to the fluid throws the fluid outward, and a vortex is formed around the rod. The elastic fluid, however, has a non-zero first normal stress difference, and the induced positive pressure gradient is expressed by a bulge of fluid forming about the shaft. The fluid may climb several shaft diameters above the surface, depending on the specific conditions. The material which climbs the rod necessarily lowers the level of the fluid surface in the beaker, unless the beaker is infinite in extent. Thus, rod climbing, or the Weissenberg effect (Beavers, 1975), may strongly interfere with other flows in the vicinity of the rotating rod by lowering the hydrostatic head or by altering the entrance effects. The fluid will not climb a rotating rod if the diameter of the rod is above a critical value, which depends on fluid characteristics (Beavers, 1975). 3.4 Selection Criteria for Experimental Fluids Because the centripetal pumping phenomenon arises from the fluid characteristics of elasticity and viscosity, fluids representing four combinations of elasticity and viscosity were chosen for this study. All the fluids were 24 Newtonian fluid viscoelastic fluid Figure 3.2 Rod-climbing Behavior in Viscoelastic Fluids. 25 above their melting points at room temperature, for ease of processing. Two of the fluids chosen were essentially Newtonian. They were chosen to serve as controls, and show how the pump operated with non—elastic fluids. Two additional fluids were prepared by mixing a common epoxy resin with a high molecular weight rubber oligomer. The blends showed slight rod-climbing behavior on mixing, (see Figure 3.2), and they were tested to show how a slightly elastic fluid would behave in the centripetal pump. There has been recent interest in blends of epoxy with rubber oligomers, because the addition of rubber in the uncured system tends to toughen the cured product, thus allowing its use in applications for which neat epoxy resins are considered too brittle (Raghava, 1988). A fifth fluid was chosen to be very elastic and to have a constant viscosity. It is a solution of polyisobutylene in polybutene and kerosene, and is a model fluid of the type known as a "Boger" fluid (Choplin, 1983). Another model fluid was formulated to be elastic and shear thinning, and was chosen to imitate one of the fluids studied by Good, et al. (1974). Formulations for these fluids appear in Table 5.1, and rheological constants appear in Table 5.2. Chapter 4 Mathematical Model 4.1 Introduction The rotating die is very similar to a centripetal pump when there is no tow being drawn through it. It is also identical to a plate and disk rheometer when the volumetric flow rate, Q, is zero. Therefore, the following theory parallels analyses which already exist for these flow situations. The geometry of the die is discussed in Section 4.2. The rheological model used to describe the response of the fluids to the stresses in the pre-pregger is presented in Section 3.3. Further simplifications of this theory for the rotating die pre-pregger are introduced in Section 4.3. The discussion of the velocity fields in Section 4.4 then leads to a macroscopic mechanical energy balance in Section 4.5. The energy balance provides an equation for Q as a function of the geometric scales of the rotating die, the operating parameter of the die, and the characteristics of the fluids. This equation is discussed in Section 4.6, and the results of a parametric study are described in Section 4.7 26 27 4.2 Pre-Pregger Flow Geometry The rotating die consists of two concentric disks of radius R separated by a gap of width H (see Figure 3.1). The upper disk rotates about its axis at an angular velocity 0, while the lower disk remains stationary. The stationary disk is provided with a die opening at the center, of radius 1% and length.lh. The working fluid is fed from the periphery of the rotating die and flows towards the axis with a spiralling motion. The origin of the cylindrical coordinate system is located at the center of the rotating upper disk as illustrated in Figure 3.1. The steady state flow patterns within the rotating die can be separated into three distinct regions: Region I Flow Between Two Disks: Fully-Developed, Two-Dimensional, Axisymmetric Swirling Flow uI = u Ie + “9199 (4.1) Region II Transition Flow: Fully-Developed, Three- Dimensional, Axisymmetric Swirling Flow Ogr IA 71 28 u” = u”e + ue”e + une (4.2) r 'r '9 ’ ‘2 Region III Flow in the Die Tube: Fully-Developed, One- Dimensional, Axisymmetric, Non-swirling Flow 1.111: = urn9 (4.3) 4.3 Kinematics The forced vortex flow induced by the relative motion of the two disks observed by Blyler (1974, see Chapter 3) is assumed to extend over both Regions I and II ue‘ = ueII = wr (1 - g) . (4.4) For 2 0, the tangential component of the velocity is or for 0 5 r 5 R. u; satisfies the no-slip condition on the lower stationary disk at z = H. ug‘ is also zero at the entrance of the outlet tube. The axial component of the velocity in Region III is given by (see Middleman, p. 88, 1977) 29 m: 9 3n+1 _ L? 5 u. «R: ——n+1 [1 ‘R.’ ] (4. ) where Q is the volumetric flow rate. For n = 1, Eq. (4.5) reduces to the Hagen-Poiseuelle law for fully developed laminar flow of a Newtonian fluid through a tube (see Middleman, p. 87, 1977). The continuity equation and the no-slip boundary conditions in Region I at z = 0 and z = H are satisfied by a radial velocity profile of the form 3 u,‘=-,,—,%f§(1-§-), (4.6) where Q is the steady-state volumetric flow rate. The axial and radial components of the velocity within Region II are constructed to satisfy the continuity equation II (ruru) + 3'2 = 0 (4.7) 1 1L r'ar as well as the condition that the three-dimensional flow field in Region II must provide a continuous transformation from the two-dimensional flow of Region I to the one- dimensional flow of Region III. This strategy, which was also employed by Good et al. (1974), yields the following expressions for the radial and axial components of the 30 velocity in Region II n __ _ 3Q 3n+1 r _ 2n r .. z _ z u. — wHRd n+1 Rd [1 3n+1 (Rd) ] H(1 H) (4'8) 11 .... 3Q 3n+1 r n—‘;1 2 2 2 Z n. - + ”Rd, ———n+1 [1 - (fi—d) ] (H) (1 - g g) . (4.9) Note that Eq. (4.8) reduces to Eq. (4.6) for r = R; and that Eq. (4.9) reduces to Eq. (4.5) for z = H. The local spiral structure of the two-dimensional flow in Region I can be characterized by the ratio of velocities evaluated at z = H/2: H -u,‘(r.§) 3 tan w = a r a = ———9—; . (4.10) ( ) ue‘(r,%) 21rer The angle 0 measures the transition from a purely tangential flow (a = 0 or w = 0°) to a purely radial flow (a = w or w = 90°). Eq. (4.10) implies that = (%:;)2 . (4.11) 31 Thus, if R; = 2.5 mm and r = 25 mm, the parameter 0 increases by two orders of magnitude from the periphery to the core of the flow field for all fluids and for all values of w. This purely kinematical feature provides ample motivation to explore the utility of a rotating die as a continuous means to mix different resins prior to fiber tow impregnation. In pre-pregger applications, it may be important to specify a to control either the contact time of the viscoelastic fluid between the rotating disks or to orient the macromolecules prior to contact with the fiber tow. In Section 4.5, a will be related to the operation and design of the pre-pregger by using a mechanical energy balance. However, it is beyond the scope of the present work to seek an optimal value of a based on impregnation results. Figure 4.1, however, shows that w decreases significantly as r/Rd increases. For a(Ra) E 1, the transition from a two-dimensional flow to an approximate one-dimensional flow dominated by the tangential component of the velocity occurs for 1 g r/Rd g 5 (see Figure 4.1); for a(Ra) = 10, the transition occurs for r/Rd s 10. 4.4 The Rheological Model Eqs. (3.3), (3.6), and (3.7) define the class of fluids examined in this study. Appendix A gives the components of the strain rate dyadic S and its upper convected Oldroyd derivative for the three flow domains defined by Figure 3.1. 32 .coaom me E 23025 26E 5. 939”. cm: 0.2 0.0 0.0 0N 0.0 0mm 06 0.0 0N 04 0.0 _ _ L0 _ _ _ 0.0 0.0m 0.00 0.0.v 0.00 0.00 0.0K 0.00 0.00 233 The second invariant of S (i.e., 2 S:S) is also listed in Appendix A for the three regions of the pre-pregger. Within Regions I and II, the major contribution to S arises from the axial gradient of the swirl component of the velocity. In Region III, however, the radial gradient of the axial velocity determines the local strain rate. Thus, f' 2 {3‘13} Region I (4.12) 32 2 2 s : s é < [3119" Region II (4.13) ' ' 32 an 111 k [ air ] Region III . (4.14) The elastic contribution to Eq. (3.3) will be neglected in Regions II and III (i.e., A = 0). Eq. (3.3), however, will be used to estimate the components of I in Region I (see Figure 3.1). The stress in Region II will be estimated by using the following "effective" Newtonian model 1” = 2 S” (4.15) with a volume average viscosity defined by I n dV 11: V11: a V . (4.16) III Eqs. (3.6) and (4.13) define n for Region II. 34 Eq. (3.3) with,i; = 0 is used to estimate the stress for Region III. Thus, In: = 2 TI §III . (4.17) The viscosity coefficient in Eq. (4.17) is defined by Eqs. (3.6) and (4.14). Appendix A gives the components of I for each of the flow regions. It follows from Eqs. (4.12) through (4.14) and Eqs. (4.4) and (4.5) that %§ Regions I&II(4.18) 1 Tea( 2§:§)2= 1 .2— r n [1R3] [_____3nn+1] ['12] Region III . (4.19) Eq. (4.18) anticipates that the viscosity coefficient in Regions I and II, for n < 1 (see Eq. (3.6)) can be decreased either by increasing w or by decreasing the gap width H. This action may either increase or decrease the viscosity within Region III, depending on the behavior of Q as o and H change. 4.5 The Macroscopic Mechanical Energy Balance The steady-state macroscopic mechanical energy balance 35 for the rotating die provides a means to estimate the volumetric flow rate Q. The equation can be written symbolically as D = («11 + w: (4.20) where D a 1]] gm} dV (4.21) V W1 5 -21r]: [7.9%]..0 r dr (4.22) W2 5 27R]: [Tr6u6]r-n dz (4.23) The dissipation function D represents the irreversible rate of conversion of mechanical energy into internal energy whereas W1 and W2 represent the rate of energy transfer to the fluid by shear stresses acting on the control surfaces at z = 0 and r = R. Eq. (4.20) requires that the two work terms balance the dissipation terms. The following four assumptions have been made to bring the macroscopic energy equation to this form 1. No slip at solid/fluid interfaces: 2. Gravitational work is neglected: 3. The changes in kinetic energy between the inlet and the outlet of the pre-pregger are neglected; and 4. The normal component of the stress at the inlet (i.e., p - 1;) equals the normal component of the stress at the outlet (p - rff‘). 36 The two work contributions in Eq. (4.20) can be written in terms of the velocity components by using the stress models discussed in Section 4.4 (also see Appendix C). Thus, R an W1== -21I (n-Efuefl o r dr (4.24) O I W =+21rRl (‘1' a—u—ea—ulu)L dz (4 25) 2 1 32 32 6 _R ° 0 By inserting the models for ug, ur, and.i; defined previously, Eq. (4.25) can be written as w, = Q a (“h—R)” . (4.26) Because n does not depend on the axial coordinate 2, W1 will exactly balance a term contained in D, as will be shown presently. The elastic nature of the fluid provides a means to redistribute the energy transferred across the two control surfaces at z = 0 and r = R into the pressure field. As was previously mentioned, this induces an inward radial flow toward the outlet tube located on the axis. This viscoelastic process makes D smaller than the dissipation integral for a purely viscous fluid under the same kinematic 37 constraints. In order to analyze this feature explicitly, the total dissipation integral is decomposed into three contributions associated with the three volume Regions I, II, and III: . (4.27) The dissipation integral over Region III follows from the results summarized by Appendixes A and B and Eq. (4.5). The result can be written as (see Appendix C, Eq. (C.16)) ZkL ___ a (3nn+1)n Qn+1 . (4.28) III :1 3114-1 w'Rd For n = 1, Eq. (4.28) reduces to the dissipation integral for fully developed laminar flow of a Newtonian fluid (see Bird et al. p. 188, 1960.) As the fluid passes through Region II, Eq. (4.28) assumes that the stress can be calculated as if the fluid were Newtonian with a viscosity coefficient defined by Eq. (4.16). Therefore, it follows from Eqs. (4.4) and (4.13) that n-I _ 2k _ — T4713[H] . (4.29) The dissipation integral over Region II can be written in 38 terms of the stress components summarized in Appendix B. There are five distinct contributions to this integral as indicated below: 2 2 an.” an,“ [3‘12”] r dr dz 4 30 + [ 3r + 32 J + 2 32 ] ( ) The dissipation integral over Region I can also be written in terms of the strain rates by using the stress components summarized in Appendix B. The result is o, = 2. [L [ n [ [331-] + 4‘15]: [33‘] ] u:I 3"18]a [urI )2 + ‘1'. T ‘33;- + 4 r au1 2 - [fi] ] ] r dr dz . (4.31) 39 The first term of DI and the first term of DH can be combined as Rd 2 27M [ I [%§] r dr + I 0 Rd n [%§]2 r dr ] . (4.32) Note that the above result equals W1 defined by Eq. (4.24). Thus, Eq. (4.4) together with the approximations defined by Eqs. (4.12) and (4.13) cause the sum of the first terms in D1 and Du to balance W1 exactly. Therefore, the remaining terms in DI and Du combine with Du to balance W2. I The term which arises from the primary normal stress difference in Eq. (4.31) reduces the magnitude of DI because u: < 0. This contribution to DI will be denoted as a R u: 31.191 3 -E a 21 91 r' 32 r dr dz . (4.33) Because {'1 Z 0 and urI 5 0, the integral E is always positive. Eq. (4.33) neglects the strain rates related to the radial component of the velocity in comparison to the strain rate associated with the tangential velocity. This is consistent with the approximation given by Eq. (4.12). Thus, by inserting the models for us, ur, and {'1 into Eq. (4.33), the following result obtains 40 E=%—[1-[%—]]30. (4.34) The macroscopic energy balance, defined by Eq. (4.20), can now be rewritten as D‘+D‘+D =E+w. (4.35) where the dissipation integrals D; and an contain only contributions due to the viscous stresses. wg, DIn and E are defined by Eqs. (4.26), (4.28), and (4.34), respect- ively. D; and DH] are defined as follows an ._ I ”1‘2"” “ [432—] and ‘ I‘a all n 2 uu 2 an u an II 2 Orr-521]] 2(a;]+2[;]+[a;+a’z] oo 2 2 + [33.1] ] r dr dz (4-36) 2 + 2[§%%—] ] r dr dz . (4.37) Appendix C gives explicit equations for D; and an in terms of Q, o, k, n, R, Rd, and H. Eq. (4.35) can be used to estimate the volumetric flow rate, Q. Eqs. (4.36) and (4.37), together with the equations for the velocity components, show that Df‘and 41 D ‘ are both proportional to Q2. Moreover, Eq. (4.28) shows 11 that DIn on Q'”1 and Eq. (4.26) indicates that W2 0: Q. Because E a W2, it follows that E a Q also. Thus, Eq. (4.35) has the following dependence on the flow rate 619’ + 629"” = 6,0 (4.38) where the dimensional coefficients 61, 62, and éaiare all positive. If a = 0, fig and E are zero (see Eqs. (4.26) and (4.34)). Thus 63:= 0 also, and the only solution to Eq. (4.38) is Q = 0. However, for 63 > 0, Eq. (4.38) has a unique solution. The qualitative behavior of Eq. (4.38), which represents a steady state power balance over the rotating die pre-pregger, will be summarized in the next section. 4.6 Dimensional Analysis Eq. (4.38) determines the volumetric flow rate Q in terms of four geometric parameters, (R, Ra,l; H), a single a. operating parameter (w), and four rheological coefficients (a,b,k,n). Dimensional analysis implies that Eq. (4.38) can be reduced to an equation containing seven dimensionless groups. The rheological parameters and the angular velocity can be combined to form the following three dimensionless ratios b, the dimensionless elasticity exponent: 42 n, the dimensionless viscosity exponent: and, C) III mm 49"" . (4.39) The dimensionless group G gives a measure of the relative importance of elastic and viscous effects in the rotating die. This follows by comparing a measure of the primary normal stress difference with a measure of the viscous shear stress. For instance, if Eq. (4.18) denotes a characteristic strain rate associated with the swirling flow, then a characteristic Weissenberg number can be identified as we = = — = :03“ = 6(a)“ - M...) Eq. (4.40) shows that G is closely associated with a local Weissenberg number, or equivalently, a local Deborah number (De a 196). The utility of G as an independent group arises from the fact that it does not depend on the geometric scales of the pre-pregger. For "Tanner" type fluids (b = 2n), Eq. (4.40) shows that We a (‘5!)'1 . 43 Therefore, with n = 1, the increase in the ratio of normal to viscous stresses is proportional to the strain rate. For shear thinning fluids (n < 1), the increase in We with &c is diminished. Although the exponent "b" for polymer melt is nominally less than two, (b 5 2), Eq. (4.40) predicts more than a proportional increase in We with 4° even for shear thinning fluids provided b > 1 + n. One of the constituents for the model fluids in this study (CTBN) has exponents of b = 1.9 and n = 0.5 (see Table 5.2). The four geometric scales of the pre-pregger give three dimensionless ratios based on the radius of the outlet tube: _ R 5. ' R"; (4.41) Ld 3.. '3 1Tdl (4.42) H B! E R_d (4.43) Eq. (4.10), with r = R: defines the remaining 6 dimensionless group as a velocity ratio a (5 a(Rd)). Thus, using the foregoing definitions, Eq. (4.38) can be rewritten as an equation for a, F(a) I cla + C20" - c = 0 . (4.44) 3 Hereinafter, the symbol a denotes Eq. (4.10) evaluated at r = Ra. The coefficients in Eq. (4.44) are defined as follows (see Appendix C for a derivation of c1, 1) fluids were simulated. Figure 4.2 shows the effect of the gap width on the 50 6:9”. b_oo_o> 9: co Eocodxm £885 05 to 82m. one ”N6 939“. CN U‘d‘m‘n m 30.0 wm Rownxa u a nod xxa us 4.6 51 velocity ratio a for two different "Tanner" fluids (i.e. b = Zn). The calculations show that the flow capacity of the pre-pregger can be significantly increased by decreasing the gap width H relative to the radius of the exit tube (i.e. i; << 1). The shear thinning fluid (n < 1) responds similarly, but the magnitude of a is less. For i3ll E 0.1, the velocity ratio for n = 0.9 is about a factor of two smaller than a for n 1.0. This occurs because the large increase in the normal stresses, which occurs as the gap width decreases, is moderated by the shear thinning effect. This conclusion follows directly from Eq. (4.40), which shows that the incremental increase in the Weissenberg number with the shear rate depends on the exponent n. The parameterization of the a curves in Figure 4.2 by X (see Eq. (4.44a)) shows that the dissipation of energy in Region I (and II) determines a for B. < 1 and that dissipation in Region III controls a for ql> 1. It is noteworthy that the magnitude of X is distributed along the a curves in about the same way for the two cases n = 1, and n = 0.9, although the values of a differ significantly for small gap widths. For the shear thinning fluid, a attains a maximum at BIll E 0.2, where X = 0.980. For smaller values of la, the velocity ratio decreases rapidly to zero because the balance between the viscous dissipation and the elastic work terms in the power balance cannot support a non-zero 52 flow rate. This follows by examining the behavior of Eq. (4.32) for B! e 0. Because c2 -v 0 as B. -o 0 (see Eq. (4.46)), Eq. (4.32) implies that lim 0 0c (8)““"’ . (4.60) an-oo 3 Thus, for b = 2n and n < 1, the above result shows that a -' 0 for BI -’ 0. However, for b = 2n and n = 1, the velocity ratio approaches some non-zero value as file 0. Finally, for 1+n < b, 0: becomes unbounded as 8lll -’ 0. Figure 4.3 shows that the parameter G has a significant effect on the velocity ratio a. For fixed values of G, the 0 curves are similar to the shear thinning example portrayed by Figure 4.2. Once again, because 1+n > b, the flow rate suddenly drops to zero below a critical value of ii < 0.20. Although the radial component of the velocity may be two orders of magnitude larger than the tangential velocity for BI = 0.2 and G = 3, Figure 4.1 shows that the local orientation of the flow field changes rapidly to a predominantly tangential flow once r/Ra:> 10. The effect on a of increasing the ratio of the die tube length to the die tube radius is seen in Figure 4.4. When the length of the die tube is increased, 8 increases, and 0 decreases. However, for values of 81' below 0.1, the value of a becomes insensitive to the value of B; This occurs because the flow resistance in the narrow gap dominates 53 26m 262.5 9:8 36:65 6 62m 9:. ”2. 6590 mm 0.0 0.0 No .0 0 0N 54 2me 362.5 9s .6 £96.. 6.136665 2:. “E. 659”. on m: o... 4.; m: o: so who so Nuo 6&6 0.0 0. _. or m 0... 8 0.N 0N 0.0 m. m. . m .1th m a o I m Aoom\a u 3 Hon x\o «0 H.o UQ‘CQ 06 55 0.N 206m 362% ms :6 426$ 65 ms. .6 661m 9: ”ms 659“. m a 0.: 0.: #4.: N... 0.: 0.0 0.0 V0 N.0 0.0 _ (I. _ir_/_..._i.0. 0 IV .0 -0 “Hm I: m u a Aomn\a u a you xxu us H.o u o 56 (i.e., X is close to unity). Figure 4.4 also shows that the peak value of 0 occurs at larger values of Bll as BL decreases. This illustrates the fact that the peak in 0 corresponds to a shifting of the dissipation domain from Regions I and II to Region III. Btis the dimensionless ratio of the disk radius to the die tube radius. Figure 4.5 shows that for a fixed value of 8', 0 increases as 3. increases. Doubling 3. from 25 to 50 almost triples a at low values of a" This means that the flow field becomes much more radial as glis decreased, and also as qtis increased. Note that the peak value of a occurs at larger values of B! as BR increases. This shows that slightly larger gap widths are required to shift the dissipation from Regions I (and II) to Region III. Figure 4.6 shows the effect of changing the viscosity exponent, n, on the velocity ratio, a. For fixed values of i; and G (which is also a function of n), the value of a rises as the value of n decreases. Thus, a decrease in n increases the radial nature of the flow, and the fluid takes fewer spirals around the disk on its way from the periphery of the disk to the entry of the die tube. Note that the behavior of the family of curves changes near the origin, as n increases. This behavior is due to the fact that b = 2 for these calculations. The asymptotic behavior of a for small values of 8. explains the different behavior for 57 03mm B_oo_m> 05 co Emcoaxm 36085 05 .6 5.0th 0:... ”0.6 9:9“. mm 0d 0.: 0.: 4.: N: 0“: 0.0 0.0 6.0 Nd 0.0 b _ — Oflfiah =chcwe= N 0 II II (D Q: Q'l-Q mm A00m\a u 3 now x\m "0 «.0 58 shear thinning (n < 1) and shear thickening (n > 1) fluids. It follows from Eq. (4.60) that a 4 0 as Bll -+ 0 provided (1+n) > b: and a -» on as Bll -’ 0 for (1+n) < b. Thus, the calculations presented by Figure 4.6 for b = 2 illustrate these two limiting cases. Of course, Figure 4.2 already shows that for b = 2n and n < 1, the velocity ratio is bounded. The conclusion which stems from these calculations is that the parameter'l s b-(1+n) has a dramatic effect on the behavior of the velocity ratio. For l== 0, a » constant as 13Ill .. 0. However, for l > 0, a -o an as 8‘ -* 0: whereas for l < 0, a 4 0 as file'CL The effect of the elasticity exponent, b, on the velocity ratio is shown in Figure 4.7. As b increases, a increases for a given value of 0“ However, note that the curves change their nature near the origin in a manner analogous to that shown by Figure 4.6. Once again, an understanding of the three cases shown follows from Eq. (4.40). The parametric study illustrates the interaction of the seven dimensionless groups that make up the flow equation. Practical questions about specific pre-pregger operations can now be answered. These results will be of assistance in the design and operation of the pre-pregger. 59 .9me 2.62.5 65 :6 “cocoaxm E92080 30:0 .9502 05 .6 69$ 05. K6 050E . ma 0.N 0H: 0“: E: N“: 0“: 00 00 £0 NHO 0.00 i\\ 0.: H a -N UHQHK ZHQCCGB: 0 -0 .0 flow H u c n u “a -N: mm u n 1060\H n a you x\6 my :.6 u 0 V: Chapter 5 Experiments 5.1 Introduction Experiments were conducted to determine the flow capacity of the rotating die prepregger in the absence of a fiber tow. Table 5.1 defines the model viscoelastic fluids used to simulate the rheological response of thermoplastic resins. For low strain rates, the P18 solution has a constant viscosity coefficient, whereas the two Separan AP-30 solutions show significant shear thinning behavior. These fluids also exhibit strong elastic behavior. The two blends of CTBN and Epon 828 show weak elastic behavior and constant viscosity, and the neat polybutene and neat Epon 828 showed insignificant elastic behavior and constant viscosity. The 0.3 wt% PIB solution was prepared by dissolving solid polyisobutylene rubber in a known amount of kerosene while stirring over gentle heat. This solution was then mixed into the polybutene liquid to form a clear, visibly homogeneous solution. The solution was kept in a tightly covered glass jar for six months. This solution was similar to one used by Chmielewski (1990). 60 61 .msme ofiuumsfi>mum an consumma mofiuawcmo « Houmoaam wu3 m.~v “mums omHHHumfic «#3 m.~¢ 00H x w R JR Honuma cmuwmomv maauuficoamuommaom can 300 moflsmamuomhaom mo umahaomoo «#3 m mH.H cmummmm «#3 m ooow a 3: “comm Hodouch ooofid ocmusnaflom wu3 n.mm 0G0wOHOX “#3 ¢.¢ OH x mm. H u a AmmHuH xmcmumH> mm :H :oxxm mcoHsuanomHsHoa was m. o mm.o mHm was n.o mum noun a»; on mum comm cH zmao ”a; ov mo.H . zmao was oe can u 3: Hmmm codmv Auo< mam comm :H noHucoou .m.m woumcHsumausxonumo was n~ nH.H zmao «as am EU\mv umwaamsm mucmsuwumcoo mmHmcmo mfinz panda mumuammam can musmsuwumcoo “mofisam Haucmfifiummxm ”H.m manna 62 The Separan AP-BO solution was prepared by following the protocol described by Good, at al. (1974). The solution was made by slowly mixing the Separan AP-30 powder into a mixture of glycerine and water. Care was taken to wet each particle and to avoid clumps. The solution showed very strong rod-climbing tendencies. After being allowed to stand for a week, the fluid appeared clear with no haziness or regions of dissimilar refractive index. The Separan AP- 30 solution was tested after one week, and the remainder was kept for six months in a tightly capped glass jar for subsequent testing. These solutions are referred to as ’fresh' and 'aged' respectively. The blends of CTBN and Epon 828 contained no curing agent. The two liquids mixed to form an optically clear, visibly homogeneous solution with a slight tendency to climb the shaft of the mixer. This rod-climbing behavior is a sign of an elastic nature, as discussed in Chapter 3. Both of the CTBN/Epon 828 blends were kept in tightly capped glass jars to prevent evaporation. Epon 828 and polybutene were tested as pure (neat) liquids. The primary normal stress difference and the viscosity coefficients of these fluids were measured over a range of strain rates, as discussed in Section 5.2. The design and operating procedures for the rotating die prepregger are presented in Section 5.3, while the scope and the results of the experimental work are summarized in Section 5.4. 63 Section 5.5 provides an interpretation of the results as well as comparison with earlier experimental work and the model developed in Chapter 4. 5.2 Rheological Parameters The rheological characterization experiments were done on a Rheometrics Mechanical Spectrometer Model 800 (also known as the RMS-BOO). Figure 5.1 shows a schematic of the cone and plate configuration used. The cone is kept stationary, and the torque and normal force necessary to keep it stationary are measured. The coordinate system is spherical, with its origin at the tip of the cone. The subscript r refers to the radial coordinate, while o is the rotational coordinate, and e is the azimuthal coordinate. The cone angle, 3, is made small enough that the approximation sin 5 9E 18 can be used to simplify the derivation of the flow field (see Appendix A, and Bird, et al., p. 522, 1987). To measure the rheological properties of a fluid, the fluid is loaded onto the plate, then the cone is lowered until the gap between the tip of the cone and the plate is 500 microns. Care is taken to ensure that the fluid in the tool contains no voids and that there is no excess fluid on the edges of the tool. For a steady shear test, the plate is rotated at a given radial velocity, and the signals from the torque and normal force transducers are read. 64 CONE FLUID PLATE Figure 5.1: The Cone and Plate Rheometer 65 The shear stress, the viscosity coefficient, and the primary normal stress difference are calculated from readings of the transducers, which produce signals from calibrated strain gauges. The torque transducer has a range of i2000 grams, and is accurate to 1’2 grams. The lever arm of the torque transducer is fixed at 1.0 cm, so the ’torque' reading, M from the torque transducer has units 1. of gram-cm. M.1 is multiplied by the acceleration due to gravity to obtain the torque acting on the cone (T = Mgg). The servo-motor which rotates the bottom plate is driven by a controller which reads the angular velocity to 0.1%, but is not accurate at rotation rates above 100 radians per second. The viscosity coefficient is calculated by relating the shear stress to the measured torque and the strain rate to the measured angular velocity. For the RMS-800 apparatus, the viscosity coefficient n is given by (see Figure 5.1 and p. 522 in Bird et al., 1987) [ 3M1g] Us _ 2111’ .27 _ %J (5.1) n: where, I“ is the shear stress (dyne/cm?) 9 is the strain rate (1/sec) IQ is the reading from the torque transducer (g-cm) g is the acceleration due to gravity (cm/sec”) w is the angular velocity (seed) R is the radius of the plate (1.25 cm) 3 is the cone angle (0.108 radians) 66 The reading, M from the normal force transducer has a 2, range of i2000 grams, and is accurate to i0.1%. However, readings are not accurate below 2 grams. The normal force acting on the apparatus is obtained by multiplying this reading by the acceleration due to gravity (F = Mag) . The primary normal stress difference is given by (see Figure 5.1 and p. 523 in Bird et al., 1987) 21129 1 - NR2 (5.2) where, M2 is the reading from the normal stress transducer (g) g is the acceleration due to gravity (cm/sec”) R is the radius of the plate (1.25 cm) The viscosity coefficient and the primary normal stress difference were measured for all the test fluids over a range of strain rates. Figure 5.2 shows a log-log plot of the shear stress, 1% against the strain rate, (& = %). The 23 and 40 wt% blends of CTBN and Epon 828 have very similar curves to the polybutene. The slope of the curves is the viscosity exponent, and slopes for these three fluids are equal to one. The value of the intercept gives the value of the viscosity coefficient, k, and values for all three fluids fall at about 800 dyne/cm?. The curve for Epon 828 has a constant slope of less than one, and an intercept just over 100 dyne/cm’. 67 89328 x mum Sam q 953 s8 - .053 $3 a 8922””. E0: Son. $95 895 ”Wm 939“. com: .mfim Embm 02 2 H __ _ _ _ T _ _ _r_ _ _ _ _ _ _ «00—. H mHos: q H q .. G H w ”082 u H .1 . H ”cooOOH a v wO/euAp ‘sseus JBGLIS 68 The shear stress versus strain rate data for the PIB and Separan solutions are plotted on log-log coordinates in Figure 5.3. The PIB data lie on a curve with a constant slope nearly equal to one, and an intercept at about 500 dyne/cmP. The two Separan solutions have curves which have slopes very different from one, and intercepts at about 1000 dyne/cm’. The viscosity power-law parameters k and n (see Equation 4.4) were determined from the data depicted in Figures 5.2 and 5.3, using a linear regression program, where the slope was equal to n, and the intercept was equal to log(k). The constants are tabulated in Table 5.2. Figure 5.4 shows the first normal stress difference, N1, plotted against the strain rate on log-log coordinates. Note that the curves for the two Separan solutions are mildly ’S’ shaped. The curve for the PIB solution has a constant slope of 2 for low strain rates, but the slope decreases with increasing strain rate above 10/sec. The curves for the blends of CTBN and Epon 828 contain data at only large strain rates, because the normal stress difference was below the sensitivity of the HMS-800 at the lower strain rates. Despite the deviations from linearity, all of these curves were modeled as power laws (see Equation 4.5) and the linear regression program was used to obtain (slope = b) and (intercept = log(a)). The results are summarized in Table 5.2. 69 gm 8?. 0 saw 58.”. 0 mi se 4 cowwsm 59.. $8 $25 .35 as 239“. 03> 9?. £25 0 F _. Pb—P——_ b P____—__ — COP 4 -0000 _. a v wo/eu/{p ‘ssans Jesus 70 255 $3 m .255 axomw 0 gm 8?. 0 saw 52“. 0 ma so 4 commit Eo: Ema wmmzw _manz ”in 2:9“. 08> 60mm 595 00w 0_. F L—h——P~ p ___—___ h _ mecca H a v wO/auAp ‘ssens leuuoN 71 nouns ou . me no Houomu 0 an Umuomuuoo 0009 0>0n mmsaw> 00.N 00.N 05.0 0h.H 0H.H N 0.0HN 00.000 0.0 v0.0 0.0 0005 00¢¢ one 0.5 0.0 0 v m~50\noww 0000“ o. o. OHH 0 fl' 00.0 00.0 n~.0 m~.0 00.0 00.0 00.0 00.0 0.H 0.H .AnmmHv axmuaafium Eoum ** .AommH. mezmHoHsno Soup 0 o a m s H H 6000 Scum it .HvsmHv .v 0:0 n mmusvam mm cm>wm 000000 00000 Hoauoc .000 .0H0 00.0 .OHH 0.0M .ONNH .00NH .000 .000 .000 .0050 .0N0 .ONH "~50\comwxmc>00 h0N.0 Nn.0 0.0m v00.0 0H.0 50.0 vh.H 00.0 000.0 000.0 000.0 H 0mg we mufisam Huucmafiummxm wow mucsumcoo Hoofimoaoonm .Avhmav .Hm um .0000 scum Q m$00.0...zsaHoa :H 0H0 sum oooH m05320300 0H 0H0 was n.o ««000H>sumaaom Hao 00906 0H mHm «#3 0H « «wcfifinamuonhdom «#3 m 0000 .cnuomom «#3 m smoky .cuummmm «#3 m mum #03 n.o mmo comm :H 2090 «03 cc mum comm :H 2090 «a: mm 2080 ocmusnauom mum comm uaanm “~.m «Hana 72 The rheological characterizations summarized in Table 5.2 are valid over the test range of 1 5 & 5 100 secdy The flow model assumes that the characterizations can be extra- polated to the shear rates encountered in the pre-pregger, the maximum of which is: = 550 sec‘1 . - _ R ‘9... ___ 2.5cm x 22sec‘1 7 H 0.1 cm lax .111 The mild ’S’ shape of the N1 versus '9 curves for the Separan solutions suggests that the slope decreases as 9 increases, i.e. b decreases. This could cause the flow model to overestimate Q at strain rates for which 0 is less than the value reported in Table 5.2. If n decreases with &, then the model will tend to underestimate Q. Equation (4.5) shows that the characteristic time, k, for any given strain rate can be calculated if the viscosity and the first normal stress difference of the fluid are known at that strain rate: w a .b-n-l k = 7% = ___—7,11-1 = :3? ‘y . (5'3) 2k 7 Note that A is not a constant with respect to strain rate unless the viscosity and first normal stress coefficient are both constants with respect to strain rate (i.e. n=1 and 73 b=2). Chmielewski (1990) estimated A for a similar 0.3 wt% PIB solution at strain rates sufficiently low that b = 2. However, strain rates as low as this were not investigated in this study. Instead, the values of A reported in Table 5.2 were calculated for 9 = 1/sec. A0 = 2%. (5.4) The rheological constants of the test fluids are reported in Table 5.2, along with some values from the literature. Note that Epon 828 and polybutene are Newtonian fluids, and no elasticity constants are reported for them. Also note that the values of a for the blends of CTBN and Epon are small compared to the fluids that are considered strongly. elastic. The PIB solution was formulated to imitate a solution studied by Chmielewski (1990), the constants of which appear in Table 5.2 for comparison. The values of a, k, and n are comparable, but the value of b reported for the present study is less than that reported by Chmielewski. This is because the values of a and b for the fluids of the present study were derived from a linear regression which included a part of the first normal stress difference curve that did not have a slope of 2. Chmielewski's value of b takes into account only that part of the curve where the 310pe is 2. The Separan AP-30 solution was formulated following the 74 recipe given in Good (1974) for a polyacrylamide solution. Reference to Table 5.2 will show that the rheological constants of the two fluids are different by two orders of magnitude for k, and three orders of magnitude for a. The polyacrylamide solution has values of a and k which make it more similar to the CTBN/Epon 828 blends than to the Separan solutions. This large difference in rheological characters may be attributed to the fact that Separan AP-30 is a copolymer of polyacrylamide and acrylonitrile. The addition of acrylonitrile will affect the hydrogen bonding density of the polymer molecules with the water in the solution. If hydrogen bonds can be considered as a weak cross-link, then changing the hydrogen bonding density or strength will change the effective molecular weight of the polymer (Davidson). From the discussion in Chapter 3, where a Tanner fluid was defined as a fluid where b = 2n, and a Boger fluid was defined as having b = 2, and n = 1, it is apparent that the PIB solution is a Boger fluid at very low shear rates, and the Separan solutions are nearly Tanner fluids. Figure 5.5 shows graphically the relationship between b and n for the fluids of Table 5.2. Figure 5.5 shows that b = 2n is a good approximation for the fluids of this study. 5.3 Experimental Apparatus and Procedure Figure 5.6 shows a schematic of the rotating die pre- 75 0; 890 4 2.032255 .. >85 05 a 02:: 030:? .2 a new a 5053 9:20:32“. ”0.0 059“. C 0.0 0.0 N0 0.0 0.0 N0 0.0 N0 v.0 _ _ _ _ p _ p _ cm ll .0 {DD 76 pregger used in this study, while Figure 5.7 shows a schematic cross-section of the die. The pre-pregger consists of a cylinder which is rotated about its axis, and a dish which holds the fluid and provides the outlet tube. The bottom of the rotor, and the inside bottom surface of the dish provide the shearing surfaces of the pre-pregger, while the volume of the dish outside the disk region provides a reservoir of fluid to feed the pre-pregger. The rotor is provided with 2 flights which serve to scrape the sides of the dish and keep partially melted polymer beads in motion against the heated surface of the dish. The flights were superfluous for runs with model fluids, but were not removed for these experiments. The hole in the center of the rotor, through which the tow would be drawn for impregnation runs, was plugged for these experiments. The dish is mounted on a stand of adjustable height, through the center of which the long outlet tube of the die must pass. Figure 5.7 shows the configuration of the pre-pregger exit tube in more detail. The die exit has a conical entry, a short cylindrical land of radius Ru, and a long cylindrical exit of much larger radius. This die tube configuration was designed to ease the tow into the small radius section for consolidation. It is more complex than the shape of the die tube studied in Chapter 4, and the choice of Rd and L,I is discussed in Section 5.5. The gap width, H, was a geometrical parameter which could 77 gZZChannel for tow fii §€?,.Ha—Slot for belt drive _7 § E / Bearing sue-.4 .ish support - adjustable height I ’ ”‘2' . . " j WWW/MA " W Shem Figure 5.6: The Rotating Die Pre-Pregger 78 // £9 571/ REE-’2... .. / /' l/ A I ‘ \~ . . \\\ ..u¢.R; = .24 cm Figure 5.7: Schematic of Rotating Die Pre-Pregger 79 be varied from 0 to 10 cm, and which could be measured to 3 0.0025 cm (0.001 in). However, the smallest gap that seemed to give reliable operation of the rotating die was 0.1 cm, and this was chosen as the smallest gap to be investigated in this study, with the other gaps chosen as simple multiples of 0.1 cm. At high rotation speeds, with fluid of low viscosity, the bottom dish of the rotating die tended to wobble on its stand t 0.006 cm (0.0025 in). The rotation rate, 0, could be set with the variable speed motor controller and measured to within 1 rpm. The rotation rates for this study were chosen to span the range of the motor, and were varied from 90 to 200 rpm (9.4 to 21 rad/sec). The low viscosity fluids were also allowed to flow out of the rotating die when w = 0 so that flow under hydrostatic conditions could be measured. The Separan solutions and the .3% PIB solution did not tend to flow when w = O. The gap widths (0.1 to 0.4 cm) and rotation rates (9.4 to 21 rad/sec), used for this study complement the work reported earlier by Good et al. (1974), who studied gap widths of 0.05 to 0.2 cm, and rotation rates of 4 to 10 rad/sec. The experimental procedure was as follows: 1. load fluid into the dish 2. adjust the height of the dish to set the gap 3. start the motor and adjust the rotation rate 80 4. fine-tune the gap width 5. remove the clamp from the outlet tube to allow fluid flow to start 6. wait for steady flow 7. start timer and start collecting flow into sample beakers 8. take four time and weight samples 9. stop rotation and replace clamp 10. report results as graphs of weight by time The two Separan solutions and the PIB solution had high enough viscosity that they did not flow out of the die tube when the pre-pregger was not running, so the clamp was not used with these fluids. Steady flow was judged by observing that the column of fluid running from the outlet tube to the weighing beaker was of a constant cross section, and that any pulsations in the flow were regular and consistent over a period of about 30 seconds. The total time to collect one sample was typically 2 minutes for the less elastic fluids, and about 30 seconds for the highly elastic fluids. The beakers and contents were weighed in a balance that was accurate to 30.1 mg. The dish was replenished with fluid either during a run or at the start of each run. With the gap width set at 0.2 cm, the volume of fluid contained in the gap would be 4.1 cm?, and the volume contained in the entire dish would be 69 cm?. At volume flow rates of 0.2 cmP/sec, the fluid 81 between the disks would have a residence time of about twenty seconds, and the volume in the dish would be exhausted in 300 seconds. 5.4 Experimental Results Each of the experimental runs generated a set of data of the type shown in Figure 5.8, where accumulated mass of the sample is plotted versus the elapsed time of the run. If the flow rate is constant throughout the experiment, then the four points form a straight line, which passes through the x-axis at the start of the experiment. The slope of this line is the average mass flow rate for the run. However, the graphs of the data from the test fluids show that the data do not form straight lines, and so it is concluded that the experiments were unsteady. Both Separan solutions and the PIB solution had a very great tendency to climb the rotor of the rotating die, and it is speculated that the rod-climbing flow competes with the disk flow for the available fluid. If the balance is tipped in favor of the rod-climbing flow, then the disk flow may be starved of fluid as the run progresses, as depicted in Figure 5.9. Therefore, the mass flow rate for each run was taken to be the slope of the line linking the first two data points, ignoring transient behavior at start-up, and the possibility of starvation at the end. The volume flow rate was calculated from the mass flow rate and the density of the 82 0: 0N_. — 900 05 mczmfiom .0033 "0.0 059“. 000 08$ 00000.0 8 H om 00 ow — p _ _ 0N 000. 26: 000E u 000.0 0| cm 6 ‘ssew palelnwnoov 83 fluid from Table 5.2. Experiment data and calculations are reported in Appendix F. Flow rate experiments were performed with two Newtonian fluids: Epon 828 and polybutene. Two slightly elastic fluids with constant viscosity were also run: CTBN/Epon 828 blends. The three elastic fluids included a constant viscosity fluid: 0.3wt% PIB solution: and two shear-thinning fluids: fresh and aged Separan AP-30 solutions. Flow rates for both Newtonian fluids were calculated for the case of 0 = 0, i.e. flow induced through the pre- pregger by gravity only, to compare with the flow rate produced by the action of the rotor. The polybutene had a maximum flow rate of 0.021 cm’/sec, which is small compared to the flow rates of the elastic fluids, but still an order of magnitude larger than the flow rate recorded (.0012 cmP/sec) when w = 0. Likewise, the Epon 828 had a maximum flow rate of .023 cmP/sec when the die was running, and .0046 cm?,/sec when w = 0. Neither of these fluids was pumped through the rotating die at rates that would allow sufficient fluid delivery to a tow being drawn through the pre-pregger at a rate of 20 cm/sec (see Chapter 3). Since the CTBN/Epon blends showed rod-climbing behavior while being mixed, it was expected that they would be pumped by the rotating die at greater rates than the Newtonian fluidl. The flbw rate data are summarized in Appendix F. 84 The flow rates for the 23% blend were 0.012 cm3/sec maximum, with a gravity flow rate of 0.002 cm3/sec. The flow rates for the 40% blend were 0.015 cm3/sec maximum and 0.0021.cmP/sec under gravity. These flow rates are comparable to those attained with the Newtonian fluids, and are not satisfactory rates for the task of supplying resin to a tow. There is considerable scatter in the data, and although the trend is generally to increase Q as 0 increases, there is no clear relationship between H and Q. The volumetric flow rates for the PIB solution are plotted against the rotation rate of the die in Figure 5.9. The maximum flow rate was 0.23 cmP/sec, which is nearly equal to the goal of 0.24 cma/sec. This maximum flow rate is produced at the highest 0 and the smallest H with Q generally decreasing as 0 decreases, and H increases. There is no indication of a critical gap width, as discussed in Chapter 4. Figure 5.10 shows the experimental flow rates for the fresh Separan solution. The maximum flow rate was 0.165 cmP/sec, obtained at the highest value of 0, and at the intermediate value of H = 0.2 cm. As with the PIB solution, Q decreases with decreasing 0. However, the values of Q for the case of H = 0.1 cm are less than those for H = 0.2 cm, at low rotation rates, while values of Q for H = 0.4 cm are less than either of the other cases. This would seem to show that there is a critical gap width 85 mm E0 0.0"... 4 E0 N0": I E0 to": U 0.0 .0; 0.0 5.. £00 20. 26.0 0.0 059”. 00000.. 3 00 0F 2 0 o . . _ . 0 00° 00.0 4 4 4 K +000 .0 m .. -000 v D m D D S -000 x D 00.0 00.0 86 NN Eu won: 4 .8 Non: .. so to": 0 00.0.0. 00.0000 :00: 00.. 00.00 00.0. 30$ ”000 059“. 00000. . 3 0m 0 H 0% 3 NH 0 H 0 P _ _ _ _ cone 4 4 4 .Q I 4 a a m 50.0 wOFd -0fi0 I 009/9 v UJO ‘0 H000 0.0N.0 00.0 87 between H = 0.1 and 0.2 cm. Values of the flow rate are plotted against rotation rate in Figure 5.11 for the aged Separan solution. The maximum Q is 0.27 cmP/sec, which is adequate to impregnate 20 cm/sec of tow, as discussed in Chapter 3. Again, the top flow rates are achieved at narrow gaps and high rotation rates. Like the data for the fresh Separan, Q decreases with decreasing w. However, the data for the aged Separan do not show the existence of a critical gap width, as the values of Q decrease monotonically with an increase of H. 5.5 Experimental Discussion The model proposed in Chapter 4 predicts that flow rates for Newtonian fluids will be zero. However, the model does not take flow induced by gravity into account. Two Newtonian fluids, Epon 828, and polyisobutylene were loaded into the pre-pregger and allowed to flow through by gravity (0 = 0). Small but measurable flow rates were produced, 0.0046 cm3/sec for Epon 828, and 0.0012 cm3/sec for polybutene. When the volumetric flow rate was measured with the pre-pregger running (0 > 0), flow rates produced were an order of magnitude larger, at 0.023 cma/sec for Epon 828, and 0.021 cm3/sec for polybutene. These flow rates are very small compared to the flow rates obtained with the elastic fluids, but they are not zero, as predicted for Newtonian fluids. It may be that the fluid has some elastic 88 :5 so": 4 :5 Non: - so to"... a dad? cmtmomw comm .2 Emu 9m. 26.... “In 239“. 0365 .3 mm ON mr or 3 NF 0.. w _ _ p b — _ coco 4 Sod 4 I ford D -m...o .. .86 D U ENG omd 088/8 v mo ‘0 89 character that is too small to be measured by the RMS-BOO, or it may be that flow is taking place because of secondary flows such as those reported by Blyler (1966). In either case, the model does not predict the flow rate behavior of the Newtonian fluids. The flow rate data for the CTBN/Epon 828 blends have so much scatter that the quality of the fit to the model cannot be judged, but the fit can be improved by translating the model curves upward by the amount of the gravity-induced flow rate. It must be noted that although the CTBN/Epon 828 blends have measurable elastic properties, and that they flow through the pre-pregger at rates that match the prediction of the model, these flow rates are still very small, being of the same magnitude as those of the Newtonian fluids, and an order of magnitude below those of the strongly elastic fluids. The comparison between the experimental data and the model calculations for the 0.3 wt% PIB solution is shown in Figure 5.12. Note that the value of I“ is not the length of the restriction in the die, 0.64 cm, but is taken as the full length of the die and outlet tube, 11.4 cm. As discussed in Chapter 4, the value of 1% has a very great influence upon Q. The choice of 12 and R.d for the experimental outlet die was not obvious. If Ra is chosen as the radius of the narrowest part of the outlet die, and I“ is chosen as the total die length, then the model Eo You: 4 80 won: I So no": 0 Eu 4.2 u 3 .80 wad H cm £5 md n m .91 $5 ad .2 EmEtmaxm 5:5 omEQEoo .2022 Nfim 2:9“. 0mm: . 3 om 3 9 i «F or m 90 _ _ _ _ _ o { ‘ 4 I . v.0 l rNd 5.0 .46 -md -06 Lao 039/9 v mo ‘0 :6 Non: -md _mooE m6 91 calculations are the same order of magnitude as the experimental data of this study. Model values are from 2 to 10 times as great as experimental values. The model assumes that the rheological constants a, b, k, and n are constant with strain rate, as discussed in Section 5.2. If the strain rates in the pre-pregger are such that the constants vary appreciably from those reported in Table 5.2, then the model may over or under-predict Q at high rotation rates and low gap widths. The model also does not account for the tendency of the PIB solution to climb the rotor, and so will over-predict flow rates if the disk region is being deprived of feed, as discussed in Section 5.4. The model predicts that H; for the PIB solution will fall at 0.07 cm (See Figure 5.13), which does not lie within the scope of the experimental data. Figure 5.14 shows the comparison between the model predictions and the experimental data for the fresh Separan AP-30 solution. The model for H = 0.42 cm matches the data values quite well at low 0, but begins to deviate from data above 0 = 16 rad/sec. The model for H = 0.21 cm does not match the data values, but gives values of Q a factor of 2 higher than the data. The explanation of starvation of the disk flow by the rotor-climbing tendency of the fluid may again be invoked to explain this phenomenon. An increase in 0 will tend to increase the flow rates through the disk as well as up the rotor. An increase in gap width would not 92 03:09 om n3.Eo v. E u 3 .28 $6 H Um .Eo md n m 62:: _mEmEtmuxo .2016 m2m> “mtm 059... Eu .1 mmd and mud om_.o 0 rd 0 rd mod ood :Emamw 5mm... cmtmamw comm (D OBS/8 v 1110 ‘0 93 Eo deHI 4 Eu homo": I So 49.0.1.1 D So 4.: u 3 .Eo sud u um .80 md u m . 5.3% $5 m 2mm: toe EmEtmaxo SE 83an0 Enos. ”36 pint 0mm: . 3 4 up 2 a Na em 8 8 .F . . 85 -mod .o_..o -920 -ONd TmNd Eo homdn I and OBS/F2 . . [110 ‘n 94 affect the rod-climbing flow, but would both decrease the volumetric flow rate through the disk by decreasing the strain rate that is driving flow, and increase the area by which new fluid can be taken into the disk flow around the periphery. Comparison of the model and data in Figure 5.14 suggests that a gap of 0.42 cm is wide enough to allow the disk flow to compete for fluid with the rod-climbing flow, because the model matches the data for low rotation rates. The balance apparently shifts at about 0 = 16 rad/sec, where the experimental flow rates begin to fail to keep up with the model flow rates. The rod-climbing field apparently dominates at all the narrower gap widths. Figure 5.13 shows that the optimum gap width for the fresh Separan solution is just greater than 0.10 cm, and this would seem to be confirmed by the experimental data which show He to be between 0.1 and 0.2 cm. The comparison between the aged Separan experiments and model calculations is shown in Figure 5.15. Again, the model matches the data well for high gap widths and low rotation rates, but fails to match the data for narrow gaps and high 0. The fact that the elastic coefficient, a, for the aged Separan solution is nearly twice that of the fresh solution may be the reason that the model predictions for the aged solution are better than those for the fresh solution. The higher elasticity may help the disk flow compete with the rod-climbing flow up to smaller gaps and higher rotation 95 E0 Yo": 4 E0 NdHI I E0 _..oHI D Eo 4.2 n 3 .80 vmd u om .Eo md n m .5833 5.3 m comm .2 EmEzmaxm 5? 8.8.80 Enos. him 9:9“. 0mm: .3 mm ON mF 09 v... NF or m _ _ i p _ _ p 90.0 .36 1.9.6 -ONd -mNd omd OBS/9 . . we ‘n 96 rates. Figure 5.13 shows that the model predicts that Hc = 0.09 cm for the aged Separan solution. This does not contradict the experimental data, which show that Hc'< 0.1 cm. Figure 5.16 shows the model calculation compared to data of Good, et a1. (1974) for an 18 wt% solution of PIB in motor oil. Reference to Table 5.2 will show that this fluid has a low but nearly constant viscosity (k = 110 dyne sec/cm”,11 = .96), and that the elasticity coefficient, a, is equivalent to that for the CTBN/Epon blends. Flow rates of this solution are comparable to those for the CTBN/Epon blends, being less than 0.01 cm3/sec. The model predicts volumetric flow rates lower than the data shown, especially at the large gap widths. Also, the predicted H? is less than that shown by the data. Good (1974) mentions the tendency of the fluids to climb over the top disk of his centripetal pump at large m, and other authors (Kocherov, 1973 and D'Amato, 1974) have noted that the centripetal pump runs successfully only when the problem of feeding the fluid or pellets into the shear zone is addressed. Figure 5.13 indicates that potential flow rates with the pre-pregger go as high as 0.8 cm3/sec for the Separan Ap-30 solutions, and 6 cm3/sec for the PIB solution at the critical gap width and w = 20 rad/sec. These rates would be sufficient to deliver fluid to a tow drawn through the pre-pregger at rates of from 30 to 100 97 ome u 3 4 0mm? u a: u a .50 u m .86 u c .o: u x :5 «23 u 3 .50 «NS n um so 3 n m .32 s 9": .88 go: 9% 5? 8.358 68.2 58 9:9”. 920 L 60 .I I 0mm? u 9.0 00.0 _ D 00.0 000.0 :86 .83. R86 -306 .83 -83 .83 -806 $85 .056 P as 068/8 v UJO ‘0 98 cm/sec, far in excess of the 20 cm/sec deemed sufficient for success, as discussed in Chapter 3. Chapter 6 Conclusions The model developed for the rotating die pre-pregger makes use of the Criminale-Ericksen-Fibley model with the second primary normal stress coefficient equal to zero. This representation provides a practical description of elastic effects for the problem of flow through the pre- pregger. The strain rate tensor S within Regions I and II (see Figure 4.1) was estimated by assuming a velocity field which satisfies continuity and no slip boundary conditions. The strain rate in the exit tube was predicted by solving the equation of motion for a power-law fluid. An analysis of the mechanical energy balance shows that the work transferred to the pre-pregger across the rotating disk surface by the shearing action on the fluid is entirely balanced by the viscous dissipation due to the deformation of the fluid caused by the swirling motion of the fluid between the disks in Regions I and II (see Figure 4.1). The fluid entering the gap from the reservoir is spun up by the rotor, and the work transferred to the pre-pregger across the circumferential surface together with the stored elastic energy of the fluid balances the remaining dissipation 99 100 effects between the disks and in the exit tube. Thus, greater efficiency in feeding fluid to the gap may increase the flow rate of fluid through the pump. The empirical fluid parameters that describe the viscous and elastic nature of the fluid (a, b, k, and n) can be obtained through rheological experiments. For the fluids studied, the ratio of b to n was shown to be approximately equal to two (see Figure 5.5). Because the flow model is very sensitive to the values of the rheological description, it is important that the above parameters be determined accurately for the full range of strain rates which will be encountered in the pre-pregger. The ratio, a, of the radial velocity of the fluid in the pre-pregger to the tangential velocity gives a dimension- less measure of the spiral nature of the flow field. Increasing values of a indicate that the flow is more radial, i.e., that a particle of fluid travels in fewer circles on its way through the pre-pregger. The qualitative behavior of a at small values of H/Rd was found to depend on the sign of the fluid parameter l== b-(1+n). For instance, for H/Rd-+ 0 and l > 0, the flow ratio becomes unbounded; however, for l<< 0, a e 0. For l== 0, a has a finite, non-zero value for H/Ra-+ 0. The model shows that the volumetric flow rate, Q, increases monotonically with the rotation rate, 0. The 101 model also shows that Q increases with the gap width, H, until H reaches a critical value, He, above which Q decreases with increasing H. As H increases from zero to lg, Q increases because the energy dissipation in Region I, D‘ 1, decreases. As H increases from He, Q decreases again, because the elastic term of the mechanical energy balance, E, decreases more quickly than 0; decreases. The flow model produces curves that fit the data of Good et al. (1974) reasonably well, although different constitutive equations and simplifying assumptions were used. The flow model can be made to fit the experimental data of this study by choosing the geometric parameter L6 to be the length of the entire die tube, rather than the length of the die restriction only (see Figure 5.6). The flow rates of the Newtonian fluids (Epon 828 and polybutene) through the pre-pregger were four to ten times as great while the pre-pregger was running, as when the fluid was draining by gravity only. The flow rates of the slightly elastic fluids (CTBN/Epon 828 blends) through the pre-pregger were the same magnitude as the flow rates for the Newtonian fluids. These fluids had measurable elasticity coefficients, and the flow model could predict the flow rates reasonably well. The highly elastic fluids (polyisobutylene and Separan 102 AP-30 solutions) flowed through the pre-pregger at rates which would be adequate to impregnate a 3K carbon tow drawn through the pre-pregger at 20 cm/sec (i.e. Q > 0.22 cm3/sec). The flow model predicted flow rates of 6 cm’/sec for the polyisobutylene solution, and 0.8 cm3/sec for the Separan solutions. The flow rates predicted by the model may be brought more into line with those obtained by experiment, by investigating the following three phenomena. First, the rheological experiments were performed at strain rates below 100 sec“, and the fluid characteristics a, b, k, and n were assumed to be constants for the strain rates produced by the pre-pregger, although they were as great as 550 secdu Second, the fluids’ tendency to climb the rotor may have set up resistance to fluid entering the gap of the pre-pregger. Third, the model assumed steady state operation, and the experiments showed that the operation was not at steady state on the time scale that samples were being taken. The mass flow graphs were not linear, and pulsing behavior was observed for all three elastic fluids. Chapter 7 Recommendations Although the flow model assumed a steady state of flow in the pre-pregger, experimental data (see Figure 5.8) showed that the flow had not generally reached steady state within the time frame of the run. In order to study steady state flows, the pre-pregger should be provided with a reservoir of fluid, and a means to feed the fluid directly into the gap. Some measure of whether steady state has been reached is needed. Steps should also be taken to prevent the fluid from climbing the rotor, as the rotor climbing effect may obscure experimental results. When the rotating die pre-pregger was built, the die was designed with a conical section in imitation of pre-preggers with stationary dies. The flow model assumed that the flow in the die tube was a fully-developed laminar tube flow. Thus, the geometrical parameters of the die tube were simply the radius, Ru, and the length, 13 of a cylinder. When the model was compared to the experiments, the choice of dimensions from the pre-pregger to enter in the calculation was unclear. In order to more clearly evaluate the potential of resin melts to be pumped by a rotating die, the 104 105 experiments should be designed to reflect the less complex geometry of the model. The fluids of the present study were all of the class of "Tanner" fluids (i.e., the first normal stress coefficient exponent, b, is approximately twice the viscosity exponent, n). A theoretical class of fluids was identified in the parametric study of the flow model for which the relationship b > 1+n is characteristic. Fluids of this class should be studied in the pre-pregger, because the model predicts that the behavior of the velocity ratio, a, changes drastically from that of the "Tanner" fluids at small gap widths. The pre-pregger was designed and built for the purpose of making pre-preg. The present study was undertaken in order to better understand the operation of the pre-pregger without the tow. It is recommended that future studies focus on the operation of the pre-pregger with fiber tow and polymer melt. The question of whether the tow would be impregnated or only coated by the pre-pregger is of interest, as is the question of which parameters may be manipulated to assure the tow is impregnated by the pre- pregger, rather than being merely coated. APPENDIX A KINEMATIC TENSORS EQUATIONS OF MOTION VELOCITY VECTORS STRAIN RATE TENSORS EQUATIONS OF MOTION: Cylindrical coordinates, variable viscosity. Equation of Continuity 12 b—f: = -o(V°y) 39 1 3 E + f -a—f(prvr) + "ill--I g—ewve) + 37,wa = 0 Equation of Motion Dv p 17;;- = -Vp +[V°I] + 09 r component: [avr av: ve av, v92 avr] _ P W+Vrfi+?§€”r—+Vs az - ap 1 3 1 3729 1'88 37:2] ’5? + [f 31:07...) + f as ' T + “—32 + 99. a component: 3v 3v ‘1 av \rv 3v __2 __fi _§._£_. r 9 __2 = .,[3t.vfl,r+,_.39 r+v,az] l 3P l 3 l 3799 379. f E + [g if‘rz're) + f as + 32 + 999 2 component: [avg av: v6 av: av!) __ a 37. 37.. ”Biz, + [% g—r(r'rn) +% 33 + __az ] + pg: Substantial Time Derivative %=%+W°W> 106 ROTATING DIE PRE-PREGGER, DISK REGION Cylindrical coordinate v; = vs S : v2 vr v3 v I Velocity Vector 32 fII 118 II Tre II 7:9 II T :2 Cone and Plate 1’: (M 68 68 0:- St I I _ II 192 _ I A :5 V TII 81' 6S ZnS-il 13% 796:" a? = 0 Km = 0 36 = 0 av” [—a£ + viscometer avI I r 32 J 119 APPENDIX C THE MECHANICAL ENERGY BALANCE D = III 16V: dV =DI+DH+D III Dissipation in disk region. DI = Zn 1:]: [I : Vy‘ r dr dz Dissipation in transition region. DI = Zn 1:]: I” : Vy“ r dr dz I Dissipation in die tube region. D = 21r raj“ 1‘” : VvIII r dr dz I a o " ’ II Work put in by rotating disk. W1 = -21r IR [1:9v81“o r dr 0 Work brought in by entering fluid. 1! W = 21!] [Rave].-. dz 0 2 120 (C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) . ave from Appendix A: r e “ n 32- V9 - rw(1-fi) and: n = k Frag)“ R run 33 W1 = -21r [o -k (WT) r dr _ 27R” k 13.92 .. - " n+2 (H) (C 9) W2 = 2" R I: [11:9 v9]r-R dz 0 ave av: from Appendix A: 1,9 = ‘ ‘1’: if 32 2 v9 — rw( -fi) _ _ 3Q E _.Z_ V. ‘ irrH H [1 H] b-2 and: ‘1’: = a[%] _ Rw b ° W.-'Qa[a) (C10) 121 D1 = 21 EL: “[[%J’ + 4[\%]6 + [ZEJ’Jr dr dz '5 R v ave 2 v 2 3v 2 —21r]] 15% [T77] + 4[—r‘—'] - [32'] rdr dz ORG R Note: First term of first integral balances with [W1] . Rd + D ‘ - E (C.11) I I)I = [w1 a R V: ave 2 V: 3V: 2 -E=21III ‘I’zT’ [7]+4[T]' If] rdrdz = Q g [56]” [[631” 3] “'1‘“ 122 [[:J““-1] ] — [R1] [1- [ )J + 11.53] mm 123 2 ' fl' 8v 2 v 2 av Du=21rjl [[333] +2[-r5] +2[ar‘f] OO 2 2 (av, avr) avt] + 'a—f'i'fi + 2 E ] rdr d2 “a O 0 Note: First term of integral balances with [W1 DII = [W]. R d I + D11 0 a nd V 2 3V 2 3V 3V 2 DII‘ = 21’ [I <71) [ 2[%] + 2[ arr] + [a—r' + 37:] o o from Appendix A: <1 '1 II I ;l .8 Q :1: =3 ‘5’ +-+ "’ H FIN h H I N :5: + :3 5.: H Wlfi \_l . Z L___J EIN /_\ H I um: :nN \_J 124 D . _ 2g’ 2k [Rwy-1 1_3_ n(3n+1)’ 1 ‘ II — N H3 n+1 H 70 (n+1)3 Rd + 9 + 54n + 264n= + 432n° + 231n‘ + 90n’ [31]” 5 (n+1)’ (2n+1) Re 4 + 1 + l4n +256n’ + 29113 + 31n ] ((2.15) 4 (n+1) (3n+1) (Sn-+1) “6 av, avr DIII = 21rLd 1“ 73—1,— + ?z' r dr 0 from Appendix A: Q 3 +1 22“ n r n ’ v = 1- — .. ‘ ”R62 n+1 [ [Rd] _"i and: _ [avg] _ [am]. Tr: " 3r - 3r [3“] (”'1 32 << 3r _ "*1 3n+1 " DIII - Zde R3n+1 [ fin J ((2.16) d 125 D; on Q2 Du. (I Q2 DIII a Qn+1 -E 0: Q -4% a Q Let: c“:1 Q2 = DI‘ + DH‘ 62 Qn+1 = DIII 63.: E + W2 Then: (“21 Q2 + c":2 0"” = C, Q élQ+ézQ"-63=o Non-dimensionalize with: 9 II ‘F ”U u e :1: .0 G) H ”In: ,0 ll F'CD I w 126 (positive) (positive) (positive) (positive) (positive) (C.17) (C.18) (C.19) When n y 1: C1 = 5(38-n") 3.. (145“.-.) + sat 111-1) [1'35“] 8 13 n(3n+1)2 . + 3(n+1)ql[ 3 7° (n+1)’ 6 9 + 54n + 264n2 + 432n? + 231n‘ + 90ns Bz I + 5 (n+1)’ (2n+1) 1 + 14n + 56n2 + 29113 + 31n‘ (C 20) 4 (n+1)” (3n+1) (5n+1) When n = 1 c. = :- 6. (166,") + .3— 1MB.) + 3?. [:3- 6.‘ + 9 6: + ——%31 I I (c.21) 127 APPENDIX D COMPUTER PROGRAM LISTING C 10 0.000CO0000OOOOOOOOOOOOOOOOODOOOOOO0000O Nancy Losure Oct 3, 1990 Program File: ARNOLD.FOR Data Input File: ARNOLD.DAT Data Output File: ARNOLD.OUT THIS PROGRAM SOLVES A NON-LINEAR EQUATION FOR THE DIMENSIONLESS FLOW RATE FROM A CENTRIPETAL PUMP, BY INTERVAL HALVING INPUT: AA IS THE ELASTICITY COEFFICIENT [DYNE*SEC6B/CM62] AB IS THE ELASTICITY EXPONENT AK IS THE VISCOSITY COEFFICIENT [DYNE*SEC6N/CM~2] AN IS THE VISCOSITY EXPONENT BRD IS THE RADIUS OF THE DIE [CM] BR IS THE RADIUS OF THE DISK [CM] BL IS THE LENGTH OF THE DIE [CM] BW IS THE ROTATION RATE OF THE DISK [RAD/SEC] BETAR IS DISK RADIUS/DIE RADIUS BETAH IS GAP WIDTH/DIE RADIUS BETAL IS DIE LENGTH/DIE RADIUS GG IS (a/k) (omega)**b-n ATOL IS THE TOLERANCE WITH WHICH ALPHA AND ALEPH ARE REQUIRED TO MATCH NW IS THE NUMBER OF VALUES OF OMEGA WHICH WILL BE CALCULATED NH IS THE NUMBER OF VALUES OF BH WHICH WILL BE CALCULATED BH(X) IS THE ARRAY WHICH CONTAINS A LIST OF GAP WIDTHS TO BE CALCULATED [CM] ANSW(X) IS THE ARRAY WHICH CONTAINS THE FLOW RATES CALCULATED FOR EACH BH(X) [CM63/SEC] ALPHA IS THE TRIAL VALUE OF THE DIMENSIONLESS FLOW RATE ALEPH IS THE CALCULATED VALUES OF THE DIMENSIONLESS FLOW RATE C1, C2, C3 ARE INTERMEDIATE CALCULATIONS IMPLICIT REAL*4(A-H, O-Z) DIMENSION BH(100),BW(100),ANSW(100) OPEN (UNIT=5, FILE=’ARNOLD.DAT’) OPEN (UNIT=6, FILE=’ARNOLD.OUT',STATUS=’UNKNOWN’, ACCESS='append') OPEN (UNIT=7, FILE=’CON') READ DATA AND WRITE DATA AND HEADINGS TO OUTPUT FILE. WRITE(6,9000) 128 12 14 20 60 65 7O READ(5,*)AA,AB,AK,AN,BR,BRd,BL,ATOL,NW,NH WRITE(6,9001)AA,BR,ATOL,AB,BRD,NW,AK,BL,NH,AN DO 12 I=l,NW READ(5,*)BW(I) CONTINUE DO 14 I=l,NH READ(5,*)BH(I) CONTINUE BETAR = BR/BRD BETAL = BL/BRD WRITE(6,9002)BETAR,BETAL DO 120 K=1,NW DO 100 I=1,NH ANSW(I) = 0.000001 ALPHA = 0.0001 J = 0 BETAH = BH(I)/BRD GG=AA/AK*BW(K)**(AB-AN) if(an.EQ.1)then c1a= 4*alog(pr) else Cla= 4/(1-AN)*(l-BETAR**(AN-l)) endif c1b= 1.60/(3-AN)*BETAH**2*(1-BETAR**(AN-3)) c1c= 52*(1+3*AN)**2*AN*BETAH**4/105/(1+AN)**4 c1d= 8*(9+54*AN+264*AN**2+432*AN**3+231*AN**4+90* AN**5)*BETAH**2 /15/(1+AN)**4/(1+2*AN) c1e= 2*(1+14*AN+56*AN**2+74*AN**3+31*AN**4)/3/ (1+AN)**3/(1+3*AN)/(1+5*AN) c1 = (c1a+c1b+c1c+c1d+c1e)/ph C2 = 2*BETAL*(2*BETAH**2*(1+3*AN)/3/AN)**AN C3 = AA/AK*(BW(K)/BETAH)**(AB-AN)*((AB+1)/AB-1/pR** & (AB)/AB)*BETAR**AB IF(AN.EQ.1)THEN ALEPH = C3/(C2+C1) ANSW(I)=ALEPH*2.094*BRD**2*BW(K)*BH(I) WRITE(6,9003)BW(K),BH(I),ANSW(I),GG,BETAH,ALEPH GO TO 100 ENDIF START INTERVAL HALVING FIRST FIND LOW AND HIGH GUESSES NFLAG=0 Gl=.00001 G2=1 FAl=C1*G1+C2*Gl**AN-C3 FA2=C1*GZ+C2*G2**AN-C3 IF((FA1*FA2).GE.0.AND.NFLAG.NE.1) THEN 129 0(1(3 100 120 9000 9001 9002 9003 9004 G2=G2*2 GO TO 70 ELSE NFLAG=1 ENDIF G3=(Gl+GZ)/2 FA3= C1*G3+C2*G3**AN-C3 write(7,*)'g1’,g1,’ fal',fa1 write(7,*)’g2’,g2,’ fa2’,fa2 write(7,*)’g3',g3,' fa3',fa3 IF(ABS(FA3).LT.ATOL)THEN ALEPH=G3 ANSW(I)=ALEPH*2.094*BRD**2*BW(K)*BH(I) WRITE(6,9003)BW(K),BH(I),ANSW(I),GG,BETAH,ALEPH GO TO 100 ELSE IF((FA1*FA3).LT.O)THEN Gl=G1 G2=GB G3=(Gl+GZ)/2 GO To 65 ELSE Gl=G3 cz=cz G3=(Gl+GZ)/2 GO TO 65 ENDIF ENDIF CONTINUE CONTINUE FORMAT(3X,’"NANCY LOSURE',lOX,’ARNOLD.FOR’,lOX, & ’OCt. 6, 199o"') FORMAT(/,/,3X,’" a="’,f8.2,’,’,10x,'" R="',f8.4,’,’, 10x,'"atol="',f8.6,’,',/,3x,'" b="’,f8.2, I’I'lox’lflRd=fll'f8.4'l'l’10x'lfl NW="I’ IZ,/,3x,’" k="’,f8.2,’,',10x,'" L="',f8.4, ’,',lOX,’" NH="',12,/, 3x,'" n="’,f8.2,',',10x) FORMAT(1X,'"R/Rd="',F8.2,’,’,7X,'"L/Rd="',F6.2,/,/ 2.2395232“ & 3X’IROMEGAfll’6x'IflHflI'lsx'IflQfll'3x’Incfll' & 5X,’"H/Rd"’,5X,’"ALPHA"') FORMAT(1X,F6.2,’,’,5X,F6.2,’,’,5X,F12.8,’,’,5X,F8.4, & ',’,5X,F8.4,’,',5X,F8.4) FORMAT(5x,'Gl',3x,f12.6,10x,’62’,3x,f12.6,lOX,’G3',3X, & F12.6) STOP END 130 "NANCY LOSURE " a=" 432.00, " b=" 1.73, " k=" 502.00, " n=" 0.96, "R/Rd=" 16.67, II OMEGA I! I. H II 8.00, 0.02, 8.00, 0.04, 8.00, 0.06, 8.00, 0.08, 8.00, 0.10, 8.00, 0.12, 8.00, 0.14, 8.00, 0.16, 8.00, 0.18, 8.00, 0.20, 8.00, 0.25, 8.00, 0.30, 8.00, 0.35, 8.00, 0.40, 10.00, 0.02, 10.00, 0.04, 10.00, 0.06, 10.00, 0.08, 10.00, 0.10, 10.00, 0.12, 10.00, 0.14, 10.00, 0.16, 10.00, 0.18, 10.00, 0.20, 10.00, 0.25, 10.00, 0.30, 10.00, 0.35, 10.00, 0.40, 12.00, 0.02, 12.00, 0.04, 12.00, 0.06, 12.00, 0.08, 12.00, 0.10, 12.00, 0.12, 12.00, 0.14, 12.00, 0.16, 12.00, 0.18, 12.00, 0.20, 12.00, 0.25, ARNOLD.FOR " R=" 2.5000, "Rd=" 0.1500, " L=" 0.5000, "L/Rd=" 3.33 "Q" "G" 0.0972, 4.2674, 0.2106, 4.2674, 0.3015, 4.2674, 0.3549, 4.2674, 0.3727, 4.2674, 0.3653, 4.2674, 0.3439, 4.2674, 0.3164, 4.2674, 0.2876, 4.2674, 0.2599, 4.2674, 0.2013, 4.2674, 0.1581, 4.2674, 0.1265, 4.2674, 0.1032, 4.2674, 0.1444, 5.0673, 0.3127, 5.0673, 0.4478, 5.0673, 0.5273, 5.0673, 0.5540, 5.0673, 0.5434, 5.0673, 0.5118, 5.0673, 0.4710, 5.0673, 0.4283, 5.0673, 0.3872, 5.0673, 0.3001, 5.0673, 0.2357, 5.0673, 0.1887, 5.0673, 0.1540, 5.0673, 0.1993, 5.8311, 0.4318, 5.8311, 0.6186, 5.8311, 0.7287, 5.8311, 0.7660, 5.8311, 0.7516, 5.8311, 0.7081, 5.8311, 0.6520, 5.8311, 0.5930, 5.8311, 0.5362, 5.8311, 0.4158, 5.8311, 131 MAY 4, 1991" "atol="0.001000, u NW=" 8 " NH="l4 "Ii/Rd" 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, DI ALPHA " 12.9005 13.9685 13.3320 11.7700 9.8874 8.0771 6.5170 5.2461 4.2384 3.4477 2.1366 1.3979 0.9591 0.6847 15.3193 16.5901 15.8395 13.9902 11.7589 9.6111 7.7585 6.2481 5.0497 4.1090 2.5479 1.6676 1.1444 0.8171 17.6286 19.0933 18.2344 16.1118 13.5481 11.0783 8.9465 7.2073 5.8266 4.7423 2.9419 12.00, 12.00, 12.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 14.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 16.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 18.00, 20.00, 20.00, 20.00, 20.00, 20.00, 20.00, 0.30, 0.35, 0.40, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.20, 0.25, 0.30, 0.35, 0.40, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.20, 0.25, 0.30, 0.35, 0.40, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.20, 0.25, 0.30, 0.35, 0.40, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.3267, 0.2616, 0.2135, 0.2619, 0.5673, 0.8129, 0.9580, 1.0073, 0.9888, 0.9319, 0.8583, 0.7808, 0.7062, 0.5479, 0.4305, 0.3448, 0.2815, 0.3317, 0.7187, 1.0299, 1.2141, 1.2770, 1.2540, 1.1822, 1.0890, 0.9909, 0.8965, 0.6957, 0.5468, 0.4381, 0.3576, 0.4086, 0.8853, 1.2690, 1.4963, 1.5743, 1.5463, 1.4581, 1.3435, 1.2227, 1.1064, 0.8588, 0.6752, 0.5410, 0.4416, 0.4924, 1.0669, 1.5295, 1.8039, 1.8984, 1.8651, 5.8311, 5.8311, 5.8311, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 6.5660, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.2770, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 7.9678, 8.6412, 8.6412, 8.6412, 8.6412, 8.6412, 8.6412, 132 2.0000, 2.3333, 2.6667, 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 1.9261 1.3221 0.9441 19.8506 21.5023 20.5396 18.1546 15.2715 12.4921 10.0916 8.1322 6.5760 5.3534 3.3223 2.1757 1.4937 1.0668 22.0006 23.8333 22.7708 20.1323 16.9406 13.8618 11.2014 9.0288 7.3026 5.9459 3.6913 2.4179 1.6603 1.1859 24.0895 26.0983 24.9390 22.0549 18.5635 15.1941 12.2810 9.9012 8.0097 6.5228 4.0506 2.6538 1.8225 1.3019 26.1256 28.3062 27.0530 23.9296 20.1466 16.4939 20.00, 20.00, 20.00, 20.00, 20.00, 20.00, 20.00, 20.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 22.00, 1.7591, 1.6212, 1.4757, 1.3354, 1.0369, 0.8153, 0.6533, 0.5334, 0.5828, 1.2631, 1.8110, 2.1363, 2.2487, 2.2097, 2.0846, 1.9215, 1.7493, 1.5833, 1.2296, 0.9670, 0.7750, 0.6328, 8.6412, 8.6412, 8.6412, 8.6412, 8.6412, 8.6412, 8.6412, 8.6412, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 9.2992, 133 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 0.1333, 0.2667, 0.4000, 0.5333, 0.6667, 0.8000, 0.9333, 1.0667, 1.2000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 13.3347 10.7528 8.7001 7.0860 4.4016 2.8842 1.9810 1.4152 28.1153 30.4639 29.1193 25.7625 21.6946 17.7653 14.3655 11.5862 9.3758 7.6373 4.7452 3.1098 2.1362 1.5262 APPENDIX E RHEOLOGICAL DATA FLUID: NEAT POLYBUTENE GEOHETRY: Cone and Plate RADIUS [mm]: 12.5 CONE ANGLE [rad]: 0.108 gammadot tau N1 strain log snear log normal rate strain stress shear stress [1/sec] rate [dyne/cm*2] stress [dyne/cm*2] 1.000 0.000 811 2.909 1.585 0.200 1285 3.109 2.512 0.400 2024 3.306 3.981 0.600 3196 3.505 252.2 6.310 0.800 5045 3.703 300.4 10.000 1.000 7956 3.901 229.3 15.850 1.200 12530 4.098 134.6 25.120 1.400 19540 4.291 474.7 39.810 1.600 30220 4.480 1511 Regression Output: Constant 2.912827416 Std Err of Y Est 0.003873764 R Squared 0. 999954815 No. of Observations 9 Degrees of Freedom 7 x Coefficient(s) Std Err of Cost. 0.984176586 0.002500511 LOG(TAU) 3 CONSTANT + X COEPF*LOG(GAHMADOT) k 3 10‘CONSTANT 3 818.14 n 3 X COEFP 3 0.98 Regression Output: Constant Std Err of Y Est R Squared No. of Observations ~Degrees of Freedom x Coefficient(s) Std Err of Coef. 0.607463763 0.374669108 fluid constants: 109 normal stress 2.402 2.478 2.360 2.129 2.676 3.179 a- b. k- "g 74.00 0.61 818.14 0.98 LOG(N1) 3 CONSTANT + X COEFF*LOG(GAHHADOT) a 3 10‘CONSTANT 3 74.00 b 3 X COEFF 3 0.61 134 FLUID: NEAT EPON 828 GEOMETRY: Cone and Plate RADIUS [mm]: 12.5 CONE ANGLE [rad]: 0.108 gammadot tau strain log shear log rate strain stress shear [1/sec] rate [dyne/cm32] stress 1.000 0.000 115 2.061 1.468 0.167 176 2.246 2.154 0.333 262 2.417 3.162 0.500 379 2.578 4.642 0.667 551 2.741 6.813 0.833 811 2.909 10.000 1.000 1184 3.073 14.680 1.167 1740 3.241 21.540 1.333 2543 3.405 31.620 1.500 3722 3.571 46.420 1.667 5409 3.733 68.130 1.833 7782 3.891 100.000 2.000 10970 4.040 Regression Output: Constant 2.080546032 Std Err of Y Est 0.009405885 R Squared 0.99980331 No. of Observations 13 Degrees of Freedom 11 x Coefficient(s) 0.989183198 Std Err of Coef. 0.004183258 LOG(TAU) 3 CONSTANT + X COEFF*LOG(GAHMADOT) k 3 10‘CONSTANT 3 120.38 n 3 X COEFF 3 0.99 Regression Output: Constant 1.50394871 Std Err of Y Est 0.009051788 R Squared 0.993069556 No. of Observations 3 Degrees of Freedom 1 x Coefficient(s) 0.459758815 Std Err of Cost. 0.038407918 LOG(N1) 3 CONSTANT + X COEFF*LOG(GAHHADOT) a 3 10‘CONSTANT 3 31.91 b 3 X COEFP 3 0.46 .135 N1 normal stress [dyne/cm*2] 187.9 218.5 267.4 a. b. k- n- 109 normal stress 2.274 2.339 2.427 fluid constants: 31.91 0.46 120.38 0.99 FLUID: NEAT CTBN GBONBTRY: Cone and Plate RADIUS [mm]: 12.5 CONE ANGLE [rad]: 0.108 gammadot tau strain log sbear log rate strain stress shear [1/sec] rate [dyne/cm‘Z] stress 1.000 0.000 5679 3.754 1.585 0.200 8930 3.951 2.512 0.400 14120 4.150 3.981 0.600 22250 4.347 6.310 0.800 34900 4.543 10.000 1.000 55940 4.748 15.850 1.200 84130 4.925 25.120 1.400 10500 4.021 Regression Output: Constant 3.941196936 Std Err of Y Est 0.346071786 R Squared 0.386887277 No. of Observations 8 Degrees of Freedom 6 x Coefficient(s) 0.519524755 Std Err of Coef. 0.26699834 IDGWAU) - CONSTANT + x cosmnoommmor) k 3 10‘CONSTANT 3 8733.67 n 3 x COEFP - 0.52 Regression Output: Constant 1.652959333 Std Err of Y Est 0.07415355 R Squared 0.98500733 No. of Observations 4 Degrees of Freedom 2 x Coefficient(s) 1.900693812 Std Err of Coef. 0.1658123? LOG(N1) 3 CONSTANT + X COEFF‘LOG(GAHHADOT) a 3 10‘CONSTANT 3 44.97 b 3 X COEFF 3 1.90 136 N1 normal log stress normal [dyne/cm32] stress 1315 3.119 4105 3.613 9516 3.978 18380 4.264 fluid constants: a- 8733.67 b- 0.52 X3 44.97 n- 1.90 FLUID: 23% CTBN IN EPON 828 GEOIETRY: Cone and Plate RADIUS [mm]: 25 CONE ANGLE [rad]: 0.04 gammadot tau N1 strain log shear log normal log rate strain stress shear stress normal [1/sec] rate [dyne/cm‘2] stress [dyne/cm‘2] stress 1.585 0.200 1267 3.103 2.512 0.400 2042 3.310 3.981 0.600 3233 3.510 6.310 0.800 5158 3.712 10.000 1.000 8503 3.930 15.850 1.200 13390 4.127 25.120 1.400 20860 4.319 833.3 2.921 39.810 1.600 31510 4.498 1960 3.292 63.100 1.800 45160 4.655 3993 3.601 100.000 2.000 58770 4.769 6708 3.827 Regression Output: Constant 2.931181167 Std Err of Y Est 0.040569108 R Squared 0.996378144 No. of Observations 11 Degrees of Freedom 9 X Coefficient(s) 0.962359666 , Std Err of Coef. 0.019340587 ““1" “flaunt" a3 6.88 b3 1.51 LOG(TAU) 3 CONSTANT + X COEPF*LOG(GAHHADOT) k3 853.46 x 3 10‘CONSTANT 3 853.46 n3 0.96 n 3 X COEFF 3 0.96 Regression Output: Constant 0.837736651 Std Err of Y Est 0.051782667 R Squared 0.988425183 No. of Observations 4 Degrees of Freedom 2 X Coefficient(s) 1.513226605 Std Err of Coef. 0.115790821 LOG(N1) 3 CONSTANT + X COEFP*LOG(GAHHADOT) a 3 10‘CONSTANT 3 b 3 X COEFF 3 6.88 1.51 1137 FLUID: 40% CTBN IN EPON 828 GEONETRY: Cone and Plate RADIUS [mm]: 12.5 CONE ANGLE [rad]: 0.108 gammadot tau strain log shear log rate strain stress Shear [1/sec] rate [dyne/cm‘Z] stress 1.000 0.000 882 2.945 1.585 0.200 1427 3.154 2.512 0.400 2264 3.355 3.989 0.601 3599 3.556 6.310 0.800 5647 3.752 10.000 1.000 9234 3.965 15.850 1.200 14560 4.163 25.120 1.400 22640 4.355 39.810 1.600 34300 4.535 63.100 1.800 49300 4.693 100.000 2.000 64290 4.808 Regression Output: Constant 2.979206393 Std Err of Y Est 0.037741773 R Squared 0.996817999 No. of Observations 11 Degrees of Freedom 9 x Coefficient(s) 0.95545423 Std Err of Coef. 0.017994131 LOG(TAU) 3 CONSTANT + X COEFF*LOG(GANMADOT) k 3 10‘CONSTANT 3 953.25 n 3 X COEFF 3 0.96 Regression Output: Constant 0.882956551 Std Err of Y Est 0.069510267 R Squared 0.979690909 No. of Observations 4 Degrees of Freedom 2 x Coefficient(s) 1.526697413 Std Err of Coef. 0.15543137 LOG(N1) 3 CONSTANT + X COEFF3LOG(GAHHADOT) a 3 10‘CONSTANT 3 7.64 b 3 X COEFF 3 1.53 .138 N1 normal stress [dyne/ca“ 2 l 935 2377 4775 7720 fluid constants: a- b k- n- normal stress 2.971 3.376 3.679 3.888 7.64 1.53 953.25 0.96 FLUID: .3% POLYISOBUTYLENE IN POLYBUTENE WITH 4.4% KEROSENE GEONETRY: Cone and Plate RADIUS [mm]: 25 CONE ANGLE [rad]: 0.04 gammadot tau N1 strain log shear log normal log rate strain stress shear stress normal [l/sec] rate [dyne/cmAZ] stress [dyne/cm72] stress 1.000 0.000 495 2.695 347.3 2.541 1.585 0.200 772 2.888 842.6 2.926 2.512 0.400 1216 3.085 2226 3.348 3.981 0.600 1909 3.281 6000 3.778 6.310 0.800 2965 3.472 13680 4.136 10.000 1.000 4571 3.660 26760 4.427 15.850 1.200 7009 3.846 47760 4.679 25.120 1.400 10700 4.029 82220 4.915 Regression Output: Constant 2.700781072 Std Err of Y Est 0.006415462 R Squared 0.999838921 No. of Observations 8 Degrees of Freedom 6 x Coefficient(s) 0.955193258 Std Err of Coef. 0.004949602 fluid constants: a3 432.49 b3 1.73 LOG(TAU) 3 CONSTANT + x COEFF‘LOG(GANNADOT) R3 502.09 k 3 10‘CONSTANT 3 502.09 n3 0.95 n 3 x COEFF 3 0.96 Regression Output: Constant 2.635979628 Std Err of Y Est 0.101790505 R Squared 0.987720941 No. of Observations 8 Degrees of Freedom 6 X Coefficient(s) 1.725281023 Std Err of Coef. 0.078532539 LOG(N1) 3 CONSTANT + X COEFF3LOG(GAHNADOT) a 3 10‘CONSTANT 3 432.49 b 3 X COEFF 3 1.725281023 1139 FLUID: 5% SEPARAN AP 30 IN 50/50 GLYCEROL AND WATER, FRESH GEONETRY: Cone and Plate RADIUS [mm]: 12.5 CONE ANGLE [rad]: 0.108 Winn»! 9‘7 tau N1 strain log shear log normal log rate strain stress shear stress normal [1/sec] rate [dyne/cm22] stress [dyne/cm‘2] stress 1.000 -0.000 1054 3.023 5238 3.719 1.259 0.100 1339 3.127 5600 3.748 1.585 0.200 1390 3.143 6116 3.786 1.995 0.300 1493 3.174 6867 3.837 2.512 0.400 1573 3.197 7863 3.896 3.162 0.500 1716 3.235 8980 3.953 3.980 0.600 1815 3.259 10400 4.017 5.011 0.700 1880 3.274 11950 4.077 6.308 0.800 1987 3.298 13560 4.132 7.942 0.900 2120 3.326 15540 4.191 9.998 1.000 2168 3.336 17850 4.252 12.590 1.100 2304 3.362 20910 4.320 15.850 1.200 2399 3.380 26350 4.421 19.950 1.300 2443 3.388 31110 4.493 25.110 1.400 2614 3.417 37720 4.577 31.610 1.500 2804 3.448 45580 4.659 39.800 1.600 2848 3.455 55290 4.743 50.100 1.700 3013 3.479 62360 4.795 63.080 1.800 3121 3.494 76040 4.881 Regression Output: Constant 3.097970066 Std Err of Y Est 0.022456146 R Squared 0.97263321 No. of Observations 19 Degrees of Freedom 17 x Coefficient(s) 0.231215057 Std Err of Coef. 0.009406521 fluid constants LOG(TAU) 3 CONSTANT + X COEFF*LOG(GANNADOT) a3 4357.98 x 3 10*CONSTANT 3 1253.05 b3 0.66 n 3 x COEFF 3 0.23 k3 1253.05 1‘. 0 e 23 Regression Output: Constant 3.639285409 Std Err of Y Est 0.037638737 R Squared 0.990501795 No. of Observations 19 Degrees of Freedom 17 x Coefficient(s) 0.663835021 Std Err of Coef. 0.015766266 LOG(N1) 3 CONSTANT + X COEFF*LOG(GANHADOT) a 3 10‘CONSTANT 3 4357.98 b 3 X COEFF 3 0.663835021 140 FLUID: 5% SEPARAN AP 30 IN 50/50 GLYCEROL AND HATER, AGED GEONETRY: Cone and Plate RADIUS [mm]: 12.5 CONE ANGLE [rad]: 0.108 gammadot tau N1 strain log shear log normal log rate strain stress shear stress normal [1/sec] rate [dyne/cm*2] stress [dyne/cm32] stress 1.000 0.000 1222 3.087 7754 3.890 1.585 0.200 1372 3.137 10080 4.003 2.512 0.400 1464 3.166 11930 4.077 3.980 0.600 1642 3.215 13570 4.133 6.308 0.800 1882 3.275 18470 4.266 9.998 1.000 2086 3.319 25190 4.401 15.850 1.200 2339 3.369 34520 4.538 25.110 1.400 2522 3.402 42990 4.633 39.800 1.600 2923 3.466 61330 4.788 63.080 1.800 3023 3.480 55950 4.748 Regression Output: Constant 3.086367363 Std Err of Y Est 0.010437745 R Squared 0.994948125 No. of Observations 10 Degrees of Freedom 3 X Coefficient(s) 0.228086867 Std Err of Coef. 0.005746208 fluid constants: a3 7490.78 LOG(TAU) 3 CONSTANT + X COEFF*LOG(GANNADOT) b3 0.53 k 3 10“CONSTANT 3 1220.02 k3 1220.02 n 3 X COEFF 3 0.23 n3 0.23 Regression Output: Constant 3.874527315 Std Err of Y Est 0.046190195 R Squared 0.981630064 No. of Observations 10 Degrees of Freedom 8 X Coefficient(s) 0.525762426 0.025428717 Std Err of Coef. LOG(N1) 3 CONSTANT + X COEFF*LOG(GANMADOT) a 3 10‘CONSTANT 3 7490.78 b 3 X COEFF 3 0.53 141. APPENDIX F PRE-PREGGER DATA 144 147 150 153 156 145 148 151 154 157 146 149 152 155 158 Figure F.1 Spreadsheet Diagram 142 SAMPLE CALCULATIONS Data GAP = H cm TIME 1 WEIGHT 1 ROTATION (rpm) TIME 2 WEIGHT 2 R = 2.5 cm TIME 3 WEIGHT 3 TIME 4 WEIGHT 4 rev rad ROTATION —+— * 2w rad _ _ min rev ROTATION “S—éfi — (0 - 6:511: min GAMMA.DOT g—g—g - R cm * w rad/sec H cm Least Squares Calculations For Fitting Data to a Line y = b + m x n m = SLOPE = 22w, - (EX, ) (Zyl) b = INTERCEPT n = NO.SAMPLE 2x; = SUN 1 = Ffl L< ll SUM 2 Ehgy; = SUM 3 Ehg’ = SUM 4 = STANDARD ERROR OF ESTIMATE = + OR VOLUMETRIC FLOW cm’._ 9 Q SEC ‘ 9 cm’ nZX,’ - (an’ 2y1 - m EX, = n TIME 1 + TIME 2 + TIME 3 + TIME 4 WEIGHT 1 + WEIGHT 2 + WEIGHT 3 + WEIGHT 4 = (TIME 1 * WEIGHT 1) + (TIME 2 * WEIGHT 2) +(TIME 3 * WEIGHT 3) + (TIME 4 * WEIGHT 4) (TIME 1)’ + (TIME 2)’ + (TIME 3)’ + (TIME 4)’ 2w1 - (INTERCEPT + SLOPE * 3(1)]2 n - 2 RATE * SLOPE Egg 143 \IU'IONI-‘mHCDNb-DI-‘b uh U1 INDEX fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh fresh FLUID Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan Separan 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% 23% CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon NEAT EPON NEAT EPON NEAT EPON NEAT EPON NEAT EPON GAP (cm) 0.104 0.104 0.104 0.104 0.182 0.207 0.207 0.207 0.208 0.208 0.208 0.415 0.415 0.415 0.415 0.415 0.415 0.104 0.104 0.104 0.104 0.104 0.104 0.104 0.104 0.207 0.208 0.208 0.208 0.312 0.312 0.312 0.312 0.312 0.312 0.417 0.417 0.417 0.417 0.417 0.417 (rpm) 140 92 120 100 140 200 140 160 100 90 120 200 120 90 140 160 100 0 90 110 110 120 160 160 200 200 90 120 160 0 90 110 140 160 200 0 90 110 120 160 200 0 90 110 140 160 (rad/sec) 14.658 9.632 12.564 10.470 14.658 20.940 14.658 16.752 10.470 9.423 12.564 20.940 12.564 9.423 14.658 16.752 10.470 0.000 9.423 11.517 11.517 12.564 16.752 16.752 20.940 20.940 9.423 12.564 16.752 0.000 9.423 11.517 14.658 16.752 20.940 0.000 9.423 11.517 12.564 16.752 20.940 0.000 9.423 11.517 14.658 16.752 ROTATION ROTATION GAMMA.DOT (1/sec) 352.356 231.548 302.019 251.683 201.346 252.899 177.029 202.319 125.841 113.257 151.010 126.145 75.687 56.765 88.301 100.916 63.072 0.000 226.514 276.851 276.851 302.019 402.692 402.692 503.365 252.899 113.257 151.010 201.346 0.000 75.505 92.284 117.452 134.231 167.788 0.000 56.493 69.047 75.324 100.432 125.540 0.000 117.788 143.963 183.225 209.400 INDEX 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 111 112 113 114 115 116 117 118 119 101 102 103 104 105 106 107 108 109 110 43 44 45 46 47 48 49 50 51 NEAT NEAT NEAT NEAT NEAT NEAT NEAT FLUID EPON EPON EPON EPON EPON EPON EPON discard 0.3% PIB discard discard discard discard 0.3% 0.3% 0.3% 0.3% 0.3% 0.3% disc disc 0.3% 0.3% 0.3% 0.3% 0.3% neat neat neat neat neat neat neat neat neat neat 40% 40% 40% 40% 40% 40% 40% 40% 40% PIB PIB PIB PIB PIB PIB ard ard PIB PIB PIB PIB PIB polybutene polybutene polybutene polybutene polybutene polybutene polybutene polybutene polybutene polybutene CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon GAP Q 5 0000000 fibubbuhfik) OOOOOOOOOOOOOOOOOOO NNhHHHHHb-fibbthNNNN 0 hubfihb-bNNNNN OOOOOOOOOO ocpc>o<3cao<3c> O NJNF3F*H+4F*H13 H .5 U" (rpm) 200 200 0 90 110 140 160 90 0 120 160 200 90 0 120 160 200 90 90 120 160 200 90 90 90 120 90 120 160 200 90 120 160 200 90 110 140 160 200 110 90 110 20.940 20.940 0.000 9.423 11.517 14.658 16.752 9.423 0.000 12.564 16.752 20.940 9.423 0.000 12.564 16.752 20.940 9.423 9.423 12.564 16.752 20.940 9.423 9.423 9.423 12.564 0.000 9.423 12.564 16.752 20.940 0.000 9.423 12.564 16.752 20.940 9.423 11.517 14.658 16.752 20.940 11.517 0.000 9.423 11.517 ROTATION ROTATION GAMMA.DOT (rad/sec) (l/sec) 261.750 130.875 0.000 58.894 71.981 91.613 104.700 117.788 0.000 157.050 209.400 261.750 58.894 0.000 78.525 104.700 130.875 58.894 235.575 314.100 418.800 523.500 235.575 58.894 117.788 157.050 0.000 117.788 157.050 209.400 261.750 0.000 58.894 78.525 104.700 130.875 235.575 287.925 366.450 418.800 523.500 287.925 0.000 117.788 143.963 INDEX 52 53 54 55 56 57 58 59 60 61 62 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% 40% aged FLUID CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon CTBN/Epon Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan aged Separan GAP O 3 OOOOOOOOOOOOOOOOOOOOOOOOOO 0 NNNNNNNNppseabsuuwwwummwwm OOOOOOOOOOOOOOO 0 bhbfiHHNNNNNHHF—‘b 146 (rpm) 120 160 200 0 140 110 140 160 90 200 0 90 110 140 160 0 200 120 90 110 160 200 160 160 160 160 140 140 160 200 200 160 140 120 90 90 120 120 90 160 200 (rad/sec) 12.564 16.752 20.940 0.000 14.658 11.517 14.658 16.752 9.423 20.940 0.000 9.423 11.517 14.658 16.752 0.000 20.940 12.564 9.423 11.517 16.752 20.940 16.752 16.752 16.752 16.752 14.658 14.658 16.752 20.940 20.940 16.752 14.658 12.564 9.423 9.423 12.564 12.564 9.423 16.752 20.940 ROTATION ROTATION GAMMA.DOT (l/sec) 157.050 209.400 261.750 0.000 183.225 95.975 122.150 139.600 78.525 174.500 0.000 58.894 71.981 91.613 104.700 0.000 130.875 78.525 117.788 143.963 209.400 261.750 209.400 209.400 209.400 209.400 91.613 366.450 418.800 523.500 261.750 209.400 183.225 157.050 117.788 235.575 314.100 78.525 58.894 104.700 130.875 Oman-1101400100014» ab U1 INDEX Q (cc/sec) 0.067855 0.042185 0.069296 0.049902 0.106685 0.174405 0.117018 0.112985 0.043433 0.045112 0.068042 0.051401 0.014139 0.019544 0.030875 0.036061 0.017173 0.00186 0.009406 0.007969 0.009863 0.00986 0.005977 0.005538 0.012092 0.007051 0.009193 0.011379 0.007085 0.00169 0.003259 0.002081 0.00462 0.006616 0.003567 0.00138 0.006325 0.005997 0.00423 0.006362 0.009869 0.004629 0.010738 0.009636 0.007096 0.004573 + or - (cc/sec) 0.004037 0.000402 0.005188 0.000638 0.002144 0.026116 0.002587 0.015602 0.000557 0.000534 0.001091 0.001052 0.000584 0.000145 0.001789 0.002482 0.000539 3.58E-05 0.000113 0.000118 0.000122 0.000104 9.5E-05 9.35E-05 0.000185 0 0.000197 0.000175 0.000105 1.64E-05 0.000146 2.65E-05 6.72E-05 0.000113 5.81E-05 6.7E-06 7.27E-05 6.59E-05 4.57E-05 0.000122 0.000168 9.57E-05 0.000122 0.000109 6.54E-05 3.59E-05 CORRCOEF NO.SAMPLE TIME 1 0.998 1.000 0.999 1.000 0.987 0.999 0.995 0.997 0.996 1.000 1.000 0.997 1.000 0.999 1.000 0.998 0.995 0.980 0.992 0.997 0.997 0.998 0.997 0.973 0.999 1.000 0.997 0.999 0.997 0.971 1.000 1.000 0.988 0.998 0.985 0.999 0.990 1.000 0.999 0.999 0.999 1.000 0.998 0.992 0.957 0.993 147 uuwaupbphuhwbbupu Dbbbfibkébhwhhbbwbbfibbbhw sense-L0 (sec) 120 210 120 120 90 50 100 90 150 180 120 90 170 230 150 110 270 290 130 140 100 220 180 190 160 130 90 140 100 180 220 140 100 120 100 350 200 210 130 140 240 340 100 180 210 140 TIME 2 (sec) 240 450 240 300 180 110 180 150 300 360 240 180 330 430 300 220 600 690 230 280 270 400 290 300 300 310 180 280 260 440 400 290 220 240 210 640 340 370 400 240 340 660 240 310 380 360 INDEX 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 111 112 113 114 115 116 117 118 119 101 102 103 104 105 106 107 108 109 110 43 44 45 46 47 48 49 50 51 Q (cc/sec) 0.022576 0.004143 0.005105 0.007213 0.017123 0.009531 0.015085 0.068644 0.004821 0.01732 0.013697 0.015428 0.007384 ERR 0.08412 0.05741 0.05668 0.150442 0.178536 0.142613 0.128228 0.218227 0.163342 0.08446 0.10423 0.13756 0.001183 0.019784 0.018136 0.010742 0.020123 0.001447 0.011842 0.008523 0.013662 0.020808 0.012613 0.006193 0.015027 0.016293 0.014822 0.008681 0.002082 0.006772 0.011264 + or - (cc/sec) 0.000388 3.19E-05 8.65E-05 9.44E-05 0.000239 0.000126 0.000192 0.013458 3.58E-05 0.002392 0.00028 0.000424 9.16E-05 0 0.003294 0.008557 0.012731 0.045055 0.033816 0.08542 0.024288 0.041334 0.030939 0.015998 0.019742 0.026055 1.54E-05 0.000423 0.000326 0.00015 0.000328 2.07E-05 0.000171 0.000131 0.000204 0.000342 0.000117 0.000246 0.000212 0.001218 0.000266 0.000117 3.39E-05 8.74E-05 0.000147 CORRCOEF NO.SAMPLE TIME 1 0.996 1.000 1.000 0.999 0.999 0.997 0.984 0.946 1.000 0.994 0.989 0.975 0.981 ERR 0.974 0.926 0.908 0.996 1.000 1.000 0.985 0.998 1.000 0.996 1.000 1.000 0.998 1.000 1.000 0.995 0.997 0.998 0.999 0.999 0.997 0.994 0.994 0.991 0.997 1.000 0.996 0.995 0.998 0.997 0.999 148 hbbfibbNbMWUDH-hnbbuhw bhkah-b bbbkhbbbbb newbhubus (sec) 90 390 720 140 130 210 190 60 290 30 60 50 110 350 80 110 40 60 30 30 40 20 30 30 30 30 160 90 110 150 100 280 160 130 140 110 140 190 110 210 130 230 5017 110 120 TIME 2 (sec) 210 630 1270 280 260 360 340 120 560 100 140 120 270 140 160 80 90 60 60 70 50 60 60 60 60 320 180 220 300 210 390 310 240 280 220 360 370 220 320 230 340 5707 260 240 52 53 54 55 56 57 58 59 60 61 62 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 INDEX 120 12 12 12 1 2 3 124 12 5 126 12 7 128 12 13 13 13 13 9 0 1 2 3 134 Q (cc/sec) 0.009816 0.005765 0.005598 0.00245 0.013384 0.004167 0.009396 0.012466 0.009519 0.013819 0.004088 0.005689 0.008524 0.009056 0.009332 0.002821 0.010382 0.009057 0.009694 0.009914 0.009649 0.006806 0.004849 0.006386 0.009036 0.006275 0.078345 0.225325 0.209778 0.264298 0.226527 0.19583 0.180821 0.173925 0.095443 0.156937 0.210531 0.069969 0.045682 0.096198 0.109552 + or - (cc/sec) 0.000154 9.25E-05 4.42E-05 0 0.000925 5.63E-05 0.000126 0.000195 0.000134 0.000204 1.95E-05 5.46E-05 0.000109 0.000123 0.000123 4.04E-05 0.000135 0.000109 0.000155 0.00015 0.000117 8.12E-05 4.61E-05 7.95E-05 9.4SE-05 0.000104 0.00272 0.014844 0.009286 0.029019 0.012727 0.008159 0.009417 0.007247 0.003748 0.005816 0.010965 0.002017 0.001303 0.003028 0.003804 CORRCOEF NO.SAMPLE TIME 1 0.998 0.997 0.977 1.000 1.000 1.000 0.991 1.000 1.000 0.999 0.998 1.000 1.000 1.000 0.999 0.997 1.000 0.999 1.000 1.000 0.994 0.992 0.993 0.998 0.994 0.998 0.999 0.998 0.997 1.000 1.000 1.000 1.000 1.000 0.998 1.000 1.000 1.000 1.000 1.000 0.998 149 bbbbbhbhbbwbbhbbhbbbhwwfibub #hbkbbhbkbbwbwb (sec) 120 120 160 410 170 140 130 130 160 130 260 190 140 140 170 260 250 170 130 160 180 210 170 230 180 220 60 50 40 40 30 40 40 50 60 50 40 60 70 60 60 TIME 2 (sec) 240 240 300 630 300 290 250 260 300 250 710 350 300 280 320 540 420 310 250 290 310 400 410 350 370 310 120 100 80 70 60 90 80 100 120 100 80 120 140 120 120 \lmoxpnor-aoowups ab U" TIME 3 (sec) 420 660 360 450 290 170 270 220 450 540 360 280 600 660 460 370 840 1220 400 410 370 530 420 420 420 270 390 380 580 620 450 340 350 340 1110 510 560 530 340 450 1200 430 510 560 650 TIME 4 (sec) 810 570 372 360 660 480 370 1010 600 530 580 780 540 530 540 360 520 490 790 590 500 460 450 1510 730 670 440 580 600 680 840 860 (9) 10.5768 10.2841 9.0126 7.4061 17.4316 10.4642 14.3305 11.7332 7.8269 9.6352 9.8556 7.3956 3.1132 4.0844 5.9176 5.7348 4.5768 0.8550 2.0500 2.0769 1.6829 1.3998 0.7273 2.5568 1.7455 1.5931 1.2029 2.1909 1.0571 1.6846 0.6755 0.3684 0.4818 0.7476 0.3574 0.7204 0.9599 0.8023 0.7550 1.2350 1.9504 2.6564 1.9346 3.4791 2.7988 1.3901 150 (9) 18.6168 22.0683 18.0159 17.5573 27.2709 22.1072 27.0832 20.9730 13.9549 19.0433 19.9510 13.2799 5.7258 7.7953 11.3375 10.9902 10.3212 2.0581 3.6313 3.0772 3.5905 3.6359 1.5914 3.7071 3.8798 3.0273 2.3427 4.1272 2.5330 2.3642 1.3636 0.7063 0.8046 1.7127 0.6136 1.2297 2.1845 1.9704 2.1578 2.0423 3.1570 4.5145 3.8830 5.4800 5.6571 2.9870 (9) 34.4013 32.4162 28.6373 26.9107 38.0078 35.1600 40.8655 29.1271 23.5827 28.2203 28.7994 20.0199 10.2783 13.4016 17.2108 16.8545 16.2324 2.8418 5.4601 4.4153 5.0092 5.1491 2.4905 4.0000 5.4615 3.2193 5.5659 3.4364 2.6584 2.1500 1.0804 1.5432 2.5861 1.1379 1.9743 3.6035 3.2410 2.7037 2.7338 4.3917 7.4087 6.4642 7.8685 6.9280 4.4738 WEIGHT 1 WEIGHT 2 WEIGHT 3 WEIGHT 4 (9) 40.2261 33.7450 54.2505 49.8615 35.3986 39.0221 24.1374 21.8594 7.1401 5.5408 6.9848 7.6560 3.1549 4.8460 6.9461 4.0271 7.0658 4.1755 2.8263 1.4280 2.5145 3.2699 1.7682 2.5307 4.3338 3.3364 3.4008 5.7478 8.2812 9.2191 8.3929 5.3407 INDEX 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 111 112 113 114 115 116 117 118 119 101 102 103 104 105 106 107 108 109 110 43 44 45 46 47 48 49 50 51 TIME 3 (sec) 310 880 1780 430 410 500 470 150 810 160 240 190 410 180 230 120 120 90 100 80 90 90 90 90 450 270 330 420 340 530 440 380 400 340 540 640 380 450 340 500 6107 400 420 TIME 4 WEIGHT 1 WEIGHT 2 WEIGHT 3 WEIGHT 4 (sec) 430 1140 580 540 650 650 1070 340 260 580 230 120 130 110 120 120 120 120 610 360 430 570 450 680 560 500 530 460 770 510 450 650 560 550 (9) 3.6238 1.9127 4.3459 1.2391 2.7738 3.2400 5.1001 5.7183 1.3450 2.3524 1.5027 2.5784 0.8359 0.2512 10.4936 8.8711 7.3181 6.6903 4.4195 5.6786 6.9667 2.312 4.7945 2.8368 2.8843 4.2068 0.1781 1.7443 2.1303 2.1175 1.5388 0.4166 3.2148 1.2717 2.3108 2.1431 2.3610 2.2185 2.1815 1.9546 2.3328 2.5304 6.3390 1.0421 1.8062 151 (9) 7.4719 3.1307 7.7621 2.5553 5.6772 5.2175 8.7066 7.9376 2.5149 3.2521 2.2484 2.9198 2.1243 16.3872 13.7238 10.9631 11.3783 9.384 9.4992 11.8729 8.4968 9.1669 5.4309 5.6893 8.0527 0.3485 3.4967 4.0350 3.8701 3.9788 0.5788 4.9510 2.2216 4.3307 4.1028 5.9660 3.8167 4.4194 3.8765 4.2022 3.7664 8.0299 2.3954 3.5075 (9) 10.1444 4.3487 10.7830 3.8205 8.7847 6.8123 11.3287 11.5999 3.5798 4.3727 3.9586 4.1316 3.3192 20.0362 15.2641 11.3673 14.751 14.2601 15.102 14.9852 13.6618 7.8674 8.5112 11.8157 0.4666 5.0668 5.8404 5.0825 6.3068 0.7757 6.3738 3.3287 5.8159 5.9303 8.6028 5.3041 6.9447 6.2152 6.1241 5.4209 8.7877 3.1763 5.4753 (9) 12.7622 5.6163 5.0256 11.1096 8.2284 13.3455 4.7070 4.7602 5.3891 3.8626 21.4659 18.7374 17.3411 19.6369 17.8826 9.5669 11.2514 15.2366 0.6710 6.6456 7.4388 6.2315 7.9996 0.9437 7.5296 4.1506 7.1779 8.9431 10.9867 8.7896 7.4637 6.4776 4.4443 7.1881 INDEX 52 53 54 55 56 57 58 59 60 61 62 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 TIME 3 (sec) 360 360 660 430 450 390 380 440 380 1170 580 430 410 460 1460 550 470 370 420 450 540 570 510 560 450 180 150 130 100 100 140 120 150 170 150 120 210 210 180 180 TIME 4 (sec) 490 480 840 560 560 500 570 520 1440 780 600 570 610 700 650 490 540 660 700 790 690 730 560 240 180 140 190 160 200 220 220 160 270 290 260 240 (9) 1.5862 1.1033 0.9012 0.8769 1.3772 0.9032 1.5902 2.0991 1.9839 2.3649 1.6749 1.2902 1.4685 1.6303 2.0513 1.2726 2.9362 2.0806 1.6664 2.0222 2.6252 2.7414 1.6589 1.3886 2.4575 2.4219 5.6851 14.1648 11.7241 12.4783 7.9491 9.2767 8.9111 9.3506 7.6955 9.1371 9.3038 4.9891 3.6206 6.8254 8.8273 152 (9) 2.9734 1.9960 1.4980 1.4645 3.3113 1.6191 2.9278 3.9654 3.5463 4.3533 3.9593 2.2914 2.9607 3.1330 3.7273 2.4446 4.8731 3.6754 2.9850 3.5064 4.3650 4.5044 3.2513 2.2815 4.7812 3.1464 11.8991 28.7388 23.4514 22.2388 16.2162 20.5831 17.5988 19.6624 14.8704 18.1905 19.1766 9.8923 7.4822 13.6863 17.3115 (9) 4.3726 2.7209 3.2057 5.1701 2.3471 3.8372 5.5694 4.9413 6.3090 5.9150 3.7501 4.2223 4.3556 5.1687 5.0568 6.3731 5.2225 4.2541 4.9241 5.9370 5.5023 4.0263 3.4982 6.6248 4.1112 17.4057 40.7531 33.059 31.1906 26.7777 32.1648 26.1549 29.7869 20.8918 27.5016 29.0648 17.4462 11.215 20.207 25.5164 WEIGHT 1 WEIGHT 2 WEIGHT 3 WEIGHT 4 (9) 5.5122 3.3753 5.2732 2.8067 6.1386 7.1331 6.2558 8.2492 6.9347 4.9382 5.7282 5.8971 6.5212 8.0186 6.8452 5.4699 6.1216 7.7117 6.3676 4.9451 4.5703 7.8679 4.7510 22.339 47.3648 37.4136 43.9293 34.5082 40.181 25.5498 40.5842 39.1312 22.2746 15.4953 29.6019 31.9465 \lUIONHKDHmNUH-b b U1 H+3P4H13P4 630.01402q 20 21 23 22 24 25 36 26 19 33 34 35 37 38 39 40 41 42 29 30 31 32 28 27 79 80 81 82 83 SUM 1 sum(x) 780 2130 720 1440 932 330 910 460 900 1740 1200 920 1100 2330 910 700 1710 2200 1360 1360 1320 1930 1430 1440 1420 440 900 1330 1230 1990 1240 1470 1160 1170 1100 3610 1050 1870 1730 1160 1610 2200 1370 1680 1990 2010 SUM 2 sum(y) 63.5949 104.9947 55.6658 85.6191 136.9608 67.7314 132.1407 61.8333 45.3645 92.2974 97.6281 64.8328 19.1173 47.1407 34.4659 33.5795 31.1304 5.7549 18.2815 15.1102 17.2674 17.8408 7.9641 15.1099 18.0329 4.6204 10.792 18.9498 11.202 9.5335 4.1891 3.5831 5.3441 8.3163 3.87708 6.4551 6.7479 10.3475 8.9529 9.4119 15.2469 14.5796 20.563 26.0467 23.7768 14.1916 SUM 3 SUM 4 sum(xy) sum(xAZ) 20185.79 248400 66068.23 1338300 15714.76 201600 37500.39 631800 37681.05 262984 8932.202 43500 35291.85 244900 10609.9 79000 15972.72 315000 47191.96 889200 35069.3 432000 17592.4 255800 8585.738 497800 35214.44 1693500 12205.86 324100 9284.837 197400 21063.67 1138500 5135.035 2048600 7569.799 589800 5899.279 547000 7042.313 556200 10463.02 1097700 3342.076 584500 5846.302 '583400 7487.944 583600 1145.566 113000 2848.914 243000 7307.259 520500 4116.117 462100 5118.125 1186500 2027.05 592800 1585.103 654300 2007.13 424000 2910.049 406100 1347.17 372200 7051.978 4044300 2772.495 415700 5876.165 1027500 4629.619 906700 3088.896 386400 6851.465 712100 12773.19 1991200 8873.706 612500 12606.96 851000 13667.16 1207700 8770.906 1311300 153 SUM 5 sum(yAZ) 1641.903 3261.722 1225.895 2226.02 5435.272 1834.453 5095.021 1425.923 812.1433 2504.931 2847.305 1214.461 148.1203 734.8853 459.7685 437.7466 390.9651 13.04263 98.18256 63.97801 89.60336 100.3068 19.2175 59.76353 96.17591 11.70251 33.51664 102.7386 36.77719 23.48238 6.938205 3.841026 9.583689 20.87241 4.925573 12.33344 18.67866 33.81206 23.66768 24.73532 66.09494 82.32601 129.1845 189.0396 158.274 59.39251 SLOPE (g/sec) 0.0801 0.0498 0.0818 0.0589 0.1259 0.2058 0.1381 0.1333 0.0513 0.0532 0.0803 0.0607 0.0167 0.0231 0.0364 0.0426 0.0203 0.0021 0.0106 0.0090 0.0111 0.0111 0.0068 0.0063 0.0137 0.0080 0.0104 0.0129 0.0080 0.0019 0.0037 0.0024 0.0052 0.0075 0.0040 0.0016 0.0071 0.0068 0.0048 0.0072 0.0112 0.0055 0.0128 0.0115 0.0084 0.0054 INDEX 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 111 112 113 114 115 116 117 118 119 101 102 103 104 105 106 107 108 109 110 43 44 45 46 47 48 49 50 51 SUM 1 sum(x) 1040 3040 3770 1430 1340 1720 1650 330 2730 290 780 620 1370 350 630 500 240 270 300 90 340 260 300 300 300 300 1540 900 1090 1440 1100 1880 1470 1250 1350 1130 1810 1200 1220 980 1150 1720 16831 1330 1330 SUM 2 sum(y) 34.0023 15.0084 22.891 12.6405 28.3453 23.4982 38.4809 25.2558 12.1467 9.9772 12.4699 15.0189 10.142 0.2512 68.3829 37.859 29.6485 32.8196 46.801 15.1778 51.2827 45.4309 45.5058 25.702 28.3362 39.3118 1.6642 16.9534 19.4445 17.3016 19.824 2.7148 22.0692 10.9726 19.6353 21.1193 27.9165 11.3393 22.3352 12.0463 20.1228 18.1953 23.1566 11.0581 17.9771 SUM 3 sum(xy) 10527.75 12947.73 32180.66 5446.621 11437.58 11313.31 17928.33 3035.595 9734.522 1095.414 2973.47 2665.466 4266.69 87.92 11677.37 6682.372 2533.848 3195.585 4227.522 740.31 4874.314 3829.955 4069.323 2267.052 2544.063 3501.171 759.296 4546.845 6248.049 7165.26 6733.56 1395.217 9070.226 4038.711 7666.755 7268.485 15583.57 5228.318 8333.915 4447.786 6710.629 8783.458 131295.9 4496.763 7311.625 }54 SUM 4 sum(xAZ) 333200 2623000 5299700 619300 544200 846200 795100 40500 2198700 36500 196400 120600 589500 122500 111300 90600 22400 26100 27000 4500 33400 21400 27000 27000 27000 27000 702600 243000 354300 613800 372200 973800 628900 468900 538900 387700 1033700 582600 465000 349000 387900 841000 95035587 553300 550900 SUM 5 sum(y‘2) 334.7438 63.91372 195.4101 47.9178 240.5187 151.8339 408.2577 230.2621 43.10456 35.23045 45.64343 61.2859 31.14815 0.063101 1240.89 500.0319 302.9597 391.8178 662.032 122.4813 718.2848 687.705 613.4514 190.9637 239.7218 454.3079 0.821128 85.10595 110.2654 84.12488 121.9681 2.000845 132.1675 34.86045 109.4417 136.5733 235.8832 47.62242 149.776 57.47642 116.3119 91.93415 181.8859 36.6646 97.21261 SLOPE (g/seC) 0.0269 0.0049 0.0061 0.0086 0.0204 0.0113 0.0180 0.0613 0.0043 0.0155 0.0122 0.0138 0.0066 ERR 0.0751 0.0513 0.0506 0.1343 0.1594 0.1274 0.1145 0.1949 0.1459 0.0754 0.0931 0.1228 0.0011 0.0181 0.0166 0.0098 0.0184 0.0013 0.0108 0.0078 0.0125 0.0190 0.0137 0.0068 0.0164 0.0178 0.0162 0.0095 0.0023 0.0074 0.0123 INDEX 52 53 54 55 56 57 58 59 60 61 62 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 SUM 1 sum(x) 1210 1200 1960 1040 900 1440 1330 1270 1470 1280 3580 1900 1470 1400 1560 2260 1920 1600 1240 1410 1600 1850 1940 1780 1840 1540 600 300 430 210 330 460 400 500 570 520 400 660 710 620 600 SUM 2 sum(y) 14.4444 9.1955 10.8781 2.3414 9.8586 7.6761 14.4938 18.767 16.7273 21.2764 18.4839 12.2699 14.3797 15.016 17.4685 8.774 22.201 17.8237 14.3754 16.5743 20.6389 19.1157 13.8816 11.7386 21.7314 14.4305 57.3289 83.6567 115.5993 65.9077 88.3566 105.9539 87.173 98.9809 69.0075 95.4134 96.6764 54.6022 37.8131 70.3206 83.6017 SUM 3 SUM 4 sum(xy) sum(xAZ) 5179.074 441700 3211.104 432000 7138.842 1256800 1282.164 565000 3450.657 303800 3223.934 619800 5872.8 545100 6986.809 478900 7121.292 634100 8082.766 494200 20153.1 4014200 7073.982 1103400 6346.309 654500 6252.625 591000 7896.991 715000 9033.888 2490800 11898.98 1031400 8397.031 768400 5217.15 456400 6714.194 577700 9587.058 766600 9806.016 985700 7816.666 1146000 6055.492 911600 11664.85 1015800 6018.802 660600 10263.38 108000 9695.085 35000 15168.41 57300 5174.908 16500 9127.119 34100 15073.19 65400 10424.25 48000 14938.01 75000 11418.74 95300 15329.67 83400 11655.05 48000 11164.27 135000 8149.737 152700 13385.63 118000 14867.13 108000 155 SUM 5 sum(yAZ) 60.86112 23.99723 41.13932 2.913714 39.59132 16.8237 63.50726 102.0299 80.06358 132.3968 101.5586 45.3642 61.56233 66.22061 87.34211 33.16681 137.2827 91.96873 59.70429 78.10488 120.6632 98.62653 53.98801 40.2585 134.691 55.2394 975.8983 2687.375 4023.744 1623.126 2442.976 3474.079 2264.02 2975.816 1369.609 2817.796 2830.316 923.2764 434.9726 1518.496 2049.275 SLOPE (g/seC) 0.0107 0.0063 0.0061 0.0027 0.0146 0.0045 0.0102 0.0136 0.0104 0.0151 0.0045 0.0062 0.0093 0.0099 0.0102 0.0031 0.0113 0.0099 0.0106 0.0108 0.0105 0.0074 0.0053 0.0070 0.0098 0.0068 0.0924 0.2659 0.2475 0.3119 0.2673 0.2311 0.2134 0.2052 0.1126 0.1852 0.2484 0.0826 0.0539 0.1135 0.1293 \JUIONHKOl-‘mel—‘ub 45 M INDEX +Or- 0.004764 0.000474 0.006122 0.000752 0.00253 0.030817 0.003053 0.01841 0.000657 0.00063 0.001288 0.001241 0.00069 0.000171 0.002111 0.002929 0.000636 4.05E-05 0.000128 0.000133 0.000138 0.000118 0.000107 0.000106 0.000209 0.000222 0.000198 0.000119 1.85E-05 0.000165 3E-05 7.59E-05 0.000127 6.57E-05 7.57E-06 8.22E-05 7.45E-05 5.17E-05 0.000138 0.00019 0.000114 0.000145 0.000129 7.79E-05 4.27E-05 INTERCEPT (g) 0.381 -0.258 -1.069 0.206 4.908 -0.061 1.622 0.168 -0.190 -0.082 0.320 2.258 0.255 -1.648 0.437 1.264 -l.174 0.377 0.957 0.716 0.639 -0.916 -0.424 1.525 -0.342 0.557 0.361 0.462 0.339 1.433 -0.126 0.031 -0.178 -0.108 -0.139 0.206 -0.189 -0.581 0.171 0.268 -0.677 0.820 0.764 1.695 1.743 0.813 I56 INDEX 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 111 112 113 114 115 116 117 118 119 101 102 103 104 105 106 107 108 109 110 43 44 45 46 47 48 49 50 51 +or- 0.000461 3.79E-05 0.000103 0.000112 0.000284 0.000149 0.000228 0.012018 3.2E-05 0.002136 0.00025 0.000379 8.18E-05 0.002942 0.007641 0.011369 0.040234 0.030198 0.07628 0.021689 0.036912 0.027628 0.014286 0.01763 0.023267 1.4E-05 0.000387 0.000298 0.000137 0.0003 1.89E-05 0.000156 0.00012 0.000186 0.000313 0.000128 0.000268 0.000231 0.001328 0.00029 0.000128 3.7E-05 9.53E-05 0.00016 INTERCEPT (9) 1.516 0.005 -0.003 0.092 0.260 0.997 2.215 1.676 0.098 1.831 0.732 1.619 0.277 ERR 5.264 4.075 5.834 -1.151 -0.257 1.858 3.088 -1.309 0.437 0.769 0.103 0.615 -0.000 0.170 0.344 0.791 -0.102 0.057 1.540 0.309 0.695 -0.093 0.758 1.080 0.588 -1.786 0.386 0.480 35.010 0.310 0.412 157 INDEX 52 53 54 55 56 57 58 59 60 61 62 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 +or- 0.000167 0.000101 4.82E-05 0.001008 6.14E-05 0.000137 0.000213 0.000146 0.000223 2.13E-05 5.95E-05 0.000118 0.000134 0.000134 4.4E-05 0.000147 0.000119 0.000169 0.000164 0.000127 8.85E-05 5.02E-05 8.66E-05 0.000103 0.000113 0.002965 0.01618 0.010121 0.031631 0.013872 0.008894 0.010265 0.007899 0.004085 0.006339 0.011952 0.002199 0.00142 0.003301 0.004146 INTERCEPT (9) 0.375 0.414 -0.270 -0.218 -1.090 0.284 0.218 0.377 0.369 0.499 0.633 0.122 0.180 0.299 0.400 0.608 0.119 0.507 0.318 0.335 0.953 1.348 0.907 -0.l63 0.902 0.974 0.465 1.297 2.289 0.138 0.037 -0.086 0.456 -0.909 1.203 -0.221 -0.674 0.028 -0.115 -0.014 1.510 158 LIST OF REFERENCES LIST OF REFERENCES Beavers, G.S.; and D.D. Joseph, (1975), "The rotating rod viscometer", Journal of Fluid Mechanics, vol. 69, part 3, pp. 475-511. Bird, R. Byron; R.C. Armstrong, and O. Hassager, 1987, Dynamics of Polymeric Liquids. vol 1., John Wiley 8 Sons, New York. Bird, R. Byron; W.E. Stewart, and E.L. Lightfoot, 1960, Transport Phenomena, John Wiley 6 Sons, New York. Blyler, L. L., Jr. ,1966, "Analysis of Normal Stress Extrusion of Polymer Melts", Ph.D. thesis, Princeton University, N.J.. Cattanach, J.B.; G. Guff, and F.N. Cogswell, 1986, "The Processing of Thermoplastics Containing High Loadings of Long and Continuous Reinforcing Fibers", Journal of Polymer Engineering, vol. 6, no. 1-4, pp. 345-362. Chmielewski, C.: C.A. Petty and K. Jayaraman, (1990), "Crossflow of Elastic Liquids Through Arrays of Cylinders", Journal of Non-Newtonian Fluid Mechanics, vol. 35, pp. 309-325. Choplin, L.: P.J. Carreau and A. Ait Kadi, 1983, "Highly Elastic-Constant Viscosity Fluids", Polymer Engineering and Science, Vol. 23, No. 8, pp. 459-464. Custom Scientific Instruments, Inc., Cedar Knolls, New Jersey, Bulletin: Plastics CS-194. D'Amato, Dominic A., 1975, "Processing Versatility With a New Mixing Extruder", Proceedings of 33rd. Technical Conference of the Society of Plastics Engineers, pp. 506-9, May 1975. Davidson, R.L., ed., Handbook of Water Soluble Gums and Resins, McGraw-Hill Book Co., New York, NY, pp. 16-1 -- 16326. Good, Paul A.: Arthur J. Schwartz, and Christopher W. Macosko, 1974, "Analysis of the Normal Stress Extruder", AIChE Journal, vol. 20, no. 1, pp. 67373, January 1974. 159 Goppel, J.M., 1969, "New Concepts in Processing", Plastics and Polymers, pp. 449-61, October 1969. Hull, Derek, 1987, An Introduction to Composite Materials, Cambridge University Press, 1987. Kataoka, T., S. Ohnishi, T. Kitano, K. Nakama, and H. Takayama, 1976, "Mixing effect of filler and polymer by an elastic melt extruder", Rheological Acta, vol. 15, pp. 268-270. Kocherov, V.L., and Yu. L. Lukach, 1973, "Flow of Polymer Melts in a Disc-Type Extruder and in Rotational Devices of the "Cone-Plate" and "Plate-Plate" Type", Polymer Engineering and Science, vol. 13, no. 3, pp. 194-201, May 1973. Lee, Walter J., and J.C. Seferis, 1987, "Hot-melt prepreg processing of advanced composites: A Comparison of Methods", Society of Plastics Engineers ANTEC ’88 Proceedings, 1987. Maxwell, Bryce and Anthony J. Scalora, 1959, "The Elastic Melt Extruder-3Works Without Screw, Modern Plastics, vol. 37, no. 9, October 1959, pp. 107-210. Maxwell, Bryce, 1962, U.S. Patent #3,046,603. July 31, 1962. Maxwell, Bryce, 1970, "Scaling Up the Elastic Melt Extruder", Society of Plastics Engineers Journal, vol. 26, pp. 48-50, June 1970. Maxwell, Bryce, 1973, "The Application of Melt Elasticity to 'Polymer Processing", Polymer Engineering and Science, vol. 13, no. 3, pp. 227-230, May 1973. Maxwell, Bryce, 1974, "A New Twist in Mixing Extruders", Plastics Engineering, pp. 40-45, May 1974. Middleman, Stanley, 1977, Fundamentals of Polymer Processing, McGraw-Hill Book Co., 1977. Prilutski, G., R.K. Gupta, T. Sridhar, and M.E. Ryan, 1983, "Model Viscoelastic Liquids", Journal of Non-Newtonian' Fluid Mechanics, Vol. 12, pp. 2334241. 160 Raghava, R.S.; (1988), "Development and Characterization of Thermosetting-Thermoplastic Polymer Blends for Applications in damage-tolerant Composites", Journal of Polymer Science, part b, vol. 26, pp. 65381. Remnev, V.P.: and N.V. Tyabin, 1971, "Velocity and Pressure Distribution in the Gap of a Disk Extruder" Polymer Mechanics, vol. 7, no. 3, May-June 1971, pp. 451-456. Rheometrics Mechanical Spectrometer manual. Starita, Joseph M., 1972, "Microstructure of Melt Blended Polymer Systems", Trans. Society of Rheology, vol. 16, no. 2, pp. 339-367. Tanner, Roger I.; 1973, "A Correlation of Normal Strss Data for Polyisobutylene Solutions", Transactions of the Society of Rheology, vol. 17, no. 2, pp. 365-373. Tanner, Roger I.: 1988, Engineering Rheology, Clarendon Press, Oxford, revised edition. Tomita, Yukio; and Hiroshi Kato, 1967, "A Study on the Elastodynamic Pump" Bulletin of JSME, vol. 10, no. 39, pp. 507-515. 161