MSU LIBRARIES ._:_— RETURNING MATERIALS: P1ace in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. VISCOUS HEAT DISSIPATION IN TANGENTIAL ANNULAR FLOW OF NON-NEWTONIAN FLUIDS By Jill Marie Kennedy-Tolstedt A THESIS Submitted to Michigan State University In partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Agricultural Engineering Department of Agricultural Engineering 1988 yoay-trxt ABSTRACT VISCOUS HEAT DIS SIPATION IN TANGENTIAL ANNULAR FLOW OF NON-NEWTONIAN FLUIDS By Jill Marie Kennedy-Tolstedt A mathematical model was developed to determine the level of viscous heat dissipation of non-Newtonian fluids in tangential annular flow. A finite difference model was developed to calculate the velocity profile and the constant of integration from the equation of motion. A finite element program was adapted to predict the temperature profile. The model is based on the equations of energy, motion and continuity, and the rheological model involving four fluid parameters. The model was used to evaluate the effect of varying the annular gap, angular velocity and fluid type on viscous heat dissipation. An experimental design was implemented to validate the model. The effects of annular gap, angular velocity and fluid type can be embodied in the Reynolds number. The model correctly predicts the measured temperature profile when laminar flow exists, (Re < 10). When turbulent flow occurs, (Re > 10), the model predicts temperature values that are higher than the experimental data. Approved Dr. Robert Y. Ofoli Approved Dr. Donald W. Edwards Date To my husband Mark and my parents iii ACKNOWLEDGMENTS I would like to express my gratitude to the following people for their support and assistance throughout my graduate program. Dr. Robert Ofoli for the many hours of his time spent answering questions, giving suggestions and for encouraging me to try new ideas. Dr. Larry Segerlind for sharing his ideas, his finite element knowledge and his office space. Dr. Jim Steffe for encouraging me to undertake a master’s degree and Dr. Jayaraman for his assistance during the theory development. Brian Heskitt, Kevin Rose, D’anne Larson and Kevin Mackie for assistance in programming, lab work and general encouragement. Marcia, Susan, Kelly, Mike and Nancy for their continued enthusiasm, patience and understanding throughout my degree. Especially to my parents and my husband, Mark, for sympathy when things went wrong, understanding when my patience ran out, for helping me to keep moving forward and for loving me in spite of it all. iv Table of Contents Table of Tables .................................................... vi Table of Figures .................................................... vii Nomenclature ..................................................... viii 1 INTRODUCTION ................................................ 1 2 OBJECTIVES : .................................................. 4 3 LITERATURE REVIEW .......................................... 5 4 MODEL DEVELOPMENT ....................................... 13 4.1 Relevant Mathematical Relations ----------------------------- 20 42 Rheology ................................................ 23 4.3 Numerical Solution for the Velocity Profile -------------------- 26 4.4 Finite Element Solution for the Energy Equation ----------------- 27 5 EXPERIIVIENTAL PROCEDURE ................................. 31 5.1 Description of the Test Apparatus ............................ 31 5.2 Calibration Techniques .................................... 42 5,3 Fluids Tested ............................................ 43 5.4 Testing Procedure ........................................ 46 6 RESULTS AND DISCUSSION ................................... 48 7 CONCLUSIONS ................................................ 69 8 SUGGESTIONS FOR FUTURE RESEARCH ........................ 70 APPENDICES ................................................... 7 1 Appendix A Raw Data ....................................... 71 Appendix B Rheological Data .................................. 91 Appendix C Computer Program - VELOC ........................ 122 Appendix D Computer Program - ODTIME ....................... 126 BIBLIOGRAPHY ................................................ 165 Table of Tables Table 1. Descriptions for Figure 1 ......................................................................... 32 Table 2. Specification of Test Cylinders ............................................................... 32 Table 3. Motor Controller Speed Chart ................................................................. 44 Table 4. Rheological and Physical Data on Test Fluids ........................................ 45 Table 5. Shear Rate Data ....................................................................................... 49 Table 6. Reynolds Numbers .................................................................................. 55 Table 7. Values of C1 ............................................................................................ 56 Table A.1 2% CMC, 1 mm Annulus, 240 RPM .................................................... 71 Table A2 2% CMC, 1 mm Annulus, 600 RPM .................................................... 72 Table A3 2% CMC, 1 mm Annulus, 920 RPM .................................................... 73 Table A.4 2% CMC, 2 mm Annulus, 600 RPM .................................................... 74 Table A5 2% CMC, 3 mm Annulus, 600 RPM .................................................... 75 Table A.6 Honey, 1 mm Annulus, 240 RPM ........................................................ 76 Table A.7 Honey, 1 mm Annulus, 600 RPM ........................................................ 77 Table A.8 Miracle Whip, 3 mm Annulus, 240 RPM ............................................. 78 Table A9 Miracle Whip, 3 mm Annulus, 600 RPM ............................................. 79 Table A.10 Raw Data - 2% CMC, 1 mm Annulus, 240 RPM ............................... 80 Table A.11 Raw Data - 2% CMC, 1 mm Annulus, 600 RPM ............................... 82 Table A. 12 Raw Data - 2% CMC, 1 mm Annulus, 920 RPM ............................... 83 Table A.13 Raw Data - 2% CMC, 2 mm Annulus, 600 RPM ............................... 84 Table A.14 Raw Data - 2% CMC, 3 mm Annulus, 600 RPM ............................... 86 Table A. 15 Raw Data - Honey, 1 mm Annulus, 240 RPM ................................... 87 Table A.16 Raw Data - Honey, 1 mm Annulus, 600 RPM ................................... 88 Table A.17 Raw Data - Miracle Whip, 3 mm Annulus, 240 RPM ........................ 89 Table A.18 Raw Data - Miracle Whip, 3 mm Annulus, 600 RPM ........................ 90 Table 3.1 Brookfield Standard - Test 1 ................................................................. 91 Table 8.2 Brookfield Standard - Test 2 ................................................................. 93 Table 3.3 Brookfield Standard - Test 3 ................................................................. 95 Table 3.4 2% CMC - Test 1 .................................................................................. 97 Table B5 2% CMC - Test 2 .................................................................................. 99 Table B6 2% CMC - Test 3 .................................................................................. 101 Table 3.7 2% CMC - Test 4 .................................................................................. 104 Table 3.8 2% CMC - Test 5 .................................................................................. 106 Table B9 2% CMC - Test 6 .................................................................................. 108 Table 3.10 2% CMC - Test 7 ................................................................................ 110 Table B.11 2% CMC - Test 8 ................................................................................ 113 Table 3.12 2% CMC - Test 9 ................................................................................ 115 Table 3.13 Honey .................................................................................................. 118 Table 3.14 Miracle Whip ...................................................................................... 120 vi Table of Figures Figure 1. Experimental Setup ................................................................................ 33 Figure 2. Cross-Section of Concentric Cylinders .................................................. 34 Figure 3. Inner Cylinder Assembly ....................................................................... 35 Figure 4. Platform for Apparatus ........................................................................... 37 Figure 5. Front View Apparatus on Platform ........................................................ 38 Figure 6. Thermocouple Mounting ........................................................................ 39 Figure 7. Side View Thermocouple Location ........................................................ 40 Figure 8. Top View Thermocouple Location ........................................................ 41 Figure 9. Effect of Rotational Speed 2% CMC, 1mm gap ................................... 50 Figure 10. Effect of Rotational Speed Miracle Whip, 3mm gap .......................... 51 Figure 11. Effect of Rotational Speed Honey, 1mm gap ...................................... 53 Figure 12. Effect of Annular Gap 2% CMC, 600 RPM ....................................... 54 Figure 13. Comparision of Predicted and Observed Time-Temperature History, 2% CMC .................................................................... 58 Figure 14. Comparision of Predicted and Observed Time-Temperature History, 2% CMC .................................................................... 59 Figure 15. Effect of Reynolds Number 2% CMC, 600 RPM ............................... 60 Figure 16. Effect of Reynolds Number 2% CMC, 1mm gap ............................... 61 Figure 17. Effect of Fluid Type 240 RPM, 1mm gap ........................................... 63 Figure 18. Effect of Fluid Type 600 RPM, 3mm gap ........................................... 64 Figure 19. Time-Temperature History 2% CMC, 600 RPM, 3mm gap ............... 66 Figure 20. Time-Temperature History 2% CMC, 50% of gap ............................. 67 Figure 21. Time-Temperature History 2% CMC, 99% of gap ............................. 68 vii Bil! Br {F} Gr NOMENCLATURE dimensionless group, Eqn. 3-1 Brinkman number, dimensionless Constant of integration, Eqn. 3-7 Constant of integration, Eqn. 4-16 Heat capacity at constant pressure, I kg-l C-l Capacitance matrix, Eqn. 4-39 impeller diameter, m FEM coefficient, Eqn. 4-44 FEM coefficient, Eqn. 4-45 Eckert number, dimensionless, Eqn. 3-5 Force vector, Eqn. 4-39 Gravity, m/s2 Griffith number, dimensionless, Eqn. 3-22 Graetz number, dimensionless, Eqn. 3-23 Heat transfer coefficient, W/m2°C channel width, m viii P10! Pe Q0) ‘1 Second invariant Thermal conductivity, W/m °C Stiffness matrix, Eqn. 4-39 Axial length, m Node spacing, m, Eqn. 4-48 Consistency coefficient Power law index, dimensionless Rheological parameter, dimensionless Rheological parameter, dimensionless Pressure, Pa Dimensionless pressure drop, Eqn. 3-7 Velocity independent, dimensionless pressure drop, Eqn. 3-7 Peclet number, dimensionless Prandtl number, dimensionless, Eqn. 3-5 Heat flux, W/ m2 Viscous heating term, (FEM) Radial coordinate, m FEM, midpoint of two nodes, Eqn. 4-47 ix RJ' Re BIO! FEM, location of node i, Eqn. 4-47 FEM, location of node j, Eqn. 4-47 Radius of outer cylinder, m Reynolds number, dimensionless, Eqn. 6-1 FEM, oscillation criterion, Eqn. 4-50 Time, 5 Temperature, °C Velocity, m s1 Average velocity, m 3-1 mean velocity, m 3-1 Mean rate of heat dissipated per unit volume, W/m3, Eqn. 3-3 water content, % Axial coordinate, m Dimensionless length, Eqn. 3- 13 GREEK SYMBOLS Constant, Eqn. 39 Thermal expansion coefficient, K-1 Slip correction coefficient, dimensionless “’0 max R/Ro, dimensionless Apparent viscosity, Pa-s Angular velocity, 3-1 Shear stress, Pa Yield stress, Pa Viscosity, Pa s Consistency coefficient, Pa 511 Consistency coefficient, Pan1 5“2 Shear rate, s-1 R -Ri, m Density, kg m-3 Rate of deformation tensor, Eqn. 4-18 nabla operator Dissipative factor, Eqn. 3-3 SUBSCRIPTS AND SUPERSCRIPTS maximum initial wall xi 0P rheol operating rheological mean or average thermal, Eqn. 3-13 angular coordinate xii 1 INTRODUCTION Viscous heating is important in several engineering problems such as flow of a lubricant between fast moving parts, flow of fluids in extrusion dies and boundary layer flow in re-entry problems. In food processes such as extrusion, tube flow, and mixing, heat generation due to shearing can cause a significant rise in temperature. Heat generation can also present problems in the determination of rheological properties in rotational viscometry or cone and plate viscometry. When processes are bound by critical temperatures, unaccounted for changes in temperature lead to problems in quality control of the product. A common misconception in analyzing flow situations is that viscous heat generation is negligible at low velocities,. Actually, viscous heating is a function of the velocity gradient, so that even at low velocities, viscous heating does occur. In the Newtonian case, predicting when viscous heating will be significant is fairly straight forward. For non-Newtonian fluids, complications arise due to the fact that viscosity is a function of the shear rate as well as temperature. When viscous heating is neglected, calculated temperature profiles can be underestimated. Bird, Armstrong and Hassager (1977) discussed tangential annular flow when addressing the phenomenon of rod climbing in non-Newtonian fluid behavior. For steady laminar flow they make the usual assumption that "a fluid particle will follow a circular trajectory centered on the axis of the cylinders and lying in a horizontal plane". This assumption is critical to tangential annular flow analysis. In rheological studies, sources of error are generally attributed to five main areas: 1) slip at the wall, 2) eccentricity, 3) viscous heat dissipation, 4) end effects and 5) secondary flows. Of these five, viscous heating has received the least attention due to the complications associated with its analysis in non-Newtonian fluids. Dealy (1982) discusses these sources of error in addition to rod climbing. He states that in annular flow of non-Newtonian fluids when the gap to length ratio is small, error due to end effects is approximately 2%. For a non-Newtonian fluid, secondary flow is characterized as a discontinuity in the slope of the shear stress versus shear rate curve. He developed an equation to predict the maximum temperature rise due to viscous heating of a power law fluid. 11R 292 T To 2k ( ) This is the same equation developed by Bird, Armstrong and Hassager (1977) when they discussed tangential annular flow in the estimation of temperature rise caused by viscous heating in concentric cylinder viscometry. Correct accounting for viscous heating leads to accurate temperature profiles, better estimation of rheological properties and improved product quality control. This study is concerned with viscous energy dissipation in purely tangential flow in concentric cylinders, with the inner cylinder rotating and the outer cylinder stationary. The generalized rheological model of Ofoli et al. (1987) given by a" = 02‘ + 11.17" (1-2) was used in this study. Hereafter, this model will be referred to as the OMS model. This is a four-parameter model for characterizing inelastic fluid foods. By appmpriate choice of parameters, conventional rheological models become special cases of this model. The model can, therefore, represent power law, Bingham plastic, Herschel-Bulkley, Casson, Heinz-Casson and Mizrahi-Berk fluid foods. Solving flow problems in terms of the generalized case results in equations that are applicable in a wider variety of circumstances. 2 OBJECTIVES The objectives of this study were: 1. To derive a mathematical model based on a generalized non-Newtonian fluid for determining the effects of viscous heat dissipation in an annular concentric cylinder, with the inner cylinder rotating and the outer cylinder stationary. 2. To use the model to predict the temperature profile of fluids held in rotation in the annular gap and to access the effect of annular gap width, angular velocity, rheological behavior of the fluid and Reynolds number on the level of viscous dissipation. 3. To test the model with experimental data. 3 LITERATURE REVIEW In their investigation of viscous heating in some simple shear flows, Sukanek and Laurence (1974) suggested the existence of a double-valued shear rate. This shear rate is due to viscosity dependence on temperature and the presence of viscous heat dissipation. Experimental studies were undertaken using three configurations: 1) plane Couette flow, 2) circular Couette flow and 3) circular Poiseuille flow. Using a Newtonian fluid with a strong temperature dependence, data clearly showed the double-valued shear rate but the magnitude is less than that predicted by their model. Some reasons cited for the discrepancy are the inability to attain a constant initial temperature, problems with keeping the gap full of fluid and possible eccentricity of the inner cylinder. Higgs (1974) studied the error due to ignoring slip at the wall in determining fluid flowrates for tube flow and annular flow. He found that for tube flow an error of up to 26% can be made in flow rate calculations. Using a slip correction coefficient, 13*, flowrates can be adjusted for slip conditions. It was determined that [3* was approximately constant and that assuming constant [3* produced only a 1% error in flow rate calculations. For rotational viscometer flow, he found that the error due to ignoring slip is offset by a lower value for the consistency coefficient. Because of the offsetting errors, the assumption of no slip at the walls is approximately correct. Seichter (1985) developed a method for estimating viscous dissipation effects on the power requirements and pressure drop across a screw pump. Although he did consider temperature dependent viscosity, his procedure was developed for Newtonian fluids only. For a highly viscous Newtonian fluid he calculated an 8.5% increase in temperature due to viscous heat dissipation, at a point 576 mm along the screw of the pump. In their study of viscous heating of a power law liquid in plane flow, Gavis and Laurence (1968) predicted a temperature change of 0.2 to 40°C when B‘ ranges from 0.025 to 200, where B' is given by 3' = (Ls-)3») (3-1) It and the Brinkman number for a power law fluid is defined by Br“) = ha-nvglflho (3—2) kTo The consistency coefficient was assumed to vary exponentially with temperature. Turian and Bird (1963) studied the error in viscosity measurements due to viscous heating when using a cone and plate viscometer. They reported a 15% error in viscosity measurements when considering Newtonian fluid flow. This error in viscosity was due to a 3°C change in temperature resulting from viscous heating. They preposed a means of estimating the temperature rise independent of the rheological model used. Froyshteter (1977) found that the rheological model chosen has a significant effect on temperature and velocity profile calculations. He studied the effect of an internal heat source on the heat transfer of non-Newtonian fluids. He found—that for the case of cooling, instability in the laminar flow occurs. Froyshteter (1982) studied heat transfer in tube flow of high viscosity non-Newtonian fluids and the stability of laminar flow due to heating and viscous heat dissipation. Two boundary conditions were studied: 1) constant heat flux and 2) constant average fluid temperature. In his analysis, he chose the Herschel-Bulkley rheological model and assumed an exponential temperature dependence for the plastic viscosity. He found that for constant heat flux, a temperature profile developed. When the temperature gradient was large, the viscosity was no longer constant across the cross-section of the tube. At the wall, where the temperature was at a maximum, the viscosity was reduced due to its dependence on the temperature. Near the center of the tube the temperature was not as high and therefore the viscosity is higher. Because the viscosity at the wall is reduced by the increase in temperature, the shear stress at the wall dropped sharply. This coupling of viscosity and shear stress means that the viscosity can effect the form and magnitude of the temperature profile. A viscosity change of three-fold resulted in a linear temperature rise. Froyshteter defined the dissipative factor ¢ as Wrw 2% ¢ = (3—3) Froyshteter found that for constant wall heat flux and 0.5 5 Br 5 1.0, the dissipative factor decreases with increasing length and approaches zero. However, for constant average fluid temperature in the same range of Brinkman numbers, the dissipative factor approaches infinity. Forrest and Wilkinson (1974) examined laminar heat transfer to power law fluids in tubes with constant wall heat flux. The effects of viscous dissipation and uniform internal heat generation were included. In the case of constant wall heat flux, they found viscous dissipation was a dominant factor. When viscous dissipation occurs, it dominates the heat transfer mainly through changes in the apparent viscosity of the fluid and other temperature-dependent rheological properties. In a study by Agur and Vlachopoulos (1981), heat transfer of a power law fluid in tube flow was considered. Using a finite difference solution for the problem of temperature dependent viscosity, it was found that the temperature profiles were the same for both temperature -dependent and -independent rheological properties, when fully developed flow was considered. When viscous dissipation was significant, the temperature was greatest near the wall at the point of maximum shear. Dang (1984) studied forced convection heat transfer of power law fluids at low Peclet numbers. He found that when viscous dissipation is present, the Nusselt number does vary with the Peclet number for Fe = 1, 5 or 10. The average temperature reaches equilibrium when ’JUIN IV (3—4) He also found that the average temperature increases with increasing power law index. Marner and Hovland (1973) used the product of the Eckert and Prandtl numbers as an indication of the importance of viscous dissipation. Their equation was n+1 E-Pr = my (3—5) M" As the product increases, viscous dissipation becomes important. In their study of vertical tube flow of non-Newtonian fluids, they confirmed that viscous dissipation distorts the velocity profile. They found that viscous dissipation also tends to increase the fiiction factor and decrease the N usselt number. A second geometry that has been used in many research studies, is plane Couette flow. Winter (1971) calculated the temperature and velocity fields for a Newtonian fluid with a temperature dependent viscosity, assuming constant wall temperature in plane Couette flow. He defined the Brinkman number as Br = nkT (3‘6) He showed that unless the viscosity is a function of temperature, the Brinkman number will not be influenced by the developing temperature field. A criterion, BBr, was defined to characterize the development of the temperature field and determine how rapidly the temperature field develops and whether equilibrium has been reached. If BBr = 0 then the temperature remains constant. When BBr is increasing the time 10 required for thermal development decreased, and when as: 2 x1 the temperature continued to increase and the viscous heat dissipated cannot be conducted away rapidly enough. Sukanek (1971) developed a series of relationships in order to determine the integration constant, C, that arises from the momentum equation. This constant can be expressed as 3 —2a F " C = 3(3) (3—7) where the dimensionless pressure group, P*, was defined as p»: = [Q'Br (3—8) and a = (6n: 2) (3 — 9) The Brinkman number was defined as = % (3 — 10) where the consistency coefficient was expressed as 11 T-T u = u.exp[—B( T ‘9] (3-11) By fixing the initial velocity, the Brinkman number can be determined. The following equation relates the Brinkman number and the velocity-independent, ' dimensionless pressure drop, P* and combined them to determine the value of P* u-l 1 Br m Br _‘ Br = 5(5) - 2(3) (3—12) Once P* is known, Eqn. 3-8 can be used to calculate I" and then Eqn. 3-7 can be used to find the value for the integration constant, C. Once C is known the temperature and velocity profiles can be solved. This relationship exists for Poiseuille flow of a power law fluid when viscous heating is important. Bonnett and McIntire (1975) used a modified Galerkin technique to study the problem of dissipation effects in the hydrodynamic stability of viscoelastic fluids. The stability of the flow is a means of determining the possibility of melt fracture occurring. They found that when viscous dissipation is included in the energy equation for plane Couette flow with a superimposed temperature gradient, overstability occurs. Overstability refers to the finite element solution for this problem. When the solution becomes unaffected by changes in material properties it is said to be overstable. The overstability tends to mask the effects of the material properties. They found that as the Brinkman number increased, the solution for the velocity profile became increasingly unstable. 12 Many researchers have studied viscous heating in the cone and plate viscometer. Bird and Ttuian (1962) found that for a standard cone and plate viscometer, a temperature rise of 3°C, due only to viscous heating, is possible. A model was developed that estimated the temperature rise independent of the rheological model. In a second study Turian and Bird (1963) looked at Newtonian fluids with temperature dependent viscosity and thermal conductivity. They speculated that the deviations from Newtonian flow were caused by viscous heating. In a third study, Turian (1965) investigated viscous heating of non-Newtonian fluids with temperature-dependent viscosity and thermal conductivity. He developed an analytical solution for the velocity and temperature profiles when viscous heating effects were important. The two sets of boundary conditions studied were 1) both plates at constant temperature and 2) the stationary plate at constant temperature with zero heat flux through the moving plate. This study considers two fluid models, power law and Ellis. He used a series expansion to determine the deviation in viscosity caused by viscous heating. Lindt (1980) developed a mathematical model to study the flow of a Newtonian fluid in concentric cylinders. In his model, velocity, temperature and concentration were time-dependent while viscosity was given as an exponential function of concentration. He illustrated the predictive capacity of the model by simulating annular flow of a polymer in a one mm gap with the outer cylinder, R°=10 mm, rotating at an angular velocity of one reciprocal second. For these conditions, the Brinkman number was less than one, showing that the viscous heating rate was relatively unimportant. Winter (1977) studied heat transfer for the helical flow case when gradients with respect to the z-direction are zero. He found that heat transfer is largely influenced by rheology. Newtonian and power law behavior were considered with viscosity as a 13 function of temperature, pressure and time. He defined six dimensionless parameters, four relating to geometry, the Biot number and the Griffith number. He found that as the Griffith number increases, so does the importance of viscous dissipation. In an earlier study, Winter (197 3) used an iterative finite difference method to examine helical flow in an annulus. He defined an axial coordinate Z such that z = 5 = —— (3—13) where lTis the thermal development length. The temperature field is established, for 103 < Z < 1. This criterion is useful for determining the length necessary for thermal development. Analysis of the model showed that most of the heat is dissipated in layers close to the inner wall. Heat is conducted to the center of the annular gap and to the outer wall. At thermal equilibrium, all the heat is conducted to the outer wall. The velocity field is strongly dependent on the temperature. As the temperature field develops, the velocity field becomes asymmetric with a viscous layer near the outer wall and most of the flow occurring near the inner wall. He found that the maximum shear rate occurs in the warmest layer at the inner wall. Kiparissides and Vlachopoulos (1978) investigated viscous dissipation in the calendaring of power law fluids. Temperature rise due to viscous dissipation was found to increase with increasing power law index and also with increasing consistency coefficient. 14 Bird et a1 (1977) discussed tangential annular flow for the phenomenon of rod climbing in non-Newtonian fluid behavior. They developed the following equations of motion for this case r—component . = -— tam» 1:) e-component o - —%%(rzo,e) (3—15) z-component o —:—’z’ - £200..) + pg. (3-16> For a Newtonian fluid the normal stress would be zero but for a non-Newtonian fluid the primary normal stress difference is nonzero and is postulated to be the cause of rod climbing. They concluded that the normal stresses are nonzero and that the normal stress difference is negative. 15 For tangential annular flow of a Newtonian fluid, Bird et al (1960) reduced the equations of motion by considering steady-state, laminar flow when only the velocity in the 6 -direction is nonzero. When the inner cylinder is stationary and the outer cylinder rotates at constant angular velocity, the velocity profile is given by (if-é (K-i) v, = siR (3-17) and the stress distribution is 0.9 = "Z“WIIIEX'i—ie] (3— 18) They compared the case of the inner cylinder rotating and the outer cylinder stationary to the case of the outer cylinder rotating and the inner cylinder stationary. A much higher Reynolds number is required for transition to turbulent flow when the outer cylinder rotates. The transition Reynolds number is strongly dependent on the ratio of the annular thickness to the radius of the outer cylinder. Pearson (1978) reviewed recent studies of non-Newtonian viscous fluids when high heat generation and low heat transfer dominate. He discussed the determination of dominating factors based on the Griffith and Graetz numbers. Because polymers are thermal insulators and many flow situations are largely adiabatic, mean temperature rises of 10 to 50 K are possible. The Graetz number alone does not give information on the importance of viscous heating for a given flow system. When the Griffith number is combined with the Graetz number, three main categories can be defined as 16 Gr 5; « 1 (3 - 19) when viscous heat generation is negligible Gr 6—2 — 0(1) (3 — 20) when viscous heat generation cannot be neglected and Gr a; 1 (3 — 21) when viscous heat generation is dominant In the above relations, ro’ Gr = m (3 — 22) rheol o C VH2 02 = 9 2L (3 - 23) ATM = i (3 — 24) 17 These relationships can be used to estimate the importance of viscous heating in a specific flow situation. However, the information necessary to calculate the Griffith number may not be readily available. The Brinkman number is usually smaller than the Griffith number and is defined as B — “0V2 (3 25) r ‘ kATap where M0,, = T, - T, (3—26) Large values of Griffith or Brinkman numbers arise from large values of velocity and small values of thermal conductivity; therefore, either the Brinkman or Griffith number should lead to the correct estimation of the importance of viscous heat generation. 4 MODEL DEVELOPMENT The general relationships which govern fluid flow and heat transfer are the continuity, momentum and energy equations. These equations are n'ni 53—?- + p(V-V) = o (4—1) Momentum 9%?- = -VP - Na) + p? (4—2) £11m pc. % = -Vq - rg—T’ilwfv') — (69W) <4-3) The geometry of interest in this study is an annular concentric cylinder, with the inner cylinder rotating while the outer cylinder is stationary. Only purely tangential flow is considered. The OMS model, a generalized rheological model, is incorporated into the equations of motion and energy to derive a relationship for viscous heat dissipation. The following assumptions were made: 1. Laminar flow exists, that is, there is no mixing of the fluid layers. 18 19 . Negligible slip at the wall. . The outer and inner cylinders are perfectly insulated. . Negligible inertial forces. . Incompressible fluid. V-V = 0 (4—5) . Constant density and heat capacity . No angular variation in velocity or temperature, that is, and if ._ 0 4 6 3T :36- - 0 (4-7) 8. The fluid can be characterized using the OMS model. 9. Each fluid particle follows a horizontal, circular trajectory about the axis of the cylinders. 20 4.1 Relevant Mathematical Relations With the assumptions in the previous section, the equations of continuity, motion and energy take the following forms Saarinen): V - V = 0 (4 — 8) Equations of Manon The three components of the equation of motion are acorn DQIIQII L ‘P— = -a—r (4-9) The r-component of the equation of motion cannot be solved directly since neither the pressure gradient nor the velocity profile is known. 052mm ave 1 a 2 F; _ 7500,.) (4-10) In the e-component of the equation of motion it was assumed that 21 ave ~57~0 (4—11) Within the narrow range of temperature increases expected in this study, (5-15°C), changes in density and viscosity are not expected to be significant, therefore changes in velocity with respect to time are expected to be negligible. The (El-component of the equation of motion, therefore, reduces to 1 a 2 0 — _r28r(r 0,9) Writ 3p 0 _ 32 + pg: This equation simplifies to AP = pgAz (4— 12) (4— 13) (4- 14) These equations are in agreement with those developed by Bird et a1 (1960) for tangential annular flow of a Newtonian fluid. Warm (4— 15) 22 Before the energy equation can be solved, the e-component of the equation of motion, Eqn. 4-12, must be solved for the velocity profile. Eqn. 4-12 can be written as C1 = r20“ 9 (4— 16) f where C1 is a constant of integration. The constitutive equation for shear stress is given as 3 = —nA (4— 17) - a V9 _ 0 ra—r[7] 0 K = 4 — 18 r—a- 33] 0 0 ( ) 3r r L 0 0 0_ so that a V9 A,9 -— rar(r] (4 19) The shear stress can be expressed as 6,9 = — 11A,, (4 — 20) since only one component of the stress tensor is nonzero. The shear rate is given by 23 2 W) ~51?) = Substituting Eqn. 420 into Eqn. 4-16 and rearranging, gives NIH RI l>| — = ”Are (4_22) Before Eqn. 4-22 can be solved, a constitutive equation is required for the apparent viscosity. 4.2 Rheology Using the OMS model, the shear stress distribution is n: "1 o = o, + uj" (4—23) Rearranging the constitutive equation, Eqn. 4-20, gives are = —— 4 - 24 11 l ( ) Combining Eqs. 4-21, 4-23 and 4-24, the expression for the viscosity becomes 0, "‘ ”5-.., I n = [7] + 11.0!) (4‘25) 24 Combining Eqn. 4-22 and 4—25 with Eqn. 4-16 gives Cl , "‘ . - . 7 = [ [67] + 11.6)” "‘ l7 (4-26) which reduces to C1 ‘1 "r "'2 —, = o. + 11.7 (4-27) For the annular geometry, when r goes from the inner cylinder to the outer cylinder, the slope of the velocity gradient will be negative so that the shear rate is given by . a V9 7 — —r 8r [ ] (4 - 28) Combining Eqn. 4-27 with the sign change for the shear rate gives I Q C "‘ . . [:51] _ o1 _ “any": (4 _ 2'9) Solving for the shear rate produces . C“ 3‘ . = 1 a.) H 25 Substituting Eqn. 4-28 into Eqn. 4-30 yields a V9 _ 1 C1 ”1 __:: "37(7) ' [ El?) ‘ u. I (4'3” which can be written as C "‘ 02‘ «[3] = [ lb] — —— Ila,- (4—32) r 1.1., r 11.. r Equation 4-32 must be solved numerically. From Eqn. 4- 16 Q C 0,, = —‘ (4—33) Incorporating Eqs. 4-31 and 4-33 into the energy equation,(Eqn. 4-15), it can easily be shown that _ C1 CW1 02‘ 1:29 W - all?) m - J + an 26 Equation 4-34 has a form that can be solved using the finite element method, (FEM). Before the energy equation can be solved, the value for C1 must be determined. The integration constant and the velocity profile can be determined by an iterative numerical integration of Eqn. 4-32. This procedure is outlined in the next section. 4.3 Numerical solution for the velocity profile The velocity profile and integration constant, C,, are determined using iterative numerical integration of Eqn. 4—32. The limits of integration for the velocity profile come from the boundary conditions. 1) v(0) = QR. at r = R,- (4 — 35) 2) v (0) = 0 at r = R0 (4 - 36) Calculations were made using the Fortran computer program, VELOC, developed for this study (Appendix C). Utilizing the boundary conditions, Eqn. 4-32 can be written as: 0 V9 _ R‘ 1 031,1 1., (7) - 1 [.—_(f—:)"‘ ——.: I; The left-hand side is equal to the angular velocity, 0. For a first approximation, C1 is set equal to the angular velocity and using Simpson’s rule the integral is evaluated. The procedure is then to continue to use new estimates of C1, computer generated, until 27 the left-hand side is equal to the right-hand side. Once C, is found, the values for the velocity profile across the annular gap can be determined. Once the temperature profile is known the mean temperature can be calculated by T = If T(r) v(r)9 rdr .. If v(r), rdr 4.4 Finite Element Solution For the Energy Equation (4— 38) For time-dependent, one-dimensional problems, the basic finite element equation (Segerlind, 1984) can be written as [C]{T} + [K]{T} - {F} = 0 (4-39) For this problem, the energy equation can be represented by P3: 3,2 :ar ' 9‘” (4‘40) Where Q(r) represents the viscous heating term. The relationship between the matrices in Eqn. 4-39 and the terms of the energy equation are mag,- => [cm (4—41) 321 HT k5; + 73; => [Kl{T} (4-42) and Q(r) = {F} (4-43) The computer program used has been adapted from a general one-dimensional, unsteady state, finite element program, ODTIME, written by Dr. Segerlind at Michigan State University, (Appendix D). Changes were made to convert the specific sections needed from cartesian to polar coordinates. Also, the expression for Q(r) was incorporated into the program The following changes were required for conversion from cartesian to polar coordinates Element Stiffness Matrix (a _ &[1 “1] 2750;.[1 -1] _ [k’] - = L _1 1 (4 44) Element Capacitance Matrix, lumped formulation " - ’1? ‘1 MP “1 - [ch—201:3 (445) 29 Element Force Vector Um} = QTLll} = ——Q(';2“L{: :. : 12:} (4-46) where _ R,- + R,- r = —_2— (4—47) For this particular problem D, = k D, = pCp The node spacing, L, is the same as the radial spacing. L = Ar (4—48) The use of the FEM does not insure absence of numerical problems. However, two criterion are available to reduce the possibility of numerical oscillation. They are implied in using the lumped formulation for the element stiffness matrix. The first one is At J1”— 4 49 < 4D,,(1-e) (_ ) The second criterion is At > 0 (4-50) 30 Theta is a value representing various solution methods. 9 = 0 Forward Difference e = ; Central Difference e = ; Galerkin Method 6 = lBackward Difference With these two criteria satisfied, the lumped formulation has a much larger operating range than the consistent formulation. Choices for the time and node increments which fit this criterion help to reduce the chance of numerical oscillation, thereby increasing solution accuracy. The solution to this problem is twofold. First the VELOC program was executed to determine the value of the integration constant and the velocity profile. Secondly, the ODTIME program is run to determine the temperature profile. Inputs to ODTIME include rheological data, the integration constant, geometry data and grid information. The time step used was one second with theta of two thirds. The initial temperature values are listed with the raw data (Appendix A). The element was divided into 50 nodes, to obtain a smooth curve of data, for the temperature versus annular gap profiles and 3 nodes for the temperature versus time profiles. The variables definition of all are listed in the program. 5 EXPERIMENTAL PROCEDURE 5.1 Description of the test apparatus A drawing of the test apparatus is shown in Figure 1, with Table 1 listing the descriptions that go with the figure. The outer cylinder is constructed of PVC pipe to help minimize heat losses through the wall. The inner cylinder was hollow and filled with sawdust for insulation to minimize heat loss into the core (Figures 2 and 3). Specifications are listed in Table 2. These dimensions were determined by running the model for various combinations. The combinations chosen were the ones for which the theoretical model predicts easily measurable temperature rises. The base of the apparatus was made of a 0.0127m (1/2") acrylic counter-sink which fits into a PVC endcap. The endcap was bolted to a 0.1016m x 0.1016m x 0.0254m (4" x 4" x 1") piece of gardor. The base pieces were hermetically sealed with silicone gel. A 0.0127m (1/2 ") brass bushing was inserted through this combined basepiece to assist in alignment. A ball bearing was located at the bottom of the base for the inner cylinder to rotate on. The bottom shaft of the inner cylinder had a recessed bottom to fit over the ball bearing. The top piece of the apparatus was of similar construction and was used for alignment. A PVC endcap fits over the two cylinders with the shaft of the inner cylinder protruding through the endcap. This endcap was bolted to a piece of gardor to create a greater thickness to help with alignment. A 0.0127m (1/2") brass bushing was inserted 31 32 Table 1. Descriptions for Figure 1 Lml Description i A Outer Cylinder, 0.4862 m PVC tubing 0.039 m radius B Inner Cylinder, steel tube filled with sawdust ! 3 sizes: 0.009 m, 0.019 m, 0.028 m ' 0.0762 111 DIA PVC Endcap 0.0127 m Acrylic Counter Sink for bolt head 0.0762 111 DIA PVC Endcap Washer and nut Gardor Baseplate J Dayton 1 hp Motor Table 2. Specification of Test Cylinders Description C D E P 0.0127 m Gardor alignment piece G H Recess for inner cylinder shaft, shaft rotates on ball bearing Outer Cylinder Inner Cylinders 18.0 463.0 8.98 38.1 468.0 19.07 55.8 463.0 27.88 H 33 -q:r-‘__/G W M“ E/‘L 3T _~ A B/—_ i D/‘lj L'I coir #1 H. Figure 1. Experimental Setup 34 Insulation \ Cyfinder lnner Cyhnder Sawdust Filler Figure 2. Cross—section of concentric cylinders \\\\X\ Boned Filled with Sawdust _. \_ Welded Figure 3. Inner Cylinder Assembly 36 into the opening to reduce fiictional wear on the PVC endcap and gardor. Two coupling bodies, with a spider between, connect the inner cylinder shaft with the shaft of the motor. The motor was a 1 HP Dayton constant RPM motor with a matching Dayton SCR controller that has an RPM range of 100 - 1200. This system was calibrated with two handheld tachometers 1. 770 TIF Photoelectric Tachometer 2. DT-205 Shimpo Digital Tachometer, Electromatic Equipment Company The apparatus was mounted to the platform shown in Figure 4. This platform was constructed of wood. The motor was bolted onto the apparatus as shown in Figure 5. Near the middle of the apparatus, a clamp was mounted to the platform and holds the apparatus in place. Holes were drilled through the base piece into the platform and pins were inserted through them to align the apparatus and reduce vibration. The thermocouples, used for temperature measurements, were arranged as shown in Figures 6 through 8. The second row of thermocouples were 90 degrees from the first row. Vertical spacing of 0.0381m (1 1/2") between thermocouples was intended to reduce local velocity disturbances from affecting nearby thermocouples (Figure 7). The wall of the outer cylinder was prepared for thermocouple mounting, as shown in Figure 6, by first shaving a small area to obtain a flat surface. A 0.00794m (5/16") in diamter was then recessed and in the middle of it a small hole was drilled into the cylinder. A 0.00794m (5/16") nut was mounted onto the recessed area using Epoxy. A hole was then drilled along the axis of a 0.00794m (5/16") bolt. The thermocouple was 37 Figure 4. Platform for apparatus 38 ILL 0 0 } 1““ Figure 5. Front View Apparatus on platform 39 T—type Epoxy lT\he:ocouple , \‘ :, j N t — Bolt] U Hypodermic Needle \ Outer cylinder N Figure 6. Thermocouple Mounting 40 0 1651 m 1 0.0381 m: 2 00381 m: 3 0.4826 m 0.0381 m: 4 0.0381 m: 5 T 0.1651 m l l Figure 7. Side View Thermocouple Location 41 Percent of Annulus 1. 99% 2. 75% 3. 50% 4. 25% 5. 0% 6. 7. 8. 99% 50% 0% Figure 8. Top View Thermocouple Location 42 inserted into a 0.127m (5") long surgical needle and the end of the T-type thermocouple was soldered to the tip of the needle. The needle was inserted into the bolt. A grommet on the end of the needle allows for the bolt to be tightened down while the needle is held in place. This procedure was carried out for each thermocouple mounted. After mounting the thermocouples, a piece of 0.0127m (1/2") insulation was fitted around the apparatus for further insulation. The thermocouples were then attached to the data acquisition system (Acquisitor by Dianachart, Rockaway, NJ). The system allows for up to 96 channels of data acquisition and has software which allows for data storage, calibration and many other functions. A Haake Rotovisco (RV-12) was used to determine the rheology of each test fluid using a MV cup with the MV—I sensor, (Appendix B). Both the 150 and 500 heads were used in collecting the rheological data. Newtonian and power law fits were obtained from a Hewlett Packerd software program attached to the Haake. For Miracle Whip, shear rate and shear stress data obtained from the Haake, were used in a non-linear, SAS regression using the OMS model. 5.2 Calibration techniques The Dayton motor was calibrated using handheld tachometers. Black tape was wrapped around the shaft to reduce reflection and then a small metallic sticker was placed on the shaft. The tachometer shines a light beam on the shaft and displays the RPM reading as reflected by the metallic sticker. Using the controller dial, various speeds were 43 located, measured and marked on the controller dial. A series of tests with the annular gap both full and empty were run while using both tachometers to verify the RPM obtained. A chart of scale reading versus RPM was made and is given here in Table 3. The thermocouples were calibrated using an ice water bath. Thermocouples were found to fluctuate +/- 0.5 °C in the ice water bath. A linear variance was assumed and each thermocouple was calibrated by adding or subtracting some factor which caused it to read 0.0 +/- 0.5 C. These calibration factors were then fed into the Acquisitor program. The Haake Rotovisco was calibrated a few days before use with a series of known' weights. Between tests, calibration was reaffirmed by testing with a Newtonian Standard. 5.3 Fluids tested Three fluids were chosen for testing: 1. 2% carboxymethylcellulose (CMC) 2. Miracle Whip 3. Honey (to provide a Newtonian sample) Their rheological and physical properties are given in Table 4. These properties were determined using the Haake Rotovisco. The values for density were determined by weighing a known volume of material. The specific heat data was based on the moisture content of the fluids and Table 3. Motor Controller Speed Chart Dayton Congoroller Setting Speed Remarks RPM motor pulses 30 240 40 420 50 600 60 772 70 920 80 1060 vibration occurs 90 1140 vibration occurs 45 Table 4. Rheological and Physical Data on Test Fluids l l 2% I Honey l Miracle CMC _ I“ 1 Whpi _ woo woo 0.806 f_ 0.472 moo 0.457 1 i Consistency Coefficient, Pa 5' 25.911 2.500 7.470 [_ 990-0 1390.0 984-0 1 Specific Heat, J kg1°C4 4100.0 4100.0 4100.0 ; Thermal Conductivity, W m1 C" 0.55 0.55 0.55 calculated using an equation given by Singh and Heldman (1984) C = 1.675 + 0.025W (5—1) P The 2% CMC was chosen because it is a non-Newtonian fluid which behaves as a power law fluid. It is readily available and is convenient for testing since it is a relatively stable fluid requiring no refrigeration. With 2% CMC the model can be tested for its ability to correctly predict the temperature rise due to viscous heating of a power law fluid. Miracle Whip was chosen because previous tests had shown it to exhibit a yield stress (Ofoli et al., 1987) This enables assessment of another facet of the model. The 46 Miracle Whip was tested for thixotrophy and was determined to exhibit slight thixotropic tendencies. The small changes in the parameters did not significantly affect the predicted temperature profile fiom ODTIME. Honey was chosen as a Newtonian fluid for comparison of the data. 5.4 Testing procedure The thermocouples were placed on the apparatus (Figure 8) to give readings at 99, 75, 50, 25 and 0% of the annular gap with two readings taken at 99, 50 and 0%. The duplicate readings were taken at a position 45°apart. This arrangement was chosen to check the validity of the assumption that there is no angular variation in temperature. Accordingly, the thermocouples were repositioned for each different inner cylinder to maintain the positioning at these gap percentages. Once the thermocouples were positioned, the inner cylinder was put into place. The fluid was then loaded into the apparatus while gently rotating the inner cylinder so that air pockets were not created. After loading, the top endcap was put on and the apparatus mounted on the platform and coupled to the motor. The setup was then left to equilibrate for several hours allowing for air bubbles to be released and the substance to reach room temperature. Before rotation began, three or four minutes of temperature data were taken to establish the average beginning temperature. The controller was then set at the desired speed and data was collected for a minimum of 15 minutes with readings taken at one minute intervals. 47 After the data was collected, the system was disassembled and thoroughly cleaned. The apparatus was then re-assembled, allowed to dry and re-equilibrated to room temperature. The testing on the Miracle Whip samples were all done on one day so that no refrigeration of the sample was needed. Rheological data for the sample was also taken on that day. 6 RESULTS AND DISCUSSION An experimental design was implemented to determine the ability of the model to predict the effects of three variables (annular width, angular velocity and type of fluid) on the level of viscous energy dissipation. The data was taken at one minute intervals. Even after calibration of the thermocouples, initial temperature readings across the annular gap varied. To represent the profile accurately, the raw temperatures were converted into a change in temperature. For the selected time interval, a change in temperature was calculated for each thermocouple. This value was added to the average beginning temperature of all the thermocouples to calculate an adjusted temperature profile Several temperature points for the given time interval were then averaged. This is the data that is presented. All the raw data is in Appendix A. The effect of angular velocity is shown in Figures 9 through 11. The shear rate range of this study is 30 - 330 s-1 (Fable 5). At 240 RPM in a 1 mm annular gap, the raw data on 2% CMC and model predictions show excellent results (Figure 9). At both 600 and 920 RPM, the data is significantly lower than the predicted results results (Figure 9). During the test runs at 920 RPM, a significant amount of rod climbing was observed, showing that flow is no longer purely tangential. This, in effect, creates a two-dimensional flow field. Figure 10 shows the data for miracle whip in a 3 mm annular gap. At 240 RPM, the model shows good agreement with the observed data. At 600 RPM the variation of the model from the observation is l to 2 °C, which is still within engineering accuracy. 48 49 Table 5. Shear Rate Data iY . Cylinder V l l Alpha Speed 5 Shear Rate j 7, l , rad s-1 - m- s-1 e l .009 .039 4.333 25.133 32.674 1 ; 62.832 81.683 1 94.248 122.525 2 .019 .039 2.053 25 . 133 49.001 62.832 122.502 94.248 183.752 1 3 E .028 .039 1.393 25.133 89.085 1 62.832 222.710 5 , 1 94.248 334.065 Shear Rates calculated using the simple shear approximation l Alpha = :0: Shear Rate Speed (Alpha _ l) 53 A = Iph“ Ri 50 meSEE op cote dam EC; .020 Km 88m 68:36”. .6 66.6 .m 8:9... N .95 3.32:4. 3 'aJnroJadural 0.00 F 0.05 0.3 9mm 06 P L p b 0.0 EB 93 .38 x EB 3a .782 l EB coo 6:5 0 4 E... can .78: .l E... cam .38 4 EB . l- o3 .86: 1:: a 4.0.3 x x x x l4 0 o 0 O . . 3.2.. “ I‘I r 0.3 51 mmSEE OF .33 com EEM if; 209:: 25%. 68:33. .6 .85 .2 8:9... N .95 33:5. 0.00— ads 0.0m odu 06 Pl b IF P 0.0 EA. ova .38 x EB SN .38: I. EB can .300 o 5.: com .33: .l . 19¢. a . w d m rods 0 m L J \\\\\\ O fodn fl I 0.0? 52 Figure 11 shows the effect of angular velocity on the temperature profile of honey in a 1 mm gap. At 240 RPM the model and data are in excellent agreement, while at 600 RPM the profile predicted by the model is higher than the data. During the experimental run, a small amount of the product was forced out of the annular gap at 240 RPM. This indicates the presence of an axial velocity, however, there is little noticeable difference between the data and model. At 600 RPM substantial amounts of honey was forced out of the experimental device. As a result, the model over predicted the data by 6 to 8 degrees. The axial velocity is a result of the centripetal force. The second variable considered is the effect of annular gap. In Figure 12, 2% CMC is tested at 600 RPM for three annular sizes. For the 3 mm annular gap the model and ‘ data are in good agreement. At 2 mm the model underpredicted by 1 degree at the outer wall and over predicted by 1 degree at the inner wall. For the 1 mm annular gap, the model overpredicts the temperature profile by 4 degrees. As the annular size decreases the Reynolds number increases. For the 1 mm annular gap, the Reynolds number, presented in Table 6, is 20.03 which is generally considered to be transition or turbulent flow for non-Newtonian fluids (Skelland, 1983). The model is based on the assumption of laminar flow and the overprediction of the model at the 1 mm annular gap is considered to be the result of the high Reynolds number. The values for C1 used in determining the temperature profiles are found in Table 7. Because of the presence of centripetal forces at high angular velocities, when testing honey and the rod climbing occuring for the non-Newtonian fluids, the laminar flow assumption becomes suspect. To address this problem the 53 33.9.: or cote mom EEF .xocox 668m 6.66.33. 6 6th .: 2:9... N .95 33:5 0.00. 06h odm odu od P F L L coo EE 9“ .33 x as SN .63: .I En. com .38 0 EB 8o .68: ..... fl 10.0.. . 106w 9 Todn Jul in Jfi x o o o o .w 1 i lllll r99. 3 ‘aJnioJadwa J. 54 8695.9 .36 EB com .020 Nm , goo 53:94 .6 hootm .Nr 959.”. N .aoo co_:cc< odpop ofib oven o.mN 0.0 . new EEn .300 x o 0 new EEn .2502 ll new EEN .300 o r can EEN gone: ...I com 55.. .300 4 fiodp new 5......' .3602 .. m... T w d m wodu D .19. x x +1 IIIHNHI “I H n El iiIPIiIlI. o w J ““““ I‘ll. #- M6 ”It. a < d 4 O 3.2.. .......... ...... .......... -I'- ...... .......... 55 Table 6. Reynolds Numbers .' Speed Annulus [ Reynolds 3 RPM mm . Number T— 240 1.0 5.03 — 600 1.0 20.03 _ 920 1.0 37.74 l— 600 2.0 6.35 [_ 600 3.0 1.16 240 1.0 5.38 ;— 600 1.0 13.46 240 3.0 0.23 . Miracle Whip - Apparent Viscosity, calculated using the shear rate at the inner wall, was used in the calculation of the Reynolds number 56 Table 7. Values of C1 “ 600 10 0.3602 || 920 10 03962 ll [_ 600 20 01469 ll — 6.... so c.0321 ll ;_ 240 1.. 0.2033 II 1_ 600 10 05084 II ; Miracle Whip : -: impeller Reynolds number was calculated using the expression by Ulbrecht and Patterson (1985) Re = —— (6—1) For non-Newtonian fluids, an impeller Reynolds number less than 10 is considered to represent laminar flow. For Reynolds numbers less than 20, laminar flow is possible but not certain. 57 Table 6 shows the calculated Reynolds numbers for the fluids and flow situations of interest in this study. Fluid data for calculation of Reynolds numbers is taken from Table 4. Only three questionable Reynolds numbers occur. These occur when the annular gap is 1 mm and the speed is greater than 600 RPM. The two Reynolds numbers at 600 RPM are greater than 10, but are still less than 20 which indicates that these tests should be in or near laminar flow. The two cases of Re > 10 are both situations where the model overpredicted the results by at least three degrees. For the case of annular gap of 1 mm and speed of 920 RPM, the Reynolds number indicates that turbulent flow is occurring. Even though the assumption of laminar flow is not valid for this case the model predicted the temperature profile within 8 degrees (Figure 9). Figures 13 and 14 show the observed temperature profile versus the predicted temperature profile as the profile develops with time. For the Reynolds number of 1.16 (Figure 13), the line is nearly 45°- showing that the prediction and data are in good agreement. Figure 14 shows the model versus observed data when the Reynolds number is 5.03. These lines show that the model is predicting higher temperatures than the data. Both Figures 13 and 14 show that the model predicts closer to the observed data as it nears the inner, rotating cylinder. Figures 15 and 16 show the influence of the Reynolds number across the annular gap at the 10 minute time interval. In Figure 15, as the Reynolds number increases the model overpredicts the temperature profile. Figure 16 illustrates the data when the annular gap is held constant and the Reynolds number is varied by changing the angular velocity. For this case also, the model predicts higher values than the observed data as the Reynolds number increases and reaches the transition zone. 58 020 Nm .boumf venuOcanohloEc. eoZomno eco 63050.5 *0 co_m_._anoo .n— 050E oEFlocaaOcano... 002030 0.3. odn odN r h IP =63 350 .m-é I am In one *0 53:00 .o—é I em ..I =2» 350 6'.— I oz 1.... 6 DAN Teen ewu—unmudwel peiogpud 59 0.20 NN 53$: PSHanEoHIoEfi 0030030 Uco 00550.5 “—0 co_m_.._anOO 4;. 0.59m oEPlocaquanh 69:02.0 =2» .830 .no.m I om 1... 30 .0 53:00 .n0.n I em ..| =03 cocc— .n0.n I am 1..- \ \\\ “ ‘ “‘II“ 00¢ 0.0m. 0.0N 0.0— L F 0.0? roan 5.9. a 0.0m emu—mmmadwq poiogptud 60 63:96 9 .86 ES com .020 am 80:52 mEochwm .6 yootm .mr 050E N .000 .6305. 0.00 P 0.2. 0.00 0.0m 0.0 F F L 1| 0 . 0.0 3.7:. 38 x opélom .3002 ll nndlom .300 o . and-om .63: .I modulom .300 0 60A: 3.813.. .63: 1.- . m. T w d m 0.8 D 1 x x lxl 1 '1011 .ll n PIIIIIIIIIIln‘III‘I-II‘I‘ ‘I‘IIIDIIIII-II. 1 o J . ncltl c < < 4 0 Teen rod... 61 30:5,: or 00:0 000 EEP .020 Nm 53:52 33.33. .6 0.00:0 .0_. 050E N .000 .6353 ode. 0.2. 0.0m 0.3 0.0 . t r .3 . 0.0 .5. SN 86 I 3. 38 x E... 9m .63: .l EB So .83 I 3.. .38 o . EB 08 .63: .l E... cam is... I 3. .38 4 EB . I... can 68: +0.9 .1... a w d m 10.0w m. X 8 X X n J M. 0 O . . . . a . . . . :odn r 0.0% 62 The third evaluation of the model was by looking at its accuracy for various fluids. Figures 17 and 18 show the model and data for the three fluids tested, 2% CMC, Honey and Miracle Whip. Figure 17 shows the comparison of a Newtonian fluid, honey, to a power law fluid, 2% CMC. For a 1 mm annular gap at 240 RPM, excellent agreement of the model and data were found for both the honey and 2% CMC (Figure 17). In Figure 18, 2% CMC is compared with Miracle Whip at 600 RPM and a 3mm annular gap. Again it is shown that the model and data are in excellent agreement. This would appear to supports the versatility of the model for predicting the temperature profile for Newtonian, power law and more general non-Newtonian fluids. As shown in the model development section, the temperature profile is directly related to the velocity profile when viscous heating occurs. Since the hypodermic needles are inserted into the fluid, disturbances in the velocity field are possible. To address this possibility, the thermocouples were arranged as shown in Figure 7. If no significant disturbances are introduced, the temperature readings for the following pairs of thermocouples (1 and 6, 3 and 7, 5 and 8) should be the same within the accuracy of the thermocouples. The raw data from each test is given in Appendix A. Comparison of the two data points taken at 99%, 50% and 0% of the annular gap, for each line, show nearly identical readings. These readings support two assumptions: 1. The thermocouples did not significantly disturb the velocity field. 2. The Assumption of negligible velocity variation in the angular direction is valid. 63 $035.: or cote 000 SF: .80.. oem 25 99... .6 63m .2 3:9... N .000 .6352. 0.00 _. 0.2. 0.00 0.0a 0.0 7 h _ L 0.0 «kiln: .Flpe .030 Nu .300 x 020 RN .300: It PIN: .wlpe $000... .300 0 >000: .300: .1... Tod. ll- 9 w d m 10.0w D X x x x m.- J .6 O rI0.0n d llllllll a: llllllll .1 llllllll m. llllllll a r0.0.9 64 83:02.: or .oto 000 880 .E0._ 000 . 85 66.... .6 63m .3 3:90 N .000 .2052. 0.0_0— 0.05 0.m0 0%N 0.0 Nb‘lNc .Flpc .020 RN .300 x 020 RN .3002 ll Rina: 86...... 9;; .68.: .38 o 9.; .68.: .68: .I 0.0 .I 0.0.. I.— a w d m 00.3 D 1 x x 11.011 J m Pl\\m..H11.l IIIIII .r IIIIIIII t IIIIIIII .0 tttttt O T 0.00 65 It has been shown that the model predicts the temperature profile across the annular gap very well at a specific point in time. Figures 19 through 21 illustrate its capability to predict the temperature profile with respect to time. In Figure 19, two locations were considered, 50% and 99% of the annular gap. The data shows a slowly increasing temperature at both locations which is well predicted by the model. Figures 20 and 21 compare the time-temperature profile for two different annular gaps. Figure 20 shows data at the midpoint of the annular gap while Figure 21 illustrates data at the inner wall. Again, excellent agreement is shown between the model and the observed data. The importance of the temperature transient was examined using an eigenvalue analysis. The maximum eigenvalue was 0.085 and the minimum value was 0.0025. Using these values and based on the grid chosen, it was calculated that steady state was reached after 30 minutes. The length of the transient is directly tied to the boundary conditions assumed. 66 a8 5...... .8. com .35 RN .033: 0.30.00E3IoEc. .3 0.590 «”3350. .900. 0.0— 0.0 0.0 F 3 h b €03 .055 N00 .300 x 26... 65.0 Nam .68: 1... A000 00 .3003 N00 .300 o 300 00 .3003 N00 .3002 ..l nulluuullluulllfllllfilllflllldlllnlllfillldllll 0.0 10.2 II.- a . w d m 10.8 m. . n . m O 10.8 3 rod... 67 no... .0 Rom .020 «N @302 0030000E3l06fi .ON 050$ mvufizhe .08; 0.09 o.m 0.0 b p L L — b ? b + 0.0 50.. 000 .000 EEn .300 x ES 000 .000 EEn .3002 II 80.. o¢N .000 5:: .300 o .. 80.. Ova .000 EEP .3002 II Todp o o o ||m||l|.P.|III.P.|IIIlIO.ON rodn r060 3 ‘aJnmJadfiJal 68 0.00 .6 N8 .020 RN @305 003030E£I08P .FN 0000i 0305:. 68:. 0.0.. o.m 0.0 _ b p b b p I p F . 0.0 000 SE F .300 x 000 ES F .3002 .l... 000 SE n .300 0 a 000 EE 0 .3002 an 10.09 3 ‘aJnmJaduJal 7 CONCLUSIONS A model has been developed for tangential annular flow of non-Newtonian fluids when the inner cylinder is rotating and the outer cylinder is stationary. The model was developed using the generalized rheological model of Ofoli et a1. (1987), and the equations of energy, motion and continuity. The model was used to access the effect of annular gap, angular velocity, fluid type and Reynolds number, on the level of viscous heat dissipation. The Reynolds number was used to evaluate the combined effects of annular gap, angular velocity and fluid type. An experimental design was implemented to validate the model. The model, a combination of two computer programs, (VELOC and ODTIME), predicts the temperature profile within 1°C of experimental data provided that the laminar flow criteria and the other major assumptions are met. As the Reynolds numbers approach transition, the model begins to overpredict the temperature profile. One of the major assumptions of this study was that the fluid parameters are temperature-independent. Over the narrow range of temperature encountered, this was an appropriate assumption but it does limit the capability of the model. As the Reynolds number exceeded the laminar flow range, (Re > 10),the model predicted temperature profiles higher than observed. For Reynolds numbers below 10, excellent agreement between the model and data was observed for all velocities, fluids and annular gaps considered. When the model is used to predict the temperature history, excellent agreement with experimental data is observed. 69 8 SUGGESTIONS FOR FUTURE RESEARCH 1. Development and solution of the equations for tangential annular flow with temperature-dependent fluid properties. 2. Develop the velocity and temperature profile relationships for two- and three-Dimensional flow. 3. Establish firm criteria for the transition to turbulent flow for an annulus. 4. Development of a model which incorporates viscoelastic effects. 70 APPENDICES APPENDIX A RAW DATA APPENDIX A RAW DATA Table A.1 2% CMC, 1 mm Annulus, 2&0 PPM TEMPERATURE VS. RADIUS 10 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 1 21.8 21.7 21.3 21.3 21 0 2 21.9 21.7 21.5 21.3 21 1 3 21.8 21.9 21.14 21.3 21 1 ‘4 21.7 21.8 21.5 21.3 21 1 S 21.6 21.8 21.5 21.3 21 2 6 21.7 21.8 21.5 21.2 21 3 7 21.7 21.7 21.5 21.3 21 3 8 21.2 21.14 21.2 21.0 21 1 9 20.8 21.3 21.2 21.0 21 1 10 20.6 21.1 21.0 20.8 21 0 71 72 Table A.2 2% CMC, 1 mm Annulus, 600 RPM TEMPERATURE VS. RADIUS 10 MINUTE TIME INTERVALS --—------__————.—---.—_--—--——----------_-_----—_-—-__.--. PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 1 28 7 28.7 28.5 27 8 27.5 2 28 6 28.5 28.7 28 2 27.6 73 Table A.3 2% CMC, 1 mm Annulus, 920 RPM TEMPERATURE VS. RADIUS 10 MINUTE INTERVALS .—————-—-—-—-—---—-—-——-—-c—-—-———-——---—--———.——.—————-—— PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 1 28.1 28 2 27.8 27 5 26 5 2 28.9 29 0 28.11 28 0 27 0 3 29.5 29 5 30.5 28 5 28 0 18 32.5 30 5 33.1 29 5 28 6 5 29.4 32 0 30.5 35 9 28 1 6 33.3 28 8 31¢.“ 29 5 28 3 7 32.5 30 5 31.6 28.7 27 3 8 32.3 3“ 5 33.0 29.0 28 5 9 29.3 28 1 32 3 29 2 29 0 10 31.8 105 308 29.0 3014 DATA AVG. 30.77 30.16 31.2” 119.118 28.17 ———---_—--—__.._-_-_____——_.—-__--__—...-0.-._-..__.__._..___.. —-—_ 74 Table All 2?; CMC, 2 mm Annulus, 600 PPM TEMPERATURE VS. RADIUS 10 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 1 23 9 23 9 23.9 23 0 22 it 2 23 6 214 1 214.3 23 7 22 7 3 23 5 211 1 214.1 23 3 22 8 '4 23 6 215 2 23.5 23 ll 23 0 5 23 5 214 0 23.6 23 3 23 1 6 23 ll 23 8 23.8 23 5 23 2 7 23 5 23 7 211.1 23 '4 23 2 8 23 5 23 8 24.0 23 7 23 u 9 23 5 23 6 23.7 23 9 23 U. 10 23 ’4 23 5 23.5 23 8 23 1% 11 23 6 23 7 23.5 23 7 23 14 12 23 6 23 6 23.2 23 5 23 14 13 23 7 23 5 23.5 23 '4 23 5 118 23 8 23 3 23.9 23 3 23 5 15 23 9 23 ll 214.1 23 6 23 6 16 23 9 23 ll 211.0 23 '4 23 8 17 23 9 23 5 23.9 23 ‘4 23 9 18 214 0 23 3 23.9 23 ll 23 7 19 23 8 23 14 23.7 23 1 23 7 20 23 9 23 14 23.8 23 2 23 7 21 23 8 23 ll 23.7 23 2 23 6 22 23 7 23 5 23.7 23 1 23 6 23 23 7 23 6 23.6 23 3 23 6 215 23 6 23 6 23.5 23 5 23 5 25 23 6 23 6 23.6 23 7 23 6 26 23 7 23 5 23.7 21% 0 23 5 27 23 5 23 6 23.5 23 9 23 ’4 28 23 5 23 6 23.6 23 9 23 5 29 23 6 23 5 23.7 2’4 3 23 6 30 23 6 23 7 23.6 2‘4 3 23 6 31 23 5 23 6 23.7 21} 2 23 6 32 23 6 23 5 23.9 21! u 23 6 33 23 5 23 5 23.7 214 2 23 5 31‘ 23 5 23 5 23.8 211 3 23 6 35 23 ’4 23 6 23.11 2“ 1 23 5 36 23 ll 23 6 23.3 23 9 23 5 37 23 3 23 14 23.1 23 8 23 ll 75 Table A5 2% CMC, 3 mm Annulus, 600 RPM TEMPERATURE VS. RADIUS 10 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 1 22 6 22.14 22.8 22 U. 22 2 2 22 5 22.5 22.6 22 5 22 3 3 22 7 22.5 22.7 22 6 22 3 ll 22 8 22.3 22.6 22 6 22 ll 5 22 7 22.3 22.7 22 7 22 5 6 22 9 22.” 22.6 22 5 22 5 7 22 9 22.3 22.5 22 5 22 ’4 8 22 7 22.5 22.11 22 ’4 22 ll 9 22 7 22.5 22.5 22 3 22 3 10 22.6 22.6 22.5 22.3 22.5 11 22.6 22.7 22.11 22.14 22.4 12 22.8 22.6 22.6 22.14 22.5 13 22.7 22.7 22.5 22.14 22.14 114 22.5 22.7 22.5 22.5 22.14 15 22.5 22.7 22.5 22.14 22.5 16 2214 22.6 22.5 22.5 22.1: 17 22.3 22.6 22.5 22.5 22.3 18 22.” 22.6 22.5 22.6 22.5 19 22.11 22.5 22.14 22.6 22.14 20 22.3 22.5 22.11 22.5 22.3 76 Table A.6 Honey, 1 mm Annulus, 2110 RPM TEMPERATURE VS. RADIUS 10 MINUTE TIME INTERVAL Percentage of gap width 99.0 75.0 50.0 25.0 0.0 1 31.8 32 1 32.0 31 9 31 2 2 31.9 31 8 31.8 31 7 31 2 3 31.8 31 7 31.6 31 6 31 2 it 31.6 31 7 31.5 31 6 31 2 5 31.5 31 7 31.5 31 5 31 2 6 31.5 31 6 31.11 31 ll 31 2 7 31.5 31.5 31.11 31 5 31 3 8 31.6 31.5 31 5 31.6 31 11 Q 31.5 31.6 31 5 31.5 31 14 10 31.11 31.1: 31 14 31.11 31 3 DATA AVG. 31.61 31.66 31 56 31 55 31 26 --—-----I-—---——-—----------——.---—----—--———~--—————--—— 77 Table A.7 HONEY, 1 mm Annulus, 600 RPM TEMPERATURE VS. RADIUS 10 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 C C C C C 1 3’4 9 35.6 35.3 35 3 35 0 2 3'4 2 35.0 314.8 3” 9 3M 7 3 314 2 314.9 39.5 311 2 314 11 u 313 0 311.7 311.6 33 8 3H 3 5 33.6 3&0 314.2 33 8 314 0 6 33.5 33.8 3’40 33." 314 0 7 33.” 33.5 33.6 33.5 33 8 DATA AVG. 33.98 314.147 34 H2 31! 17 34 31 —-—----—---—----—-———--.-—--——------~_--———-——-—--——--—— 78 Table A.8 Miracle 'n'hip, 3 mm ANNULUS, 2110 PPM TEMPERATURE VS RADIUS 10 MINUTE TIME INTERVALS PERCENTAGE OF ANNULUS 99.0 75.0 50.0 25.0 0.0 -——-------——----—-----¢n——-—--——~----_-—————--—-—‘0_————. --_-------——--—-——--————————----—--—-——----—---_—--—-—. 79 Table A9 Miracle Whip, 3 mm Annulus, 600 RPM TEMPERATURE VS. RADIUS 1 MINUTE TIME INTERVALS ———-—————-———__-———-—v~——-¢__———-—_—--—---“-——--——~u——_——‘—a PERCENTAGE OF ANNULUS 99.0 75.0 50.0 25.0 0.0 ————-----————————_--—~——-—--.—-—-——-——-—-———-———————-—-— DATA AVG. 2u.15 23.99 23.89 23.87 23.83 --——-----—-——-———--———-———-—------————-——————-————-——0——-— .——---——u-a—-—--—.q—-——u————--——-—--—--————————--———-—u-n——-——- 80 Table A.10 Raw Data - 2% CMC, 1 mm Annulus, 2140 RPM FILENAME = 2CMC31.PRN 09-25-1988 c:\jill\dataset.acq Time TEMP 1 TEMP 2 TEMP 3 TEMP u TEMP 5 TEMP 6 TEMP 7 TEMP 8 min. C C C C C C C C 19.2 19.6 19.14 19.2 19.7 19.7 19.6 19.2 19.6 19.7 19.6 19.2 19.7 19.7 19.5 AVERAGE BEGINNING TEMP ll .3 \0 U1 TEMPERATURE VS. RADIUS 1 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 1 198 193 19.6 19.3 192 199 197 195 2 200 195 19.7 19.11 19.3 201 198 19 6 3 20 2 19 6 19.8 19.6 19.11 20 3 19 9 19 7 ll 205 198 20.0 19.9 195 206 201 198 5 20 8 20 0 20.3 20 1 19.7 20 9 20 ll 19 9 6 20 9 20 3 20.3 20 2 19.8 21 0 20 5 20 0 7 21 2 20 5 20.6 20.3 20.0 21 3 20 7 20 2 8 21 5 20 8 20.9 20.5 20 1 21 6 21 0 20 3 9 21 7 21 0 21.0 20.7 20 3 21 7 21 1 20 5 10 21 9 21 2 21.2 20.9 20.5 22 0 21 3 20 7 11 22 2 21 5 21 5 21.1 20 7 22 3 21 5 20 9 12 22 l} 21 7 21 7 21.3 20 9 22 5 21 8 21 1 13 22 ’4 21 9 21 8 21.11 21 0 22 6 21 8 21 2 1’4 22 7 22 1 22 0 21.7 21 2 22 9 22 1 21 it 15 23 0 22 3 22 2 21.9 21 ll 23 1 22 ll 21 6 16 23 1 22 6 22 3 21.9 21 6 23 2 22 5 21 8 17 23 u 22 7 22 6 22.2 21 8 23 u 22 8 22 0 18 23 2 22 7 22 6 22.0 21 7 23 u 22 7 21 9 19 22 7 22 8 22 6 22.2 21 8 23 3 22 8 22 1 20 22 7 22 8 22 7 22.2 22 0 23 '4 22 9 22 2 21 22 8 22 8 22 7 22.2 22 0 23 3 22 8 22 3 22 22 9 22 7 22 8 22.1! 22 1 23 3 23 0 22 u 23 23 0 22 7 22 8 22.9 22 2 23 3 23 0 22 5 21! 23 0 22 7 22 9 22.5 22 2 23 ll 23 0 22 5 25 23 0 22 8 22 9 22.5 22 2 23 3 23 0 22 5 26 23 0 22 7 22 9 22.5 22 2 23 3 23 0 22 6 27 22 9 22 8 22 9 22.14 22 3 23 3 23 0 22 6 28 23 0 22 7 23 0 22.6 22 3 23 3 23 1 22 6 29 23 0 22 8 22 9 22.6 22 3 23.3 23 1 22 6 30 23 0 22 8 22 9 22.5 22 3 23.3 23 0 22 6 31 23 0 22 8 23 0 22.6 22 ll 23.3 23 1 22 7 U) N N U) D N N K) N u: D N N U1 '0 N U) N U.) N N on O N N 0‘ 81 Table A.1 0 (cont'd.) 82 Table A.11 Raw Data - 2% CMC, 1 mm Annulus, 600 RPM FILENAME = ZCMC32.PRN 04-26-1988 C:\JILL\DATASET.ACQ Time TEMP 1 TEMP 2 TEMP 3 TEMP u TEMP 5 TEMP 6 TEMP 7 TEMP 8 min. C C C C C C C C AVERAGE BEGINNING TEMPERATURE = 22.6 TEMPERATURE VS. RADIUS 1 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 83 Table A.12 Paw Data - 2": cmc, 1 mm Annulus, 920 RPM ROD CLIMBING FILENAME = 2CMC33.PRN 05-03-1988 c:~.jill\dataset.acq AVERAGE BEGINNING TEMPERATURE = 21.5 TEMPERATUPE VS. RADIUS 1 MINUTE TIME INTERVALS Time TEMP 1 TEMP 2 TEMP 3 TEMP -'1 TEMP 5 TEMP 6 TEMP T TEMP 8 min. C C C C C C C C PERCENTAGE OF GAP ZJIDTH 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 .-------—-——--——‘—---—--—--——--——-————_——~— —_———_._--_- -—_—.___.o——_—— 1 21.5 22.5 2.2.3 22.3 -2 1 22.2 23.0 2 6 22.5 22.14 22.11 2 2 22.3 23.1 3 22.5 22.0 21.9 23.2 is 21 7 22.5 22.8 22.14 22.3 22.2 22.2 23.1 5 22 5 23.3 23.5 23.2 23.1 23.0 22.9 23.5 6 23.5 214.7 211.7 211.2 211.2 23.1! 211.2 214.2 7 215.7 25.6 25.9 25.0 214.9 211.5 25.2 211.7 8 25.8 26.5 26.5 26.0 25.7 25.11 26.0 25.3 9 26.8 7.3 27.11 26.7 26.6 26.3 26.9 25.9 10 27.7 28.1! 28.2 27.5 27.3 27.9 27.6 26.14 11 28.7 29.2 29.0 28.3 28.1 28.2 28.5 27.2 12 29.5 30.0 29.8 28.9 28.8 29.2 29.1 27.8 13 31.9 33.9 31.0 28.5 29.5 29.6 31.14 28.7 1“ 33.8 29.5 32.0 33.8 15 33.9 32.5 37.6 30.9 30.1 16 35.3 32 1 37.6 30 7 31 2 17 35.7 37.7 32.2 27.2 33.6 34.0 . 3 5 33.5 37.5 19 111.3 33.9 39.6 314.14 33.3 27.3 36.3 311.1 20 140.8 36.7 39.0 35.0 37.7 32.5 84 Table A.13 Raw Data - 2% CMC, 2 mm Annulus, 600 RPM 09-28-1988 Time TEMP 1 TEMP 2 TEMP 3 TEMP 11, TEMP 5 TEMP 6 TEMP 7 TEMP 8 min. C C C C C C C C 1 21.0 21 3 21 0 20.9 20 7 21.6 21.2 21.3 2 21.0 21 3 21 2 20.9 20 7 21.7 21 3 21.2 3 20 9 21 3 21 1 20.8 20 6 21.6 21 1 21.2 TEMPERATURE VS. RADIUS 1 MINUTE TIME INTERVALS Ininut 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 1 20.9 21.3 21.1 20.8 20.6 21.7 21.2 21 2 2 21.14 21.14 21.1 20.6 20.6 22.1 21.1 21 2 3 21.7 21.6 21.-'4 21.0 20.7 22.2 21.9 21 3 ll 21.9 21.9 21.9 21.1 20.7 22.9 21.7 21 3 5 22.1 22.3 21.8 21.1 20.7 22.7 21.6 21 2 6 22.3 22.6 21.9 21.2 20.8 23.0 21.6 21 3 7 22.6 22.9 21.9 21.6 21.0 23.0 21.9 21.11 8 22.6 23.2 22.2 21.6 21.0 23.3 22.2 21.9 9 23.0 23.5 23.0 21.7 21.2 23.6 22.8 21 6 10 23.1 23.7 23.3 21.9 21.5 23.8 23.1 21.8 11 23.5 23.8 23.7 22.14 21.8 211.1 23.5 22.0 12 23.7 214.1 213.2 22.8 22.1 294.3 23.9 22.3 13 23.7 29.3 29.3 23.0 22.3 214.5 23.9 229 1'4 23.9 29.7 29.0 23.1 22.5 214.7 23.9 22.6 15 214.1 211.8 214.1 23.0 22.5 2’48 23.8 22.7 16 214.1} 25.0 211.3 23.3 22.7 25.0 214.1 22.9 17 214.6 25 2 214.8 23.7 22.8 25.2 214.5 23.1 18 211 8 25.6 25.0 23.9 23.2 25.14 214.7 23.3 19 25.1 25.7 25.11 211.2 23.4 25.7 25.1 23.5 20 25.2 25.8 25.5 214.3 23.6 25.8 25.2 23.7 21 25.7 26.1 25.8 211.7 23.9 26.2 25.6 214.0 22 25.8 26.3 26.0 214.9 214.2 26.14 25.7 214.2 23 26.1 2614 26.3 25.0 211.11 26.7 26.0 214.11 2'4 26.14 26.6 26.6 25.0 214.7 27.0 26.3 211.7 25 26.6 26.8 26.8 25.1 211.9 27.2 26.5 211.8 26 26.9 27.0 26.9 25.3 25.1 27.14 26.7 25.2 27 27.1 27.3 27.2 25.6 25.5 27.7 27.0 25.5 28 27.4 27.14 27.1% 25.9 25.5 27.9 27.3 25.6 29 27.5 27.7 27.6 26.0 25.7 28.1 27.11 25.8 30 27.7 27.9 27.9 26.1 25.9 28.3 27.7 26.0 31 28.1 28.1 28.1 26.5 26.1 28.6 27.9 26.3 85 ~Tcont'd.) Table A1}; “79135ni 6667-7-»(7- 0.. C... 0... nun/.20. 1.358129 888899nfl. 0..29.220g0... 7.03.4781 8999990 222222.. 36892.46 666677-n.- 20.. 02220.20... 693.591..” bronlnu78nu 0,..220b05029 «357.01.56 88802999 n4 0,. 7.0.. 0.. 0. 0._ ”(0802.46 8839999 0.. 9.9.2204?» 2.9604825. OuQuOQnOQ/gn) 2 0... 0,. 0L 0.. 0... 0L 9.?Jhw5r0 I8 13 «J «J «J - .- a.) 0 0.. Qu. Co 0... A5 7.4 0.. 0.. 0) 0L 0,... «J U. 0 0 «3 n1. 1 7- 8 a... 2 8 0 n6 9 a; 0.. Du 0 g 0 0.. a... 8 4.. 9 0 0.. 1-. 9 O... 0... 0... g 0 «J H. 28.7 28.8 ~.0.6 U 0.9 1.1 28.3 28.6 28.7 29.3 29.6 30.9 30.8 30.3 30.5 30.1 30.14 30.5 '42 143 1.114 145 1:6 u? 141 29.0 30.9 9.. 31.0 31.1 30.0 29.0 30.7 31.0 30.2 29.1 31.5 30.8 29.2 31.11 31.0 30.14 29.3 31.7 31.0 31.2 30.9 31.1 2 31.4 31.0 30.5 29.”. 31.7 31.0 31.2 86 Table A.11: Paw Data - 2% CMC, 3 mm Annulus, 600 RPM 05’02“1938 C:‘.JILL\DATASET.ACQ FILENAME = 2CMC12.PP.N ——--------————-——-_—--——--————-———.——--—————.—u..-—-_—— —-..--—_-—-— Time TEMP 1 TEMP 2 TEMP 3 TEMP '4 TEMP 5 TEMP 6 TEMP 7 TEMP 8 mm. C C C C C C C C 1 21.9 2.0 222 21.7 21.6 2 1 22.1 22.3 2 21.9 22.0 221 21 7 21.6 22 1 22.1 22.3 3 22.0 1 0 22 2 217 21.6 22 1 22.1 22.9 AVERAGE BEGINNING TEMPERATURE ‘ 22 0 TEMPERATURE VS. RADIUS 1 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 -------_-_—_—-——---‘---‘-----——‘--------—--—--————-———--— ———— 1 22.14 22 0 22.0 21.6 21.6 22.5 22 1 22.3 2 22.3 22.1 22.2 21.5 21 5 22.6 22 0 22.3 I 22.3 22.3 2‘2." 21.3 21 5 22.5 22 1 22.2 't .22.: 22.." 2‘..- 27,? ""3 :25 ”2.2 172’? 5 22.: N‘ 22.3 -35 1“." “If “23 22: 6 22.3 22 ’4 22 3 ’1 6 21.5 2” 6 ‘ “.3 ". 7 21.2 22 6 22 5 21 7 21.6 22 6 22 ll 22 14 8 21.6 22 ’4 22.7 21 8 21.5 22 8 22 5 22 ’4 9 21.1! 22.5 22.7 21.8 21.6 22.9 22.6 22.6 13 21 3 22 9 22.1 21.7 23 2 22 7 22 6 11¢ 21 9 22 6 23 0 22.1 21.8 23 3 22 8 22 7 15 23 0 22 2 21.9 23 ‘4 22 9 22 7 87 Table A.15 Raw Data - Honey, 1 mm Annulus, 2110 RPM 05-23-1988 c:\ji11\dataset.acq filename = honey31.wk1 Time TEMP 1 TEMP 2 TEMP 3 TEMP ll TEMP 5 TEMP 6 TEMP 7 TEMP 8 min. C C C C C C C C ‘---———-—-.‘-—------—-—-—-—-—-—-————.---—--———-¢-—----—-—-_—.- 299 29.8 29.6 29.3 . . 2 30.2 29.9 29.9 29.6 29.3 28.9 30.0 30.2 29 9 29.8 29.5 29 3 Average beginningr temperature = 29.7 Temperature vs. Radius 10 Minute Time Intervals ~---——-———-—-—_--——--———-—--—-—_—-——.-———--——-o——--..—.—-_-——-——-— Percentage of gap width Time 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 88 Table A.16 Paw Data - Honey, 1 mm Annulus, 600 RPM 05-23-1988 c:\ji11\dataset.acq FILENAME = HONEY32.PRN PERCENTAGE OF ANNULUS Time 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 —-——.——-.——_--—-.--—-—-—-—-.‘—--.-——--.-’—.._~.-———..———.-.—.—-— 29 6 29.11 29.3 29.2 28.8 29.14 29.8 2 29.1 29.6 29.14 29.2 29.2 28.7 29.14 29.8 29 6 29.11 29.2 29.2 28.8 29.14 29.8 Average beginning tempertature = 29.3 TEMPERATURE VS. RADIUS 1 MINUTE TIME INTERVALS PERCENTAGE OF GAP WIDTH Time 99.0 75.0 50.0 25.0 0.0 99.0 50.0 0.0 _Cn_OO-Jc~o~u1 89 Table A.17 Raw Data - Miracle Whip, 3 mm Annulus, 240 RPM 07-12-88 C:‘.jill'.dataset.acq PERCENTAGE OF ANNULUS Time 99.0 75.0 50.0 25.0 0.0 min C C C C C 1 23 9 24.0 24.0 23 8 23 3 2 24 0 24.1 24.1 23 9 23 5 3 24 0 24.1 24.1 23 8 23 3 4 24 0 24.0 24.0 23 8 23 3 AVERAGE BEGINNING TEMPERATURE = 23 8 TEMPERATURE VS. RADIUS 1 MINUTE TIME INTERVALS PERCENTAGE OF ANNULUS TIME 99.0 75.0 50.0 25.0 0.0 min C C C C C 1 24.1 24.1 24.1 23.8 23.3 2 24.1 24.1 24.1 23.8 23.3 3 24.3 24.2 24.2 23.9 23.4 4 24.1 24 1 24.1 23.8 23.3 5 24.3 24.3 24.2 23.9 23.4 6 24.3 24.2 24.3 23.9 23.4 7 24.2 24 2 24.1 23 8 23 3 8 24.4 24 3 24.1 23 9 23 3 9 24 5 24 3 24.2 23 9 23 4 10 24 4 24 2 24.2 23 9 23 3 11 245 243 24.2 239 234 12 24 6 24 3 24.2 23 9 23 4 13 24 5 24 4 24.2 23 9 23 4 14 24 6 24 4 24.2 23 9 23 3 15 245 244 24.2 23 9 23 3 .-—-——-——-—-------—-—_—-—-—-—-—-—————------—------ Table A.18 Raw Data — C:\jill\ dataset.acq FILENAME=MW12PRN AVERAGE BEGINNING TEMPERATURE TEMPERATURE Miracle Whip, 3 mm Annulus, 600 RPM 90 = 23.8 VS. RADIUS 1 MINUTE TIME INTERVALS PERCENTAGE OF ANNULUS —-—-—-——-—__-—-—————-————---~——---——————-~_-———-————_—— TIME 99.0 min. C ’1 93.38 2 24.05 3 24.02 4 23.96 5 24.12 6 24.12 7 24.26 8 24.14 9 24.28 10 24.27 11 24.22 12 24.38 13 24.47 14 24.43 15 24.49 16 24.55 17 24.55 18 24.63 ’19 24.53 20 24.58 75.0 23.99 24.11 24.07 24.03 24.10 24.12 24.23 24.13 24.25 24.23 24.19 24.26 24.33 24.23 24.35 24.31 24.41 24.39 24.38 24.34 50.0 25.0 IX) r.) ['0 N LA) U) U) L1.) 9.) r) m 'm m J: m m U1 L.) :0 U1 N 1".) U.) LA) LA) (a) UL) Lu 23.33 APPENDIX B RHEOLOGICAL DATA APPENDIX B RHEOLOGICAL DATA Table B.1 Brookfield Standard - Test 1 Linear fit TEST TJ CHLIBEHTE NEMT STE. 5... l—h P" m D V '3 $1.31 :3) U1 "‘ .3 U. . : :D ‘ ' 4 kg . I ‘1'! v3 . _ 1 4 Ill.) h- ,‘ I I :1, m H to S r-b "‘5; ..-. .31 ° ’ ' .. . r 1. “J “3'. 1 m 3., I In " " .-. H. . ., m .1 "i . . i m 91 Table 8.1 (cont'd.) Linear I1 .4 AOO'J‘J‘JIUI-hbll'flHit PT # KOO'J‘JU‘IUl-‘BNI‘UH 92 fit T=a+bx rOHHpHWUIJIJ-i '0 03 '-J U‘. U‘ 4“ N N M a= ". b: R Square: Std dew: .3395+BBB .458E+GBB .981E+BBB .688E+BBB .148E+BBI .393E+881 .518E+981 .865E+861 .126E+381 557E+B 1.251E+% -.' l O GUS RHN DRTR SPEED iradfsec) .273E-681 .BSBE-BBI .BBSE-BBI .374E-BBI .125E-001 .183E-BBI .237E-881 .283E-981 .328E-681 DJI'OHHHHCOU‘IF-J MNNMNHHHL-l .451E+B@1 .588E+681 .617E+BBI .118E+862 .482E+302 .782E+882 .995E+BB£ .381E+082 .SB7E+BB2 TORQUE (n-m: .TEBE-GBE .BBEE-BBZ .312E-892 .FBlE-BBE .255E-882 .5915-862 .B36E-862 .SBBE-BBZ .BISE-BBZ 93 Table 3.2 Brookfield Standard - Test 2 Power {it '= ? 329£+aae = .188E+BB@ are= .345 au dev= 7.465E-881 a b R 3 Std TEST TU CHLIBRHTE NENT STU. 5-12-38 SHEHR STRESS CN/MAED H H [‘0 h.) .3 '-J U! “m a m ~ mgfiF Q5 3 \\ 4 9% .‘fl '5'"... .3 0:34. I: n-b "4:4“ 11 a Hail _ P a}... 4 '. x v.0 ~— '--:;~ . m " m “x .1. x.- ... 5 I?”* 3x * ‘x “-1 57:. m ":5. In 3 ‘4‘. n W\, m N ” L A Table B.2 (cont'd.) 94 Power fit mor-qmmaumr‘ér PT# moo-umtmatdr...» v=ax4b a= 7.329£+aaa b= 1.18BE+BBB 9 Square: .345 Std dev= ?.4BSE-@Bi M Y 4.339E+BBB 2.451E+001 4.456E+800 6.588E+B@1 6.981E+006 3.61?E+BBI 9.888E+666 1.118E+002 1.148E+001 1.482E+832 1.393E+001 1.?02E+882 1.6185+991 1.995E+902 1.865E+001 2.3015+062 2.126E+881 2.587E+BBZ RHH DHTR SPEED TORQUE (radfisec) an-m) 2.2?3E-601 3.?38E-063 2.6995-881 1.8635-862 3.686E-601 1.3125-002 4.0?4E-391 1.?81E-BBZ 5.125E-001 2.2555-082 6.183E-BBI 2.591E-982 7.237E-881 3.036E-882 8.283E-661 3.582E-882 9.328E-681 3.815E-882 95 Table 8.3 Brookfield Standard - Test 3 TEST TO CHLIBRHTE NENT STD. 5-12-88 SHEHR STRESS (HKMPE r-s h) m N :1: UV :h'lih “\1 t0 o-s m (D ID _§ 01 . .3 .. a. T'G’L \. .. 030: D ‘i‘ 35. \_. 33 .1. :6 NH {-5. m \l ". .. t-b - P 11.; \. CD "a, m '. m "x. n ‘S v ‘u h) \\£ Table 8.3 (cont'd.) Linear {i 96 t Y=a+bx a: b: 3.868E 1.164E R Square= Std dev= ‘U .4 at m‘JU'ILflhblTIJH NhuawhuomJb PT # ( w‘Jo‘umJibthH mm’dOIUI-huh) Power fit X .794E+BBB .981E+BBD .B88E+BBB .148E+BBl .393E+991 .613E+831 .865E+081 .1265+601 RR” D SPEED rad/sec) .899E-801 .BBSE-BBI .874E-981 .125E-881 .183E-801 .237E-891 .283E-081 .328E-301 =aXAb a= 1.4?3E+991 b: 9.324E R Square= Std Bov- 4.783E-881 +830 +601 .996 6.712E NNHHHF‘ml-r. 8TH wuudwhflfiwh‘ -891 .996 +661 Y .538E+BBI .61?E+861 .1185+BBZ .482E+682 .?82E+862 .995E+882 .381E+662 .587E+882 TORQUE (n-m) .603E-882 .3125-802 .7BlE-882 .255E-BB2 .591E-682 .B36E-892 .SBZE-OBZ .3155-932 97 Table 8.4 2% CMC - Test 1 Power fit '1’=a -."b a= 2.919E+201 b= 4.547E-801 R Square= .968 Std dev= 5.1?8E-881 22 CMC T=25 JMK-T 5-14-88 SHEER STRESS (NIMAEE o-s n-o m N D co U! h.) 0‘. :9 m POT—M v 1 -. A . m .\{< o 3*; ncn ‘\ A’ a ‘ (O X I ‘. m 2% m o I! x \ a, p-s ::< ".3 ’ or '4 unn- x.“ 'N [O 9 x n *4 p 0' 4: m X X X X X Table 8.4 (cont'd$) * c-we " .1) A # HHHHHHHHHH mmummaiwmwawm-ummssumw NN H0 PT fl: Hr—fiHr-oo-H-o M-buNHOKDOD'flmLfl-bwmw . . 98 O . Y=aXAb a= 2.919E+391 b= 4.547E-691 R Square= .968 Std dev= 5.1?85-831 X Y 2.165E+386 3.216E+061 4.??2E+BOB 5.239E+681 ?.047E+806 ?.?B4E+381 9.611E+088 8.781E+861 1.189E+081 9.696E+831 1.491E+681 1.188E+802 1.732E+881 1.111E+802 1.958E+681 1.227E+882 2.2925+601 1.2515+862 2.481E+381 1.317E+882 2.989E+881 1.48?E+802 3.785E+681 1.578E+882 4.BSBE+681 1.594E+682 4.429E+801 1.?18E+802 5.138E+891 1.?7IE+682 6.343E+881 1.917E+BB2 7.669E+061 2.642E+882 8.814E+881 2.142E+682 1.813E+882 2.252E+802 1.385E+602 2.421E+882 1.585E+682 2.545E+982 RRH DRTH SPEED TORQUE (rad/sec) (n-m) 9.388E-882 4.8865-863 2.1986-961 F.974E-883 3.898E-681 1.1?3E-882 4.866E-881 1.337E-602 5.117E-861 1.4?6E-BB2 6.1665-691 1.6?4E-692 7.225E-891 1.692E-882 8.269E-881 1.867E-002 9.315E-001 1.984E-682 1.937E+080 2.885E-802 1.24GE+808 2.263E-082 1.4SSE+866 2.492E-832 1.6655+OBB 2.426E-082 1.875E+868 2.615E-892 2.083E+808 2.696E-082 2.693E+908 2.9185-802 3.131E+886 3.108E-082 3.619E+969 3.260E-802 4.14?E+088 3.428E-082 5.226E+900 3.6865-882 6.27626008 3.8?4E-682 99 Table 8.5 2% CMC - Test 2 22 CMC T=25 JNK-T 5-14-88 SHERR STRESS (NXMAE) "‘ W h) ‘0 “J I'J' M) 00 'D 5.1% . v 70 \‘(x 3 ‘xx , -o ‘\ 3r '1" \__. 1‘ x 5.533.121 O ‘\ 'fl-JIL ‘5: o 3% 1 m '5 I . m ,. I .- m . '1 5% 'K.‘ I‘- \D )i.‘ ‘3. w‘ur . W ‘x m m :1 n ‘4' .3. p. A k K Power {it Y=aXAb a= 2.483E+381 b= 5.856E-881 R Square= .893 ‘ Std dev= 5.8?BE-BQI Table 8.5 (cont' [I D- .“9 \J HHHHHHH mmbuM~®mmwmmAMMwn [\JI'QHHH H'SNDOJ‘J HHHWmemwflmmmmaawummmmHHam LN-flfi-FNNNNNNNNNNHHHHHHHWOJGum-{5" lOO ;??E-60@ .;43E+66& .164E+BBI .4895+661 .956E+391 .2?BE+6@1 .64BE+881 .F03E+881 .913E+991 .5158+081 .983E+881 .421E+861 .632E+BBI .174E+801 .549E+BBI .841E+681 .534E+681 .643E+681 .525E+BBI .508E+801 .?95E+BBI .234E+801 .628E+881 .97BE+801 .191E+882 .324E+BBE .44?E+002 Rfiu DRTQ SPEED (rad/sec) .BBBE-Bel .944E-861 .161E-661 .1525—931 .248E-661 .EBBE-Bal .83SE+886 .14BE+BBB .245E+808 .454E+988 .556E+888 .869E+98& .971E+686 .678E+808 .289E+BBB .497E+980 .7BBE+BBG .921E+889 .12?E+OBB .338E+BBO .535E+880 .?13E+BBB .924E+608 .142E+698 .6925+BBB 32215+088 .745E+000 l'-H‘~JNNN[oratorommwwHHHHHHHHHHHtfium. 3 l l l 2 2 2 2 2 2 2 2 2 2. 3 3 3 3 3 3 3 3 3 3 3 3 a”. .9?1E+861 .F16E+@61 .933E+BBl .164E+682 .342E+882 .4B7E+BBZ .452E+BBZ .483E+882 .573E+862 .661E+682 .?06£+882 .?62E+802 .332E+862 .855E+892 .9285+682 .985E+882 .B41E+302 .896E+632 .157E+BBZ .28?E+682 .228E+692 .2?6£+882 .297E+BBZ .359E+BBZ .441E+892 589E+682 .57SE+BBZ TORQUE (n-m) .6865-663 .32?E-382 .512E-382 .??1E-862 .642E-BBZ .142E-BBZ .216E-832 .ZSSE-BOZ .394E-682 .SZBE-BBZ .596E-882 .582E-982 .788E-602 .8245-632 922E-382 .BZIE-BOZ -IBSE-BBZ .ISBE-BBZ .2835-882 .359E-882 .391E-862 .464E-BBZ .496E-882 .SSIE-BBZ. .716E-302 .818E-982 .921E-882 101 Table 3.6 2% CMC - Test 3 22 CMC T=25 JHK-T 5-14-88 SHEER STRESS (N/HAZ) h. III. [‘J m 09 on M ID (A ‘1 #- mTfi—f . . m 'Qfig m _ fi _§ m D '11 U! m’ ‘ 0) I m ID 13 A h. I“ ... "’ ” :ut ‘# . U'l \ m T“ 0 -., n I‘ "J I g.— ‘1] ‘ N Power fit Y=aXAb a= 2.321E+061 b= 4.?17E-801 R Sauarot .908 Std dov- 5.228E-881 Table 3.6 (cont'd.) ’U .4 at mooummAc-JMH 18 unpupmtomcom-qNomxmmaAuwwNmNroH»Hv-smausro 102 X .283E+306 .698E+BBB .24BE+BBO .SSBE+BBG .164E+961 .469E+861 .686E+681 .8?8E+881 .2?BE+981 .64BE+861 .?B3E+881 .938E+681 .275E+001 .469E+801 .9938+881 .421E+081 .63ZE+881 .1?4E+881 .540E+881 .B41E+681 .584E+881 .B4BE+681 .525E+881 .588E+881 .795E+881 .234E+681 .6286+881 .9?8E+881 .191E+BBZ .324E+082 .433E+882 .566E+092 . 71060002 ”NNNNNNNNNNNNHHF‘HHHHHMHHhHr-tHlfimxjmg—o Y .971E+681 .218E+381 .93SE+BBI .?16E+BBI .9335+aax .164E+962 .161E+882 .342E+062 .4B?E+302 .452E+862 .483E+982 .5?3E+882 .695E+862 .661E+682 .?86£+882 .7625+882 .832E+982 .855E+982 .920E+682 .985E+882 .84lE+682 .8965+882 .157E+682 .28?E+062 .228E+882 .2?6E+882 .29?E+382 .359E+862 .44IE+882 .589E+882 .576E+982 ,64?E+882 206E+082 103 Table 8.6 (cont'd.) RRH DRTH PT# SPEED TORQUE (rad/sec) (n-m) 1 1.666E-601 3.680E-663 2 2.1965-881 9.464E-693 3 3.689E-861 1.288E-882 4 4.644E-BBI 1.32?E-802 S 5.161E-801 1.512E-882 6 5.1525-991 1.??12-382 ? ?.281E-631 1.?66E-682 3 3.248E-991 2.842E-892 9 9.383E-601 2.142E-062 18 1.835E+BBB 2.218E-892 11 1.14BE+888 2.258E-682 12 1.245E+BBB 2.394E-BBZ 13 1.346E+888 2.4438-682 14 1.454E+980 2.5285-802 15 1.SSGE+BBB 2.5965-962 16 1.869E+983 2.6825-682 1? 1.9?lE+066 2.?38E-882 13 2.973E+808 2.8246-382 19 2.289E+886 2.922E-882 28 2.49?E+BBQ 3.6215-362 21 2.?BBE+BBB 3.166E-882 22 2.921E+888 3.19BE-BBZ 23 3.12?E+086 3.283E-082 24 3.338E+BOB 3.3595-362 25 3.535E+388 3.391E-882 26 3.?13E+BOB 3.464E-BBZ 2? 3.924E+80@ 3.496E-882 28 4.14ZE+BBB 3.5915-892 29 4.692E+698 3.?16E-882 38 S.221E+080 3.318E-382 31 5.?45E+880 3.921E-882 32 6.267E+BBB 4.329E-882 33 6.?9SE+BBB 4.1185-882 104 Table 8.7 2% CMC - Test 4 Power +1: Y=aXAb a= 2.555E+BBI b= 5.313E-961 R Square: .953 Std dev= 5.248E-881 SHEER STRESS (H/MAZ) F" [‘0 IA '0 05 h.) 1‘ Eh IS a» u. I M I f M’ .u m “k“ 4 \fi. 1: 't, T, “r ., 1 (TI K '10 kl. I M' k, I E m x X I‘ m x.‘ 1 ... «o r x ‘5 s ‘ 3.0 .«< 0 n 381 x Table B.7 (cont'd.) Power {it v .4 1t '43 00 '4 (7': U! 4'3 Lu] ['0 H pT# 10 KDW‘JO‘uLfl-fiwwhe 105 Y=aXAb “‘l-flwNNF‘r-‘MH'O‘J-hm #Nl‘oi-H-‘WCD'UO‘IMJINNH a: b: R Square= Std dev= 2.555E 5.613E X .FSSE+BBO .?3BE+BBB .B99E+BBB .?45E+886 .231E+081 .435E+881 .642E+881 .947E+881 .204E+881 .483E+601 .771E+391 .977E+801 .665E+881 .6288+882 RHH O SPEED (radfsec) .2375-681 .1055-981 .084E-OOI .OSEE-OOI .117E-OOI .1755-891 .223E-OOI .275E-681 .ZSSE-OOI .B36E+OOO .561E+OOO .0?9E+OOO .127E+OGO .14?E+QOO +901 -861 .953 5.248E MNHHHHHHHWW‘JLHDI 9TH (dwhflumhueflhnithne -881 Y .132E+361 .?G?E+891 .486E+BBI .67BE+861 .ZBBE+BBI .014E+882 .111E+982 .258E+882 .265E+882 .385E+882 .556E+382 .?78E+682 .043E+802 .248E+882 TORQUE {n-m) .?67E-883 .683E-BB3 .139E-082 .329E-882 .482E-882 .543E-882 .692E-882 .915E-882 .925E-862 .IBBE-BBZ .368E-882 .786E-832 .189E-862 .489E-882 106 Table 8.8 2% CMC - Test 5 Power {it Y=axob a= 2.233E+981 b= 5.158E-881 R Square= .912 Std dev= 6.1622-881 22 cnc 7:25 JMK-T 5-14-88 SHEHR STRESS (NXMAB) N p N KO V LI ‘9 'D 0‘ I] N v V 70 ' -x -. \:< . m '3‘ «xxx 0 3'15. T! U! _ 39;; 1 'h ."l U" \ I . W! x\ I ;\ 70 'K. n- k, :2 0 r- X "1 0'0 \ fl 0 x n V 891 y. Table 8.8 (cont'd.) Power 13 A it aim-ROJNHG‘DOJMUILHRUH‘JH Hu—tO-v-HHHH PT# \Dm‘Ja‘uUIJin‘OH 18 107 {it T=aXAb a: 2.233E+BBI b: R Square: 5.153E-601 .912 Std dev= 6.162E-861 2 316E+386 9.2?4E+OOO l.189E+BBl 1.439E+381 2.319E+BBI 2.435E+831 2 999E+981 3.531E+861 4.638E+BBI 5.13BE+081 6.343E+981 7.669E+881 8.814E+981 1.813E+882 1.305E+932 1.585E+082 NNNNNr‘HHHHHHHleH RRH DRTH SPEED (rad/sec) 9.868E-OOE 4.666E-981 5.117E-681 6.166E-901 8.269E-OOI 1.83?E+988 1.246E+OOO 1.455E+988 1.8?SE+OOO 2.883E+BOO 2.603E+OOO 3.131E+BBO 3.619E+OOO 4 147E+BBO 5.226E+OO? 6.276E+088 t-J'ANNWNNNNNNF‘HF‘HN Y .9?1E+BBl .781E+881 .596E+801 .lBBE+082 .22"E+382 .31?E+682 .487E+382 .S?BE+882 .?13E+982 .771E+BBZ .917E+OBZ .842E+682 .142E+302 .252E+682 .421E+BBZ .545E+882 TORQUE (n-m) .BOBE-BB3 .333E-362 .476E-862 .674E-662 .867E-882 .BBSE-BBZ .263E-002 .482E-982 .615E-882 .696E-302 .918E-882 .lBBE-BBZ .268E-882 .428E-982 .686E-982 .874E-382 Table 8.9 2% CMC - Test 6 --—--—_ -- -----------—-----------— 22 CflC T=25 JMK-T 5-14-88 SHEHR STRESS *NKM*3' -— r0 ("J LO I'J'I M “m N CO 3:- L x . v . H" . . :0 ‘4;- 3 4t... -{ ‘5 ‘- nv x;‘ ‘5. <3 ‘\ T! (A “a on? Ea ‘ 03 \‘w 3: I "x. W! *k D ."t :v \_ .--~. 0‘. "t 4 h-s "D i’ ".__ \ :< 1 m m n .__, p. I3 L A N 3’ fi' 3 3 3 108 Table 8.9 (cont'd.) "U .4 3t wwwammummnump HHHH PT# WWMUILHJBVAF«JH la H-JJLMNHHHHm—qzsr.) $UNF‘KDCO‘JU‘ILfl-(kbll'oh‘ 109 S R Square= Std dev= .F3OE+BOO .69?E+OBO .B99E+996 .?4SE+OBO .231E+BOI .435E+OOI .64ZE+BOI .947E+OOI .433E+681 .?IOE+OOI .97?E+891 .665E+801 .OZOE+882 .f‘u, 5. 3.353E+BBI .219E-681 .938 ?53E NNHHHHHHIOCO'ulLfll'O RHH DRTR SPEED rad/sec) .23?E-681 .IOSE-BBI .684E-881 .6625-681 .117E-601 .IVSE-BBI .223E-881 .275E-891 .295E-681 .561E+908 .8?9E+888 .127E+BBB .14?E+986 'MNNNF‘HHHHHHm.§ -881 Y .635E+681 .?67E+681 .486E+BOI .6?OE+881 .2885+881 .814E+68£ .111E+862 .258E+OOZ .265E+BOZ .556E+882 .??8E+BOZ .843E+802 .24BE+362 TORQUE in-m) .813E-BB3 .58?E-633 .l39E-882 .328E-BBZ .462E-882 .543E-BBZ .692E-882 .SlSE-BBZ .925E-882 .368E-BOZ .?86E-882 .189E-982 .499E-862 110 Table 8.10 2% CMC - Test 7 ---—--”------—---v---—------------ 23 CMC T=25 JMK-T H H r.) 1.4 H ~13 '-J ., m A M '— Uni—fig.— r '0 159xH§ V 53 g§§5v "' ‘ak c: SE; 11 U3 3': -¢'. ‘ '-D F x, "1"“ '1 o) x I: 3 m 3‘ :3 fig '0 3: q H I." I . H p ' 1 '3‘ Ln . In. ( i .1) n v \- p—h N A 4 N Table 8.10 (cont'd.) 111 Power {it .0 .4 it l0 0) "-1 01 Ln 4k (.4 M H Y=a pupppmmmmmuummmmebmmwwwwwmwewwwmwew '5‘: Ab a= 3.365E+BBl b= 4.262E-OOI R Square: .9 Std dev= 4.39 R '4 q. I' 6 E R .331E+BOB .?53E+OOG .248E+BOO .SGOE+OGG .164E+BOI .469E+681 .683E+391 .8?8E+861 .279E+881 .64OE+OOI .783E+981 .938E+081 .275E+BO1 .469E+OOI .??8E+BOI .5655+661 .523E+OOI .328E+661 .632E+381 .1?4E+881 .54OE+901 .O4lE+OOI .584E+881 .843E+601 .525E+BBI .588E+OOI .?95E+Bal .234E+OOl .628E+BOI .9?OE+881 .191E+992 .324E+882 .433E+882 566E+BOE .713E+BO£ ‘Ur')NNNNNhJNNr0NNh-‘HHHHHHHF‘HHHHHHHHl‘om'\.IU‘|bj -601 IT! .61?E+BEI .218E+881 .935E+BB1 .?16E+881 .933E+961 .164E+862 .161E+882 .342E+BO£ .4B7E+BOB .452E+682 .483E+662 .5?3E+B@2 .685E+BOZ .661E+BO£ .?BSE+BBZ .?42E+602 .?215+832 .?62E+862 .332E+BBZ .855E+862 .9285+882 .985E+862 .B41E+BBZ .896E+682 .157E+BBZ .28?E+892 .228E+882 .2?6E+082 .29?E+862 .359E+082 .44IE+BOZ .589E+882 .576E+982 .64?E+682 .786E+302 Table 8.10 (cont'd.) PT# le‘JUNUI-RUJNP‘GI'L'OD‘JU'nUT-RNNH HHHHHHHHP‘H 23 0105mm##NIMOINQ‘INNI'ONIIJHfiHHHHHP‘HH[OCO‘NIU‘TULF-NT‘OH 112 RE» EFT; SPEED (rad/sec) .269E-OOI .IBSE-BOI .OBSE-OOI .844E-681 .161E-OOI .152E-OBI .261E-OOI .248E-961 .383E-681 .635E+BOO .148E+OOO .245E+BOO .346E+OGO .454E+BBB .556E+OOO .668E+OOO .768E+OOO .369E+OOB .971E+OOO .378E+BBB .289E+OOE .49?E+BOO .?OOE+OBO .921E+808 .12?E+BOO .338E+088 .535E+OOO .?13E+BOO .924E+BOO .142E+OOO .692E+OOO .221E+OBO .745E+OBO .26?E+OBO .?99E+809 hRwwufiolblblww(dfidwwNNNNNNNNNP-JNT‘OMI'OHHF-‘H“1.0L" TORQUE (n-m) .SBSE-BGE .464E-BO3 .EBBE-OBZ .32?E-962 .512E-082 .??lE-BQZ .766E-682 .B42E-BBZ .142E-882 .ZIBE-BBZ .258E-682 .394E-BBZ .443E-082 .SZBE-BBZ .596E-882 .SSIE-BBZ .626E-602 .682E-882 .788E-682 .324E-BBZ .922E-882 .BZIE-BOZ .IBSE-BBZ .198E-802 .283E-BBZ .359E-662 .391E-682 .464E-BB2 .496E-882 .591E-382 .?16E-802 .BIBE-BOZ .921E-682 .029E-982. 118E-882 Table 8.11 113 2% CMC - Test 8 T=25 C JMK-T SHEHR STRESS chnna) h) h ‘3': '1'! '3 n3 .3 19 is :3 c3 '39 '9 '3 IS) ‘9 CO .70 ID KKK“ :n ‘1 ‘*e %\ n1. , x\_ . . a ‘9 x"; '11 \-. . m ~§ 'J) ' D \__\ d at '*t :0,“L ~$1 rxb I-t “._ H tn II. In L '1 n :9 h) .9 A L l L '9 3 t 8 ‘3 2 Table 8.11 (cont'd.) Power {it P13 LIIAUJNH PT# U1 «lb OJ ['0 H 114 Y=axob runner.» a: b: R Square= Std dev= 4.225E-OO1 X .11?E+BBO .?4BE+OOD .089E+BGE .453E+088 .197E+BBl l.SZSE+BBl ?.625E-881 .945 ‘0 oo '4 01 L4 RRH DRTH SPEED (rad/sec) 1. “D b- q .3. 411E-661 IIEE-BOI 867E-881 4.364E-631 5. 111E-801 HHMCO-b- Y .241E+BBI .732E+891 .14BE+BBI .428E+801 .446E+801 TORQUE Ln-m) .933E-OB3 .325E-BB3 .687E-682 .282E-BBZ .438E-882 115 Table 8.12 2% CMC - Test 9 22 cnc T=28 c JMK-T 5-13-88 SHERR STRESS (NIHAED o-b o—s m (a H 03 m ... ID ‘0 'D 04* .. f v .v m éfig -4 ~33“ 'TT *3“ -._e . n D “a. 11m ‘fi{ [ m r x": s: ‘ £0 "f‘ I‘ k m f =3 X. :0 '- W p x...“ o-b N I. 34.1», '1 \'” xx 331 f .. in .¥ ['4 n '5: "J :K p... Ru 4 k “J Power {it Y=aXAb a= 3.134E+BOI b= 4.283E-681 R Sauarez 358 Std dew: 4.723E-681 Table 8.12 (cont'd.) '73 .4 #5 O‘F‘HP‘HP‘HHHH wmwmm¢wmwsnn.omummhump I'JNNNNN “JOIUIAOJNHQ ['0 h) (JNMNNN NMF‘QWCO (Al N010] m‘JO‘uUl-A HpuwpwuwuwwpupupmumbhkANMNMMNI’UNNHHHlD‘Jf-J 116 Oxl .BIZE+BOB .668E+BOB .393E+OOB .191E+OOI .418E+381 .646E+681 .B9SE+BOI .342E+661 .363E+901 .69?E+OOI .4BBE+981 .151E+OOI .404E+861 .988E+361 .966E+981 .191E+681 .591E+381 .647E+OOI .86?E+OOI .324E+881 .629E+601 .OBSE+OOI .838E+682 .168E+682 .33?E+BBZ .43OE+882 .492E+BOZ .5?BE+682 .668E+882 .SQOE+OOE .63OE+882 .685E+OO£ .812E+OOZ .792E+OO£ .813E+862 .9?1E+BOZ .944E+892 .9SBE+OO2 NrdNrONNNf-JthIJth'JNr'JNNNNPO-‘HHHHHMHHHHHHF‘KDW'JUILJ \r‘ .BB4E+BBI .?92E+681 .982E+961 .14OE+861 .399E+BO1 .12?E+862 .222E+BOZ .211E+882 .2965+682 .396E+OOZ .41EE+862 .434E+862 .52?E+BOZ .548E+OO£ .SBSE+BOZ .633E+OOZ .672E+BBZ .693E+682 .?62E+602 .9OOE+BOZ .936E+BOZ .137E+862 .ZIOE+BBZ .314E+BO£ .374E+882 .434E+662 .589E+662 .531E+682 .536E+BOZ .551E+882 .566E+BOZ .5?9E+982 .598E+BOZ .681E+OOZ .633E+882 .648E+882 .665E+802 .687E+882 Table 8.12 (cont'd.) PT# @‘DCD‘-lO’uLfl$-QJFJH wawwr-n-a- U‘IU‘PNNF‘ mod-HM wafl [OMNN hum» w NNMNN- H mmum uwmuuwu w mummawm a VflflflflfimmmmmmmmPPNNNNHHHHHflHHHHWmNmMPM“ 117 RH“ LfiTfi SPEED (rad/sec) .247E-OOI .BSSE-Bal .867E-681 .112E-OOI .ISOE-Oal .217E-891 .268E-681 .386E-301 .835E+OOO .14OE+BBO .245E+OOO .34?E+OOO .454E+OOQ .568E+OOO .665E+OOO ?OE+BOO .871E+OOO .973E+BBB .OB?E+OBO .664E+OOO .133E+GOO .6318+BOO .154E+OOO .FlOE+BBO .228E+BOO .756E+BOB .281E+686 .385E+BBB .439E+686 .5955+OOO .69?E+OOO .882E+BOO .9818+688 .107E+OOO .322E+BOG .528E+QOO .?28E+OOO .931E+800 e#ehwwuwwwwwuwuuwwmmwmmmmmmmmuHupuppue TORQUE (n-m) .694E-OB3 .934E-802 .283E-662 .391E-982 .587E-682 .715E-682 .BSOE-BBZ .344E-862 .972E-382 .IZSE-BBZ .148E-682 .183E-BBZ .323E-882 .356E-892 .414E-882 .485E-962 .544E-862 .576E-882 .681E-882 .392E-802 .BSSE-BBZ .252E-BBZ .364E-882 .521E-932 .613E-682 .7B4E-BBZ .819E-682 .352E-862 .BGOE-BBZ .883E-632 .906E-862 .926E-892 .943E-882 .959E-802 .BBBE-OOZ .031E-802 .856E-882 .BSBE-BBZ 118 Table 8.13 Honey HONEY T=36 C JMfi-T 4-23-83 SHEER STRESS (NfMAEE M '3’! L4 :3 I .4 r11 '3 1.1 1 ED I m I n I -- R\ .. . .1 P 35,. 1 H M h‘l‘. “5 "£53. m 3Rn : ‘3; H L11 9 1} Power {it Y=axob a= 2.98?E+BOB b= 9.?26E-OOI R Square: .998 Std dew= 9 4298-861 Table 8.13 (cont'd.) “U .4 HHHHH . #NMHGHDCO-dmmhwith-H" new 0101 Hr- (0V 0 H0)“-lU|-{h#lebll’UNNHD-‘F‘HKDU'I38H p—o l Q'omflamntstdmw Hyd- H pus 'I HP-‘H HHH (afloat. euro ‘NNNNHF‘HHHKDCO‘JU'IU#(‘JI‘JP‘ p D l 119 .391E+BBB .624E+086 .942E+BBB .256E+866 .139E+981 .394E+891 .675E+881 .863E+801 .185E+BBI .336E+881 .?S3E+981 .235E+881 .?31E+BOl .271E+881 .?23E+981 .835E+681 .355E+991 .183E+861 .861E+982 RHH DRTH SPEED (rad/sec? .551E-OOI .196E-OOI .1OOE-OOI .BSSE-BOI .117E-301 .170E-991 .ZZSE-OOI .273E-381 .333E-661 .B3SE+BBB .24?E+OOO .456E+BOO .66?E+OOO .8??E+980 .887E+BOO .682E+OOO .135E+88& .63ZE+BOB .713E+OOE NNHHHHHOO‘JCHUIUI-fl»ht~thNH"-J bulNNF‘r-DHO-‘F‘tnm‘Jmfl‘uLn43(4de ‘f' .264E+BBB .368E+861 .348E+881 .759E+681 .364E+BBI .385E+661 .569E+981 .14OE+681 .?36E+891 .249E+881 .411E+981 .9?1E+681 .349E+BB£ .168E+882 .241E+882 .542E+882 .83?E+092 .885E+OOZ .635E+862 TORQUE (n-m) .186E-BB3 .BB3E-BB3 .573E-883 .l99E-BB3 .120E-683 .674E-BB3 .955E-863 .824E-BG3 .731E-983 .512E-BB3 .128E-862 .365E-902 .597E-002 .?66E-862 .889E-802 .347E-892 .79?E-882 .173E-302 .BlBE-BBZ Table 8.14 Miracle Whip 120 t-zz-es SHEEP STRESS 'N ““3 .‘l. ['IJ 1.” "" T J I.“ .21 _ r l I-4fi'.- I 1.11.4.3 din Q: 9.. '32! SUSH 3 .“II E. .------r--mT.-_m.u._-_r.m-n--- B "i E M-_-W.JQEJE “- -. 7': 3"!" ‘ T x- .‘ a: ! ia 1 fl I .x ‘ .. i E J '1": 9 l. . :. : l_ I w @ . 1 .! ‘ z.‘ ; z=< 1 5 | n 1 ‘5; z k. 1 K i A A r" _ I 1 '-..r 'r-‘ 0' v x I . - 3'.- _s' Pl 9" C' N ’5' 1. fi 1: Table 8.14 (cont'd.) a to C0 -.1 II: at J's u m H :3: u) m «1 m an 4: m m . ‘ w ['owvdrer—‘w 9*t-‘Hr-tH PT# 9;; ' 0 GO "J U“! L" #3 DJ I‘ll 1“ GI '43 03 "-l 0". UI ~58 (.ml I"-J H "tIIMHHHHHR‘HF-‘H H H lfl "-.j II. .1 h o] {-1 N m H H H H v-a l0 U3 “-1 m U! L (d M III) I -. 4: m to o:- h r. 0”” .‘A 4.. lit] Lo] ['01 [Q "OJ H R" 5‘ '43 ”J '3] |.'-' iradf .69-E- 89E .IBSE-BOl 121 ++-++ 1 L4 [‘0 ‘13 ‘13 ‘3‘ IS. ‘3! IS! IS! IS '3! '5' 'SI 1'5." XI '3'" us: mmmmm + ISI 61 H -mw~hunm x m + SI S! H .1'34E+BBI .484E+861 .128E+BBI .5@@E+981 .2?9E+BBI .‘wE1E+@Bl 963 E+BBl .37€-E+U81 .793 E+BBI .69254’391 .945E+832 .154E+DBE REM DRTR SPEED sec} .EBSE- 981 .3195-861 .93?E+BBO .244E+OB@ .456E+OOB .665E+OOB .3?4E+008 .886E+OBE .69?E+BOB .133E+OOO .631E+BBB 149E+BBG ‘a:s~eeo m N M m m :4) M .4 M .3 H H H .4 H w H .¢ .1 u;- t'. [-1 :11] (.4 01-] [d N h) m M U [:1] "‘ '3' '33 '1} l I. I "d m IO r.) m m ... E. .0; H .24SE+981 .BFBE+BGE .lBlE+BBZ .E4SE+OBE .335E+BBE .415E+9@E .313E+BBE .586E+GBE .SF4E+BBE eaaa+aea 721s+eae .S42E+B@E 938E+992 .964E+@@E .937E+BBE .214E+@@2 342E+892 .4Fss+a@2 .S4BE+BBE .64?E+OOE TORQUE (Tin-m .4UE‘E- 992 .IE9E- SUE T '1 I allg' I31 1 5 IS! I . l a '9 01" I29 U! («1 ti- ' in] P. h.) ._n m TIMI? mm m IS! '3 T I I E2! '9 SI (30 [3.) M D J .zaaE-ééz .SB4E- 982 .937E- 882 .99BE-BBE .IBBE-BBE .3?1E-B@2 .564E-BOE .FSEE-BBE .86?E-G@2 9295—932 APPENDIX C COMPUTER PROGRAM - VELOC APPENDIX C COMPUTER PROGRAM - VELOC PROGRAM VELOC (7000 O 48 #3 L0 1") PI=3.141593 URITE(*,*)’Input READ(*,3)L URITE(*.*)’InPut READ(#,3) R1 URITE(*,*I’lnput READ(#,3)RO FORfiAT.DBC IPS,VALUE J=JGF+1PS C4 A(J)= (J)+VALUE C C OUTPUT THE HEADING FOR THE ELEMENT DATA C 95 URITE(6.100) 100 FORMAT(//10X.12HELEMENT DATA/12X,3HNEL,3X,3X, +12HNODE NUMBERS) CALCULATION OF THE ELEMENT MATRICES (WOO KL=2 IF(ILO.E0.2) KL=3 DOI25KK=1,NE OOIOSI=I.KL J1=ILOt 'ENTER: IOPTME' REAO(*.*) IOPTME IF(IOPTME.EO.2) GOTOIQO C C EVALUATION OF THE ANALYTICAL SOLUTION FOR NSTEPS C UITH A TIME INCREMENT OF DELTA C URITE(*,*) ’ENTER: ITYPE. NSTEPS, DELTA' READ(*,*) ITYPE.NSTEPS,DELTA TME=0.0 DOIISKK=1.NSTEPS IF((KK.EQ.1).OR.(((KK-1)/10*10).EQ.KK-1)) URITE(6,95) 95 FORMAT(1H1.///,SX,26HANALYTICAL SOLUTION TO THE, +33H SYSTEM OF DIFFERENTIAL EQUATIONS) DOIOOI=1.NEGV - E=-1.*EIGVL(I)#TME 100 EV(I)=EXP(E) CALL MTXVC. tttittiit DEBUG OUTPUT IF(IPTL.GE.4) URITE(*.5) NEGV FORMAT(/1X,16HEXECUTING JACOBI/IX,6HNEGV =,I5) INITIALIZE EIGENVALUE AND EIGENVECTOR MATRICES OO2OI=1.NEGV IF(A(I,I).GT.0. .AND. B“I.I).GT.O.) GOT015 URITE(*,6) I,I,A COMMON/INOUT/IPTL C Cttttttttt Cttttttttt C . C THIS SUBROUTINE CONVERTS THE GLOBAL STIFFNESS AND C CAPACITANCE MATRICES AS STORED IN THE A( ) VECTOR C INTO SQUARE BANOED MATRICES. THESE MATRICES ARE C THEN RETURNED TO THE CALLING PROGRAM AND STORED IN C COLUMN VECTORS. THE NEED FOR THIS PROGRAM ARISES C BECAUSE THE SUBROUTINE JACOBI UAS URITTEN TO USE C SQUARE MATRICES. UE ALSO UANT THE MATRICES IN C JACOBI TO HAVE A VARIABLE DIMENSION. C Cttttttttt Cttttttttt DEFINITION OF THE PARAMETERS IN THE CALL STATEMENT NP - NUMBER OF EQUATIONS NBU - BAND UIDTH OF THE GLOBAL MATRICES. THE SAME VALUE IS USED FOR BOTH MATRICES NEGV - THE NUMBER OF EIGENVALUES NEGV2 - THE SQUARE OF THE NUMBER OF EIGENVALUES NUMB - NUMBER OF NODES UHOSE VALUE REMAINS THE SAME FOR ALL VALUES OF TIME PHI( ) — VECTOR OF NODAL INITIAL VALUES GF( ) - GLOBAL FORCE VECTOR GSM( , ) - THE GLOBAL STIFFNESS MATRIX STORED IN RECTANGULAR FORM GCM( . ) - THE GLOBAL CAPACITANCE MATRIX STORED IN RECTANGULAR FORM S( . ) - THE GLOBAL STIFFNESS MATRIX STORED AS A SQUARE NP X NP MATRIX C( , ) - THE GLOBAL CAPACITANCE MATRIX STORED AS A SQUARE NP X NP MATRIX OOOOOOODOOOOOODODO0000000000000 147 C VCT1( ) AND VCT2( ) - UORKING VECTORS USED TO CONVERT C THE REDUCED MATRICES INTO VECTOR FORMS C Ctfitttfittt Ctttittttt C DEBUG OUTPUT C IF(IPTL.GE.4) URITE(*,5) NP,NBU.NEGV,NUMB FORMAT(/1X,16HEXECUTING ARRANG/IX,6HNP =,I5,/,1X.6HNBU =,15,/, +1X.6HNEGV =,IS/1X,6HNUMB =,15) Ln INITIALIZE THE S AND C MATRICES O1fifi DO1OI=1,NP DOIOJ=1.NP S(I.J)=0.0 O C(I.J)=0.0 PLACE GSM( , ) INTO THE UPPER PART OF S( , ) AND PLACE GSM( , ) INTO THE UPPER PART OF C’ , ) (VOC)()H NN=NP-NBU+1 KN=NBU DOSOI=1.NP J=I ‘ IF(I.GT.NN) KN=KN-1 DO25K=1.KN S(I.J)=GSM(I,K) C(I,J)=GCM(I,K) 25 J=J+1 30 CONTINUE C C DETERMINE THE ROU AND COLUMN NUMBERS THAT MAKE C UP THE REDUCED MATRIX IF(NUMB.EQ.O) GOTOTO KM: DC4OI=1.NP DOSSJ=1.NUMB IF(IB(J).EQ.I) GOTO40 S5 CONTINUE IGD(KK)=I KK=KK+1 40 CONTINUE IF(IPTL.EQ.5) URITE(6.42) (IGD(I),I=1,NEGV) 42 FORMAT(/1X,8HIGD( ) =,1OI5) C C GENERATE THE UPPER TRIANGULAR PART OF THE C REDUCED MATRIX C DOSOI=1.NEGV II=IGD(I) PHI(I)=PHI(II) GF(I)=GF(II) OOASJ=I,NEGV 148 J3=IGO(J) S(I.J)=S(II.JJ) as C(I.J)=C(II,JJ) 5) CONTINUE é COMPLETE THE LOUER PARTS OF S( , > AND C< , > L. 70 DO7SI=2.NEGV II=I-I DO7SJ=1.JJ S(I.I)=S(J.I) 75 C(I,J)=C(J,I) C E OUTPUT OF THE REDUCED MATRICES UHEN IPTL=5 IF(IPTL.LE.4) GOT0120 URITE(6.80) 8O FCRMAT(/1X.26HREDUCED CAPACITANCE MATRIX) DOBSI=1,NEGV 5 URITE(6.90) (C(I.J),J=1,NEGV) O FORMAT(5X.5E12.5) URITE(6.95) 95 FORMATC/IX,24HREDUCED STIFFNESS MATRIX) DOIOOI=1.NEGV 100 URITE(6,90) (S(I,J),-=1,NEGV) C C REARRANGE THE MATRIX SO THAT THE COLUMNS OF THE C REDUCED MATRIX ARE AT THE TOP OF THE STORAGE C VECTOR UHEN THE MATRIX IS RETURNED TO THE C CALLING PROGRAM C 120 IF(NUMB.EQ.O) GOT0145 K=0 DOlSOJ=1.NEGV DOI251=1,NEGV K=K+1 VCTI (K)=o(-1 .I) 125 VCT2(K)=C(J,I) 130 CONTINUE V:O DOIAOI=1.NEGV OOISSJ=1,NP K=K+1 IF(K.GT.NEGV2) GOTOI45 S(J.I)=VCT1(K) 135 C(J.I)=VCT2(K) 140 CONTINUE C C RETURN C 145 IF(IPTL.LE.3) RETURN URITE(*.150) 150 FORMAT(/1X.21HRETURNING FROM ARRANG) RETURN END C##*t****#*********#**##*#$#**+#**##**#####¢#****##**##¢*###*#*##*# 149 SUBROUTINE MODIFY(NP,NBU,NUMB,IE,PHI,FM,K,C) DIMENSION PHI(NP),FM(NP),IB(NP),C(NP,NBUE,K€NP,NBJ) REAL K COMMON/INOUT/IPTL tittitifit THIS SUBROUTINE MODIFIES THE GLOBAL CAPACITANCE AND STIFFNESS MATRICES UHEN THERE ARE NODAL VALUES THAT REMAIN CONSTANT UITH TIME. .*******#* DEBUG OUTPUT OOOI‘WI‘)(3(‘):).—)nn IF(IPTL.GE.4) URITEI6.2) NP,NBU FORMAT(1X,16HEXECUTING MODIFY/1X,5HNP =,15,/, +1X,5HNBLJ =.15) IV) MODIFY C AND K MATRICES BY DELETING ROUS AND COLUMNS non DOACI=1.NUMB J=IB(I) J=J-1 DOSOJM=2.NBU M=J+JM-1 IF(M.GT.NP)GOTOCD FM(M)=FM(M)-K(J,JM)*PHI(J) K(J.JM)=0. C(J.JM)=C. IF(N.LE.OT GOTOSO FM(N)=FM(N)-K(N,JM)*PHI(J) K(N,JM)=O. C(N,JM)=O. N=N-I 30 CONTINUE C(J.1)=1. K(J,1)=O. 0 FM(J)=O. h.) C) RETURN nnmb IF(IPTL.LE.4) RETURN IF(IPTL.EQ.4) GOTO70 URITE(#,41) 41 FORMAT(llX,21HMODIFIED FORCE VECTOR) URITE(*,42) (FM(I),I=1,NP) 42 FORMAT(4X,E12.5) URITE(*,45) 45 FORMAT(/1X,27HMODIFIED CAPACITANCE MATRIX) DOSOI=1.NP 50 URITE(*,55) (C(I.J).J=1.NBU) 5 FORMAT(SX.5E12.5) URITE(#,60) 60 FORMAT(/1X.25HMODIFIED STIFFNESS MATRIX) DO651=1.NP 65 7O 75 150 URITE(+,55) (K(I.J).J=1,NBU) URITE(*.75) FORMAT(IX,21HRETURNING FROM MODIFY‘ END Ct:PAAPUAAAwt:¢****tt¢tttt*tttttt+t¢ttt+itTAAAAAAA+AAAAA+PAAAAAP.itt F) D 00") U‘ OODOOOODDOOODD 000C) HOUOD 0 COO +1X.7HNBU =,I5./,1X.7HTHETA =,F8.4./,1X.8HDELTME =,E12.5) SUBROUTINE MATAP(NP,NBU,THETA,DELTME,K,C) DIMENSION C(NP,NBU),K(NP,NBU) REAL K COMMON/INOUT/IPTL ##IMOMHII‘MWI THIS SUBROUTINE GENERATES THE A AND P MATRICES FOR THE SINGLE STEP METHODS USING THE EQUATIONS A( , ) C( , )+(DELTA)#THETA#K( , ) P( . ) C(i, )- IF(ITYPE.GE.4) GOTO70 NPI=NP DO6SI=1,NP UEX(I)=0.0 DO65J=1.10 RADI=2.*ALP(J)¢(X(I)—0.5) SEC=I./COS(ALP(J)) E=(-1.)#4.#(ALP(J)**2)*TIME A=SEC/(3.+a.PALP(J)t¢2) UEX(I)=UEX(I)+4.*ATEXP(E)*COSTRADI) DIFF(I)=U(I)-UEX(I) GOTOSS ANALYTICAL SOLUTION, U(X,O)=O, U(O,T)=1. NPI=NP-1 DO741=1,NP1 UEX(I)=0.0 DO72J=1.100 T=J RAD=(2.*T-1.)¢PIPX(I+1) E=(-1.)*((2.tT—1.)**2)*PI¢PI*TIME A=4./((2.¢T-1.)*PI) UEX(I)=UEX(I)+A*SIN(RAD)*EXP(E) DIFF(I)=U(I)—UEX(I) CALCULATION OF THE L1 AND L2 NORMS 159 C 85 SUM1=0.0 SUM2=0.0 DO88I=1.NP1 SUMI=SUM1+ABS> 88 SUM2=SUM2+DIFF(I)**2 SUM2=SQRT(SUM2) C C OUTPUT OF THE CALCULATED VALUES C IF(IPTL.EQ.1) GOTO97 URITE(6,90) (UEX(I),I=1,NP1) 90 FORMAT(SX.5HEXACT,10F11.5) URITE(6.95) (DIFF(I),I=1,NPI) 95 FORMAT(5X,5HDIFF ,10F11.5) 97 URITE(6.100) SUV1.SUM2 100 FORMAT(SX,4HLI =,F12.8,10X,4HL2 =,F12.8) RETURN END Ct*ttitttttttttttttittttttttttttttttttttttitttttttittttttttttttttttttit SUBROUTINE NUMODE(IEAN,NUMB,NP,NBU,JPHI,JGF,JGSM,JGCM,JEND,IB,X,A) DIMENSION X(NP),A(JEND),AV(900’.PV(900),DUPI(900),DUP26900) DIMENSION ADP(500),IB 162 E=(-1-}*'T*T)*(PI*PI)*TIME A=SIN(RADI)*SIN(RAD2)*EXP(E) 5 UEX(I)=UEX(I)+A*8./(T#T¢PI*PI) 50 DIFF(I)=U(I)-UEX(I) GOT085 C ANALYTICAL SOLUTION. U(X,O)=1, DU/DX=U AT X=C, C DU/DX=O AT X=O.5 60 IF(ITYPE.GE.4) GOTO70 00681=1,NP UEX(I)=0.0 0065J=1.10 RADI=2.#ALP(J)*(X(I)-0.5) SEC=1./COS(ALPtCOS,1=1,~p> 9o FORMAT(SX,5HEXACT,10F11.5) URITE(6.95) (DIFF(1),I=1,NP) 95 FORMAT(5X,4HDIFF,10F11.5) 97 URITE(6.100) sun1,sun2 100 FORMAT(5X,4HL1 =,F12.3,1ox,aHL2 =,F12.8) RETURN END 163 C*##fittittfitt#t**#**###*#¢###tit+*¢**#t*##t&*ttt##ttitfittttfi+tfittittt SUBROUTINE DSSYTP(NP,NRC,JGF.JGSM,JGCM,JENC,NE,EF,E3M, +ECM,A) DIMENSION NS(NRC),EF(NRC).ECM7NRC,NRC),ESM(NRC,NRC) DIMENSION A(JEND) COMMON/IdOUT/IPTL C) 4|- {- i- a» t» a» ‘l- I- 1» THIS SUBROUTINE PLACES THE COEFFICIENTS OF THE ELEMENT CAPACITANCE AND STIFFNESS MATRICES INTO THE CORRECT POSITIONS IN THE A VECTOR. THIS SUBROUTINE IS TO BE USED FOR SYMMETRIC ELEMENT MATRICES AND TIME DEPENDENT PROBLEMS ##ttttttt DEFINITION OF THE VARIABLES IN THE CALL STATEMENT NP - THE NUMBER OF GLOBAL EQUATIONS NRC - THE NUMBER OF ROUS AND COLUMNS IN THE ELEMENT MATRICES JGF - POINTER FOR THE A VECTOR ONE POSITION AHEAD OF UHERE THE GLOBAL FORCE VECTOR STARTS JGSM - POINTER FOR THE A VECTOR ONE POSITION AHEAD OF UHERE THE GLOBAL STIFFNESS MATRIX STARTS JGCM - POINTER FOR THE A VECTOR ONE POSITION AHEAD OF UHERE THE GLOBAL CAPACITANCE MATRIX STARTS JEND - NUMBER OF MEMORY POSITIONS IN THE A VECTDOR NS( 3 - VECTOR CONTAINING THE ELEMENT INDICIES EF< ) - THE ELEMENT FORCE VECTOR ESM( . ) - THE ELEMENT STIFFNESS MATRIX ECM( , ) - THE ELEMENT CAPACITANCE MATRIX A( ) THE A VECTOR *tttflltttfi DEBUG OPTION IF(IPTL.GE.4) URITE(*.5) NP,NRC.JGF.JGSM.JGCM.JEND FORMAT(/1X,16HEXECUTING DSSYTP/IX,6HNP =,IS./,1X, +6HNRC =,IS./.1X.6HJGF =,I5./,1X,6HJGSM =,IS./.1X, +6HJGCM =,15./,1X,6HJEND =,IS) U‘ Donn00(30DODOODDDFTDDDOCTU-“)I)(_)OD()(‘)OOF)O(WD()D(')OFU DIRECT STIFFNESS PROCEDURE DOD 2O 30 C C C 35 40 45 164 DO3OI=1.NRC J5=JGF+NS A(J5)=A(J5)+EF(I) DO2OJ=1.NRC JJ=NS(J)-NS(I)+1 IF(JJ.LE.O) GOT020 JK=JGSH+