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RATE OF AIR ESCAPE FROM ROLLS WOUND AT HIGH SPEEDS WITH A FORCE LOADED NIP By HUMAIR AHMED MOHAMMED Bachelor of Science Osmania University Hyderabad, India 1992 Submitted to the Faculty of the Graduate School of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 1995 RATE OF AIR ESCAPE FROM ROLLS WOUND AT HIGH SPEEDS WITH A FORCE LOADED NIP Thesis Approved: ~__ ThesIs Adv~ser . ) ,. ,I / . /) .. /' 'I,/.,/5:;'.:.  ,/' .. /.I<'~~ /.. ... ~ Dean of the Graduate College 11 ACKNOWLEDGMENTS I wish to express my sincere appreciation to my advisor, Dr. James K. Good, for his intelligent supervision, constructive guidance, and invaluable assistance. His guidance and active participation throughout this project have been a tremendous asset. I would like to thank him for providing me with this research opportunity and a generous financial support. I would also like to thank Dr. Richard L. Lowery and Dr. John J. Shelton for serving on my graduate committee. Special thanks are due to Robert Taylor and Ron Markum at the web handling research center, for providing suggestions and assistance during the study. Most importantly, I would like to express my deepest gratitude and appreciation to my mother, Mrs. Naima Azmath who has supported me morally and financially to this day, and to my late father Mohammed Manzoor Ahmed. Finally, I would like to thank my loving wife Safia, for her consistent patience and understanding, and providing me with the moral support needed to complete this endeavor. 111 Chapter TABLE OF CONTENTS Page I. INTRODUCTION 1 II. LITERATURE REVIEW. 4 Hakiel's Model 6 Reduced Radial Modulus 9 Squeeze Film Damper Theory 12 III. AIR FILM THICKNESS MODELS 14 Hydrodynamic Equation 14 ElastoHydrodynamic Equation 17 Chang's Prediction Equation 19 IV. EXPERIMENTAL SETUP...... 23 V. MEASUREMENT OF ENTRAINED AIR 27 VI. EXPERIMENTAL RESULTSSELECTION OF A MODEL FOR AIR FILM THICKNESS 30 Bertram and Eshel's Hydrodynamic Solution 31 Hamrock and Dowson's Elastohydrodynamic Solution 32 Chang's Elastohydrodynamic Solution 33 Experimental Results....................................................................... 37 Comparison of Theory and Experiment 38 Experimental Observations Using a Metal Nip 41 Effect of Wound Roll Modulus (Eb) and the Modulus of the Nip (Ea) on the Film Thicknes 43 VII. RADIAL PRESSURE PROFILE 47 VIII. AIR ESCAPE OVER TIME 51 Comparison of Theory and Experiment 58 IV Chapter Page IX. SUMMARY AND CONCLUSIONS 61 Future Work..................................................................................... 62 REFERENCES 64 APPENDIX 65 v Table LIST OF TABLES Page I. Volume of Air Collected for Twenty Four Individual Winding Experiments 66 II. Comparison of Theoretical Values from Bertram and Eshel's Hydrodynamic Equation with Experimental Results.................................. 67 III. Comparison of Theoretical Values from Hamrock and Dowson's Elastohydrodynamic Minimum Air Film Equation with Experimental Results................................................................................... 69 IV. Comparison of Theoretical Values Calculated from Chang's Elastohydrodynamic Equation for Compressed Air Film with the Experimental Results................................................................................... 71 V. Comparison of Theoretical Values Calculated from Chang's Hydrodynamic Equation for Decompressed Air Film with the Experimental Results 73 VI. Comparison of Theoretical Values Calculated from Chang's Elastohydrodynamic Equation for Decompressed Air Film with the Experimental Results 75 VII. Comparison of Experimental Results Obtained from The Rubber Nip and The Hollow Aluminum Nip.................................................................. 77 VIII. Radial Pressures as a Function of Normalized Radius 78 IX. Volume of Air Collected Immediately After Winding as a Function The Length of The Roll Unwound.... 79 x. Comparison of The Air Collected From Two Rolls Unwound 3 Days After and Immediately After Winding Respectively as a Function of Their Lengths 80 XI. Air Layer Thickness as a Function of Time for Three Sets of Roll Samples......................................................................... 81 Vi XII. Comparison of Theoretical Values Calculated from Squeeze Film Damper Equation With Experimental Results 82 XIII. Comparison of the Range of Dimensionless Parameters Used by Hamrock & Dowson with Those Associated with the Experiments............ 84 XIV. Comparison of The Range of Dimensionless Parameters Used By Chang with Those Associated with the Experiment.................................... 85 XV. Individual Effect of Wound Roll Modulus Eb and Nip Modulus Ea on Air Layer Thickness (ho) Derived by Hamrock and Dowson and by Chang for their Elastohydrodynamic Equation's........................... 86 VII Figures LIST OF FIGURES Page 1. Schematic View of the Bubblerimeter 5 2. Air Layer Between Two Web Surfaces Before and After Winding 10 3. Illustration of a Squeeze Film Damper 12 4. Illustration ofNip Roller Assisted Centerwinding 15 5. Variation of Req as a Function ofRr, the Wound Roll Radius 22 6. Winder Configuration. 24 7. Theoretical Results of Bertram & Eshel's Hydrodynamic Equation 32 8. Theoretical Results of Hamrock & Dowson's Elastohydrodynamic Equation.. 33 9. Theoretical Results of Chang's Equation....................................................... 35 10. Air Film Thickness (he) Away From the Nip as Predicted by Chang 36 11. Chang's he Prediction Using Bertram and Eshel's Hydrodynamic Term 37 12. Experimental Results 38 13. Air Film Thickness (ho) for Hamrock and Dowson's and Chang's Equations, as Affected by Wound Roll/Nip Modulus 44 14. Radial Modulus, Er Computed as a Function of Pressure, P 48 15. Comparison of Radial Pressure as a Function of Normalized Radius (R1Rc) 49 16. Pressure Profile for The Roll with Entrained Air 50 17. Volume of Air Collected at Various Lengths of the Unwinding Roll 53 viii Figures Page 18. Comparison of the Volume of Air Collected Immediately and 3 Days After Winding, as the Roll Unwinds 55 19. Air Escape as a Function of Time(Experimental Results) 57 20. Air Escape as a Function of Time(Comparison of Theory and Experiment). 57 21. Bumps at The Roll Edges Obstructing The Escape of Entrained Air 59 22. NonUniform Pressure Distribution Along The Web Width 60 IX A a b C1,C2,C3 Ea Eb Ec Er Et Er air, Er air Er stack, Er stack Ereq, Er eq E, E' F G 2 g H h Kstack NOMENCLATURE Area of Entrained Air per Unit Width (in.2 ) Semiaxis in Transverse Direction of Line Contact (in.) Semiaxis in Motion Direction of Line Contact (in.) Polynomial Coefficients Modulus of the Nip (675 psi.) Modulus of the Wound Roll (psi.) Modulus of the core (30XI06 psi.) Radial Modulus of the Wound Roll (psi.) Tangential Modulus of the Wound Roll (psi.) Modulus of the Web with the Entrained Air (psi.) Modulus of the Web as a Stack of Layers (psi.) Combined Stack and Air Radial Modulus (psi.) Effective Modulus (psi.) Normal Applied Load (including the tension component) RMS of ball, race (RMS of the web substituted for both) (in.) Dimensionless Materials Parameter Ratio of Tangential Modulus to Radial Modulus of the Wound Roll Air Layer Thickness Thickness of the Web (in.) Air Film Thickness After Expansion (in.) Minimum Air Film Thickness (in.) Air Layer Thickness (in.) Stiffness of Air Layer Entrained (lbf/in.) Stiffness of a Stack ofWeb Layers (lbf/in.) x Pa Po Rx' Re Rn,Rg Ro, Rr t T u u W x v Combined Stiffness of Stack and Air Layer (lbf/in.) Ellipticity Parameter (k = 10 for line contact) Load per Unit Width to Overcome Air Entrainment effects(lbf/in.) Atmospheric or Ambient Pressure (psi.) Pressure Beneath the outer Layer (psi.) Effective Radius (1.085 in.) Radius of the Nip (2.0 in.) Radius of the Wound Roll (2.3715 in.) Time (sees.) Web line Tension (0.48 lbf/in.) Winding Tension (1000 psi.) Dimensionless Speed Parameter Winding Speed (in./sec.) Average ofNip and Roll Velocities (in./sec.) Velocity of the Nip (in./sec.) Velocity of the Wound Roll (in./sec.) Dimensionless Load Parameter Radial Compression of Entrained Air Layer During Winding (in.) Radial Strain (in./in.) Tangential Strain (in.lin.) Dimensionless Film Parameter Dynamic Viscosity of Air (2.6Xl 09 Ibfsec./in.2 ) Poisson's Ratio Poisson's Ratio of the Nip (0.4) Poisson's Ratio of Roll (0.01) Poisson's Ratio in the Radial Direction Poisson's Ratio in the Tangential Direction Stress in the Radial Direction (psi.) Stress in the Tangential Direction (psi.) Xl CHAPTER I INTRODUCTION Thin plastic films which are widely used for a variety of industrial applications such as packaging, magnetic recording etc. are usually wound in spirals to form a roll. In general any film that can be wound into a roll is called a web. The film may be paper, cloth or plastic. Perhaps the most widely used films today are plastic films used for a variety of applications ranging from packaging to magnetic recording media. The most commonly used polymers are polyethylene, polypropylene, cellulose acetate and polyethylene terephthalate. These films are wound by a variety of winding techniques, the most common being centerwinding. In centerwinding we have a core that is subjected to a torque. A motor provides torque to the core and controls the speed of winding. In centerwinding, we may have a layon roller whose function is mainly to squeeze the unwanted excess air entrained between two adjacent layers as the roll builds up. At low winding speeds an increase in the woundintension has been documented without being forced to increase the web line tension. This layon roller is sometimes called a nip roller. Alternately we have surface winding in which the torque is provided to a layon roller, which applies a 1 2 normal force to the core and drives it as a result of contact friction, at a speed at which it is driven by the motor. Trapping too much air between layers is highly undesirable because it prevents the interlayer contact, resulting in slippage in the axial direction, more commonly referred to as "telescoping". Air entrainment radically decreases the stress state within a wound roll thereby adversely affecting the integrity of the roll structure during storage and shipment. Air entrainment can be reduced by increasing the web line tension during winding but this is much less effective than the use of a nip. In addition, winding a roll too tightly may result in defects such as wrinkling and starring, therby degrading the quality of the roll. Thus it is important for the wound roll to have an optimum stress level, neither too low (e.g. as a result of air entrainment) so as to cause telescoping, nor too high so as to degrade the roll quality. A variety of factors need to be considered to assess the state of stress in wound rolls. The most relevant factors are web tension, radius of wound roll, presence of entrained air as well as the material properties of the web in consideration. A factor that affects the amount of entrained air and could thereby appreciably affect the stress state in wound rolls is the velocity at which the roll is wound (i.e. the "winding velocity"). Winding velocity is to be considered to take into account the amount of air that is carried along the web during winding, to be trapped subsequently in the adjacent layers. Rolls wound at higher velocities are prone to entrain more air than those at lower velocities. To increase the efficiency of a winding operation it is necessary to incorporate higher winding velocities and to reduce the effect of air entrainment. A nip roller serves 3 mostly to accomplish the above by allowing us to keep the winding velocity high while keeping the entrained air thickness low. This study seeks to investigate the air entrainment rates during high speed winding, and how these would affect the physical properties of the wound roll. Furthermore it will examine the rate at which the entrained air discharges from the wound rollover a period of time. CHAPTER II LITERATURE REVIEW Air entrainment experiments in centerwound rolls with a lay on roll (nip roller) were done and analyzed by Covell [5] in his thesis "The effect ofa nip roller on entrained air during high speed winding". Covell performed experiments in wich he collected the residual air which had been entrained at various web velocities and nip loads. He compared his results to a Hydrodynamic relationship which was derived by Bertram and Eshel [1] and an ElastoHydrodynamic relationship derived by Hamrock and Dowson [9]. Covell's experimental data correlated best with the Hydrodynamic Equation. Previously, related work on air entrainment was done by Bouquerel [3]. In his thesis "Theoretical and experimental study ofwinding ofthin plastic films: Aerodynamic effects", Bouquerel attempted to experimentally verify Hamrock and Dowson's [9] elastohydrodynamic equation. But due to inadequate correlation between the theoretical and experimental values he had to introduce a correction factor. He calculated the apparent density of the roll based on its mass and external diameter and finally deduced the air layer thickness assuming incompressible film of known density and that the density of air is negligible as opposed to the web material (Polyester in his experiment). 4 5 To measure the entrained air accurately a direct measuring technique was employed by Covell [5]. A bubble collection apparatus, also called as the "Bubblerimeter" (See Fig. 1), was developed. It consists of a rectangular water tank fixed with a spool and a 45° dead bar. The wound roll is placed on the spool and the web is directed out of the water tank to the winding machine by means of the 45° dead bar. The wound roll placed in the apparatus is subsequently unwound at a low speed, thereby ensuring all the air bubbles are trapped in the conical hood placed directly over the roll on side supports fixed inside the tank. The amount of air that is collected inside the hood is quantified by collecting it into a graduated cylinder fixed to the top of the hood. WATER LEVEL 45 DEGREE IDL WATER LEVEL ~Tonili WINDER ENTRAINED AIR BUBBLES 0 AMOUNT OF AIR COLLECTED AT AN INSTANT ___~ GRADUATED CYLINDER CONICAL HOOD SPOOL UNWINDING ROLL Figure 1. Schematic View of the Bubblerimeter. Covell conducted his experiments on 48 gage Type 442 Polyester, 6 inches wide and 1500 m long, at a winding tension of 1000 psi, and at various velocities and nip 6 loads. The experimental results obtained were compared with theoretical values from Hamrock and Dowson's [9] Elastohydrodynamic equation as well as Bertram and Eshel's [1] Hydrodynamic equation. Using a multiplicitive correction factor a modified version of the Elastohydrodynamic equation seemed to give a close approximation for the air layer thickness over the tested range of winding conditions. However, when compared without any modification his experimental values were closest to the theoretical results derived from Bertram and Eshel's [1] Hydrodynamic equation. Before discussing the algorithms that can be used to model the entrained air it is necessary to first understand the basic winding models that deal with the stress distribution in wound rolls and second how these stresses are affected by the presence of entrained air. One such classical model was presented by Hakiel [8] which has since been modified by Good, Wu and Fikes [7] to include a nip roller. The Hakiel model as well as the stresses in the wound rolls due to the presence of entrained air are presented briefly in the following pages. Hakiel's Model Hakiel [8] in 1987 presented a winding model which applied a finite difference method to solve a second order nonlinear differential equation in radial pressure. Young's modulus in the radial direction was made a function of radial pressure. The nonlinear orthotropic hoop model developed by Hakiel assumes: 7 1.) The wound roll to be a cylinder made by winding concentric hoops of web, and that the properties of the roll as each hoop is added are constant. 2.) The roll has a linear elastic behavior in the tangential direction and a non linear elastic behaviour in the radial direction which varies as a function of radial stress. 3.) Plane stresses which are functions of the radial positions only and axial stresses are zero. Hakiel begins with the equilibrium, constitutive and compatibility equations respectively in cylindrical coordinates: r(d;r )+O'r O't =0  Equilibrium Eq. (1) The linear orthotropic constitutive equations for radial and tangential directions are E =(_1)cr _(Vrt )cr r ErE t r t and (2A & 2B) Et=(~t )O't ( ~: )O'r respectively. Using the strain energy constraint, vtrEt = vrtEr and defining, g2 =~, we have Er and      Constitutive Eqs. (2A' & 2B') Finally from the linear definitions of strain in cylindrical coordinates we have r(dd:t )+ ~  Er=0  Compatibility Eq. (3) 8 Combining Eqs. (1) (2) & (3) yields a secondorder differential equation in terms of radial pressure as (4) This equation is solved several times during the execution of the model such that g(r) and the boundary conditions can be updated. Thus the model is solved for increments in pressure, which are summed to 8(P(r)) as shown in Eq.(5). As the roll builds up so does the pressure, assuming 8P as the incremental change in pressure at radius r, Eq.(4) can be modified as 2(d 2 r 8P) +3r(d 8P)  (g2 1)8P = 0 dr2 dr (5) The above is a second order differential equation with nonconstant coefficients and is subject to two boundary conditions, the first of which is obtained by equating the radial deformation of the first layer with the deformation of the core. In terms of the variation of pressure, 8P, due to the addition of the last layer this boundary condition becomes: 88P I =(~  1+ V)8P 8r r=1 Ec r=1 and the second is given by the hoop stress formula as After each solution ofEq.(5) the new increment in pressure 8P(r) must be summed with all previous increments to yield per) or (6) (7) n p. =~8P.. 1 L...J IJ j=l 9 (8) where Pi is the pressure in a certain radial sector and Eq.(5) would be solved n times. Solving the above boundary value problem numerically, Hakiel [8] modeled the radial stress distribution inside the wound roll. This model has been modified by Good, Wu and Fikes [7] in "The Internal Stresses in Wound Rolls with the Presence ofa Nip Roller ". Reduced Radial Modulus A reduction in radial modulus is one way in which air entrainment may be incorporated into Hakiel's model. To determine the radial modulus as a function of the air layer thickness, a radial modulus of air between the layers must be determined. Good and Holmberg [6] in "The Effect ofAir Entrainment in Centerwound rolls", describe a relationship for the radial modulus which is reduced by the entrained air. Assuming that the trapped air is an ideal gas under isothermal conditions, we can apply Boyle's law as follows (9) 10 where, hoand hI are the initial and final air layer thickness, Po and Pa are the initial pressure between the layers and atmospheric pressure respectively, and P is the pressure applied to compress the layer from hoto hI' as depicted in the illustration below: Web Layers Pa P+Pa ho Pa+Po Air Layer as the Web is Wound Initially x Air Layer as the Roll Builds up During Winding Figure 2. Air Layer Between Two Web Surfaces Before and After Winding. From Fig. 2, hI = ho x Eq. (9) can thus be written as (10) (11) Solving for x and dividing both sides by howe get a pseudo expression for the radial strain of the trapped air layer as (12) Inverting Eq. (12) after taking its derivative with respect to the radial pressure yields E . = (O'r+ po+ Pa)2 raIr (Po +Pa ) (13) 11 The modulus of the entrained air and that of the stack of web layers can be modeled as springs in series whose equivalent stiffness can be written as 1 1 1 =+ Keq Kstack Kair In terms of Er the above equation becomes 1 1 1 ( EreqA) = (ErstackAJ + (ErairA) ho+h h ho Simplification yields (14) (15) (16) Er stack is a material property and can be obtained from a material testing system. It is imperative to mention here that the difference between air layer thickness and the surface roughness of the film determine which one of the above expressions is applicable. For the winding conditions three possibilities may exist, 1) If the air layer is less than the mean surface roughness of the film the air layer is not a factor since the asperities contact the web surface and hence Er need not be modified. 2) If air layer is less than the maximum asperity height but greater than the mean surface roughness of the web then a combination of Er air and Er stack i.e., Er eq (Eq. (16)) should be used. 3) If air layer thickness is greater than the maximum asperity height then the expression for Erair (Eq. (13)) should be used. 12 Squeeze Film Damper Theory The following section is devoted to a latter part in this study which deals with the escape of entrained air over time from wound rolls. The theory developed herein would be used to model the rate of air escape. Blevins [2] in Applied Fluid Dynamics Handbook describes a squeeze film damper as shown below w Air Escapes i~_=±=""""",~~"""""",,,,,,,,,,,,,,,,,,,,,,,,,,,,"~~,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,", This Direction No Air Escape in This Direction Figure 3. Illustration of a Squeeze Film Damper. The bottom plate is assumed to be fixed and the upper one is allowed to fall down freely under its own weight W, and that the air present between the two plates squeezes out only along the width, L. No air escapes along the length, b. Assuming that the upper plate is falling down from a height hI to h2 then an expression for the time needed to close the gap from hI to h2 can be written as: t = /lbe [_1__1] 2W h2 h2 2 1 (17) 13 where, J.l is the dynamic viscosity of air, 2.7* 1091bf  sec I in.2 L is the dimension parallel to which air escapes, in. b is the dimension perpendicular to which air escapes, in. W is the weight of the plate, lbf. t is time in seconds required for the air layer to decrease from hI to h2 • The above equation can be used to predict air escape over various time periods, assuming the two plates to be a pair of overlapping web layers present in a wound roll. The roll width can be substituted in place ofL and the ratio W/bL can be simply substituted knowing the value of the radial pressure P, since P == W/bL. CHAPTER III AIR FILM THICKNESS MODELS Hydrodynamic Equation Bertram and Eshel [1] in "Recording Media Archival Attributes" published a formula in 1980 which relates the nip load to air layer thickness and winding velocity, viscosity, tension and geometry. Their experimental procedure involved an indirect method of measuring air layer thickness, wherein pull tabs were used to measure interlayer pressures and quantifying interlayer slippage during rapid deceleration to give an estimate of entrained air (The back emf generated by the decelerating roll was related to the inertia of the roll which changes with the amount of air entrapped). However their experiments do not check the validity of their formula. The present study starts off by verifying the accuracy and applicability of the Hydrodynamic Equation to the actual winding conditions using centerwinding with a nip roller. The Hydrodynamic Equation has been derived using the pressure distribution on the nip (See Fig. 4) which is governed by Reynolds equation: 14 dp dx with boundary conditions, h* x2 1+ ho 2RehO [1+ 2::hOJ at x= 0 at x = 00 (18) 15 (19) (20) where h* is the thickness of the air gap at which dp/dx = o. T , Web line tension Load to overcome air film lift, L Nip Roller ~~ liRe = lIRo + lIRg h = ho + X"'2/2Re Lt , Load to overcome web tension T Figure 4. Illustration ofNip Roller Assisted Centerwinding. 16 Integrating for pressure distribution and subsequent simplification yields the nip load required to overcome the air entrainment effects as: 00 L= f pdx o This equation does not solve for hoexplicitly instead it solves for the load (21) (22) required to overcome air entrainment effects. Thus to find hoat a given value ofL we should initially have a spreadsheet program that gives us the value of L for various values ofho. It should be noted that the second term of Eq.(22) is minute when compared to the first term, thus, (23) Assuming the expansion of entrained air at ambient temperature after it passes away from under the nip the thickness of the air film (he)' due to a change in the velocity profile, becomes 4ho/3 as reported by Chang [4]. Thus Eq. (23) can be modified to include compressibility effects as follows: Hence, h ~~/!URe c 3 L (24) (25) 17 Elastohydrodynamic Equation Hamrock and Dowson [9] in "Ball Bearing Lubrication" present a formula to calculate the film thickness between two bodies in elliptical contact. However, this equation can also be extended for use with two bodies in line (rectangular) contact as referenced in their study. Their elliptical contact theory is composed of 11 critical parameters. From these parameters they established 5 dimensionless groupings which can be related to the film thickness as follows: H= f(k,U,W,G) (26) where, k represents an ellipse of deformed contact with a major axis of dimension 2a and a minor axis of dimension 2b. For line contact, U is the dimensionless speed parameter which can be defined as (27) where (28) W represents the dimensionless load parameter and can be written as F W= E'R2 x 1 where Rx = 1 1 +R n Rr (29) and G is a dimensionless material parameter which is represented as E' G= 1/ a where E'= 2 1 y2 1 y2 __a +__b Ea Eb (30) 18 a is the pressureviscosity coefficient of lubrication (m2IN). Note that for W to be dimensionless F must have units of load (lbf or N) in Eq.(29) After the dimensionless parameters were grouped, each was varied keeping the rest constant to observe each parameter's influence on the film thickness, and the proportionality equations obtained were subsequently formulated each depicting its influence on the film thickness. hO ex:: (1 0.85e0.31k) Rx For k == 10, we have (31) hO ex:: 0.962 Rx h _0 oc UO.65 Rx (32) (33) (34) It is important to note here that G, the material property parameter did not affect ho/Rx for the range of low elastic modulus materials in Hamrock and Dowson's study. Hence G is missing in the film thickness equation which is now presented. Hamrock and Dowson set reasonable ranges of the dimensionless parameters W and U for various types of oil and grease lubricated bearings. They then solved coupled sets of Reynold's and Elasticity equation's to yield the film thickness, ho. From the results obtained a leastsquares curve fit was used to develop a formula, for film thickness ho as a function of the dimensionless parameters, called as the ElastHydrodynamic Equation which is: (36) (35) 19 ho= R x 7.43(10.85e0.31k)UO.65WO.21 and for our case where k = 10, the above equation reduces to: ho=R x 7. 43Uo.65WO.21 Since the Elastohydrodynamic Equation is derived using a curve fit, its applicability may be governed by the range of values for U and W used during the curve fit process. The range of values for U and W used by Hamrock and dowson for deriving the film thickness equation (Eq. (35)) have been presented in Table XIII in the Appendix. Before proceeding with a comparison of the theoretical models it would be appropriate to introduce a few shortcomings of the Elastohydrodynamic equation, 1) This equation was derived from a curvefit technique that was performed on a finite domain of dimensionless parameters. 2) The application of this equation to two cylinders in rolling contact is not clear even though it has been referenced in its study. Chang's Prediction Equations Chang [4] reported an air film thickness formula developed using a procedure similar to the one used to develop the Elastohydrodynamic Equation of Hamrock and Dowson, however it takes into consideration the compressibility effects of air. According to this theory the air film while passing under the forceloaded nip roller gets compressed but expands subsequently as it passes away from it. Chang has presented a set of analytical models for soft and hard materials with negligible and significant 20 compressibility effects. The present study falls into the category of soft materials since the nip used is made of rubber and since there is air entapped within the wound roll. Chang has grouped three dimensionless parameters viz., speed, load and material properties respectively as under: where, H= f(U,W,G) (37) (38) (39) E G= Pa 1 where E = 1 y2 1 y2 __a +__b Ea Eb (40) Note that for W to be dimensionless F must have units of load per unit width (pli or N/m) in Eq.(39) A procedure similar to that of Hamrock and Dowson's was employed by Chang to predict the individual dependencies of the dimensionless parameters (The range of values for U, Wand G used by Chang for deriving the film thickness equation have been presented in Table XIV in the Appendix) on the air film thickness hOe Finally a curve fit was done which in his case showed a perfect correlation with the numerical values. Based on this he defined the air film thickness hoas follows: (41) In his study Chang has also developed a relationship for an air film thickness, he which is the expanded thickness of the air film after it passes away from under the nip where it 21 gets compressed. The dependencies of the U, Wand G parameters for this case have been formulated as under: (42) Eq.'s (41) & (42) are to be used for cases where the effective modulus (equivalent modulus of nip and wound roll) ranges from 700 psi. to 5000 psi. For cases where the effective modulus ranges between 100 psi. and 700 psi. the above equations have been modified by Chang as follows: h o == R x 4. 8UO.66W0.35G0.47 he == Rx7.4UO.66WO.21G0.33 (43) (44) The three theoretical air entrainment algorithms described above will be used as a basis for comparing the results obtained experimentally. The one that can closely predict the air film thickness will be used to model the experimental values. Although the value of the air layer thickness as implied from the equations above appears to be a constant, actually it is a function of the radius of the winding roll which keeps increasing till the end of the winding operation. A change in the wound roll radius Rr will change the value of Req in the calculations for the air film thickness thereby affecting the value of hOe In an attempt to assess the variation of Req, various values ofReq were determined at various wound roll radii and the results plotted as shown in the figure below: 22 Req corresponding to Rr for 1500 mt. of roll 1.4 1.35 ,... .s 1.3  ~~ 1.25 ~ ~ 1.2 =:a 1.15 ~ ~ ..... 1.1 =~ ~ 1.05 .i:: = 1 ~ ~ 0.95 0.9 1.6 Req corresponding to Rr for 1500 ft. of roll 2.1 2.6 3.1 3.6 Wound Roll Radius, Rr (in.) Figure 5. Variation ofReq as a Function ofRr, the Wound Roll Radius. For the present study the length of the web selected for winding was 1500 ft, referring to Fig. 5 it can be observed that for this length of the roll there is a very little change in the value ofReq. Req starts off at .92 inch and rises to only 1.09 inches for 1500 ft of roll wound. Even for a 1500 m (4921 ft) long web (selected by Covell[5]) the value of Req rises to only 1.27 inches. Thus, the simplifying assumption of an average air layer thickness will be a sufficient approximation for the air layer thickness, throughout the roll. After selecting an algorithm for the air layer thickness, ho, it will be used to develop an equation for a reduced radial modulus which in turn may be used to predict the radial pressure profile using Hakiel' s [8] winding model. CHAPTER IV EXPERIMENTAL SETUP Precise measurement of entrained air is a crucial factor in air entrainment studies and in the subsequent winding model development. Measurement of entrained air has posed a subtle problem in the past. Researchers have relied on indirect techniques for measuring entrained air e.g., by using pull tabs for measuring interlayer pressures and deducing air layer thickness thereon and relating air layer thickness to interlayer slippage during rapid deceleration[I], or by using the apparent density of the wound roll to estimate the air layer thickness[3]. In contrast Good and Holmberg [6] in "The Effect of Air Entrainment in Centerwound Rolls", have employed a sophisticated technique for measuring air layer thickness directly using laser reflectance probes. Using an expression for air layer thickness derived by Knox and Sweeney [11] they were successful in correlating the theoretical and experimental results. However, this technique was limited to the cases where the entrained air layer thickness was greater than .0001 inch which is quite small for air films wound in to centerwound rolls but large for air air films wound into centerwound rolls with a nip. In this study "Air Collection" has been accomplished by a direct technique which can be used irrespective of the surface roughness of the web material. The idea is simply 23 24 to unwind a wound roll under water, trap all the air bubbles that are released and quantify them to give a measure of the entrained air. The apparatus used to accomplish this is the "Bubblerimeter" which has been described briefly in Chapter II (See Fig. I). Another factor that needs close observation during centerwinding with a nip roller is the nip load, because, even small nip loads can significantly reduce the amount of air entrained. It is important that the nip should be able to deliver a constant force throughout the winding operation to get a good estimate of the air layer thickness during air entrainment studies with a nip roller. This is especially important when the nip loads used are small because phenomenon such as nip roll bounce etc., get amplified leading to a bad estimate of the air layer thickness. Thus, we have to ensure that the nip roll and the wound roll remain in contact, and the nip load is held constant throughout winding. It was for the above mentioned facts that before proceeding with the air entrainment studies a new nip was designed on the existing winder, a schematic of which is shown below: Nip Roller Wound Roll Idler /' Tension Sensor Unwound Roll Tachometer Figure 6. Winder Configuration. 25 The new nip was designed to operate pneumatically. It was held in position exactly parallel to the wound roll by means of two arms extending on either sides, of which, one extreme (top one) held the nip, and the other (bottom one) an idler. This idler was used to direct the web from the tension sensor to the nip, the web after embracing the nip passes on to the wound roll. Since the nip arms were long and flexible in bending another idler was added in the center as a reinforcement, which later on served to redirect the web from the "Bubblerimeter" to the winder. With the existing setup it is possible to perform surface winding by adding a belt, that would drive the nip instead of the wound roll shaft. After the nip was designed it was configured with the existing winder. The next step was to install a mechanism that would deliver a constant force to the nip arms, which in turn would be delivered to the wound roll during winding. The response of the nip to the applied load had to be sensitive, especially when the loads applied were small. In other words, the mechanism that operates the nip should have the least reaction forces, which otherwise would oppose the nip force and subsequently diminish it. There were two possibilities, one was to use a dead weight to apply the nip load and the other was to use a pistoncylinder arrangement that would drive the nip pneumatically. The former option was ruled out in lieu of the high winding speeds needed for the experimentation, and the fact that nip roll bounce would make the dead weights almost weightless. This setup was used by Covell[5] and he reports of nip roll bounce problems associated with it. So, the use of a pistoncylinder arrangement was studied. One disadvantage however, was the traction force inside the cylinder due to the 26 presence of friction. This had to be minimized in order for the nip to deliver small loads consistently. After examining the working of a variety of pistoncylinder arrangements a specially designed mechanism that works with a minimal traction force was selected. This is manufactured by Origa Corp. has a stroke of eight inches and delivers a maximum force of 65 lbf at a pressure of up to 100 psi which is controlled using an air pressure regulator. The frequency response of air regulators can easily be less than 1 Hz. A wound roll with but one abnormality in its surface can provide a 150 Hz input to the nip roll. Hence, in order to improve its performance a 2 cu.ft. air tank was used to store and deliver the regulated air pressure to the cylinder. The advantage of such a configuration is, we have a large volume of compressible air whose pressure is not easily affected due to nip roll bounce during high speed winding. This is much superior to connecting the air cylinder directly to an air regulator, due to the poor frequency response of air regulators and to a relatively small volume of air (e.g., typically only that in the air supply tubing) which can be compressed. Thus,with this design the nip load will be as constant as possible throughout the winding operation. The minimum force that can be delivered is 2 lbs and 0.5 lb (0.083 pli for a 6 inch wide roll) increments have accurately been resolved using a force gage. With the nip design complete, a 5 hp AC vector drive was added to drive the winder to speeds in excess of 2000 feet per minute which were controlled by a digital controller with an accuracy of±10 feet per minute. CHAPTER V MEASUREMENT OF ENTRAINED AIR This study comprises of two parts, the first one attempts to verify a winding model that incorporates the entrained air immediately after winding. The second part deals with the discharge of entrained air over a period of time and attempts to test the applicability of a "Squeeze Film Damping Equation" [See Eq. (17), Chapter II] to estimate the rate of air escape from the wound roll. Thereafter a suitable relationship can be developed or the existing one modified to correlate the experimental results with the theoretical values predicted by the "Squeeze Film Damping Equation". For performing air entrainment tests Type 442 Polyester Gage 48, 6 in. wide, 1500 ft. long was used. It has a mean surface roughness of 0.27 J..l in., an RMS surface roughness of 0.42 J..l in., and a maximum asperity height of 0.78 J..l in.. The tangential modulus of the web is 600,000 psi.. The web was wound on a 3.375 inches O.D. steel core with an I.D. of3 inches. A typical O.D. for a 1500 ft roll was 4.7 inches and for the 1500 m rolls which were wound the O.D. was 7 inches. The reason for using polyester was the need for a nonpermeable web material that would not allow the air to diffuse in the radial direction, so that the air that would escape could only do so through the edges of the rollout into the atmosphere. Further, 27 28 the thickness of the entrained air layer should be greater than the mean surface roughness of the film, otherwise the air layer would not be a factor at all in the experiments (See Chapter II, Reduced Radial Modulus). For this reason the film chosen for this study has a very low surface roughness in order to ensure that the air collected would produce an average thickness of more than the surface roughness of the web. The experimental procedure involves winding 1500 ft. of web at various web line velocities and various nip loads at a constant winding tension of 1000 psi. and unwinding the roll in the bubblerimeter. The entrained air collected in ml. is then converted into an average air layer thickness throughout the roll by the following relation: h = O.06102(Vol. of air in ml.) o 1500*12*6 It has to be noted here that h 0 the air layer thickness is an average air layer (45) thickness and not the absolute thickness of the air entrained between each layer. The latter is dependent upon the radius of the wound roll and is not constant for each pair of overlapping layers. Thus the simplifying assumption of an average value makes other calculations less complex and avoids the present study from wandering off into an entirely different aspect of air entrainment which might need the measurement of the volume of air extracted as a function of radius of the wound roll. All materials have a finite surface roughness. Even if we use a material with very small roughness values it is unlikely that a pair of contacting layers would do so perfectly. There would always be some amount of air trapped in between their asperities. 29 This air however, would be present under any circumstances irrespective of winding conditions. Even with very large nip loads and low web velocities, conditions that are favorable for flushing out maximum air from between the two contacting faces, there always will be some trapped air called as the "baseline" amount of air. This baseline air should not be included in the same category as the entrained air, hence, from all the experimental air collected the baseline amount will be deducted to avoid bias. In an experiment to measure the baseline air 1500 ft. of roll was wound at a speed of 20 fpm and a 32lbfnip load (conditions favorable to flush out maximum air) and was subsequently unwound in the bubblerimeter. It was found that the baseline amount was about 10 ml, an average for two roll samples. Further an additional correction factor of 3% has been introduced to account for the slight vacuum present in the graduated cylinder of the bubblerimeter. This factor would also be subtracted from all experimental data along with the baseline amount. Table I in the Appendix displays the air collected from 24 individual winding experiments. CHAPTER VI EXPERIMENTAL RESULTS  SELECTION OF A MODEL FOR AIR FILM THICKNESS Experiments were performed at four different winding speeds, ranging from 500 fpm to 2000 fpm, and air was collected for six different values of nip loads corresponding to each speed. Since the intent of this study is to show how the air layer thickness would be affected even with small nip loads, a major portion of the discussion would be done for small nip loads. However the data obtained at higher nip loads would also be presented and discussed briefly. The results of the experiments were coded into a spreadsheet program and three dimensional surface plots were obtained. These plots show how the air layer thickness varies as a function of web speed and nip load. Theoretical results were obtained by substituting the material properties, winding conditions and other relevant parameters into the Hydrodynamic and ElastoHydrodynamic Equations respectively. The parameters used to model these equations can be found in the Nomenclature. The theoretical results were also plotted against the corresponding speeds and nip loads used for experimentation, and a comparison was made. Tables II, III, IV, V & VI in the Appendix 30 31 Appendix display the data calculated from the theoretical models and compare it with the experimental results. Bertram and Eshel's Hydrodynamic Solution The Hydrodynamic Equation predicts an exponentially decreasing air layer thickness as we proceed towards increasing nip loads and an almost linear decay towards low winding speeds (See Fig. 7). Refering to Eq. (22) (See Chapter III) the only parameters than need to be varied are the nip load L (psi) and the winding velocity U (inches/sec). The values of the constant parameters viz., J.l, Re, Ro and T can be found in the Nomenclature section. The values ofL were varied from 4 to 48lbf(experimental values) and those ofU from 500 to 2000 fpm (experimental values) and howas subsequently determined for 24 different combinations ofL and U. Maximum air entrainment as expected is observed for the highest speed and the lowest nip load i.e., 2000 fpm and 4 lbf. respectively and the lowest entrainment for high nip loads and low winding speed i.e., 500 fpm and 48 lbf. respectively. The range of average air layer thickness as depicted in the theoretical results was of the order of 6.7 J.l inch, corresponding to a high of 6.9 J.l inch, and a low of 0.2 J.l inch. Speed, (fpm) 7 6 5 Air Layer 4 Thickness, (micro in.) 3 Nip Load, (Ibl) 32 48 2000 .67 11I56 .45 1334 023 .12 .01 32 Figure 7. Theoretical Results of Bertram & Eshel's Hydrodynamic Equation. Hamrock and Dowson's Elastohydrodynamic Solution The Elastohydrodynamic Equation predicts a higher range of average air layer thickness, of the order of 20.411 inch, corresponding to a high of27.31l inch and a low of 6.911 inch. Referring to Eq. (36) (See Chapter III) the values of speed, u (inches/sec) and the normal applied nip load, F (lbt) were similarly varied as in the previous case. The elastohydrodynamic equation also uses the effective modulus, E' (modulus of the wound roll and the nip roll), hence the same was determined for different values of the radial modulus, Er of the wound roll under different nip loadings, and incorporated into Eq (36). The constant parameters used viz., k, 11 and Rx have been declared in the Nomenclature. Again the values of L were varied from 4 to 48 lbf and those of u from 500 to 2000 fpm 33 and howas determined for 24 different combinations of L and u. A distinctive feature of the Elastohydrodynamic equation is a far less drastic decrease in air film thickness towards higher nip loads and a much thicker air layer. It should be noted that the dimensionless parameters used by Hamrock and Dowson do not fall in the same range as the dimensionless parameters associated with these experiments (See Table XIII in Appendix). 30 25 20 Air Layer Thickness, 15 (micro in.) 10 5 Nip Load, (Ibl) 32 48 1!Il2530 E11520 01015 1105 2000 Speed, (fpm) Figure 8. Theoretical Results of Hamrock & Dowson's Elastohydrodynamic Equation. Chang's Elastohydrodynamic Solution Another theoretical model that was recently developed by Chang [4], which takes into account the compressibility effects of the entrained air has also been presented here. 34 Recall that Hamrock and Dowson's Elastohydrodynamic equation is meant for oil film lubrication in ball bearings whereas in air entrainment studies we are dealing with an air film. Chang's model takes into consideration the above disparity where, for air, its density is dominated by pressure unlike the density of oil which is almost unaffected by a wide range of pressures. This model assumes two rotating rollers without a web in between them and that the materials in consideration are isotropic. Here the formula for air layer thickness was again derived using a similar, but not identical set of nondimensional parameters used by Hamrock & Dowson. This equation too is derived using a curve fit, however the dimensionless parameters associated with these experiments fall in the same range as the ones used by Chang to derive the film thickness relationship (See Table XIV in the Appendix). The equivalent modulus of the nip roll and the wound roll as calculated by the effective modulus E' expression was 750 psi. on an average hence Eq.'s (41) and (42) would be used for the present study. The parameters used to determine the air film thickness (ho) using Eq. (41) (See Chapter III) are shown in Table XIV, again the values of speed, u (inches/sec) and the nip load per unit width of the roller, F (pli) were varied in a similar way as described previously and the results plotted for their 24 different combinations Chang's model predicts a thicker air layer than that predicted by the Elastohydrodynamic Equation but the rate of decrease of air layer as we proceed towards higher nip loads seems to relate more with the Hydrodynamic Equation than with the Elastohydrodynamic equation. Speed, (rpm) 50 45 40 35 Air Layer 30 Thickness, 25 (micro in.) 20 15 105o 4 Nip Load, (Ibf) 32 48  2000 .4550 .4045 83540 .3035 11I2530 .2025 131520 01015 .510 .05 35 Figure 9. Theoretical Results of Chang's Equation. All the results plotted above display the minimum air film thickness (ho), the thickness of the air film under the nip. Chang has also developed a relationship for the air film thickness (he)' the thickness of the air film after it passes away from the nip (See Chapter III, Eq.(42)). This is a more practical case where the air film after getting compressed under the nip expands when it moves away from it. The results obtained due to these compressibility effects are presented in Fig.! O. Note that this expression for he (Eq. (42)) depends upon a e1astohydrodynamic term. As is obvious the air film thickness (he) predicted after it expands is way too high than the experimental values however the interesting feature to note here is its similarity with the trend observed for Hamrock and Dowson's Elastohydrodynamic equation results (unlike the trend observed for ho, which resembled Bertram and Eshel's Hydrodynamic equation). Air Layer Thickness, (micro in.) Nip Load, (Ibf) 48 .8090 1!lI7080 .6070 1ilI5060 .4050 1!lI3040 02030 .1020 .010 Speed. (rpm) 36 Figure 10. Air Film Thickness (he) Away From the Nip as Predicted by Chang. The results of the he relationship developed by Chang utilizing the Elastohydrodynamic term is shown above. Chang had also developed another he relationship using a hydrodynamic term from Bertram and Eshel's Hydrodynamic equation by suitably modifying it to include the compressibility effect of the air film as it passes away from the nip (See Eq (25), Chapter III). The results of Chang's modification of hoto include the compressibility, using the hydrodynamic term are presented in Fig. 11. As is obvious referring to Eq.(25), the results obtained here are simply the results for ho predicted by the hydrodynamic equation times the factor 4/3 used to modify Eq. (23) to include the change in velocity profile as a result of the compressibility effects of the air film as it passes away from under the nip. Air Layer Thickness, (micro in.) 109 876 5 4 3 21o 4 Nip Load, (Ihf) 32 48 2000 Speed, (fpm) .910 .89 ml78 .67 1l'l56 .45 034 023 .12 .01 37 Figure 11. Chang's he Prediction Using Bertram and Eshel's Hydrodynamic Term. Experimental Results Finally, the experimental results obtained for 24 different tests are presented below with the air layer thickness plotted as a function of nip load and web velocity. There are some random values in the data taken at high speeds that do not compare with the expected results, this may be attributed to the instability of the nip roll and the nip roll bounce associated with those speeds. Repeated tests at high speeds would have produced reasonable values but the amount of material available for the same was limited. Above all, on an average, the experimental results produced consistent values which seemed to follow Chang's hopredicted values closely. The trend however resembles more like his he model and also the Elastohydrodynamic model. 38 40 30 Air Layer 25 Thickness, 20 (micro in.) Nip Load, (Ibl) 32 48 l!I3540 .3035 C2530 .2025 01520 01015 .510 2000 .05 Speed, (fpm) Figure 12. Experimental Results. Comparison of Theory and Experiment Comparing the values of the air layer obtained experimentally with the theoretical values it can be observed that Chang's model for the minimum film thickness (ho) is the closest to the experimental results. However when we observe the 3dimensional surface plots the trend appears to resemble Chang's model for the expanded air film thickness (he)' Also note the similarity with the Elastohydrodynamic equation results. Some of the other notable features are, at small values of nip loads the experimental results closely follow Chang's hoprediction (e.g., at 4lbfnip load), even the rate of decrease ofho appears to be similar. Towards very high values of nip loads the experimental results are 39 more closer to Hamrock and Dowson's Elastohydrodynamic equation results (e.g., at 48lbf nip load). This suggests a more predominant elastohydrodynamic regime at high nip loads. An interesting feature in the Elastohydrodynamic lubrication studies is the film parameter, A, which is related to the lubrication film thickness (air film thickness in our case) as follows: (46) where ff and fb both are rms surface roughness of the film and hois the film thickness. The film parameter, A is used to identify the lubrication regime. Hydrodynamic or fluidfilm lubrication occurs when the contacting surfaces are separated by a lubricating film so thick, they cease to contact. It has been reported by Hamrock and Dowson [9] that in this regime the value ofA is greater than 10 and may be as high as 100. Elastohydrodynamic lubrication occurs when the bearing materials are soft such as elastomers and rubbers, where the local elastic deformation of the bearing surfaces allows for a coherent lubricating film and surface contact is not fully avoided as in the previous case. The film parameter in this regime falls between 3 and 10. For the existing winding conditions we calculated the values of A, which were found to range from 9 to 65. This implies that we have a predominant hydrodynamic regime for most of the winding conditions with a borderline between hydrodynamic and elastohydrodynamic regimes. The above phenomenon as predicted by the film parameter A contradicts the experimental results which have predicted a predominant elastohydrodynamic regime. The validity of using 40 the film parameter, A (which is simply a ratio of film thickness to the surface roughness), to judge the regime of lubrication is questionable. The film parameter A was calculated in the context of ball bearing lubrication (where the supporting materials are rigid and do not experience significant deformation), whereas in the present case we are dealing with a rubber nip roll and an air entrained wound roll with the air beneath the nip, during winding, exerting a couple of 100 psi. pressure sufficient enough to deform the surfaces in contact and produce a predominant elastohydrodynamic regime (encountered with soft bearing materials) rather than a hydrodynamic regime (encountered with rigid rollers). Tables II, III, IV, V & VI in Appendix compare the results presented in the figures above. a)The Hydrodynamic equation predicts an air layer roughly six times less than the experimental results. b)Hamrock and Dowson's elastohydrodynamic equation slightly underestimates the experimental values at small values of nip load but at higher nip loads, (e.g., 48 lbt) it yields good results. c)For majority of the experimental results Chang's ho prediction values follow reasonably well, except of course for the extreme nip loads. d)Finally, comparing Chang's he (derived using Chang's elastohydrodynamic term) values we observe that they overestimate the experimental results almost by a factor of 2. Also it may quite be possible that we are underestimating the experimental values instead, as a result of air escape even before we can practically measure it in the procedure employed in the experiments. More insight can be had in this particular phenomenon in the next chapter where we study rate of air escape from wound rolls. Since Hamrock and Dowson's elastohydrodynamic equation and Chang's prediction equation's were formulated based on a curve fit technique using dimensionless 41 parameters corresponding to a set of input values, it was necessary to have an idea as to how close the dimensionless parameters dictated by the existing winding conditions were to the one's used to derive the above equations. With the ellipticity parameter, k, held constant, the load, Wand speed, U parameters were the only variables that needed to be compared with the range of dimensionless parameters used by Hamrock & Dowson to obtain the curve fit. Table XIII in Appendix shows a comparison between the range of dimensionless parameters used by Hamrock & Dowson to those associated with the experimental conditions. It is quite clear that the dimensionless parameters U and W corresponding to the practical case deviate far from those that were considered by Hamrock and Dowson to derive their elastohydrodynamic equation (Eq.(35)). Table XIV in the Appendix provides a similar comparison for the range ofU, W and G values used by Chang to curve fit and derive his film thickness equation. It can be seen that the range of values for the dimensionless parameteres dictated by the existing winding conditions fall within the same range of theoretical values used by Chang in his film thickness equation derivation. Thus Chang's model should provide a sound basis of comparison with the experimental results. Experimental Observations Using a Metal Nip To verify the authenticity of the experimental data one of the experiments was repeated using a new nip configuration. The new nip was made of Aluminum, unlike the 42 previous one which was made of rubber thus having different material properties. Two different sets of rolls, one 1500 ft. long and the other 1500 m long were wound at 2000 fpm, 8 lbf nip load and 1000 psi. web tension. Air was collected immediately after the winding operation. Table VII in Appendix displays the results recorded for the same. It was expected that the experiments would yield different results due to different material properties of the nip, on the contrary the values obtained were almost the same as the ones obtained with the rubber nip, for both lengths of the roll. The reason for this can be attributed to the fact that the experiment using the metal nip which was performed for the 2000 fpm winding velocity and 8 lbf nip load combination corresponds to a predominant hydrodynamic regime where the web layers tend to lose contact during winding, also the equivalent modulus corresponding to this case (in fact for all the nip loads used) confirms to the range where it is not affected much by any further increase in the modulus of the nip roll therefore the amount of air entrapped remains unaffected. (This has been proved in the next section, see page 43). During winding at high speeds (e.g. 2000 fpm) and small nip loads (e.g. 4,8 lbt) a comparitively thicker air film develops and exerts a pressure force on both the wound roll and the nip roll sufficient enough for preventing each from coming in contact with the other at times during winding thereby keeping the film thickness unaffected by the nip and the wound roll moduli. 43 Effect of Wound Roll Modulus (Eb) and the Modulus of the Nip (EJ on the Film Thickness In order to determine how important the moduli of the nip roll and the wound roll are for affecting the effective modulus and thus the air film thickness, a sensitivity study was performed on both Hamrock and Dowson's and Chang's Elastohydrodynamic equations (Eq's (35) &(41) respectively). The value ofEa was held constant and Eb was varied to calculate various values of hoand vice versa. The comparisons are made at a reference speed of2000 fpm and 8 lbfnip load (a combination quite frequently used all through the experimental study). It was observed during the calculations that a change in the radial modulus of the wound roll, Eb above a certain value, ceased to have a noticeable effect on the value of the air film thickness hoe Above a certain value for Eb the calculations started yielding a constant value for E', the effective modulus, hence leading to constant air film thickness. This phenomenon was observed in both Hamrock and Dowson's and Chang's equations. The results of these calculations are tabulated in Table XV in the Appendix and plotted as shown in the figure below: 44 + hO for Ham&Dow.,Ea const. 0 hO for Chang, Ea const. .~. hO for Ham&Dow.,Eb const. '~/1 hO for Chang, Eb const. 100.00 .c...:. 90.00 .5 .0.. 80.00 .~ S 70.00 '' ,;; 60.00 'Q".l c 50.00 ...:.:.c. :.c 40.00 Eo ... 30.00 Q.l ;... = 20.00 .....:.l 10.00 < 0.00 0 2000 4000 6000 8000 10000 12000 14000 Modulus of Roll Eb/NipEa, psi. Figure 13. Air Film Thickness (ho) For Hamrock and Dowson's & Chang's Equations, Eq's (35) & (41) respectively as Affected by Wound Roll/Nip Modulus. It is quite evident from Fig. (13) that the modulus of the wound roll and the nip roll radically affect the air film thickness, however only for small values « 1000 psi.) of Ea or Eb• There is a drastic change in the air film thickness (ho) when the modulus of the wound roll is varied below about 1000 psi.. Above about 1000 psi. hardly any change is noticed. A similar trend is observed with the nip roll modulus, Ea also. However it is the wound roll modulus which is affected by the entrained air hence the focus of this discussion would be the wound roll modulus Eb and the radial modulus of the wound roll 45 The values ofEbwere obtained by contact tests performed by Covell [5] using Tekscan which is a PC based instrument for pressure or force measurements. It consists of a force sensitive resistor and an expansion card that allows the PC to record and display nonuniform loads measured. The peak pressure under the nip was measured for different nip loads and was substituted in a theoretical expression ,Eq. (47), derived by Xu [12] for Er contact in the absence of air where 2 3 Er contact = aP + bP + cP (47) a=3138.0 b = 33.54 c=0.151 The calculated radial modulus of the winding roll, Ebwas then used to model the elastohydrodynamic equations. It has to be noted here that Ebwas measured statically, a condition which is very different from the one when the roll is wound at 2000 fpm. Various phenomenon can be encountered in the latter case, one of which that has been mentioned earlier is the possible loss of contact between the nip and the wound roll at high speeds and small nip loadings. In such cases a static determination of Ebis obsolete. It will be shown in the next chapter that air entrainment reduces the modulus of the wound roll radically and that the wound roll modulus is a dynamic quantity, varying as a function of the radius. Hence a radial modulus of the wound roll Er should be used for the modulus of the wound roll. But Er and ho, the air film thickness, are dependent on each other, this makes the determination ofEr or homore complicated since neither can be isolated from the other. Hence a safer approximation would be to use the static values of Ebobtained through contact tests, for 46 air film thickness calculations. The range of values for Eb obtained during the contact tests [5] varied from 8200 psi corresponding to a 4lbfnip load to about 47000 psi for 48 lbf nip load. All of these fall in the range where their variation does not have any noticeable effect on hoe CHAPTER VII RADIAL PRESSURE PROFILE To develop a radial pressure profile we need to know the dynamic values of Ep the radial modulus, which can be formulated simply by a stressstrain relationship. Er, which is a "pseudo material property" can be measured using the INSTRON, by placing a stack of webs and measuring the strain produced by them due to the application of a compressive force. The value stress/strain would then give the modulus, Er • However, in the realm of air entrainment studies Er, cannot be static, since, in the stack tests if an air layer comparable to the one entrained during winding is present then the value of strain would go up, thereby decreasing the value ofEr and vice versa. Thus in practicality we have different air layer thickness at different radii in a wound roll (recollect that the average air layer thickness is only a simplifying assumption ), these differences contribute to the variations in Er at various radii of the wound roll. It was for this reason that different Er values were formulated using a curve fit [10], resulting in a third degree polynomial expression which is: (46) where C1, C2, and C3 are constants derived from stack tests. For Er stack, the constants are: 47 48 C1 == 129.0 C2 == 0.037 C3 == 4.0e4 Applying Eq.(16) for Ereq, yields the following constants: C1 == 60.0 C2 == 0.416 C3 == 0.001 Simialarly, for Erair, application ofEq.(13) yields: C1 == 3.042 C2 == 0.0475 These relationships between Er, the radial modulus and P, the radial pressure can be depicted as follows: 14000 . Er stack 12000 o Er eq. 10000 ~Erair 0 8000 0 Er, (psi.) 0 6000 0 0 4000 o· 0 2000 0 0 0 0 0 0 0 0 0 0 0 0 0 N (V) ~ LO <.0 r" oo 0') 0 ~ P, (psi.) Figure 14. Radial Modulus,Er Computed as a Function ofPressure,P. Once the various expressions for Er are developed we can predict the radial pressure profiles using the software WINDER v 4.0 available at WHRC. This software takes the values of the roll radius, winding speeds, winding tensions, radius of core and 49 its material properties, dimensions of the web in consideration and its material properties and the constants obtained from the stack tests as input and plots the radial pressure profile. Fig.15 depicts the radial pressure profiles obtained using Er stack> Er eq and Er air respectively on a common axes: 1 .1 1.2 1.3 1.4 o t Ai ~ " ~ Amil 1.5 200 180 • Er,stack psi. 160 • Er,eq  psi 140 .. Er,air  psi Radial 120 Pressure, 100 (psi.) 80 60 40 20 RlRc Figure 15. Comparison of Radial Pressure as a Function of Normalized Radius (R/Rc). As is evident from the above figure the entrained air reduces the radial pressure at the midpoint of th roll to 3 psi. when compared with the stack and equivalent values, which are 85 psi. and 55 psi. respectively. Table VIn in Appendix depicts the range of pressure distribution over the radius of a wound roll. Fig. 16 shown below displays a detailed pressure profile for the roll entrained with air, the object our of experimentation, 50 6 I + Er,air  psi I 4 Radial Pressure, 3 (psi.) 2 0++++1++++++++ 1.02 1.06 1.09 1.12 1.15 1.19 1.22 1.25 1.28 1.32 1.35 1.38 1.41 R/Rc Figure 16. Pressure Profile for the Roll with Entrained Air. The above plot represents a typical charecteristic of the pressure profile when the ratio of Et/Er is high. The maximum pressure is present at the core and soon becomes a more or less constant value till it reaches the outer radii of the core,where it sharply decreases to zero for the outermost layer. CHAPTER VIII AIR ESCAPE OVER TIME It has been analytically proven by three different air entrainment models and experimentally proven in this study that as the winding speeds increase the amount of trapped air also escalates. If the amount of air trapped is very large e.g., during high speed centerwinding without a nip roller, it is quite possible that there will not be any interlayer contact of the web material, and the web layers might ride completely upon the air layers, this phenomenon is refered to as aeroplaning. Only at the roll edges, where the air has a good chance of discharging to the atmosphere does the possibility of layer to layer contact exists. This loss of contact between the adjacent web layers would lead to a decreased resistance to the slippage of the web axially, resulting in telescoping and other roll defects. In order to overcome this problem it is important that the underlying mechanism that operates during and after winding and which affects the rate of air entrainment and escape respectively, be studied. Up to this point in the study the former mechanism was studied, and now the study will concentrate on the phenomenon of air escape from rolls and how it affects the integrity of the roll. In order to evaluate the rate at which the air escapes a set of rolls were wound under similar nip loads and at constant web velocities and were unwound at 51 52 various lengths in time. The set of rolls 1500 ft. long were wound at 2000 fpm and a nip load of 8 lbf. The first set was unwound after 24 hours, the decrease in air volume was recorded and the average air layer thickness calculated. It was observed that the air layer thickness had diminished by about 10 Il inch on an average. The other set of rolls were unwound after three days and an additional decrease in air layer thickness by about 5 Il inch was observed. Some interesting observations were made while unwinding these rolls, as compared to the rolls unwound immediately after winding. The rolls that were unwound soon after winding displayed typical characteristics as soon as they were placed in the bubblerimeter, viz., a lot of air bubbles could be seen surfacing the water in the bubblerimeter by the time the conical hood was hooked up for air collection, These bubbles came out at a fast rate, that impeded their estimation. The bubblerimeter has to be calibrated after the roll is placed inside it and the fastest this could be done was in a minute, but a lot of air was seen escaping during this time, which could not be accounted for since the rate of escape was unknown. Further there is a spurt of air escape during the unwinding of the first few web layers, apparently there is a relatively larger amount of air in the outer layers but as the unwinding operation progresses the release of air bubbles becomes more consistent towards the inner layers. It would be safe to generalize that when unwinding a roll immediately after winding, ignoring the air escape from the first few layers of the web unwound, the amount of air collected as a function of time is nearly constant, indicating the consistency of the air layer thickness. Thus it can be assumed that the average air 53 layer thickness is the actual air layer thickness itself. The point to be emphasized here is that immediately after the winding operation the average air layer thickness would give a reasonable estimate of the absolute air layer thickness. The only shortcoming however is the failure of the existing apparatus to trap the air initially. Fig.17 shows the rate of air escape in the bubblerimeter immediately after the winding operation. The volume of air collected is plotted as a function of the length of the roll at various times during the unwinding operation, for two different rolls 1500 ft. and 1500 mts. long respectively. Table IX in Appendix shows the recorded results. 160 • 140 • 120 100 • Volume of 80 Air, (ml.) 60 0 IJ. 40 IJ. IJ. IJ. 20 0 0 1000 2000 3000 4000 5000 Length of Roll, (f1.) Figure 17. Volume of Air Collected at Various Lengths of the Unwinding Roll. In contrast when unwinding a roll after sufficient lengths of time (e.g., 8, 16, 24 hours etc.) the observations made were quite different. As opposed to the fast escape of air bubbles observed in the previous case, very few bubbles could be seen escaping from 54 the roll edges and hence by the time the apparatus was ready to collect the air, hardly any bubbles surfaced and most of them were seen sticking to the roll edges, thus, in this case all the air collected was accounted for. Only when the roll started unwinding could one see the air bubbles rising up, even so at a slow rate. This rate kept decreasing as the roll unwound and there were times when no air bubbles could be seen for considerable lengths of the roll, and towards the end there was hardly any air collected in the apparatus. A series of tests showed that, of all the air that was collected about 60% of it was collected during the unwinding of the first 400 ft. of the 1500 ft. long roll. This phenomenon questions the validity of using an average air layer thickness calculation for rolls kept for significant lengths of time (e.g., 8, 16, 24 hours etc.), because of the lot of air collected, which has been experimentally proven here to be present in the first few feet of the roll, is actually being split up over the entire length of the roll. It is for this reason that 1500 meters of roll unwound after 24 hours might have a smaller average air layer thickness than 1500 feet of it unwound after the same time. Moreover, the time taken to wind a roll 1500 meters long at 2000 fpm is about 2 min. and 30 secs., as opposed to the 53 secs needed to wind 1500 ft. of roll at the same speed. Thus in the former case while the roll builds up the air entrained in the lower layers has more time to escape and thus will escape unaccounted for. Added to this, the builtup of layers exerts additional pressure on the lower layers thereby squeezing out the entrained air more vigorously. Fig.18 shows the volume of air collected in the bubblerimeter when the roll (1500 ft. long) was unwound after 3 days, as a function of the length of the unwinding roll. 55 Also it shows a comparison with similar data obtained for the roll that was immediately unwound. Due to the limited availability of the material a similar comparison for 1500 mts. of roll could not be made. Table X in Appendix shows the recorded results. 70  500 1000 1500 2000 20 10 o.+++1 o • After 3 Days 60  • After Winding Volume of 40 Air, (mI.) 30 _ 50  Length of Roll, (ft.) Figure 18. Comparison of the Volume of Air Collected Immediately and 3 Days After Winding, as the Roll Unwinds. Since the amount of air collected after 24 hours showed a decrease in air layer thickness by 10 Il inch it was expected that after three days the decrease would be even more, but it turned out to be 5 Il inch, which implied that the rate of air escape had slowed down considerably. This led to reducing the time prior to unwinding and measuring the volume of air well before the 24 hour time period. New rolls were wound and unwound again, after 8 and 16 hours respectively. Surprisingly they had roughly the same amount of air that was left in the roll after 24 56 hours. This meant that a lot of air that could have escaped had done so immediately after winding and the remaining was left to discharge at very slow rates, over periods of days. Also it has been mentioned before that rolls placed in the bubblerimeter immediately after winding had a tendency to release a lot of air bubbles at a fast rate. This reinforces the fact that a lot of the entrained air escapes during and immediately after winding. Thus the rate of air escape had to be studied shortly after winding. A series of tests were performed and the air collected at intervals of 1, 3 and 10 minutes after the winding operation which took 53 seconds on an average. By the time the winding operation was terminated and the roll placed in the bubblerimeter ready to be unwound 1 minute and 53 seconds had elapsed. This time has been added to all time intervals after which the rolls have been unwound. Table XI in Appendix displays the data recorded for the above tests and Table XII compares the values derived theoretically from the squeeze film damper expression (Eq. (17), Chapter II) with the experimental values. The data obtained from all these experiments was grouped systematically and analyzed using a spreadsheet program that displayed the rate of decrease of air layer thickness as a function of time as a 2 dimensional plot. The theoretical results from the squeeze film damping equation were also modeled similarly and formed a basis for comparison with the experimental results. Figures 19 & 20 show the rate of air escape as a function of time for the two sets of experiments and a comparison of theory and experiment respectively. 57 100 1000 10000 +Expt. 1 __Expt. 2 10 10 5 0++++1 1 35 30 25 Air Layer 20 Thickness, micro inch. 15 Time in minutes Figure 19. Air Escape as a Function of Time (Experimental Results). +Expt. __Theoret. 10 5 o L+==~~~~I=:e=r=a==_._______1 1 10 100 1000 10000 35 30 25 Air Layer 20 Thickness, micro inch. 15 Time in minutes Figure 20. Air Escape as a Function of Time(Comparison OfTheory and Experiment). 58 Comparison of Theory and Experiment As can be seen in Fig. 20 the theoretical results display a drastic decrease in air layer thickness during the first 5 to 10 minutes after winding the roll. The air layer drops from 30.2 J.l inch at 1min and 53secs to about 5.1 J.l inch at 11min and 53secs. On the contrary the experimental values of the air layer thickness decay from 30.2 J.l inch to 19.2 J.l inch over the same time range, thus displaying a less drastic air escape. In other words, 83% of the air entrapped when winding a roll at 2000 fpm and 8 lbfnip load escapes 10 minutes after the winding operation according to the squeeze film damper prediction, whereas the experiments show the escape to be 36%. Even though the amount of air escape predicted by the theory and experiment fall roughly about 50% apart a promising feature is the similarity in the rates, i.e., as a function of time the experimental plot behaves like the theoretical one. A large amount of air escapes during the first 10 minutes after winding and then we have a plateau for the next 16 to 24 hours, a further decrease is then observed only over periods of days (3 days in the experimental study). The theoretical plot also depicts a similar trend where most of the air escapes during the first 10 minutes after winding and then the rate becomes more or less a plateau for large time periods. Based on the above discussion and a comparison of theory and experiment it can be said that the entrained air is somehow being obstructed on its way out after the winding operation which may not be an ideal case as that assumed by the squeeze film damper equation. 59 The squeeze film damper equation assumes the two contacting surfaces to be perfect. On the contrary the contacting surfaces used in the experiments have some irregularities on a micro inch scale, which are bound to affect the average air layer thickness, which has the same orders of magnitude. Also it is a common observation that when a web is trimmed to its exact width the edges that are slit are sligthly thicker than the web caliper depending on the type of slitting operation employed. The presence of bumps at the roll edges could obstruct the air escape significantly. This has been illustrated by the drawing below, which exaggerates two web surfaces in contact. Further, as the roll builds up these bumps overlap and press relatively harder on each other than the rest of the overlapping layers in the roll. This could possibly magnify the phenomenon of obstruction of entrained air. Figure 21. Bumps at the Roll Edges Obstructing the Escape of Entrained Air. Another factor that could possibly affect the rate of air escape is the nonuniform pressure distribution along the width of the web. It has been proven in previous studies [3] that, under its own weight and the pressure force exerted by the two ends of the roll the film deforms as shown below: 60 Figure 22. Nonuniform Pressure Distribution Along the Web Width. The above illustration explains for itself the bottlenecks created at the two extremities of the roll, which prevent the air from escaping. In fact the pressure profile of the pressure force exerted predicts a smaller value at the center of the roll and a little higher value towards the ends [3]. The two phenomenon explained above seem to have a combined effect on the rate of air escape. The bumps at the roll edges coupled with a higher pressure created there impede the air from escaping at a desired rate. CHAPTER IX SUMMARY AND CONCLUSIONS This study has attempted to verify a winding model that would incorporate the entrained air, and a squeeze film damping equation to model the escape of air over time. It has also verified the decrease in radial modulus due to the presence of entrained air witnessed by Covell [5]. In other words, it has studied the factors which are known to contribute to a decrease in radial pressure during centerwinding with a force loaded nip roller. Based on the experimental results obtained and comparing them with the results obtained from theoretical models the following conclusions can be derived: 1.) The air layer entrained during high speed winding with a nip roller is comparable to the theoretical minimum film thickness (ho) derived by Chang (Eq.(41)). A more practical case would be Chang's he prediction (Eq.(42)) since it considers the expansion of the air film after it passes away from the nip, but the air loss from the roll edges is not considered in deriving he (or any other air film thickness derivations for that matter) hence it is possibly overestimating the results. Further, there is a possible transition from a hydrodynamic regime to elastohydrodynamic conditions towards high nip loads (e.g. 32, 48 lbf). 61 62 2.) Even small nip loads can affect the air layer thickness, so to get reasonable results it is necessary that the nip load be controlled precisely so as to minimize the fluctuations encountered during nip roll bounce. Moreover, the radial modulus of the nip, if it is a hollow cylindrical roller should be used instead of the elastic modulus for estimating the air layer thickness. 3.) The presence of entrained air reduces the radial modulus, Er of the wound roll considerably by decreasing the radial pressure. Further, the radial modulus is a dynamic property of the roll which depends on radius of the roll and air film thickness. 4.) The bubblerimeter can be used as an air collection device as first documented by Covell [5], even though it may not be as sophisticated as laser reflectance probes, since it gives a simple and accurate account of air entrained for nonpermeable (plastic) films. 5.) Most of the air that is entrained escapes during and immediately after winding. 6.) Finally, the air escaping from a wound roll can be modeled using a modified version of the squeeze film damping equation that would take into account the fluid dynamics in between the two web surfaces with irregularities towards the edges. Future Work The following section explores the various possibilities that could be investigated to obtain a much more sophisticated air entrainment model, 63 1.) It has been proven in this study that air is present immediately after winding, a technique to measure the air present in the first few seconds after winding if devised would give a clear understanding of the phenomenon at work during air discharge from the wound roll. The escape of air during the winding operation is another area that needs to be closely monitored and estimated for including the fluid dynamics aspect, which plays a significant role in air entrainment experiments coupled with the solid mechanics aspect. 2.) A procedure needs to be formulated for proper determination ofEb's and Er's since the air film thickness hoand Er are interrelated, and Eb in turn is related to Er. 3.) Applying the equivalent radial modulus to study the air layer, when its value falls between the mean surface roughness and the maximum asperity height of the film. 4.) Measuring the air layer thickness as a function of the roll radius would contribute to modifying the existing models which assume an average air layer thickness throughout the wound roll. 5.) Lastly, the effect of nip roll dynamics on entrained air could also be included in the winding model. REFERENCES 1. Bertram, N. and Eshel, A. "Recording Media Archival Attributes." RADCTR80123, pp. 6873, April 1980. 2. Blevins, Applied Fluid Dynamics Handbook. pp.503. 3. Bouquerel, F. "Theoretical and Experimental Study of Winding of Thin Plastic Films: Aerodynamics Effects." Ph.D Thesis, Central University of Lyon, France. 4. Chang, Y.B. "Air Entrainment With a Force Loaded Nip Roller." Technical Review and Industry Advisory Board Meeting, WHRC., May 1994. 5. Covell, K. Scott "The Effect of a Nip Roller on Entrained Air During High Speed Winding.", Master's Thesis, Department of Mechanical and Aerospace Engineering, Oklahoma State University, 1994. 6. Good, J.K. and Holmberg, M.W. "The Effect of Air Enrainment in Centerwound Rolls." Proceedings of the Second International Web Handling Conference, April 1993. 7. Good, J.K. Wu, Z. and Fikes, M.W.R. "The Internal Stresses in Wound Rolls With the Presence of a Nip Roller." accepted for publication in the ASME Journal of Applied Mechanics, 1992. 8. Hakiel, Z. "Nonlinear model for Wound Roll Stresses." TAPPI Journal, Volume 70, No. 5,pp. 113117,1987. 9. Hamrock, B.J. and Dowson, D. Ball Rearing Lubrication. John Wiley & Sons, pp. 18889 and 28082,1981. 10. Holmberg, M.W. "Theoretical and Experimental Studies of Air Entrainment in Wound Rolls." Master's Thesis, Department of Mechanical and Aerospace Engineering, Oklahoma State University, 1992. 11. Knox, K.L. and Sweeny, "Fluid Effects Associated with Web Handling." Ind. Engr. Chern. Proc., vol. 10, pp. 201205, October 1971. 12. Xu, Y. "Computing Stress Distributions in Centerwound Rolls from Web Surface Charecteristics." Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, Oklahoma State University, 1992. 64 APPENDIX 65 66 TABLE I VOLUME OF AIR COLLECTED FOR TWENTY FOUR INDIVIDUAL WINDING EXPERIMENTS Nip Load Speed Vol. of Air Vol. of Air Air Layer Ibf fpm (uncorrected) (corrected) Thickness ml. ml. mIcro In. 4 44 32.98 18.6 8 35 24.25 13.7 12 500 27 16.49 9.3 16 23 12.61 7.1 32 20 9.7 5.4 48 20 9.7 5.4 4 57 45.59 25.7 8 54 42.68 24.1 12 1000 49 37.83 21.3 16 46 34.92 19.7 32 38 27.16 15.3 48 33 22.31 12.6 4 65 53.35 30.1 8 63 51.41 29.0 12 1500 57 45.59 25.7 16 57 45.59 25.7 32 46 34.92 19.7 48 45 33.95 19.1 4 80 67.9 38.3 8 67 55.29 31.2 12 2000 61 49.47 27.9 16 56 44.62 25.2 32 50 38.8 21.9 48 40 29.1 16.4 67 TABLE II COMPARISON OF THEORETICAL VALUES FROM BERTRAM AND ESHEL'S HYDRODYNAMIC EQUATION WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Deviation lbf fpm Layer Thickness (ho) Layer Thickness mIcro In. micro in. micro in. 4 1.8 18.6 16.8 8 0.9 13.7 12.8 12 500 0.6 9.3 8.7 16 0.5 7.1 6.6 32 0.3 5.4 5.1 48 0.2 5.4 5.2 4 3.5 25.7 22.2 8 1.8 24.1 22.3 12 1000 1.2 21.3 20.1 16 0.9 19.7 18.8 32 0.5 15.3 14.8 48 0.3 12.6 12.3 4 5.2 30.1 24.9 8 2.6 29.0 26.4 12 1500 1.8 25.7 23.9 16 1.3 25.7 24.4 32 0.7 19.7 19.0 48 0.5 19.1 18.6 4 6.9 38.3 31.4 8 3.5 31.2 27.7 12 2000 2.3 27.9 25.6 16 1.8 25.2 23.4 32 0.9 21.9 21.0 48 0.6 16.4 15.8 68 Bertram and Eshel's hydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. L == Nip Load, varied as 4,8,12,16,32 and 48 lbf. for each winding velocity used. U == Winding Velocity, varied as 500,1000,1500 and 2000 fpm. T == Web Line Tension, held constant at 0.48 lbf/in.throughout experimentation. Re == Equivalent Radius, calculated as 1.085 inch. f.l == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 Ro== Radius of Wound Roll, measured as 2.4 inch. an average 1t == 3.1414. ho== Air Film Thickness, from 24 different combinations of velocities and nip loads 69 TABLE III COMPARISON OF THEORETICAL VALUES FROM HAMROCK AND DOWSON'S ELASTOHYDRODYNAMIC MINIMUM AIR FILM THICKNESS EQUATION WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Deviation lbf fpm Layer Thickness (ho) Layer Thickness mIcro In. mIcro In. mIcro In. 4 11.1 18.6 7.5 8 9.8 13.7 3.9 12 500 9.1 9.3 0.2 16 8.6 7.1 1.5 32 7.5 5.4 2.1 48 6.9 5.4 1.5 4 17.4 25.7 8.3 8 15.4 24.1 8.7 12 1000 14.2 21.3 7.1 16 13.4 19.7 6.3 32 11.7 15.3 3.6 48 10.8 12.6 1.8 4 22.6 30.1 7.5 8 20.0 29.0 9.0 12 1500 18.5 25.7 7.2 16 17.5 25.7 8.2 32 15.2 19.7 4.5 48 14.0 19.1 5.1 4 27.3 38.3 11.0 8 24.2 31.2 7.0 12 2000 22.3 27.9 5.6 16 21.1 25.2 4.1 32 18.4 21.9 3.5 48 16.9 16.4 0.5 70 Hamrock and Dowson's elastohydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. where, h U were  ua +ub 2 and F W= E'R2 x where F = Normal Applied Load, lbf (including tesion component T sin 45),(e.g., 4lb nip load plus 2.88 sin 45 etc.) u = Average ofNip and Roll Velocities, varied as 500,1000,1500 and 2000 fpm. T = Winding Tension, held constant at 2.88 lbfthroughout experimentation. Rx == Equivalent Radius, calculated as 1.085 inch. Rn == Radius of Nip, 2 in. Rr == Radius of Roll, 2.4 in. E' = Calculated Effective Modulus, psi. Ea == Modulus ofNip, 675 psi. Eb == Modulus of Roll, (determined by Covell[5] using contact tests) 8200,12800,19800, 27600,37000 and 47000 psi corresponding to 4,8,12,16,32 and 48 lbfnip loads. ~ == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 ho== Air Film Thickness, from 24 different combinations of velocities and nip loads 71 TABLE IV COMPARISON OF THEORETICAL VALUES CALCULATED FROM CHANG'S ELASTOHYDRODYNAMIC EQUATION FOR COMPRESSED AIR FILM WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Deviation lbf fpm Layer Thickness (ho) Layer Thickness mIcro In. mIcro In. micro in. 4 17.1 18.6 1.5 8 11.9 13.7 1.8 12 500 9.6 9.3 0.3 16 8.3 7.1 1.2 32 5.8 5.4 0.4 48 4.8 5.4 0.6 4 28.1 25.7 2.4 8 19.6 24.1 4.5 12 1000 15.8 21.3 5.5 16 13.6 19.7 6.1 32 9.6 15.3 5.7 48 7.8 12.6 4.8 4 37.7 30.1 7.6 8 26.2 29.0 2.8 12 1500 21.2 25.7 4.5 16 18.3 25.7 7.4 32 12.9 19.7 6.8 48 10.5 19.1 8.6 4 46.3 38.3 8.0 8 32.3 31.2 1.1 12 2000 26.1 27.9 1.8 16 22.5 25.2 2.7 32 15.8 21.9 6.1 48 12.9 16.4 3.5 72 Chang's elastohydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. where, 1 where E == 1 y2 1 y2 __a +__b Ea Eb F == Nip Loading per Unit Width of roller, pli (e.g., 4/6,8/6,12/6 pli etc.) u == Average ofNip and Roll Velocities, varied as 500,1000,1500 and 2000 fpm. T == Winding Tension, held constant at 0.48 lb/in. throughout experimentation. Pa == Ambient Pressure, 14.7 psi. Rx == Equivalent Radius, calculated as 1.085 inch. E == Calculated Effective Modulus, psi. Ea == Modulus ofNip, 675 psi. Eb == Modulus of Roll, (determined by Covell[5] using contact tests) 8200,12800,19800, 27600,37000 and 47000 psi corresponding to 4,8,12,16,32 and 48 lbfnip loads. J.l == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 ho== Air Film Thickness, from 24 different combinations of velocities and nip loads 73 TABLE V COMPARISON OF THEORETICAL VALUES CALCULATED FROM CHANG'S HYDRODYNAMIC EQUATION FOR DECOMPRESSED AIR FILM WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Absolute lbf fpm Layer Thickness (he) Layer Thickness Error mIcro In. micro in. mIcro In. 4 2.3 18.6 16.3 8 1.1 13.7 12.6 12 500 0.8 9.3 8.5 16 0.6 7.1 6.5 32 0.3 5.4 5.1 48 0.2 5.4 5.2 4 4.5 25.7 21.2 8 2.3 24.1 21.8 12 1000 1.5 21.3 19.8 16 1.1 19.7 18.6 32 0.6 15.3 14.7 48 0.4 12.6 12.2 4 6.8 30.1 23.3 8 3.4 29.0 25.6 12 1500 2.3 25.7 23.4 16 1.7 25.7 24.0 32 0.8 19.7 18.9 48 0.6 19.1 18.5 4 9.0 38.3 29.3 8 4.5 31.2 26.7 12 2000 3.0 27.9 24.9 16 2.3 25.2 22.9 32 1.1 21.9 20.8 48 0.8 16.4 15.6 74 Chang's hydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. h =.!i !J.URe c 3 L L == Nip Load, varied as 4,8,12,16,32 and 48 lbf. for each winding velocity used. U == Winding Velocity, varied as 500,1000,1500 and 2000 fpm. Re == Equivalent Radius, calculated as 1.085 inch. IJ == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 1t == 3.1414. he == Decompressed Air Film Thickness, from 24 different combinations of velocities and nip loads 75 TABLE VI COMPARISON OF THEORETICAL VALUES CALCULATED FROM CHANG'S ELASTOHYDRODYNAMIC EQUATION FOR DECOMPRESSED AIR FILM WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Absolute lbf fpm layer Thickness (he) Layer Thickness Error he/hO mIcro In. mIcro In. mIcro In. 4 30.8 18.6 12.2 1.8 8 26.6 13.7 12.9 2.2 12 500 24.4 9.3 15.1 2.5 16 23.0 7.1 15.9 2.8 32 20.0 5.4 14.6 3.4 48 18.4 5.4 13.0 3.9 4 50.4 25.7 24.7 1.8 8 43.5 24.1 19.4 2.2 12 1000 39.9 21.3 18.6 2.5 16 37.6 19.7 17.9 2.8 32 32.7 15.3 17.4 3.4 48 30.1 12.6 17.5 3.9 4 67.2 30.1 37.1 1.8 8 58.0 29.0 29.0 2.2 12 1500 53.3 25.7 27.6 2.5 16 50.2 25.7 24.5 2.8 32 43.6 19.7 23.9 3.4 48 40.2 19.1 21.1 3.8 4 82.4 38.3 44.1 1.8 8 71.2 31.2 40.0 2.2 12 2000 65.3 27.9 37.4 2.5 16 61.5 25.2 36.3 2.7 32 53.5 21.9 31.6 3.4 48 49.3 16.4 32.9 3.8 76 Chang's elastohydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. where, and E G== Pa where 1 E== 1 v2 1 v2 __a +__b Ea Eb F == Nip Loading per Unit Width of roller, pli (e.g., 4/6,8/6,12/6 pli etc.) u == Average ofNip and Roll Velocities, varied as 500,1000,1500 and 2000 fpm. T == Winding Tension, held constant at 0.48 lb/in. throughout experimentation. Pa == Ambient Pressure, 14.7 psi. Rx == Equivalent Radius, calculated as 1.085 inch. E == Calculated Effective Modulus, psi. Ea == Modulus ofNip, 675 psi. Eb == Modulus of Roll, (determined by Covell[5] using contact tests) 8200,12800,19800, 27600,37000 and 47000 psi corresponding to 4,8,12,16,32 and 48 lbfnip loads. f.! == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 he == Decompressed Air Film Thickness, from 24 different combinations of velocities and nip loads 77 TABLE VII COMPARISON OF EXPERIMENTAL RESULTS OBTAINED FOR THE RUBBER NIP AND THE HOLLOW ALUMINUM NIP Speed Nip Air collected after one minute after winding* Load winding time: For 1500 ft. == 53 Sees., For 1500mt. == 2Min&30 Sees. Rubber Nip Hollow Aluminum Nip 1500 mts. 1500 ft. 1500 mts. 1500 ft. fpm lbf m!. Jl inch m!. Jl inch m!. Jl inch m!. Jl inch 2000 8 135.8 23.4 55.3 31.2 128 22.1 52.3 29.6 2000 8 125.1 21.6 59.2 33.4     * All data taken/or two different rolls one 1500 mts.and the other 1500ft. long. TABLE VIII RADIAL PRESSURES AS A FUNCTION OF NORMALIZED RADIUS 78 P P P RlRc (using Er stack) (using Er eq) (using Er air) pSI. pSI. pSI. 1 184.1033 111.8518 5.459071 1.032021 154.7302 88.54622 3.216585 1.064448 134.5964 74.32354 3.198831 1.096875 119.5859 67.88147 3.182084 1.129301 107.4788 63.32581 3.166257 1.161728 96.93085 59.50649 3.151277 1.194155* 85.04928 55.73399 3.137076 1.226581 77.19245 51.52646 3.123595 1.259008 66.85479 46.47226 3.11078 1.291435 55.5874 40.1336 3.098582 1.323861 42.92696 31.94269 3.086956 1.356288 28.29851 21.01056 3.07548 1.388715 10.78037 9.783308 2.93404 * Normalized radius corresponding to the midpoint ofthe wound roll. 79 TABLE IX VOLUME OF AIR COLLECTED IMMEDIATELY AFTER WINDING AS A FUNCTION OF THE LENGTH OF THE ROLL UNWOUND* 1500mts. 1500ft. Speed Nip Load Length Vol. of Air length Vol. of Air fpm lbf ft. ml. ft. ml. 1500 63 300 28 3000 100 600 38 2000 8 4500 138 900 46 4920 150 1200 55 1500 64 * Data taken for two different rolls 1500 mts. and 1500ft. long respectively. 80 TABLE X COMPARISON OF THE AIR COLLECTED FROM TWO ROLLS UNWOUND 3 DAYS AFTER AND IMMEDIATELY AFTER WINDING RESPECTIVELY, AS A FUNCTION OF THEIR LENGTHS 3 Days After Winding Immediately After Winding Speed Nip Load fpm Ibf Length Vol. of Air Length Vol. of Air ft. ml. ft. ml. 60 6   124 10   192 13   264 16   300  300 28 340 18   420 21   504 24   592 25   600  600 38 2000 8 684 26   780 28   880 29   900  900 46 984 30   1092 31   1200  1200 55 1204 32   1320 33   1440 34   1500  1500 64 81 TABLE XI AIR LAYER THICKNESS AS A FUNCTION OF TIME FOR THREE SETS OF ROLL SAMPLES Timet Air Layer Thickness (in minutes) (in hours) micro in. X* X*+l X*+3 X*+10 8 16 24 3X24 Sample Roll Number 1 29 25.2 22 17.5 20.8 17.5 16.4 14.2 2 27.4 24.1 22 21.4 16.4 20.8 19.2 12 3 30.2 24.1 20.8 19.2 17.5 17.5 17 14.2 t Data taken for 1500ft. ofweb wound at 2000 jjJm and 8 lbfnip load. *X represents 1min & 53 seconds (the time taken to set up the air collection apparatus). 82 TABLE XII COMPARISON OF THEORETICAL VALUES CALCULATED FROM SQUEEZE FILM DAMPER EQUATION WITH EXPERIMENTAL RESULTS Speed Nip Load Time Air Layer Thickness mIcro In. fpm lbf minutes Theoretical Experimental Deviation 1.9 30.2 30.2 0.0 2.9 14.4 24.1 9.7 4.9 9.1 20.8 11.7 2000 8 11.9 5.1 19.2 14.1 481.9 0.7 17.5 16.8 1441.9 0.4 17 16.6 4321.9 0.2 14.2 14.0 83 The squeeze film damper equation given below was used to obtain the theoretical decrease in the air film thickness as a function of time as tabulated on the previous page t = Ilbe [_1__1] 2W h2 h2 2 1 ~ == Viscosity of Air, 2.6E9 Ibfsec/in2 . P == W/bL == 3 psi. (calculated by Winder v4.0) L == Width ofWeb == 6 in. hI == Initial Air Film Thickness,30.2 micro in. (assumed from experiment) t == Time Period for Decrease of hI to the Next Lower Value, 1,3,10 minutes etc.(from experiment). h2 == Theoretically Calculated Corresponding to the Time Periods Used for Experiment (1,3,10 minutes etc),micro in. 84 TABLE XIII COMPARISON OF THE RANGE OF DIMENSIONLESS PARAMETERS USED BY HAMROCK & DOWSON WITH THOSE ASSOCIATED WITH THE EXPERIMENTS Hamrock & Dowson's Dimensionless Parameters Experimental Dimensionless Parameters Absolute Difference W U W U W U 2.20E04 2.20E03 5.14E09 5.14E08 8.93E03 2.60E02 4.07E04 1.67E05 8.71E03 4.07E04 2.38E02 1.66E05 85 TABLE XIV COMPARISON OF THE RANGE OF DIMENSIONLESS PARAMETERS USED BY CHANG WITH THOSE ASSOCIATED WITH THE EXPERIMENTS Chang's Dimensionless Parameters Experimental Dimensionless Parameters W 6.27E3 0.3135 U 1.304E8 1.63E7 G 6.8 340.1 W 0.0631 0.523 U 1.63E8 6.52E8 G 49.8 53.7 86 TABLE XV INDIVIDUAL EFFECT OF WOUND ROLL MODULUS EbAND NIP MODULUS Ea ON THE AIR LAYER THICKNESS (ho) DERIVED BY HAMROCK AND DOWSON AND BY CHANG FOR THEIR ELASTOHYDRODYNAMIC EQUATIONS Modulus of hofor hofor hofor hofor Roll Eb/Nip Ea Ham&Dow., Chang, Ham&Dow., Chang, psi Ea held const. Ea held const. Ebheld const. Ebheld const. micro inch micro inch micro inch micro inch 100 61.94 90.11 54.71 78.69 500 35.85 49.62 27.38 36.98 1000 30.48 41.58 20.57 27.07 1500 28.40 38.49 17.53 22.74 2000 27.28 36.84 15.72 20.19 2500 26.59 35.81 14.49 18.47 3000 26.11 35.11 13.59 17.23 3500 25.76 34.60 12.90 16.28 4000 25.49 34.21 12.35 15.52 4500 25.28 33.90 11.90 14.91 5000 25.11 33.65 11.53 14.39 5500 24.97 33.45 11.21 13.96 6000 24.85 33.28 10.93 13.58 6500 24.75 33.13 10.69 13.25 7000 24.67 33.01 10.48 12.97 7500 24.59 32.90 10.29 12.71 8000 24.53 32.80 10.12 12.49 8500 24.47 32.72 9.97 12.28 9000 24.42 32.64 9.83 12.10 9500 24.37 32.57 9.71 11.93 10000 24.33 32.51 9.59 11.78 10500 24.29 32.46 11000 24.26 32.41 11500 24.23 32.36 12000 24.20 32.32 12500 24.17 32.28 12800 24.16 32.26 VITA Humair A. Mohammed Candidate for the Degree of Master of Science Thesis: RATE OF AIR ESCAPE FROM ROLLS WOUND AT HIGH SPEEDS WITH A FORCE LOADED NIP Major Field: Mechanical Engineering Biographical: Personal Data: Born in Hyderabad, India, April 20, 1970, the son Mohammed Manzoor Ahmed and Naima Azmath. Education: Graduated from Little Flower High School, Hyderabad, India in June 1985; received Bachelor of Science degree in Mechanical Engineering from Osmania University, Hyderabad, India in June 1992; completed requirements for the Master of Science degree at Oklahoma State University, Stillwater, Oklahoma in May 1995. Professional Experience: Research Assistant, Department of Mechanical and Aerospace Engineering, Oklahoma State University, January, 1994 to March, 1995; Teaching Assistant, August 1993 to December 1993.
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Title  Rate of Air Escape from Rolls Wound at High Speeds with a Force Loaded Nip 
Date  19950501 
Author  Mohammed, Humair Ahmed 
Document Type  
Full Text Type  Open Access 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  RATE OF AIR ESCAPE FROM ROLLS WOUND AT HIGH SPEEDS WITH A FORCE LOADED NIP By HUMAIR AHMED MOHAMMED Bachelor of Science Osmania University Hyderabad, India 1992 Submitted to the Faculty of the Graduate School of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 1995 RATE OF AIR ESCAPE FROM ROLLS WOUND AT HIGH SPEEDS WITH A FORCE LOADED NIP Thesis Approved: ~__ ThesIs Adv~ser . ) ,. ,I / . /) .. /' 'I,/.,/5:;'.:.  ,/' .. /.I<'~~ /.. ... ~ Dean of the Graduate College 11 ACKNOWLEDGMENTS I wish to express my sincere appreciation to my advisor, Dr. James K. Good, for his intelligent supervision, constructive guidance, and invaluable assistance. His guidance and active participation throughout this project have been a tremendous asset. I would like to thank him for providing me with this research opportunity and a generous financial support. I would also like to thank Dr. Richard L. Lowery and Dr. John J. Shelton for serving on my graduate committee. Special thanks are due to Robert Taylor and Ron Markum at the web handling research center, for providing suggestions and assistance during the study. Most importantly, I would like to express my deepest gratitude and appreciation to my mother, Mrs. Naima Azmath who has supported me morally and financially to this day, and to my late father Mohammed Manzoor Ahmed. Finally, I would like to thank my loving wife Safia, for her consistent patience and understanding, and providing me with the moral support needed to complete this endeavor. 111 Chapter TABLE OF CONTENTS Page I. INTRODUCTION 1 II. LITERATURE REVIEW. 4 Hakiel's Model 6 Reduced Radial Modulus 9 Squeeze Film Damper Theory 12 III. AIR FILM THICKNESS MODELS 14 Hydrodynamic Equation 14 ElastoHydrodynamic Equation 17 Chang's Prediction Equation 19 IV. EXPERIMENTAL SETUP...... 23 V. MEASUREMENT OF ENTRAINED AIR 27 VI. EXPERIMENTAL RESULTSSELECTION OF A MODEL FOR AIR FILM THICKNESS 30 Bertram and Eshel's Hydrodynamic Solution 31 Hamrock and Dowson's Elastohydrodynamic Solution 32 Chang's Elastohydrodynamic Solution 33 Experimental Results....................................................................... 37 Comparison of Theory and Experiment 38 Experimental Observations Using a Metal Nip 41 Effect of Wound Roll Modulus (Eb) and the Modulus of the Nip (Ea) on the Film Thicknes 43 VII. RADIAL PRESSURE PROFILE 47 VIII. AIR ESCAPE OVER TIME 51 Comparison of Theory and Experiment 58 IV Chapter Page IX. SUMMARY AND CONCLUSIONS 61 Future Work..................................................................................... 62 REFERENCES 64 APPENDIX 65 v Table LIST OF TABLES Page I. Volume of Air Collected for Twenty Four Individual Winding Experiments 66 II. Comparison of Theoretical Values from Bertram and Eshel's Hydrodynamic Equation with Experimental Results.................................. 67 III. Comparison of Theoretical Values from Hamrock and Dowson's Elastohydrodynamic Minimum Air Film Equation with Experimental Results................................................................................... 69 IV. Comparison of Theoretical Values Calculated from Chang's Elastohydrodynamic Equation for Compressed Air Film with the Experimental Results................................................................................... 71 V. Comparison of Theoretical Values Calculated from Chang's Hydrodynamic Equation for Decompressed Air Film with the Experimental Results 73 VI. Comparison of Theoretical Values Calculated from Chang's Elastohydrodynamic Equation for Decompressed Air Film with the Experimental Results 75 VII. Comparison of Experimental Results Obtained from The Rubber Nip and The Hollow Aluminum Nip.................................................................. 77 VIII. Radial Pressures as a Function of Normalized Radius 78 IX. Volume of Air Collected Immediately After Winding as a Function The Length of The Roll Unwound.... 79 x. Comparison of The Air Collected From Two Rolls Unwound 3 Days After and Immediately After Winding Respectively as a Function of Their Lengths 80 XI. Air Layer Thickness as a Function of Time for Three Sets of Roll Samples......................................................................... 81 Vi XII. Comparison of Theoretical Values Calculated from Squeeze Film Damper Equation With Experimental Results 82 XIII. Comparison of the Range of Dimensionless Parameters Used by Hamrock & Dowson with Those Associated with the Experiments............ 84 XIV. Comparison of The Range of Dimensionless Parameters Used By Chang with Those Associated with the Experiment.................................... 85 XV. Individual Effect of Wound Roll Modulus Eb and Nip Modulus Ea on Air Layer Thickness (ho) Derived by Hamrock and Dowson and by Chang for their Elastohydrodynamic Equation's........................... 86 VII Figures LIST OF FIGURES Page 1. Schematic View of the Bubblerimeter 5 2. Air Layer Between Two Web Surfaces Before and After Winding 10 3. Illustration of a Squeeze Film Damper 12 4. Illustration ofNip Roller Assisted Centerwinding 15 5. Variation of Req as a Function ofRr, the Wound Roll Radius 22 6. Winder Configuration. 24 7. Theoretical Results of Bertram & Eshel's Hydrodynamic Equation 32 8. Theoretical Results of Hamrock & Dowson's Elastohydrodynamic Equation.. 33 9. Theoretical Results of Chang's Equation....................................................... 35 10. Air Film Thickness (he) Away From the Nip as Predicted by Chang 36 11. Chang's he Prediction Using Bertram and Eshel's Hydrodynamic Term 37 12. Experimental Results 38 13. Air Film Thickness (ho) for Hamrock and Dowson's and Chang's Equations, as Affected by Wound Roll/Nip Modulus 44 14. Radial Modulus, Er Computed as a Function of Pressure, P 48 15. Comparison of Radial Pressure as a Function of Normalized Radius (R1Rc) 49 16. Pressure Profile for The Roll with Entrained Air 50 17. Volume of Air Collected at Various Lengths of the Unwinding Roll 53 viii Figures Page 18. Comparison of the Volume of Air Collected Immediately and 3 Days After Winding, as the Roll Unwinds 55 19. Air Escape as a Function of Time(Experimental Results) 57 20. Air Escape as a Function of Time(Comparison of Theory and Experiment). 57 21. Bumps at The Roll Edges Obstructing The Escape of Entrained Air 59 22. NonUniform Pressure Distribution Along The Web Width 60 IX A a b C1,C2,C3 Ea Eb Ec Er Et Er air, Er air Er stack, Er stack Ereq, Er eq E, E' F G 2 g H h Kstack NOMENCLATURE Area of Entrained Air per Unit Width (in.2 ) Semiaxis in Transverse Direction of Line Contact (in.) Semiaxis in Motion Direction of Line Contact (in.) Polynomial Coefficients Modulus of the Nip (675 psi.) Modulus of the Wound Roll (psi.) Modulus of the core (30XI06 psi.) Radial Modulus of the Wound Roll (psi.) Tangential Modulus of the Wound Roll (psi.) Modulus of the Web with the Entrained Air (psi.) Modulus of the Web as a Stack of Layers (psi.) Combined Stack and Air Radial Modulus (psi.) Effective Modulus (psi.) Normal Applied Load (including the tension component) RMS of ball, race (RMS of the web substituted for both) (in.) Dimensionless Materials Parameter Ratio of Tangential Modulus to Radial Modulus of the Wound Roll Air Layer Thickness Thickness of the Web (in.) Air Film Thickness After Expansion (in.) Minimum Air Film Thickness (in.) Air Layer Thickness (in.) Stiffness of Air Layer Entrained (lbf/in.) Stiffness of a Stack ofWeb Layers (lbf/in.) x Pa Po Rx' Re Rn,Rg Ro, Rr t T u u W x v Combined Stiffness of Stack and Air Layer (lbf/in.) Ellipticity Parameter (k = 10 for line contact) Load per Unit Width to Overcome Air Entrainment effects(lbf/in.) Atmospheric or Ambient Pressure (psi.) Pressure Beneath the outer Layer (psi.) Effective Radius (1.085 in.) Radius of the Nip (2.0 in.) Radius of the Wound Roll (2.3715 in.) Time (sees.) Web line Tension (0.48 lbf/in.) Winding Tension (1000 psi.) Dimensionless Speed Parameter Winding Speed (in./sec.) Average ofNip and Roll Velocities (in./sec.) Velocity of the Nip (in./sec.) Velocity of the Wound Roll (in./sec.) Dimensionless Load Parameter Radial Compression of Entrained Air Layer During Winding (in.) Radial Strain (in./in.) Tangential Strain (in.lin.) Dimensionless Film Parameter Dynamic Viscosity of Air (2.6Xl 09 Ibfsec./in.2 ) Poisson's Ratio Poisson's Ratio of the Nip (0.4) Poisson's Ratio of Roll (0.01) Poisson's Ratio in the Radial Direction Poisson's Ratio in the Tangential Direction Stress in the Radial Direction (psi.) Stress in the Tangential Direction (psi.) Xl CHAPTER I INTRODUCTION Thin plastic films which are widely used for a variety of industrial applications such as packaging, magnetic recording etc. are usually wound in spirals to form a roll. In general any film that can be wound into a roll is called a web. The film may be paper, cloth or plastic. Perhaps the most widely used films today are plastic films used for a variety of applications ranging from packaging to magnetic recording media. The most commonly used polymers are polyethylene, polypropylene, cellulose acetate and polyethylene terephthalate. These films are wound by a variety of winding techniques, the most common being centerwinding. In centerwinding we have a core that is subjected to a torque. A motor provides torque to the core and controls the speed of winding. In centerwinding, we may have a layon roller whose function is mainly to squeeze the unwanted excess air entrained between two adjacent layers as the roll builds up. At low winding speeds an increase in the woundintension has been documented without being forced to increase the web line tension. This layon roller is sometimes called a nip roller. Alternately we have surface winding in which the torque is provided to a layon roller, which applies a 1 2 normal force to the core and drives it as a result of contact friction, at a speed at which it is driven by the motor. Trapping too much air between layers is highly undesirable because it prevents the interlayer contact, resulting in slippage in the axial direction, more commonly referred to as "telescoping". Air entrainment radically decreases the stress state within a wound roll thereby adversely affecting the integrity of the roll structure during storage and shipment. Air entrainment can be reduced by increasing the web line tension during winding but this is much less effective than the use of a nip. In addition, winding a roll too tightly may result in defects such as wrinkling and starring, therby degrading the quality of the roll. Thus it is important for the wound roll to have an optimum stress level, neither too low (e.g. as a result of air entrainment) so as to cause telescoping, nor too high so as to degrade the roll quality. A variety of factors need to be considered to assess the state of stress in wound rolls. The most relevant factors are web tension, radius of wound roll, presence of entrained air as well as the material properties of the web in consideration. A factor that affects the amount of entrained air and could thereby appreciably affect the stress state in wound rolls is the velocity at which the roll is wound (i.e. the "winding velocity"). Winding velocity is to be considered to take into account the amount of air that is carried along the web during winding, to be trapped subsequently in the adjacent layers. Rolls wound at higher velocities are prone to entrain more air than those at lower velocities. To increase the efficiency of a winding operation it is necessary to incorporate higher winding velocities and to reduce the effect of air entrainment. A nip roller serves 3 mostly to accomplish the above by allowing us to keep the winding velocity high while keeping the entrained air thickness low. This study seeks to investigate the air entrainment rates during high speed winding, and how these would affect the physical properties of the wound roll. Furthermore it will examine the rate at which the entrained air discharges from the wound rollover a period of time. CHAPTER II LITERATURE REVIEW Air entrainment experiments in centerwound rolls with a lay on roll (nip roller) were done and analyzed by Covell [5] in his thesis "The effect ofa nip roller on entrained air during high speed winding". Covell performed experiments in wich he collected the residual air which had been entrained at various web velocities and nip loads. He compared his results to a Hydrodynamic relationship which was derived by Bertram and Eshel [1] and an ElastoHydrodynamic relationship derived by Hamrock and Dowson [9]. Covell's experimental data correlated best with the Hydrodynamic Equation. Previously, related work on air entrainment was done by Bouquerel [3]. In his thesis "Theoretical and experimental study ofwinding ofthin plastic films: Aerodynamic effects", Bouquerel attempted to experimentally verify Hamrock and Dowson's [9] elastohydrodynamic equation. But due to inadequate correlation between the theoretical and experimental values he had to introduce a correction factor. He calculated the apparent density of the roll based on its mass and external diameter and finally deduced the air layer thickness assuming incompressible film of known density and that the density of air is negligible as opposed to the web material (Polyester in his experiment). 4 5 To measure the entrained air accurately a direct measuring technique was employed by Covell [5]. A bubble collection apparatus, also called as the "Bubblerimeter" (See Fig. 1), was developed. It consists of a rectangular water tank fixed with a spool and a 45° dead bar. The wound roll is placed on the spool and the web is directed out of the water tank to the winding machine by means of the 45° dead bar. The wound roll placed in the apparatus is subsequently unwound at a low speed, thereby ensuring all the air bubbles are trapped in the conical hood placed directly over the roll on side supports fixed inside the tank. The amount of air that is collected inside the hood is quantified by collecting it into a graduated cylinder fixed to the top of the hood. WATER LEVEL 45 DEGREE IDL WATER LEVEL ~Tonili WINDER ENTRAINED AIR BUBBLES 0 AMOUNT OF AIR COLLECTED AT AN INSTANT ___~ GRADUATED CYLINDER CONICAL HOOD SPOOL UNWINDING ROLL Figure 1. Schematic View of the Bubblerimeter. Covell conducted his experiments on 48 gage Type 442 Polyester, 6 inches wide and 1500 m long, at a winding tension of 1000 psi, and at various velocities and nip 6 loads. The experimental results obtained were compared with theoretical values from Hamrock and Dowson's [9] Elastohydrodynamic equation as well as Bertram and Eshel's [1] Hydrodynamic equation. Using a multiplicitive correction factor a modified version of the Elastohydrodynamic equation seemed to give a close approximation for the air layer thickness over the tested range of winding conditions. However, when compared without any modification his experimental values were closest to the theoretical results derived from Bertram and Eshel's [1] Hydrodynamic equation. Before discussing the algorithms that can be used to model the entrained air it is necessary to first understand the basic winding models that deal with the stress distribution in wound rolls and second how these stresses are affected by the presence of entrained air. One such classical model was presented by Hakiel [8] which has since been modified by Good, Wu and Fikes [7] to include a nip roller. The Hakiel model as well as the stresses in the wound rolls due to the presence of entrained air are presented briefly in the following pages. Hakiel's Model Hakiel [8] in 1987 presented a winding model which applied a finite difference method to solve a second order nonlinear differential equation in radial pressure. Young's modulus in the radial direction was made a function of radial pressure. The nonlinear orthotropic hoop model developed by Hakiel assumes: 7 1.) The wound roll to be a cylinder made by winding concentric hoops of web, and that the properties of the roll as each hoop is added are constant. 2.) The roll has a linear elastic behavior in the tangential direction and a non linear elastic behaviour in the radial direction which varies as a function of radial stress. 3.) Plane stresses which are functions of the radial positions only and axial stresses are zero. Hakiel begins with the equilibrium, constitutive and compatibility equations respectively in cylindrical coordinates: r(d;r )+O'r O't =0  Equilibrium Eq. (1) The linear orthotropic constitutive equations for radial and tangential directions are E =(_1)cr _(Vrt )cr r ErE t r t and (2A & 2B) Et=(~t )O't ( ~: )O'r respectively. Using the strain energy constraint, vtrEt = vrtEr and defining, g2 =~, we have Er and      Constitutive Eqs. (2A' & 2B') Finally from the linear definitions of strain in cylindrical coordinates we have r(dd:t )+ ~  Er=0  Compatibility Eq. (3) 8 Combining Eqs. (1) (2) & (3) yields a secondorder differential equation in terms of radial pressure as (4) This equation is solved several times during the execution of the model such that g(r) and the boundary conditions can be updated. Thus the model is solved for increments in pressure, which are summed to 8(P(r)) as shown in Eq.(5). As the roll builds up so does the pressure, assuming 8P as the incremental change in pressure at radius r, Eq.(4) can be modified as 2(d 2 r 8P) +3r(d 8P)  (g2 1)8P = 0 dr2 dr (5) The above is a second order differential equation with nonconstant coefficients and is subject to two boundary conditions, the first of which is obtained by equating the radial deformation of the first layer with the deformation of the core. In terms of the variation of pressure, 8P, due to the addition of the last layer this boundary condition becomes: 88P I =(~  1+ V)8P 8r r=1 Ec r=1 and the second is given by the hoop stress formula as After each solution ofEq.(5) the new increment in pressure 8P(r) must be summed with all previous increments to yield per) or (6) (7) n p. =~8P.. 1 L...J IJ j=l 9 (8) where Pi is the pressure in a certain radial sector and Eq.(5) would be solved n times. Solving the above boundary value problem numerically, Hakiel [8] modeled the radial stress distribution inside the wound roll. This model has been modified by Good, Wu and Fikes [7] in "The Internal Stresses in Wound Rolls with the Presence ofa Nip Roller ". Reduced Radial Modulus A reduction in radial modulus is one way in which air entrainment may be incorporated into Hakiel's model. To determine the radial modulus as a function of the air layer thickness, a radial modulus of air between the layers must be determined. Good and Holmberg [6] in "The Effect ofAir Entrainment in Centerwound rolls", describe a relationship for the radial modulus which is reduced by the entrained air. Assuming that the trapped air is an ideal gas under isothermal conditions, we can apply Boyle's law as follows (9) 10 where, hoand hI are the initial and final air layer thickness, Po and Pa are the initial pressure between the layers and atmospheric pressure respectively, and P is the pressure applied to compress the layer from hoto hI' as depicted in the illustration below: Web Layers Pa P+Pa ho Pa+Po Air Layer as the Web is Wound Initially x Air Layer as the Roll Builds up During Winding Figure 2. Air Layer Between Two Web Surfaces Before and After Winding. From Fig. 2, hI = ho x Eq. (9) can thus be written as (10) (11) Solving for x and dividing both sides by howe get a pseudo expression for the radial strain of the trapped air layer as (12) Inverting Eq. (12) after taking its derivative with respect to the radial pressure yields E . = (O'r+ po+ Pa)2 raIr (Po +Pa ) (13) 11 The modulus of the entrained air and that of the stack of web layers can be modeled as springs in series whose equivalent stiffness can be written as 1 1 1 =+ Keq Kstack Kair In terms of Er the above equation becomes 1 1 1 ( EreqA) = (ErstackAJ + (ErairA) ho+h h ho Simplification yields (14) (15) (16) Er stack is a material property and can be obtained from a material testing system. It is imperative to mention here that the difference between air layer thickness and the surface roughness of the film determine which one of the above expressions is applicable. For the winding conditions three possibilities may exist, 1) If the air layer is less than the mean surface roughness of the film the air layer is not a factor since the asperities contact the web surface and hence Er need not be modified. 2) If air layer is less than the maximum asperity height but greater than the mean surface roughness of the web then a combination of Er air and Er stack i.e., Er eq (Eq. (16)) should be used. 3) If air layer thickness is greater than the maximum asperity height then the expression for Erair (Eq. (13)) should be used. 12 Squeeze Film Damper Theory The following section is devoted to a latter part in this study which deals with the escape of entrained air over time from wound rolls. The theory developed herein would be used to model the rate of air escape. Blevins [2] in Applied Fluid Dynamics Handbook describes a squeeze film damper as shown below w Air Escapes i~_=±=""""",~~"""""",,,,,,,,,,,,,,,,,,,,,,,,,,,,"~~,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,", This Direction No Air Escape in This Direction Figure 3. Illustration of a Squeeze Film Damper. The bottom plate is assumed to be fixed and the upper one is allowed to fall down freely under its own weight W, and that the air present between the two plates squeezes out only along the width, L. No air escapes along the length, b. Assuming that the upper plate is falling down from a height hI to h2 then an expression for the time needed to close the gap from hI to h2 can be written as: t = /lbe [_1__1] 2W h2 h2 2 1 (17) 13 where, J.l is the dynamic viscosity of air, 2.7* 1091bf  sec I in.2 L is the dimension parallel to which air escapes, in. b is the dimension perpendicular to which air escapes, in. W is the weight of the plate, lbf. t is time in seconds required for the air layer to decrease from hI to h2 • The above equation can be used to predict air escape over various time periods, assuming the two plates to be a pair of overlapping web layers present in a wound roll. The roll width can be substituted in place ofL and the ratio W/bL can be simply substituted knowing the value of the radial pressure P, since P == W/bL. CHAPTER III AIR FILM THICKNESS MODELS Hydrodynamic Equation Bertram and Eshel [1] in "Recording Media Archival Attributes" published a formula in 1980 which relates the nip load to air layer thickness and winding velocity, viscosity, tension and geometry. Their experimental procedure involved an indirect method of measuring air layer thickness, wherein pull tabs were used to measure interlayer pressures and quantifying interlayer slippage during rapid deceleration to give an estimate of entrained air (The back emf generated by the decelerating roll was related to the inertia of the roll which changes with the amount of air entrapped). However their experiments do not check the validity of their formula. The present study starts off by verifying the accuracy and applicability of the Hydrodynamic Equation to the actual winding conditions using centerwinding with a nip roller. The Hydrodynamic Equation has been derived using the pressure distribution on the nip (See Fig. 4) which is governed by Reynolds equation: 14 dp dx with boundary conditions, h* x2 1+ ho 2RehO [1+ 2::hOJ at x= 0 at x = 00 (18) 15 (19) (20) where h* is the thickness of the air gap at which dp/dx = o. T , Web line tension Load to overcome air film lift, L Nip Roller ~~ liRe = lIRo + lIRg h = ho + X"'2/2Re Lt , Load to overcome web tension T Figure 4. Illustration ofNip Roller Assisted Centerwinding. 16 Integrating for pressure distribution and subsequent simplification yields the nip load required to overcome the air entrainment effects as: 00 L= f pdx o This equation does not solve for hoexplicitly instead it solves for the load (21) (22) required to overcome air entrainment effects. Thus to find hoat a given value ofL we should initially have a spreadsheet program that gives us the value of L for various values ofho. It should be noted that the second term of Eq.(22) is minute when compared to the first term, thus, (23) Assuming the expansion of entrained air at ambient temperature after it passes away from under the nip the thickness of the air film (he)' due to a change in the velocity profile, becomes 4ho/3 as reported by Chang [4]. Thus Eq. (23) can be modified to include compressibility effects as follows: Hence, h ~~/!URe c 3 L (24) (25) 17 Elastohydrodynamic Equation Hamrock and Dowson [9] in "Ball Bearing Lubrication" present a formula to calculate the film thickness between two bodies in elliptical contact. However, this equation can also be extended for use with two bodies in line (rectangular) contact as referenced in their study. Their elliptical contact theory is composed of 11 critical parameters. From these parameters they established 5 dimensionless groupings which can be related to the film thickness as follows: H= f(k,U,W,G) (26) where, k represents an ellipse of deformed contact with a major axis of dimension 2a and a minor axis of dimension 2b. For line contact, U is the dimensionless speed parameter which can be defined as (27) where (28) W represents the dimensionless load parameter and can be written as F W= E'R2 x 1 where Rx = 1 1 +R n Rr (29) and G is a dimensionless material parameter which is represented as E' G= 1/ a where E'= 2 1 y2 1 y2 __a +__b Ea Eb (30) 18 a is the pressureviscosity coefficient of lubrication (m2IN). Note that for W to be dimensionless F must have units of load (lbf or N) in Eq.(29) After the dimensionless parameters were grouped, each was varied keeping the rest constant to observe each parameter's influence on the film thickness, and the proportionality equations obtained were subsequently formulated each depicting its influence on the film thickness. hO ex:: (1 0.85e0.31k) Rx For k == 10, we have (31) hO ex:: 0.962 Rx h _0 oc UO.65 Rx (32) (33) (34) It is important to note here that G, the material property parameter did not affect ho/Rx for the range of low elastic modulus materials in Hamrock and Dowson's study. Hence G is missing in the film thickness equation which is now presented. Hamrock and Dowson set reasonable ranges of the dimensionless parameters W and U for various types of oil and grease lubricated bearings. They then solved coupled sets of Reynold's and Elasticity equation's to yield the film thickness, ho. From the results obtained a leastsquares curve fit was used to develop a formula, for film thickness ho as a function of the dimensionless parameters, called as the ElastHydrodynamic Equation which is: (36) (35) 19 ho= R x 7.43(10.85e0.31k)UO.65WO.21 and for our case where k = 10, the above equation reduces to: ho=R x 7. 43Uo.65WO.21 Since the Elastohydrodynamic Equation is derived using a curve fit, its applicability may be governed by the range of values for U and W used during the curve fit process. The range of values for U and W used by Hamrock and dowson for deriving the film thickness equation (Eq. (35)) have been presented in Table XIII in the Appendix. Before proceeding with a comparison of the theoretical models it would be appropriate to introduce a few shortcomings of the Elastohydrodynamic equation, 1) This equation was derived from a curvefit technique that was performed on a finite domain of dimensionless parameters. 2) The application of this equation to two cylinders in rolling contact is not clear even though it has been referenced in its study. Chang's Prediction Equations Chang [4] reported an air film thickness formula developed using a procedure similar to the one used to develop the Elastohydrodynamic Equation of Hamrock and Dowson, however it takes into consideration the compressibility effects of air. According to this theory the air film while passing under the forceloaded nip roller gets compressed but expands subsequently as it passes away from it. Chang has presented a set of analytical models for soft and hard materials with negligible and significant 20 compressibility effects. The present study falls into the category of soft materials since the nip used is made of rubber and since there is air entapped within the wound roll. Chang has grouped three dimensionless parameters viz., speed, load and material properties respectively as under: where, H= f(U,W,G) (37) (38) (39) E G= Pa 1 where E = 1 y2 1 y2 __a +__b Ea Eb (40) Note that for W to be dimensionless F must have units of load per unit width (pli or N/m) in Eq.(39) A procedure similar to that of Hamrock and Dowson's was employed by Chang to predict the individual dependencies of the dimensionless parameters (The range of values for U, Wand G used by Chang for deriving the film thickness equation have been presented in Table XIV in the Appendix) on the air film thickness hOe Finally a curve fit was done which in his case showed a perfect correlation with the numerical values. Based on this he defined the air film thickness hoas follows: (41) In his study Chang has also developed a relationship for an air film thickness, he which is the expanded thickness of the air film after it passes away from under the nip where it 21 gets compressed. The dependencies of the U, Wand G parameters for this case have been formulated as under: (42) Eq.'s (41) & (42) are to be used for cases where the effective modulus (equivalent modulus of nip and wound roll) ranges from 700 psi. to 5000 psi. For cases where the effective modulus ranges between 100 psi. and 700 psi. the above equations have been modified by Chang as follows: h o == R x 4. 8UO.66W0.35G0.47 he == Rx7.4UO.66WO.21G0.33 (43) (44) The three theoretical air entrainment algorithms described above will be used as a basis for comparing the results obtained experimentally. The one that can closely predict the air film thickness will be used to model the experimental values. Although the value of the air layer thickness as implied from the equations above appears to be a constant, actually it is a function of the radius of the winding roll which keeps increasing till the end of the winding operation. A change in the wound roll radius Rr will change the value of Req in the calculations for the air film thickness thereby affecting the value of hOe In an attempt to assess the variation of Req, various values ofReq were determined at various wound roll radii and the results plotted as shown in the figure below: 22 Req corresponding to Rr for 1500 mt. of roll 1.4 1.35 ,... .s 1.3  ~~ 1.25 ~ ~ 1.2 =:a 1.15 ~ ~ ..... 1.1 =~ ~ 1.05 .i:: = 1 ~ ~ 0.95 0.9 1.6 Req corresponding to Rr for 1500 ft. of roll 2.1 2.6 3.1 3.6 Wound Roll Radius, Rr (in.) Figure 5. Variation ofReq as a Function ofRr, the Wound Roll Radius. For the present study the length of the web selected for winding was 1500 ft, referring to Fig. 5 it can be observed that for this length of the roll there is a very little change in the value ofReq. Req starts off at .92 inch and rises to only 1.09 inches for 1500 ft of roll wound. Even for a 1500 m (4921 ft) long web (selected by Covell[5]) the value of Req rises to only 1.27 inches. Thus, the simplifying assumption of an average air layer thickness will be a sufficient approximation for the air layer thickness, throughout the roll. After selecting an algorithm for the air layer thickness, ho, it will be used to develop an equation for a reduced radial modulus which in turn may be used to predict the radial pressure profile using Hakiel' s [8] winding model. CHAPTER IV EXPERIMENTAL SETUP Precise measurement of entrained air is a crucial factor in air entrainment studies and in the subsequent winding model development. Measurement of entrained air has posed a subtle problem in the past. Researchers have relied on indirect techniques for measuring entrained air e.g., by using pull tabs for measuring interlayer pressures and deducing air layer thickness thereon and relating air layer thickness to interlayer slippage during rapid deceleration[I], or by using the apparent density of the wound roll to estimate the air layer thickness[3]. In contrast Good and Holmberg [6] in "The Effect of Air Entrainment in Centerwound Rolls", have employed a sophisticated technique for measuring air layer thickness directly using laser reflectance probes. Using an expression for air layer thickness derived by Knox and Sweeney [11] they were successful in correlating the theoretical and experimental results. However, this technique was limited to the cases where the entrained air layer thickness was greater than .0001 inch which is quite small for air films wound in to centerwound rolls but large for air air films wound into centerwound rolls with a nip. In this study "Air Collection" has been accomplished by a direct technique which can be used irrespective of the surface roughness of the web material. The idea is simply 23 24 to unwind a wound roll under water, trap all the air bubbles that are released and quantify them to give a measure of the entrained air. The apparatus used to accomplish this is the "Bubblerimeter" which has been described briefly in Chapter II (See Fig. I). Another factor that needs close observation during centerwinding with a nip roller is the nip load, because, even small nip loads can significantly reduce the amount of air entrained. It is important that the nip should be able to deliver a constant force throughout the winding operation to get a good estimate of the air layer thickness during air entrainment studies with a nip roller. This is especially important when the nip loads used are small because phenomenon such as nip roll bounce etc., get amplified leading to a bad estimate of the air layer thickness. Thus, we have to ensure that the nip roll and the wound roll remain in contact, and the nip load is held constant throughout winding. It was for the above mentioned facts that before proceeding with the air entrainment studies a new nip was designed on the existing winder, a schematic of which is shown below: Nip Roller Wound Roll Idler /' Tension Sensor Unwound Roll Tachometer Figure 6. Winder Configuration. 25 The new nip was designed to operate pneumatically. It was held in position exactly parallel to the wound roll by means of two arms extending on either sides, of which, one extreme (top one) held the nip, and the other (bottom one) an idler. This idler was used to direct the web from the tension sensor to the nip, the web after embracing the nip passes on to the wound roll. Since the nip arms were long and flexible in bending another idler was added in the center as a reinforcement, which later on served to redirect the web from the "Bubblerimeter" to the winder. With the existing setup it is possible to perform surface winding by adding a belt, that would drive the nip instead of the wound roll shaft. After the nip was designed it was configured with the existing winder. The next step was to install a mechanism that would deliver a constant force to the nip arms, which in turn would be delivered to the wound roll during winding. The response of the nip to the applied load had to be sensitive, especially when the loads applied were small. In other words, the mechanism that operates the nip should have the least reaction forces, which otherwise would oppose the nip force and subsequently diminish it. There were two possibilities, one was to use a dead weight to apply the nip load and the other was to use a pistoncylinder arrangement that would drive the nip pneumatically. The former option was ruled out in lieu of the high winding speeds needed for the experimentation, and the fact that nip roll bounce would make the dead weights almost weightless. This setup was used by Covell[5] and he reports of nip roll bounce problems associated with it. So, the use of a pistoncylinder arrangement was studied. One disadvantage however, was the traction force inside the cylinder due to the 26 presence of friction. This had to be minimized in order for the nip to deliver small loads consistently. After examining the working of a variety of pistoncylinder arrangements a specially designed mechanism that works with a minimal traction force was selected. This is manufactured by Origa Corp. has a stroke of eight inches and delivers a maximum force of 65 lbf at a pressure of up to 100 psi which is controlled using an air pressure regulator. The frequency response of air regulators can easily be less than 1 Hz. A wound roll with but one abnormality in its surface can provide a 150 Hz input to the nip roll. Hence, in order to improve its performance a 2 cu.ft. air tank was used to store and deliver the regulated air pressure to the cylinder. The advantage of such a configuration is, we have a large volume of compressible air whose pressure is not easily affected due to nip roll bounce during high speed winding. This is much superior to connecting the air cylinder directly to an air regulator, due to the poor frequency response of air regulators and to a relatively small volume of air (e.g., typically only that in the air supply tubing) which can be compressed. Thus,with this design the nip load will be as constant as possible throughout the winding operation. The minimum force that can be delivered is 2 lbs and 0.5 lb (0.083 pli for a 6 inch wide roll) increments have accurately been resolved using a force gage. With the nip design complete, a 5 hp AC vector drive was added to drive the winder to speeds in excess of 2000 feet per minute which were controlled by a digital controller with an accuracy of±10 feet per minute. CHAPTER V MEASUREMENT OF ENTRAINED AIR This study comprises of two parts, the first one attempts to verify a winding model that incorporates the entrained air immediately after winding. The second part deals with the discharge of entrained air over a period of time and attempts to test the applicability of a "Squeeze Film Damping Equation" [See Eq. (17), Chapter II] to estimate the rate of air escape from the wound roll. Thereafter a suitable relationship can be developed or the existing one modified to correlate the experimental results with the theoretical values predicted by the "Squeeze Film Damping Equation". For performing air entrainment tests Type 442 Polyester Gage 48, 6 in. wide, 1500 ft. long was used. It has a mean surface roughness of 0.27 J..l in., an RMS surface roughness of 0.42 J..l in., and a maximum asperity height of 0.78 J..l in.. The tangential modulus of the web is 600,000 psi.. The web was wound on a 3.375 inches O.D. steel core with an I.D. of3 inches. A typical O.D. for a 1500 ft roll was 4.7 inches and for the 1500 m rolls which were wound the O.D. was 7 inches. The reason for using polyester was the need for a nonpermeable web material that would not allow the air to diffuse in the radial direction, so that the air that would escape could only do so through the edges of the rollout into the atmosphere. Further, 27 28 the thickness of the entrained air layer should be greater than the mean surface roughness of the film, otherwise the air layer would not be a factor at all in the experiments (See Chapter II, Reduced Radial Modulus). For this reason the film chosen for this study has a very low surface roughness in order to ensure that the air collected would produce an average thickness of more than the surface roughness of the web. The experimental procedure involves winding 1500 ft. of web at various web line velocities and various nip loads at a constant winding tension of 1000 psi. and unwinding the roll in the bubblerimeter. The entrained air collected in ml. is then converted into an average air layer thickness throughout the roll by the following relation: h = O.06102(Vol. of air in ml.) o 1500*12*6 It has to be noted here that h 0 the air layer thickness is an average air layer (45) thickness and not the absolute thickness of the air entrained between each layer. The latter is dependent upon the radius of the wound roll and is not constant for each pair of overlapping layers. Thus the simplifying assumption of an average value makes other calculations less complex and avoids the present study from wandering off into an entirely different aspect of air entrainment which might need the measurement of the volume of air extracted as a function of radius of the wound roll. All materials have a finite surface roughness. Even if we use a material with very small roughness values it is unlikely that a pair of contacting layers would do so perfectly. There would always be some amount of air trapped in between their asperities. 29 This air however, would be present under any circumstances irrespective of winding conditions. Even with very large nip loads and low web velocities, conditions that are favorable for flushing out maximum air from between the two contacting faces, there always will be some trapped air called as the "baseline" amount of air. This baseline air should not be included in the same category as the entrained air, hence, from all the experimental air collected the baseline amount will be deducted to avoid bias. In an experiment to measure the baseline air 1500 ft. of roll was wound at a speed of 20 fpm and a 32lbfnip load (conditions favorable to flush out maximum air) and was subsequently unwound in the bubblerimeter. It was found that the baseline amount was about 10 ml, an average for two roll samples. Further an additional correction factor of 3% has been introduced to account for the slight vacuum present in the graduated cylinder of the bubblerimeter. This factor would also be subtracted from all experimental data along with the baseline amount. Table I in the Appendix displays the air collected from 24 individual winding experiments. CHAPTER VI EXPERIMENTAL RESULTS  SELECTION OF A MODEL FOR AIR FILM THICKNESS Experiments were performed at four different winding speeds, ranging from 500 fpm to 2000 fpm, and air was collected for six different values of nip loads corresponding to each speed. Since the intent of this study is to show how the air layer thickness would be affected even with small nip loads, a major portion of the discussion would be done for small nip loads. However the data obtained at higher nip loads would also be presented and discussed briefly. The results of the experiments were coded into a spreadsheet program and three dimensional surface plots were obtained. These plots show how the air layer thickness varies as a function of web speed and nip load. Theoretical results were obtained by substituting the material properties, winding conditions and other relevant parameters into the Hydrodynamic and ElastoHydrodynamic Equations respectively. The parameters used to model these equations can be found in the Nomenclature. The theoretical results were also plotted against the corresponding speeds and nip loads used for experimentation, and a comparison was made. Tables II, III, IV, V & VI in the Appendix 30 31 Appendix display the data calculated from the theoretical models and compare it with the experimental results. Bertram and Eshel's Hydrodynamic Solution The Hydrodynamic Equation predicts an exponentially decreasing air layer thickness as we proceed towards increasing nip loads and an almost linear decay towards low winding speeds (See Fig. 7). Refering to Eq. (22) (See Chapter III) the only parameters than need to be varied are the nip load L (psi) and the winding velocity U (inches/sec). The values of the constant parameters viz., J.l, Re, Ro and T can be found in the Nomenclature section. The values ofL were varied from 4 to 48lbf(experimental values) and those ofU from 500 to 2000 fpm (experimental values) and howas subsequently determined for 24 different combinations ofL and U. Maximum air entrainment as expected is observed for the highest speed and the lowest nip load i.e., 2000 fpm and 4 lbf. respectively and the lowest entrainment for high nip loads and low winding speed i.e., 500 fpm and 48 lbf. respectively. The range of average air layer thickness as depicted in the theoretical results was of the order of 6.7 J.l inch, corresponding to a high of 6.9 J.l inch, and a low of 0.2 J.l inch. Speed, (fpm) 7 6 5 Air Layer 4 Thickness, (micro in.) 3 Nip Load, (Ibl) 32 48 2000 .67 11I56 .45 1334 023 .12 .01 32 Figure 7. Theoretical Results of Bertram & Eshel's Hydrodynamic Equation. Hamrock and Dowson's Elastohydrodynamic Solution The Elastohydrodynamic Equation predicts a higher range of average air layer thickness, of the order of 20.411 inch, corresponding to a high of27.31l inch and a low of 6.911 inch. Referring to Eq. (36) (See Chapter III) the values of speed, u (inches/sec) and the normal applied nip load, F (lbt) were similarly varied as in the previous case. The elastohydrodynamic equation also uses the effective modulus, E' (modulus of the wound roll and the nip roll), hence the same was determined for different values of the radial modulus, Er of the wound roll under different nip loadings, and incorporated into Eq (36). The constant parameters used viz., k, 11 and Rx have been declared in the Nomenclature. Again the values of L were varied from 4 to 48 lbf and those of u from 500 to 2000 fpm 33 and howas determined for 24 different combinations of L and u. A distinctive feature of the Elastohydrodynamic equation is a far less drastic decrease in air film thickness towards higher nip loads and a much thicker air layer. It should be noted that the dimensionless parameters used by Hamrock and Dowson do not fall in the same range as the dimensionless parameters associated with these experiments (See Table XIII in Appendix). 30 25 20 Air Layer Thickness, 15 (micro in.) 10 5 Nip Load, (Ibl) 32 48 1!Il2530 E11520 01015 1105 2000 Speed, (fpm) Figure 8. Theoretical Results of Hamrock & Dowson's Elastohydrodynamic Equation. Chang's Elastohydrodynamic Solution Another theoretical model that was recently developed by Chang [4], which takes into account the compressibility effects of the entrained air has also been presented here. 34 Recall that Hamrock and Dowson's Elastohydrodynamic equation is meant for oil film lubrication in ball bearings whereas in air entrainment studies we are dealing with an air film. Chang's model takes into consideration the above disparity where, for air, its density is dominated by pressure unlike the density of oil which is almost unaffected by a wide range of pressures. This model assumes two rotating rollers without a web in between them and that the materials in consideration are isotropic. Here the formula for air layer thickness was again derived using a similar, but not identical set of nondimensional parameters used by Hamrock & Dowson. This equation too is derived using a curve fit, however the dimensionless parameters associated with these experiments fall in the same range as the ones used by Chang to derive the film thickness relationship (See Table XIV in the Appendix). The equivalent modulus of the nip roll and the wound roll as calculated by the effective modulus E' expression was 750 psi. on an average hence Eq.'s (41) and (42) would be used for the present study. The parameters used to determine the air film thickness (ho) using Eq. (41) (See Chapter III) are shown in Table XIV, again the values of speed, u (inches/sec) and the nip load per unit width of the roller, F (pli) were varied in a similar way as described previously and the results plotted for their 24 different combinations Chang's model predicts a thicker air layer than that predicted by the Elastohydrodynamic Equation but the rate of decrease of air layer as we proceed towards higher nip loads seems to relate more with the Hydrodynamic Equation than with the Elastohydrodynamic equation. Speed, (rpm) 50 45 40 35 Air Layer 30 Thickness, 25 (micro in.) 20 15 105o 4 Nip Load, (Ibf) 32 48  2000 .4550 .4045 83540 .3035 11I2530 .2025 131520 01015 .510 .05 35 Figure 9. Theoretical Results of Chang's Equation. All the results plotted above display the minimum air film thickness (ho), the thickness of the air film under the nip. Chang has also developed a relationship for the air film thickness (he)' the thickness of the air film after it passes away from the nip (See Chapter III, Eq.(42)). This is a more practical case where the air film after getting compressed under the nip expands when it moves away from it. The results obtained due to these compressibility effects are presented in Fig.! O. Note that this expression for he (Eq. (42)) depends upon a e1astohydrodynamic term. As is obvious the air film thickness (he) predicted after it expands is way too high than the experimental values however the interesting feature to note here is its similarity with the trend observed for Hamrock and Dowson's Elastohydrodynamic equation results (unlike the trend observed for ho, which resembled Bertram and Eshel's Hydrodynamic equation). Air Layer Thickness, (micro in.) Nip Load, (Ibf) 48 .8090 1!lI7080 .6070 1ilI5060 .4050 1!lI3040 02030 .1020 .010 Speed. (rpm) 36 Figure 10. Air Film Thickness (he) Away From the Nip as Predicted by Chang. The results of the he relationship developed by Chang utilizing the Elastohydrodynamic term is shown above. Chang had also developed another he relationship using a hydrodynamic term from Bertram and Eshel's Hydrodynamic equation by suitably modifying it to include the compressibility effect of the air film as it passes away from the nip (See Eq (25), Chapter III). The results of Chang's modification of hoto include the compressibility, using the hydrodynamic term are presented in Fig. 11. As is obvious referring to Eq.(25), the results obtained here are simply the results for ho predicted by the hydrodynamic equation times the factor 4/3 used to modify Eq. (23) to include the change in velocity profile as a result of the compressibility effects of the air film as it passes away from under the nip. Air Layer Thickness, (micro in.) 109 876 5 4 3 21o 4 Nip Load, (Ihf) 32 48 2000 Speed, (fpm) .910 .89 ml78 .67 1l'l56 .45 034 023 .12 .01 37 Figure 11. Chang's he Prediction Using Bertram and Eshel's Hydrodynamic Term. Experimental Results Finally, the experimental results obtained for 24 different tests are presented below with the air layer thickness plotted as a function of nip load and web velocity. There are some random values in the data taken at high speeds that do not compare with the expected results, this may be attributed to the instability of the nip roll and the nip roll bounce associated with those speeds. Repeated tests at high speeds would have produced reasonable values but the amount of material available for the same was limited. Above all, on an average, the experimental results produced consistent values which seemed to follow Chang's hopredicted values closely. The trend however resembles more like his he model and also the Elastohydrodynamic model. 38 40 30 Air Layer 25 Thickness, 20 (micro in.) Nip Load, (Ibl) 32 48 l!I3540 .3035 C2530 .2025 01520 01015 .510 2000 .05 Speed, (fpm) Figure 12. Experimental Results. Comparison of Theory and Experiment Comparing the values of the air layer obtained experimentally with the theoretical values it can be observed that Chang's model for the minimum film thickness (ho) is the closest to the experimental results. However when we observe the 3dimensional surface plots the trend appears to resemble Chang's model for the expanded air film thickness (he)' Also note the similarity with the Elastohydrodynamic equation results. Some of the other notable features are, at small values of nip loads the experimental results closely follow Chang's hoprediction (e.g., at 4lbfnip load), even the rate of decrease ofho appears to be similar. Towards very high values of nip loads the experimental results are 39 more closer to Hamrock and Dowson's Elastohydrodynamic equation results (e.g., at 48lbf nip load). This suggests a more predominant elastohydrodynamic regime at high nip loads. An interesting feature in the Elastohydrodynamic lubrication studies is the film parameter, A, which is related to the lubrication film thickness (air film thickness in our case) as follows: (46) where ff and fb both are rms surface roughness of the film and hois the film thickness. The film parameter, A is used to identify the lubrication regime. Hydrodynamic or fluidfilm lubrication occurs when the contacting surfaces are separated by a lubricating film so thick, they cease to contact. It has been reported by Hamrock and Dowson [9] that in this regime the value ofA is greater than 10 and may be as high as 100. Elastohydrodynamic lubrication occurs when the bearing materials are soft such as elastomers and rubbers, where the local elastic deformation of the bearing surfaces allows for a coherent lubricating film and surface contact is not fully avoided as in the previous case. The film parameter in this regime falls between 3 and 10. For the existing winding conditions we calculated the values of A, which were found to range from 9 to 65. This implies that we have a predominant hydrodynamic regime for most of the winding conditions with a borderline between hydrodynamic and elastohydrodynamic regimes. The above phenomenon as predicted by the film parameter A contradicts the experimental results which have predicted a predominant elastohydrodynamic regime. The validity of using 40 the film parameter, A (which is simply a ratio of film thickness to the surface roughness), to judge the regime of lubrication is questionable. The film parameter A was calculated in the context of ball bearing lubrication (where the supporting materials are rigid and do not experience significant deformation), whereas in the present case we are dealing with a rubber nip roll and an air entrained wound roll with the air beneath the nip, during winding, exerting a couple of 100 psi. pressure sufficient enough to deform the surfaces in contact and produce a predominant elastohydrodynamic regime (encountered with soft bearing materials) rather than a hydrodynamic regime (encountered with rigid rollers). Tables II, III, IV, V & VI in Appendix compare the results presented in the figures above. a)The Hydrodynamic equation predicts an air layer roughly six times less than the experimental results. b)Hamrock and Dowson's elastohydrodynamic equation slightly underestimates the experimental values at small values of nip load but at higher nip loads, (e.g., 48 lbt) it yields good results. c)For majority of the experimental results Chang's ho prediction values follow reasonably well, except of course for the extreme nip loads. d)Finally, comparing Chang's he (derived using Chang's elastohydrodynamic term) values we observe that they overestimate the experimental results almost by a factor of 2. Also it may quite be possible that we are underestimating the experimental values instead, as a result of air escape even before we can practically measure it in the procedure employed in the experiments. More insight can be had in this particular phenomenon in the next chapter where we study rate of air escape from wound rolls. Since Hamrock and Dowson's elastohydrodynamic equation and Chang's prediction equation's were formulated based on a curve fit technique using dimensionless 41 parameters corresponding to a set of input values, it was necessary to have an idea as to how close the dimensionless parameters dictated by the existing winding conditions were to the one's used to derive the above equations. With the ellipticity parameter, k, held constant, the load, Wand speed, U parameters were the only variables that needed to be compared with the range of dimensionless parameters used by Hamrock & Dowson to obtain the curve fit. Table XIII in Appendix shows a comparison between the range of dimensionless parameters used by Hamrock & Dowson to those associated with the experimental conditions. It is quite clear that the dimensionless parameters U and W corresponding to the practical case deviate far from those that were considered by Hamrock and Dowson to derive their elastohydrodynamic equation (Eq.(35)). Table XIV in the Appendix provides a similar comparison for the range ofU, W and G values used by Chang to curve fit and derive his film thickness equation. It can be seen that the range of values for the dimensionless parameteres dictated by the existing winding conditions fall within the same range of theoretical values used by Chang in his film thickness equation derivation. Thus Chang's model should provide a sound basis of comparison with the experimental results. Experimental Observations Using a Metal Nip To verify the authenticity of the experimental data one of the experiments was repeated using a new nip configuration. The new nip was made of Aluminum, unlike the 42 previous one which was made of rubber thus having different material properties. Two different sets of rolls, one 1500 ft. long and the other 1500 m long were wound at 2000 fpm, 8 lbf nip load and 1000 psi. web tension. Air was collected immediately after the winding operation. Table VII in Appendix displays the results recorded for the same. It was expected that the experiments would yield different results due to different material properties of the nip, on the contrary the values obtained were almost the same as the ones obtained with the rubber nip, for both lengths of the roll. The reason for this can be attributed to the fact that the experiment using the metal nip which was performed for the 2000 fpm winding velocity and 8 lbf nip load combination corresponds to a predominant hydrodynamic regime where the web layers tend to lose contact during winding, also the equivalent modulus corresponding to this case (in fact for all the nip loads used) confirms to the range where it is not affected much by any further increase in the modulus of the nip roll therefore the amount of air entrapped remains unaffected. (This has been proved in the next section, see page 43). During winding at high speeds (e.g. 2000 fpm) and small nip loads (e.g. 4,8 lbt) a comparitively thicker air film develops and exerts a pressure force on both the wound roll and the nip roll sufficient enough for preventing each from coming in contact with the other at times during winding thereby keeping the film thickness unaffected by the nip and the wound roll moduli. 43 Effect of Wound Roll Modulus (Eb) and the Modulus of the Nip (EJ on the Film Thickness In order to determine how important the moduli of the nip roll and the wound roll are for affecting the effective modulus and thus the air film thickness, a sensitivity study was performed on both Hamrock and Dowson's and Chang's Elastohydrodynamic equations (Eq's (35) &(41) respectively). The value ofEa was held constant and Eb was varied to calculate various values of hoand vice versa. The comparisons are made at a reference speed of2000 fpm and 8 lbfnip load (a combination quite frequently used all through the experimental study). It was observed during the calculations that a change in the radial modulus of the wound roll, Eb above a certain value, ceased to have a noticeable effect on the value of the air film thickness hoe Above a certain value for Eb the calculations started yielding a constant value for E', the effective modulus, hence leading to constant air film thickness. This phenomenon was observed in both Hamrock and Dowson's and Chang's equations. The results of these calculations are tabulated in Table XV in the Appendix and plotted as shown in the figure below: 44 + hO for Ham&Dow.,Ea const. 0 hO for Chang, Ea const. .~. hO for Ham&Dow.,Eb const. '~/1 hO for Chang, Eb const. 100.00 .c...:. 90.00 .5 .0.. 80.00 .~ S 70.00 '' ,;; 60.00 'Q".l c 50.00 ...:.:.c. :.c 40.00 Eo ... 30.00 Q.l ;... = 20.00 .....:.l 10.00 < 0.00 0 2000 4000 6000 8000 10000 12000 14000 Modulus of Roll Eb/NipEa, psi. Figure 13. Air Film Thickness (ho) For Hamrock and Dowson's & Chang's Equations, Eq's (35) & (41) respectively as Affected by Wound Roll/Nip Modulus. It is quite evident from Fig. (13) that the modulus of the wound roll and the nip roll radically affect the air film thickness, however only for small values « 1000 psi.) of Ea or Eb• There is a drastic change in the air film thickness (ho) when the modulus of the wound roll is varied below about 1000 psi.. Above about 1000 psi. hardly any change is noticed. A similar trend is observed with the nip roll modulus, Ea also. However it is the wound roll modulus which is affected by the entrained air hence the focus of this discussion would be the wound roll modulus Eb and the radial modulus of the wound roll 45 The values ofEbwere obtained by contact tests performed by Covell [5] using Tekscan which is a PC based instrument for pressure or force measurements. It consists of a force sensitive resistor and an expansion card that allows the PC to record and display nonuniform loads measured. The peak pressure under the nip was measured for different nip loads and was substituted in a theoretical expression ,Eq. (47), derived by Xu [12] for Er contact in the absence of air where 2 3 Er contact = aP + bP + cP (47) a=3138.0 b = 33.54 c=0.151 The calculated radial modulus of the winding roll, Ebwas then used to model the elastohydrodynamic equations. It has to be noted here that Ebwas measured statically, a condition which is very different from the one when the roll is wound at 2000 fpm. Various phenomenon can be encountered in the latter case, one of which that has been mentioned earlier is the possible loss of contact between the nip and the wound roll at high speeds and small nip loadings. In such cases a static determination of Ebis obsolete. It will be shown in the next chapter that air entrainment reduces the modulus of the wound roll radically and that the wound roll modulus is a dynamic quantity, varying as a function of the radius. Hence a radial modulus of the wound roll Er should be used for the modulus of the wound roll. But Er and ho, the air film thickness, are dependent on each other, this makes the determination ofEr or homore complicated since neither can be isolated from the other. Hence a safer approximation would be to use the static values of Ebobtained through contact tests, for 46 air film thickness calculations. The range of values for Eb obtained during the contact tests [5] varied from 8200 psi corresponding to a 4lbfnip load to about 47000 psi for 48 lbf nip load. All of these fall in the range where their variation does not have any noticeable effect on hoe CHAPTER VII RADIAL PRESSURE PROFILE To develop a radial pressure profile we need to know the dynamic values of Ep the radial modulus, which can be formulated simply by a stressstrain relationship. Er, which is a "pseudo material property" can be measured using the INSTRON, by placing a stack of webs and measuring the strain produced by them due to the application of a compressive force. The value stress/strain would then give the modulus, Er • However, in the realm of air entrainment studies Er, cannot be static, since, in the stack tests if an air layer comparable to the one entrained during winding is present then the value of strain would go up, thereby decreasing the value ofEr and vice versa. Thus in practicality we have different air layer thickness at different radii in a wound roll (recollect that the average air layer thickness is only a simplifying assumption ), these differences contribute to the variations in Er at various radii of the wound roll. It was for this reason that different Er values were formulated using a curve fit [10], resulting in a third degree polynomial expression which is: (46) where C1, C2, and C3 are constants derived from stack tests. For Er stack, the constants are: 47 48 C1 == 129.0 C2 == 0.037 C3 == 4.0e4 Applying Eq.(16) for Ereq, yields the following constants: C1 == 60.0 C2 == 0.416 C3 == 0.001 Simialarly, for Erair, application ofEq.(13) yields: C1 == 3.042 C2 == 0.0475 These relationships between Er, the radial modulus and P, the radial pressure can be depicted as follows: 14000 . Er stack 12000 o Er eq. 10000 ~Erair 0 8000 0 Er, (psi.) 0 6000 0 0 4000 o· 0 2000 0 0 0 0 0 0 0 0 0 0 0 0 0 N (V) ~ LO <.0 r" oo 0') 0 ~ P, (psi.) Figure 14. Radial Modulus,Er Computed as a Function ofPressure,P. Once the various expressions for Er are developed we can predict the radial pressure profiles using the software WINDER v 4.0 available at WHRC. This software takes the values of the roll radius, winding speeds, winding tensions, radius of core and 49 its material properties, dimensions of the web in consideration and its material properties and the constants obtained from the stack tests as input and plots the radial pressure profile. Fig.15 depicts the radial pressure profiles obtained using Er stack> Er eq and Er air respectively on a common axes: 1 .1 1.2 1.3 1.4 o t Ai ~ " ~ Amil 1.5 200 180 • Er,stack psi. 160 • Er,eq  psi 140 .. Er,air  psi Radial 120 Pressure, 100 (psi.) 80 60 40 20 RlRc Figure 15. Comparison of Radial Pressure as a Function of Normalized Radius (R/Rc). As is evident from the above figure the entrained air reduces the radial pressure at the midpoint of th roll to 3 psi. when compared with the stack and equivalent values, which are 85 psi. and 55 psi. respectively. Table VIn in Appendix depicts the range of pressure distribution over the radius of a wound roll. Fig. 16 shown below displays a detailed pressure profile for the roll entrained with air, the object our of experimentation, 50 6 I + Er,air  psi I 4 Radial Pressure, 3 (psi.) 2 0++++1++++++++ 1.02 1.06 1.09 1.12 1.15 1.19 1.22 1.25 1.28 1.32 1.35 1.38 1.41 R/Rc Figure 16. Pressure Profile for the Roll with Entrained Air. The above plot represents a typical charecteristic of the pressure profile when the ratio of Et/Er is high. The maximum pressure is present at the core and soon becomes a more or less constant value till it reaches the outer radii of the core,where it sharply decreases to zero for the outermost layer. CHAPTER VIII AIR ESCAPE OVER TIME It has been analytically proven by three different air entrainment models and experimentally proven in this study that as the winding speeds increase the amount of trapped air also escalates. If the amount of air trapped is very large e.g., during high speed centerwinding without a nip roller, it is quite possible that there will not be any interlayer contact of the web material, and the web layers might ride completely upon the air layers, this phenomenon is refered to as aeroplaning. Only at the roll edges, where the air has a good chance of discharging to the atmosphere does the possibility of layer to layer contact exists. This loss of contact between the adjacent web layers would lead to a decreased resistance to the slippage of the web axially, resulting in telescoping and other roll defects. In order to overcome this problem it is important that the underlying mechanism that operates during and after winding and which affects the rate of air entrainment and escape respectively, be studied. Up to this point in the study the former mechanism was studied, and now the study will concentrate on the phenomenon of air escape from rolls and how it affects the integrity of the roll. In order to evaluate the rate at which the air escapes a set of rolls were wound under similar nip loads and at constant web velocities and were unwound at 51 52 various lengths in time. The set of rolls 1500 ft. long were wound at 2000 fpm and a nip load of 8 lbf. The first set was unwound after 24 hours, the decrease in air volume was recorded and the average air layer thickness calculated. It was observed that the air layer thickness had diminished by about 10 Il inch on an average. The other set of rolls were unwound after three days and an additional decrease in air layer thickness by about 5 Il inch was observed. Some interesting observations were made while unwinding these rolls, as compared to the rolls unwound immediately after winding. The rolls that were unwound soon after winding displayed typical characteristics as soon as they were placed in the bubblerimeter, viz., a lot of air bubbles could be seen surfacing the water in the bubblerimeter by the time the conical hood was hooked up for air collection, These bubbles came out at a fast rate, that impeded their estimation. The bubblerimeter has to be calibrated after the roll is placed inside it and the fastest this could be done was in a minute, but a lot of air was seen escaping during this time, which could not be accounted for since the rate of escape was unknown. Further there is a spurt of air escape during the unwinding of the first few web layers, apparently there is a relatively larger amount of air in the outer layers but as the unwinding operation progresses the release of air bubbles becomes more consistent towards the inner layers. It would be safe to generalize that when unwinding a roll immediately after winding, ignoring the air escape from the first few layers of the web unwound, the amount of air collected as a function of time is nearly constant, indicating the consistency of the air layer thickness. Thus it can be assumed that the average air 53 layer thickness is the actual air layer thickness itself. The point to be emphasized here is that immediately after the winding operation the average air layer thickness would give a reasonable estimate of the absolute air layer thickness. The only shortcoming however is the failure of the existing apparatus to trap the air initially. Fig.17 shows the rate of air escape in the bubblerimeter immediately after the winding operation. The volume of air collected is plotted as a function of the length of the roll at various times during the unwinding operation, for two different rolls 1500 ft. and 1500 mts. long respectively. Table IX in Appendix shows the recorded results. 160 • 140 • 120 100 • Volume of 80 Air, (ml.) 60 0 IJ. 40 IJ. IJ. IJ. 20 0 0 1000 2000 3000 4000 5000 Length of Roll, (f1.) Figure 17. Volume of Air Collected at Various Lengths of the Unwinding Roll. In contrast when unwinding a roll after sufficient lengths of time (e.g., 8, 16, 24 hours etc.) the observations made were quite different. As opposed to the fast escape of air bubbles observed in the previous case, very few bubbles could be seen escaping from 54 the roll edges and hence by the time the apparatus was ready to collect the air, hardly any bubbles surfaced and most of them were seen sticking to the roll edges, thus, in this case all the air collected was accounted for. Only when the roll started unwinding could one see the air bubbles rising up, even so at a slow rate. This rate kept decreasing as the roll unwound and there were times when no air bubbles could be seen for considerable lengths of the roll, and towards the end there was hardly any air collected in the apparatus. A series of tests showed that, of all the air that was collected about 60% of it was collected during the unwinding of the first 400 ft. of the 1500 ft. long roll. This phenomenon questions the validity of using an average air layer thickness calculation for rolls kept for significant lengths of time (e.g., 8, 16, 24 hours etc.), because of the lot of air collected, which has been experimentally proven here to be present in the first few feet of the roll, is actually being split up over the entire length of the roll. It is for this reason that 1500 meters of roll unwound after 24 hours might have a smaller average air layer thickness than 1500 feet of it unwound after the same time. Moreover, the time taken to wind a roll 1500 meters long at 2000 fpm is about 2 min. and 30 secs., as opposed to the 53 secs needed to wind 1500 ft. of roll at the same speed. Thus in the former case while the roll builds up the air entrained in the lower layers has more time to escape and thus will escape unaccounted for. Added to this, the builtup of layers exerts additional pressure on the lower layers thereby squeezing out the entrained air more vigorously. Fig.18 shows the volume of air collected in the bubblerimeter when the roll (1500 ft. long) was unwound after 3 days, as a function of the length of the unwinding roll. 55 Also it shows a comparison with similar data obtained for the roll that was immediately unwound. Due to the limited availability of the material a similar comparison for 1500 mts. of roll could not be made. Table X in Appendix shows the recorded results. 70  500 1000 1500 2000 20 10 o.+++1 o • After 3 Days 60  • After Winding Volume of 40 Air, (mI.) 30 _ 50  Length of Roll, (ft.) Figure 18. Comparison of the Volume of Air Collected Immediately and 3 Days After Winding, as the Roll Unwinds. Since the amount of air collected after 24 hours showed a decrease in air layer thickness by 10 Il inch it was expected that after three days the decrease would be even more, but it turned out to be 5 Il inch, which implied that the rate of air escape had slowed down considerably. This led to reducing the time prior to unwinding and measuring the volume of air well before the 24 hour time period. New rolls were wound and unwound again, after 8 and 16 hours respectively. Surprisingly they had roughly the same amount of air that was left in the roll after 24 56 hours. This meant that a lot of air that could have escaped had done so immediately after winding and the remaining was left to discharge at very slow rates, over periods of days. Also it has been mentioned before that rolls placed in the bubblerimeter immediately after winding had a tendency to release a lot of air bubbles at a fast rate. This reinforces the fact that a lot of the entrained air escapes during and immediately after winding. Thus the rate of air escape had to be studied shortly after winding. A series of tests were performed and the air collected at intervals of 1, 3 and 10 minutes after the winding operation which took 53 seconds on an average. By the time the winding operation was terminated and the roll placed in the bubblerimeter ready to be unwound 1 minute and 53 seconds had elapsed. This time has been added to all time intervals after which the rolls have been unwound. Table XI in Appendix displays the data recorded for the above tests and Table XII compares the values derived theoretically from the squeeze film damper expression (Eq. (17), Chapter II) with the experimental values. The data obtained from all these experiments was grouped systematically and analyzed using a spreadsheet program that displayed the rate of decrease of air layer thickness as a function of time as a 2 dimensional plot. The theoretical results from the squeeze film damping equation were also modeled similarly and formed a basis for comparison with the experimental results. Figures 19 & 20 show the rate of air escape as a function of time for the two sets of experiments and a comparison of theory and experiment respectively. 57 100 1000 10000 +Expt. 1 __Expt. 2 10 10 5 0++++1 1 35 30 25 Air Layer 20 Thickness, micro inch. 15 Time in minutes Figure 19. Air Escape as a Function of Time (Experimental Results). +Expt. __Theoret. 10 5 o L+==~~~~I=:e=r=a==_._______1 1 10 100 1000 10000 35 30 25 Air Layer 20 Thickness, micro inch. 15 Time in minutes Figure 20. Air Escape as a Function of Time(Comparison OfTheory and Experiment). 58 Comparison of Theory and Experiment As can be seen in Fig. 20 the theoretical results display a drastic decrease in air layer thickness during the first 5 to 10 minutes after winding the roll. The air layer drops from 30.2 J.l inch at 1min and 53secs to about 5.1 J.l inch at 11min and 53secs. On the contrary the experimental values of the air layer thickness decay from 30.2 J.l inch to 19.2 J.l inch over the same time range, thus displaying a less drastic air escape. In other words, 83% of the air entrapped when winding a roll at 2000 fpm and 8 lbfnip load escapes 10 minutes after the winding operation according to the squeeze film damper prediction, whereas the experiments show the escape to be 36%. Even though the amount of air escape predicted by the theory and experiment fall roughly about 50% apart a promising feature is the similarity in the rates, i.e., as a function of time the experimental plot behaves like the theoretical one. A large amount of air escapes during the first 10 minutes after winding and then we have a plateau for the next 16 to 24 hours, a further decrease is then observed only over periods of days (3 days in the experimental study). The theoretical plot also depicts a similar trend where most of the air escapes during the first 10 minutes after winding and then the rate becomes more or less a plateau for large time periods. Based on the above discussion and a comparison of theory and experiment it can be said that the entrained air is somehow being obstructed on its way out after the winding operation which may not be an ideal case as that assumed by the squeeze film damper equation. 59 The squeeze film damper equation assumes the two contacting surfaces to be perfect. On the contrary the contacting surfaces used in the experiments have some irregularities on a micro inch scale, which are bound to affect the average air layer thickness, which has the same orders of magnitude. Also it is a common observation that when a web is trimmed to its exact width the edges that are slit are sligthly thicker than the web caliper depending on the type of slitting operation employed. The presence of bumps at the roll edges could obstruct the air escape significantly. This has been illustrated by the drawing below, which exaggerates two web surfaces in contact. Further, as the roll builds up these bumps overlap and press relatively harder on each other than the rest of the overlapping layers in the roll. This could possibly magnify the phenomenon of obstruction of entrained air. Figure 21. Bumps at the Roll Edges Obstructing the Escape of Entrained Air. Another factor that could possibly affect the rate of air escape is the nonuniform pressure distribution along the width of the web. It has been proven in previous studies [3] that, under its own weight and the pressure force exerted by the two ends of the roll the film deforms as shown below: 60 Figure 22. Nonuniform Pressure Distribution Along the Web Width. The above illustration explains for itself the bottlenecks created at the two extremities of the roll, which prevent the air from escaping. In fact the pressure profile of the pressure force exerted predicts a smaller value at the center of the roll and a little higher value towards the ends [3]. The two phenomenon explained above seem to have a combined effect on the rate of air escape. The bumps at the roll edges coupled with a higher pressure created there impede the air from escaping at a desired rate. CHAPTER IX SUMMARY AND CONCLUSIONS This study has attempted to verify a winding model that would incorporate the entrained air, and a squeeze film damping equation to model the escape of air over time. It has also verified the decrease in radial modulus due to the presence of entrained air witnessed by Covell [5]. In other words, it has studied the factors which are known to contribute to a decrease in radial pressure during centerwinding with a force loaded nip roller. Based on the experimental results obtained and comparing them with the results obtained from theoretical models the following conclusions can be derived: 1.) The air layer entrained during high speed winding with a nip roller is comparable to the theoretical minimum film thickness (ho) derived by Chang (Eq.(41)). A more practical case would be Chang's he prediction (Eq.(42)) since it considers the expansion of the air film after it passes away from the nip, but the air loss from the roll edges is not considered in deriving he (or any other air film thickness derivations for that matter) hence it is possibly overestimating the results. Further, there is a possible transition from a hydrodynamic regime to elastohydrodynamic conditions towards high nip loads (e.g. 32, 48 lbf). 61 62 2.) Even small nip loads can affect the air layer thickness, so to get reasonable results it is necessary that the nip load be controlled precisely so as to minimize the fluctuations encountered during nip roll bounce. Moreover, the radial modulus of the nip, if it is a hollow cylindrical roller should be used instead of the elastic modulus for estimating the air layer thickness. 3.) The presence of entrained air reduces the radial modulus, Er of the wound roll considerably by decreasing the radial pressure. Further, the radial modulus is a dynamic property of the roll which depends on radius of the roll and air film thickness. 4.) The bubblerimeter can be used as an air collection device as first documented by Covell [5], even though it may not be as sophisticated as laser reflectance probes, since it gives a simple and accurate account of air entrained for nonpermeable (plastic) films. 5.) Most of the air that is entrained escapes during and immediately after winding. 6.) Finally, the air escaping from a wound roll can be modeled using a modified version of the squeeze film damping equation that would take into account the fluid dynamics in between the two web surfaces with irregularities towards the edges. Future Work The following section explores the various possibilities that could be investigated to obtain a much more sophisticated air entrainment model, 63 1.) It has been proven in this study that air is present immediately after winding, a technique to measure the air present in the first few seconds after winding if devised would give a clear understanding of the phenomenon at work during air discharge from the wound roll. The escape of air during the winding operation is another area that needs to be closely monitored and estimated for including the fluid dynamics aspect, which plays a significant role in air entrainment experiments coupled with the solid mechanics aspect. 2.) A procedure needs to be formulated for proper determination ofEb's and Er's since the air film thickness hoand Er are interrelated, and Eb in turn is related to Er. 3.) Applying the equivalent radial modulus to study the air layer, when its value falls between the mean surface roughness and the maximum asperity height of the film. 4.) Measuring the air layer thickness as a function of the roll radius would contribute to modifying the existing models which assume an average air layer thickness throughout the wound roll. 5.) Lastly, the effect of nip roll dynamics on entrained air could also be included in the winding model. REFERENCES 1. Bertram, N. and Eshel, A. "Recording Media Archival Attributes." RADCTR80123, pp. 6873, April 1980. 2. Blevins, Applied Fluid Dynamics Handbook. pp.503. 3. Bouquerel, F. "Theoretical and Experimental Study of Winding of Thin Plastic Films: Aerodynamics Effects." Ph.D Thesis, Central University of Lyon, France. 4. Chang, Y.B. "Air Entrainment With a Force Loaded Nip Roller." Technical Review and Industry Advisory Board Meeting, WHRC., May 1994. 5. Covell, K. Scott "The Effect of a Nip Roller on Entrained Air During High Speed Winding.", Master's Thesis, Department of Mechanical and Aerospace Engineering, Oklahoma State University, 1994. 6. Good, J.K. and Holmberg, M.W. "The Effect of Air Enrainment in Centerwound Rolls." Proceedings of the Second International Web Handling Conference, April 1993. 7. Good, J.K. Wu, Z. and Fikes, M.W.R. "The Internal Stresses in Wound Rolls With the Presence of a Nip Roller." accepted for publication in the ASME Journal of Applied Mechanics, 1992. 8. Hakiel, Z. "Nonlinear model for Wound Roll Stresses." TAPPI Journal, Volume 70, No. 5,pp. 113117,1987. 9. Hamrock, B.J. and Dowson, D. Ball Rearing Lubrication. John Wiley & Sons, pp. 18889 and 28082,1981. 10. Holmberg, M.W. "Theoretical and Experimental Studies of Air Entrainment in Wound Rolls." Master's Thesis, Department of Mechanical and Aerospace Engineering, Oklahoma State University, 1992. 11. Knox, K.L. and Sweeny, "Fluid Effects Associated with Web Handling." Ind. Engr. Chern. Proc., vol. 10, pp. 201205, October 1971. 12. Xu, Y. "Computing Stress Distributions in Centerwound Rolls from Web Surface Charecteristics." Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, Oklahoma State University, 1992. 64 APPENDIX 65 66 TABLE I VOLUME OF AIR COLLECTED FOR TWENTY FOUR INDIVIDUAL WINDING EXPERIMENTS Nip Load Speed Vol. of Air Vol. of Air Air Layer Ibf fpm (uncorrected) (corrected) Thickness ml. ml. mIcro In. 4 44 32.98 18.6 8 35 24.25 13.7 12 500 27 16.49 9.3 16 23 12.61 7.1 32 20 9.7 5.4 48 20 9.7 5.4 4 57 45.59 25.7 8 54 42.68 24.1 12 1000 49 37.83 21.3 16 46 34.92 19.7 32 38 27.16 15.3 48 33 22.31 12.6 4 65 53.35 30.1 8 63 51.41 29.0 12 1500 57 45.59 25.7 16 57 45.59 25.7 32 46 34.92 19.7 48 45 33.95 19.1 4 80 67.9 38.3 8 67 55.29 31.2 12 2000 61 49.47 27.9 16 56 44.62 25.2 32 50 38.8 21.9 48 40 29.1 16.4 67 TABLE II COMPARISON OF THEORETICAL VALUES FROM BERTRAM AND ESHEL'S HYDRODYNAMIC EQUATION WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Deviation lbf fpm Layer Thickness (ho) Layer Thickness mIcro In. micro in. micro in. 4 1.8 18.6 16.8 8 0.9 13.7 12.8 12 500 0.6 9.3 8.7 16 0.5 7.1 6.6 32 0.3 5.4 5.1 48 0.2 5.4 5.2 4 3.5 25.7 22.2 8 1.8 24.1 22.3 12 1000 1.2 21.3 20.1 16 0.9 19.7 18.8 32 0.5 15.3 14.8 48 0.3 12.6 12.3 4 5.2 30.1 24.9 8 2.6 29.0 26.4 12 1500 1.8 25.7 23.9 16 1.3 25.7 24.4 32 0.7 19.7 19.0 48 0.5 19.1 18.6 4 6.9 38.3 31.4 8 3.5 31.2 27.7 12 2000 2.3 27.9 25.6 16 1.8 25.2 23.4 32 0.9 21.9 21.0 48 0.6 16.4 15.8 68 Bertram and Eshel's hydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. L == Nip Load, varied as 4,8,12,16,32 and 48 lbf. for each winding velocity used. U == Winding Velocity, varied as 500,1000,1500 and 2000 fpm. T == Web Line Tension, held constant at 0.48 lbf/in.throughout experimentation. Re == Equivalent Radius, calculated as 1.085 inch. f.l == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 Ro== Radius of Wound Roll, measured as 2.4 inch. an average 1t == 3.1414. ho== Air Film Thickness, from 24 different combinations of velocities and nip loads 69 TABLE III COMPARISON OF THEORETICAL VALUES FROM HAMROCK AND DOWSON'S ELASTOHYDRODYNAMIC MINIMUM AIR FILM THICKNESS EQUATION WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Deviation lbf fpm Layer Thickness (ho) Layer Thickness mIcro In. mIcro In. mIcro In. 4 11.1 18.6 7.5 8 9.8 13.7 3.9 12 500 9.1 9.3 0.2 16 8.6 7.1 1.5 32 7.5 5.4 2.1 48 6.9 5.4 1.5 4 17.4 25.7 8.3 8 15.4 24.1 8.7 12 1000 14.2 21.3 7.1 16 13.4 19.7 6.3 32 11.7 15.3 3.6 48 10.8 12.6 1.8 4 22.6 30.1 7.5 8 20.0 29.0 9.0 12 1500 18.5 25.7 7.2 16 17.5 25.7 8.2 32 15.2 19.7 4.5 48 14.0 19.1 5.1 4 27.3 38.3 11.0 8 24.2 31.2 7.0 12 2000 22.3 27.9 5.6 16 21.1 25.2 4.1 32 18.4 21.9 3.5 48 16.9 16.4 0.5 70 Hamrock and Dowson's elastohydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. where, h U were  ua +ub 2 and F W= E'R2 x where F = Normal Applied Load, lbf (including tesion component T sin 45),(e.g., 4lb nip load plus 2.88 sin 45 etc.) u = Average ofNip and Roll Velocities, varied as 500,1000,1500 and 2000 fpm. T = Winding Tension, held constant at 2.88 lbfthroughout experimentation. Rx == Equivalent Radius, calculated as 1.085 inch. Rn == Radius of Nip, 2 in. Rr == Radius of Roll, 2.4 in. E' = Calculated Effective Modulus, psi. Ea == Modulus ofNip, 675 psi. Eb == Modulus of Roll, (determined by Covell[5] using contact tests) 8200,12800,19800, 27600,37000 and 47000 psi corresponding to 4,8,12,16,32 and 48 lbfnip loads. ~ == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 ho== Air Film Thickness, from 24 different combinations of velocities and nip loads 71 TABLE IV COMPARISON OF THEORETICAL VALUES CALCULATED FROM CHANG'S ELASTOHYDRODYNAMIC EQUATION FOR COMPRESSED AIR FILM WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Deviation lbf fpm Layer Thickness (ho) Layer Thickness mIcro In. mIcro In. micro in. 4 17.1 18.6 1.5 8 11.9 13.7 1.8 12 500 9.6 9.3 0.3 16 8.3 7.1 1.2 32 5.8 5.4 0.4 48 4.8 5.4 0.6 4 28.1 25.7 2.4 8 19.6 24.1 4.5 12 1000 15.8 21.3 5.5 16 13.6 19.7 6.1 32 9.6 15.3 5.7 48 7.8 12.6 4.8 4 37.7 30.1 7.6 8 26.2 29.0 2.8 12 1500 21.2 25.7 4.5 16 18.3 25.7 7.4 32 12.9 19.7 6.8 48 10.5 19.1 8.6 4 46.3 38.3 8.0 8 32.3 31.2 1.1 12 2000 26.1 27.9 1.8 16 22.5 25.2 2.7 32 15.8 21.9 6.1 48 12.9 16.4 3.5 72 Chang's elastohydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. where, 1 where E == 1 y2 1 y2 __a +__b Ea Eb F == Nip Loading per Unit Width of roller, pli (e.g., 4/6,8/6,12/6 pli etc.) u == Average ofNip and Roll Velocities, varied as 500,1000,1500 and 2000 fpm. T == Winding Tension, held constant at 0.48 lb/in. throughout experimentation. Pa == Ambient Pressure, 14.7 psi. Rx == Equivalent Radius, calculated as 1.085 inch. E == Calculated Effective Modulus, psi. Ea == Modulus ofNip, 675 psi. Eb == Modulus of Roll, (determined by Covell[5] using contact tests) 8200,12800,19800, 27600,37000 and 47000 psi corresponding to 4,8,12,16,32 and 48 lbfnip loads. J.l == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 ho== Air Film Thickness, from 24 different combinations of velocities and nip loads 73 TABLE V COMPARISON OF THEORETICAL VALUES CALCULATED FROM CHANG'S HYDRODYNAMIC EQUATION FOR DECOMPRESSED AIR FILM WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Absolute lbf fpm Layer Thickness (he) Layer Thickness Error mIcro In. micro in. mIcro In. 4 2.3 18.6 16.3 8 1.1 13.7 12.6 12 500 0.8 9.3 8.5 16 0.6 7.1 6.5 32 0.3 5.4 5.1 48 0.2 5.4 5.2 4 4.5 25.7 21.2 8 2.3 24.1 21.8 12 1000 1.5 21.3 19.8 16 1.1 19.7 18.6 32 0.6 15.3 14.7 48 0.4 12.6 12.2 4 6.8 30.1 23.3 8 3.4 29.0 25.6 12 1500 2.3 25.7 23.4 16 1.7 25.7 24.0 32 0.8 19.7 18.9 48 0.6 19.1 18.5 4 9.0 38.3 29.3 8 4.5 31.2 26.7 12 2000 3.0 27.9 24.9 16 2.3 25.2 22.9 32 1.1 21.9 20.8 48 0.8 16.4 15.6 74 Chang's hydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. h =.!i !J.URe c 3 L L == Nip Load, varied as 4,8,12,16,32 and 48 lbf. for each winding velocity used. U == Winding Velocity, varied as 500,1000,1500 and 2000 fpm. Re == Equivalent Radius, calculated as 1.085 inch. IJ == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 1t == 3.1414. he == Decompressed Air Film Thickness, from 24 different combinations of velocities and nip loads 75 TABLE VI COMPARISON OF THEORETICAL VALUES CALCULATED FROM CHANG'S ELASTOHYDRODYNAMIC EQUATION FOR DECOMPRESSED AIR FILM WITH THE EXPERIMENTAL RESULTS Nip Load Speed Theoretical Air Experimental Air Absolute lbf fpm layer Thickness (he) Layer Thickness Error he/hO mIcro In. mIcro In. mIcro In. 4 30.8 18.6 12.2 1.8 8 26.6 13.7 12.9 2.2 12 500 24.4 9.3 15.1 2.5 16 23.0 7.1 15.9 2.8 32 20.0 5.4 14.6 3.4 48 18.4 5.4 13.0 3.9 4 50.4 25.7 24.7 1.8 8 43.5 24.1 19.4 2.2 12 1000 39.9 21.3 18.6 2.5 16 37.6 19.7 17.9 2.8 32 32.7 15.3 17.4 3.4 48 30.1 12.6 17.5 3.9 4 67.2 30.1 37.1 1.8 8 58.0 29.0 29.0 2.2 12 1500 53.3 25.7 27.6 2.5 16 50.2 25.7 24.5 2.8 32 43.6 19.7 23.9 3.4 48 40.2 19.1 21.1 3.8 4 82.4 38.3 44.1 1.8 8 71.2 31.2 40.0 2.2 12 2000 65.3 27.9 37.4 2.5 16 61.5 25.2 36.3 2.7 32 53.5 21.9 31.6 3.4 48 49.3 16.4 32.9 3.8 76 Chang's elastohydrodynamic equation given below was used to derive the theoretical values tabulated on the previous page, the following section specifies the values that were varied and those which were held constant to obtain the film thickness. where, and E G== Pa where 1 E== 1 v2 1 v2 __a +__b Ea Eb F == Nip Loading per Unit Width of roller, pli (e.g., 4/6,8/6,12/6 pli etc.) u == Average ofNip and Roll Velocities, varied as 500,1000,1500 and 2000 fpm. T == Winding Tension, held constant at 0.48 lb/in. throughout experimentation. Pa == Ambient Pressure, 14.7 psi. Rx == Equivalent Radius, calculated as 1.085 inch. E == Calculated Effective Modulus, psi. Ea == Modulus ofNip, 675 psi. Eb == Modulus of Roll, (determined by Covell[5] using contact tests) 8200,12800,19800, 27600,37000 and 47000 psi corresponding to 4,8,12,16,32 and 48 lbfnip loads. f.! == Dynamic Viscosity of Air 2.6E9 Ibfsec/in.2 he == Decompressed Air Film Thickness, from 24 different combinations of velocities and nip loads 77 TABLE VII COMPARISON OF EXPERIMENTAL RESULTS OBTAINED FOR THE RUBBER NIP AND THE HOLLOW ALUMINUM NIP Speed Nip Air collected after one minute after winding* Load winding time: For 1500 ft. == 53 Sees., For 1500mt. == 2Min&30 Sees. Rubber Nip Hollow Aluminum Nip 1500 mts. 1500 ft. 1500 mts. 1500 ft. fpm lbf m!. Jl inch m!. Jl inch m!. Jl inch m!. Jl inch 2000 8 135.8 23.4 55.3 31.2 128 22.1 52.3 29.6 2000 8 125.1 21.6 59.2 33.4     * All data taken/or two different rolls one 1500 mts.and the other 1500ft. long. TABLE VIII RADIAL PRESSURES AS A FUNCTION OF NORMALIZED RADIUS 78 P P P RlRc (using Er stack) (using Er eq) (using Er air) pSI. pSI. pSI. 1 184.1033 111.8518 5.459071 1.032021 154.7302 88.54622 3.216585 1.064448 134.5964 74.32354 3.198831 1.096875 119.5859 67.88147 3.182084 1.129301 107.4788 63.32581 3.166257 1.161728 96.93085 59.50649 3.151277 1.194155* 85.04928 55.73399 3.137076 1.226581 77.19245 51.52646 3.123595 1.259008 66.85479 46.47226 3.11078 1.291435 55.5874 40.1336 3.098582 1.323861 42.92696 31.94269 3.086956 1.356288 28.29851 21.01056 3.07548 1.388715 10.78037 9.783308 2.93404 * Normalized radius corresponding to the midpoint ofthe wound roll. 79 TABLE IX VOLUME OF AIR COLLECTED IMMEDIATELY AFTER WINDING AS A FUNCTION OF THE LENGTH OF THE ROLL UNWOUND* 1500mts. 1500ft. Speed Nip Load Length Vol. of Air length Vol. of Air fpm lbf ft. ml. ft. ml. 1500 63 300 28 3000 100 600 38 2000 8 4500 138 900 46 4920 150 1200 55 1500 64 * Data taken for two different rolls 1500 mts. and 1500ft. long respectively. 80 TABLE X COMPARISON OF THE AIR COLLECTED FROM TWO ROLLS UNWOUND 3 DAYS AFTER AND IMMEDIATELY AFTER WINDING RESPECTIVELY, AS A FUNCTION OF THEIR LENGTHS 3 Days After Winding Immediately After Winding Speed Nip Load fpm Ibf Length Vol. of Air Length Vol. of Air ft. ml. ft. ml. 60 6   124 10   192 13   264 16   300  300 28 340 18   420 21   504 24   592 25   600  600 38 2000 8 684 26   780 28   880 29   900  900 46 984 30   1092 31   1200  1200 55 1204 32   1320 33   1440 34   1500  1500 64 81 TABLE XI AIR LAYER THICKNESS AS A FUNCTION OF TIME FOR THREE SETS OF ROLL SAMPLES Timet Air Layer Thickness (in minutes) (in hours) micro in. X* X*+l X*+3 X*+10 8 16 24 3X24 Sample Roll Number 1 29 25.2 22 17.5 20.8 17.5 16.4 14.2 2 27.4 24.1 22 21.4 16.4 20.8 19.2 12 3 30.2 24.1 20.8 19.2 17.5 17.5 17 14.2 t Data taken for 1500ft. ofweb wound at 2000 jjJm and 8 lbfnip load. *X represents 1min & 53 seconds (the time taken to set up the air collection apparatus). 82 TABLE XII COMPARISON OF THEORETICAL VALUES CALCULATED FROM SQUEEZE FILM DAMPER EQUATION WITH EXPERIMENTAL RESULTS Speed Nip Load Time Air Layer Thickness mIcro In. fpm lbf minutes Theoretical Experimental Deviation 1.9 30.2 30.2 0.0 2.9 14.4 24.1 9.7 4.9 9.1 20.8 11.7 2000 8 11.9 5.1 19.2 14.1 481.9 0.7 17.5 16.8 1441.9 0.4 17 16.6 4321.9 0.2 14.2 14.0 83 The squeeze film damper equation given below was used to obtain the theoretical decrease in the air film thickness as a function of time as tabulated on the previous page t = Ilbe [_1__1] 2W h2 h2 2 1 ~ == Viscosity of Air, 2.6E9 Ibfsec/in2 . P == W/bL == 3 psi. (calculated by Winder v4.0) L == Width ofWeb == 6 in. hI == Initial Air Film Thickness,30.2 micro in. (assumed from experiment) t == Time Period for Decrease of hI to the Next Lower Value, 1,3,10 minutes etc.(from experiment). h2 == Theoretically Calculated Corresponding to the Time Periods Used for Experiment (1,3,10 minutes etc),micro in. 84 TABLE XIII COMPARISON OF THE RANGE OF DIMENSIONLESS PARAMETERS USED BY HAMROCK & DOWSON WITH THOSE ASSOCIATED WITH THE EXPERIMENTS Hamrock & Dowson's Dimensionless Parameters Experimental Dimensionless Parameters Absolute Difference W U W U W U 2.20E04 2.20E03 5.14E09 5.14E08 8.93E03 2.60E02 4.07E04 1.67E05 8.71E03 4.07E04 2.38E02 1.66E05 85 TABLE XIV COMPARISON OF THE RANGE OF DIMENSIONLESS PARAMETERS USED BY CHANG WITH THOSE ASSOCIATED WITH THE EXPERIMENTS Chang's Dimensionless Parameters Experimental Dimensionless Parameters W 6.27E3 0.3135 U 1.304E8 1.63E7 G 6.8 340.1 W 0.0631 0.523 U 1.63E8 6.52E8 G 49.8 53.7 86 TABLE XV INDIVIDUAL EFFECT OF WOUND ROLL MODULUS EbAND NIP MODULUS Ea ON THE AIR LAYER THICKNESS (ho) DERIVED BY HAMROCK AND DOWSON AND BY CHANG FOR THEIR ELASTOHYDRODYNAMIC EQUATIONS Modulus of hofor hofor hofor hofor Roll Eb/Nip Ea Ham&Dow., Chang, Ham&Dow., Chang, psi Ea held const. Ea held const. Ebheld const. Ebheld const. micro inch micro inch micro inch micro inch 100 61.94 90.11 54.71 78.69 500 35.85 49.62 27.38 36.98 1000 30.48 41.58 20.57 27.07 1500 28.40 38.49 17.53 22.74 2000 27.28 36.84 15.72 20.19 2500 26.59 35.81 14.49 18.47 3000 26.11 35.11 13.59 17.23 3500 25.76 34.60 12.90 16.28 4000 25.49 34.21 12.35 15.52 4500 25.28 33.90 11.90 14.91 5000 25.11 33.65 11.53 14.39 5500 24.97 33.45 11.21 13.96 6000 24.85 33.28 10.93 13.58 6500 24.75 33.13 10.69 13.25 7000 24.67 33.01 10.48 12.97 7500 24.59 32.90 10.29 12.71 8000 24.53 32.80 10.12 12.49 8500 24.47 32.72 9.97 12.28 9000 24.42 32.64 9.83 12.10 9500 24.37 32.57 9.71 11.93 10000 24.33 32.51 9.59 11.78 10500 24.29 32.46 11000 24.26 32.41 11500 24.23 32.36 12000 24.20 32.32 12500 24.17 32.28 12800 24.16 32.26 VITA Humair A. Mohammed Candidate for the Degree of Master of Science Thesis: RATE OF AIR ESCAPE FROM ROLLS WOUND AT HIGH SPEEDS WITH A FORCE LOADED NIP Major Field: Mechanical Engineering Biographical: Personal Data: Born in Hyderabad, India, April 20, 1970, the son Mohammed Manzoor Ahmed and Naima Azmath. Education: Graduated from Little Flower High School, Hyderabad, India in June 1985; received Bachelor of Science degree in Mechanical Engineering from Osmania University, Hyderabad, India in June 1992; completed requirements for the Master of Science degree at Oklahoma State University, Stillwater, Oklahoma in May 1995. Professional Experience: Research Assistant, Department of Mechanical and Aerospace Engineering, Oklahoma State University, January, 1994 to March, 1995; Teaching Assistant, August 1993 to December 1993. 



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