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A numerical investigation of the pressure distribution on a Fourdrinier paper machine drainage foil Lepp, Gary 1986

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A NUMERICAL INVESTIGATION OF THE PRESSURE DISTRIBUTION ON A FOURDRINIER PAPER MACHINE DRAINAGE FOIL by GARY LEPP B.A.Sc, The Un i v e r s i t y of B r i t i s h Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1986 © GARY LEPP, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or pub l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mechanical Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e A p r i l 23, 1986 A B S T R A C T A mathematical model and three numerical s o l u t i o n procedures have been developed to predict the pressure d i s t r i b u t i o n on a Four d r i n i e r paper machine drainage f o i l . The present models are based on a numerical s o l u t i o n of the f u l l laminar boundary layer equations. Previous modelling attempts have made use of the boundary layer equations i n various s i m p l i f i e d forms. The s o l u t i o n procedures used were grouped into two categories; the one-dimensional models and the two-dimensional model. The one-dimensional models were based on an approximate i n t e g r a l s o l u t i o n of the boundary layer equations. The two-dimensional model was based on a so l u t i o n of the boundary layer equations using a f i n i t e d i f f e r e n c e method. Experimental measurements of the pressure d i s t r i b u t i o n s and the drainage rates on three tapered f o i l s and two stepped f o i l s were made to provide data f o r comparison with the r e s u l t s of the models. Computed r e s u l t s obtained using the s o l u t i o n procedures generally did not agree w e l l with the experimental r e s u l t s of the present i n v e s t i g a t i o n . The r e s u l t s , however, did predict observed trends i n the pressure d i s t r i b u -tions due to v a r i a t i o n s i n the wire v e l o c i t y and f o i l geometries. Consequently, the models may be considered v i a b l e approaches to pr e d i c t i n g the pressure d i s t r i b u t i o n on a f o i l , though a great deal of further work i s necessary before the s o l u t i o n procedures may be used f o r design purposes. - i i -T A B L E OF C O N T E N T S Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v NOMENCLATURE x i ACKNOWLEDGEMENTS x i v 1. INTRODUCTION 1 1.1 General 1 1.2 Review of L i t e r a t u r e 2 1.3 Need for the Present Work 6 2. EXPERIMENT 7 2.1 General 7 2.2 Scope of the Experiment 7 2.3 Description of the Apparatus 7 2.3.1 O v e r a l l Apparatus 8 2.3.2 F o i l Tested 9 2.4 Instrumentation 10 2.4.1 F o i l Pressure D i s t r i b u t i o n s 10 2.4.2 F o i l Drainage Rates 11 2.4.3 Supply Water Flow Rate 11 2.4.4 Forming Wire Tension 11 2.4.5 Forming Wire Speed 12 2.4.6 Forming Wire Drainage Resistance 12 2.5 Experimental Procedure 13 2.6 Results and Observations 14 3. THEORETICAL ANALYSIS 17 3.1 General 17 3.2 The Governing Equations 19 3.2.1 The Equations of Motion 19 3.2.2 Wire D e f l e c t i o n Model 20 3.3 The One-Dimensional Model 22 3.3.1 The Conservation of Mass Model 22 3.3.1.1 The Model Equations 22 3.3.1.2 Method of Solution 25 3.3.1.3 V e r i f i c a t i o n of the Model 26 3.3.1.4 Results 27 3.3.2 The Momentum Integral Model 30 3.3.2.1 The Equations of Motion 30 3.3.2.2 Method of Solution 36 3.3.2.3 V e r i f i c a t i o n of the Model 38 3.3.2.4 Results 40 - i i i -Page 3.4 The Two-Dimensional Model 44 3.4.1 General 44 3.4.2 The F i n i t e Difference Scheme 45 3.4.3 The F i n i t e Difference Equations 46 3.4.4 Method of Solution 56 3.4.5 V e r i f i c a t i o n of the Model 58 3.4.6 Results 60 4. DISCUSSION 62 4.1 General 62 4.2 Experimental Results 62 4.3 Theoretical Results 65 5. CONCLUSIONS 70 5.1 Recommendations for Further Work 71 FIGURES 74 REFERENCES 146 APPENDIX A 152 APPENDIX B 155 APPENDIX C 159 - i v -FIGURES Page 1. T y p i c a l components and layout of a Four d r i n i e r Paper Machine 74 2. Comparison of the t h e o r e t i c a l r e s u l t s of Taylor [3] to the experimental r e s u l t s of F l e i s c h e r [9]. Results are f o r a 3 degree tapered f o i l , U w = 2000 ft/min 75 3. Experimental apparatus 76 4. Photograph of the experimental apparatus 77 5. Photograph of a tensioner r o l l e r support bolt showing the s t r a i n gauge l o c a t i o n 78 6. Photograph of the headbox 79 7. Photograph of the test section showing the headbox s l u i c e , the f o i l and the f o i l drainage trough 80 8. Photograph of the f o i l s tested 81 9. P r o f i l e s and dimensions of the f o i l s tested 82 10. St r a i n gauge placement and connection on the tension r o l l e r support bolt s 83 11. Measured non-dimensional pressure d i s t r i b u t i o n on f o i l 1 showing the v a r i a t i o n with the wire v e l o c i t y 84 12. Measured non-dimensional pressure d i s t r i b u t i o n on f o i l 2 showing the v a r i a t i o n with the wire v e l o c i t y 85 13. Measured non-dimensional pressure d i s t r i b u t i o n on f o i l 3 showing the v a r i a t i o n with the wire v e l o c i t y 86 14. Measured non-dimensional pressure d i s t r i b u t i o n on f o i l 4 showing the v a r i a t i o n with the wire v e l o c i t y 87 15. Measured non-dimensional pressure d i s t r i b u t i o n on f o i l 5 showing the v a r i a t i o n with the wire v e l o c i t y 88 16. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 500 ft/min 89 17. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 750 ft/min 90 18. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 1000 ft/min 91 - v -Page 19. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 1250 ft/min 92 20. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 1500 ft/min 93 21. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 2000 ft/min 94 22. Measured non-dimensional f o i l drainage rates as a function of wire speed f o r f o i l s 1,2,3,4 and 5 95 23. Typical r e p e a t a b i l i t y of pressure d i s t r i b u t i o n on f o i l 1 f or wire v e l o c i t i e s of 750 ft/min, 1000 ft/min and 1500 ft/min ... 96 24. Typical r e p e a t a b i l i t y of pressure d i s t r i b u t i o n on f o i l 3 f or wire v e l o c i t i e s of 500 ft/min, 750 ft/min and 1000 ft/min .... 97 25. Diagram of a f o i l showing the geometric nomenclature used i n the development of the mathematical models. Lengths i n the y - d i r e c t i o n are greatly exaggerated 98 26. Diagram of a f o i l showing the dynamic quantities used i n the development of the mathematical models 99 27. Diagram showing the forces on a small section of the forming wire 100 28. Flowchart of the so l u t i o n procedure for the conservation of mass model 101 29. Comparison of the calculated v e l o c i t y p r o f i l e for the flow over a f l a t p l ate to the Blasius p r o f i l e . Results obtained using the conservation of mass model 102 30. Comparison of the calculated v e l o c i t y p r o f i l e f o r the flow between moving p a r a l l e l f l a t plates to the exact s o l u t i o n . Calculated v e l o c i t y p r o f i l e obtained using the conservation of mass model 103 31. Calculated pressure d i s t r i b u t i o n s for f o i l 1 using the conservation of mass model showing the v a r i a t i o n with the wire v e l o c i t y . Wire d e f l e c t i o n e f f e c t s were not included i n the s o l u t i o n 104 32. Comparison of calculated drainage rates for f o i l 1 with and without wire d e f l e c t i o n included. Results obtained using the conservation of mass model 105 - v i -Page 33. Calculated pressure d i s t r i b u t i o n s for f o i l 1 using the conservation of mass model showing the v a r i a t i o n with the wire v e l o c i t y . Wire d e f l e c t i o n e f f e c t s were included i n the s o l u t i o n 106 34. Ex i t v e l o c i t y p r o f i l e s f o r f o i l 1 as a function of wire v e l o c i t y . Results obtained using the conservation of mass model 107 35. Calculated v e l o c i t y p r o f i l e s along f o i l 1 for U w = 500 ft/min. Results obtained using the conservation of mass model without wire d e f l e c t i o n e f f e c t s 108 36. Calculated v e l o c i t y p r o f i l e s along f o i l 1 for U w = 500 ft/min. Results obtained using the conservation of mass model with wire d e f l e c t i o n e f f e c t s included 109 37. Comparison of calculated pressure d i s t r i b u t i o n s f o r f o i l s 1,2 and 3 f o r u"w = 500 ft/min. Results obtained using the conservation of mass model without wire d e f l e c t i o n e f f e c t s ... 110 38. Typical s e n s i t i v i t y of the mass conservation model to the i n i t i a l f o i l - w i r e gap. Results are f o r f o i l 1 at a wire v e l o c i t y of 500 ft/min I l l 39. Flow chart of the pressure evaluation subroutine for the momentum i n t e g r a l model s o l u t i o n procedure 112 40. Comparison of the calculated v e l o c i t y p r o f i l e for the flow between moving p a r a l l e l f l a t plates to the exact s o l u t i o n . Calculated v e l o c i t y p r o f i l e obtained using the momentum i n t e g r a l model 113 41. Comparison of the calculated v e l o c i t y p r o f i l e for the flow over a f l a t p l a t e to the Blasius p r o f i l e . Results obtained using the momentum i n t e g r a l model with the boundary conditions x=0, p=p„, dp/dx=0 114 42. Comparison of the calculated v e l o c i t y p r o f i l e f o r the flow over a f l a t p late to the Blasius p r o f i l e . Results obtained using the momentum i n t e g r a l model with the boundary conditions x=0, p=pQ; x=L, p=p„ 115 43. Calculated pressure d i s t r i b u t i o n f o r f o i l 1 using the momentum i n t e g r a l model showing the v a r i a t i o n with the wire v e l o c i t y . Wire d e f l e c t i o n e f f e c t s were not included i n the s o l u t i o n 116 - v i i -Page 44. Comparison of calculated pressure d i s t r i b u t i o n s f o r f o i l s 1,2,3,4 and 5 f o r U w = 500 ft/min. Results obtained using the momentum i n t e g r a l model without wire d e f l e c t i o n e f f e c t s . The pressure d i s t r i b u t i o n along f o i l s 4 and 5 i s zero 117 45. Calculated pressure d i s t r i b u t i o n s for f o i l 1 using the momentum i n t e g r a l model showing the v a r i a t i o n with the wire v e l o c i t y . Wire d e f l e c t i o n e f f e c t s were included i n the s o l u t i o n 118 46. Comparison of the pressure d i s t r i b u t i o n on f o i l 1 for U = 500 ft/min with and without wire d e f l e c t i o n e f f e c t s included i n the s o l u t i o n . Results obtained using the momentum i n t e g r a l model 119 47. Calculated e x i t v e l o c i t y p r o f i l e s for f o i l 1 as a function of wire v e l o c i t y using the momentum i n t e g r a l model without wire d e f l e c t i o n e f f e c t s 120 48. Calculated e x i t v e l o c i t y p r o f i l e s f o r f o i l 1 as a function of wire v e l o c i t y using the momentum i n t e g r a l model with wire d e f l e c t i o n e f f e c t s included i n the so l u t i o n 121 49. Typical s e n s i t i v i t y of the momentum i n t e g r a l model to the i n i t i a l f o i l - w i r e gap height. Results are f o r f o i l 1 and a wire v e l o c i t y of 500 ft/min 122 50. V a r i a t i o n of the pressure d i s t r i b u t i o n s on f o i l 1 as a function of the drainage resistance. Results calculated using the momentum i n t e g r a l model with no wire d e f l e c t i o n e f f e c t s and a wire v e l o c i t y of 500 ft/min 123 51. V a r i a t i o n i n the predicted drainage rate as a function of the drainage resistance. Results obtained using the momentum i n t e g r a l model for f o i l 1 and a wire v e l o c i t y of 500 ft/min. Wire d e f l e c t i o n e f f e c t s were not included 124 52. E f f e c t of the wire tension on the pressure d i s t r i b u t i o n along f o i l 1. Momentum i n t e g r a l model, U w = 500 ft/min 125 53. E f f e c t of wire tension on the calculated drainage rate f o r f o i l 1. Momentum i n t e g r a l model, U w = 500 ft/min 126 54. E f f e c t of f l u i d kinematic v i s c o s i t y on the pressure d i s t r i b u t i o n along f o i l 1. Momentum i n t e g r a l model, U w = 500 ft/min, no wire d e f l e c t i o n 127 55. E f f e c t of specifying an i n i t i a l drainage v e l o c i t y on the pressure d i s t r i b u t i o n along f o i l 1. Momentum i n t e g r a l model, U w = 500 ft/min, no wire d e f l e c t i o n 128 - v i i i -Page 56. T y p i c a l f i n i t e difference mesh used f o r the so l u t i o n of the two-dimensional model 129 57. Flowchart for the two-dimensional model s o l u t i o n procedure ... 130 58. Comparison of the calculated v e l o c i t y p r o f i l e f o r flow over a f l a t p l a t e to the Blasius p r o f i l e . Calculated r e s u l t s obtained using the two-dimensional model 131 59. Comparison of the calculated v e l o c i t y p r o f i l e s for flow i n a rectangular duct to the exact s o l u t i o n showing the development of the calculated v e l o c i t y p r o f i l e s . Results obtained using the two-dimensional model 132 60. Comparison of the calculated v e l o c i t y p r o f i l e s f o r flow between moving p a r a l l e l f l a t plates to the exact s o l u t i o n . P r o f i l e s show the development of the calculated v e l o c i t i e s . Results obtained using the two-dimensional model 133 61. T y p i c a l pressure d i s t r i b u t i o n s along f o i l s 1,2 and 5 c a l -culated using the two-dimensional model. U = 500 ft/min, V w i - 0.05 f t / s e c , k d r - 2 . 5 x l 0 ~ 8 f t 134 62. V e l o c i t y p r o f i l e s f o r f o i l 2 calculated using the two-dimensional model up to the point of s o l u t i o n f a i l u r e . U w = 500 ft/min; V w i = 0.05 f t / s e c , k d r = 2 . 5 x l 0 ~ 8 f t 135 63. T y p i c a l pressure d i s t r i b u t i o n s along f o i l 1 showing the v a r i a t i o n i n the d i s t r i b u t i o n s as a function of the wire v e l o c i t y . Results obtained using the two-dimensional model. U w = 500 ft/min; V w i = 0.05 f t / s e c , k d r = 2 . 5 x l 0 ~ 8 f t 136 64. T y p i c a l e x i t v e l o c i t y p r o f i l e s for f o i l 1 showing the v a r i a t i o n i n the p r o f i l e s with the wire v e l o c i t y . Results obtained using the two-dimensional model. U w = 500 ft/min; V w i = 0 , 0 5 f t / s e c > k d r = 2 - 5 x l 0 " 8 f t 137 65. Comparison of the r e s u l t s of F l e i s c h e r [9] to r e s u l t s of the present i n v e s t i g a t i o n f o r f o i l 3 at a wire v e l o c i t y of 500 ft/min 138 66. Photograph of the flow over f o i l 2 at a wire v e l o c i t y of 1000 ft/min. Note the presence of a 'dry-out' l i n e i n d i c a t -ing a l l the f l u i d has been drawn through the forming wire .... 139 67. Photograph of the flow over f o i l 1 at a wire v e l o c i t y of 1000 ft/min. Note the continuous f i l m of water over the f o i l 140 - i x -Page 68. Comparison of the pressure d i s t r i b u t i o n on f o i l 1 f o r U w = 500 ft/min c a l c u l a t e d using the conservation of mass and the momentum i n t e g r a l models. Wire d e f l e c t i o n e f f e c t s were included i n the solutions 141 69. Comparison of the drainage rates on f o i l 1 predicted by the one-dimensional models. Wire d e f l e c t i o n e f f e c t were included i n the solutions 142 70. Comparison of the ex i t v e l o c i t y p r o f i l e s on f o i l 1 predicted by the one-dimensional models. U w = 500 ft/min; Wire d e f l e c t i o n e f f e c t s were included 143 71. Typ i c a l v e l o c i t y p r o f i l e s along f o i l 1 when an i n i t i a l drainage v e l o c i t y has been s p e c i f i e d . U w = 500 ft/min; V w £ =0.1 f t / s e c . Momentum i n t e g r a l model, wire d e f l e c t i o n e f f e c t s included 144 72. Typ i c a l v e l o c i t y p r o f i l e s along f o i l 1 when no i n i t i a l drainage v e l o c i t y has been s p e c i f i e d . U w = 500 ft/min; Momentum i n t e g r a l model, wire d e f l e c t i o n e f f e c t s included .... 145 73. Calibrated headbox s l u i c e jet v e l o c i t y as a function of the o r i f i c e pressure drop 154 74. S t r a i n versus load c a l i b r a t i o n curve for tension r o l l e r support b o l t 1 157 75. Str a i n versus load c a l i b r a t i o n curve f o r tension r o l l e r support b o l t 2 158 - x -NOMENCLATURE A c o e f f i c i e n t matrix of the system of equations for the two-dimensional model a sub-matrix within the coefficient matrix A" B vector of the r ight hand sides of the equations for the two-dimensional model S sub-matrix within the coefficient matrix 5 b sub-vector within the vector B b non-dimensional viscosity term in the equation of motion for the two-dimensional model c sub-matrix within the coefficient matrix A C j . c ^ C g constants in the equations for the one-dimensional model velocity profiles f non-dimensional streamwise velocity in the momentum integral model f velocity gradient (—) in the two-dimensional model dy h cross stream distance between the forming wire and the f o i l hj distance i n the cross stream direction between nodes ( i , j - l ) and ( i , j ) k d r drainage resistance of the forming wire distance i n the streamwise d i r e c t i o n between nodes ( i - l , j ) and ( i , j ) L f o i l length L x f o i l pitch p pressure p a , p a t m atmospheric pressure p 2 pressure gradient term (-^i) used i n the momentum integral j i d x model - x i -Q f l o w r a t e r i , r 2 ' r 3 ' r i t r * & h t hand s i d e s o f t h e e q u a t i o n o f m o t i o n f o r t h e two -d i m e n s i o n a l mode l R r a d i u s o f c u r v a t u r e o f t h e f o r m i n g w i r e „ i - l p a r a m e t e r i n t he e q u a t i o n s o f m o t i o n f o r t h e t w o - d i m e n s i o n a l K j - l / 2 m o d e l Re R e y n o l d s number s a r c l e n g t h o f a s m a l l segment o f t h e f o r m i n g w i r e s 1 > s 2 , s 3 , p a r a m e t e r s i n t h e e q u a t i o n s o f m o t i o n f o r t h e t w o - d i m e n s i o n a l s ^ . S g j S g , m o d e l S 7 > S 8 T f o r m i n g w i r e t e n s i o n u s t r e a m w i s e v e l o c i t y component v c r o s s s t r e a m v e l o c i t y component W f o i l w i d t h x s t r e a m w i s e d i s t a n c e measu red f r o m t h e f o i l l e a d i n g edge X* s o l u t i o n v e c t o r f o r t h e s y s t e m o f e q u a t i o n s o f t h e t w o -d i m e n s i o n a l mode l x s u b - v e c t o r o f t h e X v e c t o r y c r o s s s t r e a m d i s t a n c e measu red f r o m t h e u n d e f l e c t e d f o r m i n g w i r e 6 u , 6 v , 6 f , 6 p Newton i t e r a t e s r e p r e s e n t i n g s m a l l c h a n g e s i n u , v , f and p i n t h e t w o - d i m e n s i o n a l mode l ri n o n - d i m e n s i o n a l c r o s s s t r e a m d i s t a n c e \i dynamic v i s c o s i t y v k i n e m a t i c v i s c o s i t y v t t u r b u l e n t v i s c o s i t y p a r a m e t e r i n t h e e q u a t i o n s o f m o t i o n f o r t h e monentum i n t e g r a l m o d e l - x i i -p e density angle between the endpoints of a small segment of forming wire Subscripts w indicates value of quantity at the forming wire j i n d i c a t e s q u a n t i t y a t the j node i n the c r o s s stream d i r e c t i o n p indicates p a r t i a l d i f f e r e n t i a t i o n with respect to p p v indicates p a r t i a l d i f f e r e n t i a t i o n with respect to 5LP-x dx x indicates p a r t i a l d i f f e r e n t i a t i o n with respect to x Superscripts i i n d i c a t e s q u a n t i t y a t d i r e c t i o n n indicates quantity at the * indicates non-dimensional the i node i n t h e s t r e a m w i s e n t h i t e r a t i o n quantity - x i i i -ACKNOWLEDGEMENTS The author would l i k e to express h i s sincere thanks to h i s supervisor, Dr. E.G. Hauptmann. His guidance and encouragement was provided throughout a l l phases of the work, without which t h i s work would not have been p o s s i b l e . The author would also l i k e to express h i s gratitude to Dr. I.S. Gartshore f o r h i s i n t e r e s t and h i s comments regarding t h i s work. Spe c i a l thanks go to the s t a f f and fellow graduate students of the Department of Mechanical Engineering. Their assistance and advice i s greatly appreciated. F i n a l l y , the author would l i k e to thank h i s parents f o r t h e i r continuous support and encouragement provided during the course of t h i s work. - x i v -1. I . I N T R O D U C T I O N 1.1 General The primary process i n the making of paper i s the removal of water i n a c o n t r o l l e d manner from a s l u r r y of pulp and water. The removal of water takes place i n three stages. These are the formation, the pressing, and the drying of the paper sheet. The energy requirements and the corres-ponding cost of removing water increase dramatically as the paper sheet moves along the machine from the forming section to the dryers. The i n i t i a l water removal i n the forming section of the machine also has a s i g n i f i c a n t influence on the properties of the paper and la r g e l y determines the q u a l i t y of the f i n a l product. As a r e s u l t , a great deal of emphasis has been placed on improving the drainage c h a r a c t e r i s t i c s of the forming s e c t i o n of paper machines. Modern paper machines i n use are generally one of two types. These are twin wire and s i n g l e wire, or Fourdrinier, paper machines. Of the two types of machines, twin wire formers are generally superior to Four d r i n i e r paper machines. Water i s drained much quicker on a twin-wire machine due to the squeezing of the paper sheet between the wires. The q u a l i t y of paper made on a twin-wire machine i s also much better as the paper has a f i n i s h e d surface on both sides. Despite the advantages of using twin-wire machines, the cost of acquiring and i n s t a l l i n g one to replace an e x i s t i n g F o u r d r i n i e r machine i s usually p r o h i b i t i v e . As a r e s u l t , most twin-wire machines being brought into service are done so as part of an o v e r a l l expansion of production c a p a b i l i t i e s . For operators of Four d r i n i e r paper machines, there i s an emphasis on developing methods of improving the drainage c h a r a c t e r i s t i c s of the e x i s t i n g machines. 2. A F o u r d r i n i e r p a p e r m a c h i n e c o n s i s t s o f a m o v i n g f a b r i c o r w i r e mesh s c r e e n ( r e f e r r e d t o a s t h e f o r m i n g medium o r w i r e ) w h i c h i s s u p p o r t e d on r o l l e r s ( F i g u r e 1). A t one end o f t h e m a c h i n e t h e p u l p and w a t e r s l u r r y i s s p r e a d o n t o t h e f o r m i n g w i r e . As t h e s l u r r y moves down t h e m a c h i n e on t h e f o r m i n g w i r e , w a t e r i s d r a i n e d t h r o u g h t h e w i r e l e a v i n g a p a p e r mat on t h e u p p e r s u r f a c e . D r a i n a g e t a k e s p l a c e a s a r e s u l t o f g r a v i t y and a d d i -t i o n a l f o r c e s p r o v i d e d by b o t h p a s s i v e and a c t i v e d r a i n a g e d e v i c e s . A c t i v e d e v i c e s a r e u s u a l l y i n t h e f o r m o f vacuum s u c t i o n b o x e s l o c a t e d on t h e u n d e r s i d e o f t h e f o r m i n g w i r e . P a s s i v e d e v i c e s t a k e t h e f o r m o f r o l l e r s a n d s t a t i o n a r y d r a i n a g e f o i l s l o c a t e d u n d e r t h e w i r e . B e c a u s e o f t h e l o w e r e n e r g y r e q u i r e m e n t s o f p a s s i v e d r a i n a g e d e v i c e s , m a c h i n e o p e r a t o r s f a v o u r t h e i r u s e o v e r a c t i v e m e t h o d s . I n r e c e n t y e a r s , t h e s u p e r i o r d r a i n a g e c h a r a c t e r i s t i c s o f f o i l s h a v e r e s u l t e d i n t h e i r u s e a l l b u t e l i m i n a t i n g t h e u s e o f t a b l e r o l l s . D e s p i t e t h e w i d e s p r e a d u s e o f f o i l s on e x i s t i n g F o u r d r i n i e r p a p e r m a c h i n e s , v e r y l i t t l e k n o w l e d g e o r u n d e r s t a n d i n g o f t h e d r a i n a g e c h a r a c t e r i s t i c s o f f o i l s e x i s t s . F u r t h e r m o r e , no known means o f a c c u r a t e l y p r e d i c t i n g t h e d r a i n a g e p r o v i d e d by a f o i l as been p r o v i d e d . " 1*2 Review of Literature A c o m p r e h e n s i v e r e v i e w o f a v a i l a b l e l i t e r a t u r e p e r t a i n i n g t o f o i l s was c a r r i e d o u t . R e s u l t s o f t h i s s u r v e y i n d i c a t e v e r y l i t t l e work h a s b e e n done t o d e v e l o p a n a c c u r a t e a n a l y t i c a l m o d e l o f t h e p r e s s u r e d i s t r i -b u t i o n on a d r a i n a g e f o i l . Much o f t h e c u r r e n t k n o w l e d g e o f f o i l d r a i n a g e was d e v e l o p e d a s a r e s u l t o f a n a l y t i c a l and e x p e r i m e n t a l s t u d i e s o f t a b l e r o l l s . I n t h i s c o n n e c t i o n , W r i s t [1] p r o p o s e d a d r a i n a g e m o d e l f o r t a b l e r o l l s u s i n g t h e 3. analogy of a piston being withdrawn from a c y l i n d e r having a porous top. In l a t e r work, Wrist attempted to formulate a mathematical model of t h i s concept based on an analysis of the flow using B e r n o u l l i ' s theorem. An extension of t h i s concept to f o i l s was attempted by assuming a f o i l could be regarded as a stationary table r o l l . Several other independent studies by Taylor [2],[3], Meyer [4], Burkhard and Wrist [5] and V i c t o r y [6] attempted to put forward models f o r the drainage at a table r o l l . A n a l y t i c a l models f o r f o i l drainage have been suggested by Taylor [3], Meyer [7], and Ernst [8]. Taylor formulated a model based on the momentum equation f o r a boundary la y e r flow. In h i s a n a l y s i s , Taylor made the assumption the flow was f u l l y turbulent allowing him to tr e a t the flow as a plug flow i n which the v e l o c i t y was a function of x only. In doing so, Taylor neglected f l u i d f r i c t i o n e f f e c t s at the f o i l and the wire thus disregarding the no s l i p boundary conditions on the v e l o c i t y p r o f i l e . This s i m p l i f i c a t i o n permitted the reduction of the momentum equation to a simple ordinary d i f f e r e n t i a l equation which was then solved a n a l y t i c a l l y . Taylor then used the experimental conditions reported by Burkhard and Wrist [5] to c a l c u l a t e t h e o r e t i c a l r e s u l t s f o r the f o i l s at the operating conditions examined by Burkhard and Wrist. In general, Taylor obtained r e s u l t s t y p i c a l of those shown i n Figure 2. As can be seen, Taylor's r e s u l t i s i n poor agreement with the unpublished experimental r e s u l t s of F l e i s c h e r [9]. Taylor f i r s t suggested t h i s discrepancy was most l i k e l y due to the e f f e c t s of the f l u i d f r i c t i o n . To explore t h i s hypothesis, Taylor introduced f l u i d f r i c t i o n e f f e c t s i n t o h i s c a l c u l a t i o n s through the a d d i t i o n of a drag term to h i s model. The r e s u l t s of t h i s a d d i t i o n were only p a r t i a l l y successful i n that the magnitude of the maximum suction was reduced to a value consistent with the findings of Burkhard and Wrist, though the o v e r a l l shape of the pressure d i s t r i b u t i o n remained e s s e n t i a l l y unchanged. Thus Taylor suggested the discrepancy i n the r e s u l t s may be due to the e f f e c t s of d e f l e c t i o n of the forming wire over the f o i l . This 4. hypothesis, however, was not explored. Meyer [7] performed an analysis of the pressure d i s t r i b u t i o n on a f o i l by s o l v i n g the conservation of mass equation. Meyer's analysis proceeded by in t e g r a t i n g the c o n t i n u i t y equation i n the cross stream d i r e c t i o n and assuming the drainage v e l o c i t y through the forming wire obeyed the r e l a t i o n s h i p dr . v = (p-p ) w 11 a' A v e l o c i t y p r o f i l e was then c a l c u l a t e d by s o l v i n g the s i m p l i f i e d momentum equation 3 2u dp The r e s u l t a n t p r o f i l e was then used i n the c o n t i n u i t y equation and the i n t e g r a t i o n was completed. Integration of the c o n t i n u i t y equation y i e l d e d an expression f o r the pressure d i s t r i b u t i o n which was then solved using a s e r i e s s o l u t i o n based on Bessel functions. The r e s u l t s obtained by Meyer agreed i n magnitude with those of Taylor, though Meyer's s o l u t i o n d i d observe the condition that the pressure at the leading edge of the f o i l must be equal to the ambient pressure. A d i r e c t comparison of Ernst and Meyer's r e s u l t s to those of Taylor could not be made, however, due to d i f f e r e n c e s i n the operating conditions and f o i l geometries used i n t h e i r analyses. Furthermore, i t was not c l e a r exactly what conditions and geometries were used by Taylor, Ernst and Meyer. Ernst [8] attempted to model the pressure d i s t r i b u t i o n on a f o i l using an approach i d e n t i c a l to that of Meyer except that a s o l u t i o n to the r e s u l t a n t expression for the pressure d i s t r i b u t i o n was obtained numeric-a l l y as opposed to a n a l y t i c a l l y . Ernst's a n a l y s i s , however, did d i f f e r from Meyer's i n that he extended h i s analysis to include the e f f e c t s of the forming wire d e f l e c t i o n . This was done by developing an expression 5. for the forming wire d e f l e c t i o n as a function of the pressure d i s t r i b u t i o n along the f o i l assuming the forming wire had no f l e x u r a l r i g i d i t y . Form-in g wire e f f e c t s were incorporated i n Ernst's s o l u t i o n by i t e r a t i v e l y evaluating the pressure d i s t r i b u t i o n and the forming wire d e f l e c t i o n , then using the r e s u l t s to evaluate a new pressure d i s t r i b u t i o n . As would be expected, Ernst obtained r e s u l t s that agreed well with the r e s u l t s of Meyer f o r both the pressure magnitudes and the shape of the pressure d i s t r i b u t i o n s . A d i r e c t comparison of t h e i r r e s u l t s could not be made, however, due to diff e r e n c e s i n the assumed f o i l geometries and operating conditions. Detailed experimental i n v e s t i g a t i o n s on table r o l l s have been conducted by Burkhard and Wrist [5] and Bennett [10]. As part of t h e i r experimental studies, both Burkhard and Wrist, and Bennett performed measurements on stationary r o l l s . Burkhard and Wrist, to s i m p l i f y the d i f f i c u l t i e s of measuring the pressure d i s t r i b u t i o n on a stationary r o l l e r , simulated the r o l l e r using a tapered device which they r e f e r r e d to as a f o i l . As a r e s u l t of t h e i r studies, patents were taken out on the f o i l by Burkhard and Wrist. A d d i t i o n a l experimental studies have been performed by F l e i s c h e r [9], Cadieux [11], and Walser [12]. In l a t e r years, numerous experimental studies have been conducted to q u a l i t a t i v e l y compare f o i l s to table r o l l s . It has been hinted that f o i l s may produce peak suctions of up to — P^ w 2 or one h a l f that of table r o l l s , but the pressure acts over a much greater length. As a r e s u l t , net drainage rates f o r f o i l s are equal to or greater than that of table r o l l s . To t h i s date, no a d d i t i o n a l known attempts have been made to model the pressure d i s t r i b u t i o n on drainage f o i l s . Though a great deal of experimental and t h e o r e t i c a l research has undoubtedly been conducted by 6. both paper manufacturers and the manufacturers of drainage f o i l s , l i t t l e of this work has been published due to the proprietary nature of the research. Consequently, there i s very l i t t l e published information regarding studies done on the drainage properties of f o i l s . 1.3 Need for the Present Work The primary function of a paper machine i s to drain the water from a paper slurry to form a paper sheet. To properly design a paper machine and provide controlled drainage of the water from the paper sheet, designers require the a b i l i t y to accurately predict the drainage provide by the drainage devices on the machine. The purpose of the present investigation was to develop a numerical model which would permit the accurate prediction of the pressure distribu-tion, and hence the drainage provided by a f o i l . The present work attempts to improve existing analytical models by including both viscous forces and the effect of deflection of the forming wire in the model. To verify the accuracy of the models developed, comparison of the numerical results to experimental results was done. Because of a shortage of published experimental data, a simple experiment to obtain data for comparison purposes was also deemed to be a necessary part of the present investigation. 7. I I . EXPERIMENT 2.1 General Experimental in v e s t i g a t i o n s of the pressure d i s t r i b u t i o n on drainage f o i l s have been published by Burkhard and Wrist [5] and Bennett [10]. A d d i t i o n a l experimental data have also been presented i n an unpublished papers by F l e i s c h e r [10] and Cadieux and Bachand [10] . Of the studies presented, only the in v e s t i g a t i o n s of F l e i s c h e r and Cadieux and Bachand dealt e x c l u s i v e l y with drainage f o i l s . The studies of Burkhard and Wrist, and Bennett dealt primarily with table r o l l s , with an extension of t h e i r i n v e s t i g a t i o n s to drainage f o i l s by considering stationary table r o l l s . In a l l cases, i n s u f f i c i e n t d e t a i l with regard to the experimental conditions and technique used was presented to make the data of any use f o r comparison with t h e o r e t i c a l r e s u l t s of the present i n v e s t i g a t i o n . 2.2 Scope of the Experiment The purpose of the experiment was to provide data to be used f o r comparison to the r e s u l t s of the numerical a n a l y s i s . The primary aim of the experiment was to obtain pressure d i s t r i b u t i o n s on a v a r i e t y of d i f f e r e n t shaped f o i l s under various operating conditions. It was necessary that the data obtained be representative of actual f o i l s , therefore the experiment needed to simulate the operation of a f o i l as c l o s e l y as po s s i b l e . 2.3 D e s c r i p t i o n of Apparatus An experimental test r i g was designed an constructed to simulate as c l o s e l y as possible the actual operating conditions of a drainage f o i l . 8 . Design considerations included providing a means to obtain the desired data as e a s i l y as possible without s a c r i f i c i n g the goal of simulating a c t u a l operating conditions. To meet these goals, a t e s t r i g was b u i l t which f a c i l i t a t e d measurements on f u l l scale f o i l s at t y p i c a l paper machine operating speeds. The only departure made from actual machine conditions was the use of water as the working f l u i d as opposed to a pulp s l u r r y . Because of the s i g n i f i c a n t v a r i a t i o n i n the drainage resistance of a paper sheet from the wet to the dry end of a paper machine, t h i s was not considered to be a s i g n i f i c a n t departure from actual conditions. 2.3*1 Overall Apparatus The experimental apparatus consisted of a large aluminum frame work to which f i v e 26 inch wide by 4.5 inch diameter pvc r o l l e r s were mounted as shown i n Figures 3 and 4. Passing around the r o l l e r s was a 24 inch wide synthetic f a b r i c b e l t made from a piece of a t i s s u e machine forming wire. The r o l l e r s were arranged i n a manner which provided a h o r i z o n t a l t e s t section approximately 24 inches long. Drive f o r the r o l l e r - b e l t assembly was provided by a 1 horsepower e l e c t r i c motor. The motor was connected to the drive r o l l e r by means of step pulleys and a V-belt. Motor speed was c o n t r o l l e d using a Variac. This arrangement permitted b e l t speeds from 500-2000 feet per minute. Belt tension was c o n t r o l l e d by a tensioning r o l l e r . The tensioning r o l l e r was suspended from two b o l t s , one at each end of the r o l l e r (Figure 5). The support bol t s allowed the p o s i t i o n of the tensioning r o l l e r to be adjusted over a range of 6 inches. This corresponded to a b e l t tension of from 0-500 pounds per foot. To permit the measurement of the belt tension, s t r a i n gauges were mounted on the support bol t s such that the 9. load on the b o l t s could be determined. By d i f f e r e n t i a l l y adjusting the length of the b o l t s , the tensioning r o l l e r could also be used to c o n t r o l the tracking of the forming wire on the r o l l e r s . 3 A headbox having a s l u i c e of y^- x 6 inches wide was mounted at the upstream end of the t e s t s ection (Figure 6). The headbox was positioned such that a t h i n f i l m of water could be d i s t r i b u t e d onto the upper surface of the forming wire i n the t e s t section. Water was supplied to the headbox v i a a 4 inch water main. Flow rate was c o n t r o l l e d by means of a gate value mounted on the water supply pipe and measured using an o r i f i c e plate and a d i f f e r e n t i a l mercury manometer. The f o i l s tested were mounted on a support bracket beneath the t e s t s e c t i o n as shown i n Figure 7. The f o i l s were located 12 inches downstream from the headbox s l u i c e and positioned such that the f o i l was i n contact with the underside of the forming wire. To f a c i l i t a t e measurement of the drainage rates obtained by the f o i l s , water drained by the f o i l s was c o l l e c t e d i n a drainage trough mounted beneath the forming wire and j u s t downstream from the f o i l (Figure 7). Water c o l l e c t e d i n the drainage trough was channelled i n t o a 45 g a l l o n drum mounted on a scale. Flow rates were obtained by recording the time i t took to f i l l the drum u n t i l i t contained a known volume, by weight, of water. 2.3.2 F o i l s Tested Pressure d i s t r i b u t i o n s and drainage rates were obtained f o r 5 f o i l s of d i f f e r e n t cross sections. Each f o i l was machined from a block of u l t r a high molecular weight p l a s t i c . A photograph of the f o i l s tested i s given i n Figure 8. A l l f o i l s had o v e r a l l dimensions of 9 inches wide, 6 inches 1 0 . long and 1.5 inches t h i c k . F o i l s 1, 2 and 3 were tapered f o i l s having a f l a t leading edge section 1 inch long and 5.5 inch tapered sections of taper angle 1, 2 and 4 degrees r e s p e c t i v e l y . F o i l s 4 and 5 were stepped f o i l s . F o i l 4 had a f l a t leading edge section 2.5 inches long and a 4 inch stepped section having a step height of .25 inches. F o i l 5 had a 1 inch f l a t leading s e c t i o n and was stepped 1/16 inch over the remaining 5.5 inches. A summary of the f o i l dimensions may be found i n Figure 9. Along the l o n g i t u d i n a l center l i n e of each f o i l , 1/16 inch diameter pressure taps were d r i l l e d through the f o i l . The pressure taps were located at 1/4 inch centers r e s u l t i n g i n a t o t a l of 25 pressure taps on each f o i l . 2.4 Instrumentation and Measurements Because of the l i m i t e d nature and scope of the experiment, complex instrumentation was replaced wherever possible with techniques of a simple nature f o r obtaining the desired measurements. Descriptions of the measurements made and the methods used to obtain them follow. 2.4.1 F o i l Pressure Distributions The pressure d i s t r i b u t i o n on the f o i l was obtained using d i f f e r e n t i a l water manometers. A manometer tube was connected to each of the 25 pressure taps located on the f o i l allowing the pressure d i s t r i b u t i o n on the f o i l to be observed while the experiment was In progress. Pressures were determined by measuring the head of water i n each of the manometers. The head was measured to within 1/16 of an inch r e s u l t i n g i n an error of approximately 1 to 2 percent. 11. 2.A.2 Foil Drainage Rates As mentioned previously, water drained by the f o i l was c o l l e c t e d i n a drainage trough located downstream from the f o i l . Water was channelled from the drainage trough through a short length of pipe i n t o a 45 g a l l o n drum. The drum was placed on a scale allowing the weight of the drum and the water i n i t to be measured. To determine the drainage flow rate, the time required to drain a given volume, by weight, of water was measured. The measured time and volume of water drained was then used to c a l c u l a t e the flow rate. O v e r a l l measurement err o r of t h i s method was determined to be 5 percent though the technique proved to be inappropriate as w i l l be discussed l a t e r . 2.4.3 Supply Water Flow Rate The flow rate of the water del i v e r e d to the headbox i n l e t was deter-mined using a t h i n , square edge o r i f i c e p late mounted between two pipe flanges on the water supply pipe. Pressure taps, located at D and 1/2D, were connected to a d i f f e r e n t i a l mercury manometer. Flow rates were determined by measuring the d i f f e r e n t i a l pressure across the o r i f i c e p l a t e and c a l c u l a t i n g the flow rate. To eliminate the need to c a l i b r a t e the o r i f i c e p late, the o r i f i c e was designed and manufactured to the standards set out i n the ASME F l u i d Meters Handbook [13]. This technique allowed the flow rate to be determined within an e r r o r of approximately 8 percent. The o r i f i c e p l a t e s p e c i f i c a t i o n s and d e t a i l s of the flow rate c a l c u l a t i o n s can be found i n Appendix A. 2.4.4 Forming Wire Tension The forming wire tension was determined by measuring the load on the tensioning r o l l e r support b o l t s . To obtain the load on the support b o l t s , 12. each b o l t had four f l a t surfaces machined forming a square cross section (Figure 10). A s t r a i n gauge was attached to each of the f l a t surfaces and the gauges were connected to form a four gauge bridge. This arrangement permitted the d i r e c t measurement of a x i a l s t r a i n i n the bolts as s t r a i n components due to bending of the b o l t s were cancelled by the bridge arrangement. Bolt s t r a i n s were measured using a Vishay Instruments Model P-350A portable s t r a i n i n d i c a t o r and a model SB-1 Switch and Balance u n i t . To convert the b o l t s t r a i n s to loads, the b o l t s were c a l i b r a t e d by measur-ing the s t r a i n f o r known loads. The b o l t s t r a i n s could be determined with a great deal of accuracy r e s u l t i n g i n an o v e r a l l error of only 1 percent f o r t h i s measurement. Results of the tensioning b o l t c a l i b r a t i o n s can be found i n Appendix B. 2.4.5 Forming Wire Speed The forming wire speed was determined by measuring the r o t a t i o n a l speed of the drive r o l l e r using a Shimpo model DT-205 hand held d i g i t a l tachometer. The angular v e l o c i t y of the drive r o l l e r was then used to c a l c u l a t e the speed of the forming wire. Due to unsteadyness i n the motor speed, the forming wire speed could only be maintained within approxi-mately 10 percent of the desired speed. 2.4.6 Forming Wire Drainage Resistance The resistance to flow through the forming wire used on the apparatus was determined. A piece of the forming wire was glued over one end of a length of 1/2 inch diameter pvc pipe. The pipe was then f i x e d i n a v e r t i c a l p o s i t i o n with the forming wire at the lower end. To determine the drainage resistance, a constant head of water was maintained i n the 13. pipe and the flow rate was measured using the drum and scale method described f o r the f o i l drainage rate measurement. Once the flow rate was obtained, the drainage resistance was c a l c u l a t e d to be 4.0xl0~ 7 f e e t . Determination of the drainage resistance was repeated several times to given an o v e r a l l error i n the measurement of 5 percent. D e t a i l s of the drainage resistance c a l c u l a t i o n can be found i n Appendix C. 2 . 5 Experimental Procedure Because of the minimal instrumentation used, very l i t t l e setup was require f o r each experimental run. The only preparation required f o r each run was the purging of any a i r that may have been present i n the l i n e s connecting the manometer tubes to the pressure taps. The a i r was purged from the l i n e s by back f l u s h i n g the manometer tubes with water u n t i l a l l a i r was removed. A check that a l l a i r had been removed was done by ensuring a l l manometer tubes returned to t h e i r i n i t i a l l e v e l s a f t e r the experimental run was complete. For each experimental run, r e s u l t s were obtained f o r a si n g l e f o i l over a range of forming wire speeds. At the s t a r t of each run the manometer tubes were back-flushed with water and t h e i r i n i t i a l l e v e l s were recorded. The wire speed was then slowly brought up to the desired speed using the motor speed c o n t r o l . At the same time, water was supplied to the headbox. The flow rate was adjusted so that the headbox j e t v e l o c i t y was equal to the wire speed. A period of several minutes was then allowed to pass to permit the wire and j e t v e l o c i t i e s , as w e l l as the pressure d i s t r i b u t i o n on the f o i l to s t a b i l i z e . A f t e r the s t a b i l i z a t i o n period, the manometer readings were recorded and the f o i l drainage rate was measured. Measurements were made i n t h i s manner f o r each of the desired 14. wire speeds. Af t e r data had been obtained f o r a l l desired wire speeds, the water supply to the headbox was turned o f f and the forming wire was stopped. A check was then made to ensure the manometer tube l e v e l s returned to t h e i r i n i t i a l values. If a manometer tube f a i l e d to return to i t s i n i t i a l l e v e l , i t was assumed that a i r had entered the manometer l i n e and the e n t i r e run was repeated. A f t e r the run had been su c c e s s f u l l y completed, the f o i l was removed and replaced with the next f o i l . Measure-ments were then taken f o r the next f o i l and a l l subsequent f o i l s . 2.6 Results and Observations For each of the f i v e f o i l s tested, pressure d i s t r i b u t i o n s and drain -age flow rates were obtained f o r wire speeds of 500, 750, 1000, 1250, 1500 and 2000 feet per minute. A l l data were obtained f o r a wire tension of 100 pounds per l i n e a l foot. Measured pressure d i s t r i b u t i o n s f o r each f o i l , as a function of wire speed, are presented i n Figures 10-14. Both pressures and lengths are presented as non-dimensional q u a n t i t i e s . A comparison of the pressure d i s t r i b u t i o n s on each f o i l f o r each wire speed can be found i n Figures 15-20. Measured f o i l drainage rates as a function of wire speed are presented i n Figure 21. The drainage rate has been expressed non-dimensionally as: Qm u h w max where Q i s the measured drainage r a t e [ f t 3 / s / f t of f o i l l , h i s the m 1 max maximum gap between the f o i l and the wire and u w i s the wire v e l o c i t y . During the course of taking measurements, some unsteadiness i n the pressures were noted at higher wire speeds. At wire speeds of 1500-2000 ^nd " 15. feet per minute, a very s l i g h t o s c i l l a t i o n i n the pressure readings of the f i r s t four and the l a s t two pressure taps was observed. The o s c i l l a t i o n , however, consisted only of a s l i g h t j i t t e r i n the manometer tubes and was of i n s u f f i c i e n t magnitude to be measured. Attempts were made to obtain r e s u l t s f o r a range of wire tensions. It was found that small changes i n the wire tension (5-10 pounds per foot) d i d not have a measurable a f f e c t on the pressure d i s t r i b u t i o n . At increased wire tensions large enough to have a measureable e f f e c t , exces-s i v e warping of the frame, uneven s t r e t c h i n g of the wire and d e f l e c t i o n of the r o l l e r s made i t possible to keep the forming wire tracking at the center of the r o l l e r s . As a r e s u l t , data could not be obtained f o r higher wire tensions. Lower wire tensions were not attempted due to s l i p p i n g between the drive r o l l e r and the wire at speeds greater than 1000 feet per minute. Some d i f f i c u l t y was experienced maintaining the wire speed and the headbox j e t v e l o c i t i e s at the desired speeds during the recording of data. To i n v e s t i g a t e the e f f e c t t h i s had on the r e s u l t s , t e s t s were done i n which the wire and j e t v e l o c i t i e s were varied independently by small amounts. It was found that v a r i a t i o n s up to 10 feet per minute did not have a measurable a f f e c t on the pressure d i s t r i b u t i o n s or drainage rates. Since the wire and j e t v e l o c i t i e s could e a s i l y be maintained within 10 feet per minute of the desired speed, speed v a r i a t i o n s were not considered to be s i g n i f i c a n t with respect to the accuracy of the r e s u l t s obtained. To t e s t r e p e a t a b i l i t y of the r e s u l t s , a second set of runs were done on f o i l s 1 and 3. As can be seen i n Figures 23 and 24, the r e p e a t a b i l i t y of the experiment was very good. The s l i g h t v a r i a t i o n i n the r e s u l t s obtained f o r f o i l 3 can be a t t r i b u t e d to v a r i a t i o n s i n the wire tension due to f o i l changes made between the two runs. No f o i l changes were made between the two runs done on f o i l one. 1 6 . Examination of the measured drainage rates indicates a discrepancy between the drainage rates measured and what would be expected due to the s u c t i o n measured along the length of the f o i l s . In general, measured drainage rates were much greater than could be explained by the suction d i s t r i b u t i o n s . This trend was most pronounced f o r the stepped f o i l s . This problem can be a t t r i b u t e d to what turned out to be a t o t a l l y inadequate method f o r measuring the drainage rates. O v e r a l l , i t was not possible to obtain good, repeatable, drainage rate measurements and the present r e s u l t s must be considered very inaccurate. Further discussion of the r e s u l t s obtained can be found i n Chapter 4. 17. III. THE THEORETICAL ANALYSIS 3.1 General The main thrust of t h i s i n v e s t i g a t i o n was to develop a simple but accurate mathematical model of the flow i n the region between a F o u r d r i n i e r paper machine drainge f o i l and the forming wire. With t h i s model the pressure d i s t r i b u t i o n along the f o i l could then be c a l c u l a t e d thereby p r e d i c t i n g the water drainage rate that could be achieved by the f o i l . The problem under consideration consists of a high speed flow of a f l u i d i n a very t h i n region having one s o l i d , i r r e g u l a r boundary and one moving porous boundary. Due to the suction generated as a r e s u l t of the flow, flow also occurs across the porous boundary. It was assumed that because the flow region i s very t h i n compared to the length (h/L was t y p i c a l l y on the order of 0.01 to 0.04), the pressure d i s t r i b u t i o n i s a f u n c t i o n of the length only. Cross stream v e l o c i t i e s were assumed to be small compared to the wire v e l o c i t y allowing viscous terms i n the y - d i r e c t i o n to be neglected. Two-dimensional flow was a l s o assumed as the cross machine width of the f o i l i s much greater than the length (W/L i s t y p i c a l l y on the order of 100). These assumptions permitted the formulation of a model based on the t h i n shear layer equations, or the boundary layer equations. A great deal of thought was given to whether the flow i s laminar or turbulent. T y p i c a l Reynolds numbers based on the f o i l length nad the wire v e l o c i t y are on the order of 1.0x10 6 suggesting the flow would tend to be turbulent. It has been shown [14], however, that f o r flow over a f l a t p l a t e t r a n s i t i o n from laminar to turbulent flow generally occurs at a Reynolds number i n the range 18. Re ux = 3.5 to 5.0xl0 5 v It i s conceivable, therefore, that laminar flow could e x i s t over at l e a s t the f i r s t h a l f of a f o i l . In view of t h i s , and because of the lack of previous modelling of the flow over a f o i l , i t was f e l t that assuming the flow was laminar would be a reasonable approach i n the development of a model. Consequently, laminar flow was assumed for the purposes of the present i n v e s t i g a t i o n . A s o l u t i o n of the model was attempted using three separate methods. The f i r s t two methods were grouped i n t o what was c a l l e d the one-dimensional models. The s o l u t i o n of the one-dimensional models involved reducing the governing equations to a system of ordinary d i f f e r e n t i a l equations by assuming a v e l o c i t y p r o f i l e , i n t e g r a t i n g the equations of motion over the height of the flow region and s o l v i n g the r e s u l t i n g system of equations. This s o l u t i o n method was the approach taken by Taylor [3], Meyer [7] and Ernst [8] i n previous modelling attempts. The present approach d i f f e r s from those of Taylor, Ernst, and Meyer i n that a t h i r d order v e l o c i t y p r o f i l e was assumed i n the present i n v e s t i g a t i o n and the f u l l boundary layer equations were solved. Taylor's model assumed turbu-l e n t flow and a uniform v e l o c i t y p r o f i l e to make the equations t r a c t a b l e . The models of Ernst and Meyer s i m p l i f i e d the equations of motion by neglecting the i n e r t i a l terms u 3u 9y and v 3u 9y e s s e n t i a l l y reducing the equation of motion to that of the Reynolds stress equation. 19. The second s o l u t i o n technique involved the s o l u t i o n of the governing equations using f i n i t e d i f f e r e n c e formulations. This method i s r e f e r r e d to i n l a t e r sections as the two-dimensional model. 3.2 The Governing Equations 3.2.1 The Equations of Motion Using the assumptions of Section 3.1 and the notation of Figures 25 and 26, the flow between the f o i l and the forming wire may be described by the equation of motion for a t h i n shear flow u 3u , 3x" + V 3u 1 dp 3y p dx + v ay 2 (3.1) and the mass conservation equation (3.2) Boundary conditions for the problem may be given by x = 0; P = P a (3.3a) x - L; p = p a (3.3b) and y = 0; (3.4a) 20. y = - h ; u = 0 , v = 0 ( 3 . 4 b ) where L i s t h e f o i l l e n g t h , h i s t h e d i s t a n c e f r o m t h e f o r m i n g w i r e t o t h e f o i l s h o w n i n F i g u r e 2 5 , p f l i s t h e a m b i e n t p r e s s u r e , u w i s t h e f o r m i n g w i r e s p e e d , and v w i s t h e v e l o c i t y o f t h e f l u i d d r a i n e d t h r o u g h t h e w i r e . T h e d r a i n a g e v e l o c i t y , v w i s n o t e x p l i c i t l y known and i s assumed t o be a f u n c t i o n o f t h e p r e s s u r e g r a d i e n t a c r o s s t h e f o r m i n g w i r e k . v - — ( p - p ) ( 3 . 5 ) w y a v w h e r e p i s t h e p r e s s u r e i n t h e f l o w r e g i o n and k d r i s a c o e f f i c i e n t o f d r a i n a g e r e s i s t a n c e h a v i n g u n i t s o f l e n g t h . E q u a t i o n 3 .5 r e p r e s e n t s D a r c y ' s Law f o r f l o w t h r o u g h a p o r o u s m e d i a and h a s b e e n u s e d t o r e p r e s e n t v w by T a y l o r [ 3 ] , Meye r [ 7 ] , and E r n s t [ 8 ] . E q u a t i o n s 3 .1 and 3 .2 r e p r e s e n t t h e g o v e r n i n g e q u a t i o n s f o r t h e f l o w b e t w e e n t h e f o r m i n g w i r e and t h e f o i l . T h e s e e q u a t i o n s may be s o l v e d u s i n g t h e bounda ry c o n d i t i o n s 3 .3 and 3 .4 and t h e r e l a t i o n s h i p g i v e n by E q u a t i o n 3 .5 t o o b t a i n t h e p r e s s u r e d i s t r i b u t i o n a l o n g t h e f o i l . 3.2.2 Wire D e f l e c t i o n Model P r e v i o u s t h e o r e t i c a l a n a l y s i s o f t h e p r e s s u r e d i s t r i b u t i o n on a f o i l [ 3 , 7 ] have g e n e r a l l y assumed t h e f o r m i n g w i r e i s r i g i d . In t h e p a s t , f o r m i n g w i r e s were made o f b r o n z e s c r e e n s and t h e a s s u m p t i o n t h e w i r e i s r i g i d was p r o b a b l y a r e a s o n a b l e o n e . P r e s e n t day w i r e s , h o w e v e r , a r e made o f f l e x i b l e s y n t h e t i c f a b r i c s and i t c a n n o t be assumed t h e f o r m i n g w i r e i s r i g i d . Because the height of the flow region between the forming wire and the f o i l i s very small compared to the length of the f o i l , i t would be expected that any change i n the f o i l - w i r e gap height could have a s i g n i f i -cant influence on the pressure d i s t r i b u t i o n . This p r e d i c t i o n i s supported by the dependence of the pressure d i s t r i b u t i o n on the f o i l shape exhibited i n the experimental r e s u l t s shown i n Figures 16 to 21. Since d e f l e c t i o n of the forming wire a l t e r s the shape of the flow region, the e f f e c t on the pressure d i s t r i b u t i o n i s s i m i l a r to that of changing the f o i l shape. For t h i s reason, i t was decided to incorporate the e f f e c t of forming wire d e f l e c t i o n i n the evaluation of the pressure d i s t r i b u t i o n . In developing a model f o r the forming wire d e f l e c t i o n , i t was assumed the wire i s s u f f i c i e n t l y f l e x i b l e that i t has no f l e x u r a l r i g i d i t y . This assumption permits the d e r i v a t i o n of an expression f o r the forming wire d e f l e c t i o n using the force balance on the wire indicated i n Figure 27. Summing the forces i n the y - d i r e c t i o n on a small section of the wire gives (p -p)dx - 2T sinGf) = 0 (3.6) If dx i s s u f f i c i e n t l y small, then dx « ds, d6 « 1, and . ,6. d6 i n ( r) . — (3.7) Equation 3.6 may then be written (p -p)ds = T d6 3. (3.8) 22. ds Observing that the radius of curvature, R = — do R (P a-p) If the radius of curvature i s large « 1) then K ( 3 . 9 ) 1 : j d2y R " dx2 (3.10) and Equation 3.9 may be written d 2 y _ (P.-P) dx2 (3.11) Once the pressure d i s t r i b u t i o n along the f o i l i s known, the wire d e f l e c -t i o n may be determined by evaluating Equation 3.11 using the boundary conditions x = 0; y = 0 x = L x ; y = 0 (3.12a) (3.12b) 3.3 The One Dimensional Models 3.3.1 The Conservation of Mass Model 3.3.1.1 The Model Equations The conservation of mass model obtains a s o l u t i o n f o r the pressure d i s t r i b u t i o n on a f o i l using only the condition f o r mass con t i n u i t y , hence the name of the model. Derivation of the model begins by expressing the equation f o r mass co n t i n u i t y given by Equation 3.2 i n i n t e g r a l form. L e t t i n g —n o Q = / u dy (3.14) -h and s u b s t i t u t i n g the expression f o r v w given by Equation 3.5 i n t o Equation 3.13 gives § • ^ <P"Pa> (3-15) Assuming a v e l o c i t y p r o f i l e of the form u(y) = u w + c x y + c 2 y 2 + c 3y3 (3.16) with boundary conditions y = 0; u = u w (3.17a) - v . £ | _ 1 | E + v ! f u „. Q 9Y 2 w dy 1o p dx ^ 2 (3.17b) y = -h; u = 0 (3.17c) <3-17d, 24. and s o l v i n g f o r c 2 and c 3 gives -3(1 h 2 iP- + 2v u ) p d x W /o no \ c i = %u—f—u 7-\ (3.18a) 1 2h (v h - 3v) w (v c. + ± 4 E ) w 1 p dx c 2 = 2 v K (3.18b) c 3 = TThV ( 3' 1 8 c ) Substituting the expressions f o r c 1 , c 2 and into Equation 3.16 and evaluating the i n t e g r a l i n Equation 3.14 gives -v h1* + 12vh 3 + 30h2 pv u v - 72hp v 2 u 0 = L w d x 2^ ww w^_ ^ [48pv (v h - 3v)J ( J - L y ) F i n a l l y , solving for dp/dx gives [_48(^)Q(p-p ) + 144 (£^ i )Q + 3 0 ( ^ ) u ( p - p ) - 7 2 ( ^ ) u J i £ = h 2 h S h 2 (3 20) dx k, h C J , 2 0 ) where v w has been eliminated using Equation 3.5. Equations 3.15 and 3.20 represent the model equations f o r the conservation of mass model. Using the boundary conditions 25. x = 0; p = p cl (3.21a) x = L; p = p a (3.21b) these equations are solved f o r the two unknowns, p and Q. 3.3.1.2 Method of Solution The system of equations represented by Equations 3.15 and 3.20 were solved numerically f o r p and Q using the routine COLSYS [15]. COLSYS i s a general purpose software routine f o r the s o l u t i o n of boundary value problems defined by a system of ordinary d i f f e r e n t i a l equations of a r b i t r a r y order. The method used by COLSYS i s c o l l o c a t i o n at Gaussian points using B-splines of v a r i a b l e order. A s o l u t i o n to the system of equations i s obtained on a mesh which may be user s p e c i f i e d or generated by COLSYS. The number of mesh points i s increased by COLSYS u n t i l a s o l u t i o n s a t i s f y i n g a set of user s p e c i f i e d e r r o r tolerances i s obtained. A flow chart of the sol u t i o n procedure i s presented i n Figure 28. Solutions incorporating the e f f e c t s of wire d e f l e c t i o n were solved i t e r a t i v e l y as indicated i n Figure 28. A f t e r a s o l u t i o n f o r the pressure d i s t r i b u t i o n had been obtained, COLSYS was used to obtain a s o l u t i o n to the equation f o r the wire d e f l e c t i o n (Equation 3.11). The ca l c u l a t e d wire d e f l e c t i o n was then used i n the evaluation of a new pressure d i s t r i b u t i o n . A f t e r each i t e r a t i o n , the pressure d i s t r i b u t i o n was compared to the previous one. The s o l u t i o n procedure was terminated when the maximum change i n the pressure s a t i s f i e d the condition max < P. tolerance (3.22) In general, the solution converged a f t e r 3 to 4 i t e r a t i o n s . 2 6 . 3.3.1.3 Verification of the Model Before a numerical model i s used to obtain solutions to a problem, i t i s important that the model be v e r i f i e d against known r e s u l t s f o r flows having a nature s i m i l a r to the problem being studied. V e r i f y i n g a model serves two important purposes. The f i r s t purpose i s to ensure there are no errors i n the model or the s o l u t i o n procedure. The second purpose i s to gain confidence that the model adequately represents the flow being modelled and that r e s u l t s obtained using the model w i l l represent the actual flow. The conservation of mass model was v e r i f i e d by using the model to evaluate the flow over a f l a t p l a t e at zero incidence and the flow between p a r a l l e l f l a t plates i n which the upper plate was moving. These flows were used because exact solutions are a v a i l a b l e f o r them. V e r i f i c a t i o n against a couette flow was done because of i t s s i m i l a r i t y to the flow between a f o i l and the forming wire. Figure 29 shows a comparison of the calculated v e l o c i t y p r o f i l e and the Blasius p r o f i l e f o r flow over a f l a t p l a t e . As was expected, agree-ment between the two r e s u l t s i s minimal. Since the Blasius p r o f i l e i s an exact s o l u t i o n f o r the flow over a f l a t plate, exact agreement was not expected. The discrepancy i n the r e s u l t s can be accounted f o r by the two departures from the exact s o l u t i o n made by the conservation of mass model. The f i r s t and most s i g n i f i c a n t departure i s the f a i l u r e of the mass con-servation model to observe the requirement f o r conservation of momentum. Because a boundary layer on a f l a t plate grows i n the streamwise d i r e c t i o n , changes i n the momentum occur i n the streamwise d i r e c t i o n and f a i l u r e to include t h i s condition i n the model introduces large e r r o r s . The second source of error i n the r e s u l t s i s due to the approximate nature 27. of the assumed v e l o c i t y p r o f i l e . Results for the flow between moving f l a t plates are given i n Figure 30. Much better agreement with theory was obtained f o r t h i s t e s t case. In t h i s type of flow, the f u l l y developed v e l o c i t y p r o f i l e s i n the absence of a pressure gradient i s l i n e a r . Also, i f there i s no pressure gradient and the v e l o c i t y p r o f i l e i s f u l l y developed, there are no changes i n momentum i n the streamwise d i r e c t i o n . Thus, the conservation of mass model accurately models the flow enabling an exact s o l u t i o n to be obtained. As a f i n a l v e r i f i c a t i o n of the model, i t would have been desirable to compare the r e s u l t s f o r f o i l s to those obtained by Ernst [8]. Since the model developed by Ernst was e s s e n t i a l l y a conservation of mass model, i t would be expected that the r e s u l t s of the present model would be very s i m i l a r to those calculated by Ernst. Unfortunately, i n s u f f i c i e n t d e t a i l with respect to the values of key parameters, such as the f o i l length, were presented by Ernst making i t impossible to duplicate his r e s u l t s using the present model. 3.3.1.4 R e s u l t s The conservation of mass model was used to evaluate the f i v e f o i l s shown i n Figure 9. Calculated r e s u l t s were obtained for each f o i l over the range of wire v e l o c i t i e s used i n the experiment. Figure 31 shows the v a r i a t i o n the wire v e l o c i t y of the pressure d i s t r i b u t i o n on f o i l 1 i n the absence of wire d e f l e c t i o n e f f e c t s . Though r e s u l t s are only presented f o r f o i l 1, s i m i l a r trends were found f o r a l l f o i l s tested. In general the r e s u l t s i n d i c a t e a strong tendency f o r the 28. non-dimensional suction to decrease as the wire v e l o c i t y i s increased. If the pressures are converted to dimensional q u a n t i t i e s , i t i s found that the suction a c t u a l l y increases i n magnitude. In addition, the suction i s maintained over a greater length of the f o i l as the wire v e l o c i t y i s increased r e s u l t i n g i n much higher drainage rates as shown i n Figure 32. Thus, increasing the wire v e l o c i t y increases the o v e r a l l suction of the f o i l . Calculated pressure d i s t r i b u t i o n s on f o i l 1 i n c l u d i n g wire d e f l e c t i o n are given i n Figure 33. The a d d i t i o n of wire d e f l e c t i o n e f f e c t s causes the l o c a t i o n of the suction peak to s h i f t towards the t r a i l l i n g edge of the f o i l . The d e f l e c t i o n of the wire a l s o r e s u l t s i n lower peak suctions. This tendency i s c l e a r l y shown i n Figure 32 where the lower suctions cause a reduction i n the c a l c u l a t e d drainage r a t e s . The r e s u l t s shown i n Figure 33 i n d i c a t e the e f f e c t s of the wire d e f l e c t i o n are the most pronounced at the higher wire v e l o c i t i e s . Unlike the r e s u l t s obtained for the s t i f f wire case, however, the benefits of increasing the wire v e l o c i t y are not obvious. The drop i n the drainage rate at high wire v e l o c i t i e s shown i n Figure 32 suggests that unlike the s t i f f wire case, there i s an optimum wire v e l o c i t y which w i l l r e s u l t i n the maximum drainage rate obtainable f o r a given f o i l . Thus, the d e f l e c -t i o n of the wire seems to have a detrimental e f f e c t on the performance of a f o i l . This may not be the case f o r a r e a l f o i l however, due to the much la r g e r suction magnitudes measured i n the experiments than were predicted by the conservation of mass model. Examination of the v e l o c i t y p r o f i l e s shown i n Figures 34 to 36 i n d i c a t e large regions of separated flow occur i n the c a l c u l a t i o n s . Figure 34 in d i c a t e s there i s very l i t t l e e f f e c t on the v e l o c i t y p r o f i l e s 29. due to the wire v e l o c i t y . Figure 35 shows the t y p i c a l development of the v e l o c i t y p r o f i l e s along a f o i l . Including wire d e f l e c t i o n e f f e c t s i n the c a l c u l a t i o n has only a small e f f e c t on the v e l o c i t y p r o f i l e s as shown i n Figure 36. Comparing Figures 35 and 36 shows the primary e f f e c t of the wire d e f l e c t i o n i s a reduction of the separation that occurs. This can be a t t r i b u t e d to the higher drainage v e l o c i t i e s due to the increased suction at the t r a i l i n g end of the f o i l that occurs when wire d e f l e c t i o n i s included i n the c a l c u l a t i o n s . A comparison of the r e s u l t s obtained on d i f f e r e n t f o i l s shown i n Figure 37 indicates no c l e a r trends except to say that the pressure d i s t r i b u t i o n s are dependent on the f o i l geometry. Results could not be obtained f o r the stepped f o i l s ( f o i l s 4 and 5) due to a f a i l u r e of COLSYS to converge on a so l u t i o n . Because of the development of the momentum i n t e g r a l model, the reason f o r the s o l u t i o n f a i l u r e was not pursued. In obtaining the preceeding r e s u l t s , i t was found that the c a l c u l a -tions were very s e n s i t i v e to the f o i l - w i r e gap height. This s e n s i t i v i t y was caused by the 1/h terms i n Equation 3.20. As h approached 0, the rap i d growth i n the magnitude of the 1/h terms resulted i n very large truncation errors being introduced i n t o the c a l c u l a t i o n . Obviously, the s o l u t i o n would f a i l completely when h = 0. This made i t necessary to sp e c i f y an' i n i t i a l gap height before the c a l c u l a t i o n s could begin. Figure 38 shows the s e n s i t i v i t y of the r e s u l t s to the i n i t i a l gap height. As can be seen, s m a l l changes i n h^ r e s u l t e d i n l a r g e changes i n the o v e r a l l pressure d i s t r i b u t i o n . To minimize t h i s e f f e c t , the i n i t i a l gap height was made as small as possible. S e l e c t i o n of the gap height, however, was l i m i t e d by the necessity of ensuring the wire did not contact the f o i l i n ca l c u l a t i o n s including wire d e f l e c t i o n . 30. Due to the a r b i t r a r y nature of the i n i t i a l gap height s e l e c t i o n , i t was decided i t was meaningless to begin the c a l c u l a t i o n at the leading edge of the f o i l and perform the ana l y s i s over the f l a t section of the f o i l . Instead, the c a l c u l a t i o n s were started at the point where the f o i l p r o f i l e changed. This point corresponded to the beginning of the taper on f o i l s 1,2 and 3 and the l o c a t i o n of the step on f o i l s 4 and 5. The strong dependence of the pressure d i s t r i b u t i o n on the gap height a l s o resulted i n numerical problems. Because small changes i n the gap height could r e s u l t i n very large changes i n the pressure, i t was found that i n some cases the model equations were s t i f f . This condition r e f e r s to the s i t u a t i o n where a step s i z e smaller than the smallest value that can be represented by the computer i s required to obtain a s o l u t i o n meet-ing the s p e c i f i e d error tolerances. When t h i s s i t u a t i o n occurred, the s o l u t i o n generally f a i l e d . Relaxing the error tolerances occasionally permitted a s o l u t i o n to be found, however, the accuracy of the s o l u t i o n had to be considered very poor. This condition was generally confined to the a n a l y s i s of f o i l s 4 and 5, though i t was occasionally encountered with the other f o i l s . 3.3.2 The Momentum Integral Model 3.3.2.1 The Equations of Motion D e r i v a t i o n of the equations of motion f o r the momentum i n t e g r a l model proceed i n a s i m i l a r manner to the d e r i v a t i o n of the momentum i n t e g r a l form of the boundary layer equations. The de r i v a t i o n begins with the equation of motion for a boundary layer flow 31. 3u 3u 1 dp , 3 2u u — + v — = -r~ + v (3.23) 3x 3y p dx , 2 3y z and the mass conservation equation M u l t i p l y i n g Equation 3.24 by u and adding the r e s u l t ot Equation 3.23 gives n 3u i 3 f v 1 dp , 3^u o c . 2u — + — (uv) = - - - r ; + v (3.25) 3x 3y p dx 2 with the boundary conditions x = 0; p = p a x = L; p = p a y = 0; u = u w , v = v w y = -h; u = 0, v = 0 (3.26) To avoid some of the numerical d i f f i c u l t i e s encountered with the mass conservation model, Equation 3.25 i s written i n non-dimensional form. Introducing the non-dimensional parameters u L x* = — u* = f = — Re = L u v w (3.27) y = N IT v* = — p* = — h u o w pu z w Equation 3.25 may be written as 3 2 . 2 f (h> 1^ dx* + Ch> (Re" } TT ( 3 > 2 8 ) S i m i l a r l y the boundary conditions may be written as P a P a x* = 0; p* = 7- x* = 1; p* = -T r pu * r pu z w w (3.29) v n = 0; f = 1, v* = — n = -1; f = 0, v* = 0 w Integration Equation 3.28 with respect to n gives h L £ 2 d"+ ( i r ) ( f v ) Lx= - £ + <£>2<ib>€> L, (3-30> where the a s t e r i s k s (*) i n d i c a t i n g non-dimensional quantities have been dropped from Equation 3.30 f o r convenience. L e i b n i t z ' s r u l e has been used to perform the d i f f e r e n t i a t i o n under the i n t e g r a l sign. Evaluating Equation 3.30 at the l i m i t s of the i n t e g r a t i o n gives |- J° f2 dn + (h v = - p- + (1)2 (1-)(|1) |° (3.31) dx h w dx h Re 3TI 1 _ ^ As f o r the mass c o n s e r v a t i o n model, v i s r e p l a c e d w i t h the w expression v = — (p-p ) (3.32) w P a v which may be written i n non-dimensional form as Vw = ( — ) ( R e ) ( P " P a ) Substituting Equation 3.33 into 3.31 gives tz dn + ( % , ( R e ) ( P - P a ) + £ - <j>* (fexff) I If a v e l o c i t y p r o f i l e of the form f(n) = 1 + CXTI + c 2 n 2 + c 3 n 3 having boundary conditions n = 0; f = 1 n = -1; f = 0 i £ - )2 i ! i I = 0 ^ C h > (Re> 9 t i 2 l _ x i s assumed, where c., c„, and c~ are found to be 34 . c, = dx-[ 1 - 1 ( i L - ) k d r Re2(p- P f l)] (3.37a) L 2 c„ = (3.37b) 2[1 - i ( l-)Re2 k. ( p - p ) ] L2 h3. c„ = [ i ^ > R e 2 k d r ^ a > + T 2 < L l | [ l - i ( l - ) R e 2 k d r ( p - p a ) (3.37c) L2 Equation 3.34 may be evaluated g i v i n g 210- t 3 0 C 3 2 - 7 ° C 2 C 3 + 8 4 C 1 C 3 " 1 0 5 C 3 + 4 2 C 2 2 " 1 0 5 c i C 2 k , + 140c 2 + 70c 1 2 - 210c x + 210] + (_£|E.)Re(p-p ) + g (3.38) Equation 3.38 represents the equation of motion f o r the momentum i n t e g r a l model. Before proceeding with a s o l u t i o n , Equation 3.38 i s f i r s t s i m p l i f i e d . L e t t i n g * = 210" ( 3 ° C 3 2 " 7 0 c 2 C 3 + 8 4 c l C 3 " 1 0 5 c 3 + 4 2 c 2 2 (3.39) - 105c 1c 2 + 140c 2 + 70^2 - 210 C l +. 210] 35. Equation 3.38 may be written |1 = - ( % ) R e ( p - p a ) - £ + (jj)2 ( ^ ) ( 2 c 2 - 3 c 3 ) ( 3 . 4 0 ) Using the chain r u l e to evaluate 4^ - gives dx d4> _ 3iJ> 3p , 3iJ> v3x' i l i h n 4 n dx 3p 3x ,3p. 3x 3h 3x 3 W Equation 3.41 may be written *x = V * + *p Pxx + *h hx ( 3' 4 2> r r x where the subscripts x, p, and p denote d i f f e r e n t i a t i o n with respect to x,p, and r e s p e c t i v e l y . Equating Equations 3.40 and 3.42 and arranging to give an expression f o r p x x gives p — — — (3.43) XX ll> p x Equation 3.43 i s the f i n a l form of the equation of motion f o r the momentum i n t e g r a l model. Using the boundary conditions x = 0; p = p a  p uw 2 x = 1; p = (3.44) a pu 2 w thi s equation may be solved for p as a function of x. 36. 3.3.2.2 Method of Solution A s o l u t i o n to Equation 3.43 was obtained i n a manner s i m i l a r to that used f o r the mass conservation model. The s o l u t i o n techniques d i f f e r e d only i n the routine used to solve Equation 3.43. I n i t i a l l y , an attempt was made to obtain a s o l u t i o n using COLSYS. The attempt was without success. Investigation of the problem indi c a t e d Equation 3.43 was s t i f f and a s o l u t i o n to the problem was beyond the c a p a b i l i t i e s of COLSYS. To overcome t h i s problem, a routine s p e c i a l l y formulated to deal with s t i f f problems was used i n the place of COLSYS. The routine used i s known as DEBDF. This routine i s a v a i l a b l e as part of the DEPAC routines [16] and i s d i s t r i b u t e d as part of the SLATEC mathematical software l i b r a r y [17]. DEBDF i s a subroutine written i n Fortran and intended f o r use i n the so l u t i o n of i n i t i a l value problems defined by a system of f i r s t order ordinary d i f f e r e n t i a l equations. The method used by DEBDF i s a backward d i f f e r e n t i a t i o n scheme and i s s p e c i a l l y formulated to deal with s t i f f problems. Before DEBDF could be used to obtain a s o l u t i o n to the problem, i t was necessary to reduce Equation 3.43 to a system of f i r s t order ordinary d i f f e r e n t i a l equations. Introducing the dummy parameter P 2 = T S (3.45) Equation 3.43 may be written as the f i r s t order equation <*2>x = ' / * <3.46) P 2 where i|> = < K P » P 2 ) • Unlike COLSYS, DEBDF could not e x p l i c i t l y s a t i s f y the boundary condition P A x = 1; p — (3.47) p uw 2 as DEBDF can only be used to obtain a s o l u t i o n to i n i t i a l value problems. To s a t i s f y t h i s boundary condition, the s o l u t i o n was obtained by s p e c i f y -ing a second i n i t i a l condition of the form P 2 - ( P 2 ) 0 ( 3 . 4 8 ) where ( P 2 ) Q w a s a n a r b i t r a r y guess f o r p 2 at x = 0. Equations 3.45 and 3.46 were then solved f o r p and p 2 as a function of x. A f t e r the s o l u t i o n had been obtained, p at x = 1 was tested to determine i f i t s a t i s f i e d the condition given by Equation 3.47. If t h i s condition was not s a t i s f i e d , the i n i t i a l guess f o r p 2 was modified using a R e g u l i - F a l s i i t e r a t i o n technique [18]. A second s o l u t i o n was then obtained using the new guess for p 2« This procedure was repeated u n t i l the s o l u t i o n converged. Convergence of the s o l u t i o n was determined when Equation 3.47 was s a t i s f i e d to within a s p e c i f i e d e r r o r tolerance and usually occurred i n 3 to 5 i t e r a t i o n s . The remainder of the s o l u t i o n procedure, including the incorporation of wire d e f l e c t i o n e f f e c t s , was i d e n t i c a l to that of the mass conservation model. A flow chart of the o v e r a l l s o l u t i o n procedure i s given i n Figure 28 and a flow chart f o r the pressure evaluation subroutine i s given i n Figure 39. 38. 3.3.2.3 Verification of the Model V e r i f i c a t i o n of the momentum i n t e g r a l model was done by using the model to evaluate the flows over a f l a t plate at zero incidence and between moving p a r a l l e l p l a t e s . The r e s u l t s of these t e s t s were the compared to the exact solutions for the appropriate flow. As was the case f o r the mass conservation model, Figure 40 shows the cal c u l a t e d s o l u t i o n using the momentum i n t e g r a l model f o r flow between p a r a l l e l plates was i n complete agreement with the exact s o l u t i o n . This r e s u l t was expected and the tes t was done simply to check f o r errors i n the so l u t i o n procedure. Figure 41 shows a comparison of the calculated and exact solutions f o r flow over a f l a t plate at zero incidence. Comparing the r e s u l t s to those of Figure 29, i t can be seen the momentum i n t e g r a l model predicts the Blasius p r o f i l e much better than the conservation of mass model. An Inte r e s t i n g phenomena, however, was the dependence of the s o l u t i o n on the boundary conditions. In both cases the s o l u t i o n was f o r a flow having no pressure gradient but the manner i n which t h i s condition was imposed d i f f e r e d . For the f i r s t s o l u t i o n , the r e s u l t s of which are shown i n Figure 41, the boundary conditions were s p e c i f i e d as x = 0; p = , Po = 0 (3.49) pu. w As can be seen from Figure 41, reasonable agreement with the Blasius p r o f i l e was obtained. In addition, the r e s u l t s were i n exact agreement with the cubic v e l o c i t y p r o f i l e given by f(n) = 1 + | n - i n 3 39. (3.50) Examination of the pressure d i s t r i b u t i o n along the plate i n d i c a t e d a s l i g h t adverse pressure gradient, thus explaining the la r g e r displacement thickness predicted by the numerical s o l u t i o n . Closer scrutiny of the so l u t i o n procedure to determine the cause of the adverse pressure gradient indicated the problem use due to the use of the condition k v w = (-^)(Re)(p-p a) (3.51) to r e l a t e the cross stream v e l o c t i y to the pressure. For the s o l u t i o n to flow over a f l a t p late, k ^ was set equal to zero, thus e s s e n t i a l l y impos-ing a s o l i d boundary at TI = 0. This had the e f f e c t of making the flow more l i k e an i n t e r n a l flow having a parabolic shaped v e l o c i t y p r o f i l e . The lower v e l o c i t i e s near the wa l l r e s u l t e d i n an adverse pressure g r a d i -ent to conserve momentum, causing the flow to be further deaccelerated. The net r e s u l t was a much more ra p i d growth of the boundary layer than i s given by the exact s o l u t i o n . When a s o l u t i o n to the flow over a f l a t p l ate was obtained using the boundary conditions x = 0; p = pu w x = 1; p = PU w (3.52) 40 . much better r e s u l t s were obtained as shown i n Figure 42. S p e c i f i c a t i o n of the boundary conditions i n t h i s manner constrained any tendency of the s o l u t i o n to develop an adverse pressure gradient. While some erro r s t i l l occurred i n the s o l u t i o n due to the use of Equation 3.51, the majority of the error was due to the approximate nature of the assumed v e l o c i t y p r o f i l e . As a f i n a l v e r i f i c a t i o n of the momentum i n t e g r a l model an attempt to duplicate the r e s u l t s of Taylor [3] was made. In h i s sol u t i o n , Taylor assumed f u l l y turbulent flow i n which the v e l o c i t y p r o f i l e could be given by f = i L _ u w where u was a function of x only. By replacing the v e l o c i t y p r o f i l e given by Equation 3.35 with the above p r o f i l e , the momentum i n t e g r a l model should have duplicated the r e s u l t s obtained by Taylor. Unfortunately, numerical d i f f i c u l t i e s due to terms f o r 1/h i n Equation 3.43 made i t necessary to impose an a r b i t r a r y gap between the f o i l and the wire. As w i l l be discussed i n the next section, the s o l u t i o n of Equation 3.43 was highly s e n s i t i v e to t h i s i n i t i a l gap height, the r e s u l t of which was much lower suctions than those c a l c u l a t e d by Taylor. Thus, a comparison of the present r e s u l t s to those of Taylor was meaningless. 3 . 3 . 2 . 4 Results The momentum i n t e g r a l model was used to c a l c u l a t e the pressure d i s t r i b u t i o n s on f o i l s 1 to 5 given i n Figure 9. Program runs were done f o r each f o i l using i d e n t i c a l conditions f o r wire v e l o c i t y , wire tension and drainage resistance as were used i n the experiments. 41. In general, the pressure d i s t r i b u t i o n s predicted by the momentum i n t e g r a l model were very s i m i l a r to those calculated using the conserva-t i o n of mass model. Figure 43 shows the r e s u l t s obtained f o r f o i l 1 with no wire d e f l e c t i o n . As before, increasing the wire v e l o c i t y increased the peak suction. Figure 44 shows a s i m i l a r trend i n the pressure d i s t r i b u t i o n s f o r d i f f e r e n t f o i l s as that predicted by the mass conservation model. Unlike the mass conservation model, however, solutions were obtained f o r f o i l s 4 and 5, though the pressure d i s t r i b u t i o n s were e s s e n t i a l l y zero f o r each f o i l . Comparing the r e s u l t s of Figure 43 to those of Figure 45, i t can be seen that i n c l u d i n g wire d e f l e c t i o n i n the c a l c u l a t i o n has the same e f f e c t as was observed f o r the conservation of mass model. As before, there i s a tendency f o r the magnitude of the suction to decrease and the l o c a t i o n of the maximum suction to s h i f t towards the t r a i l i n g edge of the f o i l . This trend i s c l e a r l y shown i n Figure 46. An unexpected e f f e c t of i n c l u d i n g the wire d e f l e c t i o n i n the c a l c u l a -t i o n s was the tendency f o r the flow to separate. From Figure 47 i t can be seen that flow separation was only predicted at the higher wire v e l o c i t i e s when wire d e f l e c t i o n was not included. When wire d e f l e c t i o n was included, separation was predicted f o r a l l wire v e l o c i t i e s , as shown i n Figure 48. This i s probably due to the lower suction magnitudes predicted when wire d e f l e c t i o n e f f e c t s are included. Lower suction r e s u l t s i n less f l u i d being drawn through the wire causing a greater tendency f o r separation to occur due to the expansion of the gap between the f o i l and the wire. Because the momentum i n t e g r a l model had terms for 1/h i n the model equations, i t was necessary to spec i f y an i n i t i a l gap height as was done 42. f o r the mass conservation model. Figure 49 shows the dependence of the s o l u t i o n on the i n i t i a l gap height. The momentum i n t e g r a l model c l e a r l y has a strong dependence on the i n i t i a l gap height. The momentum i n t e g r a l model was al s o used to determine the e f f e c t s of other parameters i n the model equations on the predicted pressure d i s t r i -bution. Figure 50 shows the e f f e c t of the drainage resistance on the so l u t i o n . As the magnitude of the drainage resistance i s increased, the maximum suction decreases. This trend can be explained by examining the r e l a t i o n s h i p f o r the drainage v e l o c i t y given by Equation 3.33. As the v a l u e of k, i s i n c r e a s e d , the magnitude of v i s increased f o r a given dr w pressure d i f f e r e n t i a l causing greater amount of f l u i d to be drawn through the wire i n t o the flow region. Normally t h i s trend would cause an increase i n the streamwise v e l o c i t y of the f l u i d with a corresponding drop i n the pressure. This tendency, however, i s arrested by the requirement t h a t p = p at the t r a i l i n g end of the f o i l . The net r e s u l t i s a greater SL o v e r a l l drainage rate as shown i n Figure 51. Figure 52 shows t y p i c a l r e s u l t s obtained when the wire tension was varied. Increasing the wire tension reduced the amount of d e f l e c t i o n that occurs, thus the r e s u l t i s s i m i l a r to that obtained when wire d e f l e c t i o n i s not included. Figure 53 in d i c a t e s that increasing the wire tension has the b e n e f i c i a l e f f e c t of increasing the drainage rate due to the increased suction r e s u l t i n g from the s t i f f e r wire. To t r y and gain some i n s i g h t i n t o the possible e f f e c t s of performing the c a l c u l a t i o n s f o r turbulent flows, r e s u l t s were obtained f o r a range of f l u i d v i s c o s i t i e s . As can be seen i n Figure 54, changing the v i s c o s i t y has a s u b s t a n t i a l e f f e c t on the r e s u l t s . It i s d i f f i c u l t to i n f e r from these r e s u l t s how turbulence may e f f e c t the r e s u l t s except to note that there would probably be a s l i g h t increase i n the v i s c o s i t y due to turbu-lence. If t h i s were the case, i t could be expected that the maximum suction could increase. In ad d i t i o n the l o c a t i o n of the maximum suction 43 . may be s h i f t e d downstream from the l o c a t i o n predicted f o r laminar flows. At the very l e a s t , these r e s u l t s i n d i c a t e there may be some merit to performing the an a l y s i s f o r turbulent flows. An examination of the influence of the boundary conditions on the pressure was made. The r e s u l t s obtained are presented i n Figure 55 . Figure 55 shows the v a r i a t i o n i n the pressure i s diminished by s p e c i f y i n g sub-atmospheric i n l e t and e x i t pressures, though the o v e r a l l suction i s increased. In obtaining these r e s u l t s , some numerical problems were encountered due to s t i f f n e s s of the model equations. In general, a s o l u t i o n c o u l d only be o b t a i n e d when the i n i t i a l guess f o r was very c l o s e to the value that gave a convergent s o l u t i o n . If the i n i t i a l guess f o r was too s m a l l , the s o l u t i o n would not converge. If the guess was too large, the suction would r a p i d l y increase along the length of the f o i l and the s o l u t i o n would once again f a i l to converge. In some cases, i f the i n i t i a l guess was poorly chosen, a s o l u t i o n having a p o s i t i v e pressure d i s t r i b u t i o n could be obtained. Because the s p e c i f i c a t i o n of sub- atmo-spheric boundary conditions f o r the pressure i s equivalent to imposing an i n i t i a l drainage v e l o c i t y , these r e s u l t s suggest the s o l u t i o n i s strongly influenced by the drainage v e l o c i t y . F i n a l l y , an attempt was made to obtain improved r e s u l t s by performing the a n a l y s i s i n the opposite d i r e c t i o n . The s o l u t i o n proceeded as before except the s o l u t i o n was started at the t r a i l i n g edge of the f o i l . No improvement i n the r e s u l t s was obtained, however. This technique was only s u c c e s s f u l i n d u p l i c a t i n g the r e s u l t s obtained when the s o l u t i o n was started at the leading edge of the f o i l . Summarizing the r e s u l t s of the momentum i n t e g r a l model, i t was found the s o l u t i o n of the model was highly dependent on the i n i t i a l f o i l - w i r e gap s p e c i f i e d , as w e l l as the pressure boundary conditions chosen. In general, the maximum predicted suctions were much lower than those reported by Taylor [ 3 ] , or Ernst [ 8 ] . Both Taylor and Ernst predicted 44. maximum s u c t i o n s of about 4- pu 2 . The present r e s u l t s do, however, 2 w duplicate the experimental trends observed for the pressure d i s t r i b u t i o n s as a function of the wire v e l o c i t y . 3.4 The Two-Dimensional Model 3.4.1 General The two-dimensional model involved the s o l u t i o n of the equation of motion for a boundary layer flow 3u , 3u 1 dp , 92u U —  + V -r— = rr~ + V 9x 3y p dx . o 9y^ ( 3 . 53 ) and the equation for mass conservation i n f i n i t e d ifference form. The r e l a t i o n s h i p Vw - ^ < P - P « > < 3' 5 5> was used f o r the one-dimensional models to r e l a t e the drainage v e l o c i t y to the pressure d i s t i b u t i o n . F i n i t e d i f f e r e n c e approximations were written f o r Equations 3 . 5 3 to 3 . 5 5 using a non-uniform rectangular mesh f i t t e d to the flow region between the f o i l and the forming wire. The r e s u l t i n g f i n i t e d i f f e r e n c e equations were solved to obtain the pressure d i s t r i b u t i o n along the f o i l . Wire d e f l e c t i o n e f f e c t s were not included i n the two-dimensional model. This was primarily due to the d i f f i c u l t i e s of adapting the f i n i t e 45. difference mesh to the changing wire shape during the course of obtaining a s o l u t i o n . Some consideration was given to i n v e s t i g a t i n g adaptive mesh techniques but was not pursued due to the poor r e s u l t s obtained using the two-dimensional model. 3.4.2 The F i n i t e Difference Scheme Before f i n i t e d i f f e r e n c e approximations could be developed f o r Equations 3.53 to 3.55, a d i f f e r e n c i n g scheme had to be selected. In s e l e c t i n g the d i f f e r e n c i n g scheme, i t was necessary that the method be an i m p l i c i t one to ensure s t a b i l i t y of the s o l u t i o n . The scheme had to be able to support non-uniform meshes to f a c i l i t a t e f i t t i n g of a mesh to the i r r e g u l a r shape of the flow region. F i n a l l y , a proven technique was desired as i t was beyond the scope of the present i n v e s t i g a t i o n to pioneer a new f i n i t e d i f f e r e n c i n g method. Af t e r examining several f i n i t e d i f f e r e n c e schemes, a procedure known as the K e l l e r Box Method was selected. The Box Method i s a recently developed i m p l i c i t scheme described by K e l l e r [19] and K e l l e r and Cebeci [20]. It has been used to obtain solutions to a wide v a r i e t y of flows which lend themselves to d e s c r i p t i o n by the boundary layer equations. The Box Method has also been used to varying degrees of success for the a n a l y s i s of mildly separated flows. This was considered an important feature of the Box Method as i t was anticipated that some flow separation could occur over the f o i l . Features of the Box Method include: 1. i m p l i c i t formulation - unconditionally stable; 2. second order accuracy; 3. s o l u t i o n on a r b i t r a r y (non-uniform) mesh possible: 46. 4. allows very rapid v a r i a t i o n s i n the x - d i r e c t i o n ; 5. simple, easy'to solve formulation. Thus, the box method was w e l l s u i t e d to use f o r the present i n v e s t i g a t i o n . 3 . 4 . 3 The F i n i t e D i f f e r e n c e E q u a t i o n s Use of the K e l l e r Box Method involved four p r i n c i p l e steps: 1. Reduce the equations of motion to a system of f i r s t order equations; 2. Write d i f f e r e n c e equations using centered differences; 3. Line a r i z e the r e s u l t i n g system of equations ( i f non-linear); i 4. Solve the r e s u l t i n g system of l i n e a r algebraic equations. Beginning with the f i r s t step, Equations 3.1 and 3.2 are f i r s t w r i tten i n non-dimensional form au* Sv* where ju x . u . p X* = — u* = p* = —— L u w pu 2 w (3.59) y* = T " /Re v * = — /Re L u w and Re i s the Reynolds number based on the f o i l length u L Re = - 2 - ( 3 . 6 0 ) 47. In Equation 3 . 5 7 , b i s a non-dimensional v i s c o s i t y term of the form vt b = 1 + ~ ( 3 . 61 ) and v i s a t u r b u l e n t v i s c o s i t y term which i s equal to zero f o r laminar flows, v was included i n the development of the two-dimensional model to permit l a t e r expansion of the model to include turbulent flows. Continuing with the f i r s t step, i f f i s defined such that f=-g ( 3 . 6 2 ) Equations 3 .57 and 3 .58 may be written f - C3 .63) " . 6 5 , where the a s t e r i s k s have been dropped f o r convenience. This completes the f i r s t step of the K e l l e r Box Method. Equations 3 . 6 3 to 3 . 6 5 cannot be solved d i r e c t l y i n the given form as there are four unknowns, u,v,p, and f . Two methods have generally been used to overcome t h i s problem. The f i r s t method, referred to i n the l i t e r a t u r e as the Eigenvalue method [21] involves an i t e r a t i v e s o l u t i o n of the system of equations. The s o l u t i o n proceeds by assuming a pressure d i s t r i b u t i o n and then s o l v i n g Equations 3 . 6 3 through 3 . 6 5 . Results of the 48. s o l u t i o n are then used to improve the guess f o r the pressure d i s t r i b u t i o n , and the s o l u t i o n procedure i s i t e r a t e d u n t i l the pressure d i s t r i b u t i o n converges. The second method, known as the Mechul function method [21], treat s the pressure as an unknown and introduces a fourth equation ^ - 0 ( 3 . 6 6 ) The s o l u t i o n of Equations 3 . 63 through 3 .66 may then be c a r r i e d out d i r e c t l y . Though there i s generally l i t t l e d i f f e r e n c e i n the r e s u l t s obtained using the two methods, the Mechul function method has the advantage of allowing a d i r e c t s o l u t i o n without any i t e r a t i o n required. This s i m p l i -f i e s the programming of the method at the expense of one more equation to solve. The Mechul function method has also been suc c e s s f u l l y applied to problems inv o l v i n g separated flows [21]. For these reasons, the Mechul function method was used f o r the present problem. To summarize step one of the Box method, the equations of motion are written as the system of f i r s t order d i f f e r e n t i a l equations f = | f ( 3 . 6 7 ) ( 3 . 7 0 ) 49. having the boundary conditions v k y = 0; u = 1, v - /Re = (-£-•) (p-p a) (Re) 3/2 w y = - — /Re; u = 0, v = 0 (3.71) In step two of the Box method, Equations 3.67, 3.68, 3.69 and 3.70 are written i n f i n i t e d i f f e r e n c e form using centered d i f f e r e n c e s . D i f f e r e n c i n g was done on the g r i d shown i n Figure 56. Spacing of the mesh was selected such that mesh nodes f e l l on the f o i l boundary. Difference equations were written such that equations in v o l v i n g p a r t i a l d e r i v a t i v e s i n both the x and y d i r e c t i o n s were centered on (P^ ,P ,P 3,P ) shown i n Figure 56. A l l other equations were centered on ( P 1 , P 2 ) . Defining the notation i-1/2 1 , i . i - U 8j = 2 ^ 3 + Si > i 8 j - l / 2 = 2 (gj + 8 ^ ) , i i - 1 . W j k. k d y ; j h. (3.72) where g i s any quantity, the di f f e r e n c e equations f o r Equations 3.67 through 3.70 can be written _ i 1 , i i . f j - l / 2 " h j ( u J " UJ-1> (3.73) 50. . 1-1/2,rl , i i-1 n M / 2 ' V l ^ k T (uJ-l/2 " U j - l / 2 ^ + ( v f ) j - l / 2 = - k (pi - l/2 - p5hi/2> + ¥ 7 I C b f ) J " 1 / 2 " ( b f ) ^ : 1 / 2 ] (3.74) 1 , i i-1 s . 1 , i-1/2 i-1/2. n ,„ ^ < uj-i / 2 " u j - i / 2 > + hT ( v j " v j - i } = 0 ( 3 * 7 5 ) k~ ( p i " p J L i ) = ° ( 3 - 7 6 ) C o l l e c t i n g a l l terms i n ( i ) on the l e f t hand side gives \ - („i - u ^ ) - f i _ l / 2 - 0 (3.77) 7 ^ - l / 2 > 2 +1 <vf>5-l/2 + U7 (Pj-l/2> " 2ET [(bf)5 " ( b f ) j - J 2k7 (U5-l/2)2 " 1 > £ / 2 + ¥± < P £ / 2 > + 2HT [< b f> j " ' - ( b f ^ (3.78) 1 t 1 , , 1 • i i . 1 , i-1 . 1 , i-1 i - l N ,~ 11\\ k7 <uJ-l/2) + 2hT ( V J " V j-1 } = \ <*}-l/2> " 2hT ( V J " ^ - P ( 3 ' 7 9 ) ir ( p i ~ p i - i ) = 0 ( 3 - 8 0 ) 51. The next step of the K e l l e r Box Method requires the l i n e a r i z a t i o n of Equations 3.77 through 3.80. L i n e a r i z a t i o n was done using a Newton i t e r a t i o n technique. Defining the a r b i t r a r y quantity g, the value of g at the advanced step (n+1) may be found using ( g J ) n + 1 - (g^) n + (6g^) n^ (3.81) Using the notation of Equation 3.81, Equations 3.77 through 3.80 may be written at the advanced step (n+1) [(u + 6u)* - (u.+ " (f + 6 f ) j - i / 2 = 0 < 3* 8 2> 2k- [<"5-l/2 ) 2 + < 2 uS-l/2>< 6 uS-l/2>] +1 [ ( V + 6 v ) j - l / 2 < f + 6 f ) j - l / 2 ] + ^ (P + <SP>j_ 1 / 2 " 2n^ Ub(f + 6£>5] - [b(f + 6 ^ ] } = 4 ^ i / 2 - \ ^ >5hi/2 + k[ (p5-i/2>+ 2 ib r<bf>r - <M>£] (3.83) i - (u + 6 u)5_ 1 / 2 + 2KT [(v + 6v)* - (v + S v ) ^ ] = ^ < u 5 - i / 2 > = 2 h T [ v r - v 5 - i 3 ( 3 - 8 4 ) 52. JT [(P + 6P)i - (P + 6P)i-i] = 0 (3-85) where the superscript (n) has been dropped for convenience and a l l terms that are quadratic i n 6 are neglected. L i n e a r i z a t i o n was only performed on terms i n ( i ) as a l l term i n (i-1) are known q u a n t i t i e s . C o l l e c t i n g a l l terms i n <5uj, "Sv*, ^ f j» a n c * v i e l d s the l i n e a r system of equations 6 u * - S u ^ - - | h tt1 - -j h 8 f * - ( r ^ * ( 3 . 8 6 ) where (3.87) k k i i i 1 i i 1 6u. + fiu. . +T±8v:L-x±. 6V. . = (r„) . (3.88) J j-1 h j h j-1 3'j ' 6 P J " 6 P J " 1 = ( r ^ j ( 3 * 8 9 ) 'Vj = v r u j + hj 4-1/2 < 3- 9 0> + £j[0>f>} - ( b f ) ^ ] + R £ / 2 (3.91) 53. (r. )* = -p* + p* . = 0 (3.93) J J J-1 R5-i/2 = ^  <u2>5-i/2 - ( v f ) 5-1/2 + h [<bf>r - (3-94) and / \ i 1 1 , v i 1 - i = ^ U J ( S 6 } J = 2 f j b 1 (3.95) (s ) 1 = I v 1 - -L- (s )! = L / \ i 1 i I b j - l , x i 1 ( S ^ j = 2 V j - 1 + - n — ( 89>J = k " j i To complete step 3, the boundary conditions given by Equation 3.71 are written i n the form y = 0 ( j = 0); 6u* = 0, 6v* = 0 y = ^ /Re ( j - J ) ; 6uJ = 0, 6v* = 0 (3.96) 54. This simply states that u and v are known at both the f o i l and the forming w i r e . Since v w i s not e x p l i c i t l y known, but instead i s given by equation 3.55, the s o l u t i o n must be i t e r a t e d at each x node to ensure t h i s boundary condition i s s a t i s f i e d . This w i l l be discussed further i n Section 3.4.4. In the fourth and f i n a l step of the Box method, Equations 3.86, 3.87, 3.88, and 3.89 are written i n the matrix vector form 0 < i < I 0 < j < J (3.97) In Equation 3.97, A. i s the c o e f f i c i e n t matrix - i - i a c r i - i - i b . a . c . J J J 0 < j < J (3.98) where - i a o 1 0 -1 o l 0 < 86>* 6 J 0 0 -\I2 <v5 o 0 0 - i a j = 55. 1 k l/h. 0 1 0 0 0 1 1 0 0 0 0 1 0 0 - i a. J b 1 J 0 -1 o 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 1 < j < J-l 1 < j < J (3.99) - i c. J 0 0 0 0 0 0 0 0 I-I 0 0 ('1> 1J« <S7 o < j < J-l X* i s the vector J X 1 = J _ i x o _ i x. J 3" 6u J 6v S f 1 J "5 o < j < J (3.100) 56. and i s the vector J B 1 = J b 1 o b 1 J *1 < rl> j+1 ( r2>j+l b = <rl>o <r2>o 1 < j < J-1 < r3>i-i ( r2>J (3.101) The f i r s t and l a s t two rows of A^ represent the boundary conditions given by J = °J 6 u o = <r3>o " ° j = J ; 6uJ - ( r i ) J = 0 6V 1 = ( r . ) i = 0 o 'to 6 V J = ( r2>J " 0 (3.102) Equation 3.97 represents a system of l i n e a r algebraic equations i n matrix vector form. A so l u t i o n to 3.97 may be obtained by any conventional method for solving systems of l i n e a r equations. 3 . 4 . 4 Method o f S o l u t i o n The procedure used to obtain a s o l u t i o n f o r the two dimensional model was r e l a t i v e l y s t r a i g h t forward. Before beginning the sol u t i o n , a mesh having constant spacing i n the x - d i r e c t i o n was calculated. The l o c a t i o n 57. of the y nodes were c a l c u l a t e d such that for each x node, a y node was located at both the forming wire and the f o i l . The s o l u t i o n began by assuming i n i t i a l conditions f o r u,v,p, and f of the form u° = U. J J v° - V. f° - F . J j P° = P. J J 0 < j < J (3.103) Using these i n i t i a l conditions as the s t a r t i n g point, Equation 3.97 was solved at each node i n the x - d i r e c t i o n . Thus, the s o l u t i o n was marched from the leading to the t r a i l i n g edge of the f o i l . At each x node, the s o l u t i o n of Equation 3.97 yielded values for a l l q u a n t i t i e s 0 < j < J These q u a n t i t i e s were then used as the i n i t i a l guess f o r the s o l u t i o n of Equation 3.97 at the next node i n the x - d i r e c t i o n . As was mentioned i n Section 3.4.3, an i t e r a t i o n was required at each x node to s a t i s f y the condition given by Equation 3.55. A f t e r a s o l u t i o n to Equation 3.97 had been obtained, Equation 3.55 was evaluated and the computed value of v w was compared to the r e s u l t obtained from the s o l u t i o n of Equation 3.97. If the two values were not i n agreement, Equation 3.55 was u s e d t o c a l c u l a t e an updated v a l u e f o r p u s i n g the v a l u e of v 58 . obtained from the s o l u t i o n of Equation 3.97. Having done t h i s , a new s o l u t i o n to Equation 3.97 was calculated. This procedure was repeated u n t i l the boundary condition was s a t i s f i e d at which point the s o l u t i o n could proceed to the next x node. A flow chart of the s o l u t i o n procedure i s given i n Figure 57. 3 . 4 . 5 V e r i f i c a t i o n o f t h e M o d e l The two dimensional model was v e r i f i e d against the exact solutions f o r flow over a f l a t p l a t e at zero incidence, flow i n a rectangular duct, and flow between moving p a r a l l e l f l a t p l a t e s . For a l l three t e s t cases, i n i t i a l conditions were give as o u . J = 1 o V . J = 0 0 < j < J (3.104) f . = 0 o w A s o l u t i o n was then obtained f o r each of the flows. In the case of the i n t e r n a l duct flow and the couette flow, the s o l u t i o n was allowed to proceed u n t i l i t was f u l l y developed. From Figures 58, 59 and 60 i t can be seen very good agreement was obtained f o r a l l three flows. In the case of the f l a t p l ate boundary layer, some compromises had to be made to obtain a s o l u t i o n . Because the f i n i t e d i f f e r e n c e equations were formulated f o r i n t e r n a l flows i n which 59. the pressure i s unknown, an adaptation of the boundary conditions was necessary. To solve f o r the f l a t p l a t e boundary layer flow, the r e l a t i o n -ship between the drainage v e l o c i t y and the pressure given by Equation 3.55 was substituted with the r e l a t i o n h s h i p I2- = u42- (3.105) dx dx v ' The s o l u t i o n was then i t e r a t e d at each x node to s a t i s f y t h i s condition. Because t h i s procedure introduced a d d i t i o n a l error i n the c a l c u l a t i o n , exact agreement with the Blasius p r o f i l e was not obtained. The error, however, was very small and considered i n s i g n i f i c a n t as the d i f f e r e n c e equations were not intended for use on external flows. For the i n t e r n a l duct flow and the couette flow, e x c e l l e n t agreement with the exact r e s u l t s were obtained. In both cases, f u l l y developed flow was obtained within 2 percent of the length predicted by theory. Despite the excellent agreement achieved, some minor problems can be seen i n the r e s u l t s . For the duct flow, shown i n Figure 59, a s l i g h t o s c i l l a t i o n can be seen at the walls i n the developing v e l o c i t y p r o f i l e s . For the couette flow, the r e s u l t s f o r which are given i n Figure 60, the v e l o c i t y p r o f i l e s should a l l cross the centre l i n e at the same l o c a t i o n . A d d i t i o n a l l y , a small flow r e v e r s a l can be seen at the lower wall. In both flows, these small discrepancies can be a t t r i b u t e d to errors caused by a s l i g h t numer-i c a l i n s t a b i l i t y at the w a l l i n the f term (f = 4 — ) ' As the s o l u t i o n dy proceeds i n the x d i r e c t i o n , a s l i g h t o s c i l l a t i o n begins i n f . In most cases the o s c i l l a t i o n i s damped, as was observed i n the case of the duct flow. i n the couette flow the o s c i l l a t i o n i s only damped to the extent that i t s growth i s stopped, but i t i s not completely eliminated. This problem caused a great deal of trouble i n the solutions f o r the flow over f o i l s . 60. 3 . 4 . 6 R e s u l t s The r e s u l t s obtained using the two-dimensional model were generally very disappointing. Attempts to obtain r e s u l t s f or the f o i l s and wire speeds used i n the experimental t e s t s met with only minimal success. As can be seen from Figure 61, the predicted pressure d i s t r i b u t i o n s have very l i t t l e s i m i l a r i t y to the r e s u l t s obtained using the one-dimensional models. In the case of f o i l s 2, 3 and 4, r e s u l t s could not be obtained. In each of these cases, examination of the c a l c u l a t e d v e l o c i t y p r o f i l e s indicated flow separation occurred. Figure 62 shows the v e l o c i t y p r o f i l e s obtained f o r f o i l 2. Use of the Mechul function technique permited evaluation of the flows beyond the point of separation to some extent, however, the flow separation ultimately l e d to the f a i l u r e of the so l u t i o n s . Figure 63 presents the c a l c u l a t e d pressure d i s t r i b u t i o n s on f o i l 1 f o r a range of wire v e l o c i t i e s . As can be seen, these r e s u l t s are i n better agreement with the r e s u l t s of the other models, showing s i m i l a r trends i n the pressure d i s t r i b u t i o n s as the wire v e l o c i t y i s changed. Figure 6A,_however, ind i c a t e s separation was predicted at the higher wire v e l o c i t i e s . A s i g n i f i c a n t point of i n t e r e s t i n the r e s u l t s i s the f a i l u r e of the model to meet the condition that at x = L, p = p & . This l i m i t a t i o n i s due to the parabolic nature of the boundary layer equations. Attempts were made to i t e r a t e the s o l u t i o n to s a t i s f y the end boundary cond i t i o n but the s o l u t i o n was highly unstable i n the second i t e r a t i o n and f a i l e d i n a l l attempts. In obtaining the r e s u l t s presented, i t was discovered that a s o l u t i o n could often be obtained by s p e c i f y i n g an i n i t i a l drainage v e l o c i t y along 61. the length of the f o i l . This technique was used to obtain the r e s u l t s presented i n Figures 61 to 64. It was found that by s p e c i f y i n g an i n i t i a l drainage v e l o c i t y , larger suction peaks were calculated. This had the e f f e c t of drawing more f l u i d through the forming wire and i n t o the flow, thus delaying the onset of separation. If the i n i t i a l drainage v e l o c i t y was excessive, the magnitude of the suction would grow exponentially, r e s u l t i n g i n a f a i l u r e of the s o l u t i o n . As a f i n a l point of i n t e r e s t , i t should be noted that for a l l the r e s u l t s presented, c a l c u l a t i o n s were started at the leading edge of the f o i l . The s o l u t i o n procedure was started by s p e c i f y i n g an i n i t i a l gap height and the i n i t i a l conditions, and then proceeded as described i n Section 3.4.4. As was the case f o r the one-dimensional models, the i n i t i a l gap height had a s i g n i f i c a n t influence on the magnitude of the suction developed. Unlike the one-dimensional models, however, the maximum suction occurred at the t r a i l i n g edge of the f l a t s e ction of the f o i l (Figures 61 and 63). Only by s p e c i f y i n g large i n i t i a l drainage v e l o c i t i e s could the l o c a t i o n of the suction peak be s h i f t e d along the length of the f o i l . Doing so, however, usually r e s u l t e d i n an exponential growth of the suction s i m i l a r to that when an excessive i n i t i a l drainage v e l o c i t y was imposed. As before, the uncontrolled growth of the suction eventually caused the s o l u t i o n to f a i l . 62. IV. DISCUSSION 4.1 General The p r i m a r y t h r u s t o f t h e p r e s e n t i n v e s t i g a t i o n was t o d e v e l o p a m a t h e m a t i c a l m o d e l t h a t c o u l d be u s e d t o p r e d i c t t h e p r e s s u r e d i s t r i b u t i o n o n a F o u r d r i n i e r p a p e r m a c h i n e d r a i n a g e f o i l . In t h e c o u r s e o f d e v e l o p i n g a m o d e l , two d i s t i n c t a p p r o a c h e s were t a k e n r e s u l t i n g i n t h r e e s e p a r a t e m o d e l s . I n a d d i t i o n t o t h e m o d e l l i n g o f t h e p r e s s u r e d i s t r i b u t i o n , e x p e r i m e n t s we re c o n d u c t e d t o measure t h e p r e s s u r e d i s t r i b u t i o n s on a s e l e c t i o n o f f o i l s . T h e s e e x p e r i m e n t s were c o n d u c t e d u s i n g c o n d i t i o n s t y p i c a l o f a n a c t u a l p a p e r m a c h i n e w i t h t h e hope o f p r o v i d i n g r e l i a b l e d a t a w i t h w h i c h t h e r e s u l t s o f t h e n u m e r i c a l m o d e l l i n g c o u l d be c o m p a r e d . D e t a i l e d p r e s e n t a t i o n s o f t h e r e s u l t s o b t a i n e d have b e e n p r o v i d e d i n p r e v i o u s c h a p t e r s . A c o m p a r i s o n o f t h e r e s u l t s and d i s c u s s i o n o f t h e i r s i g n i f i c a n c e f o l l o w s . 4.2 Experimental Results B e f o r e a c a r e f u l e x a m i n a t i o n o f t h e e x p e r i m e n t a l r e s u l t s c a n be made, t h e p u r p o s e and t h e e x p e c t a t i o n s o f t h e e x p e r i m e n t must be e x a m i n e d . The g o a l o f t h e e x p e r i m e n t was n o t t o t r y and o b t a i n r e s u l t s w i t h w h i c h c o n c l u s i o n s r e g a r d i n g t h e p e r f o r m a n c e o f f o i l s c o u l d be made. R a t h e r , t h e i n t e n t was s i m p l y t o o b t a i n d a t a o f r e a s o n a b l e a c c u r a c y t h a t c o u l d be u s e d f o r c o m p a r i s o n t o t h e r e s u l t s o f t h e n u m e r i c a l m o d e l s . To s a t i s f y t h i s g o a l , a v e r y c r u d e e x p e r i m e n t was d e s i g n e d t o a l l o w t h e t e s t i n g o f s e v e r a l f o i l s u n d e r n o r m a l p a p e r m a c h i n e c o n d i t i o n s . W i t h t h e g o a l s o f t h e e x p e r i m e n t i n m i n d , t h e r e s u l t s o b t a i n e d c a n be c o n s i d e r e d a c c e p t a b l e f o r t h e i n t e n d e d p u r p o s e . 63. In general, the experimental r e s u l t s duplicate trends exhibited i n the r e s u l t s of previous i n v e s t i g a t i o n s [5,9,11]. For f o i l 3, very good c o r r e l a t i o n between the present r e s u l t s and those of F l e i s c h e r [9] were obtained f o r a wire v e l o c i t y of 500 ft/min (Figure 65). Unfortunately, a lack of d e t a i l e d information regarding the experimental procedure used by F l e i s c h e r made i t d i f f i c u l t to d i r e c t l y compare h i s r e s u l t s to those of the present i n v e s t i g a t i o n . Also, F l e i s c h e r ' s r e s u l t s are f o r a 3 degree f o i l whereas the r e s u l t s of the present i n v e s t i g a t i o n shown are f o r a 4 degree f o i l . Thus, the comparison i s made only to i n d i c a t e that the present r e s u l t s are of the same order of magnitude and shape. As was expected, Figures 16-21 show the measured pressure d i s t r i b u -t i o n s were s i g n i f i c a n t l y d i f f e r e n t f o r each f o i l . The r e s u l t s of Figures 11-15 v e r i f y the trend of increasing suction with increasing wire velo-c i t i e s . Because of the method used to non-dimensionalize the pressure, t h i s trend i s indicated i n Figures 11-15 by a decrease i n the non-dimensional suction. The r e s u l t s obtained f o r f o i l 3, presented i n Figure 13, best show t h i s trend. Throughout the experiments, f o i l 3 co n s i s t e n t l y provided the best r e s u l t s . Figure 13 c l e a r l y shows the r e l a t i o n s h i p between the pressure d i s t r i b u t i o n and the wire v e l o c i t y . Also evident i s the pressure f l u c t u a -t i o n s associated with the h o r i z o n t a l leading edge section of the f o i l . These f l u c t u a t i o n s are due to squeezing of the water c l i n g i n g to the lower surface of the wire between the wire and the f o i l . Figure 66 provides a possible explanation f o r the poor r e s u l t s obtained using f o i l s 1 and 2. If a close examination of the wire surface i s made at the t r a i l i n g edge of the f o i l , a d i s t i n c t i v e 'dry-out' l i n e can be seen. At t h i s boundary, a l l the water has been drawn through the wire causing the pressure to return to atmospheric. Because i t i s not cl e a r from Figure 66 where the t i p of the dry-out l i n e i s r e l a t i v e to the 6 4 . pressure tap l o c a t i o n s , and the shape of the dry-out l i n e was not l i k e l y constant, the poorly developed pressure d i s t r i b u t i o n of Figure 12 was probably due to t h i s phenomena. F o i l 3, on the other hand, always had a complete layer of water over i t s e n t i r e length as seen i n Figure 67. Though c l e a r photographs of f o i l 1 were not obtained, i t i s l i k e l y that dryout was a l s o the reason f o r the poor r e s u l t s obtained with f o i l 1. It i s also possible three dimensional e f f e c t s played a r o l e i n the poor r e s u l t s obtained on f o i l s 1 and 2. If an analogy i s drawn between a f o i l and a s l i d e r bearing, s l i d e r bearing theory suggests that f o r t y p i c a l f o i l geometries tested, the maximum suction would be approximately 0.4 times the suction obtainable on a f o i l of i n f i n i t e width [22]. If the flow was three dimensional, water could have entered i n t o the flow region from the sides of the f o i l . If t h i s were the case, the net r e s u l t would be an o v e r a l l reduction of the magnitude of the measured suction. It i s a l s o possible t h i s phenomena could have existed on f o i l 3 as the maximum suctions measured are s l i g h t l y lower than those reported i n previous i n v e s t i g a t i o n s . However, the good c o r r e l a t i o n with the r e s u l t s of F l e i s c h e r shown i n Figure 65 suggests the e f f e c t s of three dimensional flow are l e s s than that ind i c a t e d by s l i d e r bearing theory. Unfortu-na t e l y , i t i s very d i f f i c u l t to c l e a r l y determine whether three dimensional e f f e c t s did i n f a c t e x i s t as spanwise pressure measurements were not made. Further experimental work should include spanwise pressure measurements to e s t a b l i s h the possible extent of t h i s e f f e c t . The r e s u l t s obtained using f o i l s 4 and 5 (Figures 14 and 15) suggest the f l u i d flow was not established between the f o i l and the wire. Very e r r a t i c pressures were recorded over the h o r i z o n t a l section of the f o i l . At the step l o c a t i o n , however, the pressure returned to atmospheric. Only at the t r a i l i n g edge of f o i l 5, which had a much smaller step height than f o i l 4, d i d a pressure d i s t r i b u t i o n develop. These r e s u l t s , however, are 65. consistent with those of Cadieaux [11]• Cadieaux found that external su c t i o n was often necessary to i n i t i a t e and sustain the flow over a stepped f o i l . As was discussed i n Section 2.6, the measured drainage rates are considered to be very inaccurate. The measurement technique employed allowed a great deal of water to enter the sides of the drainage trough as w e l l as from the end of the f o i l . As such, the technique used was a t o t a l l y inadequate method f o r measuring the drainage rates. A much improved measurement technique needs to be devised that would l i m i t the flow of water i n t o the drain trough to that leaving the end of the f o i l . Complicating t h i s task, however, i s the need to ensure the water c o l l e c t i o n method does not r e s t r i c t the d e f l e c t i o n of the forming wire at the t r a i l i n g edge of the f o i l . Summarizing the expeirmental r e s u l t s , good suction measurements were obtained f o r f o i l 3. Suction d i s t r i b u t i o n measurements on the remaining f o i l s , however, were unsuccessful. Drainage rate measurements as a whole, were t o t a l l y unacceptable. Thus, only the pressure d i s t r i b u t i o n s measured on f o i l 3 can be considered u s e f u l f o r comparison to the r e s u l t s of the numerical modelling. Furthermore, the pressure d i s t r i b u t i o n s measured on f o i l 3 showed c l e a r trends i n the pressure as a function of the wire v e l o c i t y to look f o r i n the r e s u l t s of the numerical modelling. Though the experimental r e s u l t s were poor o v e r a l l , no attempt was made to r e f i n e the experiment to obtain improved r e s u l t s . This was due to the f a i l u r e of the numerical models to produce reasonable r e s u l t s , thus i t seemed unnecessary to improve the experiment. 4 . 3 Numerical Results Of the three numerical models presented i n Chapter 3, only the one-dimensional models provided acceptable r e s u l t s . Both the conservation of 66. mass model and the momentum i n t e g r a l model c l e a r l y reproduced the trends i n the pressure d i s t r i b u t i o n s observed i n the experimental r e s u l t s . A d i r e c t comparison of the numerical r e s u l t s to those obtained experiment-a l l y was very disappointment however. In general, the maximum suctions predicted by the models were s i g n i f i c a n t l y l e s s than those measured. Because of the s i g n i f i c a n t d i f f e r e n c e s i n the magnitudes of the c a l c u l a t e d and measured pressures, no attempt was made to d i r e c t l y compare the ca l c u l a t e d and measured pressure d i s t r i b u t i o n s . It may be pointed out that though the models of Meyer and Ernst were very s i m i l a r to the conservation of mass model, v a s t l y d i f f e r e n t r e s u l t s were obtained. This i s thought to be a r e s u l t of the excessive i n i t i a l wire gap that had to be s p e c i f i e d to begin the s o l u t i o n of the conserva-t i o n of mass model. As was mentioned i n previous sections, the f o i l - w i r e gap had a s i g n i f i c a n t e f f e c t on the r e s u l t s obtained. Meyer was able to avoid t h i s problem as he obtained an a n a l y t i c a l s o l u t i o n to h i s model. Ernst, however, l i k e l y had to deal with t h i s problem as h i s s o l u t i o n was a numerical one. It i s not c l e a r , however, how Ernst avoided t h i s problem. Figure 68 i s a comparison of t y p i c a l pressure d i s t r i b u t i o n s calcu-l a t e d using the two one-dimensional models. Under a l l conditions, the suct i o n predicted by the momentum i n t e g r a l model was l e s s than that predicted by the conservation of mass model. Consequently, predicted drainage rates f o r the momentum i n t e g r a l model were lower as shown i n Figure 69. F i n a l l y , Figure 70 in d i c a t e s much greater flow separation was predicted by the momentum Integral model. The extensive flow separation ind i c a t e d i n Figure 70 r a i s e s serious questions about the v a l i d i t y of using the boundary layer equations to model the flow over f o i l s . Though i t i s possible the flow separates, i t 67. i s highly u n l i k e l y that i t occurs to the extent suggested i n Figure 70. Instead, i t i s more l i k e l y there i s a deficiency i n the models r e s u l t i n g i n a f a i l u r e of the models to accurately predict the pressure and hence, the drainage v e l o c i t y . Consequently, i n s u f f i c i e n t f l u i d crosses the wire i n t o the flow region to prevent the tendency f o r the flow to separate due to the expansion of the flow region. In an attempt to examine t h i s p o s s i b i l i t y , several t e s t s of the e f f e c t of a r t i f i c i a l l y increasing the drainage v e l o c i t i e s were made using the momentum i n t e g r a l model. The assumption was that drainage through the wire due to g r a v i t y could account f o r a large part of the flow. To t e s t t h i s assumption, an i n i t i a l d rain-age v e l o c i t y was imposed over the length of the f o i l by sp e c i f y i n g sub-atmospheric boundary conditions f o r the pressures. The predicted v e l o c i t y p r o f i l e s f o r one of the tes t s are given i n Figure 71. If Figure 71 i s compared to Figure 72, i t can be seen that the extent of the flow separa-t i o n was i n f a c t reduced by a s u b s t a n t i a l amount. If the i n i t i a l v e l o c i t y was too large, however, the suction grew exponentially r e s u l t i n g i n a f a i l u r e of the s o l u t i o n due to the i n a b i l i t y of the s o l u t i o n to s a t i s f y the boundary condition on the pressure at the t r a i l i n g edge of the f o i l . These r e s u l t s suggest the one-dimensional models f a i l to adequately allow for the drainage v e l o c i t y . Results of c a l c u l a t i o n s that included forming wire d e f l e c t i o n e f f e c t s showed s i m i l a r e f f e c t s on the pressure d i s t r i b u t i o n as those presented by Ernst though the e f f e c t s were generally much more pronounced. The o v e r a l l e f f e c t of in c l u d i n g the wire d e f l e c t i o n i n the c a l c u l a t i o n s was a s l i g h t reduction of the maximum suction as well a s h i f t of the maximum suction towards the t r a i l i n g edge of the f o i l . These r e s u l t s tend to support the hypothesis of Taylor that wire d e f l e c t i o n e f f e c t s would s h i f t the l o c a t i o n 68. of the maximum suction towards the t r a i l i n g edge of the f o i l . Though the wire d e f l e c t i o n s were calculated, t y p i c a l d eflected wire shapes have not been presented. This i s due to the very small d e f l e c t i o n s c a l c u l a t e d as a r e s u l t of the low suctions predicted. In general, the wire d e f l e c t i o n s were on the order of about 1 to 2 percent of the maximum gap height. This does, however, suggest the s i g n i f i c a n c e of the e f f e c t of the wire d e f l e c -t i o n on the suction d i s t r i b u t i o n s . Attempts at modelling f o i l s using the approach of the two-dimensional model were generally a f a i l u r e . Though r e s u l t s were obtained that predic-ted pressures more i n l i n e with those obtained experimentally, the solu-t i o n s were unable to s a t i s f y the boundary conditions on the pressure d i s t r i b u t i o n and must be considered a f a i l u r e . If t h i s shortcoming i s momentarily overlooked, the two-dimensional solutions d i d ex h i b i t s i m i l a r trends to the one-dimensional solutions and the experimental r e s u l t s . In addi t i o n , minimal flow separation was predicted by the solutions that were obtained. The lack of wide spread separation i n the solutions using the two-dimensional model suggests that the f a i l u r e of the one-dimensional models to adequately predict the pressures i s not due to the use of the boundary l a y e r equations. Instead the separation i s probably a shortcoming of the approximations made i n assuming the v e l o c i t y p r o f i l e s . Improved r e s u l t s could perhaps have been obtained i f a higher order v e l o c i t y p r o f i l e had been used. To do t h i s , however, a d d i t i o n a l boundary conditions f o r the v e l o c i t y p r o f i l e would be required. Numerical e x p e r i m e n t a t i o n i n d i c a t e d the r e s u l t s of the one-dimensional models were highly dependent on the i n i t i a l f o i l - w i r e gap. Minimizing t h i s gap increased the magnitude of the predicted suction and 69. thus reduced the amount of separation that was predicted. Beyond a c e r t a i n point reducing the gap s i z e caused truncation errors i n the c a l c u l a t i o n s to increase to the point where the equations became s t i f f and the s o l u t i o n f a i l e d . On an a c t u a l f o i l , the wire i s i n contact with the f o i l at the leading edge. Thus, i t i s desirable to reduce the s i z e of the i n i t i a l gap s p e c i f i e d . Introduction of a large i n i t i a l gap undoubtedly accounts f o r part of the error i n the r e s u l t s . To summarize, the numerical models were a successful i n that they p r e d i c t the experimentally observed trends i n the pressure d i s t r i b u t i o n on the f o i l . The models, however, are not s u f f i c i e n t l y accurate to be of use as a t o o l f o r the optimization of a f o i l design as they f a i l to p r e d i c t the proper magnitudes and shapes of the pressure d i s t r i b u t i o n s . The major sources of e r r o r are thought to be the dependence of the s o l u t i o n on the i n i t i a l gap height as well as numerical errors introduced i n t o the c a l c u -l a t i o n s because of the i n s t a b i l i t y of the s o l u t i o n . There i s hope, though, that with further e f f o r t the momentum Integral model could be improved to the point where i t would serve as a u s e f u l design t o o l . Promise does e x i s t for the two-dimensional model though the numerical problems to be overcome are much greater. In the favour of the momentum i n t e g r a l model, the two-dimensional model i s a much more complex and computationaly d i f f i c u l t approach, though i t i s f e l t i t i s p o t e n t i a l l y the most accurate approach. 70. V. CONCLUSIONS The o b j e c t o f t h e p r e s e n t i n v e s t i g a t i o n was t o d e v e l o p a n u m e r i c a l m o d e l t h a t c o u l d be u s e d t o p r e d i c t t h e p r e s s u r e d i s t r i b u t i o n on a F o u r d r i n i e r p a p e r m a c h i n e d r a i n a g e f o i l . The p u r p o s e o f s u c h a m o d e l was t o p r o v i d e a d e s i g n t o o l t h a t c o u l d be u s e d t o e v a l u a t e t h e d r a i n a g e c h a r a c t e r i s t i c s o f a n a r b i t r a r y f o i l and p e r m i t t h e o p t i m i z a t i o n o f t h e f o i l d e s i g n . F rom t h e r e s u l t s o f t h e p r e s e n t i n v e s t i g a t i o n , t h e f o l l o w i n g c o n c l u s i o n s may be made: 1. Due t o t h e p o o r ag reemen t o f t h e m a g n i t u d e s and s h a p e s o f t h e p r e s s u r e d i s t r i b u t i o n s on f o i l s b e t w e e n t h e r e s u l t s o f t h e p r e s e n t m o d e l l i n g a t t e m p t s and t h e e x p e r i m e n t a l r e s u l t s o f t h e p r e s e n t and p r e v i o u s i n v e s t i g a t i o n s , i t mus t be c o n c l u d e d t h a t t h e m o d e l s f a i l t o a d e q u a t e l y m o d e l t h e f l o w o v e r f o i l s . In g e n e r a l , h o w e v e r , s i m i l a r t r e n d s f o r t h e r e l a t i o n s h i p be tween t h e p r e s s u r e d i s t r i b u t i o n and t h e f o r m i n g w i r e s p e e d we re o b s e r v e d i n t h e n u m e r i c a l and e x p e r i m e n t a l r e s u l t s . 2. T h r e e mode l s have b e e n d e v e l o p e d w h i c h , t hough d i f f e r e n t i n many r e s p e c t s , s h a r e t h e common f e a t u r e o f b e i n g b a s e d on a s o l u t i o n o f t h e b o u n d a r y l a y e r e q u a t i o n s . T h e s e m o d e l s a r e a n advancement o f p r e v i o u s m o d e l s a s f o r t h e f i r s t t i m e known, a l l f r i c t i o n a l e f f e c t s and w i r e d e f l e c t i o n we re i n c l u d e d i n t h e m o d e l s . D e s p i t e t h e s e i m p r o v e m e n t s , t h e p r e s e n t m o d e l s f a i l e d t o p r o d u c e good r e s u l t s . I t i s c o n c l u d e d t h a t t h e f a i l u r e o f t h e mode l s was due t o n u m e r i c a l p r o b l e m s a s s o c i a t e d w i t h o b t a i n i n g a s o l u t i o n t o t h e m o d e l s , and n o t due t o s h o r t c o m i n g s i n t h e m o d e l s . 3. While crude, the experiments of the present i n v e s t i g a t i o n were able to produce some r e s u l t s which agreed w e l l with the scant published r e s u l t s a v a i l a b l e of other experimental i n v e s t i g a t i o n s . 5.1 Recommendations for Further Work The following recommendations f o r further work are made: 1. Of the s o l u t i o n procedures developed, the momentum i n t e g r a l model warrants the most att e n t i o n f o r fur t h e r work as i t i s a much simpler modelling approach than the two-dimensional model but i t i s an advancement of previous modelling attempts, u n l i k e the conservation of mass model. Though the shapes and magnitudes of the c a l c u l a t e d pressure d i s t r i b u t i o n s do not agree w e l l with experimental r e s u l t s , very good c o r r e l a t i o n to the observed trends were obtained. Possible areas of improvement to the momentum i n t e g r a l model include reduction of the s e n s i t i v i t y of the model to the i n i t i a l f o i l - w i r e gap height. Results obtained exploring t h i s problem i n d i c a t e suction magnitudes consistent with experimental r e s u l t s could be obtained i f t h i s problem were overcome. A possible approach may be the development of a coordinate transformation that would s t r e t c h a very small gap i n t o a l a r g e r gap i n s o l u t i o n coordinates. Study i s also warranted on the e f f e c t s of the i n i t i a l contact of the wire with the f o i l as well as drainage due to g r a v i t y . 2. As an approach to improving the numerical models, i t i s recommended that a focus be placed on beginning the c a l c u l a t i o n s at the t r a i l i n g edge of the f o i l u s i n g i n i t i a l c o n d i t i o n s o b t a i n e d from e x p e r i m e n t a l measurements. This approach would avoid the problems associated with the small gap heights encountered at the leading edge of the f o i l , as the s o l u t i o n could be stopped when the gap height became excessively small. 72. 3. It i s the opinion of the author that the two-dimensional model probably has the p o t e n t i a l of being the most accurate model. Because no approximations are made with regards to the form of the v e l o c i t y p r o f i l e , the two-dimensional model has none of the shortcomings inherent i n the one-dimensional models. A great deal of development i s required, however, before the two-dimensional model can be considered a v i a b l e s o l u t i o n tech-nique. The most s i g n i f i c a n t shortcoming of the method i s the f a i l u r e of the s o l u t i o n to s a t i s f y the t r a i l i n g edge condition on the pressure. Finding a s o l u t i o n to t h i s problem w i l l not be t r i v i a l due to the para-b o l i c nature of the boundary layer equations. The t y p i c a l Reynolds numbers encountered i n paper machines are also at the upper l i m i t s of the c a p a b i l i t i e s of the d i f f e r e n c i n g scheme selected. Thus, possibly very d i f f i c u l t numerical problems may have to be overcome before an accurate s o l u t i o n can be obtained using the two-dimensional approach. As a design t o o l , the two-dimensional model may never serve a u s e f u l function due to i t s complexity and the large amounts of computer time required to obtain a so l u t i o n . 4 . Further development of the conservation of mass model i s not recommended as i t was e s s e n t i a l l y superceded by the momentum i n t e g r a l model. F a i l u r e of the conservation of mass model to conserve momentum l i m i t s the usefulness of the model and was developed mainly to gain some i n s i g h t i n t o the behaviour of the flow. The conservation of mass model i s als o i d e n t i c a l f o r a l l p r a c t i c a l purposes to the models of Meyer and Ernst and does l i t t l e to improve t h e i r models. For t h i s purpose the conserva-t i o n of mass model served a u s e f u l function but i t s shortcomings make fu r t h e r pursuit unworthwhile. 5. A great deal of further experimental work is also necessary to establish a complete set of data for comparison with the results of further numerical modelling. An important part of any further experi-mental work would be to establish the extent of possible three dimension-ality of the flow by measuring the spanwise pressure distribution. A much more appropriate method for measuring the drainage rate is also necessary. Efforts should also be made to obtain results for the effect of the wire tension to determine the significance of its effect on the pressure distribution. Finally, i t would be useful to take shear stress measure-ments along the f o i l for the purpose of determining the extent of flow separation over the f o i l i f i t does in fact exist. Shear stress measure-ments could also be useful for establishing boundary conditions for assumed velocity profiles in further numerical modelling attempts, and to establish whether or not the flow is turbulent. 6. Flow visualization experiments would be very useful. In addition to indicating whether or not flow separation occurs, such an experiment could also show the extent of three dimensional effects on the f o i l . Flow visualization would also help determine whether or not the flow is turbulent. A possible approach to this type of experiment could be the use of a clear f o i l through which photographs could be taken. Small amounts of dye could be added to the flow and photographs could be taken to trace the dye flow. Pressure Head Box Paper Slurry In Forming Board Foils Table Rolls Suction Boxes Brest Roll Forming Wire Tensioner and Guide Rollers Suction Couch Roll Paper Sheet to Presses and Dryers F i g u r e 1. T y p i c a l components and l a y o u t o f a F o u r d r i n i e r P a p e r M a c h i n e . 0.1 n i i i 1 1 1 1 1 1 f 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L F i g u r e 2 . Compar i son o f t h e t h e o r e t i c a l r e s u l t s o f T a y l o r (3) t o t h e e x p e r i -m e n t a l r e s u l t s o f F l e i s c h e r ( 9 ) . R e s u l t s a r e f o r 3 d e g r e e t a p e r e d f o i l , U = 2000 f t / m i n . 7,6. Aluminum Electric Drive Drive Roller Tensioning Roller Frame Motor and Support Bolt Figure 3 . Experimental apparatus. 77. Figure 4 . Photograph of the experimental apparatus. 7 8 . Figure 5. Photograph of a tensioner r o l l e r support b o l t showing the s t r a i n gauge l o c a t i o n . 79 . Figure 6. Photograph of the headbox. F i g u r e 7. P h o t o g r a p h o f t h e t e s t s e c t i o n s h o w i n g t h e h e a d -box s l u i c e , t h e f o i l and t h e f o i l d r a i n a g e t r o u g h . 8 1 . X L T a p e r e d Foil r ^///// YW////////////A 1 S t e p p e d Foil Foil x L X L E & (inches) (Inches) (degrees) (Inches) 1 6.5 1 1 -2 6.5 1 2 -3 6.5 1 4 -4 6.5 2.5 - 1/4 5 6.5 1 - 1/16 F i g u r e 9 . P r o f i l e s and d i m e n s i o n s o f t h e f o i l s t e s t e d . 83. To Tension Bolt Mount Gauge 1 _ (Gauge 3 opposite) Gauge 2 (Gauge 4 opposite) To Tension Roller F i g u r e 10. S t r a i n gauge p l a c e m e n t and c o n n e c t i o n on t h e t e n s i o n r o l l e r s u p p o r t b o l t s . 0 .025 0 . 0 0 0 -0.025H -0 .075 H Wire Velocity A 500 ft/min X 750 ft/min • 1000 ft/min 1260 ft/min a 1500 ft/min x- 2000 ft/min -0 .100 H 1 1 1 1 1 1 1 1 1 J 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L F i g u r e 11'. Measured n o n - d i m e n s i o n a l p r e s s u r e d i s t r i b u t i o n on f o i l 1 s h o w i n g t h e v a r i a t i o n w i t h t he w i r e v e l o c i t y . 0.05 0.00 » CL i CL -0.05-•0.10-7^ A"/ 'A' A T A ^ X A A A \ Wire Velocity A 500 ft/min X 750 ft/min • 1000 ft/min M 1260 ft/min XX !600_ft/min_ X 2000 ft/min A •0.15 H 1 -0.0 0.1 0.2 0.3 0.4 0.5 X / L 0.6 0.7 0.8 0.9 1.0 F i g u r e 12. Measured n o n - d i m e n s i o n a l p r e s s u r e d i s t r i b u t i o n on f o i l 2 show ing t h e v a r i a t i o n w i t h t h e w i r e v e l o c i t y . oa 0.05-1 0 . 0 0 Q_ i Q_ -0 .05 •0.10 - 0 . 15 -- 0 . 2 0 -^ H - i a - g r i g o-o-o-o'g Wire Velocity • 500 ft/min A 750 ft/min <0> 1000 ft/min 03 1250 ft/min V 1600 ft/min O 2000 ft/min 1 1 I 1 1 1 1 I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L F i g u r e 1 3 . Measured n o n - d i m e n s i o n a l p r e s s u r e d i s t r i b u t i o n on f o i l 3 s h o w i n g t h e v a r i a t i o n w i t h t h e w i r e v e l o c i t y . oo. 0.03-1 0.02-0.01-! 0.00 -0.01 CD GL Q_ -0.02H A - A - A - A - A - A - A - A - A - A - A - A - A - A - A - A --0.03 •0.04 -0.05-T 1 ! 1 1 1 1 1 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 A X • Wire Velocity 600 ft /min 760 ft/min 1000 ft/min 1250 ft /min X 160_0__f_t/min_ 2000 ft /min 0.8 0.9 1.0 X / L Figure 14. Measured non-dimensional pressure d i s t r i b u t i o n on f o i l 4 showing the va r i a t i o n with the wire v e l o c i t y . 00 0.050-1 0.025 i Q_ i Q_ 0.000 •0.025 ^9 tr XT A ' i — i i — i A Wire Velocity A 500 ft/min X 750 ft/min • 1000 ft/min 1260 ft/min s 1600 ft/min X 2000 ft/min •0.050-T p 0.0 0.1 — i 1 1 r— 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L F i g u r e 1 5 . Measured n o n - d i m e n s i o n a l p r e s s u r e d i s t r i b u t i o n on f o i l 5 show ing t h e v a r i a t i o n w i t h t h e w i r e v e l o c i t y . oo oo 0.05-1 0.00 * 3 Q. Q_ -0.05--0.10 -0.15 -0.20 TO? A A . \ • \ • Foil Number A Foil 1  X Foil 2  • Foil 3  Kl Foil 4 Foil 5 \ / • 1 1 1 1 1 1 1 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L Figure 16. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 500 ft/min. 00 'to: 0.05-1 0.00 CL i GL -0.05--0.10 V X / * x - x • / Foil Number A Foil 1 X Foil 2 • Foil 3 ® Foil 4 Foil 5 -0.15 -f 1 I 1 1 1 1 I 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L Figure 17. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 750 ft/min. o 0 . 0 5 - 1 » Q. « QL QL - 0 . 0 5 - 0 . 1 0 \ X - X 7 W 'X- " X" \ ' / A X Foil Number A Foil 1 X Foil 2 • Foil 3 Foil 4 Foil 5 - 0 . 1 5 H 1 1 1 1 1 1 1 1 I 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 X / L F i g u r e 18. Compar i son o f measured p r e s s u r e d i s t r i b u t i o n s on f o i l s 1 , 2 , 3 , 4 and 5 a t a w i r e speed o f 1000 f t / m i n . 1 . 0 0.05 0.00 » 3 Q. -0.05-to QL I -0.10-Foil Number A Foil 1 X Foil 2 • Foil 3 Foil 4 s Foil 5 -0.15 0.0 0.1 0.2 0.3 0.4 0.5 X / L 0.6 0.7 0.8 0.9 1.0 F i g u r e 1 9 . Compar i son o f measured p r e s s u r e d i s t r i b u t i o n s on f o i l s 1 , 2 , 3 , 4 and 5 a t a w i r e speed o f 1250 f t / m i n . to ho 0.05-1 0.00 ZD Q. Q_ i Q_ • - D -0.05 •0.10 Foil Number A Foi l 1  X Foi l 2  • Foil 3  M Foi l 4 Foi l 5 -0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 1.0 Figure 20. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 1500 ft/min. to 0.05-i 0.00 » Q. Q_ i Q_ -0.05-•0.10-Foil Number A Foil 1  X Foil 2  • Foil 3  M Foil 4 Foil 5 -0.15-r 0.0 —I 1 1 1 1 1 1 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L Figure 21. Comparison of measured pressure d i s t r i b u t i o n s on f o i l s 1,2,3,4 and 5 at a wire speed of 2000 ft/min. to Foil Number A Foil 1  X 500 750 1000 1250 1500 U w i r e ( ft/min ) 1750 2000 Figure 22. Measured non-dimensional f o i l drainage rates as a function of wire speed f or f o i l s 1,2,3,4 and 5. 0.025-1 0.000 -0.025 -0.050 Wire Velocity A 750 ft /min. Run 1 X 750 ft /min. Run 2 • 1000 f t /min. Run 1 1000 f t /min. Run 2 S 1600 ft /mi r>: Run _1_ X 1JL0J^f!/mjn^Rjjn_2 'A-•0.075-r 0.0 0.1 0.2 0.3 0.4 0.5 X / L 0.6 0.7 0.8 0.9 1.0 F i g u r e 2 3 . T y p i c a l r e p e a t a b i l i t y o f p r e s s u r e d i s t r i b u t i o n on f o i l 1 f o r w i r e v e l o c i t i e s o f 750 f t / m i n , 1000 f t / m i n and 1500 f t / m i n . 0.05-1 0.00 * -0.05 3 a CL i CL -o.ioH -0.15 Wire Velocity A 500 ft /min. Run 1 X 500 ft /min. Run 2 • 750 ft /min. Run 1 750 ft /min. Run 2 X 1000 ft /min. Run 1 1000 ft /min. Run 2 A - A ^ X -0.20-f r™ 0.0 0.1 T T 0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 F i g u r e 2k. T y p i c a l r e p e a t a b i l i t y o f p r e s s u r e d i s t r i b u t i o n on f o i l 3 f o r w i r e v e l o c i t i e s o f 500 f t / m i n , 750 f t / m i n and 1000 f t / m i n . 1.0 F i g u r e 2 5 . D iag ram o f a f o i l showing t h e g e o m e t r i c n o m e n c l a t u r e u s e d i n t h e deve lopmen t o f m a t h e m a t i c a l m o d e l s . L e n g t h s i n t h e y - d i r e c t i o n a r e g r e a t l y e x a g g e r a t e d . t h e F i g u r e 2 6 . D iag ram o f a f o i l showing t h e dynamic q u a n t i t i e s u s e d i n t h e deve lopmen t o f t h e m a t h e m a t i c a l m o d e l s . to ivO 100. F i g u r e 2 7 . D i a g r a m s h o w i n g t h e f o r c e s on a s m a l l s e c t i o n o f t h e f o r m i n g w i r e . 101. ( " s t a r t ^) Read Data ~ T ~ Initialize I = 0 Evaluate Pressure no Evaluate Wire Deflection yes 1 = 1 + 1 Output Results ( Stop ) no e s Convergence Failure I ( j r t o p j ) Figure 28. Flowchart of the so l u t i o n procedure f o r the conservation of mass model. 8.0-1 6.0-4.0 2.0 0.0 0.0 Legend Calculated Blasius Profile 0.4 0.5 0.6 U / Uo 0.7 0.8 0.9 1.0 1.1 o, N3: F i g u r e 29 . Compar i son o f t h e c a l c u l a t e d v e l o c i t y p r o f i l e f o r t h e f l o w o v e r a f l a t p l a t e t o t h e B l a s i u s p r o f i l e . R e s u l t s o b t a i n e d u s i n g t h e c o n s e r v a t i o n o f mass m o d e l . Legend 1.0 -\ Calculated • Theory >• 0.8 0.6-0.4 0.2-0.0-0.0 F i g u r e 30, 0.1 0.2 0.3 0.4 0.5 0.6 U / Uo 0.7 0.8 0.9 1.0 Compar i son o f t h e c a l c u l a t e d v e l o c i t y p r o f i l e f o r t h e f l o w be tween mov ing p a r a l l e l f l a t p l a t e s t o t h e e x a c t s o l u t i o n . C a l c u l a t e d v e l o c i t y p r o f i l e o b t a i n e d u s i n g t he c o n s e r v a t i o n o f mass m o d e l . o 0.005 0.000 -0.005 -0.010--0.015--0.020--0.025--0.030 0.1 Wire Velocity Uw f • 600 ft /min U. r« 750 ft/min u. rt 1000 ft/min u. r« 1250 ft/min Uw 1600 ft/min Uw 2000 ft /min 0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 1.0 1.1 Figure 31'. Calculated pressure d i s t r i b u t i o n s for f o i l 1 using the conservation of mass model showing the v a r i a t i o n with the wire v e l o c i t y . Wire d e f l e c -t i o n effects were not included i n the sol u t i o n . o 0.0180-1 0.0175 0.0170 0.0165-I 0.0160-o 0.0155-0.0150-0.0145 0.0140 No Wire Deflection 500 750 1000 1250 1500 U w i r e (ft/min) 1750 — i — 2000 F i g u r e 32 . Compar i son o f c a l c u l a t e d d r a i n a g e r a t e s f o r f o i l 1 w i t h and w i t h o u t w i r e d e f l e c t i o n i n c l u d e d . R e s u l t s o b t a i n e d u s i n g t h e c o n s e r v a t i o n o f mass mode. o 'Ln 0.005 3 CO Q_ I Q_ 0.000 -0.005 -0.010 -0.015--0.020--0.025 -0.030 0.1 Wire Velocity Uwlr. = 500 ft/min Uww. 750 ft/min Uwlr. 1000 ft/min Uw„. 1260 ft /min U. I , . 1500 ft/min Uwlr. 2000 ft /min 0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 1.0 Figure 33. Calculated pressure d i s t r i b u t i o n s for f o i l 1 using the conservation of mass model showing the v a r i a t i o n with the wire v e l o c i t y . Wire d e f l e c t i o n effects were included i n the solu t i o n . 1.1 o sz 0.2 0.0-•0.2--0.4--0.6 -0.8-•1.0-•1.2 •0.6 -0.4 -0.2 0.0 0.2 0.4 U / Uwire 0.6 U w lr« = 600 ft /min U wlr* 750 ft/min U wlr« 1000 ft/min U Wt f • 1250 ft /min U wl rt = 1500 ft/min Uw!,. 2000 ft /min 0.8 1.0 1.2 Figure 34. Exit v e l o c i t y p r o f i l e s for f o i l 1 as a function of wire v e l o c i t y . Results obtained using the conservation of mass model. o 0.2-1 0.0--0.2-•0.4 -0.6 •0.8 -1.0 -1.2 -0.4 -0.2 0.0 T 0.2 0.4 0.6 U / Uwire 0.8 Legend X / L - 0.164 X / L - 0.366 X / L - 0.677 X / L = 0.789 X / L = 1.000 1.0 1.2 F i g u r e 35. C a l c u l a t e d v e l o c i t y p r o f i l e s a l o n g f o i l 1 f o r U w = 500 f t / m i n . R e s u l t s o b t a i n e d u s i n g t h e c o n s e r v a t i o n o f mass m o d e l w i t h o u t w i r e d e f l e c t i o n e f f e c t s . o 03 0.2-1 0.0--0.2--0.4 -0.6 -0.8 -1.0--1.2 -0.4 -0.2 0.0 0.2 0.4 U / Uv 0.6 0.8 Legend X / L = 0.164 X / L - 0.366 X / L - 0.577 X / L = 0.789 X / L - 1.000 1.0 1.2 F i g u r e 36 . C a l c u l a t e d v e l o c i t y p r o f i l e s a l o n g f o i l 1 f o r U = 500 f t / m i n . R e s u l t s o b t a i n e d u s i n g t h e c o n s e r v a t i o n o f mass m o d e l w i t h w i r e d e f l e c t i o n e f f e c t s i n c l u d e d . o 0.005-1 X / L F i g u r e 37 . Compar i son o f c a l c u l a t e d p r e s s u r e d i s t r i b u t i o n s f o r f o i l s 1 , 2 . a n d 3 f o r U = 500 f t / m i n . R e s u l t s o b t a i n e d u s i n g t h e c o n s e r v a t i o n o f mass mode l w i t h o u t w i r e d e f l e c t i o n e f f e c t s . 0.005-1 X / L F i g u r e 38 . T y p i c a l s e n s i t i v i t y o f t h e mass c o n s e r v a t i o n mode l t o t h e i n i t i a l f o i l - w i r e gap . R e s u l t s a r e f o r f o i l 1 a t a w i r e v e l o c i t y o f 500 f t / m i n . Set up Parameters fo r DEBDF Call DEBDF (Re tu rn ) F i g u r e ' 3 9 ' . F l o w c h a r t o f t h e p r e s s u r e e v a l u a t i o n s u b r o u t i n e f t h e momentum i n t e g r a l mode l s o l u t i o n p r o c e d u r e . > 0.0 -0.2--0.4--0.6 •0.8-•1.0-Legend Calculated • Theory 0.0 F i g u r e 40. 0.1 0.2 0.3 0.4 0.5 U / Uo 0.6 0.7 0.8 0.9 1.0 C o m p a r i s o n of t he c a l c u l a t e d v e l o c i t y p r o f i l e f o r t h e f l o w be tween mov ing p a r a l l e l f l a t p l a t e s t o t h e e x a c t s o l u t i o n . C a l c u l a t e d v e l o c i t y p r o f i l e o b t a i n e d u s i n g t h e momentum i n t e g r a l m o d e l . 8.0 6.0 x S, -5 4.0 > >-2.0 0.0 Legend Calculated Q Cubic approximation Blasius Profile 0.4 0.5 0.6 U / Uo Figure 41. Comparison of the calculated v e l o c i t y p r o f i l e for the flow over a f l a t plate to the Blasius p r o f i l e . Results obtained using the momen-tum i n t e g r a l model with the boundary conditions X=0, p=p , dp/dx=0. 3. 8.0-r 6.0-x ° 4.0 2.0-0.0 Legend Calculated Cubic approximation Blasius Profile 0.0 Figure 42. Comparison of the calculated v e l o c i t y p r o f i l e f o r the flow over a f l a t plate to the Blasius p r o f i l e . Results obtained using the momen-tum in t e g r a l model with the boundary conditions X=0, p=p ; x=L, p=p , 0.005 0.000 -0.005 -0.010 •0.015 Wire Velocity u . rt 500 ft/min Uw r« = 750 ft/min Uw r« ts 1000 ft/min Uw r« = 1250 ft/min Uw tt = 1500 ft/min Uw r« - 2000 ft/min -0.020 0.1 0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 1.0 1.1 F i g u r e 43. C a l c u l a t e d p r e s s u r e d i s t r i b u t i o n f o r f o i l 1 u s i n g t h e momentum i n t e g r a l mode l show ing t h e v a r i a t i o n w i t h t h e w i r e v e l o c i t y . W i r e d e f l e c t i o n e f f e c t s were no t i n c l u d e d i n t h e s o l u t i o n . 0.005-1 X / L h-1 F i g u r e 44,. Compar i son o f c a l c u l a t e d p r e s s u r e d i s t r i b u t i o n s f o r f o i l s 1 , 2 , 3 , 4 and 5 f o r U w = 500 f t / m i n . R e s u l t s o b t a i n e d u s i n g t h e momentum i n t e g r a l mode l w i t h o u t w i r e d e f l e c t i o n e f f e c t s . The p r e s s u r e d i s t r i b u t i o n a l o n g f o i l s 4 and 5 i s z e r o . 0.005 0.000 * -0.005 Q_ i 0_ -0.010 -0.015--0.020-0.1 Wire Velocity U wlf • = 600 ft /min U wire 760 ft /min Uw„. = 1000 ft /min U w lr« = 1260 ft /min Uwlr* = 1500 ft /min U w lr« - 2000 ft /min 0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 1.0 F i g u r e 45. C a l c u l a t e d p r e s s u r e d i s t r i b u t i o n s f o r f o i l 1 u s i n g t h e momentum i n t e g r a l mode l showing t h e v a r i a t i o n w i t h t h e w i r e v e l o c i t y . W i r e d e f l e c t i o n e f f e c t s were i n c l u d e d i n t h e s o l u t i o n . 1.1 0.005-1 X / L F i g u r e 46 . Compar i son o f t h e p r e s s u r e d i s t r i b u t i o n on. f o i l 1 f o r U :-.= 500. f t / m i n w i t h and w i t h o u t w i r e d e f l e c t i o n e f f e c t s i n c l u d e d i n t h e s o l u t i o n . R e s u l t s o b t a i n e d u s i n g t h e momentum i n t e g r a l m o d e l . 0.2-i 0.0 -0.2-•0.4 -0.6--0.8--1.0 ...t X v -1.2 Legend U w t r « = 500 ft /min Uw.,. 760 ft/min U wire = 1000 ft/min U w i r « 1250 ft/min U wlr * 1600 ft/min U w l r « 83 _200JD^t/mm 1 — -15.0 -10.0 F i g u r e 47 -5.0 0.0 U / Uv 5.0 10.0 15.0 C a l c u l a t e d e x i t v e l o c i t y p r o f i l e s f o r f o i l 1 as a f u n c t i o n o f w i r e v e l o c i t y u s i n g t h e momentum i n t e g r a l mode l w i t h o u t w i r e d e f l e c t i o n e f f e c t s . M K3 O 0.2-1 -C >• 0.0--0.2 -0.4--0.6--0.8-•1.0 -1.2 -8.0 Uw.,. S 3 600 ft /min U wlr« 760 ft/min Uwlr* = 1000 ft/min U w lr« «= 1260 ft /min U w 1 r« ea 1600 ft/min Uwlr* <n 2000 ft/min -6.0 -4.0 -2.0 U / Uv, 0.0 2.0 4.0 F i g u r e 48 . C a l c u l a t e d e x i t v e l o c i t y p r o f i l e s f o r f o i l 1 as a f u n c t i o n o f w i r e v e l o c i t y u s i n g t h e momentum i n t e g r a l mode l w i t h w i r e d e f l e c t i o n e f f e c t s i n c l u d e d i n t h e s o l u t i o n . 0.005-r 0.000 -0.005 -0.010 •0.015 -0.020- Legend ho = 1/32 inch ho = 1/16 inch -0.025-0.1 0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 1.0 1.1 F i g u r e 4 9 . T y p i c a l s e n s i t i v i t y o f t h e momentum i n t e g r a l mode l t o t h e i n i t i a l f o i l -w i r e gap h e i g h t . R e s u l t s a r e f o r f o i l 1 and a w i r e v e l o c i t y o f 500 f t / m i n . 0.005 i 1 1 1 1 1 1 1 \ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 X / L ho u> V a r i a t i o n o f t h e p r e s s u r e d i s t r i b u t i o n s on f o i l 1 as a f u n c t i o n o f t h e d r a i n a g e r e s i s t a n c e . R e s u l t s c a l c u l a t e d u s i n g t h e momentum i n t e g r a l mode l w i t h no w i r e d e f l e c t i o n e f f e c t s and a w i r e v e l o c i t y o f 500 f t / m i n . 0.1 0.2 F i g u r e 50. 0.014-1 Drainage Resistance, Kdr (ft) *10"7 F i g u r e 5 1 . V a r i a t i o n i n the p r e d i c t e d d r a i n a g e r a t e as a f u n c t i o n o f t h e d r a i n a g e r e s i s t a n c e . R e s u l t s o b t a i n e d u s i n g t h e momentum i n t e g r a l mode l f o r f o i l 1 and a w i r e v e l o c i t y o f 500 f t / m i n . W i r e d e f l e c t i o n e f f e c t s were no t i n c l u d e d . 0.005-1 X / L Figure 52. Effect of the wire tension on the pressure d i s t r i b u t i o n along f o i l 1. Momentum in t e g r a l model, U = 500 ft/min. o 0.010400-1 0.010390 0.010390-0.010380 i ^ 0.010380-1 0.010370 0.010370-0.010360 400 500 600 700 800 900 Wire Tension, T (Ibs/lin.ft.) 1000 1100 Figure 53. Effect of wire tension on the calculated drainage rate f o r f o i l 1. Momentum in t e g r a l model, U = 500 ft/min. ° w O N 0.02-1 * 3 Q. E CO Q_ ol 0.00 -0.02 -0.04 -0.06H -0.08-1 -0.10 r 0.1 Viscosity • 1.088 « 10"' f t ' / i v_ m 2.000 X i o - ftV. V m 3.000 X IO"' «tV. V m 4.000 X i o 8 f t V s V m 5.000 X i o - ft*/. V - 6.000 X i o -0.2 0.3 0.4 0.5 0.6 X / L 0.7 0.8 0.9 1.0 1.1 Figure 54. Effect of f l u i d kinematic v i s c o s i t y on the pressure d i s t r i b u t i o n along f o i l 1. Momentum i n t e g r a l model, U = 500 ft/min, no wire d e f l e c t i o n . w -0.015-F i g u r e 5 5 . E f f e c t o f s p e c i f y i n g an i n i t i a l d r a i n a g e v e l o c i t y on t h e p r e s s u r e d i s t r i b u t i o n a l o n g f o i l 1. Momentum i n t e g r a l m o d e l , = 500 f t / m i n , no w i r e d e f l e c t i o n . w 129'. F i g u r e 5 6 . T y p i c a l f i n i t e d i f f e r e n c e mesh u s e d f o r t h e s o l u t i o n o f t h e t w o - d i m e n s i o n a l m o d e l . 130. S ta r t ^ Read Data I Set up Initial Conditions I Initialize u,v,f,p vectors w Evaluate and Set up A,B matrices I Solve Evaluate Ax = B Update u,v,f,p vectors F i g u r e 57. F l o w c h a r t f o r t h e t w o - d i m e n s i o n a l mode l s o l u t i o n p r o c e d u r e . 8.0 Legend Calculated U / Uo l - J Figure 58. Comparison of the calculated v e l o c i t y p r o f i l e for flow over a f l a t plate to the Blasius p r o f i l e . Calculated r e s u l t s obtained using the two-dimensional model. U / Uo F i g u r e 59 . Compar i son o f t h e c a l c u l a t e d v e l o c i t y p r o f i l e s f o r f l o w i n a r e c t a n g u l a r . M d u c t t o t h e e x a c t s o l u t i o n show ing t h e deve lopmen t o f t h e c a l c u l a t e d v e l o c i t y p r o f i l e s . R e s u l t s o b t a i n e d u s i n g t he t w o - d i m e n s i o n a l m o d e l . U / Uo H LO F i g u r e 6.0. Compar i son o f t he c a l c u l a t e d v e l o c i t y p r o f i l e s f o r f l o w be tween . w moving p a r a l l e l f l a t p l a t e s t o t h e e x a c t s o l u t i o n . P r o f i l e s show t h e deve lopment o f t h e c a l c u l a t e d v e l o c i t i e s . R e s u l t s o b t a i n e d u s i n g t h e t w o - d i m e n s i o n a l m o d e l . " i T I — i i i i 1 1 1 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L F i g u r e 6 1 . T y p i c a l p r e s s u r e d i s t r i b u t i o n s a l o n g f o i l s 1 ,2 and 5 c a l c u l a t e d u s i n g t h e t w o - d i m e n s i o n a l m o d e l . U = 500 f t / m i n , V . = 0 . 0 5 f t / s e c , k , = 2 . 5 x 10-8 f t . d r w wi Legend U / U F i g u r e 6 2 . V e l o c i t y p r o f i l e s f o r f o i l 2 c a l c u l a t e d u s i n g t h e t w o - d i m e n s i o n a l mode l f up t o t h e p o i n t o f s o l u t i o n f a i l u r e . U = 500 f t / m i n ; V . = 0 . 0 5 f t / s e c , k d r = 2 . 5 x 10-8 f t . w w l X / L F i g u r e 63 . T y p i c a l p r e s s u r e d i s t r i b u t i o n s a l o n g f o i l 1 s h o w i n g t h e v a r i a t i o n i n t h e d i s t r i b u t i o n s as a f u n c t i o n o f t h e w i r e v e l o c i t y . R e s u l t s o b t a i n e d u s i n g t h e t w o - d i m e n s i o n a l m o d e l . U = 500 f t / m i n ; V 0 .05 f t / s e c , k , = 2 . 5 x I O " 8 f t . W w i d r 0.2 0.0--0.2 -0.4 -0.6 -0.8 •1.0 -1.2--0.2 Legend Uwlrt = 600 ft /min U.i,. ° 760 ft /min Uwln = 1000 f t /min U.in = 1500 ft /min Figure 64 0.0 0.2 0.4 0.6 0.8 1.0 U / Uwire Typical exit v e l o c i t y p r o f i l e s for f o i l 1 showing the v a r i a t i o n i n the p r o f i l e s with the wire velo c i t y , " - J -------1.2 dimensional model. 2.5 x 10~ 8 f t , w Results obtained using the two-= 500 ft/min; V '. = 0.05 f t / s e c , = wi ' dr u> 0 5 0 0 0 5 -.10 .15 \ • \ \ \ • • \ / \ • Wire Velocity • 500 ft/mir> Fleischer y 2 0 -f 1 1 1 1 1 1 1 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X / L F i g u r e 6 5 . Compar i son o f t h e r e s u l t s o f F l e i s c h e r (9) t o r e s u l t s o f t h e p r e s e n t i n v e s t i g a t i o n f o r f o i l 3 a t a w i r e v e l o c i t y o f 500 f t / m i n . 139. Figure 66. Photograph of the flow over f o i l 2 at a wire v e l o c i t y of 1000 ft/min. Note the presence of a 'dry-out' l i n e i n d i c a t i n g a l l the f l u i d has been drawn through the forming wire. F i g u r e 6 7 . P h o t o g r a p h o f t h e f l o w o v e r f o i l 1 a t a w i r e v e l o c i t y o f 1000 f t / m i n . N o t e t h e c o n t i n u o u s f i l m o f w a t e r o v e r t h e f o i l . 0.005-1 X / L F i g u r e 68 . Compar i son of t h e . p r e s s u r e d i s t r i b u t i o n on f o i l 1 f o r U = 500 f t / m i n c a l c u l a t e d u s i n g t h e c o n s e r v a t i o n o f mass and t h e momentum i n t e g r a l m o d e l s . W i r e d e f l e c t i o n e f f e c t s were i n c l u d e d i n t h e s o l u t i o n s . 0.016 0 .015 -0 . 0 1 4 -Legend Conservation of Mass Model Momentum Integral Model ZD 0.013 o 0.012-0.011 0.010 ~ l 1 1 5 0 0 . 0 750.0 1000.0 1250.0 1500.0 Uwire (ft/min) 1750.0 2 0 0 0 . 0 F i g u r e 69 \ Compar i son o f t h e d r a i n a g e r a t e s on f o i l 1 p r e d i c t e d by t h e o n e -d i m e n s i o n a l m o d e l s . W i r e d e f l e c t i o n e f f e c t were i n c l u d e d i n t h e s o l u t i o n s . 0 . 2 - 1 o.o-• 0 . 2 -- 0 . 4 -- 0 . 6 - 0 . 8 -• 1 . 0 -- 1 . 2 Legend Momentum Integral Model Conservation of Mass Model • 1 2 . 0 - 1 0 . 0 - 8 . 0 • 6 . 0 - 4 . 0 U / Uv - 2 . 0 0 . 0 2 . 0 F i g u r e 70 . C o m p a r i s o n o f t h e e x i t v e l o c i t y p r o f i l e s on f o i l 1 p r e d i c t e d by t h e o n e - d i m e n s i o n a l m o d e l s . U = 500 f t / m i n ; W i r e d e f l e c t i o n e f f e c t s were i n c l u d e d . 0.2-1 0.0--0.4-x: -0.6 F i g u r e 71 . T y p i c a l v e l o c i t y p r o f i l e s a l o n g f o i l 1 when an i n i t i a l d r a i n a g e v e l o c i t y has been s p e c i f i e d . U = 500 f t / m i n ; V . = 0 .1 f t / s e c . Momentum i n t e g r a l m o d e l , w i r e d e f l e c t i o n e f f e c t s W x n c l u d e d . •EM 0.2-1 U / Uwire i—1 F i g u r e 72 . T y p i c a l v e l o c i t y p r o f i l e s a l o n g f o i l 1 when no - - i n i t i a l d r a i n a g e f v e l o c i t y has been s p e c i f i e d . U w = 500 f t / m i n ; Momentum i n t e g r a l m o d e l , w i r e d e f l e c t i o n e f f e c t s i n c l u d e d . 1 4 6 . REFERENCES 1. Wrist, P.E. "The Papermaking Process as a F i l t r a t i o n Problem." Pulp  Paper Mag. Can. V o l . 55, No. 6 (May, 1954), pg. 115. 2. Taylor, G.I. "Drainage at a Table R o l l . " Pulp Paper Mag. Can. Vol. 57, No. 3, (Conv. 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Vol. 35, B e r l i n : Springer-Verlag, 1975, pg. 445-451. 35. Cebeci, T. and Chang, K.C. "A General Method f o r C a l c u l a t i n g Momentum and Heat Transfer i n Laminar and Turbulent Duct Flows." Numerical Heat Transfer. V o l . 1, (1978), pg. 39. 36. Cebeci, T., Berkant, N. and S i l i v r i , I. "Turbulent Boundary Layers with Assigned Wall Shear." Computers F l u i d s . V o l . 3, No. 1, (March, 1975), pg. 37. 37. T e r r i l l , R.M. "Laminar Flow i n a Uniformly Porous Channel." The  Aeronautical Quarterly. V o l . 15, (August, 1964), pg. 299. 38. Wu, J.C. and Wahbah, M.M. "Numerical Solution of Viscous Flow Equtions Using I n t e g r a l Representations." Lecture Notes i n Phsyics. V o l . 59, B e r l i n : Springer-Verlag, 1976, pp. 448. 39. I s r a e l i , M. and L i n , A. "Numerical Solution and Boundary Conditions f o r Boundary Layer Like Flows." Lectures Notes i n Physics. V o l. 170, B e r l i n : Springer-Verlag, 1982, pp. 266. 40. Fletcher, C.A.J., and F l e e t , R.W. "A Dorodnitsyn F i n i t e Element Boundary Layer Formulation." Lecture Notes i n Physics. V o l. 170, B e r l i n : Springer-Verlag, 1982, pp. 189. 41. Carter, J.E., and Vatsa, V.N. "Analysis of Separated Boundary Layer F l o w s . " L e c t u r e N o t e s i n P h y s i c s . V o l . 170, B e r l i n : Springer-Verlag, 1982, pg. 167. 42. Pandolfi, M. and Zannetti, L. "Some Permeable Boundaries i n Multidimensional Unsteady Flows." Lecture Notes i n Physics. Vol. 90, B e r l i n : Springer-Verlag, 1979, pg. 439. 43. Reyhner, T.A. and Flugge-Lotz, I. "The Interaction of a Shock Wave With a Laminar Boundary Layer." Int. J . Non-Linear Mechanics. Vol. 3, pg. 173. 44. Moin, P., Mansour, N.N., Reynolds, W.C. and Ferziger, J.H. "Large Eddy Simulation of Turbulent Shear Flows." Lecture Notes i n Physics. V o l . 90, B e r l i n : Springer-Verlag, 197, pg. 400. 1 4 9 . 45. Rubin, S.G. "Predictor-Corrected Method f o r Three Coordinate Viscous Flows." Lecture Notes i n Phsyics. Vol. 18, B e r l i n : Springer-Verlag, 1973, pg. 146. 46. Wright, J.P. and Weidlinger, P. " P o s i t i v e Conservative Second and Higher Order Difference Schemes f o r the Equations of F l u i d Dynamics." Lecture Notes i n Physics. Vol. 18, B e r l i n : Springer-Verlag, 1973, pg. 169. 47. L i n , A. and Weinstein, H. "Numerical Analysis of Confined Turbulent Flow." Computers and F l u i d s . V o l . 10, No. 1, (1982), pg. 27. 48. Smith, F.T. and Merkin, J.H. "Triple-Deck Solutions f o r Subsonic Flow Past Humps, Steps, Concave or Convex Corners and Wedged T r a i l i n g Edges." Computers and F l u i d s . V ol. 10, No. 1, (1982), pg. 7. 49. Veldman, A.E.P. and D i j k s t r a , D. "A F a s t Method to Solve Incompressible Boundary-Layer Interaction Problems." Lecture Notes  i n Physics. V o l . 141, B e r l i n : Springer-Verlag, 1981, pg. 411. 50. Inoue, 0. "Separated Boundary Layer Flows With High Reynolds Numbers." Lecture Notes i n Physics. Vol. 141, B e r l i n : Springer-Verlag, 1981, pg. 224. 51. Le Feuvre, R.F. "The P r e d i c t i o n of Two-Dimensional R e c i r c u l a t i n g Flows Using a Simple F i n i t e - D i f f e r e n c e Grid f o r Non-Rectangular Flow F i e l d s . " Computers and F l u i d s . V o l . 6, No. 4, (1978), pg. 203. 52. Hirsh, R.S., Friedman, D.M. and Cebeci, T. "Solution of Turbulent Transport Equations By An Accurate Numerical Method." Lecture Notes  i n Physics. V o l . 90, B e r l i n : Springer-Verlag, 1979, pg. 282. 53. Blottner, F.G. "Non-uniform Grid Method f o r Turbulent Boundary Layers." Lecture Notes i n Physics. Vol. 35, B e r l i n : Springer-Verlag, 1975, pg. 91. 54. Rubin, S.G. and Khosla, P.K. "Turbulent Boundary Layers With and Without Mass In j e c t i o n . " Computers and F l u i d s . V ol. 5, No. 4, (1977), pg. 241. 55. Krishnaswamy, R. and Nath, G. "A Parametric D i f f e r e n t i a t i o n Version With F i n i t e - D i f f e r e n c e Scheme Applicable to a Class of Problems i n Boundary Layer Flow with Massive Blowing." Computers and F l u i d s . V o l . 10, No. 1, (1982), pg. 1. 56. Dorodnicyn, A.A. "Review of Methods f o r Solving the Navier-Stokes E q u a t i o n s . " L e c t u r e Notes i n P h y s i c s . V o l . 18, B e r l i n : Springer-Verlag, 1973, pg. 1. 57. Brandt, A. "Multi-Level Adaptive Technique f o r Fast Numerical Solution to Boundary Value Problems." Lecture Notes i n Physics. V o l . 18, B e r l i n : Springer-Verlag, 1973, pg. 82. 150. 58. Cebeci, T. and Smith, A.M.O. ANALYSIS OF TURBULENT BOUNDARY LAYERS. New York: Academic Press, Inc., 1974. 57. Cebeci, T. and Bradshaw, P. MOMENTUM TRANSFER IN BOUNDARY LAYERS. Washington: Hemisphere Publishing Corp., 1977. 58. Brady, J.F. and Acrivos, A. "Closed-Cavity Laminar Flows at Moderate Reynolds Numbers." J . F l u i d Mechanics. V o l . 115, pg. 427. 59. Brady, J.F. and Acrivos, A. "Steady Flow i n a Channel or Tube With an Accelerating Surface V e l o c i t y . An Exact Solution to the Navier-Stokes Equations with Reverse Flow." J . F l u i d Mechanics. Vol. 112, pg. 127. 60. Acrivos, A. and Schrader, M.L. "Steady Flow i n a Sudden Expansion at High Reynolds Numbers." Phys. F l u i d s . Vol. 25, No. 6, (June, 1982), pg. 923. 61. Kwon, O.K., Pletcher, R.J. and Lewis, J.P. " P r e d i c t i o n of Sudden Expansion Flows Using the Boundary-Layer Equations." J . F l u i d s  Engineering. V o l . 106, (September, 1984), pg. 285. 62. T e r r i l l , R.M. "Laminar Flow i n a Uniformly Porous Channel with Large I n j e c t i o n . " The Aeronautical Quarterly. Vol. 16, (November, 1965), pg. 323. 63. Klemp, J.B. and Acrivos, A. "A Moving-Wall Boundary Layer With Reverse Flow." J . F l u i d Mechanics. Vol. 76, part 2, (1976), pg. 363. 64. Shampine, L.F. and Gordon, M.K. COMPUTER SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS: THE INITIAL VALUE PROBLEM. Freeman, 1974. 65. Boyce, W.E. and DiPrima, R.C. ELEMENTARY DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS. 3rd ed. New York: John Wiley & Sons, Inc., 1977. 66. Housner, G.W. and Hudson, D.E. APPLIED MECHANICS - STATICS. Princeton, N.J.: D. Van Nostrand Company, Inc., 1961. 67. Goldstein, H. CLASSICAL MECHANICS. 6th ed. Reading, MA: Addison-Wesley Publishing Company, Inc., 1959. 68. Popov, E.P. INTRODUCTION TO MECHANICS OF SOLIDS. Englewood C l i f f s , N.J.: P r e n t i c e - H a l l , Inc., 1968. 69. Doebelin, E.O. MEASUREMENT SYSTEMS - APPLICATION AND DESIGN. Rev. New York: McGraw-Hill Book Company, 1975. 70. N i c o l , Tom, UBC MATRIX - A GUIDE TO SOLVING MATRIX PROBLEMS. UBC Computing Centre, Vancouver, 1982. APPENDIX A ORIFICE PLATE DETAILS 152. APPENDIX A ORIFICE PLATE DETAILS As part of the experimental procedure, i t was necessary to maintain the v e l o c i t y of the water j e t leaving the headbox s l u i c e at the same speed as the forming wire. The j e t v e l o c i t y was determined by measuring the flow rate using an o r i f i c e plate and then c a l c u l a t i n g the v e l o c i t y . The o r i f i c e p l ate was designed according to the American Society of Mechanical Engineers (ASME) recommended s p e c i f i c a t i o n s f o r o r i f i c e p l ate flow meters [13]. The o r i f i c e was sized to provide accurate flow measurements over the a n t i c i p a t e d range of flows. This corresponded to a flow range of 0.065 f t 3 / s at a wire v e l o c i t y of 500 ft/min. to 0.260 f t 3 / s at 2000 ft/min. The o r i f i c e s i z e was also selected so as to provide s u f f i c i e n t pressure changes over the range of flows to ensure accurate determination of the j e t v e l o c i t y . A f t e r a l l relevant f a c t o r s had been considered, a t h i n p late o r i f i c e having a diameter of 1.4975 inches was constructed. The o r i f i c e was i n s t a l l e d i n a s e c t i o n of pipe having an i n s i d e diameter of 4.026 inches. The pressure drop across the o r i f i c e was measured using a mercury manometer connected to D-1/2D pressure taps. The j e t v e l o c i t y was determined by measuring the flow r a t e and using the expression (A-l) to c a l c u l a t e the v e l o c i t y . In Equation A - l , A i s the area of the headbox s l u i c e ; A - 7.8 x 10~ 3 f t 2 153. (A-2) and Q i s the flow rate i n cubic feet per second. Q was determined using the expression given i n [13]; Q = a F ak/2g (-H 0 — ) (—) (A-3) where Q = flow rate ( f t 3 / s ) a = o r i f i c e area (= 1.22 x 10~ 2 f t 2 ) F a = o r i f i c e expansion c o e f f i c i e n t (= 1) k _ c c = o r i f i c e flow discharge c o e f f i c i e n t ^ l - g ^ 3 = diameter r a t i o = 3.7 x IO""1 g = 32.2 f t 2 / s p„ = 26.28 s l u g s / f t 3 Hg p R Q = 1.937 s l u g s / f t 3 h = manometer reading (inches Hg) The flow discharge c o e f f i c i e n t s f o r the o r i f i c e p l a t e were determined f o r each j e t v e l o c i t y using tables given i n reference 13. Equations A - l and A-3 were then used to c a l c u l a t e the manometer readings required to set each j e t v e l o c i t y . Figure 73 presents a c a l i b r a t i o n curve f o r the j e t v e l o c i t y versus the manometer reading. 2 5 0 0 O r 0 6 8 10 12 14 16 18 2 0 h (inches Hg) F i g u r e 73. C a l i b r a t e d headbox s l u i c e j e t v e l o c i t y as a f u n c t i o n o f ; t h e o r i f i c e p r e s s u r e d r o p . APPENDIX B CALIBRATION OF THE TENSION ROLLER BOLTS APPENDIX B 156. CALIBRATION OF THE TENSION ROLLER BOLTS C a l i b r a t i o n of the the tension r o l l e r support b o l t s was performd by loading each b o l t with a known weight and recording the s t r a i n i n d i c a t e d by the s t r a i n gauge bridge. The bolts were c a l i b r a t e d over a range of loads to ensure the l i n e a r i t y of the l o a d - s t r a i n r e l a t i o n s h i p . C a l i b r a -t i o n curves f o r each bolt are given i n Figures 74 and 7 5 . 3020 3000H A -g 2980 c c 2960 c 'CD CO 2940 2920 2900-1 A. A. 'A. A. 'A 10 20 30 40 50 60 70 Load ( pounds ) 80 90 100 F i g u r e Ik. S t r a i n v e r s u s l o a d c a l i b r a t i o n c u r v e f o r t e n s i o n r o l l e r s u p p o r t b o l t 1. 11020 11000-/-N 10980 JC u c \ 2 10960 _c u c '5. — 10940 c CD 0 0 10920-10900-10880 \ -10 20 30 40 50 60 Load ( pounds ) 70 80 90 100 F i g u r e 75 . S t r a i n v e r s u s l o a d c a l i b r a t i o n c u r v e f o r t e n s i o n r o l l e r s u p p o r t b o l t 2. Ln CO APPENDIX C DETERMINATION OF THE FORMING WIRE DRAINAGE RESISTANCE 160. APPENDIX C DETERMINATION OF THE FORMING WIRE DRAINAGE RESISTANCE In order to do a meaningful comparison of the calcuated and the experimental r e s u l t s , i t was necessary to determine the drainage resi s t a n c e of the forming wire used i n the experiments. The measured value of the drainage resistance, k^ r, was then used i n the c a l c u l a t i o n s . The drainage resistance was determined by gluing a sample of the forming wire over one end of a short length of pipe. The pipe was supported v e r t i c a l l y with the forming wire at the bottom end. Water was introduced at the top end of the pipe and the flow rate required to maintain a constant head was measured. Knowing the flow rate of the water, the head, and the cross s e c t i o n a l area of the pipe, the drainage resistance was calculated using the expressions Combining Equations C - l and C-2 and s u b s t i t u t i n g the pressure d i f f e r e n c e with Q = vA (C-l) and (P-Pa) (C-2) v = P-Pa = Pgh (C-3) gives the expression for the drainage resistance k (C-4) dr pghA 161. For the forming wire used i n the experiments, i t was found a flow r a t e of 1.2 gallons/minute was required to maintain a head of 28 inches i n a pipe having an i n s i d e diameter of 15/32 inches. Using Equation C-4, the drainage resistance of the forming wire was found to be k, - 4.0 x I O - 7 f t (C-5) dr v ' Several t r i a l s were done and the above value was found to be repeatable to within f i v e percent. 

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