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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 . S c , The U n i v e r s i t y o f 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  in THE FACULTY OF GRADUATE STUDIES Department o f Mechanical  We accept  this  t h e s i s as  to t h e r e q u i r e d  Engineering  conforming  standard  THE UNIVERSITY OF BRITISH COLUMBIA April,  1986  © GARY LEPP, 1986  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the  requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t  the L i b r a r y s h a l l make  it  and study.  f r e e l y a v a i l a b l e f o r reference  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . understood t h a t  copying or p u b l i c a t i o n of t h i s t h e s i s  f o r f i n a n c i a l gain  s h a l l n o t be allowed without my  permission.  Department  o f Mechanical  Engineering  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date  It i s  A p r i l 23, 1986  Columbia  written  A  A  mathematical  been developed machine  model  to predict  drainage  attempts  have  use  of  C  T  numerical  models  the boundary  The s o l u t i o n  categories;  t h e one-dimensional  procedures  solution  procedures  a r e based  equations.  layer used  on a  have paper  numerical  Previous modelling  equations were  i n various  grouped  models and the two-dimensional  models were based  layer  A  The p r e s e n t  forms.  boundary  R  and t h r e e  simplified  the  T  l a m i n a r boundary l a y e r e q u a t i o n s .  made  one-dimensional  S  the 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  foil.  s o l u t i o n o f the f u l l  B  into  two  model.  The  on an approximate i n t e g r a l s o l u t i o n o f  The two-dimensional  model was based  on a  s o l u t i o n o f t h e boundary l a y e r e q u a t i o n s u s i n g a f i n i t e d i f f e r e n c e method. Experimental drainage  rates  on t h r e e t a p e r e d  provide  data  results  obtained  well  with  results, tions  measurements  f o r comparison using  due  to  Consequently,  the p r e s s u r e  foils  with  results  d i d predict  variations  necessary  procedures  i n the w i r e  trends  foils  generally  Computed  d i d not agree  investigation.  i n the p r e s s u r e  velocity  and the  were made t o  o f the models.  o f the p r e s e n t  observed  and  foil  The  distribugeometries.  the models may be c o n s i d e r e d v i a b l e approaches t o p r e d i c t i n g  the 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 , is  distributions  and two stepped  the r e s u l t s  the s o l u t i o n  the e x p e r i m e n t a l however,  of  before  the s o l u t i o n  though a g r e a t d e a l o f f u r t h e r procedures  purposes.  - i i-  may  be  used  work  f o r design  TABLE  OF  CONTENTS  Page  ABSTRACT TABLE OF CONTENTS LIST OF FIGURES NOMENCLATURE ACKNOWLEDGEMENTS 1.  2.  INTRODUCTION  1  1.1 1.2 1.3  1 2 6  General Review o f L i t e r a t u r e Need f o r the P r e s e n t Work  EXPERIMENT 2.1 2.2 2.3  2.4  2.5 2.6 3.  i i i i i v xi xiv  7  General Scope o f t h e Experiment D e s c r i p t i o n of the Apparatus 2.3.1 O v e r a l l Apparatus 2.3.2 F o i l Tested Instrumentation 2.4.1 F o i l P r e s s u r e D i s t r i b u t i o n s 2.4.2 F o i l Drainage Rates 2.4.3 Supply Water Flow Rate 2.4.4 Forming Wire T e n s i o n 2.4.5 Forming Wire Speed 2.4.6 Forming Wire Drainage R e s i s t a n c e E x p e r i m e n t a l Procedure R e s u l t s and O b s e r v a t i o n s  7 7 7 8 9 10 10 11 11 11 12 12 13 14  THEORETICAL ANALYSIS  17  3.1 3.2  17 19 19 20 22 22 22 25 26 27 30 30 36 38 40  3.3  General The Governing E q u a t i o n s 3.2.1 The E q u a t i o n s o f Motion 3.2.2 Wire D e f l e c t i o n Model The One-Dimensional Model 3.3.1 The C o n s e r v a t i o n o f Mass Model 3.3.1.1 The Model Equations 3.3.1.2 Method o f S o l u t i o n 3.3.1.3 V e r i f i c a t i o n o f the Model 3.3.1.4 Results 3.3.2 The Momentum I n t e g r a l Model 3.3.2.1 The E q u a t i o n s o f Motion 3.3.2.2 Method of S o l u t i o n 3.3.2.3 V e r i f i c a t i o n o f t h e Model 3.3.2.4 R e s u l t s  - i i i-  Page 3.4  4.  5.  The Two-Dimensional Model 3.4.1 G e n e r a l 3.4.2 The F i n i t e D i f f e r e n c e Scheme 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 3.4.4 Method o f S o l u t i o n 3.4.5 V e r i f i c a t i o n o f t h e Model 3.4.6 R e s u l t s  44 44 45 46 56 58 60  DISCUSSION  62  4.1 4.2 4.3  62 62 65  General Experimental Results T h e o r e t i c a l Results  CONCLUSIONS  70  5.1  71  Recommendations f o r F u r t h e r Work  FIGURES  74  REFERENCES APPENDIX A APPENDIX B APPENDIX C  146 152 155 159  - iv -  FIGURES Page 1.  Typical  components and l a y o u t o f a F o u r d r i n i e r  Paper Machine  2.  Comparison of the t h e o r e t i c a l r e s u l t s of T a y l o r [3] t o 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 = 2000 f t / m i n  75  3.  E x p e r i m e n t a l apparatus  76  4.  Photograph of the e x p e r i m e n t a l apparatus  77  5.  Photograph o f a t e n s i o n e r r o l l e r  w  support b o l t  74  showing t h e  s t r a i n gauge l o c a t i o n  78  6.  Photograph of the headbox  79  7.  Photograph o f the t e s t s e c t i o n  showing the headbox s l u i c e ,  the f o i l and t h e f o i l d r a i n a g e trough  80  8.  Photograph of the f o i l s  tested  81  9.  P r o f i l e s and dimensions  o f the f o i l s  10.  S t r a i n gauge placement and c o n n e c t i o n on the t e n s i o n r o l l e r support b o l t s Measured non-dimensional 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 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  11.  12.  13.  14.  15.  16.  17.  18.  tested  82 83 84  Measured non-dimensional 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 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  85  Measured non-dimensional 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 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  86  Measured non-dimensional 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 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  87  4  Measured non-dimensional 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 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  88  Comparison o f measured p r e s s u r e d i s t r i b u t i o n s 1,2,3,4 and 5 a t a w i r e speed o f 500 f t / m i n  on f o i l s 89  Comparison of measured p r e s s u r e d i s t r i b u t i o n s 1,2,3,4 and 5 a t a w i r e speed o f 750 f t / m i n  on f o i l s  Comparison o f measured p r e s s u r e d i s t r i b u t i o n s 1,2,3,4 and 5 a t a w i r e speed o f 1000 f t / m i n  on f o i l s  - v -  90  91  Page 19.  20.  21.  22.  23.  24.  25.  26.  27.  28.  29.  30.  31.  32.  Comparison 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  92  Comparison 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 1500 f t / m i n  93  Comparison of 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 2000 f t / m i n  94  Measured non-dimensional f o i l drainage of w i r e speed f o r f o i l s 1,2,3,4 and 5  95  r a t e s as a f u n c t i o n  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 ...  96  T y p i c a l r e p e a t a b i l i t y of 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 of 500 f t / m i n , 750 f t / m i n and 1000 f t / m i n ....  97  Diagram o f a f o i l showing the geometric nomenclature used i n t h e development o f the mathematical models. Lengths i n the y - d i r e c t i o n a r e g r e a t l y exaggerated  98  Diagram o f a f o i l showing the dynamic q u a n t i t i e s used i n the development o f t h e mathematical models  99  Diagram showing the f o r c e s on a s m a l l s e c t i o n of the forming w i r e  100  Flowchart of the s o l u t i o n procedure f o r the c o n s e r v a t i o n of mass model  101  Comparison o f 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 f o r the f l o w over 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 . Results obtained u s i n g the c o n s e r v a t i o n of mass model  102  Comparison 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 f o r the f l o w between moving p a r a l l e l f l a t p l a t e s t o t h e exact 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 the c o n s e r v a t i o n of mass model  103  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 the c o n s e r v a t i o n o f mass model showing t h e v a r i a t i o n w i t h t h e 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 i n c l u d e d i n the s o l u t i o n  104  Comparison o f c a l c u l a t e d drainage 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 the c o n s e r v a t i o n of mass model  105  - vi -  Page 33.  34.  35.  36.  37.  C a l c u l a t e d pressure d i s t r i b u t i o n s f o r f o i l 1 using the c o n s e r v a t i o n o f mass model showing t h e v a r i a t i o n w i t h t h e 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 i n c l u d e d i n the s o l u t i o n  106  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 of w i r e velocity. 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 model  107  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 the c o n s e r v a t i o n of mass model 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  108  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 ft/min. R e s u l t s o b t a i n e d u s i n g the c o n s e r v a t i o n of mass model 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  109  w  w  Comparison of 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 and 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 model 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 ... 110 w  38.  39.  40.  41.  42.  T y p i c a l s e n s i t i v i t y of the mass c o n s e r v a t i o n model t o the 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  I l l  Flow c h a r t of the 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 o r t h e momentum i n t e g r a l model s o l u t i o n procedure  112  Comparison 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 f o r the f l o w between 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 . 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 the momentum i n t e g r a l model  113  Comparison of 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 the f l o w over 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 the momentum i n t e g r a l model w i t h the boundary c o n d i t i o n s x=0, p=p„, dp/dx=0  114  Comparison 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 f o r the f l o w over 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 the momentum i n t e g r a l model w i t h the boundary c o n d i t i o n s x=0, p=p ; x=L, p=p„  115  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 t h e v a r i a t i o n w i t h t h e 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 i n c l u d e d i n the s o l u t i o n  116  Q  43.  - vii-  Page 44.  Comparison 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 = 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 the momentum i n t e g r a l model without 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 zero  117  Calculated pressure d i s t r i b u t i o n s f o r f o i l 1 using the momentum i n t e g r a l model showing t h e v a r i a t i o n w i t h t h e 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 i n c l u d e d i n the s o l u t i o n  118  Comparison of the 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 without wire 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 the 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 model  119  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 of 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 model w i t h o u t wire d e f l e c t i o n e f f e c t s  120  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 of 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 model 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 the s o l u t i o n  121  T y p i c a l s e n s i t i v i t y of the momentum i n t e g r a l model to 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  122  V a r i a t i o n o f the 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 function of the drainage r e s i s t a n c e . Results calculated u s i n g the momentum i n t e g r a l model w i t h no wire 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  123  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 the d r a i n a g e r e s i s t a n c e . Results obtained using the momentum i n t e g r a l model f o r f o i l 1 and a wire v e l o c i t y o f 500 f t / m i n . Wire d e f l e c t i o n e f f e c t s were n o t i n c l u d e d  124  E f f e c t o f the wire t e n s i o n on 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 1. Momentum i n t e g r a l model, U = 500 f t / m i n  125  E f f e c t o f wire t e n s i o n on the c a l c u l a t e d f o r f o i l 1. Momentum i n t e g r a l model, U  126  w  45.  46.  47.  48.  49.  50.  51.  52.  w  53.  54.  w  drainage r a t e = 500 f t / m i n  E f f e c t o f f l u i d k i n e m a t i c v i s c o s i t y on 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 1. Momentum i n t e g r a l model, U = 500 f t / m i n , no w i r e d e f l e c t i o n  127  E f f e c t of 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 model, U = 500 f t / m i n , no wire d e f l e c t i o n  128  w  55.  w  - viii  -  Page 56.  T y p i c a l f i n i t e d i f f e r e n c e mesh used f o r the s o l u t i o n o f the two-dimensional model  129  57.  Flowchart f o r the two-dimensional model s o l u t i o n procedure  58.  Comparison o f 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 f o r f l o w over 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 . Calculated r e s u l t s o b t a i n e d u s i n g the two-dimensional model  131  Comparison 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 f o r f l o w i n a r e c t a n g u l a r duct t o t h e exact s o l u t i o n showing t h e development o f 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 . R e s u l t s o b t a i n e d u s i n g t h e two-dimensional model  132  Comparison 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 f o r f l o w between 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 the development of the 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 two-dimensional model  133  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 two-dimensional model. U = 5 0 0 f t / m i n , V - 0.05 f t / s e c , k - 2.5xl0~ f t  134  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 the twod i m e n s i o n a l model 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 = 0.05 f t / s e c , k = 2.5xl0~ f t  135  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 showing 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 velocity. R e s u l t s o b t a i n e d u s i n g the two-dimensional model. U = 500 f t / m i n ; V = 0.05 f t / s e c , k = 2.5xl0~ f t  136  59.  60.  61.  ...  8  w i  62.  d r  8  w  63.  w i  d r  8  w  64.  w i  d r  130  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 f o r f o i l 1 showing t h e 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 o b t a i n e d u s i n g the two-dimensional model. U = 500 f t / m i n ; w  V  65.  wi  =  0  ,  0  5  f t  /  s e c  >  k  dr  =  2  -5xl0"  8  ft  137  Comparison o f the 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 of 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 ft/min  138  66.  Photograph of the f l o w over f o i l 2 a t a wire v e l o c i t y o f 1000 f t / m i n . Note t h e presence o f 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 w i r e .... 139  67.  Photograph o f the f l o w over f o i l 1 a t a wire v e l o c i t y o f 1 0 0 0 ft/min. Note t h e c o n t i n u o u s f i l m o f water over t h e f o i l  - ix -  140  Page 68.  Comparison o f the 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 models. Wire 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 the s o l u t i o n s  141  Comparison o f the d r a i n a g e r a t e s on f o i l o n e - d i m e n s i o n a l models. Wire d e f l e c t i o n i n the s o l u t i o n s  142  w  69.  70.  1 p r e d i c t e d by t h e e f f e c t were i n c l u d e d  Comparison of the 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 models. U = 500 f t / m i n ; Wire 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  143  T y p 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 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 model, wire d e f l e c t i o n effects included  144  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 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 ; 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 i n c l u d e d  145  w  71.  w  w  72.  w  73.  74.  75.  ....  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 the o r i f i c e p r e s s u r e drop  154  S t r a i n v e r s u s load support b o l t 1  c a l i b r a t i o n curve f o r t e n s i o n  157  S t r a i n v e r s u s load support b o l t 2  c a l i b r a t i o n curve f o r t e n s i o n  roller  roller 158  - x -  NOMENCLATURE A  c o e f f i c i e n t m a t r i x of dimensional model  a  sub-matrix within the coefficient  B  v e c t o r of the r i g h t hand sides of the equations for the twodimensional model  S  sub-matrix within the c o e f f i c i e n t  b  sub-vector within the vector B  b  non-dimensional v i s c o s i t y the two-dimensional model  c  sub-matrix within the coefficient  Cj.c^Cg  constants i n the velocity profiles  f  non-dimensional streamwise model  f  v e l o c i t y gradient (—) dy  h  cross stream distance between the forming wire and the  foil  hj  d i s t a n c e i n the c r o s s stream d i r e c t i o n between nodes and ( i , j )  (i,j-l)  k  drainage resistance of the forming wire  d r  the system of equations for the two-  L  foil  equations  matrix A  for  velocity  the  one-dimensional  model  i n the momentum i n t e g r a l  i n the two-dimensional model  d i r e c t i o n between nodes  (i-l,j)  length  f o i l pitch  x  p  pressure  p ,p a  p  matrix 5  term i n the equation of motion for  d i s t a n c e i n the streamwise and ( i , j ) L  matrix A"  2  a t m  atmospheric pressure pressure gradient  i model j  term (-^i) d  x  - xi -  used i n the momentum i n t e g r a l  Q  flow  i  r  , r  2' 3' r  r i  t  rate  * & h t hand s i d e s d i m e n s i o n a l model  r  of  the  equation  motion  for  the two-  R  radius of curvature  „K i - l  parameter i n the equations of motion f o r the two-dimensional  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  j-l/2  s  1  >  3  s^.SgjSg, S  wire  model  s ,s , 2  of the forming  of  parameters model  i n the equations of motion  f o r the two-dimensional  7> 8 S  T  forming wire  u  streamwise v e l o c i t y  v  cross stream v e l o c i t y  W  foil  x  streamwise d i s t a n c e measured from t h e f o i l  X*  solution vector d i m e n s i o n a l model  x  sub-vector of the X vector  y  cross wire  tension component component  width  stream  distance  ri  non-dimensional cross  \i  dynamic  v  kinematic  viscosity  turbulent  viscosity  t  l e a d i n g edge  f o r the system of equations  measured  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 the t w o - d i m e n s i o n a l model  v  wire  from  of  the undeflected  the two-  forming  s m a l l changes i n u , v , f and p i n  stream d i s t a n c e  viscosity  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 model  -  xii -  integral  p  density  e  a n g l e between the endpoints of a s m a l l segment of forming w i r e  Subscripts w  i n d i c a t e s v a l u e o f q u a n t i t y a t the forming w i r e  j  indicates direction  p  i n d i c a t e s 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  indicates partial differentiation  with respect  to 5LPdx  indicates partial differentiation  with  to x  p  v x  x  quantity  at  the  j  node i n t h e c r o s s  respect  stream  Superscripts i  indicates direction  quantity  at  n  i n d i c a t e s q u a n t i t y a t the n  *  i n d i c a t e s non-dimensional q u a n t i t y  the  - xiii  i  node  iteration  t h  -  i n the  streamwise  ACKNOWLEDGEMENTS  The  author  supervisor,  would  like  to  express  Dr. E.G. Hauptmann.  p r o v i d e d throughout  h i s sincere  H i s guidance  thanks  to h i s  and encouragement  was  a l l phases o f t h e work, without which t h i s work would  not have been p o s s i b l e . The  author  would  also  like  t o express  h i s gratitude  t o Dr. I.S.  G a r t s h o r e f o r h i s i n t e r e s t and h i s comments r e g a r d i n g t h i s work. Special Department  thanks  go t o t h e s t a f f  of Mechanical  and f e l l o w  Engineering.  Their  graduate assistance  s t u d e n t s o f the and a d v i c e i s  greatly appreciated. Finally, continuous  t h e author  support  would  like  and encouragement  t o thank  h i s parents  f o r their  p r o v i d e d d u r i n g t h e course o f t h i s  work.  - xiv -  1. I.  1.1  INTRODUCTION  General The primary p r o c e s s i n t h e making o f paper  i s t h e removal o f water i n  a c o n t r o l l e d manner from a s l u r r y o f pulp and water. takes p l a c e i n t h r e e s t a g e s . the  d r y i n g o f the paper  ponding moves  sheet.  initial  t h e machine  water  significant determines  removal influence  from  The energy  requirements  and the c o r r e s -  i n c r e a s e d r a m a t i c a l l y as the paper the forming  i n the forming on  o f water  These a r e the f o r m a t i o n , t h e p r e s s i n g , and  c o s t o f removing water along  The removal  the  section  section  properties  the q u a l i t y of the f i n a l product.  sheet  t o the d r y e r s .  o f t h e machine a l s o  of  the  paper  and  The has a  largely  As a r e s u l t , a g r e a t d e a l o f  emphasis has been p l a c e d on improving 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 t h e f o r m i n g s e c t i o n o f paper machines. Modern paper  machines i n use a r e g e n e r a l l y one o f two t y p e s .  a r e t w i n w i r e and s i n g l e w i r e , o r F o u r d r i n i e r , paper machines.  These  Of the two  t y p e s o f machines, t w i n w i r e formers a r e g e n e r a l l y s u p e r i o r t o F o u r d r i n i e r paper machines. to  Water i s d r a i n e d much q u i c k e r on a twin-wire machine due  the s q u e e z i n g  paper  o f the paper  sheet  between t h e w i r e s .  The q u a l i t y o f  made on a twin-wire machine i s a l s o much b e t t e r as t h e paper has a  f i n i s h e d s u r f a c e on both s i d e s .  D e s p i t e t h e advantages o f u s i n g twin-wire  machines, t h e c o s t o f a c q u i r i n g and i n s t a l l i n g F o u r d r i n i e r machine i s u s u a l l y machines  being  brought  into  prohibitive.  service  expansion  of production c a p a b i l i t i e s .  machines,  there  i s an emphasis  one t o r e p l a c e an e x i s t i n g  As a r e s u l t ,  a r e done  so as p a r t  most  twin-wire  o f an  overall  For operators of Fourdrinier  on d e v e l o p i n g  methods  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 the e x i s t i n g machines.  o f improving  paper the  2.  A Fourdrinier screen  (referred  rollers  (Figure  is  spread onto  to  as the At  1). the  the forming wire, the  paper machine c o n s i s t s of  forces  one end o f  rollers  of  the  superior  all  but  by  the  of  the  drainage  1*2  Review of L i t e r a t u r e comprehensive  been done t o  result  out.  of  of  connection,  review  of of  of g r a v i t y  active  Wrist  devices.  take  l o c a t e d under the of  foils  rolls.  survey  form  of  wire.  passive drainage devices, In  have r e s u l t e d  recent in  D e s p i t e the wide  predicting  available  the  little  years,  their  use  spread use  knowledge  of  foils  the  d r a i n a g e p r o v i d e d by  literature indicate  exists.  or  pertaining very  little  Further  to  foils  work  the pressure  a  has  distri-  foil.  current  analytical  on  and a d d i -  drainage  devices  develop an a c c u r a t e a n a l y t i c a l model of  the  slurry  vacuum s u c t i o n b o x e s l o c a t e d on  characteristics  this  on  moves down t h e m a c h i n e o n  paper machines, very  accurately  Results  on a d r a i n a g e  Much  and  Passive  table  supported  and water  use over a c t i v e methods.  on e x i s t i n g F o u r d r i n i e r  carried  form of  wire.  use of  as been provided."  bution  the  their  foil  was  in  o r w i r e mesh  t h e w i r e l e a v i n g a p a p e r mat  passive  drainage f o i l s  n o k n o w n means o f  A  slurry  drainage c h a r a c t e r i s t i c s of  understanding more,  both  forming  favour  eliminating  foils  As t h e  the lower energy requirements  machine operators  of  the  and s t a t i o n a r y  Because of  which i s  Drainage takes p l a c e as a r e s u l t  provided  underside  wire)  the machine the pulp  water i s drained through  Active devices are usually the  medium o r  forming wire.  upper s u r f a c e .  tional  forming  a moving f a b r i c  and  knowledge  of  experimental  foil  drainage  studies  of  [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  was d e v e l o p e d a s  table table  rolls.  In  rolls  using  a  this the  3. analogy In  o f a p i s t o n b e i n g withdrawn from a c y l i n d e r h a v i n g a porous t o p .  l a t e r work, W r i s t  concept  based  attempted  to formulate  on an a n a l y s i s o f t h e f l o w u s i n g B e r n o u l l i ' s theorem.  e x t e n s i o n of t h i s concept  t o f o i l s was attempted by assuming a f o i l  be r e g a r d e d a s a s t a t i o n a r y t a b l e r o l l . by  Taylor  a mathematical model o f t h i s  [2],[3],  Meyer  An  could  S e v e r a l o t h e r independent s t u d i e s  [ 4 ] , Burkhard  and W r i s t  [5] and V i c t o r y [6]  attempted t o p u t forward models f o r t h e d r a i n a g e a t a t a b l e r o l l . Analytical [3],  Meyer  models  for foil  [ 7 ] , and E r n s t  drainage  [8].  Taylor  have  formulated  momentum e q u a t i o n f o r a boundary l a y e r f l o w . t h e assumption t h e f l o w was f u l l y as  a plug  flow  been  suggested  by T a y l o r  a model based on t h e  In h i s a n a l y s i s , T a y l o r made  t u r b u l e n t a l l o w i n g him t o t r e a t t h e f l o w  i n which t h e v e l o c i t y was a f u n c t i o n o f x o n l y .  so, T a y l o r n e g l e c t e d f l u i d  f r i c t i o n e f f e c t s a t the f o i l  disregarding  boundary  t h e no  slip  T h i s s i m p l i f i c a t i o n permitted  conditions  In d o i n g  and t h e w i r e  on t h e v e l o c i t y  thus  profile.  t h e r e d u c t i o n o f t h e momentum e q u a t i o n  to a  simple  o r d i n a r y d i f f e r e n t i a l e q u a t i o n which was then s o l v e d a n a l y t i c a l l y .  Taylor  then  Wrist  reported  by Burkhard and  [5] t o 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 t h e f o i l s  at the operating  conditions results result  the experimental  examined by Burkhard  typical  o f those  shown  [9].  Taylor  to the e f f e c t s  first  conditions  and W r i s t . i n Figure  i s i n poor agreement w i t h  Fleischer due  used  2.  Taylor  obtained  As c a n be seen,  Taylor's  the unpublished  suggested  of the f l u i d  In g e n e r a l ,  this  friction.  experimental  discrepancy To e x p l o r e  r e s u l t s of  was most this  likely  hypothesis,  T a y l o r i n t r o d u c e d 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 t h e a d d i t i o n of a drag only p a r t i a l l y reduced  term t o h i s model.  The r e s u l t s o f t h i s a d d i t i o n were  s u c c e s s f u l i n t h a t t h e magnitude o f t h e maximum s u c t i o n was  to a value  t h e f i n d i n g s o f Burkhard  and W r i s t ,  though t h e o v e r a l l shape 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 remained  essentially  unchanged. due  consistent with  Thus T a y l o r  suggested  the discrepancy  i n t h e r e s u l t s may be  t o t h e e f f e c t s o f d e f l e c t i o n o f t h e forming w i r e over t h e f o i l .  This  4. hypothesis,  however, was  Meyer foil  by  [7]  performed  solving  proceeded  by  direction  and  not  the  explored.  an  a n a l y s i s of  conservation  integrating assuming  the  the  of  the  mass  continuity  drainage  pressure  distribution  equation.  equation  velocity  Meyer's  on  a  analysis  in  the  cross  stream  through  the  forming  wire  obeyed the r e l a t i o n s h i p dr v  A v e l o c i t y p r o f i l e was  . (p-p  =  w  ) a'  11  then c a l c u l a t e d by  s o l v i n g t h e s i m p l i f i e d momentum  equation 3u  dp  2  The  resultant profile  i n t e g r a t i o n was an  expression  was  then used  completed.  in  the  f o r the p r e s s u r e  i n magnitude  observe  the  must be  equal  Meyer's  results  with  d i s t r i b u t i o n which was  those  condition that  the  of  Taylor,  pressure  t o the ambient p r e s s u r e . to  those  of  Taylor  at  Furthermore,  i t was  could  not  clear  geometries were used by T a y l o r , E r n s t and [8]  Ernst  attempted  to  model  the  The  the  though the  not  then s o l v e d u s i n g a  r e s u l t s obtained Meyer's  by Meyer  solution did  l e a d i n g edge of  A direct  d i f f e r e n c e s i n the o p e r a t i n g c o n d i t i o n s and analyses.  and  I n t e g r a t i o n o f the c o n t i n u i t y e q u a t i o n y i e l d e d  s e r i e s s o l u t i o n based on B e s s e l f u n c t i o n s . agreed  c o n t i n u i t y equation  foil  comparison of E r n s t  be  foil  the  made, however,  due  ally  as  expression  opposed  f r o m Meyer's the  forming  to  f o r the  analytically.  i n t h a t he wire  pressure  extended  deflection.  exactly  what  conditions  pressure  distribution  d i s t r i b u t i o n was  on  obtained  a n a l y s i s , however,  h i s a n a l y s i s t o i n c l u d e the was  and  Meyer.  Ernst's  This  to  geometries used i n t h e i r  a  foil  u s i n g an approach i d e n t i c a l t o t h a t o f Meyer except t h a t a s o l u t i o n t o resultant  and  done by  developing  an  the  numeric-  did  differ  effects  of  expression  5. for  t h e forming w i r e d e f l e c t i o n a s a f u n c t i o n o f t h e p r e s s u r e  along the f o i l ing  wire  distribution  assuming t h e f o r m i n g w i r e had no f l e x u r a l r i g i d i t y .  effects  were  incorporated  i n Ernst's  solution  by  Form-  iteratively  e v a l u a t i n g 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 d e f l e c t i o n , using  the r e s u l t s  expected, Meyer  Ernst  f o r both  t o e v a l u a t e a new p r e s s u r e d i s t r i b u t i o n .  obtained  results  the pressure  distributions.  A direct  that  magnitudes  comparison  however, due t o d i f f e r e n c e s  agreed  w e l l with  and t h e shape  of t h e i r  results  i n t h e assumed f o i l  then  As would be  the r e s u l t s  of  o f the p r e s s u r e  c o u l d n o t be made,  geometries  and o p e r a t i n g  conditions. Detailed conducted  experimental  by Burkhard  and W r i s t  experimental  studies,  measurements  on s t a t i o n a r y  difficulties roller, as  rolls. the  performed  of t h e i r  pressure  acts  drainage rates f o r f o i l s  the  this  pressure  experimental  date,  been  As p a r t o f t h e i r  and W r i s t ,  performed  to s i m p l i f y the on  a  stationary  s t u d i e s , p a t e n t s were taken out on t h e  Additional  experimental  s t u d i e s have  been  s t u d i e s have been conducted t o  to table r o l l s .  may produce peak s u c t i o n s o f up t o — P ^  To  have  and Bennett  distribution  l a t e r y e a r s , numerous e x p e r i m e n t a l  the pressure  [10].  and W r i s t , Burkhard  rolls  [ 9 ] , Cadieux [11], and Walser [ 1 2 ] .  q u a l i t a t i v e l y compare f o i l s  but  table  u s i n g a t a p e r e d d e v i c e which they r e f e r r e d t o  and W r i s t .  by F l e i s c h e r  on  [5] and Bennett  Burkhard  measuring  As a r e s u l t  by Burkhard  In  both  simulated the r o l l e r  a foil.  foil  of  investigations  over  a much  2 w  I t has been h i n t e d t h a t  o r one h a l f t h a t o f t a b l e  greater  length.  distribution  on  and t h e o r e t i c a l  known attempts  drainage  foils.  rolls,  As a r e s u l t , n e t  a r e e q u a l t o o r g r e a t e r than t h a t o f t a b l e  no a d d i t i o n a l  foils  rolls.  have been made t o model Though  r e s e a r c h has undoubtedly  a  great  deal of  been conducted  by  6. both paper manufacturers of  this  work has  research.  and the manufacturers  been published due  Consequently,  there  is  of drainage f o i l s ,  to the proprietary very  little  nature  little of  the  published information  regarding studies done on the drainage properties of f o i l s .  1.3  Need f o r the Present Work The primary function of a paper machine i s to drain the water from a  paper and  slurry provide  to form a paper sheet.  To properly design a paper machine  controlled  the  drainage  of  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 d i s t r i b u tion,  and  attempts  hence  the  drainage  provided  by  a  foil.  The  present work  to improve existing a n a l y t i c a l models by including both viscous  forces and the effect of d e f l e c t i o n of the forming wire i n the model. To  verify  the accuracy  of the models developed,  numerical r e s u l t s to experimental r e s u l t s was done. of  published experimental data, a simple experiment  comparison  comparison  of  the  Because of a shortage to obtain data f o r  purposes was also deemed to be a necessary part of the present  investigation.  7. II.  2.1  EXPERIMENT  General Experimental  foils  have  been  Additional papers  investigations published  experimental  by  Fleischer  presented,  only  the  by  data  [10]  of the p r e s s u r e d i s t r i b u t i o n on  Burkhard  and  have a l s o  and  Bachand  of F l e i s c h e r  d e a l t e x c l u s i v e l y with drainage f o i l s .  The  and  rolls,  Bennett  dealt  investigations In  for  2.2  and  with  drainage  foils  to  a l l cases,  conditions  primarily  insufficient technique  Scope o f the purpose  experiment  of  was  different  shaped  necessary  that  therefore  the  c l o s e l y as  closely  Of  the s t u d i e s  Cadieux and  w i t h an  regard  presented  the  experiment of  was  Bachand  and W r i s t ,  e x t e n s i o n of  to  table  the  their rolls.  experimental  t o make the d a t a of any  pressure  foils  under  various  the  data  experiment  to  the n u m e r i c a l  obtain  use  investigation.  obtained needed  be  to  provide  data  analysis.  The  distributions operating  the  on  be  used  primary a  of  for  aim  variety  conditions.  representative  simulate  to  It  actual  o p e r a t i o n of a  of of was  foils, foil  as  constructed to simulate  as  possible.  experimental as  unpublished  s t u d i e s o f Burkhard  with  to  D e s c r i p t i o n of An  i n an  [10].  Experiment  comparison t o the r e s u l t s  2.3  was  and  Bennett  considering stationary  detail  used  and  [10] .  comparison w i t h t h e o r e t i c a l r e s u l t s of the p r e s e n t  The  the  table by  [5]  been p r e s e n t e d  Cadieux and  investigations  Wrist  drainage  Apparatus test  p o s s i b l e the  r i g was actual  designed  an  o p e r a t i n g c o n d i t i o n s of  a drainage  foil.  8. Design data  considerations  as  actual  easily  as  operating  which  possible  slurry.  To  speeds.  the use  Because of  The  considered  2.3*1  the  26  wide s y n t h e t i c wire.  The  significant  was  connected  to  speed  fabric  4.  end  simulating  at  built  typical  from  actual  paper machine  drainage  resistance  of a paper machine, t h i s  b e l t made from a p i e c e  was  conditions.  provided  1 horsepower  was  drive  a  roller  inches  by  c o n t r o l l e d using  The  adjusted  rollers  was  a  a 24  inch  Drive  for  a horizontal  the  roller-belt  electric  motor.  The  motor  of  pulleys  and  a  step  Variac.  This  was  V-belt.  arrangement  permitted  minute. a  tensioning  b o l t s , one  roller.  a t each end  The  tensioning  o f the r o l l e r  (Figure  s u p p o r t b o l t s a l l o w e d the p o s i t i o n of the t e n s i o n i n g r o l l e r to  be  o v e r a range of 6 i n c h e s .  of  0-500  tension,  r o l l e r s were mounted  of a t i s s u e machine forming  long.  means  c o n t r o l l e d by  suspended from two  the  aluminum frame work  i n a manner which p r o v i d e d  24  by  a large  around  approximately  the  of  i n c h diameter pvc  Passing  were arranged  t e n s i o n was  r o l l e r was  from  3 and  4.5  speeds from 500-2000 f e e t per Belt  5).  made  of  desired  t e s t r i g was  foils  v a r i a t i o n i n the  to the dry  i n c h wide by  rollers  section  assembly  belt  a  the  the working f l u i d as opposed t o a p u l p  e x p e r i m e n t a l apparatus c o n s i s t e d  shown i n F i g u r e s  Motor  goal  to be a s i g n i f i c a n t d e p a r t u r e from a c t u a l  t o which f i v e  test  the  scale  departure  obtain  O v e r a l l Apparatus The  as  only  to  these g o a l s ,  full  of water as  o f a paper sheet from the wet not  on  a means  sacrificing  meet  measurements  operating  c o n d i t i o n s was  providing  without  conditions.  facilitated  machine  included  pounds  strain  per  foot.  T h i s corresponded t o a b e l t t e n s i o n  To  gauges were mounted  permit on  the  the  measurement  support  bolts  of such  the that  belt the  9. load  on  the  l e n g t h of  bolts  could  the b o l t s ,  be  determined.  By  the t e n s i o n i n g r o l l e r  t h e t r a c k i n g of the forming w i r e on the  differentially  a d j u s t i n g the  c o u l d a l s o be used to  control  rollers.  3  A headbox h a v i n g upstream end  of  the  o f y^- x 6 i n c h e s wide was  a sluice  test  section (Figure 6).  The  mounted a t the  headbox was  positioned  such t h a t a t h i n f i l m of water c o u l d be d i s t r i b u t e d onto the upper s u r f a c e of  the  forming  wire  in  the  test  headbox v i a a 4 i n c h water main.  section.  Flow r a t e was  g a t e v a l u e mounted on the water supply p l a t e and The  t e s t e d were mounted on  s e c t i o n as shown i n F i g u r e 7. from the headbox s l u i c e w i t h the u n d e r s i d e  foils,  Water  measurement  drained  by  the  it  to  the  c o n t r o l l e d by means of a  collected  in  took  to  fill  the  the  a  support  orifice  beneath the  test  i n c h e s downstream  t h a t the f o i l was  i n contact  wire. of  the was  drainage collected  rates  obtained  in a  trough  was  by  drainage  j u s t downstream from the f o i l  drainage  drum  bracket  f o i l s were l o c a t e d 12  foils  g a l l o n drum mounted on a s c a l e . time  supplied  p i p e and measured u s i n g an  p o s i t i o n e d such  of the forming  facilitate water  and  The  mounted beneath the forming w i r e and 7).  was  a d i f f e r e n t i a l mercury manometer. foils  To  Water  channelled  the  trough (Figure  into  a  45  Flow r a t e s were o b t a i n e d by r e c o r d i n g the until  i t contained  a  known  volume,  by  weight, of water.  2.3.2  Foils  Tested  Pressure of  and  d i f f e r e n t cross sections.  high molecular in  distributions  F i g u r e 8.  had  r a t e s were o b t a i n e d  Each f o i l was  weight p l a s t i c . A l l foils  drainage  for 5  foils  machined from a b l o c k of u l t r a  A photograph of the f o i l s  tested i s given  o v e r a l l dimensions o f 9 i n c h e s wide, 6 i n c h e s  10.  l o n g and 1.5 flat  inches t h i c k .  l e a d i n g edge  taper angle foils. inch  section  Foils  1 i n c h l o n g and 5.5  1, 2 and 4 degrees  Foil  4 had a f l a t  stepped  s e c t i o n having  1, 2 and 3 were t a p e r e d f o i l s  respectively.  i n c h tapered  Foils  l e a d i n g edge s e c t i o n  having a  s e c t i o n s of  4 and 5 were  stepped  2.5 i n c h e s l o n g and a 4  a s t e p h e i g h t o f .25 i n c h e s .  Foil  5 had a 1  i n c h f l a t l e a d i n g s e c t i o n and was stepped 1/16 i n c h over t h e remaining inches.  A summary of the f o i l  Along pressure located  the l o n g i t u d i n a l taps  were  a t 1/4  dimensions  center l i n e  drilled  through  each  foil.  2.4  Instrumentation and Measurements Because  of the l i m i t e d  nature  may be found i n F i g u r e 9. o f each f o i l ,  the f o i l .  inch centers r e s u l t i n g  5.5  The  i n a total  and scope  1/16  inch  pressure  diameter  taps  were  o f 25 p r e s s u r e  taps on  of the experiment,  complex  i n s t r u m e n t a t i o n was r e p l a c e d wherever p o s s i b l e w i t h t e c h n i q u e s o f a s i m p l e nature  f o r obtaining  the  desired  measurements.  measurements made and the methods used  2.4.1  Descriptions  of  the  to o b t a i n them f o l l o w .  F o i l Pressure Distributions The p r e s s u r e d i s t r i b u t i o n on the f o i l was o b t a i n e d u s i n g d i f f e r e n t i a l  water  manometers.  A  pressure  taps  the f o i l  t o be observed  l o c a t e d on the f o i l  were determined The  manometer  while  tube  connected  t o each  a l l o w i n g the p r e s s u r e  the experiment  of  the 25  distribution  was I n p r o g r e s s .  on  Pressures  by measuring the head o f water i n each o f the manometers.  head was measured t o w i t h i n 1/16  approximately  was  1 to 2 p e r c e n t .  o f an i n c h r e s u l t i n g i n an e r r o r of  11.  2.A.2  F o i l Drainage Rates As mentioned p r e v i o u s l y , water d r a i n e d by the f o i l was  drainage from  trough  located  the d r a i n a g e  drum.  The  downstream from  trough  drum was  through  The  the flow r a t e . be  5 percent  discussed  2.4.3  mined  To determine  the drainage flow r a t e ,  weight,  volume o f water d r a i n e d was  the  technique  proved  f l o w r a t e o f the water d e l i v e r e d  on  a  plate,  out  square  to  a  the  edge  orifice  pipe.  differential  by measuring  calculating  orifice  thin,  the water supply  connected  determined  set  o f the drum and  o f water was t h e n used  O v e r a l l measurement e r r o r o f t h i s method was though  gallon  t o be  the  measured.  to c a l c u l a t e determined  to  i n a p p r o p r i a t e as w i l l  be  later.  using  flanges  and  and  channelled  Supply Water Flow Rate The  were  Water was  a s h o r t l e n g t h o f p i p e i n t o a 45  r e q u i r e d t o d r a i n a g i v e n volume, by measured time  foil.  p l a c e d on a s c a l e a l l o w i n g the weight  t h e water i n i t t o be measured. time  the  collected i n a  t o t h e headbox i n l e t was  deter-  p l a t e mounted between two  P r e s s u r e t a p s , l o c a t e d a t D and  1/2D,  mercury  were  manometer.  Flow  rates  the d i f f e r e n t i a l p r e s s u r e a c r o s s the o r i f i c e  flow  rate.  the o r i f i c e was  i n the ASME F l u i d  To  e l i m i n a t e the need  d e s i g n e d and manufactured  Meters  pipe  Handbook  [13].  plate  to c a l i b r a t e t o the  the  standards  T h i s technique  allowed  t h e f l o w r a t e t o be determined w i t h i n an e r r o r o f a p p r o x i m a t e l y 8 p e r c e n t . 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 o f the f l o w r a t e 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  f o r m i n g w i r e t e n s i o n was  t e n s i o n i n g r o l l e r support b o l t s .  determined  by measuring  the l o a d on  To o b t a i n the l o a d on t h e support  the  bolts,  12. each b o l t  had  ( F i g u r e 10).  four  flat  s u r f a c e s machined f o r m i n g a square c r o s s s e c t i o n  A s t r a i n gauge was  t h e gauges were connected permitted  the  components  direct  due  arrangement.  to  t o form a f o u r  measurement  bending  Bolt  a t t a c h e d t o each o f the f l a t  of  gauge b r i d g e .  of a x i a l  the  bolts  strain were  s u r f a c e s and  T h i s arrangement  i n the b o l t s  cancelled  by  as  strain  the  bridge  s t r a i n s were measured u s i n g a V i s h a y Instruments  P-350A p o r t a b l e s t r a i n i n d i c a t o r and a model SB-1  Model  Switch and Balance  unit.  To c o n v e r t the b o l t s t r a i n s t o l o a d s , the b o l t s were c a l i b r a t e d by measuring a  the s t r a i n f o r known l o a d s .  The b o l t s t r a i n s c o u l d be determined  great d e a l of accuracy r e s u l t i n g  for  t h i s measurement.  found i n Appendix  2.4.5  with  i n an o v e r a l l e r r o r o f only 1 p e r c e n t  R e s u l t s o f the t e n s i o n i n g b o l t  c a l i b r a t i o n s can be  B.  Forming Wire Speed The  speed  forming  of  wire  the d r i v e  tachometer.  The  speed  roller  angular  was  determined  the  forming  wire  velocity  speed  of  the  drive  Due  could only  mately 10 percent of the d e s i r e d  2.4.6  measuring  the  rotational  u s i n g a Shimpo model DT-205 hand h e l d  c a l c u l a t e the speed o f the forming w i r e . speed,  by  be  roller  was  t o unsteadyness maintained  digital  then used  to  i n the motor  within  approxi-  speed.  Forming Wire Drainage R e s i s t a n c e The r e s i s t a n c e t o f l o w through the forming w i r e used on the apparatus  was  determined.  length  of  vertical the  1/2  A p i e c e of the forming w i r e was inch  position  drainage  diameter  with  the  resistance,  pvc  pipe.  forming w i r e  a c o n s t a n t head  The  g l u e d over one  pipe  was  a t the lower of water was  then end.  end o f a  fixed To  in a  determine  maintained  i n the  13. pipe  and  the  described  flow  f o r the f o i l  obtained,  the  an  of  the  overall  was  measured  drainage  drainage  Determination given  rate  the  drum  r a t e measurement.  r e s i s t a n c e was  drainage  error  using  to  be  repeated  Experimental Because  i n the measurement o f 5 p e r c e n t .  was  the  connecting from air  the  of  the  minimal  was  i n s t r u m e n t a t i o n used,  any  For  by  Details  to  of  the  setup  was  C.  A  check  little  taps.  The  i n the  a i r was  lines purged  the manometer tubes w i t h water u n t i l a l l  that  a l l a i r had  returned  to  been  their  removed  initial  was  levels  done after  by the  complete.  each e x p e r i m e n t a l of  very  have been p r e s e n t  t o the p r e s s u r e  back f l u s h i n g  run was  range  feet.  7  The o n l y p r e p a r a t i o n r e q u i r e d f o r each  a i r t h a t may  a l l manometer tubes  experimental  a  of  the manometer tubes  removed.  ensuring  over  purging  lines  4.0xl0~  was  Procedure  r e q u i r e f o r each e x p e r i m e n t a l run. run  method  s e v e r a l times  d r a i n a g e r e s i s t a n c e c a l c u l a t i o n can be found i n Appendix  2.5  scale  Once the f l o w r a t e  calculated  r e s i s t a n c e was  and  forming  run, wire  results speeds.  were o b t a i n e d At  the  for a  start  of  single  each  run  foil the  manometer tubes were b a c k - f l u s h e d w i t h water and t h e i r i n i t i a l l e v e l s were recorded. using  The w i r e speed was  the motor  t h e headbox. was to  The  control.  pass  t o permit on  manometer  measured.  At  f l o w r a t e was  e q u a l t o the w i r e speed.  distribution the  speed  then s l o w l y brought  the w i r e  the  foil  readings  up  the same time,  t o the d e s i r e d speed  water was  a d j u s t e d so t h a t the headbox j e t v e l o c i t y  A p e r i o d of s e v e r a l minutes was and  s u p p l i e d to  jet velocities,  to  stabilize.  were  recorded  After and  the  as the  w e l l as  then a l l o w e d the  stabilization  foil  drainage  pressure period, rate  was  Measurements were made i n t h i s manner f o r each of the d e s i r e d  14. wire  speeds.  After  the water  supply  stopped.  A  d a t a had been o b t a i n e d f o r a l l d e s i r e d w i r e  t o the headbox was t u r n e d o f f and t h e forming w i r e was  check  was  then  made  t o ensure  returned to t h e i r i n i t i a l values. its  initial  and  the e n t i r e  completed,  level,  the manometer  levels  i t was assumed t h a t a i r had e n t e r e d t h e manometer l i n e  r u n was r e p e a t e d .  After  the r u n had been  successfully  t h e f o i l was removed and r e p l a c e d w i t h t h e next f o i l .  Measure-  foils.  R e s u l t s and O b s e r v a t i o n s For  each o f t h e f i v e f o i l s  tested,  p r e s s u r e d i s t r i b u t i o n s and d r a i n -  age f l o w r a t e s were o b t a i n e d f o r w i r e speeds and  2000 f e e t  100  pounds  foil,  p e r minute.  per l i n e a l  as a f u n c t i o n  o f 500, 750, 1000, 1250, 1500  A l l d a t a were o b t a i n e d f o r a w i r e  foot.  of wire  Measured speed,  pressure  comparison  o f the p r e s s u r e d i s t r i b u t i o n s on each f o i l  wire  speed  are presented  are presented  f o r each  a r e p r e s e n t e d i n F i g u r e s 10-14.  and l e n g t h s  can be found i n F i g u r e s 15-20.  tension of  distributions  pressures  of  tube  I f a manometer tube f a i l e d t o r e t u r n t o  ments were then t a k e n f o r t h e next f o i l and a l l subsequent  2.6  speeds,  as non-dimensional  Both  quantities.  f o r each w i r e  A speed  Measured f o i l d r a i n a g e r a t e s as a f u n c t i o n i n F i g u r e 21.  The d r a i n a g e  rate  has been  expressed n o n - d i m e n s i o n a l l y a s :  ^nd  "  Qm u h w max  where Q i s t h e measured d r a i n a g e r a t e [ f t / s / f t of f o i l l , h i s the m max maximum gap between t h e f o i l and t h e w i r e and u i s t h e w i r e v e l o c i t y . 3  1  w  During  the course  p r e s s u r e s were noted  of taking  measurements,  a t higher wire  speeds.  some u n s t e a d i n e s s  At w i r e speeds  i n the  o f 1500-2000  15. f e e t p e r minute, a v e r y s l i g h t o s c i l l a t i o n i n t h e p r e s s u r e r e a d i n g s o f t h e first  f o u r and t h e l a s t  two p r e s s u r e taps was observed.  however, c o n s i s t e d o n l y o f a s l i g h t of  jitter  i n t h e manometer tubes and was  i n s u f f i c i e n t magnitude t o be measured. Attempts were made t o o b t a i n r e s u l t s  It  The o s c i l l a t i o n ,  f o r a range o f w i r e  tensions.  was found t h a t s m a l l changes i n t h e w i r e t e n s i o n (5-10 pounds p e r f o o t )  did  n o t have  a  measurable  affect  on  the pressure  distribution.  i n c r e a s e d w i r e t e n s i o n s l a r g e enough t o have a measureable e f f e c t ,  At exces-  s i v e warping o f t h e frame, uneven s t r e t c h i n g o f t h e w i r e and d e f l e c t i o n o f the  rollers  made i t p o s s i b l e t o keep  center of the r o l l e r s . wire  tensions.  t h e forming  wire  tracking  a t the  As a r e s u l t , d a t a c o u l d n o t be o b t a i n e d f o r h i g h e r  Lower w i r e  t e n s i o n s were n o t attempted  due t o s l i p p i n g  between t h e d r i v e r o l l e r and t h e w i r e a t speeds g r e a t e r than 1000 f e e t p e r minute. Some d i f f i c u l t y  was e x p e r i e n c e d  maintaining  the wire  speed  and t h e  headbox j e t v e l o c i t i e s a t t h e d e s i r e d speeds d u r i n g t h e r e c o r d i n g o f d a t a . To  investigate  which  the wire  amounts.  the e f f e c t  I t was found  the wire  had on t h e r e s u l t s ,  and j e t v e l o c i t i e s  have a measurable a f f e c t Since  this  were  that v a r i a t i o n s  varied  tests  were  independently  up t o 10 f e e t  done i n by  p e r minute d i d n o t  on t h e p r e s s u r e d i s t r i b u t i o n s o r drainage  and j e t v e l o c i t i e s  could  easily  small  be m a i n t a i n e d  rates.  w i t h i n 10  f e e t p e r minute o f t h e d e s i r e d speed, speed v a r i a t i o n s were n o t c o n s i d e r e d to  be s i g n i f i c a n t w i t h r e s p e c t t o t h e a c c u r a c y o f t h e r e s u l t s o b t a i n e d . To  on f o i l s of  1 and 3.  t h e experiment  obtained due  t e s t r e p e a t a b i l i t y o f t h e r e s u l t s , a second  for foil  to f o i l  s e t o f runs were done  As c a n be seen i n F i g u r e s 23 and 24, t h e r e p e a t a b i l i t y was v e r y  good.  The s l i g h t  3 c a n be a t t r i b u t e d  to variations  changes made between t h e two r u n s .  between the two runs done on f o i l one.  variation  i n the r e s u l t s  i n the wire  tension  No f o i l changes were made  16. Examination  of  the  between the d r a i n a g e suction drainage  measured  problem  along  This can  drainage  r a t e s measured and the  length  r a t e s were much g r e a t e r  distributions. This  measured  be  trend  was  of  i n a d e q u a t e method f o r measuring the possible present  to  good,  d i s c u s s i o n of  the  be  drainage  results  for  general,  the  out  rates.  drainage  very  In  turned  discrepancy  expected due  explained  pronounced what  repeatable,  r e s u l t s must be c o n s i d e r e d  Further  4.  obtain  foils.  than c o u l d  to  indicates a  what would be the  most  attributed  rates  to  by  the  measured  the  stepped be  to  a  suction foils. totally  O v e r a l l , i t was  not  r a t e measurements and  the  inaccurate. obtained  can  be  found  i n Chapter  17.  III. THE THEORETICAL ANALYSIS 3.1  General The  main  accurate  mathematical  Fourdrinier model  the  thereby  t h r u s t of  this  investigation  model  of  paper machine d r a i n g e  pressure  distribution  predicting  was  the  flow  foil  and  along  the  the water d r a i n a g e  to develop  in the  the  region  forming  foil  a  could  simple between  wire.  then  r a t e t h a t c o u l d be  but  With  be  a  this  calculated  achieved  by  the  foil. The fluid  problem under c o n s i d e r a t i o n c o n s i s t s of  i n a very  t h i n region having  moving porous boundary. flow,  flow  because  a l s o occurs  the  typically  on  f u n c t i o n of small  flow  Due  to the  across  the  region  the  order  i s very  of  0.01  the l e n g t h o n l y .  compared  to  the  typically formulation  on  the  of  boundary l a y e r A  a  of  the  order model  porous boundary. thin  velocity  plate  as a r e s u l t I t was  to  the  of  allowing  based  on  These  the  thin  one  of  the  (h/L  was  distribution  is a  layer  terms  in  be the  a l s o assumed as  the  the l e n g t h (W/L  is  assumptions  shear  a  assumed t h a t  length  viscous  i s much g r e a t e r than  100).  of  v e l o c i t i e s were assumed to  Two-dimensional f l o w was  permitted  equations,  the  or  the  equations. was  g i v e n t o whether the flow i s l a m i n a r  T y p i c a l Reynolds numbers based on the f o i l  v e l o c i t y a r e on the o r d e r of 1.0x10 turbulent.  compared  stream  speed flow  i r r e g u l a r boundary and  t o 0.04), the p r e s s u r e  foil  g r e a t d e a l o f thought  turbulent.  solid,  s u c t i o n generated  Cross  wire  y - d i r e c t i o n t o be n e g l e c t e d . c r o s s machine w i d t h  one  a high  I t has  transition  been shown  from  laminar  Reynolds number i n the range  6  the w i r e  s u g g e s t i n g the flow would tend t o be  [14], however, to  l e n g t h nad  or  turbulent  that flow  f o r flow generally  over occurs  a  flat at  a  18. ux v  Re  It  =  3.5  to  5.0xl0  i s conceivable, t h e r e f o r e , that laminar  the  first  half  of  a  foil.  In  flow c o u l d e x i s t over a t l e a s t  view of t h i s ,  p r e v i o u s m o d e l l i n g o f the flow over a f o i l , flow  was  laminar  would  model.  Consequently,  present  investigation.  A The  solution  first  of  two  dimensional  the  equations  by  a  reasonable  laminar  flow  the model was  methods  models.  reducing  be  were  The  governing assuming  equations.  Meyer  [7]  approach order full  and  assumed  grouped of  equations  to  velocity  differs  velocity  profile  in  those  into  was  models  of  a  n e g l e c t i n g the i n e r t i a l  and  of  t h a t assuming  purposes  called  of  a  the  methods. the  one-  the  differential equations  s o l v i n g the r e s u l t i n g  of  system  the approach taken by T a y l o r [ 3 ] , modelling  attempts.  i n the  present  The  present  Meyer i n t h a t a investigation  and  T a y l o r ' s model assumed  t o make the e q u a t i o n s  simplified  the  equations  of  third the  turbu-  tractable. motion  by  terms  u  3u 9y  and  v  3u 9y  e s s e n t i a l l y r e d u c i n g the e q u a t i o n o f motion t o t h a t of the Reynolds equation.  the  models i n v o l v e d  ordinary  integrating  were s o l v e d .  Meyer  was  T a y l o r , E r n s t , and  assumed  the l a c k of  three separate  what  system  a uniform v e l o c i t y p r o f i l e Ernst  f o r the  the one-dimensional  profile,  previous of  boundary l a y e r e q u a t i o n s  l e n t f l o w and The  from  [8]  felt  attempted u s i n g  T h i s s o l u t i o n method was Ernst  because of  i t was  m o t i o n over the h e i g h t of the flow r e g i o n and of  and  approach i n the development of  was  solution  a  5  stress  19. The  second s o l u t i o n technique  equations  using  finite  i n v o l v e d the s o l u t i o n of the  difference formulations.  governing  T h i s method i s r e f e r r e d  t o i n l a t e r s e c t i o n s as the two-dimensional model.  3.2  The Governing Equations  3.2.1  The Equations of Motion Using  and  26,  the  assumptions of  S e c t i o n 3.1  the f l o w between the f o i l and  and  3u , 3x"  the mass c o n s e r v a t i o n  +  V  the n o t a t i o n of F i g u r e s  the forming w i r e may  the e q u a t i o n of motion f o r a t h i n shear  u  and  25  be d e s c r i b e d by  flow  3u 3y  1 dp + p dx  v  (3.1)  ay  2  equation  (3.2)  Boundary c o n d i t i o n s f o r the problem may  x = 0;  P =  x - L; p =  be g i v e n  P  p  a  a  by  (3.3a)  (3.3b)  and  y =  0;  (3.4a)  20. y = -h;  where L i s  the f o i l  foil  shown i n  wire  speed,  The  drainage  function  of  length,  Figure  and v  is  w  h is  25,  p  v  i s not  w  pressure gradient  v  where  p is  drainage  the  v  by T a y l o r  w  using  the  the  boundary to  = 0  (3.4b)  the  -  the  ambient  of  pressure, u  the f l u i d  explicitly a c r o s s the  k. — y  (p-p  flow  the  v  wire.  (3.5)  is a coefficient  a porous media and has been used to  wire  length.  d r  through  represent  the  )  units  and 3.2  forming  wire  having  and E r n s t  the  known a n d i s a s s u m e d t o b e a  r e g i o n and k  of  is  drained through  forming  a  w  to  Equation  3.5  of  represents represent  [8].  the governing equations f o r  and  the  foil.  conditions  3.3  and  These 3.4  and  o b t a i n the p r e s s u r e d i s t r i b u t i o n  equations the  may  relationship  along the  the be  flow  solved  given  by  foil.  Wire D e f l e c t i o n Model Previous  [3,7]  have  forming rigid of  3.1  w  in  Meyer [7],  forming  E q u a t i o n 3.5  3.2.2  flow  [3],  Equations between  pressure  resistance  D a r c y ' s Law f o r  is  f l  v  the d i s t a n c e from the forming wire  the v e l o c i t y  velocity,  the  u = 0,  generally  wires  a n a l y s i s of  assumed t h e  w e r e made o f  bronze  synthetic  f a b r i c s and i t  the pressure d i s t r i b u t i o n  forming  wire  s c r e e n s and  was p r o b a b l y a r e a s o n a b l e o n e .  flexible  rigid.  theoretical  is  rigid.  In  on a the  foil past,  the assumption the wire  P r e s e n t day w i r e s , however,  is  a r e made  c a n n o t be assumed t h e f o r m i n g w i r e  is  Because the  foil  the h e i g h t o f the f l o w  i s very  s m a l l compared  r e g i o n between  the forming w i r e and  t o t h e l e n g t h o f the f o i l ,  i t would be  e x p e c t e d t h a t any change i n t h e f o i l - w i r e gap h e i g h t c o u l d have a s i g n i f i c a n t i n f l u e n c e on t h e p r e s s u r e d i s t r i b u t i o n .  T h i s p r e d i c t i o n i s supported  by the dependence o f the p r e s s u r e d i s t r i b u t i o n on t h e f o i l shown i n F i g u r e s 16 t o 21.  shape e x h i b i t e d  in  the e x p e r i m e n t a l r e s u l t s  of  t h e f o r m i n g w i r e a l t e r s t h e shape o f the f l o w r e g i o n , t h e e f f e c t on t h e  pressure d i s t r i b u t i o n  i s similar  this  decided  reason,  deflection  i t was  t o t h a t o f changing  Since  the f o i l  t o i n c o r p o r a t e the e f f e c t  deflection  shape.  o f forming  For wire  i n the e v a l u a t i o n of the pressure d i s t r i b u t i o n .  In d e v e l o p i n g a model f o r t h e f o r m i n g w i r e d e f l e c t i o n ,  i t was assumed  the w i r e i s s u f f i c i e n t l y f l e x i b l e t h a t i t has no f l e x u r a l r i g i d i t y .  This  assumption  permits  wire  deflection  using  Summing  the d e r i v a t i o n  the f o r c e  the forces  o f an e x p r e s s i o n f o r the forming  balance  on the w i r e  i n the y - d i r e c t i o n  on  a  indicated  small  section  i n F i g u r e 27. o f the w i r e  gives  (p -p)dx - 2T sinGf) = 0  If  dx i s s u f f i c i e n t l y s m a l l , then dx « ds, d6 «  . in  ,6. d6 ( ) . —  (3.6)  1, and  r  (3.7)  (p - p ) d s = T d6  (3.8)  E q u a t i o n 3.6 may then be w r i t t e n  3.  22. Observing  that the radius of curvature, R =  R  If  ( 3 . 9 )  (P -p) a  the r a d i u s o f c u r v a t u r e i s l a r g e  1 R  and  ds —  do  K  :j  «  1) then  d2y  (3.10)  " dx2  E q u a t i o n 3.9 may be w r i t t e n  d  2  y  _  (P.-P) (3.11)  dx2  Once t h e p r e s s u r e tion  may  be  distribution  determined  by  along  the f o i l  i s known, t h e w i r e  evaluating Equation  3.11  using  deflec-  the boundary  conditions  x = 0;  y = 0  (3.12a)  x = L ; y = 0  (3.12b)  x  3.3  The One Dimensional Models  3.3.1  The Conservation of Mass Model  3.3.1.1 The  The Model Equations c o n s e r v a t i o n o f mass model  obtains  a solution  f o r the 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 u s i n g o n l y t h e c o n d i t i o n f o r mass c o n t i n u i t y , hence the name o f the model.  Derivation  o f the model begins  by e x p r e s s i n g  the e q u a t i o n  f o r mass  c o n t i n u i t y g i v e n by E q u a t i o n 3.2 i n i n t e g r a l form.  —n Letting o / u dy -h  Q =  and  s u b s t i t u t i n g the expression f o r v  (3.14)  g i v e n by E q u a t i o n 3.5 i n t o  w  Equation  3.13 g i v e s  §  •  ^  <P"Pa>  (3-15)  Assuming a v e l o c i t y p r o f i l e o f the form  u(y)  =  u  w  + c y + c y x  + c y3  2  2  3  (3.16)  w i t h boundary c o n d i t i o n s  y = 0;  u = u  -v.  w  y = -h;  u = 0  (3.17a)  w  £  |  dy o 1  _ 1|E p  dx  +  v  ! f u ^  „.  Q  (3.17b)  2  9Y  2  (3.17c)  <3-17d,  24. and s o l v i n g f o r  c  and c  2  gives  3  - 3 ( 1 2 iP- + 2v u ) h  c  i  p d x %u—f—u  =  (v  c  c  7-\  =  2  3  c.  w 1 2  v  K  p dx  w  Finally,  h * + 12vh 1  d  x  2  where v  and  1 8 c )  i n t o E q u a t i o n 3.16  and  + 30h2 pv u v ww [48pv (v h - 3 v ) J  3  ^2  - 72hp v  2 u  w^_ (  J  -  L  y  )  s o l v i n g f o r dp/dx g i v e s  h  2  h  (p-p  ) -  S  72(^)uJ h  2  k, h  w  \  gives  [_48(^)Q(p-p ) + 144(£^i)Q + 3 0 ( ^ ) u i£ = dx  '  ( 3  e v a l u a t i n g the i n t e g r a l i n E q u a t i o n 3.14  -v  no  (3.18b)  = TThV  1  L  /o  (3.18a)  + ± 4 E )  S u b s t i t u t i n g the e x p r e s s i o n s f o r c , c  0 = ^  W  2h (v h - 3v) w  1  C  has been e l i m i n a t e d u s i n g E q u a t i o n  Equations  3.15  and  3.20  c o n s e r v a t i o n of mass model.  (3 20)  represent  J  ,  2  0  )  3.5. the  model  equations  Using the boundary c o n d i t i o n s  f o r the  25. x = 0; p = p x = L; p = p  these equations  3.3.1.2  a r e s o l v e d f o r t h e two unknowns, p and Q.  system o f equations  solved numerically general  purpose  software by  arbitrary  order.  The method  equations  B-splines  i s obtained  COLSYS.  a  routine  defined  using  represented  by E q u a t i o n s  system  f o r the  of  used  solution  ordinary  points  s o l u t i o n s a t i s f y i n g a s e t of user  iteratively  i s increased  the e f f e c t s  as i n d i c a t e d i n F i g u r e 28.  of wire  by COLSYS u n t i l  A f t e r a s o l u t i o n f o r the p r e s s u r e  COLSYS was used  to obtain a s o l u t i o n to The c a l c u l a t e d w i r e  d e f l e c t i o n was then used i n t h e e v a l u a t i o n o f a new p r e s s u r e  one.  the p r e s s u r e  distribution  The s o l u t i o n procedure  change i n the p r e s s u r e  satisfied  a  d e f l e c t i o n were s o l v e d  f o r t h e w i r e d e f l e c t i o n ( E q u a t i o n 3.11).  previous  a t Gaussian  i n F i g u r e 28.  the e q u a t i o n  iteration,  of  s p e c i f i e d e r r o r tolerances i s obtained.  had been o b t a i n e d ,  each  equations  s p e c i f i e d o r generated  distribution  After  value  A s o l u t i o n t o the system o f  A flow c h a r t of the s o l u t i o n procedure i s presented incorporating  COLSYS i s a  boundary  differential  on a mesh which may be u s e r of mesh  of  by COLSYS i s c o l l o c a t i o n  of v a r i a b l e o r d e r .  The number  Solutions  3.15 and 3.20 were  f o r p and Q u s i n g t h e r o u t i n e COLSYS [ 1 5 ] .  problems  by  (3.21b)  a  Method o f S o l u t i o n  The  points  (3.21a)  cl  was  was  terminated  distribution.  compared when  t o the  t h e maximum  the c o n d i t i o n  max < P.t o l e r a n c e  In g e n e r a l , the s o l u t i o n converged a f t e r 3 t o 4 i t e r a t i o n s .  (3.22)  2 6 .  3.3.1.3  V e r i f i c a t i o n of the Model  B e f o r e a n u m e r i c a l model i s used is  important  t h a t t h e model be v e r i f i e d  having  a nature  serves  two important  no  errors  to  gain  similar  confidence and t h a t  t o t h e problem b e i n g  purposes.  The f i r s t  studied.  that  t h e model  results  obtained  V e r i f y i n g a model  purpose i s t o ensure  adequately using  The second  represents  t h e model  f o r flows  there are purpose i s  t h e flow  will  being  represent the  flow.  The  c o n s e r v a t i o n o f mass model was v e r i f i e d  e v a l u a t e t h e f l o w over a f l a t parallel were  a g a i n s t known r e s u l t s  i n t h e model o r t h e s o l u t i o n procedure.  modelled actual  t o o b t a i n s o l u t i o n s t o a problem, i t  flat  used  against  a couette  exact flow  t h e model t o  p l a t e a t z e r o i n c i d e n c e and t h e f l o w between  p l a t e s i n which  because  by u s i n g  t h e upper  solutions was done  plate  was moving.  are a v a i l a b l e  because  These  flows  f o r them.  Verification  of i t s s i m i l a r i t y  t o the f l o w  between a f o i l and t h e forming w i r e . F i g u r e 29 shows a comparison o f t h e c a l c u l a t e d the B l a s i u s p r o f i l e  f o r f l o w over  a flat  plate.  velocity  p r o f i l e and  As was expected,  agree-  ment between t h e two r e s u l t s  i s minimal.  S i n c e t h e B l a s i u s p r o f i l e i s an  exact  over  plate,  solution  expected.  f o r the flow  a flat  exact  agreement was not  The d i s c r e p a n c y i n t h e r e s u l t s can be accounted  f o r by t h e two  d e p a r t u r e s from t h e exact s o l u t i o n made by t h e c o n s e r v a t i o n o f mass model. The  first  and most s i g n i f i c a n t  s e r v a t i o n model t o observe Because  a  direction, failure The  boundary  layer  i s the f a i l u r e o f the mass  t h e requirement on  a  flat  changes i n t h e momentum o c c u r  to include this  second  departure  con-  f o r c o n s e r v a t i o n o f momentum.  plate  grows  i n the  streamwise  i n t h e streamwise d i r e c t i o n and  c o n d i t i o n i n t h e model i n t r o d u c e s l a r g e  errors.  source o f e r r o r i n t h e r e s u l t s i s due t o t h e approximate n a t u r e  27. of the assumed v e l o c i t y Results  for  profile.  the  flow  between  moving  Figure  30.  Much b e t t e r agreement w i t h  case.  In t h i s type of f l o w , the f u l l y  in  developed  the v e l o c i t y p r o f i l e i s f u l l y  momentum i n the streamwise d i r e c t i o n .  model  accurately  models  the  flow  plates  theory was  absence of a p r e s s u r e g r a d i e n t i s l i n e a r . g r a d i e n t and  flat  are  given  obtained f o r t h i s  test  v e l o c i t y p r o f i l e s i n the  A l s o , i f t h e r e i s no developed, Thus,  enabling  in  pressure  t h e r e a r e no  changes  the c o n s e r v a t i o n of mass  an  exact  solution  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 d e s i r a b l e t o compare the r e s u l t s model developed would  be  similar  for f o i l s  by E r n s t was  expected  that  the  t o those c a l c u l a t e d  with  respect  to  were  presented  the by  values  Ernst  t o those  o b t a i n e d by E r n s t [ 8 ] .  essentially results  of key  making  the  a c o n s e r v a t i o n of mass model, i t  of  the  by E r n s t .  Since  present  model would  be  very  Unfortunately, i n s u f f i c i e n t d e t a i l  parameters,  i t impossible  such to  as  the  foil  length,  duplicate his  results  u s i n g the p r e s e n t model.  3.3.1.4 The  Results c o n s e r v a t i o n of mass model was  shown i n F i g u r e 9.  Calculated results  the range of wire v e l o c i t i e s used Figure distribution results foils  31  shows  on f o i l  are only  tested.  used  the  were o b t a i n e d  i n the  variation  t o e v a l u a t e the f i v e f o r each f o i l  for f o i l  the  1,  In g e n e r a l the r e s u l t s  over  experiment. wire  velocity  1 i n the absence of w i r e d e f l e c t i o n  presented  foils  of  the  pressure  effects.  Though  similar  t r e n d s were found  for a l l  indicate  a s t r o n g tendency  f o r the  28. non-dimensional  s u c t i o n t o d e c r e a s e as the w i r e v e l o c i t y i s i n c r e a s e d .  the pressures are  converted  the s u c t i o n a c t u a l l y maintained  over  increased Thus,  a  increasing  i n c r e a s e s i n magnitude.  greater  resulting  to dimensional q u a n t i t i e s ,  l e n g t h of  the  In a d d i t i o n ,  foil  as  the  i n much h i g h e r d r a i n a g e r a t e s as  the w i r e  velocity  i t i s found  If that  the s u c t i o n i s  wire  velocity  is  shown i n F i g u r e 32.  i n c r e a s e s the o v e r a l l  suction  of  the  foil. 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 on f o i l are  g i v e n i n F i g u r e 33.  the  location  the f o i l .  of  the  The  suction  a d d i t i o n of wire d e f l e c t i o n peak t o s h i f t  deflection  effects  i s clearly  towards the t r a i l l i n g edge o f  shown i n F i g u r e 32 where the lower s u c t i o n s cause  a r e d u c t i o n i n the c a l c u l a t e d d r a i n a g e results  shown  in  Figure  33  rates. indicate  the  effects  of  the  d e f l e c t i o n a r e the most pronounced a t the h i g h e r w i r e v e l o c i t i e s . the  results  increasing  obtained the w i r e  f o r the velocity  r a t e at high wire v e l o c i t i e s stiff  causes  The d e f l e c t i o n o f the w i r e a l s o r e s u l t s i n lower peak s u c t i o n s .  T h i s tendency  The  1 i n c l u d i n g wire  wire  case,  t h e r e i s an  stiff are  wire  not  case,  obvious.  wire  Unlike  however,  the b e n e f i t s  The  i n the  drop  drainage  shown i n F i g u r e 32 suggests t h a t u n l i k e optimum w i r e v e l o c i t y which w i l l  t h e maximum d r a i n a g e r a t e o b t a i n a b l e f o r a g i v e n f o i l .  result  larger  T h i s may  the in  Thus, the d e f l e c -  t i o n o f the w i r e seems t o have a d e t r i m e n t a l e f f e c t on the performance a foil.  of  of  not be the case f o r a r e a l f o i l however, due t o the much  s u c t i o n magnitudes measured i n the experiments  than were p r e d i c t e d  by the c o n s e r v a t i o n of mass model. Examination indicate Figure  34  large  of  the  regions  indicates  velocity of  profiles  separated  there i s very  flow  little  shown  in  occur  in  effect  on  Figures the  34  to  36  calculations.  the v e l o c i t y  profiles  29. due  t o the wire  velocity.  v e l o c i t y p r o f i l e s along c a l c u l a t i o n has only Figure  36.  F i g u r e 35 shows the t y p i c a l development o f the  a foil.  I n c l u d i n g wire  a small e f f e c t  Comparing F i g u r e s  d e f l e c t i o n e f f e c t s i n the  on the v e l o c i t y p r o f i l e s as shown i n  35 and 36 shows the primary e f f e c t  wire d e f l e c t i o n i s a reduction of the separation that occurs. attributed at  t o the higher  the t r a i l i n g  end  drainage  of  o f the  T h i s c a n be  v e l o c i t i e s due t o the i n c r e a s e d  the f o i l  that  occurs  when  wire  suction  deflection i s  i n c l u d e d i n the c a l c u l a t i o n s . A  comparison  Figure  37  indicates  distributions obtained to  o f the r e s u l t s no  clear  a r e dependent  on a  solution.  i n t e g r a l model, t h e reason In  caused  rapid  (foils  by  growth  truncation solution  e r r o r s being fail  different  foils  t o say t h a t  geometry.  shown i n  the  Results  pressure  c o u l d not be  4 and 5) due t o a f a i l u r e o f COLSYS o f the momentum  f o r t h e s o l u t i o n f a i l u r e was not pursued.  terms  results,  i t was found  i n Equation  i n the magnitude  would  except  s e n s i t i v e to the f o i l - w i r e the 1/h  on  Because o f the development  o b t a i n i n g the preceeding  t i o n s were very was  trends  on the f o i l  f o r t h e stepped f o i l s  converge  obtained  introduced  completely  gap h e i g h t . 3.20.  o f the 1/h  that the c a l c u l a -  terms  sensitivity  As h approached r e s u l t e d i n very  i n t o the c a l c u l a t i o n .  when h = 0.  This  This  0, t h e large  Obviously,  made i t n e c e s s a r y  the to  s p e c i f y an' i n i t i a l gap h e i g h t b e f o r e t h e c a l c u l a t i o n s c o u l d b e g i n .  Figure  38 shows t h e s e n s i t i v i t y o f t h e r e s u l t s t o t h e i n i t i a l gap h e i g h t .  As can  be  seen,  pressure  small  changes  distribution.  i n h ^ r e s u l t e d i n l a r g e changes i n t h e o v e r a l l To m i n i m i z e  was made as s m a l l as p o s s i b l e .  effect,  the i n i t i a l  gap  height  S e l e c t i o n o f the gap h e i g h t , however, was  l i m i t e d by the n e c e s s i t y o f e n s u r i n g c a l c u l a t i o n s i n c l u d i n g wire  this  the wire  deflection.  d i d not c o n t a c t t h e f o i l i n  30. Due was  to the a r b i t r a r y  decided  i t was meaningless  edge  o f the f o i l  foil.  Instead,  and perform  the c a l c u l a t i o n  t h e a n a l y s i s over  a t the l e a d i n g  the f l a t  s e c t i o n of t h e  T h i s p o i n t corresponded  t o t h e b e g i n n i n g o f t h e t a p e r on  s t r o n g dependence 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 t h e gap h e i g h t  resulted  height  could  i n numerical result  t h a t i n some cases to  to begin  gap h e i g h t s e l e c t i o n , i t  1,2 and 3 and the l o c a t i o n o f the step on f o i l s 4 and 5. The  also  of the i n i t i a l  t h e c a l c u l a t i o n s were s t a r t e d a t t h e p o i n t where t h e f o i l  p r o f i l e changed. foils  nature  the s i t u a t i o n  i n very  problems.  Because  s m a l l changes i n t h e gap  l a r g e changes i n t h e p r e s s u r e ,  t h e model e q u a t i o n s  where a s t e p  size  were s t i f f .  s m a l l e r than  i t was  found  This condition refers the smallest value  that  can be r e p r e s e n t e d by t h e computer i s r e q u i r e d t o o b t a i n a s o l u t i o n meeting  the s p e c i f i e d  solution  generally  permitted had  error  tolerances.  failed.  a solution  Relaxing  t o be found,  t o be c o n s i d e r e d very poor.  When  this  the e r r o r  however,  situation  occurred, the  tolerances occasionally  the accuracy  of the s o l u t i o n  T h i s c o n d i t i o n was g e n e r a l l y c o n f i n e d t o  t h e a n a l y s i s o f f o i l s 4 and 5, though i t was o c c a s i o n a l l y encountered the o t h e r  3.3.2  with  foils.  The Momentum Integral Model  3.3.2.1  The Equations of Motion  D e r i v a t i o n o f t h e e q u a t i o n s o f motion f o r t h e momentum i n t e g r a l model proceed form  i n a similar  o f t h e boundary  manner layer  to the d e r i v a t i o n equations.  e q u a t i o n of motion f o r a boundary l a y e r  o f t h e momentum  The d e r i v a t i o n flow  begins  integral w i t h the  31. 3u 3u + v — 3x 3y  and  1 dp , 3 u -r~ + v p dx , 3y 2  u —  =  (3.23)  z2  the mass c o n s e r v a t i o n e q u a t i o n  Multiplying  Equation  3.24  by u  and adding  the r e s u l t  ot Equation  3.23  gives  i  v  n 3u 3 f 2u — + — (uv) 3x 3y  =  1 dp , 3^u - - -r; + v p dx  . (3.25) o c  2  w i t h the boundary c o n d i t i o n s  x = 0; p = p  x = L; p = p  a  y = 0; u = u , v = v w  To  avoid  some  conservation  o f the n u m e r i c a l  model,  Equation  y = -h; u = 0, v = 0  w  difficulties  3.25  I n t r o d u c i n g the non-dimensional  a  encountered  i s written  with  (3.26)  the mass  i n non-dimensional  form.  parameters  u L x* = — L  u* = f = — u w  Re =  v (3.27)  y  =  N  IT  h  v* = — u w  E q u a t i o n 3.25 may be w r i t t e n as  p* =  — o pu w z  32.  h> 1^  2f  dx* h> Re" TT  (  + C  S i m i l a r l y the boundary c o n d i t i o n s may  P  x* = 0; p * =  a  }  ( 3 > 2 8 )  be w r i t t e n as  x* = 1; p* =  7-  pu * w  r  (  r  P pu  a z  -T  w  (3.29) v  —  n = 0; f = 1, v* =  n = -1; f = 0, v * = 0  w  Integration Equation  3.28 w i t h r e s p e c t to n g i v e s  h L " d  £  where  2  +  (  ir  the a s t e r i s k s (*)  dropped from E q u a t i o n to perform  )  (  f  v  L= - £  )  indicating  3.30  +  x  <£><ib>€> L, 2  non-dimensional  f o r convenience.  ->  (3 30  q u a n t i t i e s have  been  L e i b n i t z ' s r u l e has been used  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 s i g n .  E v a l u a t i n g E q u a t i o n 3.30 a t t h e l i m i t s o f t h e i n t e g r a t i o n g i v e s  |- J° f2 dn + (h dx  As  f o r t h e mass  h  v  w  = - p- + (1)2 (1-)(|1) |° dx  conservation  h  model,  Re  v  w  3TI  1  _^  (3.31)  i s replaced with  the  expression  v  w  =  —  P  (p-p ) a  v  (3.32)  which may  be w r i t t e n i n n o n - d i m e n s i o n a l form as  V  Substituting  z  t  E q u a t i o n 3.33  dn  +  (%,(  R e )  w  (—)(  =  i n t o 3.31  (P-P ) a  +  R e  )(P"P ) a  gives  £  - <jj>* (fexff) I  I f a v e l o c i t y p r o f i l e of the form  f(n) = 1 +  having boundary  n = 0;  c n  C TI +  2  X  2  +  c n  conditions  f = 1  n = -1; f = 0  i£ ^ h C  )2  >  (  Re>  i ! iI l _  9 t i 2  =  0  x  i s assumed, where c., c„, and c~ are found to be  3  3  34.  dx-  c, =  [1-1  (iL-) L  k d r  (3.37a)  Re2(p-  Pfl  )]  2  (3.37b)  c„ = 2[1 - i  ( l - ) R e 2 k. L2  (p-p)]  h3. [ i ^ >  R  e  2  d r ^ a >  k  +  T2  <  Ll|  (3.37c)  c„ =  [l-i(l-)Re2k L2  d r  E q u a t i o n 3.34 may be e v a l u a t e d  210-  t  3 0 C  3  - ° 2 3  2  7  C  C  +  (p-p ) a  giving  8 4 C  1 3  "  C  1 0 5 C  3  +  4  2  C  2  "  2  k  + 140c  Equation  + 70c  2  3.38  i n t e g r a l model.  - 210c  2  1  represents  x  0  (_£|E.)Re(p-p  + 210] +  the  1  equation  5  c  i 2 C  ,  ) +g  (3.38)  o f motion f o r the momentum  B e f o r e p r o c e e d i n g w i t h a s o l u t i o n , E q u a t i o n 3.38 i s f i r s t  simplified.  Letting  *  =  210"  ( 3  ° 3 C  "  2  7 0 c  2 3 C  +  8  4  c  l 3 C  "  1 0 5 c  3  +  4  2  c  2  2  (3.39) - 105c c 1  2  + 140c  2  + 70^2  - 210  C l  +. 210]  35.  E q u a t i o n 3.38 may be w r i t t e n  |1  = -  (%)Re(p-p )  - £  a  (jj)2  +  (^)(2c  2  -  3c )  (3.40)  3  U s i n g t h e c h a i n r u l e t o e v a l u a t e 4^- g i v e s dx  d4> _ 3iJ> 3p , dx 3p 3x  Equation  x,p,  V *  =  r  t h e s u b s c r i p t s x, p, and  x  x  W  3x' 3x  i l i h 3h 3x  n  4  n  +  *p  r  Pxx  +  x  *h x  and p denote  h  ( 3  differentiation  with  '  4 2  >  respect to  respectively.  Equating for p  v  3.41 may be w r i t t e n  *x  where  3  3iJ> ,3p.  E q u a t i o n s 3.40 and 3.42 and a r r a n g i n g t o g i v e an e x p r e s s i o n  gives  p  XX  —  —  ll>  p  —  (3.43)  x  E q u a t i o n 3.43 i s t h e f i n a l form o f t h e e q u a t i o n o f motion f o r t h e momentum i n t e g r a l model.  Using  the boundary c o n d i t i o n s  p  x = 0;  p = p u  a w  2  (3.44) x = 1;  a  p = pu  2  w  t h i s e q u a t i o n may be s o l v e d f o r p as a f u n c t i o n o f x.  36.  3.3.2.2  Method o f S o l u t i o n  A s o l u t i o n t o Equation used  3.43 was o b t a i n e d  f o r t h e mass c o n s e r v a t i o n  model.  i n a manner s i m i l a r t o t h a t  The s o l u t i o n techniques  differed  only i n the r o u t i n e used t o s o l v e E q u a t i o n 3.43. Initially, The  attempt was w i t h o u t  Equation  formulated The  was made t o o b t a i n  success.  3.43 was s t i f f  capabilities  part  an attempt  To overcome  t o d e a l with s t i f f routine  used  DEBDF i s a s u b r o u t i n e of i n i t i a l  ordinary  differential  differentiation  this  problem,  value  a routine  specially  problems was used i n the p l a c e of COLSYS. This  routine  i s a v a i l a b l e as  [17].  w r i t t e n i n F o r t r a n and i n t e n d e d problems  equations.  scheme  was beyond t h e  [16] and i s d i s t r i b u t e d as p a r t o f t h e SLATEC  mathematical software l i b r a r y  solution  t o t h e problem  i s known as DEBDF.  o f t h e DEPAC r o u t i n e s  COLSYS.  I n v e s t i g a t i o n o f t h e problem i n d i c a t e d  and a s o l u t i o n  o f COLSYS.  a s o l u t i o n using  defined  by a system  f o r use i n the of f i r s t  order  The method used by DEBDF i s a backward  and i s s p e c i a l l y  formulated  to deal  with  stiff  problems. Before was  DEBDF c o u l d  be used  n e c e s s a r y t o reduce E q u a t i o n  differential  Introducing  t o o b t a i n a s o l u t i o n t o t h e problem, i t 3.43 t o a system o f f i r s t  ordinary  equations.  the dummy parameter  P =TS  (3.45)  2  Equation  order  3.43 may be w r i t t e n as t h e f i r s t  <*2>x  =  '  /  P  * 2  order  equation  <3.46)  i|> = < K P » P ) •  where  2  Unlike  COLSYS,  DEBDF  could  not  explicitly  satisfy  the  boundary  condition P  x = 1;  p  A  — p u  w  (3.47) 2  as DEBDF can o n l y be used t o o b t a i n a s o l u t i o n t o i n i t i a l v a l u e To ing  satisfy  t h i s boundary c o n d i t i o n , t h e s o l u t i o n was o b t a i n e d  a second i n i t i a l  w  a  s  a  arbitrary  n  2  been o b t a i n e d ,  condition the  given  initial  technique for  This  procedure  Convergence  to  2  A second  2  satisfied  for p  [18].  p «  of  ( P  2  guess f o r p 2  )  2  (3.48)  0  a t x = 0.  as a f u n c t i o n o f x.  Equations  3.45 and  A f t e r the s o l u t i o n  p a t x = 1 was t e s t e d t o determine i f i t s a t i s f i e d t h e  by E q u a t i o n  guess  -  2  3.46 were then s o l v e d f o r p and p had  by s p e c i f y -  c o n d i t i o n of the form  P  where ( P ) Q  problems.  the  3.47. was  If this  modified  c o n d i t i o n was not s a t i s f i e d ,  using  a  Reguli-Falsi  s o l u t i o n was then o b t a i n e d was  solution  to within a specified  repeated was  until  determined  using  the new guess  the s o l u t i o n when  iteration  Equation  converged. 3.47  e r r o r t o l e r a n c e and u s u a l l y o c c u r r e d  was in3  5 iterations. The  remainder o f the s o l u t i o n procedure, i n c l u d i n g t h e i n c o r p o r a t i o n  o f w i r e 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 t o t h a t o f t h e mass c o n s e r v a t i o n model. 28  A flow  and a flow  F i g u r e 39.  c h a r t o f t h e o v e r a l l s o l u t i o n procedure i s g i v e n i n F i g u r e chart  f o r the p r e s s u r e  evaluation  subroutine  i s given i n  38.  3.3.2.3  V e r i f i c a t i o n of the Model  Verification model  of the momentum  to evaluate  between  moving  t h e flows  parallel  integral  over  a  plates.  model was done by u s i n g t h e  flat  plate  The r e s u l t s  a t zero  o f these  i n c i d e n c e and tests  were t h e  compared t o the exact s o l u t i o n s f o r the a p p r o p r i a t e f l o w . As was t h e case f o r t h e mass c o n s e r v a t i o n model, F i g u r e 40 shows t h e calculated parallel result  solution  was expected  Figure flow  those  t h e momentum  integral  p l a t e s was i n complete agreement w i t h  the s o l u t i o n  for  using  model  f o r flow  t h e exact  and t h e t e s t was done simply  between  solution.  This  t o check f o r e r r o r s i n  procedure. 41 shows a comparison  over  a flat  of the c a l c u l a t e d  p l a t e a t zero i n c i d e n c e .  o f F i g u r e 29, i t can be seen  and exact  Comparing  solutions  the r e s u l t s to  t h e momentum i n t e g r a l model  the B l a s i u s p r o f i l e much b e t t e r than  predicts  t h e c o n s e r v a t i o n o f mass model.  An  I n t e r e s t i n g phenomena, however, was t h e dependence o f t h e s o l u t i o n on t h e boundary c o n d i t i o n s . pressure  gradient  differed.  In both c a s e s t h e s o l u t i o n was f o r a f l o w h a v i n g no  but t h e manner  F o r the f i r s t  i n which  solution,  this  the r e s u l t s  c o n d i t i o n was o f which  imposed  a r e shown i n  F i g u r e 41, t h e boundary c o n d i t i o n s were s p e c i f i e d as  x = 0;  p =  ,  (3.49)  Po = 0  pu. w  As  can be  profile  seen  from  was o b t a i n e d .  Figure  41, r e a s o n a b l e  In a d d i t i o n ,  agreement  the r e s u l t s  w i t h the c u b i c v e l o c i t y p r o f i l e g i v e n by  with  the B l a s i u s  were i n exact  agreement  39. f(n) = 1 + | n - i n  Examination slight  adverse  thickness  3  (3.50)  of the pressure d i s t r i b u t i o n along the p l a t e i n d i c a t e d a p r e s s u r e g r a d i e n t , thus e x p l a i n i n g t h e l a r g e r  p r e d i c t e d by t h e n u m e r i c a l  s o l u t i o n procedure  t o determine  solution.  displacement  Closer scrutiny  o f the  t h e cause o f t h e adverse p r e s s u r e g r a d i e n t  i n d i c a t e d the problem use due t o the use of the c o n d i t i o n  v  to  relate  w  k = (-^)(Re)(p-p )  (3.51)  a  t h e c r o s s stream v e l o c t i y  t o the pressure.  For the solution to  f l o w over a f l a t p l a t e , k ^ was s e t e q u a l t o z e r o , thus e s s e n t i a l l y ing more  a solid like  boundary  an i n t e r n a l  a t TI = 0. flow  T h i s had t h e e f f e c t  having  a p a r a b o l i c shaped  impos-  o f making t h e f l o w velocity  t h e w a l l r e s u l t e d i n an adverse  profile.  The  lower v e l o c i t i e s near  ent  t o conserve  The  n e t r e s u l t was a much more r a p i d growth o f the boundary l a y e r than i s  momentum, c a u s i n g  g i v e n by the exact  the flow  t o be f u r t h e r  pressure g r a d i deaccelerated.  solution.  When a s o l u t i o n t o t h e f l o w over a f l a t  p l a t e was o b t a i n e d u s i n g t h e  boundary c o n d i t i o n s  x = 0;  p = pu  w  (3.52) x = 1;  p = PU  w  40.  much b e t t e r r e s u l t s were o b t a i n e d as shown i n F i g u r e 42. the  boundary  conditions  s o l u t i o n t o develop occurred the  i n this  an adverse  manner  pressure  constrained gradient.  was  due  t o the approximate  nature  any tendency  of the  While some e r r o r  i n t h e s o l u t i o n due t o t h e u s e o f E q u a t i o n  error  S p e c i f i c a t i o n of  still  3.51, t h e m a j o r i t y o f  o f t h e assumed  velocity  profile. As  a f i n a l v e r i f i c a t i o n o f t h e momentum i n t e g r a l model an attempt t o  duplicate  the r e s u l t s  assumed f u l l y  of Taylor  t u r b u l e n t flow  [3] was made.  In h i s s o l u t i o n ,  i n which t h e v e l o c i t y p r o f i l e  Taylor  c o u l d be g i v e n  by =iL_  f  where u was a f u n c t i o n o f x o n l y . by  Equation  should  have  t h e above  duplicated  the r e s u l t s  difficulties  necessary  t o impose an a r b i t r a r y  be d i s c u s s e d  highly  due  profile,  the momentum  obtained  t o terms  i n t h e next  w  By r e p l a c i n g t h e v e l o c i t y p r o f i l e  3.35 w i t h  numerical  will  u  by T a y l o r .  f o r 1/h  gap between t h e f o i l  gap h e i g h t ,  the r e s u l t  lower s u c t i o n s than those c a l c u l a t e d by T a y l o r .  3.3.2.4 The  i n Equation  model  Unfortunately, 3.43 made i t  and t h e w i r e .  s e c t i o n , the s o l u t i o n of Equation  sensitive to this i n i t i a l  present  integral  given  As  3.43 was  o f which was much  Thus, a comparison o f t h e  r e s u l t s t o those o f T a y l o r was meaningless.  Results momentum  integral  distributions  on f o i l s  for  each f o i l  using  and  drainage  model  was  used  to c a l c u l a t e  1 t o 5 g i v e n i n F i g u r e 9.  identical  c o n d i t i o n s f o r wire  the  pressure  Program runs were done v e l o c i t y , wire  r e s i s t a n c e as were used i n the experiments.  tension  41. In  general,  the  pressure  i n t e g r a l model were v e r y t i o n o f mass model.  distributions  similar  t o those  predicted  by  calculated using  F i g u r e 43 shows the r e s u l t s o b t a i n e d  no w i r e d e f l e c t i o n .  the momentum the  conserva-  for f o i l  1 with  As b e f o r e , i n c r e a s i n g the w i r e v e l o c i t y i n c r e a s e d t h e  peak s u c t i o n . Figure different  44  shows  foils  a similar  trend  i n the p r e s s u r e  distributions for  as t h a t p r e d i c t e d by the mass c o n s e r v a t i o n model.  the mass c o n s e r v a t i o n model, however, s o l u t i o n s were o b t a i n e d and  5,  though the p r e s s u r e  distributions  were  essentially  Unlike  for foils 4  zero  f o r each  foil. Comparing the r e s u l t s  of Figure  43 t o those  o f F i g u r e 45, i t can be  seen t h a t i n c l u d i n g w i r e d e f l e c t i o n i n t h e c a l c u l a t i o n has t h e same e f f e c t as was observed f o r t h e c o n s e r v a t i o n o f mass model.  As b e f o r e ,  there i s a  tendency f o r t h e magnitude of the s u c t i o n t o d e c r e a s e and the l o c a t i o n o f t h e maximum s u c t i o n t o s h i f t trend i s c l e a r l y  towards the t r a i l i n g  edge o f the f o i l .  This  shown i n F i g u r e 46.  An unexpected e f f e c t o f i n c l u d i n g t h e w i r e 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 t h e tendency f o r t h e f l o w t o s e p a r a t e .  From F i g u r e 47 i t can be  s e e n t h a t flow s e p a r a t i o n was o n l y p r e d i c t e d a t the h i g h e r w i r e when w i r e d e f l e c t i o n was not i n c l u d e d . s e p a r a t i o n was This  predicted f o r a l l wire  i s probably  deflection  When w i r e d e f l e c t i o n was velocities,  as shown i n F i g u r e 48.  are included. causing  Lower  suction  results  being  drawn through the w i r e  occur  due to the expansion of the gap between the f o i l  equations,  i n less  necessary  fluid  a g r e a t e r tendency f o r s e p a r a t i o n t o and the w i r e .  the momentum i n t e g r a l model had terms f o r 1/h  i t was  included,  due t o the lower s u c t i o n magnitudes p r e d i c t e d when w i r e  effects  Because  velocities  t o s p e c i f y an i n i t i a l  i n the model  gap h e i g h t as was  done  42.  for  t h e mass c o n s e r v a t i o n  solution has  on t h e i n i t i a l  model.  Figure  gap h e i g h t .  49 shows t h e dependence o f the  The momentum  i n t e g r a l model c l e a r l y  a s t r o n g dependence on t h e i n i t i a l gap h e i g h t . The momentum i n t e g r a l model was a l s o used t o determine t h e e f f e c t s o f  other  parameters i n t h e model e q u a t i o n s  bution.  Figure  solution. maximum  s u c t i o n decreases.  This  f o r the drainage  o f k, dr  pressure the  the e f f e c t  i s increased,  d i f f e r e n t i a l causing  wire  o f the d r a i n a g e  As t h e magnitude o f t h e d r a i n a g e  relationship value  50 shows  on t h e p r e d i c t e d p r e s s u r e  into  the flow  trend  velocity  r e s i s t a n c e on t h e  r e s i s t a n c e i s increased, the  c a n be e x p l a i n e d given  w  3.33.  As t h e  i s increased f o r a given  g r e a t e r amount o f f l u i d Normally  by examining t h e  by E q u a t i o n  t h e magnitude o f v  region.  distri-  this  t o be drawn through  trend  would  cause  i n c r e a s e i n t h e streamwise v e l o c i t y o f t h e f l u i d w i t h a c o r r e s p o n d i n g in  the pressure.  that  p = p  This  an  drop  tendency, however, i s a r r e s t e d by the requirement  at the t r a i l i n g  end o f t h e f o i l .  The n e t r e s u l t i s a g r e a t e r  SL  o v e r a l l drainage Figure varied. occurs,  r a t e as shown i n F i g u r e 51.  52 shows t y p i c a l  results  obtained  thus t h e r e s u l t  i s similar  to that obtained  when wire d e f l e c t i o n  F i g u r e 53 i n d i c a t e s t h a t i n c r e a s i n g t h e w i r e t e n s i o n has  the b e n e f i c i a l e f f e c t o f i n c r e a s i n g t h e d r a i n a g e s u c t i o n r e s u l t i n g from the s t i f f e r t r y and g a i n some i n s i g h t  r a t e due t o t h e i n c r e a s e d  wire. i n t o the p o s s i b l e e f f e c t s of performing  t h e c a l c u l a t i o n s f o r t u r b u l e n t f l o w s , r e s u l t s were o b t a i n e d fluid has  viscosities.  As can be seen i n F i g u r e  a substantial effect  these  results  how  If this  suction could  on t h e r e s u l t s .  turbulence  t h e r e would probably lence.  t e n s i o n was  I n c r e a s i n g t h e w i r e t e n s i o n reduced t h e amount o f d e f l e c t i o n t h a t  i s not included.  To  when t h e w i r e  may  be a s l i g h t  were  increase.  t h e case,  effect  f o r a range o f  54, changing t h e v i s c o s i t y  It i s d i f f i c u l t the r e s u l t s  increase i t could  to i n f e r  from  t o note  that  except  i n t h e v i s c o s i t y due t o t u r b u be expected  that  the maximum  In a d d i t i o n t h e l o c a t i o n o f the maximum  suction  43.  may be s h i f t e d At  the very  performing An  downstream from t h e l o c a t i o n  least,  results  indicate  there  may  flows.  be some m e r i t  to  the analysis f o r turbulent flows.  examination  pressure  these  p r e d i c t e d f o r laminar  was  o f the i n f l u e n c e o f t h e boundary  made.  The r e s u l t s  obtained  c o n d i t i o n s on t h e  are presented  55.  i n Figure  F i g u r e 55 shows t h e v a r i a t i o n i n t h e p r e s s u r e i s d i m i n i s h e d by s p e c i f y i n g sub-atmospheric  inlet  increased.  In  encountered  due  solution  and e x i t  obtaining  these  to s t i f f n e s s  could only  pressures, results,  be o b t a i n e d  the o v e r a l l numerical  equations.  solution.  distribution  chosen,  c o u l d be o b t a i n e d .  a solution  drainage  velocity,  i n f l u e n c e d by the drainage  In g e n e r a l ,  a  was v e r y guess  I f t h e guess was  In some c a s e s , i f t h e  having  a positive  Because t h e s p e c i f i c a t i o n  s p h e r i c boundary c o n d i t i o n s f o r t h e p r e s s u r e initial  were  i n c r e a s e along the length of the f o i l  t h e s o l u t i o n would once a g a i n f a i l t o converge. guess was p o o r l y  problems  I f the i n i t i a l  was t o o s m a l l , t h e s o l u t i o n would n o t converge.  initial  suction i s  when t h e i n i t i a l guess f o r  t o o l a r g e , t h e s u c t i o n would r a p i d l y and  some  o f the model  c l o s e t o t h e v a l u e t h a t gave a convergent for  though  pressure  o f sub- atmo-  i s e q u i v a l e n t t o imposing  t h e s e r e s u l t s suggest  an  the s o l u t i o n i s strongly  velocity.  F i n a l l y , an attempt was made t o o b t a i n improved r e s u l t s by p e r f o r m i n g the a n a l y s i s i n the opposite d i r e c t i o n . except  the s o l u t i o n  was  started  The s o l u t i o n proceeded as b e f o r e  a t the t r a i l i n g  edge  o f the f o i l .  No  improvement i n t h e r e s u l t s was o b t a i n e d , however.  T h i s t e c h n i q u e was o n l y  successful  when  i n duplicating  the r e s u l t s  obtained  the s o l u t i o n  was  s t a r t e d a t the l e a d i n g edge of the f o i l . Summarizing t h e r e s u l t s o f t h e momentum i n t e g r a l model, i t was the  solution  gap  specified,  general, reported  o f the model was h i g h l y dependent as w e l l as the p r e s s u r e  the maximum by T a y l o r  predicted  [3],  or Ernst  boundary  suctions [8].  on t h e i n i t i a l  were  foil-wire  c o n d i t i o n s chosen. much  lower  found  than  In those  Both T a y l o r and E r n s t p r e d i c t e d  44.  maximum  s u c t i o n s of  a b o u t 4- pu  the e x p e r i m e n t a l  as a f u n c t i o n of the wire 3.4  t r e n d s observed  results  do,  however,  f o r the p r e s s u r e  distributions  velocity.  two-dimensional  model  motion f o r a boundary l a y e r  U  i n v o l v e d the  solution  3u 3y  +  of  the  equation  of  flow  ——  3u , 9x  1 dp , rr~ p dx  + V -r— =  9u 2  (3.53)  V  . o  9y^  the e q u a t i o n f o r mass c o n s e r v a t i o n  i n f i n i t e d i f f e r e n c e form.  The  relationship  w  - ^ < P - P « >  V  used f o r the one-dimensional  the p r e s s u r e Finite 3.55  present  General The  was  The  The Two-Dimensional Model  3.4.1  and  .  w  2  duplicate  2  a  between the  3  models t o r e l a t e the d r a i n a g e v e l o c i t y t o  approximations  non-uniform  foil  and  the  were w r i t t e n f o r E q u a t i o n s  rectangular forming  wire.  mesh The  fitted  to  resulting  the  flow  finite  model.  deflection  T h i s was  effects  primarily  due  were  not  included  t o the d i f f i c u l t i e s  in  the  3.53  to  region  difference  e q u a t i o n s were s o l v e d t o o b t a i n 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 the Wire  55  distibution. difference  using  < ' >  foil.  two-dimensional  of a d a p t i n g the  finite  45. d i f f e r e n c e mesh to the a  solution.  changing w i r e shape d u r i n g  Some c o n s i d e r a t i o n  t e c h n i q u e s but  was  not  was  pursued due  given to the  to  the  c o u r s e of  obtaining  i n v e s t i g a t i n g a d a p t i v e mesh  poor r e s u l t s o b t a i n e d u s i n g  the  two-dimensional model.  3.4.2  The  F i n i t e Difference  Before Equations  finite  3.53  to  s e l e c t i n g the implicit able  difference 3.55,  a  approximations  differencing  d i f f e r e n c i n g scheme, i t was  one  to  ensure  stability  of  the  could  scheme had  shape  of  d e s i r e d as i t was a new  finite After  the  [20].  flow  region.  solution.  Box  implicit has  Method  analysis feature could  of of  selected.  In  the method be  an  scheme had  be  be  to  f i t t i n g o f a mesh t o  Finally,  a  proven  technique  the was  Method  was  d i f f e r e n c e schemes, a procedure known  selected.  scheme d e s c r i b e d  been used  to  obtain  by  has  also  mildly the Box  used  separated Method as  occur over the  F e a t u r e s of the Box  been  to  flows. i t was  The  Keller  solutions  which l e n d themselves t o d e s c r i p t i o n by Box  The  for  beyond the scope of the p r e s e n t i n v e s t i g a t i o n t o p i o n e e r  examining s e v e r a l f i n i t e  It  to  developed  d i f f e r e n c i n g method.  Keller  developed  the  be  necessary that  t o s u p p o r t non-uniform meshes t o f a c i l i t a t e  irregular  as  Scheme  Box [19] to  Method and  a  recently  K e l l e r and  Cebeci  a wide v a r i e t y o f  flows  the boundary l a y e r e q u a t i o n s .  varying This  degrees was  of  success  considered  a n t i c i p a t e d that  the  important  some f l o w  separation  include: - unconditionally  for  The  an  foil.  Method  is  1.  i m p l i c i t formulation  stable;  2.  second order a c c u r a c y ;  3.  s o l u t i o n on a r b i t r a r y (non-uniform) mesh p o s s i b l e :  46. 4.  a l l o w s v e r y r a p i d 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 t o use f o r the p r e s e n t  3.4.3  The F i n i t e  Difference  investigation.  Equations  Use o f t h e K e l l e r Box Method i n v o l v e d f o u r p r i n c i p l e s t e p s : 1.  Reduce  the  equations  of  motion  to  a  system  of  first  order  equations; 2.  Write d i f f e r e n c e equations using centered d i f f e r e n c e s ;  3.  L i n e a r i z e the r e s u l t i n g  4.  S o l v e the r e s u l t i n g system o f l i n e a r a l g e b r a i c e q u a t i o n s .  system of equations  ( i f non-linear); i  B e g i n n i n g w i t h the f i r s t non-dimensional  step, Equations  3.1 and 3.2 a r e f i r s t w r i t t e n i n  form  au*  Sv*  where ju x X* = — L  . u u* = u w  . p p* = —— pu w  2  (3.59) y* =  and  T "  L  /Re  v  *  =  — /Re u w  Re i s the Reynolds number based on the f o i l  length  u L Re = - 2 -  (3.60)  47. In  E q u a t i o n 3 . 5 7 , b i s a n o n - d i m e n s i o n a l v i s c o s i t y term o f t h e form  v t  b  and  v  flows,  i s a turbulent v  = 1 + ~  (3.61)  v i s c o s i t y term which i s e q u a l t o z e r o f o r l a m i n a r  was i n c l u d e d i n t h e development  o f t h e two-dimensional model t o  permit l a t e r expansion o f the model t o i n c l u d e t u r b u l e n t f l o w s . Continuing with the f i r s t  s t e p , i f f i s d e f i n e d such t h a t  (3.62)  f=-g  E q u a t i o n s 3 . 5 7 and 3 . 5 8 may be w r i t t e n  f  -  C3.63)  ".65,  where t h e a s t e r i s k s have been dropped f o r convenience. first  T h i s completes t h e  step of the K e l l e r Box Method. E q u a t i o n s 3 . 6 3 t o 3 . 6 5 cannot be s o l v e d d i r e c t l y i n t h e g i v e n form as  there used  are four  unknowns, u,v,p, and f .  t o overcome  this  problem.  Two methods have g e n e r a l l y  The f i r s t  method,  referred  been  to i n the  l i t e r a t u r e as t h e E i g e n v a l u e method [21] i n v o l v e s an i t e r a t i v e s o l u t i o n o f the  system  of equations.  The s o l u t i o n  proceeds  by assuming  d i s t r i b u t i o n and then s o l v i n g E q u a t i o n s 3 . 6 3 through 3 . 6 5 .  a pressure  Results of the  48. s o l u t i o n a r e then used t o improve t h e guess f o r t h e p r e s s u r e 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 iterated  until  the pressure  distribution  method, known as t h e Mechul f u n c t i o n method  [21], t r e a t s  converges. The  second  t h e p r e s s u r e as an unknown and i n t r o d u c e s a f o u r t h e q u a t i o n  ^ - 0  The  solution  of Equations  (3.66)  3 . 6 3 through  3 . 6 6 may  then  be c a r r i e d out  directly. Though using  i s generally l i t t l e  the two methods,  allowing fies  there  a direct  s o l u t i o n without  i n the r e s u l t s  obtained  f u n c t i o n method has the advantage o f any i t e r a t i o n  required.  This  simpli-  t h e programming o f t h e method a t t h e expense o f one more e q u a t i o n t o  solve. problems  The Mechul f u n c t i o n method has a l s o involving  separated  f u n c t i o n method was used of  t h e Mechul  difference  t h e Box method,  flows  f o r the present  the equations  f i r s t order d i f f e r e n t i a l  [21].  been s u c c e s s f u l l y  F o r these problem.  reasons,  applied to t h e Mechul  To summarize s t e p one  o f motion a r e w r i t t e n as t h e system o f  equations  f = | f  (3.67)  (3.70)  49. h a v i n g t h e boundary  conditions  k  v y = 0;  u = 1,  v -  /Re = (-£-•) ( p - p ) (Re) 3/2 a  w (3.71) y = - — /Re;  In are  step  written  u = 0, v = 0  two o f the Box method, E q u a t i o n s 3.67, in  finite  difference  form  using  3.68,  centered  D i f f e r e n c i n g was done on t h e g r i d shown i n F i g u r e 56. was  selected  such t h a t mesh nodes f e l l  e q u a t i o n s were w r i t t e n in  both  Figure  such  that  on the f o i l  equations  A l l other  3.70  differences.  Spacing o f the mesh  boundary.  Difference  involving p a r t i a l  derivatives  the x and y d i r e c t i o n s were c e n t e r e d  56.  3.69 and  e q u a t i o n s were centered  on (P^ ,P ,P ,P 3  on  ) shown i n  (P ,P ). 1  2  D e f i n i n g the n o t a t i o n  i-1/2 8j  1 , i . = 2 ^ 3  i-U i  +  S  >  8  i j-l/2  = 2 (gj +  8^) (3.72)  , i  where  g  i-1.  W j  k.  i s any  quantity,  k  dy j  h.  ;  the d i f f e r e n c e  equations  f o r Equations  3.67  through 3.70 can be w r i t t e n  _i f  j-l/2  1 " hj  , i (  u  i  J " J-1> U  .  (3.73)  50. . 1-1/2, l ' V l ^ k T  ,i i-1 ( J-l/2 " j - l / 2 ^  r  u  =-k 1 ^  U  (pi-l/2 - p5hi/2>  , i i-1 s . 1 < j-i/2 " j-i/2> hT u  k~  (  +  u  i " p Li  p  J  )  (  v  +  ( v f )  ¥7 ICbf)J"  v  l  vf  2k7 ( 5-l/ ) U  2  2  "  (3.74)  2  ,„  n 0  ( 3  *  7 5 )  -  ( 3  /  2  hand s i d e  (3.77)  +  1  > £ / 2+  7 6 )  gives  - 0  +1 <>5-l/2 U7 (Pj-l/2>  2  =  1  °  =  \ - („i - u ^ ) - f i _  ^-l/2>  }  " (bf)^: / ]  1 / 2  , i-1/2 i-1/2. j " j - i  C o l l e c t i n g a l l terms i n ( i ) on the l e f t  7  M/2 j-l/2 n  +  ¥  ±  " 2ET  < P £ / 2 >  +  [  (bf)  5  "  (  b  2HT [< > j " ' bf  f  (  j - J  )  b  ^  f  (3.78)  , , 1 •i i . k7 < J-l/2 2hT J " j - 1 t  1  1  u  ir  (  p  ) +  i ~ i - i p  (  ) =  0  V  V  } =  1  , i-1 .  \ <*}-l/2>  1 , i-1 i - l " 2hT J " ^-P (  V  N  ,~ ' (  (  11\\  3  3  -  7  8  9  0  )  )  51. The next Equations  s t e p o f t h e K e l l e r Box Method r e q u i r e s t h e l i n e a r i z a t i o n o f  3.77  3.80.  through  i t e r a t i o n technique.  Linearization  was done  using  a  Newton  D e f i n i n g the a r b i t r a r y q u a n t i t y g, the v a l u e o f g a t  the advanced step (n+1) may be found u s i n g  ( J)  n + 1  g  -  (g^)  n  + (6g^) ^  (3.81)  n  Using the n o t a t i o n of Equation 3.81,  Equations  3.77 through  3.80 may  be w r i t t e n a t the advanced s t e p (n+1)  [(u + 6u)* - (u.+  2k- [ < " 5 - l / 2  + ^  )2 +  " (f +  6  < S-l/2>< S-l/2>] 1 2u  (P + <SP>j_  1/2  6u  +  f  )j-i/  [  ( V+  2  =  6 v )  < * >  0  3  j-l/2  " 2 n ^ Ub(f + 6£>5] - [ b ( f +  <  f  +  6 f  82  )j-l/ ] 2  6 ^ ] }  = 4 ^ i / 2 - \ ^>5hi k[ 5-i/ > ib r<>r - < >£] +  (p  /2  +2  M  bf  2  (3.83)  i -  (u + 6 u ) 5 _  = ^< 5-i/2> u  1 / 2  =  +  6v)* - (v + S v ) ^ ]  2KT [(v +  2hT[ r- 5-i3 v  v  ( 3  -  8 4 )  52.  JT [(P + P ) i - (P P)i-i] = 6  +  6  where t h e s u p e r s c r i p t that  (- )  0  3  85  (n) has been dropped f o r convenience and a l l terms  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 ) a r e known q u a n t i t i e s . Collecting  a l l terms  i n <5uj, "Sv*, ^ j» f  a n c  *  v  i  e  lds  the l i n e a r  system o f e q u a t i o n s  6u*  -  S u ^ - - | h tt  1  - -j h 8 f *  -  (r^*  (3.86)  (3.87)  k i i i 1 6u. + fiu. . + ±8v - ±. J j-1 h j :L  T  6 P  J  "  6 P  J"1  =  (  r  x  k  i h  i 1 6V. . = (r„) . j-1 3'j  (3.88) '  ^ j  ( 3  *  8 9 )  where  'Vj v r j j 4-1/2 =  u  +h  + £j[0>f>} - ( b f ) ^ ]  < - > 3  + R £  /  2  90  (3.91)  53.  = 0  (r. ) * = -p* + p* . J J J-1  5-i/ = ^ <>5-i/2  R  u 2  2  (3.93)  -  (vf)  5-1/2  h [< >r bf  +  (3  -  94)  and  /  \  i 1 ^ =  1 U  J  1  vi  , ( S  6 J }  =  2  f  - i  j  (3.95)  (s )  / (  S  To  =I v  1  \ ^ j  i =  1  1 i 2 j-1 V  b  1  -L-  -  I j - l n — j b  +  -  complete s t e p  3,  (s )! =  L  x i  1  , ( 8  9>J  =  k" i  the boundary  conditions  given  by E q u a t i o n  3.71  are w r i t t e n i n the form  y = 0 ( j = 0);  6u* = 0,  6v* = 0 (3.96)  y = ^  /Re  (j - J);  6uJ = 0,  6v* = 0  54.  T h i s simply s t a t e s t h a t u and v a r e known a t both t h e f o i l and t h e f o r m i n g wire.  Since v  w  i s n o t e x p l i c i t l y known, but i n s t e a d i s g i v e n by e q u a t i o n  3.55, the s o l u t i o n must be i t e r a t e d condition i s s a t i s f i e d .  a t each x node t o ensure t h i s boundary  T h i s w i l l be d i s c u s s e d f u r t h e r i n S e c t i o n 3.4.4.  In the f o u r t h and f i n a l s t e p o f the Box method, E q u a t i o n s  3.86, 3.87,  3.88, and 3.89 a r e w r i t t e n i n the m a t r i x v e c t o r form  0 < i < I  In E q u a t i o n 3.97, A. i s t h e c o e f f i c i e n t  0 < j < J  (3.97)  matrix  -i - i a c ri - i - i b . a. c . J J J  0 < j < J  where  (3.98)  55. o l 0  0 0 -\I2  < 6>*  <v5  1 0 -1  -i a o  8  6  J  o 0 0  1 -i a  j  =  k l  /h.  0  1  0  0  0  1  1 0  0 1  0 0  0 0  0 0 0  0 -1  -i a. J  0  1 0  1 < j < J-l  (3.99) 0 b  o 0  1  J  0 0  0 0  -1  1 < j <J  0 0  0 0  I-I  -i c. J  0 0  (  '1> J« 1  0 0  0 0  0  o < j < J-l  0 <7 S  X* i s the v e c t o r J  _i x o  X  1  J  =  _i x. J  6u  J  6v  3"  o < j <J Sf  1  J  "5  (3.100)  56. and  i s the v e c t o r  J  b  B  1  J  b  =  1  o  1  b  J  = < l>o r  *1  < 2>o r  < 3>i-i r  1 < j < J-1  (3.101)  < l> j+1 r  ( r  The  first  ( 2>J  2>j+l  r  and l a s t two rows o f A^ r e p r e s e n t the boundary c o n d i t i o n s g i v e n  by J = °J  6 u  6V  o = < 3>o " ° r  1  o  = (r.) =0 'to i  (3.102) j = J;  6uJ -  (  r  i  ) J =0  6  V  J  = ( 2>J " r  0  E q u a t i o n 3.97 r e p r e s e n t s a system of l i n e a r a l g e b r a i c e q u a t i o n s vector  form.  A solution  t o 3.97 may be o b t a i n e d  method f o r s o l v i n g systems o f l i n e a r  3.4.4  Method o f  relatively  having  constant  by any c o n v e n t i o n a l  equations.  Solution  The procedure was  i n matrix  used t o o b t a i n a s o l u t i o n f o r t h e two d i m e n s i o n a l model  straight  forward.  Before  beginning  the solution,  s p a c i n g i n the x - d i r e c t i o n was c a l c u l a t e d .  a mesh  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  l o c a t e d a t both  such  the forming w i r e and  t h a t f o r each x node, a y node the  was  foil.  The s o l u t i o n began by assuming i n i t i a l c o n d i t i o n s f o r u,v,p, and f o f the  form u° = J  U. J  v° - V. 0 < j < J f° J  - F .  j  P° = J  Using was  these  solved  marched  at  from  initial  each  the  (3.103)  P. J  c o n d i t i o n s as  node  in  leading to  the the  the  starting  x-direction. trailing  node, the s o l u t i o n of E q u a t i o n 3.97  point, Equation  Thus,  edge  of  the  the foil.  3.97  solution At  was  each  x  y i e l d e d values f o r 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 E q u a t i o n 3.97 As was  was  Equation  3.97  E q u a t i o n 3.97. used  guess f o r the s o l u t i o n of  mentioned i n S e c t i o n 3.4.3, an i t e r a t i o n was the c o n d i t i o n g i v e n by E q u a t i o n 3.55. had  computed v a l u e of v of  the i n i t i a l  at the next node i n the x - d i r e c t i o n .  x node t o s a t i s f y to  as  to  w  been o b t a i n e d , was  3.55  was  After a  an  v a l u e s were not  updated  value  solution  e v a l u a t e d and  compared t o the r e s u l t o b t a i n e d from the  I f the two  calculate  Equation  r e q u i r e d a t each  solution  i n agreement, E q u a t i o n  for p using  the  the v a l u e  of  3.55 v  58.  obtained  from  solution  to Equation  until  the  the  solution  boundary  c o u l d proceed  3.97  of  Equation  was  3.97.  calculated.  c o n d i t i o n was  two  the  procedure  a t which  point  this,  a  new  was  repeated  the  solution  dimensional  i s g i v e n i n F i g u r e 57.  Model  model was  for  flow over a f l a t  and  flow between moving p a r a l l e l  initial  done  to the next x node.  V e r i f i c a t i o n of  The  This  satisfied  A f l o w c h a r t of the s o l u t i o n procedure  3.4.5  Having  verified  a g a i n s t the  exact  solutions  p l a t e a t z e r o i n c i d e n c e , flow i n a r e c t a n g u l a r d u c t , flat  plates.  For a l l t h r e e t e s t  cases,  c o n d i t i o n s were g i v e as  o u.  =  1  o  =  0  J  V .  J  (3.104)  0 < j < J  f.  =  0  o w  A  s o l u t i o n was  internal proceed  duct  then o b t a i n e d  f o r each of the  flow  couette  u n t i l i t was  From obtained  and  the  fully  F i g u r e s 58, f o r a l l three  59  flow,  flows.  the  In  solution  the case was  of  allowed  the to  developed. and  60  flows.  layer,  some compromises had  finite  difference  equations  i t can In  the  be  seen  case  of  very the  good agreement flat  plate  t o be made t o o b t a i n a s o l u t i o n . were f o r m u l a t e d  for internal  was  boundary  Because the  flows  i n which  59. the  pressure  necessary.  i s unknown,  To  an  adaptation  boundary  conditions  was  relation-  the p r e s s u r e g i v e n by E q u a t i o n  3.55  s u b s t i t u t e d w i t h the r e l a t i o n h s h i p  Idx  s o l u t i o n was  Because exact  this  iterated  procedure  agreement  however,  then  was  with  very  small  additional  Blasius profile and  considered  (3.105) '  v  a t each x node to s a t i s f y  introduced  the  u42dx  =  2  The  the  s o l v e f o r t h e f l a t p l a t e boundary l a y e r f l o w , t h e  s h i p between the d r a i n a g e v e l o c i t y and was  of  was  error not  in  this condition. the  obtained.  insignificant  as  calculation, The  error,  the d i f f e r e n c e  e q u a t i o n s were not i n t e n d e d f o r use on e x t e r n a l f l o w s . F o r the i n t e r n a l duct  flow and  the c o u e t t e f l o w , e x c e l l e n t agreement  w i t h the e x a c t r e s u l t s were o b t a i n e d . was  obtained  the  e x c e l l e n t agreement a c h i e v e d ,  results.  w i t h i n 2 percent  For the duct  In both c a s e s , f u l l y developed  of the l e n g t h p r e d i c t e d by  some minor problems can  f l o w , shown i n F i g u r e 59,  f l o w , the r e s u l t s f o r which a r e g i v e n i n F i g u r e 60, a l l cross  s m a l l flow  the  reversal  centre  can  be  line  a t the  s m a l l d i s c r e p a n c i e s can be a t t r i b u t e d ical  instability  proceeds cases  x  i n the  couette  i t s growth  problem caused foils.  the  a slight  i s damped, as was flow  i s stopped,  f term  observed  i t i s not  For the  couette profiles  In both f l o w s , by  begins  a  these  a s l i g h t numerAs the  solution  in f.  In most  i n the case of the  i s only damped t o the completely  can  Additionally,  (f = 4 — ) ' dy  oscillation  the o s c i l l a t i o n but  wall.  t o e r r o r s caused  w a l l i n the  direction,  the o s c i l l a t i o n  flow. that  i n the  at  seen i n the  the v e l o c i t y  same l o c a t i o n .  seen a t the lower  be  Despite  a slight oscillation  be seen a t the w a l l s i n the d e v e l o p i n g v e l o c i t y p r o f i l e s .  should  theory.  flow  duct  extent  eliminated.  a g r e a t d e a l o f t r o u b l e i n the s o l u t i o n s f o r the f l o w  This over  60. 3.4.6  Results  The very  results  obtained  disappointing.  using  Attempts  the two-dimensional  to  obtain results  speeds used i n the e x p e r i m e n t a l t e s t s met As have  can be  very  seen  little  from  similarity  d i m e n s i o n a l models. obtained.  F i g u r e 61, to  the  results  obtained  permited  e v a l u a t i o n of  foil  2.  the  Use  flows  and  wire  success. distributions  using  the  one-  2, 3 and 4, r e s u l t s c o u l d not  p r o f i l e s i n d i c a t e d flow s e p a r a t i o n occurred. for  foils  the p r e d i c t e d p r e s s u r e  In each of these c a s e s , examination  obtained  f o r the  w i t h o n l y minimal  In the case o f f o i l s  profiles  model were g e n e r a l l y  of  beyond  of the c a l c u l a t e d  be  velocity  F i g u r e 62 shows the v e l o c i t y  the the  Mechul p o i n t of  e x t e n t , however, the f l o w s e p a r a t i o n u l t i m a t e l y  function  technique  s e p a r a t i o n t o some  l e d t o the f a i l u r e of the  solutions. Figure for  a  63  range  presents  of  wire  better  agreement  trends  in  the  the  calculated  velocities.  with  the  pressure  pressure  As  can  be  of  the  other  results  distributions  as  seen,  the  F i g u r e 6A,_however, i n d i c a t e s s e p a r a t i o n was  distributions these  models,  wire  on  foil  results showing  velocity  is  are  1 in  similar changed.  p r e d i c t e d a t the h i g h e r  wire  velocities. A significant  p o i n t of i n t e r e s t  i n the r e s u l t s  model t o meet the c o n d i t i o n t h a t a t x = L, p = p . &  to  the  p a r a b o l i c nature  of  the  boundary  layer  i s the f a i l u r e o f the T h i s l i m i t a t i o n i s due  equations.  Attempts were  made t o i t e r a t e the s o l u t i o n t o s a t i s f y the end boundary c o n d i t i o n but solution  was  highly unstable  i n the  second  iteration  and  failed  the  in a l l  attempts. In could  o b t a i n i n g the r e s u l t s p r e s e n t e d ,  o f t e n be  o b t a i n e d by  s p e c i f y i n g an  i t was  discovered that a solution  initial  drainage  velocity  along  61. the  length  of  the  foil.  presented i n Figures drainage effect  of  excessive,  onset  the  I t was  suction  drawing more f l u i d the  t e c h n i q u e was  61 t o 64.  velocity, larger  thus delaying was  This  of  peaks were the  separation. of  to  found t h a t by  through  magnitude  used  the  results  s p e c i f y i n g an  initial  calculated.  forming w i r e and If  the  obtain  the  suction  initial would  This  had  i n t o the  the flow,  drainage v e l o c i t y  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 results  a  final  point  presented,  of  i n t e r e s t , i t should  c a l c u l a t i o n s were s t a r t e d  foil.  The  s o l u t i o n procedure  height  and  the  Section  3.4.4.  initial  gap  suction  initial As  height  was had  developed.  (Figures  velocities  61  the a  could  the  l e n g t h of the f o i l . growth of  the  v e l o c i t y was eventually  started  case  and for  significant  Unlike  and  was  conditions,  maximum s u c t i o n o c c u r r e d foil  be  at  63).  the the  l o c a t i o n of  by  then the  the  the  f o r a l l the edge of  the  s p e c i f y i n g an  initial  gap  described  in  models,  the  magnitude of  the  proceeded  as  one-dimensional on  one-dimensional  by  that  leading  influence  trailing  Only  at  noted  edge of  models, the f l a t  specifying suction  the  large  peak be  however,  the  s e c t i o n of  the  initial  s h i f t e d along  Doing so, however, u s u a l l y r e s u l t e d i n an  suction imposed.  s i m i l a r to As  before,  caused the s o l u t i o n to  t h a t when an the fail.  excessive  uncontrolled  drainage the  exponential  initial  growth of the  drainage suction  62. IV. DISCUSSION  4.1  General The  primary  thrust  of  the  present  investigation  m a t h e m a t i c a l model t h a t c o u l d be u s e d t o p r e d i c t on a F o u r d r i n i e r paper machine drainage f o i l . a  model,  two  distinct  In  addition  models.  experiments selection typical  were  of  of  approaches were  machine  results  presentations  previous  chapters.  of  measure  of  hope  of  distribution,  using  on  a  conditions  providing  reliable  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 compared.  the  results  A comparison of  the  obtained results  have  been  provided  and d i s c u s s i o n o f  in  their  follows.  Experimental Results  t h e purpose and the goal  of  the  expectations  experiment  was  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,  of  not  the  to  try  c o n c l u s i o n s r e g a r d i n g the performance of 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 comparison to  goal, foils  the  results  of  experiment and foils  must be e x a m i n e d .  obtain  results  c o u l d be made.  with  under  normal  i n mind,  intended  paper the  purpose.  machine  results  The which  Rather,  the  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 the  numerical models.  To  satisfy  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  experiment the  the  separate  distributions  conducted  a  developing  three  pressure  pressure  were  with  Before a c a r e f u l examination of  for  the  the  in  develop  distribution  the course of  resulting  of  experiments  paper  Detailed  modelling  to  These  actual  data w i t h which the  4.2  the  conducted  foils.  an  significance  to  to  the pressure  In  taken  was  conditions.  With  the  goals  this  several of  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  the for  63.  In the  general,  results  correlation  the e x p e r i m e n t a l  of previous  results  investigations  between t h e p r e s e n t  d u p l i c a t e trends  [5,9,11].  results  For f o i l  and t h o s e  of d e t a i l e d  Fleischer  foil  whereas  degree  investigation. the r e s u l t s  foil.  Thus,  to d i r e c t l y  Unfortunately, a  compare h i s r e s u l t s  Also, Fleischer's o f the p r e s e n t  t h e comparison  results  only  used by  t o those o f  a r e f o r a 3 degree  investigation  i s made  good  [9] were  i n f o r m a t i o n r e g a r d i n g t h e e x p e r i m e n t a l procedure  made i t d i f f i c u l t  the present  3, v e r y  of F l e i s c h e r  o b t a i n e d f o r a w i r e v e l o c i t y o f 500 f t / m i n ( F i g u r e 6 5 ) . lack  exhibited i n  shown a r e f o r a 4  to indicate  that the  p r e s e n t r e s u l t s a r e o f t h e same o r d e r o f magnitude and shape. As  was expected,  F i g u r e s 16-21 show t h e measured p r e s s u r e  t i o n s were s i g n i f i c a n t l y 11-15  verify  cities. this  different  f o r each f o i l .  the trend of i n c r e a s i n g  Because of t h e method used  trend  i s indicated  dimensional suction.  The r e s u l t s o f F i g u r e s  suction with  increasing  t o non-dimensionalize  i n Figures  11-15  by  distribu-  a  decrease  wire  velo-  the pressure, i n t h e non-  The r e s u l t s o b t a i n e d f o r f o i l 3, p r e s e n t e d i n F i g u r e  13, best show t h i s t r e n d . Throughout results.  t h e experiments,  F i g u r e 13 c l e a r l y  foil  shows t h e r e l a t i o n s h i p  d i s t r i b u t i o n and t h e w i r e v e l o c i t y . tions  associated with  3 consistently  provided  the b e s t  between t h e p r e s s u r e  Also evident i s the pressure f l u c t u a -  the h o r i z o n t a l  l e a d i n g edge  section  o f the f o i l .  These f l u c t u a t i o n s a r e due t o s q u e e z i n g o f t h e water c l i n g i n g t o t h e lower s u r f a c e o f t h e w i r e between t h e w i r e and t h e f o i l . Figure  66  provides  obtained using f o i l s  seen.  causing from  possible  1 and 2.  i s made a t t h e t r a i l i n g be  a  explanation  f o r the poor  I f a c l o s e examination  edge o f t h e f o i l ,  of the wire surface  a distinctive  'dry-out' l i n e c a n  At t h i s boundary, a l l t h e water has been drawn through the pressure  Figure  66 where  t o r e t u r n t o atmospheric. the t i p o f t h e d r y - o u t  results  the wire  Because i t i s n o t c l e a r line  i s relative  t o the  64.  pressure  tap l o c a t i o n s ,  constant,  the  poorly  probably  due  complete  layer  Though  It  the shape of  developed  of  water  over  photographs  of  the dry-out  pressure  t o t h i s phenomena.  clear  d r y o u t was  and  Foil  distribution  3,  of  i s a l s o possible three  length  1 were not  dimensional 1 and  as  seen  obtained,  times  the  tested,  the  2.  three dimensional,  from the  s i d e s of  an  maximum  s u c t i o n o b t a i n a b l e on  f l o w was  be  overall  a  effects  played  I f an analogy  s u c t i o n would  foil  of  infinite  r e d u c t i o n of  approximately [22].  are  reported  However,  Fleischer  shown i n F i g u r e  flow  less  nately,  it  dimensional  than  is  very  effects  were not made.  that  did  the 65  than  good  correlation  suggests  indicated  difficult i n fact  those  by to  the  effects  slider  exist  as  the  erratic  flow  the  It i s  the maximum in  the  previous  results  three  Unfortu-  whether  spanwise p r e s s u r e  of  dimensional  theory.  determine  would  three  measurements  F u r t h e r e x p e r i m e n t a l work s h o u l d i n c l u d e spanwise p r e s s u r e  r e s u l t s obtained using f o i l s  fluid  of  bearing  clearly  3 as  with  measurements to e s t a b l i s h the p o s s i b l e e x t e n t of t h i s The  If  the net r e s u l t  suctions  lower  0.4  i n t o the flow r e g i o n  foil  investigations.  i n the  the magnitude of the measured s u c t i o n .  slightly  that  that f o r t y p i c a l  phenomena c o u l d have e x i s t e d on  measured  67.  1.  a role  width  I f t h i s were the case,  a  i s drawn between a  be  water c o u l d have e n t e r e d  the f o i l .  was  i n Figure  also possible this  are  12  i t is likely  f o i l and a s l i d e r b e a r i n g , s l i d e r b e a r i n g t h e o r y suggests geometries  likely  Figure  a l s o the reason f o r the poor r e s u l t s o b t a i n e d w i t h f o i l  poor r e s u l t s o b t a i n e d on f o i l s  foil  not  on the o t h e r hand, always had  i t s entire  foil  l i n e was  was  not  4 and  5 ( F i g u r e s 14  e s t a b l i s h e d between the  p r e s s u r e s were r e c o r d e d  foil  and  and 15)  suggest  the w i r e .  foil.  At the s t e p l o c a t i o n , however, the p r e s s u r e r e t u r n e d t o atmospheric.  Only  at  than  foil  4,  edge of f o i l  the h o r i z o n t a l s e c t i o n of  Very  the  the t r a i l i n g  over  effect.  5, which had  a much s m a l l e r s t e p h e i g h t  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, a r e  65. consistent  with  suction  was  stepped  foil.  As  often  was  considered  those  o f Cadieaux  necessary  discussed t o be  to i n i t i a t e  i n Section  very  [11]•  2.6,  inaccurate.  Cadieaux  found  and s u s t a i n  t h e measured  The measurement  that  external  the flow  drainage  over  a  rates are  technique  employed  a l l o w e d a g r e a t d e a l o f water t o e n t e r t h e s i d e s o f t h e d r a i n a g e trough as well  as from  totally  t h e end o f t h e f o i l .  inadequate  method  improved measurement  f o r measuring  technique  f l o w o f water i n t o  the d r a i n  Complicating  task,  this  As such,  needs trough  the t e c h n i q u e  the drainage  t o be d e v i s e d  used  rates.  t h a t would  was a A  much  l i m i t the  t o t h a t l e a v i n g t h e end o f t h e f o i l .  however,  is  c o l l e c t i o n method does n o t r e s t r i c t  the  need  to  ensure  the  water  t h e d e f l e c t i o n o f the forming w i r e a t  t h e t r a i l i n g edge o f t h e f o i l . Summarizing t h e e x p e i r m e n t a l obtained  for foil  3.  results,  S u c t i o n d i s t r i b u t i o n measurements on the r e m a i n i n g  f o i l s , however, were u n s u c c e s s f u l . were t o t a l l y u n a c c e p t a b l e . on  foil  Furthermore,  foil  trends  clear  to look  f o r comparison t o the r e s u l t s  the  numerical  i n the pressure  f o r i n the r e s u l t s  unnecessary  4.3  to  produce  as a f u n c t i o n o f t h e w i r e  of the numerical modelling.  t o o b t a i n improved r e s u l t s . models  o f the  t h e p r e s s u r e d i s t r i b u t i o n s measured on  t h e e x p e r i m e n t a l r e s u l t s were poor o v e r a l l , t h e experiment  r a t e measurements as a whole,  Thus, o n l y t h e p r e s s u r e d i s t r i b u t i o n s measured  numerical modelling.  velocity  Drainage  3 c a n be c o n s i d e r e d u s e f u l  3 showed  good s u c t i o n measurements were  no attempt  Though  was made t o r e f i n e  T h i s was due t o t h e f a i l u r e o f  reasonable  results,  thus  i t seemed  t o improve the experiment.  Numerical Results Of  t h e t h r e e n u m e r i c a l models p r e s e n t e d  i n Chapter  d i m e n s i o n a l models p r o v i d e d a c c e p t a b l e r e s u l t s .  3, o n l y t h e one-  Both t h e c o n s e r v a t i o n o f  66.  mass model and t h e momentum i n t e g r a l model c l e a r l y in  the pressure  direct ally  distributions  comparison  was v e r y  predicted  of the numerical  disappointment  by  observed  t h e models  were  i n the experimental  results  however.  reproduced  t o those  less  results.  obtained  In g e n e r a l ,  significantly  the trends A  experiment-  t h e maximum s u c t i o n s than  those  measured.  Because o f t h e s i g n i f i c a n t d i f f e r e n c e s i n t h e magnitudes o f t h e c a l c u l a t e d and  measured  pressures,  no  attempt  c a l c u l a t e d and measured p r e s s u r e It very  compare t h e  distributions.  This  i s thought  t o be a r e s u l t  gap t h a t had t o be s p e c i f i e d  t i o n o f mass model.  this  problem  different  results  of the excessive  initial  t o begin the s o l u t i o n of the conserva-  As was mentioned i n p r e v i o u s s e c t i o n s , t h e f o i l - w i r e  had a s i g n i f i c a n t  avoid  to directly  t o t h e c o n s e r v a t i o n o f mass model, v a s t l y  were o b t a i n e d .  gap  made  may be p o i n t e d out t h a t though t h e models o f Meyer and E r n s t were  similar  wire  was  effect  on t h e r e s u l t s o b t a i n e d .  as he o b t a i n e d  an a n a l y t i c a l  Meyer was a b l e t o  solution  t o h i s model.  E r n s t , however, l i k e l y had t o d e a l w i t h t h i s problem as h i s s o l u t i o n was a n u m e r i c a l one. Figure lated  68 i s a comparison  using  suction  by  by  t h e momentum  the c o n s e r v a t i o n  rates  F i g u r e 69.  of typical  t h e two one-dimensional  predicted  predicted drainage  I t i s not c l e a r , however, how E r n s t avoided t h i s problem. pressure  models. integral  Under model  o f mass model.  f o r t h e momentum  integral  distributions  calcu-  a l l c o n d i t i o n s , the was  less  Consequently,  model were  lower  than  that  predicted  as shown i n  F i n a l l y , F i g u r e 70 i n d i c a t e s much g r e a t e r f l o w s e p a r a t i o n was  p r e d i c t e d by the momentum I n t e g r a l model. The questions  extensive about  flow  separation indicated  the v a l i d i t y  model t h e f l o w over  foils.  of using  i n F i g u r e 70 r a i s e s  t h e boundary  layer  serious  equations  to  Though i t i s p o s s i b l e t h e f l o w s e p a r a t e s , i t  67.  is  highly  Instead, in the  u n l i k e l y that  i t i s more l i k e l y  a failure  region  the expansion  possibility, drainage  there  t o t h e extent  velocities  fluid  were  region.  In an attempt  t h e momentum  the wire  t o s e p a r a t e due t o examine  of t h e e f f e c t o f a r t i f i c i a l l y  made u s i n g  and hence,  crosses  t o p r e v e n t t h e tendency f o r t h e flow  tests  i n F i g u r e 70.  p r e d i c t the pressure  Consequently, i n s u f f i c i e n t  o f the flow  several  suggested  i s a d e f i c i e n c y i n t h e models r e s u l t i n g  o f t h e models t o a c c u r a t e l y  drainage v e l o c i t y .  i n t o the flow to  i t occurs  i n c r e a s i n g the  i n t e g r a l model.  assumption was t h a t d r a i n a g e through t h e w i r e due t o g r a v i t y c o u l d a large part o f the flow.  To t e s t t h i s assumption, an i n i t i a l  age  v e l o c i t y was imposed  the length  of the f o i l  atmospheric boundary c o n d i t i o n s f o r t h e p r e s s u r e s . profiles  f o r one o f t h e t e s t s a r e g i v e n  compared t o F i g u r e  was  too l a r g e ,  failure the  the s u c t i o n  grew  of the flow  I f the i n i t i a l  71 i s separa-  velocity  resulting i n a  of the s o l u t i o n t o s a t i s f y  a t the t r a i l i n g  These r e s u l t s suggest t h e o n e - d i m e n s i o n a l models f a i l for  If Figure  exponentially  o f t h e s o l u t i o n due t o t h e i n a b i l i t y  boundary c o n d i t i o n on t h e p r e s s u r e  by s p e c i f y i n g sub-  71.  72, i t can be seen t h a t t h e e x t e n t  however,  drain-  The p r e d i c t e d v e l o c i t y  i n Figure  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.  The  account  for  over  this  edge o f t h e f o i l .  t o adequately  allow  the d r a i n a g e v e l o c i t y . Results  o f 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 t h e p r e s s u r e  d i s t r i b u t i o n as those p r e s e n t e d by  E r n s t though t h e e f f e c t s were g e n e r a l l y much more pronounced. effect  The o v e r a l l  o f i n c l u d i n g t h e w i r e d e f l e c t i o n i n t h e c a l c u l a t i o n s was a s l i g h t  reduction  o f t h e maximum  towards t h e t r a i l i n g  suction  as w e l l a s h i f t  edge o f t h e f o i l .  o f t h e maximum  suction  These r e s u l t s tend t o s u p p o r t t h e  h y p o t h e s i s o f T a y l o r t h a t w i r e d e f l e c t i o n e f f e c t s would s h i f t t h e l o c a t i o n  68.  of t h e maximum s u c t i o n towards t h e t r a i l i n g edge o f the f o i l . wire  deflections  were c a l c u l a t e d ,  been p r e s e n t e d . result  t y p i c a l d e f l e c t e d wire  Though t h e  shapes have n o t  T h i s i s due t o t h e v e r y s m a l l d e f l e c t i o n s c a l c u l a t e d as a  o f t h e low s u c t i o n s p r e d i c t e d .  In g e n e r a l , t h e w i r e  deflections  were on t h e o r d e r o f about 1 t o 2 p e r c e n t o f t h e maximum gap h e i g h t . 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  This  deflec-  t i o n on t h e s u c t i o n d i s t r i b u t i o n s . Attempts a t m o d e l l i n g f o i l s u s i n g t h e approach of t h e two-dimensional model were g e n e r a l l y a f a i l u r e .  Though r e s u l t s were o b t a i n e d t h a t p r e d i c -  t e d p r e s s u r e s more i n l i n e w i t h  t h o s e o b t a i n e d e x p e r i m e n t a l l y , the s o l u -  tions  were  unable  distribution momentarily  to s a t i s f y  and must  be c o n s i d e r e d  overlooked,  c o n d i t i o n s on  a failure.  t h e two-dimensional  t r e n d s t o t h e one-dimensional a d d i t i o n , minimal  t h e boundary  If this  the p r e s s u r e  shortcoming i s  solutions d i d exhibit  similar  s o l u t i o n s and t h e e x p e r i m e n t a l r e s u l t s .  In  f l o w s e p a r a t i o n was p r e d i c t e d by the s o l u t i o n s t h a t were  obtained. The  l a c k o f wide spread  d i m e n s i o n a l model suggests t o adequately  predict  l a y e r equations. approximations could  s e p a r a t i o n i n t h e s o l u t i o n s u s i n g t h e two-  t h a t t h e f a i l u r e o f t h e one-dimensional  t h e p r e s s u r e s i s n o t due t o t h e use o f t h e boundary  I n s t e a d the s e p a r a t i o n i s probably  made i n assuming t h e v e l o c i t y  perhaps have been o b t a i n e d  been used.  models  To do t h i s ,  i f a higher  however,  a shortcoming  profiles.  Improved  order v e l o c i t y  of the results  p r o f i l e had  additional  boundary c o n d i t i o n s f o r t h e  indicated  the r e s u l t s  v e l o c i t y p r o f i l e would be r e q u i r e d . Numerical dimensional Minimizing  experimentation  models this  were  highly  dependent  on t h e i n i t i a l  of  t h e one-  f o i l - w i r e gap.  gap i n c r e a s e d t h e magnitude o f the p r e d i c t e d s u c t i o n and  69.  thus  reduced  certain  the  point  amount  reducing  solution failed.  gap  actual f o i l ,  truncation  Beyond  errors  a  and  specified.  caused  predicted.  to the p o i n t where the e q u a t i o n s became s t i f f an  size  was  the  On  gap  that  in  f o i l a t the l e a d i n g edge. initial  separation  the  c a l c u l a t i o n s to increase the  of  with  the  Thus, i t i s d e s i r a b l e t o reduce the s i z e of  the  Introduction  the w i r e i s i n c o n t a c t  of  a  large  initial  gap  undoubtedly  accounts f o r p a r t o f the e r r o r i n the r e s u l t s . To  summarize,  the  numerical  p r e d i c t the e x p e r i m e n t a l l y the f o i l . as  a  optimization  the p r o p e r magnitudes and s o u r c e s of  e r r o r are  gap  height  lations  because  though,  that  improved Promise  observed trends  The models, however, a r e not  t o o l f o r the  initial  models were a  to  with  does t o be  integral  model,  to be  foil  design  as  further point for  effort  where the  of  the  the  two-dimensional  overcome are much g r e a t e r . the  two-dimensional  model  a  model  is  the a  predict  The  major  There  is  model  could  useful though  favour much  use  the  i n t o the c a l c u -  Integral  as  In  to  the s o l u t i o n on  solution.  serve  they  t o be o f  distributions.  momentum  i t would  that  d i s t r i b u t i o n on  they f a i l  the dependence o f  instability  in  s u f f i c i e n t l y accurate  shapes of the p r e s s u r e  thought  the  exist  problems  a  i n the p r e s s u r e  as w e l l as n u m e r i c a l e r r o r s i n t r o d u c e d  of  the  of  successful  hope,  design the  of  more  be  tool.  numerical  the momentum complex  and  c o m p u t a t i o n a l y 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 a c c u r a t e  approach.  70.  The model  object  that  of  could  V.  CONCLUSIONS  the present  investigation  be  used  to  predict  Fourdrinier  paper machine d r a i n a g e  to  a  provide  design  characteristics foil  of  tool  that  the  foil.  was t o pressure  a  numerical  distribution  on  a  The p u r p o s e o f s u c h a m o d e l was  could  be u s e d  foil  and permit  an a r b i t r a r y  develop  to  evaluate  the  drainage  the optimization  of  the  design. From  the  results  of  the  present  investigation,  the  following  c o n c l u s i o n s may b e made:  1.  Due  to  pressure  the  poor  agreement  distributions  modelling  attempts  previous  on  and  foils  the  investigations,  of  it  the  between  the  experimental must  results  of  of  that  the  wire  speed were 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  2.  Three  respects,  share  boundary models  layer as  deflection present  obtaining models.  were  included  the a  first  failed  to  of  which, being  These models time  known,  i n the models.  t o produce  models  solution  developed  feature  equations. the  of  been  t h e common  for  models  failure  have  was  due  distribution  the models,  fail  similar  Despite  numerical  these It  to  forming  results.  different  frictional  and  trends  and the  a r e a n advancement  all  present  in  based on a s o l u t i o n  good r e s u l t s . to  though  the  present  t h e models  In g e n e r a l , however,  the pressure  of  the  the  for  models  between  and shapes  results  be concluded  adequately model the f l o w over f o i l s . relationship  magnitudes  of  effects  of  many the  previous and  wire  improvements,  the  i s concluded that  the  problems  and n o t due t o  associated  shortcomings  in  with the  3. to  While  crude,  the experiments  of  the p r e s e n t  produce some r e s u l t s which agreed w e l l w i t h the s c a n t p u b l i s h e d r e s u l t s  a v a i l a b l e of o t h e r e x p e r i m e n t a l  5.1  investigations.  Recommendations f o r Further Work  1.  The  f o l l o w i n g recommendations f o r f u r t h e r work a r e made:  Of  the  warrants  solution  the  most  procedures  attention  for  developed, further  previous  modelling  attempts,  the  work  m o d e l l i n g approach than the two-dimensional of  i n v e s t i g a t i o n were a b l e  momentum as  integral  i t is a  c o n s e r v a t i o n of mass model.  Though the shapes and magnitudes of the c a l c u l a t e d p r e s s u r e  observed  trends  were  obtained.  simpler  model but i t i s an advancement  u n l i k e the  do not agree w e l l w i t h e x p e r i m e n t a l  much  model  distributions  r e s u l t s , v e r y good c o r r e l a t i o n t o the  Possible  areas  of  improvement  to  the  momentum i n t e g r a l model i n c l u d e r e d u c t i o n of the s e n s i t i v i t y of the model to  the  initial  problem  foil-wire  indicate  gap  height.  s u c t i o n magnitudes  Results  c o n s i s t e n t with  c o u l d be o b t a i n e d i f t h i s problem were overcome. be  the  very  development  s m a l l gap  warranted  of  into  a  a larger  As  gap  the  measurements. small  gap  using  this  results  A p o s s i b l e approach  may  stretch  Study i s a l s o  c o n t a c t of the w i r e w i t h the  foil  the n u m e r i c a l models, i t i s recommended  initial  the c a l c u l a t i o n s a t the t r a i l i n g  conditions  obtained  from  edge  experimental  T h i s approach would a v o i d the problems a s s o c i a t e d w i t h  heights  a  to g r a v i t y .  an approach t o improving  foil  experimental  i n solution coordinates.  t h a t a f o c u s be p l a c e d on b e g i n n i n g of  exploring  t r a n s f o r m a t i o n t h a t would  on t h e e f f e c t s of the i n i t i a l  as w e l l as drainage due  2.  coordinate  obtained  encountered  at  the  l e a d i n g edge  of  the  foil,  as  s o l u t i o n c o u l d be stopped when the gap h e i g h t became e x c e s s i v e l y s m a l l .  the the  72. 3.  I t i s the o p i n i o n  probably  of  the author  that  t h e two-dimensional  has t h e p o t e n t i a l o f b e i n g t h e most a c c u r a t e model.  Because no  a p p r o x i m a t i o n s a r e made w i t h r e g a r d s t o t h e form o f t h e v e l o c i t y the  two-dimensional  model has none  o n e - d i m e n s i o n a l models.  o f t h e shortcomings  A g r e a t d e a l o f development  model  profile,  inherent i n the  i s r e q u i r e d , however,  b e f o r e t h e two-dimensional model c a n be c o n s i d e r e d a v i a b l e s o l u t i o n nique. the  The most s i g n i f i c a n t  solution  Finding bolic  to satisfy  a solution  nature  the t r a i l i n g  to this  problem  o f t h e boundary  numbers encountered capabilities difficult  shortcoming  o f t h e method i s t h e f a i l u r e o f  edge  will  layer  condition  on t h e p r e s s u r e .  n o t be t r i v i a l  equations.  The  due t o t h e p a r a typical  i n paper machines a r e a l s o a t t h e upper  of the d i f f e r e n c i n g  n u m e r i c a l problems  scheme  may have  selected.  Reynolds  l i m i t s of the  Thus,  possibly  very  t o be overcome b e f o r e an a c c u r a t e  s o l u t i o n c a n be o b t a i n e d u s i n g t h e two-dimensional approach. tool,  tech-  t h e two-dimensional model may never  serve a u s e f u l  As a d e s i g n  f u n c t i o n due t o  i t s c o m p l e x i t y and t h e l a r g e amounts o f computer time r e q u i r e d t o o b t a i n a solution.  4.  Further  recommended model. limits  development as i t was  Failure  of  the  conservation  essentially  superceded  of  mass  model  by t h e momentum  i s not integral  o f t h e c o n s e r v a t i o n o f mass model t o conserve momentum  t h e u s e f u l n e s s o f t h e model and was developed m a i n l y t o g a i n some  i n s i g h t i n t o t h e behaviour of the flow.  The c o n s e r v a t i o n o f mass model i s  a l s 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 t o t h e models o f Meyer and E r n s t and tion  does l i t t l e o f mass  t o improve  model  further pursuit  served  t h e i r models. a useful  unworthwhile.  F o r t h i s purpose  function  the conserva-  but i t s shortcomings  make  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 dimensiona l i t y of the flow by measuring the spanwise pressure distribution.  A much  more appropriate method for measuring the drainage rate i s also necessary. Efforts should also be made to obtain results for the effect of the wire tension to determine distribution.  the significance  of i t s effect  on  the pressure  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 i n fact exist. ments could also  be  useful  for establishing  Shear stress measure-  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 . visualization turbulent.  would also  help  determine  whether or not  the  Flow  flow i s  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  Forming Board  Foils  Table Rolls  Suction Boxes  Suction Couch Roll Paper Sheet to Presses and Dryers  Paper Slurry In  Brest Roll  Forming Wire  F i g u r e 1.  Tensioner and Guide Rollers  T y p i c a l c o m p o n e n t s and l a y o u t o f  a F o u r d r i n i e r Paper  Machine.  0.1  n 0.0  i  i  0.1  0.2  i  1  0.3  0.4  1  0.5  1  0.6  1  1  1  f  0.7  0.8  0.9  1.0  X / L Figure 2.  C o m p a r i s o n 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 mental r e s u l t s of F l e i s c h e r ( 9 ) . R e s u l t s are f o r 3 degree tapered f o i l , U = 2000 f t / m i n .  7,6.  Aluminum Frame  Electric Drive Motor  Figure 3.  Drive Roller  Experimental  apparatus.  Tensioning Roller and Support Bolt  77.  Figure 4.  Photograph  of the e x p e r i m e n t a l  apparatus.  78.  F i g u r e 5.  Photograph of a t e n s i o n e r r o l l e r support showing the s t r a i n gauge l o c a t i o n .  bolt  79.  F i g u r e 6.  Photograph of the headbox.  F i g u r e 7.  Photograph of the t e s t s e c t i o n showing the h e a d b o x 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 .  81.  XL  Tapered  Foil  r  ^/////  1  YW////////////A Stepped  Foil  &  Foil  x (inches)  (Inches)  (degrees)  (Inches)  1  6.5  1  1  -  2  6.5  1  2  -  3  6.5  1  4  -  4  6.5  2.5  5  6.5  1  F i g u r e 9.  L  X L E  -  P r o f i l e s and dimensions of t h e f o i l s  1/4 1/16  tested.  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 r o l l e r support b o l t s .  a n d c o n n e c t i o n on t h e  tension  0.025  0.000  -0.025H  Wire Velocity  -0.075 H  A X  5 0 0 ft/min  •  1000 ft/min 1260 ft/min  a x-0.100 H 0.0  750 ft/min  1500 ft/min 2 0 0 0 ft/min  1 0.1  1 0.2  1 0.3  1 0.4  1 0.5  1 0.6  1 0.7  1 0.8  1 0.9  X / L F i g u r e 11'. M e a s u r e d 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 v a r i a t i o n w i t h the wire v e l o c i t y .  on f o i l  1 showing t h e  J 1.0  0.05  0.00  7^  A"/  A  'A'  T  »  A ^ X  A  -0.05-  A  A  CL i CL  \  Wire Velocity •0.10-  •0.15 H 0.0  A  5 0 0 ft/min  X  750 ft/min  •  1000 ft/min  M  1260 ft/min  XX  !600_ft/min_  X  2 0 0 0 ft/min  A  1-  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  X / L oa  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 v a r i a t i o n with the wire v e l o c i t y .  on f o i l  2 showing t h e  0.05-1  0.00  ^H-ia- grig  o-o-o-o'g -0.05  Q_ i  •0.10  Q_  -0.15-  -0.200.0  Wire Velocity •  500 ft/min  A  750 ft/min  <0>  1000 ft/min  03  1250 ft/min  V  1600 ft/min  O  2 0 0 0 ft/min  1  1  0.1  0.2  I  0.3  1  0.4  1  1  1  0.5  0.6  0.7  I  0.8  I  0.9  1.0  X / L Figure 13.  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 v a r i a t i o n w i t h the w i r e v e l o c i t y .  oo. on f o i l  3 showing  the  0.03-1  0.02-  0.01-  !  A-A-A-A-A-A-A-A-A-A-A-A-A-A-A-A-  0.00  -0.01 CD  GL  Q_  -0.02H  Wire Velocity  -0.03  A  600 ft/min  X  760 f t / m i n  •  1000 f t / m i n 1250 f t / m i n 160_0__f_t/min_  •0.04 X  1  -0.05-T 0.0  0.1  !  1  1  1  1  1  r  0.2  0.3  0.4  0.5  0.6  0.7  2 0 0 0 ft/min  0.8  0.9  1.0  X / L 00  F i g u r e 14.  Measured non-dimensional 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 v a r i a t i o n w i t h the w i r e v e l o c i t y .  4 showing t h e  0.050-1  0.025  i  XT A  tr  ' i—i  ^9  0.000 Q_ i  A  Q_  i—i  Wire Velocity A X  •0.025  •  5 0 0 ft/min 750 f t / m i n 1000 ft/min 1260 f t / m i n  s X  •0.050-T 0.0  p 0.1  Figure 15.  — i 0.2  1  1  0.3  0.4  r—  0.5  0.6  0.7  1600 ft/min 2 0 0 0 ft/min  0.8  0.9  1.0  X / L  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 variation with the wire v e l o c i t y .  on f o i l  5 showing t h e  oo oo  0.05-1  TO?  0.00  3  *  \•  -0.05-  \  Q.  Q_  A.  A  -0.10  Foil Number -0.15  A  Foil 1  X  Foil 2  •  Foil 3  Kl  Foil 4  •  \  •  /  Foil 5  -0.20 0.0  1  1  0.1  0.2  1  0.3  1  0.4  1  0.5  1  0.6  1  1  1  0.7  0.8  0.9  1.0  X / L F i g u r e 16.  Comparison of measured p r e s s u r e d i s t r i b u t i o n s at a w i r e speed of 500 f t / m i n .  on f o i l s 1,2,3,4 and 5  00 'to:  0.05-1  0.00 *x-x  V  /  -0.05-  •  /  X  CL i GL  Foil Number -0.10  A X  Foil 1  •  Foil 3  ®  Foil 4  Foil 2  Foil 5  -0.15 -f 0.0  1 0.1  I  0.2  1 0.3  1 0.4  1 0.5  1 0.6  I  0.7  1 0.8  1  0.9  1.0  X / L o F i g u r e 17.  Comparison of measured p r e s s u r e d i s t r i b u t i o n s at a wire speed of 750 f t / m i n .  on f o i l s  1,2,3,4 and 5  0 . 0 5 - 1  - 7W X" X  'X- "  \'  \  »  Q.  X  -  0  .  0  X A  /  5  «  QL QL  Foil Number - 0 . 1 0  A  Foil 1  X  Foil 2  •  Foil 3 Foil 4 Foil 5  -  0  .  1  5  H  1  0 . 0  0 . 1  1 0 . 2  1 0 . 3  1 0.4  1  1  1  1  0 . 5  0 . 6  0 . 7  0 . 8  I 0 . 9  X / L F i g u r e 18.  C o m p a r i s o n o f m e a s u r e d 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 a n d 5 a t a w i r e s p e e d o f 1000 f t / m i n .  1 . 0  0.05  0.00  3  »  Q.  -0.05to  QL  I  Foil Number -0.10-  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  0.6  0.7  0.8  0.9  X / L Figure 19.  Comparison of measured p r e s s u r e d i s t r i b u t i o n s a t a w i r e s p e e d o f 1250 f t / m i n .  on f o i l s  1 , 2 , 3 , 4 and 5  1.0 to ho  0.05-1  0.00 •  -  D  ZD Q.  -0.05 Q_  i Q_  Foil Number •0.10  A  Foil 1  X  Foil 2  •  Foil 3  M  Foil 4 Foil 5  -0.15 0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  X / L to  F i g u r e 20.  Comparison of measured p r e s s u r e d i s t r i b u t i o n s at a w i r e speed of 1500 f t / m i n .  on f o i l s 1,2,3,4 and 5  0.05-i  0.00  » Q.  -0.05Q_  i Q_  Foil Number •0.10-  A  Foil 1  X  Foil 2  •  Foil 3  M  Foil 4 Foil 5  -0.15-r 0.0  —I  0.1  F i g u r e 21.  1  1  1  1  1  1  1  0.2  0.3  0.4  0.5 X / L  0.6  0.7  0.8  Comparison of measured p r e s s u r e d i s t r i b u t i o n s at a w i r e speed of 2000 f t / m i n .  on f o i l s  1  0.9  1,2,3,4 and 5  1.0 to  Foil Number A  Foil 1  X  500  750  1000  1250 Uwire  F i g u r e 22.  1500  1750  2000  ( ft/min )  Measured non-dimensional f o i l drainage r a t e s as a f u n c t i o n of w i r e speed f o r f o i l s 1,2,3,4 and 5.  0.025-1  'A-  0.000  -0.025  Wire Velocity -0.050  A  750 f t / m i n . Run 1  X  750 f t / m i n . Run 2  •  1000 f t / m i n . Run 1 1000 f t / m i n . Run 2  •0.075-r 0.0  S  1600 ft /mi r> Run _1_  X  1JL0J^f!/mjn^Rjjn_2  :  0.1  Figure 23.  0.2  0.3  0.4  0.5 X / L  0.6  0.7  0.8  0.9  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 a n d 1500 f t / m i n .  1.0  0.05-1  0.00  3  *  a CL  i  CL  -0.05  -o.ioH  Wire Velocity A  500 f t / m i n . Run 1  X  5 0 0 f t / m i n . Run 2  •  -0.15  750 f t / m i n . Run 1  A-A^X  750 f t / m i n . Run 2 1000 f t / m i n . Run 1  X  -0.20-f 0.0  1000 f t / m i n . Run 2  r™ 0.1  0.2  0.3  T 0.4  T  0.5  0.6  0.7  0.8  0.9  X / L 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 v e l o c i t i e s o f 500 f t / m i n , 750 f t / m i n a n d 1000 f t / m i n .  3 for  wire  1.0  Figure 25.  Diagram of a f o i l showing the geometric n o m e n c l a t u r e used i n t h e development mathematical models. Lengths i n the y - d i r e c t i o n are g r e a t l y exaggerated.  of the  F i g u r e 26.  Diagram of a f o i l showing the dynamic q u a n t i t i e s of the mathematical models.  used i n the  development  to ivO  100.  F i g u r e 27.  Diagram showing the f o r c e s of the forming w i r e .  on a s m a l l  section  101.  ( " s t a r t ^) Read  Data  ~T~  Initialize I = 0  Evaluate  Pressure  no  Evaluate Wire Deflection  yes  Output Results ( Stop )  1 = 1+  1  no es  Convergence Failure  I (jrtopj)  F i g u r e 28.  Flowchart of t h e s o l u t i o n procedure f o r t h e c o n s e r v a t i o n of mass model.  8.0-1  Legend Calculated Blasius Profile  6.0-  4.0  2.0  0.0 0.0  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1  U / Uo o, N3:  F i g u r e 29.  Comparison 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 f o r the f l o w o v e r a f l a t p l a t e to the B l a s i u s p r o f i l e . R e s u l t s obtained u s i n g the 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  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  U / Uo F i g u r e 30,  Comparison 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 f o r t h e f l o w between moving p a r a l l e l f l a t p l a t e s to the exact s o l u t i o n . Calculated velocity p r o f i l e 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 .  o  0.005  0.000  -0.005  -0.010-  -0.015Wire Velocity -0.020-  -0.025-  Uw f •  600 ft/min  U . r«  750 f t / m i n  u. rt u. r«  1000 f t / m i n  Uw  1600 f t / m i n  Uw  2 0 0 0 ft/min  1250 f t / m i n  -0.030 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1  X / L F i g u r e 31'.  Calculated pressure d i s t r i b u t i o n s f o r f o i l 1 using the conservation of mass model showing the v a r i a t i o n w i t h the w i r e 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 i n c l u d e d i n t h e s o l u t i o n .  o  0.0180-1  0.0175  0.0170  0.0165-  I  o  0.0160-  0.0155-  0.0150No Wire D e f l e c t i o n  0.0145  0.0140  —  500  750  1000  1250 Uwire  Figure 32.  1500  1750  i  —  2000  (ft/min)  Comparison of c a l c u l a t e d drainage r a t e s f o r f o i l 1 w i t h and without wire deflection included. Results obtained using the conservation o f mass mode.  o  'Ln  0.005  0.000  -0.005  3  CO  Q_  -0.010  -0.015-  I  Q_  Wire Velocity Uwlr.  -0.020-  -0.025  =  500 ft/min  Uww.  750 f t / m i n  Uwlr.  1000 f t / m i n  Uw„.  1260 f t / m i n  U.I,.  1500 f t / m i n  Uwlr.  2 0 0 0 ft/min  -0.030 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1  X / L F i g u r e 33.  C a l c u l a t e d pressure 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 c o n s e r v a t i o n of mass model showing the v a r i a t i o n w i t h the w i r e 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 i n c l u d e d i n t h e s o l u t i o n .  o  0.2  0.0-  •0.2-  -0.4-  sz -0.6  -0.8-  U w lr«  600 ft/min  U wlr*  750 f t / m i n  U wlr«  1000 f t / m i n  U Wtf  •1.0-  =  U wl rt Uw!,.  1250 f t / m i n  •  =  1500 f t / m i n 2 0 0 0 ft/min  •1.2 •0.6  -0.4  -0.2  0.0  0.2 U  F i g u r e 34.  0.4 /  0.6  0.8  1.0  1.2  Uwire  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 of w i r e v e l o c i t y . R e s u l t s obtained u s i n g the c o n s e r v a t i o n o f mass model.  o  0.2-1  0.0-  -0.2-  •0.4  -0.6  •0.8  Legend  -1.0  -1.2 -0.4  -0.2  0.0  T  0.2  0.4 U  F i g u r e 35.  /  0.8  0.6  X/L  - 0.164  X/L  - 0.366  X/L  - 0.677  X/L  = 0.789  X/L  = 1.000  1.0  1.2  Uwire  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 . Results 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 effects. w  o  03  0.2-1  0.0-  -0.2-  -0.4  -0.6  Legend  -0.8  -1.0-  X/L  = 0.164  X/L  - 0.366  X/L  - 0.577  X/L  = 0.789  X/L  - 1.000  -1.2 -0.4  -0.2  0.0  0.2  0.4  0.6  0.8  1.0  1.2  U / Uv Figure 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 deflection effects included.  o  0.005-1  X / L F i g u r e 37.  Comparison of 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.and 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 m o d e l without wire deflection effects.  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 m o d e l t o t h e initial 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 for DEBDF  Call DEBDF  (Return) Figure'39'.  Flow chart of the pressure e v a l u a t i o n subroutine t h e momentum i n t e g r a l m o d e l s o l u t i o n p r o c e d u r e .  f  Legend Calculated  0.0 •  Theory  -0.2-  -0.4-  > -0.6  •0.8-  •1.0-  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  U / Uo F i g u r e 40.  Comparison 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 f o r t h e f l o w between moving p a r a l l e l f l a t p l a t e s 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 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 Legend Calculated Q  Cubic approximation Blasius Profile  6.0  x  S,  -5  >>-  4.0  2.0  0.0 0.4  0.5  0.6  U / Uo F i g u r e 41.  Comparison 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 f o r the f l o w over a f l a t p l a t e to the 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 the momentum i n t e g r a l model w i t h the boundary c o n d i t i o n s X=0, p=p , dp/dx=0. 3.  8.0-r Legend Calculated Cubic approximation Blasius Profile  6.0-  x °  4.0  2.0-  0.0  0.0 F i g u r e 42.  Comparison 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 f o r t h e flow over a f l a t p l a t e to the 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 momentum i n t e g r a l model w i t h the boundary c o n d i t i o n s X=0, p=p ; x=L, p=p ,  0.005  0.000  -0.005  -0.010 Wire Velocity u . rt  500  ft/min  Uw r« = 750 ft/min Uw r« ts 1000  •0.015  ft/min  Uw r« = 1250 ft/min Uw  tt  Uw r«  = 1500 ft/min  -  2000 ft/min  -0.020 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 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 model 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 . Wire d e f l e c t i o n e f f e c t s were n o t 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 h1  F i g u r e 44,.  Comparison of 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 = 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 model 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 . w  0.005  0.000  *  Q_  i  -0.005  -0.010 Wire Velocity U wlf • = 6 0 0 ft/min  0_  U wire  760 ft/min  = 1000 ft/min U w lr« = 1260 ft/min Uwlr* = 1500 ft/min Uw„.  -0.015-  U w lr«  -0.0200.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  -  0.9  2 0 0 0 ft/min  1.0  X / L Figure  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 model showing the v a r i a t i o n w i t h the w i r e 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 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.  C o m p a r i s 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 on. f o i l 1 f o r U :-.= 5 0 0 . 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  ...t  -0.2-  X  v  •0.4  -0.6-  Legend -0.8-  U wtr«  =  500 ft/min 760 ft/min  Uw.,. U wire  -1.0  =  1000 ft/min 1250 f t / m i n  U wir«  1600 ft/min  U wlr* U wlr«  -1.2  1 —  -15.0  -10.0  -5.0  0.0  5.0  83  _200JD^t/mm  10.0  U / Uv F i g u r e 47  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 of wire 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 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 effects.  15.0 M K3 O  0.2-1  0.0-  -0.2  -0.4-C >•  -0.6-  -0.8-  Uw.,.  S3  U wlr«  600 ft/min 760 ft/min  Uwlr* = 1000 ft/min U w lr«  •1.0  «=  1260 ft/min  U w 1 r«ea 1600 ft/min Uwlr* <n 2000 ft/min  -1.2 -8.0  -6.0  -4.0  -2.0  0.0  2.0  U / Uv, Figure  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 of 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 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 i n the s o l u t i o n .  4.0  0.005-r  0.000  -0.005  -0.010  •0.015  Legend  -0.020-  ho = 1/32 inch ho = 1/16 inch  -0.0250.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1  X / L Figure 49.  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 m o d e 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 a n d a w i r e v e l o c i t y o f 500 f t / m i n .  0.005  0.1  0.2  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 Figure  50.  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 a s 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 m o d e 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 a n d a w i r e v e l o c i t y o f 500 f t / m i n .  ho u>  0.014-1  Drainage Resistance, Kdr (ft) Figure 51.  *10"  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 a s a f u n c t i o n o f t h e d r a i n a g e resistance. 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 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 . Wire d e f l e c t i o n e f f e c t s were not i n c l u d e d .  7  0.005-1  X / L F i g u r e 52.  E f f e c t of the wire t e n s i o n on the p r e s s u r e Momentum i n t e g r a l model, U = 500 f t / m i n .  distribution  along  foil  1.  0.010400-1  0.010390  0.010390-  0.010380 i ^  0.010380-1  o  0.010370  0.010370-  0.010360 400  500  600  700 W i r e Tension, T  F i g u r e 53.  800  900  1000  1100  (Ibs/lin.ft.)  E f f e c t of wire t e n s i o n on the c a l c u l a t e d d r a i n a g e r a t e f o r f o i l 1. Momentum i n t e g r a l model, U = 500 f t / m i n . ° w  ON  0.02-1  0.00  -0.02 3  *  Q.  -0.04 E CO  Q_  ol  Viscosity  -0.06H  • 1.088 « 10"' f t ' / i v_ m 2.000  -0.08-1  V  m  3.000  V  m  4.000  V  m  5.000  -  6.000  V  -0.10  r  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  Xi o - ftV. XIO"' «tV. Xi o f t V s Xi o - ft*/. Xi o 8  1.0  X / L F i g u r e 54.  E f f e c t of f l u i d k i n e m a t i c v i s c o s i t y on 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 1. Momentum i n t e g r a l model, U = 500 f t / m i n , no w i r e d e f l e c t i o n . w  1.1  -0.015-  Figure 55.  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 o 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 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 , w no w i r e d e f l e c t i o n .  129'.  Figure  56.  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 of the t w o - d i m e n s i o n a l m o d e l .  the  solution  130.  Start  ^  Read Data  I  Set up Initial Conditions  I  Initialize u,v,f,p v e c t o r s  w  Evaluate and Set up A,B matrices  I Solve Ax = B  Evaluate  Update u,v,f,p vectors  F i g u r e 57.  Flowchart f o r the t w o - d i m e n s i o n a l model procedure.  solution  8.0  Legend Calculated  U / Uo l-  Figure  58.  Comparison 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 f o r f l o w over a f l a t p l a t e to the B l a s i u s p r o f i l e . Calculated r e s u l t s obtained using the two-dimensional model.  J  U / Uo F i g u r e 59.  Comparison 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 f o r f l o w i n a r e c t a n g u l a r duct to the exact s o l u t i o n showing the development of the c a l c u l a t e d velocity profiles. R e s u l t s obtained u s i n g the t w o - d i m e n s i o n a l model.  .  M  U / Uo F i g u r e 6.0.  Comparison 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 f o r f l o w between moving p a r a l l e l f l a t p l a t e s to the e x a c t s o l u t i o n . P r o f i l e s show the development of the c a l c u l a t e d v e l o c i t i e s . Results obtained using the two-dimensional model.  H LO  .  w  "i  0.0  T 0.1  I  0.2  —  i  0.3  i  i  i  0.4  0.5  0.6  1  0.7  1  0.8  1  0.9  r  1.0  X / L Figure 61.  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 the two-dimensional model. U = 500 f t / m i n , V . = 0 . 0 5 f t / s e c , wi w k, = 2 . 5 x 10-8 f t . dr  using  Legend  U / U F i g u r e 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 c a l c u l a t e d u s i n g the t w o - d i m e n s i o n a l model 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 = 2 . 5 x 10-8 f t . w  d r  w  l  f  X / L F i g u r e 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 the d i s t r i b u t i o n s as a f u n c t i o n of the w i r e v e l o c i t y . Results obtained u s i n g the two-dimensional model. U = 500 f t / m i n ; V 0.05 f t / s e c , k, = 2.5 x I O " f t . dr 8  W  w  i  in  0.2  0.0-  -0.2  -0.4  -0.6  -0.8  Legend Uwlrt  =  600  ft/min  U.i,. ° 760 f t / m i n  •1.0  -1.2-0.2  0.0  0.2  0.4  0.6 U  F i g u r e 64  /  Uwln  =  1000  ft/min  U.in  =  1500  ft/min  0.8  1.0  Uwire  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 f o r f o i l 1 showing t h e v a r i a t i o n i n the p r o f i l e s w i t h the 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 the twodimensional model. = 500 f t / m i n ; V '. = 0.05 f t / s e c , = wi ' dr w 2.5 x 1 0 ~ f t , J  8  1.2 u>  05  00  \  \•  05-  \  • .10  •  •  /  •  y  Wire Velocity  .15  \ \ \  5 0 0 ft/mir> Fleischer  2 0 -f 0.0  1 0.1  1 0.2  1 0.3  1 0.4  1  1  1  1  1  0.5  0.6  0.7  0.8  0.9  X / L Figure 65.  Comparison of investigation  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 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 .  present  1.0  139.  F i g u r e 66.  Photograph of the flow over f o i l 2 at a w i r e v e l o c i t y of 1000 f t / m i n . Note the p r e s e n c e 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 w i r e .  Figure 67.  Photograph of the flow over f o i l 1 at a w i r e v e l o c i t y o f 1000 f t / m i n . Note the continuous f i l m of water over the f o i l .  0.005-1  X / L F i g u r e 68.  C o m p a r i s o n 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 a n d t h e momentum i n t e g r a l models. 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  Legend Conservation of Mass M o d e l 0.015-  Momentum Integral Model  0.014-  0.013 ZD  o  0.012-  0.011  0.010  ~l  500.0  1  750.0  1  1000.0  1250.0  Uwire F i g u r e 69 \  1500.0  1750.0  2000.0  (ft/min)  C o m p a r i s o n 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 b y t h e o n e dimensional models. Wire 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 solutions.  0.2-1  o.o-  •  0  .  2  -  -  0  .  4  -  8  -  - 0 . 6  -  0  .  Legend  • 1 . 0 -  Momentum Integral Model Conservation of Mass Model - 1 . 2  • 1 2 . 0  - 1 0 . 0  - 8 . 0  • 6 . 0  - 4 . 0  U  F i g u r e 70.  /  - 2 . 0  0 . 0  Uv  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 b y t h e one-dimensional models. 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 .  2 . 0  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 x n c l u d e d . W  •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 n o - - i n i t i a l d r a i n a g e v e l o c i t y h a s b e e n s p e c i f i e d . U = 500 f t / m i n ; Momentum i n t e g r a l model, wire d e f l e c t i o n effects included. w  f  146.  REFERENCES 1.  W r i s t , P.E. "The Papermaking P r o c e s s as a F i l t r a t i o n Problem." Paper Mag. Can. V o l . 55, No. 6 (May, 1954), pg. 115.  Pulp  2.  T a y l o r , G.I. "Drainage a t a T a b l e R o l l . " 57, No. 3, (Conv. 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APPENDIX A ORIFICE PLATE DETAILS  As  p a r t of  the e x p e r i m e n t a l  procedure,  i t was  necessary  to  maintain  t h e v e l o c i t y o f the water j e t l e a v i n g the headbox s l u i c e a t t h e same speed as  the  forming  wire.  The  jet velocity  flow r a t e u s i n g an o r i f i c e p l a t e and The  orifice  Mechanical flow  p l a t e was  Engineers  meters  3  2000  ft/min.  sufficient  according  orifice  The  was  specifications sized  to  pressure  a t a w i r e v e l o c i t y of 500  orifice  changes  size  over  by  was  the  also  range  measuring  the  velocity.  t o the American S o c i e t y of  the a n t i c i p a t e d range of f l o w s .  flow range of 0.065 f t / s at  designed  The  determined  then c a l c u l a t i n g the  (ASME) recommended  [13].  measurements over  was  of  for orifice  provide  plate  accurate  flow  T h i s corresponded  to a  f t / m i n . t o 0.260 f t / s 3  s e l e c t e d so flows  to  as  to  ensure  provide accurate  d e t e r m i n a t i o n of the j e t v e l o c i t y . A f t e r a l l r e l e v a n t f a c t o r s had having  a  installed The  diameter  drop  manometer connected  the  1.4975  inches  was  a thin plate orifice  constructed.  i n a s e c t i o n of p i p e h a v i n g an i n s i d e diameter  pressure  The  of  been c o n s i d e r e d ,  across  the  orifice  to D-1/2D p r e s s u r e  j e t v e l o c i t y was  determined  was  measured  The  orifice  was  of 4.026 i n c h e s . using  a  mercury  taps.  by measuring the f l o w r a t e and  using  expression  (A-l)  to  c a l c u l a t e the v e l o c i t y .  sluice;  In E q u a t i o n A - l , A i s the a r e a of the headbox  153. A - 7.8 x 1 0 ~ f t 3  and  Q i s the flow  (A-2)  2  r a t e i n c u b i c f e e t p e r second.  Q was determined  using  the e x p r e s s i o n g i v e n i n [ 1 3 ] ;  0  H  Q  =  a F k / 2 g (a  —)  (—)  (A-3)  where  Q  = flow r a t e  a  = o r i f i c e a r e a (= 1.22 x 1 0 ~ f t )  F  3  2  a  k  g  = o r i f i c e expansion  coefficient  _  Hg R  Q  h  (= 1)  c  =  o r i f i c e flow d i s c h a r g e  ^ l-g^  3  =  diameter  ratio  =  coefficient  3.7 x IO""  1  ft /s 2  = 26.28 = 1.937  slugs/ft slugs/ft  3  3  = manometer r e a d i n g ( i n c h e s Hg)  The f l o w d i s c h a r g e c o e f f i c i e n t s for  2  c  = 32.2  p„ p  (ft /s)  each j e t v e l o c i t y  f o r t h e o r i f i c e p l a t e were  u s i n g t a b l e s g i v e n i n r e f e r e n c e 13.  determined  Equations A - l  and A-3 were t h e n used t o c a l c u l a t e t h e manometer r e a d i n g s r e q u i r e d t o s e t each  jet velocity.  velocity  versus  Figure  73 p r e s e n t s  the manometer r e a d i n g .  a calibration  curve  f o r the j e t  2500  O r 0  6  10  8  12  14  16  18  h (inches Hg) Figure  73.  C a l i b r a t e d headbox s l u i c e j e t pressure drop.  velocity  as a f u n c t i o n o f ; t h e  orifice  20  APPENDIX B  CALIBRATION OF THE TENSION ROLLER BOLTS  156. APPENDIX B CALIBRATION OF THE TENSION ROLLER BOLTS  Calibration loading by  o f the the tension r o l l e r  each b o l t w i t h a known weight and r e c o r d i n g  the s t r a i n gauge b r i d g e .  loads  support b o l t s  t o ensure t h e l i n e a r i t y  The b o l t s  the s t r a i n  were c a l i b r a t e d  o f the l o a d - s t r a i n  was performd by  over a range o f  relationship.  t i o n curves f o r each b o l t a r e g i v e n i n F i g u r e s 74 and 7 5 .  indicated  Calibra-  3020  3000H  A A.  -g c  2980  A.  'A.  c  2960  A.  c 'CD  2940  CO  2920 'A  2900-  1  10  20  30  40  50  60  70  80  90  Load ( pounds ) F i g u r e Ik.  Strain versus load c a l i b r a t i o n b o l t 1.  curve f o r tension r o l l e r  support  100  11020  11000-  /-N  \  10980  -  JC  u c  \  2 _c u c  10960  —  10940  '5. c CD  0 0  10920-  10900-  10880 10  20  30  40  50  60  70  80  90  100  Load ( pounds ) Figure 75.  Strain versus load c a l i b r a t i o n b o l t 2.  curve f o r tension r o l l e r  support  Ln CO  APPENDIX C  DETERMINATION OF THE FORMING WIRE DRAINAGE RESISTANCE  160. APPENDIX C DETERMINATION OF THE FORMING WIRE DRAINAGE RESISTANCE  In  order  experimental resistance  t o do  a meaningful  results,  i t was  o f the f o r m i n g  comparison  necessary  wire  used  of the calcuated  to  determine  the  i n the experiments.  and t h e drainage  The measured  v a l u e o f t h e d r a i n a g e r e s i s t a n c e , k ^ , was then used i n t h e c a l c u l a t i o n s . r  The forming  drainage wire  supported  water,  over  one end o f a  vertically  introduced maintain  r e s i s t a n c e was  with  determined short  t h e head,  head  length  sample  of pipe.  and t h e f l o w  was measured.  and t h e c r o s s  Knowing  rate  the flow  o f the  The p i p e  t h e forming w i r e a t the bottom end.  a t t h e t o p end o f the p i p e  a constant  by g l u i n g a  was  Water was required to  rate  o f the  s e c t i o n a l area of the pipe, the drainage  r e s i s t a n c e was c a l c u l a t e d u s i n g the e x p r e s s i o n s  Q  =  vA  (C-l)  and  Combining  Equations  (C-2)  (P-P )  v =  a  C - l and C-2 and s u b s t i t u t i n g  the pressure  difference  with  P-P  a  =  Pgh  (C-3)  g i v e s the e x p r e s s i o n f o r the drainage r e s i s t a n c e  k  dr  pghA  (C-4)  161. For  the f o r m i n g  wire  used  i n the experiments,  i t was  found  a  flow  r a t e o f 1.2 g a l l o n s / m i n u t e was r e q u i r e d t o m a i n t a i n a head o f 28 i n c h e s i n a p i p e h a v i n g an i n s i d e diameter  o f 15/32 i n c h e s .  d r a i n a g e r e s i s t a n c e of the forming wire was found  k, dr  -  4.0 x I O  - 7  ft  U s i n g E q u a t i o n C-4, the to be  v  (C-5) '  S e v e r a l t r i a l s were done and t h e above v a l u e was found t o be r e p e a t a b l e t o within five  percent.  

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