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Fiber orientation in a headbox Zhang, Xun 2001

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FIBER O R I E N T A T I O N IN A  H E A D B O X  by  Xun  B. Eng. Eng.  A  Zhang  Northwestern Polytechnical University, China,  1985  Beijing University of Aeronautics and Astronautics, China,  THESIS  S U B M I T T E D IN P A R T I A L F U L F I L L M E N T  T H E R E Q U I R E M E N T S  OF T H E D E G R E E  M A S T E R OF APPLIED  OF  SCIENCE  in T H E F A C U L T Y OF G R A D U A T E D E P A R T M E N T OF M E C H A N I C A L  We  a c c e p t t h i s t h e s i s as  STUDIES  ENGINEERING  conforming  to the r e q u i r e d s t a n d a r d  T H E UNIVERSITY  OF BRITISH  January © X u n  2001  Zhang,  2001  C O L U M B I A  OF  1988  In p r e s e n t i n g this thesis i n partial f u l f i l l m e n t o f the r e q u i r e m e n t s f o r a n a d v a n c e d at  the  University  of  British  Columbia,  I  agree  that  the  library  shall  make  it  degree freely  a v a i l a b l e f o r r e f e r e n c e a n d study. I further agree that p e r m i s s i o n f o r e x t e n s i v e c o p y i n g  of  this thesis f o r s c h o l a r l y p u r p o s e s m a y be granted b y the h e a d o f m y d e p a r t m e n t or b y h i s or  her  representatives.  It  is u n d e r s t o o d  that c o p y i n g  financial gain shall not be a l l o w e d without m y written  Department  of Mechanical  Engineering  T h e University of British Columbia 2324 M a i n  Mall  Vancouver,  B C  V6T  1Z4  Date: January  2001  or  publication  permission.  o f this  thesis  for  ABSTRACT  The prediction of fiber orientation is a critical parameter for papermakers who wish to control the quality of their paper products. The wet end processes, especially the headbox and the drainage stage on the forming wire, play important roles in determining the fiber orientation characteristics. The current thesis is focused on the headbox flow effect on fiber orientation. It summarizes a mathematical method, which has been developed by other researchers, for predicting the orientation of rigid fibers in dilute suspensions. This method, composed of a turbulent flow simulation model and a fiber motion model, has been applied to predict fiber motion in a headbox. To validate the method, experiments have been conducted by measuring the orientation of dyed nylon fibers moving in a pilot plexiglass headbox. Comparison of experiments and the present numerical simulations of the fiber orientation shows that the simulation method proposed can predict the trend of the statistical orientation distribution of dilute suspensions in headboxes, although there exists obvious deviations between the simulations and experiments. The fibers are seen to be more strongly oriented by the predictions than is observed in the experiments. The anisotropy of the fiber orientation in the headbox flow is caused not only by the mean flow field characteristics, but also by the turbulence characteristics, and the explicit effects of the turbulence are not yet included in the predictions. The simulation method is applied to predict fiber orientations for different headbox geometry, fiber aspect ratio and flow rate. From the prediction method, using only the mean flow effects, a larger contraction ratio is found to produce more concentrated fiber orientation in the flow direction at the exit of the headbox. The channel length, the flow velocity and the fiber aspect ratio within the range of study have little influence on the fiber orientation properties.  ii  TABLE OF CONTENTS  ABSTRACT  ii  T A B L E OF CONTENTS  iii  LIST OF T A B L E S  v  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  viii  1. INTRODUCTION  1  2. L I T E R A T U R E R E V I E W  3  2.1  Fiber Orientation and Paper Quality  3  2.2  The Definition of Fiber Orientation  4  2.3  Factors Affecting Fiber Orientation  5  2.3.1  Headbox  6  2.3.2  Jet to Wire Speed Difference  8  2.3.3  Forming Wire  8  2.3.4  Fiber Suspension Consistency  9  2.4  Headbox Flow Simulations to Investigate Fiber Orientation  11  2.5  Fiber Suspension Simulation  12  2.6  The Scope of This Thesis Work  15  3. E X P E R I M E N T A L A R R A N G E M E N T S  17  3.1  Objectives of the Experimental Work  17  3.2  Fiber Suspensions  17  3.3  Flow Loop  18  3.4  Image Analysis System  19  iii  3.5  Measurement  20  4. C O M P U T E R S I M U L A T I O N O F F L O W A N D F I B E R O R I E N T A T I O N  28  4.1  The Headbox Flow Model  29  4.2  Fiber M o d e l  32  5. R E S U L T S A N D D I S C U S S I O N  36  5.1  A n a l y s i s o f the H e a d b o x F l o w F i e l d  36  5.2  C o m p a r i s o n o f Simulation and Experimental Results  39  5.3  Factors Affecting Fiber Orientation  44  5.3.1  T h e Effect o f Contraction Ratio o n Fiber Orientation  44  5.3.2  T h e Effect o f F l o w Rate on Fiber Orientation  45  5.3.3  T h e Effect o f Channel Length on Fiber Orientation  46  5.3.4  T h e Effect o f Fiber Aspect Ratio on Fiber Orientation  47  5.3.5  T h e Effect of F l o w Elongation  47  5.4  Symmetric Channel  48  5.5  Statistical E r r o r E s t i m a t i o n  49  6. S U M M A R Y A N D C O N C L U S I O N S  60  7. R E C O M M E N D A T I O N S F O R F U T U R E W O R K  62  8. N O M E N C L A T U R E  64  9. R E F E R E N C E S  66  iv  LIST OF TABLES  T a b l e 3.1.  T h e G e o m e t r y o f the H e a d b o x C o n v e r g i n g S e c t i o n  18  T a b l e 3.2.  T h e S i g n o f the O r i e n t a t i o n A n g l e s  21  T a b l e 3.3.  T h e N u m b e r o f F i b e r s at E a c h M e a s u r e m e n t P o i n t  22  T a b l e 5.1.  Orientation Parameters Obtained  from  Experiments and Simulations: T a b l e 5.2.  43  F i b e r O r i e n t a t i o n P a r a m e t e r s f o r D i f f e r e n t Rc  (Uo = 0.24 m / s , L = 0.225 m )  45  c  T a b l e 5.3.  T h e O r i e n t a t i o n P a r a m e t e r s f o r D i f f e r e n t Uo  (Rc= 10, L = 0.225 m )  46  c  T a b l e 5.4.  Fiber Orientation Parameters for Different L  c  (Rc= 10, U o - 0 . 2 4 m / s ) Table  5.5. T h e  46  Orientation Parameters for Different A  r  T a b l e 5.6.  T h e E l o n g a t i o n o f F l o w at t h e C h a n n e l E x i t f o r D i f f e r e n t Rc  T a b l e 5.7.  O r i e n t a t i o n P a r a m e t e r s at E x i t o f A Headbox forDifferent U :  ....47 48  Symmetric 48  0  v  LIST O F  F I G U R E S  F i g u r e 2.1.  F i b e r orientation distribution pattern i n a piece o f paper  16  F i g u r e 3.1.  T h e length distribution o f n y l o n fibers  23  F i g u r e 3.2.  I m a g e s o f f i b e r s : (a) d r y d y e d n y l o n f i b e r s , ( b ) f i b e r s u s p e n s i o n  23  F i g u r e 3.3.  T h e f l o w l o o p i n the e x p e r i m e n t  24  F i g u r e 3.4.  T h e s c a l e d p l e x i g l a s s h e a d b o x u s e d i n the e x p e r i m e n t  24  F i g u r e 3.5.  C r o s s s e c t i o n a l v i e w o f the s c a l e d h e a d b o x ( d i m e n s i o n s i n c m )  25  F i g u r e 3.6.  T h e p h o t o g r a p h i c a r r a n g e m e n t f o r (a) s i d e v i e w a n d ( b ) b o t t o m v i e w  25  F i g u r e 3.7.  T y p i c a l p i c t u r e o f f i b e r s i n t h e f l o w : (a) b e f o r e a n a l y s i s ; ( b ) a f t e r a n a l y s i s . .  26  F i g u r e 3.8.  T h e m e a s u r e m e n t points a l o n g the h e a d b o x c h a n n e l  27  F i g u r e 4.1. A fiber i n three-dimensional coordinates F i g u r e 4.2.  T h e initial r a n d o m distribution o f 1000  35 fibers  35  F i g u r e 5.1. T h e p h y s i c a l m e s h o f the a s y m m e t r i c c o n v e r g i n g s e c t i o n  52  F i g u r e 5.2.  T h e streamlines o f the f l o w i n the h e a d b o x c o n v e r g e n t c h a n n e l  52  F i g u r e 5.3.  T h e pressure a n d u - v e l o c i t y c h a n g e s a l o n g the central streamline  53  F i g u r e 5.4.  T h e u - v e l o c i t y c o n t o u r s o n the central s y m m e t r y p l a n e  53  F i g u r e 5.5.  T h e v - v e l o c i t y contours o n the central s y m m e t r y p l a n e  53  F i g u r e 5.6.  T h e e l o n g a t i o n o f the f l o w c h a n g e s a l o n g the central s t r e a m l i n e  54  F i g u r e 5.7.  T h e fiber o r i e n t a t i o n d i s t r i b u t i o n at x = 4 . 5  54  F i g u r e 5.8.  T h e fiber o r i e n t a t i o n d i s t r i b u t i o n at x =  12.2  cm,  55  F i g u r e 5.9.  T h e fiber o r i e n t a t i o n d i s t r i b u t i o n at x =  15.7  cm,  55  cm,  F i g u r e 5.10.  T h e fiber o r i e n t a t i o n d i s t r i b u t i o n at x =  19.2  cm,  56  F i g u r e 5.11.  T h e f i b e r o r i e n t a t i o n d i s t r i b u t i o n at x = 2 2 . 7  cm,  56  F i g u r e 5.12.  T h e fiber o r i e n t a t i o n d i s t r i b u t i o n at x = 2 6 . 2  cm,  57  F i g u r e 5.13.  The  F i g u r e 5.14.  T h e orientation parameters a l o n g the central streamline,  F i g u r e 5.15.  F i b e r o r i e n t a t i o n d i s t r i b u t i o n s at t h e c h a n n e l e x i t f o r v a r i o u s  fiber  o r i e n t a t i o n d i s t r i b u t i o n at t h e c h a n n e l e x i t ,  C o n t r a c t i o n r a t i o s , (a) i n x - y p l a n e , ( b ) i n x - z p l a n e  vi  57 58  58  F i g u r e 5.16.  C r o s s s e c t i o n a l v i e w o f the s y m m e t r i c h e a d b o x ( d i m e n s i o n s i n m m )  59  F i g u r e 5.17.  T h e p h y s i c a l m e s h o f the s y m m e t r i c h e a d b o x  59  A C K N O W L E D G E M E N T S  I  express  Gartshore, colleagues,  sincere for  gratitude  their  helpful  M o h a m m a d R.  to  my  advice  supervisors, and  Dr.  suggestions.  Shariati and S u q i n  Dong.  Martha I  would  Their  Salcudean also  effort  like  and to  Dr.  Ian  thank  my  a n d help has  things m u c h easier b o t h i n m y experimental w o r k a n d m y s i m u l a t i o n w o r k . I a m for the f i n a n c i a l support p r o v i d e d b y F R B C R e s e a r c h  years.  viii  grateful  Award.  F i n a l l y , I w i s h to t h a n k m y w i f e , Y i n g h u i , f o r h e r constant d u r i n g the past t w o  made  support and  encouragement  1. I N T R O D U C T I O N  Paper is a heterogeneous three-dimensional composite of fibers and other materials. Its mechanical properties are highly dependent on the microstructure characteristics such as fiber properties, and the formation and orientation distribution of the fibers. The demand for high quality paper and paperboard has focussed the attention of papermakers on how to control these critical characteristics in the papermaking processes.  The fiber orientation distribution in a piece of paper determines the distribution of strength, permeability and absorbency, and affects the dimensional stability, runability and printability of the paper.  The fiber orientation in paper is determined by the processing conditions in the wet-end stage of the headbox and in the forming process. Experimental evidence has shown that fibers have some preferred orientation direction depending on the specific flow field. The headbox has a significant effect on the orientation of fibers leaving the slice. The elongation and shear in the flow leading to the slice tend to orient fibers in the machine direction. If the fiber orientation can be predicted for a given set of processing conditions, manufacturing paper with optimum mechanical properties will become much easier.  The general objective of this thesis is to investigate, both numerically and experimentally, the three-dimensional fiber orientation produced by a dilute headbox flow. In the numerical simulations, both symmetric and asymmetric headboxes are studied.  The numerical simulation method introduced here provides a quantitative methodology for the prediction of the fiber orientation resulting from the fluid kinematics. It can be used to predict fluid-fiber interactions and provide paper manufacturers  a better  knowledge of fiber orientation distribution and sheet properties. In this research work, several elements which affect the fiber orientation in a headbox, such as the headbox  1  g e o m e t r y , f l o w c o n d i t i o n s a n d f i b e r p r o p e r t i e s , are i n v e s t i g a t e d w i t h the p r e d i c t i v e a b i l i t y o f this s i m u l a t i o n m e t h o d a n d the results are a n a l y z e d .  Following  this  experimental in  a  chapter,  the  relevant  literature  flow  s i m u l a t i o n o f the  are  fiber  presented  in  Chapter  5 presents the  s i m u l a t e d results.  studies,  research,  2.  The  detailed  Parametric  of headbox the  major  3.  Chapter  4  describes  the  numerical  orientation distribution b y a combination o f a f l o w m o d e l and  fiber m o t i o n m o d e l . Chapter  summarize  in Chapter  c o n d i t i o n s a n d m e t h o d s o f m e a s u r i n g the o r i e n t a t i o n d i s t r i b u t i o n o f f i b e r s  headbox  influence  is r e v i e w e d  geometry, conclusions  comparison o f measured  o b t a i n e d u s i n g the  flow velocity o f this thesis  respectively.  2  and  fiber  and numerically  numerical method, property.  a  Chapters  and give recommendations  show 6 for  and  the 7  future  2. LITERATURE REVIEW  2.1  A  Fiber Orientation and Paper Quality  piece  o f paper  is c o m p o s e d o f n u m e r o u s  f i b e r s w h i c h are  located  w i t h i n the  paper  p l a n e a n d o r i e n t e d i n different directions. Statistically, h o w e v e r , m o s t o f the fibers b e a l i g n e d i n o n e d i r e c t i o n . T h i s a n i s o t r o p y o f f i b e r orientation is p r o d u c e d b y the  may paper  m a n u f a c t u r e p r o c e s s a n d is c l o s e l y related to several critical p a p e r properties.  T h e o r i e n t a t i o n pattern r e t a i n e d i n the  final  the  i n d i c a t e d that d e p e n d i n g o n the  s h e e t . N o r d s t r o m a n d N o r m a n [1]  degree o f  fiber  p a p e r controls the m e c h a n i c a l properties grade,  a  of  certain  o r i e n t a t i o n a n i s o t r o p y i n the p a p e r is d e s i r e d . F o r n e w s p r i n t , a rather h i g h  a n i s o t r o p y is r e q u i r e d f o r g o o d r u n n a b i l i t y i n the p a p e r m a c h i n e a n d the p r i n t i n g p r e s s . B u t f o r w o o d - f r e e sheet grades, o n the other h a n d , a l o w e r a n i s o t r o p y is d e s i r e d to isotropic dimensional changes  with variations i n moisture  and temperature.  Nordstrom  a n d N o r m a n also p o i n t e d out that the strength i n the p a p e r t h i c k n e s s d i r e c t i o n is positively orientation  by  the  i n that  degree  of  direction.  fiber-to-fiber bonding and The  strength  i n the  paper  also  by  the  thickness  ensure  degree  direction  affected of  fiber  must  s u f f i c i e n t to a v o i d d e l a m i n a t i o n i n c o l d s e t offset p r i n t i n g , o r b l i s t e r i n g i n heatset  be  offset  printing.  A  sheet is stronger  a n d stiffer i n the  direction in w h i c h most  f i b e r s are  oriented,  and  w e a k e r a n d m o r e c o m p l i a n t i n the d i r e c t i o n o f least orientation. W e c a n u n d e r s t a n d this b y s t u d y i n g the properties o f the p r i n c i p a l constituents o f paper, the w o o d the  fiber  p r o p e r t i e s a n d p a p e r p r o p e r t i e s are c l o s e l y c o r r e l a t e d  flexibility  and  coarseness  are  connected  with  the  [2,  mechanical  3].  fibers,  because  Fiber dimensions,  properties,  structural  v a r i a b l e s a n d f o r m a t i o n o f p a p e r , s u c h as t e n s i l e s t r e n g t h , t e a r i n g s t r e n g t h , b u r s t i n g a n d b o n d i n g strength, p o r o s i t y a n d sheet density.  3  A  w o o d f i b e r h a s q u i t e d i f f e r e n t p r o p e r t i e s a l o n g its a x i s c o m p a r e d t o t h o s e a c r o s s it. F o r  example,  the  strength  of a  w h e r e a s the w e t - e x p a n s i v i t y  f i b e r is m u c h  greater  and  following  the  minor  conclusions  the  fiber axis  than  i s g r e a t e r a c r o s s t h e f i b e r a x i s t h a n a l o n g i t [4].  d i r e c t i o n is d e f i n e d as the d i r e c t i o n i n the p a p e r aligned  along  direction about  as  the  paper  surface  direction  property  be  I f the  toward which most  normal  can  across  to  the  inferred  major from  are  higher  i n the  major  direction,  but  the  tear  strength  are  direction,  the  the  and  major  fibers  fiber-paper  relationship. T h e tensile stiffness, tensile energy absorption, b e n d i n g stiffness a n d strength  it,  crush  wet  expanding  experimentally  correlated  t e n d e n c y are h i g h e r i n the m i n o r d i r e c t i o n .  Directional with  fiber  differences  in mechanical  o r i e n t a t i o n [5,  6, 7].  properties  have  been  S i g n i f i c a n t fiber m i s a l i g n m e n t m a y c a u s e s e r i o u s  l e a d i n g t o p o o r d i m e n s i o n a l s t a b i l i t y [8]  a n d r e d u c e d strength.  L o e w e n [6]  defects,  summarized  t h e p a p e r q u a l i t y p r o b l e m s t h a t a r e r e l a t e d t o p o o r fiber o r i e n t a t i o n a s f o l l o w s :  •  Twist, warp, curl and  •  W e b wandering, misregistering i n multi-pass printing and colour printing.  •  Paper feed path j amming.  •  Multi-part forms debonding.  •  L o w tensile strength, l o w tear strength a n d w e a k stiffness.  •  W r i n k l e s o n l o n g lead presses and dryer wrinkles.  2.2  stack-lean.  The Definition of Fiber Orientation  F i b e r o r i e n t a t i o n refers to the angular d i s t r i b u t i o n o f fibers relative to the d i r e c t i o n ( M D ) . T h i s c a n b e v i s u a l i z e d i n the p o l a r d i a g r a m  o f F i g . 2.1.  paper-machine The  distance  f r o m t h e o r i g i n at a g i v e n a n g l e i s p r o p o r t i o n a l t o t h e n u m b e r o f f i b e r s o r i e n t e d i n t h a t direction. T h e polar diagram describes fiber  two  c o m m o n l y used  fiber  orientation terms:  the  o r i e n t a t i o n a n g l e a n d fiber o r i e n t a t i o n i n d e x . T h e fiber o r i e n t a t i o n a n g l e , 9 a s s h o w n  i n the d i a g r a m , is the angle f r o m the m a c h i n e d i r e c t i o n i n w h i c h m o s t o f the  fibers  are  o r i e n t e d . T h e f i b e r o r i e n t a t i o n i n d e x is the ratio o f the f i b e r s o r i e n t e d i n the M D  over  t h o s e o r i e n t e d i n the c r o s s m a c h i n e d i r e c t i o n ( C D ) , w h i c h is o f t e n d e f i n e d as t h e r a t i o M D  to  C D  strength  based  on  the  knowledge  that  the  fiber  orientation  of  distribution  c o r r e s p o n d s t o t h e d i s t r i b u t i o n o f s t r e n g t h . T h e f i b e r o r i e n t a t i o n i n d e x o f F i g . 2.1  is e q u a l  to the ratio o f lengths a / b .  2.3  Factors Affecting Fiber Orientation  P a p e r is m a d e i n a c o n t i n u o u s process. f r o m the slice o f a h e a d b o x  T h e s u s p e n s i o n o f fibers a n d fillers is d i s c h a r g e d  a n d d i s t r i b u t e d at h i g h s p e e d o n t o t h e f o r m i n g w i r e . O n t h e  w i r e a sheet is f o r m e d t h r o u g h d e - w a t e r i n g . T h e sheet thus f o r m e d is w e t a n d w e a k  and  n e e d s to b e f u r t h e r p r o c e s s e d i n p r e s s e s a n d d r y e r s .  T h e p r i m a r y m e c h a n i s m o f o r i e n t i n g f i b e r s i n the sheet is the h y d r o d y n a m i c s h e a r f l o w s i n the early f o r m i n g section or wet e n d operations discharge  and  formation  process.  The  headbox  o f the p a p e r m a c h i n e , design  can  have  an  i.e.,  headbox  effect  on  the  o r i e n t a t i o n o f f i b e r s l e a v i n g the slice. T h e e l o n g a t i o n a n d shear l e a d i n g to the s l i c e t e n d to orient the fibers i n the m a c h i n e d i r e c t i o n .  Many  researchers  [4,  9,  10]  have  a n a l y z e d the  factors that affect f i b e r o r i e n t a t i o n  and  h a v e a g r e e d that the p r i m a r y m e c h a n i s m i n the sheet is the h y d r o d y n a m i c p r o c e s s i n the h e a d b o x d i s c h a r g e a n d f o r m a t i o n o p e r a t i o n s o f the p a p e r m a c h i n e . W r i s t [11]  studied the  f i b e r o r i e n t a t i o n i n the jet a n d o n the f o r m i n g w i r e a n d c o n c l u d e d that the relative arrangement  o f the fibers i n a m a c h i n e - m a d e  sheet o f p a p e r is v e r y l a r g e l y  b e t w e e n the h e a d b o x a n d the e n d o f the f o r m i n g table. W i t h i n this s p a c e , the  spatial  determined orientation  o f the fibers, the d e g r e e o f f l o c c u l a t i o n , the relative d i s t r i b u t i o n o f materials t h r o u g h  the  thickness  o f the  the  sheet are  all laid  sheet a n d the m a c r o down.  Subsequent  and micro-mass processes,  d i s t r i b u t i o n i n the p l a n e o f  like pressing  effects o n fiber orientation w i t h only some micro-rearrangement  and  drying,  have  minor  a n d c o n s o l i d a t i o n o f the  w e b . N e x t w e w i l l s u m m a r i z e the m a j o r f a c t o r s i n the w e t e n d p r o c e s s e s that l e a d to n o n -  5  u n i f o r m i t y o f f i b e r o r i e n t a t i o n . B e c a u s e this w o r k is f o c u s e d o n the h e a d b o x , w e w i l l start first w i t h the h e a d b o x wire types and  2.3.1  A  fiber  e f f e c t o n the  fiber  orientation, then consider relative wire  speed,  consistency.  Headbox  cross  machine  d i r e c t i o n a l u n i f o r m i t y . T w o areas that are f a c i n g i n c r e a s i n g l y stringent q u a l i t y  demands  are  fundamental  function o f any  u n i f o r m i t y o f basis  headbox  is to  weight profiles o n  ensure  finer  the  machine  scales a n d the  and  controllability o f  fiber  orientation profiles.  A h e a d b o x c a n b e d i v i d e d into three sections b y the p r i n c i p a l f l o w patterns i n v o l v e d f l u i d distribution, f l o w rectification a n d jet d e v e l o p m e n t .  In the  first  section,  a  [12]:  tapered  h e a d e r is u s e d to a c h i e v e i d e a l l y u n i f o r m f l o w i n t o the d i s t r i b u t o r . T h e n the f l o w f r o m the distributor is i m p r o v e d t h r o u g h the rectification p r o c e s s e s . I n a h y d r a u l i c h e a d b o x , a tube bank  is  often  disturbances  used  o r i g i n a t i n g i n the  n e e d e d to p r e v e n t the tube  fiber  bank may  eliminating  i n these processes. stock  flocculation  The  wall  approach  friction i n the  system  tubes  dampens  a n d creates turbulence  which  is  i n the p a p e r - m a c h i n e f o r m i n g z o n e . T h e p r o c e s s e s i n  i n c l u d e m i x i n g a n d b l e n d i n g o f separate f l o w s f r o m a  undesirable  flow  cross-flow  and  eddies,  improving  d e v e l o p i n g turbulence o f desired scale a n d intensity  [13].  the  T h e jet  velocity  distributor, profile  development  c a n b e d e s c r i b e d as d e l i v e r i n g the s t o c k to the s h e e t f o r m i n g s e c t i o n . A n i d e a l  and  process headbox  s h o u l d p r o d u c e a u n i f o r m a n d stable jet o v e r the w i d t h o f the m a c h i n e , w i t h o u t  lateral  v e l o c i t i e s a n d m a c h i n e - d i r e c t i o n perturbations. In brief, the h e a d b o x spreads the f l o w o f p u l p out o f the  stock  a p p r o a c h p i p i n g a l o n g the w i d t h o f the p a p e r m a c h i n e ,  provides  t u r b u l e n c e " b l e n d i n g " a n d d e l i v e r s the f u r n i s h to the m a c h i n e f o r m i n g s e c t i o n .  K y o s t i et a l . [ 1 4 ]  p o i n t e d out that i n a h e a d b o x ,  fiber  orientation c a n b e i n f l u e n c e d b y the  r e c i r c u l a t i o n rate, h e a d e r pressure  distribution, f l o w distribution units a n d h e a d b o x  p a t t e r n s . M a n y r e s e a r c h e r s [4,  16,  15,  tube  17] h a v e a g r e e d t h a t t h e a d j u s t m e n t o f t h e s l i c e l i p  6  p r o f i l e n o t o n l y d o m i n a t e s the basis w e i g h t p r o f i l e o f p a p e r i n the C D d i r e c t i o n , b u t a l s o s i g n i f i c a n t l y affects the the  fiber  orientation distribution profile. In a conventional headbox,  slice l i p shape is g o v e r n e d b y the basis w e i g h t p r o f i l e controller, w h i c h k e e p s  b a s i s w e i g h t at t h e r e e l a s f l a t a s p o s s i b l e . H o w e v e r , t h i s d e m a n d f o r a u n i f o r m  the  basis  w e i g h t c o m p e t e s w i t h the d e m a n d f o r u n i f o r m i t y o f fiber orientation p r o f i l e s , b e c a u s e  a  c h a n g e i n the shape o f the slice l i p m a y result i n s i g n i f i c a n t cross f l o w , w h i c h leads to  a  v a r i a t i o n i n f i b e r orientation i n the cross m a c h i n e d i r e c t i o n .  For a conventional headbox, fiber  orientation  revolutionary dilution  profiles  headbox  control  it i s i m p o s s i b l e t o a d j u s t b a s i s w e i g h t a n d  independently  design,  headbox,  has  which been  is  of  each  called  introduced  other.  the [9,  To  solve  consistency 18,  19,  20,  cross-machine  the  problem,  profiled 21,  22,  headbox  23,  24].  a or  This  h e a d b o x enables independent control o f C D basis weight and fiber orientation profiles. The  b a s i s w e i g h t p r o f i l e is c o n t r o l l e d b y v a r y i n g the  stock  consistency  profile in  the  h e a d b o x a n d t h e s l i c e l i p i s t h e n u s e d i n t h e c o n t r o l o f fiber o r i e n t a t i o n .  N o r d s t r o m and N o r m a n [1,7]  f o u n d that a h i g h h e a d b o x n o z z l e c o n t r a c t i o n ratio, w h i c h  is the ratio b e t w e e n the inlet area a n d the outlet area, c a n not o n l y generate a h i g h  degree  o f a n i s o t r o p y o f fiber o r i e n t a t i o n , b u t c a n a l s o i m p r o v e t h e f o r m a t i o n . T h e y a t t r i b u t e d t h i s effect  to  changes  the  enhanced  strength  o f the  elongational strain field  nozzle and  the  i n t u r b u l e n c e intensity. T h e a m o u n t o f e d d y d e f o r m a t i o n is d e p e n d e n t o n  the  d e g r e e o f c o n t r a c t i o n . U l l m a r a n d N o r m a n [25]  i n the  i n d i c a t e d that the c o n t r a c t i o n ratio o f the  j e t d e v e l o p i n g s e c t i o n p l a y s a n i m p o r t a n t r o l e i n fiber o r i e n t a t i o n at t h e n o z z l e e x i t . T h e i r results  i n d i c a t e d that the  effect  of  contraction  ratio  is m o r e  significant o n  the  fiber  o r i e n t a t i o n t h a n that o f the f l o w v e l o c i t y . T h e f i b e r s h a v e b e e n f o u n d to b e m o r e s t r o n g l y o r i e n t e d i n the m a c h i n e d i r e c t i o n f o r h i g h e r contraction ratio. B a n d h a k a v i a n d A i d u n  [26]  r e p o r t e d that the a c c e l e r a t i n g f l o w i n the c o n v e r g i n g s e c t i o n tends to o r i e n t the f i b e r s i n the m a c h i n e d i r e c t i o n , a n d stretch a n d rupture the floes. T h e t u r b u l e n c e i n the f l o w d e c r e a s e fiber o r i e n t a t i o n b u t m a y a l s o i m p r o v e t h e s u s p e n s i o n d i s p e r s i o n s .  7  may  Lee  and  Pantaleo  contributes  to  the  [17]  i n d i c a t e d that b e s i d e s  resultant  fiber  orientation  the  headbox,  the  depending on  the  forming process type  of  also  former,  and  o p e r a t i n g c o n d i t i o n s s u c h as j e t to w i r e s p e e d d i f f e r e n c e , w i r e t e n s i o n a n d d r a i n a g e rate. S e v e r a l o f t h e s e effects are s u m m a r i z e d i n the f o l l o w i n g  2.3.2  The  sections.  Jet to Wire Speed Difference  most  significant  factor  determining  the  fiber  orientation  is  usually  the  speed  d i f f e r e n c e b e t w e e n the jet a n d the f o r m i n g w i r e . Ideally, the jet is a s s u m e d to b e i n the m a c h i n e direction, but i n practice, the cross f l o w s varies  from  there exist s m a l l transverse  flows. T h e magnitude  l a y e r to l a y e r w i t h i n the jet, a n d a l s o v a r i e s a c r o s s the w i d t h  o f the jet. T h e d i f f e r e n c e b e t w e e n the jet s p e e d a n d w i r e s p e e d is u s u a l l y s m a l l . B u t a small cross f l o w m a y  of  cause a significant change  even  in fiber orientation angle w h e n  s u s p e n s i o n is d e l i v e r e d o n t o the f o r m i n g w i r e . T h i s is the r e a s o n w h y , i n the  the  industry,  f i b e r o r i e n t a t i o n is p r i m a r i l y c o n t r o l l e d b y c h a n g i n g the jet to w i r e s p e e d d i f f e r e n c e . A s the d i f f e r e n c e b e t w e e n the M D c o m p o n e n t o f jet v e l o c i t y a n d the w i r e s p e e d is i n c r e a s e d , the a v e r a g e f i b e r orientation angle is r e d u c e d , a n d the f i b e r orientation i n d e x is i n c r e a s e d [4].  2.3.3  Forming Wire  B e c a u s e the jet d i s c h a r g e d f r o m the slice m a y h a v e a n o n - u n i f o r m v e l o c i t y p r o f i l e d u e to b o u n d a r y effects a n d w a k e effects, a n d also has a n i m p i n g e m e n t a n g l e w h e n the stock is s p r e a d o n t h e w i r e , it i s i m p o s s i b l e t o e l i m i n a t e t h e d i f f e r e n c e i n t h e v e l o c i t y b e t w e e n  the  jet  the  a n d the  wire. T h e velocity difference m a y  cause shear  forces  i n the  region of  s t o c k - w i r e i n t e r f a c e , a n d p r o d u c e f u r t h e r v a r i a t i o n s i n t h e f i b e r o r i e n t a t i o n . E r i k k i l a et a l . [27]  p o i n t e d out that the  fiber  orientation f o r e a c h i n d i v i d u a l l a y e r o f the sheet is f i n a l l y  settled d o w n i n the d r a i n a g e p r o c e s s a n d is a f f e c t e d b y the shear, d e - w a t e r i n g v e l o c i t y the s u s p e n s i o n , c o n s i s t e n c y a n d the t u r b u l e n c e d u r i n g the p r o c e s s .  8  of  T h e d i f f e r e n c e i n the  manner  of  directions  de-watering, s u c h as  in one  in gap  direction  such  forming, produces  as  i n the  Fourdrinier  different orientation  case  or  in  two-sidedness  two  in  the  fiber orientation.  I n a d d i t i o n to  the  h y d r o d y n a m i c effect o n the  fiber orientation,  the  turbulence  effect  s h o u l d a l s o b e c o n s i d e r e d . T u r b u l e n c e is g e n e r a t e d i n the h e a d b o x a n d m a i n t a i n e d d u r i n g drainage  by  the  drainage  elements.  In  addition, turbulence  is  induced by  the  speed  d i f f e r e n c e b e t w e e n the s u s p e n s i o n a n d the w i r e . A s the t u r b u l e n t e n e r g y is i n c r e a s e d , average  fiber  orientation  angle  is  not  changed,  but  the  in-plane  a n i s o t r o p y is d e c r e a s e d , a n d the fiber orientation i n d e x is r e d u c e d  2.3.4  fiber  the  orientation  [1].  Fiber Suspension Consistency  T h e o r i e n t a t i o n p r o d u c e d at t h e function  o f the  fiber network  s l i c e is f o u n d to b e c o n s i s t e n c y strength.  Fiber-fiber interactions  sensitive, determine  i n d i v i d u a l f i b e r s c a n b e rotated w i t h the o r i e n t e d shear f i e l d . A t h i g h e r fibers  are  Kerekes  less  and  concentration  a l i g n e d i n the  S c h e l l [28] C  v  ,  the  f l o w direction, presumably  as  a  result  d e f i n e d a c r o w d i n g factor N , w h i c h is b a s e d fiber  length  L  and  diameter  d,  to  a n d to be  represent  how  a  many  concentrations, of  flocculation.  o n the the  volume  degree  of  flocculation:  N  As  the  consistency  experiments increases.  According  =  increases,  -C  (2.1)  v  3  the  w h i c h s h o w e d that the  crowding  factor  fiber alignment  will  increase.  d e c r e a s e s as  the  Ullmar  [29]  crowding  C u r l y f i b e r s w e r e a l s o f o u n d to b e less a l i g n e d t h a n straight f i b e r s [29,  to  their  concentrations,  regimes: dilute, semi-concentrated  fiber  suspensions  are  and h i g h l y concentrated.  9  usually  classified  did factor  30].  into  three  I f f i b e r s are c o n s i d e r e d to  be  rigid cylinders w i t h length L a n d diameter d, a n d o c c u p y a fraction C  v  o f the total v o l u m e  o f the s u s p e n s i o n , D i n h [31] s h o w s that t h e d i l u t e r e g i m e i s d e f i n e d w h e n C semi-concentrated  r e g i m e is d e f i n e d as ( d / L )  r e g i m e is d e f i n e d as C nearest  neighbor  between  fibers  less t h a n L  v  2  < C  v  fraction C  v  concentrated  > ( d / L ) . I n t h e d i l u t e r e g i m e , t h e d i s t a n c e b e t w e e n a fiber a n d i t s  is greater  than  L , so the fibers  are rare. I n the s e m i - c o n c e n t r a t e d  are free  to rotate,  and  diameter d. Three regimes  interactions  regime, the spacing b e t w e e n  b u t greater than d , a n d interaction b e t w e e n  fiber  2  < (d/L), a n d the h i g h l y  fibers  are frequent.  suspension falls into the h i g h l y concentrated regime, the spacing between order o f  < ( d / L ) , the  v  is  W h e n the  fibers  c a n also be d e f i n e d i n terms o f  fibers  fiber  is o n the volume  a n d fiber a s p e c t r a t i o A , w h i c h e q u a l s L / d [ 3 2 ] : r  T h e dilute r e g i m e is w h e n :  A,  C„  «  2  1  (2.2)  the s e m i - c o n c e n t r a t e d r e g i m e is g i v e n b y :  A;  <  2  C  V  <  v  (2.3)  a n d the concentrated r e g i m e is d e f i n e d as:  CA v  In  headboxes  o f conventional  b e t w e e n 0.1  a n d 1.5%,  between  and 200  30  suspension C um,  »  r  v  fiber [33].  1  paper  (2.4)  machines,  the  lengths vary between F o r example,  fiber  weight  consistencies  1 a n d 5 m m a n d aspect ratios  i f the v o l u m e  concentration  o f the  vary vary fiber  is 1%, fibers h a v e a u n i f o r m l e n g t h o f 3 m m a n d a u n i f o r m d i a m e t e r o f 4 0  then the aspect  ratio  regime, because A " < C r  2  v  A  is 75. T h e suspension is then  r  < A  r  _ 1  (that i s 0 . 0 1 8 % <  10  1% <  i n the  semi-concentrated  1.3%). T h e r e w o u l d t h e n exist  f r e q u e n t f i b e r - f i b e r interactions i n the h e a d b o x f l o w . T h e d i l u t e s u s p e n s i o n a s s u m p t i o n i n the current s t u d y is therefore a s i m p l i f i c a t i o n o f the actual p r o b l e m .  2.4  Headbox Flow Simulations to Investigate Fiber Orientation  Computer  s i m u l a t i o n has  engineering  equipment.  been The  widely used  simulation  i n the  study  investigations  of  not  processes only  meet  that the  u n d e r s t a n d i n g a n d prediction, but also have large e c o n o m i c benefits. Several have conducted headbox  f l o w s i m u l a t i o n s i n o r d e r to investigate  occur need  in for  researchers  the f l o w i n d u c e d f i b e r  orientation.  Aidun  [34,  35]  s t u d i e d the  secondary  f l o w s i n the  headbox  a n d their  effects o n  u n i f o r m fiber orientation and mass formation b y using a non-linear k-s turbulence to  investigate  the  characteristics o f turbulent f l o w i n a l o w consistency  non-  model  headbox.  The  a u t h o r i n d i c a t e d that the c a u s e o f n o n - u n i f o r m i t y i n f i b e r o r i e n t a t i o n i n the c r o s s m a c h i n e d i r e c t i o n is the s e c o n d a r y  f l o w s that are g e n e r a t e d  i n s i d e the h e a d b o x  i n d u c e d either  by  the g e o m e t r i c e f f e c t s a n d the k i n e m a t i c s , or b y the a n i s o t r o p y o f t u r b u l e n t f l o w s .  Lee  and Pantaleo  used a standard k-s turbulence  headbox  flow  w h e n d i f f e r e n t f l o w c o n t r o l d e v i c e s w e r e e m p l o y e d , s u c h as s l i c e p r o f i l i n g , e d g e  valve  [17]  m o d e l to a n a l y z e  c o n t r o l , b l e e d controls, tube inserts a n d header r e - c i r c u l a t i o n v a l v e s . T h e y e x a m i n e d relationship correlated  between  the  the h e a d b o x  headbox  flow  characteristics  and  the  fiber  orientation,  the and  f l o w characteristics i n terms o f f l o w angle obtained f r o m C F D  s o l u t i o n s w i t h t h e m e a s u r e d f i b e r o r i e n t a t i o n . T h e y t r i e d t o u s e t h e f l o w a n g l e [3, w h i c h i s defined  by  the  M D  velocity,  u,  P =  tan"  and  C D  velocity,  v,  to  represent  the  average  fiber  orientation angle:  ( -1  (2.5)  11  S h i m i z u a n d W a d a [36] a p p l i e d the k -  s  turbulence m o d e l a n d a  finite  difference method  to s t u d y the i n f l u e n c e w h i c h e l e m e n t s o f a n i m a g i n a r y h e a d b o x , s u c h as a t a p e r e d  header,  side w a l l , contracting part a n d slice l i p , have o n paper quality, e s p e c i a l l y the u n e v e n basis w e i g h t p r o f i l e s a n d fiber o r i e n t a t i o n . T h e r e s e a r c h e r s m e n t i o n e d a b o v e h a v e t h e c o m m o n p r o b l e m t h a t t h e y t r i e d t o s t u d y fiber o r i e n t a t i o n i n t h e f l o w w i t h o u t a s p e c i f i c s i m u l a t i o n o f fiber b e h a v i o r .  2.5  The  Fiber Suspension Simulation  first  fundamental study o f the orientation o f a rigid ellipsoidal particle i n a dilute  v i s c o u s N e w t o n i a n l i q u i d w a s c o n d u c t e d b y Jeffery [37]. H e s o l v e d the f l o w  field  around  a r o t a t i n g e l l i p s o i d b y s o l v i n g S t o k e s e q u a t i o n s , u s i n g a n o - s l i p b o u n d a r y c o n d i t i o n at t h e surface the  o f the particle. T h e angular velocity vector  requirement  that  the total  torque  acting  o n the particle  Jeffrey's equation describes a simplified case o f a field  fiber  be zero.  (  1  ^  .  ,  du  _ du dx  .  2  sin  - s i n ^ c o s ^ — + cos  dx  fiber.  du  tp—  dy  i , du <p  T h e a n g l e ^ , w h i c h is the angle b e t w e e n  T h e following  lying i n a two-dimensional flow  dv  2  (p—  + cos  dx  . i ,dv  s i n <p —  dy  dx  fiber  +  sin^cos^  dv_ dy,  . ,  (2.6)  , 5v  -fsin^cost?  axis a n dx-axis, describes the orientation o f  W h e n the aspect ratio is greater t h a n unity, the orientation o f the  fiber  changes  m a i n l y i n response to deformation or rotation o f the fluid. Besides rotation, fibers translate w i t h the v e l o c i t y that the u n p e r t u r b e d fiber.  from  [38].  . -smtpcostp  the  o f the particle w a s then f o u n d  fluid  also  w o u l d h a v e at t h e c e n t r o i d o f t h e  Jeffery's theory has been verified i n the experimental w o r k b y M a s o n a n d B a r t o k  [39].  12  Since Jeffery's w o r k , several constitutive m o d e l s have been d e v e l o p e d  from  which flow  i n d u c e d o r i e n t a t i o n c a n b e p r e d i c t e d f o r the dilute o r s e m i - c o n c e n t r a t e d  or  concentrated  s u s p e n s i o n s . R a o et a l . [ 4 0 ]  i n d i c a t e d that i n a c o m p l e x f l o w , i f the spatial stress g r a d i e n t s  d u e to f i b e r s are v e r y s m a l l c o m p a r e d to spatial v i s c o u s stress g r a d i e n t s , t h e n the b e h a v i o r is N e w t o n i a n , i.e. the p r e s e n c e  o f f i b e r s d o e s n o t alter the  flow  fluid  kinematics.  C o n s e q u e n t l y , the i m p l e m e n t a t i o n o f a n a n i s o t r o p i c m o d e l is not n e e d e d a n d the sole use o f J e f f r e y ' s e q u a t i o n is s u f f i c i e n t to c h a r a c t e r i z e the o r i e n t a t i o n f i e l d . O n the o t h e r h a n d , i f stress  gradient  suspending  contributions f r o m  fluid  contributions, the  the  particles  are  comparable  suspension exhibits  or  larger  non-Newtonian  than  the  characteristics  w i t h d i r e c t i o n a l l y d e p e n d e n t properties. T h i s necessitates the s i m u l t a n e o u s s o l u t i o n o f the flow  and  orientation  fields b y  using a  proper  anisotropic  constitutive  model,  or  by  function of  its  a c c o u n t i n g for particle/particle interaction i n s o m e other w a y .  The  behavior  o f an  individual  fiber i n a  dilute s u s p e n s i o n is  only  a  o r i e n t a t i o n a n d o f the f l o w f i e l d , s i n c e the f i b e r ' s o r i e n t a t i o n w i l l n o t b e a f f e c t e d b y other fibers.  Givler  orientation equation  et  [10]  have  developed a  i n dilute suspensions  along  streamlines  al.  was  the  streamlines.  numerical  to  solve  and i n confined geometries  by  The  order  velocity  field  o b t a i n e d b y a s s u m i n g that the  s h o w n i n their w o r k that shear  scheme  flows  fibers  used  in  f o r the  integrating to  fiber  Jeffery's  determine  d o not disturb the  flow.  a l w a y s i n d u c e a p e r i o d i c rotation o f the  It  the was  particles,  a n d particles w i t h large aspect ratios s p e n d m o s t o f the t i m e i n a p e r i o d a l i g n e d w i t h the streamlines o f the  a l t h o u g h t h e y are subject to a c y c l i c t u m b l e . F o r e x p a n s i o n  flow,  the fibers w i l l a s s u m e a transverse orientation w i t h respect to the streamlines o f the  flow.  Conversely, direction. flow.  flow  flow,  i n a c o n v e r g e n t g e o m e t r y w i l l orient fibers closer to the  Stable e q u i l i b r i u m orientation exists for elongational  In an elongation  flow,  direction o f stretching. A angle ^ = All  other  it i s w e l l k n o w n t h a t s t r e t c h i n g  rotate  flows  streamline  and not for align  fibers  shear i n the  f i b e r o r i e n t e d i n the p r i n c i p a l stretching d i r e c t i o n , o r i e n t a t i o n  0, i s i n s t a b l e e q u i l i b r i u m , a n d a fibers  flow  fluid  toward  the  fiber  stable  at § =  ±  /2  n  changing  m o n o t o n i c a l l y . T h e e v e n t u a l o r i e n t a t i o n d i s t r i b u t i o n is p e r f e c t l y a l i g n e d i n the  stretching  13  position  with  equilibrium. ^  direction.  equilibrium  is i n unstable  A k b a r a n d A l t a n [41]  use a c o m b i n a t i o n o f analytical solutions a n d statistical m e t h o d s  study fiber orientation behavior i n arbitrary t w o - d i m e n s i o n a l h o m o g e n e o u s  to  flows. T h e y  u s e a n o r i e n t a t i o n d i s t r i b u t i o n f u n c t i o n , w h i c h is g e n e r a t e d statistically b y c o n s i d e r i n g the f r e q u e n c y d i s t r i b u t i o n c u r v e o f the orientation o f a large n u m b e r o f fibers, a n d t h e y f o u n d that the a c c u r a c y o f the orientation d i s t r i b u t i o n f u n c t i o n is d e p e n d e n t o n the n u m b e r  of  fibers u s e d i n the a n a l y t i c a l s o l u t i o n .  As  the  suspension  concentration  increases towards  the  semi-concentrated  regime,  b e h a v i o r o f fibers changes because o f interactions between fibers. T h e interactions  the  cause  c h a n g e s i n the angles o f b o t h interacting fibers. In a concentrated  suspension each  fiber  interacts w i t h m a n y other fibers s i m u l t a n e o u s l y , so a m e c h a n i s t i c  m o d e l w o u l d be  very  d i f f i c u l t to  create.  mathematical concentrated  There  model and  to  are  some  predict  concentrated  studies  the  directly  orientation  suspensions.  relevant  to  distribution  of  Rao  [40]  the  development  rigid  provides  an  fibers  in  approach  of  a  semi-  for  the  s i m u l t a n e o u s s o l u t i o n o f the f l o w a n d o r i e n t a t i o n f i e l d s . In h i s r e s e a r c h , the o r i e n t a t i o n  of  the f i b e r s is first c o m p u t e d b y a s s u m i n g that the stresses g e n e r a t e d d u e to the p r e s e n c e  of  fibers  is  are  zero.  Then,  c o u p l e d b a c k to the  the  orientation  field computed  governing equations  f r o m the  Newtonian solution  o f f l o w to s o l v e the a n i s o t r o p i c f l o w o f f i b e r  suspensions.  Folgar  and Tucker  [38]  developed a m o d e l for concentrated  fiber suspensions,  fiber-fiber interactions  are t a k e n into a c c o u n t b y a d d i n g a d i f f u s i o n t e r m to the  equations.  a  describe  the  approach tensors. of  T h e y used  orientational  approach state.  and introduced  Advani  et  al.  [42]  an  orientation  proposed  a  orientation  Armstrong and  concentrated  in  rheological Tucker  [38]  43,  44]  efficient  orientation  investigated the t w o - a n d t h r e e - d i m e n s i o n a l  description  [45]  p o i n t e d out  or concentrated  to  of  semi-concentrated model  Jeffery's  function  more  f o r n u m e r i c a l s i m u l a t i o n o f f i b e r o r i e n t a t i o n w h i c h u s e s a set  A l t a n et a l . [ 4 1 ,  fiber  Folgar  fibers'  statistical  where  to  homogeneous  describe that  fiber  flow  motion  fields  14  fiber  using  in non-dilute  all o f these researchers  suspensions have observed  by  who  Dinh-  solutions.  studied  semi-  orientation behavior w h i c h  is q u a l i t a t i v e l y s i m i l a r to the d i l u t e s u s p e n s i o n m o d e l s . W h i l e m o s t  o f the studies  have  f o c u s e d o n suspensions o f rigid fibers, flexible fibers w e r e also investigated b y R o s s K l i n g e n b e r g [46],  W h e r r e t t et a l . [ 4 7 ] ,  a n d D o n g et a l .  [48].  T h e p r e s e n t f i b e r m o d e l u s e s the m e t h o d o f R o s s a n d K l i n g e n b e r g [46] s c h e m e o f D o n g [48],  2.6  and  a n d the n u m e r i c a l  b o t h o f w h i c h are b a s e d o n J e f f e r y ' s o r i g i n a l a s s u m p t i o n s  [37].  The Scope of This Thesis Work  T h e c u r r e n t r e s e a r c h i s p a r t o f a n e f f o r t at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a t o  develop  computational  directly  methods  to  s i m u l a t e the  m o t i o n o f fibers i n a w a y  that c a n b e  a p p l i c a b l e a n d b e n e f i c i a l to the p u l p a n d p a p e r i n d u s t r y . T h i s thesis is l i m i t e d to the o f dilute fiber suspensions  in a headbox  converging  section.  Two  models  study  have  been  d e v e l o p e d i n the U B C r e s e a r c h g r o u p : a turbulent f l o w m o d e l is u s e d to calculate headbox flow  field,  a n d a f i b e r m o d e l is u s e d to s i m u l a t e  fiber  motion. These two  the  models  h a v e b e e n c o m b i n e d t o g e t h e r a n d a p p l i e d t o t h e s t u d y o f fiber o r i e n t a t i o n i n a h e a d b o x i n this thesis research.  B y a p p l y i n g the s i m u l a t i o n m e t h o d , reasons f o r the  fiber  orientation  c a n b e i d e n t i f i e d a n d the h e a d b o x o r f l o w c o n d i t i o n s n e e d e d to e n h a n c e p a p e r q u a l i t y a n d increase production efficiency can be  A  scaled plexiglass headbox,  present  experiments  [49].  recommended.  b u i l t at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , i s u s e d i n t h e  Photos  o f d y e d n y l o n fibers i n the f l o w are  t a k e n at  several  l o c a t i o n s a l o n g the central streamline o f the c h a n n e l f r o m side a n d b o t t o m d i r e c t i o n . A n image  analysis  method  is  applied  c o m p a r i s o n b e t w e e n the statistical The  numerical  simulation method  to  measure  the  fiber  results o f e x p e r i m e n t s is  d i f f e r e n t f l o w rates, h e a d b o x g e o m e t r i e s  further  used  to  orientation  Direct  a n d p r e d i c t i o n s is p e r f o r m e d .  predict  a n d fiber a s p e c t r a t i o s .  15  angles.  the  fiber orientation  for  F i g u r e 2.1. F i b e r orientation distribution pattern i n a p i e c e o f paper.  16  3. EXPERIMENTAL ARRANGEMENTS  3.1  Objectives of the Experimental Work  T h e o b j e c t i v e o f the e x p e r i m e n t a l w o r k is to o b t a i n data to validate the s i m u l a t i o n m o d e l b y c o m p a r i s o n b e t w e e n the e x p e r i m e n t a l results a n d the s i m u l a t i o n results.  3.2  Fiber Suspensions  T h e f i b e r s a r e m a d e o f n y l o n a n d h a v e a n o m i n a l l e n g t h o f 3 m m a n d c o a r s e n e s s o f 15 d e n i e r (1 d e n i e r =  1 g / 9 0 0 0 m ) . A s i m p l e c a l c u l a t i o n gives the w i d t h o f the fiber as 4 4 ^ m  (the d e n s i t y o f n y l o n f i b e r s is 1,140 k g / m ) . A s a result, the f i b e r a s p e c t ratio, w h i c h is 3  the ratio b e t w e e n experiment  fiber length a n d fiber w i d t h , is 68. N y l o n fibers w e r e c h o s e n  because they  can be colored, have  a density close  f o r the  to water, c a n b e c u t to  s p e c i f i c l e n g t h s a n d c a n b e c o n s i d e r e d as r i g i d r o d s . T h e f i b e r s w e r e d y e d w i t h R i t ® m a r i n e 30 b y s o a k i n g t h e m overnight. In the experiment,  blue  water w a s u s e d as the w o r k i n g  f l u i d . B e t w e e n 2 0 0 0 a n d 3 0 0 0 fibers w e r e p l a c e d i n e a c h liter o f w a t e r f o r t r a c k i n g s i n g l e fibers b y means was  o f photos. A s the consistency  w e l l w i t h i n the dilute regime,  is n o m o r e t h a n 0 . 0 0 1 % , the s u s p e n s i o n  which means  there w a s little i n t e r a c t i o n  between  fibers.  T h e lengths o f d r y n y l o n fibers w e r e tested w i t h the i m a g e analysis system. I n a  sample  o f 2 0 0 f i b e r s , 9 5 % o f the f i b e r s l a y i n the r a n g e o f 2.4 m m to 3.2 m m (the m e a n  fiber  l e n g t h w a s 2.8 m m ) . T h e d i s t r i b u t i o n o f the f i b e r l e n g t h is s h o w n i n F i g . 3.1. M o s t o f the fibers w e r e straight o r n e a r l y straight w h e n the fibers w e r e i n the d r y c o n d i t i o n o r i n the s u s p e n s i o n as s h o w n i n F i g . 3.2.  17  Flow Loop  3.3  T h e e x p e r i m e n t a l s e t - u p [49] u s e d a c l o s e d f l o w s y s t e m d i a g r a m m a t i c a l l y s h o w n i n F i g . 3.3.  E x p e r i m e n t s w e r e c o n d u c t e d i n a transparent plexiglass h e a d b o x , s h o w n i n F i g . 3.4,  to a l l o w f o r v i s u a l i n s p e c t i o n o f the f l o w . T h i s h e a d b o x headbox  w i t h the size  reduced  by a  factor  o f 5.  is a s c a l e d m o d e l o f a t y p i c a l  In the f l o w  loop, the dilute  fiber  s u s p e n s i o n is p u m p e d f r o m the reservoir tank, w h i c h c a n contain a total v o l u m e o f 3 m  3  o f f l u i d , to the h e a d b o x through the pipes a n d rectifier tubes. T h erectifier tubes are r o u n d at t h e i n l e t a n d r e c t a n g u l a r at t h e o u t l e t w i t h s l o w l y i n c r e a s i n g c r o s s s e c t i o n a l a r e a s a n d are t y p i c a l l y u s e d b o t h to p r o v i d e the t u r b u l e n c e e n e r g y n e e d e d f o r f i b e r d i s p e r s i o n a n d t o g e n e r a t e a f a i r l y u n i f o r m v e l o c i t y p r o f i l e at t h e c o n v e r g i n g s e c t i o n i n l e t . A t t h e o u t l e t o f these tubes,  it i s a s s u m e d  (and verified b y observation)  that the fibers are o r i e n t e d  r a n d o m l y b e c a u s e o f the turbulence effects o n the fibers. A l t o g e t h e r there are 4 0  rectifier  tubes, t w o r o w s i n the h e a d b o x height direction a n d 20 i n the s p a n direction. T h e f l o w t h r o u g h e a c h t u b e i s m e t e r e d a n d a d j u s t e d i n d i v i d u a l l y , s o that t h e f l o w rate at e a c h  tube  e x i t i s 0 . 3 4 l i t e r p e r s e c o n d . A s a r e s u l t , t h e a v e r a g e v e l o c i t y at t h e i n l e t o f t h e c o n v e r g i n g section is 0.24 m / s .  A f t e r travelling t h r o u g h the asymmetric c o n v e r g i n g section, the f l o w is finally d i s c h a r g e d at t h e n o z z l e o r " s l i c e " . T h e c o n v e r g i n g s e c t i o n s t a r t s w i t h a r e c t a n g u l a r remains constant  channel w h i c h  i n cross sectional area until the c h a n n e l length reaches 0 . 0 9 2 2 m (the  o r i g i n i s at t h e e n t r a n c e ) . D o w n s t r e a m o f t h i s p o i n t , t h e c h a n n e l c o n v e r g e s t o t h e n o z z l e w i t h a c o n t r a c t i o n r a t i o o f 10. D e t a i l s o f t h e h e a d b o x g e o m e t r y a r e g i v e n i n T a b l e 3.1 a n d F i g . 3.5 p r e s e n t s t h e c r o s s s e c t i o n a l v i e w .  T a b l e 3.1. T h e G e o m e t r y o f the H e a d b o x C o n v e r g i n g Section.  parameters  values  width  0.76 m (constant)  inlet height  0.075 m  slice height  0.0075 m  contraction length  ratio  10 0.3172 m 18  It  is important  to  avoid  entrapment  o f air bubbles  i n the s u s p e n s i o n  when  taking  photographs, because the b u b b l e s m a y deteriorate the quality o f pictures. A i r b u b b l e s are avoided  b y circulating the s u s p e n s i o n f l o w t h r o u g h the h e a d b o x  f o r at l e a s t 2  hours  before taking pictures.  3.4  Image Analysis System  A n i m a g e analysis m e t h o d w a s u s e d i n this study o f fiber orientation. T o detect the fibers in the f l o w , a n O p t i k o n M o t i o n S c o p e C C D (charge-coupled Cosmicar/Pentax  device)  video system  with  T V lens ( 3 . 7 m m , 1:1.6) a n d a S o n y D C R - T R V 3 2 0 d i g i t a l v i d e o c a m e r a  w e r e m o u n t e d a n d connected together. T h eO p t i k o n system captures u p to 5 0 0 frames p e r second with a resolution o f 336  x  243 pixels for each picture. T h e S o n y c a m e r a has a  s h u t t e r s p e e d o f 1/4 t o 1 / 4 0 0 0 s e c o n d .  W h i l e the O p t i k o n c a m e r a w a s u s e d to capture  pictures o f fibers i n the h i g h - s p e e d f l o w z o n e , i.e. v e r y close to the exit, the S o n y c a m e r a was  used  where  experiment,  the f l o w speed  w a s not too high (channel  t h e c a m e r a s w e r e m o u n t e d at s e v e r a l  locations  length  <  26  c m ) . In the  either b e l o w o r beside the  h e a d b o x (as s h o w n i n F i g . 3.6) i n o r d e r to o b t a i n v i e w s f r o m b e n e a t h o r f r o m the side o f the h e a d b o x . machine  F i g . 3.6 also defines the three d i m e n s i o n a l coordinates:  direction, the y-axis  represents the paper  represents the headbox span direction (cross-machine  L i g h t i n g is extremely  thickness  the x-axis is i n the  direction, a n d the z-axis  direction).  important f o r obtaining g o o d pictures. B a c k lighting w a s p r o v i d e d  w i t h a 150 w S y l v a n i a f l o o d l i g h t b u l b . A sheet o f fine g r o u n d glass, 4 m m i n t h i c k n e s s , w a s u s e d to scatter the l i g h t f o r better p h o t o g r a p h qualities.  In the experiment, v i d e o pictures were taken o f fibers i n m o t i o n i n the plexiglass channel. T o establish the f i e l d o f focus, a tube w a s temporarily inserted into the channel.  Cameras  w e r e f o c u s e d at t h e c e n t e r o f t h e c h a n n e l w h i l e t a k i n g p i c t u r e s f r o m t h e b o t t o m a n d i n a p l a n e 6 c m next to the side c h a n n e l w a l l w h e n t a k i n g pictures  f r o m s i d e v i e w s . It w a s  difficult to control the c a m e r a to ensure a very shallow depth o f field, so i n each  19  picture  the fibers may have been located at a different distance from the camera, from the boundary near the wall to the far inside of the flow, about 20 cm away from the wall.  The Sony PictureGear 4.1 Lite software was used to download the video pictures from the recorder to the P C . Matrox Inspector software was then used to evaluate fiber orientations from the recorded images. Matrox Inspector has the power to automatically recognize a target and measure the required parameters, such as length, width, angle, area, etc. When this software is used to deal with a picture of fibers, automatic measurement becomes difficult. The prime problem is contrast. Fibers have a large length-to-width ratio and the width of a fiber is only a very small fraction of the field of view, so that the contrast of the fiber in the picture is very poor. There were also some fine scratches on the plexiglass plates, and the channel width was too large to have clear fiber pictures from the side views. The resolution of the cameras also needs to be improved. A l l of these made it difficult to distinguish fibers from their background automatically. If the automatic function of the software is used, a single fiber is often viewed as several segments and treated as several separate fibers by the computer, or some fibers are not recognized at all. Therefore, the measurements were conducted with the software but the recognition of the fibers was completed manually to ensure the quality of measurement. The quantitative results were obtained by further processing the measured data.  3.5  Measurement  In order to collect enough data for statistical analysis, hundreds of pictures were taken at each experimental point. There are between 5 and 30 fibers in each picture. The resulting sample size of 1600 to 2400 fibers at each measurement point represents a compromise, from a statistical viewpoint, between accuracy and effort. The orientation of fibers was evaluated by measuring the angle between the line connecting the two ends of a fiber and the machine direction (x-axis). This definition is reasonable when almost all the fibers are straight or nearly straight. Fiber orientation angles vary between - 9 0 ° and + 9 0 ° with 0 ° corresponding to the paper machine direction. The determination of the sign of an  20  orientation angle depends upon the location of the fiber-end in the Cartesian coordinate system with the origin located at the mid-point of the fiber, as shown i n Table 3.2. During the measurement, blurry fibers, fibers located outside the border of interest, and highly curved fibers were ignored. A typical picture of fibers in the flow is shown in Figure 3.7.  Table 3.2. The Sign of the Orientation Angles view or plane  the quadrant  sign of angles  side view or  1  -  on x-y plane  IV  +  bottom view or  1  +  on x-z plane  IV  -  In order to compare the experimental results with the simulation results, for the side view case, these studies were restricted to measurements close to the central streamline of the converging section, because fiber orientations on the central streamline were simulated i n the computational study. It is important not to bring the effect of the tubes upstream of the headbox on fiber orientation into the measurement. This can not be avoided entirely for the side view case, because the central streamline is located i n the wake of a tube wall. One can assume that the wake effect on fiber orientation is quickly lost as the fibers enter the converging channel and are subject to strong stretching of the flow [50]. W h i l e taking pictures from the bottom view, the measurements are made along both the centerline o f one rectifier tube and the line extended downstream from a tube wall. The results from these two categories of measurements are then mixed together to produce the final averaged results.  The measurements were taken at several points along the headbox channel as shown i n Figure 3.8. The first point is selected very close to the inlet of the channel where x = 4.5 cm. This point is still within the flat section so that the fiber orientation situation should not change as the fiber moves downstream in the constant area section. The second point, x = 12.2 cm, is located near the beginning of the converging section. The following points are further downstream i n the converging section where x = 15.7 cm, 19.2 cm, 22.7 cm, 26.2 c m and 31cm. The point x = 31 cm is very close to the exit of the channel  21  and therefore o n l y bottom v i e w measurements were conducted. side c o u l d not b e obtained  at t h i s p o i n t . D u r i n g  specific  i.e. the fibers  area  are counted,  in a  measurements, square  area  C l e a r pictures  f r o m the  o n l y the fibers i n the  of 2  x  2  cm  2  when  the  m e a s u r e m e n t is c l o s e to the inlet (x < 2 0 c m ) a n d l x l c m w h e n the m e a s u r e m e n t is c l o s e 2  t o t h e e x i t . T h e n u m b e r o f f i b e r s c o u n t e d at e a c h m e a s u r e m e n t p o i n t i s l i s t e d i n T a b l e 3 . 3 .  T a b l e 3 . 3 . T h e N u m b e r o f F i b e r s at E a c h M e a s u r e m e n t  x-positions  Point  number of fibers in x-y plane  in x-z plane  1  1325  1638  2  2031  1793  3  2066  2885  4  2056  2325  5  1554  2333  6  1823  1356  7  N/A  1586  22  1.4-1.6  1.6-1.8  1.8-2  2-2.2  2.2-2.4  2.4-2.6  2.6-2.8  2.8-3  3-3.2  3.2-3.4  fiber length (mm)  F i g u r e 3.1. T h e length distribution o f n y l o n fibers.  F i g u r e 3.2. Images o f fibers: (a) d r y d y e d n y l o n fibers, (b) fiber suspension.  23  Y A  Fluid Tank  Figure 3.3. The flow loop in the experiment.  F i g u r e 3.5.  C r o s s sectional v i e w o f the s c a l e d h e a d b o x ( d i m e n s i o n s i n c m ) .  "I  z  4  Headbox  *x  Headbox  [k  Lighting  Camera  Lighting  Camera  (a)  F i g u r e 3.6.  (b)  T h e p h o t o g r a p h i c a r r a n g e m e n t f o r (a) s i d e v i e w a n d ( b ) b o t t o m v i e w .  25  (a)  (b) F i g u r e 3.7.  T y p i c a l p i c t u r e o f f i b e r s i n t h e f l o w : (a) b e f o r e a n a l y s i s ; ( b ) a f t e r a n a l y s i s . ( X d i r e c t i o n is the m a c h i n e d i r e c t i o n . )  26  0.1  -  —  0.08  central streamline 4> focus point  0.06 0.04  - -  0.02 -  0 -0.02  0  , , i , , , , i , 0.05  1  0.1  1  0.15  , , , , 1, 0.2  0.25  X  Figure 3.8. The measurement points along the headbox channel.  27  0.3  4. C O M P U T E R S I M U L A T I O N O F F L O W A N D F I B E R O R I E N T A T I O N  The  fiber orientation  especially  i n the  in a piece  headbox  o f paper  a n d o n the  is d e t e r m i n e d  forming wire.  b y the  Efforts  papermaking  must  be  made  process, to  derive  quantitative relationships between processing conditions a n d fiber orientations. T h e intent is to  learn h o w  orientation  to  states,  design and control so  as  to  o b t a i n the  m a n u f a c t u r i n g p r o c e s s e s to best  possible  p r e d i c t i o n o f f i b e r o r i e n t a t i o n that is r e q u i r e d f o r the  paper  generate  products.  To  favorable  perform  the  design and process control,  one  m u s t h a v e a n a c c u r a t e q u a n t i t a t i v e m o d e l o f the w a y f i b e r s c h a n g e o r i e n t a t i o n as  they  m o v e i n the f l o w .  To  s i m u l a t e f i b e r m o t i o n , t w o m o d e l s n e e d to be d e v e l o p e d a n d e f f e c t i v e l y  combined  t o g e t h e r . T h e first m o d e l is u s e d to d e s c r i b e the f l u i d m o t i o n i n a 3 - d i m e n s i o n a l d o m a i n , as  c o n s t r a i n e d b y the  specific boundary conditions. T h e second m o d e l describes  m o t i o n a n d o r i e n t a t i o n i n the f l o w f i e l d . T h e i m p o r t a n t p a r t o f the m e t h o d is to  fiber  combine  these t w o m o d e l s . T h e s i m u l a t i o n m e t h o d is u s e d here f o r the s o l u t i o n o f f i b e r o r i e n t a t i o n in  a  flow  field  of  a  Newtonian  fluid  where  the  fibers  do  not  alter  the  flow.  For  suspensions w i t h higher v o l u m e fractions, these s o l u t i o n techniques c a n be u t i l i z e d w i t h some modifications.  C u r r e n t l y , t h e a b o v e t w o m o d e l s are u n c o u p l e d i n that the f i b e r o r i e n t a t i o n state w i l l  not  alter the g o v e r n i n g e q u a t i o n s f o r the f l o w . F i b e r o r i e n t a t i o n s are c a l c u l a t e d s u b s e q u e n t the v e l o c i t y f i e l d d e t e r m i n a t i o n , h e n c e , the t w o m o d e l s m a y b e s o l v e d c o n s e c u t i v e l y .  The  d e t a i l e d a p p r o a c h c a n b e d e s c r i b e d as f o l l o w s . F i r s t l y , the f l o w f i e l d is p r e d i c t e d b y solution  o f the  Reynolds averaged  rotation  o f a r i g i d f i b e r is d e s c r i b e d b a s e d o n N e w t o n ' s  angular  momentum.  The  angular  Navier-Stokes  velocity  of  a  equations.  fiber  Then  the  Second L a w  depends  upon  translation a n d the the  c o n d i t i o n s , s u c h as v o r t i c i t y a n d the c o m p o n e n t s o f the rate o f d e f o r m a t i o n t e n s o r .  28  the and  law  local  to  of  flow  4.1  In  The Headbox Flow  the  simulation  suspension.  study,  Model  the  T h e consistency  liquid o f the  is  assumed  suspension  to  be  is v e r y  pure low,  water so  or  a  dilute  there is n o  fiber  interaction  b e t w e e n f i b e r s , a n d the f i b e r s d o not affect the f l o w f i e l d . T h e f i b e r s are u n i f o r m a n d l o n g e n o u g h s o that the B r o w n i a n m o t i o n c a n b e i g n o r e d [51].  T h e suspension can  be  fluid.  viewed  as  a  uniform  incompressible  Newtonian  Based  on  therefore  the  a s s u m p t i o n s , the available c o n t i n u u m theories c a n be u s e d for the dilute s u s p e n s i o n  The  numerical  dimensional  simulation  incompressible  of  the  flow  Reynolds  has  been  averaged  carried  out  Navier-Stokes  by  solving  equations.  the  above [32].  three-  Turbulence  c l o s u r e is o b t a i n e d b y the use o f the s t a n d a r d k - s m o d e l w i t h the w a l l f u n c t i o n treatment. W e c a n w r i t e the c o n t i n u i t y e q u a t i o n i n the f o r m of:  V u = 0  (4.1)  T h e e q u a t i o n o f conservation o f m o m e n t u m is:  =  u-Vu  -Vpl p + V -r I p  (4.2)  where: u = instantaneous  fluid velocity vector  p = m o d i f i e d pressure i n c l u d i n g the gravitational p =  forces  density  x = f l u i d stress tensor.  F o r a N e w t o n i a n fluid w i t h constant viscosity,  V-T  =  //V u 2  (4.3)  where:  29  ^ = d y n a m i c viscosity o f the fluid.  I n t u r b u l e n t f l o w , the v e l o c i t y a n d pressure c a n b e e x p r e s s e d as a m e a n a n d a f l u c t u a t i n g part:  u  =  u + u'  (4-4)  p  =  p + p'  (-) 4  5  In the equations, the over-bar denotes the m e a n , a n d the p r i m e indicates the f l u c t u a t i n g component.  A f t e r substituting E q u a t i o n s (4.3), (4.4) a n d (4.5) into E q u a t i o n s (4.1) a n d  (4.2), a n d t a k i n g a t i m e average, the g o v e r n i n g equations o f the turbulent f l o w b e c o m e :  V-u  =  uVu  N o w the stress tensor  0  (4-6)  = -Vp/p  + V-r/p  (-) 4  7  i n c l u d e s b o t h the v i s c o u s a n d turbulent R e y n o l d s stress tensors:  x  =  V T  /J  V u +V 2  - ( - p  u\u'  j  )  (4-8)  where:  r,  can  be  used  to  =  simplify  uTV;  Equation  (i,j = l , 2 , 3 )  (4.8).  (4.9)  The Reynolds  stress  tensor  T  ..  introduces  a d d i t i o n a l u n k n o w n s f o r the turbulent f l o w . In order to describe the m e a n v e l o c i t y a n d pressure  fields,  Reynolds linear k -  a  closure  stress tensor s  f o r m u l a t i o n is necessary  to the m e a n  to  relate  f l o w velocity or velocity  m o d e l [52] is e m p l o y e d to s o l v e the c l o s u r e p r o b l e m .  30  the c o m p o n e n t s gradients.  o f the  T h e standard  T h e R e y n o l d s stress t e n s o r c a n t h e n b e e x p r e s s e d as:  T,J  =  v S t  --kS  u  (4.10)  v  where:  du,  +  dx,  k  is the turbulent  kinetic  i  (4.11)  dx,  (4.12)  — u'u'  =  2  k  du  '  '  energy,  5..  is the K r o n e c k e r  Delta, and  k i n e m a t i c v i s c o s i t y w h i c h , u n l i k e its l a m i n a r c o u n t e r p a r t ,  varies  V  t  is the turbulent  spatially a n d is not a  property o f the f l u i d . T h e transport equation f o r k is:  lr-  5  dk  9  ku,  +  dx  =  (4.13)  G-E  where:  G  =  E  =  (4.14)  -u] Uj S  a  and  G  and E  are the rates  respectively.  o f kinetic  T h e transport  (4.15)  VS'iS'g  energy  production  a n d dissipation per unit  mass,  equation for the kinetic energy dissipation E is g i v e n i n the  f o r m of:  zrdx, EU:1 5  8  =  dx  v+-  31  ^ dx,  +  ( C l  G-c  2  E)^  K  (4-16)  A  f i n a l c o r r e l a t i o n , b a s e d o n a n isotropic v i s c o s i t y a s s u m p t i o n f o r the turbulent  i n t e r m s o f k a n d E , as g i v e n b y  =  V {  k / E , c l o s e s the s y s t e m o f e q u a t i o n s . 2  T h e u s u a l v a l u e s o f the constants are: K  2  / [ ( C - C , ) C ],  A  where  m  2  finite v o l u m e  K  method  = 0.41  viscosity  =  0.09,  Cj =  1.44,  C  =  1.92,  curvilinear  grids  2  a  =  k  1.0,  a  E  =  is the V o n K a r m a n constant.  in conjunction  w i t h general  is  used  c o m p u t a t i o n a l c o d e , w h i c h w a s d e v e l o p e d i n o u r r e s e a r c h g r o u p b y N o w a k [53].  in  the  Use and  v a l i d a t i o n o f t h i s c o d e f o r h e a d b o x f l o w s h a s b e e n r e p o r t e d i n t h e w o r k o f S h a r i a t i et a l . [49]  andHuaetal.  4.2  Fiber  [50].  Model  T h e m a i n c o n c e r n o f this w o r k is the p r e d i c t i o n o f f i b e r orientation i n the h e a d b o x  flow  f i e l d . I n this w o r k , the f i b e r orientation is c a l c u l a t e d f o r a g i v e n v e l o c i t y f i e l d , d e c o u p l i n g the f l o w a n d f i b e r o r i e n t a t i o n c a l c u l a t i o n s . T h e o r e t i c a l l y , the s o l u t i o n s are o n l y v a l i d f o r zero  fiber  consistency,  because  the  fibers  influence  the  solutions, this m o d e l s h o u l d not give significant errors.  At  flow. fiber  However,  for  locations, the  dilute  velocity  a n d v e l o c i t y g r a d i e n t s a r e c a l c u l a t e d b y i n t e r p o l a t i n g t h e v a l u e s at n e i g h b o r i n g n o d e s  and  are  fiber  used  to  determine  the  translation  and  rotation  o f the  fiber.  The  calculated  p o s i t i o n a n d o r i e n t a t i o n are u s e d as initial c o n d i t i o n s f o r the c a l c u l a t i o n o f the  fiber's  new  l o c a t i o n a n d o r i e n t a t i o n at t h e n e x t t i m e s t e p (t + A t ) .  The  fiber  [46,  54]  m o t i o n m o d e l u s e d i n this w o r k w a s a n d a d a p t e d b y D o n g [48].  this m o d e l . D o n g  s h o w e d the  first  developed by Ross and Klingenberg  J e f f r e y ' s e q u a t i o n [37]  i d e n t i t y o f the  results  is u s e d for the v e r i f i c a t i o n o f  f r o m both approaches.  f l e x i b l e f i b e r s c a n b e s i m u l a t e d w i t h this m o d e l , o n l y r i g i d f i b e r s are thesis. A s d e s c r i b e d b y D o n g , a this thesis  research,  a rigid  fiber  fiber  Although  s i m u l a t e d i n this  is represented b y o n e o r m o r e prolate s p h e r o i d s . I n  is represented  l e n g t h L a n d d i a m e t e r d . T h e d e n s i t y o f the  fiber  32  b y one prolate  spheroid with uniform  is the s a m e as the f l u i d . T h e m o t i o n o f a  fiber  is d e t e r m i n e d b y s o l v i n g the t r a n s l a t i o n a n d r o t a t i o n e q u a t i o n s w h i c h are b a s e d  Jeffery's original work  T h e orientation o f a expressed fiber  [37].  fiber  i n F i g u r e 4.1.  is t h r e e - d i m e n s i o n a l a n d c a n b e d e t e r m i n e d b y t w o  angles  as  T h e a z i m u t h a l a n g l e <j, i s t h e a n g l e b e t w e e n t h e p r o j e c t i o n o f t h e  a x i s o n the x - y p l a n e a n d the y - a x i s  with 0 <  <j, <  (one  n  e n d o f the  d i s t i n g u i s h a b l e f r o m the other). T h e p o l a r a n g l e 0 is the a n g l e b e t w e e n the the z - a x i s w i t h 0 < Q <  on  N  fiber  fiber  is  not  axis and  .  T h e m o t i o n o f the f i b e r relative to the s u r r o u n d i n g f l u i d is so s m a l l that the i n e r t i a l f o r c e is n e g l i g i b l e . T h e center o f the  fiber  translates w i t h the l o c a l f l u i d v e l o c i t y , a n d the f i b e r  o r i e n t a t i o n is g o v e r n e d b y the c o m p o n e n t s o f the rate o f d e f o r m a t i o n tensor, the o f the  f l o w a n d the  previous  fiber  orientation.  In order  to p r e d i c t  a  fiber  v e l o c i t y , s t r a i n t e n s o r , a n d v o r t i c i t y v e c t o r o f t h e f l o w at t h e c e n t e r o f a k n o w n . T h e a p p r o a c h d e v e l o p e d is f i r s t l y to d e t e r m i n e i n w h i c h c e l l the a c c u r a t e v a l u e s o f t h e v e l o c i t y , s t r a i n a n d v o r t i c i t y at t h e from  the  surrounding nodal  position  and  orientation  points.  obtained  In  each  from  the  time  step,  fiber  the  motion, fiber  fiber  center are  the  must  lies.  be  Then  interpolated the  fiber  local  flow  a m e a n velocity associated  with  preceding  calculation  vorticity  time  step  and  uses the  kinematics information.  F i b e r particles suspended i n a turbulent f l o w experience the  mean  f l o w field  and a random  velocity  due  to  the  fluctuating  component  of  the  turbulent f l o w . T h e orientation o f fibers depends o n a combination o f orienting effects the  mean  velocity  gradients  and  the  randomizing  effects  of  turbulence.  Only  of  mean  v e l o c i t y g r a d i e n t s are c o n s i d e r e d i n the p r e s e n t s i m u l a t i o n .  I n this s t u d y , r i g i d f i b e r s are c o n s i d e r e d w i t h u n i f o r m l e n g t h L a n d d i a m e t e r d , a n d a s p e c t ratio  L/d.  coordinates of  fiber  The (x,  position  and  orientation  y , z ) a n d a n g l e s (Q,  orientation  angles,  §).  of  a  single  can  be  described  T h e o r i e n t a t i o n f i e l d that is e x p r e s s e d  is s p e c i f i e d f r o m the  a l o n g the particle trajectory. A  fiber  s o l u t i o n o f the  i n the  orientation  large n u m b e r o f fibers is c o n s i d e r e d b y u s i n g  33  by  the form  equations statistical  expressions  d e v e l o p e d to describe the orientation  deals w i t h each individual is characterized fiber  fiber  separately.  w i t h the statistical  orientation parameter f  state. I n b r i e f , t h e n u m e r i c a l  scheme  T h e n the behavior o f large n u m b e r s o f fibers  methods.  T h e fiber o r i e n t a t i o n d i s t r i b u t i o n a n d t h e  are b a s e d o n the statistical  data.  T h e f i b e r s a r e i n i t i a l l y i n j e c t e d at t h e c e n t e r o f t h e i n l e t o f t h e c o n v e r g i n g s e c t i o n r a n d o m orientations, initial fiber  fiber angles  a l t h o u g h it i s p o s s i b l e to set u p a n o n - r a n d o m d i s t r i b u t i o n f o r t h e  state. A t t h e inlet, r a n d o m  fiber  sphere,  orientation is i m p l e m e n t e d b y c h o o s i n g the  ^ a n d 9 f o r a large n u m b e r o f fibers. T h e angles  r a n d o m l y f r o m u n i f o r m distributions Q unit  with  the area element  d Q =  [0, ]  E  n  singd^dQ  a n d <j)  G  [ 0 , ], n  Q a n d ^ are not selected since o n the surface o f a  i s a f u n c t i o n o f Q. T O d i s t r i b u t e  fibers  e q u a l l y i n a l l p o s s i b l e directions, the correct selection m e t h o d s h o u l d b e f o l l o w e d [55]: c h o o s e aj a n d a  2  t o b e r a n d o m v a r i a b l e s o n [0, 1], t h e n  9  =  na  =  c o s -  (4-17)  x  1  ( 2 a  2  - l )  (4-18)  T h e initial o r i e n t a t i o n o f 1 0 0 0 fibers is s h o w n i n F i g u r e 4.2. O b v i o u s l y , as the n u m b e r o f fibers is increased,  a more  accurate representation  o f the orientation distribution w i l l b e  obtained. F r o m such a n orientation distribution, the preferred orientation angle  a n d the  d e g r e e o f a l i g n m e n t f o r that angle c a n b e o b t a i n e d .  A l t h o u g h the a c c u r a c y o f the statistical considered, accuracy based  solution is dependent  o n e s h o u l d consider the available computational  level. Hence,  the n u m b e r  o n the n u m b e r o f fibers resources a n d the desired  o f fibers u t i l i z e d i n the statistical  o n the desired accuracy a n d the available  computational  solution can be  resources.  After trying  different n u m b e r s o f fibers i n the s i m u l a t i o n , f r o m 500 to 100,000, a s a m p l e size o f 3 0 0 0 f i b e r s w a s c h o s e n t o r e p r e s e n t t h e b u l k fiber b e h a v i o r a s t h i s g i v e s a r e a s o n a b l e expression.  34  statistical  F i g u r e 4.1. A fiber i n t h r e e - d i m e n s i o n a l  F i g u r e 4.2.  coordinates.  T h e initial r a n d o m distribution o f 1000  35  fibers.  5. R E S U L T S A N D D I S C U S S I O N  5.1  A n a l y s i s of the H e a d b o x F l o w F i e l d  The  quality o f any  chosen  grid.  asymmetric headbox  In  C F D analysis depends,  the  flow  converging  among  s i m u l a t i o n , the  geometric  section o f the h e a d b o x  s p a n d i r e c t i o n is r e d u c e d f r o m 0.76  i n F i g . 5.1  is generated  o v e r the  other  dimensions  i n the  m to 0.8  converging  factors, o n the are  experiment,  quality o f  the  same  o n l y the  as  section.  By  x  m a p p i n g the  shape  p h y s i c a l d o m a i n to a c o m p u t a t i o n a l d o m a i n , the g o v e r n i n g e q u a t i o n s c a n b e  the  size i n  c m . T h e 8 0 3 2 8 m e s h as x  the  the  shown  from  the  transformed  a n d s o l v e d i n the latter d o m a i n , u s i n g the f i n i t e - v o l u m e t e c h n i q u e , a n d t h e n the  solutions  at e v e r y n o d e a r e m a p p e d b a c k o n t o t h e p h y s i c a l d o m a i n .  T h e Cartesian coordinates direction  and  headbox  span  cross  x,  machine  is r e d u c e d  y a n d z represent the m a c h i n e direction  i n the  z  respectively.  d i r e c t i o n i n the  As  we  direction, paper have  computation.  noted The  thickness  already,  the  simulation  only  m o d e l s a s p a n w i s e s e c t i o n o f t h e l a b o r a t o r y h e a d b o x , b e c a u s e it i s a s s u m e d t h a t t h e r e i s n o s i g n i f i c a n t v a r i a t i o n i n v e l o c i t y a n d v e l o c i t y gradients i n the z d i r e c t i o n . T h e tubes  are  n o t m o d e l e d . T h e m e s h is r e f i n e d i n the r e g i o n c l o s e to the w a l l s a n d the n o z z l e b e c a u s e the  velocity  gradients  are  larger  there than  i n other  place.  The mesh  d i r e c t i o n ( x - a x i s ) is g r a d u a l l y r e f i n e d s i n c e the v e l o c i t y a n d p r e s s u r e  i n the  change  machine  r a p i d l y as  the f l o w a p p r o a c h e s the exit. B y r e f i n i n g the m e s h , w e c a n o b t a i n c o m p a r a t i v e l y  accurate  s o l u t i o n s o n the c o r r e s p o n d i n g n o d e s near the w a l l s a n d the n o z z l e . T h e m a x i m u m s i z e s at t h e  inlet i n x  a n d y d i r e c t i o n s are  around 8 m m  a n d 2.8  mesh  m m respectively;  m i n i m u m m e s h s i z e s at t h e n o z z l e i n x a n d y d i r e c t i o n s a r e a r o u n d 2 . 4  m m a n d 0.2  the m m  r e s p e c t i v e l y . T h e m e s h size i n z d i r e c t i o n is 1 m m .  T h e b o u n d a r y c o n d i t i o n s a r e u n i f o r m v e l o c i t y p r o f i l e at t h e i n l e t ( w i t h i n i t i a l k = 4 m /s , 2  2  E  =  3  x  10"  4  m /s 2  3  for all cases), zero  36  velocity  gradient  at t h e  x  slice exit for  10"  3  all  v e l o c i t i e s i n t h e x d i r e c t i o n , a n d t h e w a l l c o n d i t i o n s at t h e u p p e r a n d l o w e r c h a n n e l w a l l s . U s u a l l y , z e r o v e l o c i t y g r a d i e n t c o n d i t i o n is a p p l i e d to f u l l y d e v e l o p e d f l o w . H o w e v e r , f i n d i n t h i s s t u d y that i f the m e s h g r i d s c l o s e to the c h a n n e l e x i t are s m a l l e n o u g h , velocity  gradient  outflow  of  the  can  be  approximately  converging  channel  accepted  case.  The  as  more  the  boundary  condition  physically correct  exit  we zero  for  the  boundary  c o n d i t i o n o f c o n s t a n t p r e s s u r e has b e e n t r i e d e l s e w h e r e . T h i s c o n d i t i o n leads to n u m e r i c a l simulations w h i c h converge reason.  The symmetry  components,  and  w i t h great d i f f i c u l t y , a n d has  c o n d i t i o n (zero gradients  zero  value  of  the  velocity  not been used here for  that  n o r m a l to the b o u n d a r y o f a l l v e l o c i t y  component  normal  to  the  boundary)  is  i m p o s e d o n b o t h sides o f the c h a n n e l b e c a u s e o n l y a section o f the c h a n n e l is m o d e l l e d and  there  is  no  w a l l effect.  Appropriate velocity  gradients  are  set  to  zero  and  a  no  p e n e t r a t i o n b o u n d a r y c o n d i t i o n i s i m p o s e d at e a c h s y m m e t r y p l a n e .  The  f l u i d u s e d i n t h i s s i m u l a t i o n i s w a t e r . T h e u n i f o r m v e l o c i t y at t h e i n l e t i s 0 . 2 4  m/s,  w h i c h is the s a m e as the a v e r a g e i n f l o w v e l o c i t y i n the e x p e r i m e n t . T h e r e are n o y - a n d z direction velocity components are s h o w n i n F i g u r e 5.2.  at t h e i n l e t . T y p i c a l s t r e a m l i n e s i n t h i s c o n v e r g i n g  section  T h e fiber o r i e n t a t i o n s t u d y i s f o c u s e d o n t h e f i b e r s t r a v e l l i n g o n  the central streamline o n l y .  The  variations o f the x - d i r e c t i o n v e l o c i t y a n d pressure  s h o w n i n F i g u r e 5.3.  a l o n g the central  streamline  It c a n b e s e e n f r o m t h e p l o t s t h a t a l a r g e p r e s s u r e d r o p o c c u r s  the n o z z l e exit w h e r e a s the f l u i d v e l o c i t y increases r a p i d l y near the exit o f the  are near  channel  a n d r e a c h e s i t s m a x i m u m at t h e e x i t .  F i g u r e 5.4 5.5  gives u-velocity (x-direction) contours  g i v e s the v - v e l o c i t y (y-direction)  contours  o n the central s y m m e t r y p l a n e .  o n the  same symmetry  Figure  plane. T h e flow,  b e g i n n i n g w i t h a u n i f o r m v e l o c i t y p r o f i l e at t h e i n l e t , g r a d u a l l y i n c r e a s e s i t s s p e e d a s channel becomes the  headbox  narrower,  nozzle.  The  a n d the v e l o c i t y gradients increase fiber  orientation  will  e l o n g a t i o n a n d shear stresses.  37  be  strongly  and reach a m a x i m u m affected  by  the  the at  increased  As  some  effect  in  researchers have a  convergent  stated [25],  channel,  fiber orientation is g o v e r n e d b y the  and  both  the  elongation  of  the  flow  elongation  and  the  fiber  a l i g n m e n t i n t h e f l o w d i r e c t i o n r e a c h t h e i r m a x i m u m at t h e c h a n n e l e x i t . F o r t h i s h e a d b o x c h a n n e l , the f l o w e l o n g a t i o n a l o n g the central streamline c a n b e e x p r e s s e d  s  = J  where u  s  as:  \*±dt ds •  (5-D  is the f l o w v e l o c i t y o n the central s t r e a m l i n e a n d s is the d i s t a n c e  central streamline.  along  the  Because  dt  ds —  =  the f l o w e l o n g a t i o n c a n a l s o b e c a l c u l a t e d as:  1 du  f -1 u  s  The  velocity u  s  a n d distance  s  ds  ds  s o n the  (5-2)  central  s t r e a m l i n e are  r e l a t e d to the x  direction  v e l o c i t y u a n d x, i n the starting flat s e c t i o n , b y :  u  =  u„  x  =  s  a n d i n the c o n s e q u e n t contraction section b y :  u  =  u coscp  x  =  x  s  t  +  (s - x ) x  cos  q>  w h e r e ^ is the a n g l e b e t w e e n the central s t r e a m l i n e a n d x - a x i s i n the c o n t r a c t i o n  section,  Xj is the l e n g t h o f the flat section. S u b s t i t u t i n g the a b o v e relations into (5.2), the  flow  elongation becomes:  s  -  r 1 du , J — — dx u dx  (5.3)  38  I f u i s d e p e n d e n t o n x o n l y , e q u a t i o n (5.3) c a n b e further s i m p l i f i e d as:  du  £  (5.4)  U or  £ (x)  =  In  u(x)  (5.4')  A t the c h a n n e l exit, the f l o w e l o n g a t i o n is therefore:  (5.5)  where U  0  andU  e  a r e t h e v e l o c i t y c o m p o n e n t i n t h e x d i r e c t i o n at t h e c h a n n e l i n l e t a n d  outlet. B e c a u s e the ratio o f U / U e  0  is a p p r o x i m a t e l y e q u a l to the c o n t r a c t i o n ratio ( R . ) o f  t h e c h a n n e l , t h e n t h e f l o w e l o n g a t i o n at t h e e x i t i s o b t a i n e d a s :  (5.6)  O n e c a n f i n d f r o m (5.6) that the f l o w e l o n g a t i o n contraction  ratio  of a  convergent  channel.  s  at t h e c h a n n e l e x i t o n l y d e p e n d s o n t h e  From  (5.4)  the f l o w  elongation  can be  c a l c u l a t e d v a r y i n g w i t h c h a n n e l l e n g t h as s h o w n i n F i g . 5.6. T h e f l o w e l o n g a t i o n c a n also be calculated b y equation (5.4'), w h i c h is essentially identical to the n u m e r i c a l plot o f Fig.  5.2  A  5'.6.  Comparison of Simulation and Experimental Results  convenient  graphical representation  is n e e d e d to present  the three-dimensional fiber  orientation results. T h e projections o f the orientation o f large n u m b e r s o f fibers c a n b e  39  o b t a i n e d o n three different planes.  T h e projections  on two  o f the p l a n e s ,  p l a n e , are u s e d here, b e c a u s e the results c a n be c o m p a r e d d i r e c t l y w i t h the results for the c o r r e s p o n d i n g planes. T h e fiber orientation angle  x-y  and  x-z  experimental  , either o n the x - y or o n  a  the x - z p r o j e c t i o n p l a n e , is d e f i n e d to b e the a n g l e b e t w e e n the p r o j e c t i o n o f the f i b e r a x i s o n that p l a n e a n d the m a c h i n e d i r e c t i o n ( x - a x i s ) . T h e x - y p l a n e c o r r e s p o n d s view  i n the  measurements  measurements.  a n d the  x-z  plane  corresponds  to  the  bottom  view  side  in  the  T h e f i b e r o r i e n t a t i o n p r o j e c t i o n o n the y - z p l a n e is n o t c o n s i d e r e d b e c a u s e  it i s d i f f i c u l t to o b t a i n the f i b e r i m a g e s  o n that p r o j e c t i o n p l a n e i n the  would  orientation  be  to the  possible  to  obtain  the  fiber  results  on  the  y-z  experiments. plane  It  from  the  s p e c i f i c p o i n t o n the p r o j e c t i o n p l a n e s  are  s i m u l a t i o n , but this analysis has not b e e n done.  When  a l l t h e o r i e n t a t i o n a n g l e d a t a at e a c h  a v a i l a b l e , the  results  o f the  fiber orientation  distributions f r o m experiments  s i m u l a t i o n s c a n b e c o m p a r e d . A n a n g u l a r i n t e r v a l m u s t b e c h o s e n to p r o v i d e a p i c t u r e o f fiber o r i e n t a t i o n d i s t r i b u t i o n . F o r t h e  fiber  degrees each,  f r o m - 9 0 ° to + 9 0 ° ,  a  p(a)da  w i t h 0 ° i n d i c a t i n g the m a c h i n e  =  reasonable  . It i s s e p a r a t e d i n t o 18 z o n e s  T h e v e r t i c a l a x i s r e p r e s e n t s t h e s t a t i s t i c a l p r o b a b i l i t y d e n s i t y p(a)>  fj  from  orientation distribution diagram used  here, the h o r i z o n t a l a x i s represents the orientation angle 10  and  direction  (x-axis).  such t h a t :  1  (5-7)  F i b e r o r i e n t a t i o n d i s t r i b u t i o n s c l o s e to the inlet (x = 4.5  cm)  as s e e n f r o m the s i d e  a n d the b o t t o m v i e w o f the h e a d b o x are s h o w n i n F i g u r e 5.7.  O n e c a n see that the  view fibers  a r e a l m o s t r a n d o m l y d i s t r i b u t e d at t h i s l o c a t i o n . I f m o r e f i b e r s h a d b e e n u s e d i n e i t h e r measurements  or the  numerical  a p p r o a c h i n g a v a l u e e q u a l to  l/  n  simulations, p ( ) a  (~  of  would  have  become  more  the  constant,  0.318) for a v e r y large n u m b e r o f fibers. A s  fibers  enter the c o n v e r g i n g s e c t i o n o f the h e a d b o x , t h e y g r a d u a l l y e x h i b i t the t e n d e n c y to a l i g n i n t h e f l o w d i r e c t i o n . F i g u r e 5.8 12.2 the  c m a n d 15.7 fiber  a n d 5.9  s h o w the  fiber  o r i e n t a t i o n d i s t r i b u t i o n s at x  c m . A n interesting p h e n o m e n o n was f o u n d  from  these two  =  diagrams:  orientation distributions f r o m n u m e r i c a l s i m u l a t i o n s c h a n g e faster t h a n that f r o m  40  experiments.  W h e n fibers m o v e  further towards  m o r e s i g n i f i c a n t as s h o w n i n F i g u r e 5.10 (x  =  26  cm).  A t the  exit where  d i r e c t i o n as s h o w n i n F i g . 5.13.  x  =  31  (x =  the n o z z l e , this p h e n o m e n o n  19 c m ) , F i g . 5.11  c m , the  f i b e r s are  becomes  (x = 22 c m ) a n d F i g .  h i g h l y a l i g n e d i n the  5.12 flow  W e d o not h a v e the e x p e r i m e n t a l data o f f i b e r o r i e n t a t i o n  i n t h e x - y p l a n e , b e c a u s e t h e c h a n n e l i s t o o n a r r o w a n d t h e f l o w s p e e d i s t o o h i g h at t h a t l o c a t i o n to o b t a i n c l e a r f i b e r i m a g e s . T h e s e d i a g r a m s s h o w that the s i m u l a t i o n c a n p r e d i c t the  trend  o f the  fiber  orientation  in a  dilute headbox  flow,  d i f f e r e n c e s b e t w e e n the n u m e r i c a l data a n d the e x p e r i m e n t a l  Another  phenomenon  shown  from  these  diagrams  was  but  there  exist  obvious  data.  that  the  fiber  orientation  d i s t r i b u t i o n s o r the a l i g n m e n t s i n the x - y p l a n e are s t r o n g e r t h a n that i n the x - z p l a n e . I n o t h e r w o r d s , t h e o r i e n t a t i o n state i s c h a n g e d m o r e i n t h e s h e a r a n d e x t e n s i o n a l p l a n e p l a n e ) t h a n i n the neutral p l a n e (x-z plane). T h i s is c a u s e d b y different v e l o c i t y  (x-y  gradient  effect. I n this h e a d b o x structure, the v e l o c i t y gradients d u / d x , d u / d y , d v / d x a n d d v / d y a l l a f f e c t the f i b e r o r i e n t a t i o n i n the x - y a n d y - z p l a n e s , but o n l y d u / d x a n d d u / d y a f f e c t f i b e r o r i e n t a t i o n i n the x - z p l a n e . T h a t is w h y the fiber a l i g n m e n t i n x - y p l a n e is  the  stronger  t h a n that i n the x - z p l a n e .  I n g e n e r a l , the s i m u l a t e d f i b e r o r i e n t a t i o n s t e n d to a l i g n m o r e w i t h the f l o w c o m p a r e d  to  the e x p e r i m e n t a l results. T h e m a j o r r e a s o n f o r this p h e n o m e n o n is a l m o s t c e r t a i n l y that the t u r b u l e n c e effect is not c o n s i d e r e d i n o u r f i b e r s i m u l a t i o n . In the e x p e r i m e n t , the f l o w enters the h e a d b o x c h a n n e l t h r o u g h the rectifier tubes. A s w e h a v e m e n t i o n e d earlier,  the  rectifier tubes h a v e t w o f u n d a m e n t a l f u n c t i o n s o n the i n c o m i n g f l o w . First, t h e y p r o v i d e t u r b u l e n c e o f d e s i r e d i n t e n s i t y a n d s c a l e to b r e a k u p f i b e r f l o e s . S e c o n d l y , t h e y p r o d u c e  a  f a i r l y u n i f o r m i n f l o w at t h e c h a n n e l i n l e t . T h e t u r b u l e n c e c r e a t e d i n t h e r e c t i f i e r  tubes  will  fiber  affect  the  fiber behavior the  downstream.  T u r b u l e n c e tends  fiber orientation  to  distribution have  randomize  orientation,  w h i c h makes  appearance.  T h e current n u m e r i c a l fiber m o d e l simulates o n l y the fiber m o t i o n i n a m e a n  f l o w f i e l d , so the p r e d i c t e d f i b e r orientation d i s t r i b u t i o n is m o r e observed situation, where turbulence plays an important role.  41  a  the  comparatively  flat  o r g a n i z e d t h a n i n the  The  fiber  orientation  symmetrical  distribution  i n the  side  view  about zero because o f the asymmetric  (x-y  projection  plane)  structure o f the h e a d b o x  is not  i n the x - y  p l a n e . T h e c h a n n e l is n o t s y m m e t r i c so that z e r o d e g r e e o r i e n t a t i o n w o u l d i n d i c a t e a f i b e r b e i n g p a r a l l e l to the x - a x i s o r the l o w e r c h a n n e l w a l l . T h e angle b e t w e e n the u p p e r a n d l o w e r c h a n n e l w a l l is 1 6 . 7 ° . A s a result, the preferred orientation i n the side v i e w s h o u l d be between  Oo a n d 1 6 . 7 ° .  O n the other h a n d , the s i m u l a t i o n results  give a  symmetrical  d i s t r i b u t i o n o n x - z p r o j e c t i o n plane. T h i s is because there is n o restriction i n this p l a n e , and fibers are a l l o w e d to freely m o v e i n all possible directions.  Another  concise  w a y to  describe  fiber  orientation  parameter". T h e plane orientation parameter f  f  p  =  2  p(a)  to  use  a  fiber  "orientation  u s e d b y M c C u l l o u g h [56] i s d e f i n e d b y  cos (a-a ) 2  is  da  0  -1  (5-8)  2  where  a  o  is the m e a n o f the distributed angles o r the preferred a l i g n m e n t angle, a n d p ( )  represents  a  the p r o b a b i l i t y density  function. F o r the case  o f a finite n u m b e r  o f fiber  o r i e n t a t i o n s , n , t h e a b o v e e q u a t i o n h a s b e e n a p p r o x i m a t e d b y Y o r k [57] a s :  2  f  =  P  n  - S  c  o  s  2  ( i~ a  a  o)  (5.9)  n , i =  T h e parameter f  p  p r o v i d e s a c o n v e n i e n t m e t h o d t o d e s c r i b e a p a r t i c u l a r state o f i n - p l a n e  fiber orientation. F o r perfect alignment, f distribution, f  = 0. T h e m e a n d i r e c t i o n  a  o  =  1, a n d f o r a c o m p l e t e l y r a n d o m  c a n be calculated w i t h the f o l l o w i n g  orientation equations  [58]:  -i  a  n 0  =  cos  C  (5.10)  —  R  or  42  a  =  0  s i n "  1  -  (5.H)  R  0  where:  1  S  =  R  Table  5.1  presents  1 "  = S  c  =  the  1  "  —V  n  cos  sin  a,  a,  M  (C  2  observed  +S  2  and predicted  orientation  parameters  at  several  m e a s u r e m e n t p o i n t s a l o n g the central streamline (see F i g . 3.8).  T a b l e 5.1. O r i e n t a t i o n Parameters O b t a i n e d f r o m E x p e r i m e n t s a n d S i m u l a t i o n s :  x-positions  Fig.  Experiments  Simulations  x-y plane  x-z plane  x-y plane  x-z plane  1  0.04  -0.01  0.01  -0.00  2  0.10  0.02  0.17  0.07  3  0.14  0.05  0.33  0.15  4  0.18  0.05  0.50  0.26  5  0.22  0.15  0.68  0.38  6  0.33  0.20  0.82  0.51  7  N/A  0.53  0.96  0.76  5.14(a)  a n d F i g . 5.14(b) present the fiber orientation parameters a l o n g the central  streamline i n the x - y a n d x - z p r o j e c t i o n planes r e s p e c t i v e l y . T h e results s h o w that there are d i f f e r e n c e s b e t w e e n the m e a s u r e m e n t data a n d the s i m u l a t i o n data, a l t h o u g h t h e t r e n d is the s a m e . develops  F o r both measurements  from  a random  a n d simulations, the fiber orientation d i s t r i b u t i o n  initial c o n d i t i o n to a m u c h m o r e  a l i g n e d s t a t u s at t h e e x i t ,  a l t h o u g h t h e d i s t r i b u t i o n at t h e e x i t i s n o t f u l l y a l i g n e d . A s w i l l b e d i s c u s s e d i n t h e n e x t  43  section, the d e g r e e o f a l i g n m e n t d e p e n d s o n the h e a d b o x g e o m e t r y ,  or more specifically,  the c o n t r a c t i o n ratio o f the c h a n n e l .  5.3  Factors Affecting Fiber Orientation  W h e n s u b j e c t e d t o t h e p l a n e r a t e o f s t r a i n , QU/QX, i n t h e c o n v e r g e n t  c h a n n e l , the  h a v e the t e n d e n c y to a l i g n i n the d i r e c t i o n o f the f l o w . F o r different h e a d b o x  fibers  geometries  a n d f l o w c o n d i t i o n s , the d e g r e e o f a l i g n m e n t o f fibers i n the f l o w d i r e c t i o n is d i f f e r e n t and  is d e t e r m i n e d b y the  exposed  to  the  forces.  rates o f strain i n the  High  f l o w , a n d the t i m e  rate o f strain a n d l o n g t i m e  that the  fibers  duration w o u l d be  likely  are to  p r o d u c e h i g h l y concentrated fiber orientations.  I n o r d e r to headbox  detect what  geometry,  factors m a y  i n f l u e n c e the  fiber orientation characteristics,  the  the f l o w v e l o c i t y a n d the fiber aspect ratio h a v e b e e n c h a n g e d  and  their effects o n fiber orientation have been studied. W h e n one parameter  is c h a n g e d ,  the  others are k e p t the s a m e . T h i s e x p l o r a t i o n o f the effects o f selected p a r a m e t e r s is m u c h easier to d o i n the s i m u l a t i o n studies t h a n i n the e x p e r i m e n t a l w o r k . I n the s i m u l a t i o n , w e investigate  the f i n a l fiber orientation d i s t r i b u t i o n f o r different v a l u e s o f the  contraction  ratio ( R . ) , w h i c h is the ratio b e t w e e n inlet area a n d exit area, the c h a n n e l l e n g t h ( L ) , c  i n f l o w rate U  5.3.1  0  a n d t h e fiber a s p e c t r a t i o ( A ) . r  The Effect of Contraction Ratio on Fiber Orientation  In this  part  o f the  present  study,  o n l y the  channel's  contraction  ratio  is c h a n g e d  a d j u s t i n g the e x i t area, w h i l e the c h a n n e l l e n g t h a n d the f l o w v e l o c i t y r e m a i n the The  the  fiber  o r i e n t a t i o n p a r a m e t e r s at t h e c h a n n e l e x i t f o r d i f f e r e n t R  5.2.  44  c  by  same.  are s h o w n i n T a b l e  T a b l e 5.2. F i b e r O r i e n t a t i o n Parameters f o r D i f f e r e n t R  fiber  = 0.225 m )  Orientation parameter  Rc  The  ( U = 0.24 m / s , L  x-y plane  x-z plane  6.7  0.96  0.73  10  0.98  0.81  15  0.98  0.86  orientation parameter (which corresponds  in the x - y a n d x - z projection planes  t o t h e fiber o r i e n t a t i o n  at t h e c h a n n e l e x i t a r e i n c r e a s e d  distribution)  with  increasing  c o n t r a c t i o n ratio, a l t h o u g h the i n c r e m e n t i n the x - y p l a n e is s l o w e d d o w n as it a p p r o a c h e s i t s m a x i m u m v a l u e o f 1. T h e c o n c l u s i o n i s t h a t h i g h e r R fiber  5.3.2  c  corresponds  to m o r e  aligned  orientation i n the f l o w direction.  The Effect of Flow Rate on Fiber Orientation  Keeping  other  parameters the same,  o n l y t h e f l o w r a t e at t h e i n l e t o f t h e c h a n n e l  is  c h a n g e d . I n e v e r y c a s e t h e c o n t r a c t i o n r a t i o i s 1 0 . T h e fiber o r i e n t a t i o n p a r a m e t e r s a t t h e c h a n n e l e x i t f o r d i f f e r e n t f l o w r a t e s a r e s h o w n i n T a b l e 5;. 3 . T h e R e y n o l d s n u m b e r R  e  is  calculated b y the f o l l o w i n g equation:  R  e  where U  0  =  U  °  H  v  °  (5.12)  i s t h e v e l o c i t y at t h e c h a n n e l i n l e t , H  k i n e m a t i c viscosity o f water.  45  Q  i s t h e c h a n n e l h e i g h t at i n l e t ,  v  is the  T a b l e 5.3. T h e Orientation Parameters f o r Different I L ( R  Reynolds number  Uo  = 10, L  = 0.225 m )  Orientation parameter  (m/s)  x-y plane  x-z plane  0.16  12,000  0.98  0.81  0.24  18,000  0.98  0.81  0.36  27,000  0.98  0.80  A s already noted, the experimental  results w e r e o b t a i n e d f o r a v a l u e o f U  Q  o f 0.24 m / s .  F r o m t h e s i m u l a t i o n r e s u l t s , i t a p p e a r s t h a t t h e f i b e r o r i e n t a t i o n at t h e c h a n n e l e x i t i s n o t a f f e c t e d b y the f l o w v e l o c i t y . I n c r e a s i n g the f l o w rate o f c o u r s e i n c r e a s e s the n o r m a l a n d s h e a r r a t e s o f s t r a i n i n t h e f l o w , b u t at t h e s a m e t i m e , i t a l s o s h o r t e n s t h e f i b e r  residence  time  velocities  needs  i n the f l o w . T h e present further  investigation,  results  because  o f fiber orientation in our simulation  for different f l o w the turbulence  effect  is not  specifically i n c l u d e d , a n d the f l o w velocity has close relationship w i t h the turbulence.  5.3.3  The Effect of Channel Length on Fiber Orientation  B e s i d e s the effect o f R on  fiber orientation.  c  and U  Q  , it i s i m p o r t a n t t o d e t e r m i n e t h e e f f e c t o f c h a n n e l  T h e fiber orientation  p a r a m e t e r s at t h e c h a n n e l  exit f o r different  c h a n n e l l e n g t h s are s h o w n i n T a b l e 5.4. T h e i n i t i a l u n i f o r m c r o s s s e c t i o n c h a n n e l is the s a m e . L  is the l e n g t h o f the c o n v e r g e n t  section.  T a b l e 5.4. F i b e r O r i e n t a t i o n P a r a m e t e r s f o r D i f f e r e n t L  (R  = 10, U  = 0.24 m / s )  Orientation parameter (m)  x-y plane  x-z plane  0.1500  0.98  0.82  0.2250  0.98  0.81  0.3375  0.99  0.82  46  length  length  A l t h o u g h a shorter contraction  corresponds  to h i g h e r rates o f strain i n t h e f l o w f i e l d , t h e  fiber r e s i d e n c e d u r a t i o n i n the f l o w is also r e d u c e d . T h e total effect o f c h a n g i n g length o n fiber orientation  is therefore eliminated. H e r e again, o n l y the m e a n  b e e n c o n s i d e r e d i n the simulation. F o r further investigation,  turbulence  channel flow has  effect should be  included.  The Effect of Fiber Aspect Ratio on Fiber Orientation  5.3.4  F i n a l l y , the effect o f c h a n g e s to the fiber aspect ratio is investigated b y c h a n g i n g the f i b e r l e n g t h a n d d i a m e t e r . T h e p r e d i c t i o n o f f i b e r o r i e n t a t i o n p a r a m e t e r s at t h e c h a n n e l e x i t i s s h o w n i n T a b l e 5.5.  T a b l e 5.5. T h eO r i e n t a t i o n Parameters f o r D i f f e r e n t A r  (R  c  = 10, U  Q  = 0.24 m / s , L  c  = 0.225 m )  A (L/d)  orientation parameter  r  ( L and d in mm ) 133.3  From affect  x-z plane  0.98  0.81  (4/0.030)  68.2  (3/0.044)  0.98  0.81  33.3  (2/0.060)  0.98  0.81  t h e t a b l e it a p p e a r s that, w i t h i n the r a n g e o f interest, t h e c h a n g e i n A the  importance  5.3.5  x-y plane  fiber  orientation  distribution.  T h e turbulence  length  scale  r  does not  could  be  of  i n a m o r e complete simulation w h i c h includes turbulent dispersion.  The Effect of Flow Elongation  T h e f l o w elongation channel.  is also e x a m i n e d  T a b l e 5.6 s h o w s  f o r its r e l a t i o n s  the effect o f R  o n the elongation  c channel.  8  w i t h the contraction  ratio o f the  o f f l o w at t h e e x i t o f t h e  °  is c a l c u l a t e d f r o m E q u a t i o n (5.6). B e c a u s e the c h a n g e o f f l o w rate a n d c h a n n e l  47  length  does  not affect  ,  s  it i s c l e a r  determination o f the fiber orientation  that t h e e l o n g a t i o n  o f flow can be used  status i n a c o n v e r g e n t  f o r the  channel, i f o n l y the  mean  velocities are considered. H o w e v e r , turbulence dispersion almost certainly a significant factor w h i c h is n o t c o v e r e d b y the elongation o f the m e a n f l o w .  T a b l e 5 . 6 . T h e E l o n g a t i o n o f F l o w at t h e C h a n n e l E x i t f o r D i f f e r e n t R (U  =0.24 m/s, L  Conditions  R =15  Rc = 10  Rc = 6.7  2.7  2.3  1.9  c  s  5.4  = 0.225 m ) :  Symmetric Channel  T h e p r e v i o u s p r e d i c t i o n results are f o r a n a s y m m e t r i c  convergent  channel. A  c h a n n e l is n o w e x p l o r e d u s i n g the n u m e r i c a l simulations. T h e s y m m e t r i c in  Ullmar's  work  [25]  with  a  contraction  ratio  of  16.7  is adapted  symmetric  headbox  for the  s i m u l a t i o n . F i g . 5.16 gives the cross sectional d i m e n s i o n s o f this h e a d b o x ,  used  current  a n d F i g . 5.17  s h o w s the c o m p u t a t i o n a l m e s h o f this d e s i g n . T h e c h a n n e l w i d t h is constant. D u r i n g the simulation,  four  i n f l o w velocities  are tested.  T h e fiber orientation  parameters  o n the  c e n t r a l s t r e a m l i n e at t h e c h a n n e l e x i t f o r v a r i o u s i n f l o w r a t e s a r e s h o w n i n T a b l e 5 . 7 .  T a b l e 5 . 7 . O r i e n t a t i o n P a r a m e t e r s at E x i t o f A S y m m e t r i c H e a d b o x f o r D i f f e r e n t U •  Inflow rate  Reynolds  (m/s)  number  x-y plane  x-z plane  0.10  25,000  0.98  0.88  0.30  75,000  0.98  0.88  0.43  107,000  0.98  0.88  0.56  140,000  0.98  0.88  F r o m this  study o f a symmetric  Orientation parameter  c h a n n e l f l o w , the f o l l o w i n g c o n c l u s i o n s  f r o m the n u m e r i c a l simulations:  48  are  obtained  a.  Fiber orientation changes  f r o m r a n d o m at t h e c h a n n e l i n l e t t o t h e h i g h l y  o r i e n t a t i o n at e x i t as f i b e r s m o v e a l o n g t h e c e n t r a l b.  preferred  streamline;  A s i n the asymmetric c o n v e r g i n g section, a n increase o f i n f l o w velocity has n o effect o n the fiber orientation;  c.  T h e difference  i n fiber  orientation  for two projection  planes  exists  both  i n the  asymmetric a n dsymmetric convergent channel.  Statistical Error Estimation  5.5  A n  estimate  can be  made  o f the  error  which  will  be  present  i n the probability  distributions m e a s u r e d o r calculated, d u e to the l i m i t e d s a m p l e size w h i c h is b e i n g u s e d .  In general,  the standard  deviation within repeated  w h i c h a fraction r has a particular characteristic,  samples  consisting o f n objects,  of  is g i v e n a p p r o x i m a t e l y b y [59]  (5.13)  where  b is the fraction o f n not contained i n r a n d is therefore  e q u a l t o (1  - r).  This  estimate is accurate w h e n r a n d b are a p p r o x i m a t e l y e q u a l f o r s m a l l n a n d also f o r r a n d b u n e q u a l i f n is large. S i n c e n is larger than 1,000 i n a l l the cases o f o u r experiments,  we  a s s u m e this e x p r e s s i o n c a n b e u s e d as a n error estimate w h e n a p p l i e d to the p r o b a b i l i t y density function p ( ) for w h i c h a  p(a)  =  r A  (5.14)  In the equation, & is the interval o f orientation angle  a  f o r w h i c h the value o f p ( ) is to  be evaluated, s o m e t i m e s called the " b i n w i d t h " o f the angle  49  a  a  . A l t h o u g h A. c a n v a r y w i t h  a  i f the b i n w i d t h s are not constant,  independent o f  a  the a s s u m p t i o n is m a d e here that A is a  constant,  . T h e standard d e v i a t i o n o f the p r o b a b i l i t y d e n s i t y v a l u e p ( ) w i l l b e a  (5.15)  a n d therefore e q u a l to  r ( l - r )  n  1  (5.16)  A  Substituting (5.14) into (5.16), w e have  \E^-p(af  (5-17)  In general, i f the orientation is initially c o m p l e t e l y r a n d o m , p ( ) c a n b e expressed as a  p(a)  =  -  (5-18)  . n I n t h e p r e s e n t e v a l u a t i o n o f p ( ) , a c o n s t a n t " b i n w i d t h " A o f l\% a  n  has been used. F o r a  s a m p l e s i z e n o f 1 3 2 5 u s e d i n t h e e x p e r i m e n t a l e v a l u a t i o n s f o r t h e x - y p l a n e at x =  4.5  c m , the standard deviation o f every point i nthe probability density w o u l d be  ~ A r  -  0.036  F o r a s a m p l e s i z e n o f 1 6 3 8 u s e d i n t h e e x p e r i m e n t a l e v a l u a t i o n s f o r t h e x - z p l a n e at x 4.5 c m , the standard d e v i a t i o n is  50  T h e error c a n be expected  to b e w i t h i n a b a n d o f + 3  a  about the m e a n v a l u e ( e q u a l to  1/jr), w h i c h i m p l i e s a v a l u e o f t h e p r o b a b i l i t y d e n s i t y i n x - y p l a n e i n t h e r a n g e  0.318  +  0.108 a n d a value o f the probability density i n x - z plane i n the range 0.318 + 0.096. T h e o b s e r v e d v a l u e s at t h e first p o i n t ( x = 4 . 5 ) f o r t h e x - y p l a n e l i e b e t w e e n 0 . 2 2 1 a n d 0 . 4 5 8 (see  F i g . 5 . 7 ) , a little b i t o u t s i d e o f the u p p e r l i m i t o f the p r e d i c t e d scatter.  Presumably,  t h e o r i e n t a t i o n s o f f i b e r s at t h i s p o i n t a r e n o t c o m p l e t e l y r a n d o m b e c a u s e o f t h e u p s t r e a m f l o w e f f e c t . T h e o b s e r v e d v a l u e s at t h e s a m e p o i n t f o r t h e x - z p l a n e l i e b e t w e e n 0 . 2 4 5 a n d 0.392, f a l l i n g w i t h i n the predicted b a n d . In s u m m a r y , the present  evaluations s h o w the  effect o f s a m p l e size o n the p r o b a b l e scatter i n m e a s u r e d v a l u e s a n d g i v e a f a i r e s t i m a t e o f the observed  distribution o f data  i n both the x - y a n d x - z planes.  T h e simulations,  s h o w n i n F i g . 5.7, h a v e a n initial scatter w h i c h lies w e l l w i t h i n the p r e d i c t e d error estimates,  as e x p e c t e d .  51  statistical  F i g u r e 5.2. T h e s t r e a m l i n e s o f the f l o w i n the h e a d b o x c o n v e r g e n t c h a n n e l .  52  2.5  Ol 0  1  1  1  1  1 0.1  1  1  i  i  I  0.2  I  i  i  i  I  0.3  I  o 0  channel length (m)  F i g u r e 5.3.  i  1  1—••  '  1 °1  i  i  i  i  T h e p r e s s u r e a n d u - v e l o c i t y c h a n g e s a l o n g the c e n t r a l  F i g u r e 5.4.  i i  1 0.2  1  T h e u - v e l o c i t y c o n t o u r s o n the c e n t r a l s y m m e t r y  streamline.  plane.  003  F i g u r e 5.5.  1 0.3  channel length (m)  T h e v - v e l o c i t y c o n t o u r s o n the c e n t r a l s y m m e t r y  53  plane.  c h a n n e l length (m)  F i g u r e 5.6.  T h e e l o n g a t i o n o f the f l o w c h a n g e s a l o n g the c e n t r a l  F i g u r e 5 . 7 . T h e f i b e r o r i e n t a t i o n d i s t r i b u t i o n at x = 4 . 5 (a)  i n x - y p l a n e , (b) i n x - z  54  plane  streamline.  cm,  SIMULATION EXPERIMENT  •  £  0.4  JJj  0.3  cn  3  e °-  0.21-  -2  -1.5  -1  -0.5  0  0.5  1  1.5  2  angle (radians)  (a)  (b)  F i g u r e 5 . 8 . T h e f i b e r o r i e n t a t i o n d i s t r i b u t i o n at x = 12.2 c m , (a) i n x - y p l a n e , ( b ) i n x - z p l a n e .  0.5  •0.5  angle (radians)  0  0.5  angle (radians)  (a)  (b)  F i g u r e 5 . 9 . T h e fiber o r i e n t a t i o n d i s t r i b u t i o n at x = 1 5 . 7 c m , (a) i n x - y p l a n e , ( b ) i n x - z p l a n e .  55  56  57  •  E g  .2 i  o  | C  SIMULATION EXPERIMENT  0.4  02  i  E .» o  channel length (cm)  •  •  channel length (cm)  (a)  (b)  F i g u r e 5.14. T h e orientation p a r a m e t e r s a l o n g the central  streamline,  (a) i n t h e x - y p l a n e , ( b ) i n t h e x - z p l a n e .  -0.5  0  0.5  angle (radians)  angle (radians)  (a)  ( b )  F i g u r e 5 . 1 5 . F i b e r o r i e n t a t i o n d i s t r i b u t i o n s at t h e c h a n n e l e x i t f o r v a r i o u s C o n t r a c t i o n r a t i o s , (a) i n x - y p l a n e , ( b ) i n x - z p l a n e .  58  F i g u r e 5.17.  T h e p h y s i c a l m e s h o f the s y m m e t r i c  59  headbox.  6. S U M M A R Y  A N D C O N C L U S I O N S  T h i s thesis is c o n c e r n e d w i t h f i b e r orientation i n the c o n v e r g i n g s e c t i o n o f a h e a d b o x . summary  o f the w o r k a n d s o m e  f i n d i n g s b a s e d o n the  experiments  and  A  computational  s i m u l a t i o n s are r e p o r t e d as f o l l o w s .  A  mathematical  f l o w m o d e l and a fiber m o t i o n m o d e l have been c o m b i n e d for predicting  the o r i e n t a t i o n o f r i g i d f i b e r s i n d i l u t e s u s p e n s i o n s . T h e r e is n o p a r a m e t e r i n the m o d e l to be  determined  converging  by  experiment.  section  s p e c i f i e d at t h e  are  Rigid  numerically  inlet o f the  fibers  flowing in a  simulated.  Random  c h a n n e l . T h e statistical  symmetric initial  expressions  fiber  and  asymmetric  orientations  are  o f the orientation  of  a  large n u m b e r o f fibers c a n be evaluated b y c o m p u t i n g the orientation o f e a c h s i n g l e f i b e r a l o n g the central streamline. E x p e r i m e n t s w e r e p e r f o r m e d i n a s c a l e d h e a d b o x m o d e l validate  the  numerical m o d e l . T h e orientation  distributions o f d y e d n y l o n fibers  u n i f o r m length and width were studied using an image analysis method. A n h e a d b o x w i t h a c o n t r a c t i o n r a t i o o f 10 i s u s e d i n t h i s s t u d y as t h e b a s i s f o r  to  with  asymmetric experimental  a n d s i m u l a t i o n w o r k . T h e c o m p a r i s o n s h o w s that the s i m u l a t i o n m e t h o d c a n p r e d i c t  the  trend o f fiber orientation i n a dilute headbox  the  experimental distributions,  f l o w , a l t h o u g h the d i f f e r e n c e b e t w e e n  data a n d the n u m e r i c a l p r e d i c t i o n is significant. T h e p r e d i c t e d being  based  only  o r g a n i z a t i o n t h a n is o b s e r v e d , a d d e d to the current  on  the  in every  fiber m o d e l before  mean  flow  in  the  case. T h e turbulence it is c a p a b l e  headbox,  orientation  show  greater  dispersion effect must  o f accurately  p r e d i c t i n g the  be  fiber  orientation i n a given f l o w field.  The  s i m u l a t i o n m e t h o d has further b e e n u s e d to p r e d i c t f i b e r o r i e n t a t i o n s  headbox causes  geometry, changes  in  flow velocity velocity  a n d fiber aspect ratio.  gradients,  and  rates  of  Changes  strain,  and  for  different  i n contraction therefore  ratio  affect  the  a l i g n m e n t o f fibers i n the f l o w d i r e c t i o n . C h a n g e s i n c h a n n e l l e n g t h a n d f l o w v e l o c i t y  can  a l s o c h a n g e the f l o w f i e l d , but their effects o n the f i b e r o r i e n t a t i o n are e l i m i n a t e d i n this  60  s i m u l a t i o n b y the  fact that the  residence  time  is c h a n g e d .  T h e s i m u l a t i o n results  also  s h o w that f o r d i l u t e s u s p e n s i o n s , the effect o f f i b e r aspect ratio a n d f i b e r l e n g t h i n the r a n g e o f o u r interest (i.e. A  B o t h experimental  f  = 33-133, L = 2~4  a n d s i m u l a t i o n results  m m ) o n f i b e r o r i e n t a t i o n is s m a l l .  s h o w e d differences i n the f i b e r orientation  in  d i f f e r e n t p r o j e c t i o n p l a n e s . T h i s p h e n o m e n o n is c a u s e d b y the f l o w characteristics i n a c o n v e r g e n t c h a n n e l , i.e., the c o n t r i b u t i o n s o f v e l o c i t y gradients to f i b e r o r i e n t a t i o n are the s a m e i n all directions.  61  not  7. R E C O M M E N D A T I O N S F O R F U T U R E  T h e c u r r e n t w o r k h a s e s t a b l i s h e d a m e t h o d to i n v e s t i g a t e  W O R K  fiber orientation i n a  headbox  f l o w a n d h a s f o u n d i m p o r t a n t e f f e c t s o n the f i b e r o r i e n t a t i o n state r e s u l t i n g f r o m d i f f e r e n t headbox  geometries  a n d f l o w c o n d i t i o n s . H o w e v e r , there are p r o b l e m s to b e s o l v e d i n  future w o r k before reaching a complete understanding o f fiber orientation i n headboxes.  1.  C u r r e n t l y , o n l y the effect o f m e a n v e l o c i t i e s a n d their gradients h a v e b e e n c o n s i d e r e d i n the f i b e r m o d e l . T h e r e exist significant differences b e t w e e n the e x p e r i m e n t a l and  the  numerical  determination velocities  convergent tube  Clearly, turbulence  o f fiber orientation  o n the  experiments.  predictions.  i n the  f i b e r is a d d e d , the  It i s t h e r e f o r e  flow.  plays  I f the  an  important  influence  simulation could produce  o f the  results  role  data  in  the  fluctuating  closer  to  the  i m p o r t a n t to k n o w the c h a r a c t e r i s t i c s o f t u r b u l e n c e i n the  s e c t i o n w h i c h are i n h e r i t e d f r o m the u p s t r e a m f l o w c h a r a c t e r i s t i c s i n the  b a n k a n d header,  a n d h o w these c h a r a c t e r i s t i c s are  m o d i f i e d b y the  headbox  contraction.  2.  F l e x i b l e f i b e r s s h o u l d b e u s e d i n the s i m u l a t i o n i n s t e a d o f the r i g i d f i b e r s u s e d i n this research. UBC.  Fortunately  the  quantify  such  To  f l e x i b l e f i b e r m o d e l is a v a i l a b l e i n the results,  one  must  provide  a  research group  reasonable  definition  at of  o r i e n t a t i o n f o r f l e x i b l e f i b e r s . T h e c u r l or k i n k o f f i b e r s s h o u l d b e s t u d i e d as w e l l .  3.  T h e current  research has  focussed  o n fiber m o t i o n a l o n g the  central  streamline.  d e t a i l e d s t u d y o f the fiber orientation p r o f i l e s i n the m a c h i n e d i r e c t i o n , cross  A  machine  d i r e c t i o n a n d p a p e r t h i c k n e s s d i r e c t i o n at t h e h e a d b o x e x i t i s r e q u i r e d . T h e n o n e  may  f i n d the a n s w e r to q u e s t i o n s s u c h as w h a t c a u s e s f i b e r o r i e n t a t i o n n o n - u n i f o r m i t y i n the c r o s s m a c h i n e d i r e c t i o n , o r f i n d f i b e r o r i e n t a t i o n i n the p a p e r t h i c k n e s s d i r e c t i o n .  62  4.  I n o r d e r to h a v e a n o v e r a l l v i e w o f the f i b e r o r i e n t a t i o n i n a h e a d b o x , the f i b e r - w a l l interactions  must  be  included. A s shown, fiber orientation  in a headbox  is m a i n l y  d e t e r m i n e d d u r i n g the short d i s t a n c e that f i b e r s m o v e v e r y c l o s e to the e x i t . N e a r exit, the c h a n n e l o p e n i n g b e c o m e s  narrower  a n d fiber-wall interactions  the  m a y not  be  negligible.  5.  D u r i n g this w o r k o n l y the c o n v e r g e n t slice s e c t i o n o f a h e a d b o x is s i m u l a t e d w i t h the a s s u m p t i o n o f i n i t i a l u n i f o r m v e l o c i t y a n d r a n d o m f i b e r o r i e n t a t i o n d i s t r i b u t i o n at t h e inlet. A m o r e c o m p l e t e w o r k s h o u l d i n v o l v e the f l o w o f the s u s p e n s i o n i n the d i f f u s e r tubes o r e v e n t h r o u g h the m a n i f o l d . T h e p a p e r m a n u f a c t u r e r s fiber orientation  p r o f i l e s i n the  cross-machine  understanding o f f l u i d - f i b e r interactions the entire h e a d b o x . p r o c e s s e s , the  fiber  direction.  are c o n c e r n e d a b o u t  That requires  a  the  thorough  and fiber orientation characteristics w i t h i n  T o g i v e a total v i e w o f the f i b e r o r i e n t a t i o n i n the p a p e r m a k i n g orientation  i n the jet  a n d o n the  forming wire  s h o u l d also  be  s t u d i e d . A l l these studies c a n t h e n b e c o m b i n e d t o g e t h e r to g i v e a n o v e r a l l v i e w o f the fiber  6.  orientation i n the paper.  In o u r research, in  the  the  simulations  papermaking  fiber-fiber and  consistencies,  i n t e r a c t i o n is n e g l e c t e d , b e c a u s e the s u s p e n s i o n s  experiments i.e.,  i n the  are range  i n d e p e n d e n t l y i n s u s p e n s i o n , b u t are p r e s e n t fibers  freely floating about.  Each  fiber  very  dilute.  o f 0.1  to  At 1.5%,  in a network  commercially  used  fibers  exist  do  not  f o r m . T h e r e are  is i n c o n t a c t w i t h m a n y o t h e r  seldom  fibers, and  b e n t o u t o f its n a t u r a l r e l a x e d s h a p e . T h e b e n d i n g a n d p u s h i n g o f f i b e r s a g a i n s t other  gives  fiber  networks  a certain  affects the f l o w pattern. A s a result,  amount fiber-fiber  of mechanical  strength,  fiber-fiber  i n t e r a c t i o n s , it i s i m p o r t a n t t o s o l v e s i m u l t a n e o u s l y f o r t h e v e l o c i t y o r i e n t a t i o n i n o r d e r to o b t a i n a n a c c u r a t e d e s c r i p t i o n o f the  63  flow  and  field  a n d the  is  each  w h i c h in turn  i n t e r a c t i o n a n d its e f f e c t o n t h e  s h o u l d also be investigated i n the future. W h e n c o n s i d e r i n g  used  fluid/fiber  a n d the fiber  flow  fiber  motion.  8. N O M E N C L A T U R E  A  f i b e r aspect ratio  r  a  a  v  2  r a n d o m variables  b  f r a c t i o n o f f i b e r s n o t i n the r  C D  cross machine direction  C  volume  concentration  V  d  fiber  diameter  E  rate o f k i n e t i c e n e r g y d i s s i p a t i o n p e r unit m a s s  f  orientation parameter  G  rate o f k i n e t i c e n e r g y p r o d u c t i o n p e r unit m a s s  H  channel inlet height  Q  k  turbulent kinetic energy  L  fiber  L  length  channel length c  °  M D  machine direction  N  c r o w d i n g factor  n  n u m b e r o f fibers  p  pressure  p( )  the statistical p r o b a b i l i t y d e n s i t y f u n c t i o n  r  fraction o f fibers  a  R  contraction  ratio  c R  e  s  Reynolds number distance a l o n g the central  S..  streamline  rate o f strain t e n s o r  u t  time  u  h y d r o d y n a m i c v e l o c i t y c o m p o n e n t i n the  U U  Q  u - v e l o c i t y at t h e i n l e t u - v e l o c i t y at t h e  outlet  64  x-direction  instantaneous fluid velocity vector  u  u  s  f l o w v e l o c i t y a l o n g the central  streamline  v  h y d r o d y n a m i c v e l o c i t y c o m p o n e n t i n the y - d i r e c t i o n  w  h y d r o d y n a m i c v e l o c i t y c o m p o n e n t i n the z - d i r e c t i o n  x  the x - d i r e c t i o n i n a C a r t e s i a n  Xj  the l e n g t h o f the flat s e c t i o n  y  the y - d i r e c t i o n i n a Cartesian  system  z  the z - d i r e c t i o n i n a C a r t e s i a n  system  a  fiber orientation angle in a projection plane  a  Q  p  system  the m e a n o f distributed angles f l o w angle Kronecker Delta  A  interval o f orientation  angle  s  e l o n g a t i o n o f the f l o w  8 e  e l o n g a t i o n o f t h e f l o w at t h e c h a n n e l e x i t  (j,  fiber orientation  <p  the angle b e t w e e n the central streamline a n d x - a x i s  1^  d y n a m i c v i s c o s i t y o f the f l u i d  v  k i n e m a t i c v i s c o s i t y o f the f l u i d  v  turbulent kinematic viscosity  0  fiber orientation  p  fluid density  CTp  standard d e v i a t i o n o f the p r o b a b i l i t y d e n s i t y p  CTr  standard d e v i a t i o n o f the f r a c t i o n r  x  f l u i d stress tensor  X i j  R e y n o l d s stress t e n s o r  Q  solid angle  angle  angle  65  ( ) a  9.  1.  Nordstrom, Toughness  B . a n d N o r m a n , B . , "Influence a n d Tensile Stiffness  Nordic Pulp Paper Res. J. 2.  Tiikaja,  R E F E R E N C E S  E . , "Fiber  9  0)  : 5  Properties  Engineering/Process  o n Sheet A n i s o t r o p y , F o r m a t i o n , Z -  of Reduced 3  Nozzle",  (1994).  and Paper  and Product  F e e d A r e a to a H e a d b o x  Machine  Quality  jggg  Runnability",  TAPPI  & Trade Fair- 1 2 4 1  Conference  (1999). 3.  4.  19(4): J 1 7 5  Shakespeare, and  J., "Tutorial: Fiber Orientation A n g l e Profiles - Process  TAPPI Proceedings, Conference'- 5 9 3 ( 1 9 9 8 ) .  & Information  Control,  o f the Q u a n t i t y a n d Orientation  p ip  o f C h e m i c a l P u l p Fibers i n the Surfaces o f N e w s p r i n t " ,  6.  a  Principles  1998 Process  Control",  F o r g a c s , O . L . a n d S t r e l i s , I., " T h e M e a s u r e m e n t  T-3  p per  u  (1993).  Cross-Machine  Electrical 5.  j p [p  P a g e , D . H . , " A Q u a n t i t a t i v e T h e o r y o f the S t r e n g t h o f W e t W e b s " , Sa.  p per  u  Can.  a  64(1):  (1963).  L o e w e n , S . R . , " F i b e r O r i e n t a t i o n O p t i m i z a t i o n " , Pu\p  pp a  Can.  er  98(10): T 3 9 1  (1997). 7.  Nordstrom, B . a n dN o r m a n , B . , "Influence o f Headbox N o z z l e Contraction on  Sheet  Formation  Proceedings'8.  Uesaka,  7904 Engineering  Conference,  TAPPI  225 (1994).  T . , " D i m e n s i o n a l Stability o f Paper:  End Use", j 9.  and Anisotropy",  Ratio  P  u  I  p  P  a  p  e  r  S  c  l  Upgrading Paper Performance i n  17(2): J 3 9 (1991).  Pantaleo, S. a n d Shands, J., " C o n t r o l l i n g Fiber Orientation",  World Paper  219(2):  22(1994). 10.  Givler,  R . C , Crochet,  M . J. a n d Pipes,  Orientation i n Dilute Suspensions", 11.  Wrist,  P. E . , "Dynamics  formation  R. B . , "Numerical Prediction  j of Composite Materials^  o f Sheet F o r m a t i o n  and structure of paper: Transaction  1 7  ( ) 7  :  3  3  o f Fiber 0  (1983).  o n the Fourdrinier M a c h i n e " ,  S m o o k , G . A . , H a n d b o o k for Pulp & Paper Technologists, T A P P I ,  66  n  e  of the symposium held at Oxford-  839(1961). 12.  j  (1982).  13.  Dahl,  H . K . and Weiss,  H . G . ," A N e w H y d r a u l i c Principle f o r H e a d b o x e s " ,  Tappi J- 5 8 ( 1 1 ) : 7 2 ( 1 9 7 5 ) . 14.  K y o s t i , A . , Stoltz, P . a n d Gustavsson, H . , " A Study o f Fiber Orientation to  Headbox  Design  and Operating  (Stockholm), Pap. Technol15.  4  24  for Simultaneous  Profiles", 1  Proc.  (1990).  Pantaleo,  Optimization o f Basis  EcoPaperTech-  st  S.  Weight  a n dOrientation  B . , " A N e w Headbox  Design  Featuring  Consistency  S. B . , " H e a d b o x  Profiling  Tappi J. 78(11):  L e e , J. J . - G . a n d Pantaleo,  Angle  39 (1995).  Decoupled from Fiber Orientation Response", 17.  Related  Conf.  th  Shakespeare, J., Kniivila, J., K o r p i n e n , A . a n dJohansson, T . ," A n On-line Control System  16.  2  EUCEPA  Strategies",  Flow Analysis",  89(1995).  j p ip  Paper Sci-  u  25(12): 4 3 7 (1999). 18.  H o l i k , H . , H e b , H . , Tietz, M . and Drtina, P., " F l u i d M e c h a n i c s i nthe H e a d b o x the  K e y to Improving  Paper  Quality",  jgg  4  Engineering  Conference,  -  Tappi  Proceedings'- 2 4 7 ( 1 9 9 4 ) . 19.  Patrick, Control",  20.  K . L . , "Latest  pi u  p  &  P  a  p  e  r  Headboxes:  Separate  Basis  Weight,  Fiber  Orientation  9:137 (1996).  Malashenko, A . , " T h e Dilution Control Headbox",  Paper Technology  38(10): 42  (1997). 21.  Begemann, U . , "Modulejet Headbox Concept Operating Experience and Multi-layer Headboxes Processing Different Paper Grades",  with Single-  p \p p per Can. u  a  97(8)': 2 0 ( 1 9 9 6 ) . 22.  Nyberg, P. and Malashenko, A . , "Dilution Control Headbox and Solutions",  23.  J.Korea  Tappi  Choices, 1  ?  Threats  (1997).  28(3): 88 (1996).  V y s e , R . , K i n g , J., H e a v e n , M . a n d Pantaleo S., "Consistency Profiling Technique for C D Basis Weight Control",  25.  A  Bando, T . ,Makino, T . andFujiki, K . ,"Development o fMitsubishi N e w Headbox and Former",  24.  83 Annual Meeting, Technical Section CPPArd  Ullmar, M . andNorman, Nozzle  at  p\ p u  B . , "Observation  L o w Consistency"  TAPPI  p  a p e r  o f Fiber  Can.  Orientation i n a  Proceedings,  aN e w  97(9): 62 (1996). Headbox  1997 Engineering  &  Papermakers Conference- 8 6 5 ( 1 9 9 7 ) . 26.  Bandhakavi,  V . S. a n d A i d u n ,  Converging Zone o fa Headbox",  C . K . , "Analysis  o f Turbulent  jggg TAPPI Engineering/Process  Quality Conference & Trade Fair'- H 3 5 ( 1 9 9 9 ) .  67  Flow  i n the  and Product  27.  Erikkila,  A . L . , Pakarinen,  P. and Odell,  Layered Orientation Analysis", 28.  u  p  P  a  p  e  M . , "Sheet  Qan.  r  99(1):  8  Forming  Studies  Using  (1998).  1  Kerekes, R . and Schell, C , "Characterization o f Fiber Flocculation Regimes b y a Crowding Factor",  29.  p\  j p\ u  p  p  s i. c  a p e r  18(1): 332 (1992).  Ullmar, M . ," O n Fiber Alignment Mechanisms  in a Headbox  Nozzle",  Master  T h e s i s , R o y a l Institute o f T e c h n o l o g y ( 1 9 9 8 ) . 30.  N i s k a n e n , K . J., "Distribution o f Fiber  Orientations  Fundamentals  i n Paper",  Pmkg. (Baker & Punton, ed.) / Trans. 9 Fundamental Res. Symp. (Cambridge) Vth  275 (1989). 31.  Dinh,  S. M . , " O n the R h e o l o g y  o f Concentrated  Fiber  Suspensions",  Sc. D .  Thesis, Department o f C h e m i c a l Engineering, M I T (1981). 32.  Zirnsak,  M . A . , H u r , D . U . and Boger,  Suspensions", j  Non-Newtonian  D . V . , "Normal  Fluid Mech-  ( )  5 4  8  I  :  33.  Waller, M . H . , "Recent Developments in Headboxes",  34.  Aidun,  C . K . and Kovacs,  Aidun,  7095 Papermakers  Conference-  P  r  o  c  Tappi J-  7 0  (1)  :  3  (1987).  3  TappiJ-  7 8  ( 1 1 ) : 97 (1995).  2  1  5  and Board  (1995).  A  :  I  5  7  (1992).  Jeffery, G . B . , " T h e M o t i o n o f E l l i p s o i d a l Particles I m m e r s e d i n a V i s c o u s F l u i d " , A  l °  2  :  161 ( 1 9 2 2 ) .  F o l g a r , F . a n d T u c k e r III, C . L . , " O r i e n t a t i o n B e h a v i o r o f F i b e r s i n C o n c e n t r a t e d S u s p e n s i o n s " , jr  39.  Fiber  (1994).  Pan-Pacific Pulp Pap. Technol. Conf. (Tokyo),  Proceeding of the Royal Society, 38.  in  Shimizu, T and Wada, K . , "Computer Simulation and Measurement o f F l o w i n a Headbox",  37.  Stresses  C . K . , " H y d r o d y n a m i c Analysis a n d Optimization o f Paper  Forming", 36.  3  A . E . , " H y d r o d y n a m i c s o f the F o r m i n g Section: the  Origin o f N o n - u n i f o r m Fiber Orientation", 35.  5  Reinforced Plastics and Composites  3  ( ) 4  :  98 (1984).  M a s o n , S. G . , a n d Bartok, W . , R h e o l o g y o f Disperse Systems,  British Society  of  R h e o l o g y , C . C . M i l l , P e r m a g o n P r e s s , N e w Y o r k , C h a p t e r 2, ( 1 9 5 9 ) . 40.  R a o , B . N . , A k b a r , S. a n d A l t a n , M . C , " A C o m p a r a t i v e  Study o n the Solution  Techniques for Fiber Orientation i n Two-dimensional Converging and D i v e r g i n g  41.  Flows",  j Thermoplastic  Akbar,  S. a n d A l t a n ,  Composite Materials  (  1 0  )  :  3  11  (1991).  M . C , " O n the S o l u t i o n o f F i b e r  dimensional Homogeneous Flows", 42.  4  Orientation  Polymer Eng. Sci. 3 2 ( 1 2 ) :  i nT w o -  810 (1992).  A d v a n i , S . G . a n d T u c h e r III, C . L . , " T h e U s e o f T e n s o r s t o D e s c r i b e a n d P r e d i c t Fiber Orientation i n Short Fiber C o m p o s i t e s " ,  68  j Rheology  3  U ): 8  7  51  (1987).  43.  A l t a n , M . C , S u b b i a n , S . , G u c e r i , S . I. a n d P i p e s , R . B . , " N u m e r i c a l P r e d i c t i o n o f Three-dimensional  Fiber  Orientation  i n Hele-Shaw  Polymer Eng Sci-  Flows",  30(14): 848 (1990). 44.  A l t a n , M . C . , A d v a n i , S . G . , G u c e r i , S . I. a n d P i p e s , R . B . , " O n t h e D e s c r i p t i o n o f the O r i e n t a t i o n State f o r F i b e r S u s p e n s i o n s i n H o m o g e n e o u s  45.  33(7):  1129(1989).  Dinh,  S. M . a n d Armstrong,  R. C , " A Rheological  Semiconcentrated Fiber Suspensions", j 46.  Ross,  R . F . a n dKlingenberg,  47.  Wherrett, G . , Gartshore,  Equation  Rheology  o f State f o r  Rheology 2 8 ( 3 ) : 2 0 7 ( 1 9 8 4 ) .  D . J., " D y n a m i c  Composed o f Linked Rigid Bodies",  Flows", j  j chem.  Simulation  Phys-  o f Flexible  Fibers  106(7): 2 9 4 9 (1997).  I., S a l c u d e a n , M . a n d O l s o n , J . , " A N u m e r i c a l M o d e l o f  Fiber M o t i o n i n Shear", T h e 1997 A S M E  Fluids Engineering  Division  Summer  M e e t i n g (1997). 48.  D o n g , S., Salcudean,  2000  Screen Slots", 49.  Shariati,  M . and Gartshore,  TAPPI Papermakers Conference and Trade Fair-  M .R., Bibeau,  Experimental  I., " F i b e r M o t i o n i n S i n g l e a n d M u l t i p l e  E . , Salcudean,  M .a n dGartshore,  5 9 7 (2000).  I., " N u m e r i c a l a n d  M o d e l s o f the F l o w i n the C o n v e r g i n g S e c t i o n o f a H e a d b o x " ,  2000  TAPPI Papermakers Conference and Trade Fair'- 6 8 5 ( 2 0 0 0 ) . 50.  H u a , L . , H e , P . ,Salcudean, M . , Gartshore, Hydraulic Headbox",  2000  I. a n d B i b e a u , E . , " T u r b u l e n t F l o w i n a  TAPPI Papermakers Conference and Trade Fair-  695  (2000). 51.  Petrie,  C . J. S., " T h e rheology  Mech,m2-3): 52.  Launder,  j  o f fiber suspensions",  Non-Newtonian  Fluid  369(1999).  B . E . and Spalding, D . B . ," T h eNumerical Computation  o f Turbulent  F l o w s " , Computer Methods in Applied Mechanics ^'- 2 6 9 ( 1 9 7 4 ) . 53.  Nowak,  P., " A Multi-grid andMulti-block Method",  Technical  Report, T h e  U n i v e r s i t y o f British C o l u m b i a (1992). 54.  55.  Ross,  R.  F.  a n d kligenberg,  D . J.,  Suspensions",,/  p lp Paper Sci-  "Sphere  Picking"  Point  u  -  "Simulation  o f Flowing  from  M c C u l l o u g h , R. L . , "Anisotropic Treatise  o n Materials  Science  Fiber  24(12): 388 (1998). Eric  Weisstein's  World  http://mathworld.wolfram.com/SpherePointPicking.html, 56.  Wood  Elastic  Behavior  a n dTechnology:  Mathematics,  9/13/2000. o f Crystalline  Properties  Materials, A c a d e m i c Press, N e wY o r k , 1 0 B : 4 5 3 (1977).  69  of  o f Solid  Polymers", Polymeric  57.  Y o r k , J. L . , " F i b e r O r i e n t a t i o n i n C u r v i l i n e a r F l o w " , M a s t e r ' s of Delaware  58.  M a r d i a , K . V . , Statistics o f directional York  59.  Thesis,  University  (1982). data, A c a d e m i c  Press, L o n d o n a n d N e w  (1972).  M o r o n e y , M . J., Facts F r o m Figures, Penguin B o o k s  70  (1956).  

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