UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Fiber orientation in a headbox 2001

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
ubc_2001-0301.pdf [ 6.08MB ]
Metadata
JSON: 1.0080963.json
JSON-LD: 1.0080963+ld.json
RDF/XML (Pretty): 1.0080963.xml
RDF/JSON: 1.0080963+rdf.json
Turtle: 1.0080963+rdf-turtle.txt
N-Triples: 1.0080963+rdf-ntriples.txt
Citation
1.0080963.ris

Full Text

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 b y X u n Z h a n g B . E n g . N o r t h w e s t e r n P o l y t e c h n i c a l U n i v e r s i t y , C h i n a , 1985 E n g . B e i j i n g U n i v e r s i t y o f A e r o n a u t i c s a n d A s t r o n a u t i c s , C h i n a , 1 9 8 8 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S O F T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F M E C H A N I C A L E N G I N E E R I N G W e a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d 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 J a n u a r y 2 0 0 1 © X u n Z h a n g , 2 0 0 1 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l 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 d e g r e e at the 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 a g r e e that the l i b r a r y s h a l l m a k e it f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e 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 o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y the h e a d o f m y d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g 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 2 3 2 4 M a i n M a l l V a n c o u v e r , B C V 6 T 1 Z 4 D a t e : J a n u a r y 2 0 0 1 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 T A B L E O F C O N T E N T S A B S T R A C T i i T A B L E OF CONTENTS i i i LIST OF T A B L E S v LIST OF FIGURES vi A C K N O W L E D G E M E N T S vii i 1. INTRODUCTION 1 2. L I T E R A T U R E REVIEW 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 i i i 3.5 M e a s u r e m e n t 2 0 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 2 8 4.1 T h e H e a d b o x F l o w M o d e l 2 9 4 . 2 F i b e r M o d e l 3 2 5. R E S U L T S A N D D I S C U S S I O N 3 6 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 3 6 5.2 C o m p a r i s o n o f S i m u l a t i o n a n d E x p e r i m e n t a l R e s u l t s 3 9 5.3 F a c t o r s A f f e c t i n g F i b e r O r i e n t a t i o n 4 4 5.3.1 T h e E f f e c t o f C o n t r a c t i o n R a t i o o n F i b e r O r i e n t a t i o n 4 4 5 .3 .2 T h e E f f e c t o f F l o w R a t e o n F i b e r O r i e n t a t i o n 4 5 5 .3 .3 T h e E f f e c t o f C h a n n e l L e n g t h o n F i b e r O r i e n t a t i o n 4 6 5 .3 .4 T h e E f f e c t o f F i b e r A s p e c t R a t i o o n F i b e r O r i e n t a t i o n 4 7 5 .3 .5 T h e E f f e c t o f F l o w E l o n g a t i o n 4 7 5 .4 S y m m e t r i c C h a n n e l 4 8 5.5 S t a t i s t i c a l E r r o r E s t i m a t i o n 4 9 6. S U M M A R Y A N D C O N C L U S I O N S 6 0 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 6 2 8. N O M E N C L A T U R E 6 4 9. R E F E R E N C E S 6 6 i v 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. O r i e n t a t i o n P a r a m e t e r s 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 : 43 T a b l e 5.2. 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 c = 0.225 m ) 45 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 c = 0.225 m ) 46 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 c (Rc= 10, Uo-0 .24 m / s ) 46 T a b l e 5.5. 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 A r ....47 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 the C h a n n e l E x i t f o r D i f f e r e n t Rc 48 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 0 : 48 v L I S T O F F I G U R E S F i g u r e 2 . 1 . 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 p a t t e r n i n a p i e c e o f p a p e r 16 F i g u r e 3 . 1 . T h e l e n g t h d i s t r i b u t i o n o f n y l o n f i b e r s 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 2 4 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 2 4 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 ) 2 5 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 2 5 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 the f l o w : (a) b e f o r e a n a l y s i s ; (b) af ter a n a l y s i s . . 2 6 F i g u r e 3 .8 . T h e m e a s u r e m e n t p o i n t s a l o n g the h e a d b o x c h a n n e l 2 7 F i g u r e 4 . 1 . A f i b e r i n t h r e e - d i m e n s i o n a l c o o r d i n a t e s 35 F i g u r e 4 . 2 . T h e i n i t i a l r a n d o m d i s t r i b u t i o n o f 1 0 0 0 f i b e r s 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 5 2 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 5 2 F i g u r e 5 .3 . 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 s t r e a m l i n e 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 c e n t r a l 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 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 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 c e n t r a l s t r e a m l i n e 5 4 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 c m , 5 4 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 c m , 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 c m , 55 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 c m , 5 6 F i g u r e 5 . 1 1 . 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 c m , 5 6 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 c m , 5 7 F i g u r e 5 . 1 3 . 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 the c h a n n e l ex i t , 5 7 F i g u r e 5 .14 . T h e o r i e n t a t i o n p a r a m e t e r s a l o n g the c e n t r a l s t r e a m l i n e , 58 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 the 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 v i F i g u r e 5 . 1 6 . 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 ) 5 9 F i g u r e 5 . 1 7 . 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 5 9 A C K N O W L E D G E M E N T S I e x p r e s s s i n c e r e g r a t i t u d e to m y s u p e r v i s o r s , D r . M a r t h a S a l c u d e a n a n d D r . I a n G a r t s h o r e , f o r t h e i r h e l p f u l a d v i c e a n d s u g g e s t i o n s . I w o u l d a l s o l i k e to t h a n k m y c o l l e a g u e s , M o h a m m a d R . S h a r i a t i a n d S u q i n D o n g . T h e i r e f f o r t a n d h e l p h a s m a d e t h i n g s m u c h e a s i e r b o t h i n m y e x p e r i m e n t a l 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 g r a t e f u l f o r t h e f i n a n c i a l s u p p o r t p r o v i d e d b y F R B C R e s e a r c h A w a r d . 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 c o n s t a n t s u p p o r t a n d e n c o u r a g e m e n t d u r i n g the p a s t t w o y e a r s . v i i i 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 t h i s s i m u l a t i o n m e t h o d a n d the r e s u l t s are a n a l y z e d . F o l l o w i n g t h i s c h a p t e r , the r e l e v a n t l i tera ture i s r e v i e w e d i n C h a p t e r 2. T h e d e t a i l e d e x p e r i m e n t a l 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 i n a h e a d b o x f l o w are p r e s e n t e d i n C h a p t e r 3. C h a p t e r 4 d e s c r i b e s the n u m e r i c a l s i m u l a t i o n o f 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 b y a c o m b i n a t i o n o f a f l o w m o d e l a n d a f i b e r m o t i o n m o d e l . C h a p t e r 5 p r e s e n t s the c o m p a r i s o n o f m e a s u r e d a n d n u m e r i c a l l y s i m u l a t e d r e s u l t s . P a r a m e t r i c s t u d i e s , o b t a i n e d u s i n g the n u m e r i c a l m e t h o d , s h o w the i n f l u e n c e o f h e a d b o x g e o m e t r y , f l o w v e l o c i t y a n d fiber p r o p e r t y . C h a p t e r s 6 a n d 7 s u m m a r i z e the m a j o r c o n c l u s i o n s o f t h i s t h e s i s a n d g i v e r e c o m m e n d a t i o n s f o r f u t u r e r e s e a r c h , r e s p e c t i v e l y . 2 2. LITERATURE REVIEW 2.1 Fiber Orientation and Paper Quality A p i e c e o f p a p e r i s 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 l o c a t e d w i t h i n t h e p a p e r p l a n e a n d o r i e n t e d i n d i f f e r e n t d i r e c t i o n s . S t a t i s t i c a l l y , h o w e v e r , m o s t o f the f i b e r s m a y 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 o r i e n t a t i o n i s p r o d u c e d b y t h e p a p e r 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 r e l a t e d to s e v e r a l c r i t i c a l p a p e r p r o p e r t i e s . T h e o r i e n t a t i o n p a t t e r n r e t a i n e d i n the final p a p e r c o n t r o l s the m e c h a n i c a l p r o p e r t i e s o f the sheet . N o r d s t r o m a n d N o r m a n [1] i n d i c a t e d that d e p e n d i n g o n the g r a d e , a c e r t a i n d e g r e e o f fiber 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 i s d e s i r e d . F o r n e w s p r i n t , a r a t h e r h i g h a n i s o t r o p y i s 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 t h e 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 g r a d e s , o n the o t h e r 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 e n s u r e i s o t r o p i c d i m e n s i o n a l c h a n g e s w i t h v a r i a t i o n s i n m o i s t u r e a n d t e m p e r a t u r e . N o r d s t r o m a n d N o r m a n a l s o p o i n t e d o u t that the s t r e n g t h 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 i s a f f e c t e d p o s i t i v e l y b y the d e g r e e o f f i b e r - t o - f i b e r b o n d i n g a n d a l s o b y the d e g r e e o f fiber o r i e n t a t i o n i n that d i r e c t i o n . T h e s t r e n g t h 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 m u s t b e 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 o f f s e t p r i n t i n g , o r b l i s t e r i n g i n h e a t s e t o f f s e t p r i n t i n g . A s h e e t i s s t r o n g e r a n d s t i f f e r i n the d i r e c t i o n i n w h i c h m o s t f i b e r s are o r i e n t e d , a n d 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 o r i e n t a t i o n . W e c a n u n d e r s t a n d t h i s b y s t u d y i n g t h e p r o p e r t i e s o f the p r i n c i p a l c o n s t i t u e n t s o f p a p e r , the w o o d fibers, b e c a u s e t h e 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 [2, 3]. F i b e r d i m e n s i o n s , f l e x i b i l i t y a n d c o a r s e n e s s are c o n n e c t e d w i t h the m e c h a n i c a l p r o p e r t i e s , s t r u c t u r a l 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 s t r e n g t h , p o r o s i t y a n d sheet d e n s i t y . 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 to t h o s e a c r o s s it . F o r e x a m p l e , the s t r e n g t h o f a f i b e r i s m u c h greater a l o n g the f i b e r a x i s t h a n a c r o s s i t , w h e r e a s the w e t - e x p a n s i v i t y is greater a c r o s s the f i b e r a x i s t h a n a l o n g it [4]. I f t h e m a j o r d i r e c t i o n i s 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 s u r f a c e t o w a r d w h i c h m o s t f i b e r s are a l i g n e d a n d the m i n o r d i r e c t i o n as the d i r e c t i o n n o r m a l to the m a j o r d i r e c t i o n , the f o l l o w i n g c o n c l u s i o n s a b o u t p a p e r p r o p e r t y c a n b e i n f e r r e d f r o m the fiber-paper r e l a t i o n s h i p . T h e t e n s i l e s t i f f n e s s , t e n s i l e e n e r g y a b s o r p t i o n , b e n d i n g s t i f f n e s s a n d c r u s h s t r e n g t h are h i g h e r i n the m a j o r d i r e c t i o n , b u t the tear s t r e n g t h a n d w e t e x p a n d i n g 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 . D i r e c t i o n a l d i f f e r e n c e s i n m e c h a n i c a l p r o p e r t i e s h a v e b e e n e x p e r i m e n t a l l y c o r r e l a t e d w i t h fiber o r i e n t a t i o n [5, 6, 7]. 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 d e f e c t s , l e a d i n g to 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 s t r e n g t h . L o e w e n [6] s u m m a r i z e d t h e p a p e r q u a l i t y p r o b l e m s that are r e l a t e d to p o o r fiber o r i e n t a t i o n as f o l l o w s : • T w i s t , w a r p , c u r l a n d s t a c k - l e a n . • W e b w a n d e r i n g , m i s r e g i s t e r i n g i n m u l t i - p a s s p r i n t i n g a n d c o l o u r p r i n t i n g . • P a p e r f e e d p a t h j a m m i n g . • M u l t i - p a r t f o r m s d e b o n d i n g . • L o w t e n s i l e s t r e n g t h , l o w tear s t r e n g t h a n d w e a k s t i f f n e s s . • W r i n k l e s o n l o n g l e a d p r e s s e s a n d d r y e r w r i n k l e s . 2.2 The Definition of Fiber Orientation F i b e r o r i e n t a t i o n r e f e r s to the a n g u l a r d i s t r i b u t i o n o f f i b e r s r e l a t i v e to the p a p e r - m a c h i n e 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. T h e d i s t a n c e 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 to the n u m b e r o f f i b e r s o r i e n t e d i n that d i r e c t i o n . T h e p o l a r d i a g r a m d e s c r i b e s t w o c o m m o n l y u s e d fiber o r i e n t a t i o n t e r m s : t h e fiber 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 as s h o w n i n the d i a g r a m , i s the a n g l e 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 r a t i o o f the f i b e r s o r i e n t e d i n the M D o v e r 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 o f M D to C D s t r e n g t h b a s e d o n the k n o w l e d g e that the 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 c o r r e s p o n d s to the 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 i s e q u a l to the r a t i o o f l e n g t h s a / b . 2.3 Factors Affecting Fiber Orientation P a p e r i s m a d e i n a c o n t i n u o u s p r o c e s s . T h e s u s p e n s i o n o f f i b e r s a n d f i l l e r s i s d i s c h a r g e d f r o m the s l i c e o f a h e a d b o x 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 the f o r m i n g w i r e . O n the w i r e a s h e e t i s 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 t h u s f o r m e d is w e t a n d w e a k a n d 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 t h e e a r l y f o r m i n g s e c t i o n o r w e t e n d 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 , i . e . , 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 p r o c e s s . T h e h e a d b o x d e s i g n c a n h a v e a n e f f e c t o n t h e 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 s l i c e . T h e e l o n g a t i o n a n d s h e a r l e a d i n g to t h e s l i c e t e n d 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 . M a n y r e s e a r c h e r s [4, 9 , 10] h a v e a n a l y z e d the f a c t o r s that a f f e c t f i b e r o r i e n t a t i o n a n d 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 i s the h y d r o d y n a m i c p r o c e s s i n t h e 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] s t u d i e d t h e f i b e r o r i e n t a t i o n i n t h e j e t 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 r e l a t i v e s p a t i a l a r r a n g e m e n t o f the f i b e r s 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 d e t e r m i n e d 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 t a b l e . W i t h i n t h i s s p a c e , the o r i e n t a t i o n o f t h e f i b e r s , t h e d e g r e e o f f l o c c u l a t i o n , the r e l a t i v e d i s t r i b u t i o n o f m a t e r i a l s t h r o u g h t h e t h i c k n e s s o f the sheet a n d the m a c r o - a n d m i c r o - m a s s d i s t r i b u t i o n i n the p l a n e o f t h e s h e e t are a l l l a i d d o w n . S u b s e q u e n t p r o c e s s e s , l i k e p r e s s i n g a n d d r y i n g , h a v e m i n o r e f f e c t s o n f i b e r o r i e n t a t i o n w i t h o n l y s o m e m i c r o - r e a r r a n g e m e n t a n d c o n s o l i d a t i o n o f t h e 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 th is w o r k i s f o c u s e d o n the h e a d b o x , w e w i l l start f i r s t w i t h t h e h e a d b o x e f f e c t o n the fiber o r i e n t a t i o n , t h e n c o n s i d e r r e l a t i v e w i r e s p e e d , w i r e t y p e s a n d fiber c o n s i s t e n c y . 2.3.1 Headbox A f u n d a m e n t a l f u n c t i o n o f a n y h e a d b o x is to e n s u r e the m a c h i n e a n d c r o s s m a c h i n e 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 s t r i n g e n t q u a l i t y d e m a n d s are u n i f o r m i t y o f b a s i s w e i g h t p r o f i l e s o n finer s c a l e s a n d the c o n t r o l l a b i l i t y o f fiber o r i e n t a t i o n p r o f i l e s . A h e a d b o x c a n b e d i v i d e d i n t o three s e c t i o n s b y the p r i n c i p a l f l o w p a t t e r n s i n v o l v e d [12] : f l u i d d i s t r i b u t i o n , f l o w r e c t i f i c a t i o n a n d j e t d e v e l o p m e n t . I n the first s e c t i o n , a t a p e r e d 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 t h e d i s t r i b u t o r i s i m p r o v e d t h r o u g h the r e c t i f i c a t i o n 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 t u b e b a n k is o f t e n u s e d i n t h e s e p r o c e s s e s . T h e w a l l f r i c t i o n i n t h e t u b e s d a m p e n s f l o w d i s t u r b a n c e s o r i g i n a t i n g i n the s t o c k a p p r o a c h s y s t e m a n d creates t u r b u l e n c e w h i c h i s n e e d e d to p r e v e n t fiber flocculation 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 t h e t u b e b a n k m a y 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 separa te f l o w s f r o m a d i s t r i b u t o r , e l i m i n a t i n g u n d e s i r a b l e c r o s s - f l o w a n d e d d i e s , i m p r o v i n g the v e l o c i t y p r o f i l e a n d d e v e l o p i n g t u r b u l e n c e o f d e s i r e d s c a l e a n d i n t e n s i t y [13] . T h e j e t d e v e l o p m e n t p r o c e s s 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 sheet f o r m i n g s e c t i o n . A n i d e a l h e a d b o x s h o u l d p r o d u c e a u n i f o r m a n d s table j e t 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 l a t e r a l 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 p e r t u r b a t i o n s . I n b r i e f , the h e a d b o x s p r e a d s the f l o w o f p u l p o u t o f the s t o c k 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 , p r o v i d e s 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 . [14] p o i n t e d o u t that i n a h e a d b o x , fiber o r i e n t a t i o n 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 p r e s s u r e d i s t r i b u t i o n , f l o w d i s t r i b u t i o n u n i t s a n d h e a d b o x t u b e p a t t e r n s . M a n y r e s e a r c h e r s [4, 15, 16, 17] h a v e a g r e e d that the 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 b a s i s 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 a f f e c t s 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 p r o f i l e . I n a c o n v e n t i o n a l h e a d b o x , the s l i c e l i p s h a p e i s g o v e r n e d b y the b a s i s w e i g h t p r o f i l e c o n t r o l l e r , w h i c h k e e p s t h e b a s i s w e i g h t at the r e e l as f la t as 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 b a s i s 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 f i b e r o r i e n t a t i o n p r o f i l e s , b e c a u s e a c h a n g e i n t h e s h a p e o f the s l i c e l i p m a y r e s u l t i n s i g n i f i c a n t c r o s s f l o w , w h i c h l e a d s to a v a r i a t i o n 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 . F o r a c o n v e n t i o n a l h e a d b o x , it i s i m p o s s i b l e to a d j u s t b a s i s w e i g h t a n d c r o s s - m a c h i n e fiber o r i e n t a t i o n p r o f i l e s i n d e p e n d e n t l y o f e a c h o ther . T o s o l v e the p r o b l e m , a r e v o l u t i o n a r y h e a d b o x d e s i g n , w h i c h is c a l l e d the c o n s i s t e n c y p r o f i l e d h e a d b o x o r d i l u t i o n c o n t r o l h e a d b o x , h a s b e e n i n t r o d u c e d [9, 18, 19, 2 0 , 2 1 , 2 2 , 2 3 , 2 4 ] . T h i s h e a d b o x e n a b l e s i n d e p e n d e n t c o n t r o l o f C D b a s i s w e i g h t a n d f i b e r o r i e n t a t i o n p r o f i l e s . T h e 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 s t o c k c o n s i s t e n c y p r o f i l e i n the h e a d b o x a n d t h e s l i c e l i p is t h e n u s e d i n the 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 a n d 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 r a t i o , w h i c h is the r a t i o b e t w e e n the i n l e t a r e a a n d the o u t l e t a r e a , c a n n o t o n l y g e n e r a t e a h i g h d e g r e e 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 the 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 e f f e c t to the e n h a n c e d s t r e n g t h o f the e l o n g a t i o n a l s t r a i n f i e l d i n the n o z z l e a n d the c h a n g e s i n t u r b u l e n c e i n t e n s i t y . 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 i s 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 d i c a t e d that the c o n t r a c t i o n r a t i o 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 the n o z z l e e x i t . T h e i r r e s u l t s i n d i c a t e d that the e f f e c t o f c o n t r a c t i o n r a t i o i s m o r e s i g n i f i c a n t o n t h e 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 c o n t r a c t i o n r a t i o . 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 t h e 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 t e n d s 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 s t r e t c h a n d r u p t u r e the f l o e s . T h e t u r b u l e n c e i n the f l o w m a y 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 the s u s p e n s i o n d i s p e r s i o n s . 7 L e e a n d P a n t a l e o [17] i n d i c a t e d that b e s i d e s the h e a d b o x , the f o r m i n g p r o c e s s a l s o c o n t r i b u t e s to t h e r e s u l t a n t f i b e r o r i e n t a t i o n d e p e n d i n g o n the t y p e o f f o r m e r , a n d 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 e f f e c t s are s u m m a r i z e d i n the f o l l o w i n g s e c t i o n s . 2.3.2 Jet to Wire Speed Difference T h e m o s t s i g n i f i c a n t f a c t o r d e t e r m i n i n g the f i b e r o r i e n t a t i o n i s u s u a l l y the s p e e d d i f f e r e n c e b e t w e e n the j e t a n d the f o r m i n g w i r e . I d e a l l y , the j e t i s a s s u m e d to b e i n the m a c h i n e d i r e c t i o n , b u t i n p r a c t i c e , there e x i s t s m a l l t r a n s v e r s e f l o w s . T h e m a g n i t u d e o f the c r o s s f l o w s v a r i e s from l a y e r to l a y e r w i t h i n the je t , 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 t h e je t . T h e d i f f e r e n c e b e t w e e n the j e t 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 e v e n a s m a l l c r o s s f l o w m a y c a u s e a s i g n i f i c a n t c h a n g e i n f i b e r o r i e n t a t i o n a n g l e w h e n t h e s u s p e n s i o n i s 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 i s the r e a s o n w h y , i n the i n d u s t r y , 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 j e t to w i r e s p e e d d i f f e r e n c e . A s t h e 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 j e t v e l o c i t y a n d the w i r e s p e e d i s i n c r e a s e d , t h e a v e r a g e f i b e r o r i e n t a t i o n a n g l e i s r e d u c e d , a n d the f i b e r o r i e n t a t i o n i n d e x i s i n c r e a s e d [4] . 2.3.3 Forming Wire B e c a u s e the j e t d i s c h a r g e d f r o m the s l i c e 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 e f f e c t s a n d w a k e e f f e c t s , a n d a l s o h a s a n i m p i n g e m e n t a n g l e w h e n the s t o c k i s s p r e a d o n the w i r e , it i s i m p o s s i b l e to e l i m i n a t e the d i f f e r e n c e i n the v e l o c i t y b e t w e e n the j e t a n d the w i r e . T h e v e l o c i t y d i f f e r e n c e m a y c a u s e s h e a r f o r c e s i n the r e g i o n o f t h e 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 the 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 o u t that the fiber o r i e n t a t i o n 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 s h e e t i s f i n a l l y s e t t l e d 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 s h e a r , d e - w a t e r i n g v e l o c i t y o f t h e 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 . T h e d i f f e r e n c e i n the 8 m a n n e r o f d e - w a t e r i n g , i n o n e d i r e c t i o n s u c h as i n the F o u r d r i n i e r c a s e o r i n t w o d i r e c t i o n s s u c h as i n g a p f o r m i n g , p r o d u c e s d i f f e r e n t o r i e n t a t i o n t w o - s i d e d n e s s i n the f i b e r o r i e n t a t i o n . I n a d d i t i o n to the h y d r o d y n a m i c e f f e c t o n the f i b e r o r i e n t a t i o n , the t u r b u l e n c e e f f e c t 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 i s 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 d r a i n a g e b y t h e d r a i n a g e e l e m e n t s . I n a d d i t i o n , t u r b u l e n c e is i n d u c e d b y t h e s p e e d d i f f e r e n c e b e t w e e n t h e 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 i s i n c r e a s e d , the a v e r a g e f i b e r o r i e n t a t i o n a n g l e i s n o t c h a n g e d , b u t the i n - p l a n e f i b e r o r i e n t a t i o n a n i s o t r o p y is d e c r e a s e d , a n d the f i b e r o r i e n t a t i o n i n d e x i s r e d u c e d [1]. 2.3.4 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 the s l i c e i s f o u n d to b e c o n s i s t e n c y s e n s i t i v e , a n d to b e a f u n c t i o n o f the f i b e r n e t w o r k s t r e n g t h . F i b e r - f i b e r i n t e r a c t i o n s d e t e r m i n e h o w m a n y i n d i v i d u a l f i b e r s c a n b e r o t a t e d w i t h the o r i e n t e d s h e a r f i e l d . A t h i g h e r c o n c e n t r a t i o n s , f i b e r s are l e s s a l i g n e d i n the f l o w d i r e c t i o n , p r e s u m a b l y as a r e s u l t o f f l o c c u l a t i o n . K e r e k e s a n d S c h e l l [28] d e f i n e d a c r o w d i n g f a c t o r N , w h i c h i s b a s e d o n the v o l u m e c o n c e n t r a t i o n C v , the f i b e r l e n g t h L a n d d i a m e t e r d , to r e p r e s e n t the d e g r e e o f f l o c c u l a t i o n : A s t h e c o n s i s t e n c y i n c r e a s e s , the c r o w d i n g f a c t o r w i l l i n c r e a s e . U l l m a r [29] d i d e x p e r i m e n t s w h i c h s h o w e d that the f i b e r a l i g n m e n t d e c r e a s e s as the c r o w d i n g f a c t o r i n c r e a s e s . 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 s t r a i g h t f i b e r s [29, 3 0 ] . A c c o r d i n g to t h e i r c o n c e n t r a t i o n s , f i b e r s u s p e n s i o n s are u s u a l l y c l a s s i f i e d i n t o t h r e e r e g i m e s : d i l u t e , s e m i - c o n c e n t r a t e d a n d h i g h l y c o n c e n t r a t e d . I f f i b e r s are c o n s i d e r e d to b e N = -Cv 3 (2 .1) 9 r i g i d c y l i n d e r s w i t h l e n g t h L a n d d i a m e t e r d , a n d o c c u p y a f r a c t i o n C v o f the t o t a l v o l u m e o f t h e s u s p e n s i o n , D i n h [31] s h o w s that the 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 v < ( d / L ) 2 , the s e m i - c o n c e n t r a t e d r e g i m e i s d e f i n e d as ( d / L ) 2 < C v < ( d / L ) , a n d the h i g h l y c o n c e n t r a t e d r e g i m e i s d e f i n e d as C v > ( d / L ) . I n the 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 ts n e a r e s t n e i g h b o r is g r e a t e r t h a n L , s o the f i b e r s are f r e e to rotate , a n d i n t e r a c t i o n s b e t w e e n fibers are rare . I n the s e m i - c o n c e n t r a t e d r e g i m e , the s p a c i n g b e t w e e n fibers i s less t h a n L b u t g r e a t e r t h a n d , a n d i n t e r a c t i o n b e t w e e n fibers are f r e q u e n t . W h e n the s u s p e n s i o n f a l l s i n t o the h i g h l y c o n c e n t r a t e d r e g i m e , the s p a c i n g b e t w e e n fibers i s o n the o r d e r o f fiber d i a m e t e r d . T h r e e r e g i m e s c a n a l s o b e d e f i n e d i n t e r m s o f fiber v o l u m e f r a c t i o n C v a n d fiber a s p e c t r a t i o A r , w h i c h e q u a l s L / d [32] : T h e d i l u t e r e g i m e i s w h e n : C „ A,2 « 1 (2 .2) t h e 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 < Cv < V (2 .3) a n d t h e c o n c e n t r a t e d r e g i m e i s d e f i n e d as : CvAr » 1 (2 .4) I n h e a d b o x e s o f c o n v e n t i o n a l p a p e r m a c h i n e s , the fiber w e i g h t c o n s i s t e n c i e s v a r y b e t w e e n 0.1 a n d 1 .5%, fiber l e n g t h s v a r y b e t w e e n 1 a n d 5 m m a n d a s p e c t r a t i o s v a r y b e t w e e n 3 0 a n d 2 0 0 [33] . F o r e x a m p l e , i f the v o l u m e c o n c e n t r a t i o n o f the fiber s u s p e n s i o n C v is 1 % , f i b e r s 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 u m , t h e n t h e a s p e c t r a t i o A r i s 7 5 . T h e s u s p e n s i o n i s t h e n i n the s e m i - c o n c e n t r a t e d r e g i m e , b e c a u s e A r " 2 < C v < A r _ 1 (that i s 0 . 0 1 8 % < 1 % < 1 .3%). T h e r e w o u l d t h e n e x i s t 10 f r e q u e n t f i b e r - f i b e r i n t e r a c t i o n s 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 c u r r e n t s t u d y is t h e r e f o r e a s i m p l i f i c a t i o n o f the a c t u a l p r o b l e m . 2.4 Headbox Flow Simulations to Investigate Fiber Orientation C o m p u t e r s i m u l a t i o n h a s b e e n w i d e l y u s e d i n the s t u d y o f p r o c e s s e s that o c c u r i n e n g i n e e r i n g e q u i p m e n t . T h e s i m u l a t i o n i n v e s t i g a t i o n s n o t o n l y m e e t the n e e d f o r u n d e r s t a n d i n g a n d p r e d i c t i o n , b u t a l s o h a v e l a r g e e c o n o m i c b e n e f i t s . S e v e r a l r e s e a r c h e r s h a v e c o n d u c t e d h e a d b o x f l o w s i m u l a t i o n s i n o r d e r to i n v e s t i g a t e the f l o w i n d u c e d f i b e r o r i e n t a t i o n . A i d u n [34, 35] s t u d i e d the s e c o n d a r y f l o w s i n the h e a d b o x a n d t h e i r e f f e c t s o n n o n - u n i f o r m f i b e r o r i e n t a t i o n a n d m a s s f o r m a t i o n b y u s i n g a n o n - l i n e a r k - s t u r b u l e n c e m o d e l to i n v e s t i g a t e 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 t f l o w i n a l o w c o n s i s t e n c y h e a d b o x . T h e 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 i s 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 e i t h e r b y t h e 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 , o r 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 . L e e a n d P a n t a l e o [17] u s e d a s t a n d a r d k - s t u r b u l e n c e m o d e l to a n a l y z e h e a d b o x f l o w 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 v a l v e c o n t r o l , b l e e d c o n t r o l s , t u b e i n s e r t s a n d h e a d e r 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 t h e r e l a t i o n s h i p b e t w e e n the h e a d b o x f l o w c h a r a c t e r i s t i c s a n d the fiber o r i e n t a t i o n , a n d c o r r e l a t e d t h e h e a d b o x f l o w c h a r a c t e r i s t i c s i n t e r m s o f f l o w a n g l e o b t a i n e d f r o m C F D s o l u t i o n s w i t h the 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 to u s e the f l o w a n g l e [3, w h i c h i s d e f i n e d b y the M D v e l o c i t y , u , a n d C D v e l o c i t y , v , to r e p r e s e n t the a v e r a g e fiber o r i e n t a t i o n a n g l e : P = t a n " ( -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 t u r b u l e n c e m o d e l a n d a finite d i f f e r e n c e m e t h o d 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 h e a d e r , s i d e w a l l , c o n t r a c t i n g p a r t a n d s l i c e l i p , h a v e o n p a p e r q u a l i t y , e s p e c i a l l y t h e u n e v e n b a s i s 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 the c o m m o n p r o b l e m that t h e y t r i e d to s t u d y fiber o r i e n t a t i o n i n the 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 F i b e r S u s p e n s i o n S i m u l a t i o n T h e first f u n d a m e n t a l s t u d y o f the o r i e n t a t i o n o f a r i g i d e l l i p s o i d a l p a r t i c l e i n a d i l u t e 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 J e f f e r y [37] . H e s o l v e d the f l o w field a r o u n d 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 the s u r f a c e o f the p a r t i c l e . T h e a n g u l a r v e l o c i t y v e c t o r o f the p a r t i c l e w a s t h e n f o u n d f r o m the r e q u i r e m e n t that the to ta l t o r q u e a c t i n g o n the p a r t i c l e b e z e r o . T h e f o l l o w i n g J e f f r e y ' s e q u a t i o n d e s c r i b e s a s i m p l i f i e d c a s e o f a fiber l y i n g i n a t w o - d i m e n s i o n a l f l o w field [38] . . _ . 2 du 2 dv -smtpcostp s i n tp— + c o s (p— + s i n ^ c o s ^ du dx dy dx dv_ dy, ( 1 ^ (2 .6) . , i , du . i ,dv . , , - s i n ^ c o s ^ — + c o s <p s i n <p — - fs in^cost? du dx dy dx 5v T h e a n g l e ^ , w h i c h is the a n g l e b e t w e e n fiber a x i s a n d x - a x i s , d e s c r i b e s the o r i e n t a t i o n o f the fiber. W h e n the a s p e c t r a t i o is greater t h a n u n i t y , the o r i e n t a t i o n o f t h e fiber c h a n g e s m a i n l y i n r e s p o n s e to d e f o r m a t i o n o r r o t a t i o n o f the f l u i d . B e s i d e s r o t a t i o n , f i b e r s a l s o t r a n s l a t e 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 fluid w o u l d h a v e at the c e n t r o i d o f t h e fiber. J e f f e r y ' s t h e o r y h a s b e e n v e r i f i e d i n the e x p e r i m e n t a l w o r k b y M a s o n a n d B a r t o k [39] . 12 S i n c e J e f f e r y ' s w o r k , s e v e r a l c o n s t i t u t i v e m o d e l s h a v e b e e n d e v e l o p e d from w h i c h f l o w 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 d i l u t e o r s e m i - c o n c e n t r a t e d o r c o n c e n t r a t e d s u s p e n s i o n s . R a o et a l . [40] 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 s p a t i a l s tress 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 s p a t i a l v i s c o u s stress g r a d i e n t s , t h e n t h e f l u i d b e h a v i o r i s 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 al ter t h e f l o w k i n e m a t i c s . 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 i s n o t n e e d e d a n d the s o l e u s e o f J e f f r e y ' s e q u a t i o n i s 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 s tress g r a d i e n t c o n t r i b u t i o n s f r o m the p a r t i c l e s are c o m p a r a b l e o r l a r g e r t h a n the s u s p e n d i n g f l u i d c o n t r i b u t i o n s , the s u s p e n s i o n e x h i b i t s n o n - N e w t o n i a n c h a r a c t e r i s t i c s 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 p r o p e r t i e s . T h i s n e c e s s i t a t e s the s i m u l t a n e o u s s o l u t i o n o f t h e f l o w a n d o r i e n t a t i o n f i e l d s b y u s i n g a p r o p e r a n i s o t r o p i c c o n s t i t u t i v e m o d e l , o r b y a c c o u n t i n g f o r p a r t i c l e / p a r t i c l e i n t e r a c t i o n i n s o m e o t h e r w a y . T h e b e h a v i o r o f a n i n d i v i d u a l f i b e r i n a d i l u t e s u s p e n s i o n is o n l y a f u n c t i o n o f its 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 o t h e r f i b e r s . G i v l e r et a l . [10] h a v e d e v e l o p e d a n u m e r i c a l s c h e m e to s o l v e f o r the f i b e r o r i e n t a t i o n i n d i l u t e s u s p e n s i o n s a n d i n c o n f i n e d g e o m e t r i e s b y i n t e g r a t i n g J e f f e r y ' s e q u a t i o n a l o n g the s t r e a m l i n e s . T h e v e l o c i t y field u s e d i n o r d e r to d e t e r m i n e t h e s t r e a m l i n e s w a s o b t a i n e d b y a s s u m i n g that the fibers d o n o t d i s t u r b the flow. It w a s s h o w n i n t h e i r w o r k that s h e a r flows a l w a y s i n d u c e a p e r i o d i c r o t a t i o n o f the p a r t i c l e s , a n d p a r t i c l e s w i t h l a r g e a s p e c t r a t i o s 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 t h e s t r e a m l i n e s o f t h e flow, a l t h o u g h t h e y are s u b j e c t 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, t h e f i b e r s w i l l a s s u m e a t r a n s v e r s e o r i e n t a t i o n w i t h r e s p e c t to the s t r e a m l i n e s o f t h e flow. C o n v e r s e l y , 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 o r i e n t f i b e r s c l o s e r to t h e fluid s t r e a m l i n e d i r e c t i o n . S t a b l e e q u i l i b r i u m o r i e n t a t i o n e x i s t s f o r e l o n g a t i o n a l flow a n d n o t f o r s h e a r flow. I n a n e l o n g a t i o n flow, it i s w e l l k n o w n that s t r e t c h i n g flows a l i g n fibers i n the d i r e c t i o n o f s t r e t c h i n g . A 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 s t r e t c h i n g d i r e c t i o n , o r i e n t a t i o n a n g l e ^ = 0, i s i n s table e q u i l i b r i u m , a n d a fiber at § = ± n/2 i s i n u n s t a b l e e q u i l i b r i u m . A l l o t h e r fibers rotate t o w a r d the s table e q u i l i b r i u m p o s i t i o n w i t h ^ c h a n g i n g 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 s t r e t c h i n g d i r e c t i o n . 13 A k b a r a n d A l t a n [41] u s e a c o m b i n a t i o n o f a n a l y t i c a l s o l u t i o n s a n d s t a t i s t i c a l m e t h o d s to s t u d y f i b e r o r i e n t a t i o n b e h a v i o r i n a r b i t r a r y 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 f l o w s . 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 s t a t i s t i c a l l y 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 o r i e n t a t i o n o f a l a r g e n u m b e r o f f i b e r s , 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 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 i s d e p e n d e n t o n the n u m b e r o f f i b e r s 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 . A s the s u s p e n s i o n c o n c e n t r a t i o n i n c r e a s e s t o w a r d s the s e m i - c o n c e n t r a t e d r e g i m e , t h e b e h a v i o r o f f i b e r s c h a n g e s b e c a u s e o f i n t e r a c t i o n s b e t w e e n f i b e r s . T h e i n t e r a c t i o n s c a u s e c h a n g e s i n the a n g l e s o f b o t h i n t e r a c t i n g f i b e r s . I n a c o n c e n t r a t e d s u s p e n s i o n e a c h f i b e r i n t e r a c t s w i t h m a n y o t h e r f i b e r s s i m u l t a n e o u s l y , s o a m e c h a n i s t i c m o d e l w o u l d b e v e r y d i f f i c u l t to c rea te . T h e r e are s o m e s t u d i e s d i r e c t l y r e l e v a n t to the d e v e l o p m e n t o f a m a t h e m a t i c a l m o d e l to p r e d i c t 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 r i g i d f i b e r s i n s e m i - c o n c e n t r a t e d a n d c o n c e n t r a t e d s u s p e n s i o n s . R a o [40] p r o v i d e s a n a p p r o a c h f o r t h e 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 . I n h i s r e s e a r c h , the o r i e n t a t i o n o f t h e f i b e r s i s f i r s t 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 t h e p r e s e n c e o f f i b e r s are z e r o . T h e n , t h e o r i e n t a t i o n f i e l d c o m p u t e d f r o m the N e w t o n i a n s o l u t i o n i s c o u p l e d b a c k to the g o v e r n i n g e q u a t i o n s 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 s u s p e n s i o n s . F o l g a r a n d T u c k e r [38] d e v e l o p e d a m o d e l f o r c o n c e n t r a t e d f i b e r s u s p e n s i o n s , w h e r e f i b e r - f i b e r i n t e r a c t i o n s are t a k e n i n t o 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 J e f f e r y ' s e q u a t i o n s . T h e y u s e d a s ta t i s t i ca l a p p r o a c h a n d i n t r o d u c e d a n o r i e n t a t i o n f u n c t i o n to d e s c r i b e the f i b e r s ' o r i e n t a t i o n a l state. A d v a n i et a l . [42] p r o p o s e d a m o r e e f f i c i e n t a p p r o a c h 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 o f o r i e n t a t i o n t e n s o r s . A l t a n et a l . [41 , 4 3 , 44 ] i n v e s t i g a t e d the t w o - a n d t h r e e - d i m e n s i o n a l d e s c r i p t i o n o f f i b e r o r i e n t a t i o n i n s e m i - c o n c e n t r a t e d h o m o g e n e o u s f l o w f i e l d s b y u s i n g D i n h - A r m s t r o n g r h e o l o g i c a l m o d e l [45] to d e s c r i b e f i b e r m o t i o n i n n o n - d i l u t e s o l u t i o n s . F o l g a r a n d T u c k e r [38] p o i n t e d o u t that a l l o f these r e s e a r c h e r s w h o s t u d i e d s e m i - c o n c e n t r a t e d o r c o n c e n t r a t e d s u s p e n s i o n s h a v e o b s e r v e d fiber o r i e n t a t i o n b e h a v i o r w h i c h 14 i s 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 t h e s t u d i e s h a v e f o c u s e d o n s u s p e n s i o n s o f r i g i d f i b e r s , f l e x i b l e f i b e r s w e r e a l s o i n v e s t i g a t e d b y R o s s a n d K l i n g e n b e r g [46] , W h e r r e t t et a l . [47] , 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] a n d the n u m e r i c a l s c h e m e o f D o n g [48] , 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] . 2.6 The Scope of This Thesis Work T h e c u r r e n t r e s e a r c h is par t o f a n e f f o r t at the 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 to d e v e l o p c o m p u t a t i o n a l m e t h o d s to s i m u l a t e the m o t i o n o f f i b e r s i n a w a y that c a n b e d i r e c t l y 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 t h e s i s i s l i m i t e d to the s t u d y o f d i l u t e f i b e r s u s p e n s i o n s i n a h e a d b o x c o n v e r g i n g s e c t i o n . T w o m o d e l s h a v e b e e n d e v e l o p e d i n t h e U B C r e s e a r c h g r o u p : a t u r b u l e n t f l o w m o d e l i s u s e d to c a l c u l a t e t h e h e a d b o x f l o w 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 m o t i o n . T h e s e t w o m o d e l s 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 to the 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 t h i s t h e s i s r e s e a r c h . 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 , r e a s o n s f o r the fiber o r i e n t a t i o n 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 i n c r e a s e p r o d u c t i o n e f f i c i e n c y c a n b e r e c o m m e n d e d . A s c a l e d p l e x i g l a s s h e a d b o x , b u i l t at the 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 p r e s e n t e x p e r i m e n t s [49] . P h o t o s o f d y e d n y l o n f i b e r s i n the f l o w are t a k e n at s e v e r a l l o c a t i o n s a l o n g the c e n t r a l s t r e a m l i n e o f the c h a n n e l f r o m s i d e a n d b o t t o m d i r e c t i o n . A n i m a g e a n a l y s i s m e t h o d is a p p l i e d to m e a s u r e the fiber o r i e n t a t i o n a n g l e s . D i r e c t c o m p a r i s o n b e t w e e n the s ta t i s t i ca l r e s u l t s o f e x p e r i m e n t s a n d p r e d i c t i o n s i s p e r f o r m e d . T h e n u m e r i c a l s i m u l a t i o n m e t h o d is f u r t h e r u s e d to p r e d i c t the f i b e r o r i e n t a t i o n f o r 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 a n d fiber a s p e c t r a t i o s . 15 F i g u r e 2 . 1 . 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 p a t t e r n i n a p i e c e o f p a p e r . 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 t h e e x p e r i m e n t a l w o r k is to o b t a i n d a t a to v a l i d a t e t h e 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 r e s u l t s a n d the s i m u l a t i o n r e s u l t s . 3.2 Fiber Suspensions T h e f i b e r s are 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 g i v e s the w i d t h o f t h e f i b e r 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 i s 1 ,140 k g / m 3 ) . A s a resul t , the f i b e r a s p e c t r a t i o , w h i c h i s the r a t i o b e t w e e n f i b e r l e n g t h a n d f i b e r w i d t h , i s 68 . N y l o n f i b e r s w e r e c h o s e n f o r the e x p e r i m e n t b e c a u s e t h e y c a n b e c o l o r e d , h a v e a d e n s i t y c l o s e to w a t e r , 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 ® b l u e m a r i n e 3 0 b y s o a k i n g t h e m o v e r n i g h t . I n the e x p e r i m e n t , w a t e r 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 f i b e r s w e r e p l a c e d i n e a c h l i te r o f w a t e r f o r t r a c k i n g s i n g l e f i b e r s b y m e a n s o f p h o t o s . A s the c o n s i s t e n c y 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 w a s w e l l w i t h i n the d i l u t e r e g i m e , w h i c h m e a n s t h e r e w a s l i t t le i n t e r a c t i o n b e t w e e n f i b e r s . T h e l e n g t h s o f d r y n y l o n f i b e r s w e r e t e s t e d w i t h the i m a g e a n a l y s i s s y s t e m . I n a s a m p l e 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 f i b e r 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 t h e f i b e r s w e r e s t r a i g h t o r n e a r l y s t ra ight w h e n the f i b e r s 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 3.3 Flow Loop 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 t r a n s p a r e n t p l e x i g l a s s 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 is a s c a l e d m o d e l o f a t y p i c a l h e a d b o x w i t h the s i z e r e d u c e d b y a f a c t o r o f 5. I n the f l o w l o o p , the d i l u t e f i b e r s u s p e n s i o n i s p u m p e d f r o m the r e s e r v o i r t a n k , w h i c h c a n c o n t a i n a t o t a l v o l u m e o f 3 m 3 o f f l u i d , to t h e h e a d b o x t h r o u g h the p i p e s a n d r e c t i f i e r t u b e s . T h e r e c t i f i e r t u b e s 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 the 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 reas 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 to 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 the c o n v e r g i n g s e c t i o n i n l e t . A t the o u t l e t o f t h e s e t u b e s , it i s a s s u m e d ( a n d v e r i f i e d b y o b s e r v a t i o n ) that the f i b e r s 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 t u r b u l e n c e e f f e c t s o n the f i b e r s . A l t o g e t h e r t h e r e are 4 0 r e c t i f i e r t u b e s , t w o r o w s i n the h e a d b o x h e i g h t d i r e c t i o n a n d 2 0 i n the s p a n d i r e c t i o n . 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 the f l o w rate at e a c h t u b e e x i t i s 0 . 34 l i t e r p e r s e c o n d . A s a resul t , the a v e r a g e v e l o c i t y at the i n l e t o f the c o n v e r g i n g s e c t i o n i s 0 . 24 m / s . A f t e r t r a v e l l i n g t h r o u g h t h e 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 , the f l o w is f i n a l l y 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 starts w i t h a r e c t a n g u l a r c h a n n e l w h i c h r e m a i n s c o n s t a n t i n c r o s s s e c t i o n a l a r e a u n t i l the c h a n n e l l e n g t h r e a c h e s 0 . 0 9 2 2 m ( the o r i g i n i s at the 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 , the c h a n n e l c o n v e r g e s to 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 the h e a d b o x g e o m e t r y are 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 the 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 t h e H e a d b o x C o n v e r g i n g S e c t i o n . parameters values w i d t h i n l e t h e i g h t s l i c e h e i g h t c o n t r a c t i o n r a t i o l e n g t h 0 .76 m ( c o n s t a n t ) 0 .075 m 0 . 0 0 7 5 m 10 0 . 3 1 7 2 m 18 It i s i m p o r t a n t to a v o i d e n t r a p m e n t o f a i r b u b b l e s i n the s u s p e n s i o n w h e n t a k i n g p h o t o g r a p h s , b e c a u s e the b u b b l e s m a y d e t e r i o r a t e the q u a l i t y o f p i c t u r e s . A i r b u b b l e s are a v o i d e d b y c i r c u l a t i n g 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 least 2 h o u r s b e f o r e t a k i n g p i c t u r e s . 3.4 Image Analysis System A n i m a g e a n a l y s i s m e t h o d w a s u s e d i n t h i s s t u d y o f f i b e r o r i e n t a t i o n . T o d e t e c t t h e f i b e r s i n t h e 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 ( c h a r g e - c o u p l e d d e v i c e ) v i d e o s y s t e m w i t h C o s m i c a r / P e n t a x T V l e n s ( 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 c o n n e c t e d t o g e t h e r . T h e O p t i k o n s y s t e m c a p t u r e s u p to 5 0 0 f r a m e s p e r s e c o n d w i t h a r e s o l u t i o n o f 3 3 6 x 2 4 3 p i x e l s f o r e a c h p i c t u r e . T h e S o n y c a m e r a h a s a s h u t t e r s p e e d o f 1/4 to 1 /4000 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 c a p t u r e p i c t u r e s o f f i b e r s 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 c l o s e to the e x i t , the S o n y c a m e r a w a s u s e d w h e r e the f l o w s p e e d w a s n o t t o o h i g h ( c h a n n e l l e n g t h < 2 6 c m ) . I n the e x p e r i m e n t , 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 l o c a t i o n s e i t h e r b e l o w o r b e s i d e t h e 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 s i d e o f t h e h e a d b o x . F i g . 3 .6 a l s o d e f i n e s the three d i m e n s i o n a l c o o r d i n a t e s : t h e x - a x i s i s i n t h e m a c h i n e d i r e c t i o n , the y - a x i s r e p r e s e n t s the p a p e r t h i c k n e s s d i r e c t i o n , a n d t h e z - a x i s r e p r e s e n t s t h e h e a d b o x s p a n d i r e c t i o n ( c r o s s - m a c h i n e d i r e c t i o n ) . L i g h t i n g i s e x t r e m e l y i m p o r t a n t f o r o b t a i n i n g g o o d p i c t u r e s . B a c k l i g h t i n g w a s p r o v i d e d w i t h a 1 5 0 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 f i n e g r o u n d g l a s s , 4 m m i n t h i c k n e s s , w a s u s e d to scat ter the l i g h t f o r better p h o t o g r a p h q u a l i t i e s . I n t h e e x p e r i m e n t , v i d e o p i c t u r e s w e r e t a k e n o f f i b e r s i n m o t i o n i n the p l e x i g l a s s c h a n n e l . T o e s t a b l i s h t h e f i e l d o f f o c u s , a t u b e w a s t e m p o r a r i l y i n s e r t e d i n t o the c h a n n e l . C a m e r a s w e r e f o c u s e d at the c e n t e r o f the 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 the b o t t o m a n d i n a p l a n e 6 c m n e x t to the s i d e c h a n n e l w a l l w h e n t a k i n g p i c t u r e s f r o m s i d e v i e w s . It w a s d i f f i c u l t to c o n t r o l the c a m e r a to e n s u r e a v e r y s h a l l o w d e p t h o f f i e l d , s o i n e a c h p i c t u r e 19 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 in 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 f low 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 on x-y plane 1 - IV + bottom view or on x-z plane 1 + 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 in 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 in the wake of a tube wal l . 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 f low [50]. While taking pictures from the bottom view, the measurements are made along both the centerline of one rectifier tube and the line extended downstream from a tube wal l . 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 in 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 in the converging section where x = 15.7 cm, 19.2 cm, 22.7 cm, 26.2 cm and 31cm. The point x = 31 cm is very close to the exit of the channel 21 a n d t h e r e f o r e o n l y b o t t o m v i e w m e a s u r e m e n t s w e r e c o n d u c t e d . C l e a r p i c t u r e s f r o m the s i d e c o u l d n o t b e o b t a i n e d at t h i s p o i n t . D u r i n g m e a s u r e m e n t s , o n l y the f i b e r s i n the s p e c i f i c a r e a are c o u n t e d , i . e . the f i b e r s i n a s q u a r e a r e a o f 2 x 2 c m 2 w h e n the m e a s u r e m e n t i s c l o s e to the i n l e t (x < 2 0 c m ) a n d l x l c m 2 w h e n the m e a s u r e m e n t i s c l o s e to the 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 P o i n t x-positions number of fibers in x-y plane in x-z plane 1 1325 1638 2 2 0 3 1 1793 3 2 0 6 6 2 8 8 5 4 2 0 5 6 2 3 2 5 5 1554 2 3 3 3 6 1823 1 3 5 6 7 N / A 1 5 8 6 2 2 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) Figure 3.1. T h e length distribution of nylon fibers. Figure 3.2. Images of fibers: (a) dry dyed nylon fibers, (b) fiber suspension. 23 Y A F l u i d Tank Figure 3.3. The flow loop in the experiment. 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 ) . z 4 Headbox "I *x Headbox [k Lighting Lighting (a) Camera Camera (b) 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 ( 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 the f l o w : (a) b e f o r e a n a l y s i s ; (b) af ter 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 - — central streamline 0.08 0.06 4> focus point 0.04 0.02 - - - 0 -0.02 , , i , , , , i , 1 1 , , , , 1 , 0 0.05 0.1 0.15 0.2 0.25 0.3 X Figure 3.8. The measurement points along the headbox channel. 27 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 T h e f i b e r o r i e n t a t i o n i n a p i e c e o f p a p e r i s d e t e r m i n e d b y the p a p e r m a k i n g p r o c e s s , e s p e c i a l l y i n the h e a d b o x a n d o n the f o r m i n g w i r e . E f f o r t s m u s t b e m a d e to d e r i v e q u a n t i t a t i v e r e l a t i o n s h i p s b e t w e e n p r o c e s s i n g c o n d i t i o n s a n d f i b e r o r i e n t a t i o n s . T h e i n t e n t i s to l e a r n h o w to d e s i g n a n d c o n t r o l m a n u f a c t u r i n g p r o c e s s e s to g e n e r a t e f a v o r a b l e o r i e n t a t i o n states, s o as to o b t a i n the b e s t p o s s i b l e p a p e r p r o d u c t s . T o p e r f o r m 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 that i s r e q u i r e d f o r the d e s i g n a n d p r o c e s s c o n t r o l , o n e 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 t h e y m o v e i n the f l o w . T o 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 b e d e v e l o p e d a n d e f f e c t i v e l y c o m b i n e d t o g e t h e r . T h e f i r s t 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 s p e c i f i c b o u n d a r y c o n d i t i o n s . T h e s e c o n d m o d e l d e s c r i b e s f i b e r 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 c o m b i n e t h e s e 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 h e r e 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 i n a f l o w f i e l d o f a N e w t o n i a n f l u i d w h e r e the f i b e r s d o n o t a l ter the f l o w . F o r s u s p e n s i o n s w i t h h i g h e r v o l u m e f r a c t i o n s , these s o l u t i o n t e c h n i q u e s c a n b e u t i l i z e d w i t h s o m e m o d i f i c a t i o n s . 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 n o t a l ter 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 to t h e 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 . T h e 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 i s p r e d i c t e d b y t h e s o l u t i o n o f the R e y n o l d s a v e r a g e d N a v i e r - S t o k e s e q u a t i o n s . T h e n t h e t r a n s l a t i o n a n d r o t a t i o n o f a r i g i d f i b e r i s d e s c r i b e d b a s e d o n N e w t o n ' s S e c o n d L a w a n d t h e l a w o f a n g u l a r m o m e n t u m . T h e a n g u l a r v e l o c i t y o f a f i b e r d e p e n d s u p o n the l o c a l f l o w 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 4.1 T h e H e a d b o x F l o w M o d e l I n the s i m u l a t i o n s t u d y , the l i q u i d is a s s u m e d to b e p u r e w a t e r o r a d i l u t e f i b e r s u s p e n s i o n . T h e c o n s i s t e n c y o f the s u s p e n s i o n is v e r y l o w , s o t h e r e i s n o i n t e r a c t i o n 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 n o t a f f e c t 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 s u s p e n s i o n c a n t h e r e f o r e b e v i e w e d as a u n i f o r m i n c o m p r e s s i b l e N e w t o n i a n f l u i d . B a s e d o n t h e a b o v e a s s u m p t i o n s , t h e a v a i l a b l e c o n t i n u u m t h e o r i e s c a n b e u s e d f o r t h e d i l u t e s u s p e n s i o n [32 ] . T h e n u m e r i c a l s i m u l a t i o n o f the f l o w h a s b e e n c a r r i e d o u t b y s o l v i n g t h e t h r e e - d i m e n s i o n a l i n c o m p r e s s i b l e R e y n o l d s a v e r a g e d N a v i e r - S t o k e s e q u a t i o n s . T u r b u l e n c e c l o s u r e i s o b t a i n e d b y the u s e 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 t r e a t m e n t . W e c a n w r i t e t h e 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 o f : w h e r e : u = i n s t a n t a n e o u s f l u i d v e l o c i t y v e c t o r p = m o d i f i e d p r e s s u r e i n c l u d i n g the g r a v i t a t i o n a l f o r c e s p = d e n s i t y x = f l u i d stress t e n s o r . F o r a N e w t o n i a n f l u i d w i t h c o n s t a n t v i s c o s i t y , V u = 0 (4 .1) T h e e q u a t i o n o f c o n s e r v a t i o n o f m o m e n t u m i s : u-Vu = -Vpl p + V -r I p (4 .2 ) V - T = / / V 2 u (4 .3) w h e r e : 2 9 ^ = d y n a m i c v i s c o s i t y o f the f l u i d . 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 p r e s s u r e 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 p a r t : u = u + u' (4-4) p = p + p' (4-5) I n t h e e q u a t i o n s , the o v e r - b a r d e n o t e s the m e a n , a n d the p r i m e i n d i c a t e s the f l u c t u a t i n g c o m p o n e n t . A f t e r s u b s t i t u t i n g E q u a t i o n s (4 .3) , (4 .4) a n d (4 .5) i n t o 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 a v e r a g e , the g o v e r n i n g e q u a t i o n s o f the t u r b u l e n t f l o w b e c o m e : V - u = 0 (4-6) u V u = -Vp/p + V-r/p (4-7) N o w the stress t e n s o r x i n c l u d e s b o t h the v i s c o u s a n d t u r b u l e n t R e y n o l d s stress t e n s o r s : V T = /J V 2 u + V - ( - p u\u'j ) (4-8) w h e r e : r , = uTV; ( i , j = l , 2 , 3 ) (4 .9 ) c a n b e u s e d to s i m p l i f y E q u a t i o n (4 .8) . T h e R e y n o l d s stress t e n s o r T . . i n t r o d u c e s a d d i t i o n a l u n k n o w n s f o r the t u r b u l e n t f l o w . I n o r d e r to d e s c r i b e the m e a n v e l o c i t y a n d p r e s s u r e f i e l d s , a c l o s u r e f o r m u l a t i o n is n e c e s s a r y to re la te the c o m p o n e n t s o f the R e y n o l d s stress t e n s o r to the m e a n f l o w v e l o c i t y o r v e l o c i t y g r a d i e n t s . T h e s t a n d a r d l i n e a r k - s m o d e l [52] i s 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 . 3 0 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 t S u - - k S v w h e r e : du, dui + dx, dx, ( 4 . 1 0 ) ( 4 . 1 1 ) k = — u'u' 2 ' ' ( 4 . 1 2 ) k i s t h e t u r b u l e n t k i n e t i c e n e r g y , 5.. i s the K r o n e c k e r D e l t a , a n d V t i s t h e t u r b u l e n t 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 , v a r i e s s p a t i a l l y a n d is n o t a p r o p e r t y o f t h e f l u i d . T h e t r a n s p o r t e q u a t i o n f o r k i s : w h e r e : a n d 5 lr- 9 ku, = dx dk + G-E G = -u] Uj Sa (4 .13 ) ( 4 . 1 4 ) E = VS'iS'g ( 4 . 1 5 ) G a n d E a r e the rates 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 a n d d i s s i p a t i o n p e r u n i t m a s s , r e s p e c t i v e l y . T h e t r a n s p o r t e q u a t i o n f o r the k i n e t i c e n e r g y d i s s i p a t i o n E i s g i v e n i n t h e f o r m o f : 5 z r - 8 EU: = dx, 1 dx v + - ^ + ( C l G - c 2 E ) ^ (4 -16) dx, K 31 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 i s o t r o p i c v i s c o s i t y a s s u m p t i o n f o r the t u r b u l e n t v i s c o s i t y i n t e r m s o f k a n d E , as g i v e n b y V { = k 2 / 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 . T h e u s u a l v a l u e s o f the c o n s t a n t s are : = 0 .09 , C j = 1.44, C 2 = 1 .92, a k = 1.0, a E = K 2 / [ ( C 2 - C , ) Cm], w h e r e K = 0.41 i s the V o n K a r m a n c o n s t a n t . A f i n i t e v o l u m e m e t h o d i n c o n j u n c t i o n w i t h g e n e r a l c u r v i l i n e a r g r i d s i s u s e d i n t h e 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] . U s e a n d 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 the w o r k o f S h a r i a t i et a l . [49] a n d H u a e t a l . [50] . 4 .2 F i b e r M o d e l T h e m a i n c o n c e r n o f t h i s w o r k i s the 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 i n the h e a d b o x f l o w f i e l d . I n t h i s w o r k , the f i b e r o r i e n t a t i o n i s 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 t h e 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 z e r o fiber c o n s i s t e n c y , b e c a u s e t h e fibers i n f l u e n c e t h e f l o w . H o w e v e r , f o r d i l u t e s o l u t i o n s , t h i s m o d e l s h o u l d n o t g i v e s i g n i f i c a n t e r r o r s . A t fiber l o c a t i o n s , the v e l o c i t y a n d v e l o c i t y g r a d i e n t s are 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 the v a l u e s at n e i g h b o r i n g n o d e s a n d are u s e d to d e t e r m i n e the t r a n s l a t i o n a n d r o t a t i o n o f the fiber. T h e c a l c u l a t e d fiber p o s i t i o n a n d o r i e n t a t i o n a r e u s e d as i n i t i a l c o n d i t i o n s f o r t h e c a l c u l a t i o n o f t h e fiber's n e w l o c a t i o n a n d o r i e n t a t i o n at the n e x t t i m e step (t + A t ) . T h e fiber m o t i o n m o d e l u s e d i n t h i s w o r k w a s first d e v e l o p e d b y R o s s a n d K l i n g e n b e r g [46 , 5 4 ] a n d a d a p t e d b y D o n g [48] . J e f f r e y ' s e q u a t i o n [37] i s u s e d f o r t h e v e r i f i c a t i o n o f t h i s m o d e l . D o n g s h o w e d the i d e n t i t y o f the r e s u l t s f r o m b o t h a p p r o a c h e s . A l t h o u g h 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 t h i s m o d e l , o n l y r i g i d f i b e r s are s i m u l a t e d i n t h i s t h e s i s . A s d e s c r i b e d b y D o n g , a fiber i s r e p r e s e n t e d b y o n e o r m o r e p r o l a t e s p h e r o i d s . I n t h i s t h e s i s r e s e a r c h , a r i g i d fiber i s r e p r e s e n t e d b y o n e p r o l a t e s p h e r o i 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 . T h e d e n s i t y o f the fiber 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 3 2 fiber i s 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 o n J e f f e r y ' s o r i g i n a l w o r k [37]. T h e o r i e n t a t i o n o f a fiber 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 a n g l e s as e x p r e s s e d i n F i g u r e 4.1. T h e a z i m u t h a l a n g l e <j, i s 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 t h e fiber a x i s o n the x - y p l a n e a n d the y - a x i s w i t h 0 < <j, < n ( o n e e n d o f the fiber is n o t d i s t i n g u i s h a b l e f r o m the o t h e r ) . T h e p o l a r a n g l e 0 i s the a n g l e b e t w e e n the fiber a x i s a n d the z - a x i s w i t h 0 < Q < N . T h e m o t i o n o f the f i b e r r e l a t i v e to the s u r r o u n d i n g f l u i d i s so s m a l l that t h e i n e r t i a l f o r c e i s n e g l i g i b l e . T h e c e n t e r o f the fiber t ransla tes 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 t h e f i b e r o r i e n t a t i o n i s 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 t e n s o r , t h e v o r t i c i t y o f t h e f l o w a n d the p r e v i o u s fiber o r i e n t a t i o n . I n o r d e r to p r e d i c t a fiber m o t i o n , t h e 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 the f l o w at the c e n t e r o f a fiber m u s t b e 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 fiber l i e s . T h e n a c c u r a t e v a l u e s o f the 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 the fiber c e n t e r a r e i n t e r p o l a t e d f r o m the s u r r o u n d i n g n o d a l p o i n t s . I n e a c h t i m e step, the c a l c u l a t i o n u s e s the fiber p o s i t i o n a n d o r i e n t a t i o n o b t a i n e d f r o m the p r e c e d i n g t i m e s tep a n d the l o c a l f l o w k i n e m a t i c s i n f o r m a t i o n . F i b e r p a r t i c l e s s u s p e n d e d i n a t u r b u l e n t f l o w e x p e r i e n c e a m e a n v e l o c i t y a s s o c i a t e d w i t h t h e m e a n f l o w f i e l d a n d a r a n d o m v e l o c i t y d u e to the f l u c t u a t i n g c o m p o n e n t o f t h e t u r b u l e n t f l o w . T h e o r i e n t a t i o n o f f i b e r s d e p e n d s o n a c o m b i n a t i o n o f o r i e n t i n g e f f e c t s o f t h e m e a n v e l o c i t y g r a d i e n t s a n d the r a n d o m i z i n g e f f e c t s o f t u r b u l e n c e . O n l y m e a n 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 t h i s 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 r a t i o L / d . T h e p o s i t i o n a n d o r i e n t a t i o n o f a s i n g l e fiber c a n b e d e s c r i b e d b y the c o o r d i n a t e s (x , y , z ) a n d a n g l e s (Q, §). T h e o r i e n t a t i o n f i e l d that i s e x p r e s s e d i n t h e f o r m o f fiber o r i e n t a t i o n a n g l e s , is s p e c i f i e d f r o m the s o l u t i o n o f the o r i e n t a t i o n e q u a t i o n s a l o n g the p a r t i c l e t r a j e c t o r y . A l a r g e n u m b e r o f f i b e r s i s c o n s i d e r e d b y u s i n g s t a t i s t i c a l 33 e x p r e s s i o n s d e v e l o p e d to d e s c r i b e the o r i e n t a t i o n state. I n b r i e f , t h e n u m e r i c a l s c h e m e d e a l s w i t h e a c h i n d i v i d u a l fiber s e p a r a t e l y . T h e n the b e h a v i o r o f l a r g e n u m b e r s o f f i b e r s i s c h a r a c t e r i z e d w i t h the s ta t i s t i ca l m e t h o d s . 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 the fiber o r i e n t a t i o n p a r a m e t e r f are b a s e d o n the s ta t i s t i ca l d a t a . T h e f i b e r s are i n i t i a l l y i n j e c t e d at the c e n t e r o f the i n l e t o f the c o n v e r g i n g s e c t i o n w i t h r a n d o m o r i e n t a t i o n s , 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 i n i t i a l fiber state. A t the in le t , r a n d o m fiber o r i e n t a t i o n i s i m p l e m e n t e d b y c h o o s i n g the fiber a n g l e s ^ a n d 9 f o r a l a r g e n u m b e r o f f i b e r s . T h e a n g l e s Q a n d ^ a r e n o t s e l e c t e d r a n d o m l y f r o m u n i f o r m d i s t r i b u t i o n s Q E [0, n] a n d <j) G [0 , n], s i n c e o n the s u r f a c e o f a u n i t s p h e r e , t h e a r e a e l e m e n t d Q = s i n g d ^ d Q is a f u n c t i o n o f Q. TO d i s t r i b u t e f i b e r s e q u a l l y i n a l l p o s s i b l e d i r e c t i o n s , the c o r r e c t s e l e c t i o n 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 to 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 = nax (4 -17) 9 = c o s - 1 ( 2 a 2 - l ) (4-18) T h e i n i t i a l o r i e n t a t i o n o f 1 0 0 0 f i b e r s 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 f i b e r s is i n c r e a s e d , a m o r e a c c u r a t e r e p r e s e n t a t i o n o f the o r i e n t a t i o n d i s t r i b u t i o n w i l l b e o b t a i n e d . F r o m s u c h 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 , the p r e f e r r e d o r i e n t a t i o n a n g l e 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 a n g l e 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 s ta t i s t i ca l s o l u t i o n i s d e p e n d e n t o n the n u m b e r o f f i b e r s c o n s i d e r e d , o n e s h o u l d c o n s i d e r the a v a i l a b l e c o m p u t a t i o n a l r e s o u r c e s a n d t h e d e s i r e d a c c u r a c y l e v e l . H e n c e , the n u m b e r o f f i b e r s u t i l i z e d i n the s ta t i s t i ca l s o l u t i o n c a n b e b a s e d o n t h e d e s i r e d a c c u r a c y a n d the a v a i l a b l e c o m p u t a t i o n a l r e s o u r c e s . A f t e r t r y i n g d i f f e r e n t n u m b e r s o f f i b e r s i n the s i m u l a t i o n , f r o m 5 0 0 to 1 0 0 , 0 0 0 , a s a m p l e s i z e o f 3 0 0 0 f i b e r s w a s c h o s e n to r e p r e s e n t the b u l k fiber b e h a v i o r as t h i s g i v e s a r e a s o n a b l e s t a t i s t i c a l e x p r e s s i o n . 3 4 F i g u r e 4 . 1 . A f i b e r i n t h r e e - d i m e n s i o n a l c o o r d i n a t e s . F i g u r e 4 .2 . T h e i n i t i a l r a n d o m d i s t r i b u t i o n o f 1 0 0 0 f i b e r s . 35 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 o f t h e H e a d b o x F l o w F i e l d T h e q u a l i t y o f a n y C F D a n a l y s i s d e p e n d s , a m o n g o t h e r f a c t o r s , o n the q u a l i t y o f t h e c h o s e n g r i d . I n the f l o w s i m u l a t i o n , the g e o m e t r i c d i m e n s i o n s are t h e s a m e as 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 o f the h e a d b o x i n the e x p e r i m e n t , o n l y the s i z e i n t h e h e a d b o x s p a n d i r e c t i o n i s r e d u c e d f r o m 0 .76 m to 0.8 c m . T h e 8 0 x 3 2 x 8 m e s h as s h o w n i n F i g . 5.1 i s g e n e r a t e d o v e r the c o n v e r g i n g s e c t i o n . B y m a p p i n g the s h a p e f r o m t h e 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 t r a n s f o r m e d 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 t h e s o l u t i o n s at e v e r y n o d e are m a p p e d b a c k o n t o the p h y s i c a l d o m a i n . T h e C a r t e s i a n c o o r d i n a t e s x , y a n d z r e p r e s e n t t h e m a c h i n e d i r e c t i o n , p a p e r t h i c k n e s s d i r e c t i o n a n d c r o s s m a c h i n e d i r e c t i o n r e s p e c t i v e l y . A s w e h a v e n o t e d a l r e a d y , t h e h e a d b o x s p a n is r e d u c e d i n the z d i r e c t i o n i n the c o m p u t a t i o n . T h e s i m u l a t i o n o n l y m o d e l s a s p a n w i s e s e c t i o n o f the 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 that 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 g r a d i e n t s i n the z d i r e c t i o n . T h e t u b e s a r e 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 t h e v e l o c i t y g r a d i e n t s are l a r g e r t h e r e t h a n i n o t h e r p l a c e . T h e m e s h i n the m a c h i n e 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 c h a n g e r a p i d l y as t h e f l o w a p p r o a c h e s the ex i t . 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 a c c u r a t e 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 n e a r 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 m e s h s i z e s at the i n l e t i n x a n d y d i r e c t i o n s are a r o u n d 8 m m a n d 2.8 m m r e s p e c t i v e l y ; t h e m i n i m u m m e s h s i z e s at the n o z z l e i n x a n d y d i r e c t i o n s are a r o u n d 2 .4 m m a n d 0 .2 m m r e s p e c t i v e l y . T h e m e s h s i z e i n z d i r e c t i o n i s 1 m m . T h e b o u n d a r y c o n d i t i o n s are u n i f o r m v e l o c i t y p r o f i l e at the i n l e t ( w i t h i n i t i a l k = 4 x 10" 3 m 2 / s 2 , E = 3 x 10" 4 m 2 / s 3 f o r a l l cases ) , z e r o v e l o c i t y g r a d i e n t at the s l i c e e x i t f o r a l l 3 6 v e l o c i t i e s i n the x d i r e c t i o n , a n d the w a l l c o n d i t i o n s at 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 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 , w e 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 , z e r o v e l o c i t y g r a d i e n t c a n b e a p p r o x i m a t e l y a c c e p t e d as the b o u n d a r y c o n d i t i o n f o r t h e o u t f l o w o f the c o n v e r g i n g c h a n n e l c a s e . T h e m o r e p h y s i c a l l y c o r r e c t e x i t b o u n d a r y 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 h a s 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 l e a d s to n u m e r i c a l s i m u l a t i o n s w h i c h c o n v e r g e w i t h great d i f f i c u l t y , a n d h a s n o t b e e n u s e d h e r e f o r that r e a s o n . T h e s y m m e t r y c o n d i t i o n ( z e r o g r a d i e n t s 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 c o m p o n e n t s , a n d z e r o v a l u e o f the v e l o c i t y c o m p o n e n t n o r m a l to the b o u n d a r y ) i s i m p o s e d o n b o t h s i d e s o f the c h a n n e l b e c a u s e o n l y a s e c t i o n o f the c h a n n e l i s m o d e l l e d a n d t h e r e i s n o w a l l e f f e c t . A p p r o p r i a t e v e l o c i t y g r a d i e n t s are set to z e r o a n d a n o 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 is 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 . T h e 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 i s 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 - d i r e c t i o n v e l o c i t y c o m p o n e n t s at the 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 s e c t i o n a r e s h o w n i n F i g u r e 5 .2 . T h e fiber o r i e n t a t i o n s t u d y is 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 t h e c e n t r a l s t r e a m l i n e o n l y . T h e v a r i a t i o n s o f the x - d i r e c t i o n v e l o c i t y a n d p r e s s u r e a l o n g t h e c e n t r a l s t r e a m l i n e are s h o w n i n F i g u r e 5 .3 . It c a n b e s e e n f r o m the p l o t s that a l a r g e p r e s s u r e d r o p o c c u r s n e a r t h e n o z z l e e x i t w h e r e a s the f l u i d v e l o c i t y i n c r e a s e s r a p i d l y n e a r the e x i t o f the c h a n n e l a n d r e a c h e s i ts m a x i m u m at the ex i t . F i g u r e 5.4 g i v e s u - v e l o c i t y ( x - d i r e c t i o n ) 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 p l a n e . F i g u r e 5.5 g i v e s the v - v e l o c i t y ( y - d i r e c t i o n ) c o n t o u r s o n the s a m e s y m m e t r y p l a n e . T h e f l o w , 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 the i n l e t , g r a d u a l l y i n c r e a s e s its s p e e d as t h e c h a n n e l b e c o m e s n a r r o w e r , a n d the v e l o c i t y g r a d i e n t s i n c r e a s e a n d r e a c h a m a x i m u m at t h e h e a d b o x n o z z l e . T h e fiber o r i e n t a t i o n w i l l b e s t r o n g l y a f f e c t e d b y the i n c r e a s e d e l o n g a t i o n a n d s h e a r s tresses . 3 7 A s s o m e r e s e a r c h e r s h a v e s ta ted [25] , f i b e r o r i e n t a t i o n i s g o v e r n e d b y t h e e l o n g a t i o n e f f e c t i n a c o n v e r g e n t c h a n n e l , a n d b o t h the e l o n g a t i o n o f the f l o w a n d t h e f i b e r 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 the 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 c e n t r a l s t r e a m l i n e c a n b e e x p r e s s e d as : s = \*±dt (5-D J ds • w h e r e u s i s the f l o w v e l o c i t y o n the c e n t r a l s t r e a m l i n e a n d s i s the d i s t a n c e a l o n g t h e c e n t r a l s t r e a m l i n e . B e c a u s e ds dt = — t h e 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 : f -1 ds (5-2) 1 dus us ds T h e v e l o c i t y u s a n d d i s t a n c e s o n the c e n t r a l s t r e a m l i n e are r e l a t e d to t h e x d i r e c t i o n v e l o c i t y u a n d x , i n the s t a r t i n g f la t s e c t i o n , b y : u = u„ x = s a n d i n t h e c o n s e q u e n t c o n t r a c t i o n s e c t i o n b y : u = us coscp x = x t + (s - xx) c o s q> w h e r e ^ i s t h e a n g l e b e t w e e n the c e n t r a l 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 s e c t i o n , X j i s the l e n g t h o f the f la t s e c t i o n . S u b s t i t u t i n g the a b o v e r e l a t i o n s i n t o (5 .2) , the f l o w e l o n g a t i o n b e c o m e s : s - r 1 du , J — — dx u dx 38 (5 .3 ) 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 f u r t h e r s i m p l i f i e d as : du (5 .4 ) £ U o r £ ( x ) = In u(x) ( 5 . 4 ' ) A t t h e c h a n n e l e x i t , the f l o w e l o n g a t i o n is t h e r e f o r e : (5 .5) w h e r e U 0 a n d U e a re the v e l o c i t y c o m p o n e n t i n the x d i r e c t i o n at the c h a n n e l i n l e t a n d o u t l e t . B e c a u s e the r a t i o o f U e / U 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 r a t i o ( R . ) o f t h e c h a n n e l , t h e n the f l o w e l o n g a t i o n at the e x i t i s o b t a i n e d as : 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 s at the c h a n n e l e x i t o n l y d e p e n d s o n the c o n t r a c t i o n r a t i o o f a c o n v e r g e n t c h a n n e l . F r o m (5 .4) the f l o w e l o n g a t i o n c a n b e 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 a l s o b e c a l c u l a t e d b y e q u a t i o n (5 .4 ' ) , w h i c h i s e s s e n t i a l l y i d e n t i c a l to the n u m e r i c a l p l o t o f F i g . 5'.6. 5.2 Comparison of Simulation and Experimental Results A c o n v e n i e n t g r a p h i c a l r e p r e s e n t a t i o n i s n e e d e d to p r e s e n t the t h r e e - d i m e n s i o n a l f i b e r o r i e n t a t i o n r e s u l t s . T h e p r o j e c t i o n s o f the o r i e n t a t i o n o f l a r g e n u m b e r s o f f i b e r s c a n b e (5 .6 ) 3 9 o b t a i n e d o n t h r e e d i f f e r e n t p l a n e s . T h e p r o j e c t i o n s o n t w o o f the p l a n e s , x - y a n d x - z p l a n e , are u s e d h e r e , b e c a u s e the r e s u l t s c a n b e c o m p a r e d d i r e c t l y w i t h the e x p e r i m e n t a l r e s u l t s f o r t h e c o r r e s p o n d i n g p l a n e s . T h e f i b e r o r i e n t a t i o n a n g l e a , e i t h e r o n t h e x - y o r o n 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 to the s i d e v i e w i n the m e a s u r e m e n t s a n d the x - z p l a n e c o r r e s p o n d s to the b o t t o m v i e w i n t h e m e a s u r e m e n t s . 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 i s 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 e x p e r i m e n t s . It w o u l d b e p o s s i b l e to o b t a i n the f i b e r o r i e n t a t i o n r e s u l t s o n t h e y - z p l a n e f r o m the s i m u l a t i o n , b u t t h i s a n a l y s i s h a s n o t b e e n d o n e . W h e n a l l the 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 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 a v a i l a b l e , the r e s u l t s o f the 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 f r o m e x p e r i m e n t s a n d f r o m 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 r e a s o n a b l e 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 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 d i a g r a m u s e d h e r e , the h o r i z o n t a l a x i s r e p r e s e n t s the o r i e n t a t i o n a n g l e a . It i s s e p a r a t e d i n t o 18 z o n e s o f 10 d e g r e e s e a c h , f r o m - 9 0 ° to + 9 0 ° , w i t h 0 ° i n d i c a t i n g 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 v e r t i c a l a x i s r e p r e s e n t s the s ta t i s t i ca l p r o b a b i l i t y d e n s i t y p(a)> such that : fj p(a)da = 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 i n l e t (x = 4 .5 c m ) as s e e n f r o m the s i d e v i e w a n d t h e 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 t h e f i b e r s are 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 t h e m e a s u r e m e n t s o r t h e n u m e r i c a l s i m u l a t i o n s , p ( a ) w o u l d h a v e b e c o m e m o r e c o n s t a n t , 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 (~ 0 . 3 1 8 ) f o r a v e r y l a r g e n u m b e r o f f i b e r s . A s f i b e r s enter t h e 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 the f l o w d i r e c t i o n . F i g u r e 5.8 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 = 12 .2 c m a n d 15 .7 c m . A n i n t e r e s t i n g p h e n o m e n o n w a s f o u n d from t h e s e t w o d i a g r a m s : 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 s 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 f a s t e r t h a n that f r o m 4 0 e x p e r i m e n t s . W h e n f i b e r s m o v e f u r t h e r t o w a r d s the n o z z l e , t h i s p h e n o m e n o n b e c o m e s 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 = 19 c m ) , F i g . 5 .11 (x = 2 2 c m ) a n d F i g . 5 .12 (x = 2 6 c m ) . A t the e x i t w h e r e x = 31 c m , the f i b e r s are h i g h l y a l i g n e d i n t h e f l o w d i r e c t i o n as s h o w n i n F i g . 5 .13 . W e d o n o t h a v e the e x p e r i m e n t a l d a t a o f f i b e r o r i e n t a t i o n i n the x - y p l a n e , b e c a u s e t h e c h a n n e l is t o o n a r r o w a n d the f l o w s p e e d is t o o h i g h at that 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 t h e t r e n d o f the f i b e r o r i e n t a t i o n i n a d i l u t e h e a d b o x f l o w , b u t t h e r e e x i s t o b v i o u s d i f f e r e n c e s b e t w e e n t h e n u m e r i c a l d a t a a n d the e x p e r i m e n t a l d a t a . A n o t h e r p h e n o m e n o n s h o w n f r o m these d i a g r a m s w a s that the 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 o r t h e 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 , the 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 the s h e a r a n d e x t e n s i o n a l p l a n e ( x - y p l a n e ) t h a n i n the n e u t r a l p l a n e ( x - z p l a n e ) . T h i s i s c a u s e d b y d i f f e r e n t v e l o c i t y g r a d i e n t e f f e c t . I n t h i s h e a d b o x s t ruc ture , the v e l o c i t y g r a d i e n t s 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 t h e 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 , b u t o n l y d u / d x a n d d u / d y a f f e c t t h e 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 i s w h y the f i b e r a l i g n m e n t i n x - y p l a n e i s 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 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 t h e 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 r e s u l t s . T h e m a j o r r e a s o n f o r t h i s 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 t h e t u r b u l e n c e e f f e c t i s n o t 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 . I n the e x p e r i m e n t , t h e 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 r e c t i f i e r t u b e s . A s w e h a v e m e n t i o n e d e a r l i e r , the r e c t i f i e r t u b e s 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 . F i r s t , 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 the 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 the r e c t i f i e r t u b e s w i l l a f f e c t the f i b e r b e h a v i o r d o w n s t r e a m . T u r b u l e n c e t e n d s to r a n d o m i z e t h e f i b e r o r i e n t a t i o n , w h i c h m a k e s the 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 h a v e a c o m p a r a t i v e l y f l a t a p p e a r a n c e . T h e c u r r e n t n u m e r i c a l f i b e r m o d e l s i m u l a t e s o n l y the f i b e r m o t i o n i n a m e a n f l o w f i e l d , s o the p r e d i c t e d 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 i s m o r e o r g a n i z e d t h a n i n the o b s e r v e d s i t u a t i o n , w h e r e t u r b u l e n c e p l a y s a n i m p o r t a n t r o l e . 41 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 i n the s i d e v i e w ( x - y p r o j e c t i o n p l a n e ) i s n o t s y m m e t r i c a l a b o u t z e r o b e c a u s e o f the a s y m m e t r i c s t r u c t u r e o f the h e a d b o x i n t h e 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 t h e 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 a n g l e b e t w e e n 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 i s 1 6 . 7 ° . A s a r e s u l t , the p r e f e r r e d o r i e n t a t i o n i n the s i d e v i e w s h o u l d b e b e t w e e n Oo a n d 1 6 . 7 ° . O n the o t h e r h a n d , the s i m u l a t i o n r e s u l t s g i v e a s y m m e t r i c a l 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 p l a n e . T h i s i s b e c a u s e there i s n o r e s t r i c t i o n i n t h i s p l a n e , a n d f i b e r s a r e a l l o w e d to f r e e l y m o v e i n a l l p o s s i b l e d i r e c t i o n s . A n o t h e r c o n c i s e w a y to d e s c r i b e f i b e r o r i e n t a t i o n i s to u s e a 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 " . T h e p l a n e o r i e n t a t i o n p a r a m e t e r f 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 fp = 2 p(a) cos2 (a-a0) da - 1 (5-8) 2 w h e r e a o i s the m e a n o f the d i s t r i b u t e d a n g l e s o r the p r e f e r r e d a l i g n m e n t a n g l e , a n d p ( a ) r e p r e s e n t s the 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 . F o r the c a s e o f a f i n i t e n u m b e r o f f i b e r o r i e n t a t i o n s , n , the 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] as : 2 n fP = - S c o s 2 ( a i ~ a o ) n , = i (5.9) T h e p a r a m e t e r 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 to 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 f i b e r o r i e n t a t i o n . F o r p e r f e c t a l i g n m e n t , f = 1, a n d f o r a c o m p l e t e l y r a n d o m o r i e n t a t i o n d i s t r i b u t i o n , f = 0. T h e m e a n d i r e c t i o n a o c a n b e c a l c u l a t e d w i t h the f o l l o w i n g e q u a t i o n s [58]: - i C an = c o s — 0 R (5.10) o r 42 a0 = s i n " 1 - ( 5 . H ) 0 R w h e r e : 1 " c = 1 S c o s a , 1 " S = — V s i n a, n M R = ( C 2 +S2 T a b l e 5.1 p r e s e n t s the o b s e r v e d a n d p r e d i c t e d o r i e n t a t i o n p a r a m e t e r s at s e v e r a l m e a s u r e m e n t p o i n t s a l o n g the c e n t r a l s t r e a m l i n e (see F i g . 3 .8) . T a b l e 5 .1 . O r i e n t a t i o n P a r a m e t e r s 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 Experiments Simulations x-y plane x-z plane x-y plane x-z plane 1 0 .04 -0 .01 0.01 - 0 . 0 0 2 0 . 1 0 0 .02 0 . 1 7 0 . 0 7 3 0 .14 0 .05 0 .33 0 .15 4 0 .18 0 .05 0 .50 0 . 2 6 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 . 7 6 F i g . 5 .14 (a ) a n d F i g . 5 . 1 4 ( b ) p r e s e n t the 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 a l o n g the c e n t r a l s t r e a m l i n e i n the x - y a n d x - z p r o j e c t i o n p l a n e s r e s p e c t i v e l y . T h e r e s u l t s s h o w that t h e r e 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 d a t a a n d the s i m u l a t i o n d a t a , a l t h o u g h t h e t r e n d is t h e s a m e . F o r b o t h m e a s u r e m e n t s a n d s i m u l a t i o n s , the 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 d e v e l o p s f r o m a r a n d o m i n i t i a l 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 status at the 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 the 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 s e c t i o n , 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 , o r m o r e s p e c i f i c a l l y , t h e c o n t r a c t i o n r a t i o 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 to the p l a n e rate o f s t r a i n , QU/QX, i n the c o n v e r g e n t c h a n n e l , the f i b e r s 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 d i f f e r e n t h e a d b o x g e o m e t r i e s 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 f i b e r s i n the f l o w d i r e c t i o n i s d i f f e r e n t a n d is d e t e r m i n e d b y the rates o f s t r a i n i n the f l o w , a n d the t i m e that the f i b e r s are e x p o s e d to the f o r c e s . H i g h rate o f s t r a i n a n d l o n g t i m e d u r a t i o n w o u l d b e l i k e l y to p r o d u c e h i g h l y c o n c e n t r a t e d f i b e r o r i e n t a t i o n s . I n o r d e r to d e t e c t w h a t f a c t o r s m a y i n f l u e n c e the f i b e r o r i e n t a t i o n c h a r a c t e r i s t i c s , the h e a d b o x g e o m e t r y , the f l o w v e l o c i t y a n d the f i b e r a s p e c t r a t i o h a v e b e e n c h a n g e d a n d t h e i r e f f e c t s o n f i b e r o r i e n t a t i o n h a v e b e e n s t u d i e d . W h e n o n e p a r a m e t e r i s c h a n g e d , t h e o t h e r s 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 e f f e c t s o f s e l e c t e d p a r a m e t e r s i s m u c h e a s i e r to d o i n the s i m u l a t i o n s t u d i e s 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 i n v e s t i g a t e the f i n a l 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 f o r d i f f e r e n t v a l u e s o f the c o n t r a c t i o n r a t i o ( R . ) , w h i c h i s the r a t i o b e t w e e n i n l e t a r e a a n d e x i t a r e a , the c h a n n e l l e n g t h ( L c ) , t h e i n f l o w rate U 0 a n d the fiber a s p e c t r a t i o ( A r ) . 5.3.1 The Effect of Contraction Ratio on Fiber Orientation I n t h i s p a r t o f the p r e s e n t s t u d y , o n l y the c h a n n e l ' s c o n t r a c t i o n r a t i o i s c h a n g e d b y a d j u s t i n g t h e e x i t a r e a , 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 s a m e . 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 at the c h a n n e l e x i t f o r d i f f e r e n t R c are s h o w n i n T a b l e 5.2. 44 T a b l e 5 .2 . 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 R ( U = 0 .24 m / s , L = 0 . 2 2 5 m ) Rc Orientation parameter x-y plane x-z plane 6.7 0 .96 0 .73 10 0 .98 0.81 15 0 .98 0 . 8 6 T h e fiber o r i e n t a t i o n p a r a m e t e r ( w h i c h c o r r e s p o n d s to 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 ) i n the x - y a n d x - z p r o j e c t i o n p l a n e s at the c h a n n e l e x i t are i n c r e a s e d w i t h i n c r e a s i n g c o n t r a c t i o n r a t i o , 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 its 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 that h i g h e r R c c o r r e s p o n d s to m o r e a l i g n e d fiber o r i e n t a t i o n i n the f l o w d i r e c t i o n . 5.3.2 The Effect of Flow Rate on Fiber Orientation K e e p i n g o t h e r p a r a m e t e r s the s a m e , o n l y the f l o w rate at the i n l e t o f the c h a n n e l i s c h a n g e d . I n e v e r y c a s e the c o n t r a c t i o n r a t i o i s 10. 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 at the 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 rates are 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 i s c a l c u l a t e d b y the f o l l o w i n g e q u a t i o n : Re = U ° H ° ( 5 .12 ) v w h e r e U 0 i s the v e l o c i t y at the c h a n n e l i n l e t , H Q is the c h a n n e l h e i g h t at i n l e t , v i s t h e k i n e m a t i c v i s c o s i t y o f w a t e r . 45 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 I L ( R = 10, L = 0 . 2 2 5 m ) U o (m/s) Reynolds number Orientation parameter x-y plane x-z plane 0 . 1 6 1 2 , 0 0 0 0 .98 0.81 0 . 2 4 1 8 , 0 0 0 0 .98 0.81 0 . 3 6 2 7 , 0 0 0 0 .98 0 . 8 0 A s a l r e a d y n o t e d , the e x p e r i m e n t a l r e s u l t s 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 , it a p p e a r s that the f i b e r o r i e n t a t i o n at the 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 rates o f s t r a i n i n the f l o w , b u t at the s a m e t i m e , i t a l s o s h o r t e n s the f i b e r r e s i d e n c e t i m e i n the f l o w . T h e p r e s e n t r e s u l t s o f f i b e r o r i e n t a t i o n f o r d i f f e r e n t f l o w v e l o c i t i e s n e e d s f u r t h e r i n v e s t i g a t i o n , b e c a u s e i n o u r s i m u l a t i o n the t u r b u l e n c e e f f e c t i s n o t s p e c i f i c a l l y i n c l u d e d , a n d the f l o w v e l o c i t y h a s c l o s e r e l a t i o n s h i p w i t h the t u r b u l e n c e . 5.3.3 The Effect of Channel Length on Fiber Orientation B e s i d e s the e f f e c t o f R c a n d U Q , it i s i m p o r t a n t to d e t e r m i n e the e f f e c t o f c h a n n e l l e n g t h o n f i b e r o r i e n t a t i o n . T h e 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 the c h a n n e l e x i t f o r d i f f e r e n t 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 l e n g t h i s t h e 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 s e c t i o n . 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 . 1 5 0 0 0 .98 0 . 8 2 0 . 2 2 5 0 0 .98 0 .81 0 . 3 3 7 5 0 .99 0 .82 4 6 A l t h o u g h a s h o r t e r c o n t r a c t i o n c o r r e s p o n d s to h i g h e r rates o f s t r a i n i n t h e f l o w f i e l d , the f i b e r 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 a l s o r e d u c e d . T h e t o t a l e f f e c t o f c h a n g i n g c h a n n e l l e n g t h o n f i b e r o r i e n t a t i o n i s t h e r e f o r e e l i m i n a t e d . H e r e a g a i n , o n l y the m e a n f l o w h a s b e e n c o n s i d e r e d i n the s i m u l a t i o n . F o r f u r t h e r i n v e s t i g a t i o n , t u r b u l e n c e e f f e c t s h o u l d b e i n c l u d e d . 5.3.4 The Effect of Fiber Aspect Ratio on Fiber Orientation F i n a l l y , t h e e f f e c t o f c h a n g e s to the f i b e r a s p e c t r a t i o is i n v e s t i g a t e d b y c h a n g i n g t h e 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 the 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 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 A r ( R c = 10, U Q = 0 .24 m / s , L c = 0 .225 m ) A r ( L / d ) ( L and d in mm ) orientation parameter x-y plane x-z plane 133.3 ( 4 / 0 . 0 3 0 ) 0 .98 0.81 6 8 . 2 ( 3 / 0 . 0 4 4 ) 0 .98 0 .81 3 3 . 3 ( 2 / 0 . 0 6 0 ) 0 .98 0.81 F r o m the 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 interes t , the c h a n g e i n A r d o e s n o t a f f e c t the 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 . T h e t u r b u l e n c e l e n g t h s c a l e c o u l d b e o f i m p o r t a n c e i n a m o r e c o m p l e t e s i m u l a t i o n w h i c h i n c l u d e s t u r b u l e n t d i s p e r s i o n . 5.3.5 The Effect of Flow Elongation T h e f l o w e l o n g a t i o n is a l s o e x a m i n e d f o r its r e l a t i o n s w i t h the c o n t r a c t i o n r a t i o o f t h e c h a n n e l . T a b l e 5 .6 s h o w s the e f f e c t o f R o n the e l o n g a t i o n o f f l o w at the e x i t o f the c ° c h a n n e l . 8 i s 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 4 7 l e n g t h d o e s n o t a f f e c t s , it i s c l e a r that the e l o n g a t i o n o f f l o w c a n b e u s e d f o r the d e t e r m i n a t i o n o f the f i b e r o r i e n t a t i o n status i n a c o n v e r g e n t c h a n n e l , i f o n l y t h e m e a n v e l o c i t i e s are c o n s i d e r e d . H o w e v e r , t u r b u l e n c e d i s p e r s i o n a l m o s t c e r t a i n l y a s i g n i f i c a n t f a c t o r w h i c h i s n o t c o v e r e d b y the e l o n g a t i o n 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 the 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 . 2 4 m / s , L = 0 .225 m ) : Conditions R c=15 Rc = 10 Rc = 6.7 s 2 .7 2.3 1.9 5.4 Symmetric Channel T h e p r e v i o u s p r e d i c t i o n r e s u l t s are f o r a n a s y m m e t r i c c o n v e r g e n t c h a n n e l . A s y m m e t r i c c h a n n e l i s 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 s i m u l a t i o n s . T h e s y m m e t r i c h e a d b o x u s e d i n U l l m a r ' s w o r k [25] w i t h a c o n t r a c t i o n r a t i o o f 16 .7 i s a d a p t e d f o r t h e c u r r e n t s i m u l a t i o n . F i g . 5 . 16 g i v e s the c r o s s s e c t i o n a l d i m e n s i o n s o f t h i s h e a d b o x , a n d F i g . 5 . 1 7 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 t h i s d e s i g n . T h e c h a n n e l w i d t h i s c o n s t a n t . D u r i n g t h e s i m u l a t i o n , f o u r i n f l o w v e l o c i t i e s are tes ted . T h e 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 o n t h e c e n t r a l s t r e a m l i n e at the 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 rates are 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 (m/s) Reynolds number Orientation parameter x-y plane x-z plane 0 . 1 0 2 5 , 0 0 0 0 .98 0 .88 0 . 3 0 7 5 , 0 0 0 0 .98 0 .88 0 .43 1 0 7 , 0 0 0 0 .98 0 .88 0 . 5 6 1 4 0 , 0 0 0 0 .98 0 .88 F r o m t h i s s t u d y o f a s y m m e t r i c 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 are o b t a i n e d f r o m the n u m e r i c a l s i m u l a t i o n s : 4 8 a. F i b e r o r i e n t a t i o n c h a n g e s f r o m r a n d o m at the c h a n n e l i n l e t to the h i g h l y p r e f e r r e d 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 the c e n t r a l s t r e a m l i n e ; b . A s i n t h e 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 , a n i n c r e a s e o f i n f l o w v e l o c i t y h a s n o e f f e c t o n t h e f i b e r o r i e n t a t i o n ; c. T h e d i f f e r e n c e i n f i b e r o r i e n t a t i o n f o r t w o p r o j e c t i o n p l a n e s e x i s t s b o t h i n t h e a s y m m e t r i c a n d s y m m e t r i c c o n v e r g e n t c h a n n e l . 5.5 Statistical Error Estimation A n e s t i m a t e c a n b e m a d e o f the e r r o r w h i c h w i l l b e p r e s e n t i n the p r o b a b i l i t y d i s t r i b u t i o n s m e a s u r e d o r c a l c u l a t e d , d u e to the l i m i t e d s a m p l e s i z e w h i c h i s b e i n g u s e d . I n g e n e r a l , the s t a n d a r d d e v i a t i o n w i t h i n r e p e a t e d s a m p l e s c o n s i s t i n g o f n o b j e c t s , o f w h i c h a f r a c t i o n r h a s a p a r t i c u l a r c h a r a c t e r i s t i c , i s g i v e n a p p r o x i m a t e l y b y [59] w h e r e b i s the f r a c t i o n o f n n o t c o n t a i n e d i n r a n d is t h e r e f o r e e q u a l to (1 - r ) . T h i s e s t i m a t e i s a c c u r a t e 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 a l s o f o r r a n d b u n e q u a l i f n i s l a r g e . S i n c e n i s l a r g e r t h a n 1 ,000 i n a l l the c a s e s o f o u r e x p e r i m e n t s , w e a s s u m e t h i s e x p r e s s i o n c a n b e u s e d as a n e r r o r e s t i m a t e w h e n a p p l i e d to the 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 p ( a ) f o r w h i c h I n t h e e q u a t i o n , & is the i n t e r v a l o f o r i e n t a t i o n a n g l e a f o r w h i c h t h e v a l u e o f p ( a ) i s to b e e v a l u a t e d , s o m e t i m e s c a l l e d the " b i n w i d t h " o f the a n g l e a . A l t h o u g h A. c a n v a r y w i t h (5 .13 ) p(a) = r ( 5 . 1 4 ) A 4 9 a i f the b i n w i d t h s are n o t c o n s t a n t , the a s s u m p t i o n i s m a d e h e r e that A i s a c o n s t a n t , i n d e p e n d e n t o f a . T h e s t a n d a r d 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 ( a ) w i l l b e ( 5 . 1 5 ) a n d t h e r e f o r e e q u a l to r ( l - r ) 1 n A (5 .16 ) S u b s t i t u t i n g (5 .14 ) i n t o (5 .16) , w e h a v e \E^-p(af (5 -17) I n g e n e r a l , i f t h e o r i e n t a t i o n i s i n i t i a l l y c o m p l e t e l y r a n d o m , p ( a ) c a n b e e x p r e s s e d as 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 ) , a c o n s t a n t " b i n w i d t h " A o f nl\% h a s b e e n u s e d . F o r a s a m p l e s i z e n o f 1325 u s e d i n the 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 the x - y p l a n e at x = 4 .5 c m , t h e s t a n d a r d d e v i a t i o n o f e v e r y p o i n t i n the p r o b a b i l i t y d e n s i t y w o u l d b e ~ A r - 0 . 0 3 6 F o r a s a m p l e s i z e n o f 1638 u s e d i n the 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 the x - z p l a n e at x 4 .5 c m , t h e s t a n d a r d d e v i a t i o n is 50 T h e e r r o r c a n b e e x p e c t e d to b e w i t h i n a b a n d o f + 3 a a b o u t 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 the 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 the r a n g e 0 . 3 1 8 + 0 . 1 0 8 a n d a v a l u e o f the p r o b a b i l i t y d e n s i t y i n x - z p l a n e i n the r a n g e 0 . 3 1 8 + 0 . 0 9 6 . T h e o b s e r v e d v a l u e s at the first p o i n t (x = 4 .5) f o r the x - y p l a n e l i e b e t w e e n 0 .221 a n d 0 . 4 5 8 (see F i g . 5 . 7 ) , a l i t t le 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 . P r e s u m a b l y , the 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 are 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 the 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 the s a m e p o i n t f o r the 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 . 3 9 2 , f a l l i n g w i t h i n the p r e d i c t e d b a n d . I n s u m m a r y , the p r e s e n t e v a l u a t i o n s s h o w t h e e f f e c t o f s a m p l e s i z e 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 t h e o b s e r v e d d i s t r i b u t i o n o f d a t a i n b o t h the x - y a n d x - z p l a n e s . T h e s i m u l a t i o n s , s h o w n i n F i g . 5 .7 , h a v e a n i n i t i a l scatter w h i c h l i e s w e l l w i t h i n the p r e d i c t e d s t a t i s t i c a l e r r o r e s t i m a t e s , as e x p e c t e d . 51 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 . 5 2 2.5 Ol 1 1 1 1 1 1 1 i i I I i i i I I o i 1 1 — • • ' 1 i i i i 1 i i 1 1 - 0 0.1 0.2 0.3 0 ° 1 0.2 0.3 channel length (m) channel length (m) F i g u r e 5.3. 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 s t r e a m l i n e . 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 c e n t r a l s y m m e t r y p l a n e . 0 0 3 F i g u r e 5.5. 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 p l a n e . 53 channel 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 s t r e a m l i n e . 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 c m , (a) i n x - y p l a n e , (b) i n x - z p l a n e 54 £ 0.4 JJj 0.3 cn 3 e ° - 0.21- SIMULATION • EXPERIMENT -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 angle (radians) •0.5 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 = 15.7 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 .2 i E .» o E g 0.4 o | 02 C i SIMULATION • EXPERIMENT • • channel length (cm) channel length (cm) ( a ) ( b ) F i g u r e 5 .14 . T h e o r i e n t a t i o n p a r a m e t e r s a l o n g the c e n t r a l s t r e a m l i n e , (a) i n the x - y p l a n e , (b) i n the 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 the 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 . 1 7 . 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 . 5 9 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 t h e s i s i s c o n c e r n e d w i t h f i b e r o r i e n t a t i o n 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 . A s u m m a r y 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 e x p e r i m e n t s a n d c o m p u t a t i o n a l 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 m a t h e m a t i c a l f l o w m o d e l a n d a f i b e r m o t i o n m o d e l h a v e b e e n c o m b i n e d f o r p r e d i c t i n g t h e 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 b e d e t e r m i n e d b y e x p e r i m e n t . R i g i d f i b e r s f l o w i n g i n a s y m m e t r i c a n d 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 are n u m e r i c a l l y s i m u l a t e d . R a n d o m i n i t i a l f i b e r o r i e n t a t i o n s are s p e c i f i e d at t h e i n l e t o f the c h a n n e l . T h e s ta t i s t i ca l e x p r e s s i o n s o f t h e o r i e n t a t i o n o f a l a r g e n u m b e r o f f i b e r s c a n b e e v a l u a t e d b y c o m p u t i n g the o r i e n t a t i o n o f e a c h s i n g l e f i b e r a l o n g the c e n t r a l s t r e a m l i n e . 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 to v a l i d a t e the n u m e r i c a l m o d e l . T h e o r i e n t a t i o n d i s t r i b u t i o n s o f d y e d n y l o n f i b e r s w i t h u n i f o r m l e n g t h a n d w i d t h w e r e s t u d i e d u s i n g a n i m a g e a n a l y s i s m e t h o d . A n a s y m m e t r i c 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 the b a s i s f o r e x p e r i m e n t a l 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 t h e t r e n d o f f i b e r o r i e n t a t i o n i n a d i l u t e h e a d b o x 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 the e x p e r i m e n t a l d a t a a n d the n u m e r i c a l p r e d i c t i o n i s s i g n i f i c a n t . T h e p r e d i c t e d o r i e n t a t i o n d i s t r i b u t i o n s , b e i n g b a s e d o n l y o n the m e a n f l o w i n the h e a d b o x , s h o w g r e a t e r 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 , i n e v e r y c a s e . T h e t u r b u l e n c e d i s p e r s i o n e f f e c t m u s t b e a d d e d to t h e c u r r e n t f i b e r m o d e l b e f o r e it is c a p a b l e o f a c c u r a t e l y p r e d i c t i n g t h e f i b e r o r i e n t a t i o n i n a g i v e n f l o w f i e l d . T h e s i m u l a t i o n m e t h o d h a s f u r t h e r 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 f o r d i f f e r e n t h e a d b o x g e o m e t r y , f l o w v e l o c i t y a n d f i b e r a s p e c t r a t i o . C h a n g e s i n c o n t r a c t i o n r a t i o c a u s e s c h a n g e s i n v e l o c i t y g r a d i e n t s , a n d rates o f s t r a i n , a n d t h e r e f o r e a f f e c t the a l i g n m e n t o f f i b e r s 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 c a n a l s o c h a n g e the f l o w f i e l d , b u t t h e i r 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 are e l i m i n a t e d i n t h i s 6 0 s i m u l a t i o n b y the f a c t that the r e s i d e n c e t i m e is c h a n g e d . T h e s i m u l a t i o n r e s u l t s a l s o 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 e f f e c t o f f i b e r a s p e c t r a t i o a n d f i b e r l e n g t h i n t h e r a n g e o f o u r in teres t ( i . e . A f = 3 3 - 1 3 3 , L = 2 ~ 4 m m ) o n f i b e r o r i e n t a t i o n i s s m a l l . B o t h e x p e r i m e n t a l a n d s i m u l a t i o n r e s u l t s s h o w e d d i f f e r e n c e s i n the f i b e r o r i e n t a t i o n i n 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 c h a r a c t e r i s t i c s 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 g r a d i e n t s to f i b e r o r i e n t a t i o n are n o t t h e s a m e i n a l l d i r e c t i o n s . 61 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 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 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 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 h e a d b o x g e o m e t r i e s 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 f u t u r e w o r k b e f o r e r e a c h i n g a c o m p l e t e u n d e r s t a n d i n g o f f i b e r o r i e n t a t i o n i n h e a d b o x e s . 1. C u r r e n t l y , o n l y the e f f e c t o f m e a n v e l o c i t i e s a n d t h e i r g r a d i e n t s 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 e x i s t s i g n i f i c a n t d i f f e r e n c e s b e t w e e n the e x p e r i m e n t a l d a t a a n d the n u m e r i c a l p r e d i c t i o n s . C l e a r l y , t u r b u l e n c e p l a y s a n i m p o r t a n t r o l e i n the d e t e r m i n a t i o n o f f i b e r o r i e n t a t i o n i n the f l o w . I f the i n f l u e n c e o f the f l u c t u a t i n g v e l o c i t i e s o n the f i b e r i s a d d e d , the s i m u l a t i o n c o u l d p r o d u c e r e s u l t s c l o s e r to t h e e x p e r i m e n t s . It i s t h e r e f o r e 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 c o n v e r g e n t 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 t u b e b a n k a n d h e a d e r , 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 h e a d b o x c o n t r a c t i o n . 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 t h i s r e s e a r c h . F o r t u n a t e l y the 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 r e s e a r c h g r o u p at U B C . T o q u a n t i f y s u c h r e s u l t s , o n e m u s t p r o v i d e a r e a s o n a b l e d e f i n i t i o n o f 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 o r 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 c u r r e n t r e s e a r c h h a s f o c u s s e d o n f i b e r m o t i o n a l o n g the c e n t r a l s t r e a m l i n e . A d e t a i l e d s t u d y o f the f i b e r o r i e n t a t i o n 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 , c r o s s m a c h i n e 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 the 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 m a y 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 . 6 2 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 , t h e f i b e r - w a l l i n t e r a c t i o n s m u s t b e i n c l u d e d . A s s h o w n , 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 i s m a i n l y d e t e r m i n e d d u r i n g the s h o r t 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 t h e e x i t , the c h a n n e l o p e n i n g b e c o m e s n a r r o w e r a n d f i b e r - w a l l i n t e r a c t i o n s m a y n o t b e n e g l i g i b l e . 5. D u r i n g t h i s w o r k o n l y the c o n v e r g e n t s l i c e s e c t i o n o f a h e a d b o x i s s i m u l a t e d w i t h t h e 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 i n l e t . 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 t h e d i f f u s e r t u b e s 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 are c o n c e r n e d a b o u t t h e f i b e r o r i e n t a t i o n p r o f i l e s i n the c r o s s - m a c h i n e d i r e c t i o n . T h a t r e q u i r e s a t h o r o u g h u n d e r s t a n d i n g o f f l u i d - f i b e r i n t e r a c t i o n s a n d f i b e r o r i e n t a t i o n c h a r a c t e r i s t i c s w i t h i n the e n t i r e h e a d b o x . T o g i v e a to ta 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 t h e p a p e r m a k i n g p r o c e s s e s , the fiber o r i e n t a t i o n i n the j e t a n d o n the f o r m i n g w i r e s h o u l d a l s o b e s t u d i e d . A l l t h e s e s t u d i e s 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 t h e fiber o r i e n t a t i o n i n the p a p e r . 6. I n o u r r e s e a r c h , the fiber-fiber 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 u s e d i n the s i m u l a t i o n s a n d e x p e r i m e n t s are v e r y d i l u t e . A t c o m m e r c i a l l y u s e d p a p e r m a k i n g c o n s i s t e n c i e s , i . e . , i n the r a n g e o f 0.1 to 1 .5%, f i b e r s d o n o t e x i s t 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 i n a n e t w o r k f o r m . T h e r e are s e l d o m fibers f r e e l y f l o a t i n g a b o u t . E a c h fiber is i n c o n t a c t w i t h m a n y o t h e r f i b e r s , a n d is 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 e a c h o t h e r g i v e s fiber n e t w o r k s a c e r t a i n a m o u n t o f m e c h a n i c a l s t r e n g t h , w h i c h i n t u r n a f f e c t s the f l o w p a t t e r n . A s a r e s u l t , fiber-fiber 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 flow s h o u l d a l s o b e i n v e s t i g a t e d i n the f u t u r e . W h e n c o n s i d e r i n g fiber-fiber a n d fluid/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 to s o l v e s i m u l t a n e o u s l y f o r the v e l o c i t y field a n d t h e fiber 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 flow a n d t h e fiber m o t i o n . 63 8. N O M E N C L A T U R E A r f i b e r a s p e c t r a t i o av a 2 r a n d o m v a r i a b l e s 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 c r o s s m a c h i n e d i r e c t i o n C v o l u m e c o n c e n t r a t i o n V d fiber d i a m e t e r 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 u n i t m a s s f o r i e n t a t i o n p a r a m e t e r 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 u n i t m a s s H Q c h a n n e l i n l e t h e i g h t k t u r b u l e n t k i n e t i c e n e r g y L fiber l e n g t h L c h a n n e l l e n g t h c ° M D m a c h i n e d i r e c t i o n N c r o w d i n g f a c t o r n n u m b e r o f f i b e r s p p r e s s u r e p ( a ) the 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 f u n c t i o n r f r a c t i o n o f f i b e r s R c o n t r a c t i o n r a t i o c R e R e y n o l d s n u m b e r s d i s t a n c e a l o n g the c e n t r a l s t r e a m l i n e S.. rate o f s t r a i n t e n s o r u t t i m e 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 x - d i r e c t i o n U Q u - v e l o c i t y at the i n l e t U u - v e l o c i t y at the o u t l e t 64 u i n s t a n t a n e o u s f l u i d v e l o c i t y v e c t o r u s f l o w v e l o c i t y a l o n g the c e n t r a l s t r e a m l i n e 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 s y s t e m X j the l e n g t h o f the f la t s e c t i o n y t h e y - d i r e c t i o n i n a C a r t e s i a n s y s t e m z the z - d i r e c t i o n i n a C a r t e s i a n s y s t e m a f i b e r o r i e n t a t i o n a n g l e i n a p r o j e c t i o n p l a n e a Q t h e m e a n o f d i s t r i b u t e d a n g l e s p f l o w a n g l e K r o n e c k e r D e l t a A i n t e r v a l o f o r i e n t a t i o n a n g l e 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 the f l o w at the c h a n n e l e x i t (j, f i b e r o r i e n t a t i o n a n g l e <p the a n g l e b e t w e e n the c e n t r a l s t r e a m l i n e a n d x - a x i s 1̂  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 t u r b u l e n t k i n e m a t i c v i s c o s i t y 0 f i b e r o r i e n t a t i o n a n g l e p f l u i d d e n s i t y CTp s t a n d a r d 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 ( a ) CTr s t a n d a r d 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 t e n s o r X i j R e y n o l d s stress t e n s o r Q s o l i d a n g l e 65 9. R E F E R E N C E S 1. N o r d s t r o m , B . a n d N o r m a n , B . , " I n f l u e n c e o n S h e e t A n i s o t r o p y , F o r m a t i o n , Z - T o u g h n e s s a n d T e n s i l e S t i f f n e s s o f R e d u c e d F e e d A r e a to a H e a d b o x N o z z l e " , Nordic Pulp Paper Res. J.  9 0) : 5 3 ( 1 9 9 4 ) . 2. T i i k a j a , E . , " F i b e r P r o p e r t i e s a n d P a p e r M a c h i n e R u n n a b i l i t y " , jggg TAPPI Engineering/Process and Product Quality Conference & Trade Fair- 1241 ( 1 9 9 9 ) . 3. 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 " , j pu[p paper Sa. 19(4) : J 1 7 5 ( 1 9 9 3 ) . 4. S h a k e s p e a r e , J . , " T u t o r i a l : F i b e r O r i e n t a t i o n A n g l e P r o f i l e s - P r o c e s s P r i n c i p l e s a n d C r o s s - M a c h i n e C o n t r o l " , TAPPI Proceedings, 1998 Process Control, Electrical & Information Conference'- 593 ( 1 9 9 8 ) . 5. 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 o f the Q u a n t i t y a n d O r i e n t a t i o n o f C h e m i c a l P u l p F i b e r s i n the S u r f a c e s o f N e w s p r i n t " , puip paper Can. 6 4 ( 1 ) : T - 3 ( 1 9 6 3 ) . 6. 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 paper Can. 9 8 ( 1 0 ) : T 3 9 1 ( 1 9 9 7 ) . 7. N o r d s t r o m , B . a n d N o r m a n , B . , " I n f l u e n c e o f H e a d b o x N o z z l e C o n t r a c t i o n R a t i o o n S h e e t F o r m a t i o n a n d A n i s o t r o p y " , 7904 Engineering Conference, TAPPI Proceedings'- 2 2 5 ( 1 9 9 4 ) . 8. U e s a k a , T . , " D i m e n s i o n a l S t a b i l i t y o f P a p e r : U p g r a d i n g P a p e r P e r f o r m a n c e i n E n d U s e " , j P u I p P a p e r S c l 17(2) : J 3 9 ( 1 9 9 1 ) . 9 . P a n t a l e o , S . a n d S h a n d s , J . , " C o n t r o l l i n g F i b e r O r i e n t a t i o n " , World Paper 2 1 9 ( 2 ) : 2 2 ( 1 9 9 4 ) . 10 . G i v l e r , R . C , C r o c h e t , M . J . 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 F i b e r O r i e n t a t i o n i n D i l u t e S u s p e n s i o n s " , j of Composite Materials^ 1 7 ( 7 ) : 3 3 0 ( 1 9 8 3 ) . 11 . W r i s t , P . E . , " D y n a m i c s o f S h e e t F o r m a t i o n o n the F o u r d r i n i e r M a c h i n e " , j n e formation and structure of paper: Transaction of the symposium held at Oxford- 8 3 9 ( 1 9 6 1 ) . 12. S m o o k , G . A . , H a n d b o o k f o r P u l p & P a p e r T e c h n o l o g i s t s , T A P P I , ( 1 9 8 2 ) . 6 6 13. D a h l , H . K . a n d W e i s s , H . G . , " A N e w H y d r a u l i c P r i n c i p l e 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 . , S t o l t z , P . a n d G u s t a v s s o n , H . , " A S t u d y o f F i b e r O r i e n t a t i o n R e l a t e d to H e a d b o x D e s i g n a n d O p e r a t i n g S t r a t e g i e s " , EUCEPA 24th Conf. Proc. (Stockholm), Pap. Technol- 2 4 ( 1 9 9 0 ) . 15. S h a k e s p e a r e , J . , K n i i v i l a , J . , K o r p i n e n , A . a n d J o h a n s s o n , T . , " A n O n - l i n e C o n t r o l S y s t e m f o r S i m u l t a n e o u s O p t i m i z a t i o n o f B a s i s W e i g h t a n d O r i e n t a t i o n A n g l e P r o f i l e s " , 1st EcoPaperTech- 3 9 ( 1 9 9 5 ) . 16. P a n t a l e o , S . B . , " A N e w H e a d b o x D e s i g n F e a t u r i n g C o n s i s t e n c y P r o f i l i n g D e c o u p l e d f r o m F i b e r O r i e n t a t i o n R e s p o n s e " , Tappi J. 7 8 ( 1 1 ) : 8 9 ( 1 9 9 5 ) . 17. L e e , J . J . - G . a n d P a n t a l e o , S . B . , " H e a d b o x F l o w A n a l y s i s " , j puip Paper Sci- 2 5 ( 1 2 ) : 4 3 7 ( 1 9 9 9 ) . 18. H o l i k , H . , H e b , H . , T i e t z , M . a n d D r t i n a , P . , " F l u i d M e c h a n i c s i n the H e a d b o x - the K e y to I m p r o v i n g P a p e r Q u a l i t y " , jgg4 Engineering Conference, Tappi Proceedings'- 2 4 7 ( 1 9 9 4 ) . 19. P a t r i c k , K . L . , " L a t e s t H e a d b o x e s : S e p a r a t e B a s i s W e i g h t , F i b e r O r i e n t a t i o n C o n t r o l " , puip & P a p e r 9 : 1 3 7 ( 1 9 9 6 ) . 2 0 . M a l a s h e n k o , A . , " T h e D i l u t i o n C o n t r o l H e a d b o x " , Paper Technology 3 8 ( 1 0 ) : 4 2 ( 1 9 9 7 ) . 2 1 . B e g e m a n n , U . , " M o d u l e j e t H e a d b o x C o n c e p t O p e r a t i n g E x p e r i e n c e w i t h S i n g l e - a n d M u l t i - l a y e r H e a d b o x e s P r o c e s s i n g D i f f e r e n t P a p e r G r a d e s " , pu\p paper Can. 97(8)': 2 0 ( 1 9 9 6 ) . 2 2 . N y b e r g , P . a n d M a l a s h e n k o , A . , " D i l u t i o n C o n t r o l H e a d b o x - C h o i c e s , T h r e a t s a n d S o l u t i o n s " , 83rd Annual Meeting, Technical Section CPPA- A 1 ? ( 1 9 9 7 ) . 2 3 . B a n d o , T . , M a k i n o , T . a n d F u j i k i , K . , " D e v e l o p m e n t o f M i t s u b i s h i N e w H e a d b o x a n d F o r m e r " , J.Korea Tappi 28 (3 ) : 88 ( 1 9 9 6 ) . 2 4 . V y s e , R . , K i n g , J . , H e a v e n , M . a n d P a n t a l e o S . , " C o n s i s t e n c y P r o f i l i n g - a N e w T e c h n i q u e f o r C D B a s i s W e i g h t C o n t r o l " , pu\p p a p e r Can. 9 7 ( 9 ) : 6 2 ( 1 9 9 6 ) . 2 5 . U l l m a r , M . a n d N o r m a n , B . , " O b s e r v a t i o n o f 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 N o z z l e at L o w C o n s i s t e n c y " TAPPI Proceedings, 1997 Engineering & Papermakers Conference- 865 ( 1 9 9 7 ) . 2 6 . B a n d h a k a v i , V . S . a n d A i d u n , C . K . , " A n a l y s i s o f T u r b u l e n t F l o w i n t h e C o n v e r g i n g Z o n e o f a H e a d b o x " , jggg TAPPI Engineering/Process and Product Quality Conference & Trade Fair'- H 3 5 ( 1 9 9 9 ) . 6 7 2 7 . E r i k k i l a , A . L . , P a k a r i n e n , P . a n d O d e l l , M . , " S h e e t F o r m i n g S t u d i e s U s i n g L a y e r e d O r i e n t a t i o n A n a l y s i s " , pu\p P a p e r Qan. 9 9 ( 1 ) : 8 1 ( 1 9 9 8 ) . 2 8 . K e r e k e s , R . a n d S c h e l l , C , " C h a r a c t e r i z a t i o n o f F i b e r F l o c c u l a t i o n R e g i m e s b y a C r o w d i n g F a c t o r " , j pu\p p a p e r sci. 18(1) : 3 3 2 ( 1 9 9 2 ) . 2 9 . U l l m a r , M . , " O n F i b e r A l i g n m e n t M e c h a n i s m s i n a H e a d b o x N o z z l e " , M a s t e r T h e s i s , R o y a l Insti tute o f T e c h n o l o g y ( 1 9 9 8 ) . 3 0 . N i s k a n e n , K . J . , " D i s t r i b u t i o n o f F i b e r O r i e n t a t i o n s i n P a p e r " , Fundamentals Pmkg. (Baker & Punton, ed.) / Trans. 9th Fundamental Res. Symp. (Cambridge) V- 2 7 5 ( 1 9 8 9 ) . 3 1 . D i n h , S . M . , " O n the R h e o l o g y o f C o n c e n t r a t e d F i b e r S u s p e n s i o n s " , S c . D . T h e s i s , D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g , M I T ( 1 9 8 1 ) . 3 2 . Z i r n s a k , M . A . , H u r , D . U . a n d B o g e r , D . V . , " N o r m a l S t r e s s e s i n F i b e r S u s p e n s i o n s " , j Non-Newtonian Fluid Mech- 5 4 ( 8 ) : I 5 3 ( 1 9 9 4 ) . 3 3 . W a l l e r , M . H . , " R e c e n t D e v e l o p m e n t s i n H e a d b o x e s " , Tappi J- 7 0 ( 1 ) : 3 3 ( 1 9 8 7 ) . 3 4 . A i d u n , C . K . a n d K o v a c s , 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 S e c t i o n : the O r i g i n o f N o n - u n i f o r m F i b e r O r i e n t a t i o n " , TappiJ- 7 8 ( 1 1 ) : 9 7 ( 1 9 9 5 ) . 3 5 . A i d u n , C . K . , " H y d r o d y n a m i c A n a l y s i s a n d O p t i m i z a t i o n o f P a p e r a n d B o a r d F o r m i n g " , 7095 Papermakers Conference- 2 1 5 ( 1 9 9 5 ) . 3 6 . S h i m i z u , T a n d W a d a , K . , " C o m p u t e r S i m u l a t i o n a n d M e a s u r e m e n t o f F l o w i n a H e a d b o x " , P r o c Pan-Pacific Pulp Pap. Technol. Conf. (Tokyo), A : I 5 7 ( 1 9 9 2 ) . 3 7 . J e f f e r y , G . B . , " T h e M o t i o n o f E l l i p s o i d a l P a r t i c l e s I m m e r s e d i n a V i s c o u s F l u i d " , Proceeding of the Royal Society, A l ° 2 : 161 ( 1 9 2 2 ) . 3 8 . 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 Reinforced Plastics and Composites 3 ( 4 ) : 98 ( 1 9 8 4 ) . 3 9 . M a s o n , S . G . , a n d B a r t o k , W . , R h e o l o g y o f D i s p e r s e S y s t e m s , B r i t i s h S o c i e t y o f 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 ) . 4 0 . 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 S t u d y o n t h e S o l u t i o n T e c h n i q u e s f o r F i b e r O r i e n t a t i o n i n T w o - d i m e n s i o n a l C o n v e r g i n g a n d D i v e r g i n g F l o w s " , j Thermoplastic Composite Materials 4 ( 1 0 ) : 3 1 1 ( 1 9 9 1 ) . 4 1 . A k b a r , S . a n d A l t a n , M . C , " O n 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 i n 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 F l o w s " , Polymer Eng. Sci. 3 2 ( 1 2 ) : 8 1 0 ( 1 9 9 2 ) . 4 2 . 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 to D e s c r i b e a n d P r e d i c t F i b e r O r i e n t a t i o n i n S h o r t F i b e r C o m p o s i t e s " , j Rheology 3 U 8 ) : 7 5 1 ( 1 9 8 7 ) . 68 4 3 . 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 T h r e e - d i m e n s i o n a l F i b e r O r i e n t a t i o n i n H e l e - S h a w F l o w s " , Polymer Eng Sci- 3 0 ( 1 4 ) : 8 4 8 ( 1 9 9 0 ) . 4 4 . 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 the 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 F l o w s " , j Rheology 3 3 ( 7 ) : 1 1 2 9 ( 1 9 8 9 ) . 4 5 . D i n h , S . M . a n d A r m s t r o n g , R . C , " A R h e o l o g i c a l E q u a t i o n o f Sta te f o r S e m i c o n c e n t r a t e d F i b e r S u s p e n s i o n s " , j Rheology 2 8 ( 3 ) : 2 0 7 ( 1 9 8 4 ) . 4 6 . R o s s , R . F . a n d K l i n g e n b e r g , D . J . , " D y n a m i c S i m u l a t i o n o f F l e x i b l e F i b e r s C o m p o s e d o f L i n k e d R i g i d B o d i e s " , j chem. Phys- 106(7 ) : 2 9 4 9 ( 1 9 9 7 ) . 4 7 . W h e r r e t t , G . , G a r t s h o r e , 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 F i b e r M o t i o n i n S h e a r " , T h e 1 9 9 7 A S M E F l u i d s E n g i n e e r i n g D i v i s i o n S u m m e r M e e t i n g ( 1 9 9 7 ) . 4 8 . D o n g , S . , S a l c u d e a n , M . a n d G a r t s h o r e , 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 S c r e e n S l o t s " , 2000 TAPPI Paper makers Conference and Trade Fair- 5 9 7 ( 2 0 0 0 ) . 4 9 . S h a r i a t i , M . R . , B i b e a u , E . , S a l c u d e a n , M . a n d G a r t s h o r e , I., " N u m e r i c a l a n d E x p e r i m e n t a l 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 ) . 5 0 . H u a , L . , H e , P . , S a l c u d e a n , M . , G a r t s h o r e , 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 H y d r a u l i c H e a d b o x " , 2000 TAPPI Papermakers Conference and Trade Fair- 6 9 5 ( 2 0 0 0 ) . 5 1 . P e t r i e , C . J . S . , " T h e r h e o l o g y o f f i b e r s u s p e n s i o n s " , j Non-Newtonian Fluid Mech,m2-3): 3 6 9 ( 1 9 9 9 ) . 52 . L a u n d e r , B . E . a n d S p a l d i n g , D . B . , " T h e N u m e r i c a l C o m p u t a t i o n o f T u r b u l e n t F l o w s " , Computer Methods in Applied Mechanics ^'- 2 6 9 ( 1 9 7 4 ) . 5 3 . N o w a k , P . , " A M u l t i - g r i d a n d M u l t i - b l o c k M e t h o d " , T e c h n i c a l R e p o r t , 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 ( 1 9 9 2 ) . 5 4 . R o s s , R . F . a n d k l i g e n b e r g , D . J . , " S i m u l a t i o n o f F l o w i n g W o o d F i b e r S u s p e n s i o n s " , , / pulp Paper Sci- 2 4 ( 1 2 ) : 3 8 8 ( 1 9 9 8 ) . 5 5 . " S p h e r e P o i n t P i c k i n g " - f r o m E r i c W e i s s t e i n ' s W o r l d o f M a t h e m a t i c s , h t t p : / / m a t h w o r l d . w o l f r a m . c o m / S p h e r e P o i n t P i c k i n g . h t m l , 9 / 1 3 / 2 0 0 0 . 56 . M c C u l l o u g h , R . L . , " A n i s o t r o p i c E l a s t i c B e h a v i o r o f C r y s t a l l i n e P o l y m e r s " , T r e a t i s e o n M a t e r i a l s S c i e n c e a n d T e c h n o l o g y : P r o p e r t i e s o f S o l i d P o l y m e r i c M a t e r i a l s , A c a d e m i c P r e s s , N e w Y o r k , 1 0 B : 4 5 3 ( 1 9 7 7 ) . 6 9 5 7 . 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 T h e s i s , U n i v e r s i t y o f D e l a w a r e ( 1 9 8 2 ) . 5 8 . M a r d i a , K . V . , S t a t i s t i c s o f d i r e c t i o n a l d a t a , A c a d e m i c P r e s s , L o n d o n a n d N e w Y o r k ( 1 9 7 2 ) . 5 9 . M o r o n e y , M . J . , F a c t s F r o m F i g u r e s , P e n g u i n B o o k s ( 1 9 5 6 ) . 7 0

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 4 1
Japan 3 0
City Views Downloads
Tokyo 2 0
Shepherdsville 2 0
Unknown 1 9
Sunnyvale 1 0
Ashburn 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items