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
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 = K2 / [ ( 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