UBC Theses and Dissertations

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UBC Theses and Dissertations

Design of a capillary artificial kidney Davis, Harold Robert 1970

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THE DESIGN OF A CAPILLARY ARTIFICIAL KIDNEY by HAROLD ROBERT DAVIS B.A. S c . , 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 , V a n c o u v e r , B r i t i s h C o l u m b i a , 1964 M.A. S c . , 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 , V a n c o u v e r , B r i t i s h C o l u m b i a , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e 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 We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA December, 1970 In presenting th i s thesis in pa r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee l y ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Depa rtment The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT An e n g i n e e r i n g study was made of an a r t i f i c i a l k i d n e y — one of the major components used i n the treatment of p a t i e n t s s u f f e r i n g from c h r o n i c uremia. As the c a p i l l a r y a r t i f i c i a l kidney o f f e r e d the most advantages over other systems, an a n a l y s i s of the mass t r a n s f e r and blood flow c h a r a c t e r i s t i c s of t h i s c o n f i g u r a t i o n was attempted w i t h the view towards o p t i m i z a t i o n . I t was determined t h e o r e t i c a l l y and v e r i f i e d exper-i m e n t a l l y t h a t a Newtonian f l u i d mass t r a n s f e r a n a l y s i s i s a good approximation even though blood, the f l u i d d i a l y z e d i n the a r t i f i c i a l kidney, i s non-Newtonian. Measurements taken from a p a t i e n t undergoing h e m o d i a l y s i s i n d i c a t e d t h a t s t e a d y - s t a t e c o n d i t i o n s f o r mass t r a n s f e r can a l s o be assumed. From the mass t r a n s f e r a n a l y s i s f o r a s i n g l e c a p i l l a r y , the c h a r a c t e r i s t i c s of a composite d i a l y z e r were determined i n terms of c e r t a i n p h y s i o l o g i c a l parameters. The system was optimized by maximizing the mass t r a n s f e r e f f i c i e n c y w i t h r e s p e c t t o the membrane area. The a n a l y s i s p r e d i c t e d the best c a p i l l a r y diameter i n terms of the o p e r a t i n g parameters. I t was found t h a t the optimized system had a constant mass t r a n s f e r e f f i c i e n c y of approximately 70 percent f o r most o p e r a t i n g c o n d i t i o n s . i i i A m a n i f o l d system which would d i s t r i b u t e blood to the c a p i l l a r i e s u n i f o r m l y and which would have no r e g i o n s of s t a g n a t i o n or t u r b u l e n c e was d e v i s e d . The m a n i f o l d had a c a p i l l a r y outflow r e g i o n of r e c t a n g u l a r dimensions which was normal to the i n l e t d i r e c t i o n . I t was determined t h a t the b e s t m a n i f o l d shape was c i r c u l a r i n c r o s s - s e c t i o n and had a l i n e a r decrease i n c r o s s - s e c t i o n a l area w i t h m a n i f o l d l e n g t h . U t i l i z i n g the p r e d i c t e d system c o n f i g u r a t i o n f o r optimum mass t r a n s f e r and blood d i s t r i b u t i o n , a s m a l l v e r -s i o n of the a r t i f i c i a l kidney was designed and t e s t e d . TABLE OF CONTENTS C h a p t e r P a ge 1 INTRODUCTION 1 1.1 I n t r o d u c t i o n 1 1.2 B a c k g r o u n d . 9 2 THEORY 17 2.1 I n t r o d u c t i o n 17 2.2 Mass T r a n s f e r 21 2.3 S y s t e m P a r a m e t e r s 27 2.4 M a n i f o l d s 36 3 EXPERIMENTS 47 3.1 P a t i e n t P a r a m e t e r s 47 3.2 Mass T r a n s f e r . 51 3.3 F l o w D u c t i n g 54 4 DISCUSSION 67 5 CONCLUSIONS 7 8 REFERENCES 82 APPENDIX I - Time R e q u i r e d t o D i a l y z e a P a t i e n t . . . 85 APPENDIX I I - D i a l y z e r D e s i g n by O p t i m i z e d E q u a t i o n s 87 APPENDIX I I I — F r i c t i o n L o s s F a c t o r s f o r M a n i f o l d s . . 89 APPENDIX I V - C o r r e c t i o n s f o r R e f r a c t i o n i n M a n i f o l d s 92 APPENDIX V - Head L o s s T h r o u g h C a p i l l a r i e s 96 V LIST OF TABLES Table Page I Output of Waste Products f o r Chr o n i c Uremic P a t i e n t on Hemodialysis ( R e s t r i c t e d Diet) . . 4 I I E i g e n v a l u e s and F l u i d Bulk C o e f f i c i e n t s f o r Va r i o u s Wall R e s i s t a n c e s 98-99 I I I E i g e n v a l u e s and F l u i d Bulk C o e f f i c i e n t s f o r Various Wall R e s i s t a n c e s , p Q = 0.5 100 IV Optimized Dimensionless Length and E f f i c i e n c y f o r V a r i o u s Wall Sherwood Numbers 101 LIST OF FIGURES F i g u r e Page 1 O v e r a l l Sherwood Number as a F u n c t i o n of Dimensionless Length and Wall Sherwood Number 102 2 E i g e n v a l u e s f o r Casson F l u i d 103 3 Bulk C o e f f i c i e n t s f o r Casson F l u i d 104 4 O v e r a l l Sherwood Number and Mass T r a n s f e r E f f i c i e n c y as a F u n c t i o n of Dimensionless Length 105 5 T h e o r e t i c a l D i a l y z e r E f f i c i e n c y Graph . . . 106 6 One Dimensional Free Body f o r M a n i f o l d . . . 107 7 F r i c t i o n Loss F a c t o r as a F u n c t i o n of M a n i f o l d Shape 108 8 Schematic of Experimental Equipment f o r M o n i t o r i n g Hemodialysis . . 109 9 Blood Flow Rate and Systemic Pressures from P a t i e n t Undergoing Hemodialysis . . . . HO 10 Schematic of Equipment f o r Mass T r a n s f e r T e s t s H I 11 Blood O u t l e t C o n c e n t r a t i o n s from Mass T r a n s f e r Experiment 112 12 Turbulence Onset i n B l a s i u s Expansion S e c t i o n 113 13 Schematic of Flow C i r c u i t f o r M a n i f o l d T e s t s 114 14 T e s t M a n i f o l d C r o s s - s e c t i o n s (Models A,B) ; H 5 v i i F i g u r e Page 15 T e s t M a n i f o l d C r o s s - s e c t i o n s (Models C,D,E) 1 1 6 16 Comparison of M a n i f o l d P r e s s u r e and C a p i l l a r y Flow Rates i n Model A . . . . . . . 1 1 7 17 S t r e a m l i n e s i n Model A 1 1 8 18 F l u i d D i s t r i b u t i o n i n Model A M a n i f o l d 1 1 9 19 P r e s s u r e D i s t r i b u t i o n i n Model A M a n i f o l d . . 1 2 0 20 T o t a l P r e s s u r e Drop f o r Model A M a n i f o l d 1 2 1 21 M a n i f o l d P r e s s u r e D i s t r i b u t i o n f o r Models B,C,D,E • • • 1 2 2 22 T o t a l : P r e s s u r e Drop f o r Model B M a n i f o l d 1 2 3 23 T o t a l P r e s s u r e Drop f o r Model C M a n i f o l d 1 2 4 24 T o t a l P r e s s u r e Drop f o r Model D M a n i f o l d 1 2 5 25 S t r e a m l i n e D i s t r i b u t i o n i n Model B M a n i f o l d 1 2 6 26 S t r e a m l i n e D i s t r i b u t i o n i n Model C M a n i f o l d 12 6 27 S t r e a m l i n e C u r v a t u r e a t C a p i l l a r y E n t r a n c e 1 2 7 28 F i r s t E i g e n v a l u e as a F u n c t i o n o f W a l l Sherwood Number 12 8 29 D i s p l a c e m e n t o f Observed Dye S t r e a k Due t o R e f r a c t i o n 1 2 9 30 P r o t o t y p e C a p i l l a r y A r t i f i c i a l K i d n e y . . . 1 3 0 31 C r o s s - s e c t i o n of P o t t e d C a p i l l a r i e s . . . . 1 3 1 LIST OF SYMBOLS Area E i g e n f u n c t i o n constants M a n i f o l d s l o t width C o n c e n t r a t i on Bulk c o n c e n t r a t i o n Diameter Mass t r a n s f e r e f f i c i e n c y F l u i d bulk c o e f f i c i e n t s M a n i f o l d energy l e v e l Wall f r i c t i o n f o r c e / u n i t l e n g t h Pressure drop across m a n i f o l d (Ap/pg) Constant K i n e t i c energy c o r r e c t i o n f a c t o r Momentum f l u x c o r r e c t i o n f a c t o r O v e r a l l mass t r a n s f e r c o e f f i c i e n t M a n i f o l d l e n g t h Rate of s o l u t e t r a n s f e r T o t a l r a t e of s o l u t e t r a n s f e r Number of c a p i l l a r i e s Pressure i n ma n i f o l d Pressure at entrance to c a p i l l a r i e s Membrane p e r m e a b i l i t y P e c l e t number (UD/V) Flow r a t e T o t a l f l o w r a t e Core r a d i u s f o r Casson f l u i d O v e r a l l mass t r a n s f e r r e s i s t a n c e E i g e n f u n c t i o n Reynolds number (UD/v) O v e r a l l Sherwood number (K oD/2P) W a l l Sherwood number (PD/2P) A x i a l v e l o c i t y a t e n t r a n c e t o c a p i l l a r i Average v e l o c i t y F l u i d v e l o c i t y i n c a p i l l a r i e s Hold-up volume o f b l o o d L e ngth System parameter (2fTSh oNxt?) Momentum l o s s f a c t o r F r i c t i o n l o s s f a c t o r M a n i f o l d shape c o n s t a n t D i f f u s i v i t y c o e f f i c i e n t System parameter ( Q ^ / a ) D i m e n s i o n l e s s l e n g t h parameter (2x/DPe) E i g e n v a l u e A b s o l u t e v i s c o s i t y V i s c o s i t y i n Casson's e q u a t i o n X v K i n e m a t i c v i s c o s i t y p D e n s i t y T S h e a r s t r e s s x Y i e l d s t r e s s i n C a s s o n ' s e q u a t i o n o / I n t e g r a l e v a l u a t e d on s u r f a c e S u b s c r i p t s b B l o o d d D i a l y s a t e i I n l e t £ B l o o d f e e d e r l i n e o O u t l e t m Membrane. w W a l l ACKNOI^FJX^MENT T h e a u t h o r w i s h e s t o t h a n k D r . J . D . E . P r i c e o f t h e V a n c o u v e r G e n e r a l H o s p i t a l f o r h i s s u g g e s t i o n o f t h e p r o j e c t a n d f o r h i s h e l p i n t h e m e d i c a l a s p e c t s o f t h e t h e s i s . S p e c i a l t h a n k s a r e d u e t o D r . G . V . P a r k i n s o n f o r h i s a i d i n t h e t h e o r e t i c a l a n a l y s i s a n d D r . C . A . B r o c k l e y f o r h i s a i d i n t h e e x p e r i m e n t a l work and f o r t h e i r g u i d a n c e a n d e n c o u r a g e m e n t t h r o u g h o u t t h e r e -s e a r c h p r o g r a m . T h a n k s m u s t b e e x p r e s s e d t o numerous o t h e r members o f t h e f a c u l t y and g r a d u a t e s t u d e n t s o f M e c h a n i c a l E n g i n e e r i n g f o r t h e i r s u g g e s t i o n s , e s p e c i a l l y D r . E . G . Hauptmann whose a d v i c e c o n c e r n i n g t h e d e s i g n o f t h e m a n i f o l d p r o v e d v e r y h e l p f u l . T h a n k s a r e d u e t h e t e c h n i c i a n s i n t h e d e p a r t m e n t who c o n s t r u c t e d m o s t o f t h e e x p e r i m e n t a l a p p a r a t u s . A l s o g r a t e f u l l y a c k n o w l e d g e d i s t h e work o f t h e G . F . S t r o n g l a b o r a t o r i e s i n c a r r y i n g o u t t h e a n a l y t i c a l d e t e r m i n a t i o n s . T h e e x p e r i m e n t a l work was c a r r i e d o u t i n t h e T r i b o l o g y L a b o r a t o r y o f t h e 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 , 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 . F i n a n c i a l a s s i s t a n c e was r e c e i v e d f r o m t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a u n d e r G r a n t Numbers A - 5 8 6 a n d ' A - 1 0 6 5 a n d f r o m t h e M e d i c a l R e s e a r c h C o u n c i l o f C a n a d a u n d e r G r a n t Numbers MA-2322 a n d M A - 2 7 9 9 . C H A P T E R I I N T R O D U C T I O N 1.1 I n t r o d u c t i o n The a p p l i c a t i o n o f e n g i n e e r i n g t e c h n i q u e s t o p r o b l e m s i n t h e m e d i c a l and b i o l o g i c a l f i e l d s i s a f a i r l y new c o n c e p t b u t i t i s one w h i c h i s g a i n i n g r a p i d a c c e p t a n c e as more o f t h e p r o b l e m s r e q u i r e t e c h n i c a l s o l u t i o n s b e y o n d t h o s e a v a i l -a b l e w i t h i n t h e f i e l d . T h i s i n t e r d i s c i p l i n a r y r e q u i r e m e n t has l e d t o t h e c r e a t i o n o f b i o e n g i n e e r i n g and w i t h i n t h i s c o n t e x t t h e a r t i f i c i a l k i d n e y has c a p t u r e d t h e i n t e r e s t o f many e n g i n e e r s . A l t h o u g h a l a r g e amount o f work has a l r e a d y been done by o t h e r s t h e r e i s s t i l l room f o r a s i g n i f i c a n t c o n t r i b u t i o n i n t h e d e s i g n o f t h e a r t i f i c i a l k i d n e y . The p u r p o s e o f t h i s t h e s i s i s t o a t t e m p t t o c l a r i f y some o f t h e r e m a i n i n g p r o b l e m s i n k i d n e y d e s i g n . The a r t i f i c i a l k i d n e y i s a m a c h i n e w h i c h removes w a s t e p r o d u c t s f r o m t h e b l o o d o f a p a t i e n t who i s s u f f e r i n g f r o m c h r o n i c u r e m i a . The p r o c e s s i s known as h e m o d i a l y s i s . E s s e n t i a l l y , b l o o d i s removed f r o m an a r t e r y i n t h e p a t i e n t and i s d u c t e d t o t h e a r t i f i c i a l k i d n e y where i t i s d i s t r i b u t e d t o one s i d e o f a s e m i p e r m e a b l e membrane. On t h e o p p o s i t e s i d e o f t h e membrane i s a f l u i d c a l l e d d i a l y s a t e . The i m p u r i t i e s d i f f u s e f r o m t h e b l o o d , t h r o u g h t h e membrane, and i n t o t h e d i a l y s a t e due t o t h e c o n c e n t r a t i o n g r a d i e n t t h a t e x i s t s . The r a t e o f d i f f u s i o n depends on t h e c o n c e n -2 t r a t i o n g r a d i e n t and the r e s i s t a n c e t o d i f f u s i o n caused by the blood, membrane, and d i a l y s a t e . O b viously the molecular s i z e of the i m p u r i t y w i l l a f f e c t the d i f f u s i o n r a t e . In f a c t , the membrane w i l l l i m i t t r a n s f e r of macro-molecules n e a r l y completely. By c o n t r o l l i n g the c o n c e n t r a t i o n of the d i a l y s a t e , e q u i l i b r i u m can be maintained with the concen-t r a t i o n of v a r i o u s components i n the blood and thus prevent the removal of important ions or molecules. The p u r i f i e d b lood i s c o l l e c t e d and r e t u r n e d t o a v e i n i n the p a t i e n t . Since the a r t i f i c i a l kidney must d u p l i c a t e the f u n c t i o n s of the human kidney i t i s necessary t o s p e c i f y the nature of those f u n c t i o n s . The kidney has the task of m a i n t a i n i n g the i n t e r n a l environment of the body; b a s i c a l l y , s i x main f u n c t i o n s . F i r s t , the volume of f l u i d i n the body must remain constant. By c o n t r o l l i n g the volume of u r i n e (and thus i t s c o n c e n t r a t i o n ) a water balance can be maintained. Second, the kidney r e g u l a t e s the c o n c e n t r a t i o n of ions such as potassium, sodium, calcium, c h l o r i d e , phosphate, b i c a r b o n -ate, and s u l f a t e so as to maintain an e l e c t r o l y t e balance. T h i r d , as most end products of metabolism are a c i d i c i n nature, the kidney e x c r e t e s a c i d r a d i c a l s i n a d d i t i o n to ammonia which, i n the form of ammonium i o n s , removes excess hydrogen i o n s . T h i s maintains an acid-base balance. F o u r t h , the nitrogenous end products of metabolism such as urea and c r e a t i n i n e are excreted while other products of metabo-l i s m such as glucose and amino a c i d s are s e l e c t i v e l y reabsorbed 3 i n t h e k i d n e y . F i f t h , c e r t a i n t o x i c o r g a n i c compounds s u c h as b e n z o i c a c i d a r e d e t o x i f i e d i n t h e k i d n e y . And s i x t h , t h e k i d n e y s y n t h e s i z e s some hormones and e n z y m e s . I n o r d e r t o a c c o m p l i s h t h e s e p h y s i o l o g i c a l f u n c t i o n s t h e anatomy o f t h e k i d n e y i s v e r y s p e c i a l i z e d . A s i m p l i f i e d d e s c r i p t i o n o f t h e s t r u c t u r e w i l l s u f f i c e . B l o o d e n t e r s a c a p s u l e , c a l l e d t h e g l o m e r u l u s , i n w h i c h a r e a l a r g e number o f s h o r t , p a r a l l e l - c o n n e c t e d c a p i l l a r i e s . A l l s m a l l m o l e -c u l e s a n d a l a r g e q u a n t i t y o f f l u i d a r e f i l t e r e d i n t o t h e c a p s u l e by b l o o d p r e s s u r e and t h e n t h r o u g h a t u b u l e w h i c h l o o p s b a c k on i t s e l f b e f o r e e x h a u s t i n g i n t o c o l l e c t i n g d u c t s . The b l o o d e x i t s t h e g l o m e r u l u s and t r a v e l s i n a c o n v o l u t e d p a t h a r o u n d t h e , l o o p e d t u b u l e b e f o r e b e i n g r e t u r n e d t o a v e i n . By a p r o c e s s o f b o t h a c t i v e and p a s s i v e r e a b s o r p -t i o n a l l m o l e c u l e s r e q u i r e d by t h e b o d y t o m a i n t a i n t h e w a t e r , i o n i c , a c i d i c o r e n e r g y b a l a n c e a r e r e m o v e d f r o m t h e t u b u l e and r e p l a c e d i n t h e b l o o d . The r e m a i n i n g w a s t e p r o d u c t s c o l l e c t i n t h e b l a d d e r . I t h a s b e e n e s t i m a t e d t h a t t h e c a p i l l a r i e s i n a l l t h e g l o m e r u l i o f b o t h k i d n e y s h a v e an e f f e c t i v e f i l t e r i n g s u r f a c e o f a b o u t 2 sq.m. and number a b o u t 30 m i l l i o n . I n a d d i t i o n , t h e k i d n e y c o n t i n u o u s l y c i r c u l a t e s b l o o d a t a r a t e o v e r 1 l i t e r p e r m i n u t e and i n t h e p e r i o d o f 24 h o u r s ha s removed and r e a b s o r b e d a b o u t 180 l i t e r s o f w a t e r and p r o p o r t i o n a l l y l a r g e amounts o f o t h e r s u b s t a n c e s . A l t h o u g h t h e m e c h a n i s m o f t h e k i d n e y i s c o n s i d e r a b l y more c o m p l i c a t e d t h a n t h i s d e s c r i p t i o n t h e b a s i c f e a t u r e s o f t h e 4 method o f o p e r a t i o n have been o u t l i n e d . S m i t h [1] d i s c u s s e s i n d e t a i l t h e p r i n c i p l e s o f r e n a l p h y s i o l o g y . I f , due t o d i s e a s e o r a c c i d e n t , t h e f u n c t i o n s o f t h e k i d n e y a r e i m p a i r e d o r s t o p p e d t h e p a t i e n t s u f f e r s f r o m u r e m i a . I n t h e c a s e o f c h r o n i c u r e m i a t h e k i d n e y s have p e r m a n e n t l y c e a s e d t o f u n c t i o n and t h e p a t i e n t w i l l d i e i n a s h o r t t i m e b e c a u s e o f t h e b u i l d - u p i n t h e b l o o d o f t o x i c m e t a b o l i c w a s t e p r o d u c t s . The a r t i f i c i a l k i d n e y , i f p r o p e r l y d e s i g n e d and o p e r a t e d , c a n t a k e t h e p l a c e o f t h e r e a l k i d n e y and m a i n t a i n t h e l i f e o f t h e p a t i e n t . A summary o f t h e m a j o r s u b s t a n c e s t h a t a r e removed f r o m a p a t i e n t u n d e r g o i n g h e m o d i a l y s i s i s g i v e n i n T a b l e 1. S u b s t a n c e D a i l y O u t p u t (gm.) U r e a 12 C r e a t i n i n e 2 U r i c a c i d 0.4 Sodium 0.5 P o t a s s i u m 0.5 P h o s p h a t e 1 . 8 H y d r o g e n I o n 0 .07 Water 300-1000 T a b l e I O u t p u t o f Waste P r o d u c t s f o r C h r o n i c U r e m i c P a t i e n t on H e m o d i a l y s i s ( R e s t r i c t e d D i e t ) The a r t i f i c i a l k i d n e y has become an a c c e p t e d s o c i a l phenomenon w i t h t e c h n i c a l , e c o n o m i c , c u l t u r a l , p o l i t i c a l and p h i l o s o p h i c o v e r t o n e s . B e c a u s e o f t h i s , a c r i t i c a l e v a l u a t i o n o f t h e p r o b l e m i s n e c e s s a r y . R e c e n t s t u d i e s by 5 the U.S. Department of Hea l t h , E d u c a t i o n , and Welfare i n d i c a t e t h a t from 28,000 t o 50,000 people d i e each year i n the U.S. from kidney d i s e a s e and about 10,000 are s u i t e d to long term support e i t h e r by hem o d i a l y s i s or kidney t r a n s p l a n t . The success of d i a l y s i s i n m a i n t a i n i n g the p a t i e n t ' s l i f e i s f a i r l y good wi t h an estimated death r a t e of 10 percent per year of the pre v i o u s years d i a l y s i s p a t i e n t s . However, the co s t of m a i n t a i n i n g t h i s many p a t i e n t s on hemo d i a l y s i s i s very l a r g e and perhaps o n l y 10 percent of the number of people r e q u i r i n g d i a l y s i s a c t u a l l y r e c e i v e the treatment i n the U n i t e d S t a t e s . The f a c t o r s which i n f l u e n c e the c o s t i n c l u d e ; a m o r t i z a t i o n expense of the equipment, c o s t of d i s p o s a b l e s u p p l i e s and equipment maintenance, c o s t of l a b o r , c o s t of l a b o r a t o r y d e t e r m i n a t i o n s , and charges f o r h o s p i t a l expenses. The o v e r a l l c o s t i s extremely v a r i a b l e and depends upon many other f a c t o r s but ranges between $2,000 and $18,000 per p a t i e n t - y e a r . Home d i a l y s i s can lower the co s t c o n s i d e r a b l y . Thus the d e s i g n of an a r t i f i c i a l kidney must be concerned not only with t e c h n i c a l s o p h i s t i c a t i o n but a l s o with the social- e c o n o m i c c o n d i t i o n s i n h e r e n t i n the problem. There are a number of f a c t o r s , p r i m a r i l y p h y s i o -l o g i c a l , which provide c o n s t r a i n t s on the de s i g n of the a r t i f i c i a l kidney. Although, at bes t , the a r t i f i c i a l kidney can only approximate the a b i l i t y of the human kidney to remove water and a v a r i e t y of met a b o l i c by-products, to 6 r e g u l a t e s e v e r a l i o n s , and t o a d j u s t pH, i t must a l s o be compatible w i t h the body. That i s to say, the a r t i f i c i a l kidney must not impose on undue s t r e s s on the p a t i e n t . Of primary importance i s the treatment of blood while i t i s c i r c u l a t i n g through the a r t i f i c i a l kidney. Only a l i m i t e d volume of bloo d can be removed from the p a t i e n t f o r use i n the d i a l y z e r and a l l of i t must be r e p l a c e d . At the same time, donor blood f o r the p r e p a r a t i o n of the a r t i f i c i a l kidney i s u n d e s i r a b l e . At no time can the blood be allowed to stagnate nor can there be areas o f turbul e n c e or h i g h shear r a t e s . The red c e l l s are e a s i l y destroyed by high shear r a t e s and stagnant areas of blood l e a d to c l o t f ormations. A l l s u r f a c e s i n c o n t a c t w i t h blood must be non-t o x i c and, i f p o s s i b l e , b i o c h e m i c a l l y compatible to prevent f i b r i n b u i l d - u p on the s u r f a c e s . As pumps g e n e r a l l y w i l l hemolize blood, a r t e r i a l p r e s s u r e alone must p r o v i d e the f o r c e t o move blood through the d i a l y z e r . The a r t i f i c i a l kidney must be e f f i c i e n t i n removing the waste m a t e r i a l s . Only a l i m i t e d blood flow r a t e i s a v a i l a b l e from the p a t i e n t so n e a r l y a l l the i m p u r i t i e s must be removed i n one pass through the d i a l y z e r to prevent e x c e s s i v e l y long d i a l y z i n g times. At the same time a l a r g e volume of water must be removed by u l t r a - f i l t r a t i o n from the blood (induced by transmembrane pressure g r a d i e n t ) . The e n t i r e a r t i f i c i a l kidney must be capable of being e a s i l y 7 s t e r i l i z e d . A d d i t i o n a l l y , the d i a l y z e r should be c o n s t r u c t e d of low c o s t , y e t hig h r e a l i a b i l i t y p a r t s which w i l l be sa f e and simple i n o p e r a t i o n . I t would be p r e f e r a b l e to have a system of in e x p e n s i v e , d i s p o s a b l e d i a l y s i s c a r t r i d g e s which would minimize to a l a r g e degree the l a b o r i n v o l v e d i n r e b u i l d i n g an a r t i f i c i a l kidney a f t e r every d i a l y s i s . The s a f e t y of the system i s a prime c o n s i d e r a t i o n as the h e a l t h of the p a t i e n t cannot be j e o p a r d i z e d . W i t h i n t h i s context t h e r e are many a l t e r n a t i v e approaches to the de s i g n and o p t i m i z a t i o n of an a r t i f i c i a l kidney. The problem may be broken down i n t o three separate areas; mass t r a n s p o r t of i m p u r i t i e s from blood t o d i a l y s a t e , f l u i d d i s t r i b u t i o n , and system o p t i m i z a t i o n . E x t e n s i v e work has been done p r e v i o u s l y on two c o n f i g u r a t i o n s of hemodialyzers, the f l a t p l a t e and the c o i l e d tube. Some a n a l y s i s e x i s t s f o r a t h i r d type of d i a l y z e r - - t h e c a p i l l a r y d i a l y z e r . The f l a t p l a t e d i a l y z e r c o n s i s t s of two membranes s t r e t c h e d across s u p p o r t i n g backs and s e a l e d at the edges to form a sandwich s t r u c t u r e w i t h a narrow space between the two faces of the membranes. A f l a t sheet of blood moves through the narrow channel between the membranes while d i a l y s a t e i s c i r c u l a t e d through channeled r e g i o n s i n the s u p p o r t i n g backs t o scour the other s i d e s of the membranes. The c o i l e d tube d i a l y z e r u t i l i z e s a long 8 c e l l o p h a n e tube of f a i r l y l a r g e diameter f l a t t e n e d and c o i l e d about a porous su p p o r t i n g s t r u c t u r e which i s immersed i n d i a l y s a t e . Blood passes through the tube w h i l e d i a l y s a t e i s c i r c u l a t e d on the o u t s i d e . The c a p i l l a r y d i a l y z e r c o n s i s t s of a l a r g e number of s m a l l diameter tubes connected i n p a r a l l e l w i t h blood f l o w i n g through the tubes and d i a l y s a t e c i r c u l a t i n g on the outside'. As the c a p i l l a r y d i a l y z e r most c l o s e l y approximates the operation, of the human kidney t h i s c o n f i g u r a t i o n was chosen f o r f u r t h e r study. A comparison of d i a l y z e r systems showed t h a t c e r t a i n problems i n h e r e n t i n the f l a t p l a t e and c o i l e d tube d i a l y z e r s c o u l d be minimized or e l i m i n a t e d by u s i n g the c a p i l l a r y c o n f i g u r a t i o n . The problem then r e s o l v e d i t s e l f f i r s t i n t o an examination of mass t r a n s f e r f o r Newtonian and non-Newtonian f l u i d s i n c a p i l l a r i e s where the r e s i s t a n c e of the c a p i l l a r y to molecular t r a n s p o r t i s of the same order of magnitude as the d i f f u s i o n r e s i s t a n c e i n b l o o d , but where the d i a l y s a t e f i l m r e s i s t a n c e can be n e g l e c t e d . The a n a l y s i s was c a r r i e d out f o r s i n g l e component d i f f u s i o n although i n f a c t many components would be d i f f u s i n g s i m u l t a n e o u s l y . As the c o n c e n t r a t i o n • o f these components i s r e l a t i v e l y low the i n t e r a c t i o n of d i f f u s i o n c o e f f i c i e n t s w i l l probably be s m a l l . U l t r a f i l t r a t i o n was n e g l e c t e d even though the removal of water through the c a p i l l a r y w a l l s w i l l a f f e c t the c o n c e n t r a t i o n g r a d i e n t s w i t h i n the tube. The mass t r a n s f e r a n a l y s i s was then used to o p t i m i z e the c o n f i g u r -9 a t i o n o f t h e s y s t e m . O p t i m i z a t i o n c a n be a c h i e v e d f o r one component o n l y , h o w e v e r . T r a d i t i o n a l l y t h i s c o m p o nent h a s b e e n u r e a b e c a u s e o f t h e l a r g e c o n c e n t r a t i o n o f u r e a t h a t e x i s t s i n a u r e m i c p a t i e n t . H a v i n g e s t a b l i s h e d a s y s t e m c o n f i g u r a t i o n , t h e method o f s u p p l y i n g b l o o d and d i a l y s a t e was a n a l y z e d w i t h a v i e w t o m e e t i n g t h e r e s t r i c t i o n s p l a c e d on b l o o d t r a n s p o r t . 1.2 B a c k g r o u n d As e a r l y as 1912 A b e l , R o w n t r e e and T u r n e r [2] c o n s t r u c t e d an , i n s t r u m e n t d e s i g n e d t o i s o l a t e compounds p r e s e n t i n b l o o d . I t u s e d n i t r o c e l l u l o s e t u b e s b a t h e d i n s a l i n e a nd c o u l d be c o n s t r u e d as t h e f o r e r u n n e r t o t h e a r t i f i c i a l k i d n e y . However, t h e l a c k o f a d e p e n d a b l e , r e p r o -d u c i b l e membrane t o g e t h e r w i t h t h e l a c k o f a s a f e a n t i c o a g u -l a n t f o r u s e . i n p a t i e n t s p r e v e n t e d any s i g n i f i c a n t a d v a n c e f o r t h e n e x t 25 y e a r s . I n 1937 T h a l h i m e r [3] u s e d a c e l l o p h a n e membrane and h e p a r i n as an a n t i c o a g u l a n t f o r t h e b l o o d d u r i n g d i a l y s i s ; b o t h m a t e r i a l s r e m a i n i n u s e t o d a y . The f i r s t c l i n i c a l l y s u c c e s s f u l h e m o d i a l y z e r was d e v e l o p e d by K o l f f and B e r k [4] i n 1 9 4 3 . T h i s a r t i f i c i a l k i d n e y c o n s i s t e d o f a l o n g , c o i l e d c e l l o p h a n e t u b e t h r o u g h w h i c h b l o o d f l o w e d by g r a v i t y . The t u b e was wound on a drum w h i c h r e v o l v e d i n a d i a l y s a t e b a t h . S k e g g s and L e o n a r d s i n 1948 i n t r o d u c e d t h e f i r s t f l a t p l a t e d i a l y z e r [5] i n w h i c h the blood and d i a l y s a t e flowed i n t h i n l a y e r s on e i t h e r s i d e of a f l a t sheet of c e l l o p h a n e . The K o l f f Twin C o i l , the M i n i -c o i l , and C h r o n i c C o i l were new v e r s i o n s of d i a l y z e r s which appeared i n 1955, 1960 and 1963 r e s p e c t i v e l y , u s i n g f l a t t e n e d c ellophane t u b i n g immersed i n a l a r g e , r e c i r c u l a t i n g d i a l y s a t e bath. These u n i t s r e q u i r e d a l a r g e priming volume of blood, a blood pump to overcome the high flow r e s i s t a n c e , and a w a s t e f u l l y l a r g e amount of d i a l y s a t e . Although they were r e l i a b l e , the e f f i c i e n c y of d i a l y s i s was low f o r the c o i l e d tube d i a l y z e r s except the l a t e s t v e r s i o n s of the K o l f f Twin C o i l . By 1960, K i i l [6] d e s c r i b e d a new f l a t p l a t e d i a l y z e r which managed to maintain a c o n s i s t e n t l y t h i n b l o o d f i l m t o improve the d i a l y s i s e f f i c i e n c y . Two membranes were separated by spacers and supported by grooved, f l a t backing p l a t e s . Blood flowed between the membranes and d i a l y s a t e flowed between the membrane and the backing p l a t e s . Problems wi t h c h a n n e l l i n g of the t h i n blood f i l m and low e f f i c i e n c y r e s u l t -i n g from the membrane area l o s t due to the s u p p o r t i n g s t r u c t u r e of the backing p l a t e s l e d to many m o d i f i c a t i o n s to the K i i l d i a l y z e r . S u c c e s s i v e l a y e r s of membranes and backing p l a t e s stacked and connected i n p a r a l l e l are f e a t u r e s of the Klung [7] and Dialung [8] hemodialyzers i n t r o d u c e d i n 1962 and 1966 r e s p e c t i v e l y i n an e f f o r t to i n c r e a s e the membrane area without i n c r e a s i n g the blood flow r e s i s t a n c e so as to a v o i d the use of a blood pump. T h i s method u n f o r t u n a t e l y i n c r e a s e s 11 the volume of blood i n the d i a l y z e r . Of the aforementioned a r t i f i c i a l kidneys the K o l f f Twin C o i l , the Skeggs-Leonards C o i l , and the K i i l d i a l y z e r s have proven the most s u c c e s s f u l and are i n use today i n most r e n a l c e n t r e s which c a r r y out hem o d i a l y s i s . In g e n e r a l the c o i l d i a l y z e r s proved most convenient as they are manufactured i n pre-packaged, s t e r i l e , d i s p o s a b l e c o n t a i n e r s . However they r e q u i r e a l a r g e priming volume of blood and a bloo d pump. The f l a t p l a t e d i a l y z e r s must be r e b u i l t a f t e r every d i a l y s i s but do not r e q u i r e a priming volume of blood or blood pump. A l s o , the volume of d i a l y s a t e r e q u i r e d i s l e s s f o r the c o i l d i a l y z e r s but m a i n t a i n i n g s t e r i l e c o n d i t i o n s i n r e - c i r c u l a t e d d i a l y s a t e i s troublesome and most users p r e f e r the once-through o p e r a t i o n a v a i l a b l e with the f l a t - p l a t e d i a l y z e r s . In a d d i t i o n , i t has been found t h a t a l l of the above mentioned d i a l y z e r s c l o t blood t o some extent over a d i a l y s i s procedure. The e f f i c i e n c y of d i a l y s i s has been q u i t e low f o r a l l the d i a l y z e r s . To a l l e v i a t e some of the short-comings of the t r a d i t i o n a l hemodialyzers many m o d i f i c a t i o n s , based both on theory and p r a c t i c e , were t r i e d by workers i n the f i e l d and some new designs r e s u l t e d . Leonard and Bluemle [9] showed how the o v e r a l l mass t r a n s f e r c o e f f i c i e n t c ould be r e l a t e d to membrane area and blood flow r a t e and t h a t the o v e r a l l mass t r a n s f e r r e s i s t a n c e c o u l d be broken down i n t o blood, membrane, and d i a l y s a t e components. However, the i n t e r -r e l a t i o n s h i p between the v a r i o u s r e s i s t a n c e s was not analyzed. Babb and Grimsrud [10] presented a complete mass t r a n s f e r a n a l y s i s f o r p a r a l l e l p l a t e d i a l y z e r s w i t h r i g i d membrane supports i n 1964. T h e i r a n a l y s i s r e l a t e d the v a r i o u s f i l m r e s i s t a n c e s and presented an optimized d e s i g n which p r e d i c t e d t h a t maximum e f f i c i e n c y would be obtained from a d i a l y z e r having a very s h o r t but extremely wide blo o d channel. A f e a t u r e of the Babb-Grimsrud d i a l y z e r was the porous n i c k e l 'Foametal' membrane support which e l i m i n a t e d t o a l a r g e degree the membrane area l o s t i n other support backing p l a t e concepts. Wolf and Zaltzman [11] generated a s e r i e s of design c h a r t s f o r optimum d i a l y z e r geometries f o r both c i r c u l a r and r e c t a n g u l a r blood passages based on the p r i n c i p l e s s t a t e d by Babb and Grimsrud y e t n e g l e c t i n g the inter-dependence between the blo o d , membrane and d i a l y s a t e r e s i s t a n c e s . Stewart e t a l . [12] d e s c r i b e d an a r t i f i c i a l kidney c o n s t r u c t e d with c a p i l l a r y tubes made from d e - a c e t y l i z e d c e l l u l o s e a c e t a t e f i b r e s . The smal l bundles of c a p i l l a r i e s proved d i f f i c u l t to assemble and h e p a r i n i z e d b l o o d had a tendency to c l o t because of improper blood flow c h a r a c t e r i s t i c s but the e f f i c i e n c y and s i z e of the u n i t showed promise. F u r t h e r work produced a commercial model manufactured by Dow Chemical Company. A new type of a r t i f i c i a l kidney based upon u l t r a - f i l t r a t i o n as i n the human kidney, i n s t e a d of d i f f u s i o n as i n other d i a l y z e r s , has been designed about a membrane r e c e n t l y developed and d e s c r i b e d by B i x l e r e t a l . [13]. Recent work i n a r t i f i c i a l kidney d e s i g n has centered around a process known as r e v e r s e osmosis i n which transmembrane pressures s u f f i c i e n t l y l a r g e t o overcome osmotic p r e s s u r e are employed. New membranes which have good e l e c t r o l y t e r e j e c t i o n c h a r a c t e r i s t i c s have made t h i s p o s s i b l e . Brown and Kramer [14] suggested a d e s i g n , based on r e v e r s e osmosis, f o r a c o n t i n u o u s l y o p e r a t i n g , wearable a r t i f i c i a l kidney but as y e t no membranes are a v a i l a b l e which o f f e r s e l e c t i v e r e j e c t i o n of compounds and a non-clogging membrane s u r f a c e . While there have been many mathematical models presented t o p r e d i c t the o v e r a l l performance of hemodialyzers very seldom has the model been complete and o f t e n the a n a l y s i s was i n c o r r e c t . C o l t o n [15] presented a complete h i s t o r y and b i b l i o g r a p h y of the development of hemodialyzers up to 1967 and a l s o gave an a n a l y s i s of the performance of common e x i s t i n g d i a l y z e r s . Although the r a p i d removal of waste products from blood by d i a l y s i s i s d e s i r a b l e , the r a t e at which these products can be removed i s governed by p h y s i o l o g i c a l con-d i t i o n s . ' D i s e q u i l i b r i u m Syndrome 1 r e s u l t s when l a r g e i n t r a - c e l l u l a r - t o e x t r a - c e l l u l a r c o n c e n t r a t i o n g r a d i e n t s , due p r i m a r i l y to urea, are s e t up by r a p i d removal of waste products from the r e a d i l y d i a l y z a b l e body f l u i d s . King e t a l . [16] optimized the r a t e at which a p a t i e n t can be d i a l y z e d without i n t r o d u c i n g trauma, thus s e t t i n g a maximum 14 l i m i t on i m p u r i t y c l e a r a n c e . The f a c t t h a t b l o o d i s a m i x t u r e , a p p r o x i m a t e l y 40 p e r c e n t by v o l u m e , o f s o l i d p a r t i c l e s n e a r l y n e u t r a l l y b u o y a n t i n a l i q u i d c o m p l i c a t e s a n a l y s i s f o r a r t i f i c i a l k i d n e y s b e c a u s e o f t h e non-Newtoni'an p r o p e r t i e s t h a t r e s u l t f r o m s u c h a m i x t u r e . Of t h e r e d and w h i t e c e l l s and p l a t e l e t s w h i c h made up t h e s o l i d p a r t i c l e s , 99 p e r c e n t a r e r e d c e l l s . The l i q u i d , c a l l e d p l a s m a , i s a b o u t 7 p e r c e n t by w e i g h t p r o t e i n m o l e c u l e s d i s s o l v e d i n s o m e t h i n g l i k e ' s a l t w a t e r . ' M e r r i l l [20] s t a t e d t h a t t h e p r o t e i n o f m a j o r i m p o r t a n c e i s f i b r i n o g e n w h i c h r e a c t s w i t h t h e r e d c e l l s t o a f f e c t t h e v i s c o s i t y o f t h e m i x t u r e . I n l a m i n a r t u b e f l o w b l o o d a c t s a s a n o n - N e w t o n i a n , non-homogeneous f l u i d . The n o n - h o m o g e n e i t y r e s u l t s b e c a u s e o f t h e t e n d e n c y o f t h e c e l l s t o m i g r a t e t o r e g i o n s o f l e a s t s h e a r s t r e s s t h u s c a u s i n g a c o n c e n t r a t i o n g r a d i e n t o f c e l l s a c r o s s t h e t u b e . T h i s e f f e c t was n o t i c e d by F a h r a e u s and L i n d q u i s t [17] i n 1 9 3 1 . T hey a l s o n o t i c e d t h a t t h e a p p a r e n t v i s c o s i t y o f b l o o d d e c r e a s e d as t h e d i a m e t e r o f t h e c a p i l l a r y t h r o u g h w h i c h t h e bloo'd f l o w e d d e c r e a s e d . S i n c e t h a t t i m e c o n s i d e r a b l e e f f o r t h a s b e e n d i r e c t e d t o f i n d i n g a m o d e l f o r t h e v i s c o s i t y o f b l o o d . I t h a s b e e n f o u n d t h a t p l a s m a i s n e a r l y a N e w t o n i a n f l u i d and t h a t t h e f i b r i n o g e n - r e d c e l l i n t e r a c t i o n c a u s e s t h e n o n - N e w t o n i a n p r o p e r t i e s o f w h o l e b l o o d . The v i s c o s i t y o f b l o o d d e p e n d s on t h e h e m a t o c r i t , 15 plasma v i s c o s i t y , a p p l i e d shear s t r e s s , shear r a t e , f i b r i n o g e n content i n the plasma, and tube diameter f o r constant temperature c o n d i t i o n s . Over a wide range of shear r a t e s C a s s o n 1 s equation [20] has been found to approximate the r h e o l o g i c a l p r o p e r t i e s of blood. Charm e t a l . [18] presented a g e n e r a l i z e d equation accounting f o r the v a r i a b l e s which a f f e c t blood v i s c o s i t y based on Casson's equation. Approximate val u e s f o r the y i e l d s t r e s s i n Casson's equation have been gi v e n by Charm [19] and M e r r i l l [20] and o t h e r s . Hemolysis of the red c e l l s by flow c o n d i t i o n s such as s t a g n a t i o n and high shear areas i s important but a l s o to be c o n s i d e r e d i s the w a l l i n t e r a c t i o n w i t h the c e l l s . B l a c k -shear et a l . [21] concluded t h a t the hemolysis index of blood i n laminar flow (mg of Hemoglobin r e l e a s e d per 100 cc of blood handled) was n e a r l y v e l o c i t y independent but v a r i e d d i r e c t l y as the l e n g t h and i n v e r s e l y as the cube of the tube diameter. They a l s o d i s c o v e r e d t h a t by c o a t i n g the tube w a l l s with s i l i c o n e the hemolysis index decreased g r e a t l y . The flow s u r f a c e s must be chosen so t h a t the c l o t t i n g mechanism i n blood w i l l not be t r i g g e r e d . T h i s has been accomplished i n . t h e past by c o a t i n g the s u r f a c e s with h e p a r i n , an a n t i c o a g u l a n t , but the same e f f e c t can be achieved by a j u d i c i o u s c h o ice of m a t e r i a l s , e i t h e r s i l i c o n e or t e f l o n , both of which are i n e r t t o blood. 16 C o n s i d e r i n g the importance of h a n d l i n g blood i n a manner so as not to traumatize i t , very l i t t l e work has been c a r r i e d out on e f f i c i e n t m a n i f o l d d e s i g n . Of p r a c t i c a l importance i s a b l o o d d i s t r i b u t i o n system which p r o v i d e s even flow throughout the d i a l y z e r y e t does not have s t a g n a t i o n p o i n t s or areas of t u r b u l e n c e . Schneck and G u t s t e i n [22] examined blood flow i n branched c y l i n d r i c a l tubes by r a d i o -opaque dye f i l a m e n t i n j e c t i o n to determine the p o i n t s of boundary l a y e r s e p a r a t i o n . They concluded t h a t flow separated from the w a l l s at branch angles g r e a t e r than 60° even f o r extremely low flow r a t e s and y e t most commercial d i a l y z e r s have manifolds which t u r n blood through 90° bends. Ob v i o u s l y s t a g n a t i o n and a c l o t t i n g s i t e would r e s u l t . Hence a major c o n s i d e r a t i o n i n hemodialyzer d e s i g n w i l l be e f f i c i e n t blood d i s t r i b u t i o n . In the non-medical f i e l d c o n s i d e r a b l e work has been done i n m a n i f o l d design f o r heat exchanger heads and pipe flow f o r uniform d i s t r i b u t i o n of f l u i d s , u n f o r t u n a t e l y , p r i m a r i l y i n the t u r b u l e n t flow regime. The concept of o b l i q u e flow manifolds where pressure l o s s e s are balanced a g a i n s t momentum l o s s e s has been s t a t e d by L o e f f l e r and P e r l m u f f e r [23]. T h e i r a n a l y s i s i s amenable to laminar flow c o n d i t i o n s and the method allows a design i n which the m a n i f o l d has a s m a l l i n t e r n a l volume. C H A P T E R 2 T H E O R Y 17 2.1 I n t r o d u c t i o n Hemodialysis i s e s s e n t i a l l y a mass t r a n s f e r o p e r a t i o n i n v o l v i n g the removal of a number of unwanted s o l u t e s , such as urea, c r e a t i n i n e and u r i c a c i d , and m a i n t a i n i n g a balance of important body e l e c t r o l y t e s w h i l e s i m u l t a n e o u s l y a l l o w i n g r a p i d removal of excess water by u l t r a f i l t r a t i o n of the blood. T h i s complex process takes p l a c e through a porous membrane which has blood on one s i d e and d i a l y s a t e on the oth e r . The membrane may take v a r i o u s c o n f i g u r a t i o n s but g e n e r a l l y blood i s completely surrounded by membrane i n some d e f i n i t e g e o m e t r i c a l manner while d i a l y s a t e i s allowed t o f r e e l y c i r c u l a t e about the membrane. The c a p i l l a r y hemodialyzer i s one c o n f i g u r a t i o n which allows maximum c o n t r o l over the blood flow p a t t e r n s . The t r a n s f e r of i m p u r i t i e s i s c o n c e n t r a t i o n - g r a d i e n t c o n t r o l l e d by d i f f u s i o n through the blood f i l m to the c a p i l l a r y w a l l , through the c a p i l l a r y w a l l , and i n t o the d i a l y s a t e . Thus there are three separate r e s i s t a n c e s to mass t r a n s f e r which must be r e s o l v e d . For the sake of s i m p l i c i t y of a n a l y s i s multi-component d i f f u s i o n w i l l be n e g l e c t e d and i n s t e a d the a n a l y s i s w i l l d e a l only with the main component to be e l i m i n a t e d ; t h a t i s , urea. I t i s f e l t t h a t the e r r o r i n h e r e n t i n t h i s assumption i s small because of the smal l c o n c e n t r a t i o n s of i m p u r i t i e s t h a t are normally present i n uremic p a t i e n t s . 1 8 The a n a l y s i s f o r mass t r a n s f e r w i l l be c a r r i e d o u t f o r a s i n g l e t u b e f o r b o t h N e w t o n i a n and n o n - N e w t o n i a n f l u i d s ( b l o o d ) f l o w i n g t h r o u g h t h e t u b e s u b j e c t t o t h e f o l l o w i n g a s s u m p t i o n s . I t i s assumed t h a t t h e i n l e t c o n c e n t r a t i o n i s u n i f o r m , t h e f l o w has a f u l l y d e v e l o p e d v e l o c i t y p r o f i l e and i s l a m i n a r , a x i a l d i f f u s i o n may be n e g l e c t e d , and t h e p h y s i c a l p r o p e r t i e s o f t h e f l u i d r e m a i n c o n s t a n t . F o r t y p i c a l hemo-d i a l y s i s p a r a m e t e r s none o f t h e s e a s s u m p t i o n s r e p r e s e n t s a s i g n i f i c a n t e r r o r . A d d i t i o n a l l y i t i s assumed t h a t t h e i m p u r i t y c o n c e n t r a t i o n i n t h e d i a l y s a t e and t h e w a l l p e r m e a b i l i t y o f t h e c a p i l l a r i e s a r e c o n s t a n t . T u r b u l e n t f l o w c o n d i t i o n s and an e x c e s s o f d i a l y s a t e w i l l i n s u r e t h a t t h e d i a l y s a t e c o n c e n -t r a t i o n r e m a i n s c o n s t a n t w h i l e t h e t u r b u l e n c e w i l l l o w e r t h e d i a l y s a t e - s i d e f i l m r e s i s t a n c e t o an i n s i g n i f i c a n t l e v e l . T h i s c o n d i t i o n i s a p p r o a c h e d i n e x i s t i n g d i a l y z e r s so t o s i m p l i f y t h e a n a l y s i s t h e a s s u m p t i o n i s made. E x p e r i e n c e w i t h h e m o d i a l y z e r membranes shows t h a t , i n f a c t , t h e w a l l p e r m e a b i l i t y d e c r e a s e s w i t h i n c r e a s i n g d i a l y s i s t i m e due t o l e u k o c y t e and p r o t e i n b u i l d u p on t h e membrane s u r f a c e s . However, no c o r r e l a t i o n e x i s t s t h a t i s m e a n i n g f u l so i t w i l l be assumed t h a t t h e p e r m e a b i l i t y o f t h e membrane does r e m a i n c o n s t a n t . T h i s a s s u m p t i o n w i t h p r o b a b l y r e s u l t i n t h e l a r g e s t e r r o r i n t h e a n a l y s i s . C l a s s i c a l l y , s o l u t e t r a n s p o r t due t o c o n c e n t r a t i o n -g r a d i e n t f o r c e s has been d e s c r i b e d by M i c h a e l s [ 2 4 ] as f o l l o w s . The r a t e o f s o l u t e t r a n s f e r f r o m b l o o d t o d i a l y s a t e t h r o u g h a d i f f e r e n t i a l a r e a i s , i dm = K (c. - c , ) d A ( 2 . 1 ) o b d 19 Assuming t h a t K q i s constant over the e n t i r e area, the equation may be i n t e g r a t e d to g i v e , M = K A(Ac) (2.2) o 'mean v 1 where (Ac) mean i s the a p p r o p r i a t e mean c o n c e n t r a t i o n d r i v i n g f o r c e f o r the flow geometry under c o n s i d e r a t i o n and i s based on the d i f f e r e n t i a l i n l e t and o u t l e t c o n c e n t r a t i o n s . For a l l normal flow c o n f i g u r a t i o n s (Ac) mean i s the mean l o g concen-t r a t i o n g r a d i e n t [24]. E s s e n t i a l l y equation 2.2 d e s c r i b e s K , the o v e r a l l mass t r a n s f e r c o e f f i c i e n t , which i s not a o . tru e constant but i s dependent on the geometry of the system. The r e c i p r o c a l of K q i s con s i d e r e d to r e p r e s e n t the average o v e r a l l mass t r a n s f e r r e s i s t a n c e which i s made up of the sums of the i n d i v i d u a l mass t r a n s f e r r e s i s t a n c e s (blood, membrane and d i a l y s a t e ) Thus, R = R, + R + R, (2.3) D m a or, K = = = (2.4) / K , + / K + X/K, ' b ' m ' d G e n e r a l l y the mass t r a n s f e r r e s i s t a n c e s are interdependent and cannot be added but f o r d e s c r i p t i v e purposes equation 2.3 may be used. From a s o l u t e balance i t f o l l o w s t h a t , 20 M = Q b ( G b i - C b o ) = Q d ( c d o - c d i ) ( 2 ' 5 ) For p a r a l l e l flow c o n d i t i o n s a combination of equation 2.2 and 2.5 y i e l d s , u s i n g the a p p r o p r i a t e d e f i n i t i o n of (Ac) mean, c, . -c , . K A l n ( ) bo do (2.6) Q b 1 + ° b / Q d The e f f i c i e n c y of mass t r a n s f e r i s d e f i n e d as the r a t i o of m a t e r i a l removed to the maximum t h a t c o u l d be removed and i s given the symbol E. M E = - (2.7) Q. (c, . -c , . ) b b i d i The e f f i c i e n c y of mass t r a n s f e r f o r p a r a l l e l flow i s d e t e r -mined by combining equations 2.5, 2.6 and 2.7 to g i v e , 1 - exp E K A G, ° - d + A 1 + Qb/Q d -± ^ — . (2.8) In a l l hemodialyzer systems the i n l e t c o n c e n t r a t i o n of urea i n the d i a l y s a t e i s zero and the mass t r a n s f e r e f f i c i e n c y i s s u f f i c i e n t l y near u n i t y with a d i a l y s a t e flow r a t e much l a r g e r than blood flow r a t e t h a t f o r e i t h e r p a r a l l e l or mixed d i a l y s a t e c o n d i t i o n s the mass t r a n s f e r e f f i c i e n c y i s given . by, 21 M = 1 - exp(-K A o (2.9) Equation 2.9 w i l l be used f o r comparison purposes with a more exact theory to be developed. 2.2 Mass T r a n s f e r The s p e c i f i c problem of mass t r a n s f e r from c a p i l l a r i e s w ith w a l l r e s i s t a n c e i n the laminar flow regime has a p a r a l l e l i n the m o d i f i e d Graetz problem i n heat t r a n s f e r . Sideman, Luss and Peck [25] d e a l t i n d e t a i l w i t h the case of heat t r a n s f e r from a tube of c i r c u l a r c r o s s - s e c t i o n w i t h f u l l y developed laminar flow and constant w a l l r e s i s t a n c e . The theory to be developed w i l l extend t h e i r s o l u t i o n s to the mass t r a n s f e r case and w i l l i n c l u d e s o l u t i o n s f o r a non-Newtonian f l u i d which resembles blood. s u b j e c t to the assumptions s t a t e d i n the i n t r o d u c t i o n reduces t o , where the dimensionless v a r i a b l e s are d e f i n e d as f o l l o w s , The steady s t a t e d imensionless d i f f u s i o n equation i 3_c _ u ( p) 8c p 3p 2 H (2.10) 3p c = / P = D 2r K = DPe 2x u(p) = u(r) U (2.11) 22 The d i m e n s i o n l e s s b o u n d a r y c o n d i t i o n s f o r t h i s p r o b l e m a r e , c ( p ) = l , 0 « J p < l 5 = 0 c ( p ) = 0 , 0 4 p 4 1 E, -y oo 3 c ( p ) _ _ ( 2 - 1 2 ) 0 , p - 0 3 c ( p ) _ S h c ( p ) , p = 1 8 p w The l a s t b o u n d a r y c o n d i t i o n e q u a t e s t h e mass f l u x t h r o u g h t h e b o u n d a r y l a y e r a t t h e w a l l t o t h e mass f l u x t h r o u g h t h e membrane w a l l . By s e p a r a t i o n o f v a r i a b l e s t h e g e n e r a l s o l u t i o n b e c omes, c ( p ) .= E A n R n ( p ) e x p ( - X 2 ? ) ( 2 . 1 3 ) t h w h e r e A n i s t h e n e i g e n v a l u e o f t h e f o l l o w i n g e q u a t i o n a n d i t s b o u n d a r y c o n d i t i o n s . d 2 R -. dR X 2 u ( p ) + - — - + — R = 0 ( 2 . 1 4 ) d p 2 P dp 2 d R ( p ) —-2 = 0 , p = 0 dp ( 2 . 1 5 ) dR (p) — - = - S h R (p) , P = 1 , w n K / ' K dp 23 The i n l e t boundary c o n d i t i o n y i e l d s , £ A R (p) - 1 n=l n n (2.16) U t i l i z i n g the c o n d i t i o n of o r t h o g o n a l i t y , the c o e f f i c i e n t s A^ may be determined from, A = n ^ R n 2 ( p ) p H ( P ) d p d R n ( p j dp P=l n , dR (p) dR (p)dR(p)-n dA dp n K dp dA n J p=l (2.17) The f l u i d bulk c o n c e n t r a t i o n may be c a l c u l a t e d from, m 2pu(p)c(p)dp (2.18) But, c p = E A R p exp -A £) n=l (2.19) And, Q p u ( p ) R n ( p ) d p _2 d R n ( p )  X n d p (2.20) Thus the f l u i d bulk c o n c e n t r a t i o n can be expressed as, c = I E exp(-XE) m , n n n=l (2.21) where, n 4 A dR (p) n n V M / X2 dp n M P=l (2.22) The l o c a l Sherwood number, based on f l u i d bulk c o n c e n t r a t i o n , i s , _ 9c(p)_ (2.f23) Sh = 3p m Then, I x 2 E n e x P ( - X ^ ) S h r = i Z E e x p ( - ^ ) n=l n n (2.24) The o v e r a l l Sherwood number i s d e f i n e d by, Sh = ±-o £ Sh^d? (2.25) And by i n t e g r a t i o n i t can be shown t h a t , ~ l n ( c ) Sh = o m 4? The asymptotic l i m i t s of the Sherwood numbers are, S h o = !/ 4 ' Sh = Sh , o w ' t- = 0 (2.27) 25 The equations developed above are g e n e r a l and t h e i r s o l u t i o n i s c o n t i n g e n t upon s t i p u l a t i n g an a p p r o p r i a t e f l u i d v e l o c i t y p r o f i l e . For a Newtonian f l u i d the laminar v e l o c i t y p r o f i l e i s d e s c r i b e d by, The s o l u t i o n f o r the e i g e n f u n c t i o n s i n equation 2.14 y i e l d s the eigenvalues l i s t e d i n Table 2 f o r v a r i o u s w a l l Sherwood numbers. The corresponding f l u i d bulk c o e f f i c i e n t s E n c a l c u l a t e d from equation 2.22 are a l s o l i s t e d . The f i r s t seven eigenvalues are c a l c u l a t e d from the s e r i e s s o l u t i o n t o R n (p ) as i n r e f e r e n c e [25], but u s i n g a double p r e c i s i o n r o u t i n e , and higher order eigenvalues r e s u l t from d i r e c t numerical i n t e g r a t i o n of equation 2.14 u s i n g a m o d i f i e d Adams-Bashfor t h [26] i n t e g r a t i o n scheme. A l l computations were c a r r i e d out on an IBM 7044 d i g i t a l computer. F i g u r e 1 shows the v a r i a t i o n of the o v e r a l l Sherwood number with the dimension-l e s s c a p i l l a r y l e n g t h as c a l c u l a t e d from equations 2.21 and 2.2 6 f o r v a r i o u s w a l l Sherwood numbers. r h e o l o g i c a l behavior of blood i s one d e s c r i b e d by Casson's equation, u(p) = 2 ( l - p 2 ) (2.28) A non-Newtonian f l u i d which p a r a l l e l s c l o s e l y the (2.29) 26 where T i s the shear s t r e s s i n the f l u i d . During f u l l y developed flow there w i l l be a core moving at a constant v e l o c i t y and an annulus w i t h a laminar v e l o c i t y p r o f i l e . W i t h in the core, T = - — —£ , 0 < r ^ r 2 dx (2.30.) In the annulus, x = - — , r < r ^ R 2 dx o A f t e r s u b s t i t u t i n g the shear r e l a t i o n s , equation 2.30, i n t o Casson's equation and i n t e g r a t i n g , the non-dimensional flow v e l o c i t i e s become, 2 1+ 2p - p 8, 0 /p u(p) = °- 2/3 11 2 _ , 0 .< p ^ p Q (2.31) 1/2 + 2 / 3 P o ~ Po/42" 8/7 ^ 3 1- p 2 - 8,^ /p~ (1-P / 2 ) + 2p (1-p) U ( p ) = '-I _ ° ° f o < p < 1 V?+ 2 / 3 P°" P°/ 4 2" ^ (2.32) By i n t r o d u c i n g the flow f i e l d i n t o equation 2.14 the eigen-values f o r a g i v e n value of p Q can be c a l c u l a t e d by d i r e c t numerical i n t e g r a t i o n . F i g u r e 2 demonstrates the v a r i a t i o n of the eigenvalues w i t h p Q and F i g u r e 3 shows the v a r i a t i o n of the f l u i d bulk c o e f f i c i e n t s with p . The v a r i a t i o n of o the o v e r a l l Sherwood number with the dimensionless c a p i l l a r y 27 l e n g t h f o r a v a l u e o f p Q = 0.5 i s a l s o shown i n F i g u r e 1. E i g e n v a l u e s and f l u i d b u l k c o e f f i c i e n t s a r e l i s t e d i n T a b l e 3 f o r a v a l u e o f p = 0 . 5 . o 2.3 S y s t e m P a r a m e t e r s The f l u i d b u l k c o n c e n t r a t i o n h a v i n g b e e n p r e d i c t e d as a f u n c t i o n o f d i m e n s i o n l e s s l e n g t h and w a l l S h e r w o o d number, an a n a l y s i s o f t h e mass t r a n s f e r f r o m a c o m p o s i t e a r t i f i c i a l k i d n e y c a n be s t a t e d . F r om e q u a t i o n 2.5, t h e mass t r a n s f e r f r o m b l o o d i n a s i n g l e c a p i l l a r y i s g i v e n b y , m = ^ b i ^ b o * ( 2 ' 5 ) B u t t h e d i m e n s i o n l e s s f l u i d b u l k c o n c e n t r a t i o n a t t h e o u t l e t f r o m t h e c a p i l l a r y i s c = bo/c, . and f r o m c J m ' b i e q u a t i o n 2.26, c = e x p ( - 4 S h £ ) . Thus t h e mass t r a n s f e r ^ m o f r o m a c a p i l l a r y i s d e s c r i b e d b y , — ^ — = 1 - e x p ( - 4 S h £) (2.33) . q b c b i E q u a t i o n 2.33 may be c o m p a r e d w i t h e q u a t i o n 2.9 whence i t w i l l be s e e n t h a t a l t h o u g h t h e f o r m i s t h e same what has b e e n assumed c o n s t a n t i n e q u a t i o n 2.9 i s r e a l l y d e p e n d e n t on t h e o p e r a t i n g p a r a m e t e r s o f t h e s y s t e m . F o r t h e p u r p o s e s 28 of the f o l l o w i n g d i s c u s s i o n F i g u r e 1 has been r e p l o t t e d w i t h equation 2.33 on F i g u r e 4. I t w i l l be noted from F i g u r e 4 t h a t f o r E g r e a t e r than 70 percent the l i m i t i n g o v e r a l l Sherwood number i s n e a r l y independent of £. A l s o f o r E, > 0.3 the o v e r a l l Sherwood number may be d e s c r i b e d w i t h i n 5 percent by, Sh s h o = h~ <2'34> 1 + W 1.828 For a.system which has balanced flow i n each c a p i l l a r y , Q b = Nq b , M = Nm (2.35) Then the dimensionless l e n g t h parameter becomes, u s i n g the d e f i n i t i o n of.Pe, S = ^ (2.36) z y b I f £ > 0.3, S h Q i s n e a r l y independent of £ and hence i s n e a r l y independent of Q^, a l l other parameters remaining constant L e t , a = 2TTShoNxP (2.37) n = Q b / a (2.38) With the s u b s t i t u t i o n of equations 2.36, 2.37 and 2.38, 29 equation 2.3 3 becomes, M = n [ l - exp (- /n)] (2.39) b i which i s p l o t t e d i n F i g u r e 5. Since a i s a constant f o r any given d i a l y z e r , F i g u r e 5 M r e p r e s e n t s e s s e n t i a l l y c l e a r a n c e ( /c-^^) versus blood flow M r a t e which shows the e f f i c i e n c y of the u n i t . I d e a l l y / c k i p r o p o r t i o n a l t o i f the u n i t i s 100 percent e f f i c i e n t . I t can be shown t h a t to o b t a i n an e f f i c i e n c y c l o s e to 100 per-cent, with normal blood flow r a t e s , the d i a l y z e r dimensions become unmanageably l a r g e . From F i g u r e 5, when n = 0.8, the e f f i c i e n c y i s 71 percent. Smaller v a l u e s of n y i e l d h igher e f f i c i e n c i e s . As an e f f i c i e n c y g r e a t e r than 70 percent i s r e q u i r e d t o d i a l y z e a t y p i c a l uremic p a t i e n t a t reasonable blood flow r a t e s w i t h i n 6 hours (Appendix I demonstrates d i a l y s a n c e r a t e s ) then the de s i g n c r i t e r i o n must be n ^ 0.8. l i n e s (with the manifold represented by an e q u i v a l e n t feed l i n e length) must not exceed a predetermined amount i n order to prevent e x c e s s i v e blood from being removed from the p a t i e n t . T h i s volume i s , The t o t a l volume of blood i n the d i a l y z e r and feed V = J ( X £ D | + NxD 2) (2.40) A p p l y i n g the P o i s e u i l l e equation to the system as a whole, assuming t h a t n e g l i g i b l e p r essure l o s s e s occur i n the mani f o l d s , and s o l v i n g f o r the pressure drop across the system y i e l d s , y b 71 D* ND The feeder l i n e dimensions are g e n e r a l l y predetermined by p r a c t i c a l c o n s i d e r a t i o n s and the maximum hold-up volume of blood can be s p e c i f i e d . Each p a t i e n t has a maximum a v a i l a b l e p ressure drop across the canulae as determined by the mean a r t e r i a l and venous p r e s s u r e s . Equations 2.37 and 2.3 8 may be combined to g i v e , N X = 2^h?rT <2'42> o But from equation 2.40, x D 2 Nx = -  lJL (2.43) TTD D P D • With Sh = equation 2.34 becomes, w 2V ^ ' P D ' S h ° = i • ( 2 ' 4 4 ) 1 + irsw Combining equations 2.42, 2.43 and 2.44 and s i m p l i f y i n g y i e l d s 31 p 2 + 3 ^ 5 6 £ p . 3 . 6 5 W , ( j v . ^ 2 , . 0 ( 2 _ 4 5 ) A f u r t h e r s i m p l i f i c a t i o n r e s u l t s i f m a s s t r a n s f e r f r o m t h e c a p i l l a r i e s i s r e s t r i c t e d m a i n l y b y t h e w a l l . T h e n S h Q - S h ^ a n d b y c o m b i n i n g t h i s a s s u m p t i o n w i t h e q u a t i o n s 2 . 4 2 a n d 2 . 4 3 a n a p p r o x i m a t e e x p r e s s i o n r e s u l t s , TTT]P , 4 V ~2> = -tr  { ~ ~ K l D l ] (2'46) E q u a t i o n 2 . 4 6 i s v a l i d o n l y i f S h ^ << 1 . T h e d i a m e t e r o f t h e c a p i l l a r i e s c a n b e c a l c u l a t e d f r o m e q u a t i o n 2 . 4 5 a f t e r s p e c i f y i n g t h e s y s t e m p a r a m e t e r s , a n d a f t e r c h o o s i n g a n a p p r o p r i a t e v a l u e f o r n . E q u a t i o n s 2 . 4 1 a n d 2 . 4 3 t h e n may b e s o l v e d f o r t h e c o r r e s p o n d i n g n u m b e r a n d l e n g t h o f t h e c a p i l l a r i e s . A s n $ 0 . 8 f o r r e a s o n a b l e e f f i c i e n c y t h e r e w i l l b e a maximum d i a m e t e r w h i c h w i l l s a t i s f y t h e d i a l y s i s r e q u i r e m e n t . A n e x a m i n a t i o n o f e q u a t i o n s 2 . 4 1 a n d 2 . 4 3 s h o w s t h a t t h e l e n g t h o f t h e c a p i l l a r y r e q u i r e d i s p r o p o r t i o n a l t o i t s d i a m e t e r w h i l e t h e n u m b e r r e q u i r e d i s i n v e r s e l y p r o p o r t i o n a l t o t h e c u b e o f t h e d i a m e t e r . H e n c e t o m a i n t a i n a m a n a g e a b l e s i z e f o r t h e s y s t e m t h e c a p i l l a r y d i a m e t e r m u s t b e n e a r t h e maximum s i z e a l l o w e d b y e f f i c i e n c y c o n s i d e r a t i o n s . A s c a n b e s e e n f r o m t h e a b o v e d i s c u s s i o n t h e r e a r e many p o s s i b l e c o m b i n a t i o n s o f t u b e d i a m e t e r s , t u b e l e n g t h s and number o f t u b e s w h i c h w i l l s a t i s f y t h e p h y s i o l o g i c a l c r i t e r i o n f o r d i a l y s i s d e p e n d i n g upon t h e i n i t i a l s p e c i f i -c a t i o n s o f t h e s y s t e m p a r a m e t e r s . Some j u d g m e n t i s r e q u i r e d t o e s t a b l i s h a u s a b l e s y s t e m w h i c h w o u l d n o t s u f f e r f r o m i m p a i r e d e f f i c i e n c y f o r v a r i o u s p a t i e n t s . o f t u b e d i a m e t e r s , t u b e l e n g t h s and number o f t u b e s f o r g i v e n p a r a m e t e r s o f membrane p e r m e a b i l i t y , h o l d - u p v o l u m e , and a v a i l a b l e p r e s s u r e d r o p , i s n o t p o s s i b l e i f t h e b l o o d f l o w r a t e i s n o t s p e c i f i e d . I n t h e p a s t many a u t h o r s h a v e a t t e m p t e d t o o p t i m i z e t h e d e s i g n o f a r t i f i c i a l k i d n e y s b a s e d on some c h o i c e o f d i m e n s i o n a l p a r a m e t e r s . H o w e v e r , t h e r e s u l t h a s a l w a y s b e e n t h a t f o r o p t i m u m mass t r a n s f e r t h e s y s t e m must h a v e z e r o b l o o d c h a n n e l l e n g t h and an i n f i n i t e i n l e t a r e a , a c o n d i t i o n t h a t i s a p p r o x i m a t e d i n t h e human k i d n e y . Hence a more r e a l i s t i c o p t i m i z a t i o n p r o c e d u r e i s one i n w h i c h t h e mass t r a n s f e r p e r u n i t membrane a r e a i s o p t i m i z e d f o r t h e d e s i r a b l e s y s t e m p a r a m e t e r s . O p t i m i z a t i o n o f t h e s y s t e m ; t h a t i s , t h e b e s t c h o i c e U t i l i z i n g e q u a t i o n s 2.5, 2.21 and t h e f a c t t h a t 00 M = 1 - I E e x p ( - ^ ) (2.47) n = l t h e c a p i l l a r y mass t r a n s f e r a r e a i s , 33 = D i ~ X £ ° £ } (2.48) By c o m b i n i n g e q u a t i o n s 2.36 and 2.40 t o e l i m i n a t e Nx a r e l a t i o n b e t w e e n t h e c a p i l l a r y d i a m e t e r and d i m e n s i o n l e s s l e n g t h may be f o u n d , ^2 T T P .4V _ 2 . = 2Q T T ( — - * i D i > (2.49) Thus t h e c a p i l l a r y a r e a b e comes, r2^Qb ,4V _2H 2,1 V 2 l / 2 (2.50) The e q u a t i o n t o be o p t i m i z e d i s , M 1 - Z E exp(-A*£) -, n n n = l Q , c, .A T 2 T T Q . 'b b i 'b ,4V ^2> V 2 l / 2 (2.51) F o r a s y s t e m i n w h i c h o ptimum e f f i c i e n c y o f mass t r a n s f e r p e r u n i t membrane a r e a i s d e s i r e d e q u a t i o n 2.51 shows t h a t f o r some v a l u e o f £ t h e r e i s a maximum t o be a c h i e v e d . D i f f e r e n t i a t i n g e q u a t i o n 2.51 w i t h r e s p e c t t o £ and s e t t i n g t h e r e s u l t e q u a l t o z e r o y i e l d s , I E (1 + 2 A 2 O e x p ( - A 2 £ ) = 1 i n n * n n = l (2.52) 34 The s o l u t i o n t o t h i s t r a n s c e n d e n t a l equation i s gi v e n i n Table 4 f o r v a r i o u s w a l l Sherwood numbers. Having c a l c u -l a t e d the optimum valu e s f o r the dim e n s i o n l e s s l e n g t h parameter from equation 2.52 the e f f i c i e n c y of the system may be c a l c u l a t e d from equation 2.47. These e f f i c i e n c e s are a l s o g i v e n i n Table 4. I t can be seen t h a t optimum mass t r a n s f e r per u n i t membrane area i s obtained when the system mass t r a n s f e r e f f i c i e n c y i s approximately 70 percent. The r e l a t i o n s h i p between the w a l l Sherwood number and the dimensionless l e n g t h parameter i n Table 4 may be approximated by the equation, £ = 0.1192 + 0.3159/Sh, (2.53) w The e r r o r i n h e r e n t i n t h i s equation i s l e s s than 3 percent f o r a l l w a l l Sherwood numbers l e s s than 6. A 34 percent e r r o r e x i s t s a t the extreme case of no w a l l r e s i s t a n c e (Sh = °°) . For l a r g e w a l l Sherwood numbers a b e t t e r • w approximation i s , 5 = 0.1389 + 0.3144/Sh - 0.0477 exp(-3.408/Sh ) (2.54) w w The e r r o r i n equation 2.54 i s l e s s than 1 percent f o r a l l w a l l Sherwood ;numbers, however, the equation i s n o n - l i n e a r and as most p r a c t i c a l mass t r a n s f e r systems have a p p r e c i a b l e 35 w a l l r e s i s t a n c e e q u a t i o n 2.53 w i l l s u f f i c e f o r t h e p u r p o s e o f c a l c u l a t i n g t h e c a p i l l a r y d i m e n s i o n s . By r e p l a c i n g t h e w a l l S h e r w o o d number and d i m e n s i o n -l e s s l e n g t h p a r a m e t e r b y t h e i r a p p r o p r i a t e t e r m s , e q u a t i o n 2.53 e x p r e s s e s t h e d i a m e t e r o f t h e c a p i l l a r y a s , n 2 VD 4.195T T P ,4V 2* _ , . D + 5.30 -p ( — - x £ D £ ) - 0 (2.55) T h i s e q u a t i o n i s s i m i l a r i n f o r m t o e q u a t i o n 2.45 a l t h o u g h i t was d e r i v e d f r o m an o p t i m i z a t i o n p r o c e d u r e i n s t e a d o f a s p e c i f i e d o p e r a t i n g e f f i c i e n c y . E q u a t i o n 2.45 and 2.55 a r e n o t i d e n t i c a l , h o w e v e r , b e c a u s e t h e e f f i c i e n c y o f mass t r a n s f e r was n o t h e l d c o n s t a n t when d e r i v i n g e q u a t i o n 2.55 w h e r e a s t h e e f f i c i e n c y p a r a m e t e r i s a c o n s t a n t i n e q u a t i o n 2.45. To compare t h e two e q u a t i o n s an o p e r a t i n g p o i n t was c h o s e n w h e r e t h e c a p i l l a r y d i a m e t e r was s m a l l s o t h a t s e c o n d o r d e r d i a m e t e r t e r m s c o u l d be n e g l e c t e d . A mass t r a n s f e r e f f i c i e n c y o f 71.5 p e r c e n t c o r r e s p o n d s t o n = 0.799. From e q u a t i o n 2.55, D = 3.11 V s p / Q k a n d f r o m e q u a t i o n 2.46, D = 3.20 V P/Q, w h e r e V = V - — . . Thus f o r s m a l l s ' b s 4 w a l l Sherwood- numbers t h e two e q u a t i o n s a r e n e a r l y i d e n t i c a l . A p p e n d i x I I c o n t a i n s a d e s i g n f o r a h y p o t h e t i c a l s y s t e m u t i l i z i n g t h e p r e v i o u s l y s t a t e d o p t i m i z e d a n a l y s i s w h i c h w i l l d e m o n s t r a t e t h e u s e o f t h e e q u a t i o n s . 36 2.4 M a n i f o l d s The a n a l y t i c a l s t u d y o f mass t r a n s f e r f r o m a s y s t e m o f c a p i l l a r i e s i s i n d e p e n d e n t o f t h e f l u i d d i s t r i b u t i o n t o t h e c a p i l l a r i e s o n l y i f e v e r y c a p i l l a r y h a s an e q u a l f l o w r a t e . To a c h i e v e e q u a l f l o w r a t e s s p e c i a l c o n s i d e r a t i o n o f t h e b l o o d d i s t r i b u t i o n s y s t e m must be made. A b a s i c r e c t a n g u l a r c o n f i g u r a t i o n f o r t h e b u n d l e o f c a p i l l a r i e s was c h o s e n b e c a u s e a f l o w d i r e c t i o n n o r m a l t o t h e t u b e b u n d l e a l l o w s s h a p i n g o f t h e m a n i f o l d p r o f i l e s o t h a t e q u a l d i s t r i b u t i o n p f b l o o d t o e a c h c a p i l l a r y c a n be r e a l i z e d . The p r o b l e m i s t h e n t o d e t e r m i n e t h e s h a p e o f a m a n i f o l d t h a t w i l l c r e a t e a c o n s t a n t p r e s s u r e a t t h e e n t r a n c e t o t h e c a p i l l a r i e s , i t b e i n g assumed t h a t s i n c e a l l c a p i l l a r -i e s a r e i d e n t i c a l , c o n s t a n t p r e s s u r e w i l l y i e l d e q u a l f l o w r a t e s . S i n c e t h e b l o o d f l o w i n t h e m a n i f o l d w i l l be t h r e e -d i m e n s i o n a l w i t h l i t t l e s ymmetry a s i m p l i f i e d a n a l y s i s o f t h e p r o b l e m i s n e c e s s a r y . The o n 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 , i s o t h e r m a l c o n t i n u i t y , e n e r g y and momentum e q u a t i o n s w i l l s p e c i f y t h e r e q u i r e d m a n i f o l d s h a p e i f s u f f i c i e n t i n f o r m a t i o n c o n c e r n i n g t h e s y s t e m l o s s e s c a n be p r o v i d e d . F i g u r e 6 r e p r e s e n t s a c r o s s - s e c t i o n o f t h e m a n i f o l d w i t h t h e c o n t r o l v o l u m e f o r w h i c h t h e f l o w e q u a t i o n s a r e w r i t t e n . A f i r s t a p p r o x i m a t i o n f o r t h e m a n i f o l d s h a p e c a n be d e t e r m i n e d f r o m t h e c o n t i n u i t y and e n e r g y e q u a t i o n s . The c o n t i n u i t y e q u a t i o n f o r t h e c o n t r o l v o l u m e i s , 37 d(AU) + vbdx = 0 (2.56) where f l u i d flows from the m a n i f o l d i n t o the outflow r e g i o n of width b w i t h an a x i a l v e l o c i t y u and a r a d i a l v e l o c i t y v. For i n c o m p r e s s i b l e f l u i d flow i n the c o n t r o l volume the mechanical energy equation i s , K PU 2 K p AU [p + — ] = [AU + d(AU)] [p + dp + -2- (U + dU) ] + vbdx [p T +' £• ( u 2 + v 2 ) ] + K dx t £ T i 2 V (2.57) where K v i s the v i s c o u s energy l o s s f u n c t i o n f o r the c o n t r o l volume and K g i s the k i n e t i c energy c o r r e c t i o n f a c t o r t o compensate f o r the v e l o c i t y p r o f i l e . G r a v i t a t i o n a l p o t e n t i a l energy i s unimportant here, and i s n e g l e c t e d . The energy equation then becomes, A K p U 2 0 . . a_ [ A U ( p + _£___)] + v b + £ ( u ^ + v ^ } ] + K y = 0 (2.59) The energy f l u x through face 1-2 of the c o n t r o l volume must enter the c a p i l l a r i e s which make up the l a t e r a l . An energy balance f o r the l a t e r a l y i e l d s . P L + | ( u 2 + v 2 ) - p e + + E v (2.60) where P i s the pressure a t the e x i t from the l a t e r a l e which may be taken as zero without l o s s of g e n e r a l i t y and where E v i s the energy l o s s due t o v i s c o u s d i s s i p a t i o n along the l a t e r a l . S ince the l a t e r a l c o n s i s t s o f s m a l l diameter tubes the laminar v i s c o u s l o s s may be expressed by, E v = Ap = H i L l v = K v ( 2 > 6 1 )  v d^ The energy equation f o r the l a t e r a l w i t h a l a r g e amount of v i s c o u s d i s s i p a t i o n i s approximately, 2 P L + 1 ( u 2 + v 2 ) = ^~T~ + K V (2.62) Equation 2.62 i n d i c a t e s t h a t f o r constant outflow v e l o c i t y i n the c a p i l l a r i e s which make up the l a t e r a l the energy f l u x l e a v i n g face 1-2.of the c o n t r o l volume must be const a n t . Energy equation 2.59 then becomes K p U 2 2 | _ [ A U ( p + _ | } ] + v b r£V + . K v ] + K ^ = 0 . (2.63) For a large value of l a t e r a l viscous d i s s i p a t i o n , the viscous losses K v for the manifold can safe ly be neglec-ted. By subs t i tu t ing the cont inui ty equation 2.56 into equation 2.63 and noting that Q = UA then, h tG<P + ¥ ^ > ] - [*52+*v] i ( 2 . 6 4 ) But a constra int on the manifold design was to have constant outflow v e l o c i t y v. For constant outflow v e l o c i t y the cont inu i ty equation may be integrated to g ive , Q = Q o ( l - £)' (2.65) where Q Q i s the flow rate into a manifold of length L . The t o t a l energy into the contro l volume can be given the symbol, K pU 2 E m = P + -4j (2.66) which, when subst i tuted into equation 2.64 y i e l d s , dE • dQ pv 2 , ° r t B . s - <- J + Kv> aS ( 2 . 6 7 ) But by subs t i tu t ing equation 2.65 into 2.67 then, d E m (L-x) ~ i - E m = - ( - j + Kv) (2.68) which may be i n t e g r a t e d w i t h the boundary c o n d i t i o n s t h a t a t x = 0 the i n p u t energy l e v e l i s E . Thus, * r J mo ' 2 ( p v 1 -<£J + Kv)x E L E mo m _ E (2.69) mo 1 - *-L Examination of equation 2.69 shows t h a t the only p o s s i b l e 2 s o l u t i o n i s when E m Q = Kv + - and the energy l e v e l of the f l u i d i s constant along the m a n i f o l d . For s m a l l v i s c o u s l o s s e s i n the m a n i f o l d compared to the l a t e r a l l o s s e s the p r e s s u r e i n the m a n i f o l d i s n e a r l y constant at p = p T . But Li s i n c e the energy l e v e l i n the m a n i f o l d i s constant then t h e r e can be no v a r i a t i o n i n f l u i d v e l o c i t y along the m a n i f o l d . For constant v e l o c i t y , e q u a t i o n 2.65 g i v e s d i r e c t l y j r = 1• " £ ( 2 - 7 0 ) o A more g e n e r a l f o r m u l a t i o n of the problem of d e t e r -mining the m a n i f o l d shape can be obtained from the c o n t i n u i t y and momentum equations f o r the c o n t r o l volume shown i n F i g u r e 6. The c o n t i n u i t y equation again i s , d(AU) + vbdx = 0 (2.56) Co n s e r v a t i o n of x-monentum over the c o n t r o l volume y i e l d s , d ( K pAU ) + u p v b d x = -Adp - f d x (2.71) w h e r e f i s t h e l o c a l w a l l f r i c t i o n f o r c e a n d K i s a m momentum c o r r e c t i o n f a c t o r t o c o m p e n s a t e f o r t h e v e l o c i t y p r o f i l e i n t h e c o n t r o l v o l u m e . The v a l u e o f t h e l o c a l f r i c t i o n f o r c e d e p e n d s upon t h e v e l o c i t y g r a d i e n t o f t h e f l u i d a t t h e m a n i f o l d w a l l . The momentum c o r r e c t i o n f a c t o r v a r i e s b e t w e e n a v a l u e o f u n i t y f o r u n i f o r m ( s l u g ) f l o w and 4/3 f o r f u l l y d e v e l o p e d l a m i n a r f l o w . The a x i a l v e l o c i t y o f t h e f l u i d e n t e r i n g t h e t u b e s l o t may be r e l a t e d t o t h e a v e r a g e f l u i d v e l o c i t y by a n e x p e r i m e n t a l l y d e t e r m i n e d momentum c o r r e c t i o n f a c t o r w h i c h v a r i e s b e t w e e n z e r o a n d u n i t y . u = 3U (2.72) To m a i n t a i n c o n s t a n t p r e s s u r e i n t h e m a n i f o l d , dp = 0. The f l o w r a t e i n t o t h e m a n i f o l d i s Q Q and t h e o u t f l o w v e l o c i t y , v , i s c o n s t a n t b e c a u s e o f t h e c o n s t a n t p r e s s u r e . T h e r e f o r e t h e f l o w r a t e a t t h e c o n t r o l v o l u m e i s , UA = Q Q ( 1 - £) (2.65) By a s s u m i n g t h a t t h e momentum c o r r e c t i o n f a c t o r s a r e i n d e p e n d e n t o f t h e c o n t r o l v o l u m e p o s i t i o n t h e momentum 42 e q u a t i o n 2.71 becomes, 2 A | U d A + j^ o + f = dx dx K L pK U m m (2.73) And t h e c o n t i n u i t y e q u a t i o n b e c o m e s , A d U + u dA + ^ = o dx dx L (2.74) U s i n g e q u a t i o n s 2.65 and 2.74 t o e l i m i n a t e t h e v e l o c i t y t e r m s , e q u a t i o n 2.73 becomes, dA - 3_ dx * : K pK Q U' L-x m m o f L ) JL- _ o (2.75) The f l u i d f r i c t i o n l o s s e s c a n be e x p r e s s e d a s , f = u 3u 3 r ds (2.76) B u t t h e a v e r a g e f l u i d v e l o c i t y a t t h e c o n t r o l v o l u m e i s , U = udA (2.77) And t h e f l o w r a t e c a n be e x p r e s s e d i n t e r m s o f t h e i n l e t R e y n o l d s number and i n l e t d i a m e t e r D q a s , Re = _4Q< T T V D (2.78) Thus t h e f r i c t i o n l o s s t e r m i n e q u a t i o n 2.75 b e c o m e s , f L 4 L P K m Q o U TTK D Re m o 9u c 8 r ds (2.79) udA H o w e v e r , t h e w a l l f r i c t i o n l o s s e s may be e x p r e s s e d b y , f = c T c d s = - (2.80) w h e r e d p / d x i s t h e l a m i n a r f r i c t i o n p r e s s u r e d r o p o v e r t h e a p p r o x i m a t e l y e q u i v a l e n t c o n t r o l v o l u m e o f c o n s t a n t c r o s s -s e c t i o n i n w h i c h s u r f a c e 1-2 ( F i g u r e 6) i s w i t h o u t e f f l u x . By c o m b i n i n g e q u a t i o n s 2.76, 2.77 a n d 2.80 a n e x p r e s s i o n f o r t h e l a m i n a r l o s s e s c a n be d e t e r m i n e d . 3u c 9 r d s udA uQ dx Y (2.81) The p a r a m e t e r y c a n be e v a l u a t e d f r o m e l e m e n t a r y f l u i d m e c h a n i c s f o r any c r o s s - s e c t i o n s h a p e . W i t h t h i s p a r a m e t e r t h e momentum e q u a t i o n 2.75 b e c o m e s , dx v K TTK D R e ' L - x m m o (2.82) F o r e x a m p l e , f l u i d f l o w i n g t h r o u g h a c i r c u l a r p i p e i n t h e l a m i n a r r e g i m e h a s a p a r a b o l i c v e l o c i t y p r o f i l e w h i c h g i v e s 44 a v a l u e f o r t h e f r i c t i o n p a r a m e t e r o f y = 8TT - A p p e n d i x I I I g i v e s t h e v a l u e s o f y f o r v a r i o u s c r o s s - s e c t i o n a l s h a p e s . The m a n i f o l d s h a p e i s d e t e r m i n e d b y f l o w c o n s i d e r -a t i o n s a n d by t h e f a c t t h a t t h e r e i s an o u t f l o w s l o t o f f i n i t e w i d t h . Minimum p r e s s u r e l o s s e s o c c u r i n c i r c u l a r s e c t i o n s b u t t h e r e w i l l be some p o i n t i n t h e m a n i f o l d a t w h i c h t h e c r o s s - s e c t i o n d i a m e t e r i s e q u a l t o t h e s l o t w i d t h and b e y o n d t h i s p o i n t , t o m a i n t a i n an a r e a r e d u c t i o n r e l a t i o n as r e q u i r e d b y t h e momentum e q u a t i o n , a c r o s s - s e c t i o n s h a p e o t h e r t h a n c i r c u l a r must be u s e d . The mos t p r a c t i c a l s h a p e i s e l l i p t i c a l w i t h t h e m a j o r a x i s b e i n g e q u a l t o t h e s l o t w i d t h . The s l o t i s f u l l y o c c u p i e d w i t h c a p i l l a r y t u b e s a n d , t o t h e c r o s s - s e c t i o n , r e s e m b l e s a s o l i d w a l l . The e f f e c t o f t h i s e n c r o a c h m e n t o f t h e s l o t i n t o t h e c r o s s -s e c t i o n a r e a m u s t be a c c o u n t e d f o r . S i n c e t h e r e i s a v e l o c i t y c omponent p a r a l l e l t o t h e m a n i f o l d a x i s a t t h e s l o t , t h e c o n d i t i o n o f no f l u i d s l i p a t a s o l i d w a l l i s n o t v a l i d . T h u s , i n t h e c a l c u l a t i o n o f t h e p a r a m e t e r y, two l i m i t i n g c a s e s h a v e b e e n d e t e r m i n e d ; one f o r a n o - s l i p c o n d i t i o n a t t h e s l o t , and t h e o t h e r f o r a s h e a r - f r e e s u r f a c e a t t h e s l o t . The v a i u e s f o r y f o r t h e s e two c a s e s a r e p r e s e n t e d i n F i g u r e 7 f o r v a r i o u s c r o s s - s e c t i o n s h a p e s . As c a n be s e e n f r o m F i g u r e 7, y i s n o t a c o n s t a n t b u t becomes v e r y d e p e n d e n t on t h e c r o s s - s e c t i o n a r e a when t h e r a t i o o f t h e t u b e a r e a t o t h e s l o t w i d t h becomes s m a l l w h i c h o c c u r s a t t h e end o f t h e m a n i f o l d s e c t i o n . H o w e v e r , f o r s l o t w i d t h s w h i c h a r e much l e s s t h a n t h e i n l e t d i a m e t e r t o t h e m a n i f o l d , y r e m a i n s n e a r l y c o n s t a n t o v e r m o s t o f t h e m a n i f o l d l e n g t h . A l s o , i t w o u l d be e x p e c t e d t h a t y w o u l d l i e b e t w e e n t h e two e x t r e m e s o f a s h e a r e d and s h e a r - f r e e s u r f a c e a t t h e s l o t a n d h e n c e be n e a r l y c o n s t a n t f o r A / b 2 * 0.4. The momentum e q u a t i o n c a n be i n t e g r a t e d t o g i v e t h e c r o s s - s e c t i o n a r e a o f t h e m a n i f o l d f o r a g i v e n l e n g t h , a s s u m i n g y i s c o n s t a n t , w i t h e r r o r b e i n g i n t r o d u c e d o n l y when x - L. I n t h i s r e g i o n a t t h e e n d o f t h e m a n i f o l d , t h e a p p r o x i m a t i o n s o f o n e - d i m e n s i o n a l f l u i d a n a l y s i s p r o b a b l y become v e r y p o o r . I n t e g r a t i n g e q u a t i o n 2 . 8 2 w i t h (3» y, and K M h e l d c o n s t a n t y i e l d s , 2 ( 1 - - J 2 L ^ ) A r, x, v 2 K TTK D Re' / 0 o o X ^ — = [ l - j - j m m o ( 2 . 8 3 ) o The a r e a f r o m x = 0 t o t h e p o i n t a t w h i c h t h e a r e a becomes a s e m i - c i r c l e o f d i a m e t e r b i s d e s c r i b e d b y t h e e q u a t i o n , A = -'iDi ( i - i + S i S i i ) (2.84) •.. 4 TT 2TT w h e r e s i n 0 = j | a n c j w h e r e D i s t h e d i a m e t e r o f t h e c r o s s -s e c t i o n . The r e m a i n i n g a r e a up t o x = L i s e l l i p t i c a l i n s h a p e g i v e n b y t h e e q u a t i o n , 46 A = ^ ( 2 . 8 5 ) w h e r e h i s t h e m i n o r s e m i - a x i s d i m e n s i o n . F o r t h e e a s e i n d e s c r i p t i o n o f m a n i f o l d s h a p e s e q u a t i o n 2 . 8 3 i s s t a t e d a s , f - = [ 1 - g ] 2 6 ( 2 . 8 6 ) o w h e r e 6 = 1 - 2 L ^ . „ a n d h a s p r a c t i c a l l i m i t s 2 K TT K D R e c m m o b e t w e e n t h e v a l u e s o f z e r o a n d u n i t y . C H A P T E R 3 E X P E R I M E N T S 47 3.1 P a t i e n t Parameters In order to determine some of the p h y s i o l o g i c a l c o n d i t i o n s p r e s e n t d u r i n g hemodialysis i t was decided to moni-t o r the blood flow r a t e and p r e s s u r e s on a p a t i e n t undergoing d i a l y s i s on a two l a y e r , m o d i f i e d K i i l d i a l y z e r a t the Renal U n i t i n the Vancouver General H o s p i t a l . The f i r s t step was to o b t a i n and c a l i b r a t e an u l t r a -s o n i c Doppler flow meter. The Doppler flowmeter was chosen because i t i s the only method of determining flow r a t e s c u r r e n t l y a v a i l a b l e which does not r e q u i r e c o n n e c t i o n i n the flow c i r c u i t . I t was not d e s i r a b l e to stop the flow of blood a t anytime d u r i n g d i a l y s i s and y e t i t was a l s o not d e s i r a b l e to leave the flowmeter permanently connected i n the c i r c u i t . Hence the d e v i c e had to be an e x t r a - c o r p o r a l one which c o u l d be removed and r e p l a c e d from the blood l i n e s at w i l l . The u l t r a s o n i c doppler flowmeter c o n s i s t e d of a h i g h frequency o s c i l l a t o r (5 Mc.) which e x c i t e d a p i e z o - e l e c t r i c c r y s t a l i n i n t i m a t e c o n t a c t w i t h the o u t e r w a l l of the flow channel. The u l t r a s o n i c energy was conducted through the channel w a l l and i n t o the flow medium where b a c k - s c a t t e r i n g of energy oc c u r r e d from p a r t i c l e s i n the f l u i d . A second c r y s t a l , mounted i n a f i x e d g e o m e t r i c a l r e l a t i o n t o the f i r s t , r e c e i v e d both the t r a n s m i t t e d and b a c k - r e f l e c t e d s i g n a l s from a l l the p a r t i c l e s moving i n the f l u i d . S i nce 48 i n laminar flow the p a r t i c l e s move at v e l o c i t i e s v a r y i n g between zero and twice the average flow v e l o c i t y , the back-r e f l e c t e d s i g n a l c o n s i s t e d of the o r i g i n a l frequency p l u s the composite doppler v e l o c i t y e f f e c t . The r e c e i v i n g c r y s t a l v i b r a t e d a t the d i f f e r e n c e frequency of the t r a n s m i t t e d s i g n a l and the' s t r o n g e s t of the b a c k - r e f l e c t e d s i g n a l s which normally corresponded to the average v e l o c i t y . The s t r e n g t h of the b a c k - r e f l e c t e d s i g n a l depended upon the u l t r a s o n i c a b s o r p t i v i t y of the p a r t i c l e s . However, a i r bubbles, which have very low a b s o r p t i v i t y , r e f l e c t e d a s i g n a l of s u f f i c i e n t s t r e n g t h t o o v e r r i d e a l l other s i g n a l s . Hence, t o o b t a i n reason-able f l u i d v e l o c i t y s i g n a l s , a l l a i r bubbles had t o be removed from the system. The equation AF = 2FV cos 6/C, where AF i s the d i f f e r e n c e frequency, F i s the e x c i t i n g frequency, V i s the average p a r t i c l e v e l o c i t y , 8 i s the angle between the a c o u s t i c a l axes and flow, C i s the v e l o c i t y of sound i n the "5 f l u i d (1.5 x 10 cm/sec i n b l o o d ) , governs the doppler s h i f t frequency f o r a g i v e n system. To determine the flow r a t e the diameter of the tube must be known. A Ward A s s o c i a t e s Doppler U l t r a s o n i c Flowmeter (Model 1503) and two t r a n s d u c e r s , Model B-50 08 f o r j i n . O.D. t u b i n g and Model D-5006 f o r 0.20 i n O.D. t u b i n g were purchased from.the A l l e n L. Wolff Company. The system was p r e - c a l i b r a t e d but a check of the c a l i b r a t i o n was never-49 t h e l e s s deemed d e s i r a b l e . T h i s was accomplished by c o n s t r u c t -i n g a constant head tank w i t h o u t l e t t u b i n g of the diameter r e q u i r e d by the t r a n s d u c e r . Evaporated m i l k was used as the c a l i b r a t i n g f l u i d . Flow r a t e s were determined by stop-watch and graduate. At t h i s time i t was d i s c o v e r e d t h a t the Ward Doppler Flowmeter was n o n - l i n e a r and extremely s e n s i t i v e t o a i r bubbles i n the l i n e s . As w e l l , the method of u l t r a -s o n i c a l l y c o u p l i n g the t r a n s d u c e r s to the f l o w l i n e s was not r e p r o d u c i b l e and c o n s i d e r a b l e a c o u s t i c l o s s r e s u l t e d i n an u n s t a b l e flow v e l o c i t y r e a d i n g . For low flow r a t e s the non-l i n e a r i t i e s were not severe but as the system was to measure p u l s a t i l e flow c o n d i t i o n s c o r r e c t i o n s c o u l d not be p r o v i d e d . Normally there are no a i r bubbles i n b l o o d . I t was decided t h a t any improvement i n the flowmeter would be obtained only at a d i s p r o p o r t i o n a t e expenditure of time so i t was used with some r e s e r v a t i o n s . Blood pressure measurements on h o s p i t a l p a t i e n t s i n v o l v e d a unique p r a c t i c e . Two Statham p r e s s u r e t r a n s d u c e r s (Model P23AA) were obt a i n e d . With 100 cm of connecting t u b i n g (#7) these u n i t s had a n a t u r a l frequency of 15 cps which was s u f f i c i e n t l y high to monitor p u l s a t i l e flow. The Statham pressure t r a n s d u c e r s had s t r a i n - g a g e monitored f l e x i b l e diaphragms which r e q u i r e d e x c i t a t i o n and m i c r o v o l t D.C. a m p l i f i c a t i o n . The a s s o c i a t e d i n s t r u m e n t a t i o n f o r p r e s s u r e measurement was p r o v i d e d by a Cambridge Instrument 50 Company four channel a m p l i f i e r and c h a r t r e c o r d e r which a l s o recorded the flow r a t e s . The pressure t r a n s d u c e r s were c a l i b r a t e d a g a i n s t pressure p r o v i d e d by a sphygmomanometer re a d i n g i n mm Hg. u n i t s . The accuracy was ± 2 mm Hg. For a c c u r a t e p u l s a t i l e measurements there must be no a i r bubbles i n the pressure c i r c u i t and, f o r measuring blood p r e s s u r e , no blood must come i n c o n t a c t with the transducer membrane. These r e s t r i c t i o n s r e q u i r e d s t e r i l e s a l i n e p e r f u s i o n which was accomplished by f i l l i n g the pressure measuring c i r c u i t with s a l i n e b e f o r e c o n n e c t i o n with the blood l i n e s . When a pressure r e a d i n g was r e q u i r e d the s a l i n e p e r f u s i o n was switched from the c i r c u i t so t h a t blood pressure was recorded. At a l l other times p r e s s u r i z e d s a l i n e flowed through the system and i n t o the p a t i e n t ! s blood l i n e s t o prevent blood from e n t e r i n g the t r a n s d u c e r . F i g u r e 8 g i v e s the t e s t c i r c u i t f o r o b t a i n i n g the blood p r e s s u r e s and flow r a t e s of a p a t i e n t on h e m o d i a l y s i s . A t y p i c a l r e s u l t obtained w i t h the flowmeter and pressure t r a n s d u c e r s i s given i n F i g u r e 9 f o r a male p a t i e n t whose blood pressure was nominally 130 mm Hg. s y s t o l i c and 80 mm Hg. d i a s t o l i c . The pressure t r a n s d u c e r s were l o c a t e d very near the cannulae on the d i a l y z e r blood l i n e s and the flowmeter t r a n s d u c e r was downstream from the a r t e r i a l p ressure t r a n s d u c e r . Although the pressure measurements were r e p r o d u c i b l e the flow measurements had only an estimated accuracy of 25,percent. The flow r a t e wave form, though, 51 was c o n s i s t e n t t h r o u g h o u t t h e e x p e r i m e n t . 3.2 Mass T r a n s f e r As a c h e c k on t h e v a l i d i t y o f t h e t h e o r y f o r mass t r a n s f e r , e s p e c i a l l y f o r t h e n o n - N e w t o n i a n a s p e c t s o f b l o o d , some e x p e r i m e n t s w e r e c o n d u c t e d u s i n g l a r g e d i a m e t e r c o m m e r c i a l d i a l y s i s t u b i n g . C e l l u l o s e d i a l y s i s t u b i n g o f 1/4 i n . d i a m e t e r was o b t a i n e d f r o m t h e V i s k i n g D i v i s i o n o f U n i o n C a r b i d e o f C a n a d a . F o r t h i s t u b i n g , r e a s o n a b l e mass t r a n s f e r r a t e s c o u l d be r e a l i z e d o n l y w i t h a v e r y l o n g s e c t i o n and l o w f l o w r a t e s . A c c o r d i n g l y a 2 1/2 i n I.'D. p l e x i g l a s s t u b e 100 i n . i n l e n g t h was f a b r i c a t e d w i t h end p l a t e s and a d j u s t a b l e end s u p p o r t s t o c o n t a i n t h e d i a l y s i s t u b i n g . D i a l y s a t e was r e c i r c u l a t e d f r o m a c o n s t a n t t e m p e r a t u r e b a t h t h r o u g h t h e p l e x i g l a s s t u b e a t a s u f f i c i e n t r a t e t o e n s u r e t u r b u l e n t m i x i n g . B l o o d was d e l i v e r e d t o t h e d i a l y s i s t u b i n g f r o m a c o n s t a n t h e a d t a n k and f l o w r a t e s w e r e c h e c k e d by s t o p w a t c h and g r a d u a t e . C a r e was t a k e n t o e n s u r e t h a t t h e b l o o d p r e s s u r e i n t h e d i a l y s i s t u b i n g was e q u i v a l e n t t o t h e d i a l y s a t e p r e s s u r e on t h e o u t s i d e i n o r d e r t o p r e v e n t c o l l a p s e o f t h e t u b i n g o r u l t r a f i l t r a t i o n . F o r a l l e x p e r i m e n t s t h e d i a l y s a t e was p u r e w a t e r m a i n t a i n e d a t 37°C and c i r c u l a t e d a t a p p r o x i m a t e l y 4£/min. A s c h e m a t i c Of t h e s y s t e m i s g i v e n i n F i g u r e 10. 52 B l o o d was d r a w n f r o m t h e Red C r o s s B l o o d Bank a t t h e h o s p i t a l and c o n s i s t e d o f 450 m l o f w h o l e b l o o d and 67.5 m l o f ACD b u f f e r s o l u t i o n p e r b o t t l e . No a t t e m p t was made t o s t a n d a r d i z e b l o o d t y p e s o r h e m a t o c r i t s as t h e 1 l i t e r w o r k i n g s o l u t i o n s w e re p r e p a r e d . The p r o c e d u r e f o r p r e p a r i n g a b l o o d s o l u t i o n was t o m i x t o g e t h e r two b o t t l e s o f b l o o d a t room t e m p e r a t u r e . A s a m p l e o f 10 c c was d r a w n f r o m t h i s m i x t u r e . Then a p p r o x i m a t e l y 2 gm o f u r e a and 300 mg o f h e p a r i n w e re d i s s o l v e d i n 25 c c s a l i n e a nd a d d e d t o t h e b l o o d m i x t u r e . A s e c o n d 10 c c s a m p l e was d r a w n f r o m t h i s m i x t u r e . The w o r k i n g s o l u t i o n was p l a c e d i n t h e h e a d e r t a n k and an e x p e r i m e n t a l r u n was s t a r t e d . F r o m t h e o r i g i n a l s o l u t i o n f o u r c o n s t a n t f l o w r a t e r e s u l t s c o u l d be o b t a i n e d . As t h e b l o o d f l o w r a t e s v a r i e d b e t w e e n 1 c c / m i n and 15 c c / m i n , a b o u t 200 c c o f b l o o d a t a g i v e n f l o w r a t e w o u l d g u a r a n t e e t h a t s t e a d y - s t a t e mass t r a n s f e r h a d b e e n a c h i e v e d . F o r e v e r y f l o w r a t e a 10 c c s a m p l e was t a k e n a t t h e end o f t h e r u n . An a n a l y s i s f o r b l o o d - u r e a - n i t r o g e n c o n c e n t r a t i o n (BUN) was c a r r i e d o u t a t t h e G.F. S t r o n g l a b o r a t o r i e s i n t h e h o s p i t a l . .. A s p e c i a l p r o g r a m f o r a c c u r a t e d e t e r m i n a t i o n o f BUN was s e t up b e c a u s e t h e s t a n d a r d h o s p i t a l method was n o t s u f f i c i e n t l y p r e c i s e . S m a l l e r r o r s i n a n a l y s i s w o u l d be m a g n i f i e d when t h e d i f f e r e n c e b e t w e e n t h e i n l e t and o u t l e t c o n c e n t r a t i o n s o f u r e a was d e t e r m i n e d . E v e n w i t h r e p e t i t i o n , h o w e v e r , t h e a c c u r a c y o f BUN was ± 5 p e r c e n t . 53 F u r t h e r e r r o r s of unknown magnitude r e s u l t e d because the a n a l y s i s of BUN i s one which determines the l e v e l of f r e e n i t r o g e n i n the blood and e x t r a n i t r o g e n can be r e l e a s e d from p r o t e i n d e n a t u r a t i o n which i s p r o g r e s s i v e with traumatized blood. Thus the a n a l y s i s f o r BUN was done as soon a f t e r the experimental run as p o s s i b l e . The f i r s t sample determined the urea l e v e l (approx-i m a t e l y 3 0 mg %) of normal blood and the second sample determined the combined normal and added urea l e v e l s . Since the added urea was a c c u r a t e l y known these two samples formed a check on each other. For a l l experimental runs the i n i t i a l c o n c e n t r a t i o n of urea was approximately 110 mg %. The g r e a t e s t problem and one which probably gave the l a r g e s t e r r o r i n the experiment was the maintenance of constant flow r a t e s . Even with the l a r g e amount of h e p a r i n added to the blood mixture, c l o t formations at the o u t l e t t h r o t t l i n g v a l v e r e q u i r e d constant readjustment to the flow r a t e i n order to maintain s t e a d y - s t a t e c o n d i t i o n s . The e f f e c t of t h i s readjustment i s u n c e r t a i n and the only s o l u t i o n t o the problem would have been to have higher flow v e l o c i t i e s through the v a l v e . Otherwise c l o t t i n g i n the apparatus was :not a problem d u r i n g the experiment and no unusual f i b r i n b u i l d - u p was n o t i c e d on the d i a l y s i s tubes. Three d i f f e r e n t blood batches were t e s t e d at four d i f f e r e n t f l o w _ r a t e s f o r each batch. In a l l cases the 5 4 a c t i v e l e n g t h o f t h e d i a l y s i s t u b e was 90 i n . and t h e d a t a was r e d u c e d t o d i m e n s i o n l e s s f o r m u s i n g a v a l u e o f 2.2 x - 5 2 10 cm / s e c f o r t h e d i f f u s i o n c o e f f i c i e n t o f u r e a i n b l o o d . The e x p e r i m e n t a l r e s u l t s a r e g i v e n i n F i g u r e 11. The d a t a p o i n t s s c a t t e r a b o u t t h e l i n e c o r r e s p o n d i n g t o a w a l l S h e r w o o d number o f 3.33 w h i c h r e d u c e s t o a membrane perme-a b i l i t y o f 1/70 cm/min f o r t h e V i s k i n g c e l l u l o s e membrane. 3.3 F l o w D u c t i n g Any e x p e r i m e n t a l w o r k w i t h m a n i f o l d s and f l o w d u c t i n g m ust be done u t i l i z i n g a t r a n s p a r e n t and r e p r o d u c i b l e medium t h a t c a n be e a s i l y f a b r i c a t e d i n t o t h e d e s i r e d s h a p e s . A l t h o u g h p l e x i g l a s s has t r a d i t i o n a l l y b e e n u s e d f o r f l o w d u c t i n g t h i s . m a t e r i a l c a n be f o r m e d o n l y w i t h g r e a t d i f f i c u l t y . A f t e r some e x p e r i m e n t a t i o n a m a t e r i a l was f o u n d w h i c h met a l l t h e r e q u i r e m e n t s . A w a t e r c l e a r , p o l y e s t e r c a s t i n g r e s i n ( H a l l c r a f t P l a s t i c s L t d . - E s t e r e x 103) t h a t r e q u i r e d one c a t a l y s t a n d c u r e d r a p i d l y , g a v e c a s t i n g s o f e x c e l l e n t f i n i s h and t o l e r a n c e . The f i n i s h e d m a t e r i a l c o u l d be m a c h i n e d and g l u e d as r e a d i l y as p l e x i g l a s s . I t was f o u n d t h a t b y u s i n g h i g h l y p o l i s h e d a l u m i n u m s p e c i m e n s a s c o r e s f o r t h e m o l d and b y u s i n g a c o m m e r c i a l r e s i n m o l d r e l e a s i n g a g e n t , c o m p l e x s h a p e s c o u l d be c a s t and s e p a r a t e d w i t h e a s e i f a l a y e r o f r e s i n no more t h a n 1/2 i n . t h i c k was a l l o w e d t o c u r e a t one t i m e . T h i c k e r s e c t i o n s i n v a r i a b l y c r a c k e d due to the heat r e l e a s e d i n the exothermic s e t t i n g r e a c t i o n . For t h i c k e r s e c t i o n s l a y e r a f t e r l a y e r of r e s i n was poured w i t h e x c e l l e n t bonding o c c u r r i n g at the i n t e r f a c e between l a y e r s . The f i r s t type of m a n i f o l d t h a t was t r i e d was based on the blood flow arrangement i n the Dow c a p i l l a r y kidney. In t h i s kidney the c a p i l l a r i e s are i n l i n e with the blood flow d i r e c t i o n i n the m a n i f o l d and an abrupt expansion chamber d i s t r i b u t e s blood from a 3/16 i n . m a n i f o l d l i n e t o 10,000 c a p i l l a r i e s of approximately 200 microns diameter. N a t u r a l l y t h i s system a c t s as a t u r b u l e n t mixing chamber. To maintain laminar flow c o n d i t i o n s an expansion s e c t i o n was c o n s t r u c t e d with a w a l l taper based on the B l a s i u s c r i t e r i o n t o prevent s e p a r a t i o n of the f l u i d from the w a l l s of a dr c i r c u l a r duct. The B l a s i u s c r i t e r i o n s t a t e s t h a t /dx ^ 12 /Re f o r no s e p a r a t i o n . The w a l l shape then becomes — < e x p ( ^ | 2 i ) (3-D r ^ r Re o ' o where r i s the r a d i u s of the expansion duct at any d i s t a n c e x from the entrance and Re i s the i n l e t Reynolds number. An expansion m a n i f o l d w i t h maximum w a l l taper p e r m i t t e d by equation 3.1 was t e s t e d at d i f f e r e n t i n l e t Reynolds numbers by i n j e c t i n g a t h i n , continuous dye s t r e a k upstream from the entrance to the expansion s e c t i o n and n o t i n g the down-stream l e n g t h where t u r b u l e n c e commenced. A very d e f i n i t e 56 and r e p r o d u c i b l e l e n g t h c o u l d be d e t e r m i n e d b e c a u s e a t t h e o n s e t o f t u r b u l e n c e t h e t h i n dye s t r e a k a b r u p t l y d i f f u s e d i n t o a c l o u d o f dye w h i c h c o v e r e d t h e e n t i r e c r o s s - s e c t i o n o f t h e m a n i f o l d . The r e s u l t s a r e g i v e n i n F i g u r e 12 f o r t h i s m a n i f o l d . A t t h e same t i m e t h e dye s t r e a k was v a r i e d f r o m i t s m i d s t r e a m p o s i t i o n t o w a r d s t h e w a l l s i n an e f f o r t t o e s t i m a t e t h e v e l o c i t y p r o f i l e i n t h e e x p a n s i o n s e c t i o n . By t i m i n g t h e dye s t r e a k o v e r a measured l e n g t h a v e l o c i t y b u l g e i n m i d s t r e a m was a p p a r e n t . However, t h e m a g n i t u d e o f t h e m i d s t r e a m v e l o c i t y t o edge v e l o c i t y was so g r e a t t h a t , f o r any r e a s o n a b l e e x p a n s i o n l e n g t h , a v e r y n o n - u n i f o r m f l u i d d i s t r i b u t i o n t o t h e c a p i l l a r i e s w o u l d r e s u l t . Any u s e o f e x p a n s i o n s e c t i o n s i n t h e m a n i f o l d s y s t e m was t h e r e f o r e abandoned. An a l t e r n a t i v e method o f d i s t r i b u t i n g f l u i d t o a number o f d u c t s i s t o i n c l i n e t h e m a n i f o l d a t an a n g l e t o t h e d u c t s and t h u s t u r n t h e f l u i d ' s d i r e c t i o n o f t r a v e l . The e n e r g y b a l a n c e t h e o r y p r e d i c t e d a m a n i f o l d s h a p e , f o r a r e c t a n g u l a r o u t f l o w a r e a , w h i c h had a l i n e a r d e c r e a s e i n a r e a w i t h l e n g t h . T h e r e f o r e t h e f i r s t m a n i f o l d was c o n s t r u c -t e d w i t h t h i s s hape and w i t h e q u i v a l e n t i n l e t and o u t l e t a r e a s so as t o m i n i m i z e v e l o c i t y c hanges i n t h e m a n i f o l d . An a s p e c t r a t i o o f 10:1 was c h o s e n f o r t h e o u t f l o w a r e a b e c a u s e t h i s r a t i o o f f e r e d t h e b e s t compromise between d u c t l e n g t h and number o f c a p i l l a r i e s w h i c h c o u l d be c o n t a i n e d 57 i n the area. An aluminium core was hand f i l e d to the r e q u i r e d shape (which i n c l u d e d area compensation f o r the tube s l o t ) with an i n l e t diameter of 1 / 2 i n . and an o u t l e t area of dimensions 1 / 8 by 1 1 / 4 i n . The core was p l a c e d i n a c i r c u l a r mold and the m a n i f o l d c a s t u s i n g the p o l y e s t e r c a s t i n g r e s i n . To model the e f f e c t of the c a p i l l a r i e s i n the a r t i f i c i a l kidney s p e c i a l low t a p e r , g l a s s , microhematocrit tubes were chosen to be i n s e r t e d i n the r e c t a n g u l a r outflow area of the m a n i f o l d . These tubes ranged between 0 . 1 1 3 cm and 0 . 1 1 8 cm i n t e r n a l diameter, had a w a l l t h i c k n e s s of 0 . 0 2 cm, and were 7 . 5 cm long. A bundle of 44 tubes arranged i n a double row was potted i n t o the m a n i f o ld area with c a u l k i n g s i l a s t i c i n a manner so as to ensure t h a t the entrance to the c a p i l l a r i e s was f l u s h with the i n t e r i o r of the m a n i f o l d . For "a complete study of the m a n i f o l d i t was decided to check the pressure d i s t r i b u t i o n and flow d i s t r i b u t i o n by d i r e c t measurement. T h i s r e q u i r e d a source of constant pressure f l u i d and a method to monitor flow r a t e s . Because of the l a r g e volume of f l u i d r e q u i r e d to reach s t e a d y - s t a t e c o n d i t i o n s , r e c i r c u l a t e d tap water was used. Some p r e l i m i n -ary work showed t h a t a i r bubbles and s u r f a c e t e n s i o n e f f e c t s caused problems with the flow i n the g l a s s c a p i l l a r y tubes.. The f l u i d p ressure i n the m a n i f o l d was g e n e r a l l y i n s u f f i c i e n t to d i s l o d g e a i r bubbles caught i n the c a p i l l a r i e s . A l s o , i f the c a p i l l a r i e s exhausted t h e i r flow to f r e e a i r , s u r f a c e t e n s i o n was s u f f i c i e n t t o stop the flow through some c a p i l l a r -i e s f o r low flow r a t e s . A c c o r d i n g l y , the e n t i r e m a n i f o l d 58 s e c t i o n was immersed i n a water bath and a bubble t r a p was employed i n the flow c i r c u i t . Water was allowed to s e t t l e f o r one day bef o r e proceeding w i t h an experimental run and the r e c i r c u l a t i n g system was designed t o e l i m i n a t e bubble p r o d u c t i o n as much as p o s s i b l e . A twelve tube i n c l i n e d manometer bank was c o n s t r u c t e d to read m a n i f o l d p r e s s u r e s . One tube i n d i c a t e d the l e v e l of water i n the bath ( t h i s datum l e v e l was standard through-out the t e s t s ) w h i l e the others were connected to 21 ga. hypodermic needles mounted f l u s h w i t h the inner s u r f a c e of the m a n i f o l d . The tubes i n the manometer bank were c a l i -b r a t e d over t h e i r e n t i r e l e n g t h t o account f o r s u r f a c e t e n s i o n e r r o r s . Flow r a t e s were measured by two rotameters b u i l t to cover the range of r a t e s r e q u i r e d i n the experiments. These flowmeters c o n s i s t e d of a s t a i n l e s s s t e e l b a l l mounted i n a tapered tube and covered the range from 10 cc/min to 1500 cc/min. The u n i t s were c a l i b r a t e d by stopwatch and graduate and were found to g i v e good r e p e a t a b i l i t y with very l i t t l e temperature dependence. The f l o w r a t e s were a d j u s t e d by t h r o t t l i n g downstream of the flowmeters. Lower flow r a t e s were checked every run by c o l l e c t i n g the d i s c h a r g e d f l u i d over a long p e r i o d of time. Flow v i s u a l i z a t i o n was accomplished by i n j e c t i n g a very weak f l u o r e s c e i n dye upstream from the m a n i f o l d . A very f i n e g l a s s i n j e c t i o n tube was drawn from the hematocrit tube g l a s s . T h i s tube was i n v e r t e d i n t o the water stream through the connecting tube w a l l approximately 20 tube diameters upstream from the m a n i f o l d . A f l e x i b l e s e a l i n the connecting tube w a l l allowed the dye i n j e c t o r to be t r a v e r s e d across the flow. Dye was fed at a constant r a t e through the i n j e c t o r t o cause a continuous dye s t r e a k downstream to the m a n i f o l d . I t was found t h a t the b e s t dye s t r e a k r e s u l t e d i f the metered r a t e of dye i n j e c t i o n was constant. T h e r e f o r e , a v a r i a b l e speed p e r f u s i o n pump u t i l i z i n g a standard 2 ml s y r i n g e f e e d i n g d i r e c t l y to the i n j e c t o r was c o n s t r u c t e d and i t worked i n a s a t i s f a c t o r y manner. The flow c i r c u i t c o n s i s t e d of a constant head tank h o l d i n g approximately 3 g a l . of water, a flow meter, a bubbl t r a p , approximately 50 i n . of 1/2 in . diameter p l e x i g l a s s t u b i ng f o r flow s t r a i g h t e n i n g and c o n t a i n i n g the dye i n j e c -t i o n apparatus, the m a n i f o l d w i t h c a p i l l a r i e s immersed i n a constant head water bath, the pressure measuring apparatus, and a 3 g a l . r e s e r v o i r from which water was pumped back to the constant head tank. The water temperature was monitored i n the m a n i f o l d bath. The flow c i r c u i t i s g i v e n i n F i g u r e 13. As o n l y one dye s t r e a k at a time c o u l d be produced i n the m a n i f o l d , to o b t a i n a complete flow p a t t e r n i t was necessary to photograph each s t r e a k and then to superimpose the photographs. The dye s t r e a k was v i s i b l e t o the camera 60 through a s e t of i n t e r f a c e s ; a i r - w a t e r , w a t e r - c a s t i n g r e s i n , and again c a s t i n g r e s i n - w a t e r . At a l l times the camera was kept normal to the f r e e water s u r f a c e i n the m a n i f o l d bath. The m a n i f o l d i t s e l f was r o t a t e d to o b t a i n the r e q u i r e d t h r e e - d i m e n s i o n a l p i c t u r e of the dye s t r e a k s . The e f f e c t of the index of r e f r a c t i o n at the combined i n t e r f a c e s i s given i n Appendix IV. F o r t u n a t e l y , very l i t t l e c o r r e c t i o n to the photographed dye streak was necessary i n order to determine i t s a c t u a l p o s i t i o n i n the m a n i f o l d . Photographic p a r a l l a x was n e g l i g i b l e . The f i r s t m a n i f o l d t h a t was t e s t e d completely, a f t e r a number of prototypes had been c o n s t r u c t e d , had the c a p i l l a r -i e s i n c l i n e d at 9° to the flow d i r e c t i o n as d e s c r i b e d i n F i g u r e 14, Model A. T h i s i n c l i n a t i o n gave a uniform upper s u r f a c e to the m a n i f o l d and i t was thought t h a t t h i s would r e q u i r e l e s s t u r n i n g of the f l u i d . There was p h y s i c a l space f o r only s i x pressure taps so a check was performed to see t h a t the m a n i f o l d pressure corresponded to the i n d i v i d u a l c a p i l l a r y flow r a t e s . To o b t a i n the c a p i l l a r y flow r a t e s the m a n i f o l d was connected to the constant head tank but removed from the water bath. C a p i l l a r i e s , taken i n p a i r s , were allowed to exhaust a t atmospheric p r e s s u r e i n t o c o l l e c t i n g ducts and the f l u i d c o l l e c t e d from each p a i r of c a p i l l a r i e s was timed f o r a measured volume. The pressure f i e l d was then obtained by r e p l a c i n g t h e m a n i f o l d i n t h e w a t e r b a t h a n d r u n n i n g t h e s y s t e m a t t h e same t o t a l f l o w r a t e . I t must be n o t e d t h a t s t a b l e p r e s s u r e r e a d i n g s w e r e o b t a i n e d o n l y a f t e r one h a l f h o u r a t c o n s t a n t f l o w r a t e . The p r e s s u r e f i e l d i n t h e m a n i f o l d a n d c o r r e s p o n d i n g c a p i l l a r y f l o w r a t e s a r e g i v e n i n F i g u r e 16. As t h e c o r r e s p o n d e n c e was f a i r l y g o o d and b e c a u s e o f t h e d i f f i c u l t y i n o b t a i n i n g t h e c a p i l l a r y f l o w r a t e s i n d i v i d u a l y ( p r i m a r i l y due t o s u r f a c e t e n s i o n e f f e c t s w h i c h made many r e p e t i t i o n s n e c e s s a r y f o r c o n s i s t e n c y ) o n l y one r u n was made a t f a i r l y h i g h i n l e t R e y n o l d s number. H a v i n g e s t a b l i s h e d t h a t f o r M o d e l A t h e f l o w d i s t r i -b u t i o n was a p p r o x i m a t e l y u n i f o r m t h e s t r e a m l i n e p a t t e r n i n t h e m a n i f o l d was d e t e r m i n e d by d y e i n j e c t i o n . To p r e v e n t e x c e s s i v e d i f f u s i o n o f t h e d y e s t r e a k i n t h e l o w v e l o c i t y r e g i o n s n e a r t h e w a l l s o f t h e m a n i f o l d an i n l e t R e y n o l d s number o f a p p r o x i m a t e l y 750 was u s e d . Dye was i n j e c t e d a t p o i n t s e n c o m p a s s i n g t h e e n t i r e i n l e t a r e a and p h o t o g r a p h s w e re t a k e n o f e a c h dye s t r e a k . A c e n t r e l i n e c r o s s - s e c t i o n t a k e n i n t h e p l a n e o f t h e c a p i l l a r i e s i s g i v e n i n F i g u r e 17. D i a g r a m m a t i c r e p r e s e n t a t i o n o f t h r e e d i m e n s i o n a l s t r e a m l i n e d i s t r i b u t i o n i s d i f f i c u l t . I n s t e a d , F i g u r e 18 d e m o n s t r a t e s t h e p o s i t i o n i n t h e e n t r a n c e t o t h e m a n i f o l d o f t h e s t r e a m -l i n e s as a f u n c t i o n o f t h e c a p i l l a r i e s u s e d t o e x i t t h e m a n i f o l d , b a s e d on a p e r c e n t a g e o f t h e m a n i f o l d l e n g t h . N o t e t h a t t h e f l u i d n e a r t h e w a l l s e x i t s f i r s t . D e t e r m i n i n g t h e s t r e a m l i n e p a t t e r n v e r y n e a r t h e m a n i f o l d w a l l was i n -c o n c l u s i v e as the i n j e c t o r c o u l d not be placed very near the w a l l s and the dye tended t o d i f f u s e b e f o r e i t reached the c a p i l l a r i e s . Otherwise the dye s t r e a k l i n e s were w e l l d e f i n e d and remained i n t a c t u n t i l they e x i t e d the c a p i l l a r i e s and entered the water bath. There were no areas i n t h i s m a n i f o l d where s t a g n a t i o n or turbu l e n c e occurred. Next, pressure measurements i n the m a n i f o l d of Model A were made at d i f f e r e n t i n l e t Reynolds numbers to check i f the flow d i s t r i b u t i o n was a f f e c t e d by flow r a t e . The p r e s -sure d i s t r i b u t i o n s are given i n F i g u r e 19. The o v e r a l l p ressure drop of the ma n i f o l d and c a p i l l a r i e s i s given by the f i r s t p ressure tap (opposite the s t a r t of the outflow a r e a ) . T h i s o v e r a l l pressure drop was checked by tapping i n t o the feeder l i n e approximately four inches upstream from the f i r s t p ressure tap. No d i s c e r n i b l e p r essure d i f f e r e n c e was n o t i c e d between the feeder l i n e p r e ssure and the f i r s t p ressure tap f o r the complete Reynolds number range. The system pressure drop i s p l o t t e d as a r a t i o of *VQ 2 versus Reynolds number on l o g a r i t h m i c s c a l e s i n F i g u r e 2 0 where h i s the pressure drop i n cm.w.G. f o r the flow r a t e Q . h 2 1 Although t h e o r e t i c a l l y f o r laminar flow / Q = C(Re) ; t h a t i s , the pressure drop i s p r o p o r t i o n a l to the flow r a t e f o r a given system, i n the case of Model A m a n i f o l d , which had 44 ou t l e t , c a p i l l a r i e s , the pressure drop was determined 1*1. 2 0 87 2 5 to be, /Q = 20(Re) ' (sec /cm ). Appendix V c o n t a i n s 63 a d e r i v a t i o n f o r t h e p r e s s u r e d r o p i n t h e c a p i l l a r i e s . Momentum t h e o r y p r e d i c t e d a m a n i f o l d s h a p e w h i c h d e p e n d e d p r i m a r i l y on some e x p e r i m e n t a l d a t a c o n c e r n i n g t h e momentum and f r i c t i o n l o s s e s i n t h e s y s t e m . The m a n i f o l d A x 2 6 a r e a r e l a t i o n s h i p was e x p r e s s e d g e n e r a l l y a s /A = (1 -=-) O Li w h e r e 0 ^ 6 ^ 1 (6 = ^ for t h e M o d e l A m a n i f o l d ) . I n o r d e r t o accommodate more c a p i l l a r y t u b e s i n t h e a r t i f i c i a l k i d n e y f o r a g i v e n m a n i f o l d s i z e i t was d e c i d e d t o c o n s t r u c t f u r t h e r m a n i f o l d s o f 1/2 i n . d i a m e t e r b u t w i t h o u t f l o w a r e a s o f 1/4 by 2 1/2 i n . i n s t e a d o f 1/8 by 1 1/4 i n . a s i n M o d e l A, t h u s p r e s e r v i n g t h e l e n g t h t o w i d t h r a t i o b u t g i v i n g f o u r t i m e s t h e o u t f l o w a r e a . By c o n s t r u c t i n g m a n i f o l d s w i t h 6 = 0, 0.25, 0.5 and 1.0 some i d e a o f t h e v a l i d i t y o f momentum t h e o r y as a p p l i e d t o m a n i f o l d d e s i g n c o u l d be o b t a i n e d . A c c o r d i n g l y , f o u r m a n i f o l d s w e r e c a s t w i t h t h e p o l y e s t e r c a s t i n g r e s i n a s d e s c r i b e d i n F i g u r e s 14 and 15 ( M o d e l s B, C, D, E ) . E a c h m a n i f o l d h a d 160 c a p i l l a r y h e m a t o c r i t t u b e s p o t t e d w i t h s i l a s t i c i n t o t h e o u t f l o w a r e a i n f o u r p a r a l l e l r o w s . E l e v e n p r e s s u r e t a p s w e r e p l a c e d i n e a c h m o d e l . The c o r e s f o r t h e m a n i f o l d c a s t i n g s w e r e m a c h i n e d f r o m a l u m i n i u m s t o c k by p r o g r a m m i n g a n u m e r i c a l l y c o n t r o l l e d m i l l i n g m a c h i n e t o sh a p e t h e p r o f i l e s t h u s s a v i n g t i m e and e l i m i n a t i n g e r r o r s a s s o c i a t e d w i t h hand f i l i n g t o t e m p l a t e s . I n f i n i s h e d f o r m t h e c o r e s w e r e a c c u r a t e t o a p p r o x i m a t e l y 0.005 i n . o f t h e t h e o r e t i c a l s h a p e . U n l i k e M o d e l A, t h e o u t f l o w r e g i o n i n t o t h e c a p i l l a r i e s was s e t n o r m a l t o t h e i n l e t . 64 The p r e s s u r e d i s t r i b u t i o n i n e a c h m a n i f o l d f o r a r a n g e o f i n l e t R e y n o l d s numbers i s g i v e n i n F i g u r e 2 1 and t h e r e d u c e d t o t a l p r e s s u r e d r o p s f o r e a c h m o d e l e x c e p t M o d e l E a r e g i v e n i n F i g u r e s 2 2 , 2 3 a n d 2 4 . A u n i f o r m p r e s s u r e d i s t r i b u t i o n was o b s e r v e d i n a l l m a n i f o l d s e x c e p t M o d e l E (6 = 1 . 0 ) w h i c h h a d a c o n s i d e r a b l e p r e s s u r e d r o p n e a r t h e end o f t h e m a n i f o l d . B e c a u s e o f t h i s l a r g e p r e s s u r e d r o p M o d e l E was n o t i n c l u d e d i n t h e t o t a l p r e s s u r e d r o p v e r s u s f l o w r a t e t e s t s . From t h e p l o t s o f * V Q 2 v e r s u s Re f o r M o d e l s B, C and D t h e r e was no d i s c e r n i b l e d i f f e r e n c e i n e f f i c i e n c y a s t h e p r e s s u r e d r o p s f o r a l l m a n i f o l d s was e q u a l t o , h / Q 2 = 1 2 ( R e ) " 1 ( s e c 2 / c m 5 ) . A l l p r e s s u r e m e a s u r e m e n t s made on M o d e l s A, B, C, D and E w e r e t a k e n on t h e u p p e r s u r f a c e o f t h e m a n i f o l d o p p o s i t e t h e c a p i l l a r y e n t r a n c e s . To c h e c k t h e m a g n i t u d e o f t h e p r e s s u r e g r a d i e n t a c r o s s t h e d i a m e t e r o f t h e m a n i f o l d e l e v e n more p r e s s u r e t a p s - w e r e i n s t a l l e d i n M o d e l C i n a manner s u c h t h a t p r e s s u r e m e a s u r e m e n t s w o u l d be o b t a i n e d a t t h e e n t r a n c e t o t h e c a p i l l a r i e s . F o r a number o f d i f f e r e n t f l o w r a t e s n o . d i s c e r n i b l e p r e s s u r e g r a d i e n t was o b s e r v e d . As t h e l i m i t o f r e s o l u t i o n o f t h e manometer was 0 . 0 2 cm. W . G . t h e p r e s s u r e g r a d i e n t was l e s s t h a n t h a t amount. W i t h t h e m a n i f o l d s b a s e d on M o d e l s B, C, a n d D s e e m i n g t o h a v e t h e same p r e s s u r e c h a r a c t e r i s t i c s i t was i n s t r u c t i v e t o o b s e r v e t h e s t r e a m l i n e p a t t e r n i n e a c h by dye i n j e c t i o n . M o d e l B had a v e r y l a r g e r e g i o n o f s t a g n a t i o n a t the end of the m a n i f o l d where the flow r e v e r s e d d i r e c t i o n from the ce n t r e of the man i f o l d and flowed down the w a l l s t o the c a p i l l a r y e ntrances. F i g u r e 25 di a g r a m m a t i c a l l y i l l u s t r a t e s t h i s e f f e c t . A l l s t r e a m l i n e s above (a) i n t h i s diagram stagnated and a l l those below (a), such as s t r e a m l i n e (b), e x i t e d i n a normal f a s h i o n . Model C a l s o had a r e g i o n of s t a g n a t i o n at the end of the man i f o l d although not as l a r g e as i n Model B and no r e v e r s e flow was observed. Model C i s de s c r i b e d i n F i g u r e 26. However, the s t r e a m l i n e s i n the upper r e g i o n of the ma n i f o l d tended to be s l i g h t l y u n s t a b l e as i f a f f e c t e d by an adverse pressure g r a d i e n t . Model D gave s t r e a m l i n e s which resembled e x a c t l y those obtained f o r Model A. There were no re g i o n s of s t a g n a t i o n and no areas of t u r b u l e n c e or i n s t a b i l i t y . The s t r e a m l i n e s i n Model E were s t a b l e and no s t a g n a t i o n was observed but the uneven d i s t r i b u t i o n of f l u i d t o the c a p i l l a r i e s was e v i d e n t as i t was i m p o s s i b l e to get a dye st r e a k to e x i t the ma n i f o l d i n the end c a p i l l a r i e s . The dye st r e a k s t u d i e s showed c o n c l u s i v e l y t h a t Model A or Model D (6 = 0.5) gave the bes t f l u i d d i s t r i b u t i o n f o r use i n a r t i f i c i a l kidneys. One f a c t o r t h a t determines the m a n i f o l d shape i n the momentum-balance theory i s the amount of momentum t r a n s f e r r e d from the m a n i f o l d t o the c a p i l l a r i e s . T h i s amount i s expressed as a f r a c t i o n of the maximum amount of momentum th a t c o u l d be t r a n s f e r r e d and was given the symbol 3 i n the t h e o r y . By o b s e r v i n g t h e s t r e a m l i n e s i n the m a n i f o l d as they e n t e r e d the c a p i l l a r i e s an e s t i m a t e o f t h e momentum t r a n s f e r r e d c o u l d be made. I f the s t r e a m l i n e s c u r v e d com-p l e t e l y b e f o r e e n t e r i n g t h e c a p i l l a r i e s t h e n a l l t h e momentum was g i v e n up t o t h e m a n i f o l d and 3 = 0 . However, i f the c u r v a t u r e o f t h e s t r e a m l i n e s o c c u r r e d i n the c a p i l l a r i e s t hen a l l t h e momentum was t r a n s f e r r e d from t h e m a n i f o l d and 3 = 1 . A s i m p l e l i n e a r t a p e r m a n i f o l d was machined from a p i e c e o f p l e x i g l a s s h a v i n g an i n l e t d i a m e t e r o f 1 / 2 i n . and t a p e r i n g t o a d i a m e t e r o f 3 / 1 6 i n . over a 3 i n . l e n g t h . Seven 3 / 1 6 i n . d i a m e t e r h o l e s were d r i l l e d , e q u a l l y spaced, a l o n g t h e m a n i f o l d . T h i s m a n i f o l d was mounted i n the water b a t h and dye i n j e c t e d upstream so t h a t t h e s t r e a m l i n e s e n t e r i n g the l a t e r a l h o l e s c o u l d be photographed. F i g u r e 2 7 shows some o f t h e s e s t r e a m l i n e s . As can be seen, most o f the c u r v a t u r e o f the s t r e a m l i n e s took p l a c e i n the l a t e r a l s . The same e f f e c t was n o t i c e d on a l l t h e c a p i l l a r y tube m a n i f o l d s e x c e p t t h a t t h e c u r v a t u r e was more pronounced t h a n i n the 3 / 1 6 i n . l a t e r a l h o l e s . Thus, f o r l a m i n a r f l o w c o n d i t i o n s , a l l the momentum o f each s t r e a m l i n e was t r a n s -f e r r e d from the m a n i f o l d t o t h e l a t e r a l s and 3 = 1 . C H A P T E R 4 D I S C U S S I O N 67 4. DISCUSSION The experimental data taken from a p a t i e n t undergoing h e m o d i a l y s i s i n d i c a t e d t h a t the a r t e r i a l p r e ssure t o the d i a l y z e r had approximately a 10 percent p u l s a t i l e component on the mean p r e s s u r e . The flow r a t e v a r i e d by n e a r l y the same amount. The venous pressure was constant which i n d i -cated t h a t the K i i l d i a l y z e r damped out the p u l s a t i l e components of p r e s s u r e . The magnitude of the p u l s a t i l e components was judged t o be s m a l l enough t h a t s t e a d y - s t a t e a n a l y s i s of mass t r a n s f e r would be s u f f i c i e n t l y accurate f o r design purposes. M e r r i l l [20] gave an approximate r e l a t i o n f o r the y i e l d s t r e s s term i n Casson*s equation f o r blood v i s c o s i t y based on the weight percent of f i b r i n o g e n p r e s e n t i n the plasma (C^) and the hematocrit of the blood (H), T q = 3 x l 0 ~ 5 C 2 (H-5) 3 (dynes/cm 2) (4.1) which f o r normal blood i n d i c a t e s t h a t the y i e l d s t r e s s i s : 2 approximately;. 0.05 dynes/cm . Other authors r e p o r t e d v a l u e s 2 2 f o r the y i e l d s t r e s s from 0.04 dynes/cm to 0.10 dynes/cm . During these experimental d e t e r m i n a t i o n s of the y i e l d s t r e s s 2 the observed w a l l shear s t r e s s v a r i e d between 1 dyne/cm 2 and 1000 dynes/cm f o r d i f f e r e n t flow r a t e s i n v a r i o u s c a p i l l a r i e s . As the s l u g r a d i u s p i n the non-Newtonian theory may be expressed, u s i n g equations 2.30, by 68 (4.2) i t i s obvious t h a t p Q < 0.1 f o r any i > r a c t i c a l flow r a t e w i t h the upper l i m i t being reached o n l y a t very low flow r a t e s . For low valu e s of p Q the theory p r e d i c t e d l i t t l e d i f f e r e n c e between the mass t r a n s f e r r a t e s f o r a Newtonian and non-Newtonian f l u i d . However, i t must be noted t h a t f o r a Casson model f l u i d , which degenerates i n t o a Newtonian f l u i d when p Q = 0, the pressure induced flow r a t e i s given by, 4 p 4 TrApR ., , 4 K o 16 , — , . . „. = STJV- ( 1 + 3 po - 21 " ^  ^ > ( 4' 3 ) a I t can be seen from t h i s equation t h a t the flow r a t e i s v e r y s e n s i t i v e t o the value of p Q and thus, although the non-Newtonian aspects of the f l u i d can be n e g l e c t e d f o r mass t r a n s f e r , the complete Casson equation must be used f o r f l u i d flow. The experimental r e s u l t s f o r mass t r a n s f e r from a s i n g l e tube as p l o t t e d i n F i g u r e 4 show a reasonable c o r r e l -a t i o n w i t h the t h e o r e t i c a l p r e d i c t i o n f o r the bulk c o n c e n t r a -t i o n . The da t a , except f o r three p o i n t s , c l u s t e r s c l o s e l y about the l i n e f o r a Newtonian f l u i d and from the gen e r a l 6 9 f o r m o f t h e d a t a i t c a n be c o n c l u d e d t h a t , f o r mass t r a n s f e r a n a l y s i s , t h e u s e o f a N e w t o n i a n f l u i d a p p r o x i m a t i o n i s j u s t i f i e d . The t h e o r e t i c a l p r e d i c t i o n o f p Q t e n d s t o c o n f i r m t h i s . H a v i n g e s t a b l i s h e d t h a t t h e s t e a d y - s t a t e , N e w t o n i a n f l u i d a s s u m p t i o n s w i l l i n d e e d p r e d i c t mass t r a n s f e r r a t e s f o r h e m o d i a l y s i s , a t o t a l s y s t e m a n a l y s i s f o r a l a r g e number o f c a p i l l a r i e s c o n n e c t e d i n p a r a l l e l was p r o p o s e d . F i r s t , i n t e r m s o f e f f i c i e n c y i n r e m o v i n g i m p u r i t i e s , t h e number o f c a p i l l a r i e s r e q u i r e d and t h e i r s i z e was c o r r e l a t e d t o t h e s y s t e m f i x e d p a r a m e t e r s s u c h as b l o o d f l o w r a t e , a v a i l a b l e p r e s s u r e d r o p , h o l d - u p v o l u m e o f b l o o d , and membrane p e r m e a b i l -i t y . T h i s method o f d e t e r m i n i n g t h e d i a l y z e r d i m e n s i o n s ha d b e e n p r e v i o u s l y u s e d by Babb and G r i m s r u d [27] f o r a f l a t p l a t e d i a l y z e r b u t t h e a n a l y s i s s u f f e r s f r o m t h e n e c e s s i t y o f s t i p u l a t i n g t h e e f f i c i e n c y o f mass t r a n s f e r b e f o r e c a l c u l a t i n g t h e o t h e r d i a l y z e r d i m e n s i o n s . An i m p r o v e m e n t on t h i s m ethod o c c u r r e d when t h e mass t r a n s f e r e f f i c i e n c y was o p t i m i z e d w i t h r e s p e c t t o t h e membrane a r e a . I t was t h o u g h t t h a t b y o p t i m i z i n g t h e mass t r a n s f e r -a r e a r e l a t i o n s h i p a r e a l i s t i c o p e r a t i n g c o n d i t i o n f o r t h e d i a l y z e r c o u l d be o b t a i n e d ; t h a t i s , a maximum amount o f i m p u r i t y t r a n s p o r t e d f o r a g i v e n d i a l y z e r s i z e . I n t e r e s t -i n g l y , t h e o p t i m i z a t i o n p r o c e d u r e i n d i c a t e d t h a t t h e b e s t mass t r a n s f e r , e f f i c i e n c y p e r u n i t a r e a e x i s t e d when t h e 70 mass t r a n s f e r e f f i c i e n c y was c o n s t a n t a t 70 p e r c e n t f o r t h e c o m p l e t e r a n g e o f w a l l S h e r w o o d n u m b e r s . The d e s i g n o f a d i a l y z e r f r o m t h e o p t i m i z e d t h e o r y , as d e m o n s t r a t e d i n A p p e n d i x I I , r e q u i r e d t h a t t h e c l e a r a n c e ( t h e i m p u r i t y r e m o v a l r a t e ) be e s t a b l i s h e d a s an o p e r a t i n g c o n d i t i o n . U t i l i z i n g t h e f a c t t h a t t h e e f f i c i e n c y i s n e a r l y c o n s t a n t f o r any v a l u e s o f t h e s y s t e m p a r a m e t e r s , t h e b l o o d f l o w r a t e t o t h e d i a l y z e r i s f o u n d b y t h e e q u a t i o n , Qk = M / c ^ E , w h i c h i s s i m p l y t h e c l e a r a n c e d i v i d e d by t h e s y s t e m e f f i c i e n c y . W i t h t h i s b l o o d f l o w r a t e and t h e o t h e r s y s t e m p a r a m e t e r s , s u c h a s b l o o d h o l d - u p v o l u m e and membrane p e r m e a b i l i t y , t h e r e q u i r e d d i a m e t e r o f t h e c a p i l l a r i e s i s d e t e r m i n e d . G i v e n t h e p r e s s u r e d r o p a c r o s s t h e d i a l y z e r , t h e number and l e n g t h o f t h e c a p i l l a r i e s c a n be c a l c u l a t e d . I n p r a c t i c e , t h e d i a l y z e r w o u l d be d e s i g n e d f o r t h e w o r s t p o s s i b l e p a t i e n t c o n d i t i o n s . I f t h e minimum c l e a r a n c e t h a t w o u l d be r e q u i r e d by any p a t i e n t i s e s t a b l i s h e d and t h e minimum a v a i l a b l e p r e s s u r e d i f f e r e n c e ( a r t e r i a l - v e n o u s ) i s known t h e n a d i a l y z e r f o r t h o s e c o n d i t i o n s w o u l d d e l i v e r a t l e a s t t h e minimum c l e a r a n c e f o r any p a t i e n t e v e n t h o u g h t h e mass t r a n s f e r e f f i c i e n c y w o u l d d e c r e a s e as t h e b l o o d f l o w r a t e i n c r e a s e d . A s i m p l e method o f d e t e r m i n i n g t h e t h e o r e t i c a l o p e r a t i n g e f f i c i e n c y f o r any g i v e n d i a l y z e r i s a v a i l a b l e f r o m t h e t h e o r y . G i v e n t h e d i a l y z e r d i m e n s i o n s , f i r s t c a l c u l a t e the w a l l Sherwood number, Sh = PD/2P. Then w determine the f i r s t eigenvalue f o r t h i s w a l l Sherwood number from F i g u r e 28. The o v e r a l l Sherwood number i s approximately, S h Q = I/4 • T n e t h e o r e t i c a l mass t r a n s f e r e f f i c i e n c y then becomes, 2TrSh NxP E = 1 - exp [ 2 ] (4.4) u b A l t e r n a t i v e l y the same i n f o r m a t i o n c o u l d have been determined from F i g u r e 4 . The mass t r a n s f e r a n a l y s i s n e g l e c t e d d i a l y s a t e f i l m r e s i s t a n c e and assumed t h a t s u f f i c i e n t d i a l y s a t e e x i s t e d to have uniform d i a l y s a t e c o n c e n t r a t i o n . For t u r b u l e n t flow c o n d i t i o n s over the c a p i l l a r i e s the d i a l y s a t e f i l m r e s i s -tance w i l l be n e g l i g i b l e compared to the blood f i l m r e s i s -tance and membrane r e s i s t a n c e . But, i n p r a c t i c e , d i a l y s a t e flow r a t e s are u s u a l l y only about f i v e times the blood flow r a t e which i s not s u f f i c i e n t to ensure uniform d i a l y s a t e c o n c e n t r a t i o n s . Michaels [24] gave the formulae f o r three common d i a l y s a t e flow c o n d i t i o n s — p a r a l l e l flow, f u l l y mixed flow, and counterflow. An estimate of the magnitude of e r r o r i n h e r e n t i n n e g l e c t i n g the d i a l y s a t e flow r a t e can be determined from the f a c t t h a t f o r any d i a l y s a t e flow c o n d i t i o n s the e f f i c i e n c y w i l l never d e t e r i o r a t e to a value worse than t h a t given by. 72 1 - e x p ( - 4 S h E = _ ° _ ( 4 < 5 ) 1 + y b / Q d I f t h e d i a l y s a t e f l o w r a t e i s f a i r l y l o w some i m p r o v e m e n t i n o p e r a t i n g e f f i c i e n c y c a n be r e a l i z e d b y u s i n g c o u n t e r f l o w c o n d i t i o n s . The p r o b l e m o f d i a l y s a t e d i s t r i b u t i o n t o a l a r g e number o f b u n d l e d c a p i l l a r i e s t o r e a l i z e t u r b u l e n t f l o w c o n d i t i o n s p r o m p t e d t h e c h o i c e o f t h e r e c t a n g u l a r c o n f i g u r a -t i o n t e s t e d i n t h e e x p e r i m e n t a l m a n i f o l d s . L i p p s e t a l . [28] r e p o r t e d t h a t u n d e r medium d i a l y s a t e f l o w r a t e s t h e c a p i l l a r i e s i n t h e i r h o l l o w f i b r e a r t i f i c i a l k i d n e y t e n d e d t o f a n o u t i n t o t h e d i a l y s a t e chamber s u c h t h a t e a c h c a p i l l a r y was s e p a r a t e d f r o m i t s n e i g h b o u r s . Thus t h e c h o i c e o f 10:1 a s p e c t r a t i o r e c t a n g u l a r b u n d l e s o f c a p i l l a r i e s a r r a n g e d j u d i c i o u s l y i n t h e d i a l y s a t e chamber w o u l d p r o b a b l y a l l o w o p t i m u m d i a l y s a t e d i s t r i b u t i o n w h i l e s i m u l t a n e o u s l y a l l o w i n g t h e b l o o d d i s t r i b u t i o n m a n i f o l d t o be d e s i g n e d f o r t h e b e s t l a m i n a r f l o w c o n d i t i o n s . B e c a u s e i t was d e t e r m i n e d e x p e r i m e n t a l l y t h a t any e x p a n s i o n s e c t i o n s i n t h e b l o o d f l o w l i n e s w o u l d i n d u c e s e p a r a t i o n and t u r b u l e n c e , t h e m a n i f o l d i n l e t d i a m e t e r m u s t be r e s t r i c t e d t o a p p r o x i m a t e l y t h e d i a m e t e r o f t h e b l o o d f e e d l i n e s t o t h e a r t i f i c i a l k i d n e y ( b e t w e e n 3/16 i n . a n d 1/4 i n . d i a . ) . Due t o t h e l a r g e number o f c a p i l l a r i e s r e q u i r e d t h e r e must be a number o f m a n i f o l d s c o n n e c t e d i n p a r a l l e l t o accommodate t h e c a p i l l a r i e s w i t h i n t h e d i m e n s i o n s g i v e n i n t h e t h e o r y . T h e s e m a n i f o l d s w o u l d r e c e i v e b l o o d f r o m a s i n g l e b l o o d l i n e by u t i l i z i n g s t a n d a r d Y - c o n n e c t o r s w h i c h h a v e b e e n u s e d s u c c e s s f u l l y i n t h e two l a y e r K i i l d i a l y z e r w i t h o u t c a u s i n g c l o t f o r m a t i o n s a t t h e c o n n e c t o r s . H a v i n g e s t a b l i s h e d an o u t f l o w s i z e a nd s h a p e and k n o w i n g t h e i n l e t d i a m e t e r o n l y t h e w a l l s h a p e o f t h e m a n i f o l d was u n d e t e r m i n e d . F o r c o n v e n i e n c e i n m a n u f a c t u r i n g t h e m a n i f o l d s i t was d e c i d e d t o u s e a c i r c u l a r c r o s s - s e c t i o n w h e r e v e r p o s s i b l e . The c r o s s - s e c t i o n a r e a was t h e n t h e f a c t o r t o be d e t e r m i n e d as a f u n c t i o n o f t h e m a n i f o l d l e n g t h . S t a r t i n g w i t h t h e i n l e t t o t h e m a n i f o l d t h e r e c t a n g u l a r c a p i l l a r y b u n d l e r e p r e s e n t e d a l o s s i n e f f e c t i v e f l o w a r e a w h i c h i n c r e a s e d as t h e c r o s s - s e c t i o n d i a m e t e r d e c r e a s e d u n t i l t h e l i m i t was r e a c h e d w h e r e t h e d i a m e t e r e q u a l l e d t h e w i d t h o f t h e c a p i l l a r y b u n d l e . F rom t h a t p o i n t i n t h e m a n i f o l d l e n g t h t h e c r o s s - s e c t i o n s h a p e was c h a n g e d f r o m c i r c u l a r t o e l l i p -t i c a l t o m a i n t a i n t h e a r e a - l e n g t h r e l a t i o n s h i p . The r e s u l t i n g m a n i f o l d s h a p e r e p r e s e n t e d t h e b e s t , s m o o t h f l o w s u r f a c e b e t w e e n t h e c i r c u l a r b l o o d l i n e and t h e r e c t a n g u l a r c a p i l l a r y b u n d l e . To d e t e r m i n e t h e v a r i a t i o n i n c r o s s - s e c t i o n a r e a w i t h m a n i f o l d l e n g t h w h i c h w o u l d d i s t r i b u t e b l o o d u n i f o r m l y t o t h e c a p i l l a r i e s and y e t e l i m i n a t e a r e a s o f s t a g n a t i o n o r t u r b u l e n c e a t h e o r e t i c a l s t u d y w i t h e x p e r i m e n t a l . v e r i f i c a t i o n was c o n d u c t e d . An e l e m e n t a r y o n e - d i m e n s i o n a l e n e r g y b a l a n c e i n d i c a t e d t h a t t h e b e s t s y s t e m w o u l d o c c u r when t h e a r e a d e c r e a s e d l i n e a r l y w i t h l e n g t h , I - i - 1  (4-6> o However, a momentum t r a n s f e r b a l a n c e g a v e a more g e n e r a l a r e a - l e n g t h r e l a t i o n , A x 2 6 | : - (1 - £) (4.7) o where t h e t e r m 5 i s a f u n c t i o n o f t h e f l o w g e o m e t r y and f l o w r a t e . P r e s s u r e m e a s u r e m e n t s made on f i v e m o d e l s w i t h d i f f e r -e n t v a l u e s o f 6 ( f r o m z e r o t o one) i n w h i c h t h e o u t f l o w f r o m t h e m a n i f o l d p a s s e d t h r o u g h a s i m u l a t e d c a p i l l a r y b u n d l e i n d i c a t e d l i t t l e a b o u t t h e opt i m u m c h o i c e f o r t h e m a n i f o l d s h a p e e x c e p t t h a t u n i f o r m f l o w t h r o u g h t h e c a p i l l a r i e s e x i s t e d f a i r l y i n d e p e n d e n t l y o f t h e a r e a - l e n g t h r e l a t i o n s h i p . H owever, t h e s t r e a m l i n e s t u d y by means o f dye i n j e c t i o n showed c o n c l u s i v e l y t h a t t h e b e s t s y s t e m r e s u l t e d when 1 ' 2 * A f u r t h e r c h e c k on t h e e f f i c i e n c y o f t h e m a n i f o l d s was a t t e m p t e d b y m e a s u r i n g t h e t o t a l p r e s s u r e d r o p o f t h e m a n i f o l d and c a p i l l a r i e s . The m a n i f o l d w i t h t h e l e a s t l o s s e s w o u l d g i v e t h e l o w e s t p r e s s u r e d r o p . To make m e a n i n g f u l c o m p a r i s o n s w i t h d a t a o b t a i n e d o v e r a p e r i o d o f t i m e i t was h 2 n e c e s s a r y t o p l o t /Q v e r s u s Re on l o g a r i t h m i c s c a l e s w h e r e h was t h e p r e s s u r e d r o p i n cm. W.G. and Re was t h e i n l e t R e y n o l d s number. T h i s was done f o r a l l m o d e l s e x c e p t M o d e l E w h i c h d i s p l a y e d s u c h p o o r p r e s s u r e d i s t r i b u t i o n t h a t i t was e x c l u d e d f r o m f u r t h e r a n a l y s i s . F o r M o d e l s B, C a n d D w h i c h a l l h a d 160 c a p i l l a r i e s , and by u s i n g t h e e q u a t i o n d e v e l o p e d i n A p p e n d i x V, t h e p r e s s u r e d r o p due t o t h e c a p i l l a r i e s a l o n e w as, w h i c h was e x a c t l y t h a t o b t a i n e d e x p e r i m e n t a l l y f o r a l l t h r e e m a n i f o l d s . I t must be c o n c l u d e d t h a t t h e p r e s s u r e l o s s i n t h e m a n i f o l d i t s e l f was s o much l e s s t h a n t h e l o s s e s i n t h e c a p i l l a r i e s t h a t no d i f f e r e n c e was d e t e c t a b l e e x p e r i m e n t a l l y . The M o d e l A m a n i f o l d e x h i b i t e d a p e c u l i a r i t y i n t h a t a n o n - l i n e a r r e l a t i o n b e t w e e n t h e p r e s s u r e d r o p a n d f l o w r a t e was o b s e r v e d . . T h e o r e t i c a l l y t h e p r e s s u r e d r o p due t o t h e c a p i l l a r i e s a l o n e was, — = 12(Re) Q -1 ( s e c /cm ) h = 43.6(Re) - 1 ( s e c /cm ) Q 2 b u t t h e o b s e r v e d p r e s s u r e d r o p f o r t h e s y s t e m was, 76 ~ = 2 0 ( R e ) " ° - 8 7 ( s e c 2 / c m 5 ) Q The two e q u a t i o n s , as p l o t t e d i n F i g u r e 2 0 , d e m o n s t r a t e t h a t b e l o w an i n l e t R e y n o l d s number o f 4 0 0 t h e s y s t e m h a d a l o w e r p r e s s u r e d r o p t h a n t h e o r e t i c a l l y p r e d i c t e d . I t i s p o s t u l a t e d t h a t t h e o b s e r v e d e f f e c t was due t o t h e 9° i n c l i n a t i o n o f t h e c a p i l l a r y t u b e bank t o t h e i n l e t f l o w d i r e c t i o n . T h i s i n c l i n a t i o n p r o b a b l y d e c r e a s e d t h e e f f e c t i v e c a p i l l a r y l e n g t h a t l o w R e y n o l d s numbers a nd t h u s l o w e r e d t h e o v e r a l l p r e s s u r e d r o p f o r t h e m a n i f o l d . The power t e r m i n t h e momentum b a l a n c e e q u a t i o n f o r t h e a r e a - l e n g t h r e l a t i o n c o n s i s t s o f t e r m s i n v o l v i n g t h e amount o f momentum t r a n s f e r r e d f r o m t h e m a n i f o l d , a s h a p e f a c t o r f o r t h e v e l o c i t y p r o f i l e , a f r i c t i o n f a c t o r , a n d t h e R e y n o l d s number, a l l g i v e n i n t h e f o r m , 5 = 1 - J§ 2 L X _ 2 K TTK D Re m m o T h e s e t e r m s w ere a l l assumed c o n s t a n t when d e r i v i n g t h e a r e a -l e n g t h e q u a t i o n y e t t h e y c e r t a i n l y d e p e n d on t h e p o s i t i o n i n t h e m a n i f o l d . The d y e s t r e a k s t u d i e s i n d i c a t e d t h a t f o r l a m i n a r f l o w -the momentum t r a n s f e r f a c t o r 6 had a v a l u e n e a r l y e q u a l t o u n i t y . The p r e s s u r e m e a s u r e m e n t s showed v e r y l i t t l e p r e s s u r e d r o p a l o n g t h e m a n i f o l d . T h e r e a l s o was l i t t l e d e p e n d e n c e o f 6 o n t h e f l o w r a t e . T h e s e o b s e r v a t i o n s i n d i c a t e d t h a t t h e f r i c t i o n l o s s t e r m y was n e g l i g i b l e c o m p a r e d t o t h e o t h e r v i s c o u s l o s s e s i n t h e s y s t e m . F o r a p a r a b o l i c v e l o c i t y p r o f i l e t h e s h a p e f a c t o r 4 i s = j b u t , b e c a u s e o f t h e f l o w p a t t e r n s i n t h e m a n i f o l d , a w e d g e - s h a p e d v e l o c i t y p r o f i l e p r o v e d more s u i t a b l e w h i c h has a v a l u e n e a r e r K = 1 . F o r n e g l i g i b l e f r i c t i o n l o s s m 3 3 and o b s e r v e d v a l u e s o f 3 and n e a r u n i t y , t h e power t e r m 6 w i l l t a k e a v a l u e n e a r o n e - h a l f o r s l i g h t l y l a r g e r . Thus t h e momentum b a l a n c e a p p r o a c h t o d e t e r m i n i n g t h e m a n i f o l d s h a p e y i e l d s w h i c h a g r e e s w e l l w i t h t h e e n e r g y b a l a n c e e q u a t i o n and t h e e x p e r i m e n t a l o b s e r v a t i o n s . H o w e v e r , t h e s e f i n d i n g s a r e f o r m a n i f o l d s w h i c h h a v e a p p r e c i a b l e p r e s s u r e l o s s i n t h e o u t f l o w r e g i o n w h i c h , a l t h o u g h q u i t e s u i t a b l e f o r h e m o d i a l y z e r d e s i g n , d o e s n o t a l l o w g e n e r a l p r e d i c t i o n s t o be made a b o u t m a n i f o l d s h a p e s . V e r y l i k e l y , t h e g e n e r a l m a n i f o l d s h a p e w o u l d d e p e n d upon f r i c t i o n l o s s , f l o w r a t e s , and t h e g e o m e t r i c a l r e l a t i o n s h i p s b e t w e e n t h e o u t f l o w a r e a and i n l e t a r e a , a l l o f w h i c h h a v e b e e n shown t o be n e g l i g i b l e i n t h e above a n a l y s i s C H A P T E R 5 C O N C L U S I O N S 7 8 5. CONCLUSIONS I n t h e t r e a t m e n t o f p a t i e n t s s u f f e r i n g f r o m u r e m i a t h e c l i n i c a l p r o c e s s n o r m a l l y u s e d i s c a l l e d h e m o d i a l y s i s . The a r t i f i c i a l k i d n e y i s one o f t h e m a j o r c o m p o n e n t s o f t h e p r o c e s s and an e n g i n e e r i n g s t u d y o f t h i s u n i t was made. A c o n s i d e r a t i o n o f t h e p h y s i o l o g i c a l c o n d i t i o n s i n d i c a t e d t h a t t h e d i a l y z e r w h i c h w o u l d o f f e r t h e m o s t a d v a n t a g e s was t h e c a p i l l a r y a r t i f i c i a l k i d n e y . T h e r e f o r e an a n a l y s i s o f t h e mass t r a n s f e r and b l o o d f l o w c h a r a c t e r i s t i c s f o r a c a p i l l a r y d i a l y z e r was a t t e m p t e d w i t h a v i e w t o o p t i m i z a t i o n o f t h e s y s t e m . A l t h o u g h t h e w o r k i n g f l u i d i n t h e d i a l y z e r i s b l o o d , w h i c h h a s h i g h l y n o n - N e w t o n i a n r h e o l o g i c a l p r o p e r t i e s , i t was d e t e r m i n e d t h e o r e t i c a l l y and v e r i f i e d e x p e r i m e n t a l l y t h a t a N e w t o n i a n f l u i d a p p r o x i m a t i o n f o r mass t r a n s f e r f r o m c a p i l l a r i e s d o e s n o t i n v o l v e a g r e a t l o s s i n a c c u r a c y . H o w e v e r , f o r p r e s s u r e - i n d u c e d f l o w o f b l o o d , C a s s o n ' s e q u a t i o n , w h i c h c l o s e l y a p p r o x i m a t e s t h e r h e o l o g i c a l p r o p e r -t i e s o f b l o o d , must be e m p l o y e d t o a v o i d l a r g e e r r o r s . By m e a s u r e m e n t s t a k e n on a p a t i e n t u n d e r g o i n g h e m o d i a l y s i s i t was a l s o d e t e r m i n e d t h a t the. s t e a d y - s t a t e a n a l y s i s f o r mass t r a n s f e r was a r e a s o n a b l e a p p r o x i m a t i o n U t i l i z i n g t h e a n a l y s i s f o r a s i n g l e c a p i l l a r y , t h e mass t r a n s f e r . c h a r a c t e r i s t i c s o f a c o m p o s i t e s y s t e m were d e t e r m i n e d i n t e r m s o f c e r t a i n p h y s i o l o g i c a l p a r a m e t e r s s u c h as t h e h o l d - u p v o l u m e o f b l o o d and a v a i l a b l e b l o o d p r e s s u r e . O p t i m i z a t i o n o f t h e s y s t e m was a c c o m p l i s h e d b y o p t i m i z i n g t h e mass t r a n s f e r e f f i c i e n c y w i t h r e s p e c t t o t h e membrane a r e a . The r e s u l t i n g a n a l y s i s p r e d i c t e d t h e b e s t c a p i l l a r y d i a m e t e r i n t e r m s o f t h e d e s i r e d o p e r a t i n g p a r a m e t e r s . The o p t i m i z a t i o n p r o c e d u r e y i e l d e d d e s i g n e q u a t i o n s f o r t h e d i a l y z e r s u c h t h a t f o r a g i v e n d i a l y z e r s i z e , maximum i m p u r i t y r e m o v a l r a t e i s a t t a i n e d . The mass t r a n s f e r e f f i c -i e n c y r e m a i n e d e s s e n t i a l l y c o n s t a n t a t 70 p e r c e n t f o r t h e o p t i m i z e d s y s t e m . The e f f e c t o f d i a l y s a t e f l o w r a t e s was e s t i m a t e d t o d e c r e a s e t h e o p t i m i z e d e f f i c i e n c y t o 60 p e r c e n t a t v e r y w o r s t . A method o f d i s t r i b u t i n g b l o o d t o t h e c a p i l l a r i e s s o t h a t e a c h c a p i l l a r y h a s e q u a l b l o o d f l o w r a t e s was d e v i s e d . The p r i m a r y c o n s i d e r a t i o n i n t h e m a n i f o l d d e s i g n was t o m a i n t a i n l a m i n a r f l o w c o n d i t i o n s w i t h no a r e a s o f s t a g n a t i o n w h i l e , a t t h e same t i m e , p r o v i d i n g t h e c a p i l l a r i e s w i t h e q u a l f l o w r a t e s . I t was d e t e r m i n e d t h a t t h e most s u c c e s s f u l m a n i f o l d w o u l d h a v e t h e c a p i l l a r y o u t f l o w d i r e c -t i o n n o r m a l t o t h e i n l e t d i r e c t i o n and w o u l d be c i r c u l a r i n s h a p e . T h i s , w o u l d g i v e a minimum v o l u m e o f b l o o d i n t h e m a n i f o l d . F o r optimum d i a l y s a t e c i r c u l a t i o n a r o u n d t h e c a p i l l a r i e s and t o a i d i n b l o o d d i s t r i b u t i o n , r e c t a n g u l a r b u n d l e s o f c a p i l l a r i e s w i t h an a s p e c t r a t i o o f 10:1 w o u l d be i n s e r t e d i n t h e o u t f l o w r e g i o n o f t h e m a n i f o l d . A t h e o r e t i c a l s t u d y i n d i c a t e d t h a t t h e m a n i f o l d s h a p e w o u l d d e p e n d on t h e f l o w c o n d i t i o n s a t t h e e n t r a n c e t o t h e c a p i l l a r y b u n d l e . T h e s e f l o w c o n d i t i o n s w e r e d e t e r m i n e d e x p e r i m e n t a l l y w h i c h t h e n p r e d i c t e d t h a t a l i n e a r d e c r e a s e i n a r e a w i t h m a n i f o l d l e n g t h w o u l d d i s t r i b u t e b l o o d u n i f o r m l y . E x p e r i m e n t a l p r e s s u r e m e a s u r e m e n t s and an e x a m i n -a t i o n o f t h e s t r e a m l i n e s by d y e s t r e a k i n j e c t i o n p r o v e d t h a t t h e l i n e a r a r e a d e c r e a s e m o d e l f o r t h e m a n i f o l d i n d e e d met t h e f l o w c o n d i t i o n s when a l a r g e p r e s s u r e g r a d i e n t was p r e s e n t i n t h e o u t f l o w r e g i o n o f t h e m a n i f o l d . From t h e mass t r a n s f e r a n a l y s i s and t h e r e s u l t i n g s y s t e m o p t i m i z a t i o n p r o c e d u r e t h e s i z e o f t h e d i a l y z e r r e q u i r e d f o r c l i n i c a l u s e was e s t a b l i s h e d . A m a n i f o l d f o r b l o o d d i s t r i b u t i o n t o t h e c a p i l l a r i e s w h i c h s h o u l d n o t h e m o l y z e t h e b l o o d was d e v i s e d . W i t h t h e s e c r i t e r i a d e t e r m i n e d a c o m p l e t e d e s i g n o f an a r t i f i c i a l k i d n e y i s p o s s i b l e . B a s e d on t h e t h e o r y d e v e l o p e d i n t h i s t h e s i s a s m a l l p r o t o t y p e a r t i f i c i a l k i d n e y was b u i l t t o d e t e r m i n e t h e f e a s i b i l i t y o f c o n s t r u c t i o n o f a c a p i l l a r y h e m o d i a l y z e r . A m e t hod o f ' p o t t i n g ' a l a r g e number o f v e r y s m a l l c a p i l l a r i e s ( a p p r o x i m a t e l y 280 u I.D.) was d e v i s e d and a m e thod o f a s s e m b l i n g t h e s y s t e m p r o v e d s u c c e s s f u l . F i g u r e 30 shows t h e c o m p l e t e d u n i t and F i g u r e 31 shows an e n l a r g e m e n t o f a s e c t i o n o f t h e p o t t e d c a p i l l a r i e s . A l t h o u g h t h e s y s t e m p r e s s u r e d r o p f o r w a t e r was n e a r t h a t p r e d i c t e d by t h e P o i s e u i l l e e q u a t i o n , t h e c a p i l l a r i e s u t i l i z e d w e r e t o o o l d t o b e u s e d f o r . m a s s t r a n s f e r d e t e r m i n a t i o n s . F u t u r e w o r k must i n c l u d e a t h o r o u g h s t u d y ' i n v i v o * o f t h e mass t r a n s f e r c h a r a c t e r i s t i c s o f a number o f t e s t h e m o d i a l y z e r s . As w e l l , t h e e f f e c t o f t h e m a n i f o l d i n g on b l o o d must be e x a m i n e d t o o b s e r v e any h e m o l y s i s t h a t may be p r e s e n t . The a b i l i t y o f t h e s y s t e m t o remove w a t e r b y u l t r a f i l t r a t i o n and t h e e f f e c t o f t h e d i a l y s a t e f l o w r a t e s m ust be d e t e r m i n e d . REFERENCES 1 . S m i t h , H.W. " P r i n c i p l e s o f R e n a l P h y s i o l o g y , " O x f o r d U n i v e r s i t y P r e s s , New Y o r k ( 1 9 5 6 ) . 2 . A b e l , J . J . , R o w n t r e e , L.G., T u r n e r , B.B. "On t h e R e m o v a l o f D i f f u s i b l e S u b s t a n c e s f r o m C i r c u l a t i n g B l o o d b y Means o f D i a l y s i s , " T r a n s . A s s o c . Am. P h y s i c i a n s , _28, 5 ( 1 9 1 3 ) . 3 . T h a l h i m e r , W. " E x p e r i m e n t a l E x c h a n g e T r a n s f u s i o n s f o r R e d u c i n g A z o t e m i a . Use o f A r t i f i c i a l K i d n e y f o r t h i s P u r p o s e , " P r o c . S o c . E x p t l . B i o l . Med., 3 7 , 6 4 1 ( 1 9 3 8 ) . 4 . K o l f f , W.I., B e r k , H.T. "The A r t i f i c i a l K i d n e y : A D i a l y z e r w i t h G r e a t A r e a , " A c t a . Med. S c a n d . , 1 1 7 , 1 2 1 ( 1 9 4 4 ) . 5 . S k e g g s , L.T., L e o n a r d s , J.R. " S t u d i e s on an A r t i f i c i a l K i d n e y : P r e l i m i n a r y R e s u l t s w i t h a New Type o f D i a l y z e r , " S c i e n c e , 1 0 8 , 2 1 2 ( 1 9 4 8 ) . 6 . K i i l , F. " D e v e l o p m e n t o f a P a r a l l e l F l o w A r t i f i c i a l K i d n e y i n P l a s t i c s , " A c t a . C h i r . S c a n d . , Supp. 2 5 3 , 1 4 3 ( 1 9 6 0 ) . 7 . G a l l e t t i . , - P.M., H o p f , M.A. , P i e r c e , E.C. "A Membrane. L u n g - K i d n e y , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 8 , 4 7 ( 1 9 6 2 ) . 8 . Esmond, W.G. , C l a r k , H. " M a t h e m a t i c a l A n a l y s i s and Mass T r a n s f e r O p t i m i z a t i o n o f a Co m p a c t , Low C o s t , P u m p l e s s S y s t e m f o r H e m o d i a l y s i s ( D i a l u n g ) , " B i o m e d i c a l F l u i d M e c h a n i c s Symposium, A.S.M.E., D e n v e r , C o l o r a d o , ( 1 9 6 6 ) . 9 . L e o n a r d , E . F., B l u e m l e , L.W. " F a c t o r s I n f l u e n c i n g P e r m e a b i l i t y i n E x t r a C o r p o r e a l H e m o d i a l y s i s , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 4, 4 ( 1 9 5 8 ) . 1 0 . B a b b , A . L . , G r i m s r u d , L. "A New C o n c e p t i n H e m o d i a l y z e r Membrane S u p p o r t , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 1 0 , 3 1 ( 1 9 6 4 ) . 1 1 . W o l f , L., Z a l t z m a n , S. "Optimum G e o m e t r y f o r A r t i f i c i a l K i d n e y D i a l y z e r s , " C h e m i c a l E n g i n e e r i n g P r o g r e s s Sym-p o s i u m , S e r i a l No. 8 4 , 6 4 , 1 0 4 ( 1 9 6 8 ) . 83 12. S t e w a r t , R.D., C e r n y , J . C . , Mahon, H . I . "A P r e l i m i n a r y R e p o r t on t h e C a p i l l a r y A r t i f i c i a l K i d n e y , " U n i v . M i c h . Med. C e n t e r . J . , 3_0, 116 ( 1 9 6 4 ) . 13. B i x l e r , H . J . , N e l s o n , L.M., B e s a r a b , A. "The D i a p h r o n H e m o d i a f i l t e r : An A l t e r n a t i v e t o D i a l y s i s f o r E x t r a -c o r p o r e a l B l o o d P u r i f i c a t i o n , " C h e m i c a l E n g i n e e r i n g P r o g r e s s Symposium, S e r i a l No. 84, 6_4, 90 (1968) . 14. B r o w n , C.E., K r a m e r , N.C. " F a c t o r s i n Membrane D e s i g n and S e l e c t i o n T o w a r d a W e a r a b l e A r t i f i c i a l K i d n e y , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 1 4 , 36 ( 1 9 6 8 ) . 15. C o l t o n , C.K. "A R e v i e w o f t h e D e v e l o p m e n t and P e r f o r m a n c e o f H e m o d i a l y z e r s , " U.S. D e p t o f H e a l t h , E d u c a t i o n and W e l f a r e , W a s h i n g t o n , D.C. ( 1 9 6 7 ) . 16. K i n g , P.H., B a k e r , W.R., G i n n , H.E., F r o s t , A.B. "Computer O p t i m i z a t i o n o f H e m o d i a l y s i s , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 14, 3 8 9 . ( 1 9 6 8 ) . 17. F a h r a e u s , R., L i n d q v i s t , T., Am. J . o f P h y s i o l . , 96, 562 ( 1 9 3 1 ) . 18. Charm, S;E.# K u r l a n d , G.S., B r o w n , S.L. "The F l o w C h a r a c t e r i s t i c s o f B l o o d S u s p e n s i o n s , " B i o m e d i c a l F l u i d M e c h a n i c s Symposium, A.S.M.E., D e n v e r , C o l o r a d o ( 1 9 6 6 ) . 19. Charm, S.E. "The Y i e l d S t r e s s o f B l o o d b y a S i m p l e M e t h o d , " P r o c e e d i n g s o f t h e A n n u a l C o n f e r e n c e on E n g i n e e r i n g i n M e d i c i n e and B i o l o g y , 9_, B o s t o n , M a s s . (1967) . 20. M e r r i l l , .E.W. " R h e o l o g y i n M e d i c a l E n g i n e e r i n g - Human B l o o d , " P r o c e e d i n g s o f t h e A n n u a l C o n f e r e n c e on E n g i n e e r -i n g i n M e d i c i n e a n d B i o l o g y , 9_, B o s t o n , M a s s . (1967) . 21. B l a c k s h e a r , P.L., Dorman, F.D., S t e i n b a c h , J . H . , May-b a c h , E . J ; , S i n g h , A., C o l l i n g h a m , R.E. " S h e a r W a l l I n t e r a c t i o n and H e m o l y s i s , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 1_2, 113 ( 1 9 6 6 ) . 22. S c h n e c k , D . J . , G u t s t e i n , W.H. " B o u n d a r y L a y e r S t u d i e s i n B l o o d Flow'; A.S.M.E. p a p e r . n o . 66-WA/BHF-4 ( 1 9 6 6 ) . 23. L o e f f l e r , : A.L. , P e r l m u t t e r , M. " T u r b u l e n t F l o w T h r o u g h P o r o u s R e s i s t a n c e s S l i g h t l y I n c l i n e d t o t h e F l o w D i r e c t i o n , " NACA T e c h . N o t e 4221 ( 1 9 5 8 ) . 84 24. M i c h a e l s , A.S. " O p e r a t i n g P a r a m e t e r s a nd P e r f o r m a n c e C r i t e r i a f o r H e m o d i a l y z e r s a nd O t h e r M e m b r a n e - S e p a r a -t i o n D e v i c e s , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 12, 387 ( 1 9 6 6 ) . 25. S i d e m a n , S., L u s s , D., P e c k , R.E. " H e a t T r a n s f e r i n L a m i n a r F l o w i n C i r c u l a r a n d F l a t C o n d u i t s w i t h ( C o n s t a n t ) S u r f a c e R e s i s t a n c e , " A p p l . S c i . R e s . , A 1 4 , 157 ( 1 9 5 4 ) . 26. Hamming, R.W. " N u m e r i c a l M e t h o d f o r S c i e n t i s t s and E n g i n e e r s , " McGraw H i l l Book Co. I n c . , New Y o r k , pp. 1 8 3 - 2 2 2 , ( 1 9 6 2 ) . 27. G r i m s r u d , L., B a b b , A . L . , "The D e v e l o p m e n t o f a New H i g h E f f i c i e n c y A r t i f i c i a l K i d n e y , " B i o m e c h a n i c a l a nd Human F a c t o r s Symposium, A.S.M.E. ( 1 9 6 7 ) . 28. L i p p s , B . J . , S t e w a r t , R.D., P e r k i n s , H.A., H o l m e s , G.W., M c L a i n , E.A., R o l f s , M.R., O j a , P.D. "The H o l l o w F i b r e A r t i f i c i a l K i d n e y , " T r a n s . Amer. S o c . A r t i f . I n t . O r g a n s , 1 3 , 200 ( 1 9 6 7 ) . A P P E N D I C E S APPENDIX I The t i m e n e c e s s a r y f o r a p a t i e n t t o c o m p l e t e a h e m o d i a l y s i s p r o c e d u r e depends on t h e amount o f i m p u r i t i e s i n t h e b l o o d t o be removed and t h e e f f i c i e n c y o f t h e d i a l y z e r . The e f f i c i e n c y i s e x p r e s s e d b y, M (1.1 Q b C b i B u t t h e c o n c e n t r a t i o n o f i m p u r i t i e s i n t h e b l o o d v a r i e s as a f u n c t i o n o f t i m e o f d i a l y s i s . I f f o r a g i v e n i m p u r i t y t h e r e i s an amount n i n s o l u t i o n i n t h e e x c h a n g e a b l e body f l u i d V n a t any g i v e n t i m e , t h e n t h e c o n c e n t r a t i o n i n t h e b l o o d i s c, . = n / V and t h e mass t r a n s f e r r a t e i s d n / d t = -b i n ' Th u s , f r o m e q u a t i o n 1.1, dn ^ b E . T „ — = — d t (1.2 n I n t e g r a t i n g e q u a t i o n 1.2 w i t h t h e c o n d i t i o n t h a t a t t = 0 t h e i n i t i a l i m p u r i t y c o n c e n t r a t i o n i s c , t h e n t h e b l o o d c o n c e n t r a t i o n a t any t i m e i s , Q b E c b i = C o e x p ( " — t ] ( I ' 3 n A l t e r n a t i v e l y t h e t i m e r e q u i r e d t o d i a l y z e a p a t i e n t f r o m i n i t i a l t o a s p e c i f i e d f i n a l b l o o d i m p u r i t y c o n c e n t r a t i o n 86 t = -V n l n ( — ) o (1.4) T y p i c a l p a r a m e t e r s f o r a n o r m a l p a t i e n t o n h e m o d i a l y s i s a r e , To d i a l y z e a p a t i e n t w i t h i n 6 h o u r s w i t h t h e ab o v e p a r a m e t e r s t h e d i a l y z e r e f f i c i e n c y must be a t l e a s t 73 p e r c e n t . t h a t t h e b o d y m a i n t a i n s i m p u r i t y c o n c e n t r a t i o n e q u i l i b r i u m . I f t h e b l o o d i m p u r i t y l e v e l i s l o w e r e d f a s t e r t h a n t h e t i s s u e i m p u r i t y l e v e l t h e n t h e p r o c e s s s l o w s down when r e m o v i n g t h e same i m p u r i t y mass. A l t e r n a t i v e l y , an i m p u r i t y r e b o u n d phenomena o c c u r s a f t e r d i a l y s i s . V n = 34 £. Q b = 180 c c / m i n C q = 120 mg% ( u r e a ) c, . = 3 0 mg% ( u r e a ) T h i s t i m e assumes t h a t d i a l y s i s i s s l o w enough APPENDIX I I S u p p o s e t h a t f o r a t y p i c a l p a t i e n t t h e d i a l y s a n c e d e s i r e d i s 120 c c / m i n . F o r t h e o p t i m i z e d h e m o d i a l y z e r t h e e f f i c i e n c y i s a p p r o x i m a t e l y 7 0 p e r c e n t . T h u s , f o r t h e r e q u i r e d c l e a r a n c e r a t e , t h e b l o o d f l o w r a t e must b e , M = 0.70 Q, c, . b b i M = 120 c c / m i n C b i = 170 c c / m i n A l s o , t h e e s t a b l i s h e d p h y s i c a l p a r a m e t e r s f o r t h e s y s t e m a r F o r t h e c e l l u l o s e a c e t a t e t u b i n g - P = 5 x l 0 ~ 4 c m / s e c F o r b l o o d - p = 2 c p . V = 2x10-5 c m 2 / s e F o r t h e f e e d e r l i n e s - D £ = 0.476 cm X £ = 300 cm Maximum a v a i l a b l e p r e s s u r e d r o p - Ap = 100 mm Kg Maximum h o l d - u p v o l u m e o f b l o o d - V = 100 c c E q u a t i o n 2.55 i n t h e t h e o r y g i v e s t h e c a p i l l a r y d i a m e t e r , D 2 + 5 . 3 0 ( 2 x l 0 " 5 ) D _ 4 . 1 9 5 T V ( 2 X 1 0 " 5 ) ^ ( 1 0 0 ) 3 0 0 ( 0 . 4 7 6 ) 2 ] 5 x l 0 ~ 4 2. 84 IT D = 0.0234 cm 88 E q u a t i o n 2.43 g i v e s , N x = 4 ( 1 0 0 ) 3 0 0 ( 0 . 4 7 6 ) 2 T T ( 0 . 0 2 3 4 ) 2 ( 0 . 0 2 3 4 ) 2 1.085 x 10 5cm. E q u a t i o n 2.41 y i e l d s t h e p r e s s u r e d r o p f o r a g i v e n s y s t e m w here Nx = 1.085 x 10 5cm. . 1 2 8 ( 0 . 0 2 ) (2. 84) . 300 ^ x , Ap = 5 - [ j + j ] * (0.476) N ( 0 . 0 2 3 4 ) Ap = 1.35 x 1 0 4 + 7.75 x 1 0 6 x N Ap = 1.35 x 1 0 4 + 71.4 x 2 ( d y n e s / c m 2 ) I f x = 15 cm, N = 7240 t u b e s , a n d , 4 2 Ap = 2.96 x 10 d y n e s / c m = 22.2 mm Hg, APPENDIX I I I I f t h e v e l o c i t y p r o f i l e f o r l a m i n a r f l o w i n a c o n d u i t c a n be d e t e r m i n e d t h e n t h e l a m i n a r f r i c t i o n l o s s c a n be e x p r e s s e d a s , Y = 9u c 8r d s Q F o r c e r t a i n e l e m e n t a r y s h a p e s t h e l o s s e s a r e , d i a m e t r a l s h e a r -f r e e s u r f a c e u = 4u d x v A = TTR u = A = ^ ^ ( 1 4y d x v TTR2 R y = 8TT 2 r) R y = 4TT A = H H 2 ( 1 - i + S i ? 2 9 ) TT 2TT ' 90 s h e a r - f r e e s u r f a c e ,c , d on m a j o r a x i s • ' d c F o r i n t e r n a l t u b u l a r f l o w o v e r f u l l y s h e a r e d , w e t t e d s u r f a c e s t h e c o n c e p t o f h y d r a u l i c r a d i u s w i l l g i v e some e s t i m a t i o n o f t h e l a m i n a r f r i c t i o n l o s s e s . The l a m i n a r f r i c t i o n l o s s 2 2C i s a p p r o x i m a t e l y , y = /A w here C i s t h e w e t t e d s u r f a c e l e n g t h and A i s t h e t u b u l a r a r e a . C = b + 2 ( T T-0 ) R s h e a r e d s u r f a c e (Tr-8 + sin8cos6) on c h o r d 91 s h e a r e d s u r f a c e on m a j o r a x i s C = b + TTV b' 2 + 8" A = Tthb Y = 2 b 2 + T T 2 ( h 2 + ^ ) + 4rrb | ? + | i TThb 4 APPENDIX IV The c o r r e c t i o n f o r t h e o b s e r v e d p o s i t i o n o f a d y e s t r e a k t h r o u g h c u r v e d i n t e r f a c e s due t o t h e i n d e x o f r e f r a c t i o n i s as f o l l o w s : c = o b s e r v e d p o s i t i o n f r o m c e n t e r l i n e b = a c t u a l p o s i t i o n s i n e • l l s i n O . (IV.1) sin6^ l2 = s i n e . (IV.2) c = ^ s m u ^ (IV.3) i 2 + 6 = e 3 r - ^ s i n f i = d s i n 0 2 r 2 s i n 0 2 = a s i n y r 1 s i n 0 3 = a s i n y r - ^ s i n O ^ = b s i n g a+3+0„ = 180° a+6 + 0-j^  = 90° C o m b i n i n g e q u a t i o n s I V . 4 and I V . 1 1 y i e l d s a = 9'O-0 1+0 2-0 3 C o m b i n i n g e q u a t i o n s I V . 1 2 and I V . 1 0 y i e l d s = 9 0 + 6 ^ 6 2 + 93-94 C o m b i n i n g e q u a t i o n s I V . 7 and I V . 8 y i e l d s r 2 s i n 0 2 = r^sin©^ 94 9 5 W h e r e , s i n 8 , = — 1 r 2 s i n 6 _ = c s i n 8 2 c s i n 8 ^ = T h e i n d e x o f r e f r a c t i o n f o r a w a t e r - c a s t i n g r e s i n i n t e r f a c e w a s d e t e r m i n e d t o b e u = 1 . 1 2 . F o r t h i s v a l u e o f t h e i n d e x , w h i c h i s t h e same f o r b o t h i n t e r f a c e s , a n d f o r t w o d i f f e r e n t v a l u e s o f t u b e t h i c k n e s s e s t o e n c o m p a s s t h e e x p e r i m e n t a l m a n i f o l d d i m e n s i o n s , t h e o b s e r v e d d y e s t r e a k p o s i t i o n w a s c a l c u l a t e d a s a f u n c t i o n o f t h e a c t u a l p o s i t i o n . T h e r e s u l t s a r e g i v e n i n F i g u r e 2 9 . T h e n e c e s s a r y c o r r e c t i o n t o t h e p o s i t i o n i s e x t r e m e l y s m a l l a n d w a s n e g l e c t e d . APPENDIX V To make v a l i d c o m p a r i s o n s b e t w e e n t h e e f f i c i e n c i e s o f v a r i o u s m a n i f o l d s y s t e m s a s t o t h e r e s i s t a n c e t o f l o w w h ere t h e e x p e r i m e n t a l i n f o r m a t i o n i s g i v e n i n t e r m s o f p r e s s u r e d r o p v e r s u s i n l e t R e y n o l d s number and w h e r e t h e m a j o r i t y o f t h e p r e s s u r e l o s s o c c u r s i n t h e o u t l e t c a p i l l a r i e s , i t i s n e c e s s a r y t o d e f i n e t h e c a p i l l a r y l o s s e s s e p a r a t e l y f r o m t h e m a n i f o l d l o s s e s . F o r t u b u l a r f l o w t h e t r a d i t i o n a l f r i c t i o n f a c t o r i s d e f i n e d b y , 64 and f o r l a m i n a r f l o w i n c i r c u l a r d u c t s X = /Re, w h e r e Re i s t h e R e y n o l d s number b a s e d on t h e d u c t d i a m e t e r d. I n t h e s e e x p e r i m e n t s f o r t h e c a p i l l a r i e s , -dp = pgh h = p r e s s u r e d r o p (cm. W.G.) dx = L L = c a p i l l a r y l e n g t h u = q/A Q = t o t a l f l o w r a t e q = Q/N N = number o f c a p i l l a r i e s 4'Q Re - — — d = c a p i l l a r y d i a m e t e r TT VD * J D = m a n i f o l d i n l e t d i a m e t e r S u b s t i t u t i n g t h e s e r e l a t i o n s i n t o e q u a t i o n V . l g i v e s t h e p r e s s u r e d r o p f o r t h e c a p i l l a r i e s i n t e r m s o f t h e t o t a l f l o w r a t e and i n l e t R e y n o l d s number. hN _ 512L 1 . . ~2 2~~~4~ ~ ^v.^; Q IT gDd Re T h u s f o r a s y s t e m i n w h i c h t h e c a p i l l a r i e s r e m a i n e d c o n s t a n t f r o m m a n i f o l d t o m a n i f o l d d i r e c t c o m p a r i s o n c o u l d be made by h 2 1 p l o t t i n g /Q v e r s u s /Re f o r e a c h m a n i f o l d . F o r t h e c a p i l l a r i e s , d - 0.114 cm D = 1.27 cm L = 7.5 cm E q u a t i o n V.2 b ecomes, ^—- = 1 920(Re) 1 ( s e c 2 / c m 5 ) (V.3) Q w h i c h i s t h e p r e s s u r e l o s s due t o t h e c a p i l l a r i e s a l o n e . TABLE 2 EIGENVALUES AND FLUID BULK COEFFICIENTS FOR VARIOUS WALL RESISTANCES 98 Sh w X E n n 2. 704364 0. 819050 6. 679031 0. 097527 10. 673379 0. 032504 14. 671078 0. 015440 18. 669872 0. 008788 22. 669144 0. 005584 2.6. 668674 0. 003820 30. 668336 0. 002756 34. 668087 0. 002070 38. 667897 0. 001604 42. 667747 0. 001275 46. 667628 0. 001034 50. 667530 0. 000854 54. 667449 0. 000715 58. 667380 0. 000606 62. 667321 0. 000520 66. 667271 0. 000450 70. 667227 0. 000393 74. 667189 0. 000345 78. 667155 0. 000306 Sh =5.0 E ' rv . 90.-2S 2. 356653 0 9 : 932 6. 135040 0 .0 67.602 10. 013477 0 .016334 13. 931760 0 .006023 17. 871795 0 .002778 21. 825391 0 .001.474 25.788154 0 . 000,861 29. 757434 0 .00.0540 33. 731560 0 .000357 37. 709400 0 .000247 41.690160 0 .000176 45. 673262 0 .000130 49. 658276 0 .000098 53 . 644875 0 .000.075 57. 632804 0 .000059 Sh = w =20 X n E n 2. 606889 0. 846644 6. 509851 0. 090993 10. 449987 0. 028082 14. 402808 0. 012484 18. 362927 0. 006692 22. 328075 0. 004023 26. 297023 0. 002613 30. 268956 0. 001796 34. 243344 0. 001287 38. 219794 0. 000954 42. 198004 0. 000727 46. 177741 0. 000566 50. 158815 0. 000449 54. 141072 0. 000362 58. 124385 0. 000296 62. 108644 0. 000245 66. 093759 0. 000205 70. 079650 0. 000173 74. 066248 0. 000147 78. 053494 0. 000126 Sh = w = 3. 3.3 X n E n 2. 219839 0. 925943 5. 966837 0. 054406 9. 844982 0. 011559 13. 769764 0. 003933 17. 717692 0. 001716 2 1 . 679061 0. 000875 25. 649037 0. 000495 29. 624878 0. 000303 33. 604938 0. 000197 37. 588142 0. 00.0134 41. 573761 0. 00 :0094 45. 561279 0. 000069 49. 550323 0. 000051 53. 540613 0. 000039 57. 531937 0. 000030 Sh = w =1C X n E n 2. 516752 0. 869227 6. 364594 0. 083203 10. 270684 0. 023580 14. 200204 0. 009769 18. 143661 0. 004930 22. 096624 0. 002810 26. 056554 0. 001742 30. 021801 0. 001147 33. 991259 0. 000792 37. 964125 0. 000567 4 1 . 939800 0. 000418 45. 917828 0. 000316 49. 897852 0. 000244 53. 879587 0. 000192 57. 862803 0. 000153 6 1 . 847311 0. 000124 65. 832954 0. 000102 69. 819603 0. 000084 73. 807146 0. 000071 77 . 795488 0. 0000,59. Sh = w =2. 0 X n E n 2. 000000 0. 953623 : 5. 743923 0. 036078 9. 645060 •0. 006421. 13. 590328 0. 001988 17. 554802 0. 000818 2 1 . 529545 0. .00.04.01. 25. 510511 0. 000220; 29. 495539 0. 000132'^ 33. 483400 0. 000084 37. 473322 0. 000056^ 4 1 . 464795 0. 000039-45. 457468 0. 000028: 49. 451091 0. 000021 53. 445480 0. 000016 57. 440498 0. 000012 TABLE 2 ( c o n t i n u e d ) 99 Sh = w = 1. 0 X n E n 1. 641250 0. 980566 5. 478309 0. 015926 9. 435963 0. 002296 13. 415243 0. 000648 17. 402591 0. 000253 21. 393922 0. 000120 25. 387561 0. 000064 29. 382640 0. 000038 33. 378706 0. 000024 37. 375476 0. 000016 41. 372766 0. 000011 45. 370455 0. 000008 49. 368457 0. 000006 53. 366708 0. 000004 57. 365162 0. 000003 Sh = w = 0. 5 X E n n 1. 271627 0. 993471 5. 295097 0. 005501 9. 306337 0. 000690 13. 311938 0. 000185 17. 315343 0. 000070 2 1 . 317663 0. 000033 25. 319377 0. 000017 Sh = w 0. 33 X E n n 1. 073799 0. 996782 5. 224506 0. 002736 9. 258937 0. 000326 13. 274984 0. 000086 17. 284519 0. 000032 21 . 290937 0. 000015 25. 295657 0. 000008 Sh = w = 0. 2 • X n E n 0. 855525 0. 998740 5. 164284 0. 0 01079 9. 219455 0. 000123 13. 244498 0. 000032 17. 259226 0. 000012 21. 269086 0. 000006 25. 276311 0. 000003 Sh = w = 0. 1 X E n n 0. 618339 0. 999664 5. 116875 0. 000289 9. 188925 0. 000032 13. 221093 0. 000008 17. 239886 0. 000003 2 1 . 252420 0. 000001 25. 261641 0. 000001 Sh w = 0. 05 X E n n 0. 442157 0. 999913 5. 092436 0. 000075 9. 173364 0. 000008 13. 209216 0. 000002 17. 230097 0. 000001 2 1 . 243997 0. 000000 25. 254048 0. 000000 100 TABLE 3 EIGENVALUES AND FLUID BULK COEFFICIENTS FOR VARIOUS WALL RESISTANCES, p =0.5 Sh = w : oc X n E n 3. 008249 0. 796389 7. 106151 0. 109824 1 1 . 335489 0. 036397 15. 563675 0. 017399 19. 797775 0. 009896 24. 033820 0. 006295 28. 270248 0. 004306 Sh = w = 5. 0 X ' n E n 2. 552906 0. 904151 6. 415308 0. 069022 10. 525737 0. 015398 14. 670173 0. 005440 18. 848070 0. 002421 23. 039625 0. 001254 27. 242088 0. 000718 Sh = w = 20 X n E n 2. 878381 0. 833571 6. 880384 0. 100932 11. 042957 0. 030282 15. 213966 0. 013290 19. 401134 0. 006997 23. 595533 0. 004148 27. 795625 0. 002656 Sh = w = 3. 33 X n E n 2. 381006 0. 930334 6. 224600 0. 052740 10. 346869 0. 010248 14. 504408 0. 003341 18. 694907 0. 001412 22. 896987 0. 000705 27. 108653 0. 000393 Sh =10 w X n E n 2. 759642 0. 862863 6. 693804 0. 089980 10. 821026 0. 024227 14. 968367 0. 009747 19. 141256 0. 004773 23. 325915 0. 002659 27. 520098 0. 001614 Sh = * w --2. 0 X n E n 2. 114187 0. 959296 5. 988714 0. 032380 10. 148188 0. 005294 14. 331656 C. 001584 18. 541716 0. 000636 22. 758403 0. 000307 26. 981779 0. 000167 Sh w =1. 0 X n E ; '; n • . 1. 701124 0. 984383 5. 730370 0. 012951 9. 953853 0. 001762 14. 172335 0. 000488 18. 405310 0. 000188 22. 637906 0. 000088 26. 873344 0. 000047 Sh = w =0. 2 X n E n : 0. 863296 0. 999088 5. 451372 0. 000780 9. 764078 0. 000089 14. 023909 0. 0000-23 18. 281565 0. 000009 22. 530502 0. 000004 26. 777897 0. 000002 Sh = w =0. 5 X n E • n 1. 298190 0. 995069 5. 564564 0. 004175 9. 839048 0. 000509 14. 081833 0. 000135 18. 329530 0. 000051 22. 571945 0. 000024 26. 814608 0. 000013 Sh = w = 0. 1 X n E n 0. 621227 0. 999759 5. 411272 0. 000205 9. 738060 0. 000023 14. 003995 0. 000006 18. 265160 0. 000002 22. 516376 0. 000001 26. 765415 0. 000001 Sh = w = 0. 3 3: X n E n 1. 089462 0. 997627 5. 502991 0. 002023 9. 797969 0. 000237 14. 049991 0. 000062 18. 303115 0. 000023 22. 549095 0. 000011 26. 794349 0. 000006 Sh -w = 0. 05 X n E n 0. 443203 0. 999937 5. 390778 0. 000053 9. 724858 0. 000006 13. 993924 0. 000001 18. 256879 0. 000001 22. 507298 0. 000000 26. 756479 0. 000000 101 TABLE 4 OPTIMIZED DIMENSIONLESS LENGTH AND E F F I C I E N C Y FOR VARIOUS WALL SHERWOOD NUMBERS Sh w 5 E oo 0.0889 0.5705 20 0.1154 0.6129 10 0.1383 0.6376 5 0.1788 0.6654 3.33 0.2158 0.6803 2 0.2853 0.6953 1 0.4492 0.7076 0.5 0.7675 0.7128 0.33 1.0832 0.7141 0.2 1.7126 0.7149 0.1 3.2841 0.7152 0.05 6.4257 0.7153 10. 0.1 Sh 0 . 1 0.2 0 . 0 1 0 . 0 0 1 _L 0 . 0 1 N e w t o n i a n f l u i d - & = 0 N o n - N e w t o n i a n f l u i d - = 0 . 5 ? -0 . 1 2x DPe 1.0 10 F i g u r e 1 O v e r a l l Sherwood Number as a F u n c t i o n o f D i m e n s i o n l e s s L e n g t h and W a l l Sherwood Number o F i g u r e 2 E i g e n v a l u e s f o r C a s s o n F l u i d l . C C l E n C.C1 C . 001 1.0 C . l C.C1 C.CC1 C.CCC1 C . l -C.C1-, 0 . 0 0 1 -C 5 e0 0.C001 1.0 c C 5 F i g u r e 3 B u l k C o e f f i c i e n t s f o r C a s s o n F l u i d 0.001 0.01 0 . 1 1.0 10 S HPe g F i g u r e 4 O v e r a l l Sherwood.Number and Mass T r a n s f e r E f f i c i e n c y as a F u n c t i o n o f D i m e n s i o n l e s s L e n g t h F i g u r e 6 One D i m e n s i o n a l F r e e Body f o r M a n i f o l d o P r e s s u r i z e d S a l i n e Venous A r t e r i a l Blood L i n e Blood Line i #21 Needles i n C u f f s X X Stopcocks Pressure Transducers Flow ^ T r a n s d u c e r Flow Meter A m p l i f i e r s K i i l D i a l y z e r Chart Recorder F i g u r e 8 S c h e m a t i c o f E x p e r i m e n t a l E q u i p m e n t f o r M o n i t o r i n g H e m o d i a l y s i s o I60r Qv 140-( c c / m i n ) Flow Rate 120h 80 A r t e r i a l Pressure 60 mmHg 40 Venous Pressure 20 1 sec F i g u r e 9 B l o o d F l o w R a t e and S y s t e m i c P r e s s u r e s f r o m P a t i e n t U n d e r g o i n g H e m o d i a l y s i s Blood Tank V Heater and Mixer Blood D i a l y s a t e D i a l y s a t e Tank D i a l y s i s Tubing i g u r e 10 S c h e m a t i c o f E q u i p m e n t f o r Mass T r a n s f e r T e s t s 400 800 1200 1600 2000 Re F i g u r e 12 • Turbulence Onset i n B l a s i u s Expansion S e c t i o n Manometer Board Water Bath with Manifold Dye Injector Bubble Trap Flow Meter 0=^ Pump Constant Head Tank F i g u r e 13 S c h e m a t i c o f F l o w C i r c u i t f o r M a n i f o l d T e s t s i—1 F i g u r e 14 T e s t M a n i f o l d C r o s s - s e c t i o n s (Models A, B) F i g u r e 15 T e s t M a n i f o l d C r o s s - s e c t i o n s ( M o d e l s -C,D,E) =0> Model A. Re = 850 1.5 1.0 A p (In.W.I.) 0.5 J 3 -Pressure M f f e r e n c -O Between M a n i f o l d and Bath x/L 1.0 0 1.0 x/L F i g u r e 16 C o m p a r i s o n o f M a n i f o l d P r e s s u r e and C a p i l l a r y F l o w R a t e s i n M o d e l A F i g u r e 17 S t r e a m l i n e s i n M o d e l A 119 S e c t i o n A - A . F i g u r e 18. F l u i d D i s t r i b u t i o n i n M o d e l A M a n i f o l d 120 \ 6 3.0 -2.0 H A p 1.0 -I 6 O rj-1: 2 1 r 3 4 S t a t i o n Re 0 — — — 0 o o O O 2130 •O O 1700 -O 1280 •O 850 -O O 450 i 1 F i g u r e 19 P r e s s u r e D i s t r i b u t i o n i n M o d e l A M a n i f o l d F i g u r e 21 M a n i f o l d P r e s s u r e D i s t r i b u t i o n f o r M o d e l s B,C,D,E to to F i g u r e 22 T o t a l P r e s s u r e D r o p f o r M o d e l B M a n i f o l d F i g u r e 23 T o t a l P r e s s u r e D r o p f o r M o d e l C M a n i f o l d 125 F i g u r e 24 T o t a l P r e s s u r e D r o p f o r M o d e l D M a n i f o l d 126 Stagnant Region Streamlines Model B F i g u r e 25 S t r e a m l i n e D i s t r i b u t i o n i n M o d e l B M a n i f o l d Stagnant Region Streamlines Model C F i g u r e 26 S t r e a m l i n e D i s t r i b u t i o n i n M o d e l C M a n i f o l d L i n e a r Taper M a n i f o l d F i g u r e 27 S t r e a m l i n e C u r v a t u r e a t C a p i l l a r y E n t r a n c e F i g u r e 28 F i r s t E i g e n v a l u e as a F u n c t i o n o f W a l l S h e r w o o d Number gure 30 Prototype C a p i l l a r y A r t i f i c i a l Kidney F i g u r e 31 C r o s s - s e c t i o n o f P o t t e d C a p i l l a r i e s I 1 100^ w 

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