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Finite ele m ent analysis o f axial f low withheat transfer in a square ductMatiur RahmanDepartment of Applied Mathematics, Technical University of Nova Scotia, Halifax, N.S.,B3J 2)(4, CanadaS. Y. AhmadAdvance Engineering Branch, Atomic Energy of Canada Limited, Chalk River, Ontario, Canada,KOJ 1JO(Received July 1981; revised January 1982)Temperature distr ibution within the rod bundle of a nuclear reactor is of majorimportance in nuclear fuel design. However, a knowledge of the hydrodynamicbehaviour of the coolant is a prerequis ite to the determinat ion of the temperaturedistr ibut io n. I t is believed that a knowledge of the hydrod ynam ic behaviour of thecoolant is the most d i f f ic u l t part of the problem due to the comp lexi ty of the turbu-lent phenomena. In the present work, however, three-dimensional heat transfercalculations have been perform ed subject to an assumption of axial unifor m f low alongthe square duct. An analytical solution for this simple mathematical model has beenobtained and provides a basis for comparison w ith th e finit e elem ent solution of theheat transfer equation. Com puter results appear to agree wit h the true solu tion t owith in 1%.Key w ords: Finite elements, square duct, heat transfer, nuclear reactors

I n t r o d u c t i o nPreliminary background

An atomic reactor in a nuclear power plant is s imply asource of hea t , w hich does the same job as coal , o i l o rnatural gas in a conventional electr ical generating powerp lan t . The hea t f rom the nuc lear r eac tor conver ts o rd inarywate r in to s team to ro ta te tu rb ines jus t as the hea t f romburning coal, oil or gas is used to turn ordinary water intos team to do the same job . Turb ines ro ta te e lec tr ica lgenera tor s and the energy produced is f ed in to the pow erlines.In a nuc lear r eac tor , a compo und, u ran ium oxide (uran iumplus oxygen ), is used in fuel rods because, unlike pureuranium, metal resis ts corrosion and does not distor t underpro longed bombardment by neutrons . Each fue l bundle ,consisting of a maximum of 37 fuel rods in the PicketingNuclear Genera t ing Sta t ion , Onta r io , C anada, is approxima-tely 19.5 in (49.5 cm) long and contains 49 lb (24 kg) ofuranium oxide. A full fuel charge for one reactor consists of4680 bundles weighing a to ta l o f about 116 tons . The hea tproduced b y these 116 tons of u ran ium oxide equa ls thehea t tha t could be ob ta ined f rom about 3 mil l ion tons ofcoal. Heavy water passes over hot fuel bundles and transfersheat to the s team generators where the heat is applied toord inary wate r to tu rn i t to s team. The s team is then fedto the turbine which drives an electr icity generators .There are different atomic power reactor systems avail-able but the one developed and used in Canada is calledCANDU (CANada D eute r ium Uranium) charac te riz ing th ree

of the reactor 's dis tinguishing features: the system isCanadian ; i t uses heavy wate r (deu te r ium oxide) as themod era tor ; and the fue l is na tura l u ran ium. CAN DU fue lbundles are contained in each reactor and held in horizontalpressure tubes th rough which coolan t (heavy w ate r ) f lows.Also , a heavy wate r modera tor , w ithout which the reac t ionwould not take place, surrounds the pressure tubes.Previous work

The pred ic t ion of the tem pera ture d is t r ibu t ion in a fue lbundle is impor tan t for the sa fe and econom ic opera t ion ofthe nuc lear cores . The a tomic fuel e lements , composed ofuran ium oxide pa l le ts , genera l ly cons is t o f rod bundles withthe coolan t (heavy wate r ) f lowing ax ia l ly th rough thebundles in the space between the rods, thereby transportingthe therm al energy of the fue l to hea t exchangers andturb ines which ro ta te to produ ce e lec tr ic i ty . There fore , athorough unders tanding of the hydrodynamic behaviour ofthe coolant under normal conditions is essential for thedesign of a fuel bundle. Heat transfer calculations form animportant par t in the design of such fuel elements , whichcan be ca r ried ou t on ly i f suff ic ien t in formation a bout thevelocity f ield is available. At the sam e time, there is an eco-nomic incentive to increase the rating of fuel bund lesJ '2

Rod bundles have become the pr incipal fuel configura-tion for most atomic power reactors . Thus the investigationof f luid f low with heat transfer around rods has receivedmuch attention from nuclear scientis ts . The axial f luid f lowaround rods has been shown to be s imilar to that in ducts .0307-904X/82/060481-10/$03.00 1982 Butterworth & Co. (Publishers) Ltd Ap pl. Math . Mod elling, 1982, Vo l. 6, Decem ber 481

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Research N oteH o w e v e r , t h e u s e o f n o n c i r c u l a r d u c t s i n t h e t r a n s p o r t o ff l u i d a n d h e a t i s c o m m o n i n a t o m i c r e a c t o r c o o l a n tpassages .R a m a c h a n d r a a n d S p a l d in g 3 h a v e a p p l ie d t h e K - e t u r b u -l e n c e m o d e l , c o u p l e d w i t h t h e a l g e b r ai c s tr e s s m o d e l o fL a u n d e r a n d Y i n g ,4 t o t u r b u l e n t f l o w w i t h h e a t t r a n s f e r i na r e c t a n g u l a r - se c t i o n e d d u c t . T h e y u s e d t h e f i n it e d i f f e r e n c em e t h o d o f P a t a n k a r a n d S p a l d i n g s t o s o l v e th e s e v e n s t ro n g l yc o u p l e d n o n l i n e a r p a r t i a l d if f e r e n t i a l e q u a t i o n s f o r t u r b u -l e n c e q u a n t i t ie s . T h e i r p r e d i c t i o n s w e r e i n c l o s e a g r e e m e n tw i t h m e a s u r e m e n t s o f s e c o n d a r y f l o w p r o f i l e s a n d t h e i re f f e c t o n a x i a l v e l o c i t y a n d t h e p r i n c i p a l d i s t ri b u t i o n o fs h e a r s t r es s a n d h e a t t r a n s f e r .E m e r y e t a l . ~ h a v e c o m p u t e d t h e v e l o c it y a n d t e m p e r a -t u r e p r o f i l e s f o r d e v e l o p i n g t u r b u l e n t f l o w i n a sq u a r e d u c tw i t h c o n s t a n t w a l l t e m p e r a t u r e a n d c o n s t a n t w a l l h e a t f lu x ..T h e c o m p u t a t i o n u t i li z e d a n e x p l i c i t n u m e r i c a l d i f f e r e n c i n gs c h e m e a n d a n a l g e b r a ic c l o s u re m o d e l b a s e d o n a t h r e e -d i m e n s i o n a l m i x i n g l e n g t h . T h e c o m p u t e d l o c a l an d f u l l yd e v e l o p e d s h e a r s tr e s s e s a n d h e a t t r a n s f e r a r e k n o w n t o b ei n a g r e e m e n t w i t h m e a s u r e d d a t a a n d w i t h p r e d i c t i o n u s i n gth e K - e c l o su r e m o d e l . T h e m e t h o d h a s b e e n e x t e n d e d b yG e s s n e r a n d E m e r y 7 t o t h e d e v e l o p i n g t u r b u l e n t f l o w i nr e c t a n g u l a r d u c t s o f a r b i t r a r y a s p e c t r a t i o .C a r a j il e s c ov a n d T o d r e a s s h a v e c o n s i d e r e d a o n e - e q u a t i o nm o d e l o f t u r b u l e n c e f o r a p p l i c a t i o n t o a n i n t e r io r s u b -c h a n n e l o f a b ar e r o d b u n d l e w i t h s m o o t h r o d s . T h e yu t i li z e d t h e f ' m i t e d i f f e r e n c e m e t h o d t o c o m p u t e t h ed e t a i le d d e s c r i p t i o n o f v e l o c i t y f i e ld ( a x i a l a n d s e c o n d a r yf l o w s ) a n d t h e w a l l sh e a r s t r e s s d is t r i b u t io n o f s t e a d y ,f u l l y d e v e l o p e d t u r b u l e n t f l o w s t h r o u g h t r i a n g u l a r a r r a y s o fr o d s w i t h v a r i o u s a s p e c t r a ti o s . A c o m p a r i s o n o f t h ea n a l y t i c a l r e s u l ts a n d t h e e x p e r i m e n t a l m e a s u r e m e n t s s h o w sg o o d a g r e e m e n t .

N e t i 9 h a s p e r f o r m e d m e a s u r e m e n t s o f f u l ly d e v e l o pe dt u r b u l e n t f l o w in a s i m u l a t e d n u c l e a r f u e l r o d b u n d l e .U s i n g t h e f i n i te d i f f e r e n c e c o m p u t a t i o n s c h e m e w h i c h i sc a p a b l e o f c a l c u l at i n g t h r e e - d i m e n s i o n a l p a r a b o l i c t u r b u l e n tf l o w s , n u m e r i c a l r e s u l t s a r e c o m p a r e d w i t h t h e m e a s u r e -m e n t s a n d s h o w g o o d a g r e e m e n t . H i s r e s u l ts a r e a ls o i ng o o d a g r e e m e n t w i t h th e c o m p u t a t i o n s o f R e e c e ) e

R e c e n t l y , C h e r n a n d C h a t o n h a v e d e v e l o p e d a f i n it ee l e m e n t n u m e r i c a l m e t h o d t o p r e d i c t f r i c t i o n f a c t o r s a n dl a m i n a r h e a t t r a n s f e r c o e f f i c i e n t i n a p i p e . T h e p r o b l e m i so f a t y p i c a l h i g h p r e s s u r e , o i l -f 'd l e d p i p e - t y p e c a b l e s y s t e mu s e d f o r u n d e r g r o u n d p o w e r t r a n s m i ss i o n c o m p o s e d o ft h r e e c a b l e s e n c l o s e d i n a s t e e l p ip e . T h e h e a t g e n e r a t e dw i t h i n t h e c a b l e s y s t e m c a n b e d i s s ip a t e d l o c a l ly t o t h es u r r o u n d i n g e a r t h . A l t h o u g h t h e i r w o r k i s n o t d i r e c t e d t on u c l e a r r e a c t o r s , t h e m a t h e m a t i c a l e q u a t i o n s a r e s i m i l a rt o t h o s e u s e d i n a t o m i c r e a c t o r s , e s c e p t t h a t i n t h e l a t t e re a s e t u r b u l e n t f l o w s w i t h h e a t t r a n s f e r m u s t b e c o n s i d e r e d .S l ag t er ~2 has de ve lop ed a f 'm i t e e l em ent me thod t o so lvet h e m o m e n t u m e q u a t i o n f o r th e c e n t r a l su b c h a n n e l o f af u e l r o d b u n d l e w i t h t h e a s s u m p t i o n t h a t s e c o n d a r y f l o w( c r o s s - s t r e a m ) v e lo c i t ie s a r e s m a l l i n c o m p a r i s o n w i t h t h em a i n a x i a l f lo w . T h i s a s s u m p t i o n , w h i c h w a s a ls o e m p l o y e db y S p a r r o w a n d L o e f f l e r , 13 M e y d e r , 14 a n d B a r t z is a n dT o d r e a s , ~ s i n t h e i r r e s p e c t i v e a n a l y s e s o f s u b c h a n n e l f l o w s ,w i ll n o t b e r e q u i r e d w h e n t h e s o l u t i o n s f o r t h e f u l l s e t o ft u r b u l e n t e q u a t i o n s o f m o t i o n a r e c o n s i d e r e d . I n t h e s a m es t u d y , B a r t z i s a n d T o d r e a s s a l so e x a m i n e d s e c o n d a r y f l o w sa n d f o u n d t h a t t h e e f f e c t s o f s e c o n d a r y f l o w s a re o f m i n o ri m p o r t a n c e f o r f u l l y d e v e l o p e d f l o w i n a n i n f i n i te a r r a y o fb a r e r o d s . H o w e v e r , f o r d e v e l o p i n g f lo w s i n s u b c h a r m e l s

w i t h f l o w d i s t u r b e r s t h e s e c o n d a r y f l o w e f f e c t s m a y b ei m p o r t a n t . A s s u m i ng t h a t e f f e c t s o f s e c o n d a r y f l o w a r en e g l ig i b le , W o n g a n d A l p ~6 e m p l o y e d t h e f i n it e e l e m e n tm e t h o d t o s o lv e th e m o m e n t u m e q u a t i o n s t h a t g o v er n f u ll yd e v e l o p e d s i n g le p h a s e t u r b u l e n t f l o w s i n t h e s u b c h a n n e lf o r m e d b y r o d s a r r a n g e d i n t r ia n g u l a r a r r a y s .I n t h e p r e s e n t s t u d y , u n i f o r m a x i a l f l o w t h r o u g h a s q u a r ed u c t i s e m p l o y e d a s a c o m m e r c i a l c o n f i g u r a t i o n f o r h e a tt r a n s f e r p u r p o s e s i n a n a t o m i c r e a c t o r . M u c h o f t h e s t u d yo f c o m p u t a t i o n a l f l u i d f l o w a n d h e a t t r a n s f e r d e p e n d s o nf i n i te - d i f f e r e n c e n u m e r i c a l a n a l y s is . T h e u s e o f f i n i te -e l e m e n t m e t h o d s f o r t h i s t y p e o f p r o b l e m i s s ti ll i n adeve lopmenta l s t age . I t i s , i ndeed , essen t i a l t o deve lop f i n i t ee l e m e n t c o m p u t e r c o d e f o r a p r o b l e m s u c h as a n a t o m i cr e a c t o r w i t h c o m p l e x g e o m e t r i e s . H o w e v e r , in t h i s s tu d y w eh a v e s i m p l if i e d t h e p r o b l e m c o n s i d e r a b l y a n d a f i n i t ee l e m e n t m e t h o d h a s b e e n d e v e l o p e d f o r t h r e e - d i m e n s i o n a la x i a l fl o w w i t h h e a t t r a n s f e r . A n a n a l y t i c a l s o l u t i o n o f t h i ss i m p l e m a t h e m a t i c a l m o d e l i s r e a d i ly a v a i l a b le , a n d t h i ss o l u t io n p r o v i d e s a b a s is f o r c o m p a r i s o n w i t h t h e f i n i tee l e m e n t s o l u t i o n . F u t u r e w o r k w i l l d e a l w i t h t h e m u c hm o r e c o m p l i c a t e d t u r b u l e n t f l o w w i t h h e a t t r a n s f e r i n t h er o d b u n d l e s .M a t h e m a t i c a l a n a l y s i sT h e p a r t i a l d i f f e r e n t ia l e q u a t i o n s f o r d e v e l o p i n g i n c o m p r e s -s i b le s t e a d y t u r b u l e n t f l o w in a s q u a r e a n d / o r r e c t a n g u l a rd u c t a r e f o u n d i n t h e l i t e r a t u r e ) ' 6' 17 F o r t h e c a s e o f a x i a lf l o w in a c h a n n e l o f s q u a r e d u c t , w h e r e t h e x - a x i s c o r r e -s p o n d s t o t h e m a i n f l o w d i r e c t i o n , t h e e q u a t i o n s m a y b eg r e a t l y s i m p l if i e d a n d r e d u c e d t o t h e f o l lo w i n g :U - m o m e n t u m

1 a f t [ a 2 U 8 2 U ~ a , , a , ,F + - - } -p a x a z ( 1)a n d t h e h e a t t r a n s f e r c h a r a c t e r i s t ic s o f t h e f l o w a r e o b t a i n e df r o m t h e t e m p e r a t u r e e q u a t io n :a T [ a 2 T a 2 T ~ a ~ aU - - - - ~a X [ - - ~ - Y - 2 + a Z 2 / - - - ~ - Y ( V t ) - - - ~ - z ( W t ) (21- I itwith the algebraic Reynolds st ress mode l developed byGessner and Em ery, 18 nam ely:

= - / : ( 3 )

, , , a u : ' a u \

together vyi th the temperature Reynolds st resses:

= - - < t ~ - Z / \ \~ -Y I + ~ - ~ ! l ( 6 )U, u t, / )t , W IXY .ZT, t ' k(2~ - -pC p

s t r e a m w i s e v e l o c i t y , f l u c t u a t i n g c o m p o n e n tt r a n s v e r s e f l u c t u a t i n g v e l o c i t y c o m p o n e n ts t r e a m w i s e c o o r d i n a t et r a n s v e rs e c o o r d i n a t e st e m p e r a t u r e , f l u c t u at i ng c o m p o n e n t st h e r m a l d i f f u s i v it y

482 App l . Math. Mod el ling, 1982, Vo l . 6, Decem ber

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PkCpPpf ft .Pr t

d e n s i t yt h e r m a l c o n d u c t i v i t ys p e c i f ic h e a t a t c o n s t a n t p r e s s u r ela[p k i n e m a t i c v i s c o s i tyd y n a m i c v i s c o s i t yP ( X ) p r e s s u r e a f u n c t i o n o f X a l o n eP r a n d t l ' s m i x i n g l e n g tht u r b u l e n t P r a n d tl n u m b e r

I n t h e a b o v e e q u a t i o n s , s e c o n d a r y f l o w ( c r o s s - s t r e a m )v e l o c it i e s V a n d W a r e a s s u m e d t o b e s m a l l a n d a r e t h e r e -f o r e n e g l e c t e d . T h i s a s s u m p t i o n w i ll b e i n v al id w h e n t h es o l u t i o n s f o r t h e f u l l se t o f g o v e r n i n g e q u a t i o n s o f m o t i o na r e d e v e l o p e d .

B a r t z is a n d T o d r e a s I s e x a m i n e d t h e s e c o n d a r y f l o w s a n df o u n d t h a t t h e e f f e c t s o f s e c o n d a r y f l o w s a r e o f m i n o ri m p o r t a n c e f o r f u l l y d e v e l o p e d f l o w i n a n in f i n i te a r r a y o fb a r e r o d s . H o w e v e r , f o r d e v e l o p i n g f lo w s i n s u b c h a n n e l st h e se e f f e c t s m a y b e i m p o r t a n t .

I f i t is f u r t h e r a s s u m e d t h a t t h e a x i a l f l o w is u n i f o r m l yd i s t r i b u t e d ( i. e . p lu g f l o w c a s e ) in a c h a n n e l o f s q u a r e d u c t ,t h e n t h e e q u a t i o n s ( 1 ) - ( 6 ) m a y b e v e r y g r e a t ly s i m p l if i e d ,s o t h a t o n l y t h e t e m p e r a t u r e e q u a t i o n ( 2 ) s u r v i v e s a n dr e d u c e s t o :a T [ a 2 T a 2 T ]U - - = u +ax ~a-~ az~! (7 )

E q u a t i o n ( 7 ) d e s c r i b e s a n i d e a l s it u a t i o n o f a x i a l f l o w w i t hh e a t t r a n s f e r . I n p r a c t i c e , in a n a t o m i c r e a c t o r t h e f l u idf l o w w i t h h e a t t r a n s f e r is a v e r y c o m p l e x p r o c e s s . H o w e v e r ,t h e i d e a l i z e d e q u a t i o n w i l l b e e x a m i n e d t h o r o u g h l y i n o r d e rt o d e v e l o p a f i n i t e e l e m e n t m e t h o d , b e c a u s e t h e a n a l y t i c a ls o l u t i o n o f t h i s s i m p l e m a t h e m a t i c a l e q u a t i o n i s r e a d i l ya v a i la b l e a n d w i ll p r o v i d e a b a s i s f o r c o m p a r i s o n w i t h t h ef i n it e e l e m e n t s o l u t i o n f o r l a t e r c o r r e c t n e s s o f t h e m e t h o d .I n o r d e r t o s o l v e e q u a t i o n ( 7 ) , t h e f o l l o w i n g b o u n d a r yc o n d i t i o n s m a y b e u s e d :

T = 0 a t wal l duc t (8)T = To a t X = 0 (9)

F r o m p h y s i c a l c o n s i d e r a t i o n s t h e w a l l t e m p e r a t u r e s m a yb e a s s u m e d t o b e k e p t a t a c o n s t a n t t e m p e r a t u r e ( 0 C ) , a n dt h e i n le t t e m p e r a t u r e s a r e k e p t a t T o . I n f a c t , t h e b o u n d a r yc o n d i t i o n s ( 8 ) a n d ( 9 ) a r e f o r a n i d e al s i t u a ti o n . E m e r y e tal . 6 h a v e u s e d t h i s s et o f b o u n d a r y c o n d i t i o n s i n e v a lu a t i n gt h e t u r b u l e n t h e a t t r a n s f e r i n a s q u a r e d u c t . L a u n d e r a n dY i n g 4 h a v e u s e d s o m e c o m p l i c a t e d b o u n d a r y c o n d i t i o n sc o m p a t i b l e t o t h e t u r b u l e n c e p h e n o m e n a . S i m i l a r l y ,R a m a c h a n d r a a n d S p a l d i n g 3 f o l l o w e d t h e s a m e p r a c t i c e sa s d e s c r ib e d b y L a u n d e r a n d Y i n g . 4 I n o u r f u t u r e s t u d i e s,w h e n w e c o n s i d e r th e t u r b u l e n c e f l o w i n r o d b u n d l e s , t h ea p p r o p r i a t e b o u n d a r y c o n d i t i o n s w i l l b e a d o p t e d f o rb e t t e r c o r r e l a t i o n w i t h t h e a v a i la b l e e x p e r i m e n t a l d a t a .S i n c e w e a r e i n t e r e s t e d i n t h e g e n e r a l a p p l i c a t i o n o f t h ef u l l s e t o f t u r b u l e n c e e q u a t i o n s t o t h e c o m p l e x p r o c e s s o fa t o m i c r e a c t o r s o f w i d e l y v a r y i n g p a r a m e t e r s , i t i s u s e fu lt o r e w r i t e t h e e q u a t i o n s i n d i m e n s i o n le s s f o r m . T h ed i m e n s i o n a l a n a l y s i s o f C h e r n a n d C h a t o u h a s b e e n c l o s e l yf o l l o w e d i n d e d u c i n g t h e f o l lo w i n g d i m e n s i o n l e s s te m p e r a -t u r e e q u a t i o n .E q u a t i o n s ( 7 ) , ( 8 ) a n d ( 9 ) a r e m a d e d i m e n s i o n l e s s b yin t roducing t he fo l l owing charac t e r i s t i c var i ab l es :

R e s e a r ch N o t eX Y Zx = - - y = - - z = - -L a a ( l o )U T - - T b # au = - - 0 = - - A T = -A T k

w h e r e , L i s t h e c h a r a c t e r i s ti c a x i a l l e n g t h o f t h e r e a c t o r , at h e w i d t h o f t h e r e a c t o r , ~ t h e a v e r a g e a x i a l v e l o c i t y , a n dT b = T b ( X ) t h e r e f e r e n c e t e m p e r a t u r e a s s u m e d t o b e af u n c t i o n o f X . T h i s r e f e r e n ce t e m p e r a t u r e m a y b e d e f i n e dt o b e t h e a v e r a g e b u l k t e m p e r a t u r e , a n d ~ t h e a v e r a g e h e a tt r a n s fe r p e r u n i t a r e a . T h e t e m p e r a t u r e d i f f e re n c e A T m a yb e d e f i n e d a s A T = qa /k ~-- To -- T6 b y u s i n g th e F o u r i e rl a w o f h e a t c o n d u c t i o n , w h e r e k i s t h e c o n s t a n t f l u i dc o n d u c t i v i t y . 19

Using t hese d imensionl ess var i ab l es i n equa t ion (7) , t het e m p e r a t u r e e q u a t i o n y i e l d s :- - + - - = a 2 02 l u a 2 ~ ( u a O ] + ( a 2 "~ a T b ( 11 )

I t i s w o r t h m e n t i o n i n g t h a t e a c h t e r m i n e q u a t i o n ( 1 1 )m u s t b e o f t h e s a m e o r d e r , n a m e l y u n i t y , w h i c h i m p l i e st h a t :

/ ~ a 2= 0 ( 1 )e L

f r o m w h i c h o n e c a n d e f 'm e t h e c h a r a c t e r i s ti c l e n g t h L t o b ei n t h e f o r m :L = aR e "Pr ( 1 2 )

w h e r e : R e = a a / v = R e y n o l d s n u m b e r a n d P r = v / a =P r a n d t l n u m b e r .T h e n e q u a t i o n ( I 1 ) m a y b e w r i t t e n a s :a20 ~20 aO / a2 \ a T~- - + - - = u - - + l /- - - ( 1 3 )a y ax \ . / , r / V a x

T h e s e c o n d t e r m o n t h e r i g h t - h a n d s i d e m a y b e r e p l a c e di f w e a p p l y t h e c o n s e r v a t i o n o f e n e r g y t o a c r o s s - s e c t i o n o ft h e s q u a r e d u c t .B y t h is l a w w e k n o w t h a t t h e r a t e o f h e a t c o n v e c t e d b y

t h e f l u id m u s t b e e q u a l to t h e r a t e o f h e a t c o n d u c t e d i n t ot h e f l u i d f r o m a l l h e a t e d s o u r c e s , a n d s o :arbA p c p U = ~ lL q ( 1 4 )a X

i n w h i c h A i s t h e n e t c r o s s - s e c ti o n a l a re a o f t h e d u c te x c l u d i n g t h e a r e a o f f u e l r o d s , a n d L q i s t h e h e a t e d c i r c u m -f e r e n c e o f f u e l r o d s .T h u s , s u b s t i t u t i n g ( 1 4 ) i n t o ( 1 3 ) y i e l d s t h e f o l l o w i n gs i m p l e d i f f e r e n t i a l e q u a t i o n :a20 a20 a0- - + - - = u - - + g ( 1 5 )a y 2 b z 2 a x

w h e r e Q = a L q ] A .I t is to b e n o t e d h e r e t h a t t h e d i m e n s i o n l e s s p a r a m e t e rQ o b t a i n e d t h r o u g h d i m e n s i o n a l a n a l y s i s i s a f u n c t i o n o fa x i a l g r a d i e n t o f t h e r e f e r e n c e t e m p e r a t u r e , i .e .. a T b / a x a n di s re c o g n i z e d a s a d i m e n s i o n l e s s h e a t s o u r c e o r s i n k a c c o r d -i n g to w h e t h e r Q i s n e g at i v e o r p o s i ti v e . I t m u s t b e o f o r d e ru n i t y , f o r t h e m a t h e m a t i c a l a n a l y s i s t o b e v a l i d .T h e b o u n d a r y c o n d i t i o n s ( 8 ) a n d ( 9 ) m a y b e r e s t a t e d i nd i m e n s i o n l e s s f o r m a s f o ll o w s :

Appl . Math . Model l ing , 1982, Vo l . 6 , December 483

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R e s e a r c h N o t e0 = 0 a l o n g t h e s o l i d w a l l s

( i .e .y = 0 , 1 , z = 0 , 1 ) (16 )0 = 0 o a t x = 0 ( 1 7)

A n a l y t i c a l s o l u t io n o f t e m p e r a t u r e e q u a t i o n ( 1 5 )T h e s i mp l e ma t h e ma t i c a l mo d e l ( 1 5 ) , ( 1 6 ) a n d ( 1 7 ) l e n d si t se l f a c losed fo rm ana ly t i ca l so lu t ion and a rb r i e f sum m aryof th i s so lu t ion i s g iven be low.B y t h e me t h o d o f s e p a r a t i o n o f v a r i a bl e s, t h e g e n e r a ls o l u t i o n o f e q u a t i o n ( 1 5 ) in t h e h o m o g e n e o u s ( Q = 0 ) c as ema y b e wr i t t e n a s :

O ( x , y , z ) = ( a cosXy + b s inX y) (c cos/. tz + d s in gz) Ax e l(xa+u2)lulx (18)

wh e r e : a , b , c , d , A , X a n d / a a r e a r b i t r a r y c o n s t a n t s .Us in g t h e b o u n d a r y c o n d i t i o n s ( 1 6 ) , a n d b y s u p e r-p o s i t i o n , t h e s o l u t i o n b e c o me s :f -O (x ,y , z ) = Oo ~ Am, , s inmTry s inn~z

m = l n = lx e -[(m'+n2)lul~x ( 1 9 )

Then app ly ing the in i t i a l cond i t ion (17 ) , y ie lds :' 6 f . f . i o(x , y , z ) - 7r s inml ry sinmrz

m = l n = l mx e -[(m'+n')lul~r~x ( 2 0 )

wh ere : m , n r ange ove r a ll odd pos i t ive in tege r s ( i .e . ,m , n = 1 , 3 , 5 , 7 . . . . ) . T h e n o n h o m o g e n e o u s c a s e ( i .e . ,Q # : 0 ) i s more invo lved , and the so lu t ion in th i s case i sg iven be low.I n t h e n o n h o mo g e n e o u s c a s e , a s su me t h e s o l u t i o n t o b ei n t h e f o l l o wi n g f o r m w h i c h s a ti s fi e s t h e b o u n d a r y c o n d i t i o n( 1 6 ) :

O ( x , y , z ) = Z ~ X m n ( X ) s i n m lr y s in m r z ( 2 1 )m = l n = lEq uat ion (2 1 ) sa t isf ies (15 ) giving:

~ [uXm n + (m 2 + n 2 ) 7r2Xmn]m = l n - - 1

x sinmTry s inm rz = - - Q ( 2 2 )Us ing the concep t o f doub le Four ie r se r i e s , the coe f f i c i en t so f (sinm~ry s i n mr z ) ma y b e e x t r a c t e d a s :

, [ m 2 + n 2 \X r n n " 4" [ - ' - " ' - f f ~ ) fT t X m n = - - Q * ( 2 3 )wh e r e :

a * - 1 6 0 ( m , /7 = 1 , 3 , 5 , 7 . . . . ) ( 2 4 )u m m r 2T h e r e f o r e , t h e s o l u t i o n ( 2 1 ) m a y b e r e w r i t t e n , a ft e rin se r t ing the so lu t ion o f (23 ) , a s :

m = 1 n = l ( m 2 + / 7 2 ) Ir a+ C m n e - I ( m ' + n ' ) , ' / u l x ]

Jx sinmTry s inmrz (25 )

and app ly ing the in i t ia l co :~ .di tion (17 ) , the doub le F our ie rcoe f f i c i en t s Cmn a re g iven by :Cmn = + Q*uOo( m ~ + n 2 ) r ( 2 6 )

Thus the f ina l so lu t ion in th i s case becomes :

I(x , y , z) = 7 m=l --1 m n ( m 2 + n2 ) ~ 2m n ( m 2 + / 7 2 ) r r 2l

x e ~ ( " + " ' ) / " l " '~ 1 s in m ~ y s in , ., z ( 2 7 )No t e t h a t f o r Q = 0 , e q u a t i o n ( 2 7 ) r e d u c e s t o t h e h o m o -g e n e o u s s o l u t i o n ( 2 0 ) .I t is wo r t h me n t i o n i n g t h a t w h e n u = 0 t h e t h r e e -d i me n s i o n a l t e mp e r a t u r e e q u a t i o n ( 1 5 ) r e d u c e s t o t h e t wo -d i me n s i o n a l o n e :

a2o a2o- - + - - = Q ( 2 8 )ay 2 az 2ind ica t ing tha t u = 0 is a s ingu la r pa ram e te r (because weh a v e l o s t t h e l o we r o r d e r d e r iv a t iv e t e r m) o f t h e f u ll p a r t ia ld i f f e ren t i a l equ a t ion , z The re fo re , in such a s i tua t ion , thea n a l y ti c a l s o lu t i o n ( 2 7 ) mu s t b e m o d i f i e d t o y i e l d :

O ( x , y , z ) = 7r,~ m = l =1 m n ( m 2 + n 2 )x sinm~ry sinmrz ( 2 9 )

I t i s to be no ted tha t (29 ) sa t i s f i e s the d i f f e ren t i a le q u a t i o n ( 2 8 ) t o g e t h e r w i t h b o t h b o u n d a r y c o n d i t i o n s(16) and (17 ) . Also , when Q = 0 , we a r r ive a t the t r iv ia lso lu t ion 0 = 0 .

F i n i te - e le m e n t m e t h o dIn th i s sec t ion we d i scuss the f in i t e -e lem en t m e tho d fo r at r ia n g u l a r e l e me n t c o n s is t in g o f t h r e e n o d e s .B e f o r e th e mi n i mi z a t i o n is p e r f o r me d , e q u a t i o n ( 1 5 )ma y b e r e wr i t t e n a s :

a2o a2o ( + u ~O ~=+ - - - - Q 0 ( 3 0)a y 2 a z ~ a x /T h e s o l u t i o n t o t h e p h y s i c a l p r o b l e m i s o b t a i n e d b y mi n i -mi z i n g ( 3 0 ) f o r e v e r y p o i n t o f x .T h e f u n c t i o n a l e q u i v a l e n t o f ( 3 0 ) i s :

1 0 1s = J 2 t a y l x z / ]v ( 3 1 )

E q u a t i o n ( 3 1 ) m u s t b e mi n i mi z e d wi t h r e s p e c t t o t h e s e t o fnoda l va lues {0} .Us in g ma t r i x n o t a t i o n , ( 3 1 ) m a y b e wr i t t e n a s :

1 + 2 0 ( += f T [ { g } T [ D ] { g } _ Q U ~ x 0 ) ] d y d z ( 32 )/)

wh e r e :

{ g } T = L a y a z l ( 3 3 )

4 8 4 A p p l . M a t h . M o d e l l i n g , 1 9 8 2 , V o l . 6, D e c e m b e r

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O [ o o ]We know tha t the func t ions for 0 a re p iecewise con-t inuous over the reg ion but , ins tead a re def ined overindiv idua l e lements , 0 (e ), the in tegra ls in (32 ) m us t besepara ted in to in tegra ls over the ind iv idua l e lemen ts g iv ing:

E " 1J = ~ ~ "~{g(e)}T[D(e)]{g(e)} y dze = 1 v ( e ) z .

+ ~ o (~) a + u - - dy dze=:t e) a x ( 3 5 )w h e r e : E i s t h e n u m b e r o f e l e m e n t s .T h e m i n i m u m o f J o c c u r s w h e n :

M = 0 ( 3 6 )a{0}T h e e l e m e n t a l c o n t r i b u t i o n , aJ(e)]a{o} , o t h e t o t a lm i n i m i z a t i o n p r o c e s s, aJ/a{o } , i s:

a j ( e ) f{0} -- [B(e)]T[De)] [B(e)]{O}d y d zv ( e )

a { 0 }+ u [ N ( e ) ] T [ N ( e ) ] d y d z a ' - -x-v(e)

w h e r e :

+ r e ) Q [ N ( e ) ]T d Y d z( 3 7 )

( g ( e ) } = [ B ( e ) ] ( 0 } ( 3 8 )

- - g - y . - . a y /[B(e) ] = a/v~le aN~e aN(pC) ( 3 9 )

L "'" Z ]T o c o m p u t e t h e m i n i m i z a t i o n p r o c e s s, t h e f o l l o w i n g s y s t e mof equa t ions resu l t s :

a { 0 } + [ K ] { 0 } + { F } = 0 ( 4 0 )c] a- -~- -wh e r e th e e le m e n t c o n t r i b u t i o n s t o [ K ] , [ C ] a n d { F } a r e :

= ( u [ N ] T [N ] d y d z ( 4 1 a )c (e) ]v ( e )

= f u [ B ] 7 " [ D ] [ B ] d y d z ( 4 1 b )k ( e ) ]v ( e )

f Q [ N ] T d y d z ( 4 1 c )r e ) } =I #v(e)

All the in tegra ls in (41a) , (41b ) and (41 c) a re eva lua tedover a s ingle e lement . The e lement cont r ibut ions a re thens u m m e d .Equ at ion (40) i s a sys tem o f f i r s t o rder l inear d i f fe ren-

R e s e a r c hN o t et i a l equa t ion s . Us ing the rm i te d i f fe rence so lu t ion fora{ o} / ax , yie lds :

( [ K ] + ~ x [ C ] ) { O } l = ( - ~ x [ C ] - - [ K ] ) { e ' }o - - 2 { F }( 4 2 )

Equ at ion (42 ) ma y be so lved to y ie ld the nodal va lues a tt h e p o i n t x + A x g i ve n th e n o d a l v a lu e s o f x . T h i s s o l u ti o ntechn ique impl ies tha t the ia i t ia l noda l va lues a re known .T h e c o l u m n v e c t o r { F } c o n si s ts o f k n o w n p a r a m e t e r s .E q u a t i o n ( 4 2 ) m a y b e r e w r i t t e n i n c o m p a c t f o r m a s:[ .4]{0} = {G} (43 )

w h e r e :2[ A ] = [ K ] + - - [ C ] ( 4 4 a )A x

{ G } = ( ~ x [ C ] - - [K ] ) { O } o - - 2 { F } ( 4 4 b )T h e d e t a il e d c a l c u la t i o n s o f t h e s e m a t r i c e s m a y b e f o u n di n t h e w o r k o f S e g e r l i n d Y

S o l u t i o n o f t h e e q u a t i o n sT h e t e m p e r a t u r e e q u a t i o n ( 3 0 ) h a s b e e n s o lv e d s u b je c t t ot h e b o u n d a r y c o n d i t i o n s ( 1 6 ) a n d ( 1 7 ) , n a m e l y 0 = 0 a tt h e w a l l o f t h e d u c t a n d 0 = 1 at t h e e n t r a n c e ( x = 0 ) o fthe d uc t , for a ver t ica l c ross -sec t ion of the square du c t ( seeF i gure 1 ) b y m e a n s o f t h e f i n i t e e l e m e n t t e c h n i q u e d e s-cr ibed in the previous sec t ion . In th i s s tudy , the coord ina teso f t h e c e n t r e s o f t h e t h r e e f u e l r o d s i n t h e d u c t s a r e a s s um e dt o b e s i t u a t e d a t t h e n o d a l p o i n t s , 3 0 , 3 4 a n d 5 9 ; a n d t h ever t ica l c ross -sec t iona l a rea of the duc t has been d iscre t izedi n t o 1 2 8 e l e m e n t s w i t h 8 1 n o d a l p o i n t s , a n d n u m b e r e d i ns u c h a w a y a s t o p r o d u c e a m i n i m u m b a n d w i d t h .

73 74 7 5 7 6 7 7 7 8 7 9 6 0 8 1

( 6 , ) . ~ ( 8 3 , . 8 " , , ( e 6 ) , ~ ' , ,,

, 6 5 , 3 8 - , , ( 6 , ) \ ( ~ ) . ~ (. ,, . , \

1 1 N ~ 1 2 N 1 3 ~ , 1 4 N ,

( ' " .~ \ "~ ) .3 " . . 1 .6 , . \

( 4 3 ) ~ ( 4 5 ) _ ~ _ ( 4 7 ) 2 7 ~( 4 1 ) ~ , ~ 5 ~ , 2 6 ~ J6 ) ( 2 e ) ~ ( 3 0 ) ( 3 , )

1 6 N 1 7 % 1 8 N

1 2 3 4 5 6 7 e 9D yF i g u r e 1 Schematicdiagram of FE discretization of a squareduc tw i t h three fuel elem ents arranged in triangular arrayA p p l . M a t h . M o d e l l in g , 1 9 8 2 , V o l . 6 , De c e m b e r 4 8 5

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Research NoteT h e t e m p e r a t u r e d i s t r i b u t i o n a n d i ts g r a d i e n t a t d i f f e r e n t

n o d a l p o s i t i o n s o f t h e s q u a r e p l a n e y - - z ( i .e . , x = A xp l a n e ) a r e c a l c u l a t e d b y s o l v i n g t h e m a t r i x e q u a t i o n ( 4 3 )w i t h t h e b o u n d a r y c o n d i t i o n s d e s c r ib e d a b o v e . A s u i t a bl ec o m p u t e r c o d e h a s b e e n d e v e l o p e d t o s o l v e t h e s e t o fa l g e b r a i c e q u a t i o n s , a n d h a s b e e n m o d i f i e d t o a c c o m m o d a t et h e t h r e e - d im e n s i o n a l h e a t t r a n s f e r e q u a t i o n . T h i s c o m -p u t e r c o d e m a y b e u s e d o n l y f o r t h e s o l u t i o n o f a b a n d e ds y m m e t r i c m a t r i x . T o s t a r t t h e n u m e r i c a l p r o c e s s t h e i n i ti a lv a l u e o f 0 , w h i c h i s u n i t y a t e v e r y n o d a l p o i n t , i s i n p u t t ot h e c o m p u t e r , a l o n g w i t h t h e z e r o w a l l t e m p e r a t u r e s a t t h ea p p r o p r i a t e n o d a l p o i n t s , t o s o lv e e q u a t i o n ( 4 3 ) w i t h a ni n c r e m e n t a l v a l u e A x ; a n d t h e n e w v a l u es o f t h e t e m p e r a t u r ef i e ld a t t h e p l a n e x = A x a r e o b t a i n e d . T h i s p r o c e s s i sc o n t i n u e d s t e p - b y -s t e p , m a r c h i n g f o r w a r d t o o b t a i n a l ld e s i r e d r es u l t s . I t i s w o r t h m e n t i o n i n g h e r e t h a t , w h e n an e w s e t o f v a l u e s o f 0 is o b t a i n e d s u p p o s e a t x = x + A x ,t h e n t h e s e v a l u es a re m o d i f i e d b y t h e p r e s c r ib e d b o u n d a r yv a l ue s a n d ar e t h e n i n p u t t o c o m p u t e t h e n e x t t e m p e r a t u r ef i el d a t t h e f a c e x = x + 2 A x . T i f fs m e a n s t h a t t h e s p e c i f i e dv a l u e s o f 0 m u s t b e r e s e t a f t e r e v e r y i t e r a t i o n . 2~

I n t h is s t u d y , w e h a v e u s e d t h e m a r c h i n g s t e p s i ze o f x a sA x = 0 . 0 0 5 . T h e m a j o r i ty o f c o m p u t a t i o n s w e r e p er f o r m e dw i t h t h e p a r a m e t e r s Q = - 1 , 0 , I , a n d u = I , 0 . 5 , 0 . 1 , a n dw i t h x r an g i n g f r o m 0 . 0 0 5 t o 0 . 1 0 0 . T h e c o m p u t i n g t i m ef o r t h e C D C C y b e r 1 7 0 is v e ry m o d e r a t e a n d i n f a c t i tr e q u i r e s le s s t h a n 1 8 s t o o b t a i n a s o l u t i o n f o r a g iv e n s e t o fp a r a m e t r i c v a l ue s t o i te r a t e u p t o 2 0 s t a t io n s . T h e c o m p u t e rp r o g r a m t a k e s a d v a n t a g e o f t h e b a n d e d n e s s o f a s y m m e t r i cm a t r i x w i t h b a n d w i d t h b e i n g t e n i n t i f f s i n v e s t i g a t i o n .

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Figure 2 T e m p e r a t u r e d i s t r i b u t i o n a t n o d e 1 4 f o r Q = 1 a n d U = 1 .( - - ) , a n a l y t i c a l s o l u t i o n ; ( - - - - - - ) , F E M s o l u t i o n

Dis cus s ion o f r e s u l t sI n t h is p a p e r , w e h a v e s t u d i e d a s i m p l e m a t h e m a t i c a l m o d e lo f t h r e e -d i m e n s i o n a l h e a t t r a n s fe r i n a d u c t - t y p e r e a c t o r . T h ee n t i r e c o n t i n u u m i s d i s c r e t i z e d i n t o t r i a n g u l a r f i n i t e e l e m e n tf o r m w i t h 1 2 8 e l e m e n t s w i t h 8 1 n o d a l p o i n t s ( s e e Figure 1) .

Figures 2-11 s h o w a c o m p a r i s o n o f t h e F E M ( F i n i teE l e m e n t M e t h o d ) s o l u t i o n s w i t h t h e a n a l y t i c a l s o l u ti o n s a td i f f e r e n t n o d e s f o r t h e p a r a m e t e r s u = 1 a n d Q = 1 . C o m -p a r i s o n o f t h e s o l u t i o n s is m a d e a t t h e n o d a l p o i n t s 1 4 , 1 5 ,1 7 , 2 2 , 2 4 , 2 5 , 3 2 , 3 3 , 4 1 a n d 4 3 . I t i s o b s e r v e d t h a t t h e r e i sg o o d a g r e e m e n t . U s i ng t h e F E M s o l u t i o n s , tw o s e ts o fi s o t h e r m a l c u r v e s a r e d r a w n i n Figures 12 a n d 13 f o r Q = 1a n d u = 1 a t x = 0 . 0 1 a n d 0 . 0 4 . T h r o u g h t h i s a n a l y s is it h a sb e e n f o u n d t h a t t h e h o t t e s t p o i n t i n s i de t h e d u c t a p p e a r st o b e a t t h e n o d a l p o i n t 4 1 w h i c h h a p p e n s t o b e t h e m i dp o i n t o f e a c h c r o s s -s e c t io n o f t h e d u c t .Figures 14 a n d 15 s h o w t h e v a r i at i o n s o f t h e F E M s o lu -t i o n s o f t e m p e r a t u r e d i s t r ib u t i o n s a t t h e n o d a l p o i n t 4 1 .Figure 14 s h o w s t h e c o m p a r i s o n b e t w e e n F E M s o l u t io n sa n d a n a l y t i c a l s o l u t i o n s f o r th e p a r a m e t e r s u = 1 , a n dQ = - 1 , 0 , 1 w h e r e a s Figure 15 s h o w s t h e b e h a v i o u r o ft e m p e r a t u r e p r o f d e s a t 4 1 w h e n Q = - 1 f o r d i f f e r e n t v a lu e so f u = 1 , 0 . 5 , 0 . 1 . D u e t o th e s i n gu l ar n a t u r e o f th e d i f f e re n -t i a l e q u a t i o n w h e n u - ~ 0 , t h e n u m e r i c a l s o l u t i o n g i v e so s c i l l a to r y t y p e s o f s o l u t i o n s a s w e t r y t o c o m p u t e f o rs m a l l e r v a l u e s o f u ( u < 0 . 1 ) a n d w h e n u = 0 . T h e r e f o r e ,i n t h a t c a s e t h e n u m e r i c a l s c h e m e h a s t o b e m o d i f i e d t oc o m p u t e w i t h a t w o - d i m e n s i o n a l e q u a t i o n i n s t e a d o f at h r e e - d i m e n s i o n a l e q u a t i o n . A s m e n t i o n e d e a r l i e r, in t h a ts i t u a ti o n , w e h a v e c o m p l e t e l y l o s t o n e d i m e n s i o n w i t h t h ei n i ti a l c o n d i t i o n . C o m p u t a t i o n s i n t h a t s i t u a t i o n p r o v i d eg o o d a g r e e m e n t w i t h t h e a n a l y t i c a l s o lu t i o n s.

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Although it is implicit that the three-rod bundle is thesource of heat inside the reactor, in this study, we have notconsidered their effect on the heat transfer. We still studythis important question in our future work.

AcknowledgementThis work was made possible by a research contractbetween the Technical University of Nova Scotia and theAtomic Energy of Canada Ltd.

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1 A l u n a d , S . Y . a n d M e r il o , M . ' E f f e c t o f o r i e n t a t i o n o n c ri t i c alh e a t f l u x i n a 3 6 - e l e m e n t r o d b u n d l e c o o l e d b y F r e o n - 1 2 ' ,C A N C A M S i x th , V a n c o u v e r , 2 9 M a y - 5 J u n e , 1 9 7 7 , p p . 8 5 5 -8 5 6 , V o l . 22 A l i m a d , S . Y . ' F l u i d t o f l u i d m o d e l l i n g o f c r i t i c a l h e a t f l u x : ac o m p e n s a t e d d i s t o r t i o n m o d e l ' , h ~ t . J . Heat Mass Transfer1 9 7 3 , 1 6 , 6 4 13 R a m a c h a n d r a m V . a n d S p a l d i n g , D . B. ' F l u i d fl o w a n d h e a tt r a n s f e r i n r e c t a n g u l a r - s e c t i o n e d - d u c t s ', l m p e i i a l C o l l e g e o fS c i e n c e a n d T e c h n o l o g y , M e c h a n i ca l E n g i n e e r i n g D e p a r t m e n t ,N o v e m b e r 1 9 7 6 , H T S / 7 6 / 2 14 L a u n d e r , B . E . a n d Y i n g , W . M . 'P r e d i c t i o n o f f l o w a n d h e a tt r a n s f e r i n d u c t s o f s q u a r e c r o s s - s ec t i o n ' , P~'oc. hlstn Mech.En g r s 1 9 7 3 , 1 8 7 , 3 75 P a t a n a k a r , S. V . a n d S p a l d i n g , D . B . ' A f i n i t e d i f f e r e n c e p r o c e -d u r e f o r s o l v in g t h e e q u a t i o n s o f t h e t w o d i m e n s i o n a l b o u n -d a r y l a y e r ' , h i t . J . Heat Mass Transfer 1 967, I O , 13896 E m e r y , A . F . et al. ' T h e n u m e r i c a l p r e d i c t i o n o f d e v e l o p i n gt u r b u l e n t f l o w a n d h e a t t r a n s f e r i n a s q u a re d u c t ' , p r e s e n t e d a tH e a t T r a n s f e r D i vi s i o n o f A S M E , 1 9 7 9 , 2 - 7 D e c e m b e r , 1 9 7 9W A / H T - I 2 , N e w Y o r k7 G a s s n e r , F . B . a n d E m e r y , A . F . ' A l e n g t h s c a l e m o d e l f o rd e v e l o p i n g t u r b u l e n t f l o w i n a r e c t a n g u l a r d u c t ' , Fl u i d s En g n gS u m m e r C o n f . A S M E , Y a l e U n i v e r s i t y , N e w H a v e n , C o n n . ,1 5 - 1 7 J u n e 1 9 7 7 , P a p e r N o . 7 7 - F E - 48 C a r a j i l e s co v , P . a n d T o d r e a s , N . E . ' E x p e r i m e n t a l a n d a n a l y t i c a ls t u d y o f a x i al t u r b u l e n t f l o w s i n a n i n t e r i o r s u b c h a n n e l o f ab a r e r o d b u n d l e ' , W i n t e r An n . M e e t h l g AS M E, H o u s t o n , T e x a s ,3 0 N o v e m b e r - 5 D e c e m b e r 1 9 7 5 , P a p e r N o . 7 5 - W A ] H T - 5 19 N e f f, S. 'M e a s u r e m e n t s a n d a n a l y s i s o f f l o w i n d u c t s a n d r o db u n d l e s ' , PhD Diss. , U n i v e r s i t y o f K e n t u c k y , L e x i n g t o n ,K e n t u c k y , 1 9 7 7

1 0 R e c c e , G . J . ' A g e n e r a li z e d R e y n o l d s s tr e s s m o d e l o f t u r b u -l e n c e ' , PhD Thesis , U n i v e r s i t y o f L o n d o n , L o n d o n , 1 9 7 61 1 C h e r n , S . Y . a n d C h a t o , J . C . 'A f i n i t e e l e m e n t t e c h n i q u e t od e t e r m i n e t h e f r i c t i o n f a c t o r a n d h e a t t r a n s f e r f o r l a m i n a rf l o w s i n a p i p e w i t h i r r e g u l a r c r o s s - s e c t i o n ' . N u m e r . H e a tTransfer 1 9 7 8 , I , 4 5 31 2 S l a g te r , W . ' F i n i t e e l e m e n t a n a l y s is fo r t u r b u l e n t fl o w s i ni n c o m p r e s s i b l e f l u id s i n f u e l r o d b u n d l e s ' , Nuclear ScL Engng1 9 7 8 , 6 6 , 8 41 3 S p a r r o w , E . M . a n d L o e f f l e r , A . L . J r . ' L o n g i t u d i n a l l a n f i n a rf l o w b e t w e e n c y l i n d e r s a rr a n g e d i n r e c t a n g u l a r a r r a y ' , A I C h EJ . 1 9 5 9 , 5 ( 3 ) , 3 2 51 4 M e y d e r , R . ' S o l v i n g t h e c o n s e r v a t i o n e q u a t i o n s i n fu e l r o db u n d l e s e x p o s e d t o p a r a l l el f l o w b y m e a n s o f c u r v i l in e a ro r t h o g o n a l c o o r d i n a t e s ' ,J . Comp. Phys . 1 9 7 5 , 1 7 , 5 61 5 B a r t zi s , J . G . a n d T o d r e a s , N . E . ' T u r b u l e n c e m o d e l l i n g o f a x i a lf l o w i n a b a r e r o d b u n d l e ' , J . Heat Transfer 1 9 7 9 , 1 0 , 6 2 81 6 W o n g , H . H . a n d A l p , E. 'I n t e r i m r e p o r t o n s u b c h a n n e l fl o wm o d e l l i n g i n r o d b u n d l e s : t u r b u l e n t f l o w s o l u t i o n ' , W e s t i ng -h o u s e C a n a d a L t d , H a m i l t o n , C W A R D - 3 5 8 , M a r c h 1 9 8 01 7 R a h m a n , M . a n d H e a p s , H . S . ' T u r b u l e n t f lo w w i t h h e a tt r a n s f e r in a 3 - r o d b u n d l e ' , T e c h n i c a l U n i v e r s i t y o f N o v a S c o t i a ,H a l i f a x , C a n a d a . R e p o r t s u b m i t t e d t o A E C L , D e c e m b e r 1 9 8 01 8 G e s s n e r , F . B . a n d E m e r y , A . F . 'A R e y n o l d s s tr e s s m o d e l f o rt u r b u l e n t c o r n e r fl o w s - p a r t I : d e v e l o p m e n t o f t h e m o d e l ' ,J . Fluids Eng. , Trans . ASME, Ser ies I 1 9 7 6 , 9 8 ( 2 ) , 2 6 11 9 C a r l s l a w , H . S . a n d J a e g e r , J . C . ' C o n d u c t i o n o f h e a t i n s o l id s "( s e c o n d e d n ) , C l a r e n d o n P r e s s , O x f o r d , 1 9 6 7 , p . 72 0 V a n D y k e , M . ' P e r t u r b a t i o n m e t h o d s in f l u id m e c h a n i c s ' ,A c a d e m i c P r e ss , N e w Y o r k , 1 9 6 42 1 S e g e r l i n d , L . J . ' A p p l i e d f i n i t e e l e m e n t a n a l y s i s ' , W i l e y I n c . ,N e w Y o r k , 1 9 7 6

490 App l . Math. Model l ing, 19 82, Vol . 6, December