UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Effects of viscous dissipation on combined free and forced convection through vertical ducts and passages Rokerya, M. Shafi 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1970_A7 R64.pdf [ 3.45MB ]
Metadata
JSON: 831-1.0080680.json
JSON-LD: 831-1.0080680-ld.json
RDF/XML (Pretty): 831-1.0080680-rdf.xml
RDF/JSON: 831-1.0080680-rdf.json
Turtle: 831-1.0080680-turtle.txt
N-Triples: 831-1.0080680-rdf-ntriples.txt
Original Record: 831-1.0080680-source.json
Full Text
831-1.0080680-fulltext.txt
Citation
831-1.0080680.ris

Full Text

EFFECTS OF VISCOUS DISSIPATION ON COMBINED FREE AND FORCED CONVECTION THROUGH VERTICAL DUCTS AND PASSAGES by M. SHAFI ROKERYA B.E. (Mech.), Univers i ty of Karach i , Karachi , Pak is tan , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. in the Department of Mechanical Engineering We accept t h i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1970 In presenting t h i s thes is in p a r t i a l f u l f i l m e n t of the requ i re -ments for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r e e l y ava i lab le for referance and study. I fur ther agree that permission fo r extensive copying of t h i s thes is fo r scho la r l y purposes may be granted by the Head of the Department or by h is representat ives. I t i s understood that copying or pub l i cat ion of t h i s thes is for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permission. M. SHAFI ROKERYA Department of Mechanical Engineering The Univers i ty of B r i t i s h Columbia Vancouver 8, B r i t i s h Columbia Canada Date M0^ 1970 . i i ABSTRACT The e f fec ts of viscous d i s s i p a t i o n on the flow phenomena and heat t ransfer rate fo r f u l l y developed laminar flow through v e r t i c a l ducts and passages has been analysed under the con-d i t i o n of combined free and forced convection. The f l u i d propert ies are considered to be constant except for the va r ia t ion of density in the buoyancy term of the momentum equation. The thermal boundary condit ion of uniform heat f l u x per unit length in the flow d i rec t ion has been considered. The invest igat ion i s ca r r ied out fo r two geometries; (a) C i r cu la r ducts and (b) Concentric a n n u l i . The governing momentum and non - l inear energy equations are solved for the c i r c u l a r duct by three methods; ( i ) Power Series Method ( i i ) Ga lerk in 's Method and ( i i i ) Numerical Integration Method. The solut ions for the concentric annuli are obtained by Numerical Integration Method. Results for the v e l -oc i t y and temperature d i s t r i b u t i o n in the flow f i e l d are obtained, and information of engineering in te res t l i k e Nusselt numbers have been evaluated. For combined free and forced convect ion, the momentum and energy equations are coupled, and hence viscous d i s s i p a t i o n a f fec ts both the ve loc i t y and temperature f i e l d s . The e f f e c t of viscous d i s s i p a t i o n on the ve loc i t y f i e l d i s to reduce the flow ve loc i t y near the heated wa l l ( s ) and thus i t counteracts the e f f e c t of f ree convection on the ve loc i t y f i e l d fo r the present study of heating in upflow. The e f fec t of viscous d i s s i p a t i o n on the temperature f i e l d i s to act as a heat source in the f l u i d and reduce the temperature d i f ferences in the system. Viscous d i s -s ipat ion opposes the externa l l y impressed heating and reduces the heat t rans fe r rate when the surface t ransfers heat to the f l u i d . Consequently, lower Nusselt number values are obtained when viscous d i s s i p a t i o n i s taken into considerat ion. The quan-t i t a t i v e e f fec t of viscous d i s s i p a t i o n on Nusselt number i s found to be small f o r the case of c i r c u l a r ducts. However, fo r flow through annular passages and for the corresponding values of the same parameters, the e f fec t of viscous d i s s i p a t i o n on the heat t ransfer rate may not be ignored. i v TABLE OF CONTENTS Chapter Page ABSTRACT . .' i t LIST OF TABLES ' . v i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS . . . . . . . i x NOMENCLATURE . 1 I INTRODUCTION . 3 II SECTION I: C i r c u l a r Ducts 6 2.1 Formulation of the Problem 7 III SOLUTIONS. 11 3.1 Exact Solut ion Without Viscous D iss ipat ion Term . 12 3.2 Solutions With Viscous D iss ipat ion Term . . 3 .2 .1 Power Series Method 14 3 .2 .2 Ga lerk in 's Method. . 20 3 . 2 . 3 Numerical Integration Method . . . . 24 IV SECTION I I : Concentric Annuli 26 4.1 Formulation of the Problem 27 V SOLUTIONS 33 5.1 Exact Solut ion Without Viscous D iss ipat ion Term 34 5.2 Solutions With Viscous D iss ipat ion Term . . 36 VI DISCUSSION OF RESULTS 37 6.1 C i r c u l a r Ducts. 37 6.1 .1 Solut ion Deta i ls 37 V C h a p t e r Page 6.1.2 V e l o c i t y F i e l d 39 6.1.3 T e m p e r a t u r e F i e l d . 40 6.1.4 N u s s e l t Numbers . 41 6.2 C o n c e n t r i c A n n u l i . . . . • 41 6.2.1 S o l u t i o n D e t a i l s . . . . . . . . . . 41 6.2.2 V e l o c i t y F i e l d . 42 6.2.3 Temperature F i e l d . . . 43 6.2.4 N u s s e l t Numbers 45 6.2.5 Radius R a t i o . . 46 VII CONCLUSIONS 47 REFERENCES 72 APPENDICES 75 A DERIVATION OF NUSSELT NUMBER EXPRESSION FOR CIRCULAR DUCTS 76 B DERIVATION OF NUSSELT NUMBER EXPRESSION FOR CONCEN-TRIC ANNULI 78 C DETAILS OF GOVERNING EQUATIONS AND LIMITATIONS . 82 vi LIST OF TABLES Table Page I Velocities and temperature differences at the centre of a vertical circular duct due to viscous dissipation effects 48 II Effect of viscous dissipation parameter on Nusselt number for a vertical circular duct 49 III Nusselt number values for M=0 obtained by Exact solution and Runge-Kutta method for concentric annul us with radius ratio 0.5 50 IV Velocity distribution and temperature differences due to viscous dissipation effects for concentric annul us with outer wall heated, inner wall insul-ated for Ra=l, x= 0.75 51 V Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with outer wall heated, inner wall insul-ated for Ra-1000, x= 0.75 52 VI Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with inner wall heated, outer wall insul-ated for Ra=l, X= 0.75 53 VII Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with inner wall heated, outer wall insul-ated for Ra=1000, x= 0.75 54 VIII Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with both walls heated for Ra=l, x= 0.75 . 5 5 IX Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with both walls heated for Ra=2000, X= 0.75 56 vn LIST OF FIGURES Figure Page 1 Coordinate system for flow through a v e r t i c a l c i r c u l a r duct . . . 8 2 Coordinate system for flow through a v e r t i c a l concentr ic annulus 28 3 Ve loc i ty p r o f i l e s for concentric annulus with outer wall heated, inner wall insulated for radius r a t i o 0.25 57 4 Ve loc i ty p r o f i l e s fo r concentric annulus with outer wall heated, inner wall insulated fo r radius r a t i o 0 . 5 . 58 5 Ve loc i ty p r o f i l e s for concentric annulus with inner wall heated, outer wall insulated fo r radius r a t i o 0.25 59 6 Ve loc i ty p r o f i l e s for concentric annulus with inner wall heated, outer wall insulated fo r radius r a t i o 0.5 60 7 Ve loc i ty p r o f i l e s for concentric annulus with both wal ls heated for radius r a t i o 0.25 61 8 Ve loc i ty p r o f i l e s for concentric annulus with both wal ls heated fo r radius r a t i o 0.5 62 9 Temperature p r o f i l e s for concentric annulus with outer wal l heated, inner wall insulated fo r radius r a t i o 0.25 63 10 Temperature p r o f i l e s fo r concentr ic annulus with outer wall heated, inner wall insulated fo r radius r a t i o 0.5 64 11 Temperature p r o f i l e s fo r concentric annulus with inner wall heated, outer wall insulated for radius r a t i o 0.25 65 12 Temperature p r o f i l e s for concentric annulus with inner wall heated, outer wall insulated fo r radius r a t i o 0.5 66 13 Temperature p r o f i l e s for concentric annulus with both wal ls heated for radius r a t i o 0.25 67 14 Temperature p r o f i l e s for concentric annulus with both wal ls heated for radius r a t i o 0.5 68 v i i i Figure Page 15 Ef fect of viscous d i s s i p a t i o n parameter on Nusselt number fo r concentr ic annulus with outer wall heated,inner wall insulated 69 16 Ef fect of viscous d i s s i p a t i o n parameter on Nusselt number for concentric annulus with inner wall heated, outer wal l insulated . . . . ?0 17 Ef fect of viscous d i s s i p a t i o n parameter on Nusselt number for concentric annulus with both wal ls heated . . . 71 i x ACKNOWLEDGEMENTS The author wishes to express h is deep grat i tude to Dr. M. Iqbal who devoted considerable time on advice and guidance throughout a l l phases of the present study. Sincere thanks are also extended to Dr. B. D. Aggarwala of the Mathematics Depart-ment, Univers i ty of Calgary and Dr. M. Flower of the Department of Computer Science, B r i s t o l Univers i ty fo r t h e i r valuable suggestions. Use of the Computing Centre f a c i l i t i e s at the Univers i ty of B r i t i s h Columbia and the f i n a n c i a l support of the National Research Council of Canada are g r a t e f u l l y acknowledged. NOMENCLATURE Area of cross-section Specific heat of the fluid at constant pressure aT | j , temperature gradient in the flow direction 4 x area of cross-section . n e a t e d perimeter ' e c l u i v a l e n t diameter U2 r—7T, Eckert number, dimensionless CpAT Gravitational acceleration . ( d | + P w 9 ) D 2 ^-y — . pressure drop parameter dimensionl Eck P -^* viscous dissipation parameter, dimensionless hD. » Nusselt number, dimensionless K Wall heat flux Radial coordinate 2r PJ—» for circular ducts, dimensionless uh £-» for concentric annuli, dimensionless r 'o p^ ggc CD.* yg^- » Rayleigh number, dimensionless UD.p — — . Reynolds number, dimensionless Temperature Average axial velocity Axial velocity v z -Q-. dimensionless axial velocity V = p dimensionless Z = Ax ia l coordinate in flow d i rec t ion 3 = Coef f i c ien t of volumetric expansion n • ( R a ) 1 / 4 K = Thermal conduct iv i ty of the f l u i d r i x = — » radius r a t i o , dimensionless o y = Dynamic v i s c o s i t y of the f l u i d p = Density of the f l u i d (T - T \ <f> = pUc CD ^/4<' t e m P e r a t u r e f unc t ion , dimensionless J = dimensionless v2 = _d£ + 1 d_ d R2 + R dR Subscripts i ins ide o outside w wal l 3 1. INTRODUCTION In t h e f l o w o f a l l r e a l f l u i d s , v i s c o s i t y p l a y s an im-p o r t a n t r o l e and when v i s c o u s f l u i d s f l o w on s o l i d s u r f a c e s by and l a r g e v e l o c i t y g r a d i e n t s e x i s t . These v e l o c i t y g r a d i e n t s g i v e r i s e t o s h e a r s t r e s s e s w hich r e s u l t s i n t h e d i s s i p a t i o n o f f r i c t i o n a l e n e r g y i n t o h e a t . C o n s e q u e n t l y i n a h e a t t r a n s f e r p r o c e s s f o r t h e f l o w o f a r e a l f l u i d , t h e o m i s s i o n o f v i s c o u s d i s s i p a t i o n i n t h e t h e r m a l e n e r g y b a l a n c e o f a moving f l u i d e l e m e n t would be u n r e a l i s t i c f r o m t h e p h y s i c s o f f l u i d s . H a l l man [ 7 ] * and Morton [ 1 4 ] have i n v e s t i g a t e d t he e f f e c t o f f r e e c o n v e c t i o n on f o r c e d c o n v e c t i o n and have shown t h a t t h e e f f e c t o f f r e e c o n v e c t i o n on t h e f o r c e d v e l o c i t y f i e l d i s t o i n c r e a s e t h e v e l o c i t y g r a d i e n t s n e a r t h e w a l l s o f t h e d u c t i n up f l o w when h e a t i s t r a n s f e r r e d from t h e s u r f a c e t o t h e f l u i d . From t h e r e s u l t s o f t h e s e i n v e s t i g a t i o n s i t seems t h a t t h e s t u d y o f t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n which i s a s s o c i a t e d w i t h v e l o c i t y g r a d i e n t s c o u l d be q u i t e i n t e r e s t i n g i n t h e f i e l d o f combined f r e e and f o r c e d c o n v e c t i o n . The s t u d y o f t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n can be d i v -i d e d i n t o two b r o a d c a t e g o r i e s , ( i ) E x t e r n a l f l o w s and ( i i ) I n t e r n a l f l o w s . A b r i e f s u r v e y o f t h e a v a i l a b l e l i t e r a t u r e u n der t h e s e two c a t e g o r i e s i s p r e s e n t e d below. E x t e r n a l Flows F o r e x t e r n a l f l o w s , t h e e f f e c t o f v i s c o u s d i s s i p a t i o n i s f o u n d t o be q u i t e s i g n i f i c a n t b e c a u s e o f t h e e n e r g y g e n e r a t e d i n •Numbers i n b r a c k e t s d e s i g n a t e r e f e r e n c e s a t t h e end o f t h e t h e s i s . 4 t h e boundary l a y e r , and t h e s k i n t e m p e r a t u r e s t h a t a r e a t t a i n e d a t v e r y h i g h v e l o c i t i e s [ 8 ] . S e v e r a l s t u d i e s have been made i n t h i s r e g a r d b e c a u s e t h e phenomena o f 'Aerodynamic H e a t i n g 1 a t h i g h Mach numbers can ca u s e s e v e r e problems due t o the t e m p e r a t u r e l i m i t a t i o n s o f s t r u c t u r a l m a t e r i a l s commonly used i n t h e man-u f a c t u r e o f a i r c r a f t p a r t s and m i s s i l e s . S t u d i e s i n t h e a r e a o f Aerodynamic H e a t i n g have been r e p o r t e d by S c h l i c h t i n g [ 2 2 ] , S h a p i r o [ 2 3 ] and T r u i t t [ 2 4 ] among o t h e r s . The s t u d y o f t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n i n n a t u r a l c o n v e c t i o n was c a r r i e d o u t by G e b h a r t [ 5 ] f o r f l o w o v e r a s e m i - i n f i n i t e p l a t e p a r a l l e l t o t h e body f o r c e d i r e c t i o n . He used t h e p e r t u r b a t i o n method and has c a l c u l a t e d t h e f i r s t temp-_2 e r a t u r e p e r t u r b a t i o n f u n c t i o n f o r P r a n d t l numbers from 10 t o 1 0 4 . He has shown t h a t t h e magnitude o f t h e v i s c o u s d i s s i p a t i o n e f f e c t depends upon t h e d i s s i p a t i o n p a r a m e t e r which i s s m a l l f o r most e n g i n e e r i n g d e v i c e s w i t h common f l u i d s f o r t h e g r a v i t a t i o n a l f i e l d s t r e n g t h o f t h e e a r t h . I n t e r n a l Flows The s t u d y o f t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n i n i n t e r n a l l a m i n a r f l o w s can be d i v i d e d i n t o t h r e e p a r t s , ( i ) F o r c e d c o n v e c t i o n , ( i i ) F r e e c o n v e c t i o n and ( i i i ) Combined f r e e and f o r c e d c o n v e c t i o n , ( i ) F o r c e d C o n v e c t i o n T y a g i [ 2 5 , 26, 27, 28] i n a s e r i e s o f papers has s t u d i e d t h e e f f e c t o f v i s c o u s d i s s i p a t i o n i n f o r c e d c o n v e c t i o n t h r o u g h non-c i r c u l a r c h a n n e l s . He has used t h e method o f complex v a r i a b l e s and . has o b t a i n e d s o l u t i o n s f o r b o t h Neumann and D i r i c h l e t t y p e t h e r m a l boundary c o n d i t i o n s showing t h a t v i s c o u s d i s s i p a t i o n has s i g n i f i c a n t 5 e f f e c t on t h e N u s s e l t number. Cheng [ 3 ] has s t u d i e d t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n f o r f l o w t h r o u g h r e g u l a r p o l y g o n a l d u c t s u s i n g t h e method o f p o i n t - m a t c h i n g . E x a c t s o l u t i o n s were o b t a i n e d f o r t h e g o v e r n i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s and t h e boundary c o n d i t i o n s were s a t i s f i e d o n l y a t s e l e c t e d p o i n t s . He has a l s o o b t a i n e d r e s u l t s f o r a c i r c u l a r d u c t and has shown t h a t t h e e f f e c t o f v i s c o u s d i s -s i p a t i o n i s g r e a t e r f o r c i r c u l a r d u c t s t h a n f o r n o n - c i r c u l a r d u c t s . ( i i ) F r e e C o n v e c t i o n O s t r a c h [ 6 , 15, 16, 17, 18] has i n v e s t i g a t e d t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n i n n a t u r a l c o n v e c t i o n f l o w s t h r o u g h c h a n n e l s formed by two p a r a l l e l l o n g p l a n e s u r f a c e s and has shown t h a t t h e f l o w and h e a t t r a n s f e r a r e n o t o n l y f u n c t i o n s o f P r a n d t l and G r a s h o f numbers b u t a l s o depend on t h e d i m e n s i o n l e s s f r i c t i o n a l h e a t i n g p a r a m e t e r which may a p p r e c i a b l y a f f e c t t h e mode o f h e a t t r a n s f e r . ( i i i ) Combined F r e e and F o r c e d C o n v e c t i o n The o n l y a v a i l a b l e work i n t h e f i e l d o f combined f r e e and f o r c e d c o n v e c t i o n i s t h a t o f O s t r a c h [ 1 9 , 2 0 ] . He has u s e d t h e method o f s u c c e s s i v e a p p r o x i m a t i o n s t o a n a l y s e t h e p r o b l e m o f t a k i n g i n t o a c c o u n t t h e e f f e c t s o f f r i c t i o n a l h e a t i n g i n f l o w between v e r t i c a l p a r a l l e l p l a n e s u r f a c e s and has o b t a i n e d r e s u l t s s i m i l a r t o h i s f r e e c o n v e c t i o n a n a l y s i s . No work seems t o have been done t o s t u d y t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n f o r f l o w t h r o u g h c i r c u l a r d u c t s and a n n u l a r p a s s a g e s and i s t h e s u b j e c t o f t h e p r e s e n t t h e s i s . In t h e n e x t s e c t i o n , t h e f o r m u l a t i o n o f t h e p r o b l e m and t h e methods o f s o l u t i o n f o r t h e c i r c u l a r d u c t a r e p r e s e n t e d . SECTION I 2.1 Formulation Of The Problem Consider a vertical straight circular duct of constant cross-section as shewn in Fig. 1. The flow is considered to be laminar and fully developed both hydrodynamically and thermally, and is in the vertical upward direction along the positive Z-axis. The thermal boundary condition of uniform heat flux per unit length in the direction of flow is considered. The fluid properties are considered to be constant except for the variation of density in the buoyancy term of the equation of motion. The pressure work term in the energy equation has been neglected. Under the above mentioned conditions, the differential form of the continuity equation is identically equal to zero. The governing momentum and energy equations can be written as [1 ]* For the condition of uniform heat input in the flow dir-ection and constant fluid properties, the axial temperature grad-ient at the wall and for the fluid are constant and equal. Thus 8j_= C, where C is a constant. 3Z In the above equations density is to be considered var-iable only in the buoyancy term of the momentum equation ( 1 ) . This assumption is known to be valid as long as the density var-iations in the flow field are small [ 9 ] . Under this condition the equation of state in the linear form can be written as, * For details see Appendix C q N l ...1=0 FIGURE 1 C o o r d i n a t e System f o r Flow Through a V e r t i c a l C i r c u l a r Duct 9 C- C [ , - ^ T - T W ) (3) where p w d e n o t e s t h e d e n s i t y o f t h e f l u i d a t t h e c o r r e s p o n d i n g a x i a l p o i n t on t h e d u c t w a l l . The w a l l t e m p e r a t u r e i s d e f i n e d by, where T Q i s t h e r e f e r e n c e t e m p e r a t u r e a t Z = 0. By c h o o s i n g t h e f o l l o w i n g n o n - d i m e n s i o n a l p a r a m e t e r s , <p* (T-Tj/CPUCfCDt/M) , and i n s e r t i n g e q u a t i o n (3) i n e q u a t i o n ( 1 ) , t h e f o l l o w i n g non-d i m e n s i o n a l forms o f t h e momentum and e n e r g y e q u a t i o n s a r e o b t a i n e d . + L - 0 where V V + R.<L <f> v dKl R clfl (4) (5) In e q u a t i o n s (4) and ( 5 ) , R a y l e i g h number Ra and t h e v i s c o u s d i s s i p a t i o n p a r a m e t e r M a r e p r e s c r i b e d q u a n t i t i e s w h i l e V, <j> and L a r e t h e t h r e e unknown q u a n t i t i e s t o be d e t e r m i n e d . From t h e p r i n c i p l e o f c o n t i n u i t y , f o r c o n s t a n t f l u i d p r o p e r t i e s , t h e i n t e g r a l f o r m o f t h e c o n t i n u i t y e q u a t i o n can be w r i t t e n a s , 1 0 o r (6) In t h e p r e s e n t a n a l y s i s f o r t h e c a s e o f c i r c u l a r d u c t , e q u a t i o n s (4), (5) and (6) have been s o l v e d f o r t h e f o l l o w i n g boundary c o n d i t i o n s : Boundary C o n d i t i o n s (7) In o r d e r t o compare t h e r e s u l t s w i t h v i s c o u s d i s s i p a t i o n e f f e c t s t o t h o s e w i t h o u t i t , t h e a v a i l a b l e s o l u t i o n f o r t h e l a t t e r c a s e [7] i s f i r s t p r e s e n t e d h e r e b r i e f l y . i 3. SOLUTIONS 12 3.1 E x a c t S o l u t i o n W i t h o u t V i s c o u s D i s s i p a t i o n Term When t h e v i s c o u s d i s s i p a t i o n term i s n e g l e c t e d from t h e e n e r g y e q u a t i o n ( 2 ) , t h e p r o b l e m does n o t remain n o n - l i n e a r any more, and an e x a c t s o l u t i o n i s a v a i l a b l e [ 7 ] . T h i s e x a c t s o l u t i o n i n t h e f o r m o f K e l v i n f u n c t i o n s i s p r e s e n t e d i n a more s i m p l i f i e d manner below. By n e g l e c t i n g t h e v i s c o u s d i s s i p a t i o n t e r m , e q u a t i o n s (4) and (5) can be r e w r i t t e n a s , S i n c e t h e p r e s s u r e drop p a r a m e t e r L i s i n d e p e n d e n t o f t h e c o o r d i n a t e s y s t e m , e q u a t i o n s (4) and (8) can be d i v i d e d by L t o g i v e t h e f o l l o w i n g e q u a t i o n s : ^ (4) (8) V V + Rex (j) + | = 0, (9) 0 do) where E q u a t i o n s (9) and (10) can be combined t o g e t h e r t o g i v e , (11) 13 A g e n e r a l s o l u t i o n o f e q u a t i o n (11) can be w r i t t e n [ 1 3 ] a s , V = A,U (it) + fit ire^q -f hkai ft*)*/}*kcQ (Q • (12) The n o n - d i m e n s i o n a l t e m p e r a t u r e f u n c t i o n can be o b t a i n e d f r o m e q u a t i o n (9) a s , ($> - -± i + v V K.O. L (13) \ where In t h e p r e s e n t c a s e o f f l o w t h r o u g h a c i r c u l a r d u c t , t h e k e r and k e i terms d r o p o u t from e q u a t i o n s (12) and ( 1 3 ) . The r e m a i n i n g c o n s t a n t s A^ and A 2 a r e o b t a i n e d by a p p l y i n g t h e boundary c o n d i t i o n s V" = J = 0 a t t h e w a l l . Once V" i s known, t h e p r e s s u r e d r o p p a r a m e t e r L i s o b t a i n e d f r o m t h e c o n t i n u i t y e q u a t i o n , J{VdA The n o n - d i m e n s i o n a l v e l o c i t y and t e m p e r a t u r e f u n c t i o n s a r e t h e n d e t e r m i n e d f r o m , 14 H a v i n g o b t a i n e d t h e v e l o c i t y and t e m p e r a t u r e f u n c t i o n s , t h e N u s s e l t numbers can be e v a l u a t e d from t h e f o l l o w i n g e x p r e s s i o n , M b e i n g z e r o f o r t h i s c a s e . N u s s e l t Number* 0 0 3.2 S o l u t i o n s With V i s c o u s D i s s i p a t i o n Term Now we w i l l d e a l w i t h t h e methods o f s o l u t i o n o f t h e pr o b l e m when t h e v i s c o u s d i s s i p a t i o n t e r m i s i n c l u d e d i n t h e e n e r g y e q u a t i o n . S i n c e t h e p r o b l e m i s n o n - l i n e a r , an e x a c t s o l u t i o n does n o t seem p o s s i b l e a t p r e s e n t . T h e r e f o r e t h e s o l u t i o n f o r t h e p r e s e n t p r o b l e m was o b t a i n e d by t h r e e a p p r o x i m a t e b u t f a i r l y a c c u r a t e methods. The t h r e e methods used were, 1 . Power S e r i e s Method 2. G a l e r k i n ' s Method 3. N u m e r i c a l I n t e g r a t i o n Method 3.2 .1 Power S e r i e s Method In t h e t h e o r y o f b e n d i n g o f c i r c u l a r p l a t e s w i t h l a r g e d e f l e c t i o n , e q u a t i o n s somewhat s i m i l a r t o e q u a t i o n (5) o c c u r and Way [ 2 9 ] has used t h e power s e r i e s method t o s o l v e such a p r o b l e m . The e s s e n c e o f t h i s method i s t h a t an i n f i n i t e s e r i e s i s assumed F o r d e t a i l s see A p p e n d i x A 15 f o r t h e f u n c t i o n , and a f t e r s u b s t i t u t i n g t h i s s e r i e s e x p r e s s i o n i n t h e d i f f e r e n t i a l e q u a t i o n , t h e unknown c o e f f i c i e n t s a r e lumped t o g e t h e r i n t h e f o r m o f a r e c u r s i o n e x p r e s s i o n . Now a s s i g n i n g a n u m e r i c a l v a l u e t o t h e f i r s t c o e f f i c i e n t , a l l t h e r e m a i n i n g c o e f f i c i e n t s o f t h e s e r i e s can be d e t e r m i n e d f r o m t h i s r e c u r s i o n e x p r e s s i o n . The v a l u e s o f t h e s e c o -e f f i c i e n t s a r e t h e n improved upon by i t e r a t i o n t o s a t i s f y t h e boundary c o n d i t i o n s . The above method was used t o o b t a i n s o l u t i o n s f o r V and <(>. S i n c e V and <f> a r e s y m m e t r i c a l f u n c t i o n s o f R, t h e y can be expanded i n s e r i e s o f even powers o f R. L e t t h e d i m e n s i o n l e s s v e l o c i t y and t e m p e r a t u r e f u n c t i o n s V and <f> be e x p r e s s e d i n t h e f o r m o f i n f i n i t e power s e r i e s w i t h unknown c o e f f i c i e n t s a s , V = C0 + C,R2 + Q R % , ( d)» D. + ARa + + + •'••••» ( where C , C ] , C 2 C n and D Q, D p D 2 D n a r e t h e unknown c o e f f i c i e n t s . S u b s t i t u t i n g t h e power s e r i e s e x p r e s s i o n s (15) and (16) i n e q u a t i o n (4) and p e r f o r m i n g t h e r e q u i r e d d i f f e r e n t i a t i o n s t h e 16 f o l l o w i n g e x p r e s s i o n i s o b t a i n e d , + U(Do+D,R2+ M*1* D s £ * + ) + L _ 0 . (i?) Now e q u a t i n g t h e c o e f f i c i e n t s o f terms o f l i k e powers o f R, t h e f o l l o w i n g e x p r e s s i o n s r e s u l t , 4 C , + L = 0 / o r (18) = 0 t , (19) 36C 5 = 0 (20) = 0 (21) From t h e above e x p r e s s i o n s i t can be s e e n t h a t e x c e p t f o r t h e c o e f f i c i e n t s o f R°, t h e c o e f f i c i e n t s o f t h e r e m a i n i n g powers o f R can be w r i t t e n a s , W C ^ + R a D ^ =0 {or n-- - . ( 2 2 ) Now s u b s t i t u t i n g t h e power s e r i e s e x p r e s s i o n s (15) and (16) i n t h e e n e r g y e q u a t i o n (5) and p e r f o r m i n g t h e r e q u i r e d d i f f e r e n t i a t i o n s t h e f o l l o w i n g e x p r e s s i o n i s o b t a i n e d , 17 (4/), + /SD a£*+ 3 6 D , £ * + £ 4 / ) ^ + ; (23) In e q u a t i o n ( 2 3 ) , t h e l a s t t e r m w i t h i n t h e p a r e n t h e s i s can be w r i t t e n a s , = Z (inC„R ; Z H U • ik- G, £ R (24) L e t i n + 3k - 2 = <2.S T h e r e f o r e , n. + k - / =. S or n - S + I - k S i n c e ft ^> / , t h e r e f o r e , . Thus e x p r e s s i o n (24) becomes k < S I S=i L k-.i * R AS (25) 18 Now s u b s t i t u t i n g e x p r e s s i o n (25) i n e q u a t i o n ( 2 3 ) , t h e f o l l o w i n g e x p r e s s i o n i s o b t a i n e d , (4 A + K>DxRX+ 360 3RV *4Z)V+ ) - (C0 + CR 1* + rf+ ) (26) The c o e f f i c i e n t s o f terms o f l i k e powers o f R a r e now e q u a t e d t o g i v e t h e f o l l o w i n g s e t o f e q u a t i o n s , AD, - C0 = 0 -for k ° , ( 2 7 ) \ibx-C,+ 4M yZAk(s+i-k)C^,_kC -.0 -fa-ft, 4 >)%, - Cs f M Z <r K (5, I- k)C t i_ KC k ,0 ,28) o r f o r 1 , 3 00 k ^ S. C o l l e c t i n g e q u a t i o n s ( 1 8 ) , ( 2 2 ) , (27) and (28) t o g e t h e r , we h a v e , H , + RcDo + L - 0 (18) 4 YlZ Cn + &-Dn_, = 0 {or n = 3 , V* (22) 4D, - C0 - 0 19 (27) 4 M \ r Q^MDk(^k)(: + i_.c k - o (28) f o r ^ = I, 3 . . . . . o « From e q u a t i o n s ( 1 8 ) , ( 2 2 ) , (27) and ( 2 8 ) , i t can be seen t h a t knowing t h e v a l u e s o f C Q , D Q and L , a l l t h e s u c c e s s i v e c o e f f i c i e n t s C n and D n can be c a l c u l a t e d f o r any p r e s c r i b e d v a l u e s o f R a y l e i g h number Ra and v i s c o u s d i s s i p a t i o n p a r a m e t e r M. A p p l y i n g t h e boundary c o n d i t i o n s (7) on e q u a t i o n s (15) and ( 1 6 ) , t h e f o l l o w i n g e x p r e s s i o n s a r e o b t a i n e d , A t R = 1, (29) (30) r\=o S u b s t i t u t i n g t h e power s e r i e s e x p r e s s i o n (15) i n t h e i n t e g r a l f o r m o f t h e c o n t i n u i t y e q u a t i o n (6) and p e r f o r m i n g t h e r e q u i r e d i n t e g r a t i o n , t h e f o l l o w i n g e x p r e s s i o n i s o b t a i n e d , 20 In o r d e r t o e v a l u a t e t h e s e c o e f f i c i e n t s , t h e i n i t i a l e s t i m a t e s o f C and D„ were made f r o m t h e r e s u l t s o f t h e e x a c t o o s o l u t i o n as C Q and D Q a r e t h e v e l o c i t y and t e m p e r a t u r e d i f f e r -ence a t t h e c e n t r e o f t h e d u c t . T hese v a l u e s were t h e n improved by i t e r a t i o n so t h a t t h e c o e f f i c i e n t s o b t a i n e d f r o m e q u a t i o n s ( 1 8 ) , ( 2 2 ) , (27) and (28) s a t i s f y t h e boundary c o n d i t i o n s (29) and (30) and e q u a t i o n ( 3 1 ) . D e t e r m i n a t i o n o f t h e r e q u i r e d c o e f f i c i e n t s g i v e s t h e s o l u t i o n f o r t h e v e l o c i t y and t e m p e r a t u r e f i e l d . Knowing t h e v e l o c i t y and t e m p e r a t u r e f u n c t i o n s , N u s s e l t numbers were t h e n e v a l u a t e d f r o m e q u a t i o n ( 1 4 ) . 3.2.2 G a l e r k i n ' s Method The s e c o n d method used f o r t h e s o l u t i o n o f t h e p r o b l e m i s t h e G a l e r k i n ' s Method [ 2 , 1 0 ] . By t h i s method an a p p r o x i m a t e s o l u t i o n o f a d i f f e r e n t i a l e q u a t i o n can be o b t a i n e d by c h o o s i n g an e x p r e s s i o n w i t h a c e r t a i n s y s t e m o f f u n c t i o n s f o r t h e unknown q u a n t i t y s a t i s f y i n g t h e boundary c o n d i t i o n s and u s i n g t h e o p t i m i z -a t i o n t e c h n i q u e , t h e r e s u l t i n g e q u a t i o n s a r e s o l v e d s i m u l t a n e o u s l y t o d e t e r m i n e t h e unknown c o e f f i c i e n t s o f t h e e x p r e s s i o n . L e t t h e d i m e n s i o n l e s s v e l o c i t y and t e m p e r a t u r e f u n c t i o n s be e x p r e s s e d a s , (32) (33) 21 where C Q , C-j, and D Q, D-], D 2 a r e t h e unknown c o e f f i c i e n t s . The f a c t o r (1-R2) i n e x p r e s s i o n s (32) and (33) ensures s a t i s f a c t i o n o f t h e boundary c o n d i t i o n s ( 7 ) . E x p r e s s i o n s (32) and (33) a r e n o t t h e e x a c t s o l u t i o n s f o r V and <}> and s u b s t i t u t i n g t h e s e e x p r e s s i o n s i n e q u a t i o n s (4) and (5), we o b t a i n t h e f o l l o w i n g e x p r e s s i o n s which a r e a measure o f t h e a c c u r a c y o f t h e a p p r o x i m a t i o n s , + L ' (34) + C ^ - CX- C.R*- C ^ ) + 4 M C^ c, R I f e x p r e s s i o n s (32) and (33) were e x a c t s o l u t i o n s f o r V and <(> r e s p e c t i v e l y , t h e n Y-j and Y 2 would be i d e n t i c a l l y e q u a l t o z e r o . Now m u l t i p l y i n g Y-j w i t h t h e f i r s t , s e c o n d and t h i r d t e r m r e s p e c t i v e l y o f ( 3 2 ) , and i n t e g r a t i n g o v e r t h e d u c t c r o s s - s e c t i o n , t h e f o l l o w i n g e q u a t i o n s a r e o b t a i n e d , (36) 22 (37) (38) P r o c e e d i n g i n a s i m i l a r manner and u s i n g t h e e x p r e s s i o n f o r Y2 and e q u a t i o n (33) we o b t a i n , Y4(i-R*)fcfilR = 0 , (39) (40) VI 0- 0 (41) A f t e r p e r f o r m i n g t h e r e q u i r e d i n t e g r a t i o n s , t h e f o l l o w i n g c o m b i n a t i o n o f l i n e a r and n o n - l i n e a r a l g e b r a i c e q u a t i o n s a r e o b t a i n e d , 6 L + k(^K^).0, («, 23 i° 4 4 U 5 To) ~ u > 6 30 ' 5* 24 3o V 4 7 / (43) (44) _C - C, _ C, - J ) 0 . D, +LMfC^+C^ 6 14 £0 3 £ n \ 3 /5 + - ^ ° C J . + C ^ \ - 0 , (45) 36 IS 15 / -24 £o /AO 3 3 30 W + + Cl + CC +ic, c j - 0 , (46) 30 £0 /S" ./of y -Co - C_ - - Z A - A + ^ / M ^ £o /ao 210 6 so <r i o 43. /5 /o5" *5 J • (47) In t h e s e s i x e q u a t i o n s (42) t o (47) t h e r e a r e s e v e n unknowns C Q , C p C 2 , Dq, D p D 2 and L t o be d e t e r m i n e d . T h e r e f o r e an a d d i t i o n a l e q u a t i o n i s r e q u i r e d w hich i s o b t a i n e d by s u b s t i t u t i n g e q u a t i o n (32) i n 24 t h e c o n t i n u i t y e q u a t i o n (6) t o g i v e , Co •+ Cl + Ci - 3. • (48) 3 £ E q u a t i o n s (42) t o (48) a r e s o l v e d s i m u l t a n e o u s l y by P o w e l l ' s method t o o b t a i n t h e v a l u e s o f t h e unknown c o e f f i c i e n t s and t h e p r e s s u r e d r o p p a r a m e t e r L. Once t h e v a l u e s o f t h e unknown c o e f f i c i e n t s a r e d e t e r m i n e d t h e y can be s u b s t i t u t e d i n e x p r e s s i o n s (32) and (33) t o g i v e t h e v a l u e s o f t h e v e l o c i t y and t e m p e r a t u r e f u n c t i o n s and N u s s e l t numbers can t h e n be e v a l u a t e d . 3.2.3 N u m e r i c a l I n t e g r a t i o n The n u m e r i c a l i n t e g r a t i o n method o f Runge-Kutta o f o r d e r f o u r was used t o o b t a i n t h e s o l u t i o n s f o r t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s (4) and ( 5 ) . The method r e q u i r e s t h e com p l e t e s e t o f f u n c t i o n a l v a l u e s V, <j> and t h e i r g r a d i e n t s a t t h e s t a r t i n g boundary p o i n t and t h e e s t i m a t e s o f t h e m i s s i n g i n i t i a l b oundary c o n d i t i o n s were made from t h e e x a c t s o l u t i o n r e s u l t s . The r e s u l t i n g s o l u t i o n s were t h e n improved by i t e r a t i o n t o o b t a i n t h e d e s i r e d s o l u t i o n s s a t i s f y i n g t h e boundary c o n d i t i o n s (7) and t h e c o n t i n u i t y e q u a t i o n (6) s i m u l t a n e o u s l y . The e r r o r i n v o l v e d i n t h e f o u r t h o r d e r R-K method i s 5 o f t h e o r d e r o f h where h i s t h e s t e p s i z e . A s t e p s i z e o f 0.01 was t a k e n i n t h i s c a s e and t h e s o l u t i o n s o b t a i n e d were c h e c k e d by r e d u c i n g t h e s t e p s i z e t o 0.001. I t was seen t h a t t h e two s o l u t i o n s d i d n o t d i f f e r up t o s i x s i g n i f i c a n t f i g u r e s . T h i s c o m p l e t e s t h e methods o f s o l u t i o n s f o r t h e c i r -c u l a r d u c t . In t h e n e x t s e c t i o n , t h e f o r m u l a t i o n and s o l u t i o n s f o r t h e c o n c e n t r i c a n n u l u s a r e p r e s e n t e d . 4. SECTION I I CONCENTRIC ANNULI 27 4.1 F o r m u l a t i o n Of The Problem C o n s i d e r t h e 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 o f a f l u i d i n t h e v e r t i c a l upward d i r e c t i o n t h r o u g h t h e a n n u l a r p a s s a g e as shown i n F i g u r e 2. The a s s u m p t i o n s made i n t h e f o r m u l a t i o n o f t h e p r o b l e m , and t h e g o v e r n i n g e q u a t i o n s i n t h e d i m e n s i o n a l f o r m f o r t h e c o n c e n t r i c a n n u l u s remain t h e same as f o r t h e c i r c u l a r d u c t ( S e c t i o n I ) , and a r e n o t r e p e a t e d h e r e . In a d d i t i o n , t h e boundary c o n d i t i o n o f no s l i p a t t h e w a l l s w i l l s t i l l a p p l y . However, t h e t h e r m a l boundary c o n d i t i o n w i l l depend on t h e f o l l o w i n g t h r e e s i t u a t i o n s , Case I: O u t e r w a l l h e a t e d and i n n e r w a l l p e r f e c t l y i n s u l a t e d . Case I I : I n n e r w a l l h e a t e d and o u t e r w a l l p e r f e c t l y i n s u l a t e d . Case I I I : Both w a l l s h e a t e d w i t h e q u a l w a l l t e m p e r a t u r e s a t a g i v e n a x i a l p o s i t i o n . C h o o s i n g t h e f o l l o w i n g n o n - d i m e n s i o n a l p a r a m e t e r s , <: o o _ i _J N t t z = o FIGURE 2 C o o r d i n a t e System f o r Flow Through a V e r t i c a l C o n c e n t r i c A n n u l u s 29 t h e n o n - d i m e n s i o n a l f o r m o f t h e g o v e r n i n g e q u a t i o n s and t h e boundary c o n d i t i o n s f o r Case I , I I and I I I r e s p e c t i v e l y a r e as f o l l o w s : Case I : O u t e r W a l l H e a t e d , I n n e r Wall I n s u l a t e d . F i r s t o f a l l we r e d e f i n e t h e e q u i v a l e n t d i a m e t e r and e v a l u a t e i t . E q u i v a l e n t D i a m e t e r F o r t h i s c a s e , t h e e q u i v a l e n t d i a m e t e r i s g i v e n by, D^ _ 4 x A r e a o f c r o s s - s e c t i o n Heated p e r i m e t e r - a n , ( 4 9 ) where r Q = r a d i u s o f t h e o u t e r t u b e A = r a d i u s o f i n n e r t u b e / r a d i u s o f o u t e r t u b e . S u b s t i t u t i n g t h e n o n - d i m e n s i o n a l p a r a m e t e r s and e q u a t i o n (49) i n t h e momentum and e n e r g y e q u a t i o n s (1) and (2) r e s p e c t i v e l y , t h e f o l l o w i n g n o n - d i m e n s i o n a l e q u a t i o n s a r e o b t a i n e d , (l- yfVlV + Ra C|) + L = 0 , (50) 30 ( [ - > • ) - V > - V - H ^ Z - ^ M ^ O . ( 5 1 ) E q u a t i o n s (50) and (51) a l o n g w i t h t h e c o n t i n u i t y e q u a t i o n (6) a r e t o be s o l v e d f o r t h e f o l l o w i n g boundary c o n d i t i o n s : Boundary C o n d i t i o n s At R- > , V - dA . 0 , fit R-l , V - <{> -. o . N u s s e l t Number (52) (53) The N u s s e l t number i s g i v e n by* N a = L- LAdKi J (54) Case I I : I n n e r Wall H e a t e d , O u t e r Wall I n s u l a t e d  E q u i v a l e n t D i a m e t e r F o r t h i s c a s e t h e e q u i v a l e n t d i a m e t e r i s g i v e n by, A - a r . ( i - >VA • U s i n g t h i s v a l u e f o r D h i n e q u a t i o n s (1) and (2), t h e n o n - d i m e n s i o n a l momentum and e n e r g y e q u a t i o n s a r e o b t a i n e d a s , L i i ) l v 4 v + R<xcp + L - o, (56) ( L^V>-V +^Mg ) l o . (57) * F o r d e t a i l s see Appendix B 31 F o r c a s e II e q u a t i o n s (56) and (57) a l o n g w i t h t h e c o n t i n u i t y e q u a t i o n (6) a r e t o be s o l v e d f o r t h e f o l l o w i n g boundary c o n d i t i o n s : Boundary C o n d i t i o n s ftt'/U>, V-- ( p - 0 , (58) flt H- I ,' .V« d4 = 0 . (59) N u s s e l t Number The N u s s e l t number e x p r e s s i o n i s g i v e n b y * Nu - K (50) X A Case I I I : Both W a l l s Heated  E q u i v a l e n t D i a m e t e r For t h i s c a s e t h e e q u i v a l e n t d i a m e t e r i s o b t a i n e d a s , A - a r 0 ( i - > ) (61) U s i n g t h i s v a l u e f o r D^, t h e n o n - d i m e n s i o n a l momentum and e n e r g y e q u a t i o n s a r e o b t a i n e d a s , ( l - ^ V ' V + Ro. ct + L = 0, {62) ( l - ^ V ^ - V + 4f/->)Mg|)l=0. (63) E q u a t i o n s (62) and (63) a l o n g w i t h t h e c o n t i n u i t y e q u a t i o n (6) a r e t o be s o l v e d f o r t h e f o l l o w i n g boundary c o n d i t i o n s : 32 Boundary C o n d i t i o n s At R= > , (p * o, At R - i , V = $ = o. (64) (65) N u s s e l t Number The N u s s e l t number e x p r e s s i o n i s o b t a i n e d a s * , i Nu = A (66) * F o r d e t a i l s see Appendix B 5. SOLUTIONS 34 5.1 E x a c t S o l u t i o n W i t h o u t V i s c o u s D i s s i p a t i o n A g e n e r a l f o r m o f t h e e x a c t s o l u t i o n w i t h o u t v i s c o u s d i s s i p a t i o n [ 1 1 ] f o r t h e c o n c e n t r i c a n n u l u s i n t h e form o f K e l v i n f u n c t i o n s i s p r e s e n t e d i n a more s i m p l i f i e d manner f o r c a s e I o n l y . t h e a p p r o a c h f o r t h e o t h e r two c a s e s b e i n g s i m i l a r . F o r c a s e I , o u t e r w a l l h e a t e d , i n n e r w a l l i n s u l a t e d e q u a t i o n s (50) and (51) r e d u c e t o , V*V + (b -H _L_ » 0 , (67) L e t 1 Jg^ (69) S u b s t i t u t i n g e q u a t i o n (69) i n (67) and (68) and d i v i d i n g e q u a t i o n s (67) and (68) by t h e p r e s s u r e d r o p p a r a m e t e r L we o b t a i n t h e f o l l o w i n g e q u a t i o n s , V * V + BRa $ + 8 = 0 , (70) V 4 $ - 8 7 = 0 , ( 7 i ) where V = V / L , "$ = (J)/L. Combining (70) and (71) we o b t a i n , V 4 V + ^ V - 0, (72) where Yj^ = B^ Ra (73) E q u a t i o n (72). i s i d e n t i c a l t o (11) and t h e s o l u t i o n i s 35 g i v e n by (12) i n S e c t i o n I and i s r e p e a t e d h e r e . V= CMo ftft)+C>d0ft/?)+^ (74) from e q u a t i o n (70) we o b t a i n , B&x L (75) where The unknowns C-j, C 2 , Cg and i n (74) and (75) a r e o b t a i n e d by a p p l y i n g t h e boundary c o n d i t i o n s (52) and ( 5 3 ) . T h i s r e s u l t s i n t h e f o l l o w i n g f o u r e q u a t i o n s : 0- C,l*v-.ft>) + £K ( 7 A ) + C,ter . fyA)+ C ^ t e i . f y X ) , (77) 0 - - H'(V) + £ k-J ft>)-C3te^>)*C, taj'ft>)], (78) 0 = C, tw.ft) + Q K + C 3 for. + C< tec„C?), ( 7 9 ) 0= -^-|M-C 1K(7)+QK(?)-^ t eS,(' ,l)+^/cer.^j.(80) E q u a t i o n s ( 7 7 ) , ( 7 8 ) , (79) and (80) a r e s o l v e d s i m u l t a n -e o u s l y t o d e t e r m i n e t h e v a l u e s o f t h e unknown c o e f f i c i e n t s C-j, C 2 , and C^. Thus V" and J can be e v a l u a t e d and t h e p r e s s u r e d r o p p a r a m e t e r L i s o b t a i n e d from t h e c o n t i n u i t y e q u a t i o n , 36 The n o n - d i m e n s i o n a l v e l o c i t y and t e m p e r a t u r e f u n c t i o n s a r e t h e n d e t e r m i n e d f r o m V = V • L , <$ r (j) • L , and N u s s e l t numbers can be e v a l u a t e d f r o m e q u a t i o n ( 5 4 ) . The s o l u t i o n s f o r c a s e I I , i n n e r w a l l h e a t e d and o u t e r w a l l i n s u l a t e d and f o r c a s e I I I , both w a l l s h e a t e d were o b t a i n e d i n a s i m i l a r manner. 5.2 S o l u t i o n s With V i s c o u s D i s s i p a t i o n Term Now we w i l l d e a l w i t h t h e p r o b l e m t a k i n g i n t o a c c o u n t t h e v i s c o u s d i s s i p a t i o n term i n t h e e n e r g y e q u a t i o n . A power s e r i e s method s i m i l a r t o t h a t f o r c i r c u l a r d u c t was a t t e m p t e d w i t h o u t s u c c e s s . I t a p p e a r s t h a t G a l e r k i n ' s method c o u l d be a p p l i e d f o r c a s e I I I where i t i s e a s i e r t o s e t up t h e t e m p e r a t u r e f u n c t i o n t o s a t i s f y t h e w a l l c o n d i t i o n s . However, f o r c a s e s I and II where one o f t h e w a l l s i s i n s u l a t e d , i t i s d i f f i c u l t t o s e t up s u i t a b l e e x p r e s s i o n s f o r t h e t e m p e r a t u r e f u n c t i o n . Thus t h e numer-i c a l i n t e g r a t i o n method o f Runge-Kutta o f o r d e r f o u r was used t o o b t a i n t h e s o l u t i o n s . The g e n e r a l p r o c e d u r e f o r t h e Runge-Kutta method i s g i v e n i n s e c t i o n ( 3 . 2 . 3 ) . The s t e p s i z e t a k e n f o r t h i s p r o b l e m was h = 0.01 (1 - x) where h i s t h e s t e p s i z e and X i s t h e r a d i u s r a t i o ( r j / r 0 ) - S o l u t i o n s were a l s o o b t a i n e d by r e d u c i n g t h e s t e p s i z e but no d i f f e r e n c e was o b s e r v e d i n t h e s o l u t i o n s up t o s i x s i g n i f i c a n t f i g u r e s . 37 6. DISCUSSION OF.RESULTS The e f f e c t s o f v i s c o u s d i s s i p a t i o n on t h e f l o w phen-omena and heat t r a n s f e r r a t e as s t u d i e d f r o m t h e r e s u l t s o b t a i n e d a r e d i s c u s s e d under two s e c t i o n s , ( i ) C i r c u l a r d u c t s and ( i i ) C o n c e n t r i c a n n u l i . 6.1 C i r c u l a r Ducts F o r t h e c i r c u l a r d u c t s , we w i l l f i r s t d i s c u s s b r i e f l y t h e s o l u t i o n d e t a i l s and t h e n p r e s e n t t h e r e s u l t s f o r t h e v e l o c i t y and t h e t e m p e r a t u r e f i e l d s and t h e N u s s e l t numbers. 6.1.1 S o l u t i o n D e t a i l s A l l c a l c u l a t i o n s were made on an IBM d i g i t a l computer. F o r t h e e x a c t s o l u t i o n w i t h o u t v i s c o u s d i s s i p a t i o n e f f e c t s (M=0), t h e K e l v i n f u n c t i o n terms b e r and b e i were e v a l u a t e d i n Double P r e c i s i o n A r i t h m e t i c g i v i n g an a c c u r a c y up t o f o u r t e e n s i g n i f i c a n t f i g u r e s . These f u n c t i o n s were e v a l u a t e d f r o m t h e e x p r e s -s i o n s i n t h e form o f i n f i n i t e s e r i e s g i v e n i n Mc L a c h l a n [ 1 3 ] . In t h e e v a l u a t i o n o f t h e f u n c t i o n s , t h e c o n v e r g e n c e was v e r y r a p i d f o r th e v a l u e o f t h e argument up t o e i g h t . In t h e power s e r i e s method, seven s e t s o f i n i t i a l e s t i m a t e s v e r y c l o s e t o t h e v a l u e s o b t a i n e d from t h e e x a c t s o l u t i o n r e s u l t s were us e d and e m p l o y i n g t h e m i n i m i z i n g and i t e r a t i o n p r o c e d u r e t h e f i n a l c o e f f i c i e n t s were o b t a i n e d . The c o e f f i c i e n t s o f t h e s e r i e s were v e r y f a s t c o n v e r g i n g and t h e maximum number o f terms i n t h e s e r i e s t o be c a l c u l a t e d d i d not ex c e e d more t h a n t h i r t y f i v e . The G a l e r k i n ' s method i n v o l v e d t h e s o l u t i o n o f s i m u l t a n e o u s 3 8 n o n - l i n e a r a l g e b r a i c e q u a t i o n s and c l o s e enough i n i t i a l g u e s s e s o f t h e s o l u t i o n were v e r y e s s e n t i a l f o r r a p i d c o n v e r g e n c e . These 'e d u c a t e d ' g u e s s e s were e s t i m a t e d f r o m t h e r e s u l t s o b t a i n e d w i t h t h e e x a c t s o l u t i o n f o r M=0. In t h e Runge-Kutta's f o u r t h o r d e r m e t h o d , e s t i m a t e s o f t h e i n i t i a l g u e s s e s o f t h e m i s s i n g boundary c o n d i t i o n s and t h e p r e s s u r e d r o p p a r a m e t e r L were made f r o m t h e r e s u l t s o f t h e e x a c t s o l u t i o n , a n d were t h e n i t e r a t e d upon t o o b t a i n t h e d e s i r e d s o l u t i o n s a t i s f y i n g t h e boundary c o n d i t i o n s a t t h e end p o i n t . The e r r o r i n v o l v e d i n t h e Runge-Kutta's f o u r t h o r d e r method i s o f t h e o r d e r o f h where h i s t h e s t e p s i z e f o r i n t e g r a t i o n . R e s u l t s f o r t h e p r e s e n t c a s e were o b t a i n e d by t a k i n g a s t e p s i z e o f 0.01 and i t was n o t e d t h a t t h e r e d u c t i o n i n s t e p s i z e t o 0.001 d i d not a l t e r t h e s o l u t i o n up t o s i x s i g n i f i c a n t f i g u r e s . Coming t o t h e a c c u r a c y o f t h e methods u s e d , a f i r s t check on t h e a c c u r a c y was c a r r i e d o u t by c a l c u l a t i n g r e s u l t s f o r M=0 (no v i s c o u s d i s s i p a t i o n e f f e c t s ) by t h e t h r e e methods and c o m p a r i n g them w i t h t h e e x a c t s o l u t i o n r e s u l t s as shown i n T a b l e s 1 and 2. T a b l e 1 i s f o r v e l o c i t y and t e m p e r a t u r e f u n c t i o n s and T a b l e 2 shows t h e N u s s e l t number v a l u e s . From t h e s e t a b l e s i t can be seen t h a t t h e r e s u l t s o b t a i n e d by t h e t h r e e methods a r e i n good agreement w i t h t h e e x a c t s o l u t i o n . T h e s e t a b l e s a l s o show t h a t f o r n o n - z e r o f i n i t e v a l u e s o f t h e d i s s i p a t i o n p a r a m e t e r M, t h e agreement between t h e t h r e e methods i s v e r y good. In u p f l o w h e a t i n g o f a f l u i d t h e e f f e c t o f f r e e c o n v e c t i o n 39 i s t o a c c e l e r a t e t h e v e l o c i t y n e a r t h e w a l l [ 1 2 ] . To s a t i s f y c on-t i n u i t y , t h e v e l o c i t y n e a r t h e tube c e n t r e i s r e d u c e d . I f t h e buoyancy r a t e i s i n c r e a s e d s u f f i c i e n t l y , t h e n i t i s t h e o r e t i c a l l y p o s s i b l e t o c r e a t e f l o w r e v e r s a l a t t h e c e n t r e o f t h e d u c t . However, i t i s known [ 7 , 2 £ ] t h a t j u s t b e f o r e n e g a t i v e v e l o c i t y c o u l d o c c u r , t h e f l o w becomes u n s t a b l e and e v e n t u a l l y t u r b u l e n t . We, t h e r e f o r e , need t o l i m i t o u r a t t e n t i o n o n l y up t o t h a t v a l u e o f R a y l e i g h number which c r e a t e s f l o w r e v e r s a l . R a y l e i g h number as d e f i n e d i n t h e n o m e n c l a t u r e f o r t h e p r e s e n t s t u d y s h o u l d n o t ex c e e d 625 t o m a i n t a i n l a m i n a r f l o w . Thus f o r t h e p r e s e n t a n a l y s i s t h e maximum v a l u e o f R a y l e i g h number us e d was 625 f o r t h e c a s e o f c i r c u l a r d u c t . The v i s c o u s d i s s i p a t i o n p a r a m e t e r i s d e f i n e d as M=Eckert number/Reynolds number. The maximum v a l u e o f t h i s p a r a m e t e r used -4 i n t h e p r e s e n t a n a l y s i s was 5 x 10 . Now t h e e f f e c t o f v i s c o u s d i s s i p a t i o n on t h e v e l o c i t y and t e m p e r a t u r e f i e l d s and t h e N u s s e l t numbers w i l l be d i s c u s s e d . 6.1.2 V e l o c i t y F i e l d F o r t h e c a s e o f pure f o r c e d c o n v e c t i o n (Ra=0), t h e v e l o c i t y f i e l d i s i n d e p e n d e n t o f t h e t e m p e r a t u r e f i e l d and hence v i s c o u s d i s s i p a t i o n has no e f f e c t on t h e v e l o c i t y f i e l d . However, f o r t h e c a s e o f combined f r e e and f o r c e d c o n v e c t i o n , t h e momentum and e n e r g y e q u a t i o n s (4) and (5) r e s p e c t i v e l y a r e c o u p l e d and hence v i s c o u s d i s s i p a t i o n n o t o n l y a f f e c t s t h e t e m p e r a t u r e f i e l d b u t a l s o t h e v e l o c i t y f i e l d . The measure o f f r e e c o n v e c t i o n i s t h e 40 n o n - d i m e n s i o n a l p a r a m e t e r R a y l e i g h number and as R a y l e i g h number i n c r e a s e s t h e c o u p l i n g becomes more and more s t r o n g and hence t h e d i s s i p a t i o n e f f e c t becomes more p r o n o u n c e d . From t h e r e s u l t s o b t a i n e d i t i s seen t h a t t h e e f f e c t o f v i s c o u s d i s s i p a t i o n on t h e v e l o c i t y f i e l d i s t o r e d u c e t h e f l o w v e l o c i t y n e a r t h e d u c t w a l l s and c o n s e q u e n t l y i n c r e a s e i t n e a r t h e c e n t r e . T a b l e 1 shows t h e i n c r e a s e i n v e l o c i t y a t t h e c e n t r e o f t h e d u c t under t h e i n f l u e n c e o f v i s c o u s d i s s i p a t i o n f o r v a r i o u s v a l u e s o f R a y l e i g h number. T h i s t r e n d becomes more pronounced w i t h t h e i n c r e a s e i n R a y l e i g h number. As s t a t e d , t h e r e d u c t i o n i n t h e v e l o c i t y n e a r t h e d u c t w a l l s has been o b s e r v e d , however, t h i s d a t a i s n o t p r e s e n t e d h e r e f o r b r e v i t y . From Hallman's [ 7 ] i n v e s t i g a t i o n i t i s known t h a t f o r t h e c a s e o f u p f l o w h e a t i n g , t h e e f f e c t o f f r e e c o n v e c t i o n on t h e v e l o c i t y f i e l d i s t o i n c r e a s e t h e f l o w v e l o c i t y n e a r t h e d u c t w a l l s and t o r e d u c e i t n e a r t h e c e n t r e . Thus v i s c o u s d i s -s i p a t i o n a c t s c o n t r a r y t o t h e f r e e c o n v e c t i o n (buoyancy) e f f e c t on t h e f l o w f i e l d . From t h i s i t t h e r e f o r e f o l l o w s , t h a t t h e e f f e c t o f f r e e c o n v e c t i o n i s t o i n c r e a s e t h e s h e a r s t r e s s a t t h e w a l l whereas t h e e f f e c t o f v i s c o u s d i s s i p a t i o n i s t o r e d u c e t h e same. 6.1.3 T e m p e r a t u r e F i e l d The e f f e c t o f v i s c o u s d i s s i p a t i o n i s t o c o n v e r t f r i c t i o n a l e n e r g y i n t o h e a t and hence i t r e d u c e s t h e t e m p e r a t u r e d i f f e r e n c e s i n t h e syste m when t h e t r a n s f e r o f h e a t t a k e s p l a c e from t h e s u r f a c e t o t h e f l u i d . T a b l e 1 shows t h e t e m p e r a t u r e d i f f e r e n c e s a t t h e c e n t r e o f t h e d u c t f o r v a r i o u s v a l u e s o f R a y l e i g h number t a k i n g i n t o a c c o u n t 41 v i s c o u s d i s s i p a t i o n e f f e c t , and i t can be seen t h a t t h e t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d . T h i s t r e n d i s a l s o o b s e r v e d a t a l l p o i n t s a l o n g t h e tube r a d i u s , however, t h i s d a t a i s n o t p r e s e n t e d h e r e . 6.1.4 N u s s e l t Numbers One o f t h e main p a r a m e t e r s o f e n g i n e e r i n g i n t e r e s t i s t h e N u s s e l t number which i s a measure o f t h e h e a t t r a n s f e r r a t e , and t h e e f f e c t o f v i s c o u s d i s s i p a t i o n on N u s s e l t number i s an i m p o r t a n t a s p e c t o f t h e p r e s e n t a n a l y s i s . As m e n t i o n e d e a r l i e r , due t o t h e c o n v e r s i o n o f f r i c t i o n a l e n e r g y i n t o h e a t t h e i m p r e s s e d e x t e r n a l h e a t i n g i s opposed and t h e h e a t t r a n s f e r r a t e i s r e d u c e d . C o n s e q u e n t l y because o f v i s c o u s d i s s i p a t i o n e f f e c t , l o w e r N u s s e l t number v a l u e s a r e o b t a i n e d . T a b l e 2 shows t h e v a l u e s o f N u s s e l t numbers f o r d i f f e r e n t R a y l e i g h numbers t a k i n g i n t o a c c o u n t v i s c o u s d i s s i p a t i o n e f f e c t s . From t h i s t a b l e i t can be seen t h a t t h e N u s s e l t number v a l u e s a r e r e d u c e d and t h e r e d u c t i o n becomes more pronounced a t h i g h e r R a y l e i g h numbers. 6.2 C o n c e n t r i c A n n u l i Now we w i l l d i s c u s s t h e s o l u t i o n s o b t a i n e d and the v i s c o u s d i s s i p a t i o n e f f e c t s f o r t h e t h r e e c a s e s o f t h e a n n u l a r f l o w . 6.2.1 S o l u t i o n D e t a i l s The e x a c t s o l u t i o n s w i t h M=0 f o r t h e c o n c e n t r i c a n n u l u s a l s o i n v o l v e d t h e d e r i v a t i v e s o f K e l v i n f u n c t i o n s because o f t h e t h e r m a l boundary c o n d i t i o n o f one w a l l b e i n g i n s u l a t e d . These f u n c t i o n s were e v a l u a t e d i n Double P r e c i s i o n f r o m M c L a c h l a n [ 1 3 ] . 42 The number o f terms r e q u i r e d f o r c o n v e r g e n c e was o f t h e o r d e r o f 20. The n o n - l i n e a r p r o b l e m (M>0) was s o l v e d by Runge-Kutta f o u r t h o r d e r method i n Double P r e c i s i o n and t h e a c c u r a c y o f R-K method was j u d g e d by o b t a i n i n g r e s u l t s f o r M=0 and comparing them w i t h t h e e x a c t s o l u t i o n r e s u l t s . T a b l e 3 shows t h e N u s s e l t number v a l u e s as o b t a i n e d by t h e e x a c t s o l u t i o n and Runge-Kutta method f o r d i f f e r e n t v a l u e s o f R a y l e i g h number. From t h i s t a b l e i t can be seen t h a t t h e r e s u l t s o b t a i n e d by t h e two methods a r e i n good agreement. A f u r t h e r check on t h e a c c u r a c y was made by c o m p a r i n g t h e r e s u l t s o b t a i n e d by t h e two methods f o r Ra=l which a p p r o x i m a t e s t o f o r c e d c o n v e c t i o n f l o w w i t h t h e r e s u l t s o f Cheng [ 4 ] s i n c e no r e s u l t s seem t o be a v a i l a b l e i n p u b l i s h e d l i t e r a t u r e f o r combined f r e e and f o r c e d c o n v e c t i o n t h r o u g h a n n u l a r p a s s a g e s . T hese r e s u l t s were a l s o f o u n d t o be i n v e r y c l o s e agreement. Now we w i l l d i s c u s s t h e v e l o c i t y and t e m p e r a t u r e f i e l d s and t h e e f f e c t o f v i s c o u s d i s s i p a t i o n , f o r t h e t h r e e c a s e s s t u d i e d . 6.2.2 V e l o c i t y F i e l d F i r s t o f a l l we w i l l d i s c u s s t h e v e l o c i t y p r o f i l e s f o r M=0 as shown i n f i g u r e s 3 t o 8. F i g u r e 3 shows t h e v e l o c i t y p r o f i l e s f o r t h e c a s e o f o u t e r w a l l h e a t e d , i n n e r w a l l i n s u l a t e d ( c a s e I) f o r x=0.25. From t h i s f i g u r e i t can be seen t h a t as R a y l e i g h number i n c r e a s e s , t h e v e l o c i t y g r a d i e n t s n e a r t h e o u t e r w a l l ( h e a t e d w a l l ) i n c r e a s e . T h i s i n c r e a s e i n v e l o c i t y near t h e o u t e r w a l l r e d u c e s t h e same ne a r t h e i n n e r w a l l and e v e n t u a l l y f l o w r e v e r s a l t a k e s p l a c e a t Ra=2000. In F i g u r e 4 a r e shown t h e v e l o c i t y p r o f i l e s f o r x=0.5. 43 A s i m i l a r t r e n d i s o b s e r v e d here by i n c r e a s i n g Ra w i t h f l o w r e v e r s a l t a k i n g p l a c e now a t Ra=4000. The v e l o c i t y p r o f i l e s f o r t h e c a s e o f i n n e r w a l l h e a t e d , o u t e r w a l l i n s u l a t e d ( c a s e I I ) a r e shown i n F i g u r e 5 f o r A=0.25. From t h i s f i g u r e i t can be seen t h a t by i n c r e a s i n g R a y l e i g h number, t h e v e l o c i t y g r a d i e n t s n e a r t h e i n n e r w a l l ( h e a t e d w a l l ) a r e i n c r e a s e d w i t h f l o w r e v e r s a l o c c u r i n g a t R a = 2 8 x l 0 4 . F i g u r e 6 shows t h e v e l o c i t y p r o f i l e s f o r A=0.5 and f o r t h i s c a s e f l o w 4 r e v e r s a l o c c u r s a t Ra=5xl0 . Fo r t h e c a s e o f both w a l l s h e a t e d ( c a s e I I I ) , t h e v e l o c i t y p r o f i l e s f o r A=0.25 a r e shown i n F i g u r e 7. T h i s f i g u r e shows t h a t as Ra i n c r e a s e s , t h e v e l o c i t y g r a d i e n t s n e a r both t h e w a l l s i n c r e a s e . T h i s i n c r e a s e i n v e l o c i t y near both t h e w a l l s r e d u c e s t h e same n e a r t h e c e n t r a l r e g i o n and e v e n t u a l l y a r e v e r s a l o f f l o w o c c u r s a t Ra-6500. F i g u r e 8 shows t h e v e l o c i t y p r o f i l e s f o r A=0.5. The same e f f e c t o f R a y l e i g h number i s o b s e r v e d h e r e on t h e v e l o c i t y f i e l d w i t h f l o w r e v e r s a l now o c c u r i n g a t Ra=7000. 6.2.3 T e m p e r a t u r e F i e l d Now we w i l l d i s c u s s t h e t e m p e r a t u r e p r o f i l e s f o r M=0. F i g u r e s 9 and 10 show t h e t e m p e r a t u r e p r o f i l e s f o r o u t e r w a l l h e a t e d , i n n e r w a l l i n s u l a t e d f o r A=0.25 and 0.5 r e s p e c t i v e l y . From t h e s e f i g u r e s i t can be seen t h a t t h e t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d by i n c r e a s i n g t h e R a y l e i g h number. F i g u r e s 11 and 12 show t h e t e m p e r a t u r e p r o f i l e s f o r t h e ca s e o f i n n e r w a l l h e a t e d and o u t e r w a l l i n s u l a t e d w i t h x=0.25 and 0.5 r e s p e c t i v e l y . F o r t h i s c a s e a l s o i t can be seen t h a t w i t h 44 t h e i n c r e a s e i n Ra, t h e t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d . In F i g u r e s 13 and 14 a r e shown t h e t e m p e r a t u r e p r o f i l e s f o r t h e c a s e o f both w a l l s h e a t e d f o r x=0.25 and 0.5 r e s p e c t i v e l y . For t h i s c a s e t o o , t h e t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d w i t h i n c r e a s i n g R a y l e i g h number. S i n c e t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n on t h e v e l o c i t y and t e m p e r a t u r e f i e l d i s f o u n d t o be v e r y s m a l l , i t i s n o t conven-i e n t t o p r e s e n t t h e r e s u l t s g r a p h i c a l l y a nd, t h e r e f o r e , a g e n e r a l t r e n d i s r e p r e s e n t e d by t h e f o l l o w i n g t a b l e s . T a b l e s 4 and 5 show t h e e f f e c t o f v i s c o u s d i s s i p a t i o n on t h e v e l o c i t y and t e m p e r a t u r e f i e l d s f o r t h e c a s e o f o u t e r w a l l h e a t e d , i n n e r w a l l i n s u l a t e d w i t h x=0.75 f o r Ra=l and 1000 r e s -p e c t i v e l y . T a b l e 4 f o r Ra=l i s a l m o s t a pure f o r c e d c o n v e c t i o n c a s e and i t can be seen t h a t t h e r e i s no s i g n i f i c a n t e f f e c t o f v i s c o u s d i s s i p a t i o n on t h e v e l o c i t y f i e l d as t h e v e l o c i t y f i e l d i s i n d e p e n d e n t o f t h e t e m p e r a t u r e f i e l d . However, i t can be seen t h a t t h e t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d . T a b l e 5 shows t h a t as Ra has i n c r e a s e d , t h e e f f e c t o f v i s c o u s d i s s i p a t i o n on t h e v e l o c i t y f i e l d becomes more pronounced. The v e l o c i t y n e a r t h e o u t e r w a l l ( h e a t e d w a l l ) i s r e d u c e d w h i l e i t i s i n c r e a s e d n e a r t h e i n n e r w a l l . The t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d t h r o u g h o u t . T a b l e 6 and 7 show t h e d i s s i p a t i o n e f f e c t s f o r i n n e r w a l l h e a t e d and o u t e r w a l l i n s u l a t e d . From T a b l e 6 i t can be seen t h a t f o r R a=l, d i s s i p a t i o n p a r a m e t e r M has no e f f e c t on t h e v e l o c i t y f i e l d b u t t h e t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d . As Ra i n c r e a s e s , 45 i t can be seen f r o m T a b l e 7 f o r Ra=1000, t h a t v i s c o u s d i s s i p a t i o n r e d u c e s t h e f l o w v e l o c i t y n e a r t h e i n n e r w a l l ( h e a t e d w a l l ) . The t e m p e r a t u r e d i f f e r e n c e s a r e a l s o r e d u c e d w i t h i n c r e a s i n g M. For t h e c a s e o f both w a l l s h e a t e d t h e e f f e c t o f M i s shown i n T a b l e s 8 and 9. From T a b l e 8 f o r Ra=l, i t can be seen t h a t t h e r e i s no s i g n i f i c a n t e f f e c t o f M on t h e v e l o c i t y f i e l d though t h e t e m p e r a t u r e d i f f e r e n c e s a r e r e d u c e d . T a b l e 9 f o r Ra=2000 shows t h a t f o r h i g h e r v a l u e s o f R a , v i s c o u s d i s s i p a t i o n r e d u c e s t h e f l o w v e l o c i t y n e a r b o t h t h e w a l l s and t h e t e m p e r a t u r e d i f f e r e n c e s . 6.2.4 N u s s e l t Number As m e n t i o n e d e a r l i e r v i s c o u s d i s s i p a t i o n opposes t h e i m p r e s s e d e x t e r n a l h e a t i n g and r e d u c e s t h e h e a t t r a n s f e r r a t e r e s u l t i n g i n l o w e r v a l u e s o f N u s s e l t numbers. F i g u r e 15 shows t h e e f f e c t o f v i s c o u s d i s s i p a t i o n on N u s s e l t numbers f o r o u t e r w a l l h e a t e d , i n n e r w a l l i n s u l a t e d w i t h x=0.25 and 0.5. From t h i s f i g u r e i t can be seen t h a t N u s s e l t numbers d e c r e a s e w i t h i n c r e a s e i n t h e d i s -s i p a t i o n p a r a m e t e r M. The r e d u c t i o n i n N u s s e l t numbers becomes more pronounced a t h i g h e r R a y l e i g h numbers. F i g u r e 16 shows t h e e f f e c t o f M on N u s s e l t numbers f o r i n n e r w a l l h e a t e d , o u t e r w a l l i n s u l a t e d . F or t h i s c a s e t o o , i t can be seen t h a t lower v a l u e s o f N u s s e l t numbers a r e o b t a i n e d when v i s c o u s d i s s i p a t i o n i s t a k e n i n t o a c c o u n t . The e f f e c t o f M on N u s s e l t numbers f o r t h e c a s e o f both w a l l s h e a t e d i s shown i n F i g u r e 17. As a n t i c i p a t e d t h e N u s s e l t numbers a r e a g a i n r e d u c e d w i t h i n c r e a s i n g M and t h i s r e d u c t i o n becomes more pronounced at higher Rayleigh numbers. 6 . 2 . 5 Radius Ratio* The e f f e c t of radius r a t i o x on the Nusselt numbers can be seen from Figures 15, 16 and 17. Figures 15 and 17 show that fo r outer wall heated and inner wall insulated or for both wal ls heated, high values of Nusselt numbers are obtained by increasing x whereas from Figure 16 i t can be seen that for inner wall heated, outer wall insulated the Nusselt number values are reduced. A comparison of the reduction in Nusselt numbers for the same value of the d i s s i p a t i o n parameter M has also been studied . It i s found that the maximum reduction occurs for the case of inner wall heated, outer wall insulated and the minimum reduction occurs for the case of both wal ls heated. For deta-ils see Appendix C 7. CONCLUSIONS The e f f e c t s o f v i s c o u s d i s s i p a t i o n on t h e f l o w phen-omena and h e a t t r a n s f e r r a t e f o r combined f r e e and f o r c e d c o n -v e c t i o n t h r o u g h v e r t i c a l c i r c u l a r d u c t s and c o n c e n t r i c a n n u l i has been s t u d i e d . From t h e r e s u l t s o b t a i n e d i t i s c o n c l u d e d t h a t t h e e f f e c t s o f v i s c o u s d i s s i p a t i o n on t h e f l o w f i e l d i s t o r e d u c e t h e v e l o c i t y n e a r t h e h e a t e d w a l l ( s ) t h e r e b y c o u n t e r a c t i n g t h e e f f e c t o f f r e e c o n v e c t i o n on t h e v e l o c i t y f i e l d i n u p f l o w when t h e t r a n s f e r o f h e a t t a k e s p l a c e from t h e s u r f a c e t o t h e f l u i d . Thus i t f o l l o w s t h a t due t o v i s c o u s d i s s i p a t i o n e f f e c t s , t h e s h e a r s t r e s s a t t h e w a l l ( s ) i s r e d u c e d . V i s c o u s d i s s i p a t i o n r e d u c e s t h e t e m p e r a t u r e d i f f e r e n c e s i n t h e s y s t e m and hence t h e e f f e c t o f buoyancy i s d e c r e a s e d . The d i s s i p a t i o n o f f r i c t i o n a l e n e r g y i n t o h e a t r e d u c e s t h e h e a t t r a n s f e r r a t e when h e a t i s t r a n s f e r r e d from t h e s u r f a c e t o t h e f l u i d and r e s u l t s i n l o w e r N u s s e l t number v a l u e s . TABLE I V e l o c i t i e s and Temperature D i f f e r e n c e s a t t h e C e n t r e o f a V e r t i c a l C i r c u l a r Duct due t o V i s c o u s D i s s i p a t i o n E f f e c t s . E x a c t S o l u t i o n Power S e r i e s Methoc i G a l e r k i n ' s Method Runge-Kutta Method R a y l e i g h Number Ra V i s c o u s D i s s i p a -t i o n P a r a -meter M V e l o c i t y V Temp-e r a t u r e D i f f e r e n c e V e l o c i t y V Temp-e r a t u r e D i f f e r e n c e • V e l o c i t y V Temp-e r a t u r e D i f f e r e n c e V e l o c i t y V Temp-e r a t u r e D i f f e r e n c e * 1 0 0.0001 0.0005 1.9913 -0.3742 1.9912 1.9913 1.9913 -0.3742 -0.3739 -0.3723 1.9913 1.9913 1.9913 -0.3742 -0.3739 -0.3723 1.9924 1.9924 1.9924 -0.3744 -0.3740 -0.3725 10 0 0.0001 0.0005 1.9152 -0.3681 1.9152 1.9153 1.9155 -0.3681 -0.3677 -0.3663 1.9152 1.9153 1.9154 -0.3681 -0.3677 -0.3663 1.9163 1.9164 1.9165 -0.3682 -0.3679 -0.3665 50 0 0.0001 0.0005 1.6139 -0.3432 1.6139 1.6139 1.6143 -0.3432 -0.3432 -0.3420 1.6131 1.6132 1.6135 -0.3433 -0.3430 -0.3420 1.6149 1.6150 1.6154 -0.3434 -0.3432 -0.3422 100 0 0.0001 0.0005 1.3061 -0.3173 1.3061 1.3061 1.3065 -0.3173 -0.3173 -0.3163 1.3035 1.3035 1.3038 -0.3173 -0.3171 -0.3163 1.3069 1.3070 1.3073 -0.3175 -0.3173 -0.3165 500 0 0.0001 0.0005 0.1564 -0.2091 0.1564 0.1583 0.1660 -0.2091 -0.2087 -0.2069 0.1346 0.1370 0.1465 -0.2079 -0.2073 -0.2052 0.1563 0.1583 0.1660 -0.2093 -0.2089 -0.2071 625 0 0.0001 0.0005 0.0123 -0.1921 0.0123 0.0149 0.0248 -0.1921 -0.1915 -0.1895 0.0123 -0.0090 0.0035 -0.1904 -0.1898 -0.1873 0.0121 0.0146 0.0246 -0.1923 -0.1918 -0.1897 TABLE II E f f e c t o f V i s c o u s D i s s i p a t i o n P a r a m e t e r on N u s s e l t Numbers f o r a V e r t i c a l C i r c u l a r Duct N u s s e l t Number Nu • —T R a y l e i g h V i s c o u s D i s s i p a -Number t i o n Parameter E x a c t S o l u t i o n Power S e r i e s G a l e r k i n ' s R unge-Kutta Ra M Method Method Method 1 0 4.3743 4.3734 4.3742 4.3713 0.0001 4.3653 4.3665 4.3633 0.0005 4.3329 4.3354 4.3308 10 0 4,4688 4.4679 4.4689 4.4658 0.0001 4.4591 4.4604 4.4568 0.0005 4.4237 4.4267 4.4214 50 0 4.8735 4.8721 4.8735 4.8694 0.0001 4.8542 4.8619 4.8572 0.0005 4.8105 4.8156 4.8079 100 0 5.3429 5.3407 5.3428 5.3375 0.0001 5.3181 5.3270 5.3204 0.0005 5.2552 5.2633 5.2519 500 0 7.9516 7.9445 7.9518 7.9369 0.0001 7.8782 7.8925 7.8705 0.0005 7.6122 7.6538 7.6040 625 0 8.4911 8.4827 8.4934 8.4739 0.0001 8.3998 8.4196 8.3908 0.0005 8.0665 8.1228 8.0579 50 TABLE I I I N u s s e l t Number V a l u e s f o r M=0 O b t a i n e d by E x a c t S o l u t i o n and Runge-Kutta Method f o r C o n c e n t r i c A n n u l u s w i t h R a d ius R a t i o 0.5 N u s s e l t Number Nu Ray-l e i g h Num-b e r Case I : O u t e r Wal' In n e r WaV a t e d . H e a t e d , I n s u l -Case I I : I n n e r Wall H e a t e d , O u t e r Wall I n s u l -a t e d . Case I I I : Both W a l l s H e a t e d . Ra E x a c t S o l u t i o n Runge-K u t t a Method E x a c t S o l u t i o n Runge-K u t t a Method E x a c t S o l u t i o n Runge-K u t t a Method 1 7.556 7.557 18.546 18.545 8.117 8.117 500 8.923 8.927 18.747 18.750 9.318 9.334 1000 10.078 10.086 18.949 18.953 10.362 10.396 TABLE IV V e l o c i t y D i s t r i b u t i o n and Temperature D i f f e r e n c e s due t o V i s c o u s D i s s i p a t i o n E f f e c t s f o r C o n c e n t r i c A n n u l u s w i t h O u t e r Wall Heated, I n n e r Wall I n s u l a t e d f o r R a = l , x=0.75 D i s s i p a t i o n P a r a m e t e r M= 0.0 M= 0.0003 M= 0.0005 Ra d i u s R 0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 0.97 1.0 V e l o c i t y V Temperature D i f f e r e n c e 0.0 0.5625 0.9893 1.2852 1.4543 1.4873 1.4271 1.2376 0.9348 0.5214 0.0 V e l o c i t y V -0.1517 -0.1514 -0.1493 -0.1442 -0.1350 -0.1215 -0.1036 -0.0816 -0.0563 -0.0286 0.0 0.0 0.5625 0.9894 2852 4543 4873 4271 2376 0.9348 0.5214 0.0 Te m p e r a t u r e D i f f e r e n c e A V e l o c i t y V Temperature D i f f e r e n c e — * _ 0.1451 0.1450 0.1434 0.1388 1304 1175 1003 0790 0543 0275 0 0.0 0.5625 0.9893 1.2852 1.4542 1.4872 1.4270 1.2375 0.9348 0.5214 0.0 -0.1406 -0.1406 -0.1394 -0.1352 -0.1273 -0.1149 -0.0981 -0.0772 -0.0530 -0.0268 0.0 52 TABLE V V e l o c i t y D i s t r i b u t i o n and Te m p e r a t u r e D i f f e r e n c e s due t o V i s c o u s D i s s i p a t i o n E f f e c t s f o r C o n c e n t r i c A n n u l u s w i t h O u t e r Wall H e a t e d , I n n e r Wall I n s u l a t e d f o r Ra=1000, x=0.75 -D i s s i p a t i o n P a r a m e t e r M M=0.0 M=0.0003 M=0.0 005 Ra d i u s R V e l o c i t y V Temper-a t u r e Di f f e r -ence * V e l o c i t y V Temper-a t u r e D i f f e r -e nce • V e l o c i t y V Temper-a t u r e D i f f e r -e nce <i> 0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 0.97 1.0 0.0 0.3472 0.6693 0.9630 1.2182 1.3055 1.5237 1.4993 1.2851 0.8117 0.0 -0.1327 -0.1325 -0.1312 -0.1278 -0.1214 -0.1113 -0.0969 -0.0781 -0.0550 -0.0284 0.0 0.0 0.3540 0.6788 0.9718 1.2238 1.3094 1.5201 1.4919 1.2761 0.8045 0.0 -0.1287 -0.1286 -0.1275 -0.1243 -0.1183 -0.1085 -0.0945 -0.0760 -0.0533 -0.0273 0.0 0.0 0.3586 0.6852 0.9777 1.2276 1.3120 1.5176 1.4869 1.2701 0.7997 0.0 -0.1260 -0.1260 -0.1250 -0.1220 -0.1162 -0.1066 -0.0929 -0.0747 -0.0522 -0.0266 0.0 53 TABLE VI VELOCITY DISTRIBUTION AND TEMPERATURE DIFFERENCES DUE TO VISCOUS DISSIPATION EFFECTS FOR CONCENTRIC ANN-ULUS WITH INNER WALL HEATED, OUTER WALL INSULATED FOR Ra=l, A=0.75 D i s s i p a t i o n P a rameter M M=0.0 M=0.0003 M=0.0005 R a d i u s R V e l o c i t y V Temper-a t u r e D i f f e r -ence <J> V e l o c i t y V Temper-a t u r e D i f f e r -ence • V e l o c i t y V Temper-a t u r e D i f f e r -ence • 0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 0.97 1.0 0.0 0.5629 0.9899 1.2857 1.4546 1.4875 1.4269 1.2372 0.9343 0.5210 .0.0 0.0 -0.0208 -0.0401 -0.0570 -0.0710 -0.0820 -0.0901 -0.0953 -0.0982 -0.0993 -0.0995 0.0 0.5630 0.9900 1.2859 1.4548 1.4877 1.4271 1.2374 0.9344 0.5211 0.0 0.0 -0.0194 -0.0375 -0.0535 -0.0667 -0.0770 -0.0843 -0.0888 -0.0910 -0.0917 -0.0917 0.0 0.5627 0.9896 1.2854 1.4542 1.4871 1.4265 1.2369 0.9341 0.5209 0.0 0.0 -0.0184 -0.0358 -0.0512 -0.0638 -0.0736 -0.0803 -0.0844 -0.0862 -0.0866 -0.0865 54 TABLE V I I VELOCITY DISTRIBUTION AND TEMPERATURE DIFFERENCES DUE TO VISCOUS DISSIPATION EFFECTS FOR CONCENTRIC ANN-ULUS WITH INNER WALL HEATED, OUTER WALL INSULATED FOR Ra=TOOO, X=0.75 D i s s i p a t i o n P a r a m e t e r M M=0.0 M=0.0003 M=0.0005 Radius R V e l o c i t y V Temper-a t u r e D i f f e r -ence • V e l o c i t y V - Temper-a t u r e D i f f e r -ence V e l o c i t y V Temper-a t u r e D i f f e r -ence • 0.75 0.77 0.79 1 0.82 j 0.84 | 0.87 ! 0.89 0.92 0.94 0.97 1.0 0.0 0.6896 1.1438 1.4046 1.5076 1.5113 1.3490 1.1257 0.8226 0.4462 0.0 0.0 -0.0208 -0.0398 -0.0561 -0.0695 -0.0797 -0.0870 -0.0917 -0.0942 -0.0951 -0.0953 0.0 0.6820 1.1341 1.3964 1.5031 1.5086 1.3534 1.1332 0.8308 0.4521 0.0 0.0 -0.0194 -0.0375 -0.0531 -0.0657 -0.0754 -0.0821 -0.0862 -0.0882 -0.0888 -0.0888 0.0 0.6768 1.1274 1.3908 1.5000 1.5067 1.3564 1.1382 0.8364 0.4561 0.0 0.0 -0.0185 -0.0359 -0.0510 -0.0632 -0.0724 -0.0787 -0.0824 -0.0842 -0.0846 -0.0845 1 55 TABLE V I I I VELOCITY DISTRIBUTION AND TEMPERATURE DIFFERENCES DUE TO VISCOUS DISSIPATION EFFECTS FOR CONCENTRIC ANNULUS WITH BOTH WALLS HEATED FOR Ra=l, X=0.75 D i s s i p a t i o n P a rameter M M=0.0 M=0. 0003 M=0.0005 Temper- Temper- Temper-a t u r e a t u r e a t u r e R a d i u s V e l o c i t y D i f f e r - V e l o c i t y D i f f e r - i / e l o c i t y Di f f e r -R V ence V ence V ence • • 0.75 0.0 0.0 0.0 0.0 0.0 0.0 0.77 0.5632 -0.0520 0.5632 -0.0515 0.5630 -0.0511 0.79 0.9900 -0.0969 0.9900 -0.0961 0.9896 -0.0955 0.82 1.2853 -0.1308 1.2853 -0.1299 1.2849 -0.1293 0.84 1.4539 -0.1511 1.4539 -0.1502 1.4534 -0.1495 0.87 1.4868 -0.1566 1.4867 -0.1557 1.4863 -0.1550 0.89 1.4265 -0.1472 1.4265 -0.1463 1.4269 -0.1456 0.92 1.2373 -0.1241 1.2373 -0.1232 1.2369 -0.1226 0.94 0.9349 -0.0895 0.9349 -0.0888 0.9346 -0.0882 0.97 0.5217 -0.0467 0.5217 -0.0462 0.5215 -0.0458 1.0 0.0 0.0 0.0 0.0 0.0 0.0 56 TABLE IX VELOCITY DISTRIBUTION AND TEMPERATURE DIFFERENCES DUE TO VISCOUS DISSIPATION EFFECTS FOR CONCENTRIC ANNULUS WITH BOTH WALLS HEATED FOR Ra=2000, X=0.75 D i s s i p a t i o n Parameter M M=0.0 M=0.0003 M=0.0005 Radius R V e l o c i t y V Temper-a t u r e D i f f e r -ence V e l o c i t y V Temper-a t u r e D i f f e r -ence V e l o c i t y V Temper-a t u r e D i f f e r -ence • 0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 • 0.97 1.0 0.0 1.2565 1.3567 1.0294 0.7093 0.6360 0.7721 1.1313 1.4545 1.3050 0.0 0.0 -0.0483 -0.0835 -0.1047 -0.1151 -0.1180 -0.1146 -0.1036 -0.0817 -0.0463 0.0 0.0 1.2510 1.3537 1.0306 0.7135 0.6409 0.7769 1.1333 1.4523 1.3002 0.0 0.0 -0.0472 -0.0822 -0.1032 -0.1136 -0.1165 -0.1131 -0.1021 -0.0803 -0.0453 0.0 0.0 1.2475 1.3517 1.0313 0.7162 0.6440 0.7799 1.1346 1.4509 1.2972 0.0 0.0 -0.0466 -0.0814 -0.1023 -0.1127 -0.1156 -0.1122 -0.1011 -0.0795 -0.0446 0.0 57 DISTANCE R FIGURE 3 V e l o c i t y P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h O u t e r Wall H e a t e d , Inner Wall I n s u l a t e d f o r Radius R a t i o J I 1 -I 0.5 0.6 0-7 0.8 0.9 1.0 DISTANCE R FIGURE 4 V e l o c i t y P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h O u t e r Wall H e a t e d , I n n e r Wall I n s u l a t e d f o r R a d i u s R a t i o 0.5 i 1 r — r DISTANCE R FIGURE 5 V e l o c i t y P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h I n n e r Wall H e a t e d, O u t e r Wall I n s u l a t e d f o r R a d i u s R a t i o 0.25 60 61 0-3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 DISTANCE R FIGURE 7 V e l o c i t y P r o f i l e s f o r C o n c e n t r i c Annulus w i t h Both W a l l s Heated f o r Radius R a t i o 0.25 62 0.5 0.6 0.7 0.8 0.9 1.0 DISTANCE R FIGURE 8 V e l o c i t y P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h Both W a l l s Heated f o r Radius R a t i o 0.5 63 0.4 0-6 0.8 1.0 DISTANCE R FIGURE 9 Temperature P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h O u t e r Wall H e a t e d , Inner Wall I n s u l a t e d f o r Radius R a t i o 0.25 FIGURE 10 Temperature P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h O u t e r Wall H e a t e d , I n n e r Wall I n s u l a t e d f o r R a d i u s R a t i o 0.5 65 T M=0 04 0.6 0.8 1.0 DISTANCE R FIGURE 11 Temperature P r o f i l e s f o r C o n c e n t r i c Annulus w i t h Inner Wall H e a t e d , O u t e r Wall I n s u l a t e d f o r Radi R a t i o 0.25 i 1 r M=0 0.5 0.6 0.7 0.8 0.9 1.0 DISTANCE R FIGURE 12 T e m p e r a t u r e P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h I n n e r Wall H e a t e d , O u t e r Wall I n s u l a t e d f o r Radius R a t i o 0.5 • 67 T M=0 0.4 0.6 0.8 1-0 DISTANCE R FIGURE 13 T e m p e r a t u r e P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h Both W a l l s Heated f o r Radius R a t i o 0.25 FIGURE 14 T e m p e r a t u r e P r o f i l e s f o r C o n c e n t r i c A n n u l u s w i t h Both W a l l s Heated f o r Radius R a t i o 0.5 10.0 -FIGURE 15 0.0001 0.0003 DISSIPATION PARAMETER M E f f e c t o f V i s c o u s D i s s i p a t i o n P arameter on N u s s e l t Number f o r C o n c e n t r i c A n n u l u s w i t h O u t e r Wall H e a t e d , I n n e r Wall I n s u l a t e d 0.0005 70 40.0 Ra=iooo X= 0 . 2 5 X= 0 . 5 30.0 L U CD 20C Ra=iooo L U CO CO 10.0 1.0 0.0 0.0001 0 .0003 DISSIPATION PARAMETER M 0.0005 FIGURE 16 E f f e c t o f V i s c o u s D i s s i p a t i o n Parameter on N u s s e l t Number f o r C o n c e n t r i c A n n u l u s w i t h I n n e r Wall H e a t e d , O u t e r Wall I n s u l a t e d 71 10d cr LU CD 9.0 LU CO CO 8.0 7.0 X=0.25 X= 0.5 Ra=i000 Ra=500 Ra=l ao o.oooi O0003 DISSIPATION PARAMETER M FIGURE 17 E f f e c t o f V i s c o u s D i s s i p a t i o n P a r a m e t e r on N u s s e l t Number f o r C o n c e n t r i c A n n u l u s w i t h Both W a l l s Heated 0.0005 72 REFERENCES 1. B i r d , R.B., W.E. S t e w a r t and E.N. L i g h t f o o t , " T r a n s p o r t Phenomena", John W i l e y & Sons I n c . , New York (1960) 2. B l a q u i r e , A., "Non L i n e a r System A n a l y s i s " , Academic P r e s s , New Y o r k , N. Y. (1966) 3. Cheng, K. C., " D i r i c h l e t Problems f o r L a m i n a r F o r c e d C o n v e c t i o n w i t h Heat S o u r c e s and V i s c o u s D i s s i p a t i o n i n R e g u l a r P o l y g o n a l D u c t s " , J o u r n a l A . I . C h . E . s V o l . 13, No. 6, pp. 1175-1180 (1967) 4. Cheng, K.C. and G. J . Hwang, "Laminar F o r c e d C o n v e c t i o n i n E c c e n t r i c A n n u l i " , J o u r n a l A.I.Ch.E., V o l . 14, No. 3, pp 510-512 (1968) 5. G e b h a r t , B., " E f f e c t s o f V i s c o u s D i s s i p a t i o n i n N a t u r a l C o n v e c t i o n " , J . o f F l u i d M e c h a n i c s , V o l . 14, pp 225-232 (1962) 6. G o r t l e r , H., " G r e n z s c h i c h t f o r s c h u n g " , IUTAM SYMPOSIUM FREIBERG/BR. (1957) 7. H a l l m a n , T.M., "Combined F o r c e d and F r e e L a m i n a r Heat T r a n s f e r i n V e r t i c a l Tubes w i t h U n i f o r m I n t e r n a l Heat G e n e r a t i o n " , T r a n s . A.S.M.E., V o l . 78, pp 1831-1841 (1956) 8. Howarth, L., "Modern Developments i n F l u i d Dynamics - High Speed Flow", V o l . I I , C l a r e d o n P r e s s , O x f o r d , E n g l a n d (1953) 9. I q b a l , M., " I n f l u e n c e o f Tube O r i e n t a t i o n i n L a m i n a r C o n v e c t i v e Heat T r a n s f e r " , Ph.D. T h e s i s , M c G i l l U n i v e r s i t y (1965) 10. K a n t o r o v i c h , L.V. and V . I . K r y l o v , "Approximate Methods o f H i g h e r A n a l y s i s " , I n t e r s c i e n c e P u b l i s h e r s , I n c . , New Y o r k , N.Y. (1958) 11. L u , P.C., "A T h e o r e t i c a l I n v e s t i g a t i o n o f Combined F r e e and F o r c e d C o n v e c t i o n Heat G e n e r a t i n g Laminar Flow I n s i d e V e r t i c a l P i p e s w i t h P r e s c r i b e d Wall T e m p e r a t u r e s " , M.S. T h e s i s , Kansas S t a t e C o l l e g e , M a h a t t a n , Kansas (1959) 12. M a r t i n e l l i , R.C. and L.M.K. B o e l t e r , "The A n a l y t i c a l P r e d i c t i o n o f S u p e r p o s e d F r e e and F o r c e d V i s c o u s C o n v e c t i o n i n a V e r t i c a l P i p e " , U n i v e r s i t y o f C a l i f o r n i a ( B e r k e l e y ) P u b l i c a t i o n s i n E n g i n e e r i n g , V o l . 5, No. 2, pp 23-58 (1942) 13. M c L a c h l a n , N.W., " B e s s e l F u n c t i o n s f o r E n g i n e e r s " , O x f o r d U n i v e r s i t y P r e s s , • E n g l a n d (1934). 73 14. M o r t o n , B.R., "Laminar C o n v e c t i o n i n U n i f o r m l y Heated V e r t i c a l P i p e s " , J . o f F l u i d M e c h a n i c s , V o l . 8, pp 227-240 (1960) 15. O s t r a c h , S., "New A s p e c t s o f N a t u r a l C o n v e c t i o n Heat T r a n s f e r " , T r a n s . A.S.M.E.,Vol. 75, No. 7, pp 1287-1290 (1953) 16. O s t r a c h , S., " U n s t a b l e C o n v e c t i o n i n V e r t i c a l C h a n n e l s w i t h H e a t i n g f r o m Below and I n c l u d i n g t h e E f f e c t s o f Heat S o u r c e s and F r i c t i o n a l H e a t i n g " , NACA TN 3458 (1955) 17. O s t r a c h , S., "Laminar N a t u r a l C o n v e c t i o n Flow and Heat T r a n s f e r o f F l u i d s w i t h and w i t h o u t Heat S o u r c e s i n Ch a n n e l s w i t h C o n s t a n t Wall T e m p e r a t u r e s " , NACA TN 2863 (1952) 18. O s t r a c h , S., "Theory o f L a m i n a r F l o w s " , S e c t i o n F, Ed. F. K. Moore, P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , New J e r s e y , (1964) 19. O s t r a c h , S., "Combined N a t u r a l and F o r c e d C o n v e c t i o n L a m i n a r Flow and Heat T r a n s f e r o f F l u i d s w i t h and w i t h o u t Heat S o u r c e s i n C h a n n e l s w i t h L i n e a r l y V a r y i n g Wall T e m p e r a t u r e s " , NACA TN 3141 (1954) 20. O s t r a c h , S., "On P a i r s o f S o l u t i o n s o f a C l a s s o f I n t e r n a l V i s c o u s Flow Problems w i t h Body F o r c e s " , NACA TN 4273 (1958) 21. S c h e e l e , G.F., "The E f f e c t o f N a t u r a l C o n v e c t i o n on T r a n -s i t i o n t o D i s t u r b e d Flow i n a V e r t i c a l P i p e " , Ph.D. T h e s i s i n C h e m 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 I l l i n o i s , 1962 22. S c h l i c h t i n g , H., "Boundary L a y e r T h e o r y " , M c G r a w - H i l l Book Co. I n c . , F o u r t h E d i t i o n (1960) 23. S h a p i r o , A.H., "The Dynamics and Thermodynamics o f a C o m p r e s s i b l e F l u i d " , V o l . I I , The Ronald P r e s s Co., New Y o r k , N. Y. (1954) 24. T r u i t , R.W., "Fundamentals o f Aerodynamic H e a t i n g " , The Ronald P r e s s Co., New Y o r k , N.Y. (1960) 25. T y a g i , V.P., " F o r c e d C o n v e c t i o n o f a D i s s i p a t i v e L i q u i d i n a Channel w i t h Neumann C o n d i t i o n s " , T r a n s . A.S.M.E., J o u r n a l o f A p p l i e d M e c h a n i c s , pp 18-24 (1966) 26. T y a g i , V.P., "Laminar F o r c e d C o n v e c t i o n o f a D i s s i p a t i v e F l u i d i n a C h a n n e l " , T r a n s . A.S.M.E., J o u r n a l o f Heat T r a n s f e r , pp 161-169 (1966) 74 27. T y a g i , V.P., "A G e n e r a l N o n - C i r c u l a r Duct C o n v e c t i v e Heat T r a n s f e r Problem f o r L i q u i d s and G a s e s " , I n t . J . Heat and Mass T r a n s f e r , V o l . 9, pp 1321-1340 (1966) 28. T y a g i , V.P., " G e n e r a l Study o f a Heat T r a n s m i s s i o n Problem o f a Channel-Gas Flow w i t h Neumann-Type Thermal Boundary C o n d i t i o n s " , P r o c . Comb. P h i l . S o c , V o l . 62, pp 555-573 (1966) 29. Way, S., "Bending o f C i r c u l a r P l a t e s w i t h L a r g e D e f l e c t i o n " , T r a n s . A.S.M.E., V o l . 56, pp 627-636 (1934) APPENDICES 76 APPENDIX A DERIVATION OF NUSSELT NUMBER EXPRESSION FOR CIRCULAR DUCTS The N u s s e l t number e x p r e s s i o n f o r c i r c u l a r d u c t s i n terms o f t h e d i m e n s i o n l e s s v a r i a b l e s i s o b t a i n e d as shown below, where q = av e r a g e h e a t f l u x T w = t e m p e r a t u r e o f t h e w a l l T b = b u l k t e m p e r a t u r e o f t h e f l u i d . The b u l k t e m p e r a t u r e can be w r i t t e n a s , X = J J T \ ^ A / / K C ( A ( A - l ) (A-2) S u b s t i t u t i n g (A-2) i n ( A - l ) , we o b t a i n , T w -s (A-3) Energy Balance Between Sections1&2 77 Now c o n s i d e r a f l u i d f l o w i n g between s e c t i o n s 1 and 2 o f a c i r c u l a r d u c t as shown i n t h e f i g u r e . By making an ene r g y b a l a n c e , we o b t a i n , CucF A (T 4 -T.) ^ ^PAZ+M[//gap/iJAz, ( A-4) where T-| and 1^ a r e t h e b u l k t e m p e r a t u r e s a t s e c t i o n s 1 and 2 r e s p e c t i v e l y and P i s t h e h e a t e d p e r i m e t e r o f t h e d u c t . 9 Z S u b s t i t u t i n g f i - = C i n e q u a t i o n (A-4), we o b t a i n , fl= ecFukC - ZML'ffdvfzdll. ( A - 5 ) N u = Now s u b s t i t u t i n g (A-5) i n (A-3), we o b t a i n , k L e c r u ^ C - ^ j ^ m ^ (A.7) + g M ' / f ^ ^ / e ' ( A" 8 ) '[<J> VRdR / J V R^R 78 APPENDIX B DERIVATION OF.NUSSELT NUMBER EXPRESSIONS FOR CONCENTRIC ANNULI The N u s s e l t number e x p r e s s i o n s f o r t h e c o n c e n t r i c a n n u l i a r e o b t a i n e d as shown below, Case I: O u t e r Wall H e a t e d , I n n e r Wall I n s u l a t e d N u s s e l t number i s g i v e n by t h e e x p r e s s i o n N a .. ^ - £ • • The e q u i v a l e n t d i a m e t e r f o r t h i s c a s e i s g i v e n by, J \ = Zr0 ( i - Xx) , ( B-2) where r Q i s t h e r a d i u s o f t h e o u t e r t u b e and x i s t h e r a d i u s r a t i o r - j / r 0 -By making an e n e r g y b a l a n c e as shown f o r t h e c i r c u l a r d u c t we o b t a i n t h e f o l l o w i n g e x p r e s s i o n , C C p U A ^ - T , ) . %P l& +JL[ti(*$di\l& , (B-3) where P i s t h e he a t e d p e r i m e t e r . From e q u a t i o n (B-3) we o b t a i n , U s i n g t h e v a l u e o f D^ from (B-2) and s u b s t i t u t i n g (B-4) i n ( B - l ) we o b t a i n , 79 Nu = k 4 i _ v0 j U g / J k .eCpUA c - ^ tu* l/dvfRM - 1 + 8('-^*)M/^) (UK Case I I : Inner Wall H e a t e d , O u t e r Wall I n s u l a t e d (B-5) (B-6) (B-7) N u s s e l t Number f \ | u _ kbu _ Dk 9/ The e q u i v a l e n t d i a m e t e r f o r t h i s c a s e i s g i v e n by, ( B - l ) (B-8) By making an e n e r g y b a l a n c e t h e f o l l o w i n g e x p r e s s i o n i s o b t a i n e d , eCpUfifo-T,), l?LSI A 2 . (B-9) From e q u a t i o n ( B - 9 ) , s u b s t i t u t i n g (B-8) f o r we o b t a i n . (B-10) S u b s t i t u t i n g (B-10) i n ( B - l ) , we o b t a i n , 80 I ( B - l l ) A (B-13) Case I I I : Both W a l l s Heated Nu = U> = A -4- • ( B - D The e q u i v a l e n t d i a m e t e r f o r t h i s c a s e i s g i v e n by, D h = 2 r Q (1-x) By making an e n e r g y b a l a n c e we o b t a i n , where q^ and q Q a r e t h e a v e r a g e h e a t f l u x a t i n n e r and o u t e r w a l l r e s p e c t i v e l y . From e q u a t i o n ( B - l 4 ) we o b t a i n , (B-15) 81 where q = q a v e r a g e S u b s t i t u t i n g (B-15) i n ( B - l ) t h e f o l l o w i n g e q u a t i o n i s o b t a i n e d , N k k k ?cFuhc -AMI \(&)\JLR\ - 1 •+ '{ (pvRcLK/'fvRjLg (B-16) (B-17) (B-18) 82 APPENDIX C DETAILS OF GOVERNING EQUATIONS AND LIMITATIONS The f i n a l form o f t h e g o v e r n i n g e q u a t i o n s as g i v e n by (1) and (2) were o b t a i n e d i n t h e f o l l o w i n g manner: On t h e b a s i s o f t h e a s s u m p t i o n s on page 7, t h e e q u a t i o n s o f m o t i o n i n r and e d i r e c t i o n s can be i g n o r e d . The b a s i c momen-tum e q u a t i o n i n Z - d i r e c t i o n f o r c o n s t a n t p and y i s g i v e n by [ 1 ] , C(m +vr dVz+v^dVz +vk-m) = -2t + u,[ii(rm V 3 t sr r s© 3 H / 35 f [ r a n 3rJ 8 v z F o r s t e a d y f l o w j^- = 0 and beca u s e o f symmetry t h e com-ponent o f v e l o c i t y i n o - d i r e c t i on v a n i s h e s . F o r f u l l y d e v e l o p e d 3V l a m i n a r f l o w , v,. = - r - — = 0 and s i n c e p r e s s u r e i s o n l y a f u n c t i o n r o z o f Z, e q u a t i o n ( C - l ) r e d u c e s t o °--~^+^&^rW)±Cfe (C-2' S i n c e Z i s measured p o s i t i v e i n t h e upward d i r e c t i o n , t h e n e g a t i v e s i g n b e f o r e g z i s t a k e n . Thus we have, 0 = -dp +jufdh& +±d^x)-C%- (c-3) The b a s i c d i f f e r e n t i a l e n e r g y e q u a t i o n f o r c o n s t a n t K, y 83 and p can be w r i t t e n a s , ?ct$r„kVlr+ a-L +Jx $ + T £ Ik , (c.4) Dt where = I n t e r n a l h e a t g e n e r a t i o n s o u r c e e n e r g y $ = V i s c o u s d i s s i p a t i o n f u n c t i o n * i s g i v e n by [ 1 ] , (C-5) (C-6) R e - w r i t i n g (C-4) i n an expanded form we h a ve, Pr {21+ Vy-2E+ )& 2 r + t f z 2 n - K n 2 (r?T\+± 21". E l i m i n a t i n g t h e terms which a r e e q u a l t o z e r o f o r c o n d i -t i o n s mentioned e a r l i e r , and f o r no i n t e r n a l h e a t g e n e r a t i o n s o u r c e , e q u a t i o n (C-6) r e d u c e s t o 32 The r e l a t i v e s i g n i f i c a n c e o f c o m p r e s s i o n work t o t h a t o f v i s c o u s d i s s i p a t i o n can be seen by comparing t h e l a s t two terms on t h e r i g h t hand s i d e o f e q u a t i o n (C-7). E q u a t i o n (C-7) i n t h e non-d i m e n s i o n a l f o r m can be w r i t t e n a s , vVv + ^^V^f^^^V^^^- (c-8) 84 D i v i d i n g t h e c o e f f i c i e n t o f c o m p r e s s i o n work term by t h a t o f v i s c o u s d i s s i p a t i o n , we o b t a i n t h e f a c t o r (1/16) Pe Re 3 C D h „ T h i s f a c t o r shows t h a t f o r s m a l l v a l u e s o f P e c l e t number, R e y n o l d s number, 3 t h e c o e f f i c i e n t o f v o l u m e t r i c e x p a n s i o n and t h e temper-a t u r e r i s e i n the f l o w d i r e c t i o n , t h e c o m p r e s s i o n work term can be n e g l e c t e d . Thus e q u a t i o n (C-7) r e d u c e s t o * (C-9) A d i s c u s s i o n on t h e i n c l u s i o n o f c o m p r e s s i o n work term has a l s o been g i v e n by T y a g i [ 2 7 ] . The v a r i a b i l i t y o f t h e p h y s i c a l p r o p e r t i e s w i t h temper-a t u r e makes t h e prob l e m h i g h l y n o n - l i n e a r and thus e x t r e m e l y d i f f i -c u l t t o s o l v e . Hence f o r t h i s r e a s o n , t he p r e s e n t s t u d y d e a l s w i t h c o n s t a n t p r o p e r t i e s e x c e p t f o r t h e v a r i a t i o n o f d e n s i t y i n t h e buoyancy term o f t h e momentum e q u a t i o n . To e n s u r e t h i s , t h e temp-e r a t u r e d i f f e r e n c e s i n t h e system s h o u l d be s m a l l s i n c e a l l t h e p h y s i c a l p r o p e r t i e s a r e a f u n c t i o n o f t e m p e r a t u r e . M o r e o v e r , t h e d u c t l e n g t h has t o be s m a l l t o a v o i d v a r i a t i o n o f p r o p e r t i e s a l o n g t h e d u c t l e n g t h . LIMITATIONS OF THE RADIUS RATIO FOR ANNULUS F o r t he c o n c e n t r i c a n n u l u s , t he range o f r a d i u s r a t i o X i s from 0 t o 1. F o r x = 0, t h e a n n u l u s r e d u c e s t o a c i r c u l a r d u c t w i t h a w i r e i n the c e n t r e p a r a l l e l t o t h e a x i s , whereas f o r x = 1, 85 t h e c o n f i g u r a t i o n o f p a r a l l e l p l a t e s i s o b t a i n e d . The range o f x used f o r t h e p r e s e n t a n a l y s i s i s from 0.25 t o 0.75. I f t h e a n n u l a r gap i s t o o s m a l l , t h e p h y s i c a l p r o p e r t i e s may n o t r e m a i n c o n s t a n t due t o l a r g e v i s c o u s h e a t i n g e f f e c t s . On t h e o t h e r hand, i f t h e a n n u l a r gap i s t o o l a r g e , the s y s t e m a p p r o x i m a t e s a l m o s t t o f l o w a l o n g a s i n g l e v e r t i c a l c y l i n d e r i n which c a s e a f u l l y d e v e l o p e d f l o w i s o b t a i n e d o n l y beyond a v e r y l a r g e e n t r a n c e l e n g t h and i s n o t o f i n t e r e s t f o r t h e p r e s e n t s t u d y . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080680/manifest

Comment

Related Items