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.

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.), U n i v e r s i t y of K a r a c h i , K a r a c h i , P a k i s t a n , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc.  i n the Department of Mechanical Engineering We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA March, 1970  In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e ments f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r referance and study.  I f u r t h e r agree that permission f o r  extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of the Department or by h i s  representatives.  I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  M. SHAFI ROKERYA  Department of Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, B r i t i s h Columbia Canada Date M ^ 0  1970.  i i ABSTRACT The e f f e c t s of viscous d i s s i p a t i o n on the flow phenomena and heat t r a n s f e r rate f o 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 cond i t i o n of combined free and forced convection.  The f l u i d  properties are considered to be constant except f o r the v a r i a t i o n of density i n the buoyancy term of the momentum equation.  The  thermal boundary condition of uniform heat f l u x per u n i t length i n the flow d i r e c t i o n has been considered. i s c a r r i e d out f o r two geometries; (a) Concentric a n n u l i .  The i n v e s t i g a t i o n  C i r c u l a r ducts and (b)  The governing momentum and n o n - l i n e a r energy  equations are solved f o r the c i r c u l a r duct by three methods; Power Series Method ( i i ) Integration Method.  G a l e r k i n ' s Method and ( i i i )  (i)  Numerical  The s o l u t i o n s f o r the concentric annuli are  obtained by Numerical Integration Method.  Results f o r the v e l -  o c i t y and temperature d i s t r i b u t i o n i n the flow f i e l d are o b t a i n e d , and information of engineering i n t e r e s t l i k e Nusselt numbers have been evaluated. For combined f r e e and forced c o n v e c t i o n , the momentum and energy equations are coupled, and hence viscous d i s s i p a t i o n a f f e c t s both the v e l o c i t y and temperature f i e l d s .  The e f f e c t o f  viscous d i s s i p a t i o n on the v e l o c i t y f i e l d i s to reduce the flow v e l o c i t y near the heated w a l l ( s ) and thus i t counteracts the e f f e c t of f r e e convection on the v e l o c i t y f i e l d f o r the present study of heating i n upflow.  The e f f e c 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 f e r e n c e s i n the system.  Viscous d i s -  s i p a t i o n opposes the e x t e r n a l l y impressed heating and reduces the heat t r a n s f e r rate when the surface t r a n s f e r s heat to the fluid.  Consequently, lower Nusselt number values are obtained  when viscous d i s s i p a t i o n i s taken i n t o c o n s i d e r a t i o n .  The quan-  t i t a t i v e e f f e c 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, f o r flow  through annular passages and f o r the corresponding values of the same parameters, the e f f e c t of viscous d i s s i p a t i o n on the heat t r a n s f e r rate may not be ignored.  iv TABLE OF CONTENTS Chapter  Page  ABSTRACT . .'  it  LIST OF TABLES  ' .  LIST OF FIGURES  vi viii  ACKNOWLEDGEMENTS . . . . . .  .  ix  NOMENCLATURE  .  1  I II  INTRODUCTION SECTION I: 2.1  III  . 3  C i r c u l a r Ducts  6  Formulation of the Problem  7  SOLUTIONS. 3.1  11  Exact S o l u t i o n Without Viscous D i s s i p a t i o n Term  3.2  IV  V  5.2 VI  3.2.1  Power Series Method  14  3.2.2  G a l e r k i n ' s Method. .  20  3.2.3  Numerical Integration Method . . . .  24  Concentric Annuli  Formulation of the Problem  SOLUTIONS 5.1  12  Solutions With Viscous D i s s i p a t i o n Term . .  SECTION I I : 4.1  .  26 27 33  Exact S o l u t i o n Without Viscous D i s s i p a t i o n Term  34  Solutions With Viscous D i s s i p a t i o n Term . .  36  DISCUSSION OF RESULTS  37  6.1  C i r c u l a r Ducts.  37  6.1.1  37  Solution Details  V  Chapter  Page  6.2  VII  6.1.2  Velocity Field  39  6.1.3  Temperature F i e l d  .  40  6.1.4  N u s s e l t Numbers  .  41  Concentric Annuli . . . . •  41  6.2.1  Solution Details . . . . . . . . . .  41  6.2.2  Velocity Field  .  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 Ratio . .  46  CONCLUSIONS  47  REFERENCES  72  APPENDICES  75  A  DERIVATION OF NUSSELT NUMBER EXPRESSION FOR CIRCULAR DUCTS  B  C  76  DERIVATION OF NUSSELT NUMBER EXPRESSION FOR CONCENTRIC ANNULI  78  DETAILS OF GOVERNING EQUATIONS AND LIMITATIONS .  82  vi LIST OF TABLES Table I  II III  IV  V  VI  VII  VIII  IX  Page Velocities and temperature differences at the centre of a vertical circular duct due to viscous dissipation effects  48  Effect of viscous dissipation parameter on Nusselt number for a vertical circular duct 49 Nusselt number values for M=0 obtained by Exact solution and Runge-Kutta method for concentric annul us with radius ratio 0.5  50  Velocity distribution and temperature differences due to viscous dissipation effects for concentric annul us with outer wall heated, inner wall insulated for Ra=l, x= 0.75  51  Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with outer wall heated, inner wall insulated for Ra-1000, x= 0.75  52  Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with inner wall heated, outer wall insulated for Ra=l, X= 0.75  53  Velocity distribution and temperature differences due to viscous dissipation effects for concentric annulus with inner wall heated, outer wall insulated for Ra=1000, x= 0.75  54  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  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 f o r flow through a v e r t i c a l c i r c u l a r duct . . .  8  2  Coordinate system f o r flow through a v e r t i c a l concentric annulus  28  3  V e l o c i t y p r o f i l e s f o r concentric annulus with outer wall heated, inner wall i n s u l a t e d f o r radius r a t i o 0.25 57  4  V e l o c i t y p r o f i l e s f o r concentric annulus with outer wall heated, inner wall i n s u l a t e d f o r radius r a t i o 0.5. 58  5  V e l o c i t y p r o f i l e s f o r concentric annulus with inner wall heated, outer wall i n s u l a t e d f o r radius r a t i o 0.25 59  6  V e l o c i t y p r o f i l e s f o r concentric annulus with inner wall heated, outer wall i n s u l a t e d f o r radius r a t i o 0.5 60  7  V e l o c i t y p r o f i l e s for concentric annulus with both w a l l s heated f o r radius r a t i o 0.25  61  8  V e l o c i t y p r o f i l e s f o r concentric annulus with both w a l l s heated f o r radius r a t i o 0.5  62  9  Temperature p r o f i l e s f o r concentric annulus with outer wall heated, inner wall i n s u l a t e d f o r radius r a t i o 0.25  63  Temperature p r o f i l e s f o r concentric annulus with outer wall heated, inner wall i n s u l a t e d f o r radius r a t i o 0.5  64  Temperature p r o f i l e s f o r concentric annulus with inner wall heated, outer wall i n s u l a t e d f o r radius r a t i o 0.25  65  Temperature p r o f i l e s f o r concentric annulus with inner wall heated, outer wall i n s u l a t e d f o r radius r a t i o 0.5  66  Temperature p r o f i l e s f o r concentric annulus with both w a l l s heated f o r radius r a t i o 0.25  67  Temperature p r o f i l e s f o r concentric annulus with both w a l l s heated f o r radius r a t i o 0.5  68  10  11  12  13 14  vi i i Figure 15  16  17  Page E f f e c t of viscous d i s s i p a t i o n parameter on Nusselt number f o r concentric annulus with outer wall heated,inner wall i n s u l a t e d  69  E f f e c t of viscous d i s s i p a t i o n parameter on Nusselt number f o r concentric annulus with inner wall heated, outer wall i n s u l a t e d . . . .  ?0  E f f e c t of viscous d i s s i p a t i o n parameter on Nusselt number f o r concentric annulus with both w a l l s heated . . .  71  ix ACKNOWLEDGEMENTS The author wishes to express h i s deep gratitude 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 Department, U n i v e r s i t y of Calgary and Dr. M. Flower of the Department of Computer S c i e n c e , B r i s t o l U n i v e r s i t y f o 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 University  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 . perimeter ' l diameter e  n e a t e d  c  u  i  v  a  l  e  n  t  U C AT Eckert number, dimensionless 2  r—7T, p  Gravitational acceleration .(d| P 9 +  ^-y  w  ) D  2  — . 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 h £-» r'o for concentric annuli, dimensionless p^ggc CD.* yg^» Rayleigh number, dimensionless UD.p — — . Reynolds number, dimensionless Temperature u  Average axial velocity Axial velocity v  z  -Q-. dimensionless axial velocity  V  =  p  dimensionless  Z  =  A x i a l coordinate i n flow d i r e c t i o n  3  =  C o e f f i c i e n t of volumetric expansion  n  •  K  =  (Ra)  1 / 4  Thermal c o n d u c t i v i t y of the f l u i d  x  =  y  =  i — » radius r a t i o , dimensionless o 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  <> f  =  (T - T \ pUc CD ^/4<'  J  =  r  v  2  =  t  e  m  P  e  r  a  t  u  r  dimensionless _d£ 2 dR  + +  1 R  d_ dR  Subscripts i  inside  o  outside  w  wall  e  f u n c t i o n , dimensionless  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 imp 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 . T h e s e v e l o c i t y g r a d i e n t s give r i s e t o shear s t r e s s e s which r e s u l t s i n the d i s s i p a t i o n o f f r i c t i o n a l energy i n t o h e a t .  Consequently  i n a heat t r a n s f e r  process f o r the flow o f a real f l u i d , the omission o f viscous d i s s i p a t i o n i n t h e thermal energy b a l a n c e o f a moving f l u i d e l e m e n t w o u l d be u n r e a l i s t i c from 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 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 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 viscous d i s s i p a t i o n which i s a s s o c i a t e d with 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 c a n 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 flows.  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 d e r 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 For external flows, the e f f e c t o f viscous 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 b o u n d a r y 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 at very high v e l o c i t i e s [ 8 ] .  that are attained  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 high  Mach numbers c a n cause s e v e r e p r o b l e m s due t o t h e 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 manu 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 A e r o d y n a m i c 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 10 . 4  to  He has shown t h a t t h e m a g n i t u d e 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 w h i c h 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 strength o f the earth. 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 c a n 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  convection,  ( 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)  Forced  Convection  T y a g i [ 2 5 , 2 6 , 2 7 , 28] i n a s e r i e s o f papers has s t u d i e 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 n f o r c e d convection through nonc i r c u l a r channels.  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 both Neumann and D i r i c h l e t t y p e  thermal  b o u n d a r y 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 point-matching.  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 b o u n d a r y 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 ducts than f o r n o n - c i r c u l a r ducts. (ii)  Free Convection 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  viscous d i s s i p a t i o n i n natural convection flows through channels 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 w h i c h 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 transfer. (iii)  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  forced convection i s t h a t of Ostrach [19, 20].  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 to his free convection 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 the c i r c u l a r duct are presented.  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 direction and constant fluid properties, the axial temperature gradient 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 variable 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  Nl  ...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 )  (3)  W  where p 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 w  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  where T  Q  i s d e f i n e d by,  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* (T-Tj/CPUCfCDt/M)  parameters,  ,  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 nond 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 .  VV  + R.<L  <f> +  L  -  0  (4) (5)  where v  In e q u a t i o n s  dK R clfl (4) and ( 5 ) , R a y l e i g h number Ra and t h e l  viscous d i s s i p a t i o n parameter M are p r e s c r i b e d q u a n t i t i e s while 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 c a n be written as,  10  (6)  or  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 to those without i t , the a v a i l a b l e s o l u t i o n f o r the l a t t e r c a s e [7]  i s f i r s t presented here 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 problem 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 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 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 , ^  (4)  (8) 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 following equations:  VV  +  Rex  (j)  +  |  0  =  0,  (9)  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) c a n be w r i t t e n [ 1 3 ] a s ,  V = A,U  (it) + fit ire^q  -f hkai  ft*)*/}*kc  Q  (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 K.O.  L  V  (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 f r o m 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 b o u n d a r y  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  drop parameter L i s o b t a i n e d from the 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 then determined from,  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 being zero f o r t h i s case. N u s s e l t Number*  0  3.2  0  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 p r 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 equation.  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 .  Therefore the 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 a c c u r a t e methods.  3.2.1  but f a i r l y  The t h r e e methods used w e r e ,  1.  Power S e r i e s Method  2.  G a l e r k i n ' s Method  3.  Numerical  I n t e g r a t i o n Method  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  For d e t a i l s see Appendix 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  assigning a numerical value to the 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 from 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 the 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 = C  0  + C,R  2  + QR%  ,  d)» D. + AR +  +  a  where C , C , C ]  2  C  n  and D , D p Q  D  2  (  + •'••••» ( D  n  are the  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 following expression i s obtained,  + U(Do+D,R + 2  M* * 1  D £* +  )+L_0.  s  (i?)  Now e q u a t i n g 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, the f o l l o w i n g expressions r e s u l t , 4C,  36C  5  +  L  =  0  =  0  =  =  0  (18)  /or t  (19)  ,  (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--  -.(22)  Now s u b s t i t u t i n g the 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 the r e q u i r e d d i f f e r e n t i a t i o n s the f o l l o w i n g expression i s obtained,  17  (4/),  + /SD £*+ 3 6 D , £ * + £ 4 / ) ^ +  ;  a  (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 written as,  = Z (inC„R  Z H  ;  • ik-  U  G, £  R (24)  Let  i n + 3k - 2  Therefore,  n. + k -  or  n  Since  ft  -  /  =  <2.S  =.  S  S+ I- k  ^> / ,  therefore,.  k <  S  Thus e x p r e s s i o n (24) becomes  I  S=i L k-.i  AS *  R  (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 following expression i s obtained,  (4 A  +  K>DxR + X  - (C +  CR * 1  0  3 6 0 R V *4Z)V+  )  3  +  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 to give the f o l l o w i n g s e t of equations,  AD, - C \ib -C,+ x  or  4  0  4M  0  =  -for  k ° ,  2  7  )  -.0 -fa-ft,  yZAk(s+i-k)C^,_ C k  >)%, - Cs f  (  M Z <r K (5, I- k)C _ C ,0 ,28) ti  for  K  k  00  1 , 3  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 ,  + RcDo + L - 0  H,  4 Yl Cn + &-D _, = 0 Z  n  {or n = 3, V*  (18)  (22)  19  4D,  C  -  4 M\r  0  - 0  (27)  Q^MDk(^k)(: _.c - o +i  for ^ = I,  )  k  (28  3 . . . . . o«  From e q u a t i o n s ( 1 8 ) , ( 2 2 ) , (27) and ( 2 8 ) , i t can be s e e n t h a t knowing t h e v a l u e s o f C , D and L, a l l t h e s u c c e s s i v e Q  Q  c o e f f i c i e n t s C and D 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 n  n  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 b o u n d a r y 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 integration,the following expression is obtained,  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 o and D„o 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 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  ence a t t h e c e n t r e o f t h e d u c t .  differ-  T h e s e 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 b o u n d a r y c o n d i t i o n s (29) and (30) and e q u a t i o n ( 3 1 ) . Determination of the required c o e f f i c i e n t s gives the 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 from equation ( 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 problem  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  approximate  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 c a n 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 b o u n d a r y 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 ation technique, the resulting equations are solved simultaneously 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  functions  be e x p r e s s e d a s ,  (32) (33)  21 where C , C-j,  and D , D-], D  Q  Q  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 the 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 w h i c h a r e a measure o f t h e a c c u r a c y of the approximations,  +  L  '  +  C ^ -  (34)  CX-  C.R*-  C ^ ) +  4M  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 w o u l d be i d e n t i c a l l y e q u a l t o z e r o . 2  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 following equations are obtained,  (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 ,  Y (i-R*)fcfilR = 0 , 4  (39)  (40)  VI 0-  0  (41)  A f t e r performing the r e q u i r e d i n t e g r a t i o n s , the 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°  6  _C  30  4  '  5*  - C, _ C, 6 14 £0  +  ^° J.  -  36  +  30  -Co  £o  24  4  £  n  >  u  15  LMfC^ C^ \ 3 /5 +  +  6  43.  /5  30  W  ./of  /S"  210  3  +ic, c j -  CC  £0  (45)  /  3  - ZA so  (43)  (44)  7 /  C^ \ - 0 ,  + Cl  /ao  V  +  3  /AO  -  3o  ~  To)  5  0  £o  - C_  U  - J ) . D,  +  C  IS  -24  4  0  ,  (46)  y  -A  +^/M^  <r  /o5"  io  *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 , Cp Q  C , D, 2  q  Dp  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 h i c h 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 3  + Ci £  -  3.  (48)  •  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 c a n 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 evaluated. 3.2.3  Numerical 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 R u n g e - K u t t a 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 complete 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 b o u n d a r y 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 o u n d a r y 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 results.  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 checked  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 not 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 duct.  In the next s e c t i o n , t h e f o r m u l a t i o n and  f o r the c o n c e n t r i c annulus are  presented.  solutions  4.  SECTION I I  CONCENTRIC ANNULI  27  4.1  F o r m u l a t i o n O f The P r o b l e m  Consider the f u l l y developed laminar flow 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  b o u n d a r y 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 b o u n d a r y c o n d i t i o n w i l l depend on t h e following three situations, 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 insulated.  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 insulated.  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 at a given a x i a l p o s i t i o n . Choosing the f o l l o w i n g non-dimensional  parameters,  <: o  _i  t  FIGURE 2  o  _J  t  N  z=o  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 Concentric Annulus  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 b o u n d a r y 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 W a l l 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 it. E q u i v a l e n t Diameter 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  -  where  r A  Q  an  ,  =  radius of the o u t e r tube  =  radius of inner tube/radius of outer tube.  (49)  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 non-dimensional equations are obtained,  (l- yfV V + Ra C|) + L = 0 , l  (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  Conditions  dA . 0 ,  At R- > , V -  (52)  fit R-l , V - <{> -. o .  (53)  N u s s e l t Number The N u s s e l t number i s g i v e n by* Na  =  J  LAdKi  L-  (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 Equivalent  Diameter 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 -  -  >V  •  A  U s i n g t h i s v a l u e f o r D i n e q u a t i o n s (1) and (2), t h e h  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 ) v v + R<xcp l  4  + L  - o,  (L^V>-V ^Mg)lo. +  * F o r d e t a i l s see Appendix B  (56)  (57)  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  flt  (p-  V--  ftt'/U>,  I ,' .V«  H-  d4  =  0,  (58)  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 by*  Nu  -  K  (50) A  X  Case I I I : Both W a l l s Equivalent  Heated  Diameter 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  (61)  ( i - > )  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  energy equations are obtained as,  ( l - ^ V ' V + Ro. ct + L = 0,  ( l - ^ V ^  {62)  - V + 4f/->)Mg|) =0. (63) l  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  (p * o,  At R= > ,  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  *  =  For d e t a i l s see Appendix B  A  (66)  5. SOLUTIONS  34  5.1  Exact S o l u t i o n Without Viscous D i s s i p a t i o n  A g e n e r a l form 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 [11] f o r t h e c o n c e n t r i c annulus  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 . For case I , outer wall heated, i n n e r wall 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 Let  +  (b  -H  _L_  » 0,  Jg^  1  (67)  (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 the f o l l o w i n g e q u a t i o n s ,  V*V V $ V  = 0,  -87  4  where  + 8 = 0 ,  + BRa $  =  V/L  ,  "$ =  (70) (7i)  (J)/L.  C o m b i n i n g (70) and (71) we o b t a i n ,  + ^ V -  V V 4  where  Yj^  =  B^Ra  0,  () 72  (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 b y (12) i n S e c t i o n I a n d i s r e p e a t e d V=  here.  ftft)+C>d ft/?) ^  CMo  0  (74)  +  from e q u a t i o n (70) we o b t a i n , (75)  B&x L where  The unknowns C-j, C , Cg and  i n (74) a n d (75) a r e  2  o b t a i n e d by a p p l y i n g t h e b o u n d a r y c o n d i t i o n s (52) a n d ( 5 3 ) . T h i s results i n the following four equations:  0- C,l*v-.ft>) 0  £ K ( 7 A ) + C,ter.fyA)+  +  H'(V) + £ k-J  --  0 = C, tw.ft) + Q K  C^tei.fyX) ,  ft>)-C te^>)*C, 3  + C  3  for.  +  taj'ft>)], C<  tec„C ), ?  (77)  (78)  (79  )  0= -^-|M-C K(7)+QK(?)-^ S,(' l)+^/cer.^j.(80) te  ,  1  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 are 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 c a n 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 , b o t h 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 without success.  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 numeri c a l i n t e g r a t i o n method o f R u n g e - K u t t a 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 R u n g e - K u t t a 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 the r a d i u s r a t i o ( j / ) r  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 b u t 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 six s i g n i f i c a n t figures.  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 phenomena a n d heat t r a n s f e r r a t e as 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 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 a n d ( i i ) Concentric annuli. 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  Solution Details 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 c o m p u t e r .  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 . T h e s e f u n c t i o n s were e v a l u a t e d from 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 M c 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 t h 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  estimates  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 used and e m p l o y i n g t h e m i n i m i z i n g a n d 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 n o t e x 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  38  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  guesses  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  'educated' g u e s s e s were e s t i m a t e d from t h e r e s u l t s o b t a i n e d with the exact s o l u t i o n f o r  M=0.  In t h e R u n g e - K u t t a ' 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  of  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 b o u n d a r y 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 b o u n d a r y 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 R u n g e - K u t t a ' s f o u r t h o r d e r method i s o f the order of 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 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  0.01  did  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  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 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 non-zero 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  is t o a c c e l e r a t e t h e v e l o c i t y near the wall [12].  To s a t i s f y con-  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 the  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 possible to create flow reversal at the centre of the duct. 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 of R a y l e i g h number w h i c h 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 the present study should not  e x 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 used was 625 f o r t h e c a s e o f c i r c u l a r duct. 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 6.1.2  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 .  Velocity Field For 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 independent o f the temperature  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 a n d 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 but a l s o t h e v e l o c i t y f i e l d .  field  The measure o f f r e e c o n v e c t i o n i s t h e  40  non-dimensional  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 near 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 centre.  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  duct under the i n f l u e n c e o f viscous d i s s i p a t i o n f o r v a r i o u s values o f R a y l e i g h number.  T h i s t r e n d becomes more p r o n o u n c e d 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 near  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 here f o r b r e v i t y .  From H a l l m a n ' 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 case o f upflow h e a t i n g , the 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 near 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 ( b u o y a n c y ) e f f e c t on the flow 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  of f r e e convection i s t o increase the shear s t r e s s a t the wall 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  Temperature 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 heat 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  differences  i n t h e system 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 to the 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  differences at the centre  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 c a n 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 reduced.  This trend i s also observed 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 here. 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  earlier,  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 dissipation effects.  From t h i s t a b l e i t c a n 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  Concentric Annuli  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 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 s f o r the three cases o f the annular flow. 6.2.1  Solution Details 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 b o u n d a r y 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 . T h e s e 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 problem (M>0) was s o l v e d by R u n g e - K u t t a  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 R u n g e - K u t t a 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 approximates  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  passages.  T h e s e 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 a g r e e m e n t . 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  fields  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  Velocity Field 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 , the v e l o c i t y g r a d i e n t s near the outer wall (heated 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 n e 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  heated,  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 c a n 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 are increased with flow reversal occuring a t Ra=28xl0 .  wall) Figure  4  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 . F o 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.  This figure  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 the w a l l s i n c r e a s e .  both  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 o r A=0.5.  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  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  here  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  Temperature 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  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 M=0.  p r o f i l e s f o r outer wall  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  differences  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 rthe  c a 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 c a n 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 b o t h 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 c o n v e n ient to present the r e s u l t s g r a p h i c a l l y and, t h e r e f o r e , a general 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 pectively.  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 c a n 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 independent o f the temperature f i e l d .  However, i t c a n be seen  that the temperature d i f f e r e n c e s are reduced.  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 p r o n o u n c e d .  The v e l o c i t y n e a r t h e  outer wall (heated w a l l ) i s reduced while i t i s i n c r e a s e d near 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 c a n 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 but the temperature d i f f e r e n c e s are reduced.  As Ra i n c r e a s e s ,  45 i t c a n 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 reduces t h e flow v e l o c i t y near the i n n e r wall (heated 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 b o t h 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 R a = 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 c a n 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  p r o n o u n c e d 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 o r t h i s c a s e t o o , i t can be seen t h a t l o w e r 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 viscous d i s s i p a t i o n i s taken i n t o account. 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 b o t h 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 f o r outer wall heated and inner wall i n s u l a t e d or f o r both w a l l s heated, high values of Nusselt numbers are obtained by i n c r e a s i n g x whereas from Figure 16 i t can be seen that f o r inner wall heated, outer wall i n s u l a t e d the Nusselt number values are reduced. A comparison of the reduction i n Nusselt numbers f o r 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 f o r the  case of inner wall heated, outer wall i n s u l a t e d and the minimum reduction occurs for the case of both w a l l s 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 phenomena 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 near t h e heated w a l l ( s ) thereby c o u n t e r a c t i n g the 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 shear s t r e s s a t the w a l l ( s ) i s reduced.  Viscous 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 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 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 . Exact Viscous DissipaVelocity t i o n ParaV meter M  Solution  Power S e r i e s Methoc  G a l e r k i n ' s Method  Runge-Kutta Method  i  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  Temperature Difference * -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  Rayleigh Number Ra  Velocity Temperature V Difference  TempVelocity erature V Difference •  Velocity Temperature V Difference  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 Rayleigh Number Ra  Viscous Dissipat i o n Parameter  •  —T  Exact Solution  Power S e r i e s Method  Galerkin's Method  Runge-Kutta Method  M  1  0 0.0001 0.0005  4.3743  4.3734 4.3653 4.3329  4.3742 4.3665 4.3354  4.3713 4.3633 4.3308  10  0 0.0001 0.0005  4,4688  4.4679 4.4591 4.4237  4.4689 4.4604 4.4267  4.4658 4.4568 4.4214  50  0 0.0001 0.0005 0 0.0001 0.0005  4.8735  4.8721 4.8542 4.8105  4.8735 4.8619 4.8156  4.8694 4.8572 4.8079  5.3429  5.3407 5.3181 5.2552  5.3428 5.3270 5.2633  5.3375 5.3204 5.2519  500  0 0.0001 0.0005  7.9516  7.9445 7.8782 7.6122  7.9518 7.8925 7.6538  7.9369 7.8705 7.6040  625  0 0.0001 0.0005  8.4911  8.4827 8.3998 8.0665  8.4934 8.4196 8.1228  8.4739 8.3908 8.0579  100  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 R u n g e - K u t t a 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 i u s R a t i o 0.5  N u s s e l t Number Nu Rayleigh Number  Case I : O u t e r Wal' H e a t e d , I n n e r WaV I n s u l ated.  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 ated.  Case I I I : Both W a l l s H e a t e d .  Exact Solution  RungeKutta Method  Exact Solution  RungeKutta Method  Exact Solution  RungeKutta 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  Ra  TABLE IV V e l o c i t y D i s t r i b u t i o n and T e m p e r a t u r e D i f f e r e n c e s due to Viscous Dissipation E f f e c t s f o r C o n c e n t r i c Annulus 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 = l , x=0.75  Dissipation M= 0.0 Radius R 0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 0.97 1.0  Velocity V 0.0 0.5625 0.9893 1.2852 1.4543 1.4873 1.4271 1.2376 0.9348 0.5214 0.0  Parameter  M= 0.0003  Temperature Difference -0.1517 -0.1514 -0.1493 -0.1442 -0.1350 -0.1215 -0.1036 -0.0816 -0.0563 -0.0286 0.0  Velocity V 0.0 0.5625 0.9894 2852 4543 4873 4271 2376 0.9348 0.5214 0.0  M= 0.0005  Temperature Difference A  0.1451 0.1450 0.1434 0.1388 1304 1175 1003 0790 0543 0275 0  Velocity V  Temperature Difference  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 T e m p e r a t u r e D i f f e r e n c e s due to Viscous 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 Annulus 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 Parameter M  -  M=0.0003  M=0.0  Radius R  TemperV e l o c i t y aDit ufrfee r - V e l o c i t y V V 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 0.3472 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  Temperature Difference • -0.1287 -0.1286 -0.1275 -0.1243 -0.1183 -0.1085 -0.0945 -0.0760 -0.0533 -0.0273 0.0  M=0.0 005 TemperV e l o c i t y aD ti uf rf ee r V ence <i> 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 ANNULUS WITH INNER WALL HEATED, OUTER WALL INSULATED FOR R a = l , A=0.75  D i s s i p a t i o n Parameter M M=0.0  Radius R  0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 0.97 1.0  M=0.0003  M=0.0005  TemperV e l o c i t y aD ti uf rf ee r V ence <J>  TemperVelocity ature DifferV ence •  TemperV e l o c i t y aD ti uf rf ee r V ence •  0.0 0.5629 0.9899 1.2857 1.4546 1.4875 1.4269 1.2372 0.9343 0.5210 .0.0  0.0 0.5630 0.9900 1.2859 1.4548 1.4877 1.4271 1.2374 0.9344 0.5211 0.0  0.0 0.5627 0.9896 1.2854 1.4542 1.4871 1.4265 1.2369 0.9341 0.5209 0.0  0.0 -0.0208 -0.0401 -0.0570 -0.0710 -0.0820 -0.0901 -0.0953 -0.0982 -0.0993 -0.0995  0.0 -0.0194 -0.0375 -0.0535 -0.0667 -0.0770 -0.0843 -0.0888 -0.0910 -0.0917 -0.0917  0.0 -0.0184 -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 ANNULUS WITH INNER WALL HEATED, OUTER WALL INSULATED FOR Ra=TOOO, X=0.75  D i s s i p a t i o n Parameter M M=0.0  Radius R  Velocity V  Temperature Difference •  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  M=0.0003 -  M=0.0005  TemperVelocity ature DifferV ence  TemperVelocity ature DifferV ence •  0.0 0.6820 1.1341 1.3964 1.5031 1.5086 1.3534 1.1332 0.8308 0.4521 0.0  0.0 0.6768 1.1274 1.3908 1.5000 1.5067 1.3564 1.1382 0.8364 0.4561 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 -0.0185 -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 R a = l , X=0.75  D i s s i p a t i o n Parameter M M=0.0  Radius R 0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 0.97 1.0  M=0. 0003  Temperature Velocity Difference V •  0.0 0.5632 0.9900 1.2853 1.4539 1.4868 1.4265 1.2373 0.9349 0.5217 0.0  0.0 -0.0520 -0.0969 -0.1308 -0.1511 -0.1566 -0.1472 -0.1241 -0.0895 -0.0467 0.0  M=0.0005  Temperature Velocity D i f f e r - i/elocity ence V V  Temperature Di f f e r ence  0.0 0.5632 0.9900 1.2853 1.4539 1.4867 1.4265 1.2373 0.9349 0.5217 0.0  0.0 -0.0511 -0.0955 -0.1293 -0.1495 -0.1550 -0.1456 -0.1226 -0.0882 -0.0458 0.0  0.0 -0.0515 -0.0961 -0.1299 -0.1502 -0.1557 -0.1463 -0.1232 -0.0888 -0.0462 0.0  0.0 0.5630 0.9896 1.2849 1.4534 1.4863 1.4269 1.2369 0.9346 0.5215 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  Radius R  0.75 0.77 0.79 0.82 0.84 0.87 0.89 0.92 0.94 • 0.97 1.0  M=0.0  M=0.0003  M=0.0005  Temperature Velocity DifferV ence  Temperature Velocity DifferV ence  Temperature Velocity DifferV ence •  0.0 1.2565 1.3567 1.0294 0.7093 0.6360 0.7721 1.1313 1.4545 1.3050 0.0  0.0 0.0 -0.0483 1.2510 -0.0835 1.3537 -0.1047 1.0306 -0.1151 0.7135 -0.1180 0.6409 -0.1146 0.7769 -0.1036 1.1333 -0.0817 1.4523 -0.0463 1.3002 0.0 0.0  0.0 0.0 -0.0472 1.2475 -0.0822 1.3517 -0.1032 1.0313 -0.1136 0.7162 -0.1165 0.6440 -0.1131 0.7799 -0.1021 1.1346 -0.0803 1.4509 -0.0453 1.2972 0.0 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 FIGURE 3  R  V e l o c i t y P r o f i l e s f o r Concentric Annulus with Outer 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  0.5  J  I  1  -I  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 Annulus with Outer 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 Annulus with 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 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  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 A n n u l u s w i t h Both W a l l s Heated f o r R a d i u s R a t i o 0.25  0.9  1.0  62  0.5 FIGURE 8  0.6  0.7  0.8  DISTANCE  R  0.9  1.0  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 R a d i u s R a t i o 0.5  63  0.4  0-6  DISTANCE FIGURE 9  0.8  R  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 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  1.0  FIGURE 10 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 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  DISTANCE FIGURE 11  0.8  R  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  1.0  i  1  r  M=0  0.5  0.6  0.7  0.8  0.9  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 Annulus 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 •  1.0  67 T  M=0  0.4  0.6  DISTANCE FIGURE 13  0.8  R  Temperature P r o f i l e s f o r C o n c e n t r i c Annulus with Both W a l l s Heated f o r R a d i u s R a t i o 0.25  1-0  FIGURE 14  Temperature P r o f i l e s f o r C o n c e n t r i c Annulus with Both W a l l s Heated f o r R a d i u s R a t i o 0.5  10.0 -  0.0001  0.0003  DISSIPATION PARAMETER M FIGURE 15  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 Annulus 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 X=0.25  Ra=iooo  X=0.5  30.0  LU CD  20C  Ra=iooo  LU  CO CO 10.0  1.0 0.0  0.0001  0.0003  DISSIPATION PARAMETER M 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 Annulus 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  0.0005  71 X=0.25 X= 0.5  Ra=i000 10d  cr  LU  Ra=500  CD 9.0  LU  CO CO  Ra=l 8.0  7.0 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 " , A c a d e m i c 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 . V o l . 13, No. 6, pp. 1175-1180 (1967) s  4.  Cheng, K.C. and G. J . Hwang, " L a m i n a r 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 . C h . 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 , " A p p r o x i m a t e 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 L a m i n a r 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 in a Vertical Pipe", University of California (Berkeley) 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 227240 (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 from 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 C h 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 a n d 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 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 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)  Book  H e a t i n g " , The Ronald  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 P r o b l e m 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 S t u d y o f a Heat T r a n s m i s s i o n P r o b l e m o f a C h a n n e l - G a s 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., " 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 " , 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 a s shown b e l o w , (A-l) where  q = average heat f l u x T  w  = temperature o f the wall  T  b  = bulk temperature o f the f l u i d .  The b u l k t e m p e r a t u r e c a n be w r i t t e n a s ,  X  =  J J T \ ^ A / / K C ( A  (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  e n e r g y b a l a n c e , we o b t a i n ,  Cuc A F  (T -T.) 4  ^^PAZ+M[//gap/iJAz,  (A-4)  where T-| and 1^  a r e  the bulk temperatures 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 of the duct. 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 , 9Z  fl= ec ukC  - ZML'ffdvfzdll.  F  (A-5)  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 ,  Nu = k L  e c  r  u ^  C  - ^ j ^  + gM'/f^^/e '[<J> VRdR / J V R^R  m  '  ^  .  (A 7)  (A  "  8)  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 b e l o w , 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 b y , J\  where  =  Zr  0  (i-  X) x  (B-2)  ,  r 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 Q  ratio - j / r  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 ,  CCpUA^-T,).  where P i s t h e h e a t e d  %P l&  +JL[ti(*$di\l&  ,  (B-3)  perimeter.  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 obtain,  79 i  Nu =  k  v  4  0  _  j Ug/  J  (B-5)  k  .eCpUA  c - ^ t u * l/dvfRM  (B-6)  - 1 + 8('-^*)M/^) (UK  Case I I :  (B-7)  Inner Wall H e a t e d , O u t e r Wall I n s u l a t e d  N u s s e l t Number  f \ | _ kbu _ u  Dk  9/  (B-l)  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-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 obtained,  eCpUfifo-T,),  l?LSI  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  A2.  (B-9)  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-ll)  A  (B-13)  Case I I I : Both W a l l s Heated  = U>  Nu  = A  -4-  •  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 D  h  = 2r  Q  (B-D by,  (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 heat f l u x a t i n n e r and o u t e r w a l l  respectively. 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 is obtained,  k  N  (B-16)  k  k  - 1  ? uhc c  F  -AMI  •+  '{  \(&)\JLR\  (B-17)  (B-18)  (pvRcLK/'fvRjLg  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 a s g i v e n b y (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 c a n b e 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 b y [ 1 ] ,  C(m +v  r  V3t  dVz+v^dVz  sr r  s© 8 v  For steady flow  +vk-m) = -2t + u,[ii(rm / 35 f [ r a n 3rJ 3  H  z  j^- = 0 and because 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 .  For f u l l y  developed  3V  l a m i n a r f l o w , v,. r = -o zr - — = 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 o f Z, e q u a t i o n ( C - l ) r e d u c e s t o  °--~^ ^&^rW) fe ±C  +  -'  (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 , the negative s i g n before g i s taken. z  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 ,  ?c $r„k r+ l  t  where  V  a- x $ T £ Ik , . L  +J  +  = I n t e r n a l heat generation source  Dt  (c  4)  energy  $ = Viscous d i s s i p a t i o n function * i s g i v e n by [ 1 ] ,  (C-5) R e - w r i t i n g (C-4)  Pr  i n an expanded form we h a v e ,  {21+ Vy-2E+ )& 2 r + t f z 2 n - K n 2 (r?T\+± 21". (C-6) 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 r i g h t hand s i d e o f e q u a t i o n (C-7).  t h e l a s t two terms  E q u a t i o n (C-7)  on  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 the c o e f f i c i e n t of compression  work t e r m 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-  ature r i s e i n the flow d i r e c t i o n , the compression  work term can be  neglected. Thus e q u a t i o n (C-7) r e d u c e s t o * 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  (C-9)  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 tempera t u r e makes t h e p r o b 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 c u l t to solve.  diffi-  Hence f o r t h i s r e a s o n , t h e p r e s e n t s t u d y d e a l s w i t h  constant p r o p e r t i e s except f o r the v a r i a t i o n o f d e n s i t y i n the 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 s y s t e m s h o u l d be s m a l l s i n c e a l l t h e physical p r o p e r t i e s a r e a f u n c t i o n o f temperature.  Moreover, the  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 the d u c t l e n g t h . LIMITATIONS OF THE RADIUS RATIO FOR ANNULUS F o r t h e c o n c e n t r i c a n n u l u s , t h e 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 t h e 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  the 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 obtained. 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 to 0.75.  The range o f x I f the annular  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 remain 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 .  constant  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 system a p p r o x i m a t e s  almost to flow  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  developed  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 not of i n t e r e s t f o r the present study.  

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}]}"
                            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:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080680/manifest

Comment

Related Items