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Conjugate natural convection between two concentric spheres. Lau, Meng Hooi 1971

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CONJUGATE NATURAL TWO  CONVECTION  BETWEEN  CONCENTRIC SPHERES  .  . by LAU MENG HOOI  A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS .  accept  required  this  SCIENCE  t h e Department o f  Mechanical  We  FOR THE DEGREE OF  MASTER OF APPLIED in  OF  thesis  Engineering  as c o n f o r m i n g  to t h e  standard  THE UNIVERSITY  OF BRITISH COLUMBIA  November,  19 71  In p r e s e n t i n g requirements British freely  available  I agree t h a t f o r reference  permission  scholarly  in partial  f u l f i l m e n t of the  f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f  Columbia,  agree t h a t for  this thesis  the L i b r a r y  shall  and S t u d y .  I further  f o rextensive  copying of t h i s  thesis  p u r p o s e s may be g r a n t e d b y t h e Head o f my  Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood  that  copying or p u b l i c a t i o n  gain  s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n  of this thesis  f o rf i n a n c i a l  LAU MENG HOOI  Department of,-Mechanical The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date  make i t  /<?7/  Engineering Columbia  permission.  ABSTRACT  T h i s work c o n s i d e r s t h e c o n j u g a t e c o n v e c t i v e h e a t transfer  between a s p h e r e c o n t a i n i n g h e a t  s o u r c e s and a  c o n c e n t r i c envelope maintained at a s p e c i f i e d temperature. essentially  The s p a c e  between t h e two i s f i l l e d  incompressible f l u i d .  rotationally  symmetrical  p l a c e o v e r t h e gap w i d t h  free  leading  large  Two  case o f s m a l l c o n d u c t i v i t y  leading  ratio  of conductivities,  conductivity, and o f t h e  to a constant  leads to the s o l u t i o n perturbation  radius  ratio  and P r a n d t l  The  number  parameters.  Streamlines,  isovorticity  c u r v e s and i s o t h e r m s  obtained f o r v a r i o u s combinations  velocity  cases, of  w i t h t h e G r a s h o f number as m a i n p a r a m e t e r .  appear as secondary  are  transport  are considered separately.  the governing equations through r e g u l a r  expansions  heat  interface;  The a n a l y s i s o f h e a t t r a n s p o r t of  to take  limiting  relative  t o an i s o t h e r m a l c o r e t o f l u i d  interface  l a m i n a r and  c o n v e c t i o n i s assumed  the core.  sphere o f i n f i n i t e l y  converse flux  Steady,  w i t h an  and c o n d u c t i o n i s t h e s o l e  mechanism c o n s i d e r e d i n s i d e an i n n e r  constant  distribution  i s determined  of the parameters.  and b o t h  local  The  and o v e r a l l  v a l u e s o f t h e N u s s e l t number a r e o b t a i n e d . A flow v i s u a l i z a t i o n core  s u r f a c e temperature  mentally. is  found.  Reasonable  t e s t was u n d e r t a k e n  distribution  qualitative  and t h e  was d e t e r m i n e d  experi-  agreement w i t h t h e a n a l y s i s  iii  ACKNOWLEDGEMENTS  The sincere advice  author wishes  t o e x p r e s s h i s g r a t i t u d e and  thanks t o P r o f e s s o r  Zeev  Rotem f o r h i s g u i d a n c e and  throughout a l l t h e s t a g e s o f t h e program.  Special  t h a n k s a r e due t o D r s . E.G. Hauptmann and M. I q b a l s u g g e s t i o n s and d i s c u s s i o n s . thank  the e n t i r e s t a f f  P. H u r r e n during  the construction  and L. Dery  and o p e r a t i o n  Depart-  assistance.  J . Hoar and  for their  assistance  o f the experiment.  p r o g r a m was p r o v i d e d  by t h e N a t i o n a l  o f Canada t h r o u g h G r a n t No. 67-2772 f o r  thanks a r e due.  The a u t h o r i s g r a t e f u l f o r t h e award  of a U n i v e r s i t y o f B r i t i s h 1969 t o 1971.  for their  t h a n k s a r e due t o M e s s r s .  Support o f t h i s  which  Columbia,  (chief technicians)  Research C o u n c i l  the author wishes t o  of the Mechanical Engineering  ment, U n i v e r s i t y o f B r i t i s h Special  Also,  for their  Columbia  Graduate  Fellowship  from  iv  TABLE OF CONTENTS Page ABSTRACT  i i  ACKNOWLEDGEMENTS LIST  OF TABLES  LIST  OF FIGURES  i i i v i v i i  NOMENCLATURE  x  Chapter 1  INTRODUCTION  2  ANALYSIS  3  1 8  2.1  Formulation o f Conjugate  2.2  Governing  2.3  Method o f S o l u t i o n  2.4  Solution  2.5  Solution  Equations  Problem  . . . .  . •  !0  .  o f Constant F l u x Problem  8  . . . .  1  5  2  1  30  EVALUATION OF ANALYTICAL RESULTS  33  3.1  Range o f V a l i d i t y  34  3.2  Streamlines  38  3.3  Velocity Distribution  47  3.4  Vorticity  52  3.5  T e m p e r a t u r e D i s t r i b u t i o n and  3.6 4  . . . . .  of Solutions  Contours  Contours  59  H e a t - T r a n s f e r Rates  7-L  EXPERIMENT 4.1  Experimental Apparatus  8  8  3  3  V  Page  Chapter  5  4.2  Experimental  4.3  Experimental  96  Results  103  CONCLUSIONS  REFERENCES  . .  APPENDIX  I - Calculation Effect  APPENDIX  94  Procedure  of Viscous  Dissipation  I I - Conjugate Problem w i t h Inner Sphere C o n t a i n i n g D i s t r i b u t e d Sources . .  APPENDIX I I I - A B r i e f Review on S m a l l G r a s h o f Numbers N a t u r a l C o n v e c t i o n A b o u t a Heated Sphere .  vi  L I S T OF  TABLES  Table I II  Page O v e r a l l N u s s e l t Numbers f o r C o n j u g a t e 3 = 1.15, 2.0, and 3.0  Case, 82  E x p e r i m e n t a l Surface Temperature D i s t r i b u t i o n o f Two C o n c e n t r i c S p h e r e s , G = 1 . 3 2 x l 0  101  E x p e r i m e n t a l S u r f a c e Temperature D i s t r i b u t i o n o f Two C o n c e n t r i c S p h e r e s , G = 3 . 7 2 x l 0  101  E x p e r i m e n t a l S u r f a c e Temperature D i s t r i b u t i o n o f Two C o n c e n t r i c S p h e r e s , G = 5 . 7 8 x l 0  102  R a t i o o f V i s c o u s D i s s i p a t i o n Term t o Cond u c t i o n Term; R a t i o o f V i s c o u s D i s s i p a t i o n Term t o C o n v e c t i o n Term ( A i r )  109  4  III  5  IV  5  V  VI  R a t i o o f V i s c o u s D i s s i p a t i o n Term t o Cond u c t i o n Term; R a t i o o f V i s c o u s D i s s i p a t i o n Term t o C o n v e c t i o n Term (Water)  .  109  vii  L I S T OF FIGURES Figure 2.1.1 .3.1.1  Page Physical Configuration Approximate various  3.2.1  Upper Bound o f G f o r  a/3  37  S t r e a m l i n e s f o r C o n j u g a t e C a s e , to=10, 6=2.0, a=.72, G=10  3.2.2  3.2.4  42  Streamlines f o r Constant Flux 3=2.0, a=.72, G=103 Streamlines f o r Conjugate  Case, 43  C a s e , co=10, 44  3  S t r e a m l i n e s f o r C o n j u g a t e C a s e , to=10, 3=2.0, o= 10  3.2.6  co=10l^,  3  6=1.15, a=.72, G=10 3.2.5  41  3  S t r e a m l i n e s f o r Conjugate Case, 3=2.0, a=.72, G=10  3.2.3  10  G=72  45  S t r e a m l i n e s f o r Conjugate Case,  w=10,  3=2.0, a=.72, G=2100 3.3.1 3.3.2 3.4.1  Radial Velocity, V Angular Positions)  R  . . .  v s . Radius  (Various 50  T a n g e n t i a l V e l o c i t y , V Q v s . Radius (Various Angular P o s i t i o n s ) .  51  V o r t i c i t y Contours, Conjugate w=10, 3=2.0, o=.72, G=10  Case, 55  V o r t i c i t y Contours, Conjugate w = 1 0 , 3=2.0, a=.72, G=10  Case,  3  3.4.2  15  3.4.3  56  3  V o r t i c i t y Contours, Constant Flux 3=2.0,, o=.72, G=10 .  Case,  3  3.4.4 3.5.1  46  V o r t i c i t y Contours, Conjugate w=10, 3=2.0, o=10, G=72  57  Case,  Temperature D i s t r i b u t i o n , Conjugate Case,,'; co=10, 3=2.0, a=.72, G=103  58 62  viii Figure 3.5.2  Page Temperature D i s t r i b u t i o n o f Inner Sphere, C o n j u g a t e C a s e , UJ=10, 6=2.0, a=.72, G=10  63  3  3.5.3 3.5.4 3.5.5  Temperature D i s t r i b u t i o n , w=10, 3=2.0, a=10, G=72  Conjugate  Case, 6  Temperature D i s t r i b u t i o n , Conjugate w=10l5, 3=2.0, a=.72, G=103  4  Case, 65  Temperature D i s t r i b u t i o n , C o n s t a n t F l u x C a s e , 3 = 2.0, a=.72, G=10  66  3  3.5.6  Isotherms, Conjugate a=.72, G=10  Case,  Isotherms, Conjugate a=10, G=72 .  Case,  Isotherms, Conjugate a=.72, G=10  Case,  u)=10, 3=2.0, 67  3  3.5.7 3.5.8  w=10 , 3=2.0, 68 w=10l5, 3=2.0, 69  3  3.5.9  Isotherms, Constant F l u x Case, a=.72, G=10  3=2.0, 70  3  3.6.1 3.6.2 3.6.3  N u s s e l t Number A g a i n s t A n g u l a r P o s i t i o n , C o n j u g a t e C a s e , OJ=10, 3=2.0, a=.72, G=10  . .  77  N u s s e l t Number A g a i n s t A n g u l a r P o s i t i o n , C o n j u g a t e C a s e , oo=10, 3=2.0, a=10, G=72 . . .  78  N u s s e l t Number A g a i n s t A n g u l a r P o s i t i o n , C o n j u g a t e C a s e , w = 1 0 l , 3=2.0, a=.72, G=10  79  3  5  3  3.6.4  N u s s e l t Number A g a i n s t A n g u l a r P o s i t i o n , C o n j u g a t e C a s e , OJ=10 , 3=2.0, c=.72, G=1400 . .  8  0  3.6.5  O v e r a l l N u s s e l t Number as F u n c t i o n o f R a y l e i g h Number, C o n j u g a t e C a s e  4.1.1  Experimental Apparatus  8  4.1.2  Plexiglas  8.5  4.1.3  L o c a t i o n s o f Thermocouples I n n e r Sphere  Inner Sphere  81 4  on S u r f a c e o f 87  ix Figure  Page  4.1.4  Support  Stem w i t h G l a s s H e m i s p h e r e  4.1.5  Top V i e w o f S u p p o r t Hemisphere Thermocouple  88  Stem and G l a s s 89  4.1.6  Typical  Calibration  4.1.7  Platinum-Wire Temperature-Sensing C a l i b r a t i o n Curve  Curve  . . . .  91  Probe, 92  4.1.8  Layout o f Experimental Apparatus  4.4.1  Smoke P a t t e r n  a t G = 1 . 3 2 x l 0 , AT=.024°C  . . . .  4.4.2  Smoke P a t t e r n  a t G = 3 . 7 2 x l 0 , AT=7.21°C  . . . .  9  4.4.3  Smoke P a t t e r n  a t G = 5 . 7 8 x l 0 , AT=10.3°C  . . . .  99  4  5  5  93 97 8  X  NOMENCLATURE  A(R,6,<f>)  dimensionless  B(R')  heat  c  specific  P .  c  V E  source  heat  source  function per unit  heat  o f the f l u i d  , E  4  volume  at constant  R K .-AT _— f — ref , 2  number,  6  Grashof  g  acceleration  k  thermal c o n d u c t i v i t y  Nu^  l o c a l N u s s e l t number  " o  v  . . operators 3  Nu  number,  (  gyAT  r e f  2  R'  / v  of gravity  / 3-1 ,• ~3~> X  f o r the inner  30  l  Tm  l o c a l N u s s e l t number i  3-1 \  sphere,  i • R  =  1  f o r the outer  80  1  Q  sphere,  2  K-p Q  r e f e r e n c e r a t e o f heat  flux 3  Ra  R a y l e i g h number,  R  radial  T  (eq.  pressure viscous dissipation  2  f u n c t i o n ( e q . 2.2.3)  ^  Y  ^  T r  e  f  R  i  /  ^  v a  coordinate  temperature a.  T r e fr-  r e f e r e n c e t e m ptre r a t u r e ,r  V  dimensionless  velocity  Q x (3-D — 4TT3K^R' . f I  ^  B-l)  XI  a  thermal  diffusivity  6  radius ratio,  R' / R!  ' o ' 1 expansivity at constant  y e  ratio term  5  dimensionless  n  cos 0  pressure  of viscous dissipation  vorticity  term  vector,  t o conduction  [V x V ] ^  2 V  Laplace  operator, spherical  coordinates  0  angular  coordinate  0 X  dimensionless temperature r a t i o of viscous dissipation term  u  dynamic  v  kinematic  p  density of  a  P r a n d t l number,  cj)  circumferential  $ v  dissipation function  ¥  dimensionless  term  viscosity viscosity fluid v/a coordinate  (longitude)  c  stream  function, '  OJ  thermal  conductivity ratio,  Superscripts, subscripts ' refers to dimensional refers  to  average  f  refers  to  fluid  1 , ~  refers  t o inner  sphere  o  refers  to outer  sphere  ref s  to convection  /"refers t o r e f e r e n c e refers  V'/vR'.  k /k g  quantities  values  to s o l i d material  '  f  I  1  1.  The closed  study  spaces  has  recent years. play  a role  of thermally induced been r e c e i v i n g  The  reason  from  reactor  c o r e and  increasing  for this  insulation  which i s of i n t e r e s t Batchelor  description  o f such  in  rectangular cavity.  t h a t such  flows  of a c a v i t y  The  m o d e l he  at d i f f e r e n t  infinite  The  i n the  F l u i d motion i s generated  a problem  leads  differential  the  to  equations  cavity  that at a s u f f i c i e n t l y  number t h e  flow would  by  with  in a  two  vertical  a narrow  He  the  be  heat  also  h i g h v a l u e of the of constant  a continuous  and  i t i s assumed t o  cavity.  of a core  top  (the b r e a d t h ) .  Batchelor estimated  postulated  of  doubly-glazed  i s c l o s e d a t the  direction  f o r v a r i o u s flow regimes i n the  surrounded  the problem  c o n s i d e r e d had  by b u o y a n c y and  consist  by  flow regimes  temperatures,  third  two-dimensional.  temperature  between a n u c l e a r  Further-  motivated  analytically  a i r - s p a c e between them.  and  over  windows.  of b u i l d i n g s ( p a r t i c u l a r l y  windows) i n v e s t i g a t e d  flux  en-  itself.  [1], o r i g i n a l l y  insulation  essentially  attention  space  of doubly-glazed  a set of n o n - l i n e a r coupled p a r t i a l  b o t t o m and  in  i t s pressure v e s s e l to advection i n lakes  more, t h e m a t h e m a t i c a l  walls  motion  seems t o be  c o n v e c t i o n i n the a n n u l a r  to thermal  thermal  fluid  i n a wide a r e a o f t e c h n o l o g i c a l a p p l i c a t i o n ,  ranging  and  INTRODUCTION  boundary  Rayleigh vorticity layer.  Poots  [ 2 ] , u s i n g a doubly  function  and  orthogonal expansion  the temperature,  conclusions previously arrived Subsequent i n v e s t i g a t o r s  [3],  Elder  [4  ] and  de V a h l D a v i s  various  finite difference  In  [3]  both  and  a problem. retention boundary  [4  conditions.  iately  The  approach s i m i l a r p r o b l e m by at  imately  found  for this  At  sufficiently  of  a solid  et  a l . [6,7,8,9,10]  natural  core  reviewed  inside  convection of  cylinders  and  to achieve  the  indeed one  the  numerical  setting  the  boundaries  i s not  overcame t h e from  was  unknown i n t h e  immed-  i n an instability  stream  large values  but not  t o o as p r e v i o u s l y p o s t u l a t e d by papers  a p p a r e n t l y from  assumption  ] t h a t t h e r e was  results.  scheme  a t the h o r i z o n t a l  the v o r t i c i t y  constant v o r t i c i t y ,  The  Churchill  They a l l e m p l o y e d  t o a p p r o x i m a t e by  to that of Wilkes,  [5  and  t h e o t h e r hand de V a h l D a v i s ,  calculating  the boundary.  number, he  On  ].  Elder, i n order  justification  obvious.  [5  as an e x p l i c i t  o f h i s scheme, h a d  zero.  are Wilkes  o f the n u m e r i c a l  normal g r a d i e n t of the v o r t i c i t y to  verified  at.  d i f f i c u l t i e s arose  o f the v o r t i c i t y  stability  t o some e x t e n t  schemes t o o b t a i n n u m e r i c a l  ] instability  Wilkes'  stream  computed B a t c h e l o r ' s model f o r  s e v e r a l v a l u e s o f R a y l e i g h number and the  f o r the  function  of the  Rayleigh  a core of  of constant  approxtemperature  Batchelor.  above do  the e n c l o s e d  investigated  c o n s i d e r the  space.  the  f l u i d i n a gap  c o n c e n t r i c spheres,  not  cases  Recently of steady  presence Mack laminar  between c o n c e n t r i c  both  experimentally  and  3 analytically. temperature  The  cylinders  w i t h the i n n e r  Flow v i s u a l i z a t i o n data  f o r t h e two  reported  are kept at constant  c y l i n d e r o r sphere b e i n g the  f o r both  concentric  t h a t the  or spheres  configurations  and h e a t  i n t h e gap  Similar  cylinders.  Three  'kidney-shaped (c)  r e s u l t s were f o u n d  'falling  1  type  vortices';  (b) a  ratio  'crescent-eddy'  o f 3.14,  1.19.  Mack and  Mack and  Hardee  [10] o b t a i n e d a n a l y t i c a l  solutions  concentric  cylinders  coefficients number. was  i n powers o f t h e R a y l e i g h and  seem t o y i e l d  Their  convergent r e s u l t s  experimental studies r a t h e r higher, than e x p e r i m e n t a l and  is  to  Bishop [ 9 ] , i n the  number f o r  expansions  1600.  up  the  expansions  for concentric  spheres the  o f R a y l e i g h numbers u s e d i s  analytical results  comparison  i s perhaps  between  not  i n h e a t t r a n s f e r work t o f o r both  i n order to a r r i v e boundary  Prandtl  t o a maximum v a l u e o f  Thus d i r e c t  boundary c o n d i t i o n s  the  perturbation  solutions  form  (based on gap w i d t h ) , w h i l e i n t h e i r  the range  i s customary  flow v e l o c i t y  of their  analytical  R a y l e i g h number o f 1600  That  In t h e i r  (a) a  o f h i g h e r - o r d e r terms a l s o depend upon t h e  not given.  idealized  spheres.  A p r o o f of convergence  It  concentric  type  ratios  expansions  and  the  were o b s e r v e d ;  diameter  of s e r i e s  of  these correspond r e s p e c t i v e l y 1.72,  authors  two  f o r the case o f  types of flow-regimes  eddy  The  between t h e  c o n c e n t r i c s p h e r e s d e p e n d e d upon t h e d i a m e t e r spheres.  transfer  s p h e r e s were o b t a i n e d .  f l o w regime  hotter.  specify  the temperature  at a w e l l - s e t  conditions  meaningful.  are  and  problem.  usually  constant;  temperature,  constant f l u x o r a combination  ever, i n actual p r a c t i c e conducting enclosing  t h e boundary  of these.  How-  conditions a tthe  surfaces are r a r e l y  known b e f o r e h a n d .  They depend upon t h e c o u p l e d mechanisms o f c o n d u c t i o n i n t h e solid In  b o u n d a r y and c o n v e c t i o n o f t h e f l u i d  technologically  ('film') This  place  coefficient  variation  solid,  interesting  geometries  of  leads to a r e d i s t r i b u t i o n  i n the f l u i d .  The e f f e c t  transfer:  coupled but i n addition  and t h e s o l i d  attention.  flow around  HH  the term  'conjugate' f o r t h i s  analysis  the v e l o c i t y  of this  uncoupled  i n the  which  takes  conduction/  fields  fields  of the  Due t o t h a t  t y p e have o n l y r e c e i v e d  scant  o f p u b l i s h e d work.  c o n s i d e r e d two e x a m p l e s o f two d i m e n s i o n a l  a body w i t h l i n e  o f the s l i p - f l o w  the energy  i s a review  uniform.  i n free convective  a r e c o u p l e d as w e l l .  The f o l l o w i n g  Perelman  view  flux  n o t o n l y a r e t h e momentum and e n e r g y  c o m p l i c a t i o n problems o f t h i s  or  o f heat  o f the coupled  transfer  never  i n some b a l a n c e w i t h t h e c o n v e c t i v e m o t i o n  the f l u i d  fluid  the heat  at the surface i s v i r t u a l l y  c o n v e c t i o n mechanism i s most p r o n o u n c e d heat  over t h e boundary.  and p l a n e h e a t  s o u r c e s and u s e d  type o f problem.  profiles  In h i s  were assumed t o be e i t h e r  type, u n i n f l u e n c e d by c o n v e c t i o n .  assumption,  from t h e energy  t h e momentum f i e l d field.  linear  In  i n the f l u i d i s  Hence t h e s o l u t i o n  tothe  momentum e q u a t i o n i s known s e p a r a t e l y o f t h e c o n v e c t i o n o f heat.  Some t i m e  later  Rotem  [12] d e v e l o p e d  method f o r t h e e v a l u a t i o n o f i n t e r f a c e  an  approximate  temperature  profiles  and  the t r a n s f e r  coefficient  laminar boundary applies  f o r heat  layer with w a l l  t o the case Kelleher  conduction.  of uncoupled  and  Yang  f o r the conjugate problem over a two-dimensional  transfer  equations  to a  forced  Again  t h e method  only.  [13] e m p l o y e d a G o e r t l e r f o r the  free  convection of  c o n d u c t i n g body w i t h can  series  sources.  Here t h e v e l o c i t y  advance:  i t i s coupled to the temperature  internal  i n d e e d n o t be  fluid heat  specified  in  distribution  on  s u r f a c e o f t h e s o l i d , w h i c h i n t u r n d e p e n d s upon b o t h source d i s t r i b u t i o n  and  convection.  The  region of  the  the heat  interest  .e  of t h e i r  a n a l y s i s was Lock  vection tion  from  and  heat  Gunn  the  leading  [14] c o n s i d e r e d t h e  a downward p r o j e c t i n g  completed  by m a t c h i n g  edge.  laminar free  f i n , but  conduction p a r t s o f the problem  s o l u t i o n was and  and  p l a c e d near  s o l v e d the  separately.  the i n t e r f a c i a l  conconvec-  The  temperature  flux. The  study of a conjugate heat t r a n s f e r  problem  in a  * confined  space  i s apparently not a v a i l a b l e  Also, previous investigators problems,  both  of conjugate heat  i n e x t e r n a l and  in internal  considered the conjugate e f f e c t s The  present study  i n one  c o n s i d e r s steady  dimensional natural  i n the  convection  transfer  flows  o r two  conjugate  literature.  [ 1 5 ] , have  dimensions  only.  laminar three-  (with r o t a t i o n a l  symmetry) i n  * An e x c e p t i o n t o t h i s i s t h e r e c e n t a n a l y s i s o f Rotem [16] o f t h e c o n j u g a t e c o n v e c t i o n f o r t h e c a s e o f h o r i z o n t a l concentric cylinders.  6 a fluid  b e t w e e n two  c o n c e n t r i c spheres.  the q u a s i - l i n e a r coupled  governing  vorticity  t r a n s p o r t are obtained  expansion  o f the v a r i a b l e s stream  in  terms o f a s c e n d i n g  conductivity  ratio.  the expansion cases  The  flux  o r an  cases  the  limiting  e x p e r i m e n t was  into  The  experimental  the n a t u r e  analytical An  of the  convective  f o r the  which The  and  with  an  present  problem.  visualization  conjugate  provide flow  a p p l i c a t i o n which t h i s For  of  concentric  further insight supplement  the  convection  gravitational  field  a x i s as  value.  r e g i m e may t o be  be  a n a l y s i s w o u l d model i s  g l o b a l  convection occurs  number e x c e e d s a c r i t i c a l  vertical  thermal  i n the  conjugate  between two  v e c t o r a c c e l e r a t i o n of g r a v i t y  local  the  results.  t h e b o u n d a r y and  in  o f the  results  geophysical circulations. the  problem  in detail.  also considered  designed  f l o w p a t t e r n i n t h e gap  spheres.  The  i s o t h e r m a l i n n e r sphere  work:  An  asymptotic  number.  and  and  temperature  in this  i s evaluated  are  are  f u n c t i o n and  an  range o f each parameter over  i s o t h e r m a l outer envelope these  o f energy  form of  radius r a t i o ,  scheme i s v a l i d  of a constant  i n the  of importance  P r a n d t l number, t h e  solutions to  equations  powers o f t h e G r a s h o f  o t h e r parameters which are are the  The  geophysical  circulations  i s everywhere normal o n l y when a c e r t a i n However, i n s i g h t  g a i n e d by  to  to Rayleigh  the  c o n s i d e r i n g the  o r i e n t e d everywhere' p a r a l l e l  shown i n F i g u r e  :  2.1.1, i n t h i s  some g e o p h y s i c a l m o d e l s , s m a l l e r r e g i o n s  to  thesis.  of c l o s e d  the For  circu-  lation  are  not  d i s s i m i l a r to those considered  be  shown t o e x i s t  [ 7 ] , i n which  Another a p p l i c a t i o n i s to insulating  f l a s k to minimize the  containers  i t i s often shells,  not  concentric  and  bending s t r e s s , instead supports.  retained  between t h e  is  obtained  design loss:  of  a  spherical  for large  t h i s would i n t r o d u c e  spherical  buckling  problems  o f membrane s t r e s s e s , p a r t i c u l a r l y  Thus a s p h e r i c a l gap s p h e r i c a l envelope  f o r which the  would  f e a s i b l e t o have a vacuum between  the  n e a r the  as  conditions  here.  the  heat  the  present  filled and  analysis  a is  with  gas  is  configuration applicable.  8  2.  2.1  Formulation  of  ANALYSIS  Conjugate  The a n a l y s i s c o n s i d e r s three-dimensional  Problem  steady  natural convection  between two c o n c e n t r i c s p h e r e s . that  shown i n F i g u r e  2.1.1.  e i t h e r by a c o n s t a n t  source  sources.  conjugate  in a fluid  laminar contained  The p h y s i c a l c o n f i g u r a t i o n i s  A solid  i n n e r sphere  a t i t s c e n t r e o r by  I t i s c o o l e d by l a m i n a r  i s heated  distributed  n a t u r a l convection of the  fluid  inside  t h e gap b e t w e e n t h e c o n c e n t r i c s p h e r e s .  The  outer  sphere  i s kept  heat  transfer the  a t the i n t e r f a c e  fluid  inside  at a constant  depends upon t h e c o u p l e d  the i n n e r sphere  definition type.  between  the s o l i d  of heat  Thus  i n n e r sphere  mechanisms  and c o n v e c t i o n  the t r a n s f e r  Some p r e v i o u s  temperature.  of  and  conduction  i n the f l u i d .  By  i s t h e r e f o r e of the  conjugate  i n v e s t i g a t o r s [10] have c o n s i d e r e d t h e  natural  convective problem f o r the s p h e r i c a l c o n f i g u r a t i o n  without  the a d d i t i o n a l  along  the boundaries  The t e m p e r a t u r e the  i n n e r sphere  applications  complication of conductive  and t h i s work w i l l  and h e a t  follows:  be d i s c u s s e d  flux distributions  are of great importance  (i) determination  later.  on t h e s u r f a c e o f  i n the  f o r which the problem t r e a t e d here  The o b j e c t i v e s o f t h e p r e s e n t  effects  numerous i s a model.  investigation  o f the temperature  a r e as  and h e a t  flux  variations  on t h e i n n e r s p h e r e ;  a t i o n on t h e s u r f a c e  of the outer sphere;  of the temperature, v e l o c i t y the f l u i d  the a n a l y s i s  P r a n d t l number conductivity (d) 6, the  are  that  sphere  i n Figure  2.1.1.  assumed t h a t  o f the thermal  to that of the  A l lphysical properties  first  a r e assumed  This  assumption  i n Appendix I o f t h i s flow p a t t e r n  The l a t t e r  thesis.  It  dissipation  Hence t h e c o r r e s p o n d i n g t e r m s w i l l  checked  be  assumption i s  An a x i s y m m e t r i c a l  i s assumed i . e . t h e r e i s no a z i m u t h a l  Hence a l l q u a n t i t i e s c£> .  t h e apex  i n t r o d u c e d by B o u s s i n e s q .  from the energy e q u a t i o n .  longitude,  to that of  i n as f a r as i t s  i s concerned.  deleted  convective  fluid;  c o o r d i n a t e s a r e used, the  t h e c o m p r e s s i o n work and v i s c o u s  are a l s o n e g l i g i b l e .  the  (b) a, t h e  c o o r d i n a t e 8 b e i n g measured c l o c k w i s e from  i t s i m p l i c a t i o n s were  swirl.  o f importance  o f the r a d i u s o f the o u t e r sphere Spherical  lastly  spheres.  (c) GO, t h e r a t i o  inner  d e p e n d e n c e on t h e t e m p e r a t u r e  is  and  vari-  distributions of  the parameters  constant, except the d e n s i t y o f the f l u i d  and  flux  (a) G , t h e G r a s h o f number;  of the s o l i d  inner sphere.  as shown  and v o r t i c i t y  of the f l u i d ;  the r a t i o  angular  (iii)  i n t h e gap b e t w e e n t h e c o n c e n t r i c I t w i l l be shown  in  ( i i ) of the heat  a r e t a k e n t o be i n d e p e n d e n t o f  10  CONSTANT TEMPERATURE OUTER ENVELOPE  SOLID INNER SPHERE  CENTRAL HEAT SOURCE MATCHING OF TEMPERATURE AND FLUX AT SOLID/FLUID INTERFACE  Figure  2.2  Governing All  values)  c  Physical Configuration  Equations  dimensional  are primed  quantities sphere.  2.1.1  quantities  used  (except f o r p r o p e r t y  i n what f o l l o w s and a l l d i m e n s i o n l e s s  a r e unprimed. R = R'/R-  1  and  L e t R| be t h e r a d i u s o f t h e i n n e r 3 = R'/ ' o 1 R  a  ^ e the dimensionless  11 radial  c o o r d i n a t e and d i m e n s i o n l e s s r a d i u s  respectively. outer spheres  T and T a r e t h e t e m p e r a t u r e s respectively;  sphere  o f t h e i n n e r and  g, t h e a c c e l e r a t i o n o f g r a v i t y ;  Y, t h e v o l u m e t r i c c o e f f i c i e n t pressure;  o f the outer  o f thermal expansion  v, t h e k i n e m a t i c v i s c o s i t y ;  at constant  and a , t h e t h e r m a l  diffusivity. A r e f e r e n c e temperature are  introduced:  AT  ref  = AT  =  Q x  ref  and t e m p e r a t u r e  and  (3-D/(47T3R fK ) ;  where Q i s r a t e o f h e a t f l u x ,  f  assumed c o n s t a n t .  Let Ra  = gyAT  ref  i  d e n o t e t h e R a y l e i g h number;  a  the  Prandtl  G  the  Grashof  9 the  =  v / a  number;  =  gyAT  2 ref  i  number;  =  (T-T J/AT ref ' r e f  dimensionless f l u i d  c  temperature;  difference  12 0  =  (T-T  the dimensionless The  -)/AT ref ' ref  i n n e r sphere  equations  temperature.  of c o n t i n u i t y , motion  and  energy  Newtonian, i n c o m p r e s s i b l e c o n s t a n t p r o p e r t y f l u i d w i l l be  stated here;  stream  t h e y may  function  ¥ will  be  be  found  i n many t e x t b o o k s .  i n t r o d u c e d , so t h a t  continuity  is fulfilled  quantities  i n t r o d u c e d above t h e s t e a d y  tions  reduce  to the  identically.  for a not Stokes  the e q u a t i o n  Then i n t h e state  dimensionless  governing  equa-  following,  2  P U/ 4  =  L. R  9 ( ¥  '  ^  E  2  2 >(R,n)  + R  2  z  =  (l-n ) G  a  ~  R  V 0 2  The  =  ,  2  {1  n .  31 . i I I . 2  T] )  3n  ;  8  (  W  '  3(R,n)  0  (2.2.1)  )  (2.2.2)  -3(R,n)  A(R,9,(j))  .  (2.2.3)  boundary .conditions s u b j e c t t o which  h a v e t o be  R  9(Rxn,0)  0  +  V 0  2E f  +  solved w i l l  be  of  (2.2.1) t o  (2.2.3)  ^  3¥ 3R  =  =  0  at  R  =  p  1,  (2.2.4)  3m y  or,  =  0  w  as  =  alternatively  <F  Here  JQ  =  0  6  =  0,  at  n  =  ±  -(l- )I | I = o 2  n  at  R  =  (2.2.5)  =  39 9R  at  R  =  1  0  =  -1  at  R  =  3  cos 6  and the o p e r a t o r s  1  (2.2.5a)  1  30 3R  n =  7T  stated  ©  =  at  E , E 2  4  (2.2.6)  (2.2.7)  .  (2.2.8)  and V  are  2  defined  follows:-  E  +  2 =  (i-  2 n  3  )  3R  E  4  =  E  a  2  2  3n  2 [E^]  4(i-n ) 2  4  3R  R"  , 2 ( i-R  2 n  )  ,n R  3 _ 4  3R  3  .  3  3n  3  ) 3R3n  R  2  d-n )  3  R  3R  2  3n  4(i-  +  4  4  2 n  )  3  2  3n  2  14  v  =  2  a 9  =  E  I 2 d  2  R  2  R  4  2  +  The  2  ~^r-  8 R  |^  (n/R)  ^  while  the  The  third  - ^  2  n  (n/R)  effects, effects  I i t will  be  always a p e r m i s s i b l e Lastly, for  the  solid  For the  case  sphere,  the  except  The  first  be  no  flow  Conditions  | ^ )  .  term  two  effect.  represents  ( 2 . 2 . 2 )  convective  and  compression In  ( 2 . 2 . 2 ) .  shown t h a t f o r c l o s e d s p a c e s  this  is  assumption.  through  conduction  A(R,6,c)>) i s t h e source be  boundary c o n d i t i o n s  ensure  i s the  ( 2 . 2 . 3 )  of a s i n g l e heat  ( 2 . 2 . 5 )  equation  right-  of  r e p r e s e n t s the buoyancy  of viscous dissipation  equation  and  the  convection  the r i g h t - h a n d s i d e the  inner sphere.  along  represents  ( 2 . 2 . 1 )  terms on  g i v e the  ( 2 . 2 . 1 )  s o u r c e f u n c t i o n may i * a t the c e n t e r p o l e . The  n  the e q u a t i o n  work were n e g l e c t e d i n t h e e n e r g y Appendix  9  l e f t - h a n d s i d e o f the e q u a t i o n  conduction  effects.  the  2 n a_.  2  §=;  +  of v o r t i c i t y .  hand s i d e o f t h e e q u a t i o n  The  3  l e f t - h a n d side of  the d i f f u s i o n  vorticity  3  . (l-n )  the  a t the  set to  ( 2 . 2 . 4 )  source  flow  function:  c e n t r e o f the  zero  everywhere  state  that there  s u r f a c e s o f the  t h a t the  equation  is  two  inner  will  spheres.  axisymmetrical  * The c a s e o f t h e i n n e r s p h e r e c o n t a i n i n g u n i f o r m l y d i s t r i b u t e d heat sources i s c o n s i d e r e d i n Appendix I I .  15 about  the v e r t i c a l  Q = 0 , IT ( i . e . a t  axis  Boundary c o n d i t i o n s  (2.2.6) and (2.2.7) s t a t e  conservation p r i n c i p l e , f l u x on b o t h  t o a problem  normalization  2.3  i . e . the e q u a l i t y  sides of the interface  t h a t both temperature lead  n = ± 1). the energy  o f temperature  solid/fluid.  The  and f l u x a r e s p e c i f i e d w i l l  of the conjugate type.  condition  and  fact  be seen t o  (2.2.8) i s a  f o r the temperature.  Method o f S o l u t i o n In  the present a n a l y s i s  i t i s assumed t h a t t h e i n n e r  sphere has a c e n t r a l h e a t source o n l y i . e . in  equation  be  i n e q u i l i b r i u m when t h e r e i s a n o n - z e r o  difference  (2.2.3).  A(R,0,(f>)  i s zero  In s p h e r i c a l geometry the f l u i d  between t h e i n n e r s p h e r e  cannot  temperature  and t h e b u l k o f t h e f l u i d ,  no m a t t e r how s m a l l t h a t d i f f e r e n c e may be as t h e v e c t o r acceleration  of gravity  to  fluid  of  t h e buoyancy  be  a t r e s t , b u t buoyancy  which  interface.  i s everywhere  after  only  i n the f l u i d . forces w i l l  I t s h o u l d be n o t e d t h a t  equations inner  boundary  Initially cause  conditions  may  The g o v e r n i n g  then apply t o t h i s  steady  the convective flow i s the  available.  respectively  component  c o n v e c t i v e motion  As m e n t i o n e d  (2.2.2) and (2.2.3) f o r t h e f l u i d  sphere  solid  the f l u i d  become q u a s i - s t e a d y .  equations w i l l  source o f v o r t i c i t y  energy solid  forces  to the  i s due t o t h e t a n g e n t i a l  some t i m e w i l l  time-independent motion.  This  inclined  above, t h e and t h e  are coupled through the  (2.2.6) and ( 2 . 2 . 7 ) .  On t h e o t h e r h a n d  16 the  vorticity  the  fluid  0,0  As  only,  (2.2.1),  the motion  o f the f l u i d  i t will  natural  (2.2.2) .-and  i n terms  This  be  assumption  characterizing  variables,  t o seek  e . g . Ra.  The  3  G r a s h o f numbers.  obviously  3>1;  becomes a s i n g l e this  latter  lined w i l l  the  inner  envelope  the range  V,  to  number.  suitable viscous of the  the problem  dealt  some o t h e r  are obtained f o r a  v a l u e o f G, a, O J and  of radius  sphere  range  combinations of  various  3 are considered. ratio  as 3 t e n d s t o i n f i n i t y  There  admissible:  the i n n e r  i n an unbounded e x p a n s e  case the r e g u l a r p e r t u r b a t i o n  sphere  of f l u i d .  t h e o r y t o be  To  out-  [19,20].  t h e o t h e r hand, i f oo t e n d s t o z e r o o r  s p h e r e has  either  respectively.  singularity  any  scaling  involving  solutions  F o r each  not apply  On  o f buoyancy  [18] . M o r e o v e r ,  v a l u e s o f the o t h e r parameters on  variables  OJ .  and  perturbation  a limitation  to equations  T h a t i s , an e x p a n s i o n o f t h e p r e s e n t  w i t h h e r e i s a m u l t i - p a r a m e t e r one a,  that  with suitable  type i s not n e c e s s a r i l y unique  parameters,  buoyancy  i n the Grashof  i n the sense  the r a t i o  term i n  solutions  (2.2.3) f o r t h e d e p e n d e n t  f o r c e s might have been used  is  equation for  i s s e t - u p by  of p e r t u r b a t i o n expansions  i s an ad hoc  parameter  of  the energy  are a l s o c o u p l e d through the buoyancy  (2.2.1). forces  t r a n s p o r t e q u a t i o n and  and must be  The  a c o n s t a n t f l u x o r an  isothermal  former o f these cases leads to  solved  h e a t t r a n s f e r between two  infinity  separately.  concentric  isothermal  The  case of spheres  has  17 b e e n s o l v e d a n a l y t i c a l l y by 2.5 an  the  case o f  isothermal  present  no  V/a  envelope  (the s o l u t i o n f o r w h i c h had  has  follows:  difficulties.  than  However, as  temperature p r o f i l e  1  /v  with not  may  be  parameter p e r t u r b a t i o n as  the  dimensionless  not  values a  °° t h e  a l l boundary  fulfilled. starting  with  stream f u n c t i o n ,  used. It  (2.2.2) and  i s proposed here to solve equations (2.2.3) f o r c o n j u g a t e  p a r a m e t e r s G,  a,  3 and  GO i n f i n i t e l y  neither of  s m a l l or very  r e g u l a r p e r t u r b a t i o n expansions f o r the assumed t o  (2.2.1),  natural-convective  t r a n s f e r between c o n c e n t r i c s p h e r e s w i t h  are  small  t e n d s t o become s i n g u l a r and  a different  t o be  sphere s u r f a c e  considered.  i n f l u e n c e o f a i s as  f o r the  rather  section  outer  energy equation  Therefore  In  flux  be  inner  [10].  constant  particular  conditions  Hardee  the  been a v a i l a b l e ) w i l l The  Mack and  heat the  large.  v a r i a b l e s ¥,0  and  The 0  be oo  00  (2.3.1)  00  00  (R, 1=0  00  m=0  n)  (2.3.2)  00  (2.3.3)  0 1=0  m=0  18 I t may  be  shown t h a t f o r t h e c a s e  t r a n s c e n d e n t a l i n the expansion they  invariably  give r i s e  considered here,  p a r a m e t e r G do n o t  Thus s u c h  t e r m s w o u l d a t most be  complex e i g e n m o t i o n s  which are not  S u b s t i t u t i n g equations into 0 and  the g o v e r n i n g e q u a t i o n s 0 are  ¥  =  any  further.  (2.3.3)  and  f o r *F,  the f o l l o w i n g expansions  0°  0°  (R)  + Ga0^  2  (R)  2  2  2  n)  R  2  Equating c o e f f i c i e n t s  2  (2.3.4)  (R,n)  + G a02  (R,n)  2  (R,n)  + Ga0^  + G a 0  reduce  + G a^2 ( ,  ,  2  differential  (2.3.1), (2.3.2)  2  + G a 0  equations  trivial  associated with  c o n s i d e r e d here  (R,n) + G ¥° (R,n)  G\ +  0  only a  obtained.  =  0  occur;  t o homogeneous e q u a t i o n s w i t h homo-  geneous b o u n d a r y c o n d i t i o n s w h i c h w i l l y i e l d solution.  terms  (2.3.5)  ,  (R,n)  + G a02  (R,n)  2  (R,n)  (2.3.6)  o f e q u a l powers o f G a  t o an  equations,  J  , the  i n f i n i t e s e t of uncoupled  governing  linear  0  (2.3.7)  (2.3.8) 3-(Rxn,0 )  9  o  (2.3.9)  3(R,n)  (2.3.10) ! R  3(^ /6 ) 1  0  (2.3.11)  3(R,n)  2  i R  2  1  3R  3n  i _  <JI  2E¥  +  9n  l-n  R'  2.3n  a 3R  4 °  R 3n  1  (2.3.12)  (2.3.13)  (2.3.14)  3(^ /  a  R  0 )  2  2  0  (2.3.15)  3(R,n)  (2.3.16)  1  3<Y°,  R  3(R,n)  oj)  3<^,  a  6°) (2.3.17)  R  3(R,n)  20 The above s e t o f e q u a t i o n s the  f o l l o w i n g boundary  0^  i s solved  i n sequence s u b j e c t  conditions:  0,  at  R  =  1  (2.3.18)  -1  at  R  =  3  (2.3.19)  must  n o t have a s i n g u l a r i t y  inner 1/R,  sphere g r e a t e r  a t the centre  than the zeroeth  of the  term,  f o r 1, m > 1  .m  at  m 0'  at  R  =  1;  R  =  3R  k  (2. 3.22)  30 m at  3R  R  =  1;  1, m > 1  j  (2.3.21)  3;  1, m _> 1 m  namely (2.3.20)  1, m > 1  90  to  3^ 1 3R  at  2  i  3  V  i  at  1, 3;  R  =  j,  k > 1  n =  (2.3.23)  (2.3.24)  ±1;  j , k > 1  (2.3.25)  21 2.4  Solution  The e q u a t i o n s  (2.3.7) and  They a r e t h e c o n d u c t i o n e q u a t i o n s and f o r t h e s o l i d  inner  ature d i s t r i b u t i o n conditions  sphere.  o  E ¥° 4  function  The s o l u t i o n s  f o r pure conduction  =  ( i - 1) x  Upon s u b s t i t u t i o n  is a  f o r the f l u i d  subject  first.  assumed i m m o b i l e , a r e t h e tempert o boundary  (2.3.18) and ( 2 . 3 . 1 9 ) .  0°  This  (2.3.8) a r e s o l v e d  1  1  =  (^y)  of  (l-n )/R x 2  creeping  (2.4.2)  (2.4.2) i n t o  (^j)  flow' equation.  (2.3.9),  ,  (2.4.3)  A solution  i s stream  t  ¥°  =  (1-n ) (~ 2  + B R X  + C^R  2  -• | i  +  R )X(^ ) 4  D ] L  T  (2.4.4) which has t o s a t i s f y  the boundary c o n d i t i o n s  c o n s t a n t s o f i n t e g r a t i o n A^, by a p p l y i n g c o n d i t i o n s form,  B-^ ,  and  (2.3.25).  The  are determined  (2.3.24) and a r e g i v e n h e r e i n c l o s e d  22  h  =  (6  -  9  B  1  =  (-3B  C  ±  =  (23  =  (23  D  1  46  + 86  9  9  7  + 66  8  -  123  -  6 3  8  7  6  -  -  ?  53  4B  -  7  + 103  + 43  +  6  5B  5  + 83  5  + 103  6  •+ 4 3  5  3 )/8A  -  5  -  4  63  -  4  123  +  3  33 )/8A 3  4  +  23 )/8A 2  23 )/8A 2  where A = 43  -  8  93  7  + 103  5  -  93  The n e x t two e q u a t i o n s simultaneously (2.3.11).  =  0  V^GJ  =  - 2 n ( - 4 + -4 + - 4 - —  solutions  and  are s o l v e d  (2.4.4)  into  (2.4.5)  R  R  8R  + D )x(-^-) 3-1 1  ,  (2.4.6)  x  o f which are 2  F =  0, x  (2.3.11)  are  R the  (2.4.2)  V 0^ 2  ,  2  (2.3.10),  after substituting  They  + 43  3  n (E,R +  .  ) R  3-1  + C  + E R  ,  (2.4.7)  and 1 A 0 7 = n (—K 2R 1  F + -4 R  B + — R  1  x  D R i —+ — 2 12  In R ) x ( - 2 - ) 3-1  2  .  (2.4.8)  The  boundary  determine  conditions  directly  from the boundary  c o n d i t i o n s by  n e x t e q u a t i o n t o be  substituting  into  (2.4.4)  o  o  %  E^, E^,  =  n d - n  (12B  ) ( — ^  +  2  +  computer. i s o b t a i n e d by  A,)  12B,C,  + • — H  3  R  R  R  B -  calculated  (2.3.12),  12A,B. 2  solved  to  and F., .  l o n g e r g i v e n e x p l i c i t l y h e r e b u t were  The  E  are used  (2.3.23)  the constants o f i n t e g r a t i o n ;  They a r e no  ,  to  (2.3.20)  2  —  +  12B D 1  C  +  1  -  1  |  +  D R )X(^| )  .  2  I  T  (2.4.9)  2R  The  solution  o  =  for  n(i-n  i -  ) (-4-  2  R  ^1^1  +  a +  G  2  R  which of  F T r  satisfies  2  + H  1  r  "nT  2  R  a  R  a  6  +  +  3  6T  R  24  1 - ^4  l  B  .(12B: + A, ) R 1  (12B D 1  3  " IW  ln R  +  1  R  1 R  4  5  ln  R)  2  '  )  (2.4.10)  boundary G , a  H , a  conditions  (2.3.25).  I  a r e d e t e r m i n e d by  a  and  J  a  The  constants applying  (2.3.24).  The be  2  D  +  integrations  conditions  now  (  - -=-^ + ir  2  5  R  8 X  s  next h i g h e r - o r d e r term  determined  i n the expansion of ¥  upon s u b s t i t u t i n g e q u a t i o n  (2.4.8)  into  can  24 (2.3.13):  4 1  2  2 A  1  1 — R  1 + —~ + C  3 F  R  2 B  D  1 ' +  6  R  R  2  (2.4.11) The s o l u t i o n i s ,  ^  + l5  = n ( i - n ) (% ^ R 2  - — R + 12  , , D,R + -G, R + -= ^ 288 B  + t£  8  B^ =— 15  6  R  ^L24  -  3  z  R  3  5  ln R  l n R) 840  2  '  (2.4.12)  which s a t i s f i e s  boundary c o n d i t i o n s  of  G^/ H2/ I ^ <3  integration  conditions  temperature  i s now p o s s i b l e terms.  (2.4.10) i n t o  2  2  determined  by a p p l y i n g  t o determine  the higher-order  s i m u l t a n e o u s l y upon  (2.3.15).  (2.3.14) and  substituting  (2.4.2)  The e q u a t i o n s a r e  = 0  9 1 v 0^  e  The n e x t two e q u a t i o n s  (2.3.15) a r e s o l v e d  V e^  r  The c o n s t a n t s  (2.3.24).  It  and  a  an  (2.3.25).  (2.4.13)  D  ••• 7 =  3n -D  (-4^  2  (  • 2 H  ' R  1  B  1 1 C  2R  2  r  a  2  -  (  ;  G  1  2  (12B D R  B  +  i  +  A  24R  ^  i> 3  z  + C ) -  2  i A  R  4  -  1 1 B  2R  J  5  2 R  6  25  + _  The  solutions  m  R _ _  R j x ^ )  m  .  (2.4.14)  are L + -|) (^ ) R a  9*  =  (3n  2  =  <^  2  - 1) (K* R  2  "  R 2  X  ,  T  (2.4.15)  and  0  3  {5 K  B +  (T=- +  4  320' i 2W  ^  +  - 2 W  B.C. R + -4^=- +  6R  +  R — In R 3360 ~" 3  The  (12B  12  m 1  R  l  n  A  R  l  +  l ) i-4 ln R 3 (  1 Q R  of  the h i g h e r - o r d e r s i n g u l a r i t y which  at  the c e n t r e o f the i n n e r sphere, L  following are  condition  (2.3.20).  determined u s i n g boundary The  last  terms  "2 calculated and  are  (2.3.17).  The  l  a  4 R  1  l  L  2  2  .  ^  *  a n <  2  In R  Q 3 ( i ) 8-l  L  2  would w i l l be  a  +  + C ) R  1  x ~  , ^ ,  are  ) R 3  + A,) — —  2  B  constants of i n t e g r a t i o n  2W~  +  IT4R  (12B D  B  T  - Hr  (2.4.16)  I  n  v  ^  give r i s e  r e m a i n i n g unknown c o n s t a n t s  conditions  (2.3.21) t o  o f the temperature  •  T  n  e  to  set to zero  (2.3.23).  expansions  2 and  e w  equations are  (2.3.16)  26  0  , ( 2 . 4 . 1 7 )  •  ( l - i) { i ( 2R  (  (  ^  3 n  2  I  "  A  2  +  1  B  1  F  -  ,  2  A  1  1  R  , (  5  -  .  1  4  2  7 B  D  5 C  (  H  •  1 2  1  ±  l E l  M  R  2  n  l  n  (  4 A  j  ~1  '  2  6 A  R  1  2 C  F  1  1  1 ) i  L  r  .  F  L  1  )  2 C ?  1  1  J  -)  R  n  R  8  6  i  l  A l R  )  2  R  l  i ]_ F  n  2  R  A  1  ft  ( B  3  ( ]_ ]_ 2B  F  +  A  3  R  R  <FT>  x  1  n  i i ) C  4"  R°  R  A i  6R  ( 4 B  + J  ( 3 B  1  1  D  1  C  -  2  A  D  1)  - ^ i ,  +  ( 4 C  1  D  1  - E  L.  ( 4 B . E .  +  R  4  1  1  - ^ ) R  1  C  1  +  4  +  ( « +  -  1 0  -A  3  1  +  4  D  1  F  1 ^  D, D E )R 1  1  2  T  R  1  3 B _  - J : + B-E. +  4 D  X  -  2  6  R  +  8  n  -  l  - r  R  ^  B  3 E  )  D  1 5  1  1 2  1 0 9  R  1  * -  + 3C D,  Z  + 4D  D  1  +  E  2  ,  i)-(G°  2  1  2  f  (—±-±  B  2  A  -  1  !  6  . )  A  +  4 8  3 A D F - — L I - - 1  C  3  1  A  R  ( 3 B  F  7  1 > _ _  C  g  i i )  A  2  3  27  + -  In R - ^  3R  +  j 2C  In R + (^1) R 6  - 2D  l E ; L  2  R  2  3  + ^1 i  n  In R  |  R  Ix  (JL)  3  P -L  l  —  (J^-) . 3  X  (2.4.18)  The s o l u t i o n s a r e  and  e  2  I  = (3n - i) 2  n  R  h 2  +  2  I  ( 3 I  2  +  5 B  4  —ig. 84R  +  R  1 1 " F  5  A  l l ) C  2 +  8D  l F l  )  1  R " ~  6  2  +  5C  1  1  1  240  2  E  3  OG$  R  i  16  36  2C  3LL-) i  -?  l  A  12R^  +  (  5  l D l  +  n 1 1 6R  1  - g ^ ,  1  28 4 + §r  73D ( 448  A B, ~ 6R  - D E )  17B  B D  (  in  R  L)  R  , 1  R  )  . )• ln R \  R  252  (2B F  + A,C  n  1  X  X  6 R  ' x  1  B  l  (—)  2  permissible  1 1  A  )  \'  F  5  0  A  T  4  8  R  J I  N  R  0  1 1 B  6R  (4C,D 1  3  9R  + 4D,F.)  J  £  - i r e  - E  1  X  X  + 36  D R  —)  ,  2  F l  (4B..C 15  l  (—)  1  2  R  - 2A D  1  r  3  ln R + 432  b 2 / (P° + •-=•) } 2  R  since  at the centre  -) 4  X  ln R  B  Q  i s s e t to zero  (  - i  36  In R +  l  6  1  1  In R +  A  N  A  5B - + 4C;  — i " i  1  C  R  +  3  18  i 18RR  x  (4B. E,  4 D + £i_ ( _ i + D..E..) 6 24-  +  j  + 2C..E,) + — 36  3  X  R  6  + j < 15R  +  C (B D 1  in  R  45  . 3 ,3 (^) p  ) +  1  2  2 + — 6  2C  2  . 4  D  ~ ln R 72R  (-^-i - _ i )  -  720  (  A In R  x B  _  3  3  °1  (—)  4  ln R  R  72  .  (2.4.20)  X  a singularity o f the inner  of t h i r d sphere  order  i s not  (2.3.20).  With-  '"b out  loss  of g e n e r a l i t y  P  i s a l s o s e t t o zero s i n c e from  2  the  t e m p e r a t u r e m a t c h i n g c o n d i t i o n s a t t h e i n t e r f a c e o n l y one o f the c o n s t a n t s , o r P^ n e e d be r e t a i n e d . The o t h e r *"b b b b b '"b remaining constants, M, M, N , P / Q and Q are determined 2  by  the  application  2  2  2  2  o f boundary c o n d i t i o n s  2  (2.3.21)  to  (2.3.23). It the  i s seen t h a t the  expressions  harmonics, while  for in  0™  o r 0™  functions of are the  n  set of  Gegenbauer p o l y n o m i a l s  appearing spherical appear.  in  30 2  ,  Solution  5  In  of Constant  this  section,  t r a n s f e r b e t w e e n two constant an  f l u x on  Flux  the case of f r e e  concentric  3  T  convective heat  spheres with a  the s u r f a c e o f the s o l i d  i s o t h e r m a l outer sphere  o - - k  Problem  x  4TTR!  prescribed  i n n e r sphere  i s considered.  The  heat  and  flux,  2  R! 1  on  the s u r f a c e o f the i n n e r sphere  be  constant.  Thus t h e c o n j u g a t e  between t h e s o l i d The in  section  The no  energy  i n n e r sphere  are a p p l i c a b l e  fields  of the  longer coupled.  the  i n n e r sphere  one  parameter,  However, t h e in  section  jugate case. and  be  fluid  (2.2.1) and  and  involving  the s o l i d  still  on  than  the ranges  is  lost.  (2.2.2) as  case.  T h i s case o f G,  are  (2.2.3) f o r is  still  o and  the conjugate 3  3.  It  case.  outlined  hold.  non-linear coupled governing equations constant flux  (2.2.1)  c a s e a r e s o l v e d by  scheme i n t h e s i m i l a r manner as f o r t h e The  given  i n n e r sphere  o f a and  to  interface  equation  parameters  namely OJ , l e s s  i n advance  constant flux  considered here.  expansions  (2.3.5) r e s p e c t i v e l y .  subject will  one  (2.2.2) f o r t h i s  perturbation  the  for this  fluid  limitations  2.3  The and  and  a t the  Thus t h e t e m p e r a t u r e  i s not  a multi-parameter has  effect  governing equations, 2.2  i s specified  f o r ¥ and  The  con-  0 are those o f  simplified  a  (2.3.4)  boundary c o n d i t i o n s  t o w h i c h t h e g o v e r n i n g e q u a t i o n s h a v e t o be  solved  31  90, =  1  at R  =  0  at R =  1  at R  0  a t R = 3;  = 0  at R =  =  (2.5.1)  9R  90  ra  9R  0m  j  2 i ) 2  3  K  ¥  3  The  boundary  the  surface  condition  zeroeth  Therefore t o the  ization  1  (2.5.2)  B  1,  (2.5.3)  1, 6;  m  > 1  (2.5.4)  j,• k  (2.5.1) s t i p u l a t e s t h a t t h e sphere remains  The  order  constant  term of  no f h i g h e r o r d e r  f l u x on  condition  >  >_ 1  (2.5.5)  i — = 0 a t n = ± l ; j , k > l . 3n  of the'inner  after normalization. the  =  1, m  9R  ¥ . = - (1-n  by  1;  the  The  c o n d i t i o n o f the  (2.5.4) s t a t e s t h a t t h e are  zero  at the  outer  and  (2.5.6) h a v e t h e  temperature,  of  the  temperature higher  envelope.  give  inner sphere  boundary c o n d i t i o n  order The  same meaning as  a t the  0,  on  i.e. unity  condition i s  terms i n 0 s h o u l d  surface  (2.5.2).  the  flux  constant  flux  (2.5.6)  fulfilled  expansion. a contribution  i . e . boundary  (2.5.3) i s a outer  terms i n the  sphere. 0  expansion  boundary c o n d i t i o n s previously.  normal-  (2.5.5)  The given for  set of uncoupled  i n s e c t i o n 2.3  0 are not  linear  (with the  considered  differential  exception  here) are  equations  t h a t the  equations  solved subject to  the o  boundary c o n d i t i o n s given  o°  which  fulfilled The  0 and  ~~  1  R  ~  solution  are R  factors  (g^y) /  solutions.  The  are  determined d i r e c t l y (2.5.6) i n t h e  for 0  O  is  (B+D 6  s o l u t i o n s f o r a l l the  the  to  The  the boundary c o n d i t i o n s  ¥ expansions  constant  above.  those  constants using  computer.  higher  given  B 2 (3^1")  a n c  of  (2.5.1) and  ^  order  terms i n  i n s e c t i o n 2.4 R  ^8^T^  3  (2.5.3).  except  w i l l not  the the  appear i n  i n t e g r a t i o n s i n the  the boundary c o n d i t i o n s  solutions (2.5.1)  33  3.  EVALUATION OF  In the p r e v i o u s for  the  conjugate  has  been c o n s i d e r e d .  before,  these  combinations 6 .  and  heat  ANALYTICAL RESULTS  chapter,  analytical solutions  t r a n s f e r between c o n c e n t r i c Subject  s o l u t i o n s may of the  the  t o the  be  limitations outlined  assumed t o be  four c h a r a c t e r i s t i c  It i s anticipated  spheres  valid  for various  parameters  t h a t t h e r e w i l l be  a ,  G,  limiting  w, values  f o r each of the parameters beyond which the p e r t u r b a t i o n expansions limit  the  the 2.1  radius of  of  the  laminar  and  to  criterion  of the  the  convergence o f the  a i d o f the velocity  between the  two  differentiating  heat  results.  The  limitations  transfer  rates  lines  on  components o f t h e  stream the  the  a l s o be and  computer were o b t a i n e d .  Finally,  as  assumed i n  approximately  p e r t u r b a t i o n expansions  c o n c e n t r i c spheres the  flow  i n determining  s o l u t i o n s as m e n t i o n e d above w i l l  tangential  the  s o l u t i o n s are  may  turbulence.  Contours of s t r e a m l i n e s , v o r t i c i t y with  f a c t o r s which  l e a d i n g t o a f l o w o f more  used here  i n s e c t i o n 3.1.  Other  axisymmetrical  become u n s t a b l e  nature  The  outlined  longer converge.  steady  may  complicated  the  no  range o f a p p l i c a b i l i t y  follows: section  will  fluid  applicability discussed.  isotherms The  were c a l c u l a t e d  given.  and  gap by  and  temperature d i s t r i b u t i o n s  ( N u s s e l t Numbers) a r e  plotted  radial  i n the  function analytically  is  computing and  the  3  •1  Range o f V a l i d i t y The  of Solutions  perturbation expansions f o rthe v a r i a b l e s o f  V, 0 a n d 0 a r e o b t a i n e d up t o t h e s e c o n d o r d e r t e r m s . series through  e x p a n s i o n s f o r f , 0 and 0 a r e e q u a t i o n s (2.3.6).  asymptotic  for  (2.3.4) such  expansions i s o f the order o f the f i r s t  neglected. parameter  The e r r o r made b y t r u n c a t i n g  This  term w i l l  tend t o zero r a p i d l y  6 i s reduced t o zero.  finite,  non z e r o G , . a l l  were r e t a i n e d  f o ractual  convergence  term  as t h e  In the numerical e v a l u a t i o n s  t h e terms  computation.  o f h i g h e r o r d e r . t e r m s no e x t r a  The  f o r V, 0 a n d 0 d e r i v e d Due t o t h e c o m p l e x i t y  t e r m was d e r i v e d  f o r each o f t h e expansions.  t o check  Thus t h e n u m e r i c a l  c o n v e r g e n c e o f t h e e x p a n s i o n s c a n n o t be g u a r a n t e e d . An  alternative  determine the p r a c t i c a l series and  criterion  u p p e r bound o f t h e c o n v e r g e n c e  e x p a n s i o n s as f o l l o w s :  3 and a t a ' t y p i c a l '  o r d e r terms  for  i s proposed here t o o f the  F o r g i v e n v a l u e s o f G, a , to  location  t h e v a l u e o f any h i g h e r -  0 and 0 c o n s i d e r e d must n o t b e g r e a t e r  than t h e p r e v i o u s term. more h i g h e r - o r d e r  terms  In a d d i t i o n  t h e sum o f any two o r  must n o t have a v a l u e g r e a t e r o  than o  the p r e v i o u s term o r t h e fundamental terms, i . e . ^ , 0  O  and  ~ o 0o  .  The ent  from t h a t  criterion  o f convergence d e f i n e d here  u s e d by Mack e t al_.  considered that  [9, 10],  These  i s differworkers  as l o n g a s t h e maximum m a g n i t u d e o f any  35 h i g h e r o r d e r term i n e i t h e r s e r i e s f o r ¥ and 0 d i d n o t exceed the  maximum v a l u e o f the a p p r o p r i a t e l o w e s t o r d e r term a t a  ' t y p i c a l ' l o c a t i o n , convergence c o u l d be assumed.  On  this  b a s i s , they found from t h e i r a n a l y t i c a l s o l u t i o n s a t low a t h a t i t was p o s s i b l e t o have double c e l l s i n the f l o w  field.  T h e i r s e r i e s f o r ¥ and 0 were o b t a i n e d by expanding i n a s c e n d i n g powers o f R a y l e i g h number r a t h e r than i n the Grashof number.  Thus the c o e f f i c i e n t e x p r e s s i o n s f o r the  v a r i o u s terms i n the s e r i e s a r e dependent on a : constants with c  x  they i n c l u d e  .  Double c e l l s cannot o c c u r on the b a s i s on which the r a d i u s o f convergence i s d e f i n e d i n t h i s t h e s i s .  Now,  the  e x i s t e n c e o f a secondary c e l l i n a d d i t i o n t o the p r i m a r y c e l l had been observed e x p e r i m e n t a l l y by B i s h o p e t a l . [7] f o r a Rayleigh  number 3  mately 45 x 10 of  (as d e f i n e d i n t h i s t h e s i s ) o f a p p r o x i a t a v a l u e o f B o f 1.19.  However, the range  R a y l e i g h numbers ( f o r which the s e r i e s w i l l  'converge')  c o n s i d e r e d b o t h by Mack e t al_. and i n the p r e s e n t t h e s i s i s below t h a t quoted by B i s h o p f o r the o c c u r r e n c e o f a secondary cell.  I f the, r a d i u s o f convergence proposed by Mack had  been adopted here a secondary c e l l would i n d e e d e x i s t f o r a v a l u e o f Ra l a r g e enough (but n o t n e c e s s a r i l y a t low a o r B).  An example o f t h i s i s i l l u s t r a t e d i n s e c t i o n 3.2.  There-  f o r e i t can be c o n c l u d e d t h a t a n u m e r i c a l ' d e m o n s t r a t i o n o f double c e l l s i s r e l a t e d t o the f a c t t h a t the s e r i e s i s no l o n g e r c o n v e r g e n t i n the work c i t e d .  36 For both sidered, on  numerical  G o r Ra  0.01  the  conjugate  investigations  decreases  t o 100  and/or  3 increasing The  constant  i n the  case  v a l u e s o f a and the  conjugate  temperature  The  compressive the  of  the  at  both  flux  case  solutions  G i s lower  case  I f the  for  ratio  the boundary and  i n the  fixed  of  only f o r  thermal  can  This i n effect  and  the  suffic-  conditions i n  2.3.23)  no corres-  'conjugateness'  n o t be  steady  either  term  and  of  and  show t h e term (A)  range  effects;  ratios  to either  o f the  the  i n the energy  positions.  equation  more  In  magnitude  conduction equation  concentric I t i s seen  i s at l e a s t  of  over,  axisymmetrical.  o f the v i s c o u s d i s s i p a t i o n  i n the energy  of  the n o n - c o n s i d e r a t i o n  between t h e two  angular  ratio  laminar  VI  c o n v e c t i o n term  radial  are  t o the  viscous dissipative  f l u i d - i n t h e gap  .  i s 3.  simultaneously.  of the  t h a t the  —8  This i s  lost.  tables  10  t o 2.5.  conjugate  ( i . e . 2.3.21  the v i s c o u s d i s s i p a t i o n  of  con-  from  l o w e s t v a l u e o f OJ ( a p p l i c a b l e  I , T a b l e s V and  (e) o r t h e  of  field  work and  f l o w may  Appendix  1.15  other p o s s i b l e l i m i t a t i o n s  applicability  a increasing  u p p e r bound on  of i n t e r e s t  constant  the problem i s  cases  f o r w h i c h t h e upper bound o f G i s  fulfilled  ponds t o t h e  flux  show t h a t t h e u p p e r bound  from  us s h o u l d v a n i s h t h e n  conductivity  l o n g e r be  The  case)  i e n t l y h i g h t o be  the  than  3.  constant  monotonically with  shown i n F i g u r e 3.1.1. flux  and  term  (2.2.2)  spheres from  term  of the  the to order  Hence n e g l e c t i n g v i s c o u s h e a t i n g as assumed i n  1.0  1.5 Figure  3.1.1  Approximate  2.0  3 Upper  Bound o f G f o r V a r i o u s  2.5 a/3  38 section  2.2  is justified.  the experimental between t h e two axisymmetrical  3.2  I t w i l l be  investigation  shown l a t e r  t h a t the  c o n c e n t r i c spheres about the v e r t i c a l  flow i n the  i s steady  gap  laminar  and  axis.  Streamlines In  ations  view o f the q u a l i t a t i v e  o f G,  contour  plots  a,  to and  s i m i l a r i t y f o r a l l combin-  3 f o r which the expansions  of streamlines, i s o v o r t i c i t y l i n e s  a r e g i v e n f o r a f i x e d v a l u e o f Ra=720 f o r b o t h and  the  constant  equation sented  A single  i n n e r sphere) i s o b t a i n e d .  the outer sphere. rotational  ation  The  of the  symmetry.  fluid  space  cell  f o r the The  into  i t from  fluid  t h e c i r c u l a t o r y m o t i o n , where *F has  in  the upper h a l f  p a s t t h e mid  flows  and  is  iso-  upward i n along  'toroidal'  flow  polariz-  The  lower  region  with  fluid  withdrawn  and  fluid The  flowing  center  a maximum v a l u e , i s  f l o w r e g i o n a t 6 - 82°  p o i n t of the annular  pre-  downward  space  i n n e r sphere  of  ¥  'crescent-  the boundary o f the o u t e r sphere.  of the  isotherms  c a s e o f an  fluid  a reservoir  i t i n the v i c i n i t y o f the  and  OJ=10 a r e  o f the  inner sphere.  from  convergent,  conjugate  There i s a j e t l i k e  acts l i k e  are  s t r e a m l i n e s as g i v e n by  i n n e r sphere  flow i n the  above t h e t o p o f t h e  the  The  ( o b s e r v e d p r e v i o u s l y [10]  immediate v i c i n i t y  with  cases.  i n F i g u r e 3.2.1.  thermal the  flux  (2.3.4) f o r a=.72, G=1000, 3=2.0 and  eddy' t y p e  of  through  space.  and  slightly  39 Similar values inner  configurations are obtained  o f Ra=720 ,. 6=2.0 f o r u>=10^ s p h e r e ) and t h e c o n s t a n t  Figures  annular  (approximately  f l u x case  3.2.2, 3.2.3 r e s p e c t i v e l y .  streamlines  with  isothermal  as shown i n  Figure  3.2.4 shows t h e  f o r a=.72, G=1000, OJ=10 a n d 6=1.15 w i t h t h e  s p a c e b e t w e e n t h e two c o n c e n t r i c s p h e r e s  stretched that  streamline  i n t o a r e c t a n g u l a r form f o r c l a r i t y .  the center of the streamlines  to u n i t y , the flow i s e s s e n t i a l l y  plotted  I t i s seen  i s a t 8=90°. 'creeping  here  flow  As 6 t e n d s i n a narrow  1  o  gap.  The c r e e p i n g f l o w  solution  ( i . e . f^) places the center  of the streamlines  a t 8=90°.  10 w i t h  o f Ra=720, 6=2.0 f i x e d ,  case  the values  I n c r e a s i n g the value  f o r the conjugate  ( w i t h oo=10 , 10"^) and f o r t h e c o n s t a n t  streamline  of the streamline  values  as t h e b u o y a n c y e f f e c t s  decreases Figure  f l u x case, the  c o n f i g u r a t i o n s remain e s s e n t i a l l y  values  contours  when Ra i s m a i n t a i n e d  3.2.5,  approximately  of a to  unchanged.  a r e lower than  the previous  are r e l a t i v e l y  reduced  constant while  a i s increased),  The c e n t e r o f t h e eddy c r o s s s e c t i o n i n t h e same p o s i t i o n  as b e f o r e  The  upward and o u t w a r d d i s p l a c e m e n t s  the  value  o f Ra when 6 i s l a r g e r  (G  remains  f o r a l l cases.  o f t h e c e n t e r depends on  than  terms i n W t h e n become i m p o r t a n t ) .  1.15  ( i . e . higher  Also the angular  i . e . OJ=10 , 8 - 8 2 ° ; and f o r t h e c o n s t a n t  8 = 75°.  flux  case  order  position  o f t h e c e n t e r o f t h e s t r e a m l i n e s depends on t h e v a l u e o f OJ  The  40 As of  stated  previously  the asymptotic expansion  i n section in ¥  3.1, f o r c o n v e r g e n c e  (with a l l t h e terms  actually o  derived  used  must a l w a y s  i n the computations) be t h e p r e d o m i n a n t  occur i n the annular space, values o f the stream  the  flow f i e l d .  term.  I f multiple  the adjacent c e l l s  function  However, t h e f u n d a m e n t a l  the fundamental  term  i s always  secondary  cell  would  positive  Thus i n o r d e r t o have m u l t i p l e  term.  cells  of opposite a l g e b r a i c  h i g h e r o r d e r t e r m s w o u l d have t o become larger fundamental  term should  have sign  throughout c e l l s the  than the  F i g u r e 3.2.6 shows t h e p r e s e n c e o f a  a t the lower r e g i o n  o f t h e gap f o r t h e c o n j u g a t e  c a s e w i t h OJ=10, a=.72, G=2100 and 6=2.0.  The s e c o n d a r y  cell  does, n o t e x t e n d  a c r o s s t h e gap and i s much weaker t h a n t h e  primary  I t may be c o n c l u d e d , as e x p l a i n e d a b o v e ,  cell.  here the expansion  f o r ¥ i s no l o n g e r c o n v e r g e n t .  B i s h o p e t a l . [7] o b s e r v e d e x p e r i m e n t a l l y t h a t secondary  cell  was f o r m e d  cell  c a n n o t be a g e n u i n e  Moreover,  the f i r s t  near the top of the i n n e r  Hence, i n t h e p r e s e n t a n a l y s i s  that  sphere.  the occurrence o f a secondary  feature.  42  45  F i g u r e 3.2.5 Streamlines  f o r Conjugate  Case,  co=10, 3=2.0, a=10, G=72  3.3  Velocity  Distributions  I t was pattern  i s axisymmetrical.  velocity, of  V^.  the f l u i d  obtained  assumed p r e v i o u s l y  V. R  G  the convective  Thus t h e r e  The r a d i a l  i s no  and t a n g e n t i a l  i n t h e R- and 8- d i r e c t i o n s  analytically  equation  that  flow  latitudinal  velocity  components  respectively  by d i f f e r e n t i a t i n g  the stream  are function,  (2.3.4):  1  3¥  1  R sine  38  R  2n  x  + G  x  2  (  I  2R  D R + _ i  144  l  B  _ ± . +  _  R  R  -j  +  (3.3.1)  3n  +  C  R  ,  -  -  G  R  a  3  (  1 2 B  -i  560  2  + H  i*) <^r 2 x  R + ~% +' % R R  a  2  B C.  £ + A ) + -i-i - — 2  1  24  (12B D 1  )  i  B  £_ R  D  *  9  24R  +  8  1  2  + G a x (3n -D 2  ±  (3n - 1)  B.  A-i  l  A  3¥ Z  R  l n R + —— R l n R > x  60  R  3 +  )  H* R  +  I ^ K  2  (^-) P  J  b 2 +  K  b  1  +  C) ±  48  A  F + -± 12R 8  C R i 24  3  \  - — ln R 840  V f i  =  =  _ _ L _ l i R s i n O 3R  x  D R + —— 288  2  (12BJ +  =  -  (3.3.1a)  ik^Lll  2 J  - 5G: R  -  3  + G a 2  3H  A ) - 1 1 C  4 D l  2  A  B  R^~  2R  (12B D  + C)  3  D^R 1  ±  1  -  )  R  ±— (1 + 3 l n R)  60  D R  BR + - ± - (1 + 3 I n R) 15  48  2  1 1  3  4  R X  5G^ R  I  P C R — + — 4 . 6  -  24  (1 + 5 l n R)  \-  R +  1  +  B  x n  a  R  x  3  560  (3.3.2)  B, ± - 2C, + H R 8  ,3  2 4R  + 2—  31 , 3R  R  B  R  2  (J^r)  R"  x n  R In R  15  P  (l-n )?  + G  B  - 3H^ R + — £ R  +  — 12R  x  R (^fy)  49 R+  (1 + 5 l n R)  840  The  radial  using  the expressions GJ=10,  i s plotted  R  6  (giiy)  and t a n g e n t i a l v e l o c i t i e s  For V  x  (3.3.1a) and  6=2.0, G=1000,  (3.3.2a)  were e v a l u a t e d (3.3.2a).  a=.72; t h e r a d i a l  against radius R f o r various  as shown i n F i g u r e  3.3.1.  numerically  At a given  velocity  angular p o s i t i o n s  radial  p o s i t i o n the i  velocity  Each p r o f i l e  a maximum o r a minimum v a l u e  at  has e i t h e r  an a p p r o x i m a t e  radial  0°<6<.60°, t h e f l u i d < 6 < 180°.  90° is  about twice  the  radial  (i.e.  outer zero  that of the r a d i a l  Finally,  g r a d i e n t near the inner sphere's than  t h a t near  the outer  i n f l o w when  a t 0=0°  outflow  i n f l o w at 0=180°.  t h e same v a l u e s  of OJ,  surface  sphere's  6/ G and a as b e f o r e , t h e  of the t a n g e n t i a l v e l o c i t y ,  position,  R, f o r v a r i o u s  i s higher sphere.  a t R=1.5  flow. than  V , Q  against the r a d i a l  0 a r e shown i n F i g u r e  shows a maximum v a l u e  o f downward  sphere  and a r a d i a l  For  ( i . e . R=2.0).  profiles  value  outflow  occuring  1.4 j< R <^ 1.55.  I t i s seen t h a t t h e r a d i a l  velocity  For  profile  p o s i t i o n between  has a r a d i a l  R=1.0) i s h i g h e r  surface  increases with  decreasing 0 .  magnitude o f the r a d i a l  f o r upward f l o w  The upward f l o w  t h e downward  flow  3.3.2.  and a minimum  speed near t h e i n n e r speed near the  For 0=90°, the t a n g e n t i a l v e l o c i t y , indicating  Each  V , is Q  t h a t t h e upward and t h e downward  flow  are  approximately  still  equal.  g r e a t e r then  the  However, t h e  upward v e l o c i t y  downward v e l o c i t y .  f r o m upward f l o w  t o downward f l o w o c c u r s  1.4  This crossover  <_ R <_ 1.55.  The  Vorticity  fluid  vorticity  region  p o i n t moves t o w a r d s t h e  f e a t u r e s o f the  are  mentioned due  similar  radial  f o r other  t o buoyancy  and  gap  convective only  between the vorticity  source  o f to,  vector  ]  R  2 I ( l - n )2 G  2B L  •T- ~  R  1  0  D  l  R  +  I  x  (  3^T  )  of  of  i n the  by:  6  motion  concentric  9R  =  inner  tangential  combinations  the  f o r c e s i s the  symmetry, t h e  i s given  [ V x V  9V  i n s e c t i o n 2.3,  a v a i l a b l e i n the  (J) - d i r e c t i o n  =  i n the  point  Contours  With r o t a t i o n a l  ?  crossover  a.  As the  general  profiles  3, G and  3.4  The  0 increases.  s p h e r e as  velocity  is  spheres. positive  53  1 + - ~ 4R  2 (12B^ + A 1  Z  1 - ± — D  1  4  G  ^  O R  2  (9 + 14 I n R)  I  A  0  - i  -  The Figures  may  may  vary  3.4.1  as:  vorticity  lines  C  1  R  D  JL + - i 2R  nR  1  r  3  L_  4  12  (3.4.1)  ))  3.4.4  are s i m i l a r .  extent.  The m e r i d i o n a l  and t h e i r  G, o, O J and 6.  radial  contours The  i s generated;  vorticity  of the f l u i d  ment  three regions  regions are  o f t h e i n n e r sphere  i s dissipated.  sphere  diffusion  where  Note t h a t t h i s  resembles q u a l i t a t i v e l y  where  r e g i o n where  i s t r a n s f e r r e d by  the boundary a t the o u t e r  vorticity  which  depend upon t h e  the c e n t r a l vortex  from the i n n e r sphere  shown  length o f the  position  These t h r e e  t h e immediate v i c i n i t y  convection;  into  +  1  features o f the v o r t i c i t y  through  in radial  the v o r t i c i t y and  + c x  be c o n v e n i e n t l y d i v i d e d i n t o t h r e e r e g i o n s  four parameters  the  F  2R  (9 + 14 i  overall  isovorticity  defined  C.) 1  R-  B  + _± + 3 840  field  T  i  R  in  + 1  12  6  +  (12B,D 1  B. "1  2  K (_  1  R + £ — 560  R  2B,C, R — - + R 4  ) +  arrange-  that described  54 by  Batchelor  for free  c o n v e c t i o n between p a r a l l e l  plane  boundaries. In the  immediate v i c i n i t y  vorticity  o f the  set  the heated  up by  lines in  fluid  of the  i s generated  by  i n n e r sphere.  the  i n the  other  particles  are r o t a t i n g  i n the  clockwise  vorticity  v e c t o r d e f i n e d by  angles for  and  this  into  the plane  r e g i o n s by  The  most n o t a b l e  the  two  The  here fluid  direction.  The  (3.4.1) i s a t  right  o f the paper  isovorticity  feature i n this  i n each o f the  figures  tori  s p a c e between t h e  considered  as  and  as  'vortex r i n g ' .  The  in  upper r e g i o n o f the  flow  gap.  field  This position  the  c e n t r e o f the  streamlines  of  the  corresponding  cases.  is  a r e g i o n o f nearly magnitude o f the  The  sense of r o t a t i o n  With  whole They  can  vortex i s  q u i t e near t o  3.2),  from  centre of the e.g.  i s opposite  the that  f o r each vortex  Figure  i s the h i g h e s t i n t h i s  fluid  be  central  o f the  is different  vorticity  vorticity o f the  and  (section  Near the  constant  line  zero.  rotational  the whole o f the centre  of  The  sheets.  other  'toroidal-  c o n c e n t r i c spheres.  'vortex tubes'  o f the  of value  extends around the  region  mid-point  a  lines  from the  r e g i o n i s the  s u r f a c e s of the v o r t i c i t y  symmetry, e a c h o f t h e s e  the  regions.  c e n t r a l vortex region i s separated  two  annular  equation  extent  region. The  shaped'  two  motion  isovorticity  i n angular  to those  the  convective  Thus t h e  h a v e t h e h i g h e s t m a g n i t u d e and  comparison  i n n e r sphere  3.4.3. region.  ( i . e . counter-  56  F i g u r e 3.4.3 Vorticity  Contours, Constant F l u x Case,  6=2.0, o=.72, G=10  3  ' F i g u r e 3.4.4 Vorticity  Contours, Conjugate  Case,  OJ=10 , 3=2.0, a=10, G=72  59 clockwise)  t o t h a t i n the r e g i o n a t the i n n e r sphere  as t h e v o r t i c i t y The  i s transferred  vorticity  i s transferred  boundary r e g i o n o f the o u t e r vector  from  o f the inner sphere.  contours  are there q u a l i t a t i v e l y  Hence t h e v o r t i c i t y  similar  t o those  But f o r each  the i n d i v i d u a l  h a v e t h e same a n g u l a r e x t e n t  a t the  The f e a t u r e s o f t h e v o r t i c i t y  the inner sphere.  magnitude o f t h e v o r t i c i t y  t o and d i s s i p a t e d  as t h a t i n t h e i m m e d i a t e  vicinity  r e g i o n near  there.  sphere.  i s i n t h e same d i r e c t i o n  surface  i n this  i n the  corresponding  contour  does n o t  region,at the outer  sphere. As  i n the streamline c o n f i g u r a t i o n , the features o f  the v o r t i c i t y provided of  contours  change  Ra., 8 and OJ be k e p t  little  with  constant.  a change o f a,  While  the p o s i t i o n  t h e c e n t r e o f t h e v o r t e x d o e s n o t change w i t h  a,  i t is  a f f e c t e d b y Ra., 8 and O J .  3.5  Temperature D i s t r i b u t i o n The  temperature  annular  space  equation  (2.3.5).  Contours  distribution  of the f l u i d  between the c o n c e n t r i c spheres For the conjugate  yields  the temperature  sphere  with  distribution  a s i n g l e heat  of the temperature f o r both  and  inside  the constant  flux  source  case?  case  i s g i v e n by  equation  i n the s o l i d  at the centre.  the i n n e r sphere  i n the  (2.3.6)  inner Calculation  i s irrelevant  and when OJ i s v e r y  large  60 e.g.  OJ=10  15  ( i . e . an i s o t h e r m a l i n n e r s p h e r e )  conjugate case. that in  f o r .72  <_ o <_ 10  3.5.6  distribution of  I t i s seen from the F i g u r e s  a does not a l t e r  Figures  and and  a, a t l e a s t  influence  of  distribution given  later The  similar 3.5.1  even i n the  and w i t h Ra,  3.5.7.  This  i n that range.  a i s a higher  profiles  indicates  the heat t r a n s f e r  effect  and h e a t t r a n s f e r r a t e s . in section  that  rates  temperature  are q u i t e  independent  expressed,  on b o t h t h e  the  temperature  A proof of this i s  3.6.  f o r the conjugate and.constant The  a change  the  g e n e r a l f e a t u r e s o f the temperature  t o 3.5.5.  3.5.3  or the c o n t o u r s ,  Alternatively  order  and  OJ f i x e d ,  3 and  the temperature  3.5.1  temperature  profiles  are  flux cases, Figures  distribution  due  t o pure  o  conduction  (i.e.0 ) O  i s g i v e n by  the dashed  F o r t h e c o n j u g a t e c a s e , a t any p a r t i c u l a r 1 _< R <_ 3 r t h e t e m p e r a t u r e w i t h d e c r e a s i n g 9. the curve f o r pure 150°  close  indicates  curve.  The  f o r comparison. position  from the f i g u r e s )  f o r 0=0°  profiles  t o g e t h e r compared  and  60°  f o r 120°  t o any  two  that the l o c a l heat t r a n s f e r  t h e i n n e r s p h e r e and of  profiles  radial  c o n d u c t i o n w h i l e the p r o f i l e s  l i e above t h i s  relatively This  The  (as s e e n  line  l i e below  f o r 90° and  and  150°  are  profiles.  rates  the o u t e r sphere i s q u i t e  increases  f o r both  independent  0 f o r 120° 1 0 f . 1 8 0 ° . In t h e i r  e x p e r i m e n t a l work, S c a n l a n e t a l . [21]  postulated  that  a reversal  of the o r d e r i n g  occur  (see t h e i r  f o r the occurrence of m u l t i c e l l u l a r  Figures  of the temperature  8 t o 10).  Direct  flow,  profiles  comparison  would of  their  r e s u l t s with  since their  those  r a n g e o f Ra  a b l y h i g h e r and  with  However, i t i s s e e n here,  t h e r e i s no  given here used  reversal of t h e i r  temperature  varies with  on  the  where O J = 1 0 ^  ( i . e . an  variation  the  on  constant  3.2  flux  assumed t h e h e a t  flux  order.  the  here.  the  occur here  i n accord  above.  except  inner  sphere  i n the conjugate  case  The  temperature  s u r f a c e i s most p r o n o u n c e d f o r  For i t i s i n t h i s  on  consider-  illustrated  Hence  isothermal inner sphere).  case.  was  profiles  s u r f a c e of the  i n n e r sphere's  meaningful  that considered  f l o w does n o t  angular p o s i t i o n  X  experiments  t h a t the temperature  the d i s c u s s i o n i n s e c t i o n The  the  in their  a s m a l l e r 3 than  existence of m u l t i c e l l u l a r with  i s perhaps not  case  s u r f a c e of the  that i t i s  i n n e r sphere  u n i f o r m even though c o n v e c t i o n i s t a k i n g p l a c e i n the On  the  o t h e r hand  bution of heat convection at  the  i.e.  flux  i n the  a t R=l,  fluid.  the  temperature  A l s o the  contours  and will  redistri-  when t h e r e i s o f the  i n n e r sphere  fluid  temperature  3.5.6. be  essentially  f o r pure conduction  of heat  those .  from  the  source, a t the c e n t r e o f the c o n c e n t r i c s p h e r e s . c o n v e c t i v e motion of the  isotherms  upwards, F i g u r e s 3.5.6  both  fluid.  there i s a  temperature  matched w i t h  resulting  the  case,  i n n e r sphere  F i g u r e s 3.5.1, 3.5.2  concentric circles  single  conjugate  inside  i n t e r f a c e must be  The of  f o r the  is  relatively conduction  cold and  fluid  from  fluid  d i s p l a c e s the  t o 3.5.9.  T h i s i s due  the bottom b e i n g heated  c o n v e c t i o n as  The  i t rises  t o the  to up  top o f  by the  F i g u r e 3.5.6 Isotherms, Conjugate Case,  w=10, 3=2.0, a=.72, G=10  3  6 8  gap.  The shape o f t h e i s o t h e r m s  s i n c e pure conduction is  i s predominant.  shown t h a t a t t h e b e g i n n i n g  equation  the inner  constant  flux  isotherm  coincides with  convective  the s o l i d  Figure  place,  3.5.6.  The concentric e.g.  convection inner  3.6  3.5.9.  of the inner  The  KT  Nu. i  /o\ (8)  =  there  a r e ' d i s t o r t e d ' from t h a t o f f o r 6=120°  approximately  region  vigorous  i n t h e immediate v i c i n i t y  Rates  l o c a l Nusselt  of flux,  flux.  sphere.  Heat T r a n s f e r  iso-  o f the inner  even though  shows t h a t i n t h i s  i s taking place  When  i s displaced  large the surface  around t h e r e g i o n This  sphere.  i s a redistribution  t o be i s o t h e r m a l  of heat  the zero  The l o w e r p o r t i o n i s d i s p l a c e d  temperature contours circles  Figure  space.  When w i s v e r y  and/or o f  the p o r t i o n o f the zero  s p h e r e where t h e r e  a redistribution  alone,  i n t h e upper h a l f r e g i o n  s p h e r e c a n be c o n s i d e r e d is  i s isothermal  the surface  upwards i n t o t h e a n n u l a r  to circles  o f the s o l u t i o n o f the energy  sphere's surface  motion takes  similar  I n s e c t i o n 2.2, i t  thus f o r pure conduction  therm a p p r o x i m a t e l y  into  isstill  numbers  - —(3-D x  30 —  8  3R  R=l  o f the  72 and  (6-1) 6  Nu (0)  — — -  o  62  Q  x  80 — 8R  x  R=6  are  defined  f o r the inner  The  Nusselt  numbers d e f i n e d h e r e h a v e t h e r a d i u s  inner  s p h e r e as t h e common l e n g t h  transfer Nusselt or  rate  i s then obtained  numbers  the outer  inner  s p h e r e and o u t e r  sphere.  sphere are given  °  It  separately the  26  b o t h Nu^  o v e r a l l Nusselt The  OJ  - -fcii  i s s e e n t h a t Nu^  and  o f Nu^  6 Q  Nusselt  f  — 8R  2 3  J  , will  heat  numbers  sphere  f o r the  r e s p e c t i v e l y by:  6 d9  (3.6.1)  s i n 8 d6  (3.6.2)  R=l  must be e q u a l q  of the inner  sin 8R  J  o  =  The o v e r a l l  The o v e r a l l N u s s e l t  26  1  o f the  by i n t e g r a t i n g t h e l o c a l  over the e n t i r e s u r f a c e  s p h e r e and o u t e r  Nu~  scale.  sphere r e s p e c t i v e l y .  serve  R=6  to N U  q  .  Evaluating  as a c h e c k on t h e v a l u e  of  number. numbers, Nu^  Q  ,  aire d e p e n d e n t upon 0,  and as shown i n F i g u r e s  3.6.1  show t h a t  t o the pure  i n comparison  to  3.6.4.  The  conduction  Ra,  curves value  73 (shown as a d a s h e d has  sphere  reduced rate  and  transfer  i s t h e same.  3.5  from this  the  d e p e n d on inner  0 - 1 3 0 ° and  tours  As  the  sphere  for NU .  acts  as  i . e . OJ. 125°  half  the  For  0 as  Ra  increases,  inferred and  in  and section  150°.  However, and  a  conjugateness  the  NU  q  3.5  respectively from  transfer As  the  of  the  shows t h e t u r n i n g p o i n t s  the  the i n n e r sphere,  annular space  as  as  a and  t a k i n g p l a c e i n the  decreases  and  the  of these t u r n i n g p o i n t s  i n section  i s a maximum.  outer  locations  and  r e g i o n near  spheres  0 <_ 180°  f o r 120°  F i g u r e 3.6.5  for  case.  moves t o w a r d s  a maximum v a l u e f o r Nu^  i f i t were a r e s e r v o i r .  rate  transfer  of the  i n n e r and  120°  rate  mentioned  with  Ra-1000,  temperature  Thus t h i s  (or f l u x )  hot  fluid  is  from  the  before i n section  between the c o n c e n t r i c The  con-  there i s vigorous  fluid.  fluid  f l o w s downwards a l o n g t h e o u t e r s p h e r e .  transfer  and  of the heat  OJ d e c r e a s e s  profiles  The  q  the bottom o f the  from both  3.6.4.  l o c a t i o n where t h e h e a t  inner  it  and  temperature  c o n v e c t i v e motion  i n the upper  rate  the  o u t e r sphere  deviation  n o t v a r y much w i t h -*  inferred  in this  transfer  two-thirds of  o f the upper  i . e . 0 = 0 ° as  t h e v a l u e s o f Ra,  sphere  OJ=10 .  rate  region there e x i s t s  minimum v a l u e  lower  This angular p o s i t i o n  3.6.1, 3.6.3  Ra=720, Nu. do i ,o  the  the heat  c o n d u c t i o n v a l u e i s a t 0=0° i n each  of the spheres  Figures  figures)  greatest  angular p o s i t i o n  where t h e h e a t  top  The  the pure  i s an  spheres  nearly half  elsewhere.  from  There  at  i n the  been i n c r e a s e d f o r about  inner  in  line  spheres  i s c o o l e d as  Hence, the  flows i n t o  3.3,  the  heat  reservoir.  In  the  the  region  fluid  near  the  hotter cold the  a r o u n d .0-130°  near  the  outer  rising  fluid heat  sphere fluid  will  Nu  will  So  some  of  e n t r a i n e d by inner  the  to be  be  the  into  rate  i s vigorous  sphere.  will  near  flow  transfer  conduction.  inner  there  than  a minimum n e a r  fluid  relatively remaining  This  1  of  'cold'  The  'reservoir . lower  the  the  sphere.  a value  convection  will  reduce  that of  0=125°  pure  s i n c e most  of  o the  heat  flux  i s convected  sphere.  This  results  section  3.5. The  for  the  OJ,  low  the  Ra  of  Ra  Ra=720,  a  Figures  3.6.1  that  of  and  more c o n d u c t i v e  as  as  G  displace  the  fore  will  a  NU  q  on  low  a does  Nu. i ,o  for a  For  a  to  increase near  the  of  top  significant ^ For  a  fixed plays  OJ and  of  a=.72, N U a  fixed  in this the  is  q  fluid  increases.  of  to  low  be  Ra,  This  will  case.  outer  greater  There-  sphere  as  decreases. It  higher  will  order  now  effect  be  shown  f o r the  that  the  overall  see  significantly,  for a  upwards  inner  conduction  value  consider  Thus  the  isotherms,  very  thus  Nu.  effects)  further  the  thesis.  fixed  value  a decreases.  near  1  alter  I t i s usual  isotherms  of  i s not  in this  not  (buoyancy  fluid  convection  Hence  3.6.2.  f o r a=10.  a decreases,  to  role.  change  a  the  distortion  considered  corresponds  predominant  than  i n the  influence of  range  away b y  influence of  Nusselt  a  numbers.  is  a  75 Let  1®. 3R  =  -  § (0-1)  + TNI X c o s 0 +  (TN2 + T N 3 ) x ( 3  cos 0-l) 2  "+ TN4  where  TNI x c o s 0  =  30° ~=-  (6a) x  aR  TN2  x  2 (3 c o s ' 0 - 1 )  TN3  x  (3 c o s  0  =  8  9  0  (G^a) x  2 2  0 - 1)  + TN4  =  (G a ) x 2  2  « i oR  i.e.  30 —  30° =  3R  Hence  30°o 30^ + Ga — - + G a — - +  3R  3R  (6-1)  Nu. l  26  80 G o — 2  3R  30  sin  8R  0 d0  3R R=l  =  1 -  TN4  (3.6.3) R=l  76 Similarly NU  1 - 6  =  Q  2  x TN4  R=3  Equations  (3.6.3) and (3.6.4) show t h a t t h e v a l u e s  o v e r a l l N u s s e l t numbers a r e a l t e r e d  from  o f the  t h a t o f pure 2  conduction  a l o n e by TN4.  independent f o r e Nu. 2  is  of 8  TN4 comes f r o m  i . e . the zeroeth s p h e r i c a l harmonic.  i,o 2 Ra . increases with  i n c r e a s e o f 3 and Ra, and as '  F i g u r e 3.6.5 and T a b l e  3 and Ra, and as OJ t e n d s  I.  to unity, this  t h e gap o f t h e c o n c e n t r i c s p h e r e s  the s o l i d  inner sphere.  increases.  there  There-  •*  OJ d e c r e a s e s ,  heat  and i s  a r e f u n c t i o n s o f Ra o n l y s i n c e t h e c o e f f i c i e n t o f  Nu. ^ i, o  in  The t e r m  I f 3 i s small i . e .  transfer  values of  implies that the f l u i d  becomes as c o n d u c t i v e  the heat  transfer  rate  t h a t o f p u r e c o n d u c t i o n as  no c o n v e c t i v e m o t i o n i n t h e f l u i d .  same v a l u e o f 1.12 f o r t h e o v e r a l l N u s s e l t  number  as t h a t g i v e n by Mack and H a r d e e f o r 3=2.0, Ra=10 0 0 i s obtained here  as  1 _< 3 ^ 1.25, t h e o v e r a l l  rate i s essentially  i s relatively The  Therefore  For fixed  f o r computing the conjugate  c a s e when OJ=10  15  0° Figure  30° 3.6.1  60°  90°  e  N u s s e l t Number A g a i n s t A n g u l a r a=.72, G=10  120° Position,  150°  180°  C o n j u g a t e C a s e OJ=10 , 3=2.0,  00  Figure  3.6.4  N u s s e l t Number A g a i n s t a=.72, G=1400  Angular P o s i t i o n ,  Conjugate  C a s e , u>=10, 6=2.0. ' ' '  00  o  500 Figure  3.6.5  R  "  O v e r a l l N u s s e l t Number as F u n c t i o n  1000 of Rayleigh  1500 Number, C o n j u g a t e  Case •  00  TABLE I OVERALL NUSSELT NUMBERS FOR CONJUGATE CASE, 3 = 1.15, 2.0, and 3.0  Ra  Ra  *  *  O V E R = Ra x (  3  A L L  NU M B E R  N U S S E L T  )  s-r G  0  co=10 1.000  8=3.0  8=2.0  8=1. 15 OJ=10  w=10  OJ=5  co=10  w=5  1.000  1.00  1.00  1.00  1.00  1.00  1.000  1.00  1.00  1.00  1.06  1.08  1.000  1.000  1.01  1.01  1.01  0.72  1.000  1.000  1.02  1.02  1.03  0.10  1.000  1.000  1.02  1.04  1.06  10  1.000  1.000  1.03  1.04  1.06  10  1.001  1.002  1.12  1.17  1.26  1.001  1.002  1. 12  1.17  1.26  1.001  1.004  1.26  1.36  1.57  0.72  1.001  1.004  1.26  1.36  1.57  5  1.004  1.008  1.74  10  10  0.01  100  10  0.10  500  10  0.50  720  10  1000  10  1000  10  2000  200  2000  2800  3000  300  3000  4000  5000  1000  0.72 10  • 1.000  oo to  83  4.  The  purpose  of the experimental  flow v i s u a l i z a t i o n  study  p a t t e r n b e t w e e n two e q u i p m e n t was possible The  butions  on  that  steady,  a l . [6,7]  isothermal  on  t y p e had  experimental  o f two  4.1.2  and  as  to give a  better  temperature  distri-  They were t o  between the c o n c e n t r i c  axisymmetrical. been performed  apparatus,  by  Bishop  shown i n F i g u r e 4.1.1-,  of.,;two c o n c e n t r i c s p h e r e s . 6"  diameter  The  plexiglas  a 3" d i a m e t e r  i n n e r sphere  hemispheres.  hemispherical  f o u r thermocouples  the e i g h t  sphere.  The  thermocouples  positions  o f the  inside on  the  was  Each  of  cavity.  shows t h e a r r a n g e m e n t o f t h e c y l i n d r i c a l  element with the  inner  of t h i s  as c l o s e l y  Apparatus  the hemispheres had  cavity  and  experimental  i n the previous chapters.  f l o w i n t h e gap  laminar  convection  spheres.  The  Figure  the  a  the c o n v e c t i o n p a t t e r n s between c o n c e n t r i c  Experimental  made o u t  model o u t l i n e d  r e s u l t s were e x p e c t e d  Experiments  consisted  The  the s u r f a c e o f the i n n e r sphere.  s p h e r e s was  4.1  natural  i t resembled  o f t h e f l o w p a t t e r n s and  c o n f i r m as w e l l  et  of the conjugate  so t h a t  the a n a l y t i c a l  understanding  i n v e s t i g a t i o n was  c o n c e n t r i c spheres.  designed  experimental  EXPERIMENT  heating  spherical  the s u r f a c e of  thermocouples  are  the shown  Figure  4.1.1  Experimental  Apparatus  Figure  4.1.2  Plexiglas  Inner  Sphere  86 in  F i g u r e 4.1.3.  glas for  The i n n e r s p h e r e  s t e m , a c e n t r a l h o l e was d r i l l e d  cavity). from  The s i l i c o n e  polyethylene  of silicone  o i l into  paint  t o reduce  facilitate  o u t e r sphere  half  o f the o u t e r sphere  hole  a 2" d i a m e t e r  silicone  polystyrene concentric  support  of light  from  heat  of the cavity.  watts. a flat  black  i t and t h u s  work.  a .2 81" w a l l t h i c k n e s s . hole.  Corning The  Through  fitted,  g l u e d and s e a l e d w i t h  stem extended T h i s p a r t was  into  t h e gap between t h e  1 3/4" l o n g , 1" i n d i a m e t e r  a 1/2" d i a m e t e r  hole  drilled  a l o n g t h e whole l e n g t h o f t h e s u p p o r t  stem.  thermocouple  leads to the h e a t i n g  e l e m e n t were p a s s e d stems. was  then  ensured  this  o f 1/8" h i g h , m a c h i n e d on t h e 3"  i n t e r n a l l y with  wires  lower  As shown i n F i g u r e 4.1.4, p a r t o f t h e  spheres.  threaded  sprayed with  h a d a 2" d i a m e t e r  collar  stem, was  compound.  the s p h e r i c a l  c o n s i s t e d o f two 10" d i a m e t e r  g l a s s hemispheres with  outer support  was  the r e f l e c t i o n  the photographic  The pyrex  sphere  stem  tubes  o i l was r e q u i r e d t o t r a n s f e r  h e a t i n g e l e m e n t h a d an o u t p u t o f up t o 50 plexiglas  by a p l e x i -  the support  the h e a t i n g element evenly t o the envelope  The  and  supported through  t h e p a s s i n g o f two .066" d i a m e t e r  (used f o r t h e i n t r o d u c t i o n  The  was  and t h e c o n n e c t i n g through  the c e n t r a l hole of both  The i n n e r stem w i t h a 1" d i a m e t e r , screwed onto  t h e o u t e r stem.  the c o n c e n t r i c i t y  The  o f t h e two  This  1/4" h i g h  support collar  arrangement  spheres.  87  gure  4.1.3  L o c a t i o n s o f Thermocouples  on S u r f a c e  o f Inner  Sphere  Figure  4.1.4  Support  Stem w i t h  Glass  Hemisph  Figure  4.1.5  Top View o f S u p p o r t Stem and G l a s s  Hemisphere  90 Smoke c o u l d be concentric running  s p h e r e s t h r o u g h one  through  stem, F i g u r e  4.1.5.  o u t e r s p h e r e was 0 ° , 3 0 ° , 90° The  The  and  glue.  4.1.1.  temperature bath.  The  The  The w a t e r  held  t a n k was  controller  University  taped to the side  'Brisket' h e a t i n g tape  connected t o the temperature the p l e x i g l a s base, F i g u r e  ing  thermocouple Figure  i n the tank. of the  the  c o n t r o l l e r was  4.1.1.  The  The  watts) around  a  inlet  was  'Randolph' pipes a t the  tank.  w i r e s u s e d were o f 36 gauge  4.1.6  support  i n the tank  o f i t by and  60  placed  water  was  temperature  (output c a p a c i t y  connected t o the o u t l e t  aluminium b a c k - p l a t e of the  for  shown  Columbia,  o f the o u t e r sphere  from the top t o the bottom  type.  as  d e s i g n e d by  of B r i t i s h  s e n s i n g probe  Alumel  30")  brass  *  platinum wire temperature  The  x  two  a  used t o p r o v i d e a constant  temperature  pump w h i c h was  mounted o n t o  (30" x 28"  used t o c o n t r o l the water  circulated  of the  at positions  i n p o s i t i o n by  tank  A temperature  G e o p h y s i c s Department,  A  support  were s e a l e d t o -  thermocouples  s u p p o r t stem was  o f a water  ;  stem.  diameter holes  s u r f a c e temperature  four  b a s e was  :  c o n t r o l l e r was  5/8"  the  120°.  o u t e r sphere  t o t h e frame  in Figure  o f t h e two  Pyrex hemispheres  m e a s u r e d by  p l e x i g l a s s base. bolts  t o t h e gap b e t w e e n  the whole l e n g t h o f t h e o u t e r sphere  gether with s i l i c o n e  of  introduced  shows a t y p i c a l  the thermocouples. As t h e c a l i b r a t i o n The h e l p o f D r . R u s s e l and Mr. B. this item i s g r a t e f u l l y acknowledged.  Chromel-  calibration  curve  data f o l l o w e d the Goldberg i n design-  9  v  Manufacturer's Experimental  I  I  1  40  60  80  data data  TEMPERATURE / °C T y p i c a l Thermocouple C a l i b r a t i o n  Curve  gure  4.1.8  Layout  of Experimental  Apparatus  94 manufacturers' couples was  data  very  were c a l i b r a t e d .  The  also calibrated, Figure The  l i n e d with enclosure  s i d e s o f the black  was  lining  illuminating  of  light  On  the  into  graphic vertical  t a n k were e i t h e r  s i d e of the light.  Two  t r a v e r s e d the  adjacent  tank  observation  of the  lining  between the  Experimental  i n n e r s p h e r e was  heating-rates t i m e was  increased. heating D.C.  until  allowed A L-C  current  the black the  projectors  that a v e r t i c a l  of the  paper  concentric  to allow v i s u a l  and  plane  spheres. was  photo-  flow p a t t e r n i n the i l l u m i n a t e d c o n c e n t r i c spheres. apparatus  Figure  4.1.8  assembled.  Procedure  Owing t o t h e the  so  slide  A  tank a r e c t a n g u l a r opening  shows t h e w h o l e e x p e r i m e n t a l  4.2  slit  mm  or  whole  work.  cut i n t o  5 0 0 - w a t t s 35  of the  probe  painted black  for collimating  centre plane  paper  thermo-  sensing  photographic  w i d t h was  s i d e o f the  the b l a c k  plane  wire  of the  4.1.8.  1/8"  were p o s i t i o n e d i n f r o n t of  three  platinum  dark t o f a c i l i t a t e  a t one  only  c o n s t r u c t i o n p a p e r so t h a t t h e  c e n t r a l narrow s l i t  cut  closely,  low  thermal  heated the  up  slowly  u n i t was  ( s u p p l i e d by  power s o u r c e ) .  At  i n s m a l l increments  d e s i r e d v a l u e was  to elapse before filter  c o n d u c t i v i t y of p l e x i g l a s  the  achieved.  the h e a t i n g r a t e  Sufficient was  u s e d t o smooth o u t  a Heathkit same t i m e ,  r e g u l a t e d low the water  of  the voltage  temperature  95 in  t h e t a n k was k e p t c o n s t a n t .  a period  o f a p p r o x i m a t e l y two d a y s  steady-state,  such t h a t  constant over t h i s Cigar the c o n c e n t r i c (Figure sphere  Before each e x p e r i m e n t a l r u n , allowed to achieve  the thermocouples'  r e a d i n g s were  period.  smoke was  gently  introduced into  spheres through e i t h e r  4.1.1) w h i c h s u p p o r t stem  the tank.  was  connected  one o f t h e two t u b i n g s  t h e two 1/8" h o l e s i n t h e o u t e r  t o t h e two o p e n i n g s  Sufficient  t i m e was  t h e gap between  a t the back p l a t e o f  allowed f o r the  of the flow p a t t e r n b e f o r e photographs  stabilization  were t a k e n .  camera and Kodak TRI-X f i l m w i t h a s p e e d o f 400 used  f o r photography.  j e c t o r s was This  source  Focusing  from t h e l i g h t  the camera d i r e c t l y  p l a n e was d i f f i c u l t  taken.  sphere  surface  source.  on t h e i l l u m i n a t i n g  and u n s u c c e s s f u l .  u s e d w h e r e b y t h e c a m e r a was  ASA were  were  heating of the inner  absorption of radiation  mm  f r o m t h e two p r o -  s w i t c h e d on o n l y when p h o t o g r a p h s  avoided supplementary  through  was  The l i g h t  A 35  So an i n d i r e c t  focused onto  a finely  method printed  surface p l a c e d adjacent t o the outer sphere, i n the plane o f the path o f the i l l u m i n a t i n g attached tions  A polarizing  filter  was  t o t h e camera l e n s t o e l i m i n a t e most o f t h e r e f l e c -  from the inner  1/2  second  and  F8 r e s p e c t i v e l y .  Kodak D-19 speed  light.  sphere.  Exposure  o r 1 second w i t h lens  apertures setting  The e x p o s e d  f i l m was  d e v e l o p e r f o r 10 m i n u t e s  t o 800 ASA.  c o n t r a s t paper  t i m e s were  The p h o t o g r a p h s  for better contrast  either a t F5.6  developed using  so as t o r a i s e were p r i n t e d o f t h e smoke  the f i l m  on h i g h patterns.  96  4.4  Experimental Figures  pattern  i n the  under t h r e e the  outer  tor. of  4.4.1 gap  t o 4.4.3  between t h e  g l a s s sphere,  T h i s caused  There are sphere  photographs o f the  c o n c e n t r i c spheres  t h e b r i g h t and  sphere.  Due  There are  two  two  some d u s t p a r t i c l e s  halves  on  the  line  the r e f l e c t i o n  of the  The  of l i g h t  reflection  starting  outer by  o f the  spots  surface  outer  seen t h a t the spheres axis.  and  the  and  axisymmetrical  diameter  ratio  temperature d i f f e r e n c e s , the eddy  1  type  can  essentially  (a) t h e  support  (b)  the  slowly.  The  The  regions  stem  i t is  vertical f o r a l l three  velocity  flow  as  fluid  'crescentpatterns  follow:  of each  of high v e l o c i t y  fluid  is  observations  (3=1.67) and  c e n t r a l eddy r e g i o n where t h e  relatively  photographs.  gap.  about the  immediate v i c i n i t y  i s a thin-layer  outer  between t h e c o n c e n t r i c  analysis. two  sphere.  o f gap  flow p a t t e r n i s of the  divided into  r e g i o n i n the  where t h e r e  used  as p r e d i c t e d i n t h e be  gap  the  thermocouples.  sphere  photographic  flow p a t t e r n i n the  i s steady At  visual  surface  the  i n the top  of  reflec-  the  outer  a p p e a r s as a w h i t e r e g i o n i n t h e b o t t o m o f t h e From b o t h  a  across  shadows o f  from the  sphere the  as  s u r f a c e o f the  w h i c h a p p e a r as b r i g h t s c a t t e r the white  thickness  dark l i n e s the  flow  obtained  dark r e g i o n s over  These are  s e a l between t h e  Similarly,  t o the  i t s inner surface acted  i n the photographs.  horizontal  are  operating conditions.  the outer  gap  Results  sphere  flow; i s moving  near the  inner  sphere  98  100  i s h i g h e r than t h a t near the o u t e r sphere.  At the top o f the  i n n e r sphere there i s a j e t - l i k e flow p o l a r i z a t i o n .  The  f l u i d separates from the i n n e r sphere near the top and flows downward along the o u t e r sphere i n t o the bottom o f the gap. T h i s r e g i o n i s r e l a t i v e l y stagnant and a c t s as i f i t were a reservoir.  The c e n t r e of the eddy i s i n the upper h a l f o f  the gap between the c o n c e n t r i c spheres.  Its position  r e l a t i v e l y s t a t i o n a r y f o r a l l three temperature  remains  differences  tested. Bishop e t a l . [7] had r e p o r t e d t h a t i n t h e i r  experi-  ment on n a t u r a l convention between c o n c e n t r i c i s o t h e r m a l spheres, as the h i g h speed flow separated from the top of the i n n e r sphere, a corner eddy was  observed i n the 'corner'  formed by the i n t e r s e c t i o n o f the s u r f a c e o f the i n n e r sphere w i t h the v e r t i c a l a x i s of symmetry.  However, t h i s phenomenon  i s not observed here f o r n a t u r a l conjugate c o n v e c t i o n . Tables I I , I I I and IV show the temperature  distributions  on the s u r f a c e o f the i n n e r sphere, the temperature o f the o i l i n various positions  (Figure 4.1.2) i n the c a v i t y  inside  the i n n e r sphere and the s u r f a c e temperature o f the o u t e r sphere.  In each case, the temperature on the s u r f a c e of the  i n n e r sphere decreases as 6 i n c r e a s e s , as p r e d i c t e d  theoretically.  The f o l l o w i n g Tables (II to IV) show the temperature d i s t r i b u t i o n on the s u r f a c e s of the c o n c e n t r i c spheres a t the p o s i t i o n s ;  i n d i c a t e d i n F i g u r e 4.1.2.  101 TABLE I I  6 =  1. 4 w a t t s Position  AT =  G =  „024°C  Temperature  1.32 x  Position  10  4  Temperature  (1)  0°  28.3  (9)  A  33°C  (2)  30°  28°C  (10)  B  34.2°C  (3)  45°  28°C  (11)  C  -  (4)  60°  27.5°C  (12)  D  36.6°C  (5)  90°  26.5°C  (13)  0°  (6)  120°  -  (1.4)  30°  ( )  150°  25.5°C  (15)  90°  (8)  165°  25.3°C  (16)  120°*  * * *  24.5°C 24.5°C 24.5°C 24.5°C  *  Outer Sphere. TABLE I I I  • Q = 4. 29  watts  Position  AT =  7.21°C  G =  /  3.72 x  Position  Temperature  10  5  Temperature  (1)  0°  35.5°C  (9)  A  48.5°C  (2)  30°  35.4°C  (10)  B  49.5°C  (3)  45°  35.1°C  (11)  C  36.2°C  (4)  60°  32.5°C  (12)  D  54 °C  (5)  90°  31°C  (13)  (6)  120°  -  (14)  (7)  150°  28.8°C  (15)  90°  (8)  165°  28.3°C  (16)  120°*  * Outer  Sphere.  * 0°  *  30°  *  25.3°C 25.3°C 25.3°C 25.3°C  1 0 2  TABLE IV • Q = 6.14 w a t t s  AT = 10. 3°C ,  Position  Temperature  G = 5.78 x 10'  5  Position  Temperature  (1)  0°  37.5°C  (9)  A  61°C  (2)  30°  37.0°C  (10)  B  68.5°C  (3)  45°  37.0°C  (11)  C  -  (4)  60°  36.8°C  (12)  D  72.5°C  (5)  90°  36.5°C  (13)  0°  (6)  120°  -  (14)  30°  (7)  150°  31.4°C  (15)  90°  (8)  165°  31.2°C  (16)  120°  * * * *  24.5°C 24.5°C 24.5°C 24.5°C  * Outer sphere.  Although for  the experimental r e s u l t s  t h e v a l u e s o f w=15,  h i g h e r G numbers  6=1.67, a=0.72 and a p p r e c i a b l y  than those f o r which the p e r t u r b a t i o n  e x p a n s i o n s w i l l be v a l i d , distributions similar  obtained here are  the flow p a t t e r n  on t h e s u r f a c e o f t h e i n n e r  and t h e t e m p e r a t u r e sphere are q u a l i t a t i v e l y  t o those p r e d i c t e d by the a n a l y s i s .  experimental  results  the c o n c e n t r i c assumptions  Moreover, the  confirm that  t h e f l o w i n t h e gap b e t w e e n  spheres i s steady,  l a m i n a r and a x i s y m m e t r i c a l ,  used i n the a n a l y s i s .  103  5.  The  CONCLUSIONS  p r e s e n t i n v e s t i g a t i o n has l e d t o the f o l l o w i n g  results: 1.  Theoretical for  steady  solutions  f o r the governing  equations  laminar axisymmetrical conjugate  c o n v e c t i o n b e t w e e n two c o n c e n t r i c s p h e r e s obtained.  The c a s e o f a c o n s t a n t f l u x  natural  were  inner  s p h e r e w i t h an i s o t h e r m a l o u t e r s p h e r e was s o l v e d separately. 2.  The l i m i t s  of the a p p l i c a b i l i t y  o f the s o l u t i o n s  were d e f i n e d . 3.  The s t r e a m l i n e c o n f i g u r a t i o n was f o u n d the crescent-eddy  type.  t o be o f  The e x i s t e n c e o f s e c o n d a r y  c e l l s was f o u n d n o t t o be a g e n u i n e f e a t u r e o f either  the conjugate or the non-conjugate  considered 4.  Contours  here.  of isovorticity  distributions fluid  cases  lines,  of velocity  i s o t h e r m s and  and t e m p e r a t u r e  i n t h e gap b e t w e e n t h e c o n c e n t r i c  of the spheres  were o b t a i n e d a n d d i s c u s s e d . 5.  L o c a l heat  transfer  s p h e r e s were 6.  The i n f l u e n c e  rates  from both  i n n e r and o u t e r  determined. o f P r a n d t l number on t h e o v e r a l l  t r a n s f e r r a t e was f o u n d  heat  t o be a h i g h e r - o r d e r e f f e c t .  10 4 7.  The  flow p a t t e r n obtained experimentally i s  steady,  laminar,  crescent-eddy 8.  The  least  and  of  the  type.  experimental  the v e r y  axisymmetrical  results  support  qualitatively.  the  analysis  at  REFERENCES  G.K. B a t c h e l o r , H e a t t r a n s f e r by f r e e c o n v e c t i o n a c r o s s a c l o s e d c a v i t y between v e r t i c a l b o u n d a r i e s a t d i f f e r e n t temperatures, Q u a r t . J . A p p l i e d M a t h . 3, 1954, 209-233. G. P o o t s , H e a t t r a n s f e r by l a m i n a r f r e e c o n v e c t i o n i n e n c l o s e d p l a n e gas l a y e r s , Q u a r t . J . Mech. A p p l i e d M a t h . 11, 1958, 257-273. J.O. W i l k e s and S.W. C h u r c h i l l , The f i n i t e - d i f f e r e n c e computation o f n a t u r a l convection i n a r e c t a n g u l a r e n c l o s u r e , A . I . Ch. E . 12_, 161-166. J.W. E l d e r , N u m e r i c a l e x p e r i m e n t s w i t h f r e e c o n v e c t i o n i n a v e r t i c a l s l o t , J . F l u i d Mech. 24, 1966, 823-843. G. de V a h l D a v i s , L a m i n a r N a t u r a l c o n v e c t i o n i n an enclosed rectangular cavity, Intern. J . Heat/Mass T r a n s f e r 11, 1968, 1675-1693. E.H. B i s h o p , R.S. K o l f a t , L.R. Mack and J.A. S c a n l a n , P h o t o g r a p h i c s t u d i e s o f c o n v e c t i o n p a t t e r n s between c o n c e n t r i c s p h e r e s , S . P . I . E . J . 3^ 1964, 47-49. E.H. B i s h o p , R.S. K o l f a t , L.R. Mack and J.A. S c a n l a n , C o n v e c t i v e h e a t t r a n s f e r between c o n c e n t r i c s p h e r e s , P r o c . 1964 H e a t T r a n s f e r / F l u i d Mech. I n s t . , ( S t a n f o r d U.P. 1 9 6 4 ) , 69-80. E.H. B i s h o p , L.R. Mack and J.A. S c a l a n , H e a t t r a n s f e r by n a t u r a l c o n v e c t i o n b e t w e e n c o n c e n t r i c s p h e r e s , I n t e r n . J . Heat/Mass T r a n s f e r 9_, 1966, 6949-662. L.R. Mack-and E.H. B i s h o p , N a t u r a l c o n v e c t i o n b e t w e e n h o r i z o n t a l c o n c e n t r i c c y l i n d e r s f o r low R a y l e i g h numbers, Q u a r t . J . Mech. A p p l i e d M a t h . 2_1, 19 68, 223-24.1. L.R. Mack and H.C. H a r d e e , N a t u r a l c o n v e c t i o n b e t w e e n c o n c e n t r i c s p h e r e s a t low R a y l e i g h numbers, I n t e r n . J . Heat/Mass T r a n s f e r 11_, 1968, 387-396. T.L. P e r e l m a n , On C o n j u g a t e d p r o b l e m s o f h e a t t r a n s f e r , I n t e r n . J . o f Heat/Mass T r a n s f e r 3, 1961, 293-303.  106  12.  Z. Rotem, The e f f e c t o f t h e r m a l c o n d u c t i o n o f t h e w a l l upon c o n v e c t i o n f r o m a s u r f a c e i n a l a m i n a r b o u n d a r y l a y e r . I n t e r n . J . Heat/Mass T r a n s f e r 10, 1967, 461466.  13.  M.D. K e l l e h e r and K-T. Yang, A s t e a d y c o n j u g a t e h e a t t r a n s f e r p r o b l e m w i t h c o n d u c t i o n and f r e e c o n v e c t i o n , A p p l i e d S c i . R e s . 1 7 , 1967, 249-269.  14.  G.S.H. L o c k and J . C . Gunn, L a m i n a r a downward-projecting f i n , J . Heat S e r i e s C, 1968, 63-70.  15.  E . J . D a v i s and W.N. G i l l , The e f f e c t s o f a x i a l c o n d u c t i o n i n t h e w a l l on h e a t t r a n s f e r w i t h l a m i n a r f l o w , I n t e r n . J . Heat/Mass T r a n s f e r 1 3 , 1970, 459-470.  16.  Z. Rotem. C o n j u g a t e f r e e c o n v e c t i o n f r o m h o r i z o n t a l c o n d u c t i n g c i r c u l a r c y l i n d e r s , I n t . J . o f Heat/Mass T r a n s f e r 19, 1971, i n p r e s s .  17.  L . K n o p o f f , The c o n v e c t i o n c u r r e n t h y p o t h e s i s , R e v i e w s o f G e o p h y s i c s 2_, 1964, 89-122.  18.  M. Van Dyke, P e r t u r b a t i o n methods i n f l u i d Academic. P r e s s 196 4.  19.  F . E . F e n d e l l , L a m i n a r n a t u r a l c o n v e c t i o n a b o u t an i s o t h e r m a l h e a t e d s p h e r e a t s m a l l G r a s h o f number, J . F l u i d Mech. 34, 1968> 163-176.  20.  C.A. H i e b e r and B. G e b h a r t , M i x e d c o n v e c t i o n f r o m a s m a l l s p h e r e a t s m a l l R e y n o l d s and G r a s h o f numbers, J . F l u i d Mech. 3_8, 1969 , 137-159.  21.  J.A. S c a n l a n , E.H. B i s h o p and R.E. Powe, N a t u r a l c o n v e c t i o n h e a t t r a n s f e r between c o n c e n t r i c s p h e r e s , I n t e r n . J . Heat/Mass T r a n s f e r 1_3, 1970 , 1857-1872.  22.  J . Proudman and J.R.A. P e a r s o n , E x p a n s i o n s a t s m a l l R e y n o l d s : numbers f o r t h e f l o w p a s t a s p h e r e and a c i r cular cylinder. J . F l u i d Mech. 2_, 1957, 237-262.  23.  J . J . Mahoney, Heat t r a n s f e r a t s m a l l G r a s h o f P r o c . Roy. S o c . A238, 1957, 412-423.  24.  Md. A. H o s s a i n and B. G e b h a r t , N a t u r a l c o n v e c t i o n a b o u t a s p h e r e a t low G r a s h o f number, F o u r t h I n t e r n . Heat T r a n s f e r C o n f e r e n c e , P a r i s , 1970, V o l . 4, 1-12.  f r e e c o n v e c t i o n from T r a n s f e r ASME  mechanics,  number.  107 APPENDIX I  CALCULATION  The  energy e q u a t i o n o f the f l u i d  dissipation spherical  OF VISCOUS DISSIPATION EFFECT  terms  (with r o t a t i o n a l  V T - pC  (V« ^ ~ 3R'  2  f  P  where y $  R  21  + -1 R« 36  i s the d i s s i p a t i o n  v  viscous effects  I $  =  2  8 V  per unit  R  (  <  I  2  —  V  expressed i n  3  V  The  3R  R  V  e  1  =  0  (A-l)  V  The d i s s i p a t i o n  38  function,  R e + (_£. + J ± c o t 6) R R' V  2  S.)  v  ^R  R  R'  1  38  dimensionless r a d i a l  = -v T  y $  1  are introduced  V  +  volume.  i ( J * + ± R' • R'  +  )  t e r m o f m e c h a n i c a l e n e r g y by  R  3R  R'  velocity  symmetry)  the viscous  coordinates i s  V  K  including  as  and t a n g e n t i a l  components o f  follows,  R! V  R  ;  V,  Rendering the energy e q u a t i o n  V  1  (A-l) dimensionless,  10 8  y 0 - a (V_ l i + l i 1®) + 3R R 86  =  2  where t h e v i s c o u s d i s s i p a t i o n  V  0  (A-2)  V  number i s  2 y v R Kj, AT _ i f ref  =  2  Let  e  v  =  V 0 2  denote t h e d i m e n s i o n l e s s to  the conduction  r a t i o of the viscous d i s s i p a t i o n  term  term.  Let  v  X =  a (V 30_ 3R  denote t h e d i m e n s i o n l e s s to the convection For  ^0 30) R 39  r a t i o o f the viscous d i s s i p a t i o n  term.  a i rwith  G=1000, a=0.72, R | = l f t , 3=2.0, OJ=10  and  P=1.146 x 10 ^ ° , t h e v a l u e s o f |e| and |x| a t b o t h  and  radial positions  spheres  term  angular  i n t h e gap b e t w e e n t h e two c o n c e n t r i c  a r e g i v e n i n T a b l e V.  109  TABLE V RATIO OF VISCOUS DISSIPATION TERM TO CONDUCTION TERM: RATIO OF VISCOUS DISSIPATION TERM TO CONVECTION TERM (AIR)  R  1.3  1.1  1 1 e  6. 5 8 x l 0  1 1  3. 1 6 X 1 0 "  9  9. 2 5 x l 0 -  1 0  1. 8 4 x l 0 ~  9  9. l O x l O "  1 0  7. 9 5 x l 0 ~  1 0  2. 2 8 x l 0 ~  1 0  3. 5 4 x l 0  1 0  1. 6 4 x l 0 ~  1 0  1. 4 5 x l 0 ~  1 0  4. 9 4 x l 0 ~  1 0  1. 4 3 x l 0 ~  1 1 1. 8 9 x l 0 ~ e  1 1 5. 4 2 x l 0 " A  1 1 e  3. 2 3 x l 0 ~  1 1 8. 0 0 x l 0 ~ x  e  1.8  3. 5 6 X 1 0 "  - 1 1  |X| . 2. 7 2 x l 0 ~  1.5 1 1  8  1 0  - 9  1 0  9  3. 2 7 x l 0  _ 1 1  15°  5. O O x l O "  1 0  15°  8. I S x l O -  1 0  60°  1. 8 0 x l 0 ~  1 0  60°  1. 4 9 x l 0 "  1 0  135°  4. l O x l O "  1 0  135°  F o r w a t e r w i t h G=62, a=11.6, R|=l f t , 6=2.0, to=10 and -12 t?=4.42x10  , t h e v a l u e s o f |e| and | x | a t b o t h  radial positions  a n g u l a r and  i n t h e gap between t h e two c o n c e n t r i c  spheres  are given i n Table V I . TABLE V I RATIO OF VISCOUS DISSIPATION TERM TO CONDUCTION TERM: RATIO OF VISCOUS DISSIPATION TERM TO CONVECTION TERM (WATER) R  1 1 e  6. 33x10  1 1 8. 98x10 X  4. 65x10  -13 X  -13 J  x  e  J  -12  1 1 2. 1 9 x l 0 1 1  1.5  . 1.3  1.1  - 1 1  6. 7 8 x l 0 ~  1 4  4. 21x10  4. 1 4 x l 0 ~  1 2  8. 99x10  1. 22x10 6. 63x10  -12 -12 x  z  9. 45x10  1  x  -13 x  1. 3 3 x l 0 ~  - 1 3  1. 8 1 x l 0 ~  -11 2. 58x10 -  1. 0 5 x l 0  _ 1 1  7. 6 0 x l 0  | x\  L  3. 0 7 x l 0 ~ 9. 0 5x10 1. 17x10  1 3  - 1 1  -12 ' X  1. 3 7 x l 0 5. 62x10  1 4  -12  2. 8 0 x l 0 "  9  2. 5 7 x l 0  X  4  -12  -13 •  5. 12x10  e  1.8  15° 15° 60°  1 2  60°  - 1 3  135°  -12  135°  110 T a b l e s V and V I show t h a t  the r e l a t i v e order of  magnitude o f the v i s c o u s d i s s i p a t i o n duction  term  term t o e i t h e r  (| e | ) o r t h e c o n v e c t i o n t  e  r  m  the con-  (I ^ I ) i s at least  ~8 of  t h e o r d e r o f 10  dissipation  .  o f energy  t o be n e g l i g i b l e  Thus t h e a s s u m p t i o n i n the f l u i d  that  the v i s c o u s  assumed i n s e c t i o n  i s f o u n d t o be v a l i d .  2.1  Ill  APPENDIX I I CONJUGATE PROBLEM WITH INNER SPHERE CONTAINING DISTRIBUTED SOURCES  C o n s i d e r now t h e i n n e r s p h e r e in  section  2 with uniform d i s t r i b u t e d  of a s i n g l e heat  x V i  s  2  section  T.  =  source  % TT  x  ^ ref  function per unit  4TT  the energy  2  3  x  3 R[ K  ^  T r  e  f '  volume  assumed  defined i n  8 x — TT R 3 i 1  3  (3-D f  (3-D K  f  equation  ~ Q  equation  0;  38  ~  R!  3  B R|  y 2  The e n e r g y  (B-l)  The r e f e r e n c e t e m p e r a t u r e ,  B  Rendering  instead  B (R)  2.2, w i t h Q r e p l a c e d h e r e by  AT  sources  problem  t h e n becomes  where 8(R) i s t h e h e a t constant.  heat  source a t i t s centre.  of the i n n e r sphere  K  o f the conjugate  ( B - l ) d i m e n s i o n l e s s , i t becomes  36 = u  (3-D  (  B  ~  = c o n s t a n t . f o r g i v e n v a l u e s o f w and 6 .  2  )  112  Then t h e e n e r g y inner  sphere  conditions exactly the  equation  i s r e p l a c e d here  by e q u a t i o n  and t h e method o f s o l u t i o n  (B-2).  for this  2.2 f o r t h e The b o u n d a r y  case are  t h e same as p r e v i o u s l y d e s c r i b e d i n s e c t i o n  s e t o f uncoupled  section  (2.2.3) g i v e n i n s e c t i o n  linear  differential  2.3 r e m a i n s e s s e n t i a l l y  exception that the equation  equations  t h e same w i t h  2.  given i n  the sole  (2.3.7) i s r e p l a c e d h e r e b y  38  (B-3)  (6-D  OJ  Equation  (B-3) i s s o l v e d s i m u l t a n e o u s l y w i t h  (2.3.8) s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s (2.3.19).  Also  equation  (2.3.18) and  The s o l u t i o n s a r e  ~o OJ  (6-1)  o  (6-1) As the  the equations  0, and 0 e x p a n s i o n s  unchanged; t h e i r  for^  a  n  d t h e h i g h e r o r d e r terms o f  and t h e b o u n d a r y c o n d i t i o n s r e m a i n  solutions  are those  given i n section  2.4 .  113 Hence f o r t h e c o n j u g a t e p r o b l e m the  temperature  two  concentric  distribution  function,  0 of the f l u i d  i n t h e gap  Y  and o f the  s p h e r e s do n o t d e p e n d on what f o r m o f h e a t  source d i s t r i b u t i o n inner  the stream  inside  sphere temperature  source d i s t r i b u t i o n  the i n n e r  distribution  inside  the inner  sphere.  However, t h e  does depend on t h e h e a t sphere.  1 1 4  APPENDIX I I I A BRIEF REVIEW ON SMALL GRASHOF NUMBERS NATURAL CONVECTION ABOUT A HEATED SPHERE  In  the a n a l y s i s o f conjugate n a t u r a l c o n v e c t i o n  between c o n c e n t r i c ratio,  3"*  s p h e r e s , i t i s seen  the problem  00  reduces  source. the  The  resulting  radius sphere i n  flow f i e l d i s  plume above t h e  heat  v a n i s h e s o u t s i d e t h e plume and  s h o u l d v a n i s h everywhere above t h e s p h e r e , w h e r e i n  regular p e r t u r b a t i o n expansion  thesis For  temperature  velocity  wake r e g i o n  The  confined to a v e r t i c a l  The  as t h e  to a s i n g l e heated  an unbounded e x p a n s e o f f l u i d . essentially  that  i s inadequate except  i t i s i n the s o l u t i o n  except i n the i t s h o u l d be  ° f o r ¥^  near  narrow bounded.  scheme e m p l o y e d  i n the region  as R->-°°  in this  the  sphere.  3 as R-+°° t h e 0 (R )  (eq. 2 . 4 . 4 ) ,  o  term o f (eq.  ¥^ c o r r e s p o n d s t o an 0(R)  behaviour  [22]  V  R  3.3.1a). Thus t h e v e l o c i t y b o u n d a r y  not  in  satisfied.  This  i s analogous  f o r s m a l l Reynolds  d i m e n s i o n a l body  number  condition  at i n f i n i t y  to the Whitehead  flow past a f i n i t e  ( i . e . t h e c o n v e c t i v e e f f e c t must  c o n s i d e r e d a t the d i s t a n t  region  although d i f f u s i v e  i s predominant  effect  Therefore, there exists  away f r o m t h e  an o u t e r r e g i o n  near  paradox size threebe  sphere, the s p h e r e ) .  i n which  is  the  con-  115 vective,  d i f f u s i v e and b o u y a n c y e f f e c t s  of magnitude. expansions field.  An  inner-and-outer  w i l l be  are o f the  matched  r e q u i r e d f o r the  same o r d e r  asymptotic  solution  o f the  flow  N o t e t h a t a t a l a r g e d i s t a n c e away f r o m t h e  although sphere,  there  i s conjugate  i t will  effect  a p p e a r as a h e a t  In o r d e r  t o o b t a i n the  r e g i o n where t h e  convective  same o r d e r , Mahoney  [23]  a t the  point  surface of  equations  diffusive  i n t r o d u c e d an  the  source.  governing  and  sphere,  effects  i n the are  appropriate  outer  of.the  length  -1 scale  ( i . e . R=R.G 2) - l  together with  an  asymptotic  expansion  1  in  t e r m s o f G 2.  solutions ity  He  f o r the  noted  equations  a v e r t i c a l plume i n t h i s  expansion  outer  i n the  ing  by  the  seeking  Fendell  equations  i s based  t h e plume.  [19]  i n the The  constant magnitude.  Gebhard  [20]  by  obtained  exact  similar-  assuming the e x i s t e n c e  of  However, i t i s i m p o s s i b l e the  regular perturbation .  results  obtained solution,  approximate  and  t h e n by  solution,  lineariz-  assumed  uniform  coordinate perturbation solution  velocity  above t h e  sphere  d i s t a n c e from the  T h i s procedure only.  an  r e g i o n i n t h e manner o f  magnitude o f the  upon t h e  Hence t h e  outer  f r o m unbounded g r o w t h w i t h  qualitative  to  a similarity  Oseen's e q u a t i o n . stream  I n s t e a d he  region.  solution  of obtaining  inner region.  Recently, first  complexity  outer region.  s o l u t i o n s t o the  t o match t h e  the  In t h e i r  can be  is  reduced  sphere  to  a.  expected  to  yield  conjectures, Hieber  showed t h a t i t seems p l a u s i b l e  in  and  t o assume t h a t  116 velocity  i n t h e wake b e h a v e as a u n i f o r m  matching r e g i o n .  T h i s i s b a s e d on t h e i r  forced  from a sphere  convection  stream  only i n the  results  on f r e e a n d  a t low R e y n o l d s and  Grashof  numbers. Hossain perturbation Grashof  is  i s used  analysis to this  as t h e e x p a n s i o n  exponentially decaying  functions w i l l R->°°.  [24] e m p l o y e d a s i n g l e  scheme i n t h e i r  number  ation with  and G e b h a r t  However, t h e d i s a d v a n t a g e s  only v a l i d  condition  f o r very  s m a l l G,  problem.  The  parameter i n combin-  functions.  ensure the v e l o c i t y  parameter  These  decaying  and t e m p e r a t u r e v a n i s h of this  scheme a r e  (b) t h e t e m p e r a t u r e  a t the s u r f a c e o f the i n n e r sphere  as  (a) i t boundary  i s not s a t i s f i e d  completely. The  difficulties  numbers f r o m a h e a t e d divided  i n the a n a l y s i s o f small  sphere  seem t o be t h a t t h e f l o w  vicinity  varies  o f the sphere  f o r each r e g i o n i . e . the d i f f u s i v e  effect  (a) i n t h e  i s predominant,  (b)  i n t h e m a t c h i n g r e g i o n o f t h e i n n e r and o u t e r  the  convective, diffusive  importance, limits  very  a r e known.  of these  the v e l o c i t y  regions  i n the flow  Hence n o t a l l s c a l i n g  The t e c h n i q u e s  qualitatively.  field  natural  interesting  s h o u l d be  field  flows  zero.  are apparently  lengths or v e l o c i t i e s success-  are not s u c c e s s f u l here  Therefore  from a s i n g l e heated  and c h a l l e n g i n g .  field  are of equal  T h i s i s due t o t h e e n t i r e l y  being encountered.  convection  flow  [20] w h i c h a r e e m p l o y e d  f o r the forced convection  except flow  and b o u y a n c y e f f e c t s  (c) a t i n f i n i t y  complicated.  fully  field i s  i n t o r e g i o n s where t h e p r e d o m i n a n c e o f p a r t i c u l a r  physical effects  The  Grashof  a study sphere  of  different conjugate  w o u l d be  

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