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Conductive and convective heat transfer with radiant heat flux boundary conditions Sikka, Satish 1969

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CONDUCTIVE AND C0NVECTIVE HEAT TRANSFER WITH RADIANT HEAT FLUX BOUNDARY CONDITIONS  by  SATISH SIKKA B.Sc.  Mech. E n g . , U n i v e r s i t y Delhi,  India,  of Delhi,  1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS  FOR THE DEGREE OF  M.A.Sc. in  t h e Department of  Mechanical  We  accept  required  THE  this  thesis  Engineering  as c o n f o r m i n g  t o the  standard  UNIVERSITY OF BRITISH April  COLUMBIA  1969  r  In p r e s e n t i n g  this thesis  in partial  f u l f i l m e n t o f the requirements 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 B r i t i s h C o l u m b i a , t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e I further agree that permission  f o r extensive  I agree  that  and Study.  copying of this  thesis  f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s .  It i s understood that copying or publication  of t h i s t h e s i s f o r f i n a n c i a l written  n o t b e a l l o w e d w i t h o u t my  permission.  D e p a r t m e n t o f 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  gain shall  April  25, 1969  ENGINEERING  Columbia  ABSTRACT Some with  conductive  radiative  Three  boundary  specific  therefore,  solid  I  the  and  interplanetary  space  to  surroundings  physical  vection with  are  squares  appears  by to  The  results  a variational be  most  is  point are  linear  The  radiation-conduction parameter  absorptivity. is  effect  of  appear  to  semi-grayness  by  two  of  approximate, least-  earlier  on  the  results  fit  method  simplicity  depend or  con-  equation  (ii)  and  -con-  in.  appreciably  the  surface  receiving  surface  analysed. In  lar  fin  the base t  Part  II  the  heat  d i s s i p a t i n g heat  radiation  are  The  and  accuracy  either  the  not  state,  loss  least-squares  Its  energy  conduction by  in  receive  Steady  obtained  in-long,  sphere  radiant  and no h e a t  matching  regarding does  has,  produced  bodies  computation. on  accuracy  study  solid  compared w i t h  method.  suitable  a  Rankine.  the  conditions  and  solid  properties,  (i)  The  and emanate  zero degree  methods,  theoretically.  distribution  The  of  problems  parts.  side  Solution  boundary  fitting.  obtained  on one  at  transfer  analysed  cylinders  studied.  surface  assumed.  semi-analytical  three  temperature  flux  and  nonlinear  into  is  radiation  are  heat  been.analysed.  rectangular  parallel  stant  conditions  divided  Part  circular  their  convective  problems have  been  In  and  are of  assumed.  analysed.  the  fin  There  and is  transfer  from i t s The  characteristics  surface  temperature  constant radiant  by is  physical  interaction  of  convection  a and  assumed u n i f o r m  and  surface  between  the  circu-  along  properties fin  and  iii i t s base.  Two  separate s i t u a t i o n s are c o n s i d e r e d .  In the  first  s i t u a t i o n heat t r a n s f e r from the end of the f i n i s n e g l e c t e d . S o l u t i o n of the l i n e a r conduction equation with n o n l i n e a r boundary C o n d i t i o n s has been obtained by the l e a s t - s q u a r e s f i t method. A s o l u t i o n has a l s o been o b t a i n e d by the f i n i t e d i f f e r e n c e method and the r e s u l t s compared.  R e s u l t s are presented f o r a wide range  of  environmental  c o n d i t i o n s and p h y s i c a l and s u r f a c e p r o p e r t i e s  of  the f i n . In the second  of  the f i n i s a l s o i n c l u d e d i n the a n a l y s i s .  s i t u a t i o n heat t r a n s f e r from the  o b t a i n e d by a f i n i t e d i f f e r e n c e procedure.  end  The s o l u t i o n i s I t i s shown t h a t  n e g l e c t i n g heat t r a n s f e r from the end i s a good approximation  for  long f i n s o r f o r f i n s of h i g h thermal c o n d u c t i v i t y m a t e r i a l . In P a r t I I I the problem of laminar heat t r a n s f e r i n a c i r c u l a r tube under r a d i a n t heat f l u x boundary c o n d i t i o n s has been analysed.  F u l l y developed v e l o c i t y p r o f i l e i s assumed and  the tube i s c o n s i d e r e d s t a t i o n a r y .  A steady r a d i a n t energy  flux  i s being i n c i d e n t on one h a l f of the tube circumference w h i l e the f l u i d emanates heat through the w a l l on a l l s i d e s by  radiation  to  finite  a zero degree temperature  d i f f e r e n c e procedure  environment.  has been o b t a i n e d .  A s o l u t i o n by The  temperature  distri-  b u t i o n and the N u s s e l t number v a r i a t i o n are presented f o r a wide range of the governing p h y s i c a l  parameters.  TABLE OF CONTENTS Page ABSTRACT  . . . . . . .  L I S T OF TABLES V  v i i  L I S T OF FIGURES  viii X  ACKNOWLEDGEMENTS GENERAL PART I  INTRODUCTION  i i  . . . . . . . .  Temperature D i s t r i b u t i o n Interplanetary  . . . .  xi  i n S o l i d Bodies i n  Space  . . . .  1  ABSTRACT  2  NOMENCLATURE  3  INTRODUCTION  4  FORMULATION  AND SOLUTION OF THE PROBLEM  8  a) C i r c u l a r C y l i n d e r  9  b) R e c t a n g u l a r C y l i n d e r  13  c) S p h e r e  16  DISCUSSION OF RESULTS  19  a) C i r c u l a r C y l i n d e r  20  b) R e c t a n g u l a r C y l i n d e r  23  c) S p h e r e  . . . . .  Temperature D i s t r i b u t i o n E f f e c t o f Semi-Grayness o f S u r f a c e Properties I n c l u s i o n o f C o n v e c t i o n Heat T r a n s f e r i n the A n a l y s i s CONCLUSIONS . . . . . . . . . . .  24 25 26 27 28  V  Page PART I I  Temperature D i s t r i b u t i o n and E f f e c t i v e n e s s of a R a d i a t i n g and Convecting C i r c u l a r F i n . .  44  ABSTRACT  45  NOMENCLATURE  46  INTRODUCTION  48  FORMULATION AND SOLUTION OF THE PROBLEM  51  S e c t i o n A - No Heat T r a n s f e r from the End o f the F i n  52  DISCUSSION OF RESULTS  59  S e c t i o n B - Heat T r a n s f e r from the End o f the F i n i s Not Neglected  63  COMPARISON OF SECTION A WITH SECTION B . . . . . . .  .  66  CONCLUSIONS PART I I I Laminar Heat T r a n s f e r i n a C i r c u l a r Tube  67  under S o l a r R a d i a t i o n i n Space  81  ABSTRACT  . . . . . . . . .  82  NOMENCLATURE  83  INTRODUCTION  . .  86  FORMULATION OF THE PROBLEM  88  METHOD OF SOLUTION  92  The F i n i t e D i f f e r e n c e Procedure  92  Determination o f N u s s e l t Number  94  Transformation  o f Coordinates  . . . . . .  96  O v e r a l l Energy Balance  97  DISCUSSION OF RESULTS Average W a l l Temperature N u s s e l t Number  100 .  100 102  vi Page Angular Wall Temperature D i s t r i b u t i o n  107  O v e r a l l Energy Balance  108  CONCLUSIONS  109  BIBLIOGRAPHY  129  APPENDICES .  135  APPENDIX A — E v a l u a t i o n o f Expressions Energy Balance E r r o r  for Overall  APPENDIX B - E v a l u a t i o n o f C o n f i g u r a t i o n F  *  V  A  2  136 Factor .  .  .  .  .  .  .  .  APPENDIX C - The F i n i t e D i f f e r e n c e Procedure . . . . .  139  142  L I S T OF  TABLES  Table 1-1  1-2  1-3  1-4  1-5  1-6  1-7 1-8 1- 9  2- 1 3- 1  3-2  Page E f f e c t o f A and a on i e n t s a. o f E q u a t i o n Cylinder  the v a l u e s of the c o e f f i c (1-8) - - - c i r c u l a r 29  Comparison o f v a l u e s of the c o e f f i c i e n t o b t a i n e d by d i f f e r e n t methods - - Circular Cylinder  a„ 30  Comparison o f the o v e r a l l percentage e r r o r (Eq. l - 3 3 a ) o b t a i n e d by d i f f e r e n t methods - Circular Cylinder E f f e c t o f A and a on t h e v a l u e s o f t h e i e n t s A i o f E q u a t i o n (1-21) Rectangular C y l i n d e r . . .  31  coeffic32  Comparison of v a l u e s bf the c o e f f i c i e n t A o b t a i n e d by d i f f e r e n t methods - - Rectangular C y l i n d e r . . . . . .  33  Comparison of the o v e r a l l percentage e r r o r (Eq. l - 3 3 b ) o b t a i n e d by d i f f e r e n t methods - - Rectangular C y l i n d e r . . .  34  E f f e c t o f A and a on i e n t s a^ o f E q u a t i o n  35  0  the v a l u e s o f the c o e f f i c (1-30) - - - S p h e r e  Comparison of v a l u e s of the c o e f f i c i e n t a o b t a i n e d by d i f f e r e n t methods - - - S p h e r e Q  . . . .  36  Comparison o f the o v e r a l l percentage e r r o r (Eq. l - 3 3 c ) o b t a i n e d by d i f f e r e n t methods - - Sphere  37  Comparison of f i n e f f e c t i v e n e s s v a l u e s t h o s e o b t a i n e d by S p a r r o w and N i e w e r t h  68  with [ 3 9 ] . . . .  V a l u e s o f t h e d i m e n s i o n l e s s forms o f h e a t f l u x , a v e r a g e w a l l t e m p e r a t u r e and b u l k temperature f o r y=5 Overall error  i n energy  balance  (Eq.3-36)  . . . .  110 Ill  L I S T OF FIGURES Figure 1-1  Page Coordinate Cylinders  System  Rectangular  and C i r c u l a r  . . . .  38  1-2  Spherical Coordinate  1-3  Isothermals f o r Rectangular (B=0.5, A=0.1, a=0.5) Isothermals f o r Rectangular (B=2, A=0.1, a=0.5)  Cylinder  Isothermals f o r Rectangular (B=2, A = 1 0 ~ , a=0.5)  Cylinder  Isothermals  Cylinder  1-4 1-5  System  •  39  .  41 42  3  1- 6  (B=2,  f o r Rectangular  A=10 , - 3  a=l)  .  2- 1  C i r c u l a r F i n Geometry  2-2  Variation of Configuration Length  40  Cylinder  43 69 Factor  with  Axial  and F i n Base R a d i u s  70  2-3  E f f e c t o f A on A x i a l T e m p e r a t u r e D i s t r i b u t i o n . .  71  2-4  E f f e c t o f N on A x i a l T e m p e r a t u r e D i s t r i b u t i o n . .  72  2-5  E f f e c t o f Base S u r f a c e E m i s s i v i t y and A on Fin Effectiveness E f f e c t o f E n v i r o n m e n t T e m p e r a t u r e s on F i n  73  Effectiveness  74  2-7  E f f e c t o f N and A on F i n E f f e c t i v e n e s s  75  2-8 2-9  E f f e c t o f F i n L e n g t h on F i n E f f e c t i v e n e s s E f f e c t o f F i n B a s e R a d i u s on F i n Effectiveness  2-6  2-10 2-11  . . . .  76 77  Comparison o f A x i a l Temperature D i s t r i b u t i o n o b t a i n e d f o r Two C o n d i t i o n s  78  F i n E f f e c t i v e n e s s A g a i n s t F i n Length f o r Both C o n d i t i o n s  7.9  Figure  Page  2- 12  Finite  3- 1  Tube Nomenclature  3-2  Difference  Variation  Representation andGeometry  o f Average  w i t h E, (y=0.5) 3-3  Variation with  Critical  3-5  Variation  3-6  3-7  .  112  Temperature  113  o f Average  Wall  Temperature  • 114  Relationship o f Average  between Wall  y a n d \\>  115  Temperature  5 U=25)  Variation Y  80  5 (y=5)  3-4  with  Wall  . . .  =0.5  116  of Nusselt  Number w i t h  5for  (<|><* )  i  cr  Variation  of Nusselt  Y=0.5 U>^ )  Number w i t h  3-9  3-10  Variation  Y=0.5  • • 118  of Nusselt  (^-<P ) cr  Variation  with  Variation  for  E;  119  o f D i m e n s i o n l e s s Heat o f Wall  Temperature 3-11  Number w i t h  .  £ f o r Y=0-5  Variation  with  Flux  (^^^> ) . . . .  . 120  cr  Temperature  B, f o r Y=0 .5  of Nusselt  and Bulk (ty-ty  Number w i t h  )  . . . . . . .  122  cr  Variation (ib>\b  of Nusselt  Number w i t h  £ f o r y=5  )  123  3-13  Variation  of Nusselt  Number w i t h  £: f o r  3-14  Variation  of Nusselt  Number w i t h  £ f o r ^=25  3-15  Angular Wall  Temperature  Distribution  3-16  Angular Wall  Temperature  Distribution  (Y=0.5, 5=0.1) (^=25,  3-17  Angular {ip=25,  121  £ f o r .Y=5  (^<^ ) 3-12  7  £f o r  cr  3-8  1  Y=5)  Wall Temperature £ = 0.1)  ij>=0 . . . . 124 . . .  125  126 • • 127 Distribution 128  ACKNOWLEDGEMENTS  The  author wishes  t o e x p r e s s h i s deep g r a t i t u d e  t o Dr.  M. I q b a l who d e v o t e d c o n s i d e r a b l e t i m e on t h e g u i d a n c e o f t h i s study.  S i n c e r e thanks  f o r h i s many u s e f u l  are a l s o extended  suggestions.  t o P r o f e s s o r Z. Rotem  In a d d i t i o n , the author  also  wishes  t o acknowledge t h e v e r y u s e f u l  Fowler  and Mr. K. Teng o f t h e C o m p u t i n g C e n t r e , U n i v e r s i t y o f  British  Columbia;  Department,  a n d D r . B.D. A g g a r w a l a  University  Financial  a d v i c e g i v e n b y Mr. A.G.  o f the Mathematics  of Calgary.  support of the  N a t i o n a l Research C o u n c i l o f  Canada and use o f t h e Computing C e n t r e f a c i l i t i e s University  of British  Columbia  are g r a t e f u l l y  a t the  acknowledged.  GENERAL In  spacecraft  INTRODUCTION  applications conductive  and  convective  heat t r a n s f e r problems w i t h r a d i a t i v e boundary c o n d i t i o n s a r e frequently  encountered.  Spacecraft  components e x p o s e d t o f r e e  s p a c e d i s s i p a t e h e a t m o s t l y b y r a d i a t i o n and may a l s o solar energy  radiation.  Extended surfaces  from the s p a c e c r a f t  a circulating  often  f o r maintaining  Some o f t h e e x t e n d e d s u r f a c e s radiators,  are  fluid  being  used t o c a r r y heat  bution  i n and h e a t t r a n s f e r f r o m t h e s e s u r f a c e s  to the r a d i a t o r s .  and  circular  distri-  are of great  s p e c i f i c problems, a l l i n the  b u t o f i n c r e a s i n g c o m p l e x i t y , have b e e n  rectangular Part  been d i v i d e d  i n t o three  c y l i n d e r s and s o l i d  spheres  parts. solid  Part  I  circular  i n interplanetary ,  and r a d i a t i o n .  the problem o f laminar heat t r a n s f e r i n a  tube under r a d i a n t heat Each p a r t introduction  analyzed.  I I analyzes the heat t r a n s f e r c h a r a c t e r i s t i c s o f a  f i n d i s s i p a t i n g h e a t by c o n v e c t i o n  analyzes  parts.  The t e m p e r a t u r e  w i t h the temperature d i s t r i b u t i o n i n long,  space.  III  work t h r e e  study has, t h e r e f o r e ,  deals  from  importance.  In the p r e s e n t  The  equilibrium  may be i n t h e f o r m o f f i n - t u b e  the spacecraft  same a r e a ,  used t o d i s s i p a t e  a thermal  within  engineering  receive  f l u x boundary  contains  so t h a t  References t o published  however, combined t o g e t h e r  circular  conditions.  i t s own n o m e n c l a t u r e , a b s t r a c t  i t c a n be r e a d  Part  and  independently o f the other  literature  a t the end.  and A p p e n d i x e s a r e ,  PART  I  T E M P E R A T U R E DISTRIBUTION IN SOLID BODIES IN INTERPLANETARY SPACE  ABSTRACT The t e m p e r a t u r e d i s t r i b u t i o n circular  and r e c t a n g u l a r  cylinders  planetary  space i s s t u d i e d .  radiation  flux  surroundings physical  on one s i d e  fitting.  conditions  bodies receive  Steady  The r e s u l t s  The l e a s t - s q u a r e s  parallel  state, constant  i s o b t a i n e d by two  approximate, ( i i ) least-  are presented i n s e r i e s  results fit  convection  conduction equation with  ( i ) p o i n t m a t c h i n g and  a r e compared w i t h e a r l i e r method.  sphere i n i n t e r -  and no h e a t l o s s by  S o l u t i o n of the l i n e a r  s e m i - a n a l y t i c a l methods,  solid  and emanate r a d i a n t e n e r g y t o t h e i r  a t zero degree Rankine.  n o n l i n e a r boundary  squares  and a s o l i d  The s o l i d  and s u r f a c e p r o p e r t i e s ,  a r e assumed.  produced i n l o n g ,  o b t a i n e d by a  f o r m and  variational  method a p p e a r s t o be most  r e g a r d i n g a c c u r a c y and s i m p l i c i t y  i n computation.  suitable  I t s accuracy  d o e s n o t a p p e a r t o d e p e n d a p p r e c i a b l y e i t h e r on t h e r a d i a t i o n c o n d u c t i o n p a r a m e t e r o r on t h e s u r f a c e a b s o r p t i v i t y . of semi-grayness of the r e c e i v i n g  surface  i s analysed.  The  effect  NOMENCLATURE 2a  diameter o f c i r c u l a r c y l i n d e r , diameter or width o f r e c t a n g u l a r c y l i n d e r , f t .  o f sphere  a ^ , A^  c o e f f i c i e n t s i n assumed t e m p e r a t u r e p r o f i l e s  A  3 4 4 eaG a /k , r a d i a t i o n - c o n d u c t i o n p a r a m e t e r , dimensionless  2 b  height of rectangular cylinder, f t .  B  b / a , one h a l f o f t h e d i m e n s i o n l e s s rectangular cylinder  G  incident  k  thermal conductivity ft. °R.  R  r / a , dimensionless  T  t e m p e r a t u r e a t any p o i n t i n t h e s o l i d ,  x, y  rectangular  X, Y  x/a,y/a, dimensionless  height of  2 radiation flux,  BTU/hr.ft.  of the material,  BTU/hr.  radius °R.  coordinates rectangular  coordinates  Greek Symbols a  coefficient of absorptivity,  e  coefficient of emissivity,  dimensionless  o  Stefan-Boltzmann constant, BTU/hr.ft.2 °R4  0.1714 x 10  X  kT/Ga, d i m e n s i o n l e s s  r,  6 , 4>  dimensionless  temperature  c i r c u l a r and s p h e r i c a l  coordinates  —8  INTRODUCTION  With  advances  i n space e x p l o r a t i o n ,  interest in  problems  d e a l i n g w i t h t h e e f f e c t o f s o l a r r a d i a t i o n on s p a c e c r a f t i s becoming o f i n c r e a s i n g level out in  Not  only  i s the  o f importance, but a l s o the temperature  t h e body.  The  s e v e r a l ways.  mation the  importance.  temperature  temperature  v a r i a t i o n through-  v a r i a t i o n a f f e c t s the  I t induces thermal stresses  and  spacecraft  causes  defor-  o f t h e s t r u c t u r a l members o f t h e s p a c e c r a f t , w h i c h  motion  and  stability  of the s a t e l l i t e .  affects  Furthermore, i t  i affects  t h e i n s t r u m e n t and b a t t e r y  duce h i g h - t e m p e r a t u r e r e g i o n s cryogenic  fuel  The t o two  thermal energy  and  t h a t may  i n a spacecraft  will  r a d i a t i o n from p l a n e t s  that  a body w i l l  acquire  ation within  t h e body w i l l ,  i s , therefore, for  and  or  The  these energy  sources  surface  temperature  vari-  a f f e c t e d by t h e  thickness  level  shape  t o the source of the w a l l  and  of i n t e r n a l r a d i a t i o n .  Knowledge o f t e m p e r a t u r e  analysis  The  due  generated  itself.  and t h e  i n a d d i t i o n , " be  the c o n d u c t i v i t y  p r e s e n c e o r absence  stress  be g e n e r a l l y  r o t a t i o n o f t h e body w i t h r e s p e c t  of r a d i a n t energy,  craft  under  energy  and e m i s s i v i t y o f t h e body.  the  intro-  boiling in  the energy  the s p a c e c r a f t  d e p e n d upon t h e amount o f t h i s  the o b j e c t ,  will  and  absorptivity  of  local  i t may  a b s o r b e d by d i r e c t i n c i d e n c e  by e l e c t r o n i c i n s t r u m e n t s w i t h i n of temperature  cause  and  tanks.  s o u r c e s , s o l a r energy  reflections  performance  desired design.  distribution within  a  f o r thermal c o n t r o l and/or  spacethermal  5 In  f r e e space, heat t r a n s f e r by c o n v e c t i o n from the  s u r f a c e w i l l be n e g l i g i b l e .  T h e r e f o r e , the thermal problem i n -  v o l v e s o n l y conduction w i t h i n s o l i d m a t e r i a l and r a d i a t i o n a t its  surfaces.  For many s o l i d m a t e r i a l s t h a t can be used f o r  the  type o f a p p l i c a t i o n under c o n s i d e r a t i o n and under the temper-  ature range g e n e r a l l y encountered, i t can be s a f e l y assumed t h a t t h e i r p h y s i c a l p r o p e r t i e s are i n v a r i a n t w i t h temperature.  There-  f o r e , the d i f f e r e n t i a l equation governing heat flow reduces t o the  l i n e a r form.  However, the s u r f a c e boundary  condition  be n o n l i n e a r due t o the fourth-power law c o n s i d e r a t i o n .  Exact  a n a l y t i c a l s o l u t i o n o f such a problem appears t o be q u i t e c u l t i f at a l l possible. such problems  will  diffi-  A standard approach t o s o l u t i o n o f  i s by d i r e c t f i n i t e - d i f f e r e n c e procedure.  The t r a n s i e n t , one-dimensional conduction problem w i t h r a d i a t i v e boundary c o n d i t i o n s has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n . * Mann and Wolf  [1] c o n s i d e r e d the problem o f h e a t i n g a semi-..  i n f i n i t e s o l i d , whose i n i t i a l heat-transfer process. of  a finite initial  temperature  Abarbanel  temperature.  i s zero, by a n o n - l i n e a r  [2] extended t h i s f o r the case He employed L a p l a c e transforms  technique and s o l v e d the r e s u l t i n g i n t e g r a l equations n u m e r i c a l l y . Jaegar  [3] used a s e r i e s method o f s o l u t i o n w h i l e Chambre[4]  s o l v e d the V o l t e r r a i n t e g r a l equations by the method of s u c c e s s ive  approximations. Goodman [5] developed an approximate mathematical  nique i n t r o d u c i n g the concept of 'heat balance i n t e g r a l ' a p p l i e d i t t o problems heat f l u x .  i n v o l v i n g a change o f phase and  techand  radiation  I t i s s i m i l a r t o the w e l l known Karman-Pohlhausen  •Numbers i n square b r a c k e t s r e f e r t o r e f e r e n c e s i n the B i b l i o g r a p h y .  6 method f o r boundary  l a y e r flow.  Later,  [ 6 ] , he used the heat  balance i n t e g r a l t o s o l v e the problem o f t r a n s i e n t c o n d u c t i o n i n a semi-infinite s o l i d slab a f u n c t i o n o f temperature  s u b j e c t e d t o a heat f l u x which i s and time.  He  method t o i n c l u d e temperature-dependent the a n a l y s i s .  [7] then extended  the  thermal p r o p e r t i e s i n  Many authors used Goodman's method i n t h e i r work.  Schneider  [8] used Goodman's method t o extend the work o f  reference  [6] f o r f i n i t e b o d i e s .  Roberts  [9] used i t t o study  the temperature d i s t r i b u t i o n i n c y l i n d e r s heated by  radiation.  F a i r a l l , e t a l . [ 1 0 ] used f i n i t e - d i f f e r e n c e techniques t o s o l v e the one-dimensional t r a n s i e n t problem f o r a s l a b of f i n i t e thickness. Biot  [11, 12] i n t r o d u c e d the concept o f p e n e t r a t i o n depth  and t r a n s i t time and a p p l i e d the v a r i a t i o n a l technique t o s o l v e one-dimensional t r a n s i e n t conduction problems w i t h r a d i a t i v e boundary  conditions.  Lardner  [13] a p p l i e d B i o t ' s  variational  p r i n c i p l e t o a number of d i f f e r e n t one-dimensional heat t i o n problems w i t h r a d i a t i v e boundary  conditions.  conduc-  Richardson  a l s o employed B i o t ' s method t o s o l v e the problem o f unsteady  [14] one-  d i m e n s i o n a l conduction i n a s e m i - i n f i n i t e s l a b w i t h the heat th f l u x a t the s u r f a c e p r o p o r t i o n a l t o the n  power o f the s u r f a c e  temperature. Problems  p e r t a i n i n g d i r e c t l y t o s o l a r h e a t i n g have been  t r e a t e d by r e f e r e n c e s [15-30], among o t h e r s . g i v e n approximate  Most o f them have  a n a l y t i c a l or n u m e r i c a l s o l u t i o n s o f l i n e a r -  i z e d governing e q u a t i o n s .  Charnes  and Raynbr [15] used p e r t u r -  b a t i o n a n a l y s i s t o determine the approximate, l i n e a r i z e d  solution  to  the problem o f s o l a r heating o f a r o t a t i n g ,  circular  cylinder.  distribution jected the  Nichols  radiation  s p h e r e s , cones  i n space.  solutions  o f problems  references  and c y l i n d e r s  The s o l u t i o n s  equations.  i n this  [17-24].Schmidt  a nonrotating  radiation.  sub-  by.Nichols f o r  pointed  shell  In  This  analytical  i n a slab  and a s p h e r e s u b j e c t  more d i f f i c u l t  matching  to solar  [27] i n v e s t i g a t e d  of s a t e l l i t e s  and space  the problem  vehicles.  and c u r v a t u r e  c a s e s have b e e n  In the present a n a l y s i s solid  circular  to solar  and r e c t a n g u l a r  radiation  produced in a  Approximate recently  the temperature  i s studied.  than that  f o r long,  cylinders F o r long,  thin hollow c y l i n d e r s .  analysed without l i n e a r i z i n g  t h e boundary  by two a p p r o x i m a t e , s e m i - a n a l y t i c a l methods and  radiation.  t h e p r o b l e m becomes t w o - d i m e n s i o n a l a n d h e n c e  p r o b l e m h a s been ditions  subjected  of the s t a t i o n a r y  i n long,  cylinders  and e a r t h  one-dimensional  r e s u l t s when a member i s n o t r o t a t i n g .  studies  distribution  to solar  the case  i n t h e maximum t e m p e r a t u r e d i f f e r e n c e  presented[28, 29].  solid  exposed  o r d e r t o o b t a i n maximum s t r e s s  we a r e i n t e r e s t e d member.  [24] d i s c u s s e d  transient,  S a n d o r f f and P r i g g e [26] a n d H a n e l temperature c o n t r o l  p r e s e n t e d by  [25] u s e d a t h e r m a l - e l e c t r i c a l  t o study the  temperature d i s t r i b u t i o n  have b e e n  on t h e t e m p e r a t u r e o f t h e s h e l l .  and S u n d e r l a n d  a n a l o g u e computer  field  analytical  out the importance o f the r a t i o o f  to emissivity  Zerkle  Other approximate  and H a n a w a l t  cylindrical  They  absorptivity  of  temperature  o n e - d i m e n s i o n a l p r o b l e m were o b t a i n e d by n u m e r i c a l i n t e g r a -  t i o n of the d i f f e r e n t i a l  of  [16] a n a l y s e d s u r f a c e  i n thin-walled  to solar  thin-walled,  i i ) least-squares  fit.  The con-  i ) point-  FORMULATION AND  SOLUTION  Consider stationary, Figure  1-1,  and  a solid  s o l i d body i s s u b j e c t  long,  solid  s p h e r e , as  to the  OF  THE  PROBLEM  c y l i n d e r s , as  shown i n F i g u r e  following  shown i n  1-2.  Each  conditions:  2 (i) A  steady  r a d i a n t energy  incident the  on  a plane perpendicular  source of t h i s  assumed h e r e t h a t at  (ii)  zero  f l u x o f G BTU/hr. f t .  energy being the  solid  energy  from p l a n e t s  The  curvature  the  that  of  It is  being vision,  further  in perfect  i t does n o t  or other  produced i s small  line  sun.  i s placed  d e g r e e s R a n k i n e and  any  to the  is  vacuum  receive  sources. so  that  the  incident  2 radiant  (iii)  f l u x i n BTU/hr. f t .  a l s o assumed t h a t  the  volve  amount o f  appreciable  the  areas opposite  The  solid  (iv) For  the  i s of  curvature  to the  distance  Under t h i s can  be  of  neglected.  r and  side of  the  For  heat  the  material, i s assumed small  c y l i n d e r and flow  to  three  the  a b s o r b e d by  been a c h i e v e d  the  solid  and  between t h e  sun.  c y l i n d e r ends  angular be  a  cases  thermal problem reduces to a two-dimensional (v) A b a l a n c e has  large  compared  from the  s p h e r e , due  Thus f o r a l l t h e  in-  radiation in  temperature d i s t r i b u t i o n w i l l  6 only.  It is  incident radiation,  o r w i d t h and  between t h e  condition  symmetry, t h e  i n t e r a c t i o n of  cylinder length  compared w i t h i t s r a d i u s  same.  produced does not  i s o t r o p i c homogeneous  c y l i n d e r s , the  , w i t h the  remains the  the  one.  radiant  heat r e - r a d i a t e d  function  into  heat space.  ( v i ) P h y s i c a l and s u r f a c e p r o p e r t i e s o f t h e s o l i d m a t e r i a l are i n v a r i a n t with (vii)  The s o l i d  temperature,  surface i s a diffuse  e m i t t e r and d i f f u s e  reflector. (viii)  S u r f a c e p r o p e r t i e s a and e a r e t o t a l h e m i s p h e r i c However, i n g e n e r a l , The  solutions  governing  a ^ e .  e q u a t i o n s , b o u n d a r y c o n d i t i o n s and t h e i r  f o r the three s o l i d s  separately  values  considered are dealt  with  i n the f o l l o w i n g pages.  a) CIRCULAR CYLINDER Under t h e f o r e g o i n g assumptions, tial  equation o f heat  conduction  the governing  differen-  and t h e b o u n d a r y c o n d i t i o n s  may be w r i t t e n as f o l l o w s :  *r.  of  r  J  11  ,  3 ^ Q 4 £  m  (i-2)  •Ira  The rephrased  = a G-c o s Q-- e - c r t T ]  energy  equation  i n t o convenient  dimensionless  variables  and t h e b o u n d a r y c o n d i t i o n s may be  dimensionless  f o r m s by i n t r o d u c i n g  as f o l l o w s :  (1-4)  10 The  resulting  yx  ,  ^x  equations are:  '  ^>x  \  a cosS  =  -  yx  f\ [X]  _ o  , 3* 4 8 4 f  ;  (1-6)  1 4B 4  ;  d-7)  2 |\ —  where  An (1-5)  exact  analytical  [30].  solution of this  The two o t h e r  following  sections.  nature.  izing  (i) P o i n t Matching The  without  developing  a number o f unknown c o e f f i c i e n t s w h i c h  on t h e s u r f a c e .  equation  (1-5).  satisfied  exactly  The b o u n d a r y only  The number o f p o i n t s  I t c a n be shown t h a t  i s satisfied  linear-  a poly-  by a p o l y n o m i a l  conditions  a t a number o f  c h o s e n on t h e s u r -  t h e same as t h e number o f unknown c o e f f i c i e n t s  polynomial.  their  Solution  (1-7) a r e t h e n e x a c t l y  (1-5)  i n the  conditions,  (1-6)  is  i n reference  B o t h t h e s e methods a r e a p p r o x i m a t e b y  the d i f f e r e n t i a l  face  equations.  methods o f s o l u t i o n a r e d e s c r i b e d  satisfies and  serai-  these  problem i s given  p o i n t m a t c h i n g method i n v o l v e s  nomial with  points  Approximate  However, t h e s o l u t i o n s a r e o b t a i n e d  t h e boundary  parameter.  s o l u t i o n o f the system o f equations  methods a r e , t h e r e f o r e , u s e d t o s o l v e  variational  very  i s a radiation-conduction  t o (1-7) a p p e a r s t o be i m p o s s i b l e .  analytical The  E c r C^O."'/^  y *  the d i f f e r e n t i a l of the f o l l o w i n g  i n the  equation form:  11 oo  X  The  -  coefficients  (1-6)  and  (1-7)  Cos-nQ  X  are a t the  d i f f e r e n t values  of  then obtained  simultaneous a l g e b r a i c and on  a.  The  by  chosen p o i n t s  0).  This  •  on  i n s e r t i n g (1-8) the  number o f  linear  algebraic  terms chosen i n equations  Raphson method f o r g i v e n  i n a^  and  (1-8).  the  The  nonlinear  parameters  solved  f o r a^ by  values  of the  parameters A  been  Newton  have b e e n t e s t e d  and  t o symmetry o n l y  b o u n d a r y has  the  In  the  Due  depend  a.  terms of  r e s u l t s compared.  (1-8)  A  r e s u l t i n g non-  are  t h i s work s e v e n t o f i f t e e n  one  and  h a l f of  the  considered.  Least-Squares F i t Solution: While  the  p o i n t m a t c h i n g method u s e s t h e  boundary p o i n t s  as  polynomial  then e x a c t l y  and  the  more p o i n t s at the  than the  satisfies  equally  eliminates squares  the  fitting The  are  f i t to the  spaced  the  on  the  then not  b o u n d a r y as  boundary so  minimized  the  uses  boundary  The  of  conditions  satisfied  a whole.  conditions but  i t  points  that i n t h i s problem i t  technique. t o be  in  f i t method  The  exactly  need t o i n c l u d e w e i g h t i n g  expression  boundary  least-squares  unknown c o e f f i c i e n t s .  pre-chosen points  ensures a b e t t e r  same number  number o f unknown c o e f f i c i e n t s  a t these pre-chosen p o i n t s , the  are  boundary ( i . e .  r e s u l t s i n a set of  equations  in  number o f r e s u l t i n g a l g e b r a i c e q u a t i o n s w i l l  the  (ii)  d-8)  is  f a c t o r s i n t o the  least-  12  where E  2  denotes  denotes  t i o n of  summation o v e r  summation o v e r A  =  the  the  region  ^  region  |  is y  Z. OL \\ Cosiw i n t h e  above  4 | .  expression  and Substitu-  yields  -n-D  (1-10)  The f u n c t i o n t o be f i t t e d  [ 2  no.'to* 1x6  to  zero at various  -*cos§  boundary p o i n t s  K(i c^cos^V]  +  in  is  2  0  and  S i n c e the parameters  a , n  function  it  is  e x p a n s i o n and t h e n t h e i.e., the  •- =0 series  which i s is  t i -  we g e t  ,  first  is  non-linear  locally  least-squares  m  s o l v e d to determine the  an i t e r a t i v e  fit  linear  by  criteria Using only algebraic  coefficients  method and i n i t i a l  the  linearized  Tv - 0, V, 2,........oo . a system o f  in  a  n  .  unknown Taylor's applied, m terms  of  equations However,  approximate values  of  a  this n  have t o be guessed  and new values o b t a i n e d .  process was repeated  (a/*  until  _  w  |  <V0  The number o f boundary p o i n t s was seven s e r i e s b)  coefficients  RECTANGULAR The  In t h i s study the  5  ,  *.«A.I  varied  and the r e s u l t s  from 30 t o 100 w i t h compared.  CYLINDER  system o f governing equations f o r t h i s geometry may  be w r i t t e n as,  '1 B.C.I:  At  X=-rCX_, H  7  B.C.2:  B.C.3:  X -- a .  At  A t M = +b,  via =  1  B.C.4:  £(T  T\  . W  - £<r T"  (1-14)  A t M- -\>,  We i n t r o d u c e a dimensionless temperature  A = UT/Ga  as  ,  and the dimensionless lengths as,  X  =  x/a  ,  (1-13)  - i  4L  14  Y ^ ^ / O L The  equations  less  f o r m as  4Y i  ,  (1-11) t o  (1-15) c a n  ,  now  be  where  w r i t t e n i n the  *\  -o  ,  dimension-  (1-16)  (1-17)  ~bX "bX  (1-18)  7>X 7>X  (1-19)  bY  "bX "bY _  a  follows,  bx*  where  \>|  a  (1-20)  tixQ^o^/k  The  -*- ^ s  t  i e  r a  diation-conduction  p o i n t m a t c h i n g method and  method h a v e a g a i n  the  parameter.  least-squares f i t  b e e n employed t o s o l v e t h e  above s y s t e m  of  equations. (i) P o i n t Matching It  satisfies was a, A  i s known t h a t t h e  the  employed and  Solution  differential and  the  r e a l p a r t of the  equation  c o e f f i c i e n t s A^,  B were o b t a i n e d  from the  (1-16).  polynomial,  This  for various  polynomial values  boundary c o n d i t i o n s  of  (1-17)  to  15 (1-20).  The  same number of c o e f f i c i e n t s A^ were taken as  number of p o i n t s on the boundary.  the  In t h i s study the e f f e c t of  seven t o ten p o i n t s on the boundary has been examined. ( i i ) Least-Squares F i t S o l u t i o n As b e f o r e , more p o i n t s than the unknown c o e f f i c i e n t s i n the polynomial  are chosen and,  i n s t e a d of s a t i s f y i n g the boundary  c o n d i t i o n s e x a c t l y , the e r r o r i s minimised by the technique.  The  t h a t weighting The  p o i n t s are e q u a l l y spaced on the boundary so f a c t o r s do not have to be  expression  /E  included.  to be minimised i s  Y,l where  least-squares  |W-»  J  )  (1-22)  (  denotes summation over the r e g i o n  X =+l, 0 4 Y 4 ^ ' ,  Z.  denotes summation over the r e g i o n  Az  denotes summation over the r e g i o n  Y ^ ^ >-1-4^ 4 + 1.  and  S u b s t i t u t i o n of expression  yields  X •=. Real  "1^  i n the above 2  t^fuul^kxC^^Tl +ft«*Mi uwfl] j .(1.23)  16 As i n t h e c a s e o f t h e c i r c u l a r c y l i n d e r t h e n o n l i n e a r least-squares f i t t i n g  t e c h n i q u e i s employed t o d e t e r m i n e t h e  unknown c o e f f i c i e n t s  rV^  ,  ^ > ^>2,  "*> -  ,  ~0.  ' I n t h i s i n v e s t i g a t i o n 40 t o 100 p o i n t s o n t h e b o u n d a r y have been c o n s i d e r e d f o r seven c o e f f i c i e n t s i n t h e s e r i e s .  C  )  SPHERE The e n e r g y e q u a t i o n a n d t h e b o u n d a r y c o n d i t i o n s f o r t h e  sphere a r e as f o l l o w s :  \  t  -hi-1  V  "be "be  1  (1-24)  7»e '  v  (1-25)  (1-26)  7^  k  Introducing dimensionless  X  -  WT/GOL  variables  and  K=  */a  ,  the energy e q u a t i o n and boundary c o n d i t i o n s reduce t o t h e f o l l o w i n g dimensionless form:  \  K s^ l  =  •ax  Co»«  - h[jt)  be *e  (1-27)  ^e )  N  I  S  T  4  6  4  ?  J  ;  ( 1  "  2 8 )  d-29)  17 is  where The  the  (i) P o i n t Matching  polynomials  radiation-conduction  above s y s t e m o f e q u a t i o n s  p o i n t m a t c h i n g and  The  the  least-squares  been s o l v e d by  the  f i t methods.  Solution  s e r i e s expansion as  has  parameter.  f o r A i s assumed i n t e r m s o f  Legendre  follows:  (1-30) The by  coefficients the  the  a  were o b t a i n e d  n  f o r various values  p o i n t m a t c h i n g method d e s c r i b e d  effect  of  five  to f i f t e e n  earlier.  terms i n the  of A  In t h i s  s e r i e s has  and  a  study  been  examined. (ii)  Least-Squares F i t Solution The  coefficients  been o b t a i n e d  by  of the  s e r i e s expansion  t a k i n g more p o i n t s on  unknown c o e f f i c i e n t s  and  employing  the  f o r A have  boundary than  a least-squares  also  the  fitting  technique. The  5  expression  t o be  minimised i s 2  (1-31)  18 where  ^  denotes  summation o v e r  the  region  ^p- 4 § 4  ^  and  ^  denotes  summation o v e r  the  region  ^  ^£  Using the  series  expression yields  expansion f o r the  following  A from e q .  4$  (1-30) the  4  above  form:  (1-32)  The n o n l i n e a r to determine  least-squares  t h e unknown  fitting  coefficients  technique  is  employed  DISCUSSION OF In  order to obtain numerical X,  temperature  physically  be p r e s c r i b e d . A = eoG  RESULTS  Its value  radiation  dimensionless  v a l u e s o f A and  In the r a d i a t i o n - c o n d u c t i o n  3 4 4 a /k , t h e s o l a r  constant.  realistic  values o f the  the  to  parameter  f l u x G can  a t a d i s t a n c e from  a have  be  sun  considered  equal  to that of  2 the e a r t h i s approximately  442  BTU/hr. f t .  e m i s s i v i t y may  range from  considerably.  A c c o r d i n g l y the r e s u l t s — 16  using A a, has  i n t h e r a n g e o f 10 been v a r i e d The  the  0.05  results  from  t o 1.0.  t o 10.  0.05  to  o b t a i n e d by  ratio  s i n c e a l l the  an e x a c t  available,  by  absorptivity,  the  references are  results [28-30],  and of How-  approximate  o f these boundary v a l u e problems i s not  i t is difficult  t o judge  their  however, t o examine t h e a c c u r a c y  solutions  from  on  vary  t h e p o i n t m a t c h i n g method  possible,  balance  may  of  1.0.  t h r e e methods o f s o l u t i o n  solution  a/k  surface  l e a s t - s q u a r e s f i t method a r e compared w i t h  ever,  coefficient  have b e e n a n a l y z e d The  t h e v a r i a t i o n a l method as d e s c r i b e d by  and  The  The  accuracy. of the  the p o i n t of view of o v e r a l l  the b o d i e s .  The  percentage  approximate  thermal  thermal  It is  energy  energy balance  e r r o r c a n be w r i t t e n as  (1-33)  The  expression  cylinder  and  the  (1-33) f o r  the  s p h e r e can  circular  cylinder,  be w r i t t e n i n  the  rectangular  dimensionless  form  as,  and  V  L  ~  respectively. through  ^  TT-r  ^  *• V »  6  ,  D e t a i l e d d e r i v a t i o n s of the e x p r e s s i o n s  (1-33c) a r e g i v e n  i n Appendix A  (pp.  (l-33a)  136-138 ) .  R e s u l t s o f t h e v a r i o u s methods f o r t h e a r e now  (1-33C)  three  geometries  discussed.  a) CIRCULAR CYLINDER In the  series  f o r X f o r the  expansion 2  (1-8) be  the  results w i l l  values  the  while  R cos  36,  Therefore,  the magnitude of the o f a^,  a^  f  a , 2  a^/  e t c . , can the  accuracy  coefficients  a^,  a^  never  and  a^,  of  coefficients  relatively  less  a are given  i n Table  1-1.  a^ a r e h i g h l y s e n s i t i v e sensitive  t o ex.  the  a^. as  l e a s t - s q u a r e s f i t method, f o r v a r i o u s v a l u e s  t h e p a r a m e t e r s A and t h a t the  26,  i n magnitude.  depend on  Representative o b t a i n e d by  one  cylinder  3  f u n c t i o n s Rcose, R cos  g r e a t e r than  circular  of  T h i s t a b l e shows to values  For very  of  A,  small values  of  -7 A,  i . e . , A o f t h e o r d e r o f 10  the A  coefficients  < 10  are decaying  or s m a l l e r , the magnitudes extremely  deals with m a t e r i a l s of high  example, those  of s o l i d  metals.  rapidly.  thermal  Therefore  of  T h i s range  of  conductivity, for  i t appears t h a t  the  e x p r e s s i o n chosen f o r X i s e x c e l l e n t . sidering of  the f a c t  This i s fortunate  t h a t one w o u l d be d e a l i n g w i t h m e t a l s  t h e a p p l i c a t i o n s o f t h e t y p e where t h i s For values  -7  i n t h e r a n g e o f 10  shows t h a t t h e m a g n i t u d e o f t h e s e  t o 10  coefficients  , Table  rapidly.  covers  porcelain,  one  has t o take  the  cylinder.  into  o f the dimensionless  obtained  the a c t u a l l o c a t i o n of a p o i n t i n i s a t R=0,  temperature  T h i s t a b l e shows t h a t t h e r e line  values  from t h e o t h e r methods,  s e m i - a n a l y t i c a l cases.  between t h e c e n t r e  by  account  those  The p o i n t c o n s i d e r e d  f o r a l l the three  methods.  roughly  c o n d u c t i v i t y , f o r example g l a s s ,  examine t h e d i f f e r e n c e i n n u m e r i c a l  f r o m t h e v a r i a t i o n a l method w i t h  methods.  and 10  ebonite, e t c .  To  values  s o t h a t X = a^  Table a t R=0  i s very  1-2  gives the  f o r the three slight difference  t e m p e r a t u r e e v a l u a t e d by t h e  As a m a t t e r o f f a c t  the a x i a l  temperatures  three obtained  t h e v a r i a t i o n a l method and t h a t o b t a i n e d by t h e l e a s t - s q u a r e s  fit,  with  identical  seven c o e f f i c i e n t s  and s e v e n t y  up t o f o u r s i g n i f i c a n t  boundary p o i n t s , are  figures.  T h i s i s an  interest-  i n g o b s e r v a t i o n s i n c e t h e v a r i a t i o n a l method i s b a s i c a l l y o p t i m i s a t i o n method and t r i e s  to b e s t - f i t  s o d o e s t h e l e a s t - s q u a r e s f i t method. fit  still  -1  The r a n g e o f A between 10  m a t e r i a l s o f low t h e r m a l  1-1  i s decaying  -7 quite  i n most  a n a l y s i s c o u l d be  -1 applied.  con-  method i s much s i m p l e r  variational  a g i v e n e x p r e s s i o n and  However, t h e l e a s t - s q u a r e s  and more s t r a i g h t  forward  than the  method.  Since available  an  the exact value  the accuracy  of the a x i a l  temperature  i s not  o f t h e t h r e e methods i s compared by t h e  22  values o f the e r r o r i n o v e r a l l heat balance (l-33a).  T h i s i s presented  least-squares f i t s o l u t i o n energy balance  i n Table 1-3, which shows t h a t the g i v e s the  least error  i n overall  and t h i s e r r o r reduces as the number o f boundary  points i s increased. of the v a r i a t i o n a l  I t i s a l s o p o s s i b l e t o i n c r e a s e the accuracy and point-matching  more terms i n the s e r i e s (1-8). matching  as g i v e n by e q u a t i o n  solutions  by  taking  T h i s i s e a s i l y done f o r the p o i n t -  method w h i l e f o r the v a r i a t i o n a l method t h i s would, be-  s i d e s i n c r e a s i n g the number o f n o n l i n e a r a l g e b r a i c equations  t o be  s o l v e d , i n c r e a s e t h e i r s i z e c o n s i d e r a b l y and i n v o l v e c o n s i d e r a b l e labour. The  point-matching  s o l u t i o n has been t r i e d by t a k i n g  seven t o f i f t e e n terms i n the s e r i e s (1-8) and the same number of p o i n t s on one h a l f o f the c y l i n d e r boundary.  The p o i n t s were  e q u a l l y d i s t r i b u t e d over h a l f o f the circumference way t h a t  i n such a  (1,0) and (1,TT) l o c a t i o n s were not i n c l u d e d .  The decay  of the magnitudes o f the c o e f f i c i e n t s a^ under parameters A and a i s e x a c t l y s i m i l a r t o t h a t observed i n the l e a s t squares f i t solution.  The a x i a l temperature i s a l s o very c l o s e t o t h a t  o b t a i n e d by the o t h e r methods (Table 1-2).  As the number o f  p o i n t s i s i n c r e a s e d , the o v e r a l l energy balance equation  (l-33a) decreases  (Table 1-3).  e r r o r as p e r  However, t h i s  overall  e r r o r does not appear t o be a f f e c t e d much by the v a r i a t i o n o f A and a. The  l e a s t - s q u a r e s f i t method u t i l i z e s a l e s s e r number  o f terms o f the s e r i e s (1-8) and more number o f p o i n t s on the boundary.  I t i s c l e a r t h a t the accuracy  o f the s o l u t i o n w i l l  23 be  a f f e c t e d b o t h by  the  number o f c o e f f i c i e n t s  number o f  l o c a t i o n s s e l e c t e d on  ature  the  overall  The  axial  and  and  1-3..  obtained is  by  the  v a r i a t i o n a l method.  method i n c r e a s e s p o i n t s on the  as  the  terms i n the  increasing the  The  the  point-matching  accuracy  of the  may  offset  round-off  the  temper-  values balance  method o r  least-squares and  1-2  the  even  fitting  number  of  d o u b t s l o w s down  decrease i n t r u n c a t i o n  error.  seems t o be  the  i n Tables  o v e r a l l energy  i n c r e a s e d , w h i c h no  s e r i e s there  Beyond a c e r t a i n number a greater  advantage i n  r a t h e r than i n c r e a s i n g  s e r i e s terms.  Comparing the least-squares  better overall  b)  by  tabulated  axial  c l o s e to the  t h e number o f b o u n d a r y p o i n t s  number o f  the  i s very  The  number o f c o e f f i c i e n t s  the boundary are  i n c r e a s i n g the  are  methods.  The  c o n v e r g e n c e r a t e , and  e r r o r by of  two  b e t t e r than t h a t obtained  the  boundary.  energy balance  temperature  other  the  c h o s e n and  t h r e e methods c o n s i d e r e d ,  f i t method i s t h e  e n e r g y b a l a n c e ) and  best  i t appears  i n accuracy  that  (gives  a  simplicity.  RECTANGULAR CYLINDER The  true  r e m a r k s made a b o u t t h e  f o r the  latter  rectangular  case a l l the  expressions.  The  three  circular  c y l i n d e r with  the  c y l i n d e r are  emphasis t h a t i n  methods i n v o l v e h a n d l i n g  p o i n t m a t c h i n g and  the  points. that  they  number on  In g e n e r a l ,  e q u i d i s t a n t and  a l l the  sides.  may  not  the  the  larger fitting  i n the s e l e c t i o n  i t i s preferable to avoid  I t i s d e s i r a b l e to d i s t r i b u t e are  of  least-squares  methods i n a d d i t i o n r e q u i r e some m a n i p u l a t i o n of boundary p o i n t s .  also  points  i n such a  n e c e s s a r i l y be  the  corner way  same  24  Table 1 - 4 p r e s e n t s v a l u e s of the c o e f f i c i e n t s A Q , A^, etc.,  The  t a b l e again shows a r a p i d decay i n the magnitudes  the c o e f f i c i e n t s .  i.e.,  2  f o r the r e c t a n g u l a r c y l i n d e r f o r some r e p r e s e n t a t i v e v a l u e s  of A and a. of  A ,  Table 1 - 5 p r e s e n t s the v a l u e s of  the dimensionless temperature  a t X=Y=0.  *-Ag,  T h i s t a b l e shows  t h a t a l l the t h r e e methods g i v e r e s u l t s which agree very w e l l . The v a l u e s of the o v e r a l l energy balance e r r o r from presented i n Table 1 - 6 .  (l-33b)  are  Again the l e a s t - s q u a r e s f i t method  g i v e s the h i g h e s t accuracy i n terms of the o v e r a l l e r r o r . v a r i a t i o n a l method appears matching method.  The  next i n accuracy and f i n a l l y the p o i n t  In the p o i n t matching method, f o r a f i x e d  v a l u e of B, i t becomes d i f f i c u l t t o d i s t r i b u t e a s m a l l number of p o i n t s on the boundary u n i f o r m l y .  I n c r e a s i n g the number o f  p o i n t s too much t o ensure e q u i d i s t a n t p o i n t s leads t o d i f f i c u l t y i n convergence of the system of n o n l i n e a r e q u a t i o n s . e x p l a i n why  This  there seems t o be no decrease i n o v e r a l l e r r o r  may on  i n c r e a s i n g the p o i n t s from seven t o ten i n the p o i n t matching method f o r the r e c t a n g u l a r c y l i n d e r as i s e v i d e n t from Table  1-6.  c) SPHERE S i m i l a r r e s u l t s have been o b t a i n e d f o r the sphere they are t a b u l a t e d i n Tables 1 - 7 ,  1-8  and 1 - 9 .  the c o e f f i c i e n t s of the s e r i e s expansion distribution.  As observed  f o r the  f o r the other two  i e n t s are decaying i n magnitude very r a p i d l y . the v a l u e s of the d i m e n s i o n l e s s temperature sphere.  Table 1 - 7  and gives  temperature  s o l i d s , the  coeffic-  Table 1 - 8  presents  a t the c e n t r e o f the  The v a l u e s o b t a i n e d by the three methods agree  reason-  25 ably w e l l .  The  are presented squares  v a l u e s o f the o v e r a l l energy  i n Table  1-9.  f i t method i n t h i s  Although case  the e r r o r  i s h i g h e r than  t h e v a r i a t i o n a l method, i t i s c o n s i s t e n t and the sharp  rise  values of  A. and  simplicity  method i s recommended f o r use  determined  i t is relatively  F o r example,  dimensional  temperature  temperature  A(X,Y) by  A=T/(Ga/k) and t o 1-6  cylinder  i n such  show t h e  to calculate  T(x,y)  combining  i s obtained  value of A  isothermals plotted  shows t h a t as B,  1-6  exhibit  of the s e r i e s  expan-'  the temperature  dis-  from  the  (i.e.,  the  the  quantities:  a.  A l l the  a c r o s s the body.  Figures  four  plots  Comparison of F i g u r e ratio  of height  to  increases.  l a t t e r , which i s f o r  variation.  conductivity increases, there i s lesser  indicates  dimensionless  i s f o r a m a t e r i a l of higher temperature  the  f o r the r e c t a n g u l a r  of temperature  w i t h F i g u r e 1-5  c o n d u c t i v i t y ) , shows a l e s s e r  variation  by  least-squares f i t  the dimensionless  i n c r e a s e s the g e n e r a l l e v e l  the thermal  does n o t  f o r the r e c t a n g u l a r c y l i n d e r ,  f o r v a r i o u s v a l u e s o f B, A and  C o m p a r i n g F i g u r e 1-4 a lower  that obtained  "3 4 4 1/4 A=(eoG a /k ) s o t h a t T= A(AG/eo) .  w i t h F i g u r e 1-4  width,  least-  problems.  d e m o n s t r a t e symmetry a b o u t t h e X - a x i s . 1-^3  the  the c o e f f i c i e n t s  simple  tribution.  1-3  f o r the  (l-33c)  Distribution  Having sion  from  i n i n a c c u r a c y o f t h e v a r i a t i o n a l method f o r h i g h  Thus f o r a c c u r a c y  Temperature  balance  C o m p a r i s o n o f F i g u r e 1-5  thermal Thus  as  temperature with  t h a t a t a h i g h v a l u e o f a, t h e c o e f f i c i e n t  Figure of  26 absorptivity, slightly, Effect  the temperature  especially  a c r o s s t h e body  i n t h e lower temperature  o f Semi-Grayness In s p a c e c r a f t  applications,  radiant  increases  region.  of Surface Properties  a much l o w e r t e m p e r a t u r e of the i n c i d e n t  variation  the spacecraft w i l l  compared t o t h e s u n w h i c h flux  i n this  analysis.  be a t  i s the source  Therefore, the  maximum e n e r g y e m i t t e d by t h e s u r f a c e o f t h e s p a c e c r a f t w i l l be at  relatively  will  be m a i n l y a t l o w e r w a v e l e n g t h s .  emissivity  of a given surface  temperature function  h i g h e r wavelengths  lation to  solutions  radiation.  n o t be c o r r e c t  of equations  (1-26) does  ant w i t h temperature  a=e.  The f o r m u -  (1-16) t o (1-20) a n d (1-24) assumption.  are t o t a l  semi-grayness, i . e . ,  and g e o m e t r y , e increases  i s t r u e a s l o n g as we  distribution  has b e e n i n v e s t i g a t e d .  of  Therefore, i n the present  of these sets of equations are, therefore,  that the surface properties  at a fixed  isa  and t h e t e m p e r a t u r e  n o t make t h e above m e n t i o n e d  This  energy  a function of i t s '  of absorptivity  t o assume t h a t  (1-1) t o ( 1 - 3 ) ,  o f any s u c h c o n s t r a i n t .  of t h i s  i s , i n general,  o f both, the s u r f a c e temperature  i t will  solar  The c o e f f i c i e n t o f  only, while the c o e f f i c i e n t  of the source of i n c i d e n t problem  while the incident  independent consider  h e m i s p h e r i c and a r e i n v a r i -  over the surface.  a ^ e , on t h e t e m p e r a t u r e  I t i s noted that  The e f f e c t distribution  f o ra given material  v a l u e o f a , an i n c r e a s e  t h e maximum t e m p e r a t u r e  The  difference  i n the value and v i c e  versa.  27 Inclusion of Convection Although of  no  convection  t e r m c r e a t e s no p r o b l e m by fit  the heat  valid. easily  a n a l y s i s has  either  transfer, inclusion  the point-matching three  solid  r e m a i n s u n a l t e r e d and The  be  only  Analysis  been c a r r i e d  additional difficulties  method f o r t h e  equation  Heat T r a n s f e r i n the  out  of the  i n the  method o r t h e considered.  so  series  v  the  least-squares The  solution  energy is  still  change i s i n t h e b o u n d a r y c o n d i t i o n s w h i c h  incorporated.  case  convection  s o l u t i o n s of  bodies the  f o r the  can  28  CONCLUSIONS T h e o f  s o l i d  s p h e r e , T w o  l o n g  s u b j e c t e d  t o  ( i i )  r e s u l t s  t o  b o d i e s ,  a p p r o x i m a t e ,  a n d  T h e  t e m p e r a t u r e  o t h e r  w i t h  s t r u c t u r a l  a s  m e t h o d s a n d  s h a p e s .  f i t t i n g ,  m e t h o d i t  a c c u r a t e  c y l i n d e r s  a n d  i n  i n  i n v o l v e s  s p a c e  d i f f e r e n t  h a s  b e e n  ( i )  h a v e  e x a m i n e d  b e e n b y  p r o v e d m u c h  b e  s i m p l e r  a n d  a n a l y s e d .  a n d  v a s t l y  t h e s o l u t i o n .  s u p e r i o r  c o m p u t a t i o n  r e s u l t s .  p r e s e n t e d s p h e r e s  h e r e a n d  c a n  c a n  b e  b e  a  p o i n t - m a t c h i n g  a v a r i a t i o n a l t o  t y p e s  c y l i n d e r s  m e t h o d s ,  o b t a i n e d h a s  t h r e e  r e c t a n g u l a r  r a d i a t i o n  t h o s e  f i t  t w o ,  v e r y T h e  h o l l o w  s o l a r  l e a s t - s q u a r e s  c o m p a r e d  p r o v i d e s  c i r c u l a r  s e m i - a n a l y t i c a l  l e a s t - s q u a r e s t h e  d i s t r i b u t i o n  a p p l i e d  e x t e n d e d  t o  t h i c k  f o r  o t h e r  a n d  TABLE Effect  1-1  o f A and a on t h e v a l u e s o f t h e c o e f f i c i e n t s a. o f E q u a t i o n  (1-8)  Circular Cylinder (Least-squares a  A lO"  1 1  1 .0  a  o  "422 .4  f i t method f o r 7 s e r i e s a  l  2 2 1 0 .6196x10" -0 .4118x10" -0 .9173x10"  0 .2819x10  -o .6859x10" 2  0 .9572x10  0. 2496  1 2 7 0 .5300x10" -0 .2880x10" -0 .5300x10"  0 .2084x10  0. 2365  0 .5148x10" -0 .2762x10"  0. 1586  2 1 2 0 .3800x10" -0 . 1 2 3 1 x 1 0 ' -0 . 4 8 2 1 x 1 0 "  0 .6663x10  2 2 1 0 .2187x10" -0 .2985x10" -0 .4080x10"  0 .3040x10  1 .0  0 .3895 0. 1301  0 .5  5  10"  1  10 10" IO" 10  2  1 1  11 .23  0 .5  1 .105  0 .5  0 .3361 0. 9 0 7 7 x 1 0 "  0 .1  237 .5  1  5  0. 2551  1 .296  IO"  a  -0 .1628x10"  1 .0  355 .2  6  4  0 .1044  1  0 .5  a  0. 4853  IO"  1 1  3  -0 .1675x10"  1 .0  IO"  a  0 .1059  7  10  2  0. 4985  IO"  42 .23  a  t e r m s a n d 100 b o u n d a r y p o i n t s )  0. 4 9 9 7 x 1 0 "  4  1 2 0 .3048x10" -0 .6661x10"  1  1 1 1  0 .1  1 .334  0. 4 5 6 1 x 1 0 "  0 .1  0 .2340  o. 3 2 3 5 x 1 0 " 1  ]  9  1  4  0 .1061x10" -0 . 3 3 7 S x l 0 " 0 .1007x10" -0 .1469x10" 0 .7736x10"  4  2  0 .2304x10"  3  -0 .1060x10" -0 .1053x10"  -0 .5231x10"  -0 .1060x10" -0 . 1 0 3 7 x 1 0 " -0 .9681x10"  1  2  2 2 3  0 .6362x10 0 .7637x10  0 .1285x10  0 .9638x10 0 .6772x10 0 .1225x10  a  -8 -6 -3 -3 -9 -5 -4 -3 -10 -6 -4  6  0 .3026x10" 0 .3012x10" 0 .2704x10" 0 .2190x10" 0 .1513x10" 0 .1500x10" 0 .1405x10" 0 .1248x10" 0 .3027x10" 0 .2981x10" 0 .2819x10"  2 2 2 2 2 2 2 2 3 3 3  TABLE 1-2 Comparison o f v a l u e s of  the  coefficient Circular  A  a  1CT  12  Variational Method  1.0  0.7511  X  10  Least Squares Fit Method  Point Matching Method 7 Points  X  10  3  0.7511  X  10  3  2  0.4224  X  10 0.4223  X  10  2  0.4223  X  10  2  0.1296  X  10 0.1296  X  10  0.1296  X  10  10  3  0.7504  X  10  2  0.4206  X  10  2  0.4228  X  10  2  0.4220  X  10  0.1289  X  10  0.1297  X  10  0.1294  X  10  1  1.0  0.1296  X  10  1.0  0.3895  0.5  0.3552  X  10  5  0.5  0.1123  X  10  1  0.5  0.1105  X  10  0.5  0.3361  0.3901  0.3865 3  0.3537  X  10  3  2  0.1118  X  10  2  0.1100  X  10  0.1124  X  3 0.3549 lO 10 0.1122  0.1106  X  10  0.3556  X  0.3340  lO"  1 1  0.1  0.2375  X  10  lO"  2  0.1  0.1334  X  10  0.1  0.2340  3  0.1104  0.3366  0.2365  X  10  0.1329  X  10  0.2329  2  3  X  3 0.3552 10 10 0.1123  X  10  X  X  10  0.1336  X  10  0.2342  3  2  0.3895  0.3895  0.1105  10 0.3552  X  10  3  0.3552  X  X  10 0.1123  X  10  2  0.1123  X  10 10  X  10 0.1105  X  10  0.1105  X  10  0.2375  X  10  0.1334  X  10  3  2  X  10  0.1333  X  10  0.2337  * This denotes 7 unknown coefficients and 30 points on the boundary.  3  0.2375  X  10 0.2375  X  10  0.1335  X  10 0.1334  X  10  0.2340  3  0.2340  -3 2  0.3361  0.3361  0.3361  0.2373  3  X  J  2  0.3356  0.2378  3  0.3895  0.3888 13  10  10 0.7511  X  10"  10  X  0.7519  10  lO"  0.7512  3  X  10"  3  10  0.4223  7 - 100  7 - 70  7 - 30*  15 Points  10 Points  methods  Cylinder  X  1.0  11  o b t a i n e d by d i f f e r e n t  0.7479  7  10"  Q  3  lO"  10  a  3  0.2340  LO  o  3  TABLE Comparison of  the  o v e r a l l percentage  error  Circular  A  a  1-3 (Eq.  l-33a)  7 Points  lO"  1 2  1.0  - 0.0134693  1.6842149  lO"  7  1.0  - 0.0426800  1.6842179  lO"  1  1.0  0.0065672  1.6838901  1.0  - 0.7490059  1 .6793281  -  me  Cylinder  Point Matching Method  Variational Method  o b t a i n e d by d i f f e r e n t  Least Squares Fit Method 7 - 100  10 Points  15 Points  7 - 30  7 - 70  0.4123867  0.3658459  - 0 .04568/0  - 0.0083730  - 0.0940919  0.4123822  0.3658451  - 0 .0456810  - 0.0083655  - 0.0040814  0.4123792  0.3658444  - 0 .0455469  - 0.0082284  - 0.0039473  0.4123673  0.3657885  - 0 .0505567  - 0.0133947  - 0.0091359  0.4123822  0.3658473  - 0 .0456855  - 0.0083685  - 0.0040904  lO"  1 1  0 .5  0.0166636  1.6842179  -  10"  5  0 .5  0.0705664  1.6842224  -  0.4123852  0.3658451  - 0 .0456795  - 0.0083625  - 0.0040859  lO"  1  0 .5  - 0.0020015  1.6841374  0.4123867  0.3658459  - 0 .04555^8  - 0.0082254  - 0.0039473  0 .5  - 0.2901209  1.6826615  -  0.4123762  0.3658362  - 0 .0464052  - 0.0091076  - 0.0048324  0 .1  - 0.0450330  1 .6842179  -  0.4123852  0.3658511  - 0 .0456929  - 0.0083730  - 0.0041038  0 .1  - 0.1428091  1.6842157  -  0.4123867  0.3658429  - 0 .0456706  - 0.0083596  - 0.0040710  0 .1  0.0819797  1.6841456  0.4123822  0.3658481  - 0 .0455678  - 0.0082433  - 0.0039592  10  10 lO" lO" 10  1 1  2  TABLE Effect  o f A and a on t h e v a l u e s  1-4  o f t h e c o e f f i c i e n t s A^ o f E q u a t i o n  Rectangular Cylinder (Least-squares  A  a  IO"  1 2  IO"  9  IO" IO"  1. 0  A  o  562 .3  A  fit  method w i t h 7 s e r i e s  l  A  2  A  3  0 .1999  0 .2406x10"  1. 0  99 .92  0 .4941  0 .1992  0 .1337x10"  6  1. 0  17 .69  0 .4683  0 .1955  0 .7042x10"  3  1. 0  0 .3578  0 .1717  0 .2772x10"  IO"  1 2  0. 1  3 .054 316 .2  0 .4998x10"  1 1  0 .2000x10"  1 1  (B=0.25)  terms  0 .4989  0 .4287x10"  3 2 2 1 5 4  8  0. 1  IO"  4  0. 1  3 .154  0 .4818x10"  0. 1  0 .5522  0 .4102x10" 1 0 .1845x10" 1 0 .1878x10" 2  10" IO'  1  1 2  0. 05  IO"  8  0. 05  IO'  4  0. 05  265 .9 26 .59 2 .655  0 .4981x10"  0 .2499x10"  1  1  0 .1998x10" 0 .1975x10"  0 .9999x10"  1  2  0 .4269x10" 0 .4101x10"  0 .1275x10"  and 100 b o u n d a r y  •  IO"  31 .61  (1-21)  3  5  A  4  0 .5604x10 0 .3183x10 0 .1872x10 0 .1093x10 0 .9953x10 0 .9983x10 0 .1030x10 0 .6212x10 0 .2960x10  0 .2494x10" 1 0 .9993x10" 2 0 .1272x10" 4  0 .2963x10  0 .2445x10" 1 0 .9928x10" 2 0 .1241x10" 3  0 .3023x10  A  -4 -3 -2 -1 -6 -5 -3 -3 -6 -5 -4  points)  5  -0 . 2 2 9 1 x 1 0 " -0 . 1 0 2 3 x 1 0 " 0 .2099x10" 0 .1830x10" -0 . 4 2 4 6 x 1 0 " -0 . 3 9 2 7 x 1 0 " -0 . 1 1 1 4 x 1 0 " 0 .6012x10" -0 . 1 2 6 7 x 1 0 " -0 . 1 2 0 7 x 1 0 " -0 . 7 0 1 7 x 1 0 "  A  5 4 4 2 7 6 5 4 7 6 7  6  -0. 5 0 i l x l 0 " - 0 . 2816x10" - 0 . 1557x10" - 0 . 5733x10" - 0 . 8931x10" - 0 . 8907x10" - 0 . 8866x10" - 0 . 4267x10" - 0 . 2657x10" - 0 . 2649x10"  5 4 3 3 7 6 5 4 7 6  -o. 2 6 4 5 x 1 0 " 5  to to  TABLE 1-5 Comparison o f v a l u e s o f the c o e f f i c i e n t A Rectangular  a  10" icr 10  1 2  9  - 6  -  Method  o b t a i n e d by d i f f e r e n t  7 Points  methods  Cylinder  Point Matching Method  Variational  A  0  1  10 Points  9 Points  Least Squares Fit Method 7 - 10  7 - 80  7 - TOO  1.0  0 .5624  X  10  3  0. 5623  X  10  3  0.5623  X  10  3  0. 5623  X  10  3  0. 5623  X  10  3  0.5623  X  10  3  0.5623  X  10  3  1.0  0 .1000  X  10  3  0. 9990  X  10  2  0.9990  X  10  2  0. 9991  X  10  2  0. 9992  X  10  2  0.9992  X  10  2  0.9992  X  10  2  1•0  0 .1778  X  10  2  0. 1769  X  10  2  0.1769  X  10  2  0.  1769  X  10  2  0. 1769  X  10 .0.1769  X  10  2  0.1769  X  10  2  0. 3055  X  10  0.3048  X  10  0. 3055  X  10  0. 3054  X  10  0.3054  X  10  0.3054  X  10  2  10"  3  1.0  0 .3140  X  10  lO"  1 2  0 .1  0 .3162  X  10  3  0. 3175  X  10  3  0.3177  X  10"  0. 31 75 X 10  3  0. 3162  X  10 0.3162  X  10  3  0.3162  X  10  3  8  0 .1  0 .3162  X  10  2  0. 3161  X  10  2  0.3161  X  10  0. 3160  X  10  2  0. 3161  X  10  0.3161  X  10  2  0.3161  X  10  2  4  0 .1  0 .3163  X  10  •0". 3153  X  10  .0.3153  X  10  0. 3153  X  10  0. 3154  X  10  0.3154  X  10  0.3154  X  10  0 .1  0 .5612  0 .05  0 .2659  X  10  3  0. 2659  X  10  3  0.2659  X  10  3  0. 2659  X  To  0. 2659  X  10 0.2659  X  10  3  0.2659  X  10  3  0 .05  0 .2659  X  10  2  0. 2659  X  10  2  0.2659  X  10  2  0. 2657  X  10  0. 2659  X  10  0.2659  X  10  2  0.2659  X  10  2  0 .05  0 .2660  X  10  0. 2655  X  10  0.2655  X  10  0. 2653  X  10  0. 2655  X  10  0.2655  X  10  0.2655  X  10  10~ 10~ lO'  1  lO" 10  1 2  - 8  lO"  4  .0. 5522  3  2  0.5518  0. 5522  3  2  0.5522  0. 5522 3  2  3  2  0.5522  co co  TABLE Comparison of  the  o v e r a l l percentage  error  1-6 (Eq.  Rectangular  A  a  1<T  12  Variational Method  l-33b)  9 Points  by d i f f e r e n t  methods  Cylinder  1  Point Matching Method 7 Points  obtained  10 Points  Least Squares F i t Method 7 - 40  7-80  |  7 - 100  1.0  - 0.0342488  0.0025503  0.0025697  0.0025503  0.0000454  0.0000350  0.0000291  IO"  9  1.0  0.0364594  0.0422530  0.0428043  0.0422530  0.0001647  0.0000671  0.0000559  IO"  6  1.0  0.1067489  0.0161126  0.0345334  0.0161126  0.0013083  0.0007093  0.0006542  IO'  3  1.0  0.0058122  -0.0551924  0.8436181  -0.0473037  0.0267804  0.0235520  0.0231773  0.1  0.0364825  -1.6917363  -1.8311754  -1.6917363  0.0000313  0.0000253  0.0000216  0.1  0.0481263  0.0067174  0.0067689  0.1678914  0.0000603  0.0000298  0.0000276  IO" 10  12  - 8  IO"  4  0.1  - 0.0203714  0.0816956  0.0877619  0.0816956  0.0005998  0.0002799  0.0002384  IO"  1  0.1  - 0.0398651  0.0357717  0.3065333  0.0357717  0.0087924  0.0068709  0.0066407  IO"  12  0.05  0.0230677  -0.0470638  -0.0470564  -0.0470638  0.0000231  0.0000298  0.0000171  IO"  8  0.05  0.0312531  0.0056140  0.0056550  0.2616279  0.0000454  0.0000291  0.0000238  IO"  4  0.05  - 0.0515928  0.0227228  0.0248097  0.2607897  0.0003278  0.0001334  0.0001132  to  TABLE Effect  1-7  o f A a n d a on t h e v a l u e s o f t h e c o e f f i c i e n t s  a  i  of Equation  (1-30)  Sphere (Least-squares  A  a  a  0  a  f i t method f o r 7 s e r i e s  l  10"  1 6  0. 1 3976. 0  0 .4999x10"  10"  1 2  0. 1  0 .4999x10"  10"  8  0. 1  10"  4  0. 1  1.0  0. 1  lO'  1 6  10"  8  0. 5  lO"  1 6  lo"  1 2  59. 46  0 .2479  5. 941  0. 5  0. 5776 0 .1334  0 .2304  1. 0 7 0 7 1 . 0  0 .4999  1. 0  0 .4993  8  1. 0  lO"  4  1. 0  1.0  594. 6  0. 5  lO"  1. 0  1 1 1  0 .4877x10" 1 1 . 0.3954 0 . 3 9 7 1 x 1 0 " 3. 976  0 .2498  0. 5  1.0  0 .4987x10"  0 .2500  1 2  4  39. 76  0. 5 5 9 4 5 . 5  io" lO"  397. 6  a  707. 1 70. 71  0 .4930  7. 056 0 .4369 0. 6736 0 .2040  2  t e r m s and 40 b o u n d a r y p o i n t s )  a  a  4  a  5  11 4 -0 .3101x10 -10 - 0 . 3 7 3 7 x 1 0 " -0 .1367x10" 11 4 2 0. 3 8 9 7 x 1 0 " 0 .2564x10 -10 -o. 3 7 3 7 x 1 0 " -0 .2 0 2 1 x 1 0 " 11 4 2 0. 3 8 9 2 x 1 0 " -0 .3212x10 -8 - 0 . 3 7 3 5 x 1 0 " 0 .8997x10" 8 4 2 -6 0. 3 8 4 7 x 1 0 " -0 .3176x10 -0. 3714x10" 0 .1201x10" 7 2 -4 4 0. 3 3 5 0 x 1 0 " -0 .2217x10 0 .8219x10" -0. 3555x10" 10 3 1 -9 0 .1288x10" 0. 1 9 4 8 x 1 0 " -0 .3654x10 -0. 1869x10" 10 3 1 -8 0. 1 9 4 8 x 1 0 " -0 .1610x10 -0. 1868x10" 0 .1044x10" 9 3 1 -6 0 .6871x10" 0. 1 9 4 0 x 1 0 " -0. 1865x10" -0 .1817x10 8 1 -4 3 -0. 1833x10" 0 .5994x10" 0. 1 8 6 1 x 1 0 " -0 .1625x10 5 3 1 -3 0 .2700x10" 0. 1 1 6 7 x 1 0 " -0 .4999x10 -0. 1637x10" 10 1 -9 0. 3 8 9 7 x 1 0 " o .3344x10 -o. 3 7 3 7 x 1 0 " 3 -0 . 2 2 2 5 x 1 0 " 10 3 1 -8 0. 3 8 9 4 x 1 0 " -0 .9900x10 -0. 3733x10" 0 .1209x10" 8 3 1 -5 0 .3873x10" 0. 3 8 6 8 x 1 0 " -0 .1020x10 -0. 3721x10" 6 3 -4 1 0 .3064x10" 0. 3 5 9 3 x 1 0 " -0 .8397x10 -0. 3620x10" 3 4 -2 1 0 .1047x10" -.0. 2 9 8 3 x 1 0 " -0 .1402x10 0. 1 7 6 4 x 1 0 " 0. 3 8 9 7 x 1 0 "  2  3  a  6  0 .1310x10" 0 .1310x10" 0 .1310x10" 0 .1305x10" 0 .1263x10" 0 .6552x10" 0 .6551x10" 0 .6543x10" 0 .6465x10" 0 .5930x10" 0 .1310x10" 0 .1308x10" 0 .1305x10" 0 .1280x10" 0 .1117x10"  5 5 5 5 5 5 5 5 5 5 4 4 4 4 4-  to  TABLE Comparison  1-8  of values of the c o e f f i c i e n t  a  0  o b t a i n e d by d i f f e r e n t  methods  Sphere  D i m e n s i o n l e s s T e m p e r a t u r e (X = a )  a t the C e n t r e o f t h e S p h e r e  0  Point-Matching 7 Points  5 Points  Least-Squares F i t Solution  Solution 15 P o i n t s  10 P o i n t s  0 .3976  X  10  4  10"  3  0 .3976  X  10  3  10~  2  0 .3975  X  10  2  10"  0 .3970  X  10  0 .5946  X  10  4  10"  0 .5945  X  10  3  10"  0 .5943  X  10  2  0 .5913  X  10  0 .7071  X  io  4  10"  0 .7070  X  10  3  10"  0 .7065  X  10  2  10"  0 .6997  X  10  X  io  4  0.3976  X  io  4  10 2 10  0 .3969  X  10  3  0.3976  X  io  0 .'3969  X  10  2  0.3976  X  10  X  10  0 .3969  X  10  0.3976  X  10  0.5958  X  io  4  0 .5935  X  io  4  0.5946  X  10  4  3  0.5958  X  10  3  0 .5935  X  io  3  0.5946  X  10  3  2  0.5958  X  10  2  0 .5935  X  io  2  0.5946  X  10  2  10  0.5954  X  10  0 .5930  X  10  0.5941  X  10  X  10  0.7086  X  10  4  0 .7058  X  io  0.7071  X  io  4  0 .7007  X  10  3  0.7086  X  io  3  0 .7058  X  10  0.7071  X  10  3  0 .7007  X  10  2  0.7086  X  10  2  0 .7057  X  10  0.7071  X  10  2  0 .6992  X  10  0.7071  X  10  0 .7043  X  10  0.7056  X  10  X  10  4  0 .3940  X  10  4  0.3985  X  0 .3912  X  10  3  0 .3940  X  10  3  0.3985  X  0 .3912  X  10  2  0 .3940  X  10  2  0.3985  X  0 .3911  X  10  0 .3940  X  10  0.3984  0 .5850  X  10  4  0 .5892  X  io  4  0 .5850  X  io  3  0 .5892  X  io  0 .5849  X  10  2  0 .5892  X  10  0 .5845  X  10  0 .5888  X  0 .6956  X  io  4  0 .7007  0 .6956  X  10  3  0 .6956  X  10  2  0 .6941  X  10  4  10  4  3  a  A  (7-40)  0 .3969  0 .3912  Variational Solution  4  3  2  10"  10~  16 12 8 4 16 12 8  10" 4 16 12 8  10" 4  0. 1 0. 1 0. 1 0. 1 0. 5 0. 5 0. 5 0. 5 1. 0 1. 0 1. 0 1. 0 OJ  Comparison of  the  overall  percentage  TABLE  1-9  error  (Eq.l-33c)  o b t a i n e d by d i f f e r e n t  methods  Sphere Point 5  Points  7  Points  M a t c h i n g Method 10  Points  15  Points  Least-Squares F i t Method (7-40)  Variational Method  A  a  6.3037  3.5264  -0.8366  0.7385  -0.0057206  52 x  IO  - 7  IO"  6.3037  3.5264  -0.8366  0.7385  -0.0057220  52 x 1 0  - 7  io"  6.3037  3.5264  -0.8366  0.7385  -0.0057116  127 x 1 0 "  6.3037  3.5264  -0.8366  0.7385  -0.0057206  387 x 1 0  6.2973  3.5276  -0.8367  0.7390  -0.0056669  84.186  6.3037  3.5264  -0.8366  0.7384  -0.0057235  67 x  IO  - 7  io"  1 6  0.5  6.3037  3.5264  -0.8366  0.7384  -0 .0057086  52 x  IO  - 7  io"  1 2  0.5  6.3037  3.5264  -0.8366  0.7384  -0.0057235  89 x  IO  - 7  io"  8  0.5  6.3030  3.5266  -0.8366  0.7384  -0.0057012  151 x  IO  - 5  io"  4  0.5  6.2462  3.5362  -0.8369  0.7392  -0.0061736  -  6.3037  3.5264  -0.8366  0.7385  -0.0057206  60 x 1 0  - 7  io"  1 6  1.0  6.3037  3.5264  -0.8366  0.7385  -0.0057146  52 x  IO  - 7  io"  1 2  1.0  6.3037  3.5264  -0.8366  0.7385  -0.0057101  127 x  IO  - 7  io"  8  1.0  6.3017  3.5268  -0.8366  0.7385  -0.0056922  0.2468  io"  4  1.0.  6.1860  3.5472  -0.8371  0.7424  -0.0090599  -  10°  1 6  1 2  0.1 0.1  7  io"  8  0.1  - 7  io"  4  0.1 0.1  10°  0.5  10°  1.0 CO  Figure  1-1  Coordinate Cylinders  System  Rectangular  and  Circular  Figure  1-2  Spherical  Coordinate  System  Figure  1-3  Isothermals for Rectangular (B=0.5, A = 0 . 1 , a=0.5)  Cylinder  Figure  1-4  Isothermals for Rectangular (B=2, A = 0 . 1 , a=0.5)  Cylinder  Figure  1-5  Isothermals for Rectangular (B=2, A = 1 0 ~ 3 , =0.5) a  Cylinder  Figure  1-6  Isothermals f o r Rectangular (B=2, A=10~3, a = l )  Cylinder  P A R T  TEMPERATURE  DISTRIBUTION  RADIATING  AND  I I  AND  EFFECTIVENESS  CONVECTING  CIRCULAR  FIN  OF  A  ABSTRACT  A transfer its  theoretical analysis c h a r a c t e r i s t i c s of  surface  by  and  surface  properties  between t h e sidered. the  f i n and  In  the  f i n i s neglected.  with nonlinear squares finite  and  The  There i s r a d i a n t  separate s i t u a t i o n s are from the  been o b t a i n e d by  s o l u t i o n has the  properties end  of of  the the  s o l u t i o n i s o b t a i n e d by  shown t h a t  neglecting  f o r long  fins  fin.  the  f i n i s also a finite  heat t r a n s f e r or  In  for fins  of  equation the  a l s o b e e n o b t a i n e d by  r e s u l t s compared.  con-  end  has  method and  is  interaction  boundary c o n d i t i o n s A  from  physical  l i n e a r conduction  from the  material.  constant  Results  f o r a wide r a n g e o f e n v i r o n m e n t a l c o n d i t i o n s  surface  imation  f i n and  s i t u a t i o n heat t r a n s f e r Solution  heat  temperature  the  difference  transfer  Two  The  of  f i t method.  presented  assumed.  i t s base.  first  the  the  f i n d i s s i p a t i n g heat  radiation.  base of  are  been c o n d u c t e d o f  a circular  c o n v e c t i o n and  assumed u n i f o r m a l o n g t h e  has  and  leastthe are physical  second s i t u a t i o n heat  included  difference from the  i n the  procedure.  end  of high thermal  i s a good  analysis. It is approx-  conductivity  NOMENCLATURE  a  radius of c i r c u l a r  fin,  f t .  3 E^aaT  A  0  /k, r a d i a t i o n - c o n d u c t i o n  parameter,  dimensionless F  configuration  factor, dimensionless 2  h  heat  k  thermal conductivity  1  f i n length, f t .  L  1/a/ d i m e n s i o n l e s s f i n l e n g t h  N  ha/k, c o n v e c t i o n parameter,  dimensionless  Q  rate  BTU/hr.  r,z  cylindrical  R, Z  r / a , z/a, d i m e n s i o n l e s s c y l i n d r i c a l c o o r d i n a t e s  *2  f i  T  temperature  T  0  T  n  transfer  o f heat  base  f i n base  fluid  coefficient,  loss  BTU/hr. f t . °R.  o f f i n , BTU/hr. f t .  from  fin,  coordinates f o r f i n ,  °R.  f t .  radius, f t .  a t any p o i n t  i n the f i n ,  °R.  t e m p e r a t u r e , °R.  b u l k t e m p e r a t u r e , °R.  00  T*  effective  radiation  environment  t e m p e r a t u r e , °R.  Greek  Symbols  a  c o e f f i c i e n t of a b s o r p t i v i t y ,  e  c o e f f i c i e n t of e m i s s i v i t y ,  a  Stefan-Boltzmann 2 4 ft. °R.  n  f i n effectiveness,  g A  dimensionless  —8  r  2  / / a  c o n s t a n t , 0.1714 x 10  BTU/hr.  dimensionless  ' dimensionless '  (T-T ) /T , i n the f i n Q  T  A*  dimensionless  Q  dimensionless  oo/ ° ' dimensionless T  fluid  temperature  bulk  T*/T , dimensionless e f f e c t i v e environment temperature c  a t any  temperature radiation  Subscripts 1  f i n surface  2  base  ~  fluid  . !  surface bulk  Superscripts *  effective radiation  environment  point  INTRODUCTION The interest trol/  d e s i g n o f space  i n r a d i a t i o n heat  the s o l a r energy  surface  areas  tended  areas  class is  fins.  vective  model.  method w h i l e  Bartas  and S e l l e r s  [33] u s e d  the f i n e f f e c t i v e n e s s f o r r e c analyzed  fins—where  [32] a n a l y s e d t h e o n e -  f o r a rectangular f i n using  in detail  the r a d i a t o r  i n t e r f e r e n c e of r a d i a t i o n  annular  only a  is flat,  by p i p e s .  heat  Chambers  the f i n e f f i c i e n c y  transfer.  limiting  i . e . , there  f i n and compared t h e f i n e f f i c i e n c i e s  and r a d i a t i v e  analyzed  to the  most o f t h e work h a s b e e n  Lieblein  distribution  some g e n e r a l c h a r a c t e r i s t i c s [36]  fins,  [34] n u m e r i c a l l y d e t e r m i n e d  mushroom  The  In order t o p r o v i d e the l a r g e  for radiating  However, t h e y  of radiating  Somers  space  away t o t h e s u r r o u n d i n g s .  area.  methods t o c a l c u l a t e  no s p a c i a l  by  fins.  temperature  a finite-difference  tangular  i n the  law s e t s an a b s o l u t e u p p e r l i m i t  the one-dimensional  numerical  generated  s e v e r a l s t u d i e s have b e e n made on t h e s t e a d y  transfer  dimensional  direct  r e q u i r e d , i t i s n a t u r a l t o c o n s i d e r the use o f ex-  Although  for  con-  r e q u i r e d f o r t h i s p u r p o s e a r e o f t e n huge, s i n c e  surfaces—i.e.,  state heat  v e h i c l e by  equipment o r o t h e r sources  transfer rate per unit  surface  by a s p a c e  must be d i s s i p a t e d  the Stefan-Boltzmann heat  For accurate thermal  f r o m p l a n e t s and t h e e n e r g y  power p l a n t s , e l e c t r o n i c itself  transfer.  absorbed  .incidence o r r e f l e c t i o n  vehicle  v e h i c l e s has s t i m u l a t e d c o n s i d e r a b l e  for a  and flat,  f o r con-  E c k e r t et. a l . [35] s t u d i e d  of r a d i a t i n g  fins.  the e f f e c t i v e n e s s o f long, plane  Sparrow e t a l . radiating  fins  49 with mutual i r r a d i a t i o n . of i n t e g r o - d i f f e r e n t i a l n u m e r i c a l methods. the e f f e c t s  They f o r m u l a t e d equations  of mutual i r r a d i a t i o n  presence  the  f i n heat  loss  Sparrow  linearized  b l a c k and  selectively  solution  available,  [39]  reduced  by  Shouman  Among t h e v e r y  annular  [40] has  considered a one-dimensional  heat  conduction  exact  g i v e n an e x a c t area f i n .  fins.  numerical  heat  few  the  calculated  flow i n  f o r the one-dimensional  sur-  operating  have g i v e n t h e  f o r a constant cross-sectional  fin-to-base  heat  gray  between t h e b l a c k r a d i a t o r  Niewerth  solutions  iterative  considered  S p a r r o w e t a l . [38]  convecting-radiating fins.  solutions  has  and  system  o c c u r r i n g between a f i n and i t s  f i n e f f e c t i v e n e s s f o r one-dimensional  Recently,  in  [37]  is significantly  of the base s u r f a c e .  f i n s w i t h mutual i r r a d i a t i o n  and  Eckert  They showed t h a t i n t h e r a n g e o f p r a c t i c a l  conditions,  the  s o l v e d them by  L a t e r , S p a r r o w and  a d j o i n i n g base s u r f a c e f o r both faces.  and  t h e p r o b l e m as a  analytical  general  However,  f l o w model and  he  neglected  interaction.  S i n c e a minimum mass o f a l l e q u i p m e n t i s d e s i r a b l e i n space  vehicles,  a number o f s t u d i e s have b e e n made t o  t h e optimum f i n s i z e mass.  Liu  [41]  having  presented  rectangular p r o f i l e .  the  largest  r a t i o of heat  t h e optimum s i z e  Wilkins  [42]  and  radiating  N i l s o n and  o b t a i n e d t h e minimum-mass t r i a n g u l a r p r o f i l e Berggren and  [44]  suggested  Winter  and  analyzed  simple  cases  means o f m a x i m i z i n g  Schaberg  arrangement of f i n s  [45]  suggested  f o r thermal  fin.  an  rejection  f i n of  o f a space  to  a  [43]  Callinan  and  radiators  per u n i t  intermeshing  control  transfer  Curry  of nonconvective  heat  determine  surface craft.  weight.  50 Apparently the  two-dimensional  all  used  literature  heat  techniques.  f o r both  one-dimensional  transfer  investigation  are determined  been employed t o a n a l y z e circular from  fin.  the  by  the  base i s i n c l u d e d .  and  they  Holdredge  to s o l v e the  f i n s w i t h no  and  have  fin-base  two-dimensional  solution  problem  f o r the  cases. two-dimen-  two-dimensional  In a d d i t i o n , The  have b e e n compared w i t h  a series  results  o f the  heat  radiation  radiation  a finite  solution,  whose  l e a s t - s q u a r e s f i t method,  H e a t i s d i s s i p a t e d by  i t s surface.  studies for  in fin-tube radiators.  In the p r e s e n t coefficients  few  H o l s t e a d and  in trapezoidal profile  S p a r r o w e t a l . [47] have o b t a i n e d t h e s i o n a l heat  fins  a f i n i t e - d i f f e r e n c e procedure  transfer  interaction  contains very  flow i n r a d i a t i n g  finite-difference  [46] haye u s e d of heat  the  and  has  flow i n a convection  interchange with  the  l e a s t - s q u a r e s f i t method  difference  solution.  F O R M U L A T I O N AND S O L U T I O N Consider shown  in (i)  Figure The of  (ii)  (iii)  a  circular  (2-1)  There  is  no  there  is  rotational  The  has  Two c a s e s  is  those  of  not  T  length  ' / a s  conditions: along  0  the  base  are  end  sides.  neglected  Section  A.  the  surface  radiation  properties  with  the  the  In  Section the  fin  is  of  that  with  fin  are  with  B' t h e  heat  the  are  the  neglected  results  the  and  base  material.  base  they  A deals  of  and  so  temperature.  interaction  c o n s i d e r e d and Section  fin  symmetry,  interactions  sections.  from the  on  homogeneous  fin  multiple  and  radiation  isotropic  The  end  a uniform value  of  (v)  from i t s  and  following  with  is  that  the  'a'  invariant  fin  transfer  radius  are  The  two  to  incident  physical  ' (iy)  in  has  of  PROBLEM  fin.  materials  ely  subject  temperature the  fin  OF T H E  base  not  dealt  in  in  separat-  which  comparison  transfer  obtained  considered. with  case  but  are  from  heat with  the  compared  with  SECTION A No Heat T r a n s f e r from the End of the F i n The  steady s t a t e d i f f e r e n t i a l  and the boundary c o n d i t i o n s may  equation of heat  conduction  be w r i t t e n as f o l l o w s :  Energy E q u a t i o n :  Vl  •  j_.  JL.  +  J £ L =o  .  (2-D  Boundary C o n d i t i o n s a r e :  ,  0  B.C.  (i)  At  B.C.  (ii)  At  t - ^  B.C.  (iii)  At  t - 0  B.Ci  (iv)  Z  The  -  T - T -|L ,  (2-2)  =  -JLL  0  •  - Q  (2-3)  .  "bt  f o u r t h boundary c o n d i t i o n a t r=a i n v o l v e s  the energy balance equation of an elemental area dA^ The heat conducted  (2-4)  on the f i n ,  from w i t h i n the f i n t o the s u r f a c e may  d i s s i p a t e d by three modes of heat t r a n s f e r .  be  F i r s t l y , the f i n  s u r f a c e element has r a d i a n t interchange with space :  • ^ , - ^  6  .  <  r  T  t  =  a  «,<rT* k . ^ A V  —  Then there i s c o n v e c t i o n interchange w i t h  &!\  • Vi { T ^  a  —  X.)  (2-5a)  space  »  and f i n a l l y , r a d i a n t interchange with the f i n base  (2-5b)  53  K^^K^X^s  V V ' ^  A  +  The c o n f i g u r a t i o n f a c t o r the  fraction  of  the  total  i n t e r c e p t e d by s u r f a c e U s i n g the  F  m-»-n  T  ^ «  i n t r o d u c e d above  e n e r g y e m i t t e d by s u r f a c e  (2  -  5  c)  denotes  'm* t h a t  is  'n . 1  reciprocity  theorem f o r  configuration  factors (2-6a)  (2-6b) and t h e  summation t h e o r e m  (2-6c)  M , -=»oo  and a s s u m i n g g r a y body p r o p e r t i e s energy balance  Rearrangement  equation reduces  of  the  terms  to  yields  for  the  fin  (i.e.  ct^=e^) ,  the  "bT  _fc,<r  (2-7)  The g o v e r n i n g e q u a t i o n in  the  fin  and t h e b o u n d a r y  convenient variables  dimensionless as  for  the  temperature  distribution  c o n d i t i o n s may be r e p h r a s e d  forms by i n t r o d u c i n g  into  dimensionless  follows:  X = (T-T„VX  , X. = t / T . , X* = T/T. . <2-8a) •3 / (2-8b)  8c)  In  terms  of  these  the boundary  -r-  conditions  = 0  <U0  2-3,  S I  -0  _bX_  (2-2,  the energy  \ ~oX =0  X  new v a r i a b l e s ,  1  2-4,  2-7)  equation  (2-1)  and  become  (2-9)  (2-10)  (2-11)  (2-12)  (2-13)  Equation ables.  (2-9) i s s o l v e d by t h e method The method  of solution  i s outlined  L e t t h e s o l u t i o n be o f t h e X Then  e q . (2-9)  -  HQ  of separation of v a r i below.  form  • ClU)  (2-14)  becomes:  1  Thus t h e two r e s u l t i n g  <U  equations are:  (2-15)  "  5  4-  The s o l u t i o n  to  (2-15) i s o f t h e  -\-  COS Y n Z where Equation  C/  (2-16) may  4  ,  C  2  5V.r> Y n X  ,  (2-17)  are constants  be r e w r i t t e n as  C o m p a r i n g t h e above e q u a t i o n w i t h [48]  form  i t c a n be e a s i l y  seen  t h e e q u a t i o n g i v e n by  t h a t t h e above e q u a t i o n  reducible  to Bessel's equation  following  form:  and i t s s o l u t i o n  Wylie  i s of a  i s of the  type  56  (2-18)  where  / 4 C  Bessel  a  r  e  u n  known  constants,  f u n c t i o n of the f i r s t  modified  Bessel  Thus, from  I  kind of order  D  i s the modified  z e r o and  f u n c t i o n o f the second k i n d o f order  (2-14),  (2-17) and  to the energy equation  (2-18),  K  i s the  Q  zero.  the complete g e n e r a l  solution  (2-9) i s  (2-19)  Use o f e q u a t i o n  (2-10) i n (2-19) y i e l d s  C,=0  .  Then  (2-20)  where  ,  Cg  a r e new unknown c o n s t a n t s .  condition equation  [ C Since  5  T »  (2-12) t o ( 2 - 2 0 ) ,  -  (2-12) must h o l d  we  Applying  have  7  C K,W] t  .0  f o r a l l Z,  (2-21)  Therefore,  i.e.  .  while  Now  Using  boundary  t h e c o n d i t i o n o f no h e a t  transfer  for  a l l R,  from t h e end,  we  get  * W \ Cos Y A L . - 0 .  57 T h i s y i e l d s the  eigen-values  for m  -m = U T . + V . W / I L Thus t h e  general  >  s o l u t i o n to  as  ^^,1,2,  (2-9)  -o  .  becomes  X -l^L^-^i^t^^lj The m i n e d by  unknown c o e f f i c i e n t s  p o i n t matching or  C  (n=0,-l  least-squares  on  the  boundary, choosing  along  of  the  f i n and  The  length  2,... ) can  be  00  f  fitting  number o f p o i n t s the  •«-,» at a  equally  matching or  finite  spaced  fitting  to  deter-  points  eq.(2-13).  least-squares  f i t method has  been employed h e r e s i n c e i t has  been, p r o v e n t o be  more e f f i c i e n t  than the  Part  I of the  p o i n t m a t c h i n g one  thesis.  From b o u n d a r y c o n d i t i o n  (2-13) t h e  minimized  i s obtained  where  d e n o t e s summation o v e r a l l p o i n t s  Using  the  takes  to  be  as  series solution for  above e x p r e s s i o n  expression  the  X  as  following  i n the  expressed i n form  region  (2-22)  the  in  58  -v(2-23) The  f u n c t i o n t o be  /s~  .  is  Since  first  applied  the  and  linearized  a system of  solved to give upon  to zero  the  and  i n the  then the  linear  process  is  criteria which  are  iterated  7  ^  points  on  the  ^  _  b o u n d a r y were  Thus f o r known v a l u e s  of  X  I n t h i s work 7  coefficients  chosen. , X*,  A,  N,  L and  e„  2.  oo '  temperature d i s t r i b u t i o n factor F,  i t  n  least-squares  The  is  parameters C ,  a l g e b r a i c equations C^.  coefficients  until 100  a t v a r i o u s boundary p o i n t s  function i s non-linear  locally to get  fitted  a  i s known.  a  can The  be  determined i f the  the  configuration  c o n f i g u r a t i o n f a c t o r was  evaluated  1 2  and  i s given  <j> -  where  and  by  —  The  of  the  following  expression,  +\  i s dimensionless  f i n base r a d i u s .  t h e above e x p r e s s i o n  i s given  results  of  3 and  are  for various  points  shown i n F i g u r e  (2-2) .  The  detailed  i n Appendix B  c o n f i g u r a t i o n f a c t o r s were f i r s t  various values and  p  ~L  "^/(X-  derivation  the  tabulated  along  the  (p.  139)  for  f i n length  The s y s t e m o f e q u a t i o n s (2-9) t o (2-13) has a l s o b e e n As p op le vn e d ix by C.a f(p. i n i t1e4-2d)i. f f e r e n c e p r o c e d u r e w h i c h i s d e s c r i b e d i n  DISCUSSION OF RESULTS The ature  results  distribution  A x i a l Temperature The  are presented  i n the form o f a x i a l  and f i n e f f e c t i v e n e s s . Distribution:  temperature, d i s t r i b u t i o n  represented  temper-  i n Figures  (2-3) and  along  (2-4).  the axis  Figure  (R=0) i s  (2-3) shows  a t any p o i n t on t h e a x i s t h e t e m p e r a t u r e d e c r e a s e s w i t h ing values expected  of the radiation-conduction  since high  conductivity  values  p a r a m e t e r A.  of A signify  k or high values  of f i n radius  t u r e T , each o f which decreases the r a t i o 0  the  end i s zero,  as e x p e c t e d , i n d i c a t i n g  increas-  This  a low v a l u e  0  i s as  of thermal  'a' o r b a s e T/T .  that  tempera-  The s l o p e a t  no h e a t t r a n s f e r f r o m  the end. Figure the An  convection increase  fore,  (2-4) i l l u s t r a t e s  the e f f e c t  p a r a m e t e r N on t h e a x i a l  i n N c a u s e s more h e a t  lower s u r f a c e  and a x i a l  of the v a r i a t i o n o f  temperature  l o s s by c o n v e c t i o n  distribution. and, t h e r e -  temperatures.  Fin Effectiveness: The in  h e a t t r a n s f e r p e r f o r m a n c e o f t h e f i n c a n be  terms o f t h e f i n e f f e c t i v e n e s s d e f i n e d as  Gl  where t h e i d e a l h e a t fin  expressed  > (2-25)  flux  i s d e f i n e d by a s s u m i n g t h a t t h e e n t i r e  i s a t t h e b a s e t e m p e r a t u r e and r a d i a t e s w i t h o u t Non-dimensionalising  interference.  a l l the terms, the e x p r e s s i o n f o r  60 fin  effectiveness  Figures variation the  of  takes  (2-5)  of  seven  sentative  cases  effect  of  emissivity  ness.  Increasing  fin  due  to  while  Figure  ideal  heat of  of  increases  in  the  fact  an  increase  that  end  fin  the  been p r e s e n t e d . of  the  base  of  the  base  since  the  actual  transfer changes of  the for  X*  in  and  X  decreases of  shown.  This  decrease  decreases base  shown the  X ^ and  decreases  X*  is  is  the  effective-  temperature  and  case  fin  to  repre-  shows  high  in  values  few  transfer  same.  lower  the ©ue  heat  the  X*  of  emissivity  remains X^  a  (2-5)  on t h e  surface  the  X ^ and  the  only  Figure  surface  with  effect  effectiveness.  independent parameters  Increase  most p a r t  form  illustrate  on t h e  interaction  effect  (2-6).  for  (2-9)  values  increasing  The  ness  have  effectiveness  the  t©  following  some p a r a m e t e r s  presence  the  the  in  fin  effective-  X*  range  due  to  both  the  and  the actual  oo  heat  transfer  ratio, the  fin  other  observed  and  the  ideal  effectiveness, parameters.  for  will  This  3=1, w h i c h  heat  transfer.  then  The  effect  depend upon the  minimum i n  represents  value  effectiveness  the  case  of  on  was  no  their of  not  fin-base  interaction. Figure the in  convection the  greatly  value  of  (2-7)  illustrates  parameter  N on  N decreases  r e d u c i n g the  heat  the  fin  the  effect  the  effectiveness.  surface  transfer  of  by  variation An  temperature,  radiation.  of  increase thus  This  reduction  is  normally  the  greater  overall effect  than the i n c r e a s e thus b e i n g  Increasing values ature as  ation  of  (2-8). $  cause h i g h e r  f a c t o r thus r e s u l t i n g  f i n with  the base.  surroundings  A  This  effect  i s small  as c a n be s e e n f r o m t h e o r d i n a t e s  (2-9). c o m p a r i s o n h a s a l s o b e e n made i n T a b l e  by  S p a r r o w and N i e w e r t h  [39].  by t h i s work w i t h In Table  as one t o a v o i d b r i n g i n g i n t h e e f f e c t  2-1  8  range o f v a l u e s  since only  ence  [39].  high  conductivity fins  range.  dimensional  This  by t h i s  effectiveness  taken  interaction limited .  since  greater  the f i n  throughout  work c o n s i d e r s  lower s u r f a c e  since i trepresents  i s small,  a n a l y s i s i s lower  i n f i n body t e m p e r a t u r e s  f i n case.  has been  o f A) a r e c o n s i d e r e d ,  i s because the p r e s e n t  model which causes  greater variation  obtained  a few r e s u l t s were a v a i l a b l e i n r e f e r -  (low v a l u e s  as o b t a i n e d  those  f o r a very  Although the d i f f e r e n c e i n values  effectiveness  2-1 o f t h e v a l u e  of fin-to-base  c o m p a r i s o n c o u l d , however, be made o n l y  isothermal  of the  suppresses the heat t r a n s f e r t o the  f i n effectiveness obtained  the  of the configur-  i n more r a d i a n t i n t e r a c t i o n  of  The  values  and, t h e r e f o r e , d e c r e a s e s t h e f i n e f f e c t i v e n e s s .  However, t h i s of Figure  reduce t h e temper-  This decreases the f i n e f f e c t i v e n e s s  i n Figure  High values  term,  t o decrease the f i n e f f e c t i v e n e s s .  o f the f i n length  of the f i n surface.  illustrated  i n the convection  a two-  temperatures.  causes lower f i n departure  from t h e  The  62  Comparison of the Least-Squares F i t R e s u l t s w i t h Finite Difference Solution The the  two  an  high  values  very  o f the  evidenced  closely.  to four s i g n i f i c a n t  d e v i a t i o n i n the  by  Figure  (2-8).  the  least-squares  more b o u n d a r y p o i n t s  values The  the  I t has  are  d i f f e r e n c e technique.  values  obtained  t e r m s and  100  method.  increased  are  Thus t h e  finite  the  computer  is  an  as  the chosen and  f i t method,  obtained  (2-8)  is  with  When more s e r i e s t e r m s  In F i g u r e  least-squares  time.  there  f i n effectiveness  approach those  compared w i t h  difference values,  there  number o f b o u n d a r y p o i n t s  and  that  However, e v e n  least-squares  by  values  d e v i a t i o n depends upon  least-squares  boundary p o i n t s  boundary p o i n t s  of  taken i n the  f i n e f f e c t i v e n e s s values  by  The  been o b s e r v e d  figures.  f i t technique.  finite  the  (2-3).  obtained  r a d i a t i o n - c o n d u c t i o n parameter A  number o f s e r i e s t e r m s and  the  i n Figure  c l o s e agreement i n temperature d i s t r i b u t i o n ,  appreciable  in  temperature d i s t r i b u t i o n  illustrated  each other  a g r e e m e n t up  this  o f the  methods a r e  agree w i t h for  results  the  by  the  f i n effectiveness  f i t method f o r 7 s e r i e s  f o r 20 those  s e r i e s t e r m s and o f the  finite  f i t method r e s u l t s  although  a t the  can  expense  400  difference approach of  SECTION Heat T r a n s f e r from  For neglect  fins  the heat  t o t h a t from  t h e End  B  o f t h e F i n i s Not  o f s h o r t l e n g t h i t may transfer  i t s sides.  from  t h e end  This situation  n o t be of the  Neglected  very f i n as  i s analyzed  correct  to  compared in  this  section. The ditions 2-3) now  energy  equation  remain u n a l t e r e d .  i s altered. takes  the  The  and Only  the o t h e r t h r e e boundary  con-  boundary c o n d i t i o n ( i i ) (equation  b o u n d a r y c o n d i t i o n a t t h e end  of the f i n  f o l l o w i n g form:  Zrt  or Using S e c t i o n A, then  takes  the d i m e n s i o n l e s s  the the  Thus t h e e n e r g y case  above e q u a t i o n  parameters d e s c r i b e d e a r l i e r i s rendered  dimensionless  in  and i t  form  equation  and  the boundary c o n d i t i o n s f o r t h i s  are:  A \  .  » =0  2=&  *X  .  VA  o (2-10)  64  Z-_L  ^x  •  -0  (2-12)  ,  -MV xJ ^ X * \ V - ^ J +  (2-13)  The this 4^— dz  in  Use  (2-19)  general solution  derived earlier  c a s e we c a n n o t g e t any e i g e n - v a l u e s  1  I  T ^ 0 . Z=L  To s a t i s f y J  the f o l l o w i n g  this  still  for m  holds but i n since  c o n d i t i o n we t a k e t h e s o l u t i o n  form:  (2-12)  of equation  (2-29) =0 .  i n above e x p r e s s i o n y i e l d s  or I f we t r y t o s a t i s f y  then  X vanishes  Thus t h e f u l l equation fitting  (2-10), then C  N  =  0  and  i & ^ f ] . ; ^ )  x But  equation  at Z = L a l s o which v i o l a t e s equation  f o r m o f A as e x p r e s s e d  (2-10)  i n (2-29) i s r e t a i n e d and  i s not s a t i s f i e d exactly.  procedure  i s then  (2-28)  The l e a s t - s q u a r e s  a p p l i e d t o t h e boundary c o n d i t i o n s a t  65  three boundaries, z = 0, 1 = L and R = 1 as expressed by the equations  (2-10),  (2-28) and  (2-13).  However the equations became too unstable and not e a s i l y amenable t o s o l u t i o n .  T h i s was  probably due to the severe r e -  s t r i c t i o n s imposed on the l e a s t - s q u a r e s f i t t i n g procedure  i n this  case. The f i n i t e d i f f e r e n c e procedure was s o l v e the system of equations procedure used was  (2-9, 2-10,  then a p p l i e d to  2-28,  2-12,  s i m i l a r t o the one used i n S e c t i o n A  d e s c r i b e d i n Appendix C  t r a n s f e r from the end)  The  and  (p.142). The r e s u l t s o b t a i n e d were com-  pared w i t h the f i n i t e d i f f e r e n c e r e s u l t s of S e c t i o n A  The  2-13).  (no heat  and p l o t t e d i n F i g u r e s (2-10) and  (2-11).  f i n e f f e c t i v e n e s s f o r S e c t i o n B i s r e - e v a l u a t e d as  follows:  V  <W  CM.+vro})i^{t-r") +UT -T»)] ' 0  On n o n - d i m e n s i o n a l i s a t i o n the above e x p r e s s i o n takes the ing  follow-  form: -4- \  u  \.(2-31)  COMPARISON OF Figure  (2-10)  ture d i s t r i b u t i o n in  SECTION A WITH SECTION B  illustrates  F o r low v a l u e s  conductivity)  agreement  comparatively  smaller values  s t r a t e d by t h e p l o t s  in  the r e s u l t s  very high in  of  of L.  Even i f heat R "5^1  values  ( 2L+1  thermal  c o n d i t i o n s i s reached  T h i s i s more c l e a r l y  (2-11).  of A reduce the d i f f e r e n c e conditions.  However, e v e n f o r  =  a t low v a l u e s  of L,  will  still  transfer i-  R  L  s  cases  (cf. equations  2-26  from the end i s n e g l i g i b l e  neglected,  and  and  the d i f f e r e n c e i n the  1 ^  at  demon-  to the i n h e r e n t d i f f e r e n c e i n d e f i n i t i o n s  d Z  2 factors  ( i . e . high  c o n d u c t i v i t y o f t h e f i n t h e marked d i f f e r e n c e  f i n e f f e c t i v e n e s s f o r t h e two  term  tempera-  conditions stated  for f i n effectiveness i n Figure  o b t a i n e d by t h e two  thermal  of A  t h e two  o f L o r low v a l u e s  T h i s i s due  2-31). the  between  f i n effectiveness, especially  remain.  of a x i a l  o b t a i n e d by t h e two b o u n d a r y  S e c t i o n s A and B.  Both h i g h values  the r e s u l t s  A N C  ^ L  c  ^use  a marked d i f f e r e n c e i n t h e  of f i n effectiveness, especially  f o r low v a l u e s  of  L.  CONCLUSIONS  The fin  having  ysed.  problem of two-dimensional heat flow radiant i n t e r a c t i o n with  The  s o l u t i o n i s obtained  least-squares axial is  a l s o s o l v e d by  high  Results  temperature d i s t r i b u t i o n  obtained  by  the  a finite  two  methods a r e  the  are p r e s e n t e d  of  compared.  results  f i n becomes s m a l l or  in prior  b e e n s o l v e d by  demonstrate t h a t  ductivity  anal-  a s e r i e s e x p a n s i o n and i n terms  the  The  problem  results  For materials  agree c l o s e l y w i t h  h e r e i n has  of heat t r a n s f e r from the  p r o b l e m has  o f the  been  f i n effectiveness.  i n v e s t i g a t i o n reported  g e n e r a l l y been n e g l e c t e d  results  circular  with the  one-  model.  The effect  and  f i n b a s e has  d i f f e r e n c e s o l u t i o n and  t h e r m a l c o n d u c t i v i t y the  dimensional  the  f i t method.  by  the  in a  f o r long  an e f f e c t  that  has  effect  d i f f e r e n c e procedure.  thermal  This  The  o f heat t r a n s f e r from the  f o r f i n materials of high  fins.  considered  studies of r a d i a t i n g f i n s .  a finite  the  end,  also  end  con-  68  TABLE 2-1 Comparison o f f i n e f f e c t i v e n e s s values w i t h those by S p a r r o w a n d N i e w e r t h  Values o f t h i s work L  A  obtained  [39] (g = 1, e „ = 0.4)  Reduced t o v a l u e s o f [39] N  N =2AL r  2  N  C V  =2NL  2  A =6 oo  oo  A*=e* t h i s  n  work  n  [39]  io"  2  3.873 0.005  0.3  0.15  . 0.7  0.7  0.691  0.728  IO"  2  3.873 0.005  0.3  0.15  0.9  0.9  0.681  0.718  IO"  3  12.25  0.005  0.3  1.5  0.7  0.7  0.552  0.580  io"  3  12.25  0.005  0.3  1.5  0.9  0.9  0.541  0.571  io"  2  5.477 0.01  0.6  0.6  0.7  0.7  0.539  0.577  IO"  2  5.477 0.01  0.6  0.6  0.9  0.9  0.522  0.558  IO"  3  17.32  0.006  0.6  3.6  0.7  0.7  0.391  0.420  io"  3  17.32  0.006  0.6  3.6  0.9  0.9 •  0.380  0.409  1 Figure  2-4  Effect  o f N on A x i a l T e m p e r a t u r e  Distribution  " ~io* Figure  7  2-5  10'  6  10'  5  10"  A  E f f e c t of Base Surface Fin Effectiveness  4  IO*  3  Emissivity  10" and A  10  2  on  Figure  2-6  Effect  of  Environment  Temperatures  on F i n  Effectiveness  .O O i  (D to  I  Hi Hi  (D O rt  O  Hi  > 0 *J  H-  3 W  Hi Hi  (D  o  rt  <  (D 01  tn  FIN w  EFFECTIVENESS ^  6s  0 0  Figure  2-8  Effect  of  Fin  Length  on F i n  Effectiveness  *This denotes 7 terms in the series and T O O boundary points.  cr.  CO CO U J  LU >  o U J Li. U _ U J  5 Figure  2-9  Effect  of  Fin  Base  Radius  P  on F i n  6  7 Effectiveness  8  80  Figure  2-12  Finite  Difference  Representation  P  LAMINAR  A  R  III  T  HEAT T R A N S F E R  UNDER  SOLAR  IN A C I R C U L A R  RADIATION  IN S P A C E  TUBE  ABSTRACT The  problem o f laminar heat t r a n s f e r  i n a circular  tube  under r a d i a n t h e a t f l u x boundary c o n d i t i o n s has been a n a l y z e d . Fully  developed v e l o c i t y  considered stationary. incident  on one h a l f  profile  i s assumed and t h e t u b e i s  A steady r a d i a n t energy  o f the tube c i r c u m f e r e n c e w h i l e  emanates h e a t t h r o u g h t h e w a l l  bution  i s being the.fluid  on a l l s i d e s by r a d i a t i o n  zero degree temperature environment. difference  flux  A s o l u t i o n by  p r o c e d u r e has b e e n o b t a i n e d .  and t h e N u s s e l t number v a r i a t i o n  wide range o f the g o v e r n i n g p h y s i c a l  to a  finite-  The t e m p e r a t u r e  distri-  are presented f o r a  parameters.  NOMENCLATURE radius  of  incident thermal  tube,  ft.  radiation  flux,  conductivity  tube  length,  1/a,  dimensionless  of  BTU/hr. fluid,  ft.  2  BTU/hr.  ft. tube  length  2 fluid  pressure,  lb/ft.  2 heat  transfer  radial r/a, Pr.  rate,  coordinate,  dimensionless Re,  /uCp/k, r/a,  BTU/hr. ft. radial  number,  dimensionless  Prandtl  number,  dimensionless  dimensionless  ,  Reynold's  temperature average axial  coordinate  Peclet  radius  R / Pe , d i m e n s i o n l e s s coordinates 2Ua/v  ft.  at  fluid  fluid  any  radius  number, point  velocity,  velocity,  in  trans  dimensionles ,  °R.  ft./hr. ft./hr.  84  v  radial fluid  x  axial coordinate,  X  dimensionless a x i a l coordinate  velocity,  ft./hr.  ft.  Greek Symbols a  c o e f f i c i e n t of a b s o r p t i v i t y dimensionless  o f tube  wall,  Y  eoT  e  coefficient  of e m i s s i v i t y of tube w a l l ,  X  1/3 T/(k/aa) '  ,  v  kinematic viscosity  £  X/Pe  p  density of f l u i d ,  3  a  a/k  0  ,  a dimensionless  parameter  dimensionless of f l u i d ,  dimensionless  temperature 2 f t . /hr.  , dimensionless a x i a l distance 3  lb^/ft. —8  Stefan-Boltzmann . °R.  c o n s t a n t , 0.1714 x 10  2 BTU/hr.ft.  4  ip  a (G^a^a/k ) dimensionless  6  angular  4  ,  r a d i a t i o n - c o n d u c t i o n parameter,  coordinate  Subscripts 0  at entrance  b  fluid  bulk  (x =  0)  critical radial at  wall  axial  INTRODUCTION  Heat t r a n s f e r problems r e l a t i n g have been t h e s u b j e c t  and by  Klein Singh  Kuga  (although  variable)  viscous  Solutions  also included  [ 5 4 ] . Hsu  exponential  Various conditions.  temperature  include  w a l l heat  heat  reviewed con-  generation.  temperature  (although  variable)  distribution. heat-flux  t h e work o f S i e g e l , S p a r r o w and  [55] c o n s i d e r e d  and Kuga  are well  Tribus  the e f f e c t s of a x i a l heat  a sinusoidal wall  involving prescribed  distribution  field  d i s s i p a t i o n and c o n s t a n t  boundary c o n d i t i o n s  i n tubes  i n c l u d e t h e c l a s s i c a l work o f S e l l a r s ,  [53] c o n s i d e r e d  Hallman  types o f boundary  [51]. Other s o l u t i o n s i n t h i s [52] who  duction,  various  involving prescribed  boundary c o n d i t i o n s  flow  o f i n v e s t i g a t i o n f o r many y e a r s .  i n v e s t i g a t o r s have d e a l t w i t h Solutions  t o laminar  [56] s o l v e d  a s i n u s o i d a l w a l l heat  flux  t h e p r o b l e m f o r s i n u s o i d a l and  fluxes.  There i s another c l a s s o f problems i n which n e i t h e r the w a l l temperature nor the w a l l heat the w a l l heat f l u x ature.  This  attention. fer  equation  ature  t y p e o f p r o b l e m has o n l y This  i s a more d i f f i c u l t  now  involves  i s prescribed.  recently received problem since  t h e unknown v a r i a b l e  a l . [57] e x t e n d e d G r a e t z s o l u t i o n t o i n c l u d e flow  flat  the Graetz  Stein  to the concurrent  flow  [58] s o l v e d double pipe  some trans-  ( e i t h e r temper-  r a t h e r than e x p l i c i t  tance t o heat t r a n s f e r i n laminar  temper-  the heat  [57-6.0] have t r e a t e d s u c h t y p e o f p r o b l e m s .  conduits.  Instead,  as a f u n c t i o n o f t h e w a l l  o r h e a t f l u x ) i n an i m p l i c i t  References et  i s specified  flux  surface  in circular  form. Sideman resis-  t u b e s and  problem p e r t a i n i n g  h e a t e x c h a n g e r s and  introduced  87 a n  e f f e c t i v e n e s s c o e f f i c i e n t f o r  s o l v e d  t h e  t h r o u g h t h e  p r o b l e m  a  t u b e .  H e  r a d i a n t  a p p r o x i m a t e s u l t s  t h e  a n d  a l s o  f l u i d  I r v i n e  s a m e  [ 6 0 ]  p r o b l e m  i n  a n  a l s o  a n d  [ 5 9 ]  l a m i n a r  s o l u t i o n  o b t a i n e d  n e i t h e r  T h i s  r e a c t o r s  p r o b l e m  c o u p l i n g o f t i o n  a  t h e  h e a t  o f  i n  f l o w  t e r m s  o f  i t e r a t i v e p r e s e n t e d  v e r i f i e d  a n d  i n  a r i s e  t w o  s a t e l l i t e s i n  u n d e r g o i n g  t u b e  e i t h e r  r a d i a t i o n  p r o b l e m  i n  h e a t  f o r  h e a t e d c o o l i n g  b y  a  f r o m  [ 5 9 , h e a t  a n  t h e  r e -  t h e  b e e n  t h e  r a t e  i n  a p p l i c a t i o n s  s y s t e m s  o r  i n  p r e s e n t i n v e s t i g a e m p l o y e d  d e v e l o p e d  u n i f o r m  c o n s i d e r e d  t r a n s f e r  c r a f t  r e j e c t i o n  h a s  6 0 ]  a p p l i c a t i o n s  s p a c e  s p a c e . . I n  f u l l y  t h e  h a s I n  p r o c e d u r e  p r o b l e m  b e i n g  o n  s p a c e c r a f t .  m a y  t r a n s f e r a  i n v e s t i g a t o r s  p a r t i c u l a r  f i n i t e - d i f f e r e n c e  i n  t h e  i n c i d e n t r a d i a t i o n f l u x  f l u i d .  n u c l e a r  f l u i d  a  a p p r o x i m a t e  a n d  t h e  o f  C h e n  e x p e r i m e n t a l l y .  effect o f  t h e  D u s s a n  s o l x i t i o n f o r  E o w e v e r , t h e  a n  s e r i e s  s o l u t i o n .  e x c h a n g e r s .  c o o l i n g  o b t a i n e d  L i o u v i l l e - N e u m a n n  n u m e r i c a l  o f  o f  h e a t  l a m i n a r  i n c i d e n t  s u r f a c e .  f l u x  t o  s o l v e  f l o w a n d  o f a l s o  FORMULATION OF THE PROBLEM  Consider through  a constant property  a circular  tube  of radius  fluid  i n laminar  flow  'a' ( F i g u r e 3 - 1 ) . A  steady  2 r a d i a n t energy  flux of G  BTU/hr. f t .  being  i n c i d e n t on one h a l f  fluid  emanates h e a t  to  have a f u l l y T„.  developed  Heat t r a n s f e r  t h a t t h e tube flow e f f e c t s  velocity  i s not rotating from  conservation equations  The  and a u n i f o r m  a t x=0.  temperature  I t i s assumed  a b o u t any a x i s s o t h a t  situation  will  i s considered to  secondary  f o r c e may be n e g l e c t e d .  s t a t e d above t h e f o l l o w i n g  apply.  continuity  equation  since  \)^. — 0  —  Equation  solution  given  profile  centrifugal  automatically satisfied  Momentum  the f l u i d  Equation  The  is  x=0,  a t the w a l l s t a r t s  arising  circumference while the  t h e w a l l on a l l s i d e s by r a d i a t i o n  At  For the p h y s i c a l  Continuity  o f t h e tube  through  a 0°R. e n v i r o n m e n t .  of projected area i s  o f t h i s momentum  equation  below,  where U i s t h e a v e r a g e  fluid  velocity,  i s w e l l known and i s  89 Energy  Equation The  energy  e q u a t i o n f o r the system i s ,  ^ _bT_ _ _W_ ( ^  v  ~  ?C  V  F  I n t h e above energy fluid  \ "bT +  3  +  equation the a x i a l heat conduction i n the  equation i s n e g l i g i b l e  Using the s o l u t i o n the energy  equation reduces  w) J ^ -  F^  k  - -9C;L-^F  (i)  At  B.C.  ( i i ) At  B.C.  ( i i i ) At  ditions  X-  &  ,  t -  0  and  t" -z <X , we  f o r t h e two  >  + '  T  100. (3-1)  <> x » T  a r e as  follows.  T - T  (3-4)  6  § -  TV/2  ,  —  have the f o l l o w i n g  regions of the  tfT"|(3-3)  T^T+T^\  +  boundary c o n d i t i o n s f o r the system  B.C.  Pe  to  T  «L\  for  this  of the v e l o c i t y p r o f i l e from eq.  •)\\Y\ - W l ^ -  k  T ' ~ F ~  VTV . ~~W)  i s n e g l e c t e d , s i n c e i t i s known t h a t t h e e f f e c t o f  term i n the energy  The  \  -0 two  .  (3-5)  boundary  con-  circumference.  "oT  3  "Ot Convection  l o s s e s from the s u r f a c e have been i g n o r e d ,  their inclusion w i l l difficulty.  f  i n t r o d u c e no a d d i t i o n a l  •  (3-7,  although  mathematical  . The  above b o u n d a r y c o n d i t i o n s  is  very  thin  and  of  low  no  temperature drop through the  thermal c o n d u c t i v i t y  c i r c u m f e r e n t i a l heat conduction compared t o t h e Since may  consider  (Figure leads  to  there  i s symmetry  the  following  (iv)  At  Q  B.C.  (v)  At  B=T  rephrased  - ^  , — ^0  resulting  The  boundary c o n d i t i o n s  at  ^  x  ,  0  X  ana  wall  there  the  are  is  axial  and  negligible  wall. 6=0  and  i . e . , arc  This  we  6=TT,  ABC  symmetry  also  ,  •  O^t"40v  4+4  0  0L  (3-8)  .  boundary c o n d i t i o n s f o r m s by  (3-9)  may  be  introducing  follows:  energy equation  = 0  lines  77-7—v77  ^ -  »  The  X  the  that  tube  conditions.  ,  and  the  that  tube  (iii).  convenient dimensionless  =  at  and  circle,  , ^  d i m e n s i o n l e s s v a r i a b l e s as  K  the  boundary  energy equation  into  of  so  tube w a l l  about the  condition  two  B.C.  The  i n the  upper p a r t  f o r boundary  the  tube w a l l  heat t r a n s f e r normal t o the  only  3-1)  assume t h a t  90  '(3-10)  is,  are,  =  X,  S  =  ;  TV/l  (3-12)  ,  ^  =  0  J  (3-13,  91  - y co&B  fclV]  _  > cue 4^  (3-14)  .  (3-15)  at  51  8r(i  , cud 41  =0  (3-17)  at In  equation  (3-16)  (3-14),  tion-conduction  = a (G a a/k )  parameter,  3  4  4  is  a dimensionless  radia-  METHOD OF An (3-11) t o to  exact a n a l y t i c a l s o l u t i o n (3-17) a p p e a r s  develop  a series  Such a s e r i e s w i l l appeared A  t o be  solution be  o f the energy  a double  infinite  t o be v e r y cumbersome by  Finite-Difference The  X and  partial  R and  also  o f the system  impossible.  f i n i t e - d i f f e r e n c e s o l u t i o n was  The  SOLUTION  equation  a l e a s t squares therefore  attempted.  Procedure:  differential  i n X and  6.  equation  U s i n g the  (3-11) i s p a r a b o l i c forward  The  approximation  central  the remaining d e r i v a t i v e s  as  of  t h e R,  an e x p l i c i t i s used  a p p e a r i n g i n the e q u a t i o n  approximation  to the energy  method for  (3-11) .  equation  (3-11)  and  sizes  follows:  where i , j , k and for  difference  obtain  6 and  a^, b, c are the s u b s c r i p t s X directions  t e r m s t h e above e q u a t i o n  respectively.  yields  On  in  difference  of  thus  (3-11) .  f i t procedure.  we  is  possible  s e r i e s , h a n d l i n g of which  f o r the d e r i v a t i v e  finite-difference  equations  I t i s quite  approximation solution.  of  step  rearrangement  The  93  (3-18)  On the boundaries, however, we do n o t t r y t o s a t i s f y the d i f f e r e n t i a l equation but s a t i s f y the boundary c o n d i t i o n s o n l y . The  g e n e r a l approximation was d e s c r i b e d i n Appendix C o f P a r t I I  o f the t h e s i s and i s b r i e f l y repeated  here.  Using the f o r w a r d - d i f f e r e n c e approximation  f o r the d e r i v -  a t i v e o f a f u n c t i o n a t a p o i n t , one o b t a i n s , i n g e n e r a l , f o r a function  < j > a t any boundary,  (3-19)  We thus s t a r t w i t h the known temperature u t i l i z i n g boundary c o n d i t i o n equation i n the next s e c t i o n o f  X  at  X=c  d i s t r i b u t i o n a t X=0,  (3-12).  The i n t e r i o r p o i n t s  (c i s the step s i z e i n the  X - d i r e c t i o n ) are c a l c u l a t e d by the r e l a t i o n  (3-18).  F o r the  t h r e e boundaries, whose boundary c o n d i t i o n s are expressed by the equations  (3-13),  (3-16) and (3-17), the g e n e r a l e q u a t i o n  (3-19)  t o be used i n the d e t e r m i n a t i o n o f the f u n c t i o n a t the boundary reduces t o the f o l l o w i n g :  (3-20)  94 For vanish.  t h e b o u n d a r y R=l t h e n o r m a l d e r i v a t i v e 0 < 6 < 2  F o r example, o v e r t h e range  difference  relation  t  n  does n o t finite  e  used t o determine f u n c t i o n v a l u e s  at the  boundary i s  where NA The  by  i s t h e number  above r e l a t i o n  of intervals  i s nonlinear  chosen i n the R - d i r e c t i o n .  and i s s o l v e d  for  X.._ ,, . , NA+1, j , k  t h e Newton-Raphson method. When t h e f u n c t i o n v a l u e s  we  then proceed t o evaluate  in  a similar  till  g r i d was  the r e s u l t s  further increase Determination  the f u n c t i o n a t the next  first  remained a p p r e c i a b l y  i n grid  of Nusselt  c a n be e v a l u a t e d  V=  t h e same w i t h  made  any  Number the Nusselt  as f o l l o w s .  at a certain X-section  number Nu  a t a c e r t a i n value  i s defined  averaged over the  "  where b a r o v e r a q u a n t i t y section  size  fineness.  A semi-local Nusselt number  X-section  u s e d and t h e n t h e g r i d  Once t h e t e m p e r a t u r e s o l u t i o n i s o b t a i n e d number  are evaluated  manner.  A coarse finer  a t one X - s e c t i o n  ,3  .  as t h e N u s s e l t circumference.  .  (3-22,  denotes i t s average v a l u e o f x.  T^ i s t h e f l u i d  over a c r o s s -  bulk  temper-  95 ature  for  any c r o s s - s e c t i o n .  k_  Now  C  *T  and  X  -  x  =  M  (3-23)  f  1  X  (3-24)  (3-25) Substituting equation  the values  (3-22),  we  of q  w  ,  and T  or,  f r o m above  into  obtain  2* -  f e  in  [  Aft  "b^" lire  2<x  i n dimensionless form,  IT!  1\ Since  there  relation  is  symmetry a b o u t t h e  lines  9=0 and  0=IT,  the  becomes  IX*  ~M;IXU<IU*  above  96  Now  |  (3-14) and  may be e v a l u a t e d by u s i n g the boundary c o n d i t i o n s (3-15) t o y i e l d the f o l l o w i n g  On s i m p l i f i c a t i o n t h i s  yields  T r a n s f o r m a t i o n of Coordinates I f the e q u a t i o n  (3-11) were t o be s o l v e d i n i t s p r e s e n t  form, f o r h i g h v a l u e s of P e c l e t numbers t h e r e i s a danger  of  l o s i n g a l l h i g h - o r d e r d e r i v a t i v e s and thus g e t t i n g i n a c c u r a t e results.  To a v o i d t h i s we i n t r o d u c e the f o l l o w i n g t r a n s f o r m a t i o n ;  K Then equation  =  ft.^L  .  (3-27)  (3-11) takes the f o l l o w i n g form  The f i n i t e d i f f e r e n c e approximation  t o the above e q u a t i o n  then  becomes  (3-29)  Boundary c o n d i t i o n equations  (3-14) and (3-15) change t o  "ax  W ) .  •ad The r e l a t i o n  , f  46  4T  .  ,3-3i)  (3-26) f o r N u s s e l t number takes the f o l l o w i n g  O v e r a l l Energy  form  Balance  Although c o n s i s t e n c y o f r e s u l t s w i t h f i n e r g r i d s i z e i s normally used as a c r i t e r i o n o f convergence  and accuracy i n  f i n i t e d i f f e r e n c e s o l u t i o n s , a heat balance check was a l s o made i n t h i s case. E q u a t i n g the d i f f e r e n c e i n e n t h a l p i e s a t two s e c t i o n s over a c e r t a i n l e n g t h o f the tube t o the n e t heat t r a n s f e r  from  the tube w a l l , we o b t a i n the f o l l o w i n g heat balance e q u a t i o n .  x-.o e.  6  t-o.  +-o 6-0  -  \  \  SC^ T X  %  *<Jfi4^  I n s e r t i n g the known s o l u t i o n o f the v e l o c i t y p r o f i l e above e q u a t i o n changes t o  (3-33)  (3-1) the  98  XrO  0  O-K  T  X^t  rf^  On n o n - d i m e n s i o n a l i s a t i o n of a l l the terms the above e q u a t i o n s i m p l i f i e s to  \ \  tL  Aft A*  = k  i  X  X* ^ M 'X,L Use of the boundary c o n d i t i o n s ( 3 - 1 4 ) and 1  2  (3-15)  for  8X  IR=I  i n the above e q u a t i o n y i e l d s  \  \  (.Nicwft  -txtUeu - J J ex* 1  Use o f the symmetry c o n d i t i o n y i e l d s the f o l l o w i n g heat  balance  e q u a t i o n a f t e r some s i m p l i f i c a t i o n and rearrangement of the terms.  f  \  X\  ^V-^i»4.\ = ^k +^ - L J  [  X W ( 3 - 3 4 ,  In terms of the transformed c o o r d i n a t e s the heat balance  equation  becomes  (3-35)  T h e r e f o r e , percentage e r r o r i n thermal energy balance i s g i v e n as  follows:  error  \  .  0  r  L i r t, f  0  0  'lUfe  (3-36)  DISCUSSION OF  The average w a l l  results  are p r e s e n t e d i n the form o f v a r i a t i o n  temperature  and t h e a n g u l a r w a l l  Average  Wall The  (3-24).  The  distance of  a t any  ©f  distance  section.  Temperature average w a l l average  4  d  \l> (= c t ( G a a/k  temperature  wall in  dimensionless 1  and N u s s e l t number w i t h a x i a l  temperature d i s t r i b u t i o n  i s plotted  the  RESULTS  T~  i s d e f i n e d by e q u a t i o n  w  temperature v a r i a t i o n  Figures  ( 3 - 2 ) , ( 3 - 3 ) and E,,  parameters  with (3-5)  G oT  y  axial  (=  i n terms  a ^—) 3  and  "I /"}  ) '  ).  Y=0.5 and d i f f e r e n t  In F i g u r e  values of  (3-2)  .  has b e e n p l o t t e d f o r Tw expected, =— i s higher  As  this  o  for  a higher value of  a t any v a l u e o f £ .  r e d u c e s t o one w i t h no distribution are  radiation  becomes i n d e p e n d e n t o f a.  well  and  t h e maximum d e v i a t i o n  v a l u e s over which  cent.  the comparison  It is difficult  and  the  For t h i s  compared w i t h t h o s e o b t a i n e d by Chen  fairly of  incident  F o r ty=0 t h e  [59].  temperature case the  The  in results  problem  results  results  f o r the  i s made i s l e s s  agree  range  than 2 per  t o s a y w h e t h e r Chen's r e s u l t s  or the  p r e s e n t ones a r e more a c c u r a t e s i n c e b o t h a r e i t e r a t i v e n u m e r i c a l solutions. the it  The  average w a l l  results  F o r \p = 1.8  a v e r y s m a l l v a r i a t i o n w i t h K.  p a r t i c u l a r value of y there T  w  ally  show t h a t  does  a t Y=0.5 and  for  < 1.8 •  t e m p e r a t u r e d e c r e a s e s w i t h £ w h i l e f o r i/> >  i n c r e a s e s w i t h £.  has  also  This  is a critical  not vary appreciably  t h i s means t h a t  the average w a l l indicates  temperature that  for a .  v a l u e o f ip f o r w h i c h  from the i n i t i a l  t h e amount o f e n e r g y  1.8  value T .  received  c  and  Physic-  energy  101 emitted ature  are  has  adjusted  a very  Figure In  this  80. is  case  For  small  (3-3)  the  that  a way  axial value  variation  in  that  the  Figure  vemperature of  of  4> i s the  (3-2).  in  although of  the  with in  with the  tube  £ for  (3-16)  case p l o t t e d a s t o make T  in w  variation  positions ature  of  For  the  between  y  a  n  stressed,  not  values  possible of  (3-3)  the  results  of  choosing  the  critical  keep  the  net  variation  tube  length.  has  however,  T  '  happens £ for yet is  This  because  the it  w  portion  decreases  clearly  happens  shown  for  the  E, a t  different  the  average  angular wall  temper-  was  also  that  this  value  of  conclusions a  critical carried as  is  out- f o r  \p f o r  a  an  at  in  other  Figure  (3-4).  approximate  given  y since  with  £,  especially  the  non-linearity  of  the  problem.  In  attempt  has  b e e n made  to  of be  f"  w  /  \p a n T  0  f r o m 1.0  emphasized that  for  it  constant  of  a minimum o v e r to  in  relationship  a graph only  arrived  T^  values  to  the  are/plotted  critical  keep  i>, b e c a u s e  It  temperature  the v a r i a t i o n i s such c a s e o f y-5, however,  make  regarding  however,  the to  to  confirms  analysis  be  with  ,  This  later.  y-5.  £.  The  It  is  as  (3-2)  y and  for  such  Figure  $•  with  for  cr'  This  a l s o where £. In the  is  r  tube.  discussed  tube  wall  radiation  the  the  cf  relationship  is  of  temperature  values may  portion  wall  of d  incident  Figure (3-2) increase with  Figure  analysis  the  which  with  temper-  neighbourhood of  ii» > il>  rise.  increases  of  decrease Thus  initial  temperature  other  the  r  an  exposed to the  Figure  the  £ after  wall  wall  variation  average  ^ decreases  average  deviation.  shows  the  cr  to  such  critical  < 4>  similar  in  some e x t e n t  high  the the  102 v a l u e s a r e d e p e n d e n t upon t h e t u b e l e n g t h L.  critical Figure  (3-4)  From  f o r a c e r t a i n v a l u e o f y, i f ^ i s g r e a t e r t h a n  corresponding  critical  v a l u e , i t may  average w a l l temperature  be  of the f l u i d  concluded  that  the  the  at tube e x i t w i l l  exceed  F o r v a l u e s o f ij> b e l o w t h e o p t i m u m v a l u e  the i n l e t temperature.  the average v/all temperature  will  continuously decrease  with  axial distance. Figure  (3-5)  illustrates  t h e v a r i a t i o n o f T /T w  f o r t h r e e v a l u e s o f y a t a f i x e d v a l u e o f \p. I|J=25 i s g r e a t e r t h a n F o r y=5  and  ip  and,  y=50 t h e v a l u e  optimum v a l u e s o f  and  t h e r e f o r e , T /T  \p=25  A t y~0.5, the  /T  Q  value  increases with  c  i s l e s s than the  therefore T  with £  G  £.  corresponding  decreases  w i t h E,.  w N u s s e l t Number The Figures  v a r i a t i o n o f N u s s e l t number w i t h E i s p r e s e n t e d  (3-6)  variation  to  f o r two  (3-8).  F o r y=0.5  Figure  v a l u e s o f \p. I n b o t h  cases  (3-6)  illustrates  a c o m p a r i s o n i s a l s o made w i t h Chen's  noted may  t h a t a t v a l u e s o f E < 0.01  be due  t h e two  c r  For  series  an a p p r o x i m a t e s o l u t i o n g i v e n i n t e r m s o f after linearizing  In the p r e s e n t  This The  those  results obtain-  Liouville-Neumann  the boundary c o n d i t i o n a t the tube  s t u d y t h e r e s u l t s a r e o b t a i n e d by  ence s o l u t i o n w i t h o u t  It is  studies.  c o m p a r i s o n o f N u s s e l t number v a l u e s h a v e b e e n made w i t h  ed by  this  solutions diverge.  1 o f Chen's p a p e r . T h i s t a b l e p r e s e n t s  any  temperature,ie.  [59] v a l u e s .  t o t h e d i f f e r e n c e i n a p p r o a c h o f t h e two  given i n Table  the  ^ < ^ . F o r ij>=0 ,_ a t  s e c t i o n , the w a l l temperature i s l e s s than the i n s i d e 5T —) i s n e g a t i v e , s i n c e the f l u i d i s l o s i n g heat. 5 r r=a ^ ^ case  in  a  wall.  finite-differ-  l i n e a r i z i n g the.boundary c o n d i t i o n at  the  wall. It  has been n o t e d t h a t  for other values follows.  N u s s e l t number d e c r e a s e s w i t h ?  The w a l l t e m p e r a t u r e  temperature  < i|>  ij> a l s o when  of  is  f o r the p o r t i o n of  .  This is  greater  the tube  than the  f o r the remaining p o r t i o n of  temperature  less  cumference, greater  i.e.  than T  the  £  | _ r  of  tude w h i l e  (T^ - T )  £,  rate  for  from the Figure  of  In  it  ature  greater  the tube  ( )  2ir  0  ~ or  I  de  positive.  value  of  ip.  than the  higher values  For  at  of  ^=25 and ^=120.  With  increasi n magni-  41 r e d u c e s Nu a t  any  the net heat  of  i n s i d e temperature  incident radiant  is  for three  is  41.  other  trans-  less  values  From t h e  any s e c t i o n , t h e w a l l  temper-  f o r the p o r t i o n of  f l u x but f o r than the  the  inside  remaining temperature.  i n s u c h a manner as t o make in this  low v a l u e s  case. of  short distances.  Also  £, Nu i s  With i n c r e a s i n g values  r e d u c e d In v e r y  cir-  Nu d e c r e a s e s .  f o r each value  > ii  positive *  Nu i s  it  is  rate decreases  the magnitude o f  has b e e n n o t e d t h a t  f a c i n g the  r=a  so t h a t  the  Also  positive.  shows a s i m i l a r p l o t  the v a r i a t i o n  1  Nu i s  wall  However, over  negative.  the  f l u i d t o the s u r r o u n d i n g s .  p o r t i o n the w a l l temperature However,  of  , increase of  decreases  case  analysis is  , is  transfer  increases  \JJ < ij/  (3-7)  this  de  the net heat  s e c t i o n because i t fer  a  and, t h e r e f o r e ,  w  ing values  Also,  integrated value  incident  the t u b e ,  than the i n s i d e temperature.  has b e e n n o t e d t h a t  inside  f a c i n g the  radiation but, is  e x p l a i n e d as  of  > T. w o  lower  £, Nu i s  so  that  for a higher  substantially  The d e c r e a s e i n Nu i s  \p and Nu e v e n i n c r e a s e s (The a n a l y s i s  T„  towards  less  t h e end  has b e e n c a r r i e d o u t f o r  for  for  values  104 of  £ up t o  fer  rate  0.1  only).  This is  decreases with  because although the heat  £ , t h e d i f f e r e n c e between T  trans-  and T,  w als© decreases with upon t h e  two r a t e s Figure  with £ for value  of  the  5=0.02.  To o f f e r  ;  of  illustrates  particular  average w a l l  of q u a n t i t i e s temperature  at  \\> \li a  a  given  Nu d e c r e a s e s w i t h %  wildly  i n the n e i g h b o u r -  of  b e h a v i o u r we have  s u c h as n e t h e a t  (3-9)  and  for  (3-10).  of dimensionless heat  denominator of equation  (3-32)  with'!  of  analyze rate, case.  Figure  (3-9)  (3-32),  flux with  which r e p r e s e n t the  ature.  and  (3-9)  sent  the  ical  c o n s t a n t common t o a l l  (3-9)  absolute values  shows t h a t  range o f  ! values  to -0.075. 0.02, heat  for  transfer 3-10)  into  of heat of  them has b e e n d r o p p e d .  for values  so t h a t  rate  fluid. Nu i s  do n o t  T". and T. , s i n c e  indicated values  transfer the  the o r d i n a t e s  flux,  c o v e r e d , the heat  Initially,  the net heat  (Figure  the  (3-10)  is  of  y and  f l u x decreases  of  !  less  positive.  in this  the  temperreprea numer-  Figure and f o r  the  f r o m +0.29  than approximately  positive,  Also,  !.  dimensionless  and t h e d i m e n s i o n l e s s f l u i d b u l k  Figures  which  t h e two t e r m s o f  average w a l l temperature In  values.  this  the numerator of e q u a t i o n  shows t h e v a r i a t i o n  to  transfer  and b u l k t e m p e r a t u r e  figures  the v a r i a t i o n  (3-10)  that  o f N u s s e l t number  The N u s s e l t number e v e n goes t o n e g a t i v e  shows t h e v a r i a t i o n  Figure  illustrates  region but f l u c t u a t e s  These are p l o t t e d i n  represents  the v a r i a t i o n  s i t u a t i o n when  an e x p l a n a t i o n f o r t h i s  the v a r i a t i o n  depends  decrease.  This figure  y.  o v e r most o f hood o f  £ and t h e N u s s e l t number v a r i a t i o n  (3-8)  a very  b  indicating range of  With i n c r e a s i n g  !,  net T  > T^  values  105 of  £, the d i f f e r e n c e i n the v a l u e s of T  and a t 5 = 0 . 0 2 1 , T  and T, W  For £ > 0 . 0 2 1 , T  = TV . W  D  diminishes  D  < T, . W  Therefore, i n  D  the r e g i o n 5 = 0 . 0 2 0 t o 0 . 0 2 3 , because of the s m a l l d i f f e r e n c e s : i n T*  and T^,  w  the N u s s e l t number f l u c t u a t e s between very l a r g e  t i v e and negative v a l u e s . becomes n e g a t i v e  For £ > 0 . 0 2 3 5  the heat t r a n s f e r  posi-  rate  (Figure 3 - 9 ) i n d i c a t i n g t h a t net heat t r a n s f e r  now  occurs from the f l u i d to the surroundings. A l s o i n t h i s range o f v a l u e s of 5 , f > T. so t h a t Nu i s u l t i m a t e l y p o s i t i v e , w b However, i t may  be s t r e s s e d here t h a t the w i l d  i n Nu i s only over a very s m a l l r e g i o n of the tube  fluctuation  l e n g t h and i f a  mean N u s s e l t number f o r the whole l e n g t h of the tube were to be uated, i t would not be much a f f e c t e d .  A good agreement i n heat  balance f o r t h i s case i s shown i n Table  (3-2).  For y=0.5 and v a l u e s of ^ l e s s than and g r e a t e r than the v a r i a t i o n of Nu was spectively.  d e s c r i b e d by F i g u r e s  In the f o l l o w i n g s e c t i o n s we  v a r i a t i o n of Nu Figure  for  now  ( 3 - 6 ) and  (3-11)  at Y - 5 .  i s noted t h a t f o r the same v a l u e of  of  Chen  The  r e s u l t s show  (3-6).  When v a l u e s  ( 3 - 6 ) are compared with those i n F i g u r e  than f o r y=0.5. For  t/j=0  there i s good agreement w i t h the  results  [ 5 9 ] for £ > 0 . 0 0 4 .  c r i t i c a l value f o r Y = 5 i s shown i n F i g u r e £  (3-11) i t  , Nu has a lower v a l u e f o r Y = 5  V a r i a t i o n of Nu with £ f o r v a l u e s of IJJ  of  ( 3 - 7 ) re-  d e s c r i b e the  a t r e n d s i m i l a r t o t h a t of the curves i n F i g u r e i n Figure  c r  i s a p l o t of Nu w i t h £ f o r t h r e e v a l u e s of  f  Nu  ^ »  y=5.  \p each l e s s than the c r i t i c a l vlaue of  of  eval-  (3-12).  g r e a t e r than  the  At low v a l u e s  the N u s s e l t number v a l u e s are h i g h e r f o r h i g h e r v a l u e s of IJJ.  106  But f o r h i g h e r values of £ there appears a t r e n d which may  For example, a t 5 = 0 . 1 , Nu decreases w i t h i n c r e a s i n g  surprising.  up to 300 and then i n c r e a s e s w i t h i n c r e a s i n g values  values of  i|> above 300.  of  seem  The  e x p l a i n e d when we  apparently s u r p r i s i n g behaviour  observe  is^easily  the v a r i a t i o n i n heat t r a n s f e r  average w a l l temperature and  f l u i d bulk temperature  rate, 3-1).  (Table  The heat t r a n s f e r r a t e decreases w i t h 5 f o r a l l values of \\> and i s higher f o r higher values of $ at any  But the d i f f e r e n c e  i n average w a l l temperature and f l u i d bulk temperature i s a l s o h i g h e r a t any  5 f o r h i g h e r values of  e s p e c i a l l y very s m a l l f o r ^=150 and of  This d i f f e r e n c e i s tji=200 because these  4» are q u i t e c l o s e to the c r i t i c a l value f o r y=5.  r e s u l t s i n a h i g h value of (3-13)  Figures  and  (3-14)  illustrate  no i n c i d e n t r a d i a t i o n f l u x . High values of  high value of y value of e.  m a  Y  y  f i x e d values of  As expected,  a l s o decrease Nu.  to the  case with  a/k,  y=caT  a  a  be caused by a low value of k or a h i g h  With a low thermal c o n d u c t i v i t y f l u i d  the  heat  A high e m i s s i v i t y  the tube w a l l i n c r e a s e s the heat t r a n s f e r r a t e but a l s o de-  creases The  . .Figure  Nu decreases 3  Since  t r a n s f e r r a t e decreases, thus lowering Nu. of  then  the v a r i a t i o n of  (3-13) g i v e s the p l o t s f o r ^=0, which corresponds  5.  This  Nu.  Nu w i t h 5 . f o r three values of y and  of  values  T  and T. i n such a manner as to i n c r e a s e (T, - T ). w b b w r e l a t i v e i n c r e a s e i n (T. - T~ ) i s higher than the i n c r e a s e D  W  i n heat t r a n s f e r r a t e , the o v e r a l l e f f e c t being a r e d u c t i o n i n Nu. With the i n c l u s i o n of i n c i d e n t r a d i a t i o n f l u x , however,  107 the  situation is  number w i l l critical  n o t as s i m p l e .  As e x p l a i n e d e a r l i e r ,  d e p e n d upon w h e t h e r  value  is  greater  and i t s - c l o s e n e s s t o t h i s  (3-14)  gives  the p l o t s  higher  f o r a lower  for  value  ^=25.  of  For  critical  low v a l u e s  y but f o r values  Nu f i r s t  decreases  and t h e n i n c r e a s e s w i t h y.  for  is  same l i n e s  this  on t h e  Angular W a l l Temperature  (3-15)  to  (3-17).  Figure  of w a l l temperature  values  y and £.  In  Figure  £, Nu  5 near  the  is  0.1,  The e x p l a n a t i o n (3-12).  Distribution  variation of  of  than  value. of  Nusselt  as t h e d i s c u s s i o n o f F i g u r e  The a n g u l a r w a l l t e m p e r a t u r e Figures  or lower  the  all  variation  (3-15)  is  presented  shows t h e  for d i f f e r e n t values  cases  the  in  angular, o f 'ifi and f i x e d  temperature v a r i e s  .  0  f r o m a maximum a t 0=0°  and 9=180°  points  on t h e  the w a l l while of  9=0  is  zero,  6=180°  increase is  temperature  very near  Figure  Also,  slow.  There i s  gives  for different  it,decreases with any d i r e c t  c o n v e c t i o n from the  incident fluid  a c c o r d i n g to the  temperature,  the  two  o f \p,  9=0 i n c r e a s e s  rapidly  i n the neighbourhood  a s h a r p change i n t h e  the p l o t s  near  heat  at  wall  9=90°.  The w a l l t e m p e r a t u r e  receive  The s l o p e  increasing values  i n the w a l l temperature  \\i and y. 6=TT  for  i n the neighbourhood of  (3-16)  w a l l temperature  9=180°.  i n d i c a t i n g a n g u l a r symmetry a t  circumference.  temperature  the  t o a minimum a t  values near  £. flux  Near  of  angular v a r i a t i o n  of  £ and a f i x e d v a l u e  6=0 i n c r e a s e s w i t h 6=TT  (although  f a c i n g the  the it  incident  f l u i d does receives flux)  f o u r t h power law o f r a d i a t i o n .  t h e r e f o r e , d e c r e a s e s w i t h £.  Near  E,  of  but not  heat  but i t The  of  by loses  wall  9=0 t h e e f f e c t  of  1 9 8  incident  f l u x causes a slow i n c r e a s e o f  with £ for this  c a s e where  Similar value  of  <  value  values  are p l o t t e d i n F i g u r e  Y reduces the w a l l temperature of e m i s s i v i t y  of  y and a  at every  fixed  An i n c r e a s e  angular p o s i t i o n .  the tube w a l l reduces the w a l l heat  in A high  temperature  loss.  Balance  The p e r c e n t a g e e r r o r by e q u a t i o n  of  ( 3 - 1 7 ) .  a t e a c h a n g l e by i n c r e a s i n g t h e r a d i a t i v e O v e r a l l Energy  temperature  .  plots.for different  ip a n d ' Y  the w a l l  (3-36)  was  This  is  less  than 3 per cent  in overall  calculated  for  energy balance  some v a l u e s  t a b u l a t e d i n . T a b l e 3-2 w h i c h shows t h a t f o r most c a s e s .  of  as  given  y and \p.  the e r r o r  is  CONCLUSIONS The e f f e c t transfer  o f i n c i d e n t r a d i a n t f l u x on l a m i n a r  i n a c i r c u l a r tube has been s t u d i e d .  tribution  recent  Temperature d i s -  a n d N u s s e l t numbers h a v e b e e n e v a l u a t e d .  c a s e o f no i n c i d e n t f l u x t h e r e s u l t s solution  short distances  Forthe  have been compared w i t h a  and have been found t o agree w e l l , from the tube  heat  entrance.  except a t  110 TABLE Values  of  the  dimensionless  temperature  3-1  forms  and b u l k  a)  of  heat  flux,  temperature  for  average y=5  5=0.001  Dimensionless Heat F l u x  T  b  T  w  '  150  18.31  9.03  7.90  1.13  200  25.39  9.47  7.91  1.56  300  37.03  10.14  7.94  2.20  500  55.43  11.10  7.98  3.12  1000  90.53  12.61  8.04  4.57  b) Dimensionless Heat F l u x  wall  5=0.1  V  T  b  150  4.86  8.39  8.25  0.14  200  7.11  8.85  8.53  0.32  300  11.32  9.56  8.96  0.60  500  19.10  10.56  9.55  1.01  1000  37.70  12.12  10.51  1.61  T  b  TABLE Overall  Error  in  3-2 Energy  (Equation  Y  Balance  3-36)  Percentage  0.5  '""-0.44  25  0.5  -3.09  25  5  -1.97  80  5  -2.20  25  50  -1.10  1.8  Error  9=TT/2  X=0  Figure  3-1  Tube N o m e n c l a t u r e and Geometry  IO"  IO"  3  Figure  3-10  2  ?  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C . , " L a m i n a r Heat T r a n s f e r i n Tube w i t h N o n l i n e a r R a d i a n t H e a t - F l u x Boundary C o n d i t i o n , " I n t . J . H e a t Mass T r a n s f e r , V o l . 9, 1 9 6 6 , p p . 4 3 3 - 4 4 0 .  60.  D u s s a n , B.I. and T . F . I r v i n e J r . , " L a m i n a r Heat T r a n s f e r i n a Round Tube w i t h R a d i a t i n g Heat F l u x a t t h e O u t e r W a l l , " P r o c . 3 ~ I n t . Heat T r a n s f e r C o n f . , 1 9 6 6 , V o l . V , p p . 184189.  A  EVALUATION BALANCE  OF  P  P  E  N  EXPRESSIONS  D  I  FOR  C  E  S  OVERALL  ENERGY  ERROR  EVALUATION  OF  THE  DIFFERENCE  FINITE  CONFIGURATION  FACTOR  PROCEDURE  F  d A  ^  A  APPENDIX  A  EVALUATION OF EXPRESSIONS FOR OVERALL ENERGY BALANCE ERROR The accuracy o f the p r e s e n t a n a l y s i s i s i n v e s t i g a t e d from the p o i n t o f view of energy balance.  The percentage e r r o r  i n o v e r a l l thermal energy balance i s d e f i n e d as f o l l o w s :  . Percentage e r r o r =  Energy absorbed - Energy e m i t t e d ^ Energy absorbed  „ x 100 . (1-33) 1 A r t  T h i s e x p r e s s i o n i s e v a l u a t e d f o r the three bodies c o n s i d e r e d . a) CIRCULAR CYLINDER The energy balance i s c a r r i e d out over a u n i t l e n g t h o f the c y l i n d e r as f o l l o w s : Energy absorbed =  1 i\0L G  Energy e m i t t e d  1 \  =  fcff"  BTU/hr.  (A-l)  BTU/hr.  (A-2)  T h e r e f o r e , percentage e r r o r i n energy balance, from e q u a t i o n (1-33).,  137  b) RECTANGULAR  CYLINDER:  Carrying the  out the energy balance over a u n i t length o f  c y l i n d e r , we have,  1WG  Energy absorbed =  Energy emitted =  1  \ £  BTU/hr.  ^  Percentage e r r o r =  + 1  £€  £<r Ij  1^  e < 5  < ^  "^  T  (A-3)  < k  BTU/hr.  ( A -4 )  2WG  = h - £ U % ^V^V\,«]] (l-33b)  c) SPHERE: Energy a b s o r b e d =  ITO- G ^  BTU/hr.  Energy emitted  BTU/hr. (A-6)  Percentage e r r o r -  jW  G<X - W o .  UT J .^^tt ^ t  ^  (l-33c)  APPENDIX  B  EVALUATION OF CONFIGURATION FACTOR F  The  configuration  d  A  l+  A  2  f a c t o r , view f a c t o r o r shape  factor,  F dA^->- A2 /  i s d e f i n e d as t h e f r a c t i o n o f energy d i r e c t l y i n -  c i d e n t o n s u r f a c e A2 1 energy d i f f u s e l y .  f r o m s u r f a c e dA^ assumed t o be  An a n a l y s i s  emitting  a l o n g t h e l i n e s g i v e n by W i e b e l t [49] y i e l d s  (B-l) As  s u g g e s t e d by Sparrow  above a r e a i n t e g r a l verted  F  into a line  integral  to  is  [50]  conveniently  the  con-  yield  2 i ^-y^ ^ V * i - * ^ - ( ? i - Z , H x i  =1-  where 1^, m^ a n d n ^ a r e t h e d i r e c t i o n c o s i n e s o f dA^, a n d t h e c o o r d i n a t e s x^, y^,  /  r e p r e s e n t t h e l o c a t i o n o f t h e a r e a dA^.  These a r e a l l c o n s t a n t  during the integration.  The c o o r d i n a t e  140 axes a r e g e n e r a l l y so chosen so t h a t t h e normal t o d A cisely  along a coordinate l i n e .  Then two o f t h e t h r e e  c o s i n e s a r e z e r o and a l a r g e p o r t i o n o f t h e c o n t o u r equation  (B-2) v a n i s h e s .  1  Thus i n t h i s  lies  pre-  direction  integral of  case t h e element d A  1  has  i t s normal p o i n t i n g i n the p o s i t i v e x - d i r e c t i o n so t h a t 1 ^ 1 , m^= ]_ n  =  0*  Additionally,  z^= 0, x ^ = a , y ^ = 0 .  S u b s t i t u t i o n o f t h e s e v a l u e s i n (B-2) y i e l d s  The c o n t o u r  an a r c  from  X -^ %  C  w h i c h bounds t h e a r e a A  running from  -+ $vr\9 4  Co&§  Also,  Along both  ,  -e  4  0  ^  -  contours  0  e < e  4  V" Si.T\ 8  := N\ Si-O^ ,  ^  i s made up o f two p a r t s :  and a s t r a i g h t  «  a  5  0  b  .  On  w h i l e on t h e  +  ^  line  i  "  the  straight  ^ + ^  running  arc  line,  -7,)  141  On i n t e g r a t i o n ,  r  -  Since  Z  e  0  \  is  the  above e x p r e s s i o n y i e l d s  o . V z % A '  ft.  g i v e n by c o s  Nondimensionalising  z  e  0  = a/r  and r  above e x p r e s s i o n c h a n g e s  r  2  by  2  -r<b?«Vv.VtaA)Un^) 1  / therefore,  Z = z/a  and g = r / a , 2  the  to  (2-24) where  <£  -  Z  %  + f  /  + 1  .  APPENDIX THE The e l l i p t i c  C  F I N I T E DIFFERENCE PROCEDURE p a r t i a l d i f f e r e n t i a l equation  s o l v e d u s i n g a square g r i d .  Using  the standard  (2-9)  5-point  was approx-  i m a t i o n o f t h e L a p l a c i a n , t h e f i n i t e d i f f e r e n c e scheme f o r t h e i n t e r i o r points i s :  Xi,j  =  •/i.j+i'^  w h e r e i , j and a , b  +  • Hi)  +  Xi,|-.  TUT)  a r e t h e s u b s c r i p t s and s t e p  Z and R d i r e c t i o n s r e s p e c t i v e l y , ( F i g u r e 2-12, On t h e b o u n d a r i e s , h o w e v e r , d i f f e r e n t i a l equation Using  p.80).  we do n o t t r y t o s a t i s f y  b u t s a t i s f y the boundary  the forward-difference  s i z e s f o r the  approximation  conditions  a t any b o u n d a r y ,  k For example,  A coarse  !U<L  .  g r i d was  first  (C-l)  t  C-D  (R=l) f o r a  (80x10)  for a finer grid.  although  v a r i e d w i t h t h e v a r i a b l e p a r a m e t e r s A , N, e t c .  of the problem, a value suitable.  was  A number o f  ©f t h e o v e r - r e l a x a t i o n p a r a m e t e r w e r e t r i e d and  the optimum v a l u e  grid,  used t o o b t a i n a rough shape w h i c h  u s e d as t h e i n i t i a l d i s t r i b u t i o n values  for a function  -u. +\%)  f o r the boundary  only.  f o r the d e r i v a t i v e  o f a f u n c t i o n a t a p o i n t , one o b t a i n s , i n g e n e r a l , $  the  o f 1.9  seemed t o be g e n e r a l l y m o s t  

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