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Stitch weld effect on solar collector efficiency factor Lo, Andy Ka-Ming 1985

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STITCH WELD EFFECT  ON  SOLAR COLLECTOR E F F I C I E N C Y FACTOR  by Andy K. B.Sc,  LO  Queen's U n i v e r s i t y  a t K i n g s t o n , 1983  A THESIS SUBMITTED I N PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE  in FACULTY  OF GRADUATE STUDIES  DEPARTMENT OF MECHANICAL  We a c c e p t t h i s  ENGINEERING  t h e s i s as conforming  to the required  standard  UNIVERSITY OF B R I T I S H COLUMBIA  ©  September,  1985  Andy K. LO,  1985  OF  In  presenting  degree at the  this  thesis  University  in  partial  fulfilment  of  the  requirements  of British  Columbia,  I agree that the  for  an  advanced  Library shall make it  freely available for reference and study. I further agree that permission for extensive copying  of  department  this thesis for scholarly or  by  his  or  her  purposes may be  representatives.  It  is  granted by the understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department of  M.E.CMr\M\CP(L  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6(3/81)  QcfoUr  9 ,  W{  tzKJ6jfkJEER.tkgj  head of my copying  or  my written  ABSTRACT The  thermal  effects  of  c o n d u i t s of a w a t e r - c o o l e d absorber heat  plate  transfer  inside  the  have  flat  been  p r o c e s s from tube  has  stitch plate  studied. the p l a t e  been  based  on t h e d i f f e r e n c e  and  bulk  mean t e m p e r a t u r e  determining The  and  have been p r e d i c t e d conditions  spot thin  tube  wall, and  lower  configuration.  length welding  the  tends  which  case.  heat  transfer  The  two  tube  temperature factor in  F'. v a l u e o f F'  extreme  boundary  wall,  the  of s t e p c h a n g e s of the heat  spot  limit  size  i n both  condition  t h e a x i a l and  f l u x . In  this  case,  c o e f f i c i e n t and h e n c e F' a p p r o a c h are  determined found  by  that  the  welding  F' i n c r e a s e s  o f s p o t s among  fluid  has a l s o  the  their spot  with the  the spot angle; the percentage  the  and  o f 0.883. F o r a  boundary  i s being d i s t r i b u t e d . Furthermore, inside  flowing  i s s u b j e c t e d . F o r a t h i c k and  b e i n g w e l d e d ; and t h e number  distribution  fluid  i s an i m p o r t a n t  t o an upper  I t was a l s o  parameters:  model o f t h e  between bond  considering  directions  transfer  limits  following  to  F' does n o t depend on  of a s e r i e s  circumferential  this  by  and n o n - c o n d u c t i v e  heat  A physical  l i m i t s of t h e a c t u a l  t o which the f l u i d  spacing,  comprises  the  lower  the coolant  c o l l e c t o r to i t s  the c o l l e c t o r e f f i c i e n c y factor  upper  conductive  solar  presented.  coefficient fluid  welding  o f tube  which the temperature  been computed f o r  Table  of Contents  ABSTRACT  i i  L I S T OF TABLES  v  L I S T OF FIGURES  vi  L I S T OF SYMBOLS  x  ACKNOWLEDGEMENTS 1.  xiii  INTRODUCTION  1  1 .1 G e n e r a l  1  1.2 O b j e c t i v e o f t h e P r e s e n t  Work  4  1.3 L i t e r a t u r e Review 1.3.1 The E n t r y 1.3.2 G r a e t z  6  Length  Problem  •  Problem  1.3.3 P r o b l e m s w i t h  8 C i r c u m f e r e n t i a l V a r i a t i o n ...10  1.3.4 C o n j u g a t e d P r o b l e m 2.  MATHEMATICAL  12  MODEL  16  2.1 G o v e r n i n g E q u a t i o n s  3.  ..16  2.2 I d e a l i z a t i o n s  17  2.3 S i m p l i f i e d  19  Equations  2.4 T h e r m a l Boundary C o n d i t i o n s  20  ANALYTICAL SOLUTIONS  24  3.1 O v e r v i e w  of A n a l y t i c a l  3.2 S e p a r a t i o n  Solutions  3.4 G r a p h i c a l NUMERICAL  ....  o f V a r i a b l e s Method  3.3 S o l u t i o n f o r " T h i n " Tube u s i n g  4.  7  Illustration  25 Superposition  of A n a l y t i c a l  SOLUTIONS  24  Results  ....28 ....30 34  4.1 O v e r v i e w o f N u m e r i c a l M e t h o d s  34  4.2 F i n i t e  35  Difference Formulation  4.3 C a l c u l a t i o n o f N u s s e l t  Numbers  37  4.4 5.  APPLICATION 5.1 5.2  6.  Preliminary  Results  38  TO COLLECTOR EFFICIENCY FACTOR  Heat Transfer Temperature Behaviour Tube  of  Coefficient  based  42 on  Bond 43  Nu(x)  for a Continuously  Welded 45  5.3  Behaviour  5.4  Efficiency Factor Configurations  DISCUSSION  of Numerical Procedure  o f Nu(x) o f S p o t Welded Tube  AND  for  various  46 Spot 48  CONCLUSIONS of A n a l y t i c a l  50  6.1  Discussion  and N u m e r i c a l Methods  ..50  6.2  Recommendations  53  6.3  Conclusions  55  REFERENCES  56  APPENDIX A  93  APPENDIX B  .95  APPENDIX C  97  L I S T OF  TABLES Page  Table  2.1.  Thermal boundary c o n d i t i o n s f o r developed and d e v e l o p i n g f l o w s t h r o u g h s i n g l y connected ducts.  v  65  L I S T OF  FIGURES Page  FIG. FIG. FIG.  1.1.  Cross s e c t i o n collector.  1.2.  of a b a s i c f l a t  Cross  solar 58  E n e r g y f l o w i n an solar c o l l e c t o r .  1.3.  plate  section  operating  water-cooled 59  of  typical  plate  and  tube  arrangement.  60  FIG.  1.4.  Common ways of p l a t e - t u b e b o n d i n g .  FIG.  1.5.  Hydrodynamic  FIG.  1.6.  Physical  FIG.  1.7.  FIG.  and  thermal  situation  Linear velocity method. 2.1. Two extreme conditions : ( i ) A t h i c k and ( i i ) A t h i n and  profile  of  FIG.  2.3.  Axial  FIG.  3.1.  Continuously of 4 5 ° .  FIG.  3.3  thermal  boundary  distribution welded  Spot-welded tube 60% o f i t s l e n g t h .  of w a l l heat  of w a l l heat tube  with  66 flux.  flux. spot  67 67  angle 68  with  2 spots  occupying 68  Dimensionless w a l l temperature of a tube welded continuously with s p o t a n g l e 45° versus dimensionless a x i a l distance 1000x = l000X/a-Pe.  3.4.  64  i n Leveque  conductive wall. non-conductive w a l l .  Peripheral distribution  FIG.  assumed  problem.  63  64 cases  2.2.  3.2.  lengths.  f o r the Graetz  FIG.  FIG.  entry  61,62  69  Local Nusselt no. Nu and peripheral a v e r a g e N u s s e l t no. N u of continuously welded tube versus dimensionless axial d i s t a n c e 1000X = l 0 0 0 X / a » P e .  70  Dimensionless wall temperature versus dimensionless axial distance 1000X = l000X/a°Pe o f a s p o t - w e l d e d tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 45°.  71  p  FIG.  FIG.  3.5.  3 . 6 ( i ) . P e r i p h e r a l a v e r a g e N u s s e l t number Nun versus d i m e n s i o n l e s s a x i a l d i s t a n c e lOOux = l000X/a«Pe f o r a s p o t - w e l d e d tube w i t h a vi  s i n g l e spot occupying spot angle 45°. FIG.  FIG. FIG. FIG.  FIG.  FIG. FIG.  FIG.  FIG.  FIG.  60%  of  its  length, 72  3.6(ii). P e r i p h e r a l a v e r a g e N u s s e l t no. Nu versus dimensionless a x i a l distance lOOOx = l 0 0 0 X / a - P e f o r a s p o t - w e l d e d tube w i t h 4 s p o t s o c c u p y i n g 60% o f i t s length, spot angle 45°. ' p  4.1. 4.2. 4.3.  4.4.  4.5. 4.6.  4.7.  4.8.  4.9.  Grid  division  of  tube  Nodes appearing energy equation.  in  volume. finite  73 74  difference 75  D i m e n s i o n l e s s t e m p e r a t u r e t of a u n i f o r m l y heated tube versus dimensionless axial d i s t a n c e lOOOx = l000X/a-Pe at various radial distances : finite difference results.  76  L o c a l N u s s e l t no. Nu o f a u n i f o r m l y h e a t e d tube versus dimensionless a x i a l distance 1000X = l000X/a-Pe : finite difference results.  77  Angular p o s i t i o n s represented c u r v e s i n F i g u r e s 4.6 - 4.9.  78  by  various  Dimensionless wall temperature versus dimensionless axial distance 1000X = l000X/a«Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a c o n t i n u o u s l y welded tube w i t h spot a n g l e 45°.  79  Dimensionless wall temperature versus dimensionless axial distance 1000X = l000X/a«Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube w i t h 2 s p o t s o c c u p y i n g 60% o f i t s l e n g t h , s p o t a n g l e 4 5 ° .  80  D i m e n s i o n l e s s temperature i n s i d e the tube at radial co-ordinate r=4/5 versus dimensionless axial distance 1000x = 1000X/a«Pe for a spot-welded tube w i t h 2 s p o t s o c c u p y i n g 60% o f i t s length, spot angle 45°.  81  D i m e n s i o n l e s s temperature i n s i d e the tube at radial co-ordinate r=1/5 versus dimensionless axial distance 1000X = l000X/a«Pe, f o r a spot-welded tube w i t h 2 spots occupying 60% of i t s l e n g t h , spot angle 45°.  82  vii  FIG.  FIG.  FIG. FIG.  4.10(D. Peripheral average N u s s e l t no. N u versus dimensionless a x i a l distance lOODx = l000X/a«Pe : finite difference results f o r a s p o t - w e l d e d tube w i t h a s i n g l e spot occupying 6 0 % of i t s length, spot angle 45°.  83  4 . l 0 ( i i ) . P e r i p h e r a l average N u s s e l t No. Nup versus d i m e n s i o n l e s s a x i a l d i s t a n c e lOODx = l000X/a«Pe : finite difference results for a spot-welded tube with 4 spots o c c u p y i n g 60% o f i t s l e n g t h , spot angle 45°.  84  5.1. Bond temperature T] (X) a p p r o x i m a t e d by temperature i n s i d e tube T ( a , 0 , X ) .  85  5.2. N u s s e l t no. b a s e d on bond t e m p e r a t u r e Nu^ versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l000X/a-Pe f o r a continuously welded t u b e o f v a r i o u s h a l f - s p o t a n g l e s <t>  86  5.3. N u s s e l t no. b a s e d on bond t e m p e r a t u r e Nuj-, and mean Nusselt no. b a s e d on bond temperature Nu^m versus dimensionless a x i a l d i s t a n c e 1000X = l 0 0 0 X / a « P e for a spot-welded tube with 2 spots occupying 60% o f i t s l e n g t h , s p o t a n g l e 3 6 ° .  87  5.4. Mean N u s s e l t n o . b a s e d on bond t e m p e r a t u r e bm versus dimensionless axial distance 1000X = l 0 0 0 X / a - P e f o r a s p o t - w e l d e d tube with 8 s p o t s o c c u p y i n g 60% o f i t s l e n g t h , spot angle 36°.  88  5.5. Mean N u s s e l t n o . b a s e d on bond t e m p e r a t u r e bm over a d i m e n s i o n l e s s tube l e n g t h o f .088 a s a f u n c t i o n o f s p o t configurations for a spot-welded tube w i t h spot a n g l e of 36°.  89  5.6. E f f i c i e n c y F a c t o r F' a s a f u n c t i o n o f s p o t configurations for a collector with d i s t a n c e W between i t s s p o t - w e l d e d tubes of ,15m and s p o t a n g l e o f 3 6 ° .  90  6.1. E f f i c i e n c y F a c t o r F' a s a f u n c t i o n o f s p o t configurations for a collector with distance W between i t s s p o t - w e l d e d t u b e s of ,1m and s p o t a n g l e o f 3 6 ° .  91  p  3  0  FIG.  FIG.  N u  FIG.  N u  FIG.  FIG.  viii  FIG.  6.2.  E f f i c i e n c y F a c t o r . F' as a f u n c t i o n of s p o t configurations for a collector with d i s t a n c e W between i t s s p o t - w e l d e d tubes of ,2m and s p o t a n g l e of 3 6 ° .  ix  92  L I S T OF SYMBOLS  A  cross  sectional area,  a  tube  inner  B  wall  conductance  b  bond w i d t h ,  radius,  [m ] 2  [m]  parameter  ( = k 7/ka w  )  [m]  bond c o n d u c t a n c e ,  [W/m ° C ]  Cp  specific  D  tube  F  fin efficiency  F'  collector efficiency factor  g"  v e c t o r i a l body  h  c o n v e c t i v e heat  transfer  c o e f f i c i e n t , [W/m °C]  I  i n t e r n a l energy  per u n i t  mass,  k  thermal c o n d u c t i v i t y  of f l u i d ,  of f i n m a t e r i a l ,  heat  inner  capacity  diameter  forces,  g  thermal  k  w  thermal c o n d u c t i v i t y  conductivity  tube  length,  m  /(U /k 5), L  (=2a),  s  [m]  parameter  k  L  [J/kg°C]  of f l u i d ,  [N] 2  o f tube  [J/kg] [W/m°C]  wall,  [W/m°C]  [W/m°C]  [m] a  factor  appearing  in  e f f i c i e n c y F. m M  fluid r  M  mass f l o w r a t e ,  maximum r a d i a l  [kg/s]  node no.  maximum a x i a l node no.  M,  maximum c i r c u m f e r e n t i a l  N  total  N  radial  <t>  no. o f w e l d i n g  node no.  spots along a  node no.  x  tube  the f i n  N  axial  x  node no.  N,  circumferential  Nu  local  Nu^  Nusselt  <t>  N u  bm  N u s s e l t no. no. b a s e d  mean N u s s e l t  NUp  peripheral  Nu  mean N u s s e l t  m  p  pressure,  Pe  Peclet  q  wall  q  peripheral  q  no. based  average  s o l  flux  into  average  energy  incident  wall  solar  heat  heat  flux,  resistance  flux,  2  l e n g t h p e r tube  [W/m ] 2  stress tensor  T  temperature, [°C] fluid  inlet  c o - o r d i n a t e (= R/a)  coefficient  S  (= k 7 / k D )  m  fluid  b u l k mean t e m p e r a t u r e , [ ° C ]  T  w  inner  wall  peripheral  w  temperature, [°C]  T  wm  [W/m ]  c o - o r d i n a t e , [m]  R e y n o l d s no.  T  ^  2  [W/m]  Re  n  [W/m ]  gain per unit  dimensionless radial  T%  on Nu p  fluid,  r  wall  temperature  No. (=Re.Pr)  radial  w  on bond  2  R  R  temperature  N u s s e l t no.  no. b a s e d  flow d i r e c t i o n , Q  on bond  [N/m ]  heat  useful  u  node no.  temperature average  wall  xi  o f tube, [°C] temperature, [°C]  i n the  dimensionless w'  temperature  (= k(T-T\ ) / q a ) n  wm  dimensionless peak  flow  axial  c o u n t e r p a r t s of T. , T , T , T * i n ' m' w' wm  speed  (=2u f o r p a r a b o l i c p r o f i l e )  component  of flow v e l o c i t y ,  mean f l o w  speed,  [m/s]  collector  overall  heat  transfer  [m/s]  coefficient,  [W/m C] 20  fluid  velocity  distance  [m/s]  between t u b e s ,  welded percentage axial  spot  of tube  co-ordinate,  dimensionless  [m] length  [m]  axial  co-ordinate  l e n g t h and s p o t  spacing,  (= X / a . R e . P r )  [m]  X , / a . R e . P r and X /a.Re.Pr 1 s thermal tube fin  diffusivity  (= k/pc  wall thickness, thickness,  peripheral half-spot  [m]  co-ordinate, angle  [radians]  [radians] [kg/m.s]  viscosity,  time, [s] dimensionless  2  [m]  dynamic v i s c o s i t y , kinematic  ),[m /s]  temp.  [m /s] 2  ( = (T -  xii  T w  ) / (  T  i  n  ~  T w  '  '  ACKNOWLEDGEMENTS The E.G.  author c o n s i d e r s  Hauptmann  throughout  this  been c r u c i a l The wonderful  from  spiritual  Dr.  study.  M.  University  Their  of  s i t e of s t u d y i n g the author  half-way support  fortunate  Iqbal  as  t o have had D r .  his  supervisors  g u i d a n c e and a s s i s t a n c e  towards the completion  Finally, some  and  himself  British  of t h i s  work.  Columbia  and t h e c o m p u t i n g  provided  the  and encouragement  xiii  world,  this  facilities.  wishes t o thank h i s f a m i l y  across  have  who  throughout  have this  members, given work.  1.  1.1  INTRODUCTION  GENERAL Heat  transfer  collector cooling heat in  include  phenomena  solar  in  radiation,  a  building  flow  forced  The most common t y p e o f s o l a r heating  and  water  heating  in Figure  convective  collector  used  i s the f l a t  plate  water-cooled  t y p e a s shown  the  i n w h i c h t h e v a r i o u s modes o f h e a t  manner  solar  c o n d u c t i v e and c o n v e c t i v e  t o t h e a m b i e n t , and i n t e r n a l  transfer.  water-copied  1.1. F i g u r e  1.2  shows  transfer are  involved. The  black,  transferring welded  solar  energy-absorbing  the absorbed  tubes  in  energy  absorbed. C a l c u l a t i n g  away  i s t h e aim of f o r c e d  in  internal  absorber solar  f l o w s . One o r  surface.  radiation  absorber losses  The  plate,  thus  plate  positioned location  convective two  covers  transfer cover  insulation  analysis solar  t o incoming  and is  carried  the  radiation  convective  the heat  being  are transparent  reducing  through  from t h e radiative  included  to  losses [ 1 ] ,  c o l l e c t o r s are almost  i n question,  concentration,  heat  envelopes  w i t h an o r i e n t a t i o n  device  usually  t h e amount o f h e a t  t o t h e a t m o s p h e r e . A back  Flat  has means f o r  w a t e r c a r r i e s away  but opaque t o t h e t h e r m a l  reduce c o n d u c t i v e  solar  to a f l u i d ,  which f l o w i n g  being  plate  is  always s t a t i o n a r y  optimized  f o r the p a r t i c u l a r  and f o r t h e t i m e o f y e a r  intended  the flux  to  operate.  of i n c i d e n t  1  and  i n which  Without  radiation  is,  the  optical at  best,  2 1100  W/m .  Both  2  absorbed  by  beam  the  applications  and  diffuse  collector,  requiring  which  energy  radiation can  be  are  being  designed  delivery  at  for  moderate  temperatures. The the  flow  i n the welded  laminar regime. Laminar  technical cooling  importance  d e v i c e s . The  absorber in  of c o o l a n t  solar  plate  and  it  the c o o l i n g  and  hence  is  of  great  o c c u r s i n many h e a t i n g  transfer  c o l l e c t o r design since  efficiency,  i s commonly i n  flow heat t r a n s f e r  since heat  tube  coefficient  water  i s an  between  important  i t determines the  the  economic  and the  factor  collector  value  of  the  installation. The  performance  proportional the  ratio  energy gain surface  to i t s of  had  interpretation efficiency  cross-section Figure  1.3.  W{  1/[U  of  of  the  solar  at F'  useful if  the l o c a l is  design  water  that as and  the  it  (b+(W-b)F) ] +  to the  collector  is heat  a  useful  absorbing  temperature. measure exchanger  tube arrangement  Another of  the  [2].  i s shown  A in  1/C, + 7 / U D k ) + 1 / U D h ) } , b w (1.1)  where t h e d i m e n s i o n s a r e a s  indicated  other  i n the "LIST  terms, which appeared here :  represents  collector efficiency factor i s  L  repeated  which  energy gain  fluid  a  heater i s d i r e c t l y  F',  f a c t o r  result  of the p l a t e  The  T  actual  would  been  a  e f f i c i e n c y  the  that  of  in Figure OF  1.3,  and  SYMBOLS",  the are  3  U  =  r  overall  coefficient  of heat  transfer  from  the f l a t  Li  plate rear h  =  to the o u t s i d e a i r ,  coefficient  ,k  for  transfer  conductivities  materials thermal  of heat  from  t h e tube  wall  to  i n the tube,  = thermal  =  allowance  losses,  the water k  including  of  the  plate  and  tube  respectively,  conductance  o f t h e t u b e - p l a t e bond,  tanh[m(W-b)/2] F  =  = fin efficiency; m(W-b)/2  m = v/(U /k 6) . L  In  s  denominator the passage (1)  ( 1 . 1 ) the four  equation  c a n be t h o u g h t of heat  conduction  from  terms  heat  of  heat  the  of as the r e l a t i v e  the p l a t e  of  in  along  into  right  hand  resistance to  t h e w a t e r due t o :  the f l a t  plate  towards t h e  tubes, (2)  conduction  from  the p l a t e  t o t h e tube  through  t h e t u b e - p l a t e bond, (3)  conduction throughout of  (4)  the tube  wall  to the inner surface  the tube,  transfer  of  heat  from  t h e tube  inner s u r f a c e into the  water. The tube,  governing  namely,  equation, formulated  equations of f l u i d  1) the c o n t i n u i t y  and and  3)  the  energy  extensively  flow  equation, equation,  studied.  inside 2)  a  the  have  Although  circular momentum  been an  well energy  4 equation water  c a n be w r i t t e n  flowing  conditions  of  inside  the  the temperature  collector  tube,  An  important  aspect  1.4(i) used  condition series  - ( i v ) . An u l t r a s o n i c s p o t  f o r economic  i n the a x i a l  of s t e p  the  direction  changes  In c o n v e n t i o n a l  reasons.  (Figure  studies  be  of  flux.  concerning  The t e m p e r a t u r e  is  reality,  the  because of a x i a l  tube  boundary  material, condition  described The by  1.2  tube  includes  in  at  wall  flowing  the  terms of w a l l  i n t o account  t h e boundary as  a  transfer,  well or  region  in  wall  heat  which  can  the conduction  interest, c a s e . In  conduction  is  subject neither  nor w a l l with  heat  specified  of  water  temperature  defined  the  the present  s o - c a l l e d "conjugated problem" deals taking  number  described in  flux i s clearly  and p e r i p h e r a l  the  A  heat  the temperature  or the heat  inside  i n which the  considered  convective  m a t h e m a t i c a l l y a t t h e boundary of t h e which  the  1.4(iv)).  m a t h e m a t i c a l model o f t h e p r o b l e m  boundary c o n d i t i o n s  of  welding process i s  In t h i s c a s e can  the  boundary  plate).  common ways o f b o n d i n g a r e shown and b r i e f l y  often  the  the  i s the f a s h i o n  i s welded onto the f i n ( t h e absorber  Figures  of  on t h e w a t e r depend on t h e c o n f i g u r a t i o n  collector. tube  to describe  heat  in to a be  flux.  this situation  i n the s o l i d  material.  OBJECTIVE OF THE PRESENT WORK Although  laminar  convective  extensively  studied  for  conditions,  no p r a c t i c a l  various  heat  transfer  kinds  of  h a s been boundary  s o l u t i o n i s a v a i l a b l e f o r flows i n  5 the  spot-welded  objective spot  and  tube  of t h i s  work  the  tube  coefficient,  the  performance, can  The  before  and  be  set  the  the e f f e c t s transfer  of  the  coefficient  Knowing  consequential  effect  on  collector  collector  efficiency  spot  the  of  the  understanding  spacing  welding  is  efficiency  transfer  flowing should  spots  and  this  and  consumes  be  located  can  spot  economic  reduced,  spacing  can  c o s t s i n t o a c c o u n t ) . As process  inside be  i s slow  is significantly  length  welding  of . g r e a t  process  f a r a p a r t the  spot  (taking  fluid  in  from  must terms  the be  i n the  mean N u s s e l t  number, w h i c h  i s the d i m e n s i o n l e s s form  coefficient  from  tube  and  This  dimensionless  distribution  transfer  start,  spot-welded  the  an be  understood.  of  fluid,  a  temperature  mean h e a t  The  determined.  however, t h e h e a t to  and  the heat  1.4(iv).  fluid.  knowing how  determined  on  particular  the c o l l e c t o r  optimum  Figure  the  in  since  By  in  i s to i n v e s t i g a t e  wall  q u e s t i o n of  importance energy.  shown  spacing dimensions  between  factor  as  i n terms of  the c i r c u l a r  of  tube  the the  to  the  fluid. Obviously, through An  the welding  extreme  thin  so t h a t  material plate  the  case  heat  flux  spots than exists  peripheral  is negligible,  into  through  the  when t h e t u b e  and and  axial  water rest  wall  i s higher  of the  the heat  flux  the  i f the tube m a t e r i a l  tube.  is vanishingly  conduction within  e n t e r s t h e water o n l y t h r o u g h  the o t h e r extreme,  the  from  the  welding i s highly  the  wall  absorber  spots.  At  conductive  6 and  the  occurs wall  wall  i s thick,  in a l l directions will  be  reduced.  a b o u n d a r y of  uniform  neither  of t h e s e  Analytical process  temperature  In t h i s  case,  variation  the water  temperature.  In  solutions  numerical  efficiency of  factor the  in  the  i s subject  reality,  to  however,  conditions prevail. to the  c o n f i g u r a t i o n s of  against  limits  and  w i t h i n the w a l l m a t e r i a l  convective  heat  i n v o l v e d i n t h e above extreme c a s e s  various  1.3  conduction  spots  solutions. of t h e s e  realistic  and The  were s o u g h t  spacings results  extreme m o d e l s  transfer  and  of t h e can  for  checked collector  be  used  as  situation.  LITERATURE REVIEW Internal  geometries studied  flow  under  convective various  in great d e t a i l  work  in laminar  and  London  flow  [3],  fluid  1.  Entry  2.  The  scope  literature.  various of  ducts  literature  was  of  review  of  Shah  geometries  and  present flow  constant  reviewed  been  been done by  duct  the  various  has  A major  under  review  is  f o r a Newtonian  p r o p e r t i e s , passing through  non-porous The  in  conditions  to f o r c e d c o n v e c t i v e laminar  straight,  categories  transfer  f o r c e d c o n v e c t i o n has  including  with constant  section.  boundary  i n the  b o u n d a r y c o n d i t i o n s . The restricted  heat  stationary,  circular four  cross problem  : l e n g t h problems  Graetz  solution;  problem  i n duct and  flows;  its  two  classical  methods of  7 3.  Problems with c i r c u m f e r e n t i a l developed  4.  or d e v e l o p i n g  Conjugated material  1.3.1  THE In  is  problems,  i s taken  of  begin  the  fluid  i s termed  fluid  The  fully  travels  considered  to from  hydrodynamic illustrated  the  and  hydrodynamically  the  that than  fluid  flow  to  does not  The  velocity  change  where t h e  invariant  the The  the  developed  hydrodynamically  the 5,  a  as heat  flow  is  downstream flow  are  becomes  called  the  respectively,  as  developing  thermally  flow.  It  Prandtl  has  the  developed,  been Pr of  profile  d e v e l o p m e n t and flow even  at  flow  developing  number  the v e l o c i t y  profile  consider  hydrodynamically  profiles  to a tube.  where t h e  l e n g t h s ,  and  1.5.  developed  about  entrance  required before  e n t r y  than  if  temperature  accurate  tube  velocity  d e v e l o p e d .  thermally  thermal  complicated  greater  the  velocity  Similarly,  axially  entrance  in Figure  and  d e v e l o p e d  t h e r m a l l y  and  its  e s t a b l i s h e d and  is  A t h e r m a l l y and  however,  at the  downstream.  be  hydrodynamically  more  in  in ducts,  to  temperature  h y d r o d y n a m i c a l l y  coefficient  distances  and  conduction  transfer  according  to develop  i s already  transfer  axially  into consideration.  categorized profile.  profile  profile;  where h e a t  f o r c e d c o n v e c t i v e heat  temperature  flow  thermal  w i t h an  ENTRY LENGTH PROBLEM  often  the  variation,  but  shown the  [4],  fluid  development i t is  the though  is  is  leads  sufficiently entrance  as  there  no  is  8 hydrodynamic  starting  1.3.2 GRAETZ  PROBLEM  Graetz  [5]  length.  i n 1883 c o n s i d e r e d an i n c o m p r e s s i b l e  with constant p h y s i c a l  properties  tube,  fully  hydrodynamically  thermal  profile.  The t u b e  uniform  temperature. 9T  The e n e r g y  = a(  and w i t h a d e v e l o p i n g a  constant  and  ) , R 9R  2  (1.2)  boundary c o n d i t i o n s a r e : F o r X < 0, T = T for at  The  circular  equation i s  + 9R  a  1 9T  2  9X The  developed  i s maintained at  3 T  u  flowing through  fluid  physical  provided problem. obtained  T = T = w  i s a s shown  has  been  a comprehensive The  constant, constant,  R = 0, 9T/9R = 0.  situation  problem  =  X > 0 a t R = a ,  known a s t h e G r a e t z p r o b l e m . Graetz  i n  closed  primarily  i n F i g u r e 1.6, a n d i s now  A review  of e a r l i e r  done by Drew  literature  form  solution  work on t h e  [ 6 ] . Brown  survey  for  to this  problem  by two m e t h o d s : t h e  Graetz  [7] a l s o  the  Graetz  has been  method  and  the L e v e q u e method. The  G r a e t z method  uses  t e c h n i q u e and as a r e s u l t reduced obtained  the  the d i f f e r e n t i a l  t o the S t u r m - L i o u v i l l e in  the  form  separation  of  t y p e . The  an i n f i n i t e  of  equation solution  series  variables (1.2) i s is  then  expansion of  e i g e n v a l u e s a n d e i g e n f u n c t i o n s . The number o f t e r m s  required  9 for  a d e s i r e d accuracy  approaches The  increases sharply  solution  to the  temperature d i s t r i b u t i o n ,  where  R  (T - T ) / ( T w  are  n  and  obtained  the  only  the  presented The  the  point  with  u  1.7.  Leveque  8  =  first  two  [3].  Sellars ten  an  step  and  et  terms, al.  in  0.  in  the the  are  It  and  same s l o p e  constants,  similarity  to the  employs  resulting the  "flat  near  the  occurs..  The  boundary that at  is illustrated  following  and  ones. the  as  Nusselt  independently  approximation  thermal  the  respectively,  temperature  situation the  higher  (1.3)  n  [8]  and  series  ,  Graetz  employs  x =  X  the  solution  layer the  was wall  in Figure for  the  :  w  1 "  r,  i t s solution  change  obtained  in  of  infinite  n  three  asymptotic  having  (T - T ) / ( T . -  where TJ =  n  f o r the  and  as  i n an  eigenvalues  = 0 a t w a l l . The  tube  and  near  terms  2  technique only  in  constants.  formulas  distribution  circular  X/(a.Re.Pr)  c R exp(-X x)  }  method  where t h e  linear  =l  w  [9]  solution  assumed  T )  are  asymptotic  is valid  velocity  i s presented  n  Leveque  equation  -  problem,  eigenfunctions  first  transformation  plate"  c  series  determined  i n  the  eigenvalues,  of  x =  zero.  Graetz  9 =  as  n  R/a  (9x/2) ^ 1  ,  T ) w  =  [ l / r ( 4 / 3 ) ] SQ e x p ( - z ) d z 3  ,  (1.4)  10  and  T i s the Mercer  extended  r(n)  [ 1 0 ] , Worsoe-Schmidt  the  solving  :  gamma f u n c t i o n  the  Leveque energy  corresponding  solution  fg(e z " " ) d z Z  [11]  solution equation  =  and  by  a  (equ  n  .  1  Newman  [12]  have  p e r t u r b a t i o n method  1.2)  directly.  The  is  (1.5) where £ = this  series  is valid and  (9x/2)^  corresponds  solutions  Grigull finite  conditions constant;  and are  Nusselt  The  first  solution.  Tratz  not  [13]  This  method  with  two  solution  the  uniform  Graetz  using  temperature, wall,  presented  which  graphs  dimensionless  (1.2)  different  ( i i ) a u n i f o r m l y heated  of  in  accurate.  solved equation  on  term  the  boundary i.e.  T = w  i . e . q=constant. show  temperature  spatial  0(R,X),  local  mean N u s s e l t number Nu, ( X ) , m  PROBLEMS WITH CIRCUMFERENTIAL VARIATION  Leveque  the  above m e n t i o n e d  methods a r e b a s e d there  wall-fluid  is  no  studies related  on  the  i n t e r f a c e . Moreover,  For  example,  a s t e p change  simple  circumferential  ( t h e s t e p change  uniform. into  are  number N u ( X ) , and '  All  inlet  to Leveque's  : ( i ) a wall with  distributions  where  and  difference  Results  tube.  f o r i n t e r m e d i a t e v a l u e s of x where b o t h  Leveque  1.3.3  f o r the c i r c l a r  Graetz  and  boundary c o n d i t i o n s variation  fluid  at  the  temperature  at  the  i n w a l l c o n d i t i o n ) i s assumed  to  be  water  the  to  at uniform  i n w a l l temperature  or  temperature  flows  wall  flux.  heat  11  There  is axial  c o n d i t i o n s do  symmetry not  [14]  Reynolds distributions) circumference He  first  transfer  vary  i n the  tube  f o r any  the  of a c i r c u l a r the  tube,  without  solution of  the  peripheral  solution,  however, d o e s not d e s c r i b e t h e  the  the  of  arbitrary variable  wall  the  using  variation.  entrance,  to  heat as  solutions  but  temperature only the flow  a for  superposition.  thermally developed  heat  peripheral  the  This  profile  peripheral further  down  flux  problem. heat  angle,  flux  and  one  First,  further  they expressed  as a F o u r i e r  obtained  entrance  s t e p i n the  region.  i n <f>,  series  the  the  temperature  The  solution  is  form  = t,(X)  q  [ 1 5 ] took  Roy  i n the thermal  t(R,0,X) = k(T -  where q  and  circumferential  distribution of  axial  the  tube. Bhattacharyya  the  tube  the  obtained  flux  q{<p) a r o u n d  circumference  arbitrary  heat  (temperature  corresponding  series,  distribution  then  flux  Fourier  near  and  heat  boundary  direction.  solutions  prescribed  across a small portion  development  the  i n the c i r c u m f e r e n t i a l  obtained  obtained  because  T  i n  )/(qa)  + t (R) 2  i s the p e r i p h e r a l  [ 1 + |  represents  the  ( a c o s m0 m  wall  + t (R,0) 3  + t,(R,*,X)  average  wall  + b^sin  m<j>)]  = k 9T/9R = q  which  i s a function  heat  flux  a n g l e o n l y . A p p l y i n g Duhamel's  heat  (1.6)  ,  flux,  superposition  and  (1.7)  of  theorem,  the the  12  solution  f o r an a r b i t r a r y  qU,X) can  be  series  be f u r t h e r  1.3.4  distribution  ( a ( X ) c o s m 0 + b ( X ) sinmtf>) ] m  of eigenvalues  i s expressed  i n t e r m s o f an  and e i g e n f u n c t i o n s . T h i s work  discussed in Section  obtaining  using  the  a priori  solution  problem  3.2.  temperature  the  fluid  only,  f o r the e n t i r e  f o r both  e t a l . [16] s o l v e d t h e  circular  Mori  for  boundary c o n d i t i o n s  solid-fluid  the f l u i d  tube.  However,  f o r the complicated et  a l .  [17]  conjugated  and  heat  f l u x , @,  assumed  distribution  no  numerical  closed-form  considered  two  results  the in  and (2) c o n s t a n t  wall-fluid  thermal  by s u p e r p o s i n g  solution. the  Equating  interface  the temperature  of the s o l i d  unknown c o e f f i c i e n t s  ^ee  Table  2.1  obtained  for  and  and f l u i d the  power  (TX  a s a power s e r i e s the  energy  heat  1  with  equation the Graetz  fluxes  media t h e y series.  (1)  temperature  for  was t h e n  :  temperature,  to  fluid  were  boundary  tube  interface  the a x i a l d i r e c t i o n  for  solution.  unknown c o e f f i c i e n t s . The s o l u t i o n the  solid  problem  c o n d i t i o n s a t the o u t s i d e w a l l of the c i r c u l a r constant  the  i s obtained.  Luikov  presented  fluid  i s formulated  medium s y s t e m , a n d a s o l u t i o n  They  (1.8)  m  The s o l u t i o n  than  conjugated  the  flux  CONJUGATED PROBLEM Rather  and  + |  obtained.  infinite will  = q(X)[l  w a l l heat  across  obtained the A  conclusion  13  relevant  to  constant Nu(x)  the  heat  For  solution  when 7/L *  10-  that  a  <  0.0001  heat  the  for R  and  the  however,  conduction  with  the  local  @  convection  i t approaches  tends may  and  7  wall  wall,  conventional  > 2X10"  w  "thin"  outside  wall. A wall  Sparrow  wall  hydrodynamically thin-walled gradient  and  was  wall  to  be  the®  equalize  considered  when 7/L  <  symmetry a r o u n d t h e  (x >  0)  was  constant.  where t h e  difference  fluid  the thin  0.001  for  bulk  mean  Since  neglected.  The  problem  circular  (x < 0)  heat  of  flow.  and  f l u x at  the  a  the  tube  region  dimensionless* Substantial the  wall  on  an  and  the  convective T  by  iterative  and  w  the  Peclet  conductance  amounts  non-directly  only  an  of  the  was  also  heated p o r t i o n  of  of had  heating  the  tube  elliptic-finite  scheme w h i c h  dealt  t u b e w a l l . P l o t s of- t h e heat  f l u x q,  wall  and  T  respectively,  and  m  show t h a t  number,  parameter  convective  a  temperature  direct  wall  number were p r e s e n t e d . T h o s e g r a p h s depends  in  considered of  outside  were o b t a i n e d  fluid  temperatures,  conduction  of  c i r c u m f e r e n t i a l dependence because  w i t h the  distribution  conduction  radial  Solutions  axial  effect  the  method e m p l o y i n g  consecutively  the  axial  laminar  a x i s . The  region  considered  considered,  was  t o have no  insulated  [18]  developed  tube  i n the  considered  along  the  for a  [3].  5  simultaneous  Nusselt  i s that  "thick" wall,  inside  Faghri  an  for  because a x i a l  temperature  w  work  f l u x s p e c i f i e d at  approaches  problem.  R'  present  heat (x <  Pe, B  =  axial  and  k 7/ka. w  t r a n s f e r can 0)  of  the  the  occur tube  1 4 because of  wall conduction.  These e f f e c t s  propagated  downstream  substantial  i n c r e a s e i n both  all  the  along  conduction  can  conduction. Nusselt  between  of a x i a l and  assuming of  a  of  variations oscillation amplitude  heat  was  heat  Nusselt  by  flux  the  in  the  the  accurately axial  be  q(x) near wall  axial  of  the  48/11,  interaction  along  the  the  wall.  starting and  finite  tube outer  the  the  thermal  reduces  a  profiles  However,  bulk  mean solid  method.  Their  has  inlet  guessed  the  Nusselt  a  definite  no.  Nu(x)  section.  Step  produce  of Nu(x) in  with  in  conduction  and  direction.  fluid  element  thickness  by  axial  distribution  distribution  predicted  conduction.  up  w a l l heat  nevertheless  thickness  wall  (x > 0 ) ,  the  i n the  no.  temperature  especially in  of  fluid  value  conduction  set  downstream d i s t a n c e , where t h e i r uniform  of  analysed  step variations  determined  distribution  effect  They c o n s i d e r e d a w a l l whose t h i c k n e s s  local  and  on  [19]  axial  show t h a t a x i a l  influence  temperatures  heating  developed  is a  conduction.  procedure  temperature,  results  bulk  that the  of d i r e c t  fully  Pagliarimi  periodic  was  found  region  tube.  iteratie  region  t h e w a l l and  are  so t h e r e  a c o n v e c t i v e boundary c o n d i t i o n a t  the  distribution  flowing f l u i d ,  overwhelm t h e e f f e c t  c o n v e c t i o n and  possessed An  readily the  the  I t was  number a t t a i n s  Barozzi  face  tube.  In  independent  wall  by  of p r e h e a t i n g  and  periodic q ( x ) , whose  relatively  short  approach t h a t of  overall  heat  flux  a  can  o r d i n a r y methods d i s r e g a r d i n g  15  The  above  provides to  the  an  effect  that  although  idea  work.  No  directly  by  no  means  of the a v a i l a b l e work  has  been  investigates  problem,  The n e x t  complete,  results  relevant  found  in  the s t i t c h  on t h e N u s s e l t number, a n d hence on  performance.  aspects  overall  present  literature  review,  solar  the  welding  collector  c h a p t e r d e s c r i b e s the m o d e l l i n g of the  concentrating  on  the  convective  f o r t h e two e x t r e m e c a s e s m e n t i o n e d  heat  transfer  in Section  1.2.  2. MATHEMATICAL MODEL To  analyze  absorber tube,  the  plate  a  transfer  t o the water  mathematical  geometrical  situation  differential solid  heat  and  a  must  describing be  solar its  the  developed.  collector spot-welded  p h y s i c a l and  The  applicable  e q u a t i o n s and boundary c o n d i t i o n s f o r both the fluid  certain  assumptions. typical  the  f l o w i n g through  model  media  Enormous s i m p l i f i c a t i o n to  from  class Those  flat  plate  are  this  chapter.  i s o b t a i n e d by r e s t r i c t i n g  attention  of  discussed  flows  assumptions solar  and  by  can  be  collector  used  domestic  w a t e r h e a t i n g . The s i m p l i f i e d  the heat  transfer  process  in  making  certain  justified in  f o r the  building  mathematical  i s p r e s e n t e d , and o t h e r  and  model o f important  terms a r e d e f i n e d .  2.1 GOVERNING As fluid  the i n v e s t i g a t i o n  medium,  mechanics vector 1.  EQUATIONS  form  the  have  three  involves  conservation  t o be s a t i s f i e d .  i n a moving  equations  in  fluid  They a r e p r e s e n t e d h e r e i n  C o n s e r v a t i o n o f Mass :-  Conservation  o f Momentum  • V  (2.1)  :-  p -D V /Dr = p g + V 3.  transfer  :  Dp/Dr = -p V 2.  heat  Conservation  of I n t e r n a l  p DI/Dr = - V  • q + heat  Energy  «S ;  :-  s o u r c e s + S( V V  16  (2.2)  ) ;  (2.3)  17  where  standard  notations  D(  )/Dr i s t h e s u b s t a n t i a l  S  i s the s t r e s s  tensor  in  vector  derivative,  of  2  on t h e medium  It to  should  be n o t e d  any c o n t i n u u m ,  solid,  solid  that these  Solving is  reasonably  very  general  difficult.  by g . equations  are  applicable  For a stationary  + heat  sources  .  (2.4)  equations  ( 2 . 1 ) t o (2.4) a s t h e y  One  only  can  applicable solutions for special  hope  to  obtain  idealized  cases.  A number o f a s s u m p t i o n s have been made i n m o d e l i n g collector the  of  interest,  forces  becomes  = - V • q  the  time.  ) represents the  a l l body  as w e l l as l i q u i d .  the energy equation p9l/9r  stand  i s represented  a r e used .  T denotes  where  a n d t h e t e r m S( V V  c o m p l e t e c o n t r a c t i o n . The r e s u l t a n t acting  calculus  w h i c h c a n be j u s t i f i e d  the solar  by e x a m i n i n g  p h y s i c a l parameters i n v o l v e d .  2.2 IDEALIZATIONS A t a b l e of d i m e n s i o n s and r e l e v a n t p h y s i c a l for  a  typical  flat-plate  solar  collector  properties  is  given  A p p e n d i x A. The f o l l o w i n g a s s u m p t i o n s a r e made b a s e d on figures 1.  given  Since  the  fluid  incompressible  other 2. 2  medium (AT as  is  water  flow  well  S e e Appendix C  i s laminar  and  10 ° C ) , i t i s  temperature  considered  as Newtonian.  p r o p e r t i e s a r e assumed t o s t a y  The w a t e r  the  i n the t a b l e .  v a r i a t i o n s are small be  in  Furthermore,  constant.  b e c a u s e Re = uD/v  to  1,400 .  18 3.  Dissipation only  4.  No h e a t  source  tube  i s present  there  lead  moderate  already  seen  can  conservative coeff ic i e n t . Axial [18]  in  Appendix  for  water  at  4 .  at  can  be  the  profile  directly  into  the (2.1),  assumption of  be This'  the  well  in Figure  energy  1.5,  equation  (2.2) and (2.3)  results  the  to  [4].  because  a s shown  to solve equations This  assumed  entrance  simplification  estimate  heat  in  transfer  is  this  negligible.  Dimensional  i s t h e c a s e when t h e P e c l e t  (Pe = Re.Pr > 1 0 0 ) . F u r t h e r d i s c u s s i o n  analysis number i s given  B.  The h e a t t r a n s f e r c o e f f i c i e n t i s g r e a t e r w i t h a u n i f o r m v e l o c i t y p r o f i l e , w h i c h i s f o u n d i n s l u g f l o w s and a t t h e d u c t i n l e t d u r i n g s i m u l t a n e o u s d e v e l o p m e n t o f v e l o c i t y and temperature p r o f i l e s [ 2 0 ] . 3  a  3  shows t h a t  large  profile  velocity  conduction  is  significantly.  number,  *  developed  simultaneously.  6.  that,  i s known  diffusivity  enormous  having  solar  length at the  development  development  realizing  the v e l o c i t y  be s u b s t i t u t e d  without  profile  :  known p a r a b o l i c  only  diffusivity  thermal  in  as  starting  the P r a n d t l  momentum  fully  results  by  =  Therefore,  fluid  by t h e p l a t e .  the v e l o c i t y  temperature,  v — a  =  i n the  the thermal p r o f i l e be  flow v e l o c i t y i s  i s low.  i s no h y d r o d y n a m i c  entrance,  T h i s can  Pr  n e g l e c t e d because  i s being absorbed  Although  to  be  a few cm/s and Mach number  energy 5.  can  19 7.  2.3  Only  steady  the  partial  can  be  state  performance  d e r i v a t i v e s with  d i s c a r d e d from  i s c o n s i d e r e d . Thus a l l  respect to time,  the governing  9(  )/9T,  equations.  S I M P L I F I E D EQUATIONS Assuming  developed  the v e l o c i t y  at  distribution  the  to c i r c u l a r  of p e r i p h e r a l  can  i s the a x i a l  be  velocity  of  the  Poiseuille Newtonian derived As known,  in detail  steady  the  (R/a) ] 2  mass  state  of  with  axial  is  also  velocity  the and  ,  (2.5) and  rate"  profile  U,  which  i s the  peak  i s known a s  the  for incompressible  properties.  Its  form  is  [20]. velocity  distribution  being  momentum e q u a t i o n s n e e d n o t equations  concern.  conditions  i  flow  distribution  the energy  are  fully  velocity  invariant  of t h e v e l o c i t y ,  in Burmeister  continuity  media  1 -  constant  a consequence of  c o n s i d e r e d . Only solid  from  with  t o be  a n g l e . That i s ,  velocity  fluid  fluid radial  the  flow. T h i s p a r a b o l i c  flow  the  known and  component  calculated  the  the  symmetry,  V = u ( R ) i = U[ where u  of  entrance,  i s assumed t o be  d i s t a n c e . Due independent  tube  profile  For  and  f o r both the  with  the  solid  fluid  media,  constant  be and  under  thermal  conductivity k V T 2  s  For  the  * See  absorber  Appendix  plate,  A.  = sources the heat  . source  (2.6) is  equal  to  the  20 difference loss  between t h e  t o the ambient  through  For a Newtonian reduces  the  H  =  I + p/p  5  properties and  equation  further  stay  that  sources - V  the  heat  T  equation  (2.3)  constant,  dissipation reduces  to  that  is  [20]  Substituting  equation  (2.5)  directly  state  equation  f o r the  where a = k/pc Equation  -  (R/a) ] 2  2  fluid  no  heat  the  source i s  the  .  energy  (2.7) into  (2.7), the  i s found  to  steady  be (2.8)  2  i s t o be  $ is  that  3T/3X = a V T  i s the thermal (2.8)  mass, and  :  = kV T  U[1  unit  negligible,  DT/DT  energy  • q + ju* ,  A p p l y i n g the assumptions  p c  diffusivity. s o l v e d under c e r t a i n  c o n d i t i o n s as d i s c u s s e d i n the next  boundary  section.  THERMAL BOUNDARY CONDITIONS A set of  heat  flux  obtained large  5  and  1/U .  fluid,  i s the e n t h a l p y per  function .  present,  2.4  radiation  resistance  incompressible  = Dp/Dr + h e a t  DH/DT  the d i s s i p a t i o n water  solar  to p  where  incoming  specifications  c o n d i t i o n s at the  to s o l v e the  variety  S e e Appendix  of  fluid  describing  temperature  inside  of t h e d u c t must  equation  be  (equ.2.8).  A  t h e s e t h e r m a l boundary c o n d i t i o n s can  be  C for details.  energy  wall  and/or  21  specified profile  for  development.  systemize A  the c l a s s i c a l  wall,  general,  [3]  studied  i n the l i t e r a t u r e .  however, b e c a u s e o f c o n d u c t i o n temperature  encountered  exist  t u b e under wall 1.  for  or heat  study, determined  For  a  thick  i n the tube  conditions  are  of the d u c t .  Only i n  be s p e c i f i e d . Two  extreme  flowing  through  the spot-welded  by t h e " t h i c k n e s s "  and h i g h l y c o n d u c t i v e  peripheral  conduction  the  of  rest heat  2.1 ( i ) ) .  the  wall  2.1 ( i i ) ,  of the  tube  the a x i a l  and  wall  compared  Therefore,  solar  heat  flowing  until  spot  compared water  to  to the (Figure  t h e whole t u b e  wall  as  shown  p e r i p h e r a l and a x i a l with  the  radiation  welding  to a wall  wall  into spots.  the  conductance.  absorbed  water  Figure  conductance i s  raidal  being  in  by  only  radially  Thus t h e f l o w i n g  boundary which c o n s i s t s  of  the  water i s spots  of  flux.  definition  1.3.4.  occurs  plate passes the  the welding  temperature.  negligible  subject  the  and low c o n d u c t i v e the  collector  from  wall,  i s considerable  into  Conduction  For a thin  through  of heat  transfer  comes t o a u n i f o r m  The  flux  at the i n s i d e wall  t h e water  to  :  radial  2.  attempted  i n Table 2 . 1 .  extreme c a s e s can those c o n d i t i o n s cases  the thermal  London  c a s e s a r e shown  well-defined  seldom  and  the boundary c o n d i t i o n s  few w e l l - s t u d i e d In  Shah  problem concerning  of a " t h i n "  w a l l h a s been  given  in  Section  22  For be  the i d e a l i z e d  depicted  by  forwardly. angular flux  Peripherally,  = {  where a £(<t>)  m  0  S  o  0  spots  straight  occupy  < <f> < (2ir-<j> ), 0  c a n be e x p r e s s e d  the  the heat  i n terms of  1  s o l  otherwise 0oA)  [1 + | m  (a cosm^) ] ,  1  (2.9)  m  = 2 ( sinm</> )/(m<j> ) . 0  i s a s shown  the p e r i p h e r a l Axially, fixed  heating  < 4> < <j> . Where #  C  0  i n F i g u r e 2.2. I t s h o u l d be n o t e d  Q  and  the  quite  series,  = (Q  at  expressions  i s zero. This d i s t r i b u t i o n  0,  is  ( 2 ) , the boundary c o n d i t i o n s can  mathematical  range - 0  a Fourier  f(0)  case  sol  UoA) = q  mean h e a t  flux  through  the spots are of f i x e d  intervals.  The h e a t  <t>, c a n be e x p r e s s e d  that  flux,  as f o l l o w s  the w a l l .  l e n g t h and a r e l o c a t e d  as a f u n c t i o n  of  both  x  :  q(0,x) = fU)H(x) - f ( ^ ) H ( x - x ) + f U M x - U ^ X g ) ] 1  -  f(^)H[x-(2x +x )] 1  s  +  ,  (2.10)  0, x < 0 where H ( x ) = { 1, x £ 0  is  the  Heaviside  q ( x , 0 ) where -cb  0  unit  step function.  The  < <t> < <f> i s i n F i g u r e 2.3. 0  illustration  of  23 Closed be  form  solutions  of  reasonable  o b t a i n e d o n l y f o r t h e a b o v e two  boundary c o n d i t i o n s a r e c l e a r l y corresponds known. The the  to  method and  s u b j e c t of the next  checked  a g a i n s t and  which w i l l  be  the  p r e d e f i n e d . Extreme case  (1)  and  i t s solution  solutions  c h a p t e r . The  supplemented  can  e x t r e m e c a s e s , where  the G r a e t z problem  analytical  applicability  f o r case  analytical  (2)  are  results  were  with a numerical  d e s c r i b e d i n Chapter  4.  i s well  procedure  ANALYTICAL  3.  The  governing  conditions been  literature.  usefulness in this  The  problem  under  2. T h i s c h a p t e r of  solutions  of  boundary  study  describes  available  in arriving  at the  is detailed  have  briefly in  the  solution in  this  s u p e r p o s i t i o n i s shown t o be  k i n d of l i n e a r  of  problem.  ANALYTICAL SOLUTIONS  analytical  solutions flows  for hydrodynamically  thermal  entrance  following  f o u r k i n d s of methods :  1.  associated  (2) of S e c t i o n 2.4  principle  OVERVIEW OF  and  method employed  the extreme case  great  3.1  methods  The  The  transfer  i n Chapter  analytical  chapter.  equation  f o r the heat  presented  the  for  energy  SOLUTIONS  are  S e p a r a t i o n of V a r i a b l e s  obtained  and  primarily  Similarity  developed by  the  Transformation  methods; 2.  Variational  3.  Conformal  4.  Simplified  Methods;  M a p p i n g Method; Energy  T h e s e methods and been will  briefly be  w h i c h was  Equation their  described  given here employed  and  to the  by  Method.  sources  Shah and  separation  in solving  i n the  London of  the p r e s e n t  24  literature  have  [3]. Attention  variables problem.  method,  25 3.2  SEPARATION OF The  first a  separation  study  of  by G r a e t z  circular  obtained the  VARIABLES METHOD  by  variables  [5] of the  tube.  Many o f t h e  similar  methods. The  variables  i n the energy  t h e e i g e n v a l u e s and  method  thermal  equation  constants  of  of  the  is  in  of  eigenvalues,  Reynolds  and  Roy  the  [14],  work  parts.  by  First,  thermal  circular The  i s employed Sellars  et a l . [ 8 ] ,  tube  energy  and  Roy  equation  is  by  can  solution  region for developed  with v a r i a b l e  solution involving  duct  rewritten  be  described in obtained  laminar  here,  that  1.3.  was  circumferential  are  work  Bhattacharyya  in Section  and  temperature  entrance  The  series  to the c i r c u l a r  Bhattacharyya the  ordinary  c o n s t a n t s . E x a m p l e s of  [ 1 5 ] , w h i c h have been c i t e d  The two  infinite  e i g e n f u n c t i o n s , and  where t h i s method by  an  determining  resulting  by v a r i o u s a p p r o a c h e s .  terms  separating  and  equations  of  work were  involves (2.8)  i n the  region  later  differential presented  used  entrance  solutions method  was  wall in  flow  heat  for in a flux.  cylindrical  co-ordinates:  U( 1 - R / a ) 2  3T  2  2  = o(  9X where t h e a x i a l The  at  9 T 9R  2  +  1 9T R  1  +  3R  R  c o n d u c t i o n term  9 T 2  90  2  )  , (3.1)  2  9 T/9X 2  2  has  been  omitted.  boundary c o n d i t i o n s ares  R = a  : k9T/9R = q = q [ l +  |  9 U) m  ] ,  (3.2a)  26  where g (0) = a cosing + b sinmtf* , m m m  (3.2b)  3  and at  q i s t h e mean w a l l h e a t  flux  over  the  circumference;  X- = 0 : T = T. I  The  (3.2c) n  dimensionless  expressed  forms  of  the  above  equations  were  as 9t  9 t  (1-r )  =  +  9x r  1 9t  2  2  9r  1 9 t 2  +  , r d<t>  r 9r  2  2  (3.3)  2  = 1 : 9t/9r  = 1 + 1 g U) m= 1 m x = 0 : t = 0 ,  ,  (3.4) (3.5)  where t = k ( T - i ) / q a > T  n  x = aX/Ua r  = R/a  The  2  = X/a.Pe ,  .  (3.6)  equations  (3.3) t o (3.5) were s o l v e d by  of v a r i a b l e s and t h e r e s u l t t  = 4x + r  :  2, ( r / m ) g ($) m=1 m - L, Cr, R exp(-/3 x ) - !! L c' R g (0)exp(-/3 x ) , s=1 Os Os ^ 0s m=1s=0 ms ms^m * ms ,^ j 2  - r / 4 - 7/24  was  separation  ft  +  m  2  n  2  n  K  Y  F  7  where R and  c  m  ms  (r.fl ) ms  are eigenfunctions,  a r e the c o r r e s p o n d i n g  s  mathematics the reader values  ms  constants.  are  eigenvalues,  For d e t a i l s  i s r e f e r r e d t o R e f . [ l 5 ] where  of t h e e i g e n f u n c t i o n s and e i g e n v a l u e s  The  Nusselt  (3.6) i s p e r h a p s b e s t number,  i . e . the  seen  of the useful  are given.  advantage of d i m e n s i o n a l i z i n g the v a r i a b l e s  equations the  3  as  from t h e e x p r e s s i o n  dimensionless  heat  in of  transfer  27 coefficient. given  By d e f i n i t i o n ,  the  local  number  is  by : NuU,X) = { q(0,X)/[T  U , X ) - T (X) ] }(D/k) . W  The  Nusselt  term  i n braces i s the l o c a l  commonly  denoted  dimensionless  by  (3.8)  111  h.  heat  transfer  Expressing  in  coefficient, terms  of the  temperatures, with rhc AT = q(27ra)AX ==> t =4x , p m m  equation  (3.8) t a k e s t h e f o r m 2 [ 1  NuU,x)  J i  +  = 1 1  /  2 4  + m  " The  ?i s  |  peripheral  definition  9m<*  - Si 0 s 0 s  ^ ^)M] [c  R  m s  m s  c  s  m  0  )]  (1 P f  )g  m s  average N u s s e l t  of Shah and London  R  i n  (0)  where q(X) T  wm  ( X )  1  ^ 0 s  e x  )  P(^ I  x )  0  exp(-/3 |x) m  ]  number, a c c o r d i n g  to the  [ 3 ] , i s given by:  Nu (X) = { q ( X ) / [ T ( X ) - T ( X ) ] p  (  w m  m  }(D/k) ,  (3.10)  = (1/2TT) f 2 7 r q U , X ) d 0 , =  According  (l/27r);  2 7 r  T (0,X)d0 . w  t o the e x p r e s s i o n  p e r i p h e r a l average N u s s e l t  of t i n equation  ( 3 . 7 ) , the  number can a l s o be w r i t t e n as  Nu (x) = p  11/24  -  s  Z  1  c  Whereas t h e f i r s t deals  O s  R  O s  part  O,0  O s  >  exp(-/3 |x) 0  .  (3.11)  o f B h a t t a c h a r y y a a n d Roy's  w i t h a t u b e w i t h no a x i a l  variation  i n w a l l heat  work flux  28 besides  the  obtaining The  entrance  the s o l u t i o n  solution  superposition series  s t e p change, t h e second p a r t i n v o l v e s  was  obtained  formula  solution  impractical  on  3.3 SOLUTION FOR Consider differential  a  A  tube  applying (3.7).  The  resulting  integrals  calculation.  and  A  bdry  =  H  problem.  "THIN" TUBE USING SUPERPOSITION boundary  value  problem  whose  governing  equation i s  is  1  +  H  a  (t) = 0 ,  linear  operator,  and  has  derivative/ Y(*)  2  +  +  Suppose t h e r e e x i s t the  is  simpler  boundary c o n d i t i o n s t'  wall.  Duhamel's  i s a p p l i c a b l e to the present  A where  heated  complicated  engineering  technique  by  equation  involves  for  superposing  f o r an a r b i t r a r i l y  H  2N  '  2N f u n c t i o n s t , t . . . . 1  2  2N'  fc  satisfying  following: A(t.) = 0 , t  i|bdry=  H  i '  1  "  1  '  2  2  N  '  2N then it  their  sum,  satisfies  t^, will the  be t h e s o l u t i o n  differential  boundary c o n d i t i o n s : A(  .2* t . ) = i=1 i  .1" [ A ( t . ) ] i=1 i  = 0 ,  equation  as  t o (*) b e c a u s e well  as t h e  29  (  ill Vlbdry"  Substituting 9  1  2  [  a  and  letting  2  2  1  = f(0)H(x) ,  H  2  = - f(0)H(x-  H  3  = f(0)H[x-(x +x )]  X ; L  s  by e q u a t i o n s  ( * ) . The s o l u t i o n  =  t  3  = t (r,0,x-(x +x  illustrated and  expression  1  1  of  1  the  of  (2.10) i s t h e  ,  (3.7),  , etc.  s  temperature  tube  the  distance.  = f ( 0 ) H(x) ,  in equation  (2.9).  appropriate  distribution which  i t s circumference,  with  (3.13)  with  boundary c o n d i t i o n i s  qU,x)  same  (3.12)  (3.12),  3.1. A x i a l l y ,  invariant  where f(</») i s g i v e n  :  ,etc,  entering a circular  by F i g u r e  for this  •  ] ,  )) H [ x - ( x + x ) ]  equation  fraction  is  2N  1  1  describes  a  H  t o the f u n c t i o n t i n equation  1  over  +  f o r A i n (*)  t ^ r ^ x )  = - t (r,0,x-x )H(x-x ) ,  flow  +  i s then  2  laminar  2  (3.3) and  t  steady  H  ) ,  1  i s identical  constants,  +  operator  H  t ^ r ^ j x )  x=0  1  (1-r ) — 3x  t(r,0,x)  1  H  2  r 3tf>  the system  where t  =  a  +  the p r o b l e m p o s e d as  iibdry>  a2  1  r 3r  2  ( t  the f o l l o w i n g  + 3r  ill  heating The  a  of  a  i s heated situation starts  at  mathematical  30  The also  be  Nusselt found  number c o r r e s p o n d i n g  to s o l u t i o n  (3.12)  by s u p e r p o s i t i o n :  ^ <t> * <t>o,  F o r -0o  1/Nu(0,x) =.|*  (t  where N u ( 0 , x )  i s given  1  1/Nu (0,x) =  (t  1/Nu (0,x) =  <t  2  3  w  W  f  i  -  2  -  t  3  -  t  t  m  ffl  m f  .)/2  by  1"  =  [l/N  equation  U i  (0,x)]  (3.9),  ,  and  2  )/2  = -H(x-x ) / Nu (0,x-x )  3  )/2  = H[x-(  1  1  X ; L  (3.14)  ,  l  +x ) ]/NU1 ( 0 , X - ( s  X ; L  etc.  For  +XS) ) , (3.15)  other  there  values  i s no The  0,  of  heat  flux  peripheral  found  by  effect  on  conduction  3.4  can  the  Nusselt  through  average  similar  the  Nusselt  numbers  The  number on  of  boundary  extreme  situation  energy  passes  welding  s p o t s . The  distribution  into  in  the  analytical this  (3.13). With  1/8  of  continuously  along  case  the  solution  tube  by  thin  radially for  6  m  The s p o t a n g l e o f 4 5 ° , ' i . e . 0 a r b i t r a r i l y to enable comparison 6  O  (3.11).  The  axial  d e p i c t s the that  the  temperature  equations  (3.12)  being  sin(m0 )(2/m0 ) o  solar  through  the  circumference  i t s length , a =  be  RESULTS  i s so  only  i s given  can  fluid  S e c t i o n 2.4  wall  water  Nu^(x)  equation  ANALYTICAL  tube  since  B.  c o n d i t i o n (2) o f where t h e  zero  neglecting  i s d i s c u s s e d i n Appendix  GRAPHICAL ILLUSTRATION OF  is  wall.  superposition  the N u s s e l t  number  o  = TT/8 was c h o s e n with other cases.  and  welded and k> =0 m  31  in The  the  function 9 (0)  which appears  m  s o l u t i o n i s w r i t t e n here  t O , 0 , x ) = 4x  The  axial  "  sll  "  nil  value  various  c  R  of  development  positions  which are  a  e  P(^0s  x  ms n,s ^ms R  (1  1  r =  the  3.3.  heated  downstream, t h e  d i f f e r e n c e i n temperature  The the  the  local  just  i n s i d e the  one  which  i s just  above s i t u a t i o n  The  local  Nusselt  p o s i t i o n s i n s i d e the  is  f o r those  that Nu (x)  heated  tube.  p  definition case,  the  Nu(#,x) welded  This  of N u ( x ) p  thermal  is identical  directly  should  come  that  entrance  has  only  an  for  shows  the  angular angular near  no  a  is  the  a  angular  heated  region  numbers f o r Figure  since  of  its  a  be  uniformly  i s understood. of  two  value  I t might  surprise  effect  3.4.  shown f o r  region.  that  (equation(3.10))  '  x  There  in  region,  as  )  (0=2TT/1O).  x  was  to  6  distance,  between  outside  positions outside  noticed  with  were p l o t t e d a g a i n s t number  1  x ) ]  sharply  p e r i p h e r a l average Nusselt  angular zero  (  those  developed.  is  and  t ^  temperatures at a l l  relationship thermally  P ^ms  at v a r i o u s  increase  '  3  against  temperatures at  significant  ( 0 = 7 T / 1 O ) and  e x  This diagram  termed  which  {  plotted  situation  position  )  temperature  the  linear as  x  <V°™*>  )  was  wall  directly  Further  assume  )  0 in Figure  of  of  r=1:  for  ''os  expected,  entrance.  angles  [c  expression  ? [a cosm0)/m] m= 1 m  ( 1  t at of  As  +  Os Os  sSo  values  positions.  the  11/24  +  i n the  if  the  In  any  increasing  32  sharply  the  heat  transfer  t h e N u s s e l t numbers n e a r To  illustrate  (3.16),  a tube  occupying was  kept  as  60% at  the w a l l  Figure  of  be  superposition  3.2  i t s l e n g t h was  with  seen  Q  two  on  from  equation  welding  c o n s i d e r e d . The  = TT/8 . The  i . e . <I>  45°,  as can  x=0.  the e f f e c t  in  of  coefficient,  solution  spots  spot  angle  becomes,  for  temperature, 2N  t(1,0,x) where t t  t (1,0,x)  =  _  t  equation  1^ '^' ~ i^ 1  x  1  t  t =  -t (1,0,x-(2x +x ))  4  1  (3.16),  H(x-x ) ,  x  t = 3  ,  i  i s g i v e n by  1  2  = |  (1 ^ ^ - ( x j + X g ) ) H ( x - ( x + x ) ) 1  1  1  ,  s  H(x-(2x +x ))  s  1  g  ,  etc, with  2x-^=  0. 6 ( t u b e - l e n g t h ) ,  2x =  0.4(tube-length)  s  t(1,0,x) noted  that  pronounced there to  was the  i n the  downstream c a n It  was  total  heat  plate  at  i n F i g u r e 3.5.  I t can  temperature  fluctuation  range  of  is directly  heated.  i n temperature  from  also  <j> t h a t  difference  general be  trend  of  temperature  number  spot c o n f i g u r a t i o n s ,  a fixed  i n p u t . The any  Again,  0=tr/lO  increasing  observed.  welding  l e n g t h and  be  i s much more  i n t e n d e d t o compare t h e mean N u s s e l t number  between d i f f e r e n t weld  against x  axial  is a significant  0=27T/1O. The  total  plotted  .  tube of  s p o t a n g l e , and c o u l d be  equally  welded spaced  given a hence a  to the and  Nu  m  fixed fixed  absorber identical  33 stitches, length  while a s t i t c h  angle  o f 60% o f t h e t u b e  of  45°  a single be  3.6  stitch;  observed  higher  mean  "thermal  next the  that  where t h e tube  a h i g h e r number  Nusselt  number  correctness by s o l u t i o n  of  problem.  the  they  as  shown  in  It  results  introduce  can in a more  profile. results  can  be  n u m e r i c a l m e t h o d s . The  difference  formulation  difference  compared w i t h t h o s e d e s c r i b e d i n t h i s  chosen  respectively.  analytical  finite  weld  was w e l d e d a t : ( i )  to the temperature  obtained through  The  total  were  of s t i t c h e s  since  chapter d e s c r i b e s the f i n i t e same  cases  and ( i i ) 4 s t i t c h e s ,  entrance e f f e c t "  The checked  ( i ) and ( i i ) ,  a  l e n g t h were a r b i t r a r i l y  i n v a r i a n t s . R e s u l t s o f two d i f f e r e n t Figures  and  chapter.  results  of were  4.  The  analytical  chapter the  are  fluid  chapter value  solutions  those  and  f o r the  i s bounded by  p r o b l e m . The  idealized  assumptions  outlined  the  simplifications, variables  The  are  and  (3.6)  entrance can  solutions  analytical  methods have been o u t l i n e d  be  the  numerical  1.  Finite  2.  Monte C a r l o Method;  3.  Finite  obtained  methods c a n  Element  as  check  a  The  and same  and  a  basic  hence  the  dimensionless  used.  be  for by  hydrodynamically  s e v e r a l methods.  in Section 3 . 1 ,  classified  The  while  as f o l l o w s :  method u s e d formulation  and  Method.  description  literature  of  This  D i f f e r e n c e Methods;  A brief  details  the w a l l .  same b o u n d a r y  the  are  flows  iterative  last  NUMERICAL METHODS  thermal  the  the  to the  2.2,  Section  retained,  at  solutions.  developed  in  flux  both  analytical in  in  b o u n d a r y c o n d i t i o n where  approach  serve  defined in equation  OVERVIEW OF  presented  s p o t s of h e a t  results  to  SOLUTIONS  results  describes a numerical  supplement  4.1  NUMERICAL  can  of be  i n the p r e s e n t of  the  technique. this  t h e s e methods and found  i n Shah and  work s o l v e s t h e  dimensionless The  next  energy  section  formulation.  34  is  their London  finite  sources [3].  The  difference  equation devoted  using to  an the  35 4.2  F I N I T E DIFFERENCE FORMULATION A  program  equation  (3.3)  distribution Nusselt the  of  the  to  values  and  Ax,  grids,  with  t e r m s of  the  r  9  N  ... M  = 0,1,  ... M,  3-dimensional  the  temperature  t u b e was  obtained.  computed d i r e c t l y definitions  in  The from  equations  r  9  ... M  radial, circumferential, grid  spacing  shown i n F i g u r e by  a  4.1.  cylindrical  The  location  r  radial  i s the  circumferential grid  axial  grid  where  x  i s the  i s the  in  a t a l l nodes  t(N ,N^,N ),  grid  A0,  coordinate  temperature  array,  of A r ,  number,  number o f  number,  the  node.  X  the  in f i n i t e  following + t,  5  1 t  d i f f e r e n c e form, t h e  3  - t,  1 t  2  - 2t  r  equation  + t„  5  +  2  energy  form:  + (Ar)  the  i n a 3-dimensional  = 0,1,  ~ 2t  then  corresponding  =0,1,  Written  3  the  numbers. Thus t h e  X  t  was  hence  d i v i d e d by  defined  grid  stored  N  takes  inside  r e s p e c t i v e l y , as  e a c h node was  and  s o l v e the  and  using  t u b e volume was  axial  N  to  (3.10).  and  be  fluid  number d i s t r i b u t i o n  The  can  developed  numerically,  temperature  (3.8)  of  was  2Ar  t -t 0  = r  2  (1-r )  5  ,  2  (A<j>)  Ax  2  (4.1) where  t ,...,t 0  represent  5  indicated  in Figure  point  ,N  (N L  the  temperatures at the l o c a t i o n s  This equation  ,N ) whose t e m p e r a t u r e (p X  dimensionless  different  4.2.  the  cases  radial  need t o be  is  is t  coordinate considered  0  written , and  for  r = N  «Ar  the is  IT  of  that  separately:  point.  Two  36 ( i ) r * 0: t where a  =  0  [a,t, + a (t +t«) + a t 2  =  (Ar) (l-r )  a,  =  1 - Ar/2r ,  a  2  =  (Ar/rA0)  a  3  =  1 + Ar/2r ,  0  35  (ii)r  .  2  2  — 2 — 2a  =  2  2  3  / Ax  =  avg  ,  (4.2)  ,  the average  of  the tube  - t )/(Ar)  t(1,N  temperature So  (4Ax/Ar )t  +  2  imaginary  tube w a l l  to  whether  a t a node on  spot  t(M +1,N  (1 +  2  x  (1 - 4 A x / A r ) t  .  2  M  r  +  1 was  5  (4.4)  added o u t s i d e  gradient  pre-described that  surface and  axial  at the  heat  was  location  flux.  was  the wall The  determined  = n/4>  ,  0  by  within  = t(M -1,N r  ,N ) x  7  + 2 A r ( 7 r / 0 ) ,* o  a cosm</») = ir/<t> when -<f> <<f><<j> , m  neighborhood  spot:  : At/Ar  N )  r  immediate  the temperature the  (4.3)  y  the node's p e r i p h e r a l  within  at the  , +1)  9  r  by  of a w e l d i n g  - 1)/(M  surface N =  implement  determined  temperature  ,N  9  cl V  An  2  5  center-line.  =  0  = 4(t  N =0 <p  is  t  7  0  = 0 :  where t  (i)  5  .  2  that  5  ,  2  V t  as  + a t ] / a  3  0  0  0  (4.5)  37  (ii)not t(M  within r  spot  +1, N., N ) = t ( M -1, N , N ) . ' 6' x r ' <b' x  Therefore, expressed nodes.  : At/Ar = 0  t h e temperature  i n terms  After  temperatures  another,  temperature with  was  less  condition  deviation  than  considered  circular  t h e tube  two  attained  and  the a r r a y  4.3  CALCULATION OF NUSSELT NUMBERS  fluid  body, t h e h e a t  according average  form,  0  ,  e  conform the  evaluations was t h e n contain  x  distribution  the Nusselt  number,  The  definition  of  x  fx  f  a  l  l  s  within  otherwise,  throughout  h, a n d i t s were  computed  the peripheral  numbers a r e r e p r o d u c e d  wm^ ^~ m^ ^^' * t  T  After  Convergence  coefficient  and mean N u s s e l t  x  ^ " ^  t h e p r o c e s s was  transfer  to d e f i n i t i o n s .  t  0.  after  distribution.  obtained the temperature  dimensionless  =  r  solution  the  = x  t ( N , N ^ , N ) would  the  Having  temperature  N  successive  0.5% a t a n y n o d e .  of  Evaluation  kept a t 0 to  been u p d a t e d ,  between  array  cross-section  entrance  t(r,0,O)  a t a l l nodes had  until  i t s neighborhood  t o (4.6).  a t t h e e n t r a n c e was a l w a y s  temperature  at  c a n be  a new v a l u e f o r e a c h  e q u a t i o n s (4.2)  from  t h e boundary  repeated  function,  was done one  starting  node  the 3-dimensional  with a non-zero  node was o b t a i n e d u s i n g of  a t each  of the t e m p e r a t u r e s  initializing  temperatures  (4.6)  here:  a spot; (4.7)  38  N u ( x ) = O/xJ/^Nu  (x)dx ,  m  where  (4.8)  t^(<t>,x)= d i m e n s i o n l e s s t  w m  (x)  = peripheral  i  x-within-spot  finite  r  wm  x  +  V  2  t(M ,N ,N ) r  N  where UTDA(N  1  -t(M ,M ,N ) r  0  ,  x  (4.10)  x  +  2 ^ ^  UTDA(N ,N ,N ) r  2  sections,  basic  ,M.,N )] + t(0,N. N ) 7r(Ar) } , 0 x 0 x (4.11) f  those  The  obtained  •(A0/2)(r  by  program of  other  -r, ), 2  PROCEDURE  a l g o r i t h m as o u t l i n e d  results  2 2  (4.12)  r  the computational  stages.  x  2  r  RESULTS OF NUMERICAL  the  0  0  2  r  With  against  x  ,N ) = ( 1 - r ) - t ( N ,N ,N ) IT (p X IT (p X = N -Ar , r = (N +1)-Ar .  r,  several  integrals  0  r  ,N  4.4 PRELIMINARY  two  0  (2/t){ l[ [UTDA(N ,0,N ) r • + UTDA(N  and  (4.9)  0  =  x  ,  f o r m u l a t i o n , the above  2  m  ) s  follow:  x'  t (N )  I  otherwise.  t(M ,0,N ) l  and x < n(x,+x  s  n = 1,2,  difference  were computed a s  t dA ,  2  A  i f x ^ nx,+(n-1)x where  t  ,  2  r  In  temperature,  = (2Aa ) ; d-r )  A  x-outside-spot  temperature,  temperature,  = (1/UA) J ( u t ) d A  and  wall  w  = b u l k mean  m  average  wall  (1/2ir) SQ t ( 0 , x ) d 0  = t (x)  local  each  was  i n the previous developed  s t a g e were  methods,  if  in  checked  available,  39 before those  another results  First, wall  was  s t a g e was s t a g e by  the taken  as  tabulated  temperature,  plotted values was  entrance  was  due  number 4.4,  boundary  condition  to  has  been w e l l  Shah  and  s t u d i e d and London[3].  does not d e p e n d on  temperature,  t o be h i g h e r n e a r converge  be  the  i t s value at various r a d i a l  results  to  heated  in  of t h e w a l l  at  found  large  t o the c o a r s e n e s s  w i t h Shah and  test  results The  the  the have  computed peripheral  coordinates r  the  finite  difference  the e n t r a n c e , a l t h o u g h x  tube  was  Compared w i t h t h e t a b u l a t e d result  the  two  (x>80). T h i s d i s c r e p a n c y a t  i n a l l the l a t e r  ( Nu(#,x) = Nu  presents  uniformly  a g a i n s t x i n F i g u r e 4.3.  found  section  a  as e x p e c t e d ,  <t>, and  angle  This  stage.  s i m p l e s t case of  program. T h i s problem been  developed.  of t h e  (x) i n t h i s  results grid  case)  and  i t was  system.  The  i s shown  London's v a l u e s p l o t t e d  the  found  Nusselt  in  Figure  a g a i n s t the  same  axes. Next,  a  circumference insulated Figure  shown for  x,  observed this  8  directly  for  case  Shown  same as  8  3.3.  This  shows t h e  problem ,  of the  i n that  have  1/8  welding  situation wall  angular  the  to  (the  Compared w i t h  i n the case  in Figure  heated  various  as w e l l a s  chosen  else.  F i g u r e 4.6  i n F i g u r e 4.5. the  was  everywhere  3.1.  against  wall  of  spot)  its and  i s as  shown i n  temperature  plotted  positions the  entrance uniformly  at the w a l l  analytical  results  d i s c r e p a n c y can heated  as  tube.  of t h e u n i f o r m l y h e a t e d  tube,  be In a  40 linear  relationship  downstream, thermally  the  where The  and  the  be  remembered t h a t  Further introducing  complicating  in Figure  temperature profiles a short  the  step  3.2  was  obtained  radial  4.9  for  are  further  a l s o shown  r=l/5.  The  (0=0  and  gradient  i s what p e r m i t s  into  inner  r=l/5,  the  fluid.  at  Close  heat to  temperature h a r d l y  pronounced  smaller  10  inner  that  Q=Qsol #oA) Shown i n F i g u r e  there  (  3.5.  is  no  it the  conditions flux,  step  the  fluid  except  1 0  than  at The  various and  c l o s e to the  wall  that  at  positions within  tube  weld t o  pass  centre-line,  f l u c t u a t e s with transfer  as can  be  a  temperature  from t h e  thermally  wall  f o r r=4/5  radial  r i s e s or  The  tube at  temperature  This  problem  changes.  4.8  fluid  by  temperature  results  : Figure  flux  the  The  c i r c u m f e r e n t i a l temperature g r a d i e n t .  commented 9  i n the  4.6,  t depends on  4.7.  l e n g t h . F u r t h e r m o r e , c i r c u m f e r e n t i a l heat less  and  program.  those angular  0=7T/1O).  the  4.3  i n s i d e the  inner  especially  the  the  analytical  the  spot  of  cf heat  by  a s u b s t a n t i a l l y lower  welding  different  boundary  (r=4/5) has wall,  between  downstream f r o m a l l the  coordinates  Figure  t h a t when  0  i s shown i n F i g u r e  temperature d i s t r i b u t i o n s  as  on <j> .  9  the  the  far  cautioned  definition  solved  observed  described  in Figures  changes  agree w e l l with distance  is  be  is  temperatures  in turn depends  axial  x can  flow  reader  dimensionless  of q w h i c h  shown  t  f o r example between t h o s e  should value  region  developed.  comparing cases,  a  between  is  seen It  developed  at flow  much  from  the  should  be  region  in  41 this  case  because  there  conditions a l l along Finally, to  allow  of  memory  Figures the  the flow  the flow  finer  divisions  space,  variation  l e n g t h t o be c o n s i d e r e d was of the g r i d s without  N u s s e l t numbers were c a l c u l a t e d  from  from  method p r o d u c e d That  t h e same r e s u l t s  the numerical  as g r i d  work. F u r t h e r evaluation  elaborated.  spacing gets results  are  results  solutions  shown to  finer  appropriate in  and  resulted in  simple  Whereas  analytical  were now  computed  (4.7) t o (4.1.1).  ( i i ) , the  as the a n a l y t i c a l  numerical method.  converge to the a n a l y t i c a l reassures that both to  the next  collector  increase  program.  using equations  Figures 4.10(i)  halvened  much  which  using  i n S e c t i o n 3.3, t h e y  definitions,  As c a n be seen  of boundary  length.  3 . 6 ( i ) t o ( i i ) were s o l v e d by t h e  directly  those  axial  a n d t h e same two c a s e s  expressions outlined  ones  is  collector  chapter.  performance  methods  efficiency  I m p l i c a t i o n of  will  also  be  5. A P P L I C A T I O N TO COLLECTOR E F F I C I E N C Y FACTOR As  mentioned  F'  i s given  F-  =  heat  the  T  (b+(W-b)F) ] +  k 1/C.  L  transfer  expression slightly  was  various  spot  length,  and  angle  between angles total  as  than  that  this  uniform  wall  coefficient result  can  are  important limit of  this  of  factor  the  d o e s not be  case on  as  interesting  in determining actual  N)  F'.  for  equation x  was  can  for  spot  w  a l l  F'  to  heat  a and  limit  the  results  for  spot  the  transfer  configuration.  upper  will  tube  Choosing  tend  the  p e r f o r m a n c e , and  of  between  the  This result  42  calculated  illustrated.  tube since  Nu(x) The  a tube.  be  this,  (3.10).  percentage  because  chapter.  l/(irDh)} .  definition  and  "thick"  t h e o t h e r hand  more  ) +  relationship  depend  regarded  w  to  spots along  the  temperature  p e r f o r m a n c e . On tube  efficiency  appropriate a  welded  0  number N o f  for  h  in  Nu  (2<t> ) ,  invariant,  results  + 7/drDk  b  from  s p o t c o n f i g u r a t i o n s ( i . e . w and The  collector  coefficient  determined  different  relationship  spot  1,  by  •. 1/[U  W{  The  i n Chapter  This  of  actual  the  "thin"  configuration is  s e r v e s as be t h e  the  major  lower  concern  43 5.1 HEAT TRANSFER COEFFICIENT BASED ON BOND TEMPERATURE The is  common d e f i n i t i o n  based  wall  on  of N u ( x )  the d i f f e r e n c e  temperature  (e.g. equation(3.10))  p  between  the p e r i p h e r a l  a n d t h e b u l k mean t e m p e r a t u r e  A careful  analysis  of the c l a s s i c a l  a Nusselt  number b a s e d  and t h e b u l k mean t e m p e r a t u r e  conjunction  w i t h e q u a t i o n ( 1 . 1 ) . The r e s u l t  problem  "thin"  associated  tubes  The  is briefly  useful  direction  plate,  T^  the in  s  o  i  Section  transferred  in  *  s  t  n  to  terms  e  solar  t o the f l u i d .  tube  that  the  bond  s h o u l d be u s e d i n of the c l a s s i c a l  plate  collector  with  below.  per unit  s o l  -  energy  of l e n g t h  i n the flow  U (T -T )] L  b  ,  a  (5.1)  a b s o r b e d by u n i t  area of the  a t t h e p l a t e - t u b e bond, T  and t h e r e m a i n i n g t e r m s  1.1. U l t i m a t e l y ,  the  useful  gain  The r e s i s t a n c e  o f t h e bond, t h e  resistance.  The u s e f u l  gain  is  are defined  q^  t o heat  tube  &  must  be  flow t o the  wall,  and  the  c a n be e x p r e s s e d  of these t h r e e r e s i s t a n c e s as  1/C + b  It  b][Q  +  that  t u b e c a n be shown [ 1 ] t o be  temperature,  comprises  fluid  gain  i s the temperature  ambient  fluid  energy  [(W-b)F  u  Q  outlined  for a collector q =  where  with the f l a t  reveals  between  temperature  fin  of the f l u i d .  f i n problem  on t h e d i f f e r e n c e  average  7/7rDkw+  1/TrDh .  h a s been shown [ 2 ] t h a t  t h e bond r e s i s t a n c e  1/C,  is  D  negligible since  the  compared w i t h t h e o t h e r r e s i s t a n c e s . tube  wall  Furthermore,  i s assumed e x t r e m e l y t h i n ,  the r a d i a l  44 resistance The  offered  useful  energy  by  the w a l l  g a i n per  unit  T  Kb  _  T  of  = 27raq(X),  U  obtained  the a x i a l  layer  of  radial  resistance  air  can  whether X  falls  within  (5.2)  following  familiar  form  is  bm  "  bm  where no tube  welding  and  f l o w , and  (Figure over  the  i s done, a  f i n offers  b o t h q and  h^  0  " =  ( k  spot or  resistance  5.1).  large  in  (5.3)  the  applies  not.  Furthermore,  are both  negligible,  </> = 0 c a n Therefore  the e n t i r e  a  thin  be  taken  the  as  the  mean h e a t  flow l e n g t h L i s  q(X)dX  L  ; L  a welding  T  (  a  f  d i m e n s i o n l e s s form,  h  (5.3)  m  a t R = a and  coefficient  h  b  the w a l l  temperature  1  In  is  M  t a k e n as z e r o . T h u s , e q u a t i o n  temperature  transfer  tube  [T (X)-T (X)] .  the  t o heat  t h e bond and  fluid  bond  be  /  positions  between  equation  the  flow l e n g t h per  .  the  = q(X)  b  since  negligible.  : h (X)  At  so  is also  w  m  1/»Dh But Q ( X )  7/7rDk  /°  ) N u  0  f  X  _  )  T  equation  m  ( ) X  (5.4)  m  (5.4)  becomes  bm  (k/Dl) ; j N u ( x ) d x b  ,  (5.5)  45  where Nu^Cx) = 2 / [ t ( 1 , 0 , x ) - 4 x ] ,  i f x-within-spot,  = 0 , i f x-outside-spot, and  1 = L/a«Pe Solving  equation  equation  (5.2) f o r  (T - T m  T  where  and  into  ) ] ,  F' = W{l/[U (b+(W-b)F)]  same e x p r e s s i o n  (1.1)  if  f o r F' c a n  (5.7)  W(irDh )}  +  L  are  substituting  ( 5 . 1 ) , one o b t a i n s q = WF' [Q ,- U ^u sol L  The  (5.6)  bm  be  obtained  from  equation  t h e r e s i s t a n c e s due t o t h e bond and t h e t u b e  wall  neglected. The  equation  Nusselt  no.  based  on bond  (5.6) was c a l c u l a t e d  superposition efficiency comparing  method  using the series  d i s c u s s e d i n Chapter  factor  can  then  the  value  of  configurations,  easily F'  the  bahaviour  definition.  This  is illustrated  To  for  calculate  solution  3. The  different  N  m  i n t h e next  t(1,0,x)-4x  welded tube  with  is  spot angle  using equation  (3.16) :  the  20 , o  this  collector Before  chosen using  to this  section.  WELDED TUBE  dimensionless  required.  and  welding-spot  of Nu^(x) and u ^ ( x )  Nu^(x),  as i n  calculated.  OF NU(X) FOR A CONTINUOUSLY  difference  found  be  Nu^  one c o n f i g u r a t i o n was a r b i t r a r i l y  illustrate  5.2 BEHAVIOUR  temperature  For  a  temperature continuously  difference  can  be  46  Nu (x) = 2/[t(1,0,x)-4x] b  2 "  C  1  1  /  2  4  ' -mil m 3  mllslo Nu^ various well  values  equals  the  Nusselt  curve  As  (  a g a i n s t 1000X of  x )  0  average  heated  4>  the  2(j>  0  Nu (x), p  based  factor  o f more  where  tube  heated  on  bond  Thus t h e Nu ( x ) P  and a p p r o a c h e s t h e  1 1  so d o e s t h e N u s s e l t  peripheral  It the  )  to the  accepted  T h i s i s not the  case.  8  t(l,0,x)  temperature. the  '  5  5.2 f o r  corresponds  wall,  with  on  the spot angle, the higher  efficiency  Figure  case  average  w h i c h does n o t d e p e n d on  uniformly  based  ir  decreases,  on bond t e m p e r a t u r e . number  in  =  0  wall  angle  spot  '  x )  (  of a u n i f o r m l y h e a t e d  uniformly  (  appears  The c a s e  <j> .  P ^0s  e x  ]  the  coefficient  cost  P ^ms  v a l u e o f 48/11.  temperature  larger  e x  f o r <t>o = n c o i n c i d e s  Nusselt  with  a  R  asymptotic  number b a s e d the  R  peripheral  for a  expected  c  ms rns m  plotted  known c a s e  curve  C  "sli 0s 0s  / m  <j>  0  temperature  h  b  heat m  wall  but conforms  i s obvious mean  for  that the transfer  and hence t h e  F ' . T h i s i n c r e a s e i n F' i s o b t a i n e d a t t h e  welding.  5.3 BEHAVIOUR OF NU(X) OF SPOT WELDED TUBE A  larger  spot  spot-welded  as well  spot-welded  tube,  11  Shown  angle  results  i n a h i g h e r F' f o r t h e  as t h e c o n t i n u o u s l y welded tube.  the higher the percentage  i n F i g u r e 3.4  w of tube  For the length  47  being  welded t o the a b s o r b e r  through given  the w  spots),  and  number N o f more s p o t s  can the in  the h i g h e r  <j> ,  spots r e s u l t s  be  superposed  spot-welded S e c t i o n 3.3,  in a higher h ^ .  larger  is  because  effect.  f o r the c o n t i n u o u s l y  .|  N  [1/Nu. (x) b  i s g i v e n by  welded  hence h ^  argument  (5.8),  1/Nu (x) = H[x-(x +x )] / N u 3 b  outlined  and  1  s  1 b  ,  (x-(x +x )) , l  s  etc.  corresponds To  for  (5.9)  / Nu^lx-Xj)  2 b  should  tube  ] ,  equation  1/Nu (x) = - H U - x ^  It  a  obtains  b  (x)  entrance  for  into a  This  tube. F o l l o w i n g a s i m i l a r  1/Nu (x) =  1 b  welding  t o o b t a i n t h e N u ^ l x ) and  one  flows in  Furthermore,  the  i n t r o d u c e more t h e r m a l  e x p r e s s i o n (5.8)  (from which heat  is h ^ .  distributing  0  The  where N u  plate  (5.10) be  noted  that  Nu  1 b  (x)  t o the c o n t i n u o u s l y welded  calculate  the  mean  in  equation  (5.9)  tube.  Nusselt  number  Nu,  ,  the  bm integration One-third  of  (5.5)  R u l e . F i g u r e 5.3  arbitrarily Figure  in equation  5.4  chosen  case  shows N u  b m  (x)  was  computed  shows N u ( x ) and  o f 4>  0  = TI/'\§,  stabilize  and  Nu  b m  rr/ ^ x  D  Nonetheless,  (x)  approaches  i  f o r the  N=2, N=8.  0  fluctuations.  downstream and  m  W=60%  N u  Simpson's  u^ (x)  f o r </> = 7 r / l 0, w=60% and b  abrupt  N  b  the d i s c o n t i n u o u s n a t u r e of N u ( x ) ,  possess  using  n  the an  both  while Because cases  fluctuations asymptotic  48  value. that  In r e a l i t y , of  the  the f l u c t u a t i o n s of N U j ^ x )  temperature  downstream d i s t a n c e  than  profiles  as  stabilize  well  at a  as  shorter  i s shown due t o c o n d u c t i o n  in  the  tube w a l l [ 1 9 ] .  5.4 E F F I C I E N C Y FACTOR The  ultimate  illustrated lower  FOR VARIOUS  goal  of d e v e l o p i n g  in the previous  limit  of  the  collector  tube.  illustrations  o f F' f o r v a r i o u s  Peclet  number  corresponds Nusselt  This  the tube to  percentage w of tube a  tube.  l e n g t h and spot particular m, 5.5 of  1 2  See  by  the  length,  This  is  spacing  tube  distance  0  of N u  b m  Appendix A  and  length  x of  half-spot  and t h e t o t a l  are i n v a r i a n t with  of  2m The  f o u n d by  configuration angle  0  spacing  spot  for  a  W o f 0.15 in  = 36°) f o r v a r i o u s  F' was computed a n d was  spots  that  position  N  is  <t> , w e l d e d  number N o f  ( . 0 8 8 ) was p l o t t e d a g a i n s t angle  the  0.088.  b a s e d on t h e a s s u m p t i o n  = TT/10 ( i . e . s p o t  5.6.  mm,  t h e t u b e l e n g t h was  0  w. The c o r r e s p o n d i n g  Figure  over  the  1 2  graphical  configurations.  s e t o f <f> , w and N. F o r a t u b e  the value f o r <t>  spot  the  f a c t o r F' o f a  presents  a t x = 0.088. The s p o t  determined  form s o l u t i o n s  r a d i u s a t o be 4.5 5060,  averaged  NUfcrn/*)  completely  along  be  closed  efficiency  section  to the dimensionless  number  evaluating  inner  CONFIGURATIONS  two s e c t i o n s was t o p r e d i c t  spot-welded  Fixing  SPOT  Figure values  plotted  in  49 As  expected,  decreases,  and  the  the decrease  more i n t e r e s t i n g  aspect  also  that  be o b s e r v e d  i n c r e a s e F'  efficiency  i s faster  i s the  F' d e c r e a s e s as w  when w  increase  increasing  significantly.  factor  o f F'  is  small.  w i t h N.  N beyond about  The  I t can  40 does  not  6 . DISCUSSION AND The  problem  solved its 3.  by  of  two  of t h i s  same  difference  series  of  s p o t s of h e a t  i n d e p e n d e n t m e t h o d s . The  results The  a  CONCLUSIONS  problem  problem  formulation  A comparison  of  the  has  w i t h an two  analytical  have been been  iterative and  been and  i n Chapter  using scheme  has  method  described  solved  methods  flux  a  finite  (Chapter 4 ) .  their  results  is  appropriate.  6.1  DISCUSSION OF The  was  ANALYTICAL AND  analytical  o b t a i n e d by  solution  and  that  solved  boundary  t o be  Roy  condition  inverted  obtained  by  number  of  wall  of a s i n g l e  1 3  tube u s i n g  tube  whose step  the N u s s e l t  the  axial  which i s  1 3  shifted  therefore  summing  solution boundary  be  up to  a the  condition  function). numbers f o r  t ( l , # , x ) needs  2.3  and  to  (equ.(3.3))  form  can  tube  due  functions,  (the  axial  B h a t t a c h a r y y a and  in Figure  step  inverting,  to  t h e b u l k mean t e m p e r a t u r e  Illustrated  equation  solution  solution  unit  solution  p r o b l e m assumes a  same  welded  spot-welded  Furthermore,  of H e a v i s i d e u n i t  the  temperature  because  linear.  simply s h i f t i n g ,  To c a l c u l a t e welded  is  energy  a l o n g t h e x - a x i s . The  continuously consists  [ 1 5 ] . The  of t h i s  a s i m p l e summation and  to the " t h i n "  s u p e r p o s i n g the a v a i l a b l e  Bhattacharyya was  NUMERICAL METHODS  and 50  the  continuously  Roy's s o l u t i o n , be  computed.  i s simply given  depicted  only  the  This  is  by  by e q u a t i o n ( 2 . 1 0 ) .  51  t The  ( x ) = 4x .  of t(l,0,x) i n v o l v e d three  computation  which appeared  m  i n equation  (3.16)  infinite  series  :  oo  (i)  L  [a cosm0)/m] , m  where a = 2 s i n (m0 )/(m<j> ) m  s  Il  o  c  mil  0s 0s R  s=0  C c  with  occurred brackets was  0s  P ^0s  ^ms  (  )  i n the  series.  the  decayed  ^  x  )  c  o  ' s  m  <  t  >  )  e  m  P<^  x m  )  •  ]  s  s e r i e s was slow a n d sometimes  Although  third  x  of  series,  the the  e x p o n e n t i a l l y with  the  term  cos(m</>)  200 terms  h a d been  same  problem  expression  i n square  i n c r e a s i n g m and  when m^s>l0. The number o f t e r m s s e r i e s i n c r e a s e d when  x  also  s  and  required  was  small  x < 5x10 ~ ) . 3  The  superposed s o l u t i o n f o r the  increase  solution a  x  was t e s t e d o n l y a f t e r  by t h e s e c o n d and t h i r d  not  ( 1  e  of the f l u c t u a t i o n  insignificant  (i.e.  )  of the f i r s t  because  in  / 3  R  m. C o n v e r g e n c e  included  '  ms ms  Convergence uncertain  { 1  ;  0  spot-welded  the c o m p u t a t i o n a l time s i g n i f i c a n t l y  f o r the continuously  welded  tube h a d been  f u n c t i o n o f x. T h i s was t h e main a d v a n t a g e  method  over  tube d i d  directly  applying  Duhamel's  once t h e s t o r e d as  of the present superposition  formula. It is  should  relatively  because  be p o i n t e d  out that  easy c o m p u t a t i o n a l l y  the eigenvalues  and  the  analytical  f o rthe wall  eigenfunctions  solution  temperature  involved  have  52 been  tabulated  readily (i.e.  i n B h a t t a c h a r y y a and  used.  r < l ) has  The  eigenfunctions  are  not  f o r r<1  the  entire  the  tube  temperature  v o l u m e . The  i n terms  because  o f CPU  the  result  boundary  the  be tube  eigenvalues  much  computation,  distribution  program  condition grid  and M,=20 f o r a 2m <j>  equations  the  (3.8)  analytical  the numerical  entire  volume  mean t e m p e r a t u r e  t  temperature  3-dimensional  heat  understanding  and  offered  results, was  numbers  (3.10))  counterparts  the  as  a  (M =400,  definition  (as i n  with  t o the  their  correctness  temperature  in  the  i n t h e e x p r e s s i o n of the bulk of the  increased  the  transfer  by  comparing  . A picture  not  in a l l  and  very large  a check  since  involved  was  small  is  tube),  Nusselt  and  the  This  variation  t o be  was  aimed  expensive to  required.  involves  s p a c i n g had  method  throughout  d e v e l o p e d was  t h e number of n o d e s i n v o l v e d  Computing  of  inside  difference  t i m e and memory s p a c e  t h r e e d i m e n s i o n s . The  of  can  those  require  o t h e r hand, t h e f i n i t e  obtaining  M =10, r  , which  and  tabulated.  On  run  distribution  n o t been computed b e c a u s e  and  at  temperature  Roy's paper,  internal  distribution  understanding  occurring  obtained  inside  through  of  the  the f l u i d .  This  the  analytical  solution. Nevertheless, cost  required  eigenvalues solution  by and  it the  is two  unfair methods.  eigenfunctions  were a l r e a d y  given  t o compare t h e  and  This  required its  is by  computing  because  the  the  analytical  computational  effort  53 cannot  6.2  be a c c o u n t e d f o r .  RECOMMENDATIONS If  material  designing  the  cost flat  recommended  for  conduction  inside  transferred (Figure It  into  2.l(i)),  thermal  approaches  in  of  the  7  from  :  This  mean  and  spot  2.1(ii)).  the  heat  to  tube  the  The  wall  i s more p r o n e for this  case wall  for  the  uniform  Nusselt  no.  Nu (.088)  is  4.776,  and  [3]  to  m  be  F' o f 0.883. The  and depends  on  actual  the  total  wall. i s desired  and t h e t u b e  attention  for R > w  fluid  s h o u l d be  10' , 5  the heat  can  be  thought  provided  by  the  mean  Nusselt  and t h e e f f i c i e n c y  lower  be  spots alone.  wall  has  paid  to  c o n f i g u r a t i o n . When 7/L ^ 0.0001  to  Nu^  axial  b e i n g done, a s w e l l a s t h e t h i c k n e s s and  t o the passage  temperature  and  the welding  m  factor  lower  o r when 7/L. < 0.001 fin  Nu  tube i s  the e n t i r e  tube  in  copper  allows  a thick  London  and n o n - c o n d u c t i v e ,  restricted  approach  that  limit  a low m a t e r i a l c o s t  the  (Figure  of through  however,  of the tube  welding  2x10" ,  Shah  welding  be t h i n  wall  throughout  i s , of c o u r s e ,  conductivity  to  a thick  Peripheral  tube  t o an e f f i c i e n c y  o f F'  If  the  instead  case.  corresponds  amount  welding.  consideration  s t r e s s e s . The mean N u s s e l t number  temperature  value  collector,  the f l u i d  an upper  tabulated  plate  spot  s h o u l d be n o t e d ,  to  i s n o t an i m p o r t a n t  limits  given  no.  factor  for  R ^ w  transfer  of as b e i n g  welding based  on  spots bond  F' f o r t h i s  case  i n F i g u r e s 5.5 and  5.6,  54  respectively. the  The  i n c r e a s e of  actual  (7/LR ).  The  w  strong  function  Figures  6.1  and  of 6.2,  value  the  i s h i g h e r and  efficiency  tube  where t h e  increases with  factor  s p a c i n g , W,  as  F'  is also  a  illustrated  in  s p o t a n g l e have been  kept  s h o u l d be  a  at  0O = TT/1O.  In any percentage becomes  case, of  the  more  over  cost  The  for  l e n g t h and  This,  large  number  number  the  tube  of c o u r s e , has  of  tube  l e n g t h of  the t h i n  and  industrial  of  it  configuration  should  i s just  spot-welded  including thickness, spot-welded collector.  the  t o be  thin  should  separated, separated, over  40  tube,  and  weighed a g a i n s t  per  i s t o be  be  done  short  spots  long tube  consideration,  welding  large  spots. (for a  important  taken a g a i n s t  mechanism,  usually  an  robot.  Finally,  the  This  non-conductive  is  spot welding  closely  2m).  done o v e r  the c i r c u m f e r e n c e . T h i s  of s p o t s s h o u l d be  the c o n t r o l l a b i l i t y  of  when  of  o f a s m a l l number of w i d e l y  total  typical  tube  of w e l d i n g . Moreover,  a  instead  welding  important  non-conductive. the  spot  overall tube  one solar  heat  s p a c i n g and  collector  be  pointed  factor  out  d e t e r m i n i n g the  collector. transfer bond  as w e l l  as  that  Other  coefficient width,  etc,  the  spot  efficiency  ' parameters, U" ,  plate  L  affect  the c o n v e n t i o n a l l y  the  welded  55 6.3  CONCLUSIONS The  Two  objectives  ideal  models f o r t h e heat  spot-welded conditions given  o f t h e p r e s e n t work have been  collector  corresponded  solution  t o the lower  was of  obtained the  effect  collector  on  solution  available  superposition kind  of  of  tube,  inside  results  detailed the  finite  which  the e n t i r e factor  in  obtained  the  linear tube,  difference  described  tube  volume.  the The  due t o s p o t s o f h e a t  literature.  solution.  The  This  existing  technique  of  t o be o f g r e a t u s e f u l n e s s i n For  interesting  a g i v e n amount o f w e l d i n g  short spots r e s u l t e d  transfer  by s u p e r p o s i n g  problems. an  was  f o r which  heat  using the a n a l y t i c a l was  the  performance.  equation,  was d e m o n s t r a t e d  non-conductive distributing  the  efficiency  was i n v e s t i g a t e d  analytical  of  through  energy  distribution  this  limit of  temperature  solution  inside  were s e t up i n w h i c h t h e b o u n d a r y  and n o n - c o n d u c t i v e  understanding  phenomenon  input  process  c o u l d be f o r m u l a t e d m a t h e m a t i c a l l y . A t t e n t i o n  to the thin  An  tube  transfer  achieved.  the finding  over a  i n a higher e f f i c i e n c y  thin  and  was  that  large factor.  number  REFERENCES  1.  J.A. Duffie and W.A. Beckman : " S o l a r E n g i n e e r i n g of Thermal P r o c e s s e s " , John Wiley & Sons, New York. (1980)  2. A.  3.  W h i l l i e r : " T h e r m a l R e s i s t a n c e of t h e T u b e - P l a t e Bond in Solar Heat Collectors", S o l a r E n e r g y Vol.8, No.3, pp95-98. (1964)  R.K. Shah and A.L. London : C o n v e c t i o n In D u c t s " , A d v a n c e s S u p p . l , A c a d e m i c P r e s s . (1978)  4. W.M.  K a y s and Transfer",  i n  Flow  Heat  the P h y s .  Thermal Chem.  T.B. Drew : " M a t h e m a t i c a l P r o b l e m s : A Review", pp26-80. (1931)  Forced  T r a n s f e r  M.E. C r a w f o r d : " C o n v e c t i v e Heat McGraw H i l l . (1966)  5. L . G r a e t z : "On P a r t 1, A n n . 6.  "Laminar  and  :  Mass  C o n d u c t i v i t y of Liquids", V o l . 1 8 , pp79-94. (1883)  Attacks T r a n s .  On  Forced  Convection Vol.26,  A m . I n s t . C h e m . E n g .  7.  G.M.  8.  J.R. S e l l a r s , M. T r i b u s and J . S . K l e i n : "Heat Transfer to L a m i n a r F l o w i n a Round Tube o r F l a t C o n d u i t - t h e Graetz Problem Extended", T r a n s . ASME Vol.78, pp441-448. (1956)  9. M.A.  Brown : "Heat o r Mass T r a n s f e r i n a F l u i d i n L a m i n a r F l o w i n a C i r c u l a r or F l a t C o n d u i t " , A m . I n s t . C h e m . E n g . J o u r n a l V o l . 6 , p p 1 7 9 - l 8 3 . (1960)  L e v e q u e : " L e s l o i s de l a t r a n s m i s s i o n de par convection", A n n . M i n e s . M e m . , Ser.12 pp201-299,305-312,381-415. (1928)  chaleur Vol.13,  10.  A.McD. M e r c e r : "The Growth o f t h e T h e r m a l B o u n d a r y Layer a t the I n l e t to a Circular Tube", A p p .S c i . R e s . , S e c t . A V o l . 9 , pp450-456. (1960)  11.  P.M. W o r s o e - S c h m i d t : "Heat Transfer in the Thermal E n t r a n c e R e g i o n of C i r c u l a r T u b e s and A n n u l a r Passages w i t h F u l l y Developed Laminar Flow", I n t . J . Heat Mass T r a n s f e r V o l . 1 0 , pp541-551. (1967)  12.  J . Newman T r a n s f e r ,  13. U.  : " E x t e n s i o n of t h e L e v e q u e S o l u t i o n " , J T r a n s . A S M E Vol.90, pp361-363. (1969)  Grigull and ausgebildeter Mass  T r a n s f e r  H. Tratz : "Thermischer laminarer rohrstromung", V o l . 8 , pp669-678. (1965)  56  .  einlauf Int.  J .  Heat  in Heat  57  14. W.C. R e y n o l d s : "Heat T r a n s f e r t o F u l l y D e v e l o p e d L a m i n a r Flow i n a C i r c u l a r Tube w i t h A r b i t r a r y C i r c u m f e r e n t i a l Heat Flux", J . Heat Transfer, T r a n s . ASME V o l . 8 2 , p p l 0 8 - 1 1 1 . (1960) 15. T.K. B h a t t a c h a r y y a and D.N. Roy : "Laminar H e a t T r a n s f e r i n a Round Tube with Variable Circumferential or Arbitrary W a l l Heat F l u x " , I n t . J . Heat Mass T r a n s f e r , V o l . 1 3 , p p l 0 5 7 - l 0 6 0 . (1963) 16. A.V. L u i k o v , V.A. A l e k a s a s h e n k o a n d A.A. A l e k a s a s h e n k o : " A n a l y t i c a l Methods o f S o l u t i o n o f C o n j u g a t e d P r o b l e m s in Convective Heat Transfer", I n t . J . Heat Mass T r a n s f e r , V o l . 1 4 , p p l 0 4 7 - l 0 5 6 . (1971) 17.  S. M o r i , M. S a k a k i b a r a a n d A. T a n i m o t o : " S t e a d y Heat T r a n s f e r t o Laminar Flow in a Circular Tube with C o n d u c t i o n i n t h e Tube W a l l " , H e a t T r a n s f e r - J a p a n e s e R e s e a r c h V o l . 3 ( 2 ) , pp37-46. (1974)  18.  M. F a g h r i , E.M. Sparrow : " S i m u l t a n e o u s W a l l a n d F l u i d A x i a l C o n d u c t i o n i n L a m i n a r P i p e - f l o w Heat Transfer", J . Heat T r a n s f e r , T r a n s . ASME V o l . 1 0 2 , pp58-63. (1980)  19. G.S. Barozzi a n d G. Pagliarini : "Conjugated Heat Transfer in a Circular Duct with U n i f o r m and Non-uniform Wall Thickness", Heat and Technology V o l . 2 , pp72-89. (1984) 20. L.C. B u r m e i s t e r Interscience.  : "Convective (1983)  Heat  Transfer",  Wiley  21. C . J . Hsu : "An E x a c t Analysis o f Low Peclet number Thermal Entry Region Heat Transfer i n Transversely Non-uniform Velocity Fields", Am.Inst.Chem.Eng. J o u r n a l , V o l . 1 7 , No.3, p p 7 3 2 - 7 4 0 . (1971)  58  Black absorber plate  -U  0  0  Insulation  FIG  1.1  Cross  section  Inner cover  Outer cover  <•>  Fluid conduit  of a b a s i c  flat  0  0  1L  r  Collector box  plate  solar collector  59  FIG.  1 . 2 . Energy  flow  i n an  operating  solar  collector.  FIG  1.3  Cross section  of p l a t e  and  tube  arrangement  61  (ii)Tube  FIG.  embedded  i n trough  formed by  1.4. Common ways o f p l a t e - t u b e  plate  bonding.  (iii)Plate  (iv)Tube FIG  1.4  includes conduit  i s welded  within  itself  onto p l a t e at separated  Common ways of p l a t e - t u b e  bonding  spots  63  Laminar flow  Hydrodynamic entrance region Le  FIG  1 . 5 H y d r o d y n a m i c and t h e r m a l  entry  lengths  64  FIG  FIG  1.6 P h y s i c a l  1.7 L i n e a r  velocity  situation  of Graetz  p r o f i l e assumed  problem  i n L e v e q u e method  Applications  Description  Designation  Constant wall temperature  ©  Condensers, evaporators, automotive  peripherally as well as  radiators (at high flows), with negligible  axially  wall thermal resistance  Constant axial wall temperature  Same as those for (f) with finite wall thermal resistance  with finite normal wall thermal resistance  ©  Radiators in space power systems, high-  Nonlinear radiant-flux  temperature liquid-metal facilities,  boundary condition  high-temperature gas flow systems  ©  Same as those for (H4)for highly  Constant axial wall heat flux with constant peripheral wall  conductive materials  temperature Same as those f o r ( H 4 ) f o r very low-  Constant axial wall heat flux  © ©  with uniform peripheral wall  conductive materials with the duct  heat flux  having uniform wall thickness  Same as those  Constant axial wall heat flux  for(hT) with  finite normal  with finite normal wall  wall thermal resistance and negligible  thermal resistance  peripheral wall heat conduction  Electric resistance heating, nuclear heating,  Constant axial wall heat flux with finite peripheral wall  gas turbine regenerator, counterflow heat  heat conduction  exchanger with C m i „ . C m a < -  1. all with  negligible normal wall thermal resistance  ©  Exponential axial wall heat flux  Parallel and counterflow heat exchaneers  Constant axial wall to fluid  G a s turbine reeenerator  bulk temperature difference  Table  2.1 T h e r m a l  developing  boundary  conditions  flows through  singly  f o rdeveloped a  connected  ducts  ( i ) A t h i c k and  (ii)  FIG  2.1  Two  A thin  and  conductive  wall  non-conductive  e x t r e m e c a s e s of t h e r m a l  wall  boundary  conditions  67  f(4>)  -to  *o  FIG  2.2  2TT  Peripheral  distribution  of w a l l  heat  flux  q(x)  Qsol  x  FIG  2.3  i  +  x  *  Axial distribution  3Xj  of w a l l  heat  flux  +  ^x7  68  FIG  3.2  Spot  welded tube w i t h 2 s p o t s 60%  of  i t s length  occupying  69  Lf)  1000X/(a.Pe)  FIG.  3.3  wall temperature of c o n t i n u o u s l y Dimensionless with spot angle 45° versus welded tube a x i a l d i s t a n c e 1000x=1OOOX/a•Pe. dimensionless  70  FIG.  3.4.  Local Nusselt n o . Nu a n d p e r i p h e r a l a v e r a g e Nusselt n o . Nup of continuously welded tube versus dimensionless axial distance l000x=l000X/a«Pe.  71  lOOOXAa.Pe)  FIG.  3.5.  Dimensionless wall temperature versus dimensionless axial distance 1000X lOOOX/a.Pe of a spot welded tube w i t h 2 s p o t s o c c u p y i n g 60% o f i t s length, spot angle 45°.  72  CD C\J  _,  CD ZD  CO •  zz. OJ  cn  ro t_  CM' —  OJ  >  cn —i ro !_ OJ  J= tZL  CD OD  — 1  S_  aj  Q_  CD  0.0  6.0  12.0  18.0  24.0  1000X/a.Pe  FIG.  30.0  r  36.0  I ©1 42.0  3.6(i). P e r i p h e r a l average Nusselt no. Nup versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=1OOOX/a«Pe for a spot-welded tube with a s i n g l e spot occupying 60% of i t s l e n g t h , s p o t a n g l e 4 5 ° .  1 48.0  73  o * CD __ CM  CD  IQOuX/ia.Pe)  FIG.  3.6(ii). P e r i p h e r a l average N u s s e l t no. N u versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=1OOOX/a•Pe f o r a spot-welded t u b e w i t h 4 s p o t s o c c u p y i n g 60% o f i t s l e n g t h , spot angle 45°. p  74  FIG  4.1 G r i d  division  of  tube volume.  75  FIG  4.2  finite  Node p o s i t i o n s difference  appeared  energy  in  equation  76  0.0  4.0  8.0  12.0  16.0  20.0  24.0  28.0  32.0  36.0  40.0  1000X/a.Pe  FIG.  4.3. D i m e n s i o n l e s s t e m p e r a t u r e t of a u n i f o r m l y heated tube versus dimensionless axial distance lOOOx = lOOOX/a-Pe at various radial distances : finite difference results.  44.0  48.0  77  FIG.  4.4. L o c a l N u s s e l t no. Nu o f a u n i f o r m l y h e a t e d tube v e r s u s d i m e n s i o n l e s s axial distance lOOOx = lOOOX/a-Pe : finite difference results.  FIG.  4.5.  Angular p o s i t i o n s represented c u r v e s i n F i g u r e s 4.6 - 4.9.  by  various  79  CD LT)  ioocx/(a.Pe)  FIG.  4 . 6 . Dimensionless wall temperature versus dimensionless axial distance 1000X l000X/a«Pe : f i n i t e difference results for a c o n t i n u o u s l y welded tube w i t h spot a n g l e 4 5 ° .  80  CD  rsi  a. frj-  loooX/(a.Pe)  FIG.  4.7.  Dimensionless wall temperature versus dimensionless axial distance 1000X = l000X/a-Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube w i t h 2 s p o t s occupying 60% o f i t s l e n g t h , s p o t a n g l e 4 5 ° .  81  CD  FIG.  4.8.  Dimensionless temperature inside t h e tube a t radial c o - o r d i n a t e r=4/5 v e r s u s dimensionless axial d i s t a n c e 1000x=1000X/a»Pe f o r a s p o t - w e l d e d tube w i t h 2 spots occupying 60% o f i t s l e n g t h , s p o t a n g l e 45°.  82  CD  CM ro CD'  QJ  1000X/(a.Pe)  FIG.  4.9.  D i m e n s i o n l e s s temperature i n s i d e the tube at radial co-ordinate r=l/5 versus dimensionless axial distance 1000X = l000X/a«Pe f o r a s p o t - w e l d e d t u b e with 2 spots occupying 60% of i t s l e n g t h , spot angle 45°.  "1 0.0  1  1 5.33  FIG.  1  1 10.66  1  1 16.0  1  1 21.33  1  1—•—1 26.66  lOOOX/ia.Pe)  1 32.0  1  1 37.33  1  1— 42.66  4 . 1 0 ( i ) . P e r i p h e r a l average Nusselt no. Nup versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x = 1000X/a«Pe : f i n i t e difference results for a spot-welded tube with a s i n g l e spot o c c u p y i n g 60% of i t s l e n g t h , spot angle  45°.  84  FIG.  4.lO(ii). Peripheral a v e r a g e N u s s e l t Nup versus dimensionless axial d i s t a n c e 1000x=1DOOX/a•Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube with 4 spots occupying 60% o f i t s l e n g t h , s p o t a n g l e 45°.  85  FIG  5.1  Bond  temperature  T.(X) approximated  by  T(a,0,X)  86  CD  l£> — i  FIG.  5.2. N u s s e l t no. b a s e d on bond t e m p e r a t u r e N u versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=1OOOX/a•Pe f o r a continuously welded tube of v a r i o u s h a l f - s p o t a n g l e s 4> D  0  87  Cxi  FIG.  5.3.  Nusselt no. b a s e d on bond t e m p e r a t u r e Nu^ and mean N u s s e l t n o . b a s e d on bond temperature Nu^ versus dimensionless axial distance 1000x=l000X/a*Pe f o r a spot-welded tube with 2 s p o t s o c c u p y i n g 60% of i t s l e n g t h , s p o t a n g l e 3 6 ° .  88  lOOOX/a.Pe  FIG.  5.4. Mean N u s s e l t n o . b a s e d on bond temperature Nu versus dimensionless axial distance l000x=l000X/a«Pe f o r a spot-welded tube with 8 s p o t s o c c u p y i n g 60% o f i t s l e n g t h , s p o t a n g l e 3 6 ° . D m  89  FIG.  5.5. Mean N u s s e l t n o . b a s e d on bond temperature Nu over a d i m e n s i o n l e s s tube l e n g t h o f .088 as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a spot-welded tube w i t h s p o t a n g l e of 3 6 ° .  b r n  90  CD  ro  No. of S t i t c h e s , N  . 5.6.  E f f i c i e n c y F a c t o r F' a s a f u n c t i o n of s p o t configurations for a collector with distance W between i t s s p o t - w e l d e d tubes o f ,15m and s p o t a n g l e o f 3 6 ° .  0.0  4.0  8.0  12.0  16.0  No.  FIG.  6.1.  20.0  of  24.0  Stitches. N  28.0  32.0  E f f i c i e n c y F a c t o r F* a s a f u n c t i o n o f s p o t configurations for a collector with distance W between i t s s p o t - w e l d e d tubes o f ,1m and s p o t a n g l e of 3 6 ° .  92  . 6.2.  E f f i c i e n c y F a c t o r F' a s a f u n c t i o n o f s p o t configurations for a collector with d i s t a n c e W between i t s spot-welded tubes o f ,2m and s p o t a n g l e o f 36°.  APPENDIX Specifications  Collector Tube Cross  inner  radius,  a =  s e c t i o n a l area  Collector Absorber  L =  area plate  Tube w a l l Distance  a T y p i c a l Two-cover S o l a r  Dimensions  Tube l e n g t h ,  Fluid  of  4.5x1U" m 3  of  tube, A =  = 2m  x  thickness,  7 =  Thermal c o n d u c t i v i t y , Thermal d i f f u s i v i t y ,  a =  Other  5xl0  Pr  =  _ < t  m  0.15m  = v/a  4.174  kJ/kg°C  1.561x10  m /s  -7  2  3  5.62x10""kg/ms  viscosity, v =  no.,  2.54xl0-"m  k = 0.644 W / m ° C  Dynamic v i s c o s i t y , n =  Prandtl  6 =  c^=  p = 988.8 kg/m  Kinematic  2  (50°C)  heat c a p a c i t y ,  Density,  _ 5  1m  thickness,  Properties  6.36xl0 m  2m  between t u b e s , W  Specific  A  =  5.68x10" m /s 7  2  3.64  Parameters  Coolant  flow  Mean f l o w  r a t e per  tube, m =  s p e e d , u = m/pA  R e y n o l d s no.  of c o o l a n t  5.55x10~ kg/s 3  = 0.088 flow,  Re  93  m/s =  1390  Collector  Peclet's Thermal  no.  of c o o l a n t  conductivity  T  = 4  = RePr  W/m °C 2  tube  = 5060 material(copper),  W/m°C  o v e r a l l heat U  Pe  o f p l a t e and  k = k = 385 s w Collector  flow,  transfer  coefficient,  APPENDIX B  Effect  Fluid the  analysis  from  the  case,  as  equation  effect  s t e p change  seen  from  tube.  Taking  at No.  in thermal  the  axial  axial  conduction  neglected  where Pe=<=°) , and  this  Pe=5000, an  small(Pe<100).  omitted  T h i s term  is  In  any  only at a s h o r t d i s t a n c e  from  into  (which  axial  was  2  is  condition,  account, than  can  be  heated  the N u s s e l t  no.  that obtained  with  to  i n c r e a s e s as  distance X  as  uniformly  corresponds  discrepancy  superposed  spot-welded  tube  conduction  spot  l e n g t h i s of  However, t h e  has  error  Nu(x)  distribution  to  along very  caused  the  1/5000 =  solution,  makes u s e ,  corresponding  axial  Pe=w  2  for in  of  one  x  the  case  decreases.  tube  radius  f o r the  "thin"  to  the  Let  9 t/3x  beginning.  i s lower  2  x = X/a-Pe ~  values  been a c c o u n t e d  the  boundary  conduction  f o r X<10"  Since  not  of H S U [ 2 1 ] f o r t h e  results  obtained  corresponds  has  i s obvious  Nu(x)  For  Conduction  i t s r e p r e s e n t a t i v e term  o n l y when P e c l e t  its  Axial  conduction  energy  important  the  axial  of F l u i d  can  be  and  i . e . Nu(x), the e n t i r e  s m a l l x,  concerns,  same o r d e r  co  and  Nu  (x)  95  l e n g t h , of  effect  especially  of  fluid  when  the  insignificant.  represent  of a c o n t i n u o u s l y w e l d e d  respectively,  the  flow  of m a g n i t u d e a s t h e r a d i u s .  shown t o be  Nu (x)  2x10-".  and  the  tube Nu  (x)  Nusselt  with  Pe<°°  represent  no. and the  96  Nusselt  no. d i s t r i b u t i o n  Pe=<» r e s p e c t i v e l y .  o f a s p o t - w e l d e d t u b e w i t h Pe<°° and  Also l e t  1/Nu(x) - 1/Nu ( x ) = e ( x ) , then  1 / N u ( x ) = 1/Nu(x) - H ( x - x ^ ) / N u ( x - x ^ )  +  s  H(x-x^-x )/Nu(x-x -x ) s  =  1  - ....  s  1/Nu ( x ) + e ( x ) CO  - H(x-x ) [ 1 / N U O O ( X - X 1 )  + e(x-x )]  1  1  +  H(x-x,-x)[1/Nu(x~x,-x ) + e ( x - x . - x ) ] - ... i s i s i s =  1 / N U S ( X ) + e(x) - e(x-x,)H(x-x,) + co  X  e(x-x -x )H(x-x -x ) 1  Thus,  the error =  s  1  1/Nu (x) -  I  - ....  s  l/Nu^U)  s  e ( x ) - e ( x - x , ) H ( x - x , ) + e ( x - x , - x ) H ( x ~ x , - x ) - .... 1 1 I s I s  When x i s s m a l l ,  e(x) i s small  since  b o t h Nu ( x ) a n d  Nu ( x ) CO  are  l a r g e . When x i s l a r g e  e(x) approaches the  error  so t h a t  superposed  is  no  approaches  accumulating  effect  error  Nu^ a n d  terms  on e a c h  involved  in  other, i n the  solution. the discrepancy  not only a t small  However, t h e l a r g e renders  Nu  the i n d i v i d u a l  have a c a n c e l l a t i o n  In c o n c l u s i o n , exists  2  zero. Furthermore,  series  there  (X>10~ ),  that  Peclet  discrepancy  between Nu  and  x b u t a l o n g t h e whole t u b e no. o f 5060 i n negligible.  the  Nu (x) o  length.  present  case  APPENDIX C Expression  The  stress  o f S( V V ) and $ i n C a r t e s i a n  tensor  :  / S =  s s  xx  s xy  s \ xz \  yx  s yy  s yz  s  s  \ s \ zx  where s ^ on  x  zy  denotes the s t r e s s  the surface  The V e l o c i t y  /, z z /'  acting  along  the  x  direction  normal t o t h e y - a x i s .  Deformation Tensor :  V V =  9u/9x  9u/9y  9u/9z^  9v/9x  9v/9y  9v/9z  9w/9x  9w/9y  9w/9z  where u, v , w a r e t h e x-, y - , z-component, the  Coordinates  velocity V .  97  respectively,  of  The C o m p l e t e C o n t r a c t i o n :  S(  V V ) =  + s  x  x  3u/3x  + s  y  x  3u/3y + s  z  x  3u/3z  + s  x  y  3v/3x + s  y  y  3v/3y + s  z  y  3v/3z  + s  The D i s s i p a t i o n  *  =  Function  (2/3)(  3w/3z  :  2 [(3u/3x) -  3w/3x + s„_ 3w/3y + s „  2  + (9v/3y)  V • V )  2  + ( 3v/3z + 3w/3y  2  + (3w/3z)  2  ]  + ( 3u/3y + 3v/3x ) )  2  2  + ( 3w/3x + 3u/3z )  2  

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