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Combined free and forced convection in a horizontal tube under uniform heat flux Kupper, Arthur K. 1968

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COMBINED FREE AND FORCED CONVECTION IN A HORIZONTAL TUBE UNDER UNIFORM HEAT FLUX  by ARTHUR K. KUPPER d i p l . Ing. ETH 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc.  i n the Department of Mechanical  Engineering  We accept this thesis as conforming  to the  required standard  THE UNIVERSITY oft BRITISH COLUMBIA October 1968  In p r e s e n t i n g  this thesis in p a r t i a l  f u l f i l m e n t of  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y for reference extensive  and  study.  British available  I further agree that permission  for  c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may  g r a n t e d by t h e Head o f my  D e p a r t m e n t o r by h i s  gain  Department of  s h a l l not  A/^//.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date  /Z-Z/—&<8  be a l l o w e d w i t h o u t my  ^AJ<6,  Columbia  be  representatives.  It i s u n d e r s t o o d that c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s financial  the  written  for  permission  ii  ABSTRACT This t h e s i s presents experimental r e s u l t s of combined f r e e a i d forced convection laminar heat t r a n s f e r f o r water flowing through a c i r c u l a r h o r i z o n t a l tube w i t h uniform w a l l heat f l u x .  The Reynolds number  ranged from 100 to 2000, and changes i n heat t r a n s f e r rate allowed a v a r i ation, of Grashof number from 300 to 30,000. from 4 to 9.  The P r a n d t l number ranged  The e f f e c t of secondary flow created by free convection  occurring at higher Grashof number i n d i c a t e s an increase i n Nusselt number up t o 200 per cent. l a t i o n s are given. Nu  =  For the fully-developed region two t e n t a t i v e correThe expression  48/11 + 0.047 P r / 1  3  (Re R a ) / 1  5  c o r r e l a t e s 53 per cent of the data to w i t h i n ± 10 per cent.  Another  s l i g h t l y more accurate expression which c o r r e l a t e s 68 per cent of the data to w i t h i n ± 10 per cent, but does not s a t i s f y the pure forced convection, i s Nu  =  2.41 + 0.082 P r / 1  3  (Re R a ) / 1  5  ACKNOWLEDGEMENTS I wish to express my deep gratitude to Dr. M. Iqbal and to Dr. E.G. Hauptmann f o r t h e i r guidance and advice throughout my  graduate s t u d i e s .  A l s o , I wish to thank the members of the Mechanical E n g i n e e r i n g Department s t a f f f o r t h e i r assistance and the Department f o r the use of i t s facilities.  The f i n a n c i a l assistance given by the U n i v e r s i t y  of B r i t i s h  Columiba and by the National Research Council of Canada i s g r a t e f u l l y acknowledged.  iv  TABLE OF CONTENTS  ABSTRACT  i i  ACKNOWLEDGEMENT  iii •  LIST OF FIGURES  vi  NOMENCLATURE  viii  INTRODUCTION  1  LITERATURE SURVEY  4  Uniform Wall Temperature  5  Uniform Wall Heat Flux  9  PURPOSE OF EXPERIMENT  19  DESCRIPTION OF EXPERIMENTAL EQUIPMENT  19  Flow Loop  19  Test Section  21  Test Section I n s u l a t i o n  25  Power Supply  27  Temperature Measurement  27  EXPERIMENTAL PROCEDURE  .29  Test Procedure  29  Thermocouple C a l i b r a t i o n Method  29  C a l i b r a t i o n Result  30  RESULTS AND DISCUSSION  31  Entrance Length and C i r c u m f e r e n t i a l Temperature V a r i a t i o n  32  A x i a l Temperature P r o f i l e s  35  V  Contents continued  Local Nusselt  Number  Heat Transfer Correlation  . 38 47  CONCLUSION  51  REFERENCES  52  APPENDIX A  DERIVATION OF GOVERNING EQUATIONS  55  APPENDIX B  LOCATION OF TEST SECTION THERMOCOUPLES  56  APPENDIX C  SAMPLE CALCULATION  58  APPENDIX D  ERROR ANALYSIS  62  APPENDIX E  TABULATED LOCAL HEAT TRANSFER DATA  65  LIST OF FIGURES  la  V e l o c i t y P r o f i l e s f o r Heatea upward Flow  lb  Motion of F l u i d P a r t i c l e s under Buoyancy E f f e c t s  2  Coordinate System  3  Temperature V a r i a t i o n i n the A x i a l D i r e c t i o n  4  Reproduction of Figure 1 from Reference 14  5  Reproduction of Figure 3 from Reference 14  6  Reproduction of Figure 4 from Reference 14  7  Reproduction of Figure 9 from Reference 16  8  Reproduction of Figure 6 from Reference 18  9  Schematic of Heat Transfer Loop  10  P a r t i a l View of Experimental Arrangement  11  View of Test Section  12  Thermocouple P l a c i n g on Test Section Circumference  13  Schematic of Power Supply  14  Thermocouple C i r c u i t  15  Representative Temperature P r o f i l e , Run Number 3  16  Representative Temperature P r o f i l e , Run Number 42  17  Representative Temperature P r o f i l e , Run Number 1  18  Representative Temperature P r o f i l e , Run Number 43  19 •  Representative Temperature P r o f i l e , Run Number 37  vii  L i s t of Figures continued 20  Representative Temperature P r o f i l e , Run Number 27  40  21  Representative Nusselt Number P r o f i l e , Run Number 3  41  22  Representative Nusselt Number P r o f i l e , Run Number 37  43  23  Representative Nusselt Number P r o f i l e , Run Number 1  44  24  Representative Nusselt Number P r o f i l e , Run Number 43  45  25  Representative Nusselt Number P r o f i l e s , Run Numbers 1 and 43  46  26  C o r r e l a t i o n of Nusselt Number, Equation 17  48  27  C o r r e l a t i o n of Nusselt Number, Equation 18  49  viii  NOMENCLATURE A  A x i a l temperature gradient  a  Tube radius  C  A x i a l pressure gradient i n f l u i d  c  S p e c i f i c heat at constant pressure  P  D  Tube diameter  E  Voltage 8 E AT D  3  Gr  — —5  Gz  -r- Re Pr -7^-  g  A c c e l e r a t i o n of g r a v i t y  h  T~^T ' w  k  Thermal c o n d u c t i v i t y  L  C h a r a c t e r i s t i c length  m  Mass flow r a t e  Nu  . Grashof number  6  4  n e a t  k  , Graetz number  L2  t r a n s r e r  coefficient  • Nusselt number  P  Power input  Pr  — f — , P r a n d t l number k  q  Wall heat f l u x density  R  Dimensionless r a d i a l coordinate  ix  Nomenclature continued  r  R a d i a l coordinate  Ra.  Gr Pr , Rayleigh number  Re  "  T  F l u i d bulk temperature  T  w  , Reynolds number  Tube temperature  T*  Dimensionless temperature  u  Average v e l o c i t y Dimensionless a x i a l v e l o c i t y  v^  Velocity i n r a d i a l direction  VQ  V e l o c i t y i n angular d i r e c t i o n  X  Coordinate i n a x i a l d i r e c t i o n  a  Tube i n c l i n a t i o n with respect to the h o r i z o n t a l  3  Expansion c o e f f i c i e n t  AT  Temperature d i f f e r e n c e ^ f l u i d bulk to w a l l  •V  Laplacian  8  Angular coordinate  V  Dynamic v i s c o s i t y of f l u i d  V  Kinematic v i s c o s i t y of f l u i d  p  Density of f l u i d  i|)  Stokes stream f u n c t i o n  position  1  INTRODUCTION The rate of heat t r a n s f e r between a s o l i d surface and a f l u i d depend on both f r e e and forced convection mechanisms.  may  Forced convection  occurs when f l u i d motion i s produced by a pump or s i m i l a r means.  In free  or n a t u r a l convection, f l u i d motion occurs due to buoyancy forces produced from temperature (or e q u i v a l e n t l y density) d i f f e r e n c e s a r i s i n g from the heat t r a n s f e r s i t u a t i o n i t s e l f .  For laminar heat t r a n s f e r asso-  c i a t e d w i t h a c i r c u l a r tube, the buoyancy forces are characterized by the dimensionless Grashof number, and the forced flow i s characterized by the Reynolds number.  In most p h y s i c a l s i t u a t i o n s , both modes of heat t r a n s f e r  are present, and t h e i r r e l a t i v e magnitude determines whether a flow i s considered as f o r c e d , f r e e , or combined f r e e and forced convection.  The  present i n v e s t i g a t i o n o r i g i n a t e s from study of flow and heat t r a n s f e r through tubes of f l a t p l a t e s o l a r c o l l e c t o r s , where both free and forced convection mechanisms are important. In p r a c t i c e , pure forced convection i s never exactly r e a l i z e d , and use of the term merely implies that free convection e f f e c t s are n e g l i g i b l e . Where n a t u r a l convection e f f e c t s are not n e g l i g i b l e , the s o l i d surface o r i e n t a t i o n w i t h respect to the d i r e c t i o n of g r a v i t a t i o n a l force also becomes important. For heating of upward flow i n v e r t i c a l tubes, free convection aids the forced upward motion of f l u i d near the w a l l .  This creates steeper  v e l o c i t y and temperature gradients near the w a l l and hence r e s u l t s i n a higher heat t r a n s f e r c o e f f i c i e n t .  To s a t i s f y c o n t i n u i t y , the increased  v e l o c i t y near the w a l l r e s u l t s i n flow r e t a r d a t i o n at the tube centre,  2  and i n extreme cases a reversal of flow at the tube centre may even occur (Fig. l a ) .  Free convection opposes the forced flow during heating of f l u i d s flowing downward, and a decrease i n heat transfer rate r e s u l t s .  In the case of a horizontal tube, the free and forced convectlve motions are perpendicular to one another.  For pure forced flow the v e l o c i -  ty p r o f i l e i s symmetrical with respect to the tube axis.  However, when  the tube wall i s heated, the f l u i d p a r t i c l e s receive heat from the w a l l , forming a temperature gradient through the f l u i d .  The f l u i d p a r t i c l e s  near the w a l l have a lower density ( i n most f l u i d s ) than those i n the centre of the tube, and motion i s created because of gravity forces. The f l u i d p a r t i c l e s near the wall r i s e upwards, while those i n the centre move downwards.  Superimposing t h i s motion on forced flow, the f l u i d par-  t i c l e s develop s p i r a l t r a j e c t o r i e s , ascending near the wall toward the top  and descending i n the centre as they move through the tube (Fig. l b ) .  There are many possible variations of the problem stated. may be cooled or heated from the tube wall i n d i f f e r e n t ways. t i o n , i n t e r n a l heat sources may also be present.  The f l u i d In addi-  The two most i n t e r e s t -  ing and p r a c t i c a l cases of combined free and forced convection i n h o r i zontal tubes occur when either the wall temperature or wall heat f l u x are approximately uniform. case.  The present i n v e s t i g a t i o n deals with the l a t t e r  LITERATURE SURVEY  Forced convection i s independent of tube o r i e n t a t i o n and has been extensively investigated. The influence of free convection on forced flow i n h o r i z o n t a l tubes has been given soraev?hat less a t t e n t i o n . Five equations govern the temperature and v e l o c i t y f i e l d s i n t h i s problem. The continuity equation, the momentum equations (one f o r each of the three v e l o c i t y components) and the energy equation.  Using c y l i n d r i c a l  coordinates, as outlined i n F i g . 2,. and nondimensional  forms Iqbal and  Stachiewicz [12]* obtained f o r angles a other than 90°,  ' X + i ' l f e - f f!> V R  3R 36  36 3R  J  x  +  We  - RaT* s i n „ -  0  x  The derivation of the above equations i s explained i n Appendix A.  FIGURE 2  Coordinate System  l« * Numbers i n square brackets refer to references.  5  In the above equation physical properties of the f l u i d are considered constant except for density v a r i a t i o n a f f e c t i n g the buoyancy terms.  In  the energy equation, a x i a l conduction, viscous d i s s i p a t i o n and pressure work terms have been ignored.  Furthermore,  the flow i s considered hydro-  dynamically fully-developed.  Two boundary conditions for the temperature encountered  at the wall are usually  i n most p r a c t i c a l instances of laminar heat transfer i n c i r -  cular horizontal tubes:  uniform w a l l temperature  (such as a r i s i n g from  heating with a condensing vapor), and uniform wall heat flux (such as a r i s i n g from heating with e l e c t r i c currents). tions produce a x i a l temperature  a)  These two boundary condi-  variations as shown i n F i g . 3.  Uniform Wall Temperature  Apparently no t h e o r e t i c a l study has yet been made for combined free and forced convection i n horizontal tubes with constant wall  temperature.  There are, however, several reports regarding experimental investigations.  Colburn [1] has shown that natural convection may increase the rate of heat transfer by a factor of three or four.  He presented the r e l a -  tionship  Nu = 1.75 G z / am 1  3  (1 + 0.015 Gr */ ) ( u . / u j / i b r 3  1  3  (4)  where Nu i s the Nusselt number with f l u i d properties based on the a r i t h am r  metic mean temperature  r  difference between f l u i d and tube (the indices " f "  and "b" refer to f i l m and bulk temperature factor (1 + 0.015 G r , / ) (u, / u ^ ) ^ 1  3  1  3  respectively).  The correction  allows for both natural convection  D I S T A N C E A L O N G T U B E (3A)  W A L LT E M P E R A T U R E  D I S T A N C E A L O N G T U B E (3B) F I G U R E 3 5A 3B  T E M P E R A T U R E V A R I A T I O N IN T H E AXIAL D I R E C T I O N C O N S T A N T D A L L M E A T F L U X C O N S T A N T W A L L T E M P E R A T U R E  7  and r a d i a l v a r i a t i o n of f l u i d p r o p e r t i e s . Sleder and Tate [2] suggested that the v i s c o s i t y v a r i a t i o n i n equat i o n (4) might be replaced by  )  0 , 1 1  * , where u  b w  denotes the f l u i d v i s -  w  c o s i t y evaluated at the w a l l temperature.  They a l s o suggested the Gras-  hof number should be evaluated at the mean f l u i d temperature rather than f i l m temperature.  However, these changes throw no f u r t h e r l i g h t on the  n a t u r a l convection process i t s e l f . From t h e i r extensive experimental data, Kern and Othmer [3] n o t i c e d that Sieder and Tate's [2] equation p r e d i c t e d high heat t r a n s f e r c o e f f i c i e n t s at high Reynolds numbers, and the opposite at low Reynolds numbers. For  t h i s reason they introduced an a r b i t r a r y f u n c t i o n ( l n  1 0  Re)  - 1  i n the  n a t u r a l convection term, obtaining the equation Nu  am  = 1.86  (f) u  {  (-Sf) K.  A  L,  }  Kern and Othmer obtained markedly b e t t e r c o r r e l a t i o n of t h e i r r e s u l t s w i t h t h i s equation, than by using the preceeding equations. M a r t i n e l l i et a l . [4] presented a t h e o r e t i c a l approach to the problem of combined forced and free convection. The a n a l y s i s applied to the cases of heating of f l u i d s flowing v e r t i c a l l y upwards, and c o o l i n g of f l u i d s flowing downwards i n tubes with uniform w a l l temperature.  They  obtained the equation Nu  = 1.75 am  Ei 1  { Gz  + 0.0722 F m  2  (• N  The c o r r e c t i o n f a c t o r F i allows f o r use of a r i t h m e t i c rather than loga-  (6)  8  r i t h m i c mean temperature d i f f e r e n c e s , w h i l e the c o r r e c t i o n allows f o r a reduction erature  i n free convection forces  as the f l u i d  [5] m o d i f i e d e q u a t i o n  Nu  i n horizontal  (6) t o t h e form Gr  = 1.75 . Gz + 12.6 , am { m (  m  Pr  m  D •  \  0  „ 0.14  1/3  1 + 0  , V (—) w  . }  )  , (7)  W  Grashof number i s now based on mean temperature d i f f e r e n c e between  the w a l l bulk  and f l u i d ,  and the p r o p e r t i e s  of t h e f l u i d  a r e based on the mean  temperature.  McAdams  [6] recommended an e q u a t i o n s i m i l a r t o t h a t  Eubank and P r o c t o r placed  using  p r e s e n t e d by  [ 5 ] , except w i t h t h e c o e f f i c i e n t s 12.6 and 0.4 r e -  by 0.04 and 0.75 r e s p e c t i v e l y .  J a c k s o n , S p u r l o c k , and Purdy  [7] have s t u d i e d  a i r i n a c o n s t a n t temperature h o r i z o n t a l  w e l l r e p r e s e n t e d by the s e m i - t h e o r e t i c a l  Nu, 1  The  b u l k temp-  tubes, Eubank and Proc-  L  The  2  rises.  From d a t a f o r h e a t i n g o f o i l s tor  factor F  = 2.67 { Gz m m  subscript  logarithmic  Oliver latively  2  + 0.0087  "lm" denotes t h a t  mean temperature  [8] r e p o r t e d  (Gr  2  tube.  Their  rates  r e s u l t s are  equation  Pr ) w w  T  -  5  } / 1  (8)  6  the heat t r a n s f e r c o e f f i c i e n t  i s based on  difference.  an i n v e s t i g a t i o n o f the same problem u s i n g r e -  non-viscous l i q u i d s .  He p r e s e n t e d  the f o l l o w i n g T  Nu  heat t r a n s f e r  = 1.75 { Gz + 5.6 • IOT * (Gr P r am m m  D  „ ,,  1/3  0 . 7  }  1  relationship  (u / \ i w b  j ^ '  l  k  (9)  I t should be noted that the r a t i o L/D rather than D/L i s used.  The rea-  son i s that when the Grashof number i s m u l t i p l i e d by the D/L r a t i o , a term D* occurs, w i t h the r e s u l t that a small v a r i a t i o n i n D produces 1  large v a r i a t i o n s i n (Gr Pr 5).  This v a r i a t i o n i s not r e f l e c t e d by corre-  sponding changes i n heat t r a n s f e r c o e f f i c i e n t s . Brown and Thomas [9] conducted a study of combined f r e e and forced convection heat t r a n s f e r i n h o r i z o n t a l tubes, and proposed the r e l a t i o n Nu ( u / u , )°' * lL  W D  = 1.75 { Gz + 0.012  (Gz G r / ) / 1  3  4  3  }  (10)  l / 3  However, t h e i r r e s u l t s do not agree with e x i s t i n g c o r r e l a t i o n s . M a r t i n and Carmichael [10] have also reported r e s u l t s f o r l o c a l heat t r a n s f e r c o e f f i c i e n t s f o r t h i s problem, using water as the heat t r a n s f e r medium.  b)  Uniform W a l l Heat Flux Very few t h e o r e t i c a l analyses are a v a i l a b l e f o r combined free and  forced convection i n a h o r i z o n t a l tube under uniform heat f l u x . Morton [11] analysed the problem by s o l v i n g equations (1) to (3) by a p e r t u r b a t i o n method, using the Rayleigh number as a perturbation parameter.  The Nusselt number based on the mean temperature was evaluated  from the r e s u l t i n g d i s t r i b u t i o n s and was expressed by the f o l l o w i n g asympt o t i c expression v a l i d f o r small values of Ra Re and Pr Nu = 6 { 1 + (0.0586 - 0.0852 Pr +0.2686 P r ) ( f f o f ^ ) 2  The mean temperature was defined as T  = / T dA / A.  2  + .....}  (11)  No n u m e r i c a l ' l i m i t s  10  of Rayleigh number were given f o r the region of v a l i d i t y of the r e s u l t s . I q b a l and Stachiewicz  [12] analyzed the problem i n c l u d i n g tht e f f e c t  of tube o r i e n t a t i o n . They a l s o solved equation (1) - (3) by a perturbat i o n method using the Rayleigh number as a perturbation parameter.  How-  ever, the Nusselt number was based on the bulk temperature, defined as T  b  - / T v d A / / v d A  (12)  They have shown that f o r h o r i z o n t a l tubes, the Nusselt number based on equation  (12) i s a f u n c t i o n of the product RaRe as w e l l as P r , while f o r  flow through i n c l i n e d tubes Nu = f (Re, Ra, P r ) .  I t i s known [20] that  for flow through v e r t i c a l tubes, Nu * f (Ra) only. Another t h e o r e t i c a l i n v e s t i g a t i o n of combined f r e e and forced conv e c t i o h i n h o r i z o n t a l tubes w i t h uniform heat f l u x was Casal and G i l l [13].  In t h e i r a n a l y s i s they assumed the f l u i d p r o p e r t i e s  constant, except that density was ing equations.  reported by Del  allowed to vary throughout the govern-  The governing equations were a l s o solved by p e r t u r b a t i o n  a n a l y s i s , the r a t i o Gr/Re being used as perturbation parameter. 2  Del Casal  and G i l l s t a t e that the Froude number, c h a r a c t e r i z i n g density d i f f e r e n c e s , a l s o enters the problem. Nusselt number was  However, no s o l u t i o n f o r the energy equation or  given.  There are few experimental  i n v e s t i g a t i o n s a v a i l a b l e i n the l i t e r a -  t u r e f o r the case of combined free and forced convection i n h o r i z o n t a l tubes under uniform heat f l u x . Ede  [14] c a r r i e d out an experiment using seven d i f f e r e n t tube dian-  11  e t e r s , w i t h both a i r and water used as the heat t r a n s f e r medium.  Rey-  nolds numbers were v a r i e d from 300 to 100,000, while Grashof number vari e d up vo 1 0 .  The P r a n d t l number was approximately 0.7 f o r a i r , and  7  v a r i e d from 4 - 1 2  f o r water.  Ede p l o t t e d h i s r e s u l t s i n three d i f f e r e n t  ways, and h i s f i g u r e s are reproduced i n F i g s . 4 to 6.  Unfortunately, no  mention of the values of Grashof number employed at various Reynolds numbers was made.  Ede a l s o presented an e m p i r i c a l c o r r e l a t i o n f o r the Nus-  s e l t number as  „  Nu = 4.36 (1 + 0.06 G r ' ) 0  (13)  3  Considering the t h e o r e t i c a l studies of combined free and forced convect i o n i n h o r i z o n t a l tubes, one notices the lack of parameters such as P r a n d t l number and Reynolds number i n Ede's c o r r e l a t i o n . McComas and Eckert [15] reported an experimental study on combined f r e e and forced convection heat t r a n s f e r under uniform w a l l heat f l u x i n a h o r i z o n t a l tube.  A i r was used as heat t r a n s f e r medium.  The Grashof num-  ber was v a r i e d by using a i r at d i f f e r e n t pressures. I t ranged from 1 to 1000.  McComas and Eckert d i d not present any working c o r r e l a t i o n , on the  grounds that i n s u f f i c i e n t experimental data was a v a i l a b l e f o r t h i s s i t u a tion. Recently, a report was published by Mori et a l . [16] on free and forced convection i n h o r i z o n t a l tubes with constant w a l l heat f l u x , using a i r as a heat t r a n s f e r medium.  Measurements of temperature and v e l o c i t y  d i s t r i b u t i o n at c e r t a i n cross sections of the t e s t tube were made, and the l o c a l Nusselt number was c a l c u l a t e d from l o c a l w a l l temperature.  The  12  lOOOr  AIR o  WATER .  IOO  , i<\ k  if 10  IOO  F I G U R E 4  iOOO  lOOOO  lOOOOO  R E P R O D U C T I O N OF F I G U R E 1F R O M R E F E R E N C E 14  13  I Nu  i j  40-  I j  —  :  Gr «= IO  S  30  v.' .  i 20  j |  i •: • •  IO 8 7 6  • *  s  • • • *  i  •  •  .  «  • "  ' '  i• ! i ' i  5  • ii  •  1  6  7  i  !  S  9  iO  ' i  3  3 /?<?  F I G U R E 5  i  i i —  -  i  !  I  ;  4  5  l 6  l l  l  .  R E P R O D U C T I O N O F F I G U R E 3F R O M R E F E R E N C E 14  4 C  3C  2d lOf-  - THEORY 67- • O  10  s  F I G U R E 6  IO  R E P R O D U C T I O N O F F I G U R E 4 F R O M REFERENCE 14  7  14  Reynolds number was v a r i e d from 100 to 130,000.  The Nusselt number cor-  r e l a t i o n was expressed by the r e l a t i o n Nu = 0.61 (ReRa) / 1  5  1.8 (Re Ra)T7T >  { 1 +  which represents curve 2 of F i g . 7.  (14)  I t remains t o be explained why t h i s  form of c o r r e l a t i o n was chosen, since one n o t i c e s that equation (14) i s equivalent to Nu = 1.098 + 0.61 (Re R a ) / 1  (14a)  5  Although Morton's a n a l y s i s shows that Nusselt number i s a f u n c t i o n of the RaRe product, as w e l l as P r , Mori's equation does not contain the P r a n d t l number e x p l i c i t l y .  The reason f o r t h i s appears t o be due to the f a c t  that Mori used only a i r (Pr=l) i n h i s experimental i n v e s t i g a t i o n and therefore no c o r r e l a t i o n w i t h Pr could be attempted. I n a second r e p o r t , Mori and Futagami [17] present a t h e o r e t i c a l i n v e s t i g a t i o n of the same problem.  In t h e i r a n a l y s i s they d i v i d e the flow  i n the tube i n t o a t h i n l a y e r along the tube w a l l and a core region. In the t h i n l a y e r , the v e l o c i t y and temperature f i e l d s are a f f e c t e d by v i s c o s i t y and thermal c o n d u c t i v i t y .  On the other hand, i n the core region,  the v e l o c i t y and temperature f i e l d s are a f f e c t e d mainly by the secondary flow and the e f f e c t s of v i s c o s i t y and thermal c o n d u c t i v i t y are disregarded. They give an approximate s o l u t i o n f o r very l a r g e products of ReRa, and expressed the Nusselt number as Nu Nu  • 0.2189 o  (15)  F I G U R E 7  R E P R O D U C T I O N O F F I G U R E 9 F R O M R E F E R E N C E 16  16  where 10?  6  m  Nu « 48/11 and x, i s given by o m + 5c  m  P r - (25 Pr + 4)c  3  2  m  Pr + (20 Pr + I k  Pr - 5 P r = 0 m 2  Very r e c e n t l y Shannon and Depew [18] published an experimental i n v e s t i g a t i o n of flow through a h o r i z o n t a l c i r c u l a r tube w i t h uniform w a l l heat f l u x .  Water near the i c e p o i n t was used as heat t r a n s f e r medium.  The Reynolds number was v a r i e d from 120 to 2300, and the Grashof number ranged up to 2.5 • 1 0 . 5  No c o r r e l a t i o n formula was given, but they  p l o t t e d t h e i r data as (Nu - Nu„ ) versus (Gr P r ) / 1  4  / Nu„  GZ  Fig.  as shown i n  GZ  8. Nu i s the Nusselt number measured i n t h e i r experiment, and Nu„ Gz  i s a f u n c t i o n of Graetz number evaluated from Siegel's [19] s o l u t i o n . Their d i f f e r e n c e represents the p o r t i o n of Nusselt number which i s due to f r e e convection. I t was f e l t that the n a t u r a l convection i s dependent on the term (Gr P r ) / * o n l y , as suggested by M i k e s e l l [21]. 1  1  In the entrance r e g i o n , the heat t r a n s f e r r a t e depends on the r a t i o L/D, which can be taken i n t o account by i n t r o d u c i n g the Graetz number. For the f u l l y developed flow region however, the heat t r a n s f e r c o e f f i c i e n t w i l l be independent of the tube l e n g t h , and hence the Nusselt number w i l l not be a f u n c t i o n of the Graetz number. Table 1 gives a summary of the region of dimensionless parameters covered by experimental data reported by references [14] - [18]. contains the region covered by the present i n v e s t i g a t i o n .  I t also  0  1  1  1 2  1 3.  I  4  (GrPr) >  F I G U R E 8  N  u  I 5  1 / 4  Gz  R E P R O D U C T I O N O F F I G U R E 6 F R O M R E F E R E N C E 18  Re  Ede  McComas &  Gr  Pr  300 - 100,000  Eckert  Fluid  A i r / Water  100  900  1 -  1,000  0.75  Air  1,890  1,450  33 -  .57  0.75  Air  Shannon & Depew  120  2,300  up t o 2.5.10  2 - .14  Water  Present I n v e s t i g a t i o n  100  2,000  300 - 30,000  4 - 9  Water  Mori  TABLE 1  Summary of Experimental  Investigations.  5  19  PURPOSL' OF EXPERIMENT As mentioned i n the l a s t chapter, the purpose of the present e x p e r i ment was t o o b t a i n the heat t r a n s f e r c o e f f i c i e n t , or Nusselt number, as a f u n c t i o n of other dimensionless parameters.  The apparatus was a l s o de-  signed to i n v e s t i g a t e the entrance length required f o r the flow to become fully-developed.  DESCRIPTION OF EXPERIMENTAL EQUIPMENT  Flow Loop A schematic diagram of the flow loop i s shown i n F i g . 9. t r a n s f e r medium used was d i s t i l l e d and deionized water.  The heat  A constant temp-  erature bath was used to maintain a steady i n l e t temperature.  I t also  provided a constant head of eight f e e t , assuring a constant flow r a t e . Water from the head tank flowed down the stand pipe, and temperature was measured before entering the t e s t s e c t i o n . smooth i n l e t t o the hydrodynamic t o the o u t l e t .  I t then passed through a  approach, then i n t o the t e s t s e c t i o n and  The i n l e t , tube, and o u t l e t were placed i n a vacuum tube  f o r heat i n s u l a t i o n .  The hydrodynamic  approach s e c t i o n has a length of  one foot and an outside diameter of 0.25 inch.  The length to diameter  r a t i o of 52 allowed the v e l o c i t y p r o f i l e to become fully-developed. The s i x foot long t e s t s e c t i o n was separated thermally from the hydrodynamic  approach s e c t i o n by a t e f l o n d i s k .  The hydrodynamic ap-  proach and t e s t s e c t i o n are both from the same 10/1000 inch w a l l t h i c k -  C O O L E R II  C O N S T A N T " B F E R A T U R E B A T H  C A R T R I D G E S  V A C U U M P U M P  Q  VAcuuf-1  C H A M B E R  TEST S E C T I O N F L O W  •cxi—©f'ETER  S U M P T A N K  F I G U R E 9  C O O L E R I  C I R C U L A T I N G PUIT  S C H E M A T I C O F H E A T I R A N S F E R L O O P  21  ness I n c o n e l tube.  Thermocouples a t t a c h e d  to the o u t e r w a l l enabled  t e r m i n a t i o n of the l o c a l w a l l temperature. temperature was temperature. where the  measured b e f o r e b e i n g  t e s t s e c t i o n e x i t water  c o o l e d back to approximately  I t t h e n p a s s e d through a flowmeter and  flow r a t e c o u l d be a d j u s t e d , and  flow r a t e was  The  measured by  pump c i r c u l a t e d  exchanger m a t r i x ganic p a r t i c l e s  throttling  valve,  One  cartridges i n series,  of the c a r t r i d g e s c o n t a i n e d  from the water. inlet  S i n c e the c o n s t a n t  temperature was  F i g . 10 shows p a r t o f the e x p e r i m e n t a l  set  A  or  a mixed bed  ion-  to remove o r -  temperature bath con-  maintained  by  first  cooling  the water i n a heat exchanger connected to a domestic c o l d water  Test  exact  c o l l e c t i n g water over a known time i n t e r v a l .  to d e i o n i z e the water, the o t h e r a m a t r i x  tained only a heater,  room  i n t o a sump tank.. The  the water e i t h e r through two  directly into a cooler.  de-  supply.  up.  Section  The t e r and  t e s t s e c t i o n was  an I n c o n e l tube w i t h  a 0.25  a w a l l t h i c k n e s s of 10/1000 of an i n c h .  The  i n c h o u t s i d e diamel e n g t h of 72  inches  gave a l e n g t h t o d i a m e t e r r a t i o o f about 313.  The  checked f o r u n i f o r m i t y over t h e whole l e n g t h .  An o v e r a l l i n s i d e diameter  was  a l s o measured, but  checked.  The and  low  o u t s i d e diameter  the u n i f o r m i t y of the w a l l t h i c k n e s s was  was  not  F i g . 11 shows d e t a i l s o f the t e s t s e c t i o n .  t e s t s e c t i o n was  d e s i r e d t o have h i g h e l e c t r i c a l  thermal c o n d u c t i v i t y .  Hie former was  resistivity,  to o b t a i n a h i g h e r  resist-  ance and hence lower c u r r e n t by a h i g h e r v o l t a g e drop, the l a t t e r to m i n i mize heat c o n d u c t i o n  along tube a x i s .  A N i c k e l A l l o y used f o r r e s i s t a n c e  22  T E S T SECTION  24  heating s a t i s f i e s t h i s condition best.  A t h i n tube w a l l i s used since  t h i s w i l l reduce thermal conductance and r a i s e the e l e c t r i c a l r e s i s t a n c e . An "Inconel X 750" tube was found to be most s u i t a b l e f o r t h i s . Thermocouples were mounted on the t e s t tube with small non-conducting clamps i n order to maintain good contact between the tube w a l l and wire junction.  To minimize e r r o r due to thermocouple conductance the thermo-  couple wires were wound around the tube a few times. Due to secondary flow i n the t e s t s e c t i o n , higher heat t r a n s f e r rates are expected from the lower than the upper h a l f of the tube c i r cumference.  This gives r i s e to non-uniform temperature over a circumfer-  ence at any tube s e c t i o n .  In an attempt to take t h i s i n t o account,  thermocouples were placed at three points at a tube s e c t i o n as shown i n Fig.  12.  I  Fig.  12:  Thermocouple p l a c i n g on t e s t section circumference  The a x i a l spacing of thermocouples was 12 inches i n the middle part of the tube, and l e s s near each end i n order to allow a determination of the  entrance length and to detect heat loss due to conductance along  power leads and tube w a l l .  The l o c a t i o n of the thermocouples i s given  25  i n Appendix B.  C i r c u l a r copper f l a n g e s were mounted at each end of the  t e s t s e c t i o n by a s l i g h t p r e s s f i t .  The e l e c t r i c a l c o n d u c t i v i t y was im-  proved by u s i n g c o n d u c t i v e epoxy s i l v e r  The hydrodynamic  solder.  approach s e c t i o n was t h e r m a l l y i n s u l a t e d  t e s t tube by a t e f l o n tube of one i n c h l e n g t h .  The i n l e t  from the  to the hydro-  dynamic approach had a smooth c o n v e r g i n g shape, and was a l s o made o f teflon.  At the t e s t s e c t i o n e x i t , a tube o f 10 i n c h e s i n l e n g t h of the same d i a m e t e r was used b e f o r e d i v e r g i n g t o a l a r g e r d i a m e t e r .  In the l a r g e r  s e c t i o n , a f i n e w i r e mesh was used to p r o v i d e m i x i n g .  The t e s t s e c t i o n and hydrodynamic  approach were p l a c e d on a r e d  f i b r e m a t e r i a l w i t h low t h e r m a l c o n d u c t i v i t y .  Test Section  Insulation  For t h e s m a l l h e a t t r a n s f e r c o e f f i c i e n t s encountered i n l a m i n a r f l o w , the i n s u l a t i o n o f the t e s t s e c t i o n i s v e r y i m p o r t a n t .  Hallman  [20] found  t h a t use o f a vacuum chamber was f i v e times more e f f e c t i v e i n terms of i n s u l a t i o n and a t t a i n m e n t o f s t e a d y s t a t e c o n d i t i o n s than o r d i n a r y lation material.  insu-  T h e r e f o r e t h i s d e v i c e was a p p l i e d i n t h i s experiment.  A t e n f o o t l o n g s t e e l tube o f f o u r i n c h e s diameter was used as a vacuum chamber, i n which the t e s t s e c t i o n and hydrodynamic  approach were p l a c e d .  26  110  VOLT  VOLTAGE &GULATOR  UJJJJJJMi}  /rrrrrrnt  VARiAC  'juuiuxwuub prrrrrr)  TRANSFOWCR  WATTMETER  I  FIGURE  13  TEST  SCHEMATIC  OF  ROWER  SECTION  SUPPLY  1  27  Power Supply The t e s t s e c t i o n i s r e s i s t a n c e heated by a power supply as shown i n F i g . 13.  A l t e r n a t i n g current was employed i n order to eliminate voltage  pick up across the thermocouple j u n c t i o n .  Current passes through a v o l t -  age r e g u l a t o r w i t h an accuracy of 1 per cent, and was adjusted by a Variac.  A steady step-down transformer gave the required lower voltage.  The current then passed through a wattmeter, and to power-leads attached to the t e s t s e c t i o n .  Temperature Measurement Copper-Constantan thermocouple wire of 30 gauge was used throughout f o r temperature measurement. shown i n F i g . 14.  There are two groups of thermocouples, as  The f i r s t group pertains to thermocouples where less  accuracy i s r e q u i r e d , f o r example room temperature, and a u x i l i a r y temperatures of the flow loop such as temperature i n the c o o l e r , flowmeter, e t c . These temperatures were measured with a "Honeywell E l e c t r o n i c 15" i n s t r u ment having an accuracy of 1°F.  This instrument has 24 i n p u t s , i s s e l f -  balancing and does not need a reference j u n c t i o n .  The second group of  thermocouples measured temperature of the t e s t s e c t i o n w a l l and the mixing cup temperature.  They were selected by a multipole switch, measured  e i t h e r by the Honeywell instrument ( f o r rough readings), or by the K3 U n i v e r s a l potentiometer ( f o r accurate readings).  The K3 instrument en-  ables a voltage s e l e c t i o n of 0.5 m i c r o v o l t , and i s used w i t h standard c e l l , galvanometer and reference j u n c t i o n . f o r a reference j u n c t i o n .  A melting i c e bath was used  D i s t i l l e d water i c e was used f o r t h i s purpose  and was constantly s t i r r e d to maintain uniform temperature.  28  H O N E Y W E L L ELECTRONIC 1 5  S E L E C T O R S W I T C H >P L E A S U R I N G > JUNCTIONS >  C O P P E R C O N S T A N T A N  M E L T I N G ICE B A T H G A L V A N O T C T E R  F I G U R E 14  T H E R M O C O U P L E C I R C U I T  29  EXPERIMENTAL PROCEDURE  Test  Procedure  The e x p e r i m e n t a l d a t a were taken i n the f o l l o w i n g manner.  A flow  r a t e and a power l e v e l were s e t , and allowed to reach a steady v a l u e . When the i n l e t water temperature  a t t a i n e d a steady v a l u e , readings were  taken o f t h e flow r a t e , power, water i n l e t ,  and o u t l e t temperature,  b e f o r e and a f t e r r e a d i n g the w a l l temperature  both  thermocouples.  Thermocouple C a l i b r a t i o n Method  The  t e s t s e c t i o n and mixing  cup thermocouples were c a l i b r a t e d i n  o r d e r t o o b t a i n b e t t e r accuracy i n temperature  measurements.  These  thermocouples were c a l i b r a t e d by comparing them w i t h a s t a n d a r d mercury thermometer.  F o r c a l i b r a t i o n , a l l thermocouples were detached  from the t e s t  t i o n and f a s t e n e d t o the bulb o f the mercury thermometer.  The hot j u n c -  t i o n s were then put i n t o a Dewar f l a s k t o g e t h e r w i t h the s t a n d a r d meter, and water o f d e s i r e d temperature  sec-  thermo-  p l a c e d i n the Dewar f l a s k .  Each  thermocouple v o l t a g e was read s i m u l t a n e o u s l y on the p o t e n t i o m e t e r and mercury thermometer. temperatures  T h i s procedure  was repeated f o r f o u r d i f f e r e n t  ( r o u g h l y 170, 112, 89, and 55°F) c o v e r i n g the whole r e g i o n  i n which thermocouples were used t i o n t h e temperature  f o r the t e s t r u n s .  During the c a l i b r a -  o f the i c e bath was a l s o measured w i t h a mercury  thermometer, and found  t o vary by 0.1 c e n t i g r a d e .  30  C a l i b r a t i o n Result The voltage reading of a l l thermocouples v a r i e d not more than f i v e microvolts;, which corresponds to 0.25°F. The accuracy of the mercury thermometer was 0.09°F. When the galvanometer was used, one could detect a voltage v a r i a t i o n not smaller than 3 to 4 m i c r o v o l t s .  Since the ther-  mocouples agreed w i t h each other very c l o s e l y , they were not treated sepa r a t e l y but only one c a l i b r a t i o n curve was used f o r a l l thermocouples. The data from the c a l i b r a t i o n procedure were f i t t e d by method of l e a s t squares to a second degree polynomial equation of the form T  =  where  32 + C E + C E X  (16)  2  2  T  temperature, °F  E  microvolts  Ci, C  2  constants.  The constants were evaluated, g i v i n g the f o l l o w i n g equation T  =  32 + 4.763 • 10~ E - 1.507 • l O ^ E . 2  2  (16a)  31  RESULTS AND DISCUSSION Heat t r a n s f e r data were obtained f o r a h o r i z o n t a l c i r c u l a r tube with i n t e r n a l laminar flow.  The boundary condition was approximately uniform  heat f l u x over the e n t i r e t e s t s e c t i o n . 100 t o 2000.  The Reynolds number ranged from  The Grashof number was v a r i e d by applying d i f f e r e n t rates  of heat generation, and the n a t u r a l convection e f f e c t on the forced l a m i nar heat t r a n s f e r was studied by comparing the high Grashof number runs to runs at lower Grashof number at approximately the same Reynolds number.  Since water was used as the heat t r a n s f e r medium the Prandtl number  could be changed only by using a d i f f e r e n t temperature range. The flow was considered laminar when no w a l l temperature f l u c t u ations were present. While taking temperature readings, one could observe when the flow was changing from steady laminar to random eddying flow. The l i m i t of laminar flow i s dependent on Reynolds number and Grashof number.  I t was found that at a c e r t a i n Reynolds number the t r a n s i - .  t i o n flow was present i f heat f l u x was high enough, that i s , high Grashof number.  A smaller Reynolds number s t a b i l i z e d the flow.  I t was not the  purpose of the present work to i n v e s t i g a t e the l i m i t s of laminar flow. These observations were merely to omit data i n the t r a n s i t i o n or turbul e n t flow region.  32  Entrance Length and C i r c u m f e r e n t i a l Temperature V a r i a t i o n S i m i l a r to the development of a laminar v e l o c i t y p r o f i l e , the temperature p r o f i l e needs a c e r t a i n a x i a l distance to become f u l l y - d e v e l oped.  The entrance region i s determined from measurement of the a x i a l  w a l l temperature gradient. shown i n F i g . 15.  Consider the f l u i d temperature p r o f i l e as  At the entrance to the heated s e c t i o n the tempera-  ture i s uniform. As the f l u i d moves downstream, heat i s removed from the w a l l and the temperature d i f f e r e n c e between f l u i d and the w a l l i s i n creased. the  In the tube center there i s a core of f l u i d which i s s t i l l at  same temperature as i n the unheated s e c t i o n .  As the f l u i d moves  f u r t h e r downstream the core diminishes and f i n a l l y disappears. This i s the point when the temperature p r o f i l e i s considered fully-developed. For pure forced convection and laminar flow, the entrance length w i l l be d i f f e r e n t than the present case under i n v e s t i g a t i o n . F i g . 16 shows temperature p r o f i l e s f o r run number 42.  Between X/D =  100 and 300 the Reynolds number increases from 669 to 742, and the Grashof number from 1269 to 1541.  The flow can be considered fully-developed  approximately from the point X/D =75.  At X/D = 104 the d i f f e r e n c e be-  tween bottom and top w a l l temperature i s 2.6°F, while f u r t h e r downstream the d i f f e r e n c e between bottom and top w a l l temperature increases to become a maximum of 5.9°F at about X/D.=  200, and then decreases s l i g h t l y  towards the t e s t s e c t i o n end. I t may be noted from F i g .  16 that i n the entrance region the w a l l  temperature gradient i s high at the tube i n l e t , and decreases slowly i n  100  95  90  80  LU CC Z3  X/D  100  RE  129  PR  8,4  7,2  GR  309  549  RA  2592  3963  o  AVERAGE WALL  o  T O P WALL  o  SIDE  v  BOTTOM  WALL  300 .  146  TEMPERATURE  TEMPERATURE TEMPERATURE  WALL  TEMPERATURE  75  LU  s 70  50  JL  0  100  200  AXIAL  FIGURE  15  JL  DISTANCE- X/D  REPRESENTATIVETEMPERATURE PROFILE RUN  NUMBER  3  300  400  100  90  X/D  100  300  RE  668  742  PR  7.8  7.1  GR  1269  1553  RA  9871  11029  80  o LU  or  c.  50  o  AVERAGE WALL  n  T O P WALL  •  SIDE  v  BOTTOM  0  16  TEMPERATURE TEMPERATURE  WALL  TEMPERATURE  1 100  200 AXIAL DISTANCE  FIGURE  WALL  TEMPERATURE  REPRESENTATIVE TEMPERATURE RUN  NUMBER  42  300 X/D  PROFILE  400  35  the a x i a l d i r e c t i o n .  At about 100 diameters downstream i t becomes almost  constant.  A x i a l Temperature P r o f i l e s F i g . 17 shows the tube w a l l temperature and the f l u i d bulk temperature f o r run number 1. Between X/D = 100 and 300 the Reynolds number v a r i e s from 366 to 408, and the Grashof number from 703 to 965. Downstream of X/D = 100 the average w a l l temperature remains constant. The d i f f e r e n c e between f l u i d bulk and mean w a l l temperature i s 5.7°F at X/D » 100, and decreases s l i g h t l y t o 4.6°F at the t e s t s e c t i o n e x i t .  The  d i f f e r e n c e between bottom and top w a l l temperature i s approximately 1.5°F, o r nearly 30 per cent of the d i f f e r e n c e between mean w a l l and f l u i d bulk temperature. F i g . 18 shows the temperature p r o f i l e s f o r run number 43. The Reynolds number i s 331 at the i n l e t t o the heated s e c t i o n .  This t e s t run  was performed w i t h higher heat f l u x , g i v i n g a Grashof number of the order of 2000.  The gradient of the average w a l l temperature decreases u n t i l  approximately X/D = 6 0 ; from here on downstream i t remains constant. Average w a l l and f l u i d bulk temperature p r o f i l e s are not p a r a l l e l .  The  d i f f e r e n c e between f l u i d bulk and mean w a l l temperature i s 8.8°F at X/D = 100, and decreases t o 6.7°F at X/D = 300. One should not expect the f l u i d bulk temperature gradient and the mean w a l l temperature gradient to be exactly the same.  With change of  the f l u i d bulk temperature the properties of water change, and t h i s causes a change i n the dimensionless numbers such as Reynolds, Grashof, and  i  100  90  80  o LU  i  i  r~—r  X/D  100  300  RE  366  409  PR  '8.2  7.2  GR  702  965  RA  5766  6975  o  AVERAGE WALL  n  T O P WALL  o  S I D E WALL  v  BOTTOM  TEMPERATURE  TEMPERATURE TEMPERATURE  WALL  TEMPERATURE  LU LU  100  200  AXIAL  FIGURE  17  REPRESENTATIVE RUN  NUMBER  1  300  DISTANCE  TEMPERATURE  X/D  PROFILE  400  37  100  90  X/D  100  300  RE  364  440  PR  7.4  6,0  GR  1571,  RA  11628  2523 15076  80  LU 3  UJ 70  60  o  A V E R A G E WALL T E M P E R A T U R E  a  T O P WALL T E M P E R A T U R E  ^  S I D E WALL T E M P E R A T U R E  v  BOTTOM WALL T E M P E R A T U R E  1  50 0  1 0 0  2 0 0  300  A X I A L D I S T A N C E X/D  FIGURE  1 8  REPRESENTATIVE TEMPERATURE .«RUN NUMBER 4 3  PROFILE  400  38  P r a n d t l numbers.  The heat f l u x i s the same over the e n t i r e te.^t s e c t i o n ,  but a change i n the free convection e f f e c t influences the heat t r a n s f e r r a t e , therefore the temperature d i f f e r e n c e w i l l change.  In the t e s t run  considered, ( F i g . 18), the Reynolds number increases from 364 at X/D = 104, to 440 at X/D = 306.  In the same s e c t i o n the P r a n d t l number changes  from 7.4 to 6.0, and the Grashof number increases from 1571 to 2523. F i g . 19 shows the temperature p r o f i l e of run number 37, which covered Reynolds numbers In the region of 1150 to 1300. v a r i e d approximately from 2300 to 3600.  The Grashof number  The temperature d i f f e r e n c e i n  the entrance region increased r a p i d l y , then decreased s l i g h t l y towards the t e s t s e c t i o n e x i t . F i g . 20 shows the temperature p r o f i l e f o r a t e s t run w i t h high flow r a t e *and heat f l u x .  The Reynolds number v a r i e s from 1374 to 1600, and  the Grashof number from 6000 to 9600.  The temperature d i f f e r e n c e between  mean w a l l and f l u i d bulk i s nearly 20°F at X/D = 200.  A large circumfer-  e n t i a l temperature d i f f e r e n c e i s produced due to the high heat f l u x . When Figs- 17, 18, 19, and 20 are compared w i t h each other, one not i c e s a consistent droping of the average w a l l temperature near the tube exit.  This droping i s more pronounced i n higher Grashof number runs.  -Local Nusselt Number F i g . 21 shows the Nusselt number p r o f i l e f o r a run w i t h Reynolds numbers i n the order of 130, and with Grashof numbers ranging from 117 to 500.  The curve c l o s e l y resembles the p r e d i c t i o n f o r pure forced-convec-  FIGURE  19  REPRESENTATIVE TEMPERATURE , RUN  NUMBER  37  PROFILE  1 X/D 110  -  "  1  1  100  300  1374  1611  PR  6,1  5,1  GR  7129  10421  RA  43470  53039  RE  100  0  o  AVERAGE  D  T O P WALL  o  SIDE  v  BOTTOM  100  20  RUN  NUMBER  WALL  TEMPERATURE  WALL  27  TEMPERATURE  300  DISTANCE  REPRESENTATIVE TEMPERATURE  TEMPERATURE  TEMPERATURE  200  AXIAL  FIGURE  WALL  X/D  PROFILE  4(  0  I  I  I  DO  I  200  I  I  300  A X M . DISTANCE X/D FIGURE 21  frmaumrm RJN fern* 3  NUWELT NUPHER PROFILE  I  CO  42  t i o r (dashed l i n e ) .  The Nusselt numbers are s l i g h t l y higher than 4.36,  the value predicted for pure forced convection.  F i g . 22 represents a Nusselt number p r o f i l e with the Reynolds number i n the order of 1100 to 1300 and the Grashof number of about 20,000. to the high Reynolds number, the temperature  Due  p r o f i l e requires a larger  distance to become established i n s p i t e of the very high Grashof number.  F i g . 23 shows the plot of Nusselt number versus a x i a l tube distance for run number 1.  Reynolds numbers are of the order of 370, while Gras-  hof numbers vary from 700 to 970. there i s a high heat transfer rate.  At the beginning of the heated section The Nusselt number then decreases  r a p i d l y as the temperature p r o f i l e develops, and at X/D = 100 the curve reaches its. minimum.  This point coincides with the temperature  profile  becoming fully-developed ( F i g . 23). From t h i s point onwards the Nusselt number increases with approximately constant slope.  F i g . 24 represents the Nusselt number v a r i a t i o n f o r run 43. Reynolds and Grashof numbers are i n the order of 370 and 1600 respectively. At the tube i n l e t the Nusselt number i s approximately 9.  I t decreases to  a minimum at X/D = 60, and then increases at a constant rate.  Comparing run number 1 and 43 (replotted i n F i g . 25), we notice that i n the second case flow development i s achieved i n shorter distance, that i s a higher Grashof number seems to accelerate flow development. 104 i t appears that the Reynolds numbers are the same.  At X/D «*  I t i s interesting  to study the e f f e c t of Grashof number on Nusselt number at this point. I t appears that the higher Grashof number causes an increase i n the  43  12  ID  8  U  6h  2 I-  0 AXIAL DISTANCE X/D FIGURE 2 2  REWESOITATI* RUN NUMBER 37  Hussar  NUMBER PROFILE  100  0  200 AXIAL  FIGURE 23  REPRESENTATIVE RUN  NUMBER  1  DISTANCE  NUSSELT  NUMBER  300  X/D  PROFILE  400  45 12  10  8  cc LU  =3  6  _l LU CO CO  r>  X/D  100  300  RE  364  440  PR  7.4  6.0  GR  1571  2523  RA  11628  15076  I 0  100  i  200  i 300  AXIAL DISTANCE X/D FIGURE 2 *  R E P R E S E N T A T I V E NUSSELT N U M B E R PROFILE RUN S L U M B E R 43  400  22  JD  8  ^ ^ • X ' "  0  100  p  ° —  RUN NURSES 43  V  RUN NUMBER 1  °  200  300  AXIAL DISTANCE X/D FIGURE 25 tewestNTATivE NUSSELT NUMBER PROFILES RUN NUMBERS 1 AND 43  400  47  Nusselt number from 5.2 i n run 1 to 5.6 i n run 43.  Furthermore the  Nusselt number increases at a greater r a t e .  heat Transfer C o r r e l a t i o n C o r r e l a t i o n of o v e r a l l heat t r a n s f e r c o e f f i c i e n t s f o r the e n t i r e t e s t s e c t i o n length w i t h combined free and forced convection under u n i form heat f l u x i s complicated by the f a c t that the free convection e f f e c t s do not s t a r t at the tube i n l e t , but require a s t a r t i n g length to be e s t a b l i s h e d .  For uniform w a l l temperature, both the r a t i o D/L  L/D were used [8] to give f a i r c o r r e l a t i o n .  and  For uniform w a l l heat f l u x  i t appears that there i s a pronounced development region to e s t a b l i s h the secondary flow, and a region of thermally developed flow where- the heat t r a n s f e r c o e f f i c i e n t s are dependent only on the parameters Gr, Re, Pr. Several c o r r e l a t i o n s were tested on the data. Nu = Y| + 0.047 P r / 1  3  (Re R a ) / 1  The  expression (17)  5  appears to c o r r e l a t e 53 per cent of the data ( p l o t t e d i n F i g . 26) i n the fully-developed region to w i t h i n ± 10 per cent.  Another s l i g h t l y more  accurate expression which c o r r e l a t e s 68 per cent of the data to w i t h i n ± 10 per cent but does not s a t i s f y the pure forced convection case i s Nu = 2.41 + 0.082 P r / 1  3  (Re R a ) /  This data i s p l o t t e d i n F i g . 27.  1  5  (18)  A point of caution might be added here.  The above c o r r e l a t i o n s do not apply to f l u i d s of P r a n d t l numbers higher than 10, where the a d d i t i o n a l f a c t o r of v a r i a t i o n of v i s c o s i t y w i t h  48  70  60  U  50  or 40  O  i  h-  30  H  3  20  10  U  0  (RE RA)  1/5  FIGURE 2 6  C O R R E L A T I O N O F NUSSELT N U M B E R , E Q U A T I O N 17  49  l  70  l  l  l  60  50  o oo o °<>  —  •  Q_  o  o  9 OJ  °o° 3 0 K  w  —  _Q  °8oo ° oo  *<> °o  o °A«o °  i  3  o oo _ ^ oo° V °o o© o °  20h —  —  0 0 0  0  0  io-r  —  l 0  10  l  l  l  20  30  (RE RA)  1/5  FIGURE 2 7  C O R R E L A T I O N OF NUSSELT N U M B E R , E Q U A T I O N 18  40  50  50  temperature  i s dominant a l s o .  The expressions (17) and (18) should be  considered as very t e n t a t i v e . Further studies using various other f l u i d s , tube s i z e s , and lengths are required i n order to v e r i f y these c o r r e l a t i o n s , and perhaps shed f u r t h e r l i g h t on the b a s i c nature of t h i s problem.  M.  51  CONCLUSION From t h i s i n v e s t i g a t i o n experimental data was obtained f o r combined f r e e and forced convection laminar heat t r a n s f e r f o r water flowing through a c i r c u l a r h o r i z o n t a l tube with uniform heat f l u x .  Results show  that at low values of Grashof number the Nusselt number i s i n good agreement w i t h t h e o r e t i c a l p r e d i c t i o n for pure forced convection.  High Gras-  hof numbers r e v e a l that the f r e e convection e f f e c t w i l l create substant i a l increase i n Nusselt number.  Two t e n t a t i v e c o r r e l a t i o n s f o r the  Nusselt number f o r fully-developed flow are given i n equation (17) and (18) as a f u n c t i o n of Re Ra and P r .  52  REFERENCES 1.  A..°. Colburn,  "A Method of Correlating Forced Convection Heat  Transfer Data and a Comparison with F l u i d F r i c t i o n , " Trans. AIChE, v o l . 29, 1933, pp. 174 - 210.  2.  E.N. Sieder and G.E. Tate,  "Heat Transfer and Pressure Drop of L i -  quids i n Tubes," Ind. Eng. Chem., v o l . 28, 1936, pp. 1429 - 1435.  3.  D.G. Kern and D.F. Othmer,  "Effect of Free Convection on Viscous  Heat Transfer i n Horizontal Tubes," Trans. AIChE, v o l . 39, 1943, pp. 517 - 555.  4.  R.C. M a r t i n e l l i , C.J. Southwell, G. Alves, H.L. Craig, E.B. Weinberg, N.F. Lansing., and. L.M.K. Boelter,  "Heat Transfer and  Pressure Drop for a F l u i d Flowing i n the Viscous Region through a V e r t i c a l Pipe," Heat Transfer Data, Part I, Trans. AIChE, v o l . 38, 1942, pp. 493 - 530.  5.  O.C. Eubank and M.S. Proctor,  " E f f e c t of Natural Convection on  Heat Transfer with Laminar Flow in.Tubes," M.S. Thesis i n Chemical Engineering, Massachusetts I n s t i t u t e of Technology, Cambridge, 1951.  6.  W.H. McAdams,  "Heat Transmission," Third E d i t i o n McGraw H i l l Book  Co.,Inc., New York, 1954, p. 235.  7.  T.W. Jackson, J.M. Spurlock, and K.R. Purdy,  "Combined Free and  Forced Convection i n a Constant Temperature Horizontal Tube," Journal AIChE, v o l . 7, 1961, pp. 38 - 41.  53  8.  D.R. O l i v e r ,  "The E f f e c t of Natural Convection on Viscous Flow  Heat Transfer i n Horizontal Tubes," Chemical Engineering  Science,  v o l . 17, 1962, pp. 335 - 350.  9.  A.R. Brown and M.A. Thomas,  "Combined Free and Forced  Convection  Heat Transfer for Laminar Flow i n Horizontal Tubes," Journal Mechani c a l Engineering Science, v o l . 7, 1965, pp. 440 - 448.  10.  J . J . Martin and M.P. Carmichael,  "Combined Forced and Free Convec-  t i v e Heat Transfer i n a Horizontal Pipe," ASME, Paper No. 55-A-30, 1955.  11.  B.R. Morton,  "Laminar Convection  i n Uniformly Heated Horizontal  Pipes at Low Rayleigh Numbers," Quarterly Journal of Mechanics and Applied Mathematics, v o l . 12, no. 4, 1959, pp. 410 - 420.  12.  M. Iqbal and J.W. Stachiewicz,  "Influence of Tube Orientation on  Combined Free and Forced Laminar Convection Heat Transfer," Journal of Heat Transfer, Trans. ASME, Series C, 1966, pp. 109 - 116.  13.  E. Del Casal and N.N. G i l l ,  "A Note on Natural Convection  Effects  i n F u l l y Developed Flow," Journal AIChE, v o l . 8, 1962, pp. 570 - 574.  14.  A.J. Ede, "The Heat Transfer C o e f f i c i e n t for Flow i n a Pipe," International Journal of Heat and Mass Transfer, v o l . 4, 1961, pp. 105 - 110.  15.  S.T. McComas and E.R.G. Eckert,  "Combined Free and Forced Convec-  t i o n i n a Horizontal Circular Tube," Journal of Heat Transfer, Trans. ASME, Series C, 1966, pp. 147 - 153.  54  16.  Y. Mori, K. Futagami, S. Tok.ida, and M. Nakamura,  "Forced  Convec-  t i v e Heat Transfer i n Uniformly Heated Horizontal Tubes," International Journal of Heat and Mass Transfer, v o l . 9, 1966, pp. 453 463.  17.  Y. Mori and K. Futagami,  "Forced Convective  Heat Transfer i n Uni-  formly Heated Horizontal Tubes," International Journal of Heat and Mass Transfer, v o l . 10, 1967, pp. 1801 - 1813.  18.  R.L. Shannon and C.A. Depew,  "Combined Free and Forced Laminar  Convection i n a Horiz ontal Tube with Uniform Heat Flux,"  ASME Paper  No.. 67-HT-52, 1967.  19.  R. Siegel, E.M. Sparrow, and T.M. Hallman,  "Steady Laminar Heat  Transfer i n a C i r c u l a r Tube with Prescribed Wall Heat Flux," Applied S c i e n t i f i c Research, Section A, v o l . 7, 1958, p. 386.  20.  T.M. Hallman,  "Combined Forced and Free Convection i n a V e r t i c a l  Tube," PhD D i s s e r t a t i o n , Purdue University, West Lafayette,  Indiana,  1958.  21.  R.D. M i k e s e l l ,  "The E f f e c t of Heat Transfer on the Flow i n a Hori-  zontal Pipe," PhD Thesis, Chemical Engineering Department, University of I l l i n o i s ,  22.  Anon.,  1963.  "Resistance Welding of Nickel and High-Nickel A l l o y s , "  Technical B u l l e t i n T-33, The International Nickel Company.  APPENDIX A  DERIVATION OF GOVERNING EQUATIONS  Equations energy  (1) t o (3) were d e r i v e d from c o n t i n u i t y , momentum and  equations  developed,  i n c y l i n d r i c a l coordinates.  Flow i s c o n s i d e r e d  and p h y s i c a l p r o p e r t i e s of the f l u i d  fully-  are assumed constant ex-  cept t h e d e n s i t y v a r i a t i o n a f f e c t i n g t h e buoyancy terms.  In the energy  e q u a t i o n a x i a l c o n d u c t i o n , v i s c o u s d i s s i p a t i o n and p r e s s u r e terms are ignored.  The e q u a t i o n s a r e then f u r t h e r reduced w i t h help of the stream  f u n c t i o n \\i and expressed as  rv v  r  _ _aj>_ 96  _e v  and  _ il  =  v  9r  the parameters  R  =  — a  : '  V  x  =  v -2v/a  T ; '  T*  =  With these parameters the d i m e n s i o n l e s s numbers become  Re  =  -Ca /(4pv )  Gr  =  BgAaVv  Pr  =  c p/k P  Ra  =  Gr Pr  3  2  2  - T  W  A a Pr  APPENDIX B LOCATION OF TEST SECTION THERMOCOUPLES Thermocouple Number  Distance from Test Section Inlet [in]  57  1  4  top  59  1  4  bottom  60  2  8  top  61  3  13  top  62  4  17  top  63  5  22  top  64  6  26  top  65  7  30  top  66  8  35  top  67  9  39  top  68  10  43  top  69  11  48  top  70  12  52  top  71  12  52  side  72  12  52  bottom  73  24  104  top  74  24  104  side  75  24  104  bottom  76  36  157  top  Distance i n Diameters  Location on circumference  Thermocouple Number  Distance from Test Section Inlet [in]  Distance i n Diameters  Location on circumference  77  36  157  side  78  36  157  bottom  79  48  209  top  80  48  209  side  81  48  209  bottom  82  60  261  top  83  60  261  bottom  84  60  261  side  85  66  287  top  86  68  296  top  87  20  87  top  88  70.5  307  top  89  70.5  307  side  90  70.5  307  bottom  91  16  70  top  58  APPENDIX C  SAMPLE CALCULATION  The data of run number 3 are chosen f o r purposes of i l l u s t r a t i o n . The measured quantities are  Flow rate  41.6  ccm/min at 15°C  Power input  25.0  watts  Temperature d i s t r i b u t i o n as given i n F i g . 15 (p.33) The heat f l u x was calculated from the bulk temperature r i s e and flow rate. The temperatures given i n F i g . 15, were obtained using the thermocouple calibration results. the  A mean outside wall temperature was calculated from  three thermocouples placed at d i f f e r e n t circumferential locations  (Fig. 12, p.24).  The mean wall temperature i s f a i r e d by a straight l i n e ,  and shows that maximum deviation i s very small. The bulk temperature was measured before the test section i n l e t and a f t e r the test section e x i t , and i t s gradient assumed to be constant since the uniformity of heat generation over the tube length could not be checked. The flow was considered fully-developed away from the region of steep temperature gradient. X/D =75.  For this run, this occurred approximately at  At the location X/D = 104, the recorded mean wall and bulk  59  temperature were T  - 59.8°F  T  =  57.0°F  The heat t r a n s f e r c o e f f i c i e n t i s based on inner w a l l temperature, and the temperature drop through the w a l l was c a l c u l a t e d from  * - 2ir {  -  R 2 l n  w  7  }  ( ( > 1 )  where W k  heat generation rate per u n i t volume of tube material w  thermal conductivity of w a l l ( k =8.67 BTU/hr °F f t [22]) w  R  outside radius  =  0.125 inch  r  i n s i d e radius  =  0.115 inch  For t h i s apparatus (C-l) reduces t o T  =  4.19 10" P  (C-2)  4  where P i s t e s t s e c t i o n power input i n watts. Equation (C-l) was derived w i t h the assumption of uniform power density i n the tube w a l l and no heat loss at the outer surface.  Furthermore, a u n i -  form a x i a l temperature gradient i n the tube w a l l and constant p h y s i c a l properties of the tube were assumed.  The maximum c a l c u l a t e d temperature  drop across the tube w a l l was 0.24°F, and f o r t h i s run was about 0.01°F.  60  Heat T r a n s f e r C o e f f i c i e n t and N u s s e l t Number  The N u s s e l t number i s d e f i n e d as  Nu  =  (C-3) f  fc  where h stands f o r t h e heat t r a n s f e r  h  coefficent,  Y^IT w  =  (C-4)  and q i s t h e heat flow r a t e . based on t h e b u l k  The thermal c o n d u c t i v i t y o f water  temperature.  The heat f l u x was c a l c u l a t e d from the flow r a t e and the b u l k rise  q  =  was  m Cp A/area  where m  mass flow r a t e  Cp  s p e c i f i c heat o f b u l k  A  b u l k temperature g r a d i e n t  Area  s u r f a c e area per foot l e n g t h .  temperature  T h e r e f o r e t h e heat f l u x and N u s s e l t number a r e  q  =  0.061  Nu  =  4.52  BTU/ft sec 2  temperature  61  Reynolds Number  The Reynolds number i s defined as  m 4  uD Re  =  —  or  —  v  p TTVD  i f the mean v e l o c i t y u i s expressed i n terms of the flow rate so that for this case  Re  =  129  .  '  > .  The v i s c o s i t y was evaluated at water bulk temperature.  Grashof Number  The Grashof number i s here defined as  3 g AT D  Gr  3  (C-6)  where  3  c o e f f i c i e n t of thermal expansion  g  g r a v i t a t i o n a l acceleration  AT  temperature difference beteen bulk f l u i d and inner wall, as calculated  For this point c  Gr  =  309  62  APPENDIX D  ERROR ESTIMATES To give some idea of the accuracy of the data, e r r o r estimates were performed.  The accuracy v a r i e d a great deal between runs and f o r various  p o s i t i o n s on the t e s t s e c t i o n .  Because of the number of v a r i a b l e s which  a f f e c t the r e s u l t s , i t i s i m p r a c t i c a l t o present estimated e r r o r s f o r each data p o i n t , and e r r o r estimates f o r only two cases are presented below.  E r r o r i n mass flow rate  0.5% < m < 1.5%  P o s s i b l e e r r o r i n water temperature at i n l e t to t e s t s e c t i o n 0.25°F < T < 0.5°F  P o s s i b l e e r r o r i n water temperature at e x i t of t e s t s e c t i o n 0.5°F < T < 1.0°F  E r r o r i n tube w a l l temperature measurements f o r heated s e c t i o n Tw  = 0.25°F  The e r r o r s i n water temperature are noticed by s l i g h t f l u c t u a t i o n of the measured temperature.  They are due to the e r r o r i n temperature measure-  ments and due t o the inherent i n s t a b i l i t y of the flow system as w e l l as the power generation.  P o s s i b l e e r r o r s i n property values such as thermal  63  c o n d u c t i v i t y and v i s c o s i t y due to e r r o r i n temperature may be as much as 0.2%. 1  Probable Errors f o r Results For run number 3 Difference from w a l l to bulk temperature  AT  =  2.75°F  A x i a l temperature gradient of f l u i d bulk  A  =  2.42°F/ft  From e r r o r i n w a l l and bulk temperature the bulk temperature gradient may be i n e r r o r of  3.4%  Error i n temperature d i f f e r e n c e  T  =  0.5°F  -• 15.2%  Probable e r r o r of dimensionless numbers are Re  G r  =  =  ~  =  r  pvD  B AT  2.0%  18.5%  R  E r r o r estimated f o r run number 27 At l o c a t i o n X/D = 200 the temperature d i f f e r e n c e is  T  = 19.7°F  and the bulk temperature gradients i s  A  =  3.54°F/ft  64  E r r o r of d i f f e r e n t components are mass flow rate  0.5%  w a l l temperature  0.25°F  f l u i d bulk temperature at i n l e t  0.25°F  f l u i d bulk temperature at e x i t  1.0°F  temperature d i f f e r e n c e  1.25°F  bulk temperature gradient  = 6.35% = 4.7%  And the e r r o r of dimensionless numbers are Nusselt number  7.95%  Reynolds number  1.8%  Grashof number  6.35%  65  APPENDIX E  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 1, Q = 0.146  BTU/sft sec  Nu  Re  Pr  Gr  Ra  1  8.6  347  8.8  316  2767  3  7.4  350  8.7  384  3339  5  6.3  352  8.6  465  4021  7  6.0  349  8.7  460  4017  9  5.9  356  8.5  526  4483  12  6.0  357  8.5  527  4473  24  5.2  366  8.2  702  5766  36  5.5  376  7.9  770  6108  48  5.6  387  7.7  850  6524  60  5.9  399  7.4  915  6794  70.5  6.2  409  7.2  965  6975  X  66  TABULATED LOCAL HMT  Run Number 2, Q = 0.121  TRANSFER DATA  BTU/sft. sec  X  Nu  Re  Pr  Gr  Ra  1  8.1  124  8.8  275  2413  3  6.3  126  8.7  381  3304  5  5.5  127  8.6  457  3907  7  5.3  129  8.4  506  4277  9  4.9  130  8.3  583  4858  12  4.9  132  8.2  634  5187  24  4.8  141  7.6  867  6563  36  4.8  151  7.0  1146  8050  48  4.7  161  6.5  1470  9593  60  4.7  171  6.1  1847  11214  70.5  4.7  180  5.7  2203  12637  67  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 3, Q = 0.061  B T U / s f t sec  X  Nu  Re  Pr  Gr  Ra  1  8.1  120  9.1  117  1061  3  6.2  121  9.0  159  1429  5  5.0  122  8.9  206  1840  7  4.8  123  8.9  225  1987  9  4.8  124  8.8  235  2063  12  4.8  125  8.7  248  2154  24  4.5  129  8.4  309  2592  36  4.4  133  8.0  378  3039  48  4.6  137  7.7  423  3277  60  4.6  142  7.5  488  3637  70.5  4.5  146  7.2  549  3963  68  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 11, Q = 0.394  BTU/sft sec  X  Nu  Re  Pr  Gr  Ra  1  9.8  534  8.4  943  7873  3  7.2  538  8.3  1336  11049  5  6.4  543  8.2  1578  12906  7  6,1  547  8.1  1724  13955  9  5.9  552  8.0  1850  14831  12  5.4  559  7.9  2135  16844  24  5.7  590  7.4  2542  18882  36  6.3  621  7.0  2861  20030  48  6.9  654  .6.6  3155  20837  60  7.8  687  6.2  3358  20939  70.5  8.6  716  5.9  3478  20680  69  TABULATED LOCAL HE4.T TRANSFER DATA  Run Number 12, Q » 0.527  BTU/si: sec  X  Nu  Re  Pr  Gr  Ra  1  10.4  540  8.3  1209  10032  3  8.2  546  8.2  1627  13323  5  7.4  552  8.1  1916  15463  7  6.6  558  8.0  2274  18108  9  6.3  565  7.9  2498  19617  12  6.3  571  7.8  2644  20503  24  6.3  616  7.1  3629  25805  36  6.9  659  6.6  4301  28299  48  7.6  705  6.1  4884  29781  60  8.6  750  5.7  5397  30659  70.5  9.6  788  5.4  5688  30566  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 13, Q = 0.479  X  Nu  Re  1  13.1  3  BTU/sft sec  Pr  Gr  Ra  1008  8.7  695  6064  9.3  1015  8.7  1017  8815  5  8.3  1020  8.6  1169  10062  7  7.7  1025  8.6  1296  11092  9  7.3  1031  8.5  1409  11982  12  7.0  1038  8.4  1550  13051  24  6.2  1070  8.1  2046  16582  36  5.8  1104  7.8  2547  19884  48  5.7  1141  7.5  2959  22251  60  5.7  1179  7.2  3431  24859  70.5  5.7  1212  7.0  3833  26936  71  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 14, Q = 0.474  X  Nu  Re  1  15.9  3  BTU/sft sec  Pr  Gr  Ra  1000  8.8  540  4758  9.9  1007  8.7  899  7863  5  8.6  1014  8.7  1074  9329  7  7.9  1021  8.6  1209  10425  9  7.5  1026  8.6  1307  11199  12  7.4  1032  8.5  1378  11707  24  6.7  1063  8.1  1788  14629  36  6.2  1097  7.9  2262  17823  48  5.7  1133  7.6  2818  21398  60  5.3  1171  7.3  3472  25405  70. 5  5.1  1202  7.1  4083  28995  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 17, Q = 0.502  X  Nu  Re  1  16.1  3  BTU/sft sec  Pr  Gr  Ra  1425  8.9  546  4846  12.4  1430  8.8  721  6375  5  10.7  1433  8.8  842  7426  7  8.4  1448  8.7  1138  9937  9  8.0  1455  8.7  1226  10648  12  7.5  1465  8.6  1348  11625  24  6.8  1496  8.4  1696  14224  36  6.3  1530  8.2  2066  16846  48  5.8  1565  7.9  2489  19732  60  5.4  1602  7.7  2964  22873  70.5  5.2  1634  7.5  3408  25709  73  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 27, Q = 0.720  BTU/sft sec  X  Nu  Re  Pr  Gr  1  12.2  1259  6.7  3057  20618  3  9.6  1269  6.7  4005  26773  5  8.7  1278  6.6  4567  30254  12  8.0  1312  6.4  5465  35117  24  7.2  1374  6.1  7129  43470  36  7.3  1437  5.8  8134  47133  48  7.2  1497  5.5  9480  52503  60  8.0  1559  5.3  9699  51261  70.5  8.5  1611  5.1  10421  53039  Ra  74  TABULATED LOCAL Hi'AT TRANSFER DATA  Run Number 29, Q = 0.996  X  Nu  Re  1  15.5  1102  3  10.8  5  BTU/sfr. sec  Pr  Gr  Ra  8.1  1734  13989  1114  8.0  2625  20900  9.5  1126  7.9  3116  24504  7  8.9  1138  7.8  3525  27381  9  8.5  1151  7.7  3853  29553  12  8.2  1171  7.5  4309  32418  24  7.7  1252  7.0  5960  41601  36  7.9  1332  6.5  7353  47765  48  8.1  1417  6.1  8896  53839  60  8.3  1501  5.7  10545  59789  70.5  8.5  1571  5.4  11954  64383  75  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 31, Q = 0.754  BTU/sft sec  X  Nu  Re  1  15.3  1583  8.2  1223  10069  3  12.0  1591  8.2  1608  13147  5  9.7  1599  8.1  2039  16559  7  9.0  1608  8.1  2254  18188  9  8.7  1616  8.0  2385  19112  12  8.5  1631  7.9  2569  20357  24  8.8  1689  7.6  2876  21915  36  8.8  1749  7.3  3367  24653  48  8.4  2037  6.1  6166  37935  60  8.5  1872  6.8  4524  30692  70.5  8.0  1926  6.6  5328  34981  Pr  Gr  Ra  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 32, Q = 0.989  BTU/sft sec  X  Nu  Re  Pr  Re  1  11.5  1600  8.2  2218  3  10.8  1606  8.1  2406  5  9.5  1614  8.1  2816  7  8.8  1625  8.0  3119  9  8.5  1634  8.0  3329  12  8.8  1658  7.9  3437  24  8.0  1734  7.4  4581  36  9.9  1815  7.0  4445  48  9.0  1903  6.7  4395  60  8.2  1977  6.4  7353  70.5  8.1  2045  6.1  8383  77  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 34, Q - 0.254  X  Nu  Re  1  9.0  3  BTU/sft sec  Pr  Gr  Ra  1577  8.2  699  5755  8.6  1581  8.2  746  6121  5  8.3  1585  8.2  782  6391  7  8.2  1587  8.2  800  6528  9  7.9  1589  8.1  831  6769  12  7.9  1592  8.1  839  6819  24  7.2  1609  8.0  970  7777  36  6.6  1628  7.9  1125  8903  48  6.3  1646  7.8  1233  9626  60  6.5  1666  7.7  1268  9772  70.5  6.9  1685  7.6  1252  9523  78  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 35, Q = 0.231 BTU/sft sec  X  Nu  Re  Pr  Or  Ra  1  10.8  1106  8.1  564  4570  3  9.7  1109  8.1  637  5148  5  9.1  1112  8.1  686  5523  7  8.6  1115  8.0  736  5905  9  8.2  1117  8.0  786  6294  12  7.9  1120  8.0  825  6580  24  6.6  1137  7.8  1064  8341  36  5.9  1155  7.7  1268  9772  48  5.7  1173  7.6  1409  10671  60  5.8  1191  7.4  1471  10946  70.5  6.4  1210  7.3  1415  10342  79  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 36, Q = 0.258  X  Nu  Re  1  11.1  3  BTU/sft sec  Pr  Gr  Ra  1073  8.3  546  4551  10.0  1075  8.3  612  5085  5  9.3  1078  8.3  668  5533  7  9.0  1081  8.3  701  5791  9  8.4  1083  8.2  759  6255  12  8.3  1088  8.2  789  6458  24  7.1  1107  8.0  1012  8111  36  6.4  1124  7.9  1212  9530  48  6.3  1143  7.7  1323  10215  60  6.7  1164  7.6  1349  10201  70.5  7.3  1181  7.4  1307  9724  80  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 37, Q =0.500  X  Nu  Re  1  12.4  3  BTU/sfL sec  Pr  Gr  Ra  1085  8.3  956  7946  9.4  1088  8.3  1277  10577  5  8.9  1090  8.3  1379  11385  7  8.0  1093  8.2  1556  12809  12  7.8  1116  8.0  1766  14155  24  6.6  1156  7.7  2456  18895  36  6.5  1195  7.4  2850  21130  48  6.2  1235  7.1  3462  24734  60  7.4  1275  6.9  3284  22635  70.5  7.6  1311  6.7  3579  23891  81  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 38, Q = 0.497  BTU/sft sec  X  Nu  Re  Pr  1  11.3  656  8.3  1049  8720  3  8.6  660  8.2  1445  11873  5  8.2  656  8.1  1571  12780  7  7.5  672  8.0  179  14441  9  7.1  678  8.0  1994  15855  12  6.9  687  7.8  2186  17114  24  6.9  726  7.4  2784  20480  36  6.8  757  7.0  3317  23258  48  7.6  807  6.5  3768  24585  60  8.0  849  6.1  4262  26220  70.5  8.4  885  5.9  4678  27464  Gr  Ra  82  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 39, Q = 0.972  X  Nu  1  9.9  3  BTU/sft sec  Re  Pr  Gr  Ra  662  8.2  2427  20012  8.5  673  8.1  3094  24961  5  7.9  685  7.9  3603  28471  7  7.6  696  7.8  4042  31351  9  7.3  709  7.6  4540  34484  12  7.7  728  7.4  4817  35486  24  8.0  808  6.5  6973  45647  36  8.2  893  5.8  9517  55567  48  8.6  976  5.3  12221  64500  60  9.0  1065  4.8  15282  73061  70.5  9.3  1138  4.4  18064  80018  *  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 43, Q = 0.234  BTU/sft sec  X  Nu  Re  Pr  Gr  Ra  1  8.0  331  8.3  703  5826  3  6.5  334  8.2  906  7434  5  5.9  336  8.1  1035  8406  7  5.6  339  8.0  1145  9212  9  5.5  342  8.0  1207  9609  12  5.4  346  7.8  1304  10232  24  5.6  364  7.4  1571  11628  36  5.9  383  7.0  1829  12787  48  6.2  403  6.6  2086  13789  60  6.6  422  6.3  2340  14630  70.5  7.0  440  6,0  2523  15076  84  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 44, Q = 0.440  TU/sft sec  X  Nu  Re  Pr  Gr  1  9.8  335  8.2  1143  9363  3  7.7  340  8.1  1558  12548  5  6.6  344  7.9  1923  15263  7  6.3  349  7.8  2140  16728  9  6.0  354  7.7  2376  18281  12  6.2  366  7.4  2664  19733  24  6.4  401  6.7  3731  24859  36  6.5  439  6.0  5044  30308  48  6.6  477  5.5  6521  35717  60  6.7  516  5.0  8264  41325  70.5  6.9  550  4.6  9775  45448  Ra  85  TABULATED LOCAL HEAT TRANSFER DATA  Run Number 45, Q = 0.646  BTU/sft sec  X  Nu  Re  Pr  Gr  Ra  1  7.9  325  8.1  2188  17656  3  7.2  333  7.9  2699  21192  5  6.9  340  7.6  3120  23849  7  6.9  349  7.4  3465  25775  9  6.8  357  7.2  3867  28012  12  6.6  372  6.9  4618  32028  24  6.5  427  5.9  7703  45568  36  6.4  482  5.1  11679  60051  48  6.4  543  4.5  16714  75226  60  6.4  603  4.0  22745  90569  70.5  6.5  654  3.6  28120  101948  TABULATED LOCAL HEAT TRANSFER DATA  Run N;imber 46, Q = 0.642  BTU/sft sec  X  Nu  Re  Pr  1  9.1  378  3  7.6  5  Gr  Ra  6.8  3661  24730  387  6.6  4741  31223  7.0  396  6.4  5582  35820  7  6.7  404  6.3  6289  39326  9  6.7  413  6.1  6805  41497  12  7.0  429  5.8  7375  43126  24  6.8  484  5.1  11182  56983  36  6.8  . 543  4.4  15940  71191  48  6.9  604  4.0  21700  85424  60  7.0  664  3.5  27718  98218  70.5  7.1  715  3.3  33303  108565  

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