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Thermophoresis in sols. McNab, Gordon Spencer 1972

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THERMOPHORESIS IN SOLS b y  GORDON SPENCER McNAB B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia,  1970  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n the Department of Chemical  Engineering  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA August,  1972  In p r e s e n t i n g  this thesis i n p a r t i a l f u l f i l m e n t of the requirements  f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 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 a v a i l a b l e f o r r e f e r e n c e and study.  I f u r t h e r agree that permission  f o r extensive copying of t h i s  t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood that  o r p u b l i c a t i o n o f t h i s t h e s i s ' f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n  permission.  Gordon S. McNab  Department o f  Chemical  Engineering  The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada  copying  Abstract  Experiments ical  latex  The  for  phenomenon,  gases but  Dilute  not  two h o r i z o n t a l ,  whereas  the  the  function t i d e  The  from the  The of  ° K m"^  <_ vT  p a r t i c l e s i n water  spheres with  thus  The  has  and d e n s i t y ,  tested  k  spher-  temperature previously  top  or  been  n-hexane were  d i s k was  gradient  ,  absolute  fluid  light  microscope.  velocity,  vT,  trapped  heated  c r e a t i n g a temperature  a low power  gradient,  conductivity,  e q u a t i o n was  applied  p a r t i c l e m o t i o n was m e a s u r e d b y o b s e r v i n g  v.. -th  This  micron-size,  liquids.  parallel disks.  temperature  y,  of  thermophoresis,  p a r t i c l e thermophoretic  thermal  viscosity,  for  l o w e r one was c o o l e d ,  liquid.  reflected  l i q u i d s due t o  called  suspensions of  between  in  c o n d u c t e d on t h e m o t i o n  p a r t i c l e s in stagnant  gradients. reported  were  thermal  y_^,  was  found  temperature,  to T^ 5  conductivity,  be a par- •  kf,  p:  = -  0.26  9  .  2 k  in  the  <_ 30.J.00.0. ° K m " ^ .  f  kf ].  ranges  p  -Hp K  vT ~  284 ° K <_ T K  <_ 343  ° K and  No d e p e n d e n c e , on. p a r t i c l e d i a m e t e r  7,000 was  noted. Thermophoresis liquid-particle eering  in liquids  separation.  s i t u a t i o n s where  large  It  i s a weak  may,  however,  temperature  ii  effect  and  impractical  be i m p o r t a n t  gradients  occur.  in  for  engin-  Table of  Contents Page  LIST OF TABLES  vii  LIST OF FIGURES  ix  Chapter I. II.  INTRODUCTION  .  .  .  .  .  .  .  .  .  .  .  .  PREVIOUS WORK  .  THERMOPHORESIS IN GASES  .  .  .  .  .  .  .  . .  .  . .  1  . .  . .  4 4  The Small P a r t i c l e Regime (Kn »  1)  .  .  .  .  .  .  5  The Large P a r t i c l e Regime (KN «  1)  .  .  .  .  .  .  6  .  .  .  .  .  10  .  .  .  .  .  16  .  .  .  .  .  16  THERMOPHORESIS IN LIQUIDS III.  .  APPARATUS INTRODUCTION THE CELL  .  The O p t i c a l  .  .  .  .  .  .  .  .  .  .  .  .  .  .  17  Tube  19  The Base  19  The Bottom D i s k  21  The S p a c e r R i n g  21  The Top D i s k  24  THE HEATING OF THE TOP DISK  .  .  .  .  .  .  .  .  .  24  The E l e c t r i c H e a t i n g C o i l  .  .  .  .  .  .  .  .  .  24  .  .  .  26  A s s o c i a t e d E l e c t r i c a l Apparatus  iii  .  .  .  .  IV  Chapter  Page THE COOLING OF THE BOTTOM DISK  . . . . . . . .  TEMPERATURE MEASUREMENT OPTICAL EQUIPMENT The L i g h t Source  28 .  . . . . . . . . . . . .  29  . . . . . . . . . . . .  29  The Microscope IV.  30  PHYSICAL PROPERTIES OF THE EXPERIMENTAL MATERIALS THE PARTICLES  . .  . . . . .  32  Water . . . . . . . . . . . . . . . . Viscosity  33  . . . . . . .  33  Density Thermal C o n d u c t i v i t y  33 .  34  Hexane  23  Viscosity Density Thermal C o n d u c t i v i t y V. EXPERIMENTAL METHOD  35 . . . . . . . .  35  . . . . .  35  . . . . .  43  PREPARATIONS FOR A RUN  .  THE RUN  PRELIMINARY DATA TRANSFORMATIONS  43 45  METHODS OF DATA ANALYSIS  Disk Temperatures  32 32  THE LIQUIDS  VI.  28  48 . . . . . . .  48 48  V Chapter  Page Top  Disk  Bottom Local  48  Disk  49  Temperature  Thermophoresc ANALYSIS The  OF  VII.  a  D i s t r i b u t i o n  o f  the  THE  AND  t  .  .  .  50 53  DATA  54  f o r  54  Equation o f  RESULTS  AN  TRANSFORMED  Values  M o d i f i e d  Gradient  V e l o c i t y  Empirical  Block  and Temperature  55 C o e f f i c i e n t s  i n  Each  Block  .  .  Test  57  DISCUSSIONS  EXISTENCE EMPIRICAL  OF  59  THERMOPHORESIS  EQUATION  FOR  THE  IN  LIQUIDS  .  .  .  .  59  THERMOPHORETIC  VELOCITY A n a l y s i s Summary  59 of o f  the Thermophoretic the  D i s t r i b u t i o n  Block  o f  the  a's  E f f e c t  of  P a r t i c l e  The  E f f e c t  o f  L i q u i d  The  .  .  .  59 59  i n  Each  Diameter  Thermal  Block on  .  .  .  .  50 61  a  C o n d u c t i v i t y  63.  a  Final  Empirical  THERMOPHORETIC  Equation  f o r  55  FORCE  .  THE E F F E C T OF GRAVITY C E L L W A L L E F F E C T S ON T H E  A  C o e f f i c i e n t  a's  The  on  56  TEMPERATURE  GRADIENT  THEORETICAL  MODEL  OF  .  .  .  .  66 67  LOCAL  TEMPERATURE  AND .  THERMOPHORESIS  IN  LIQUIDS  .  67"  . .  .  68  vi Chapter VIII.  Page CONCLUSIONS AND RECOMMENDATIONS CONCLUSIONS  81  RECOMMENDATIONS  82  NOMENCLATURE REFERENCES  81  . . . .  83  . .  91  APPENDICES  94  I.  95  II. III.  PROPERTIES OF WATER AND n-HEXANE EXPERIMENTAL DATA  100  SAMPLE AND ERROR CALCULATIONS  120  CALCULATION OF a AND ERROR ESTIMATION  120  Water  121  Hexane  127  Observations  131  THE UPPER VELOCITY LIMIT OF STOKES LAW  131  THE TEMPERATURE DROP ACROSS EACH DISK  132  THE TEMPERATURE RISE OF COOLING WATER IN PASSING THROUGH THE BOTTOM DISK THE EFFECT OF TEMPERATURE DEPENDENT k ON T AND dT/dx  133  f  p c  IV.  133  Water  134  Hexane  134  THERMOPHORETIC FORCE AND VELOCITY FOR THE LARGE PARTICLE REGIME  136  List  of Tables  Table  Page  1.  Thermal  C o n d u c t i v i t i e s of the Apparatus M a t e r i a l s  .  .  .  17  2.  Statistical  3.  Comparison of Experimental  4.  The Thermal  5.  The V i s c o s i t y  6.  The D e n s i t y o f Hexane  7.  The Thermal  8.  Code L e t t e r s  f o r Blocks  9.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 23.1°C  .  .  .  .  102  10.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 32.7°C  .  .  .  .  103  11.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 39.7°C  .  .  .  .  104  12.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 49.2°C  .  .  .  .  105  13.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 50.2°C  .  .  .  .  106  14.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 55.7°C  .  .  .  .  107  15.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 57.3°C  .  .  .  .  108  16.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 69.0°C  .  .  .  .  109  17.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 78.0°C  .  .  .  .  110  18.  Experimental  Data o f B l o c k A ,  (Ty-Tg)  = 81.7°C  .  .  .  .  Ill  19.  Experimental  Data o f B l o c k B ,  (Ty-Tg)  = 37.7°C  .  .  .  .  112  20.  Experimental  Data o f B l o c k B ,  (Ty-Tg)  = 39.5°C  .  .  .  .  113  21.  Experimental  Data  (Ty-Tg)  = 44.9°C  .  .  .  .  114  Data f o r Each B l o c k  60  and P r e d i c t e d a ' s  C o n d u c t i v i t y of Water  .  .  64  .  .  .  .  .  .  96  o f Hexane  .  97 98  C o n d u c t i v i t y o f Hexane .  .  .  of Block B,  .  vii  . .  . .  . .  . .  .  .  .  .  .  .  99  .  .  101  vi i i Table  Page  22.  Experimental  Data o f B l o c k C,  (Ty-Tg)  = 34.6°C  .  .  .  .  115  23.  Experimental  Data  of Block C,  (Ty-Tg)  = 56.0°C  .  .  .  .  116  24.  Experimental  Data o f B l o c k C,  (Ty-Tg)  = 68.9°C  .  .  .  .  117  25.  Experimental  Data o f B l o c k D,  (TT-TB)  = 37.8°C  .  .  .  .  118  26.  Experimental  Data o f B l o c k D,  (Ty-Tg)  = 54.9°C  .  .  .  .  119  27.  Sample and E r r o r  Calculations—water  123  28.  Sample and E r r o r  Calculations—Hexane  128  29.  The Upper V e l o c i t y L i m i t  o f S t o k e s Law  132  List  of  Figures  Figure  Page  1.  An O v e r a l l  2.  The B a s e  3.  The B o t t o m D i s k  4.  The S p a c e r R i n g  23  5.  The Top D i s k  25  6.  The E l e c t r i c a l  7.  The M i c r o s c o p e  8.  Viscosity  9.  D e n s i t y o f Water as a F u n c t i o n o f Temperature  10.  Thermal  View o f  the C e l l  .  .  .  .  18 20  .  .  .  .  .  .  .  .  .  .  .  .  .  .  Apparatus  27 30  o f Water as a F u n c t i o n o f Temperature  C o n d u c t i v i t y o f Water as a F u n c t i o n  . .  .  .  .  37  .  .  38  of  Temperature 11.  Viscosity  12.  Density of  13.  Thermal  of  22  39 n-Hexane as a F u n c t i o n o f  Temperature  n-Hexane as a F u n c t i o n o f T e m p e r a t u r e  Conductivity of  of Temperature  .  .  n-Hexane as a .  .  .  .  14.  The c . d . f .  of Block A (Water,  d  15.  The c . d . f .  of Block B (n-Hexane,  .  microns)  d .  16.  The c . d . f .  of Block C (Water,  17.  The c . d . f .  of Block D (n-Hexane,  .  =  .  .  .  .  .  .  .  .  microns)  . .  . .  .  70  1.011 71  d p = 0.790 microns)  microns)  41  42  p  d  40  Function  = 1.011  p  .  .  .  .  72  = 0.790 73  18.  The c . d . f .  of B l o c k E  ( a l l Water)  74  19.  The c . d . f .  of  (all  75  Block F  n i x- H e x a n e )  X  Figure 20.  Page y/p  TpK a s a F u n c t i o n  of Temperature  for  Water  and n-Hexane 21.  22.  23.  24.  25.  T h e r m o p h o r e t i c V e l o c i t y i n Water as a of Temperature G r a d i e n t  76 Function 77  T h e r m o p h o r e t i c V e l o c i t y i n n-Hexane as a of Temperature G r a d i e n t  Function 78  G r a v i t a t i o n a l T e r m i n a l V e l o c i t y i n Water as a Function of Temperature . .  .  G r a v i t a t i o n a l T e r m i n a l V e l o c i t y i n n-Hexane as a Function of Temperature . . . . . . . . . C o - o r d i n a t e System and V a r i a b l e s f o r P a r t i c l e Regime  the  .  .  .  .  .  79  80  Large 137  Acknowledgment  The for  author wishes  the e f f o r t s  of  have  been  The of  Mineral  of  Forestry  all  Ms. this  thesis  Dr.  J.  and D r .  a r e a l s o due t o help, L.  advice,  Idler  Meisen,  without  this  project  from  the  Department  the  Faculty  Kozac  would  of  acknowledged. the  and  e n t i r e workshop  t h i s work would  staff  patience.  a s s i s t e d the author  in a recognizable  her e f f o r t s ,  Leja  A.  are also g r a t e f u l l y  K.  A.  appreciation  completed.  Engineering  their  sincere  Dr.  and d e t e r m i n a t i o n  a s s i s t a n c e of  Thanks for  express  his supervisor,  whose e n c o u r a g e m e n t never  to  form of have  in  English.  taken  presenting Without  considerably  longer. The w o r k financially of  by  a Research  reported  in this  the N a t i o n a l Assistantship.  xi  t h e s i s was  Research Council  supported i n the  form  Chapter  I  Introduction  When a t e m p e r a t u r e contains micron-size gradient  towards  the gas b u t This  i s not  causes a net  a p a r t i c l e by  imbalance gives  r i s e to  force  w h i c h moves  called  t h e Thermophoretic  a  The liquids two  of  present.  work,  thermophoresis The knowledge,  to  thermophoretic  the  phenomenon,  presence of  a temperature  imbalance i n the average  a momentum  flux,  known  convection currents  trans-  it.  a l s o known a s t h e at  in  gradient.  momentum  the f l u i d molecules which surround  This  Thermophoretic  a terminal  velocity  velocity. l i k e l y that  no a s s u r a n c e t h a t  therefore,  thermophoresis  existed  t h e many s i m i l a r i t i e s b e t w e e n  However, because of  certain differences the  was a n a t t e m p t  p r o j e c t was u n d e r t a k e n  no e x p e r i m e n t a l i n the  r e s u l t s are  to  thermal  This  the  extend  the  the  between  phenomenon w o u l d to  in  be  concept  liquids.  present  been r e p o r t e d its  a r e s u l t of  gas w h i c h  p a r t i c l e s move down  lower temperature.  as g a s e s b e c a u s e o f  t h e r e was  This  i m p o s e d on a s t a g n a n t  p a r t i c l e s , the  considered i t  fluids.  these s t a t e s ,  is  t h e p a r t i c l e down t h e g r a d i e n t  author  as w e l l  types  of  i s due e n t i r e l y  gradient  ferred to  (10"^m)  regions  as Thermophoresisj  gradient  evidence of  literature.  inconclusive.  In  One  because,  thermophoresis theoretical  addition,  the  author's  in liquids  s t u d y was f o u n d  i t was h o p e d t h a t  v e l o c i t y m i g h t be l a r g e e n o u g h  1  to  t o be i m p o r t a n t  the in  has but  of  2 engineering  problems where a l i q u i d  An e x a m p l e o f sensible  of  s i t u a t i o n i s the  heat exchangers where  sub-layer. the  t h i s type  If  has a l a r g e t e m p e r a t u r e  the p a r t i c l e v e l o c i t i e s should prove  of m i c r o n - s i z e p a r t i c l e s from The existence  initial  of  objective- of  to  by e x p e r i m e n t a l  of  the s t r u c t u r e of  g r a d i e n t s upon i t ,  is  unfortunately,  was  the of  experi-  liquids,  and  i n s u f f i c i e n t l y developed  Nevertheless,  a number  a t t e m p t s w e r e made t o d e r i v e a t h e o r e t i c a l e x p r e s s i o n f o r These,  removal  measurement  The m a j o r e f f o r t  p r o v i d e an e x a c t a n a l y t i c a l a p p r o a c h .  phoretic velocity.  the  large,  t h i s w o r k was t o p r o v e o r d i s p r o v e  because the m o l e c u l a r theory temperature  t o be v e r y  laminar  liquids.  thermophores i s i n l i q u i d s  the e f f e c t of  of  i n the  a n o v e l method f o r  the p a r t i c l e thermophoretic v e l o c i t i e s . mental  particulate fouling  l a r g e g r a d i e n t s may e x i s t  thermophoretic e f f e c t might provide  gradient.  proved  the  of  thermo-  t o be u n s u c c e s s f u l a n d  are not d e s c r i b e d i n the present t h e s i s . The tween t h e thermal  s e c o n d o b j e c t i v e was t o f i n d a n e m p i r i c a l  t h e r m o p h o r e t i c v e l o c i t y and t h e v i s c o s i t y , f l u i d d e n s i t y  conductivity,  particle  diameter.  because they  to form a s o l  temperature  These  temperature  gradient,  p a r a m e t e r s w e r e c o n s i d e r e d t o be  spherical  and  important  (1)  p a r t i c l e s were suspended i n a  w i t h a. v o i d f r a c t i o n i n e x c e s s o f  g r a d i e n t was c r e a t e d i n t h e l i q u i d  two h o r i z o n t a l , The  absolute temperature,  and  govern thermophores i s i n g a s e s .  Micron-size, liquid  r e l a t i o n s h i p be-  formation of  parallel natural  by t r a p p i n g  disks maintained at d i f f e r e n t  stagnant  0.999. it  A  between  temperatures.  c o n v e c t i o n c u r r e n t s was s u p p r e s s e d by  heating  3  the  upper  d i s k and c o o l i n g t h e b o t t o m  The velocities  experimental  was d i r e c t v i s u a l  of  scattered  of  light  light  particles  too small  phoretic  had t o  relevant meters  observation  (with  thermophoretic  a low power  particles.  The  even though  t o be d i r e c t l y v i s i b l e . a temperature  microscope)  r e s u l t i n g specks the spheres  The m o t i o n o f  gradient  them-  the  compared w i t h  a b s e n c e was c o n s i d e r e d t o be a m e a s u r e o f  the  their  thermo-  velocity. Since  it  measuring  particle positions  i n the presence of  in its  used f o r  r e f l e c t e d from the  i n d i c a t e d the  s e l v e s were  motion  method  one.  the equation  for  thermophoretic  v e l o c i t y was e m p i r i c a l ,  be b a s e d on a s t a t i s t i c a l a n a l y s i s o f parameter.  over  An e f f o r t  l a r g e ranges  was  in order  therefore to obtain  the e f f e c t  made t o v a r y  of  each  these  as v a l i d an e q u a t i o n  paraas  possible. The  s p h e r i c a l p a r t i c l e s which were used i n t h i s  polystyrene size  p a r t i c l e s were  were u s e d , water in  a s t h i s was  1.011  l i q u i d thermal  according perature  to  the  readily available.  microns  and n - h e x a n e ,  the only m a t e r i a l  were  i n which uniform,  Two  and 0 . 7 9 0 m i c r o n s . selected primarily  conductivity.  The  other  l i q u i d and t h e a b s o l u t e  g r a d i e n t s were a p p l i e d ,  study  average Two to  give  micron-  particle  different  diameters  liquids,  a large  liquid properties  temperature.  were  variation varied  Different  tem-  r a n g i n g f r o m 7 , 0 0 0 t o 3 0 , 0 0 0 °K. m ~ ^ .  Chapter  II  Previous  Although in  liquids,  the  a brief  p r e s e n t work review  i s concerned only with  this  i s done f o r  tant  c o n c e p t s , w h i c h have been w e l l  phoresis  to  reasons.  phenomenon  This  applicable  two  of  liquids.  First,  Second,  in liquids fails  to  Work  i n gases i s  the terminology  developed f o r  the e x i s t i n g  provide  thermophoresis provided.  a n d many  gases, are  l i t e r a t u r e on  an a d e q u a t e  impor-  also  thermo-  understanding  of  the  phenomenon.  Thermophoresis  1.  Due to  to  in  Gases  the aforementioned  a d i s c u s s i o n of  the most  reasons, this  important  section is  restricted  p a p e r s on t h e r m o p h o r e s i s  in  gases.  M o r e c o m p r e h e n s i v e a c c o u n t s h a v e b e e n p r e p a r e d by Waldmann  Schmitt  (2),  Springer  Basically, This  (3),  only  and Fuchs  of  (4).  one m e c h a n i s m c a u s e s t h e r m o p h o r e s i s  m e c h a n i s m c a n be r e a d i l y u n d e r s t o o d o n c e i t  mean v e l o c i t y a n d momentum o f temperature.  region of  Therefore,  experiences Two  in gases.  i s recognized that  the  f l u i d molecules are i n c r e a s i n g functions  molecules which o r i g i n a t e  the f l u i d s t r i k e a p a r t i c l e w i t h  t h o s e w h i c h come f r o m t h e  and  cooler region.  a n e t f o r c e w h i c h moves  it  4  hotter  g r e a t e r mean momentum' t h a n As  a result,  down t h e  regimes have been d e f i n e d f o r  i n the  the  the  temperature  convenient  particle gradient.  analysis  of  5 thermophoresis. parameter,  Their  definition  i s b a s e d on t h e  l i m i t s of  a single  Knudsen Number ( 3 ) , w h i c h i s d e f i n e d a s : •  the  K n = ^  [1] P  where  d  is  the  p a r t i c l e diameter  f l u i d molecules. regime  (Kn  »  The  1),  large particle  l i m i t s of  sometimes  between  these  c a l l e d the  two  0.1  < Kn < 10 and i s s o m e t i m e s  The  two r e g i m e s d i f f e r  i s t h e mean f r e e  Kn c o r r e s p o n d  regime (Kn << 1 ) ,  A zone e x i s t s  and L  to  the  referred  to  as t h e  used to  from  and  the  region.  approximately  transition  d e s c r i b e the  the  -particle  continuum  l i m i t s which ranges  of  small  free molecule region,  a l s o known as t h e  i n t h e model  path  region.  transfer  of  momentum b e t w e e n m o l e c u l e s a n d p a r t i c l e s .  a.  The  In that  Small  this  regime,  t h e y do n o t  lisions  are  P a r t i c l e Regime  affect  infrequent  the v i c i n i t y of  the  (Kn  ing  to  1)  p a r t i c l e s a r e c o n s i d e r e d t o be s o  conditions and t h u s  i n the gas.  t h e number  a n y one p a r t i c l e i s v e r y  reflected molecules interacting with Consequently,  »  of  low.  minute  Molecule-particle  col-  r e f l e c t e d molecules The  l i k e l i h o o d of  the  is  negligible.  m o l e c u l e s i m p i n g e on a p a r t i c l e w i t h momenta  correspond-  conditions Although  prevalent this  incident molecules  in  i n the bulk  of  r e g i m e was a n a l y s e d  twentieth  century  (2,5,6),  available  and i s o u t l i n e d  a more r i g o r o u s here.  The  the  gas.  i n the e a r l y  part  of  the  s o l u t i o n has r e c e n t l y  equation  predicting  become  thermophoretic  6 v e l o c i t y was  obtained  from K i n e t i c Theory.  was c o m p u t e d by summing t h e momenta t h e momenta  of  Several  of  r e f l e c t e d molecules but  and d i f f u s e  models  appear  have  to  have  thermophoretic  Derjaguin  ture  independently  velocity,  and v j  gradient  denote  far  Equation the  y_^,  - w^^ v  the a b s o l u t e  [2]  In ticle  surmomenta  tend to  e l a s t i c (specular) the  the  fluid By  tempera-  definition,  molecule-particle  reflected molecules  data  to  are  distribution  suggest  velocity within  (Kn  <<  that  5 per c e n t  of  1)  the molecules which are  interact with  r e f l e c t e d molecules  and t h e  (10).  L a r g e P a r t i c l e Regime  regime,  the  m  particle, respectively.  a s Kn -* °°  for  regime:  c a n be d e s c r i b e d b y a M a x w e l l i a n  value  Chapman  J  temperature  predicts a thermophoretic  this  and Mason a n d  particle  There are s u f f i c i e n t experimental  The  c a l c u l a t i n g the  (8),  i n the small  away f r o m t h e  speeds  experimental  b.  particle  t h e same e q u a t i o n  a n d one when t h e d i r e c t i o n s o f  random and t h e function.  minus  p r i n c i p l e o n e s a r e b a s e d on e l a s t i c  derived  the c o e f f i c i e n t " a " e q u a l s zero f o r collisions  the e n t i r e for  and Bakanov  =  w h e r e T^  transfer  incident molecules  been p o s t u l a t e d the  n e t momentum  collisions.  Waldmann(7), (9)  the  the r e f l e c t e d molecules over  face. the  of  The  r e f l e c t e d by t h e  i n c i d e n t ones s i n c e t h e  i n the v i c i n i t y of  number  a p a r t i c l e i s very  high.  of  par-  7 Therefore, in  the bulk  Since of  the  it  of  effects,  be u s e d .  partly times  the  f l u i d but  i s very d i f f i c u l t  these  not  i n c i d e n t molecules are  As  the  In  with to  the  order  gradient  point  acquires  only  the  with due  for  for  simultaneous  occurrence  employed  Kn »  Therefore,  it  is  called  temperature  a function the  of  realize gradient  colliding  momentum  the  f l u i d where the  hot the  If  the  s u r f a c e would  thermal  gradient  some-  related  T h e mean momentum o f  gradient.  the  is  to  f l u i d molecules  point.  the  and  a similar  a n e t momentum f l u x  to  1 can-  transfer,  necessary  induces  a molecule from  temperature  for  t h e momentum  model  region of  surface.  model.  fluid  that  conditions  particle  has been d e v e l o p e d  primarily  the  the  p a r t i c l e rebound w i t h  f l u i d adjacent  the  the  Furthermore,  at  i n f l u e n c e d by  near  a m o l e c u l e coming from  a velocity  to  the  p a r t i c l e whereas  momentum.  stationary,  is  the  p a r t i c l e down t h e  gradient  in  thus  and n o t  upon c o l l i s i o n ,  momentum t o  the  is  this  on t h e  temperature  temperature  Hence,  mechanics,  to understand  r e f l e c t e d molecules face  account  a new m o d e l  particle surface.  surface  those  thermal creep ( o r s l i p )  the  a particular  the  to  a result,  that a temperature along  a l s o by  only  K i n e t i c Theory approach  b a s e d on c o n t i n u u m called  not  particle they  cold  exists  sur-  originated.  region  gives  region and  forces  p a r t i c l e were flow  up  creep v e l o c i t y .  tangential  the  to  the  held,  the This  flow  particle  s u r f a c e (2). Maxwell  (11)  derived  an e q u a t i o n  for  the  thermal  creep  velocity,  V V  =  1 J L - * L  4 T P  K  ds  ri 3  L J J  8 where  ^  i s  the  tangential  Epstein v e l o c i t y duction the  i n  the  Equation  of  [3]  as  equation  to  a  obtain  f o r  the  flow  k^  and  p e c t i v e l y .  k  are  p  Both  the  temperature  and  v i c e  the  tangential  the  and  past  2  f l u i d  d i s t r i b u t i o n This  i n  e f f e c t  solved  v e l o c i t y .  P  Fourier  of  T „  r e s u l t This  the  into  v e l o c i t y  Navier-Stokes f i n a l  equation  vT  -  K  r  the  p a r t i c l e  i n  i s :  - i  4 L  thermal  present  con-  along  y_  are  becomes  heat  gradient  E p s t e i n ' s  p a r t i c l e  only  the  t h i s  s o l u t i o n  2k.+k f p  thermophoretic  temperature  sphere.  and  c o n d u c t i v i t i e s  v e r s a .  creep the  a  the  s u b s t i t u t e d  thermal i n  f o r  He  = _ A __f  v  - t h  where  regime.  c o n d i t i o n  creeping  gradient.  expression  p a r t i c l e  obtain  boundary  an  p a r t i c l e  large  to  a  derived  large  equation  surface  served  (12)  temperature  c o n d u c t i v i t i e s ,  Equation  a f f e c t s  important  at  low  J  r e s -  because  [4]  that  ^  of  the  f l u i d  Knudsen  numbers. E p s t e i n ' s an  order  of  and  kp/kj:  —th  ^  w  magnitude  <_ 1 0 n  e  r  e  a  Brock tions  f o r  c o l l i s i o n )  equation of  (13). s  i  n  (15)  the  r e a l i t y , r e f i n e d jump  hydrodynamic  _  - t h  as  the  2  kp/k^  (a  (l+3C  M  His  ,kf+C  T  (  T  f  does  of  v e l o c i t i e s provided  Equation not  d e r i v a t i o n  f u n c t i o n  s l i p .  r e s u l t s  •> =>,  v e l o c i t y  E p s t e i n ' s  3_  "  thermophoretic  experimental  However,  temperature and  p r e d i c t s  the  vanish  by  type  equation  [4]  Kn  <  0.1  p r e d i c t s (14).  i n c l u d i n g of  w i t h i n  c o r r e c -  m o l e c u l e - p a r t i c l e  i s :  Knx  k p  u  j  Kn)(2kf+kp+2CTk  p  Kn)  P  T  j K  -  (-(--i L l 5 j  9 where  Cj  and  ature  jump  Equation  are  and  [5]  hydrodynamic  i s  experimentally p a r t i c l e  and  of  l i m i t e d and  thermophoretic  Derjaguin  Fourier (2)  and  s u l t i n g  that  d e r i v a t i o n  p a r t i c l e  well  C j ,  equation as  p r e v i o u s l y  Equation  [7] Both  an  has  order  of  the  2  same  noted  reduces Equations magnitude  an  a d d i t i o n a l  i n  to  (17)  Sk.+k F  of  , P  T  P  Kn  term  w i t h i n  i n  zero  Schmidt  t h e i r  r e -  i s :  jump  at  obtained:  m  y_ VT  of  the  Equation  an  the  and  equation  and  0.  i s  nor  temperature  =  pT„ K  the  the  to  [3]  term  [6]  the  when  Waldmann  f i n a l  of  Kn  regarding  and  f l u x  of  p r e d i c t  Equation  the  weakness case  0.  Equation  2k.+k +2CTkoKn f p T p  Equation [6]  +2CTk  Kn  t h i s  Their  of  not  and  for  [4]  c o r r e c t  =  temper-  determined  composition  heat  defined  the  be  does  that  included  only  are  unless  f o r  usefulness  Equation  It  claimed  d e r i v a t i o n  = _1  v  - t h  k^/k^  poorly  can  that  (13).  understood.  the  to  j u s t i f i c a t i o n  Yalamov i n  0.1  The  chemical  (16)  i s  t h e i r  are  and  f l u x  the  reduces  high  ,  This  <  included  This  surface  at  constants  v e l o c i t i e s  Kn  Bakanov  neither  Derjaguin the  and  on  [5]  gives  provided  instead  equation.  state  Equation  accounting  r e s p e c t i v e l y .  the  dependent  v e l o c i t i e s  constants  s l i p ,  because  equation  magnitude  i n c o r r e c t  are  f l u i d .  Brock's order  dimensionless  L  experimental  [5].  When  Kn  /  J  c o n s t a n t , =  0 ,  [6]. [7]  p r e d i c t  experimental  v e l o c i t i e s values  at  Kn  which <  0.1  are  w i t h i n  (13).  10 Neither [4]  to  equation [7]  microns  predicts zero v e l o c i t i e s at high kp/k^.  have been v e r i f i e d f o r  p a r t i c l e diameters  these t h e o r e t i c a l equations 1 S  mately  ProPort"ional zero  particle  2.  (as  it  to  the  for  that  c a n be made r e g a r d i n g  thermophoretic  temperature  velocity.  gradient  is  i n l i q u i d s ) or very  in  Liquids  In a l l  all cases,  and,  if  Kn i s  v^  is  independent  large,  approxiof  diameter.  Thermophoresis  As phoresis  noted  previously,  have y e t  no e x p e r i m e n t a l  been p u b l i s h e d .  c o n s t r u c t a t h e o r e t i c a l model d e r i v a t i o n of variable  the  isothermal  for  However, this  equations  f l u i d temperatures.  His  d a t a on l i q u i d Dwyer ( 1 9 )  phenomenon  limits  the a p p l i c a b i l i t y of than a micron  h i s work  to s o l s  independent  of  Equation.  form of  include  Brownian motion.  This  p a r t i c l e diameters  to a f l u i d drag  the term  plus a rapidly f l u c t u a t i n g force  p a r t i c l e v e l o c i t y and a c c o u n t s f o r  i r r e g u l a r m o l e c u l a r bombardment particular  with  the  freely  Dwyer p o s t u l a t e d t h a t  p a r t i c l e a c c e l e r a t i o n i s equal  he a s s u m e s t o be S t o k e s i a n ) is  to  to  (20).  B a s e d on N e w t o n ' s S e c o n d L a w , dimensional  by m o d i f y i n g  a n a l y s i s i s r e s t r i c t e d to  p a r t i c l e s which e x h i b i t a l a r g e degree of  much l e s s  thermo-  attempted  f o r Brownian motion  moving  latter  l a r g e a s 30  (18). T h e r e a r e two k e y o b s e r v a t i o n s  —th  as  Equations  i s sometimes c a l l e d  (which  term.  The  the  which causes Brownian m o t i o n .  N e w t o n ' s S e c o n d Law  one-  the  This Langevin  11 Dwyer considered a small interval of time, AO, during which only the fluctuating term was allowed to vary.  The interval was chosen  to be so small that any space or time variations in temperature during this period were regarded as n e g l i g i b l e . by imagining that the f l u i d  This model can be understood  is divided into a large number of very small  regions, each of which is at a d i f f e r e n t , but i n t e r n a l l y constant, temperature. of each AG.  The p a r t i c l e jumps from one region to another at the end When the p a r t i c l e is in a region, i t is affected  the f l u i d in that p a r t i c u l a r region.  only by  Therefore, the p a r t i c l e i s in a  constant temperature f l u i d during each AO, although the value of this temperature may change between time i n t e r v a l s . Dwyer then modified Chandrasekhar's solution (21)  of Langevin's  equation, which is v a l i d for the isothermal case, and derived a "displacement p r o b a b i l i t y d i s t r i b u t i o n " for the p a r t i c l e s after time 0. A "displacement p r o b a b i l i t y d i s t r i b u t i o n " is an equation which gives the p r o b a b i l i t y of finding a p a r t i c l e at a certain displacement ( f i n a l position minus i n i t i a l  position) from i t s i n i t i a l  point.  Dwyer's d i s -  t r i b u t i o n contains temperature as a function of position and time, but does not include the temperature gradient because of the assumption of constant f l u i d temperature in each AG.  He used this d i s t r i b u t i o n to  obtain an equation for the "mean square displacement," <A >, in one 2  dimension after time Q.The l a t t e r is the mean of the square of the dis placements of an i n f i n i t e number of p a r t i c l e s evaluated after time Q. Dwyer's equation i s : <A > = <fjj- J T ( A U ) ) ( l - e x p ( B U - o ) ) ) d C > P o 2  9  2  [8]  12 where 3TTCI U  c«  * - -s 2  P and k is the Boltzmann's Constant and m denotes the p a r t i c l e mass. P The " p r o b a b i l i t y averaging" operator, symbolized by <>, takes the mean of the enclosed function over an i n f i n i t e number of i d e n t i c a l all  s t a r t i n g at the same i n i t i a l  temperature.  particles  p o i n t , i . e . , a l l at the same i n i t i a l  Dwyer never makes this d e f i n i t i o n c l e a r .  The following p a r t i c u l a r cases of varying temperature were considered by Dwyer. Case A:  The temperature is a l i n e a r function of displacement. T  NA  [10]  A = X' - X'  [11]  K  =T  K  Q  +  where 0  T^  0  is the temperature at the i n i t i a l  J  point, x^, and N is a constant  temperature gradient. Dwyer substituted Equation [10]  into Equation [8] and found  that a constant temperature gradient does not change the mean square displacement from that of the isothermal (T^ ) value, i . e . , no thermophoresis for a constant temperature gradient.  there is  He explained  this r e s u l t by claiming that the square of the p a r t i c l e displacement i s increased by the same amount when the p a r t i c l e goes up the gradient  13 as  it  i s d e c r e a s e d when t h e  postulated this  that  the  difference  Case B:  The  p a r t i c l e g o e s down  same number  vanishes  temperature  of  i n the  the  gradient.  p a r t i c l e s go up t h e  averaging  Since  gradient  as  he down,  procedure.  is a linear function  of  the square o f  the  dis-  placement.  T  Although d e s c r i b e any Equation will  this  will  indicates Again,  shows  Equation  Equation  temperature [8],  the  particle will that  [8]  If  it  isothermal  becomes u p o n  does  t h e mean s q u a r e  When  displacement  M is negative.  Dwyer  a positive M indicates that initial  always [12]  point  into  regions  t a k e s , whereas travel  into  h a s no p h y s i c a l  not  the  of  is correct,  conditions  it  higher  a negative M  colder  regions.  meaning.  should reduce to  (22).  W i t h T^  +  g Q  par-  = T^Q,  func-  Einstein's Equation  integration: 2kT  <A > = 2  Ko  (  6Q  m  As 0 a p p r o a c h e s  it  d e s c r i b e s t h e mean s q u a r e d i s p l a c e m e n t a s a  temperature. under  Equation  valid,  distribution.  and d e c r e a s e i f  by a s s e r t i n g t h a t  the  [12]  M A 2  is mathematically  positive  move f r o m  +  no m a t t e r w h a t d i r e c t i o n i t  that  this  t i o n of  M is  always  temperature  [12]  Ko  T  i s used i n Equation  increase i f  ticle  =  physically realizable  [12]  explained  [8]  Equation  K  infinity,  8  _ I  e x  P(-  )  (4-exp(-Be)))  2  Equation  [13]  tends  to  [13]  14 2 k T  Ko  0  [14]  <A > = ir-M2  Equation sional  [14]  is  identical  to  Einstein's  Equation  mean s q u a r e d i s p l a c e m e n t i n a u n i f o r m  therefore, However,  that  the  steady  the v a l i d i t y  of  state  the  component  transients  for  fluid.  of  the  one-dimen-  This  indicates,  Equation  i n Dwyer's  [13]  is  equation  correct.  has  not  been e s t a b l i s h e d . Although have is  equally  valid for  gases.  should hold for  gases,  even f o r  therefore,  is  Therefore,  both  p a r t i c l e s which e x h i b i t  to support  constant  variables  tion  temperature  incompatible with  are poorly  are d i f f i c u l t  found  in his As  affected interval, no n e t  AG.  of  fluid.  experimental  d e f i n e d a n d some o f  that  derivation final  thermophoresis  Brownian motion Dwyer's  exists  in  equation,  observations. to the  a major  previously,  by f l u i d a t However,  if  Dwyer a s s u m e d t h a t  a constant the  temperature  particle lies  imbalance can a r i s e i n the average  particle  from  force  possible.  is  of  However,  gradients.  However,  in his  h i s a n a l y s i s and h i s  a n a l y s i s appears  to f o l l o w .  conclusions  be c o r r e c t steps  although  in his  conceptual  deriva-  error  was  work.  stated  only  types  each s t e p  a l a r g e degree  Dwyer's mathematical his  [8]  a b e a r i n g on l i q u i d t h e r m o p h o r e s i s ,  equation of  Dwyer u s e s E q u a t i o n  the f l u i d m o l e c u l e s , and,  a p a r t i c l e was during  each  time  i n an i s o t h e r m a l  momentum  fluid,  transferred  as a r e s u l t ,  no  What Dwyer h a s c a l c u l a t e d , t h e r e f o r e ,  to  the  thermophoretic is  the  effect of a non-uniform f l u i d temperature upon the mean square d i s placement of p a r t i c l e s under the assumption that they are only i n f l u enced by a series of isothermal  fluids.  Chapter  III  Apparatus  1.  Introduction  The  apparatus  w h i c h a s m a l l volume between  c o n s i s t e d b a s i c a l l y of of  liquid,  two h o r i z o n t a l ,  e l e c t r i c a l l y whereas circulating (vertical)  The  t h e b o t t o m one was  kept  temperature  convection currents.  c o n t a i n i n g the  parallel disks.  cooling water  through  gradient The  a cylindrical cell  it.  This  i n the  p a r t i c l e s , was  t o p d i s k was at  remainder of  a lower temperature  sol without  the equipment  functions  and i n c l u d e d a l i g h t s o u r c e ,  and c o n t r o l s  the d i s k  temperatures.  The observing  p a r t i c l e s w e r e made  the  scattered light  arrangement  i s sometimes  for  p a r t i c l e s ranging  viewing  diameter. its  be  through  the  a low-power  from approximately  d e p t h made i t  r e g i o n between  ideal for  p o s s i b l e to the d i s k s .  creating  microscope,  microscope.  the  (23)  0.5  to  and i s 1.5  particles in  wall  effects  and This suitable  microns  present study  observe Cell  axial  i l l u m i n a t i n g them  c a l l e d an U l t r a m i c r o s c o p e  The U l t r a m i c r o s c o p e was  large focal  c e n t r e of  " v i s i b l e " by  by  f u l f i l l e d measure-  ment and s u p p o r t for  trapped  heated  e s t a b l i s h e d an  stagnant  in  in  because the  could  thus  minimized. The  reticle  into  p a r t i c l e v e l o c i t y was m e a s u r e d by the m i c r o s c o p e e y e p i e c e .  16  incorporating  S i n c e the  a  grid  d i s t a n c e between  the  17 grid  l i n e s was  required  2.  The  for  known,  velocity  a p a r t i c l e to  travel  c o u l d be c a l c u l a t e d f r o m halfway  between  the  time  them.  Cell  An o v e r a l l disks  the  and t h e  view  of  the  cell  b a s e w e r e made f r o m  spacer r i n g c o n s i s t e d of  is given  in Figure  leaded b r a s s .  The  1.  The  optical  two  tube  a,id  b o r o s i l i c a t e p y r e x g l a s s and p l e x i g l a s s ,  res-  pectively. Table 1 gives and r e p r e s e n t a t i v e are heat  given  for  the  thermal  values  conductivities  f o r water  completeness  and f o r  and h e x a n e . later  these  These  materials  conductivities  use i n e s t i m a t i n g  the  radial  flux.  Table Thermal  conductivities  of  1 the Apparatus Thermal  Material  The  W  119  1  (24)  Glass  0.880  (25)  Plexiglass  0.208  (24)  Water  0.600  Hexane  0.130  optical  a rubber  provided  Materials  Conductivity  J nf  Brass  over  of  tube  0-ring.  a tight  seal.  rested  The The  firmly  l a t t e r was top  disk  upon  the  b a s e and f i t t e d  inset  i n the  bottom d i s k  rested  on t h e  spacer r i n g .  tightly and This  18  i A  i  A'  1  c  1  Figure 1 An Overall View of the Cell  19 ring,  w h i c h was  not  p h y s i c a l l y attached  to  either  parallel  a n d s e p a r a t e d by a known d i s t a n c e .  d i s k was  parallel  to  tioned horizontally A priori Problems were diameter. effects However,  it  a.  b.  diameter provide  the  cell  Details  of  the  two  bottom  c o u l d be  posi-  difficult.  in selecting a suitable  h a d t o be l a r g e e n o u g h the  the  d i m e n s i o n s was  heating  to minimize  and c o o l i n g  the  prevent  liquid.  a r e a s o n a b l e b a l a n c e between  s u c c e s s f u l apparatus  wall  equipment. excessive  A number  these  disk  of  apparatuses  requirements  could  presented  the  are  in  subsections.  Tube  The  optical  tube  of  7/8".  the  The  had an o u t s i d e  ends were g r o u n d  The  Base  The  base, which provided  in Figure 2 with  water  channel.  smooth,  one  tube  and  inch lengths  of  1"  to  to  for  inside  1/4"  the I.D.  right  on t h e  the  cell,  i n d i c a t e the  turbulent  the base through 3/8" O.D.,  and an  sit flat  a stable foundation  was a d d e d t o e n s u r e left  of  p a r a l l e l , and a t  the bottom d i s k o u t l i n e d  The b a f f l e  water which entered  Approximately  diameter  tube a x i s which a l l o w e d the  shown  ing  the  s c a t t e r e d l i g h t by  Optical  to  disk faces  the  base.  particularly  space f o r  The  diameter angles  l e v e l l i n g the  s p e c i f i c a t i o n of  before  be a c h i e v e d . following  Hence,both  kept  face of  a l s o had t o be s u f f i c i e n t l y s m a l l t o  absorption of were b u i l t  by  encountered  This  and t o  the base.  The  disk,  flow  of  two h o l e s copper  base.  is  cooling coolshown.  tubing  20  Figure 2 The Base  21 v/ere s o l d e r e d i n t o feet  these  of polyethylene The  tubing  three  holes  levelling  screws.  This  levelling  the base.  c.  to  The Bottom  Disk  The  disk  bottom  a tolerance of less  The  h o l e s and c o n n e c t e d t o t h e w a t e r (1/2" O . D . ,  3/8"  s e t i n the t r i a n g u l a r was f o u n d  I.D.). pattern  t o be t h e m o s t  i s depicted than 0.001"  i n Figure  d.  with  t h e base s u r f a c e  The S p a c e r  Figure disks The  3.  I t s f a c e was p o l i s h e d  t o e n s u r e an even a x i a l  to within  that  r i n g was made f r o m  0 . 1 ° of a r c .  by a c o n s t a n t  distance of 0.118"  p l e x i g l a s s because t h i s m a t e r i a l  i n excess of 100°C without  i t s c o e f f i c i e n t o f thermal  was s u f f i c i e n t l y l o w t o p r e v e n t  less  than  flux.  i t s f a c e was  softening  For ease o f assembly, the i n s i d e diameter  radial  of the o p t i c a l  angles  to  0.001".  than  or axial  the r i n g ' s  r i n g were machined a t r i g h t a tolerance of less  expansion,  with-  noticeably.  8.1x10  thermal  (0.003m).  could  -5  problems.  heat  t h e s p a c e r r i n g w h o s e f u n c t i o n was t o k e e p t h e  p a r a l l e l and s e p a r a t e d  In a d d i t i o n ,  for  Ring  4 shows  stand temperatures  accommodated  s t a b l e arrangement  d i s k was s o l d e r e d t o t h e b a s e i n s u c h a m a n n e r  parallel  b a t h b y two  1 °K  (24),  expansion  o u t s i d e d i a m e t e r was 0 . 0 1 " tube.  to i t s a x i s ,  The f a c e s o f t h e  smooth,  and p a r a l l e l  22  Figure 3 The Bottom Disk  23  Figure 4 The Spacer Ring  24 e.  T h e Top  The Figure  5.  Disk  l a s t part The  face of  less  than 0 . 0 0 1 " .  I.D.  to a l l o w f o r The  coil  into  the  top  disk,  and hexane  Heating  a.  The  of  to  electrical with  an O.D.  After  and e a s e o f  not all  in  24K w i r e  shown  a tolerance  of  the  optical  t h e Top  tube's  parts  heating  5 but which had been  "Easypoxy," manufactured an a d e q u a t e  temperatures  seal  inserted by CONAP  against  i n excess of  are  water  100°C.  Disk  Coil  fluxes  w e r e d e s i r e d , i t was of  nearly  heating wire mandatory.  100°C.  The w i r e  necessary f o r  This  the  requirement,  the  use of  an  c h o s e n was p y r o t e n a x  24K  0.093". wire  has a r e s i s t a n c e a l l o y  approximately  turn,  in  assembly.  t h e s m a l l s p a c e i n s i d e t h e d i s k , made  of  i s shown  in Figure  electrical  with  twenty-five  by an e l e c t r i c a l l y i n s u l a t i n g l a y e r This,  d i s k which  s m a l l e r than  epoxy g l u e p r o v i d e d  reach temperatures  Pyrotenax resistance  which are  later.  high heat  combined w i t h  top  t h e d i s k c o n t a i n e d an e l e c t r i c a l  E l e c t r i c Heating  Since disk  of  1/16"  expansion  and c o u l d w i t h s t a n d  The  the  was  i t was f i l l e d  This  was  O.D.  part  in detail  cell  d i s k was a l s o p o l i s h e d t o  thermal  hollow  Incorporated.  top  Its  the  this  and a t h e r m o c o u p l e ,  discussed  3.  of  i s coated with  h a s a maximum  power  a thin rating  times of  that  of  (with  a specific  copper),  compressed magnesium  layer of  core  17  of  copper.  watts  surrounded oxide.  Although,  per f o o t ,  i t was  the found  Figure 5 The Top Disk  26 that,  with  care,  A flat wire,  coil,  figure  flux.  The  nected to coil  b.  the for  to  the e n t i r e  ends o f  c o u l d be  the  the c o i l  i n s e r t i o n of  Figure 6. ammeter, circuit  circuit It  delivered  the  top d i s k  used f o r  It former  to  was  reasons.  transformer  volts  amps) toring  out  of  left  This  axial  coil  heat  t h e d i s k and  con-  i n the centre  powering  the heating a Variac,  latter  two w e r e  The m a i n s c u r r e n t  coil  of  is depicted  a transformer, incorporated (117  volts,  and c o n t r o l l e d by a S u p e r i o r 117  volts. the  the  current coil  in  an  into  the  60 h e r t z )  was  Electric  Variac  controlled  power d i s s i p a t e d i n t h e  necessary to pass the primary to a l e v e l  This  through  a  coil.  trans-  could tolerate.  was a Hammond 167P6 w i t h maximum s e c o n d a r y  ratings  This  of  6.3  and 5 amps. Since  former  p r o d u c i n g an e v e n ,  upwards  by a d j u s t i n g  reduce the v o l t a g e  transformer  disk.  Apparatus  The  temperature  top  pyrotenax  thermocouple.  Company V a r i a c w h o s e r a n g e was 0 t o the  inches of  the  s p a c e was  c o n s i s t e d b a s i c a l l y of  safety  to  bent  A small  a  and a v o l t m e t e r . for  thereby  were  power s u p p l y .  six  i n s i d e face of  face,  Associated Electrical  The  exceeded.  containing approximately  was s o l d e r e d f l u s h  covered almost  the  this  the  present, were  coil  an AC v o l t m e t e r  incorporated  devices This  c o u l d e a s i l y be o v e r l o a d e d ,  to  into  provide  circuit  (0  to  10 v o l t s )  the c i r c u i t .  a measure  of  had t h e a d v a n t a g e  and ammeter  They were  t h e power of  even w i t h  the (0  employed  u s e d by  easy c o n t r o l  the  over  trans-  to as  10 moni-  coil. the  power  Lino  OSwitch  7*  Fuse  m. Ammeter Hooting • Coil Voltmotor  Neutral O  Transformer Variac  Figure 6 The E l e c t r i c a l  Apparatus  ro  28 dissipated siderably  in  The  and  Cooling  The  latter  an  internal  of  by  was  a  heating  pump,  which  and  three  litres  to  per  top  O.D.)  and  inside  the  top  adjusting  face  EMF  current  the  much  was  con-  smaller  i t  water  part  to  of  the  a  heat  through  Constant  was  passed  at  the  a  i t  and  from  disk  through  served at  this a  a  this  Bath  with  coil  and  near  7°C.  mix  the  to  rate  of  means  of  was  reservoir.  Temperature  temperature  bath,  bottom  sink  An bath  approximately  Measurement  disk  of  (in  placed was  temperature  space the the in  hot space  a  Since  within  junction left  with  in  a  by  Such the  soldered  the  heating  filled  Leeds  and  current-carrying EMF  a  top  was  flask  thermocouple  temperature.  measured  thermocouple.  thermos  the  the  was  limitations  measured  insulated, disk  as  NB-515  maintained  formed  testing,  Potentiometer. trically  Tap  Copper-Constantan  was  resulting  primary  minute.  because  junction  than  function  Lauda  thereby  circulate  Temperature  The  the  volt  Disk  to  coil.  electric water  had  l i t r e  10  was  welding  117  c i r c u l a t i n g cooling water  water  selected  the  accurate  Bottom  disk  bath  (0.040"  more  the  bottom  accomplished  5.  Varying  current.  The  the  coil.  simpler  secondary  4.  the  with  to  small disk. the  coil).  28  pyrotenax a direct  gauge  size  was  After centre The  ice water  Northrup  was  a  of  cold and  the  8690  M i l l i v o l t  wire  was  elec-  indication  of  29 It since  6.  disk  The  Light  The  p a r t i c l e s were observed therefore  the b e s t p a r t i c l e  lamp's  illumination, a small  cent of  illumination.  mately  by  consumption of  the  from the  The  found  l o n g and 1 2 5 ° o f  The  flux  uncovered  outside of  the  arc  in  the light  satisfied (Model  F6T5-  h a d an O . D .  to give  the  i n s u l a t i n g tape  (approximately  S i n c e the  tube.  ren-  obscured  white  lamp  long,  In o r d e r  initial  available after  optical  one  the  Company.  black  a r e a was  that  best source that  6 watts.  covered with  luminous  source  convection currents  l u m i n e s c e n t s e c t i o n was 7 - 1 / 2 "  s e c t i o n 1"  lumens.  and  source of  an e x c e s s i v e l y s t r o n g  Electric  the a v a i l a b l e s u r f a c e a r e a ) .  12  The  the bath water  "Coollite" florescent  the Western  t h e b u l b was  was 260 l u m e n s ,  of  An u n d e r p o w e r e d  i t was f u r t h e r m o r e  t h e s e c r i t e r i a was a s m a l l  and a power  that  required.  l i g h t and c r e a t e d s p u r i o u s  By e x p e r i m e n t a t i o n ,  1/2",  disk  cooling water.  v i s u a l l y and a l i g h t  p a r t i c l e s i n v i s i b l e whereas  scattered  of  bottom  Source  i n t e n s i t y was  The  inch  was e s t i m a t e d f r o m  a.  the  flow  i n the  later.  CW-HH) m a n u f a c t u r e d  for  the  Equipment  cell.  all  place a thermocouple  Optical  dered  gave  temperature  be d i s c u s s e d  correct  the  i m p o s s i b l e to  i t was n e c e s s a r y t o m a x i m i z e  bottom will  was  t a p i n g was  placed approximately  correct  except  4-1/2  lamp  of  per  output approxione  30 b.  The M i c r o s c o p e  The m i c r o s c o p e , w h i c h parts  s u p p l i e d by t h e G a e r t h e r  arrangement  in Figure  an M239 e y e p i e c e a n d M226 o b j e c t i v e  This  arrangement  was l i s t e d  tion  power o f 27 a n d a f o c a l lens.  These  by G a e r t h e r length  values  was a s s e m b l e d  lens.  The  an M193F u p p e r  as p r o v i d i n g  a total  o f 0.1374m between  a r e f o r an o p t i c a l  liquid.  Figure The  7  Microscope  micro-  eyepiece adapter.  magnifica-  t h e s u b j e c t and  path  i n a i r but  d i d n o t seem t o c h a n g e g r e a t l y when t h e s u b j e c t l a y b e h i n d a s m a l l of  from  I t was a m o n o c u l a r  r a c k a n d p i n i o n f o c u s i n g M101A  on an M194 9 0 ° e r e c t i n g p r i s m w i t h  the o b j e c t i v e  7,  S c i e n t i f i c Company.  b a s e d on an a d j u s t a b l e  scope tube w i t h was m o u n t e d  i s shown  they  layer  The of  m i c r o s c o p e was m o u n t e d on an M309 s u p p o r t  an M330 Rod a n d C o l l a r w h i c h  three  directions.  pounds  and,  The  entire  as a r e s u l t ,  The  grid  allowed  c o n s t r u c t i o n weighed  c o u l d be e a s i l y  reticle  i n the  p a r a l l e l , equispaced  entire  vision.  mentally  found to  The  to  a distance  focus.  The  r e t i c l e and t h e  so t h a t  the  l i n e s c o u l d be p l a c e d p a r a l l e l  It "visible" cell  was  eyepiece  f o u n d by e x p e r i m e n t  when t h e  horizontal  and t h e m i c r o s c o p e ' s  angle  optical  means  t o move i n  less  than  custom-made  l i n e s which  d i s t a n c e between  be e q u i v a l e n t  by  all  twenty  moved.  e y e p i e c e was  and had f i f t e e n f i e l d of  the microscope  stand  covered  these of  0.0004m  that  the  between  p a t h was  the  Gaerther  almost  l i n e s was  c o u l d be r o t a t e d to  by  at  the  experi-  the  point  about t h e i r  edges  of  the  incident  approximately  light 110°.  axis  disks.  p a r t i c l e s were most  the  of  to  clearly the  Chapter  Physical  1.  Properties of the Experimental  Materials  The P a r t i c l e s  The styrene  latex  extremely 1.011  spherical  p a r t i c l e s used i n t h e p r e s e n t study were  a n d s u p p l i e d b y t h e Dow C h e m i c a l Company.  ( L o t Number L S - 1 1 3 8 - B )  ( L o t Number L S - 1 1 1 7 - B ) .  per cent confidence l i m i t s According  and d  The e r r o r bounds  p  poly-  s i z e was dp =  = 0.790 ± 0.001  c o r r e s p o n d t o t h e 95  a n d w e r e c a l c u l a t e d f r o m d a t a p r o v i d e d b y Dow.  t o Dow, t h e s p e c i f i c  ( 1 0 4 8 kg m~ ) a t room t e m p e r a t u r e .  t h e a c c u r a c y o f t h i s v a l u e was g i v e n . specify  Their  u n i f o r m and two d i a m e t e r s were s e l e c t e d f o r t h i s w o r k ,  ± 0.001 microns  microns  1.05  IV  the p a r t i c l e thermal  gravity  of the p a r t i c l e s i s  Unfortunately, Furthermore,  conductivity but Perry  no i n d i c a t i o n o f  the supplier d i d not (24) g i v e s a value  o f 0 . 1 2 8 J m~ s~ K'" . 1  2.  lo  1  The L i q u i d s  Water and n-hexane number o f r e a s o n s . different  thermal  (CgH^)  Basically,  were used i n t h i s  study  they were chosen because o f t h e i r  c o n d u c t i v i t i e s , as i s e v i d e n t from Table  In a d d i t i o n , t h e i r v i s c o s i t i e s  for a  and d e n s i t i e s a r e s i m i l a r ,  damage t h e c o n s t r u c t i o n m a t e r i a l s , a n d t h e y  1 (page  17).  they w i l l  not  are non-toxic.  Measurements o f p a r t i c l e v e l o c i t y were t a k e n a t  32  very  different  33 points fore  within  the  necessary  conductivity be  found  in  available  by  the  British end  Dr.  means A.  at  different  equations of  of  Kozak  "Physical  for  an  empirical  regression  of  the  All  Faculty  graphs  Properties" of  the  temperatures.  viscosity,  temperature.  a  (26).  interruption  of  When  density,  no  equation  the  fitted  at  equations  chapter,  pages  37  there-  thermal  equation  written  Forestry  was  and  simple was  programme of  It  to  and  The are to  could  the  made  University presented  42,  at  to  avoid  were  found  text.  Water  i .  Viscosity  Three in  the  to  within  equations  literature  90°C),  1  the  equation  per  is  T^  is  responding  given  the to  i i .  Data Company  cent  =  predicting  (24,25,27).  equations  1  where  thus  literature, by  the  unnecessary  a.  obtain  Columbia  of  and  functions  the  data  of  to  as  available  cell  by  in  the  were  Although  temperature  greatly  Perry  the  (24)  viscosity the  and was  predicted  range  different  of  of  in  water  viscosities  for  this  21.482((Tc-8.435)+(8078.4+(Tc-8.435)2)  temperature  Equation  [15]  expressed is  shown  in in  degrees Figure  ( 5 ° C ±TQ  interest  complexity.  adopted  agreed  The  ±  simplest  work:  1 / 2  Celsius.  )-1200  The  curve  [15]  cor-  8.  Density  for  Handbook  the  (25).  density By  of  means  water of  the  were  found  previously  in  the  Chemical  mentioned  Rubber  programme  34 an e q u a t i o n was f i t t e d range  1 5 ° C <_ T Q  to  these data f o r  one d e g r e e i n t e r v a l s  average d e v i a t i o n s  above-mentioned data)  between  data are l e s s  i s shown i n F i g u r e  i i i .  Thermal  Data were i n Appendix  deviations  [16]  2  r e s u l t s from t h i s  than  1 per c e n t .  The  e q u a t i o n and curve  the  (without  the  Conductivity  taken from t h r e e  sources (24,25,28)  by  between  produced from Equation  [17]  listed  t h e r e g r e s s i o n programme w h i c h  Tc-(7.15006xl0"6)  the c a l c u l a t e d thermal less  and a r e  pro-  equation:  = 0.567558+0.001862289  d a t a were on t h e a v e r a g e  b.  I  9.  duced the f o l l o w i n g e m p i r i c a l  The  the  They were f i t t e d  I.  kf  the  <_ 9 5 ° C g i v i n g :  p = 999.168-0.00426  The  in  I  [17]  2  c o n d u c t i v i t i e s and  than 1 per c e n t .  a r e shown i n F i g u r e  The  d a t a and t h e  the curve  10.  Hexane  A commercial grade of was w a t e r .  This  amounted  to  h e x a n e was u s e d w h o s e g r e a t e s t less  impurity  than 0 . 0 5 per cent w h i l e a l l  i m p u r i t i e s were p r e s e n t i n the p a r t s  per b i l l i o n  range.  other  35 i.  Viscosity  Data f o r 29,30) the  and a r e  hexane  listed  v i s c o s i t y were  i n Appendix  I.  taken from The  three  sources  r e g r e s s i o n programme  (25, gave  equation:  y = 0 . 0 0 1 1 7 - 0 . 0 0 0 0 0 2 9 T^  w h e r e T^  is  the  temperature  (as  previously defined)  The  data  and E q u a t i o n  ii.  were found  [18]  i n degrees  to  average  d a t a were  i n Appendix  I.  averaged  less  the data  and E q u a t i o n  Thermal  It  was v e r y  Sherwood with  than  i i i .  for  than  Deviations 1 per  cent.  11.  t a b u l a t e d by Timmermans The  d i f f e r e n c e s between  ature  less  a r e shown i n F i g u r e  programme  defined  p = 931.5-0.928  The  Kelvin.  Density  These listed  expressed  [18]  the (31)  the  Figure  12  and a r e  also  equation:  TK  [19]  p r e d i c t e d d e n s i t y and t h e  1 per c e n t .  provides  physical  data  a comparison  between  [19],  Conductivity  d i f f i c u l t to o b t a i n c o n s i s t a n t data  thermal gave  the  (29)  conductivity  of  some e x p e r i m e n t a l  hexane.  However,  from the Reid  liter-  and  values which agree reasonably  t h o s e u s e d by S a k i a d i s a n d C o a t e s  (32)  who o b t a i n e d an  well  empirical  36 equation  f o r  Appendix  I  and  can  be  the  (with  alkane a  s e r i e s .  value  d e s c r i b e d  by  from an  k  The t i o n  d e v i a t i o n s [20]  thermal  and  averaged the  c o n d u c t i v i t y  that  the  f  derived  = 0.140-0.00044  than  According i s  groups  of  International  equation  less  d a t a .  Both  2  per to  by  T  are  C r i t i c a l the  recorded Tables  Equation 1/5  i n (33))  programme:  [20]  c  cent.  approximately  data  Figure [ 2 0 ] , to  1/4  13  shows  hexane that  Equa-  has of  a  water.  Figure 8 V i s c o s i t y of Water as a Function of Temperature  990 ro I  980  970  960  20  40  80  60  Figure 9 Density of  Water as a F u n c t i o n o f  Temperature  0-70  1  f  —I  1  Figure Thermal  Conductivity  of  :  T  10  W a t e r as a F u n c t i o n  of  Temperature  40  Figure Viscosity  of  n-Hexane  as  a  11 Function of  Temperature  41  Figure Density of  n-Hexane  12  as a F u n c t i o n o f  Temperature  42  Figure Thermal  Conductivity  of  13  n-Hexane as a F u n c t i o n  of  Temperature  Chapter  Experimental  1.  Preparations  The  for  s o l was  a  prepared  as f o l l o w s .  e i t h e r water  f o r m e r was u s e d , boil, of  it  Method  Run  Dow s u s p e n s i o n , w h i c h c o n s i s t e d o f i n one l i t r e o f  latex  o r hexane  cool  under  of  the  experimentally One  top  d i s k and e f f e c t  found  drop of  not  to  require  drop  of  the  This  When  heated  procedure  tended  the  concentrated  particles in water,  distilled,  cover.  the d i s s o l v e d gases which o t h e r w i s e  One  and w e l l m i x e d .  had b e e n p r e v i o u s l y  and a l l o w e d t o  underside  V  to  axial  to  flux.  placed  the a brisk  expelled  form bubbles  heat  was  most  on  the  H e x a n e was  degassing.  Dow s u s p e n s i o n h a d a v o l u m e  of  approximately  _5 5x10  1.  S i n c e the volume  Dow as 0 . 1 , Therefore, fraction  the a c t u a l one d r o p  between  volume  per  transfer After  to  keep o u t  of  runs  with  sol  the  l i t r e of water  before  it  from  this  because water  inhibited.  sol  had b e e n p r e p a r e d dust.  With  proper  became c o n t a m i n a t e d .  a  covered  p a r t i c l e s p e r d r o p was g i v e n  p a r t i c l e s p e r d r o p was  was  airborne  necessary during  of  0 . 9 9 9 9 9 0 and 0 . 9 9 9 9 9 9 .  was e v e n c l o s e r t o u n i t y particle  f r a c t i o n of  container  run.  43  to  5xl0~^l.  produced a s o l w i t h In h e x a n e , and h e x a n e  the  are  a  void  this  sol  fraction  make-up  covered  lasted a  A s m a l l s y r i n g e was provide  void  i n s o l u b l e so  i t was s e t a s i d e a n d care,  by  number  filled  i n the c e l l  if  44 After The  top  which  disk  was  cedure which  was  drawn  ensured was  disk's  The levelling within ment Extra  base  procedure  mained  in  the  acted  the  cell  a  Fisher  and  located the  added was  to  was  on  bottom  four and  flat  filled  on  under the  This  the  top  spacer inch  3/4  to  with  s o l .  d i s p l a c i n g some  discarded.  trapped  the  sol  prodisk  ring. above  compensate  after  one  surface.  disk  to  top  often  before  to  a  the  for  any  The  run  the  flows  disk  were  power  was  fed  the  pyrotenax  power  top  grid could  disk  essentially was  steady  switched focused  lines then  temperature  was  tap  was  set  to  heating  versus  state  re-  begun.  Variac  the  This  of  the  of  align-  apparatus  start, to  to  microscope.  the  reached  the  the  before  the  the  parallel  the  bottom  of  stability.  that  started,  means  (sensitive  disks'  increase  the  by  through  ensure  run to  m i c r o s c o p e was of  The  the  plot  the  horizontally  lines  and  cell  the  placed  grid  minutes  reading  disk.  and  reservoir  cooling water  fifteen  and  thereby  reticle  repeated  experimental  ring  a  its  bath  minutes  v/as  12-000 c i r c u l a r l e v e l  the  that  cell  approximately  as  hours  An  were  seated  kept  to  Finally, spacer  bubbles  Three  indicated fifteen  syringe  position.  pre-selected  coil.  second  correct  to  its  Approximately a  of  was  position,  was  and  the  into  p l a c e d when  checked with  weight  air  removed,  evaporation.  degree)  entire  water  no  a  level  by  was  eased  with  surface  screws  One  was  off that  liquid  disk  then  liquid  upper  of  top  correctly  The  loss  the  time  approximately  on. on  the  aligned  commence  at  region with  any  the time.  inside edge  the of  45 2.  The  Run  The periods  of  f l u o r e s c e n t lamp was time to minimize  the  turned on. radiant  It  heating  was u s e d o n l y  for  of  and  the  liquid  short  particles. The  experiments  were conducted o n l y  s c a t t e r e d b y t h e p a r t i c l e s was Some t i m e had t o be a l l o w e d f o r darkness  and t o  was u s e d t o  l o c a t e the  provide  too f a i n t  at  n i g h t s i n c e the  t o be v i s i b l e i n  particles.  A small  piece of  easier  carbon  the m i c r o s c o p e , the  individual  ticles  a p p e a r e d as s m a l l ,  shapeless specks of  light.  larger  specks with  p a r t i c l e s or agglomerates  On a c c o u n t o f  these  due t o g r a v i t y  determination of were not microns  the  (34)  latex  and s i n c e  these  ignored.  l a r g e r p a r t i c l e s had a h i g h e r  terminal  which m i g h t have i n t e r f e r e d w i t h velocity.  Similar  p a r t i c l e s w h o s e d i a m e t e r was  b e c a u s e , as p r e v i o u s l y  par-  they were  thermophoretic  necessary for  paper  Occasionally  d e f i n i t e shapes were a l s o observed  represented airborne dust  velocity  the  viewing.  When s e e n t h r o u g h  size,  daylight.  t h e e y e t o become a c c u s t o m e d t o  a black background f o r  their  light  noted,  they d i d not  the  precautions  less  than  scatter  0.5  light  and  thus were i n v i s i b l e . When some l i q u i d added w i t h whenever  the make-up  it  t i l l e d water, often,  if  syringe.  became b a d l y  agglomerates.  The  cell  from the  The  sample i n the c e l l  contaminated with was e m p t i e d ,  and r e f i l l e d w i t h  necessary.  reservoir evaporated,  more was was  replaced  airborne dust or  particle  rinsed several  fresh sol before  times with  every  run or  dismore  For  transit grid  e a c h datum p o i n t ,  a s t o p w a t c h was u s e d t o m e a s u r e  time> i . e . , t h e t i m e a p a r t i c l e t o o k t o t r a v e l o n e - h a l f o f a  space.  pressed  Also,  the  initial  and f i n a l  i n g r i d spaces from the bottom  particle grid positions,  d i s k , were  measurements were r e s t r i c t e d to such s m a l l mize  the v a r i a t i o n s  more t h a n 1/8 reduced the  inch from the  thermal  experimentally similarly  that  space of  radial  to  1/8  p o s i t i o n but  inch  It  those c l o s e r  tion,  no m e a s u r e m e n t s w e r e made f o r large p a r t i c l e s to It  was s u f f i c i e n t t i m e f o r In for  the course of  the presence of  fested themselves of  ring to  behaved  the  approximately  s e t t l e o u t and t o  this  to  a run  thermal  one  effect.  After  to agita-  three minutes  let  edge  p a r t i c l e s were  reservoir.  was f o u n d e x p e r i m e n t a l l y  to  the system again  that  three  minutes  occur. i t was  necessary to check p e r i o d i c a l l y  convection currents.  i n t h e movement  t h e u p p e r and l o w e r d i s k s .  moved h o r i z o n t a l l y  noted  t o p d i s k was g e n t l y moved up a n d down  t h e s o l and t o add f r e s h s o l f r o m t h e  reach steady s t a t e .  This  taken w i t h i n  e i t h e r d i s k t o e l i m i n a t e any h y d r o d y n a m i c  remix  allow very  was  from the  I n a d d i t i o n , no m e a s u r e m e n t s w e r e  l o c a t e , the  mini-  movement.  the spacer r i n g .  When t h e c o n c e n t r a t i o n h a d become s o l o w t h a t difficult  The  during p a r t i c l e  edge e f f e c t s .  p a r t i c l e s more t h a n  ex-  those p a r t i c l e s which were  inner surface of  and h y d r o d y n a m i c  regardless of  moved e r r a t i c a l l y .  for  recorded.  distances i n order to  i n absolute temperature  V e l o c i t i e s were measured o n l y  grid  the  If  of the  the  These  currents mani-  p a r t i c l e s near the  surfaces  p a r t i c l e s near the upper  i n one d i r e c t i o n w h i l e t h o s e n e a r t h e  lower  disk disk  47 moved i n t h e  opposite  be p r e s e n t .  T h e y c o u l d be e l i m i n a t e d b y  After was  d i r e c t i o n , convection  the c o o l i n g water  excessive operator bath  and t h e  checked p e r i o d i c a l l y during was  r e l e v e l l i n g the  30 t o 35 v e l o c i t y m e a s u r e m e n t s  e n d e d due t o  eye  thought  EMF,  The  the  temperature  w h i c h had  r e c o r d e d when  to  apparatus.  had b e e n t a k e n ,  strain.  thermocouple  a r u n , were  c u r r e n t s were  run of  been  the  experiment  over. Although  no d e f i n i t e  heating  coil  water.  All  r e m a i n i n g d e v i c e s were  The  terminal  son w i t h  the  was a l w a y s  shutdown  shut  velocity  thermophoretic  the above p r o c e d u r e s  off  due  prior  to stopping  switched off to  velocity.  but without  p r o c e d u r e was e s t a b l i s h e d ,  gravity It  applying  the  flow  i n a random  of  a temperature  cooling  order.  was m e a s u r e d f o r  was o b t a i n e d by  the  compari-  duplicating  gradient.  Chapter  Methods  1.  P r e l i m i n a r y Data  The  VI  of Data  Analysis  Transformations  experimental  data were  i n the form of  particle positions,  transit  times,  t h e r m o c o u p l e r e a d i n g s , and c o o l i n g w a t e r  tures.  T h e y c o u l d n o t be u s e d i n t h i s f o r m a n d h a d t o be  b e f o r e any f u r t h e r techniques  a.  a n a l y s i s was p o s s i b l e .  used f o r  Disk  i.  these i n i t i a l  Top  given  empirical  transformations.  the  procedure c o n s i s t e d of  temperature  values  for  t o p d i s k t e m p e r a t u r e was m e a s u r e d  by  two s t e p s .  copper-Constantan the  e q u a t i o n f o r 0 <_ T^  TQ  Second,  the  Disk  i n r e f e r e n c e (35)  related with  transformed  section outlines  a t h e r m o c o u p l e whose c a l i b r a t i o n i s b r i e f l y d i s c u s s e d This  tempera-  Temperatures  As m e n t i o n e d p r e v i o u s l y , with  This  bath  First,  here.  t h e EMF  values  thermocouples were  r e g r e s s i o n programme.  The  resulting  <^ 1 0 0 ° C i s :  = 0.140687+25.5802(EMF)-0.529102(EMF)2  the d i f f e r e n c e s between p r e d i c t e d by E q u a t i o n  be made i n E q u a t i o n  [21]  cor-  [21]  so t h a t  the experimental were f o u n d . it  tures .  48  temperatures  [21]  and  Corrections could  p r e d i c t e d the experimental  the then  tempera-  49 The  l a t t e r were o b t a i n e d u s i n g the f o l l o w i n g method.  d i s k was s u b m e r g e d  i n the c o o l i n g water  achieved.  The  v a l u e and t h e c o r r e s p o n d i n g b a t h  recorded.  This  It  EMF  p r o c e d u r e was r e p e a t e d a t d i f f e r e n t  was f o u n d  were a p p r o x i m a t e l y Therefore, results,  that  1.1  Equation  s t a t e was  temperature  temperatures.  predicted temperatures  per cent lower than  were  the experimental  which  values.  i.e.:  value  = 0.142235+25.8616(EMF)-0.534922(EMF)2  p r e d u c t e d by E q u a t i o n  temperature,  Tj,  since  t o be n e g l i g i b l y s m a l l .  given  i n Appendix  ii.  The the water  [22]  [22]  was c o n s i d e r e d t o be t h e  the temperature  found  This  [21]  steady  top  t h e e q u a t i o n was a d j u s t e d t o g i v e c o r r e s p o n d i n g l y h i g h e r  TQ  The  bath u n t i l  The  A numerical estimate of  t h i s drop  bottom  d i s k f a c e t e m p e r a t u r e was a s s u m e d t o e q u a l  i n the b a t h ,  Tw,  value accounted f o r  plus a small the  equation for  the f l o w of  litres/minute),  the  c o r r e c t i o n f a c t o r of  temperature  TB  Since  is  III.  Disk  The  disk  d r o p a c r o s s t h e b r a s s f a c e was  Bottom  bottom d i s k .  top  the bottom  rise  between  that 0.1°C.  t h e b a t h and  disk temperature,  Tg,  the  is:  = Tw+0.1  [23]  c o o l i n g w a t e r was r a p i d ( a p p r o x i m a t e l y  t h e o r e t i c a l temperature  of  rise  of  the water  3  between  50 the  inlet  Appendix the  and t h e III).  copper  ence. channel  ture  does  b.  and o u t l e t  c o u l d not  drop  not  Local  local  disk  was  tubes across  measured the  average  initial in grid  affect  outlines  the  the  of  differwater  included  in  III,  bottom d i s k  the  where tempera-  results.  Gradient  techniques  at  not  in  i n Appendix  i n the  final  ends  no n o t i c e a b l e  resistance  is given error  the  and g r a d i e n t  used f o r  a particle  calculating  position  from  the  two  conductivity.  position,  position, spaces  7.5  c o u l d be e x p r e s s e d as  drop  the  (see  PQS,  of  a particle  during  a  measurement,  as:  d i s k s was  disk,  film  than 0 . 0 3 ° C  placed at  indicated  the  large  and t h e r m a l  less  c a l c u l a t e d and was  this  P  where the  and  were  T e m p e r a t u r e and T e m p e r a t u r e  temperatures  defined  of  significantly  temperature  The  b o t t o m d i s k was  thermometers  that a f a i r l y  This "subsection the  the  be a c c u r a t e l y  An e s t i m a t e  i s a l s o shown  of  a check,  temperature  correction. it  As  inlet  The  outlet  grid  from  [24]  os  ST,  and t h e  the  of  position,  bottom d i s k .  spaces w i d e ,  i n terms  final  the  the  Since  average  dimensionless  E^,  the  particle distance  were  region  between  position from  the  top  follows:  [25]  51 where x = 0 and x = h a t t h e t o p and bottom The  local  temperature  disk,  respectively.  a t ^- was c a l c u l a t e d f r o m a d e r i v a t i o n  b a s e d on F o u r i e r ' s  Law w h i c h , f o r o n e d i m e n s i o n a n d s t e a d y  The  heat f l u x  constant axial  thermal  i s d e n o t e d by q .  state i s :  The e q u a t i o n s f o r  c o n d u c t i v i t y used i n t h e p r e s e n t work were o f t h e f o r m :  kf  Combining Equations  = A + B Tc + C Tc2  [ 2 6 ] and [ 2 7 ]  q = -  Integrating  Equation  (A  [27]  results i n :  BTC  +  CTC2)  +  £  [28]  [28] gives  /x o  qdx = - /  T c  (A+BTC+CTC2)  dT  [29]  Tj  or  q x = A(TrTc)+|{TT2-T  At  )4(TT3-Tc3)  [30]  q h = A(TT-T )4(TT2-TB2)+§{TT3-TB3)  [31]  x = h , T^ = T g , s o  c  2  that  B  52 Equations  [ 3 0 ] and [ 3 1 ]  c a n be combined t o y i e l d :  fTc3+|Tc2+ATc+[qh(^)-(ATT+|TT2+fTT3)]  It  c a n be seen t h a t  [32] ful  at x/h.  However,  a n d was f o u n d  calculated temperature  there  are three  only  to f a l l  temperatures  one o f t h e s e , T p ( . ,  within  = 0  [32]  which s a t i s f y  Equation  was p h y s i c a l l y m e a n i n g -  10 p e r c e n t o f t h e t e m p e r a t u r e ,  on t h e b a s i s o f a c o n s t a n t  temperature  gradient.  T ^ ,  This  i s defined by:  T  LIN  =V<VV®  Equation  [32] reduces to Equation  perature  (B = C = 0 ) . Although  simpler to find  Equation  C 3 3 ]  [ 3 3 ] when k^ i s i n d e p e n d e n t  of  tem-  [ 3 2 ] c a n b e s o l v e d a n a l y t i c a l l y ( 3 6 ) , i t was  t h e s o l u t i o n by an i t e r a t i v e  procedure.  The f o l l o w i n g  m e t h o d was a d o p t e d . Let values and  the left-hand  obtained f o r the ( i - l ) t h  T ( . j , $(-j)C  linear  side of Equation  ^  approximation  t  n  e  and i t h i t e r a t i o n by  d i f f e r e n c e s between  t h e s l o p e , D,  (j). T  Denote t h e  rj(-j_-|)>  these values  ^(i-1)  are small,  a  c a n b e made f o r T ^ :  Tc  where  [32] equal  = D(j)+E0  and i n t e r c e p t ,  o.  T  E  [34]  , a r e d e f i n e d by  c(D-;c(i-i)  [  3  5  ]  53  E - T 0  Since Equation  [34]  s e t equal  to  Equation  [32].  [36]  to  intercept  i s equal EQ  to  is  )  [36]  - O t „ (  numerical  the  T h e s e new v a l u e s iteration.  a preselected value  1  next  value i n the  c a l c u l a t e d by  1 S  the  (  the  zero,  and  continue  reaches T  the  c  when (J) i n  series,  substituting  are  substituted  The  process  c l o s e to  of  zero.  is Tp^  t  is in  Q^ -JJ +  in Equations  [35]  and  s t o p p e d when l ^ ^ l is  then  s e t equal  to  C(i). The  and s t a b l e  initial  values  convergence.  TUN+1.0,  and a v a l u e  Equation  [32].  since  further The  at the  each p  The  the  for  terminal  l i m i t for  temperature,  temperature  Thermophoretic  The resulted  Tp^,  [27]  forces  are  velocity  the  additive  i s given  -0-1>  and [ 3 1 ]  =  -  ^c(2)  7^(2)  i  =  n  10~5  were by  c a l c u l a t e y,  combined and  the  p,  and  rearranged,  equation:  T37]  S L -  hkf  L J / J  Velocity  thermal  v  , w h i c h was d e t e r m i n e d  and g r a v i t a t i o n a l  and a c t e d by:  =  rapid  noticeably.  was u s e d t o  g r a d i e n t was g i v e n  particle velocity,  from  ^(1)  l ^ ^ j l was c h o s e n t o be  a l t e r Tp^  dx  c.  \ l N '  =  chosen to g i v e  c a l c u l a t e d by s u b s t i t u t i n g  (J)^  r e d u c t i o n d i d not  local  i t e r a t i o n were  T h e y w e r e ^^(1)  When E q u a t i o n s  local  for  i n the  forces.  same d i r e c t i o n ,  experimentally,  Since the  these  thermophoretic  54 v  where  v^  denotes  the  th  terminal  =  v  settling velocity  (P - p ) d v V  Equation [39] exceed 0.1  (34),  The time,  is valid  x, i . e . ,  =  time  The the  this  liquids.  a.  The  The  by  the  transit  travel  one-half  -  e x  of  [40]  the  Tp^,  give  sols.  This  y,  p,  the  for  v  transform  These  resulting  as p o s s i b l e the methods  e x i s t e n c e of next  about used  thermophoresis  in in  chapter.  th  v^  Method  used to  and k^.  section presents  numerous  Rayleigh  techniques  as much i n f o r m a t i o n  Equation for  and p o s t u l a t i n g  some d e r i v e d  p a r t i c l e to  i s d i s c u s s e d i n the  Empirical  not  Data  An e m p i r i c a l e q u a t i o n programme  the  i s p r e d i c a t e d upon  latter  Number d o e s  III.  vt^,  then analysed to  a n a l y s i s and  T391 L J y j  Hence:  Tg,  of  g  p a r t i c l e Reynolds  section outlined  into TT,  thermophoresis  the  r e q u i r e d by  Transformed  previous  raw d a t a  d a t a were the  of  i.e.:  18y  v Analysis  gravity,  v e l o c i t y was c a l c u l a t e d f r o m t h e  a g r i d s p a c e , 0.0002m.  2.  2  due t o  P-  g  see Appendix  the  P-  provided  experimental  [38]  ex"vg  was o b t a i n e d  u s i n g the  possible relationships of  dimensional  regression  including  analysis  (24).  55 S i n c e an e m p i r i c a l e q u a t i o n obtained  i s s t a t i s t i c a l l y meaningful  on t h e b a s i s o f many d i f f e r e n t  could only  express  v^  i n terms  which best d e s c r i b e d the  of  data  was:  TpK  = Tpc  data  Tpc,  points, u,  dT/dx,  only  if  s u c h an  a n d p.  equation  The  equation  where  The  + 273.2  [42]  d i m e n s i o n l e s s c o e f f i c i e n t , a, i s d e f i n e d by  rearranging  Equation  [41]:  a =  "xik pT  b.  Block Values  of  1.011  For  The form  for  conducted with  the sake of  microns,  further  of  l i q u i d s and p a r t i c l e  to  as a  water  expressed  i n a more  usable  by:  1  n  a = ±r I n  =  e.g.,  dia-  block.  a i n each b l o c k were  analysis  $ 2  two  convenience each c o m b i n a t i o n ,  is referred  n values  dx  a  E x p e r i m e n t s were meters.  pK  [43]  Tn T_ Tl  (  [44]  a.  i=l  " a ? " ( " i=l 1 i=l  1  a  i  1  )  2  /  n  )  [45]  56 and s = (s2)  where s  a i s t h e mean o f  and s a r e  the v a r i a n c e  tively.  The  Equation  [43].  of  value If  of  the  However, c a r r i e d out butions  of  was a f u n c t i o n  of  Coefficients  before  any  further  kf,  i n Each  had t o  c o e f f i c i e n t s about  The  The  data  the f o l l o w i n g  respec-  point  t h e mean  by  coefficients  Block  their  c o u l d be d e s c r i b e d by a n o r m a l  added b e n e f i t  collection.  data  be o b t a i n e d  t h e s t a t i s t i c a l a n a l y s i s w o u l d be g r e a t l y  a l s o have the  ith  the a ' s ,  and  s t a t i s t i c a l a n a l y s i s c o u l d be  information  individual  d p or  of  in a block  significantly.  the  these d i s t r i b u t i o n s tion,  deviations  of  on t h e a ' s , the  coefficients  was c a l c u l a t e d f r o m t h e  v^  Distribution  individual  and s t a n d a r d  the b l o c k s would d i f f e r  c.  by  all  [46]  1 / 2  of  exposing  d i s t r i b u t i o n s were  any  found  f r o m e a c h b l o c k was  on t h e  b l o c k mean.  If  distribution  func-  simplified.  operator  bias  by t h e f o l l o w i n g  transformed  distri-  into  It  would  i n the  data  method.  a variable,  z,  equation: a--a  i, This  variable  facilitates The calculated  has t h e  for  [47]  same d i s t r i b u t i o n s  comparison of cumulative  4-  these with  the  as t h e  c o e f f i c i e n t s and  normal  distribution.  distribution function  each b l o c k .  The  c.d.f.  at  (c.d.f.)  a value  of  z  it  for  the  z's  was  is  the  fraction  57 of  the  to  that value  The  total  c.d.f.  number of  for  a straight  d.  t h a t of  Modified t  c.d.f.  the  The  the  is,  therefore,  plotted  normal  less  than  a function  on n o r m a l  of  probability  distribution function.  co-ordinate  appear  normal  or z  equal (37).  paper  for  The  latter  the  same  system.  t  later, test  of  the v a r i a n c e s  However, W e t h e r i l l  (38)  under  these circumstances.  His  u s i n g the f o l l o w i n g  t  gives  are  the hypothesis  s u b s c r i p t s 1 and 2 r e p r e s e n t  data.  calculated  w h i c h have a  Test  as w i l l  each b l o c k ,  possible.  The  line in this  Since, for  observations  e a c h b l o c k was  comparison with is  z.  of  not  a-j =  two d i f f e r e n t  is  blocks  a t e c h n i q u e w h i c h c a n be  t values  for  the  not of  applied  above h y p o t h e s i s  are  equation:  =  [48] 2,1/2  Furthermore,  the  degrees  f  where  of  freedom,  ,,2 2  f,  were d e f i n e d  by:  [49]  58 The tgg,  i.e.,  freedom)  value the  t for  calculated  greater, never  t c a l c u l a t e d from Equation  t  table.  The  t o be e q u a l . two a ' s  [48]  t  was c o m p a r e d (with  ,  the  work.  was  less  than  Conversely,  when  the e x p e r i m e n t a l  g 5  c o u l d n o t be r e g a r d e d a s e q u a l .  be a p p l i e d w i t h o u t  some a p p r e c i a t i o n o f  f  with  degrees  of  95 p e r c e n t l i m i t wa5.  i s g e n e r a l l y used i n experimental  from Equation  the  [48]  a 95 p e r c e n t c o n f i d e n c e l i m i t  taken from a s t a t i s t i c a l  chosen because i t  sidered  of  the  When t h e  two a ' s w e r e t value  This  physical  test  t;  con-  was shoula  situation.  Chapter  Results  1.  The E x i s t e n c e  The  velocity,  2.  of Thermophoresis  in Liquids  p a r t i c l e s possessed a measurable v e l o c i t y  regions of the l i q u i d  Therefore,  and D i s c u s s i o n s  f i r s t o b j e c t i v e o f t h i s w o r k was a c h i e v e d when i t was  covered that cooler  VII  thermophoresis and t h e o t h e r  An E m p i r i c a l Velocity  over  and above  does exist  f o r the  As m e n t i o n e d p r e v i o u s l y ,  were  c a n be f o u n d  thermophoretic i n Appendix  II.  i t was p o s s i b l e t o o b t a i n an e m p i r i c a l When t h e r e g r e s s i o n p r o -  t h e 342 p o i n t s o f t h e w a t e r - 1 . 0 1 1  an e x p r e s s i o n i n t h e f o r m o f E q u a t i o n T h i s was a l s o f o u n d  The  alone.  Thermophoretic  equation f o r the thermophoretic v e l o c i t y . gramme was u s e d w i t h  the  due t o g r a v i t y  in liquids.  transformed data,  Equation  that  towards  dis-  [41]  micron b l o c k ,  best described the data.  t o b e t r u e when d a t a f r o m t h e o t h e r b a s i c b l o c k s  used.  a.  A n a l y s i s of the Thermophoretic  i.  Summary  Equations  of  [44],  Coefficient  the Block a ' s  [45],  and [ 4 6 ] were u s e d t o r e d u c e t h e d a t a  e a c h b l o c k t o a f o r m more c o n v e n i e n t f o r i n t e r p r e t a t i o n . a r e shown i n T a b l e  2.  The c a p i t a l 59  in  The r e s u l t s  l e t t e r s A t o F a r e a code  to  60 identify  each b l o c k .  Table Statistical  2  Data f o r Each  Block  Fluid  d P (microns)  1.011  n  a s  2  s  H20  Hexane  Block A  Block B  = 342  n  = 94  = 0.1148  o  = 0.0856  = 0.000303  s2  = 0.000066  = 0.0174  s  =  Block D  Block C 0.790  n  a s  2  s  = 107  n  = 0.1149  a  = 0.000268  Both  n a s  2  s  ii.  Distribution Each B l o c k  The  '  = 0.0164 Block  s  = 57 0.0903 2  s  = 0.000141 = 0.0119 Block  E  = 449  n  = 0.1148  a  = 0.000294  s  = 0.0171  s  F  = 151 = 0.0874 2  = 0.000099 = 0.0099  of the a's i n  c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f the a ' s i n each  was o b t a i n e d b y t r a n s f o r m i n g a^ t o z . u s i n g c.d.f.'s  0.0081  a r e shown i n F i g u r e s  Equation  [47].  14 t o 19 i n c l u s i v e ( t h e s e  block  The r e s u l t i n g  c a n be f o u n d  at  61 t h e end o f to  Chapter  the c . d . f .  of  these figures line  except  VII).  a normal  that  for  The  the  This  z,  some d e v i a t i o n  when a i s  extreme  is  values  of  z,  particles with  than  normal  it  c.d.f.  a very  high v . ^ ,  the observations the  implies that  cu<a.  predicted  a positive  Since both than  predict.  avoid this  deviations  very  However, s i n c e l e s s  than  c.d.f.  were  low  Although  tendency,  by  z  the normal  and l e s s w i t h  z.  it  v  t h  ,  the  is  obvious  5 per cent  of  proclivity,  c o e f f i c i e n t s c a n be r e g a r d e d a s n o r m a l l y  distributed  a.  iii.  The  The  Effect  effect  Block A with t e s t e d was ci into  positive bias.  i n a n y one b l o c k w e r e a f f e c t e d by t h i s  thermophoretic  about  present.  to  the  operator  the value  less  from  by  t h a t more o b s e r v a t i o n s  d i s t r i b u t i o n f u n c t i o n would  some b i a s was  [47],  v^.  was  i s apparent  z and v e r y  Conversely, of  c a n be s e e n  to a small  lower than  value  o p e r a t o r made a c o n s c i o u s e f f o r t that  negative  r e p l a c e d by a.  the experimental  taken of the  v^  a higher experimental  were such t h a t at  very  It  corresponds  c a n be a p p r o x i m a t e d  a s c a n be s e e n f r o m E q u a t i o n  [41]  indicates  at  data  are a t t r i b u t a b l e  means t h e e x p e r i m e n t a l  Equation  l i n e on e a c h g r a p h  distribution function.  experimental  These d e v i a t i o n s A negative  straight  of  Equations  and t = 0 . 0 5 4 .  = a^. [48]  P a r t i c l e Diameter  on a  p a r t i c l e d i a m e t e r on a c a n be f o u n d by  Block C, A  of  and B l o c k B w i t h When t h e v a l u e s  to  [50],  S i n c e the  f was  latter  B l o c k D.  from Table  found  value  The 2  first  were  hypothesis  substituted  t o be a p p r o x i m a t e l y  i s much l e s s  than  comparing  tQt-  infinity =  1.96,  62 the  hypothesis  i s c o n f i r m e d a n d p a r t i c l e d i a m e t e r h a s no e f f e c t  thermophoretic The this  clusions  = 88 a n d t  the  t e s t s where  in  water.  same t e s t was c a r r i e d o u t  case, f  Although  velocity  t t  test  = 2.63,  on t h e h y p o t h e s i s  s l i g h t l y l a r g e r than  i n d i c a t e s t h a t cig and  In  addition,  because the v a r i a n c e s of  the  the  = a^.  tg^ =  are unequal,  = t g ^ c a n n o t be u s e d as a b a s i s f o r  (38).  on  In  T.99. results  definitive  from  con-  r e s u l t s m u s t be i n t e r p r e t e d w i t h  two b l o c k s a r e d i f f e r e n t ,  as shown  care  sub-  sequently.  2 An F t e s t experimental  (38)  value,  was c a r r i e d o u t  Fgx,  on t h e  and d e g r e e s o f  hypothesis  freedom,  2  Sg = s ^ .  a n d Vg a r e  The  defined  by: D e x = - T S  F  S  [52]  v = n - 1  [53]  D  g  B  the values  in Table  table,  t h e 95 p e r c e n t v a l u e  mental  value  is  F  2,  l a r g e r than  B  v = n - 1 D  Using  C51]  = 2.14,  and f r o m a s t a t i s t i c a l  is approximately the  tabulated one,  S i n c e the  1.4.  F  experi-  the v a r i a n c e s are  un-  equal . Different block  and a b s e n t  a n d 17,  variances signify i n the o t h e r .  the data of  As  t h a t some e r r o r  i s present  in  one  c a n be s e e n by c o m p a r i n g F i g u r e s  B l o c k D d e v i a t e more f r o m t h e  normal  15  distribution  63 than  t h o s e of B l o c k B.  variance  This  indicates that  i n B l o c k D may be due t o t h e  the s i g n i f i c a n t l y  p r e v i o u s l y mentioned  larger  operator  bias. There i s only  an a p p r o x i m a t e l y  five  per cent d i f f e r e n c e  cig a n d  w h i c h i s b e s t e x p l a i n e d by t h e e x t r a  the water  d a t a show no p a r t i c l e d i a m e t e r e f f e c t .  that  the  thermophoretic  velocity  i n hexane  error.  In  Thus,  between  addition,  it  is  i s a f u n c t i o n of  unlikely  particle  s i ze. Consequently, is  independent  be t r u e which  of  i t was c o n c l u d e d t h a t  p a r t i c l e diameter.  i n gases f o r  small  iv.  in  The E f f e c t o f L i q u i d C o n d u c t i v i t y on a  the a's  hexane thermal to  for water  (Block  If  conductivity,  [48]  to  t  [50]  tgg = 1.96 a n d ap.  it  gave  a f  = »  and t h e r e f o r e  This  velocity  i n k^ b e t w e e n  thermophoretic  since  the hypothesis  a^ = oip.  i s very  for  upon  there  liquids.  This  those  i s dependent  two  = 23.9.  the  all  liquid  i s a 400  Equations  much l a r g e r  v e l o c i t y m u s t be r e g a r d e d as a  than a^  function  conductivity.  effect  to  velocity,  a s i g n i f i c a n t d i f f e r e n c e e x i s t s between  thermophoretic  thermal  affect  s h o u l d become a p p a r e n t  and t  this  t h e same c o n d i t i o n  c a n be c o m p a r e d w i t h  t e s t was a p p l i e d t o  Thus,  liquid  E)  found  Thermal  thermophoretic  500 p e r c e n t d i f f e r e n c e The  of  F).  (Block  i.e.,  (17)  velocity  liquids.  S i n c e p a r t i c l e d i a m e t e r does n o t all  thermophoretic  Other workers  Knudsen Numbers,  c a n be c o n s i d e r e d t o e x i s t  the  c a n be c o m p a r e d w i t h  that  found  in gases.  The  64 ratio  ap/a£ c a n be c o n t r a s t e d w i t h  (Equation  [4])  and D e r j a g u i n  (Equation  2 2kf  Epstein:  "  the values  1  8 k  f  Epstein  f o r Kn = 0 :  [54]  kp  +  +  [6])  p r e d i c t e d by  P  k  [55]  Derjaguin:  The d ' s f o r h e x a n e The  and w a t e r  and t h e i r  ratios  v a l u e s were c a l c u l a t e d u s i n g average  Equations  [ 5 4 ] and [ 5 5 ] .  thermal  As c a n be s e e n ,  Table  are presented  = 0.600 J  s"  1  m"1 ° K  _ 1  kf  (hexane)  = 0.130 J s "  1  nf1  °K  _ 1  = 0.128 J s "  1  nf1  °K  - 1  a  water  for liquids is  very  and P r e d i c t e d a ' s  (water)  Author  conductivities in  the r a t i o  kf  P  3.  3  Comparison of Experimental  k  i n Table  a  hexane  a hexane/a  water  (%) Epstein  0.6777  0.5026  74.2  Derjaguin  1.855  1.505  81 . 2  McNab  0.1148  0.0874  76.1  close though  to that  p r e d i c t e d f o r g a s e s by t h e E p s t e i n  e s s e n t i a l l y only  been s t u d i e d ,  i t appears  two v a l u e s that  relationship.  of l i q u i d thermal  Al-  conductivity  an e x p r e s s i o n s i m i l a r t o  Epstein's  have  65 relation  b.  holds  for  The  Final  The  data  an e q u a t i o n  of  both  types  Empirical  obtained  the  of  fluid.  Equation  for  i n t h i s work can d e f i n i t e l y  t  .  h  5-J-.VT  -  f  p  where t h e more g e n e r a l analysis  figures 0.087  are  for  vT  i n Appendix  justified  for  replaces III a.  T  dT/dx.  large two  Therefore,  a equals  0.11  noted,  the  of  variances  and  the  significant for  water  and  numerical  values  coefficient  respectively.  Since  Y becomes  for  0.26  functional  relationship  Equation [54].  The  between  coefficient  as:  k  the  The  that only  may be d e s c r i b e d by a f o r m  a  [57],  PK  hexane.  a c a n h e n c e be w r i t t e n  When t h e  [56]  indicate  H o w e v e r , as p r e v i o u s l y a and k f  by  form:  v  error  be d e s c r i b e d  =  both  f + kp  .  from Table 3 are s u b s t i t u t e d  y equals only  2kf  Y  two  0 . 2 5 4 and 0 . 2 6 1 figures  fluids.  into  f o r w a t e r and  c a n be r e g a r d e d  as  [ 5 7 ]  Equation hexane,  significant,  Thus:  *th • - °- 2k "l fc p T g 26  f  pK  ™  66 In a d d i t i o n , = 17%  that  the  fact  coefficients,  thermal  of  that  the  appears thermal  Epstein  to hold f o r  v^  T p £ and t e m p e r a t u r e  gradient  shown t h a t — ^ — i s p PK  lower f o r  21  difference  both  f l u i d suggests  Note t h a t ,  due t o  types  of  different the  A more d e t a i l e d d i s c u s s i o n o f  IV. hexane  than f o r water  i s e x p l a i n e d by F i g u r e hexane  than water  velocity  a n d 22 w h i c h d e p i c t t h e m a g n i t u d e  [58].  (0.26/1.5)  a l b e i t with  for  i n thermophoretic  100  gas e q u a t i o n ,  i n Appendix  lower value of  liquids is  equation.  creep i n l i q u i d s .  creep i s given The  The  velocity for  p r e d i c t e d by t h e E p s t e i n  The  existence  thermophoretic  the e f f e c t  at  at  2 0 , where t h e same  i s a l s o apparent  of v ^  the it  vth  is  temperature.  in  Figures  a s c a l c u l a t e d by  of — ~ — ,  same  Equation  drops with  rising  temperature. T h e r e i s one f u r t h e r istence of  thermophoresis  to a case of found wall 3.  piece of  in liquids.  particulate fouling  to e x p l a i n the f a i l u r e of  evidence which supports When E q u a t i o n  [56]  the  was  applied  i n a s e n s i b l e heat exchanger, small  ex-  it  p a r t i c l e s t o d e p o s i t on a  was  hot  (39). Thermophoretic  The velocity  Force  thermophoretic  by means o f  force,  Stokes  Law.  Fp = -  w h e r e Vp  i s the speed of  the  F^,  c a n be o b t a i n e d f r o m  The S t o k e s  3 TT y  drag f o r c e ,  Fg,  the is:  dp Vp  particle relative  [59]  to  the  stagnant  fluid.  67 At  steady  state,  this  particle velocity  F  Therefore,  combining Equations  = - 0.78  4.  The  Effect  of  the magnitude  [39].  However,  m u s t be t a k e n 1.011  [60]  D  [59],  and  [60]:  [61]  into  VI,  terminal  the measured v e l o c i t y velocity,  Vg as a f u n c t i o n  for  i s very hexane,  account.  of  predicted  2 0 ° C and found  by S t o k e s  t o be w i t h i n  Fourier's  is  the  the  °A  ,  it  24  c a n be  as v ^  terminal  velocity  5 per cent of  and for  liquids  the  of  Equation  same o r d e r  i n both  sum  23 a n d  TQ a s c a l c u l a t e d by  gravitational  a  at  values  Law.  C e l l W a l l E f f e c t s on t h e L o c a l and T e m p e r a t u r e Gradient  A computer  Figures  much s m a l l e r t h a n v Vg i s o f  The  Vg.  m i c r o n p a r t i c l e was m e a s u r e d e x p e r i m e n t a l l y  approximately  5.  of  Since vn f o r water g  neglected.  and  TT  i n Chapter  and t h e g r a v i t a t i o n a l  give  [58],  F  to v ^  Gravity  As m e n t i o n e d v ^  = -  th  i s equal  Temperature  s i m u l a t i o n was b a s e d on t h e  numerical  solution  of  Equation:  A  2  T = 0  [62]  68 This  was u s e d t o s t u d y  cell  wall  upon t h e  Heat t r a n s f e r natural  to  vertical  the  convection.  and r e p r e s e n t a t i v e simulation  the e f f e c t  outside  The  computer  gradient  and t e m p e r a t u r e  cautions  mentioned  not  affect  6.  A Theoretical in Liquids  the  the  d a t a were  liquid  used,  This Equation  [3]  dense systems  Where  r  attempts  loss  d i d not This,  assured that  of  substance  i n the  tionship,  which  the  of  failed.  Maxwell's  is s t i l l  cell  from Table effect  the  develop  velocity.  Nevertheless,  thermal  by  dimensions 1.  the  radial  The  temperature the  heat  a theoretical H o w e v e r , when  one model  thermal  pre-  flux  and o f  state  at  the  the  of  the  equivalent  zero pressure.  imprecise in i t s  model  did  promise.  However,  Enskog  theory  thus  properties  thermal  for  A functional the  of  became:  v i s c o s i t y and  concept of  to  physical  showed  creep.  creep equation  complicated function  gaseous  be  Thermophoresis  w e r e made t o  liquid  the  the  temperature.  combined w i t h  was m o d i f i e d by a n a p p l i c a t i o n o f (40).  local  was c o n s i d e r e d t o  properties  was b a s e d on t h e c o n c e p t o f  is a rather  conductivity  cell  and  through  measurements.  thermophoretic  they  losses  was s u p p l i e d w i t h  heat  V,  heat  gradient  significantly.  i n Chapter  Model  the  physical  the  experimental  Several predict  of  indicated that  radial  temperature  air  values  of  the rela-  physical  69 situation, fluid  However,  d e p e n d i n g upon t h e v a l u e s o f  p r o p e r t i e s , r c o u l d be e i t h e r i m a g i n a r y o r r e a l  than one. in  was d e r i v e d .  this  The  latter  with a value  r e s u l t i s r e q u i r e d to e x p l a i n the data  imaginary but very c l o s e Therefore,  to  a n d h e x a n e w e r e u s e d , r was  the p o i n t where  i t w o u l d become  some c o r r e c t i o n s i n t h e d e r i v a t i o n a n d b e t t e r  t h i s model  to apply  i n the  liquid  case.  less  obtained  work. When p h y s i c a l d a t a f o r w a t e r  allow  the  real.  data  might  99-9  Line  99-5 99 98 h  Circles  -3  is  Normal are  Ey.pt  -  -I Figure  The c . d . f .  Dist  of  Function  Values  0  z 14  Block A (Water,  d n = 1.011  microns)  71  ~~—r—~r— 9 3 -9  Line  99-5 99 98  Circles  is N o r m a l are  Expt-  Figure The c . d . f .  of  Dist-  0,  Function  Values  15  Block B (n-Hexane,  d p = 1.011  microns)  72  -  3  -  2  -  1  0  1  2  Z Figure The  c.d.f.  of  16  Block C (Water,  d  =  0.790  microns)  3  75  -  3  -  2  -  1  0  Z Figure The c . d . f .  of  Block  19 F  (all  n-Hexane)  1  2  3  76  u/p  Tpj,  as a F u n c t i o n of  Temperature  for  Water and  n-Hexane  77  0  I £L d x  2 x  l  0  -  °I<  4  Figure  m"  3 1  21  T h e r m o p h o r e t i c V e l o c i t y i n Water as a F u n c t i o n o f T e m p e r a t u r e  Gradient  78  0  I i l dx  2  xl0" Figure  Thermophoretic  Velocity  in  n-Hexane  as  °K  4  m'  3, 1  22 a  Function  of  Temperature  Gradient  79  1-6  •  1-2  oo  O  -  0-8  dp =1-011  microns  >  0-4  dp =0-7S0  20  Gravitational  Terminal  Velocity  80  60  40  Figure  microns  23  i n Water  as a F u n c t i o n  of  Temperature  Figure Gravitational  Terminal  Velocity  24  i n n-Hexane as a F u n c t i o n  of  Temperature  Chapter  Conclusions  1.  and  VIII  Recommendations  Conclusions  Thermophoresis The  was f o u n d t o  occur i n  liquids.  p a r t i c l e thermophoretic  velocity  in a liquid  c r i b e d by t h e  c a n be d e s -  equation:  v., = - a VT —th p T|^ —  where a e q u a l s 0.11 Thermophoretic  f o r water  velocity  a n d 0.087 f o r  i s a f f e c t e d by  n-hexane.  liquid  thermal  conduc-  tivity. Evidence to  account f o r  Particle The  exists  that the equation f o r the  diameter  thermophoretic  c r i b e d by  the  thermal  conductivity  d i d not e f f e c t  the  c a n be  thermophoretic  equation:  = - 0.78  . T• 2k + k  7T 9  f  81  rewritten  effect:  f o r c e on a p a r t i c l e i n a l i q u i d  kf  F,. -th  velocity  2 d n —%r— p p TpK  -  vT  velocity. c a n be d e s -  82 -  The  form of  expressions  the equations  Both  -  The  the values  thermophoretic  important ients  velocity  and f o r c e  in gases. creep i n  effect  This  liquids.  i n l i q u i d s are  p r e d i c t e d by t h e E p s t e i n  17  is therefore  per  relationships.  i n l i q u i d s i s v e r y s m a l l and  i n s i t u a t i o n s where It  thermal  Epstein's  very high  impractical for  temperature use i n  is grad-  liquid-  separations.  Recommendations  Although phoresis the  only  exist.  particle  2.  the existence of  thermophoretic  cent of  l i q u i d s i s s i m i l a r to  used to d e s c r i b e thermophoresis  lends credence to -  for  the v e l o c i t y  and f o r c e e q u a t i o n s  derived  i n l i q u i d s may be s i g n i f i c a n t i n some e n g i n e e r i n g  thermophoretic  study  by e n g i n e e r s .  model  for  effect  is  too  However,  t h e e f f e c t may be  small  to warrant  the development  worthwhile.  of  further a useful  for  thermo-  problems, experimental theoretical  Nomenclature  dimensionless  coefficient  constant  coefficient  used  used  i n Equation  i n Equation [27],  used i n Equation  J s  - 1  [3]  m - 1 °K"^  [IV-19] 1 1  coefficient  constant  used  i n Equation  used i n E q u a t i o n  coefficient cumulative  used  [27],  J s  m  °K  [IV-19]  i n Equation [27],  distribution  2  function,  J s~  1 - 1 m  defined  3 °K~  i n Chapter  IV,  [5]  section 2(c)  constant  used  i n Equation  [IV-19]  experimental  dimensionless  constant  used i n Equation  experimental  dimensionless  constant  used  and  [7]  specific  slope  of water,  4200 J  kg"1 ° C  of Equation [ 3 4 ] , s m °C J  constant  latex  heat  used i n E q u a t i o n  particle  diameter,  - 1  [IV-19]  microns 83  _ 1  i n Equations  [5]  sub-  84 temperature  gradient  one-dimensional ordinates,  operator  final  top  temperature  ° K m"^  or  from  symbol  for  degrees  of  Equation  freedom  co-  an o b s e r v a t i o n ,  grid  disk  r e a d i n g , mV  Equation  exponent  in Cartesian  ° K m~^  [II-7]  a p a r t i c l e during  the bottom  of  particle surface,  ° C m~^  d i s k thermocouple  intercept  to  gradient  d e f i n e d by E q u a t i o n  p o s i t i o n of  spaces  tangential  [34],  of  for  °C  e  experimental  t  value,  defined  by  [49]  experimental  f  force exerted  test  value,  d e f i n e d by E q u a t i o n  [51]  by f l u i d o n p a r t i c l e , d e f i n e d by E q u a t i o n  [IV-28],  _2 kg m s Stokes  drag  force  on p a r t i c l e , d e f i n e d by E q u a t i o n  [59],  _? kg m s thermophoretic  force  on p a r t i c l e , d e f i n e d by E q u a t i o n  _2 kg m s vectorial  form  of  thermophoretic  force,  kg m s  _2  [60],  acceleration  of  gravity,  d i s t a n c e between  Boltzman's  thermal  fluid  the d i s k s , 0.003 m  constant,  conductivity  thermal  9 . 8 m s~  1.38  of  xlO  J  - 2 3  °K_1  disk brass, J  conductivity,  J  s"1  m"1  K n u d s e n N u m b e r , d e f i n e d by E q u a t i o n  dimensionless Equation  variable  thermal  mean f r e e  path  coefficient  °K-1  °K_1  [1]  used i n Equation  conductivity,  of  [49],  defined  by  used i n Equation  p a r t i c l e mass,  mass f l u x  of water  number o f  observations  coefficient  J  s"1  m-1  °K_1  f l u i d molecules, m  spherical  Equation  m-1  [50]  particle  average  s"1  [12],  °K m  kg  through  cooling water  used i n Equation  [10],  °K  channel,  g r i d spaces from  the  s"1  m"1  p a r t i c l e p o s i t i o n d u r i n g an o b s e r v a t i o n , [24],  kg  bottom  disk  defined  by  axial  heat  flux,  position  along  particle  Reynolds  i)  ii)  the  by E q u a t i o n  radial  axis,  [26],  defined  Number, d e f i n e d  standard  block  d e f i n e d by E q u a t i o n  of  data,  as a d i m e n s i o n :  Equation  initial spaces  of  the a's  s~  deviation  m~  on F i g u r e  of  25, m  III,  the a's  section 2  about  [46]  seconds  about  a i n a block of  data,  defined  p o s i t i o n of from  a p a r t i c l e during  the bottom  variable  an o b s e r v a t i o n ,  for  t  test,  defined  f r o m a 95 p e r c e n t s t a t i s t i c a l t  disk temperature,  temperature,  °C  temperature,  °K  temperature  local  by  at  Dwyer's  temperature,  grid  disk  by E q u a t i o n  table  for  f  defined  initial  determined  by E q u a t i o n  point,  from  [23],  °C  °K  Equation  [32],  [48]  degrees  freedom  bottom  a in a  [45]  dimensionless  value  J  i n Appendix  as a v a r i a b l e :  variance  of  defined  °C  87 local  temperature,  top d i s k  defined  temperature,  cooling water  flow  of  f l u i d at  gravitational  tangential  defined  temperature  measured v e l o c i t y ,  by E q u a t i o n  r =  00  terminal  component  by E q u a t i o n  i n the b a t h ,  defined  with  by E q u a t i o n  respect to  velocity,  of  [42],  [22],  °C  °C  [40],  m s"^  p a r t i c l e , m s~^  defined  fluid velocity,  °K  by E q u a t i o n  defined  [39],  on F i g u r e  25,  m s~^  upper  velocity  section  2,  Stokes  Law,  defined  i n Appendix  of  Equation  [59],  particle relative  to stagnant  fluid,  used  component  of  fluid velocity,  creep v e l o c i t y ,  thermophoretic  vectorial  in  m s"^  defined  on F i g u r e  m s"^  thermal  III,  m s"^  velocity  radial  l i m i t of  velocity,  form of  defined  by E q u a t i o n  [3],  velocity,  m s~^  m s"^  thermophoretic  m s"^  25,  88 x  = distance  x1  = Dwyer's  x  o  z  =  from  the  particle  Dwyer's  initial  = transformed  top  disk, m  position, m  particle  position, m  dimensionless  variable  for  a, d e f i n e d  by  Equation  [47]  • Greek  a  = dimensionless  Letters  thermophoretic  coefficient  defined  by  Equation  [43]  a  = mean v a l u e  3  = variable  Y  = dimensionless  coefficient  r  = dimensionless  variable  Equation  A  2  = vector  a for  of  defined  a block of  data,  by E q u a t i o n [ 9 ] ,  defined  by E q u a t i o n  [44]  s"1  used i n E q u a t i o n  used i n E q u a t i o n [63]  [57]  to  modify  [3]  operator  AT  = temperature  AX  = thickness  AG  = small  of  "del  change  disk  interval  of  2  "  used  face,  time  i n Appendix  III,  sections  1/16"  defined  by D w y e r ,  s  3 and 4 ,  °C  89 5  = dummy v a r i a b l e  used f o r time  8  = angular  9  = time used i n Dwyer's work,  A  = d i s p l a c e m e n t o f p a r t i c l e , d e f i n e d by E q u a t i o n  p o s i t i o n as d e f i n e d  i n Equation  i n Figure  [8],  s  25  s  [11],  m  v 2 <A 2 >  = mean s q u a r e d i s p l a c e m e n t ,  p  = fluid viscosity,  v  = degrees  kg  of freedom  d e f i n e d by E q u a t i o n  [8], m  s~^ used i n F t e s t s ,  defined  by E q u a t i o n s  a n d 53 TT  = 3.1416  p  = f l u i d density,  _3  p  n  P  kg m -3 k g m"  = particle density,  = summation  = transit  operator  t i m e f o r p a r t i c l e t o move o n e - h a l f  = left-hand  side of Equation  = stream f u n c t i o n ,  [32], J  s  d e f i n e d by E q u a t i o n s  grid space, s  m~^  - 1  [IV-8],  [IV-19]  coefficient  d e f i n e d by E q u a t i o n  [IV-5],  m s~^  [IV-9],  and  [52]  90 Subscripts  A to  F = code l e t t e r s  value  for  i  = i  1,2  = dummy v a r i a b l e s  LIN  = case of  each b l o c k of  i n a c o l l e c t i o n of  identifying  temperature  data,  defined  i n Table  values  two  independent  different  blocks of  k^  Miscellaneous  < >  = operator  y_  = vector  used i n Equation  operator  "grad"  or  2  [8]  "gradient"  \  data  References  1.  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Organic Solvents, I n t e r s c i e n c e P u b l i s h e r s , 1955.  31.  R e i d , R.  C.  and T.  K.  2nd e d .  Liquids, 32.  S a k i a d i s , B.  C.  and J .  33.  Washburn,  W.  (ed.).  34.  35.  New Y o r k : Coates.  2nd e d .  The Properties McGraw-Hill A.I.Ch.E.  International  New  of Gases and  Book C o . ,  J l . ,  Critical  Conversion  Tables for Thermocouples, and N o r t h r u p  Uspensky, J . V. C o . , 1948.  1950.  York:  1966.  2,-275  (1955).  Tables.  New  M c G r a w - H i l l Book C o . , 1929. F o u s t , A . S . , L. A . Wenze.l , C . W. C l u m p , L . M a u s , a n d L . Andersen. Principles of Unit Operations. New Y o r k : W i l e y and S o n s , 1960.  Leeds 36.  E.  Sherwood.  Publishing Co.,  Issue 4 .  York:  B. John  Philadelphia:  Co.  Theory of Equations.  New Y o r k :  McGraw-Hill  Book  93 37.  G u t t m a n , I . and S . S . W i l k s . Introductory New Y o r k : J o h n W i l e y a n d S o n s , 1 9 6 5 .  38.  W e t h e r i l l , G. B. Elementary Statistical and C o . , 1 9 6 7 .  39.  H o p k i n s , R. M . , U n i v e r s i t y cation (1972).  40.  H i r s c h f e l d e r , J . 0 . , C . F . C u r t i s s , a n d R. B . B i r d . Molecular Theory of Gases and Liquids. New Y o r k : J o h n W i l e y , 1 9 6 4 .  41.  B e n n e t t , C . 0 . a n d J . E. M y e r s . Momentum, Heat, and Mass Transfer. New Y o r k : M c G r a w - H i l l B o o k C o . , 1 9 6 2 .  Engineering  Methods.  of B r i t i s h Columbia,  London:  personal  Statistics.  Methuen  communi-  APPENDICES  94..  Properties  . This  Appendix  Appendix  I  of  and  Water  lists  most of  the e m p i r i c a l r e l a t i o n s h i p s between temperature. order  of  Each  gives  i n the  Experimental  number.  chapter  The  used to  f l u i d properties  and i d e n t i f i e s  empirical equations  entitled  Materials,"  the data  the experimental  i n c r e a s i n g temperature  as a r e f e r e n c e given  table  n-Hexane  page  "Physical 32.  data the  obtain and in  source  are  Properties  of  the  96 Table The  c (°c)  Thermal  Conductivity  k  T  (J  4 Water  Source  f  s ' V ^ C  of  - 1  )  0.00  0.569  (25)  0.00  0.569  (28)  6.84  0.575  (25)  10.00  0.587  (25)  0.587  (28)  16.84  0.593  (25)  20.00  0.604  (28)  24.84  0.610  (25)  30.00  0.619  (28)  36.84  0.624  (25)  37.78  0.629  (24)  37.78  0.630  (25)  40.00  0.632  (28)  46.84  0.638  (25)  50.00  0.644  (21)  56.84  0.649  (25)  10.00  .  60.00  - 0.654  (28)  65.56  0.659  (25)  66.84  0.660  (25)  70.00  0.663  (28)  76.84  0.669  (25)  80.00  0.670  (28)  86.84  0.676  (25)  90.00  0.676  (28)  93.33  0.681  (24)  93.33  0.678  (25)  96.84  0.681  (25)  97  Table 5 The V i s c o s i t y o f  Hexane y  (°C)  (°K)  Source  (kg n f V ) xlO3  0.00  273.20  0.381  (31)  15.00  288.20  0.337  (29)  15.00  288.20  0.324  (30)  16.35  289.55  0.328  (29)  18.55  291.75  0.320  (29)  20.00  293.20  0.318  (29)  20.00  293.20  0.326  (25)  21.40  294.60  0.313  (29)  23.22  296.42  0.310  (29)  25.00  298.20  0.294  (29)  25.00  298.20  0.292  (30)  25.00  298.20  0.294  (25)  30.00  303.20  0.278  (29)  40.00  313.20  0.271  (25)  40.00  313.20  0.262  (31)  50.00  323.20  0.248  (25)  Table The T^  T^  (°C)  (°K)  Density  6 of  Hexane p (kg  Source rn" ) 3  0.0  273.2  676.9  (29)  10.0  283.2  668.3  (29)  15.0  288.2  663.9  (29)  20.0  293.2  659.5  (29)  25.0  298.2  654.9  (29)  30.0  303.2  650.5  (29)  40.0  313.2  641.2  (29)  50.0  323.2  631.8  (29)  60.0  333.2  622.1  (29)  70.0  343.2  612.2  (29)  99  Table The  c (°c) T  Conductivity  k  of  Hexane Source  f  (J s ~ V l o C _ 1 )  4.0  0.1420  (33)*  20.0  0.1310  (31)  30.0  0.1270  (31)  37.8  0.1240  (32)  38.0  0.1235  (31)  57.0  0.1150  (31)  60.0  0.1135  (31)  Value  Tr  Thermal  7  c l a i m e d t o have l a r g e  possible  In a d d i t i o n , s l o p e = - . 0 0 0 4 3 6 = 32.8°C to 5 7 . 2 ° C .  J s  variation.  'm ' ° C  for  range  100  Appendix  Experimental  This obtained letters Table  8.  appendix  i n the for  contains  course of  each b l o c k of  the data  II  Data  the experimental present work. are presented  The in  data code  101  Table 8 Code L e t t e r s Letter  Liquid  for  Blocks dp (microns)  A  Water  1.011  B  Hexane  1.011  C  Water  0.790  D  Hexane  0.790  102  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  N D i / i u ^ i / > c ^ f c r t t r > p ^ f * * r * ' ^ " r - r ^ ' o o o o o o j cr c? cr cr ^ c i f ^ o c r i v o £ f » > M f * i ' " i ,  N  ( N (N  r-  *— r - r - ^ - * —  ,  r  i  ,  ~ O O O O O O O O O O O O C » C * C » O ^ C ^ O ^ C T » 0 ' C ^ C 0 3 0  coeococoa)caa3roc503cosocor*r^r^r^r»CP  0>  ON  C C" ( T D I T f J l P> O CT* O N O N O * O * O N O N O N O N O N O N O N O * C" O "  O" C ' CN ^  ON O N ©N C N  0*  1  O " O N ON  0  S  O N O N O * O N O N O N O"* O N O N O N O N O N  ^OOMnMM^lnlnln^ft^ncocori;a33aoooooonr»lr*lrl|/^ln^^^oo o ^ c r c a o c - c ^ t r c ' O ' C ' o c ^ t r r ' C ' C ^ o o o o o o o o o o o o c O O O ' - ' o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  o^u^oooor-r-r-r-r^r-'^rir  ^ J n  co'a3a>r^r^r^r^r^r^r^r^r--r»r-p*.r*r-r-  T - f - t - » - r - r - r - r - r - r - ^ r - » - r - r - r - r - t - t - r - T - r - T - « - t - r s i n j r ^ ( N r N n r ^ r ^ f ^ f N ( N  lOU^l/Ni/N'j^i/>U^in\/Nu^i/^i/N(/lL.^LOi/*^L^  r-i-r-^MNNNNMMNnnrnnnnnnfnnnncf  cr cr cr rj cr in in tn.i in  Table 10 Experimental Data of Block A , ( T - T ) = T  LIQUIO « Water T ("C) • 40.0 n » 34 Is) 1.75 1.75 1.75 2.25 2.25 2.25 2.75 2.75 2.75 2.75 2.75 2.75 2.75 3.25 3.25 3.25 3.25 3.25 3.75 3.75 3.75 3.7 5 1.25 «.25 U.25 1.25 U.75 4.75 a.75 U.75 0.75 5.25 5.25 5.25  36. 36. 5 1.  10.  U9. 19. 55. 50. uu. 10.  58. US. 52. 60. 57. US. U 1. 62. 60. 19. U5. 60. 63. 56. 60. 6 1. 65. 50. 65. 58. 6 1. 58. 70. 65.  PC (°C)  T  p  15.2 15.2 15.2 17.U 17.U 17.U 19.6 19.6 19.6 19.6 19.6 19.6 19.6 21.8 21. 8 21.8 21.8 21.8 2U. 0 2U.0 2U.0 21.0  26.2 26.2 26.2 26.2 28.3 28.3 20.3 28. 3 28.3 30.5 30.5 30.5  8  dT/dx k p u (°K nf ) (J s'V'K' ) (kg nf ) (kg nfV ) xlOf  1  11135. 11135. 11135. 11067. 11067. 110h7. 11002. 11002. 11002. 1 1002. 11002. 1 1002. 11002. 10910. 10910. 10910. 10910. 10910. 10879. 10879. 10879. 10879. 108 21. 108 21. 10821. 108 21. 10765. 10765. 10765. 10765. 10765. 107 11. 10711. 10711.  32.7°C  d (microns) » 1.011 T CO • 7.3  T  T  R  1  0.59U 0.5 91 0.5 9U 0.5 98 0.5 98 0.598 0.601 0.601 0.601 0. 6 01 0.601 0.6 01 ' 0.601 0.6 05 0.605 0.605 0.6 05 0. 6 05 0.6 08 0. 608 0.6 08 0.608 0.6 11 0.6 11 0.6 11 0.6 11 0.6 15 0.6 15 0.6 15 0.6 15 0.6 15 0.6 18 0.6 18 0.6 18  3  990.2 998. 2 998. 2 997.9 9 9 7. 9 99 7.9 997.5 99 7.5 9 9 7. 5 997. 5 99 7.5 997.5 997.5 997. 1 99 7. 1 997. 1 9 9 7. 1 9 9 7. 1 99 6. 7 996.7 996. 7 996. 7 996. 3 996.3 996. 3 996. 3 9 9 5. 8 995. 8 995. 8 995. 8 995. 8 995.2 995.2 995.2  1  11. 32 1 1. 32 11.32 10.69 • 10.69 10.69 10. 12 10.12 10. 12 10. 12 10. 12 10. 12 10. 12 9. 60 9.60 9.60 9.60 9. 60 9. 13 9.13 9.13 9 . 13 8.69 8.69 8.69 8.69 8. 29 8.29 8. 20 0. 29 8. 29 7.91 7.91 7.91  th (ms') V  1  xlO  6  5.56 5.56 3.92 5.00 1.08 1.08 3. 61 U.00 1 . 55 5. 00 3.15 U.1U 3. 85 3.33 3.5 1 U.U1 U.88 3.23 3.33 1.08 1.11  3.33 3. 17 3.57 3.33 3.28 3. 08 U. 00 3. C8 3.15 3. 28 3.15 2 . 86 3.08  T (mW ) xlO M/P  a  pK  1  9  3.93 3.93 3.93 3. 69 3.69 3.69 3.17 3.17 3.17 3.17 3.17 3.17 3.17 3.26 3.26 3.26 3.26 3.26 3.06 3.08 3.08 3.08 2.91 2. 91 2.91 2.91 2. 76 2. 76 2. 76 2. 76 2.76 2. 62 2. 62 2.62  0.1270 0. 1270 0.0395 0. 1225 C. 1000 0. 1000 0.09 55 0. 1019 0.1193 0.1311 0.09 0 5 0.1161 0. 1010 0.0933 0.098 3 0.12U3 0. 1367 0.09 0 5 0.0993 0.1217 0. 1325 0.099 3 0. 1006 0.1132 0. 1056 0. 1010 0. 1037 0.1316 0 . 1037 0.1161 0. 110U 0 . 1230 0.1020 0 . 1098  104  > * o \ C D o —  O  o  o  f  _  r  o  -  ,  " « " O r ' " 0 ^ c > c n ) ' - c D C P c  0 0 ' " ' - 0 * - * - » ~ * - O  o  o  o  o  o  o  o  o  o  0 > i T l U ^ i A O O O O O O O i ^ O r> CT>  r* C  1  0  s  C~ C7"> CT> C" CT">  *  CT  -  o  N  o  (  - 0 ' - » -  r  o  C > r  o  C  i  o  ,  >  f  o  o o v c ^ i ' * '  o  o  o  O o  ^  v n «— c - o  O O •o  O ^ O ^ ^ f n r i n n  i n v.n n  ^"i O ^  1  ~ - - 0  o  o  o  *- o  *o  r o \D \D  o  o  o  C * CT» CT* CT> C  ^ C"»  CT>  o  o  *~ O * ~ o  o  o  vD vO \ D CO CO CO CO  irt  <T> C  m \ e p . «n m  r - <- o  0*> O  n  n  n  d  <T> 0>  o n  ^  I ^ O O O O C G O O O C ' - ' - ' - ' - ' - ' - ' - ' - r - ^ r - r - r - p j ^ f N M N I N f S . ' N l N t V I N U*) vo \0 i £ o O sO « 0 \ O v O > A ' £ ' 0 ' O v O ^ ) v O \0 VO vO O vO vO O <«0 MO \ D vO \ 0 ^ s O >A  00000000000000000000000000000000000  cr^r»r^r^r*r*r*oc^oc c c c *c '*'" ' -'"'* ' >  C  vX3 vD O r  ,  ,  ,  N  r  ,  ,  T i r  O  i ' * ' ^ ' ^ \o3 lO \fi O O O O ^ O C C C l  D  o  Table Experimental  Data of B l o c k A ,  LIQUID - Water T CC) - 56.5 n •> 36  (Ty-Tg)  = 49.2°C  d (microns) - 1.011 T CO » 7.3 p  T  T (s)  12  B  T dT/dx k CC) CK m") (J s'V "^ ) (kg m') (IcgnfV ) xlO pc  f  1  1  p  y  1  3  1  4  v (rns ) xlO th  -1  6  u/p T (roW ) xlO p|<  1  9  1.75  29.  19.3  16075.  0.601  997.6  1 0 . 20  6. 90  3.49  0.  1.75  28.  19.3  16875.  0.601  997.6  10.20  7.  11  3.49  0.1211  1170  2.25  29.  22.7  16729.  0.606  997.0  9 . 11  6. 90  3.19  0 . 1293  2.25  36.  22.  7  167 2 9 .  0 . 6 06  997.0  9.11  S.56  3.  0.  2.25  39.  22.  7  16729.  0.606  997.0  9.11  5.  3.19  0.C961  2.75  53.  26.0  1659  0.0777  2.75  26.  26.0  16  2.75  30.  2.75 2.75  13  19  1012  1.  0.6  11  996.3  8.72  3.77  2.92  0.6  11  99 6 . 3  8.72  7.69  2.92  C.  1585  26.0  5 9 1. 1 6 S 9 1.  0.6  11  996. 3  8.72  6.67  2.92  0.  1375  3 1.  26.0  1659  1.  0.6  11  996. 3  8.72  2.92  0 . 1329  39.  26.0  16S91.  0.6  11  996.3  8.72  6.15 5 . 13  2.92  0.10  26.0  1659  1.  0.6  11  99b. 3  8.72  1.55  2.92  0 . 0 9 38 0 . 1 177  57  2.75  uu. 35.  26. 0  16 5 9 1.  0.6  11  996.3  8.72  5.71  2. 92  3.25  36.  29.3  16U60.  0.6  16  995.  8.  5.56  2.69  0.  3.25  U9.  29.  16U60.  0.6  16  995.5  8.11  U . 08  2. 69  0 . 0 9 20  3.25  3 1.  29.3  16U60.  0.6  16  995.5  8.  11  6.15  2. 69  0.1155  3.25  UU.  29. 3  16U60.  0.6  16  995.5  8.  11  U.55  2. 69  0.1026  3.75  52.  32.6  16 3 3 6 .  0.621  991.6  7 . 57  3 . 85  2.49  0.0917  3.75  37.  32. 6  16 3 3 6 .  0.621  991. 6  7.57  5.1 1  2.19  0 . 1330  3.75  US.  32.6  16 3 3 6 .  0.621  991. 6  7 . 57  U . UU  2.19  0 . 1092  39.  32.  6  16 3 3 6 .  0.621  7 . 57  5.13  2.19  0.1261  14.25  59.  35. 8  16 2 2 0 .  0 . 6 2S  991.6 991. 7  7.09  3.39  2 . 31  0.09 0 5  1.25  UU.  35. 8  16 2 2 0 .  0.625  993.7  7.09  U.55  2.31  0.  U.25  39.  35. 0  16 2 2 0 .  0.62S  99 3 . 7  7.09  5.  2 . 31  0 . 1369  U.75  US.  39. 1  16 1 0 9 .  0.629  992.7  6.66  U .4U  2.  15  0. 1202  1.75 1. 75  58.  39. 1  16 1 0 9 .  0.629  992.  6.66  3.15  2.  15  0.0996  U3.  39. 1  16 1 0 9 .  0.629  992.7  6 . 66  1.65  2.  15  0.  U.75  63.  39. 1  16 1 0 9 .  0.629  992.  6.66  3.  2.15  0.0915  U.75  U2.  39. 1  16 1 0 9 .  0.629  992.  7  6.66  U.76  2.  0.  -.75  58.  39. 1  15 1 0 9 .  0.629  992.  7  6.66  3.45  2.15  0.0996  5.25  6 1.  U2.3  1600U.  0.631  991.6  6.28  3 . 28  2.01  0.  5.25  63.  U2.3  1600U.  0.631  991.6  6 . 28  3.  17  2.01  0.0987  5.25  63.  U2.  1600U.  0.631  991.6  6.28  3.  17  2.01  0.0987  5.75  68.  US.5  1S90S.  0.637  990.U  5.93  2.  94  1.88  0.0904  5.75  US.  US.5  15905.  0.637  990.4  5.93  4.44  1. 8 8  0 . 113 6  5.75  65.  U5.S  15905.  0.6  990. 4  5.93  3. 08  1.88  0.  5.75  50.  U5.5  15905.  0.637  990.U  5.93  U.00  1. 8 8  0 . 1339  2.75  3.7  5  3  3  37  5  7 7  11  13  17  15  1254  1215  1313 1375 1021  1031  f^ooc'* r »c 1  ,  ,  ,  i ^'"»''' K  f  ,  'CON'  ,  J  f i u i r ? r ' O C ( * * , r M m x ^ r — cs^P^r*-**^ s  o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o  ^^ooo^^^^•^^ ^^^*^^*^^cocO(\|^Jf\l^lfN^^:cDcocD^/lL*l^/^ v  n  m o  o o o  ^  o  x  o o  o '-i  ^ u*. \P  3  in  5  J  i n m c; c. o o u~. i n c i  5  J  c  3  J  c  M  r  IN  N  C  i n CM  N  co CD  <N CN * N  •-  c  o  o  o c c o o c o c o c c o o o c o o o c o o o o o o o o o o o o o o  ci o  o c; c>  tN fN M «V (N  m  c.  u*i  ci in o  c: o c» .n c  in c: i n c i c» c i c : in i n c i o ct i n ct c» i n  107  o c e s 0 ^ ^ ^ o c * ^ ^ » A l n ^ ^ r • c o ^ • - ' - • -  o  o  o  o  o  o  o  o  o  o  t - » - o o o o o o N N f N i N ^  o  |  o  o  P ' ^ o  o  ,  o  ^ ^ o o o o o o ^ c n a l a • ^ ^ I  o  o  o  o  :  o  J ^ ^ ' [ y ^ c ^ 3 0 ( ^ 0 ' - ^  o  o  o  o  o  o  o  o  o  o  i ^  n  o  o  ^ ^ » N  ,  o  o  C ' ? 0 ' C » r * f » T - i n i f l i n i f > i n ' - » - * * o  ,  ^  s  r  1  r  '  r  ,  ^  ^ ^ ^ '  1  ,  1  ' * l  r  ,  n  w  n  ,  /  1  u  l  v  '  ,  o  o »  1  ,  "  r  cnc*coa0cocx>cOQco;ccocor^r^r^r^  C?* C" . CT"- ^*  O"*  1  C * C > CT* T » f T * CX^ CT* (T* CT* C *  d z j - o o o o o o i n u i i n c i * " * - * vO vO vO v O O  O  O  O  O  O  O  -  *  -  \C O  v 3 v O \0 » 0 v O v f i \ 0  O  O  O  C  O  C  C  O  O  0  s  o c > o o > f l  r  O  vO  vD  O  O  O  C  O  CT*  s  CT*  CT* CT*  CT*  •"•^-minmminoooiy:*' *-0  N N N N N r j M r ^ M o i ( N < N ^ n n n n n r i r r i 3  r» r - CM <N CN  0 CT*C7~> CT*  O  O  v O v O "-O O \ A  vO s D vO \0  v  O  .a-  "rfrsin*r^r^r^r^cNCNrNjr^r~r^r^r^r*rNjrMr^  O  J  O  J  O  J  C  ^  O  j  O  3  O  3  C  a  O  in in  c c t^i c o o r— o o o ^ ^ ^ o r ^ c ^ ^ ^ ^ " O * ' ' C T C - > 0 ^ ' - 0 0 0 0 - — C O ^ « ^ ^ D 0 ' ~ ' ^ > 0 ' — C O * " ' " « * " ' ' ' »— O * ' " • ,  -  o  o  r n '  o  Y  o  r  <  r  v  —  o  ' ' ^ r o  o  ,  i r  o  s  o  o  o  f ^ r *  -  o  o  o  o  o  o  o  o  o  o  r  ^ o ' - " o O u ^ c o ' ^ p * 0 » — O r ^ . ^ - » - o ^ T O »~ r - » « ^ - T — T— r -  1  o  o  o  o  o  o  o  o  o  j j r r r \ ; r N i r N j r * J o c o o c o o c j \ C T ^ ' y » r  ,  ' r *  O o  CO  r ^ r ^ r ^ x c o c o ^ ^ ^ i n s n u n i n ^ o c ^ c ^ ^ ^ — ' ^ ' ' ^ '  h C  (*•• ^ 1  C  O  a  . ' I L^. . ' I C  C ' C * > ^ C > f f i C C  ,  C  N  f f  1  a  "  ""1  ^  G O i p v ^ O f f  W  ^  N  ^  3  o  r\ o  .'N ( N ( N I N  o  c  o  o  o  o  o  C7* ff CT* CT* O"* O CTt o*i cr- o> CT o>  1  1  1  u  o r r ^ T r r O C O O v O O ^ * — ' ~ » ~ O O O O ~ ^ * ^ O v 3 *~ C O O ^ ^ ^ ^ ^ ^ r g r ^ r N r ^ r j r ^ ^ r N i r n r n r ^ f ^ M r ^ r ^ r " , f i r - ,  _Q 03  r  o  c  c  o  c  o  o  o  c  o  o  o  o c N c  c c N y  cM-ia*. i r . a c r ^ ^ o — —• f ^ o o M I IT. r r m c* c ( r c c r ' o c o c  o  o  o  o  o  -  o  o  r  o  ,  c  —  o  c  o  o  o  c  o  o  o  ra n p i a a 3 r i n L". i n o L I O U I i n n o o c c Ci c i u*» i n i n -J"; t n ^ - «-* — o o o o c c c c c : x ; c a a > c r c r c J ' O ' r r c s x s * a cC a  i n a" »- co r * •/> ca o  s co i n aj 1  c^rnr^nj<^rM^^*-<7'C*cr<?*r^r^r^r*inu^  Q-  ^ r O ^ ' - P J O C O C C ' - i . - . L*t<Nc". r*. M f s r ^ f i n n m r * . a r , r u na  o  I T  i / i I T . -.n in - i m L 1 i / i  J I  r - ^ f - r N H N f f N N l M ^ r .  -_n r  r  ,  t n m -..n  .I  n  * C C w ' l f f C O ^ C O " ( N O a n a a a j i i i n r , j i T i ^ i r t  n  n  m  r  j  a  f  in i n  . n m i n i n »n i n j  j  j  i  j  ?  j  ^  ^  tfl i n  109  C - T ' " 0 O C O ' - C o  r ri  o  ,  o  o  o  o  o  ' O ' - ' - O ' -  ,  o  o  o  o  o  o  - ' - O ' - ' - O ^ 0 '  r  o  o  o  o  o  o  o  o  N  o  ' c r o  "  r  o  ,  o  -  (  o  S  J  o  ' o  " f N ' " 0  ,  o  o  o  o  o  r  - r - " o  o  o  1 0 a ' " O ^ ^ O l / l C 3 ^ 0 ^ ^ • - c O v O M ' * c ^ 0 ^ o ' - J ( O ^ O ^ J l n ^ f t O n O O l ^ o • - c r c > c o o I N i n c ' O y ) r - i n r - / - . i - ^ c r r — i n r - r— ^ o r « *~~ c r o - m o o r ^ - m r - * ,  r  l  1  r- o  (Ti i n ^  I T I .-t c r  cr cr  3 3 f \ f N M » - ' - ' * * - » - * * ' - ' - c f ^ c > (  s  ,  r * r * ' , ' n n M « - » -  ^ c ^ < 7 A ^ v o o ^ ^ ^ ^ ^ ^ c ^ c T ^ < T ' r r cr cr cr cr ^ c crooovnduP-OOu-(<T>c> o o o •— «~ «- f ^ o j r N f N r g f s c N f N ^ t N r ' . p - r . r-. n n r n ^ cr a - c r c r c f i n i n m ' n t n \0 \£> vO iO O vO \0 O vO v O vD v o v^5 v o \ £ \ 0 vO vO \ £ \ C v C - 0 %c v O vD \ 0 SO *JD v3 r  f  >  J  r  s  J  v  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  vo n  o  "  »•  O  «n L"-  co ro ^ S 33 lA i/l I'N N <N (N tN fN f» f*^ r*^ r^* f& r^** r*> r"^i r*~>  O  O  O  O  O  O  O  O  G  O  O  O  O  O  O  O  O  ©  rr cr -=r cr cr cr  r^rMrsifMrsirsirNCMrsirMrsirsi ! M ( N ( N f N r N ( N r s i ( N t N f N C N f N  c o c o c o e o c o c o c o c r c J ' c r o c o o o o o o i n i n i n o o o i n i n o c r ^  r - N t N r N N t N f N i ' i r s n n r .  w  n  n  n  n  c  f  a  ^  crcr  crcr  c r c r c r c r t n  inininu*>vOvOvO  u ^ ^ u ^ i n i n u ^ i n t n i n x n ^ n i n u ^ u ^ u n i n i n i n u ^ i n u n u ^ r - r - r - r ^ r ^ N N ^ H ( N N . N f v f N r ^ n ^ f ' n n n n f " r n f 0 3  cr a cr cr cr t n i n u t v O ^ O  Table Experimental  Data of Block A ,  CC) •  n • os  P  T  (s) 1.75 1.75 1.75 2.25 2.2S 2.25 2.75 2.75 2.75 2.75 3.25 3.25 3.25 3.25 3.25 3.25 3. 25 3.75 3.75 3.75 3.75 3.75 1.25 U.25 U.25 U.75 U.75 l' 7 5 U.75 U.75 5.25 5.25 5.75 5.75  27. 22. 25. 27. 2U. 23. 35. 29. 28. 30. 38. 30. 35. 34. 38. 35. 36. 36. 40. 35. 38. 4 1. 43. UU. 40. 42. 40. 50. 42. 40. U8. 53. 50. 59.  PC  T  CO .  85.6  TR  = 78.-0°C  CO -  7.6  34  dT/dx  CK  (Ty-Tg)  d p (microns) - 1.011  LIQUID - Water TT  17  V  P  V  nf ) (J s ' V H - ) (kg nf ) (kg nf V ) 1  27.0 26961. 2696 1. 27.0 26961. 2 7.0 26626. 32.3 26626. 32.3 26 6 26. 32.3 26 320. 37. 6 25 320. 3 7.6 37. 6 26 320. 25 320. 37.6 26040. U2. 8 U2. 8 25040. 26040. U2. 8 26040. 42. 8 26040. U2. 8 25040. 42. 8 260 40. U2. 8 25704. 48.0 25784. 48.0 25704. 48.0 25784. 48.0 25784. 40.0 53.2 • 25551. 53.2 25551. 25551. 53. 2 25 3 38. 58.2 25 3 38. 58.2 25 338. 58.2 58.2 25 3 38. 25 3 38. 58.2 63.3 25 145. 63.3 251U5. 24969. 68.3 2U969. 68.3  1  3  1  xlO ' 4  0.6 13 0.6 13 0.6 13 0.620 0.620 0.620 0.628 0.628 0.628 0.628 0.6 34 0.63U 0.63U 0.6 34 0.6 34 0.63U 0.634 0.64 1 0.64 1 0.64 1 0.64 1 0.6U 1 0.646 0.646 0.646 0.652 0.652 0.652 0.652 0.652 0.657 0.657 0.661 0.66 1  996. 1 996. 1 996. 1 994. 7 994. 7 994. 7 99 3. 1 993. 1 993. 1 99 3. 1 • 991.3 99 1. 3 991.3 99 1.3 991.3 991.3 991.3 989.3 989.3 989. 3 989. 3 989. 3 987. 1 987. 1 9 8 7. 1 984. 7 984. 7 984. 7 984.7 984. 7 982. 1 982. 1 979.3 979.3  8.54 8.54 8. 54 7.62 7.62 7.62 6.35 6.8 5 6.85 6.8 5 6.21 6.21 6.21 . 6. 21 6.21 6. 21 6. 21 5.67 5.67 5.67 5. 67 5.67 5. 21 5.21 5. 21 4.81 U.81 4.01 U.81 U.81 U. U6 4. U6 U. 15 4. 15  P/P  th  (ms") 1  xlO  6  7.U 1 9. C9 8. 00 7.4 1 8. 33 8.70 5.7 1 6. 90 7. 14 6.67 5. 26 6.67 5.71 5. 88 5. 26 5.71 5.56 5.56 5.00 5.71 5. 26 4.88 4.65 4.55 5.00 4. 76 5.00 U. 00 U.76 5.00 U. 17 3.77 U.00 3.39  T  a  p|<  (mV "*- ) 1  xlO  1  9  2.86 2. 86 2. 86 2.51 2.51 2.51 2.22 2. 22 2.22 2.22 1.98 1.98 1.9 8 1.98 1.98 1.98 1.98 1.78 1. 78 1.78 1.78 1.78 1. 62 1. 62 1. 62 1.U7 1.47 1.47 1.47 1.47 1.3 5 1.35 1.24 1.24  0.0963 0. 1101 0.10 39 0.1111 0. 12 48 0. 1304 0.0977 0.1181 0. 1222 0.1141 0. 1018 0. 1291 0.1106 0. 11 38 0. 1018 0. 1106 0. 1076 0. 1208 0. 1086 0. 1241 0.1143 0. 1060 0. 1126 0. 1102 0. 1210 0. 1276 0. 1340 0. 1072 0. 1276 0. 1340 0. 1229 0.1112 0. 1291 0. 1094  Ill  cr- c r o CO m in f N o r - m cr I N m r\; o o  o  o  o  o  o 04 CO CO CO c r c r <N  CN  o  cr  i n O r o r » r o r o ON f N m i n vO m cr cr c r fN m CP in i n r - m fN cr r o cr- o cr. T— »— o  o  o  vO cr  o  r-  o  r-  r-  o  O  o  r-  CN  cr  cr  o  r-  o  o  CN  r->  o  o  o  o  i n i n to n c I/I M N J i n co cr r - fN co co fN vC f N o o f N c i r N r n * - O O CN r - O N f N  m fN o > CO  o  o  O  r-  m r-i C N r rf— r*- r - i n i n m  r-  o  o  o o  o  rN c j fN o i n cr cr r r m  o  o  O  O  o n  o crro  cr>  O  cr*  fN CN fN <N fN CM  oC_>  o ro O o in r-  ON  o o  cr  O O  rs in  r - vr> C O in CO cr r » o o cr  in in o o o o c r C 7 i n m IT. i n  CO  CO CO  o o  r-  CO r -  in ^r  o  o  m m c r in c r CN • — c r  in in  ro  ro co ro r - r»  r-  CO r- o in  i n i n C O C O c o 3 5 CO c r . r -  i£> r-  o N3 c r  O  m iO fN o cr  «x>  sO i n  ro CO ro f n m m in in m in  in  i n \ ^ c o f * N . i o v o O r * ~ u * » r * m c o o o i n CO *~ i n r cr in f*l i n o ,  cr  in  cr  m  i n cr  in in in in in in in in  r-  r-  r-  r-  f>  cr r o  _  rn  o  cr  m r - r » r - r - r - r - r - ( N O J N ( N f N i r N in in c O o o O O vO v o n n n o o  in i n m  r-  i n cr ro  r  r-  r-  r-N n  m  a &  r-  a  =T =t  =1 &  CT  r - r** ON o*» CT* O N Cf* O *  v£) vC •C c r \ 0 •J0 v o n r , ro o o o r* r* rcr fNl f N f N f N f N o CO =c c o CO c o CO c c C O CN O o - c - c r cr- cr- e'- c r c r ON c r CT. c c c c CO c c c c CO CO CO CO c c C O CO CC CO c o c o c c x r - r*» rcr- cr. ON O N O c r cr* O N C N ON CJN C N ON C C O N ON CT* O N O " ON O N C CN C N C N O N cr cr er- c r cr- C r  •~  »—  v D \£> o  O  O  o  O  o  cr cr cr ON ON v£> c r c r i f N . - M f N c c CO CO CO c c r~ i C N f N OJ fN r s  r-  cr  cr cr  r>  o  • h r- n < f N f N CM r o  c  c  o O  Crcr X r*. fN  v£J o NC ON cr •3 •D o CO in i n i n i n in rr - r»- >---. f N fN| f N f N f N f N  o  O  O  C  O  O  r- r-  rr o r o  NC  NO VO v£> o  c c CO CO C C  o o  cr vO fN r— fN  O  o  c - CT CO CC c c c r cr. C if> i C r N fN fN  m  in  o  o  O  o  c r CT- c r CT CO TO CO ON c r ON c r US VO fN fN fN rN  O  in in  in  in  CN  o  cr  r- \ 0  i n i n cr  cr- c r cr cr  o o  Cr P r-1 n rv£ O \n CN f N f N f N f N cr  =0 CO cr  CO c r C r cr- C r cr- c r cr cr cr cr cr  o  o  n f  o o  C  o  o  r> O r - *— O  o o  CO CO ^ O O fN ro ro sTi O k/N vO f N CN f N f N f N f N C N  r*- Ni On vOi n U 0  ON c r  o  CN  o  ro ro fN fN *~* Nf) VO O * CN  f N f N jNi i n i n i n r * r - r *  cr c r o cr- c r o rs i n i n i n o o cr ir. i n i n i n vo NO o o ^ o r- r-  o r-  o  r»"> r - O cr i n i n O r . fN fN fN fN r o C l r o m ro cr  «*0 O ro r t  r-» f N f*N r *—) m m m  •> m * r-  i nin m in in in in in in in in in in r - r - fN fN fN fN r fN r r» r»  in Li p~ r -  m un i n m i n i n m i n i n i n i n uo uo i n uo i n J O i n r— r - r - r-> r - f N f N f N I N r » r * r - f N r N f N r*» r * r »  -  --  c?  > <N O I N N  o  f N f N f N f N f N f N f N f N CO CO CO c o c r c r c r O N cr- O N c r a - c r cr cr cr cr c r CT c r c r c r c r c r u"i i n m u o i n i n N O i o v o N O i£> o vO v O i O i O s D \ C <D vO N O * 0 V O V O IJO ^ 0  O N ON ON ON O N f N fN fN fN fN fN f N fN fN VCN O <£, l O <D i O KD  f N CN f N f N f N f N f N f N f N C * m  r i cr  o c-i c r  r-i r< cr cr  c r vO f N O c o c o v o o r** i n m c r c r c r m i n c r i n N O uo  cr  cr  cr  cr cr  m  in in  m in  %n  f s j L O o « ~ « - c r o m r ~ » — f ^ u ^ r s i c c ^ c j 3 I T .? o — ^ o x> »— =7 *~ <j~-.q r*<Tr^r->cxr-o*Cv2C". ,^r^r^cr:r=Tf^inootf^:jc^fNr-fsj cccr^3' k  o  o  c  o  o  o  o  o  c  c  o  c  c  c  o  o  c  c  o  c  o  c  o  o  o  o  c  c  o  o  o  o  o  c  o  o  o  o  o  o  o  o  o  o  c  o  o  c  o  c  o  c  o  o  c  o  o  o  o  o  o  o  O  -j" lT\ \T*  O -O  I T U1 U- *n J  J =T .3  f> <— «— r» N  {M «M  c_> o  x  r - cc  i n r- i n r -  o m CC i n O r- r- L ~ i n M r— r - r-. -n .3 i n  vO a  co  r ^ r ^ ^ ^ ^ ^ ^ ^ ^ ^ c c c ^ o o x c D x c o x c c c z c o r ^ r ^ ^ ^ x C i n i n i n  I E  CO  0)  o o o  O —  CO CO O O O O  I  • co co co i n i n ir. *- « — «— « —  cc cc =j 3- ^ o  o in  o o 3 x a ) m 3 o o ^ 3 j ^ r - ^ - ~ ^ c r > r - ~ r - r - r ~ ~ r - " aN ^ o o ri C C I T L". i / i L". ITI t " l t " l -1 r in oi a 3 J 3 2 J 3 3 CT -33 r"l xO v C vO vO vD NO *0 " O VO «0 VO VO O O * C O SO NO VO vO  o  o  o  o  o  o  o  c  o  o  c  o  o  c  o  o  c  o  o  c  c  c  o  o  o  o  o  o  o  o  o  a \ C * C 3 3 3 .3 .3 3 3 3 C > C N N M C C C O O O C C 3 3 » - r - ^ r M f N U*» r g r s i r N i r N i r N j r s i P ' j r N i r N j f N i r ^ r N i f r ^ r ^ a 3 r t o IT ^ I T o J 3 c I O C N  CD  ; CC X 3 3  ininc?(T'C^cr-a rorrrncoccrNjf^r\jr~r~--i N  X  ^rOCvO  c o c c o o o o o r - . r~minincocoaocoor->(  o c o ^ o o ^ r * . o « ^ 3 inr-=5r-^oor-\T)r*cro=y r*~ co co co co co co coTOCO p~ co co co co co co c* r** cc 0  S  co r-> \c o m CN co cc co co cr*  u i i f i u i i T i r > i r i A i r t t i o o ^ o i n i n i r u o u i i n i r u n o i n i / i ^ ^ fNr^r^r^r*f^r^rs!rs;rv:r^r^f^rNjrsjp*r^ • ( N N N CM t N I  • ;r 3 3  3  3 3 i n in i n i n m  Table Experimental  Data  of  LIQUID - Hexane T (°C) - 52.1 n - 34 T  T  (s)  PC CO  20  Block  B,  B  dT/dx I 'f (kg nfV ) ("K m") (J s'V K") (kg m") xlO 1  4  1.83  1.6  1.90  1.  660.5  3.  23  2.07  1.6ft  129  655.  8  3.08  1.19  1.5  655.  8  3.OR  1.73  1.58  0.  129 129  .3.00  1.89  1.50  0.  120  3.00  1.57  1.50  0.0773  0.  1 28  3.01  1.6  0.  128  3.01  1.11  5  12899. 12099.  8 b55. 8 6 5.3.1.  0.0H55 0.0931  0.  128  653.1  3.01  5  2.01  12899.  (1.  128  653.1  .3.01  1.6  1  1301U.  0.  127  651. 0  2.9 3  1  130114.  0.  127  651.  0  1.66 1. 80  29.  1  13'.11U.  0.  127  651.  1.17  1.19  0.07  57  29.  1  110114.  0.  127  2.9.3  1.78  •1.19  0.09  16  31.7  13131.  0.126  651. 610.  0 0  2.93 2.93  6  2.06  1.75  1.15  31.7  13133.  0.  61 0. 6  2.  86  1.50  1.15  31.7  13133.  0.  126  610.  6  2.8 6  1.60  1.15  0.01111  31.7  131  3.3.  0.  126  618.  6  2.  06  1.17  0.0771  13111.  0.  126  618.6  2.0 6  1.72  1 3255.  0. 0.  125  616.  1  1.52  1.10  0.08  125  616.  2.  1.15, 1.15'  2.70 2.78  1.35  1.10  0.0726  132  660.5  12569.  132  660.  1.25  71.  18.  8  12569.  0. 0.  132  2.25  91.  23.  9  12786.  0.  2.  23.9  12706.  0.  2.25  82. 77.  23.9  12706.  2.  811.  23.9  2.75  83.  26.5  1278 6. 12099.  2.75  92.  26.5  2.75  73.  26.  2.75 1.25  81. 03.  26. 29.  3.  25  76.  29.  3.  25  90.  5  79.  3.75 3. 75  79. 05.  814.  89.  xlo'  23  0.  8  75  6  T|  23  8  3.75  P/P  1  3.  10. 18.  12569.  70.  3.  (ms") xlO  3.  80.  3. 2  1  3  1.25  25  39.5°C  dp (microns) • 1.011 T (°C) - 12.6  1.25  25  (Ty-Tg)  3.75  HO.  31.7  1.25  06.  31.  1.25  93.  .VI.  U.  76.  31.3  3 3  11255. 1 1255.  1 26  5  655.  653.1  73  8  0.0370 0.0901 0.0931  8  0.0736  1.51  0.0817  1.51  0.0728  51  0.1011  1.51 1.19  O.OfalV  1.19  0.0968  1.  8  8  68  O.Ob  51  0.0921 0.00 10  0.0907 20  0.  125  616.  1 1  1.10  0 . 0 9 0 5  31.3  11255.  0.  125  616.  1  2.78  1.6  1  1.10  07.  .31.3  0.  125  1  2.  1.5  0  1.10  0.08 65 0.0:105  Uft.  37.0  0.  121  616.  U.75  1 3 2 5 5. 13181.  1.19  1.35  0.0b21  83.  37.0  1 3 38 1.  0.  121  613.  7 7  2.71  1.  2.71  1.58  1.15  0.08 7 0  5.25  80.  1.1510.  0.  123  611.  2  2.63  1.61  1.31  0.0126  5.25  90.  39.  39.7  11510.  0.  123  63  1.  1.31  0.0769  81.  .39.7  13510.  0.  2.  5.25  123  611.2 611.2  5.25  88.  13510.  0. 123 0.121  611.2  2.  63  630.  6  2.55  0. 0.  121  630.  6  121  63 0 .  6  2. 5 5  25  it.25  81.  1.  25 75  7  5.75  89.  39.7 <I2.<I  0.75  80.  12.1  13613.  5.75  90.  12.1  136U3.  13613.  611.  78  2.6 3  2.55  1.83  36  1.6 1.11  1.35 1.19 1.33  1  1.31  0.0908  1.31 1.26  0.0737  1.26  0.0062 0.0770  1.26  0.0701  • " C O O * o  o  C  C  o  O  o  C  -  o  O  o  c  o  o  o  o  c  c  o  o  c  o  o  o  o  c  o  c  o  o  o  o  o  c  c  o  o  o  o  c  c  o  o  o  o  iT! i T  L"I I T  \C O  3  3  1  3  3  o  o  o  r - p-> n  o  o  c  o  r  o  o  o  o  ( N ( N  r *  o  r - r J r-j ( \  o  N'-fMfMWN'-fM'-'-'-IN'-M'-'-'-'-'*'- i-«-»-«-f-«-«-^-  C r C C O ^ ^ v C i O S 3 ^ - r - ^ - * — X C 0 \ C o c r * ' C " » O O ' * 3 3 3 =T r— U 1 l O l A l A t"> l " U*, L i 1 . 1 l " t " ! l i " L i i n a 3 3 3 3 3 3 3 3 I T C * f * «•*» m vO v O \ D \ D vO , o vO \ D »0 O M D V O v O "«0 \Q l O '^3 \ C v O M O «0 \0 vO r  o  o  o  o  o  o  o  o  o  c  o  o  o  o  c  c  o  c  r - r - r - c o c r i c c c o x c c " i " ^ i r i r i c o o ^ • ^ ' ^ ' " ' ^ l ^ > J I N O J f ^ i ^ ^ ^ f N i f N t r * ; * - " . o r . r - r-s 3 3 0 J O ^ c r ^ r - . &zx&=ra=jz3zTzt 3 3 3 3 3 3 3 3 v  ,  1  ,  s  c o c o c o r ^ f ^ r ^ r ^ r ^ v O i o * j ^ i n i n i n 3  o  n i J ^ c r 3  t  p  c  o  o  o  o  o  o  o  P C ' ^ c r c ( r 3 3 ^ l ^ I c : J " ' J ^ ^ c ^ o c c f N f M s o i r i n i r i n i n 1.1 m ,  3 3 3 3 3 ' 3 ' 3 3 i n ^  ,  *  o  a c v u  o  c o c c ^ o c c c o c o r » ii ni n i n  ,  * o o !  i n ^ o - 3 i n o r ^ i £ : ^ r ^ r ^ ^ o r N i ^ r o j c o ^ i n ^ c ^ o r * ' ~ i n c o f - i o i o a ) v o r * i o ^ o ^ r ^ r * r ^ c o r * c D r ^ r ^ o ) r ^ r * a ) c c c o c D f * c 3 p ^  l A i n i A O i A i n i n m i n i r i n m i n i n i n i n i n i A i r .  o i A i A i n i T i A i A i A ' A i A  fNNNrNNiNnmrrajj  3 3 i n i A i A L i m o  115  *-o*~*-'™oO'-*-  -»-c-o*-*-'-*-«-«-*-o*-'-«-«-»-o«-«-*-«-«-'-»-*-*-  o  o  o  o  o  o  o  o  0 ^ i } O 0 C O O O  iA  »—  o  ^  ^  o  r  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  ^ ^ ^ ^ C ^ ^ C ^ C ^ ( > N ( N l ^ N ^ ^ ^ •  m in 3 3 a 3 i n 3 3  «— 3 3 3  o  a 3 ( > ( ^ C  ,  r  3 3  l  n  n  r  ,  |  c  o  ^ 0  1  o  o  o  o  o  o  o  o  o  ^ X f N N f V f ^ i ( N C ^  a ^ r i a m n a n f n n n f N r r i n r n N i N P n r f f i  C O C O C O C D O  a  ' 3 ^ 3 3  3 0 0 0 0 0 0 0 l 0 0 < " 1  ^ ^ ^ o o o o o c ^ c ^ a ^ c ^ c ^ C ( ^ c o c o a 3 c o c o a 3 0 3 c o c o c o c o c o ! 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" l IT.  3 3 3 3 3 3 3 3  t l ITI ITI iT>  3 n m in in in in  116  m  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  ^» c r  o  o  r - i N i A r - a n c M N M n f N o ^ ^ u  o  o  o  o  o  o  o  r - C ^ C ' C ^ r ^ C ^ C ^ t N f v i M C D f f l C D O i O O a t O C O C O C J C C O O O O O a  o  3  o  3  o  o  o  3  O  o  o  O  o  1  o  CO c o o  o o »- o c ^ i n ^ o ^ i T i r t.'*i "ii;r* niAor^cc ^ ^ o r ^ c o i n L n ^ c o c o r n i A r v r ^ ' v o c o c ^ 0 3 0 ^ o ^ ^ ^ ( N ^ r ^ ^ ^ ' - a ^ M ^ ' - ^ r , f ^ ^ c D ^ ^ / 1 0 " f f i n n ^ r o o N t f ^ • - f N O T  -  >  r  >  f  r  l  r  ,  c o x 3 c o c o c ^ ^ ^ W ' A i ' ' \ i n i r i i n o o o o o u i t f i u i i r t i n r ' i - » * r - f * ( s f s M r t n  c^cococoa3cococccoaDr^r~r^r~r^r^r^r~  h J 5 >£ ^ ) £ O O IJI U 1 1.1 J .1 3 j * r ^ ^ ^ n r ^ ^ t N ^ ( N l N t N ' - ' > - f - ' - C < ^ a i C O C O C O c ^ ( ^ y f ^ o c l C ^ l T t ^ l C 7 0 ' £ ^ ( ^ c r ' ^ r ^ c • . r * c r . c ^ ^ c ^ c r * o - c c T * c c c ^ c T ' C O a s c o c o c o ^ o CT> CT* CTN Cr> C - CT CT> CT* CX" CT> CT> 0"" CT> CT CT> CT^ C"» C> CT. CT CT* C O " O C> C * C CT> O"* 0*> C7N C * <T 1  4  ,  s  1  \ £ «fl «C ^ ) v O vO O o  o  o  o  o  o  c  \0 O o  c  v C vD O o  o  o  O \ D vO \ 0 ^-O O o  o  N  s  1  o  o  o  o  O *0 ^  NO  o  o  o  c  c  o  o  o  MO O o  o  vO o  o  O O o  o  s O vO v O vO o  o  o  o  ^"^o*o*oovDvc*N'Nr>»oooooor>-r* ^ o o o o o o o t c a 5 r ^ r * r ^ r - - . r - r ' i / > L - . c ^ ^ c n c n o < r o c a c c c o o c c c c c D S : c 3 3  n ^ N ( N ( N N l N ( N O o o f  S  C D C C C ^ C C ' a 3 C O a 3 C O c O C O C O C O C O C O C O C O e O C O  ^ ^ ^ ^ ^ ^ ^ 3 3 3 ' - ' - r - r - ^ • c o c o c o c C l J ^ / l l / ' ,  r - » - f -  ^u^u^ininiAtncroo^rsjrNirsicMrNirNj^o^  a i T l ^ t n < ) N f f l C C N u i e ^ O N t c r r O o ^ n h O P » - v D v O N * 0 ' N C r n ( » ' h i n N f N n C N N ^ n N n n a r . n n a n a r i i / i a n ^ s r . 3 3 3 J 3 a o i / i i n ^ 3 « i > r  r ^ r ^ M r s i r ^ r i r M r ^ r - r ^ r ^ r s i r s j r s i p s i r s i r ^  117  t - O ' " *  -  tr-«— o *  -  *  -  O O O O  1  -  * " " "  -  ^ *  r  " o *  -  ' —  o «—  .— i— f— »—  OOOOOCJ>OOOOOOOOOOOOOOOOOOOOOOOOOOOOO  o o o ^ ^ o ^ c j  cr • j c r - - ' - f - » - ' - ( j  ,  c r i C  r ^ r >  ,  r ^ r ^ o ^ v O O u O L O L n t O r r c r cr c r cr  r o r o r n r*j C N C N C N f N CNI C N f N C N r ^ r s j r s i ^ r — r — r —  ( N o N O N i N r i O i n O i o i n o v f l y ; ' i n i n - n c n - j o i n ^ r ^ f N C N r - o D i / . t/i' c^uoo^ocrcr-jocoi^uocj'cr  t  r—  *— -jo c o c r  » — r » - r — r ~ ^ -  o r r o i A « - N s O L n '  ,  ,  i < N c o c o r *  crcrinrocrcrrorocroorororocTi  c r i T i i n i r t u o i r i r o c r  c ^ c ^ c ^ o o o ^ f l ^ o • a ^ l N N N N ^ ^ ^ o o o o o o o a ) N ^ J M N o o o o o c r c r c T ^ ^ ^ ^ r ^ r o r o < ^ r ^ r ~ r ^ r ^ r ^ cocococDcocDr^r^r--«r--^  v O » D v D i n i n u i f v i N f N i N f f l o ) c o c o c o r > ) N r > i - j o o o L I V . / ia ^ c r c r O C ^ O C r - f ^ O ^ f ^ f ^ C ^ C ^  cr c r c r  i n i n u o u o c r c r c r -cr C N f N f N f N f N  r N r N " N f N f N ^ - ^ ^ C T f ^ c r * ' C r - r ^ f ^ ,  C ^ O ^ C ^ o v D s o r ^ r o r o r o c r - o c ^ C ^ o o o ^ ^ ^ f N f N f N f N f N f N f N f N f N r o r o r o cr cr cr cr c r c r cr cr i T i n i - " ! ^ ^ ^ ^ / ! ^ vD ^ D v £ l i i O i O v D \ O v S v O O ^ v C i O vO vO vD \ 0 > f l \ £ v D O ^ O v D O N 5 %D \ D ' J >fl ^ 5 >fl *0 o  o  c  ^ ^ ^ r ^ r ^ r o r o r f N f N f  o r o N  o  o  o  o  o  o  o  o  o  o  o  o  o  c  o  o  o  o  o  o  o  o  c  o  o  c o l ^ . iTin^NNNNr^f-i---f-^--'-rrrN^r'. ^ L ^ j ^ u o f N ^ N r N f N o o o o o c x c ^ r o r o r ^ r o r o r o r ^ r o r ^ r o r o r o f N f N f N f N r N f ^ f N f N f N r ^ f N r N f N r N r N r N r N f N C ^  o  o  c o o l £  o  ,  o  o  " * -  o  ,  o  * ' •  ,  "  ^Dlfi^OMnf^OOOOvO^O^lI50•NNN©OCOC^^r^n^^^^^Nf,^Nl^l(N crcr.crer. cr*cr%cr:=rcrcrcocccoxCT N N N ( N f N f N f * i r . n r .r o r o r o r o r o c r c r  cr cr c r c r c r  ^ \ O ^ C ^ ^ ^ C ^ r o r v » a ; f N r o ^ v o o C r o N i ^ fNrifNrNfvj(Nr(Nriri^cf crrororomrouocr  cr  i n y o i n u o u o u o u o u o v O N O x O s O v O  cff"^c>3u iu*i-jiiiOin»OC'/i ,  v n i T u i t n o i n i / i u i i n i / i i / i i r i i n t n i r i i r r ^ r ^ r ^ r N f N - i r * * r ^ r ^ r ^ r N f N «-•-•—  f N C N ' N f N f N f N J  > cr c r cr crcrcrcrcruounuouni/iiriin«nuo  r c r c x : ( r ^ o ^ c ^ J 3 ^ ' ^ ^ ^ C T ^ c • x ^ w . • : ^ ^ ' n ^ ^ 3 3 ^ o « - o * - o o c * - c - - - c _ - o c c c c c o o o o o c : c c c c o o c : c o o o o c c o o o o o o o o c c  r»"> f\J fNJ in 3 l"l C> w  C K N I N r r-  CO r  c o o o c c o o o c o c c o o  OI -1 <~ 3 C 1  ^ ^ T - o o o o o o o o o o c ^ c ^ c r  or-r-33 3 a a  •- ^  %  c r c o c o c o c o c D c o c o r  • CO CO CD C D CD j  3 zT  ,  * p  ,  »  S O O O ' A v O v O i O N t N  noo3®cco3vDC^iei£wrr^ni-^»-^c>a 0'.'00>o\fi3tf O O C IT lA LT* i,1 iT> O U~ i/l U1 ^ J " - iT, ^ L"' 3 ? 3 3 3 3 3 3 3 <0 O O ^ v«T5 \DvO^C\SOO^^CO VO O vO ifl \G \0 \0 O ^ O \0 v £ > \0 ,  nf^^f^l^-'•'-'-'*oooooc^c^c^c^C'^^^^lOvo^D^fl^nln^n3tf m n m n n r . n n r n n n r r^rNi^rsir^rsir^Ojr^rNtr^c^cNc^CN'CNnifN o o o o o o o c o o o o o o o o o o o o o o o o o o o c o o o  •- N IN n n r n n ~ s 3  33ciinLn^ j^ ^os£op-r^r»c^c7 cr*cT»oo  • \ D x D \0 ^9 \ D '  ,  ,  ,  • > D M > \ 0 ^ > N N f M C O C O C O C D M f ^  i n i n L i i n i n i n i T i i n i / i L * i i r i i r i n i r i i r i i n i n i n i n i n i n 1/1 1/1 L I L i L I i n i n i n vn i n fN N I N M \  nj(Nnnrnrirnnnnrifns3tf3333ini/)in«n(nirt  Table 26 Experimental Data o f Block D, (Ty-Tg) = 54.9°C LIQUID • Hexane  d (microns) • 0.790  T (°C) • 62.7  Tg  T  p  CO • 7.8  n • 26 P  os  T  («) 1.75 2.25 2.25 2.25 ?.75 2.75 3.25 1.2 5 3. 25 3.25 3. 25 3.25 3.75 3.75 3.75 3.75 3.75 U.25 U.25 U.25 «. 25 U.75 U.75 5.25 5.25 5.25  65. 75. 63. 58. 75. 52. 69. 76. 77. 57. 70. 5U. 58. 69. 73. 67. 79. 75. 79. 68. 73. 75. 63. 89. 73. 81.  PC  T  (*c) 19.7 23.1 23. 1 23. 1 26. 7 26.7 30.3 30. 3 30. 3 30.3 30. 3 30.3 33.9 33.9 33.9 33.9 33.9 37.6 37. 6 37.6 37.6 U1.2 U1.2 U5.0 U5.0 U5.0  dT/dx  k  (°K m"1) (J s'V'-K"') 17 3U6. 175U8. 17548. 17548. 1776 3. 17763. 17986. 17986. 17986. 17986. 17986. 17986. 18 211. 18211. 18211. 18211. 18211. 10452. 18452. 16452. 18452. 18695. •1S695. 18956. 18956. 18956.  P  f  0. 131 0. 130 0. 130 0. 130 0. 128 0. 128 0. 127 0. 127 0. 127 0. 127 0. 127 0. 127 0. 125 0. 125 0. 125 0. 125 0. 125 0. 123 0. 123 0. 123 0. 123 0. 122 0. 122 0. 120 0. 120 0. 120  (kg  V  nf ) 3  659.7 656. 5 656. 5 656. 5 653.2 653.2 649. 8 64S. 8 649. 8 649. 8 649. 8 649. 8 646. 5 64 6. 5 64 6.5 646. 5 64 6.5 643. 1 643. 1 643. 1 64 3. 1 639. 7 639. 7 63 6.2 636.2 636.2  (kg  V  P/P  th  T  p|<  nfV )  (ms")  (mVH ')  2. 67 2.24 2.74 3.02 2.22 1.40 2. 43 2. 16 2.11 3.04 2. 39 3. 23 2.96 2. 41 2.25 2. 59 2.04 2. 16 2.02 2. 4 3 2.2 3 2. 13 2.63 1. 68 2. 17 1.90  3.08 2.67 3. 17 3.45 2.67 3. 85 2.90 2.63 2.60 3.5 1 2.86 3. 70 3.4 5 2. 90 2.74 2. 99 2.53 2.67 2.53 2.94 2.74 2.67 3. 17 2.25 2.74 2.47  1.66 1. 60 1. 60 1.60 1.53 1.53 1.47 1.47 1.4 7 1.47 1.47 1.47 1.4 1 1.41 1.U 1 1.41 1.41 1.3 4 1.34 1.34 1.3U 1.28 1.28 1.22 1.22 1.22 •  1  xlO  4  1  xlO  6  a  -  xlO  9  0.09 27 0.0300 0.0978 0.1073 0.CM 17 0.1250 0.0') 20 0.OH 18 0. 0807 0.1151 0. 0905 0. 1223 0. 1156 0. 0941 0.0879 0.0976 0.07 97 0. 08 14 o . n;j70 0.0979 0.0R90 0. 0889 0. 1097 0. 0727 0. 0939 0. 0823  120  Appendix  Sample  1.  Calculation  This  of  w a t e r and one f o r not late  the most  provides  hexane.  probable  1:  The  When v a r i a b l e s absolute  Rule  2:  The into  absolute tioned  sample  error  Two  calculations  estimates  give  s i m p l e r u l e s were  of  the  a, o n e  for  largest  employed  two  to  errors, calcu-  their  errors.  given  relative  errors  categories: The  former  tolerances, for  for  both  derived  and e q u a t i o n s .  estimates  add  their  errors. are m u l t i p l i e d  errors.  rules  a r e added o r s u b t r a c t e d ,  When v a r i a b l e s  absolute  construction The  Estimation  two  ones.  Calculations  errors:  Rule  fall  and E r r o r  a and E r r o r  section  III  or d i v i d e d ,  t h e w a t e r and hexane  absolute  errors  are  c a l c u l a t e d from  The  latter  were  i n the  and the  obtained  calibration results, coefficients  equations  programme c a n be c o n s i d e r e d a s m e a s u r e s  confidence  l i m i t of  coefficient.  subsections  stipulated previously directly  o r measurement  regression  each  add  of  obtained the  menfrom  errors. by  95 p e r  the cent  121 a.  Water  The  following  equations  are required:  „ v  Note:  since v g f o r water  th  _ 0.0002  T  i s n e g l i g i b l e , Equation  [38]  reduced t o the  above.  P x , h~ ~  •  For  purposes  qh = A ( T T - T B )  of error  +  |  os 775"  (Ty-Tg)  estimation only:  (qh)  kf  -  +  §  ( T ^ )  .  ( A T ^ T J f f i )  = A + BTpc +  C(Tpc)2  p = 999.168 - 0.0042607 ( T p c ) 2  J-= 21.482 ( ( T p c - 8.435) + (8078.4 + ( T p c - 8.435) )^ - 1200. 2  Note:  Reference  (24)  gives  no e s t i m a t e  used i n t h e v i s c o s i t y e q u a t i o n . all  of error  For purposes  t h e e r r o r was a s s u m e d t o b e i n T p Q .  f o r the coefficients  of c a l c u l a t i o n ,  therefore,  122 dT dx  TpK  = Tpc  +  V  a=  273.2  th  M  P Tni/  PK  The 13,  following  27.  pertain  to  the  first  datum  point  Pne. = OS  2.25  in  Table  Appendix I I .  d  The  values  dT dx  p  = 1.011  TT  = 57.4°C  TB  =  stipulated  on f o o t n o t e d  x  = 50s  7.2°C  sample and e r r o r The  microns  entries  calculations for absolute  errors  are given at  this  are  the  point  listed  end o f  in  the  are  shown  italics. table.  in  Table  Comments  Table Sample and E r r o r Variable  27  Calculations—Water  Value  Absolute Error  Relative  Error  0.0002  0.0002m  ± 1.42x10  T.  50s  ± Is.  ± 0.0200  4.00xl0"6ms'1  ± 0.416xl0"6ms_1  ±  2.25  ± 0.10 grid  spaces  ± 0.0445  ± 0.10 grid  spaces  ±  0.0130  V  th  P  os  7.5  7.5  os/7-  P  5  x/h T  T  T  B  grid grid  spaces spaces  ± 0.0840  m  0.1040  0.300  ± 0.0173  ±  0.0575  0.700  ±  ±  0.0247  ± 0.5°C  ±  0.0087  ± 2.7°C  ±  0.3750  ±  ± 0.00034  57.4°C  .  7.2°C  0.0173  A  0.567558Jnf V  B  1.862289xl0"3Jm"1s"1oK'2  ± 4.6xl0"5Jm"1s"loK-2  C  -7.15006xl0"6Jm-1s'loK"3  ±  4.6xlO"7Jm"1s"loK~3  ± 0.0640  50.2°C  ±  3.2°C  ± 0.0630  28.4Jm~1s~1  ±  1.74Jm"1s"1  ±  V  T  B  A(TT-TB)  *  1 o  K  _ 1  0.0019Jm"1s"loK"1 .  ± 0.0250-  0.0633  (Table Variable  1 1  27 -  continued)  Value  Absolute  Error  Relative  Error  3294.8°C2  + 57.3°C2  + 0.0174  51.8°C2  + 38.9°C2  + 0.7500  3243.0°C2  + 96.2°C2  + 0.0300  f(T2-TB2)  3.021Jm"1s"1  + 0 . 1 6 7 0 J m '. 1 , - 1  + 0.0550  1  189119°C3  + 4936°C3  + 0.0261  373°C3  + 420°C3  + 1.125  188746°C3  + 5356°C3  + 0.0284  -0.450Jm"1s"1  + 0.042Jm~  S~  + 0.0924  qh  Sl.OJnfV  + 1.946Jm~1 s "  + 0.0628  (f)(q")  21.7Jm"V  + i .899jm~1 s"  + 0.0875  AT,  32.6Jm"1s"1  + 0.346Jm~  ± 0.0106  3.069Jm_1s-1  + 0.130Jm" s~  + 0.0424  -0.451Jm"1s"1  + 0.041Jm"1 s"  + 0.0901  35.22Jm'1s"1  + 0.517Jnf1 s  + 0.0147  -13.5Jm"1s-1  + 2.416Jm" s~  + 0.1790  22.9°C  +  'B T3-T3 'T 'B 3  U  T  B  ;  1  BT2  TT  CT3 3'T AT  T4 T4 T T  T  (^)qh-[ATj-r|-T2+yT3] T  PC  1  4.2°C  ***  s  + 0.1830  ( T a b l e 27 - c o n t i n u e d ) Variable  B T  PC  C T2 U PC k  f  Value  Absolute  Error  Relative  Er  0.0427Jm"1s~1°K"1  ± 0.0089Jm"1s"loK"1  ± 0.2080  -0,0037Jm~1s"1oK~1  ±  0.0016Jm"1s"loK"1  ± 0.4300  o.eoejnfV ^  ±  0.0124Jm"1s'1oK'1  ± 0.0205  1  -1  999.168  99?.168kgm~3  ± 0.5kgm"3  ±  -0.0042607  -0.0042607kgm"3oC"2  ± 2.1xlO" kgm"3oC"2  ± 0.0490  -0.0042607T2C  2.23kgm"3  ± 0.93kgm"3  ±  0.4150  p  996.9kgm~3  ±  1.43kgm"3  ±  0.0014  Tpc-8.435  14.555°C  ±  4.2°C  ± 0.2890  (Tpc'-8.435)2  211.85°C2  ±  122.5°C2  ±  0.5780  8078.4+211.85  8290.2°C2  ±  122.5°C2  ±  0.0148  (8290.2)^  91.05°C  ±  0.674°C  ±  0.0074  Tpc-8.435+91.05  105.61°C  ±  4.874°C  ±  0.0461  21.482(105.61)  2268.6mskg_1  ± 104.7ms k g " 1  ±  0.0461  1068.6m.skg~1  ±  ± 0.0980  y  0.000936 dT  "'dx  ,  kgirfW  17073oKm_1  5  104.7mskg"1  0.0005  ± 9.2xl0'5kgm"1s"1  ± 0.0980  ±  ±  1564°Km_1  0.0916  ( T a b l e 27 Variable  T  PK  a  continued)  Value  Absolute  Error  Relative  Error  296.1°K  ± 4.2°K  ± 0.0142  0.0739  ±  ± 0.3092  A b s o l u t e e r r o r from t h e measurement o f A b s o l u t e e r r o r i s a sum o f drop  -  0.0228  one-half a grid space, ±  t h e a b s o l u t e e r r o r i n T^,  i n the f i l m r e s i s t a n c e i n the c o o l i n g water c h a n n e l .  ± 0.2°c,  .42x10" m. a n d an e s t i m a t e o f  From t h e D i t t u s - B o e l t e r  the  temperature  Equation  [41],  ± 4.1°C,  and an  this  d r o p was c o n s i d e r e d t o be 2 . 5 ° C . A b s o l u t e e r r o r i s a sum o f estimate of  that  obtained from the approximation f o r T p r ,  t h e e r r o r due t o t h e i t e r a t i o n i t s e l f , ± 0.1°C.  ro  127 b.  Hexane The b a s i c equations are those l i s t e d i n the previous s u b s e c t i o n ,  with the f o l l o w i n g changes:  th v  g  =  0,0002 _ T  > p - p > i8p  d  g p g  p = 931.5 - 0.928 T  p = 0.00117 - ( 2 . 9 x l 0 ~ ) 6  p K  T  p K  The values are the f i r s t datum p o i n t i n Table 26, dp = 0.790 microns  P  T j = 62.7°C  x  T  B  Q S  = 1.75 = 65s  = 7.8°C  The sample and e r r o r c a l c u l a t i o n s are i n Table 28 with the s t i p u l a t e d absolute errors i t a l i c i z e d .  T a b l e 28 Sample and E r r o r C a l c u l a t i o n s — H e x a n e Variable  Value  Absolute Error  Relative  Error  1.75  ± 0.1  ± 0.0571  0.233  ± 0.0163  ± 0.0701  0.767  ± 0.0163  ±  62.7°C  ± 0.5°C  ± 0.0080  7.8°C  ± 2.7°C  ±  0.3461  A  0.140Jm"1s"loC"1  ±  O.OOUm^s"1^"1  ±  0.0072  B  -4.4xlO'4Jm_1s"loC"2  ±  5.0xlO'6Jm-1s"1oC'2  ±  0.0114  54.9°C  ±  3.2°C  ±  0.0583  7.686Jm"1s'1  ± 0.5034Jm"V  ±  0.0655  3931.3°C2  ±  62.9°C2  ±  0.0160  60.8°C2  ±  42.09°C2  ± 0.6922  3870.5°C2  ±  104.99°C2  ± 0.0271  X  F T  T  T  B  V  B  T  A(T -T ) T  B  1 1 T -T 'T 'B 2  2  fU -T ) 2  qh  2  -0.8515Jnf  V  1  1  0.0213  ± 0.0328Jm"1s'1  ±  0.0385  ± 0.5362Jm"1s'1  ±  0.0784  (Table Variable  28 -  continued)  Value  Absolute  >)  5.242Jm'1s"1  ±  AT,  8.778Jm'1s~1  ± 0.1334Jm~  BT2  TT K(qh)-CATT4T2] PC  T  -0.8649Jm" s'  Error  0.5227Jm"1s"1 V  1  Relative  Error  ±  0.0997  ±  0.0152  ±  0.0237Jm"1s*1  ± 0.0274  -2.671Jm"1s"1  ±  0.6798Jm"1s"1  ±  19.7°C  ±  1  1  -0.00867Jnf  5.3°C  0.2545  ± 0.2690  ±  0.0024Jm"1s"loK"1  ±  0.2804  0.131Jm"1s"loK"1  ±  0.0034Jm"1s"loK"1  ±  0.0260  292.9°K  ±  5.3°K  ± 0.0181  931.5  931.5kgm"3  ±  5.0kgm"3  ±  -0.928  -0.928kgm"3oK'1  ±  0.01kgm"3oK_1  ± 0.0170  -0.928TpK  -271.8kgm"3  ±  9.540kgm"3  ±  0.0351  p  659.7kgm"3  ±  14.54kgm"3  ±  0.0220  0.00117  O.OOlUkgs'V  ±  10"5kgs~1m'1  ± 0.0085  -2.9xl0"6  -2.9xlO"6kgs"1m"loK'1  ±  0.15xlO'6kgs"1m'loK"1  ±  0.0052  -2.9xlO"6TpK  S^xloAgs'V  ±  l.gSxlO'^gs^m""1  ±  0.0233  V  0.000321kgs"1m"1  ± 2.98xl0"5kgs"1m'1  ±  0.0928  B T  f  k  T  PC  PK  V H "  1  1  1  0.0054  ( T a b l e 28 Variable dT "dx  Value  Absolute  p  ±  Error  0.1127  *  0.790microns  ± 0.001 microns  ± 0.0013  0.624(microns)2  ±  ±  0.0016(microns)2  *  9.8ms"2  g  Relative  *  1048kgm"3  d  Error  ± 1955°Knf1  17346°Km"1  P  P  continued)  0.0026  *  388.3kgm"3  ± 14.54kgm"3  ±  0.0374  0.411xl0~6ms-1  ±  ±  0.1328  T  65.0s  ± 1.0s  ± 0.0154  .0002/x  S.OSxloAs"  ±  0.306xl0"6ms_1  ±  0.0994  2.67xl0~6ms~1  ± o.seixio'^s" 1  ±  0.1352  0.0927  ± 0.0353  ±  0.3808  Pp-p v  V  g  1  th  a No e r r o r e s t i m a t e  available.  0.0546xl0'6ms'1  131 Observations  c.  These c a l c u l a t i o n s gave c o n s e r v a t i v e imental  errors  of  the a ' s .  s u p p r e s s e d so t h a t c a n be s a f e l y  2.  The  film  Upper V e l o c i t y  the  error  r e s i s t a n c e i n the  cooling water  Limit  of  Stokes  Stokes  Law.  This  Reynolds  Number f o r  channel  i n t h i s work are based  s e c t i o n shows  a particle d  Re =  upper  maximum  quickly  the  that a l l  applicable  on  measrange  Law. The  The  is  exper-  Law  c a l c u l a t i o n s performed  p a r t i c l e s obeying  this  i n Tg  the  Note t h a t a l a r g e  u r e d and c a l c u l a t e d v e l o c i t i e s a r e w e l l w i t h i n of  for  neglected.  Many o f the  the  estimates  l i m i t of  velocity,  Stokes v  .  Law  is defined  as:  p v J - ! ! *  i s r e a c h e d when Re = 0.1.  is defined  Thus,  the  as:  m  m  pa  p The in  the  f o l l o w i n g3  previous  values  section:  of  vm were found  m  for  the  conditions  used  132 Table The Upper  Velocity  29  Limit  of  Stokes  Law  P  Fluid (kg  y  m"3)  (kg  v  nfV ) 1  m  (m  s"1)  (microns)  (°c)  Water  1.011  22.9  996.9  0.000936  0.093  Water  0.790  22.9  996.9  0.000936  0.119  Hexane  1.011  19.7  659.7  0.000321  0.048  Hexane  0.790  19.7  659.7  0.000321  0.062  Since is  3.  the v e l o c i t i e s  clear that  in this  Stoke's  The T e m p e r a t u r e  Fourier's  Law  work were o f  of  of  10"^ m s ~ \  it  applied,  Drop A c r o s s Each  Law  the order  Disk  one-dimensional  heat conduction  states:  or  =^ "  4T  where  kg i s  Table  1);  the  AX,  the  and 5 ) ;  and AT,  of  J nf1  31.0  thermal  s  thickness of  the - 1  conductivity of  .  brass  the d i s k f a c e ' s  temperature Therefore,  drop.  Table  from the above 1  heat f l u x  a n d AT  equal  2.95 J  (119  s  (1/16  27  J nf1  s"1  °C  - 1  --  inch--Figures  g i v e s a qh f o r  equations,  the  water  axial  -5 and 4x10  C,  respectively.  3  The  133 temperature  4.  drop i s n e g l i g i b l e .  The T e m p e r a t u r e Passing through  The  R i s e of C o o l i n g Water the Bottom Disk  temperature  ated from a simple heat  rise,  AT,  of  w h e r e m,, i s t h e mass f l u x w i s the s p e c i f i c  mental  local the  the water  and 2 . 9 5 J  temperature.  through  4200 J  kg"1  the c o o l i n g channel °C_1  the c o o l i n g water  Dependent  (24).  The  and a x i a l  Therefore,  = 0 . 0 1 4 ° C , which i s n e g l i g i b l y  the  Tpr,  conductivity of  heat  flux  above  small.  k-  If  and t e m p e r a t u r e the  liquid  t h e s u b s c r i p t LIN  gradient,  T  T  denotes  "  and dT " dx, LIN  (T.  (TT  dT/dx,  the  when  i s a s s u m e d t o be i n d e p e n d e n t the case of  then:  LIN  and  experi-  p r e s e n t s e c t i o n e s t i m a t e s the changes which a r i s e i n  f l u i d temperature,  thermal  c a n be e s t i m -  AT  respectively.  s~\  The E f f e c t o f T e m p e r a t u r e upon T p c and dT/dx  The  Cw  heat of water,  e q u a t i o n p r e d i c t s AT  5.  of  volumetric flow rate of  are 3 1 m i n - 1  the c o o l i n g water  balance:  q =  Cw  in  "  Vh  constant  k^,  of  The  differences  measures  of  between  the e f f e c t  these parameters of  the  and t h e  temperature  actual  dependent  values  thermal  conduc-  tivity.  a.  Water  From s e c t i o n  1:  TT  = 57.4°C  TB  = 7.2°C  = 0.700 -  j£ = 1 7 0 7 3 ° K  Hence:  T1  41  and t h e  percentage  These v a l u e s  b.  are  small but  1:  m"  1  are:  (Tpc/TLIN  ((dT/dx)  From s e c t i o n  3 ° CU  LIN  errors  Hexane  =22  = 16733°K  100  100  LIN  /  not  -  1)  =  (dT/dxLIN)  -  negligible,  2.7%  1)  =  2.03%  are  nf1  135  and  TT  = 62.7°C  TB  = 7.8°C  = 0.767 -  •S-  percentage  pectively. with  = 17346°K  m"1  so  T  The  Q  differences  The  temperature  minus  L I N  kf  20.6°C  = 18300°K LIN  are  signs are  whereas  -  -4.4  m"  1  p e r c e n t and - 5 . 2 1  present  for water  because rises.  kf  for  per c e n t , hexane  res-  falls  136  Appendix  Thermophoretic  Force  IV  and V e l o c i t y  the Large P a r t i c l e  This etic  force  clarify  section outlines  and v e l o c i t y  the  r o l e of  discussion will Fourier gradient  heat at  the  creeping flow  stitutes  creep i n the two p a r t s : to  the s o l u t i o n of  to y i e l d the  equation  a boundary  condition for  (at  has a c o n s t a n t  polar co-ordinate its surface,  creep v e l o c i t y ,  r -»- » )  v^  are oriented  v *s  s o l u t i o n of  tangential  the Navier-Stokes  thermal  gradient,  the  4  u P  TK  the  temperature thermal  Equation  dT/dx, the  $L ds  The con-  that  is  mov-.  The  imposed on i t .  centre of  negative  for  Equation.  x-direction.  gradient,  thermal  =-  The  creep v e l o c i t y  particle in a fluid  i n the  to  f o r c e and v e l o c i t y .  the Navier-Stokes  temperature  Equation for  the  the  i t s origin at  a tangential  Maxwell's  thermophor-  p a r t i c l e regime.  i n the negative  temperature  s y s t e m has  v^,  the  use i n the Maxwell  thermophoretic  i s r e q u i r e d because the  ing at a v e l o c i t y  At  first,  obtain  F i g u r e 25 shows a s t a t i o n a r y  fluid  large  p a r t i c l e s u r f a c e and i t s second,  solution for  a s p h e r i c a l p a r t i c l e i n an a t t e m p t  thermal  c o n s i s t of  Regime  the E p s t e i n  conduction equation  creep equation;  Maxwell  of  for  dT/ds,  the and  A  particle. thermal  e direction.  creep v e l o c i t y  is:  nv-ii L i V  1 J  Hot Fluid  Cold Fluid  Figure Co-ordinate  System and V a r i a b l e s  25 for  the Large P a r t i c l e  Regime  138 Epstein  (12)  the F o u r i e r  obtained the tangential  to the f o l l o w i n g boundary  and r a d i a l fluid  heat f l u x  dT/dx.  T = 0  [IV-2]  conditions.  First,  the  temperature  a t a p o i n t on t h e p a r t i c l e s u r f a c e and i n t h e  adjacent to that  ient at r = » is  g r a d i e n t by s o l v i n g  Equation:  A2  subject  temperature  point are equal.  Second,  the temperature  i s u n a f f e c t e d by t h e p r e s e n c e o f t h e p a r t i c l e ,  Epstein's  i . e . , i t  e q u a t i o n f o r dT/ds i s : 3k  3? = Hence E q u a t i o n  grad-  [IV-1]  H7^V£ "< > si  [IV  6  "  3]  c a n be r e s t a t e d a s :  v^  = OJ s i n (8)  [IV-4]  where  = u  Since  f l u i d would f o r c e  causes  8kf + 4kp  _p TK  P  the p a r t i c l e i s s t a t i o n a r y ,  perature gradient.  fers,  9k f  therefore, particle  If  dT dx  [IV-5]  t h e f l u i d c r e e p s up t h e t e m -  t h e p a r t i c l e were f r e e  t o move,  t h e p a r t i c l e down t h e g r a d i e n t . from hydrodynamic  slip  movement.  The N a v i e r - S t o k e s  Equation i s :  i n that  however,  Thermal  the f l u i d  creep itself  the dif-  139  M  E4  where  the o p e r a t o r ,  E,  [IV-6]  = 0  i s defined as:  a n d t h e s t r e a m f u n c t i o n , \p, i s r e l a t e d t o t h e f l u i d v e l o c i t y vp  and v Q  components,  by:  v  =  _  UL  1 r  s i n (e)  a  [ I V  -8]  r  and  The method o f s o l v i n g E q u a t i o n and B r e n n e r  four  [IV-6]  boundary  i s a fourth  o u t l i n e d by Happel  c o n d i t i o n s m u s t be s p e c i f i e d . v  r + c°  v  = - v  r  vr  = 0  r  that  order p a r t i a l d i f f e r e n t i a l  r •> °°  = dp/2  T h i s meets  follows  (20).  Equation and t h u s  [IV-6]  r  = v  f  t h e r e q u i r e m e n t o f no f l o w  = dp/2  cos (e) f  s i n (e)  equation  These a r e : [IV-10] [IV-11] [IV-12]  through  the p a r t i c l e surface.  [IV-13]  140 This  means no h y d r o d y n a m i c  velocity  is present.  a p p e a r s w i t h i n o n e mean f r e e  In t h e case o f l i q u i d s meter.  slip  this  Since L i s very  However,  path,  L,  i s approximately  small,  the boundary  the thermal  creep  of the p a r t i c l e s u r f a c e .  twice the molecular  dia-  c o n d i t i o n a t r = dp/2 + L  dp/2 i s :  vQ  = - v  Using formed  Equations  [IV-8]  = - oo s i n ( e )  [IV-14]  —5  O  and [ I V - 9 ] ,  the boundary  c o n d i t i o n s c a n be t r a n s -  to:  r  *  9  |9~ e6  00  =  "  v  f  p 2  s  i  n  (0)  c o s  r -> «  9± = = - vf r sin2.(e) 9  r = dp/2  r f- = 0 |9 -8  9  (Q)  [IV-15] [IV-16]  1  ClV-17]  a r  V  =  rf  2  A general  *  "  =  u  ~ 2  S  i  n  2  s o l u t i o n of Equation  = sin  2  (e)  (-^  r  4  -  £  ( 0 )  [ I V  f-f-  and  Differentiating  Equation  = s i n ( e ) cos (e)  r  r  + CQ  [IV-19]  4  1 8 ]  [IV-6] i s :  r  2  +  [IV-19]  where A „ , B . C , and D „ a r e c o n s t a n t s d e t e r m i n e d from t h e o o o o conditions.  "  b o u n d a r yJ  gives:  - BQ r + 2 CQ r  2  + ^ )  [IV-20]  141  |JL  =  S  Substituting  in  2  2A.r3 B_ D, ( _ § — - ^ + 2 CQr - -f) r  (e)  Equation  o " V  [IV-15]  A r4 - - f - - BQr  i n Equation  [IV-20]  results  in:  2 D  9  2 CQr2  +  [IV-21]  +  [IV-22]  When r -> ° ° , t h e r e c a n be no p o w e r o f r > 2 on t h e r i g h t - h a n d s i d e Equation  [IV-22].  Therefore,  A  In a d d i t i o n , a l l terms but 2 C Q r 1/r  as r  °°.  2  = 0  o  Substituting  can be n e g l e c t e d because r  2  »  r or  Equation  = - Y~  [IV-17]  dn 0  [IV-18]  =  "  B  o 2^ "  i n Equation  Equation  [IV-25]  [IV-24]  i n Equation  dn V  f  [IV-21]  Bn u = - rr- - v . f *P  Dividing  [IV-23]  Therefore,  CQ  and E q u a t i o n  of  4  9  [IV-20]  D„  +  results  gives:  [ I V  "  2 5 ]  in:  8 Dn 2y (dn)3  2 by d p / 2 a n d a d d i n g t h e r e s u l t  [IV-26]  to  Equation  142 [IV-26]  gives:  BQ  The  = ^  vf)  [IV-27]  f o r c e e x e r t e d by t h e f l u i d on t h e p a r t i c l e ,  Ff  Combining Equation  [IV-27]  Ff  = - 4 TT  = 3 TT p d  p  v  BQ  v  and E q u a t i o n  is:  [IV-28]  [IV-28]  - 3 rr y d  f  F^,  p  yields:  (| co)  [IV-29]  or  Ff  = Fn  When t h e p a r t i c l e i s f r e e e q u i l i b r i u m between F^ = 0 .  t o move ( r a t h e r  [IV-30]  than held  stationary)  t h e d r a g and t h e t h e r m o p h o r e t i c f o r c e s e x i s t s and  Hence:  v  If  - Fth  the f l u i d i s stagnant,  f  = j  «o  [IV-31]  the thermophoretic v e l o c i t y of the p a r t i c l e  is:  v  or,  u t i l i z i n g Equation  t h  [IV-5]:  = - v  f  = - f co  [IV-32]  143 3 v  This  is Epstein's  phoretic force is  th  =  kf  " 2 2 kf  equation for  yi  + k  dT  r- w T  ~p~\ d7  thermophoretic v e l o c i t y .  -  The  ^o-i  L17-3-3-!  thermo-  then F  t h  - 3 „ „ d  p  v „ t  [IV-34a]  

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