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Determination of gas effective diffusivities in porous solids, dispersion coefficients in packed beds.. Davis, Brian Richard 1965

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i  DETERMINATION OF GAS EFFECTIVE DIFFUdlVITIES IN POROUS SOLIDS, DISPERSION COEFFICIENTS IN PACKED BEuS AND' MOLECULAR DIFFUSIVTTY OF BINARY SYSTEMS  Brian Richard Davis D.L.C., Loughborough College, 1956  M.A.Sc., University of British Columbia, 1963 A T h e s i s Submitted In P a r t i a l F u l f i l m e n t of  The Requirements For the Degree Of Doctor of Philosophy in the Department of Chemical Engineering  We accept this Thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December, 19^5  In the  requirements  British  for  Columbia,  available mission  for  for  purposes his  presenting  I  an  reference  cation without  of  this  thesis  my w r i t t e n  Department  of  and  by  It for  study.  the  is  partial  degree  the  of  in  I  of  understood  financial  Chemical  ^/j*****^  the  Library  this  Head  at  thesis my  Engineering Columbia  ftf(>(o  of  University  of  make  agree for  freely per-  scholarly  copying  shall  it  that  Department  that  gain  fulfilment  shall  further  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date  that  copying  be g r a n t e d  representatives,,  thesis  advanced  agree  extensive  may  this  not  or or  be  by publi-  allowed  The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES  PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  of  BRIAN RICHARD. DAVIS Diploma, Loughborough  College of Advanced Technology,  . UK, 1956 M.A.Sc, The University of B r i t i s h Columbia, 1962  December 21, 1965 AT  10:00 A.M.  IN ROOM .207, CHEMICAL ENGINEERING  COMMITTEE IN CHARGE Chairman: I. McT, Cowan R. M. R. Branion J . Lielmezs S. D. Cavers K. L. Pinder N. Epstein D. A. Ratkowsky J.R.Sams External Examiner: G. L, Osberg National Research Council Ottawa Research Supervisor:.D. S. Scott  ABSTRACT SECTION I AN EXPERIMENTAL METHOD FOR THE MEASUREMENT OF EFFECTIVE GAS DIFFUSIVITIES IN POROUS PELLETS, AND THE LONGITUDINAL DISPERSION COEFFICIENT IN PACKED BEDS Present methods of measurement of e f f e c t i v e d i f f u s i v i t i e s are not generally adaptable to the p e l l e t s i n a packed bed, for example a c a t a l y t i c reactor, An unsteady state pulse method has been developed employing simple gas chromatographic rate theory, The method i s generally applicable to p e l l e t sizes down to about 2mm, With homogeneous p e l l e t s reasonable agreement was obtained on comparison of e f f e c t i v e d i f f u s i v i t i e s measured by a steady state method. For anisotropic solids the unsteady state d i f f u s i v i t y can be quite different from the steady state value due to differences i n d i f f u s i o n path, ;  Pulse dispersions measured i n beds of non porous pellets have revealed a laminar flow regime where the . dispersion c o e f f i c i e n t i s dependent on the square of the v e l o c i t y . This regime was reported for flow, i n straight pipes but has not previously been demonstrated i n packed beds, SECTION II DEVELOPMENT OF AN UNSTEADY STATE FLOW METHOD . FOR MEASURING BINARY GAS DIFFUSION COEFFICIENTS Effusion measurements of one gas from a packed bed of known geometry (porosity and tortuosity) into a second flowing gas have been evaluated as a v e r s a t i l e technique for the determination of binary gas d i f f u s i o n coefficients. The molecular d i f f u s i v i t i e s measured ( t 10%) approached the scatter encountered by other methods (t 5%) and satisfactory results (+ 3%) are envisaged by optimising parameters i n the method,  GRADUATE STUDIES F i e l d of Study:  D i f f u s i o n and Kinetics  Chemical Engineering Reactor Design D. S, Scott. Industrial Kinetics and Catalysis ' D. S, Scott Mass Heat and Momentum Transfer S, D, Cavers Chemical Kinetics E.. A. Ogryzlo G. L James D , Gas Adsorption and Solvent. D, S, Scott Extraction S, D Cavers 3  a  s  Other Studies Chemical Engineering Thermodynamics Analogue Computers Mathematical Operations i n Chemical Engineering Fortran Programming D i f f e r e n t i a l Equations Industrial Relations Industrial Organisation  P. L, Silveston E. V, Bohn .N. Epstein H, Dempster S, A. Jennings N. A. Hall J . P. Van Gigch  PUBLICATIONS  DAVIS, B. R. and D. S. SCOTT, 1964. Rate of isomerisation of cyclopropane in a flow reactor.. Ind. & Eng, Chem:. Fundamentals, _3_i. 20-23  II ABSTRACT SECTION I AN EXPERIMENTAL METHOD FOR THE MEASUREMENT OF EFFECTIVE GAS DIFFUSIVITIES IN POROUS PELLETS, AND THE LOGITUDINAL DISPERSION COEFFICIENT IN PACKED BIDS Present methods f o r measuring effective: d i f f u s i v i t i e s i n small porous p a r t i c l e s arc not a p p l i c a b l e to assemblages of such p e l l e t s , f o r example, as i n c a t a l y t i c r e a c t o r s , and require s p e c i a l techniques or apparatus.  A pulse •  technique has been developed v h i c h can s u c c e s s f u l l y y i e l d a reasonable value of the d i f f u s i v i t y by a n a l y s i s of pulse d i s p e r s i o n i n terras of simple chromatographic r a t e theory. nearly i d e a l .  A non-adsorbing  pulse gas i s necessary, and hydrogen i s  Because of the high molecular d i f f u s i v i t y of hydrogen the  smallest s i z e of p a r t i c l e v h i c h can be tested v i t h t h i s gas i s about 2 mm aiEuaeter.  The unsteady s'tate pulse e f f e c t i v e d i f f u s i v i t y measurement v h i c h  should be more r e a l i s t i c f o r c a t a l y t i c studies was compared v i t h a conventional steady s t a t e method and good agreement obtained^in a s p h e r i c a l i s o t r o p i c p e l l e t ( ^ ) ; hovever, as may be expected agreement was poor w i t h a n i s o t r o p i c p e l l e t s . A regime vas found i n a study of beds of non porous p e l l e t s vhere the d i s p e r s i o n c o e f f i c i e n t i s p r o p o r t i o n a l to the' square of the v e l o c i t y .  This  regime i s reported f o r pipes but has not been r e a l i z e d as a separate regime i n packed beds.  This d i s p e r s i o n data i s compared v i t h the l i m i t e d data of other  workers although the ranges of experimental conditions do not overlap. EJECTION I I DEVELOPMENT OF AW UNSTEADY STATE FLOW METHOD FOR MEASURING BINARY C-AS DIFFUSION COEFFICIENTS E f f u s i o n of one gas from a packed bed of knovn geometry i n t o a second f l o v i n g gas has been evaluated as a v e r s a t i l e technique f o r determination of binary gas d i f f u s i o n c o e f f i c i e n t s having fev l i m i t a t i o n s of pressure, temp-v.. eraturc and a n a l y s i s method. Optimization of experimental parameters should y i e l d s a t i s f a c t o r y r e s u l t s (t 3/6).  I l l  SECTION 1 EXPERIMENTAL MEASUREMENT OF EFFECTIVE GAS DIFFUSIVITIES III POROUd PELLETS AND DISPERSION COEFFICIENTd IN PACKED BEDS  Pape  INTRODUCTION A.  THIELE MODULUS  1  B.  DIFFUSION MECHANISMS  3  Molecular D i f f u s i o n  k  Knudsen D i f f u s i o n  5  Intermediate D i f f u s i o n  6  v.  Effective Diffusion Coefficient C.  7  EXPERIMENTAL ESTIMATION OF EFFECTIVE DIFFUSIVITIES Prediction  -  Steady State Methods  D. II  8 8 9  Chemical Reaction Method  10  Unsteady State Methods  10  Frequency Response and Pulse Methods  11  Comparison of Methods  12  STATEMENT OF OBJECTIVES  13  DERIVATION OF VAN DEEM'TER'3 EQUATION  l6  THEORY A.  Height Equivalent to a 'Theoretical Plate (HETP) Measurement of HETP  l6 • 19  Input Pulse D i s t r i b u t i o n  20  Rate Theory  21  Mass Transfer C o e f f i c i e n t and E f f e c t i v e D i f f u s i v i t y 25 External Mass Transfer C o e f f i c i e n t  26  Internal Mass Transfer C o e f f i c i e n t  26  iv Page 29  Van Deciliter" s E q u a t i o n  T y p i c a l V a l u e s o f t h e E f f e c t i v e D i f f u s i v i t y Terra(C) 50 L e a s t Square E r r o r F i t t i n g o f Data t o Van Deemter Equation B.  III  IV  LONGITUDINAL DISPERSION COEFFICIENTS  3U  Velocity P r o f i l e Contribution  36 39  APPARATUS A.  DEVELOPMENT  B.  DESCRIPTION OF APPARATUS  C.  DETECTORS  39  '  39 kk  Hydrogen Flame I o n i z a t i o n D e t e c t o r  kk  Thermal C o n d u c t i v i t y D e t e c t o r  kj  EXPERIMENTAL 'PROCEDURE  k8  A.  OUTLINE OF EXPERIMENTAL INVESTIGATION  ko  B.  NON POROUS PELLETS I N PULSE APPARATUS  kd  C.  POROUS PELLETS I N PULSE APPARATUS  D.  INDEPENDENT EFFECTIVE DIFFUSIVITY MEASUREMENT  E.  PREPARATION OF TEST COLUMNS  F.  OPERATION OF PULSE APPARATUS  " 51  ' "  $k 55 55 56  RESULTS  'B.  52 ',53  Thermal C o n d u c t i v i t y D e t e c t o r  A.  ?  *  Hydrogen Flame D e t e c t o r  V  33  NON POROUS PELLETS  •• 5°"  Treatment o f Data f o r Non Porous P e l l e t s  56  R e s u l t s f o r Beds o f Non Porous P e l l e t s  60  LONGITUDINAL DISPERSION COEFFICIENT  72  Page  C.  POROUS PELLETS  J9  Porous Pellet Samples  79  Steady State Apparatus Results  o2  Treatment of Data for Pulse Apparatus  62  Porous Pellet Results  83  Comparison of Steady State and Pulse Apparatus Results  88  t  IV  DISCUSSION A.  B.  NON POROUS PELLETS  89  HETP vs. Velocity Curves  89  Axial Dispersion Coefficient  90  Correlation of the Eddy Diffusivity  93  POROUS PELLETS  9^  Inconsistency of the Steady State and Pulse Results 95 for Activated Alumina Porosity  96  Non Spherical Pellets Methane Pulse Errors  .  "'  96 97 97  VII  CONCLUSIONS  99  VIII  RECOMMENDATIONS  99  vi SECTION I I DEVELOPMENT OF AN UNSTEADY STATE FLOW METHOD FOR MEASURING BINARY GAS DIFFUSION COEFFICIENTS  I  INTRODUCTION  100  II  THEORY  102  A.  SIMPLIFIED SOLUTION OF DIFFUSION EQUATION  102  B.  MORE RIGOROUS SOLUTION  105  C.  COMPUTATION OF SLOPE OF DECAY CURVE WITH A RESIDUAL CONCENTRATION  110  III  APPARATUS  112  IV  PROCEDURE  11^  V  A.  SELECTION OF DISPLACED AND DISPLACING GAS  B.  OPERATION OF EQUIPMENT  RESULTS  llh ll6  -  117  A.  TREATMENT OF DATA  117  B.  PARALLEL TUBE PACKING  119  C.  POROUS SOLID PACKING  12k  D.  SPHERICAL PACKING  126  VI  DISCUSSION  132  VII  CONCLUSIONS  1 5 6  VIII  RECOMMENDATIONS  1 3 6  NOMENCLATURE  137  LITERATURE CITED  ikO  vii  APPENDIX I DETERMINATION OF DIFFUSIVITY I N POROUS SPHERICAL PELLETS BY A STEADY STATE METHOD INTRODUCTION THEORY Knudsen D i f f u s i o n Bulk o f Molecular D i f f u s i o n APPARATUS PROCEDURE t  Operation RESULTS C a l i b r a t i o n o f Thermal C o n d u c t i v i t y  Detectors  A c t i v a t e d A l u m i n a P e l l e t l / U " Diameter N o r t o n C a t a l y s t S u p p o r t , l / 2 " Diameter "Alundum" CONCLUSIONS APPENDIX I I RESULTS FROM PULSE APPARATUS NON POROUS Program Results POROUS Program Results FLOW METER CALIBRATION  viii Page APPENDIX I I I RATE OF DIFFUSION FROM A SPHERICAL PELLET  idk  MANUFACTURER'S DATA ON POROUS PELLET  l8U  Norton C a t a l y s t Supports l / 2 " d i a m e t e r SA 20J m i x t u r e  185  A c t i v a t e d Alumina P e l l e t s A l c o a HI51 diameter  185  and l / 8 "  POROSITY OF PACKED BEDS  l88  ESTIMATION OF THE MOLECULAR DIFFUSIVITY OF THE METHANE AIR SYSTEM  191  APPENDIX I V ADSORPTION OF GASES BY ACTIVATED ALUMINA PELLETS THEORY AND APPARATUS RESULTS  192 ,192  •  CONCLUSION  191+ L96  APPENDIX V RESULTS FOR SECTION I I , MOLECULAR DIFFUSIVITY APPARATUS  198  PROGRAM  198  RESULTS  END  P a r a l l e l 1 Tube Bed  201  Porous S o l i d Bed  2lh  S p h e r i c a l P a c k i n g Bed  226  23k  ix TABLES SECTION 1 1.1.  The e f f e c t o f p e l l e t d i a m e t e r  and e f f e c t i v e d i f f u s i v i t y on t h e  e f f e c t i v e d i f f u s i v i t y term (C) i n e q u a t i o n l.II  I.5O.  Summary o f t h e p e l l e t and tube t o p e l l e t d i a m e t e r by t h e e x p e r i m e n t a l  31  r a t i o s covered  .50  runs.  l.III  D i s p e r s i o n r e s u l t s v i t h beds o f non porous p e l l e t s .  .65  l.IV  Effect o f pulse size  l.V  F u r t h e r d i s p e r s i o n r e s u l t s w i t h beds o f non porous p e l l e t s .  68  l.VI  Values o f t h e eddy d i f f u s i v i t y term c o n s t a n t ,  70  l.VII  P r o p e r t i e s o f porous p e l l e t samples.  l.VIII  D i s p e r s i o n r e s u l t s f o r porous p e l l e t s .  l.IX  Comparison o f e x p e r i m e n t a l e f f e c t i v e d i f f u s i o n c o e f f i c i e n t s .  87  1. X  Potential  98  (peak h e i g h t ) on HETP.  67  .  80 Qk  Errors.  SECTION I I 2.1  Properties of diffusion c e l l s .  115  2.II  R e s u l t s f o r p a r a l l e l tube p a c k i n g .  2.Ill  R e s u l t s f o r porous s o l i d p a c k i n g .  125  2.IV  Results f o r s p h e r i c a l packing.  127  2. V  Comparison o f r e s u l t s w i t h p u b l i s h e d d a t a  131  '  120  APPENDICES A I I . 1 Flow meter c a l i b r a t i o n A IV.1  R e s u l t s f o r a d s o r p t i o n apparatus  181 '  197  X  FIGURES SECTION I  1.1  Pore model f o r d e r i v a t i o n o f e f f e c t i v e n e s s  1.2  Model f o r d e r i v a t i o n o f p l a t e t h e o r y .  17  1.3  Gaussian d i s t r i b u t i o n p r o p e r t i e s .  19  l.h  M a t h e m a t i c a l model f o r t h e column.  22  1.5  T y p i c a l p l o t o f HETP v s . v e l o c i t y ( e q u a t i o n 1.50).  32  1.6  Apparatus f o r e x p l o r a t o r y t e s t s .  ko  1.7  Basic experimental apparatus.  kl  1.8  Pulse i n j e c t o r s .  1+3  1.9  Hydrogen f l a m e d e t e c t o r .  1+5  1.10 HETP v s . v e l o c i t y f o r r u n 52.  61  l.li  factor.  62,  HETP v s . v e l o c i t y f o r runs 51, 69 and 70.  1.12 Eddy d i f f u s i o n t e r m , A, ( e q u a t i o n 1.50)  v s  »  p e l l e t diameter.  1.13 D i s p e r s i o n c o e f f i c i e n t , Dj, v s . i n t e r s t i t i a l v e l o c i t y , u . l.lU  2  I n v e r s e eddy P e c l e t number v s . s u p e r f i c i a l Reynolds number.  71 73  lh  1.15 I n v e r s e eddy P e c l e t number v s . m o l e c u l a r P e c l e t number.  75  1.16 I n v e r s e Eddy Schmidt number v s . h y d r a u l i c d i a m e t e r Reynolds number.  76  1.17 E m p i r i c a l d i s p e r s i o n c o e f f i c i e n t c o r r e l a t i o n  77  SECTION I I  103  2.1  Model o f t h e bed f o r proposed d i f f u s i o n  2.2  D i f f u s i o n apparatus.  2.3  R e s u l t s w i t h p a r a l l e l tube bed.  2.1+  R e s u l t s w i t h p a r a l l e l tube b e d .  Ethane-nitrogen.  122  2.5  R e s u l t s w i t h p a r a l l e l tube bed.  Butane-nitrogen.  123  2.6  Results w i t h s p h e r i c a l packing bed.  2.7  R e s u l t s w i t h s p h e r i c a l p a c k i n g bed.  Ethane-nitrogen.  129  2.8  R e s u l t s w i t h s p h e r i c a l p a c k i n g bed.  Butane-nitrogen.  130  experiment.  113 Hydrogen-nitrogen.  Hydrogen-nitrogen.  121  128  xi APPENDIX I A I.l  Sample mounting i n s t e a d y s t a t e a p p a r a t u s .  ihh  A 1.2  Steady s t a t e a p p a r a t u s .  1^-6  A I I . 1 Flow meter c a l i b r a t i o n .  '  183  A IV.1  A d s o r p t i o n measurement a p p a r a t u s  193  A IV.2  Gas volume v s . i n v e r s e p r e s s u r e f o r a d s o r p t i o n measurement  196  •' ( x i i )  ,  ACKNOWLEDGEMENT  Thanks a r e extended t o D r . F o r s y t h e and t h e f a c u l t y and s t a f f o f t h e U n i v e r s i t y - o f B r i t i s h Columbia, chemical department.  engineering  P a r t i c u l a r l y the a u t h o r w i s h e s t o t h a n k Dr. E p s t e i n  f o r h i s h e l p i n Dr. S c o t t ' s absence and Mr. E. R. R u d i s c h e r and Mr.  J . B a r a n o v s k i f o r t h e i r a s s i s t a n c e and c o - o p e r a t i o n . j  The a u t h o r i s i n d e b t e d t o Dr. S c o t t f o r h i s d i r e c t i o n  and s u p p o r t and i s g r a t e f u l f o r f i n a n c i a l a s s i s t a n c e extended by the N a t i o n a l Research C o u n c i l .  5.  - 1 -  INTRODUCTION A.  THIELE MODULUS The r a t e o f r e a c t i o n i n a porous s o l i d - c a t a l y s t can be l i m i t e d  by t h e r a t e a t w h i c h r e a c t a n t s and p r o d u c t s can d i f f u s e i n and o u t o f t h e solid.  Thiele ( l ) q u a n t i t a t i v e l y described  treatment v h i c h i s a p p l i e d t o a simple a t i o n below.  t h i s e f f e c t w i t h a mathematical  case o f an i n f i n i t e s l a b i n t h e d e r i v -  I n F i g . 1.1 a s i n g l e pore o f r a d i u s r and l e n g t h L i s shown.  A f i r s t o r d e r gas phase r e a c t i o n w i t h r a t e = k C  A  moles/(sec)(cm ) i s 2  assumed t o be t a k i n g p l a c e i s o t h e r m a l l y on t h e pore w a l l s , and a C ^ Q moles/cm  concentration pore mouth. -D dCA dx~  3  constant  i s m a i n t a i n e d a t each f a c e o f t h e s l a b a t t h e  A m a t e r i a l b a l a n c e around t h e element dx y i e l d s ,  7T r  - (-D) f* dCA +  2  L  d  x  d CA dx* 2  Sx ITT r  2  - kC  A  2 TTv  fix  = 0  (l.l)  J  w h i c h may be s i m p l i f i e d t o , d  2  C  A  =  gk  dx^"  C  (1.2)  A  rD  The boundary c o n d i t i o n s , C  A  = C A Q a t x = 0, and d C / d x = 0 when x = L , A  may be a p p l i e d t o t h e s o l u t i o n o f (1.2) t o g i v e t h e c o n c e n t r a t i o n  i n the  pore, G  A = A0 c o s h [ C  where h = L  / 2k, <\ Dr  h( " L")] cosh h 1  "  (1.3)  commonly known as t h e T h i e l e modulus.  The r a t e o f r e a c t i o n i s g i v e n by t h e r a t e o f d i f f u s i o n o f A i n t o t h e pore mouth, w h i c h i n t u r n i s g i v e n b y , rate/pore  =  =  -D [ ~ \ dx  J /  7T r  (l.h)  2  x = 0  D C A O h t a n h (h)  TT r 2  Figure  1.1  Pore Model For Derivation Of Effectiveness  Factor  - 3 I f t h e whole pore c o n t a i n s  gas  a t the surface  concentration, • 2TTrL m o l e s /  C^Q, the r a t e o f r e a c t i o n w i l l "be a maximum, g i v e n by k . C  AO sec.  The  defined  r a t i o of t h e r a t e g i v e n by e q u a t i o n (l.h)  and  t h i s maximum r a t e i s  as the e f f e c t i v e n e s s f a c t o r , E. rate/pore  = E = rD  maximum r a t e The o n l y , and  h  2kL  tanh  h  =  tanh  h  (1.5)  h  2  e f f e c t i v e n e s s f a c t o r i s a f u n c t i o n o f the T h i e l e modulus  can be used t o c a l c u l a t e t h e r a t e o f r e a c t i o n when d i f f u s i o n a l  r e s i s t a n c e s are c o n t r o l l i n g . r a t e of reaction/pore  = k C ^ Q 2TTr L E  (1.6)  A d d i t i o n a l e q u a t i o n s can be d e r i v e d r e a c t i o n , (2) o t h e r assumed pore g e o m e t r i e s (3), stoichiometry  f o r other orders of  o r f o r cases where t h e  does not a l l o w equirnolar c o u n t e r d i f f u s i o n t o o c c u r I n p r a c t i c a l cases i t i s v e r y d i f f i c u l t  t h e p o r e geometry o f a porous s o l i d , and based upon u n i t mass o f c a t a l y s t .  to define  (l)(l+).  accurately  r a t e c o n s t a n t s a r e more commonly  I t i s convenient mathematically to t r e a t  t h e porous s o l i d as a homogeneous medium h a v i n g an e f f e c t i v e d i f f u s i v i t y , r a t h e r t h a n a t t e m p t i n g t o use w i t h t h e v o i d f r a c t i o n and B.  the t r u e i n t e r s t i t i a l d i f f u s i v i t y t o g e t h e r  s u i t a b l e assumptions about t h e pore geometry.  DIFFUSION MEGIIANISMS There a r e two  b a s i c gas  t r a n s p o r t processes which occur i n  porous s o l i d s , and w h i c h obey F i c k ' s laws o f d i f f u s i o n , namely, m o l e c u l a r or bulk d i f f u s i o n which occurs through i n t e r m o l e c u l a r d i f f u s i o n , w h i c h depends o n l y upon w a l l c o l l i s i o n s . known as process.  c o l l i s i o n s , and  Knudsen  I n a d d i t i o n , a phenomenon  " s u r f a c e d i f f u s i o n " can t a k e p l a c e , b u t t h i s i s not a w e l l u n d e r s t o o d S u r f a c e d i f f u s i o n i s b e l i e v e d t o r e s u l t from m u l t i l a y e r s o f  gas  m o l e c u l e s condensed t o a l i q u i d - l i k e s t a t e , w h i c h f l o w from the' a r e a s w i t h s e v e r a l l a y e r s to those of lower surface concentration.  This process r e s u l t s  i n d i f f u s i o n r a t e s much l a r g e r t h a n t h o s e p o s s i b l e by c o l l i s i o n mechanisms.  Gases above their c r i t i c a l temperature are less likely to display this phenomenon, because of the reduction i n surface adsorption under these conditions. Molecular Diffusion i n Fores This mode of diffusion predominates when the ratio of the pore radius to mean'free path is greater than about 10. Pick's law, or the one dimensional flux equation for steady-state molecular diffusion i n a two component mixture takes the form, %  = - B D  ^GT  +  ( N  A  +  N  B)  (1.7)  CA  vm  where the last term accounts for bulk flows xrhich may be caused by nonequimolar counter diffusion rates of the two gases with respect to stationary coordinates.  In order bo apply the equation to a porous structure the  flux i s taken per total unit area of solid and pore, rather than unit area of pore only, N  X A  = N 6 A  p  = -  [£3^,J]  dCA  1  dx  i  j  +  (N  X A  + Ng ) 1  C_A  (1.8)  Pm  where X i s the "tortuosity" which corrects for the fact that the pore length is greater than the geometric length of the structure.  The terms which are  grouped with the diffusivity form the definition of an "effective diffusivity" which w i l l be discussed later. Fick's second law which describes the unsteady state diffusion process is usually expressed as,  bA _  C  C  At  4x  A  (1.9)  2  In a porous solid, introducing the concept of an effective diffusivity, this equation should be modified as shown below:  -  5  -  I n u n i t a r e a o f a porous i n f i n i t e s l a b , a mass b a l a n c e o v e r the element Sx when no c h e m i c a l r e a c t i o n i s o c c u r r i n g and a l l o w i n g f o r a net b u l k f l o w g i v e s t h e r a t e o f change o f gas c o n t e n t i n terms o f the e f f e c t i v e d i f f u s i v i t y , D^, a s ,  - ^ A - Sx o + D  <f p  E  / h  %\  Sx  -  a(  u  C  A)  Sx  (1.10)  where u i s t h e s u p e r f i c i a l b u l k v e l o c i t y , ^ i m p l i c a t i o n o f 1.10  0  C  A  gives,  =  +  $t For e q u i m o l a l  °E_ €  C $ x  P  A 2  -  __1  e  p  d(u A ) C  d  (1.11)  x  c o u n t e r d i f f u s i o n , t h i s reduces t o t h e u s u a l form o f F i c k ' s  second- law, t h a t i s , e q u a t i o n ( l . 9 ) e x c e p t t h a t t h e m o l e c u l a r d i f f u s i o n c o e f f i c i e n t i s replaced by the e f f e c t i v e d i f f u s i o n c o e f f i c i e n t d i v i d e d by the  porosity. I f e q u i m o l a r c o u n t e r d i f f u s i o n o c c u r s t h e n e q u a t i o n (1.8) f o r  the.- s t e a d y s t a t e reduces t o N  X A  =  - D  d E  C  A  (1.12)  dx The b i n a r y m o l e c u l a r d i f f u s i o n c o e f f i c i e n t o f a gas i s proportional t o the absolute  t e m p e r a t u r e t o about t h e 1.7 power, and i n v e r s e l y  proportional t o the pressure. Knudsen D i f f u s i o n T h i s mechanism predominates when t h e mean f r e e p a t h o f t h e gas m o l e c u l e s i s g r e a t e r t h a n t h e pore r a d i u s , and because w a l l c o l l i s i o n s c o n t r i b u t e p r i m a r i l y t o the process i n these circumstances, the d i f f u s i o n c o e f f i c i e n t i s independent o f t h e p r e s e n c e o f o t h e r g a s e s .  Bulk f l o w i s  not d i s t i n g u i s h a b l e from d i f f u s i o n i n t h i s c a s e , and so F i c k ' s l a w i n t h e f o r m o f e q u a t i o n (1.12) a p p l i e s .  -6 -  The Knudsen d i f f u s i o n c o e f f i c i e n t i n c y l i n d r i c a l s t r a i g h t pores i s g i v e n by,  D  K  = 2/3 r v  (1.13)  v h e r e v i s t h e average v e l o c i t y o f t h e gas m o l e c u l e s , and r t h e p o r e r a d i u s . I n consequence, t h e v a l u e o f t h i s c o e f f i c i e n t i s independent o f p r e s s u r e , and p r o p o r t i o n a l t o t h e square r o o t o f t h e t e m p e r a t u r e . I n t e r m e d i a t e o r Mixed D i f f u s i o n  Coefficient  I n t h e i n t e r m e d i a t e range between m o l e c u l a r and Knudsen d i f f u s i o n t h e r e i s a r e g i o n where b o t h t h e above d i f f u s i o n mechanisms o c c u r . The r a t i o o f p o r e r a d i u s t o mean f r e e p a t h l i e s a p p r o x i m a t e l y between t h e following  l i m i t s i n t h e i n t e r m e d i a t e zone:  Knudsen  , Intermediate  0.1  <  r A,  <  Molecular  10  By assuming round c a p i l l a r i e s , r i g i d sphere k i n e t i c s and d i f f u s e m o l e c u l a r r e f l e c t i o n from t h e w a l l s , S c o t t and D u l l i e n the following  relationship  (5)  derived  f o r t h e f l u x i n a b i n a r y gas m i x t u r e i n t h e  intermediate region. N  A  =  -  _Z  RT  d  y  A dx  (1.1M 1 - JYA + 1 D  where j = 1  + N^/N-Q,  B  D  KA  and y^ i s t h e mole f r a c t i o n o f  A .  I f t h e term i n b r a c k e t s i s c o n s i d e r e d as t h e d i f f u s i o n c o e f f i c i e n t , i t i s o b v i o u s t h a t i n t h i s r e g i o n t h e c o e f f i c i e n t i s dependent upon the c o n c e n t r a t i o n and f l u x .  A d i f f u s i o n c o e f f i c i e n t d e f i n e d by an equation  o f t h e form o f (1.12) and measured i n t h i s r e g i o n i s n o t v a l i d f o r u s e i n t h e T h i e l e modulus as d e f i n e d p r e v i o u s l y , as t h e s t o i c h i o m e t r y o f t h e c h e m i c a l r e a c t i o n imposes a f l u x r a t i o w h i c h i s u n l i k e l y t o be t h e same as t h e f l u x r a t i o • o b t a i n e d i n an independent n o n - r e a c t i v e  determination.  Effective Diffusion Coefficient  - 7 - • (1.8)  An e f f e c t i v e d i f f u s i o n c o e f f i c i e n t i s d e f i n e d i n e q u a t i o n f o r m o l e c u l a r d i f f u s i o n where two f a c t o r s a r e used t o m o d i f y t h e t r u e o r interstitial diffusivity.  The p o r o s i t y , o r v o i d f r a c t i o n , i s a  fairly  e a s i l y d e f i n e d and measured a b s o l u t e q u a n t i t y , and i n a g r a n u l a r bed may o f t h e range o f 0.3  t o 0.5.  be  However, t h e t o r t u o s i t y i s a d e r i v e d q u a n t i t y ,  and i s t h e r e f o r e u s u a l l y a l e s s w e l l - d e f i n e d p r o p e r t y , e s p e c i a l l y i n nonu n i f o r m pore s t r u c t u r e s .  A l t h o u g h a v a l u e o f around 1.5  from s i m p l e pore m o d e l s , i t can v a r y from 1 t o 100 experimental r e s u l t s .  m i g h t be  expected  when c a l c u l a t e d  Thus, a t y p i c a l s i m p l e s t r u c t u r e may  d i f f u s i v i t y about It- t i m e s l e s s t h a n t h e i n t e r s t i t i a l  from  have an  effective  value.  The l a r g e range o f t o r t u o s i t y v a l u e s can be a t t r i b u t e d t o sources.  two  F i r s t , t h e pores a r e not n e c e s s a r i l y open-ended and so t h e mass  t r a n s f e r may  be o n l y o c c u r r i n g i n a l i m i t e d number o f p a s s a g e s ,  oecond,  t h e pore r a d i u s i s l i a b l e t o v a r y a l o n g t h e l e n g t h o f t h e p o r e , and i t has been shown (6)  (7)  t h a t t h e r a t e o f d i f f u s i o n i s s m a l l e r t h r o u g h a pore  v a r y i n g radius than i t i s through a c y l i n d r i c a l to  of  pore o f e q u i v a l e n t volume  surface r a t i o . The e f f e c t i v e d i f f u s i v i t y can s e r v e as a s i m p l e c o r r e c t i o n t o  t h e d i f f u s i o n mechanism so t h a t t h e d i f f u s i o n e q u a t i o n d e s c r i b e s t h e t r a n s p o r t b e h a v i o u r i n a u n i f o r m porous s t r u c t u r e .  On t h e o t h e r hand, porous s t r u c t u r e s  r  c a n be so haphazard  t h a t any o f the mechanisms d e s c r i b e d may  same time i n s e r i e s o r i n p a r a l l e l  i n t h e same s o l i d .  occur a t the  The use o f an  e f f e c t i v e d i f f u s i v i t y i n t h i s case amounts t o f o r c i n g t h e b e h a v i o u r to  observed  f i t one o f t h e d i f f u s i o n e q u a t i o n s , and so t h e r e s u l t cannot be used t o  p r e d i c t t h e d i f f u s i v e b e h a v i o u r under o t h e r c o n d i t i o n s ' .  C.  EXPERIMENTAL ESTIMATION OF THE EFFECTIVE DIFFUSIVITY  Prediction The b a s i s  f o r t h e p r e d i c t i o n o f t h e e f f e c t i v e d i f f u s i v i t y has  been b r i e f l y o u t l i n e d i n t h e p r e v i o u s p a r a g r a p h , and c l e a r l y r e s t s on some p h y s i c a l i d e a l i z a t i o n o f t h e pore s t r u c t u r e .  P r e d i c t i o n methods based  upon p o r o s i t y and e x p e r i m e n t a l t o r t u o s i t y v a l u e s a r e o f t e n n o t t o o s a t i s f a c t o r y due t o the n o n - u n i f o r m n a t u r e o f many porous s o l i d s .  However,  a v a r i e t y o f c a t a l y s t p e l l e t s can be a p p r o x i m a t e d by t h e " p i l e o f b r i c k s " s t r u c t u r e w h i c h y i e l d s a model c o n s i s t i n g o f a honeycomb o f connected passages.  'This approach has been d e s c r i b e d i n d e t a i l b y V/heeler (2), w i t h  r u l e s f o r p r e d i c t i n g the e f f e c t i v e d i f f u s i o n c o e f f i c i e n t defined  by t h i s  model. Other s i m p l e p o r e models i n c l u d e unconnected p a r a l l e l c y l i n d r i c a l pores (8) (9), and p o r e s w i t h v h i c h a r e used t o e x p l a i n t h e h y s t e r e s i s  "ink b o t t l e " capacities  (10)  i n certain adsorption-desorption  curves. P o s s i b l y o f more g e n e r a l a p p l i c a t i o n t o t h e problems i n c a t a l y t i c k i n e t i c s i s the b i d i s p e r s e  involved  pore s t r u c t u r e model proposed by  Wakao and Smith ( l l ) and M i n g l e and Smith (12).  I n the l a t t e r paper, a  c o n c e p t o f l a r g e r macro pores i n s e r i e s w i t h m i c r o p o r e s i s u s e d .  I n the  f o r m e r , t h r e e p a r a l l e l mechanisms a r e c o n s i d e r e d ; f i r s t , d i f f u s i o n t h r o u g h the macro pores between t h e b a s i c p a r t i c l e s from w h i c h t h e p e l l e t i s p r e s s e d , s e c o n d , d i f f u s i o n i n t h e m i c r o pores o f t h e b a s i c p a r t i c l e , and f i n a l l y , s e r i e s d i f f u s i o n from m i c r o p o r e s t o macro p o r e s o r v i c e v e r s a . does n o t r e q u i r e  The model  e m p i r i c a l c o n s t a n t s , o r assumptions r e g a r d i n g t h e mode o f  d i f f u s i o n i n any o f t h e p o r e s , b u t p o r o s i t i e s and a p o r e s i z e f r e q u e n c y d i s t r i b u t i o n function are required  i n addition to t o r t u o s i t y values. •  - 9Steady State Experimental Method Cor Measurement of Diffusion i n Solids In t h i s method, a c y l i n d r i c a l catalyst p e l l e t i s f i t t e d into a tube and two test gases o f known composition arc passed contirmously across the ends.  The two exit streams are analyzed, and from an appropriate  solution o f the d i f f u s i o n equation the e f f e c t i v e d i f f u s i v i t y can be computed  (13)  (5).  This method has also been used to obtain molecular d i f f u s i v i t i e s ( l ^ ) , because c a l i b r a t i o n of the porous p e l l e t by a gas pair of known d i f f u s i v i t y allows c a l c u l a t i o n of the d i f f u s i v i t i e s o f other gas pairs by making the assumption that the t o r t u o s i t y i s independent o f the gas system. As a technique for measuring molecular d i f f u s i v i t i e s i t has the advantage that i t i s a flow method, and so analysis  i n s i t u i s not required.  other hand, care must be taken that a narrow pore size d i s t r i b u t i o n and  On the exists  that Knudsen d i f f u s i o n does not occur. \Th.en  used  as a method for measuring the e f f e c t i v e d i f f u s i v i t y i n  porous p e l l e t s one must be sure that the correct d i f f u s i o n equation has been used (e.g.  eqn. (1.8) or (1.12)).  The method can be applied to the mixed  d i f f u s i o n range i f measurements are made a t varying t o t a l pressures. However, it. has the l i m i t a t i o n of being tedious i f a representative average value i s needed, because each p e l l e t must be tested separately, and cracks and  fissures have an overwhelming influence' on the r e s u l t .  The technique  i s not convenient for use with other than c y l i n d r i c a l " shapes, and therefore other shapes must be machined to cylinders.  I f the p e l l e t i s not  i s o t r o p i c , t h i s procedure may r e s u l t i n a f a u l t y value of the d i f f u s i o n coefficient.  - 10  The  r e s u l t obtained  -  by t h e s t e a d y s t a t e method i n a b i d i s p e r s e  p e l l e t weighs t h e d i f f u s i v i t y i n f a v o u r o f t h e l a r g e r p o r e s , b u t i n t h e c h e m i c a l r e a c t i o n case most o f t h e c o n v e r s i o n pores.  takes place i n the micro  T h i s b i a s i s f r e q u e n t l y n o t s e r i o u s as t h e m i c r o p o r e s a r e g e n e r a l l y  s h o r t , and so a m i c r o p o r e e f f e c t i v e n e s s f a c t o r o f u n i t y i s common ( l l ) . Hence, t h e d i f f u s i o n a l r e s i s t a n c e t o r e a c t i o n i s i n t h e macropores, and the s t e a d y s t a t e e f f e c t i v e d i f f u s i v i t y v a l u e may be q u i t e adequate. Chemical R e a c t i o n Method I t i s obviously possible t o carry out a chemical r e a c t i o n o f knovn k i n e t i c b e h a v i o u r a t c o n s t a n t c o n d i t i o n s u s i n g  successively  s i z e s o f p e l l e t u n t i l t h e r e a c t i o n r a t e becomes c o n s t a n t , an e f f e c t i v e n e s s f a c t o r o f u n i t y has been r e a c h e d .  smaller  indicating that  From t h e s e  effectiveness  f a c t o r s t h e T h i e l e modulus, and hence e f f e c t i v e d i f f u s i o n c o e f f i c i e n t s , can be c a l c u l a t e d p r o v i d i n g t h e k i n e t i c b e h a v i o u r i s n o t complex. method i s n o t easy t o a p p l y e x p e r i m e n t a l l y ,  This  and i s s u b j e c t t o many e r r o r s .  Unsteady S t a t e Methods A t y p i c a l procedure f o r measuring the molecular d i f f u s i v i t y o f a gas  (Loschmidt method) c o n s i s t s o f f l u s h i n g two c y l i n d e r s w i t h t h e t e s t  g a s e s , and t h e n b r i n g i n g them t o g e t h e r a t t i m e z e r o w i t h 1  on t o p .  t h e l i g h t e r gas  One o r b o t h o f t h e c y l i n d e r s i s removed a t a g i v e n t i m e , and t h e  t o t a l contents analyzed.  'The d i f f u s i o n c o e f f i c i e n t i s t h e n c a l c u l a t e d  from t h e s o l u t i o n d e r i v e d from F i c k J s second l a w ( e q u a t i o n  (l.9))«  It is  d i f f i c u l t t o a c h i e v e a c c u r a c y i n t h i s experiment due t o t h e t e n d e n c y f o r e d d i e s t o be c r e a t e d  e i t h e r when t h e c y l i n d e r s a r e f i t t e d t o g e t h e r o r b y  the a c t i o n o f temperature  gradients.  - 11 For porous p e l l e t s the analog o f the above experiment cannot be readily applied due to the r a p i d i t y o f the d i f f u s i o n process i n ,^ases. For example, i f a one cm. diameter p e l l e t o f t y p i c a l pore structure i s i n i t i a l l y bathed i n one gas", ana at time zero the surface i s flushed with another gas, then  SQ.Q^o  o f the f i r s t gas i n the p e l l e t i s removed by  d i f f u s i o n i n 10 seconds i f the d i f f u s i v i t y D-g/Ep i s 0.01 cm /sec. 2  (See Appendix I I I for d e t a i l s o f t h i s calculation.)  Thus, i t i s obvious  that some means to extend the time scale i n experiments with small p e l l e t s would be very desirable. Currie (6) has developed a non-flow apparatus of t h i s type which can be used only at normal temperatures and pressures f o r measuring d i f f u s i v i t i e s i n s o i l s and other granular beds.  Only rather complex -  frequency response techniques, discussed below, are presently available f o r the measurement of e f f e c t i v e d i f f u s i o n c o e f f i c i e n t s by transient response methods. Frequency Response and Pulse Methods McHenry and Wilhelm (15) have described a method f o r measuring the eddy d i f f u s i v i t y i n packed beds, and t h i s apparatus has been used also by Deissler and Wilhelm (l6) to measure both the e f f e c t i v e d i f f u s i v i t y and the eddy d i f f u s i v i t y i n packed beds.  The method i s based on frequency  response techniques using a concentration sine wave generated i n the feed to the bed, with amplitudes and phase angles recorded a t the entrance and e x i t of a t e s t section. In the same way, Van Deemter, Zuiderweg and Klinkehberg (17) have applied the d e l t a function (that i s , an i d e a l pulse) to packed beds i n the form of gas chromatography columns and i o n exchange beds.  Hougen (l8)  - 12  -  has pointed out that there i s no r e a l d i f f e r e n c e between the  results  obtained by a d e l t a function or by a frequency response method. In the work of Van Deemter et a l the d i s p e r s i o n  e f f e c t s due  molecular d i f f u s i v i t y , eddy d i f f u s i v i t y and a mass t r a n s f e r  to  coefficient  are each found, on the basis of the theory developed, to have a d i f f e r e n t v e l o c i t y dependence, which  allo'.vs  f a c t o r on the d e l t a f u n c t i o n .  separation of the influence of each  The mass transfer c o e f f i c i e n t can be derived  i n terms of the e f f e c t i v e d i f f u s i v i t y of the porous p e l l e t , cud hence, i f t h i s q u a n t i t y can be evaluates, an e f f e c t i v e d i f f u s i o n c o e f f i c i e n t be c a l c u l a t e d  from i t .  may  The theory on which t h i s approached i s based i s  d e a l t w i t h more f u l l y i n succeeding  sections.  Comparison of Various Methods "In porous s o l i d s there i s a basic d i f f e r e n c e  between the  applicatio  of d i f f u s i o n c o e f f i c i e n t s to the steady state and the unsteady s t a t e , difference  'iliis  i s the r e s u l t of the capacitance.effects which manifest themselves  i n the unsteady s t a t e .  In other words, the time of d i f f u s i o n from a porous  s o l i d containing dead end pores would be much greater tnan the  effective  d i f f u s i v i t y measured by a steady state method would i n d i c a t e .  This e f f e c t i s  allowed f o r i n equation ( l . l l ) because instead of the e f f e c t i v e d i f f u s i v i t y alone, the e f f e c t i v e d i f f u s i v i t y divided by a capacitance term (the p o r o s i t y ) is utilized.  ' S i m i l a r l y , i f adsorption occurs on the surface of the  solid  then the volume of gas adsorbed must be added to the porous volume or p o r o s i t y i n the d i v i s o r .  (This l a s t statement regarding adsorption assumes  that the adsorption process i s e f f e c t i v e l y at e q u i l i b r i u m and  that  the  isotherm i s l i n e a r , otherwise the simple d i f f u s i o n equation would no longer hold.)  With the correct d i f f u s i o n equation, there should be  no  - 13 -  basic difference between e f f e c t i v e d i f f u s i v i t y i n an i s o t r o p i c s o l i d  determined  by a steady state or unsteady s+ate method. I f bulk d i f f u s i o n i s the transport mechanism there i s no d i f f i c u l t y i n c o r r e c t l y defining the e f f e c t i v e c o e f f i c i e n t f o r either the steady state or unsteady state methods. predominates.  However, this i s not true when Knudsen d i f f u s i o n  C o n s i d e r a simple model o f dead end pores of eg.ual length i n  p a r a l l e l i n which Knudsen d i f f u s i o n i s taking place.  The t o t a l composition of  each pore (after a step change i n surface concentration) w i l l vary according to i t s radius.  I n i t i a l l y , the large pores w i l l y i e l d the major f l u x , but  a f t e r a time the lower f l u x i n the smaller pores w i l l r e s u l t i n larger concentration gradients, which w i l l eventually r e s u l t i n the f l u x from the smaller pores equalling or exceeding that from the larger pores.  Hence, an  unsteady state experiment i n the Knudsen regime may y i e l d a d i f f u s i v i t y which varies with time. An i n t e r e s t i n g aspect o f t h i s l a t t e r conclusion arises because the majority o f a solid-surface catalyzed chemical reaction occurs i n the smaller pores (due to the large surface area), and i f these pores are long then they may not be f u l l y e f f e c t i v e .  The steady state method i s i n s e n s i t i v e  to the' resistance which may occur i n dead end pores, while the unsteady state method i s p o t e n t i a l l y capable of allowing f o r t h i s resistance.  The  unsteady state method w i l l give a d i f f u s i v i t y which i s some average value c f a l l pore resistances, and the a b i l i t y of this value to describe the rate of a d i f f u s i o n limited chemical reaction may depend upon the weighting by the experimental procedure or the experimenter.  For example, most unsteady  state methods involve an i n i t i a l period before readings are taken i n order to allow the a p p l i c a t i o n to the data of simple solutions o f the d i f f u s i o n equation applicable a t longer times.  Thus, i n t h i s case, the  - in  -  d i f f u s i v i t y obtained from such experiments may b e expccLed to be weighted i n favour c f the small pores i f Knudsen d i f f u s i o n predominates. D.  OBJECTIVES OF 'THE PRESENT WORK On the basis of the foregoing comparison of methods for obtaining  a value of the e f f e c t i v e d i f f u s i v i t y , i t i s apparent that, i n most cases, a d i f f u s i o n G o S f f i e i i n t Obtained from an unsteady-state experiment i n v h i G h a l l the pores contribute to the d i f f u s i o n a l process may well b e a better value f o r use i n chemically reacting systems.  In many instances, steady-  state experiments may also give suitable values, but t h i s cannot b e assumed without considerable knowledge of the p a r t i c u l a r porous structure. It would be u s e f u l to develop a method using a pulse technique, which would avoid most of the experimental d i f f i c u l t i e s of frequencyresponse measurements, while giving the advantages of an unsteady-state method and which could be applied to a representative sample of p e l l e t s vithout requiring s p e c i a l shaping.  I t might be possible to make use of  such a technique to follow changes i n c a t a l y s t d i f f u s i o n a l behaviour with age.  Recent advances i n the theory of transport processes i n chromatographic  columns suggest that i t might be possible to i n t e r p r e t pulse dispersion r e s u l t s i n such a way as to y i e l d an' e f f e c t i v e d i f f u s i o n c o e f f i c i e n t . The primary objective of the present work was to attempt the development  of a pulse method as a means of measuring e f f e c t i v e d i f f u s i v i t i e s  of gases i n porous p e l l e t s , a technique not previously reported.  A  secondary objective was to be the i n v e s t i g a t i o n of the use of unsteady state flow methods f o r measuring the binary d i f f u s i o n c o e f f i c i e n t of gases.  The  flow methods possess the advantage of allowing analysis outside the apparatus, by any convenient means.  Further, the use of a porous bed of u n i t t o r t u -  o s i t y would also allow such a measurement to give absolute values of the  - 15 d i f f u s i o n c o e f f i c i e n t without any c a l i b r a t i o n being necessary.  Freedom  from convective e f f e c t s would a i d i n making possible measurements a t widely varying temperatures and pressures, as does the freedom i n choice of concentration measurement.  - 16 II THEORY A.  DERIVATION OE VAN DEEMTER EQUATION  Height Equivalent t o a T h e o r e t i c a l P l a t e The performance o f a chromatograph column i s g e n e r a l l y measured I n terms o f a " h e i g h t e q u i v a l e n t t o a t h e o r e t i c a l p l a t e " ( H E T P )  v  I n a gas  chromatograph column a narrow band' o f sample gas i s i n j e c t e d i n t o a stream o f c a r r i e r gas w h i c h passes t h r o u g h t h e column t o a d e t e c t i n g d e v i c e .  The  components o f t h e sample have d i f f e r i n g r e t e n t i o n t i m e s i n t h e column depending upon t h e p r o p e r t i e s o f t h e gas component and t h e l i q u i d s t a t i o n a r y phase i n t h e column.  I t i s o b v i o u s t h a t a column w h i c h r e s u l t s i n a  b r o a d e n i n g o f t h e p u l s e i s d e t r i m e n t a l t o t h e s e p a r a t i o n d e s i r e d , and t h e h e i g h t e q u i v a l e n t , t o a t h e o r e t i c a l p l a t e (HETP) w h i c h i s d e f i n e d below i s a measure o f t h e degree o f l o n g i t u d i n a l d i s p e r s i o n . The HETP i s o b t a i n e d  by p o s t u l a t i n g t h a t t h e mechanism o f p u l s e  b r o a d e n i n g i s caused by e q u i l i b r a t i o n o f t h e s t a t i o n a r y m a t e r i a l i n a g i v e n p l a t e w i t h t h e m o b i l e gas phase w h i c h t h e n passes on t o t h e n e x t p l a t e . A l i n e a r absorption isotherm W C concentration,  = C  T  (where C  = s t a t i o n a r y phase c o n c e n t r a t i o n )  = m o b i l e phase  i s assumed and a  m a t e r i a l b a l a n c e around t h e n t h p l a t e (see F i g u r e 1.2) w i t h an i n c r e m e n t o f gas f l o w dU y i e l d s ,  du(c _ - c ) = (v +v; )dc n  1  n  v  n  (1.15)  from w h i c h i s o b t a i n e d , n dU d C  = n-1 - n V + Wv C  C  (l.l6)  - 17 -  + '8  Figure 1.2  '  Model For Derivation Of Plate Theory where V = volume of mobile phase i n plate and v = volume of plate stationary phase. Assume that a l l the pulse gas i s i n i t i a l l y i n stage 1 y i e l d i n g an i n i t i a l gas concentration and  C _i_ n  =  C .  Applying n = 1 to equation ( l . l 6 ) ,  C  n  = Ci,  0, -dCi  (1.17)  iU  V-p where Vp = Volume of plate = (V + Wv).  C i = K exp / -  U V.  \  On integration, (1.17) gives,  (1.18)  - 18 When U = 0,  , and. t h e r e f o r e K = c ' i n ( l . l 8 ) .  = C  Hence,  C i = C exp I - _U_ \  \ V  (1.19)  J  Wow a p p l y i n g t h e above r e s u l t t o e q u a t i o n ( l . l 6 ) w i t h n = 2, dCg. *  +  C ^  U  V  =  C _ e x p / - U _ \  P  P  (i.20)  w  I  V  E q u a t i o n (1.20) can be s o l v e d b y u s e o f t h e i n t e g r a t i n g  factor,  exp^ + U V  Csexp/yV  exp - U_ + U_ dU = C ' U_ + K  l* V 'j  V  V  P  V  P  V  P  When U = 0, C2 = 0, and so K = 0, y i e l d i n g  C  2  = c ' U_ exp / - U_  P  (1.21)  P  the r e s u l t ,  \  (1.22)  Kence b y c o n t i n u i n g t h i s p r o c e s s t o t h e n t h sxage  c„ = c ' i / " 1  '  , exp /- y_ \  1  (n-£)!V -l  n  p  (1.23.)  ~"*(~V ) p  This i s a Poisson d i s t r i b u t i o n f u n c t i o n ,  and f o r a l a r g e number  o f p l a t e s t h i s d i s t r i b u t i o n approaches a G a u s s i a n o r n o r m a l d i s t r i b u t i o n . The mean o f t h e above d i s t r i b u t i o n i s U_ and t h e v a r i a n c e i s U_ ( t h a t i s , V  that the ( M e a n ) / ( 2  s t a n d a r d  P  deviation)  V  2  =/U_\ / U_ = U_ 1V ] V V \ VI P p  Now U i s t h e t o t a l volume o f gas w h i c h has f l o w e d , and a theoretical the  plate,  i s t h e volume o f  so t h a t when t h e mean approaches t h e end o f t h e column  (mean) /©* = no. o f t h e o r e t i c a l 2  P  2  plates.  By d e f i n i t i o n , t h e r e f o r e , EETP = L  \ \ mean /  where L i s t h e column l e n g t h .  2  (1.2U)  19 Measurement of RTLTP For a large number of stages the output can be assumed to be a Gaussian d i s t r i b u t i o n , and the mean and variance may be read d i r e c t l y from the record of the output at the end of the column by using the properties of the Gaussian d i s t r i b u t i o n shown i n Figurel.3.  This represent-  a t i o n i s n o t s t r i c t l y c o r r e c t , i n t h a t the output i s Gaussian w i t h r e s p e c t to position i n the column, while the recorded p r o f i l e at the end of a column i s with respect to time.  However, i f the time of purge of the  pulse i s small r e l a t i v e to the time of the mean, the error i n reading t h i s time d i s t r i b u t i o n compared to the distance d i s t r i b u t i o n i s n e g l i g i b l e .  Figure  1.3  Gaussian D i s t r i b u t i o n  Properties  Inasmuch as $0% of p normal d i s t r i b u t i o n l i e s between '-d l i m i t s , then the time of purge which i s approximately k 6, must be«mean to achieve a Gaussian d i s t r i b u t i o n .  I f now both sides of t h i s i n e q u a l i t y are squared  and m u l t i p l i e d by L, on rearranging 16 Lcr « mean " 2  L  or L  »  16 HETP  2  Hence, a column must contain much more than 16 plates to sai.isfy an assumption of a Gaussian d i s t r i b u t i o n i n the output r e c o r d . Input Pulse D i s t r i b u t i o n The d e r i v a t i o n of the HETP assumed t h a t a l l the pulse i s i n ihe f i r s t stage at the s t a r t , however, i t i s obvious that i f the pulse extended over several stages an e f f e c t would be noticed i n the output. I t has been shown by Van Deemter (17) that the e f f e c t of the i n i t i a l d i s t r i b u t i o n can be ignored i f , Ar  v,7H where A  c  <  °'  5  ( 1  '  2 5 )  i s the volume of gas i n the i n i t i a l pulse and n i s the number of  t h e o r e t i c a l plat.es. Rate Theory The t h e o r e t i c a l p l a t e model does not attempt to e x p l a i n the rate processes o c c u r r i n g i n a chromatograph column, out r e l i e s on the f a c t that the sum of several d i s t r i b u t i o n s tend to approach a normal Gaussian d i s t r i b u t i o n , having a mean made up of the sum of the independent means, and having a variance made up of the sura of the independent variances (19). One obvious mechanism which occurs to cause e. pulse to broaden i s molecular d i f f u s i o n i n the mobile phase.  L o n g i t u d i n a l d i f f u s i o n i n the  - 21 -  s t a t i o n a r y phase c a n g e n e r a l l y be i g n o r e d as t h e s t a t i o n a r y phase i s d i s c o n t i n u o u s i n a packed b e d , and, i n a d d i t i o n , t h e d i f f u s i o n  coefficient  i s s m a l l i n t h i s phase. There i s a group o f l i t t l e understood  p r o c e s s e s which, cause a  p u l s e t o d i s p e r s e due t o the f l o w p a t t e r n i n t h e packed b e d .  ,  Fortunately,  i n a deep "bed t h e s e a b e r r a t i o n s a r c o f a s t a t i s t i c a l n a t u r e v h i e h tend t e r e s u l t i n a Gaussian  d i s t r i b u t i o n as o b t a i n e d f o r m o l e c u l a r d i f f u s i o n , so  t h a t t h e y c a n be grouped t o g e t h e r i n a term d e s c r i b e d as t h e eddy d i f f u s i v i t y . I n t h e work o f Van Deemter e t a l (17) t h e eddy d i f f u s i v i t y (below a p a r t i c l e Reynolds number o f 1) i s c o n s i d e r e d t o be caused by t h e d i f f e r e n c e m  f l o w p a t h s between p a r t i c l e s .  These concepts  are discussed i n the  following sections. A pulse broadening  mechanism analogous t o t h e t h e o r e t i c a l p l a t e  mechanism d e s c r i b e d e a r l i e r can a l s o o c c u r i n t h e chromatograph column. I f a r e s i s t a n c e e x i s t s p r e v e n t i n g e q u i l i b r i u m between t h e m o b i l e and s t a t i o n a r y phase, t h e n t h e degree o f p u l s e b r o a d e n i n g  caused by t h e c a p a c i t a n c e  o f t h e s t a t i o n a r y phase i s i n c r e a s e d due t o t h e f a c t t h a t a l t h o u g h  less  m a t e r i a l e n t e r s t h e s t a t i o n a r y phase t h e time t a k e n t o g e t o u t a g a i n causes t h e p u l s e t o b r o a d e n more t h a n would be t h e case f o r t h e e q u i l i b r i u m situation. L a p i d u s and Amundson (2) have d e r i v e d an e x p r e s s i o n based on a d i f f u s i o n model t o d e s c r i b e t h e c o n c e n t r a t i o n p r o f i l e f o r t h e c o n d i t i o n s where a p u l s e gas p a s s e s t h r o u g h a packed bed c o n t a i n i n g a s t a t i o n a r y phase w i t h a l i n e a r a b s o r p t i o n i s o t h e r m between t h e gas and s t a t i o n a r y phase.  'The p u l s e i s n o t assumed t o be i n e q u i l i b r i u m w i t h t h e s t a t i o n a r y  phase, due t o a r e s i s t a n c e d e f i n e d by a mass t r a n s f e r c o e f f i c i e n t ,  oc.  L o n g i t u d i n a l d i f f u s i o n i n c l u d i n g m o l e c u l a r and eddy c o n t r i b u t i o n s i s  -  -22  characterized by a dispersion c o e f f i c i e n t , D , and i s assumed to occur i n  ii the mobile phase, but not i n the stationary phase. Figure  The model i s shown i n  l.k. A material balance around the element 6x y i e l d s  F i iC F  2  =  x  at JCZ <H  F D 2  L  * CI dx  =  -  2  FiU dC±  + OC (WC - Ci) 2  (1.26)  dx  2  o « ( C i - WC )  (1.27)  2  VThere W i s the equilibrium constant between the mobile and stationary phases. replaced by  1  I f the stationary phase i s a porous s o l i d V/ can be  .  Fi  Fa  C.  dx  I'  *  t  ax  uCi  MOBILE PHASE  dpi--  • WCs  STATIONARY PHASE  Figure l.k Mathematical Model f o r the Column  - 2$ F o r a s m a l l p u l s e i n j e c t i o n t i m e , t , and a n i n i t i a l p u l s e  concentration  C , L a p i d u s and Amundson (20) o b t a i n e d t h e f o l l o w i n g s o l u t i o n t o the above  equations,  C_i = x t C 2t^|Yru t 0  L  exp / ^  -(x - u t ) h D t  2  L  -ott \ + Fi J  X t 2titf TH^t* h  exp / - ( x - u t ) \ 4 D]_t 1  n  F(t*)  dt  2  \y' /  (1.28)  1  where F(t)  =/ Wt V ^ e x p / o*W ( t - t ) - c c t i \F F (t-t ) J 1 F Fi  1  2  1  1  1  1  2  2  \ l / 2 / / \ /  2  a  V/o1 FiF  \ 2  J  (1.29)  W  where t i s t i m e , t , t i m e o f i n i t i a l p u l s e w i t h c o n c e n t r a t i o n C , x i s d i s t a n c e a l o n g t h e column, and I i i s t h e h y p e r b o l i c B e s s e l f u n c t i o n . I t has been shown by Van Deemter e t a l (17) t h a t t h e above s o l u t i o n can be reduced t o a G a u s s i a n d i s t r i b u t i o n under c e r t a i n c o n d i t i o n s . These c o n d i t i o n s a r e t h a t t h e h e i g h t o f a t r a n s f e r u n i t F j . u ^ C L , t h e h e i g h t oc o f t h e bed, and t h e l o n g i t u d i n a l m i x i n g stage 2 DL «  L.  Essentially,  u t h e s e r e q u i r e m e n t s s t a t e t h a t t h e column must c o n t a i n a l a r g e number o f t h e o r e t i c a l p l a t e s , i n w h i c h case t h e c o n c e n t r a t i o n p r o f i l e reduces t o , Ci = where  fltp  exp / / u - B t L  1 = 1 + Fg_ W, o*i = 2 DLL  £  F i  ~  3  and C^ = 2 2  \  F T, 2  FiW^u  (1.30)  T h i s i s a G a u s s i a n d i s t r i b u t i o n w i t h mean L/u o r B t and v a r i a n c e Ci  + C2 .  2  2  As mentioned above, t h e v a r i a n c e o f a G a u s s i a n d i s t r i b u t i o n  i s composed o f t h e sum o f t h e i n d i v i d u a l v a r i a n c e s , so e q u a t i n g cf mean " 2  the r a t i o  f o r t h e above s o l u t i o n y i e l d s t h e f o l l o w i n g w h i c h c a n be combined  2  w i t h t h e HETP d e r i v a t i o n o f e q u a t i o n  (1.2*0.  - 2k tfi + c ^ = 2 D| L / u \ + / 1 \ 2 F L / u \ = 02.' (L/u) ~u^ \ L j [ l + _ l 2 _ I <*FiW u ( L ] rnearr " \ FiW / 2  2  2  2  2  2  2  2  = 2 D —'{J  2  +/ ' 1 \ 2 F i u = cr x L ( l + WFi ] oc mean 2  L -  (1.31)  2  =  HETP  (1-32)  2  F y  \  2  Trie d i f f u s i v i t y Mechanism which  2  OQGUK'H  i n equation  (1.32) r e f e r s t o any a x i a l m i x i n g  i n t h e m o b i l e phase,so t h a t i t c a n be s a i l e d a  d i s p e r s i o n c o e f f i c i e n t , i n c l u d i n g t h e eddy d i f f u s i v i t y .  I t was p o i n t e d  out by Van Deemter t h a t i n t h e l a m i n a r r e g i o n t h e eddy d i f f u s i v i t y i n a packed bed i s p r o b a b l y c r e a t e d b y t h e d i f f e r e n c e i n f l o w p a t t e r n s i n t h e bed.  A p e r f e c t l y u n i f o r m bed thus c o n c e i v a b l y has no eddy term. The m o l e c u l a r d i f f u s i v i t y c o r r e c t e d f o r t h e p a t h  lengthening  i n a packed bed b y a t o r t u o s i t y f a c t o r , and t h e eddy d i f f u s i v i t y D^'are commonly assumed t o be a d d i t i v e , so t h a t t h e d i s p e r s i o n c o e f f i c i e n t D i s given by, D  L  =  + D*  v  (1.33)  L  where DjJ^ depends o n t h e a x i a l d i s p e r s i o n caused b y t h e f l o w p a t t e r n s . T h i s a s s u m p t i o n i s d i s c u s s e d b y K l i n k e n b e r g and S j e n i t z e r (19) and t h e y concluded  t h a t t h i s approach i s j u s t i f i a b l e i f t h e t h e o r y  describes the r e s u l t s .  adequately  'The abundant work on gas chromatography appears  to l e n d s u p p o r t t o t h e a s s u m p t i o n o f a d d i t i v i t y o f c o e f f i c i e n t s .  At  h i g h f l o w r a t e s , t h e m o l e c u l a r d i s p e r s i o n becomes n e g l i g i b l e compared t o t h e t u r b u l e n t d i s p e r s i o n , so t h a t t h e o v e r a l l d i s p e r s i o n i s t h e same as t h e f l o w d i s p e r s i o n and c a n be c a l l e d t h e eddy d i f f u s i v i t y . At l o w f l o w r a t e s , e.g. p a r t i c l e Reynolds numbers  1, t h e eddy  d i f f u s i v i t y c a n be r e p r e s e n t e d a c c o r d i n g t o Van Deemter e t a l by t h e expression D * = L  u d . Thus e q u a t i o n  (I.32), a f t e r i n t r o d u c i n g t h e  - 25 concept of additive HETP = 2 #a  c o e f f i c i e n t s stated i n (1.33)> takes the form,  + 2 D-n + f  1  1 The  quantity  "*  2 Fm  2  + V/Fx  E  (l.3>0  2  i s reported bo decrease with larger diameter p e l l e t s ,  having a value of about 8 f o r 200 mesh, and p r a c t i c a l l y zero f o r ~$0 mesh, particles.  '  Mass Transfer C o e f f i c i e n t and Effective D i f f u s i v i t y D i f f u s i o n a l resistance to mass transfer from the mobile phase to the i n t e r i o r of the p e l l e t s (stationary  phase) i s made up o f two parts,  the f i r s t being due to resistance i n the mobile phase and the second to the resistance within the p e l l e t s .  The solution of Lapidus and Amundsen  (2) used by Van Deemter et a l (17) (equation 1.28) treats the resistance i n terms of a mass.transfer c o e f f i c i e n t . Van Deemter et a l treated the two resistances as separate mass transfer c o e f f i c i e n t s which could be combined by the resistances-inseries r u l e .  (A mass transfer c o e f f i c i e n t i s r e a l l y a conductance  rather than a resistance hence the reciprocals•are  1 OC  =1 e<  x  +W <X  Z  =  1 k^Ap  the additive  + W kA 2  property.)  (1-35)  P  Where oft. i s the mobile phase c o e f f i c i e n t / u n i t v o l . of bed, ofe i s the stationary k  2  phase c o e f f i c i e n t and ocis the o v e r a l l c o e f f i c i e n t with k i and  being the corresponding surface mass transfer c o e f f i c i e n t s .  W i s the  p a r t i t i o n c o e f f i c i e n t , which i s necessary i n gas chromatography because the d i f f u s i o n i n the stationary  phase occurs i n a l i q u i d arid the l i q u i d -  phase concentration gradients are expressed i n terms o f equivalent equilibrium  gas phase concentrations i n order to make equation (1.35)  - 26 consistent.  In d i f f u s i o n i n porous s o l i d s , the e f f e c t i v e d i f f u s i v i t y i s  defined on the basis of the i n t e r s t i t i a l gas  concentrations and  so  the  p a r t i t i o n c o e f f i c i e n t becomes a quantity r e l a t i n g i n t e r s t i t i a l concentrations to stationary  phase concentrations, that i s 1  , where 6  i s the p e l l e t  »  r  porosity. External Mass Transfer  Coefficient  Based on the work of Ergun (21)  Van Deemter et a l suggested  the  use of the following c o r r e l a t i o n for the mass transfer c o e f f i c i e n t i n the mobile phase, k i = 2_5_  A cm. /sec. (lo6) Fi \Taeve k i i s the mass transfer c o e f f i c i e n t per unit area and A i s the P Dp.  6  9  surface area per unit volume of «*i  =  In a bed  Ap k i  sec."  bed, (1.37)  1  of spherical p a r t i c l e s of diameter d , the  area per unit volume, A^,  i s given by the following,  surface  i f the bed  porosity  is F i , A  p  = 6(1  Internal Mass Transfer  - Fi)/d  (1.38)  p  Coefficient  In t h i s work i t i s desired to obtain the e f f e c t i v e d i f f u s i v i t y i n the porous p e l l e t , and  so i t i s necessary to f i n d a  between the mass transfer c o e f f i c i e n t , cfe = k diffusivity.  Such an expression was  derivation, but  vhere C  s  and  respectively.  C  A^,  given by Van  and  the  effective  Deemter et a l without  i t can be obtained i n a p e l l e t of radius R as shown below.  *±) \ ar /  Let  2  relationship  a v g  r=R  = Ka (C  s  - C  are the surface and  )  (1.39)  average concentrations i n the p e l l e t  27 Crank (22) (page 2J3) has obtained solutions of the d i f f u s i o n equation f o r a s p h e r i c a l p e l l e t of radius R which give the concentration C  A  at any radius r and time t when the surface concentration changes step-  wise from 0 to C , s  C©  C  A  = C  +  /  V  2 RC„ ~7f~r~  (^lV^  2-  \  n  n=l  \  nTTr  sin (  R  exp /  - Dr:n TT t  [  R  J  2  \  2  /  (1.1*0)  S i m i l a r l y the average concentration i s given by: c  avg - C  - 6 C  g  exp  R  2  (1A1)  n=l I f t i s large then only the f i r s t terms o f the series solutions need be considered.  This amounts to suggesting that C  g  approaches C  a V  g  and i n view of the r a p i d i t y of gas d i f f u s i o n as demonstrated i n the example given i n the introduction, t h i s assumption would appear to be reasonable. From (1.1*0), CA_ -  1 = C  A  Cs  - C C  = -2_R_  s  i n  | 7 £ r j e x p ^ - F . rr t j  (1.1*2)  exp / - D W T t \ I ""i^ " )  (1.1*3)  2  D  7Tr  S  From ( l . 4 l ) , 1 C  =  • 6  c avs;  2  TP  s  2  Divide (1.42 by (1.4-3) = Cavg Let C  s  A  c s  - C=  =  C C  (1.44)  CA  S  s - C  a v g  , thensubstitute i n equation  r D  3 i n 77 r H  2 R TT 6 r  (1.39)  R E  R7T  3r  Sin TTr R  Taking the l i m i c as  -  ^2  (C  s  - C g) a V  Ar  A.?  limit  3k  Cc - Ca v g .  bin 2  7Tr "R  TTr R  A r  (1^5)  - 28 o  r  2  TT DE kad  1  =  2  where d  p  = 2 R  p  therefore k  = 2 ]-p  2  2  £E  3  d  (l.t6)  p  The mass transfer per u n i t volume of bed efe i s obtained multiplying k  2  by the surface area per unit volume A  given by  p  by  equation  (1.56).  'The two mass transfer c o e f f i c i e n t s could be combined using equation  (l.35)•  However, at this stage, the conditions of the present  work d i f f e r s from that of Van Deemter et a l . k, 2  If k i i s large compared to  then when the inverse i s summed i n equation  (1.35)*  ma  7  t>  e  ignored.  'The expression for k i suggested by Van Deemter et a l applies only to the laminar flow region, but i n the turbulent region the value w i l l be greater rather than l e s s , and so i f reason can be found to neglect k j i n the laminar region, i t need not be considered i n evaluating mass transfer i n the turbulent flow region. I f we as.sume the e f f e c t i v e d i f f u s i v i t y i n the porous p e l l e t i s l/5 of the molecular d i f f u s i v i t y as suggested i n the introduction, and assume a bed porosity of O.k, (I.36) and  then the r a t i o of k i / k  (1.1)5) i s around 28.  2  from  equations  Tims, at most, the resistance outside the  p e l l e t makes a yjo contribution and can be ignored. The derivation for k  2  was made on the assumption of a step  change i n the surface concentration, but i n this work a Gaussian curve i s expected to describe the surface concentration.  I t would be desirable to  have a derivation applicable to other surface functions, or at l e a s t to the Gaussian function. k  2  An attempt to obtain an alternate expression for  using a ramp surface function could not be made to reduce to the  expression of (1.46), because the exponential functions would not cancel  -29out as i s tne case i n the step y i e l d i n g equation (1.4'-:).  Thus, a f u r t h e r  degree o f approximation r e s u l t s from using ( 1 . 4 6 ) , but i t i s prooable chat an experimental constant other than 2/577" can be found which would 2  y i e l d a s a t i s f a c t o r y d i f f u s i v i t y from a pulse experiment. Van Deemter's Equation .'The expression for the mass 'i^anaf/as- ee§ffieiems eaa new ba substituted i n equation ( 1 . 3 4 ) . and  Ignoring k i , and combining  (l«30)  (1.35)  (1.46)  OC = 2 / 3 7 1 2 D£  (1.47)  6 (I- Fi)  3p  dp  S u b s t i t u t i n g (1.47) i n t o ( 1 . 3 4 ) , HETP = 2 tfd + 2 D p  2  +  3  2 2 ? id,, u  Tff* U  1 +VfFi  E  (1.43)  (l-Fit  TP J ?  Making the s u b s t i t u t i o n s ,  Fi  = e  F  =  w  HETP = 2 ^ d P  2  B  1 - F i = (1 - 6 ) 3  =  -  l  —  + 2 D  TT  2  1 +  1 £ (l p  2  JL  -6 ) B  4  6* ^  u  7T D „ U -  , (l>9) i  v  )  This i s the equation derived by Van Deemter et a l (17) and i t may be observed to be o f the general form HETP = A + B/u + Cu  (1.50)  where A, B and C would be constants f o r a given packed bed and a given sys tern. A sketch o f the behaviour o f t h i s equation i s shown i n Figure 1.5 i n d i c a t i n g the p h y s i c a l s i g n i f i c a n c e o f the constants A, B and C.  - 30 Trie magnitude of the A r,erm, vhich may he c a l l e d the eddy c i l f C u s i v i t y term, depends l a r g e l y on the value of  As poin'cea out by  Van Deemter et a l , % i s expected to be quite small for l a r ^ e packing s i z e s , e.g. 30 mesh diameter or larger.  Therefore, i t i s l i k e l y that the eddy  d i f f u s i v i t y term may not seriously mask the other terms. At low flow rates, the quantity B/u, or molecular d i f f u s i v i t y term, may be expected to dominate, while at high flow rates the e f f e c t i v e d i f f u s i v i t y , or Cu, term w i l l predominate. The e f f e c t i v e d i f f u s i v i t y , or Cu term, i s of primary i n t e r e s t and so t h i s term w i l l be considered i n more d e t a i l .  I t was mentioned  e a r l i e r that a lower mass, transfer c o e f f i c i e n t or e f f e c t i v e d i f f u s i v i t y causes the pulse to broaden, and i n the e f f e c t i v e d i f f u s i v i t y term a lover e f f e c t i v e d i f f u s i v i t y does indeed r e s u l t i n a larger HETP, which i s a measure of the amount of pulse d i s s i p a t i o n .  S i m i l a r l y , a non porous  p e l l e t w i l l have zero porosity (£p), and so the p e l l e t capacity term becomes zero.  'This implies that f o r a bed of non porous p e l l e t s the  following equation should apply: HETP = A + B/u The influence of the bed porosity ^  on the magnitude of the  e f f e c t i v e d i f f u s i v i t y term i s very small over the range of p o r o s i t i e s commonly found i n random p e l l e t packings.  On the other hand, the p e l l e t  diameter which has an exponent of 2 has a strong influence on the value of the e f f e c t i v e d i f f u s i v i t y term, and suggests that this term w i l l be much more e a s i l y evaluated f o r l a r g e r p e l l e t s . •Typical Values of the E f f e c t i v e D i f f u s i v i t y Term (C) In Table 1 . 1 following,some values of the e f f e c t i v e d i f f u s i v i t y term are calculated f o r some t y p i c a l porous p e l l e t properties and dimensions,  - 31 and f o r a range o f p o s s i b l e  effective diffusivities.  The v e l o c i t i e s where  t h e m o l e c u l a r d i f f u s i o n term e q u a l s t h e e f f e c t i v e d i f f u s i v i t y term i n  (1.4-9) ( i . e . t h e minimum shown i n F i g u r e 1.5) a-re a l s o  equation  f o r an e f f e c t i v e d i f f u s i v i t y o f 0.01  c m / s e c , a n assumed m o l e c u l a r d i f f u s i o n 2  c o e f f i c i e n t o f 0.2 c m / s e c , and a t o r t u o s i t y f a c t o r o f 1.33. 2  be shown  calculated,  I t can e a s i l y  that, v  2DR  1 1 + 6  ^ i n  p v  €.3 D  _.^  2  2  dp D  e  2  ( I  d up p  /ji,  ^  G  - e) B  Because t h e v e l o c i t y a t t h e minimum i s p r o p o r t i o n a l Reynolds number,  B  2  to l / d the p a r t i c l e  a t each o f t h e minima o b t a i n e d f o r d i f f e r e n t  p a r t i c l e s i z e s w i l l be t h e same, and has a v a l u e o f around 6 i f a v a l u e of l/6 i s taken f o r the kinematic v i s c o s i t y . TABLE I . I THE EFFECT OF PELLET DIAMETER AND EFFECTIVE DIFFUSIVITY OK THE EFFECTIVE DIFFUSIVITY TERM (C) IN EQUATION ( I • 50) * ____d cms. Eff .DifTr^--^.^^  1  (%  = 0.4, E. = p  •5  .25  0.33)  .1  .1 cm /sec  .0375  .00938  .00235  .01  • 375  .0938  .0235  •00375  .938  .235  .0375  2  .001  3-75  V e l o c i t y a t minimum from u = fB^cS/sec m  i  n  Icj  .895  1.79  From t h e f i r s t t h r e e l i n e s o f c a l c u l a t i o n s  3.16  .  . O O O 3 7 5  8.95  i n Table 1 . 1 ,  i t is  o b v i o u s t h a t f o r h i g h e f f e c t i v e d i f f u s i v i t i e s and s m a l l p e l l e t d i a m e t e r s a pulse dispersion  method may b r e a k down because t h e e f f e c t i v e  diffusion  EDDY  DIFFUSION  CONTRIBUTION  INTERSTITIAL VELOCITY u y-  *  —  Figure 1.5 Typical Plot o f HE TP Vs. Velocity (Equation 1.50)  - 33 -  term becomes too small with respect to the other terms unless extremely high flow rates can be used.  For example, assuming the figures given  above, the constant 3 = .15 and A = d  i f the quantity 2 tf= 1 i s used.  At very high flow rates, the description of eddy d i f f u s i v i t y suggested by Van Deemter et a l (17) would not be expected to hold.  However, i f the  "pea?£@§t miK@i " *aeael (CUsQusflwet i n the fellowiny seQtien ) applies, ae a  suggested by McHenry and Wilhelm ^iven by DTJ* = l/2 u d 400.  p  (15)* then the eddy d i f f u s i v i t y Djf i s  f o r s u p e r f i c i a l Reynolds numbers from about 10 to  'This expression compares favorably with that suggested by Van Deemter  et a l ( D ' = % ud ). L  However, the constant given as l / 2 a c t u a l l y varied  with Reynolds number from l / l . 8 to l / 2 . 2 i n McHenry and Wilhelm's experimental work, and t h i s d r i f t would tend to cause errors i n determining the e f f e c t i v e d i f f u s i v i t y term i n equation (l.50) i f p e l l e t s with high d i f f u s i v i t y and small diameter were used. As mentioned e a r l i e r , the v e l o c i t i e s a t the bottom of the 'fable 1.1  show the l o c a t i o n of the minimum i n Figure 1.5*  a  ^d  correspond  to a constant p a r t i c l e Reynolds number of about 6 which i s w e l l above the flow range considered by Van Deemter et a l . Nevertheless, i t i s apparent that i f the e f f e c t i v e d i f f u s i v i t y term i s to be maximised r e l a t i v e to the other terms, higher flow rates and Reynolds numbers must be used. Least Square Error f i t of data to Van Deemter Equation Consider an equation of the type H = A + B/u + Cu  (I.51a)  Given an adequate number of experimental points r e l a t i n g H to u: f o r a given packing bed and gas system, the best values of the constants A, B and C can be determined by a "least squares error" f i t .  - 34 Three simultaneous u s u a l way, 1.  Sum  equations  i n v o l v i n g A,  L) and C can "bo o b t a i n e d  that i s ,  a l l the n a a t a  points:  \ + c£u = £ l  nA +  (i.5r°)  2.  M u l t i p l y by the c o e f f i c i t n t oi' 3:  5.  M u l t i p l y by t h e c o e f f i c i e n t o f  C:  + nB + c£u = £liu  A£U  (l.5Id)  2  E l i m i n a t i n g A and B from (1.51b), (l.51c), and - n X(Hu)  T(H)  C - S-^ 'I(u)  g  1  - nj££  \  /  \  (l.51d)  2§)  - n  / Z H  V^^-/  /2u  (l.ie) 5  - nl  \WW  J  m~ir Hence,  4tr)  u  n£(|) w  ,  N  2  v  (l.51f)  Z(*)  A = Il-i - L i u - D i g n 3.  i n the  (l.51g)  LONGITUDINAL DISPERSION COEFFICIENTS 'There a r e two models g e n e r a l l y used t o d e s c r i b e the  dispersion  i n packed beds o f non porous s o l i d s .  longitudinal  The d i s p e r s e d p l u g f l o w  model superimposes c o - o r d i n a t e s moving a t t h e average s t r e a m v e l o c i t y , u, on t h e a p p r o p r i a t e s o l u t i o n o f the d i f f u s i o n e q u a t i o n .  Thus, the v a r i a t i o n  o f a x i a l c o n c e n t r a t i o n p r o f i l e C w i t h time t and a x i a l d i s t a n c e x o f a q u a n t i t y M p e r u n i t a r e a o f gas w i t h d i f f u s i v i t y D , L  i n i t i a l l y on a p l a n e  -  a t ;c = 0 i s g i v e n D y (22)  CA =  M 2  exp /  */  -x  -  5 5  (1.52)  \  2  \2(2D t)y  tfD t L  L  w h i c h i s a G a u s s i a n d i s t r i b u t i o n w i t h mean 0 and v a r i a n c e 2 D. t. f  With plug flow at a v e l o c i t y u t h i s  CA »  M..  exp / - ( L - u t ) \  (1.53)  2  JHTSSZT w i t h mean L  becomes  [  2(2D t) L  J  and v a r i a n c e 2 D j t .  A second model i s t h e " p e r f e c t m i x e r s i n s e r i e s " w h i c h can be developed by a p p l y i n g the t h e o r e t i c a l p l a t e d e r i v a t i o n d e s c r i b e d  earlier,  a g a i n y i e l d i n g a P o i s s o n d i s t r i b u t i o n , w i t h a mean o f tL and v a r i a n c e U , where U i s t h e volume o f gas w h i c h has passed t h r o u g h , and V t h e  v  L L  volume o f each m i x e r .  The number o f p e r f e c t m i x e r s i n s e r i e s i s e q u i v a l e n t  t o t h e number o f s t a g e s and i s g i v e n by U / V ^ as b e f o r e . Equating the r a t i o  (mean) = number o f m i x e r s , N, f o r b o t h variance 2  models,  V) L  = N =  L  =  2  2 D t L  uL 2 D  I f one m i x e r i s assumed t o c o r r e s p o n d  (1.5*0 L  t o each l a y e r o f p a r t i c l e s ,  then  t h e number o f m i x e r s = L/dp, and t h e r e f o r e ,  D  L  = 1/2 u d  (1.55)  p  'The o n l y work, ( a p a r t from a few d a t a p o i n t s i n t h e l a m i n a r f l o w regime o b t a i n e d b y C a r b e r r y and B r e t t o n (23))  w h i c h has been c a r r i e d  out w i t h gases f o r t h e p u r p o s e o f i n v e s t i g a t i n g d i s p e r s i o n models i n packed beds has been done b y McHenry and W i l h e l m (15), r e s p o n s e technique.'  using a  frequency  They found t h a t o v e r a p a r t i c l e Reynolds number  i  - 36 (based on s u p e r f i c i a l v e l o c i t y ) range o f 10-400 t h e above r e l a t i o n s h i p held reasonably  well.  Other facto'rs w h i c h i n f l u e n c e t h e a x i a l d i s p e r s i o n c o e f f i c i e n t a r e buoyancy e f f e c t s w h i c h may be expected when f l o w r a t e s a p p r o a c h l a m i n a r c o n d i t i o n s , and w a l l e f f e c t s w h i c h I-Iiby (2^) has shown g r e a t l y increase t h e apparent d i s p e r s i o n c o e f f i c i e n t . Velocity Profile Taylor  Contribution (25) has s e p a r a t e d  the velocity p r o f i l e contribution to  the d i s p e r s i o n c o e f f i c i e n t i n pipe flow. regimes e x i s t i n p i p e f l o w .  T a y l o r found t h a t f o u r d i s p e r s i o n  The f i r s t i s due t o m o l e c u l a r  predominates a t low f l o w r a t e s .  d i f f u s i o n which  As t h e v e l o c i t y i n c r e a s e s t h e p a r a b o l i c  p r o f i l e contributes t o the l o n g i t u d i n a l d i s p e r s i o n o f a pulse, but t h e molecular profiles.  d i f f u s i v i t y i s a b l e t o l a r g e l y remove t h e r a d i a l  concentration  T h i s y i e l d s a n eddy d i f f u s i v i t y , Dv*  = K  u R a  (1.56)  2  where u i s t h e mean v e l o c i t y , D-g t h e m o l e c u l a r pipe radius.  d i f f u s i v i t y , and R i s t h e  I t may be noted t h a t h i g h m o l e c u l a r  reduce t h e eddy d i f f u s i v i t y i n t h i s r e g i o n .  d i f f u s i v i t y gases  'The c o n s t a n t K i s i j _ g f o r p i p e s ,  b u t A r i s (26) has shown t h a t t h e c o n s t a n t depends upon t h e geometry o f t h e system.  The range o f a p p l i c a t i o n o f t h e above regime i s d e s c r i b e d b y  Turner (27) a s , 7 DB R where L  1  «  u  <  k D L R  so D * = D ^ .  diffusivity contribution i s negligible  S i n c e a G a u s s i a n d i s t r i b u t i o n i s assumed we c a n s a y t h a t  95/> o f t h e p u l s e e x i s t s i n f o u r s t a n d a r d c  (1.57)  2  i s t h e l e n g t h o f t e s t s e c t i o n c o n t a i n i n g most o f t h e p u l s e .  W i t h i n t h e above l i m i t s t h e m o l e c u l a r L  x  B  deviations.  If L L  5 7  i s d e f i n e d as k<J t h e n , K J 2D t  =  L  Substituting for D  from (I.56) and s e t t i n g t = ^ , where L i s  L  t h e l e n g t h o f column ( o r mean), t h e upper l i m i t becomes,  2 16  u«  2  For pipes K i = u  «  K  L , D X  1  B 2  g i v i n g a n upper l i m i t o f  SB"  10 L D  "I  B  R " 2  Turner (27) i n h i s d e r i v a t i o n o b t a i n e d 5 LD/R f o r t h e R.H.S. o f (1.58), 2  a p p a r e n t l y because o f t h e o m i s s i o n o f a f a c t o r o f 2 i n d e f i n i n g L  .  The r e s i d e n c e time i s i n t r o d u c e d i f t h e v e l o c i t y i s r e p l a c e d by L t h e n , t  L  = u « 10 *fs_  t  R  ort »  2  _Rf_ 10 D  (I.58)  B  I n o t h e r words, a p u l s e must be a l l o w e d t o f l o w f o r some f i n i t e time a f t e r t h e i n j e c t i o n b e f o r e t h e eddy d i f f u s i v i t y i s d e f i n e d b y equation  (1.56).  I f t h e d i s p e r s i o n i s measured v e r y s h o r t l y a f t e r  i n j e c t i o n ( a t i m e l e s s t h a n R /l0 Dg), t h e n t h e eddy d i f f u s i v i t y i s g i v e n 2  by some u n d e f i n e d  function.  'This l a t t e r f u n c t i o n does n o t i n c l u d e t h e  m o l e c u l a r d i f f u s i v i t y , and so resembles t h e f u n c t i o n f o r t h e t u r b u l e n t regime. L o n g i t u d i n a l d i s p e r s i o n i n t h e t u r b u l e n t f l o w regime i n p i p e s has been d e a l t w i t h b y T a y l o r b y u s e o f t h e u n i v e r s a l v e l o c i t y  profile.  T h i s approach y i e l d e d  D * . = 7.1^ B u ^ f " L  where f i s t h e F a n n i n g f r i c t i o n  factor.  (1.59)  -  38 -  The a p p l i c a t i o n o f t h e T a y l o r d e r i v a t i o n t o packed beds has been somewhat l i m i t e d , a l t h o u g h B i s c h o f f and L e v e n s p e i l (28) the o v e r a l l p r o f i l e i n a packed bed.  have c o n s i d e r e d  Inasmuch as t h e v e l o c i t y p r o f i l e i n  packed beds approaches p l u g f l o w t h e c o n t r i b u t i o n o f t h e o v e r a l l p r o f i l e t o a x i a l dispersion i s small. Saffman (29)  has d e v e l o p e d a model based on a network o f  c a p i l l a r i e s o f l e n g t h twhich  a r e j o i n e d i n a random manner.  Assumptions  must be made r e g a r d i n g t h e l e n g t h and d i a m e t e r o f t h e c a p i l l a r i e s ,  Saffman  d e r i v e d t h e f o l l o w i n g e x p r e s s i o n t o c o v e r t h e t r a n s i t i o n from l a m i n a r t o an eddy regime and b y assumming a c a p i l l a r y l e n g t h t o d i a m e t e r r a t i o o f 5 t h e e x p e r i m e n t a l r e s u l t s o f H i b y (2k) f o r l i q u i d were f i t t e d * uL [ L o g 6~  L  e  2  5 D  uL 3  'The v a l u e o f t h e t o r t u o s i t y X 3 t f l k,  - 17 1  2  ,2  - 1 8  I  2  D  uL  + Dg  J  X  +  k_  D  3  + 0  (I.60)  9  o b t a i n e d b y Saffman from t h i s model approaches  c o n s i d e r a b l y h i g h e r t h a n t h e t o r t u o s i t i e s n o r m a l l y encountered  i n beds o f s p h e r e s . Saffman's model appears t o show t h e most p o t e n t i a l a t p r e s e n t i n d e s c r i b i n g t h e a x i a l m i x i n g i n packed b e d s , b u t as i n T a y l o r ' s work on p i p e s assumptions made c o n c e r n i n g t h e n a t u r e o f t h e f l o w l e a d t o d i f f e r e n t solutions.  Hence, t h e b a s i c f l o w mechanism must be u n d e r s t o o d b e f o r e one  c a n a p p l y t h e a p p r o p r i a t e s o l u t i o n from t h e model.  - 39 III APPARATUS A.  DEVELOPMENT The i n i t i a l work t o t e s t t h e c a l c u l a t i o n o f e f f e c t i v e  diffusivities  by means o f t h e Van Deemter e q u a t i o n was c a r r i e d o u t on an a p p a r a t u s based on a gas chromatograph as shown i n F i g u r e 1.6.  A cyclopropane  p u l s e was  i n j e c t e d b y a chromatograph sample v a l v e i n t o a h e l i u m c a r r i e r gas and was d e t e c t e d on a GOV/ MAC model 9238-D t h e r m a l c o n d u c t i v i t y c e l l .  The f l o w  r a t e was measured w i t h a soap b u b b l e f l o w meter a t t h e c e l l o u t l e t , and t h e d e t e c t o r o u t p u t was r e c o r d e d recorder.  on a Leeds and N o r t h r u p -1 t o 10 mv  The t e s t s e c t i o n was mounted i n a v e r t i c a l p l a n e ,  although  i n i t i a l l y a s e t o f r e s u l t s were t a k e n w i t h a h o r i z o n t a l b e d , b u t t h e s e t t l ing  o f t h e p a c k i n g r e s u l t e d i n a c h a n n e l a l o n g t h e t o p o f t h e b e d . The  f i r s t v e r t i c a l a p p a r a t u s s u f f e r e d from t h e f o l l o w i n g d e f e c t s : 1.  The d e t e c t o r would o n l y o p e r a t e w i t h i n a l i m i t e d gas f l o w range (about 50 m l s / m i n . ) .  2.  The s m a l l p o r t s i n t h e sample i n j e c t o r r e s t r i c t e d t h e f l o w o f g a s .  3.  No p r o v i s i o n e x i s t e d f o r a d j u s t i n g t h e r e c o r d e r c h a r t speed, ana a t the a v a i l a b l e c h a r t speed t h e p u l s e o u t p u t was n o t broad enough t o make a c c u r a t e measurements o f s t a n d a r d d e v i a t i o n s p o s s i b l e .  B.  DESCRIPTION OF APPARATUS The a p p a r a t u s w i t h w h i c h t h e b u l k o f t h e r e s u l t s were t a k e n i s  shown i n F i g u r e 1.7.  The s h o r t c o m i n g s o f t h e e a r l i e r a p p a r a t u s were  e l i m i n a t e d i n t h i s set-up by the f o l l o w i n g m o d i f i c a t i o n s :  -  1+0 -  PULSE INJECTOR C Y C L O P R O P A N E PULSE jGAS PACKED TEST C O L U M N HELIUM C A R R I E R GAS  THERMAL CONDUCTIVITY D E T E C T O R SOAP BUBBLE F L O W METER  M O O R E * F L O W C O N T R O L  Figure 1 . 6 Apparatus For Exploratory Tests  -  Ul  Figure  -  1.7  Basic Experimental Apparatus  - 1.-2 1.  A c a r t e s i a n manostat was f i t t e d on t h e column e x i i  to maintain a  s l i g h t p o s i t i v e p r e s s u r e i n t h e column ( l / 2 to 2 i n c h e s o f m e r c u r y ) . T h i s p r e s s u r e made i t p o s s i b l e t o d i r e c t a s i d e s t r e a m t h r o u g h a c a p i l l a r y . o supply the detector a t a f i x e d flow r a t e . J  At gas f l o w  r a t e s l e s s t h a n t h e amount needed f o r t h e d e t e c t o r t h e manostat s u p p l i e d a d d i t i o n a l g a s , thus r e v e r s i n g t h e f l o w d i r e c t i o n between t h e m a n i f o l d and t h e manostat. 2.  A sample o r p u l s e i n j e c t i o n v a l v e was c o n s t r u c t e d h a v i n g l a r g e p o r t s as shown i n F i g u r e 1.8.  T h i s F i g u r e a l s o shows a n e x p e r i m e n t a l  pulse  i n j e c t i o n system w h i c h was used t o t e s t t h e e f f e c t o f v a r y i n g p u l s e size. J.  A Bausch and Lomb 0- 10, 100, 1000 mv r e c o r d e r w i t h c h a r t speed a d j u s t m e n t s f r o m 0.05 t o 20 i n c h e s / m i n . a l l o w e d t h e p u l s e s t o be r e c o r d e d i n such a way t h a t good a c c u r a c y c o u l d be o b t a i n e d i n measuring t h e d i s p e r s i o n o f the p u l s e . The a p p a r a t u s  was s e t up w i t h t h e t e s t bed mounted i n a v e r t i c a l  p l a n e , and r e s t i n g on a m a n i f o l d b l o c k a t t h e d i s c h a r g e end. The s i d e s t r e a m f o r t h e d e t e c t o r was t a k e n from t h e manifold,, and t h e main column e f f l u e n t gas d i s c h a r g e d t h r o u g h t h e m a n o s t a t . t o a mercury manometer i n d i c a t e d t h e m a n i f o l d a b s o l u t e A i r c a r r i e r gas was t a k e n a t e i t h e r s u p p l y o r from a c y l i n d e r o f d r y a i r .  3O70  A p o r t connected pressure.  RE f r o m t h e b u i l d i n g  The a i r passed t h r o u g h a  r e g u l a t o r s e t f o r 22 p s i g . downstream p r e s s u r e , and t h e n t o a f l o w meter c o n s i s t i n g o f a s p . g r . I o i l manometer and c a p i l l a r y t u b e . s e r i e s o f c a p i l l a r y tubes were c a l i b r a t e d u s i n g soap b u b b l e f l o w meters o r a wet t e s t gas m e t e r , so t h a t a wide range o f f l o w s c o u l d  A  - 43 -  /  -IrHrl-  I" POLYETHYLENE BLOCK  ELLINCER  I'D. SILVER STEEL ROD WITH SLOTS FOR jfe O 'R ' INGS. _S I'D. SAMPLE HOLES  PULSE INJECTOR CARRIER GAS { i'NPT 'TEE'^J SPRING LOADED POP VALVE TEST COLUMN  -V»||5v. 2 » ' PUSH BUTTON —  PULSE GAS SOLENOID VALVE  EXPERIMENTAL PULSE INJECTOR FOR VARYING PULSE SIZE  Figure 1.8 Pulse Injectors  - kk _be c o v e r e d .  No a t t e m p t was made t o s i z e the c a p i l l a r i e s so that t h e y  remained i n t h e i r l i n e a r r a n g e . passed through a Moore Constant  From the flow meter, t h e c a r r i e r gas D i f f e r e n t i a l gas flow c o n t r o l and  by-pass l o o p t o t h e p u l s e i n j e c t o r .  The p u l s e i n j e c t o r was mounted  on a v e r t i c a l r a i l so t h a t i t c o u l d be a d j u s t e d o v e r a s i x foot range to  a l l o w f o r v a r y i n g column l e n g t h s .  P o l y e t h y l e n e t u b i n g was used t o  s u p p l y t h e c a r r i e r gas, as w e l l as t h e p u l s e gas t o t h e i n j e c t o r .  The  f l e x i b i l i t y o f t h e p o l y e t h y l e n e t u b i n g a l l o w e d t h e i n j e c t o r t o be a d j u s t e d anywhere on t h e r a i l w i t h o u t t h e neea o f p i p i n g a l t e r a t i o n s . A m i c r o s w i t c h mounted on t h e i n j e c t o r was e i t h e r opened o r c l o s e d a t each movement o f the i n j e c t o r s .  This a c t i o n o p e r a t e d a n  event marker on t h e r e c o r d e r t o i n d i c a t e t h e s t a r t o f each r u n . C.  DETECTORS  Hydrogen Flame I o n i s a t i o n D e t e c t o r In  equation  (1.49), i t was e v i d e n t t h a t l o v e r e f f e c t i v e  i n c r e a s e d t h e magnitude o f t h e C t e r m .  diffusivities  To t a k e advantage o f t h i s , t h e  hydrogen f l a m e i o n i s a t i o n d e t e c t o r was s e l e c t e d , as i t a l l o w e d t h e use o f a i r and h y d r o c a r b o n s o f any c o n v e n i e n t m o l e c u l a r w e i g h t , as opposed t o t h e need f o r hydrogen o r h e l i u m  (which have h i g h d i f f u s i v i t i e s ) as one o f t h e  gases i n t h e r m a l c o n d u c t i v i t y d e t e c t o r s i f h i g h p r e c i s i o n i s d e s i r e d .  In  a d d i t i o n , t h e hydrogen flame d e t e c t o r i s l i n e a r o v e r a 5 decade r a n g e , and the h i g h s e n s i t i v i t y a l l o w s t h e u s e o f e x t r e m e l y s m a l l p u l s e volumes. The d e t e c t o r was c o n s t r u c t e d from t h e c i r c u i t d e s c r i b e d by H a r l e y , N e l s , and P r e t o r i u s (30) and i s shown i n F i g u r e 1.9.  A power s u p p l y was  a l s o c o n s t r u c t e d t o s u p p l y t h e d e t e c t o r , however, t h e AC f i l a m e n t s u p p l y was  found to c r e a t e e x c e s s i v e n o i s e i n t h e o u t p u t so t h e d e t e c t o r tube was  powered from a 6 v o l t  accumulator.  - 4  5  Figure  -  1.9  Hydrogen Flame D e t e c t o r  -  1,6  -  Wo o u t p u t c o u l d be o b t a i n e d i n i t i a l l y from t h e c i r c u i t as d e s c r i b e d , ana on i n v e s t i g a t i o n t h e v o l t a g e s on t h e 6 SN 7 tube were found t o be o u t s i d e t h e range i n w h i c h a response c o u l d be e x p e c t e d .  To c o r r e c t  t h i s problem i t was n e c e s s a r y t o change t h e two l o a d r e s i s t o r s from 10 Ki"L t o 100 KSL  .  I t i s c o n c l u d e d "chat a m i s p r i n t has o c c u r r e d i n t h e o r i g i n a l The a c t u a l i o n i s a t i o n o r combustion cheuribar was  publication.  constructed  t o minimize the holdup time o f the primary a i r c o n t a i n i n g the t r a c e s o f p u l s e gas from t h e m a n i f o l d .  The a i r - e n t e r e d t h r o u g h t h e a n n u l a r space i n  t h e g l a s s tubes and j o i n e d w i t h t h e hydrogen b e f o r e p a s s i n g t h r o u g h t h e s t a i n l e s s s t e e l o r i f i c e w h i c h formed one e l e c t r o d e .  Hydrogen was  supplier'  from a c y l i n d e r v i a a Moore f l o w c o n t r o l l e r and a r o t a m e t e r a t a r a t e o f about 150 mis/min.  Lower f l o w r a t e s i n c r e a s e d t h e d e t e c t o r o u t p u t b u t i n  t h e extreme, t h e flame became u n s t a b l e . t h e c a p i l l a r y a t t h e r a t e o f about 0.7  A i r and p u l s e gas a r r i v e d t h r o u g h mis/sec.  The volume o f t h e d e t e c t o r  a i r s i d e and s u p p l y t u b e s from t h e m a n i f o l d was e s t i m a t e d a t 0.2 a t i m e l a g o f about 0.3  mis, g i v i n g  seconds.  I n i t i a l l y , t h e flame o r i f i c e was made f l u s h w i t h t h e m e t a l e l e c t r o d e , b u t t h e h e a t from t h e flame caused t h e g l a s s w a r e t o c r a c k and so t h e o r i f i c e was m o d i f i e d by a d d i n g about 1 l / 2 " o f l / 8 i n c h s t a i n l e s s s t e e l t u b e . ' The whole assembly was h e l d on a r u b b e r bung, t h u s s u p p l y i n g the i n s u l a t i o n f o r t h e p l a t i n u m e l e c t r o d e w h i c h was s u p p o r t e d by a heavy wire i n s e r t e d i n the rubber.  S h i e l d e d c a b l e connected t h e d e t e c t o r t o  t h e e l e c t r i c a l system, and a grounded copper chimney s h i e l d e d t h e flame from d r a u g h t s . Other m o d i f i c a t i o n s t o t h e r e f e r e n c e c i r c u i t F i g u r e 1.9)  ( a l s o shown i n  i n c l u d e d a f o u r t h p o s i t i o n on t h e s e l e c t o r s w i t c h w i t h a  - ii7 10 meg r e s i s t a n c e t® ground, and a coarse and f i n e z e r o s o t t i n g u s i n g 20 K and  50 K v a r i a b l e r e s i s t o r s i n p a r a l l e l .  The 10 megohm p o s i t i o n was used  on a l l r u n s . F i b r e g l a s s f i l t e r s were f i t t e d i n t h e hydrogen tune and t h e m a n i f o l d t o reduce the n o i s e i n t h e d e t e c t o r caused by d u s t .  The d e t e c t o r  s t i l l gave o c c a s i o n a l c h a r a c t e r i s t i c jumps i n o u t p u t , p r o b a b l y caused by d u s t i n t h e secondary a i r , b u t no a t t e m p t was made t o c o r r e c t t h i s . Thermal C o n d u c t i v i t y D e t e c t o r A "Gow mec" model 9238D t u n g s t e n w i r e t h e r m a l c o n d u c t i v i t y d e t e c t o r was used w i t h t h e recommended c o n v e n t i o n a l a u x i l i a r y  circuits.  A 6v. b a t t e r y s u p p l i e d t h e c u r r e n t f o r t h e d e t e c t o r and t h e event marker on t h e r e c o r d e r .  'The o u t p u t t o t h e r e c o r d e r was f i t t e d t o an a t t e n u a t o r  h a v i n g 1, 2, 5, 10, 50, 100 and 500 r a t i o s , b u t o n l y t h e 1, 2.and 5 p o s i t i o n s were needed i n t h e p u l s e  apparatus.  'The r e f e r e n c e s i d e o f t h e d e t e c t o r was s u p p l i e d through a n e e d l e v a l v e from t h e 22 p s i g a i r l i n e , and a s m a l l b l e e d  maintained.  The c a p i l l a r y s u p p l y i n g t h e measuring s i d e o f t h e d e t e c t o r from t h e m a n i f o l d was s i z e d t o g i v e a p p r o x i m a t e l y gauge m a n i f o l d  pressure.  46 mls/min. o f a i r a t l / 2 " Hg  -  1*8 -  IV EXPERIMENTAL PROCEDURE A.  OUTLINE OF EXPERIMENTAL INVESTIGATION The experimental work was carried out i n three parts to (a)  test  the a p p l i c a b i l i t y of Van Deemter's equation with large p e l l e t diameters and higher flow r a t e s ; (b) measure the e f f e c t i v e d i f f u s i v i t y i n some samples o f porous p e l l e t s using the pulse method, and (c)  compare the  e f f e c t i v e d i f f u s i v i t y , obtained with the pulse experiment to those obtained by an independent method.  Part (a) was c a r r i e d out by i n j e c t i n g methane  and hydrogen pulses i n beds containing non-porous p e l l e t s , while (b) was an obvious extension of (a)  to porous p e l l e t s .  The w e l l - t e s t e d steady state  method was selected to obtain an independent e f f e c t i v e d i f f u s i v i t y value. I t was convenient, however, to develop a s p e c i f i c s o l u t i o n of the d i f f u s i o n equation to f i t p e l l e t s with curved faces.  The d e t a i l s of t h i s section  o f the work are recorded i n Appendix 1. B.  NON POROUS PELLETS IN PULSE APPARATUS A simple gas chromatograph assembly was used f o r some e a r l y  exploratory runs with a cyclopropane pulse i n an helium or a i r gas flowing through beds of 2 mm. glass spheres.  carrier  These r e s u l t s were  discarded due to l i m i t a t i o n s of the apparatus, which included a l i m i t e d supply of the 2 mm. glass beads necessitating short beds, as w e l l as the defects already l i s t e d . With the development of the more sophisticated apparatus,  a  series of runs using methane pulses i n a i r was carried out with various bed diameters and lengths packed with three kinds of non porous p e l l e t s : 0.208 cm. No. 9 lead shot, O.568 cm. glass beads, and 1 cm. diameter ceramic beads.  - k Because the value of the quantity " C " i n equation (l.50), HETP = A + B + Cu 9  u was not found to he zero i n the exploratory work with non porous p e l l e t s , runs 5 0 to 5 5 were designed to determine the magnitude of t h i s term, and to investigate ways o f minimizing i t .  To check the p o s s i b i l i t y that this  e f f e c t was caused by a high v e l o c i t y "by-pass" flow at the w a l l , run 5 0 was made with a  5  cm. diameter column packed with the  and having a maximum p a r t i c l e Reynolds number of 2 .  0 . 2 0 8  Run 5 0  cm. lead shot, w a s  different  from the other runs i n that a higher pressure was used, giving a lower diffusivity.  Run 5 1  w a s  made with a 2 . 5 cm. column packed with the lead  shot to see i f p a r t i c l e to tube diameter r a t i o had much influence on the wall e f f e c t .  Run 5 2 duplicated run 5 0 ,  range, and normal column pressure.  but used a higher Reynolds number  A run designated 51D  w a s  also made on  the 2 . 5 cm. bed, but f i v e doughnut rings were d i s t r i b u t e d evenly down the column i n an attempt to eliminate the w a l l e f f e c t . Run 5 3 was made with a 6 . 2 7 cm. diameter column packed v i t h the 1 cm. ceramic spheres.  The experimental sample i n j e c t i o n system using a  solenoid valve, which allowed varying pulse s i z e s , was introduced i n t h i s column.  Run  was made on a l / V polyethylene tube packed with 3 mm.  glass spheres, and run 5 5  w a s  made with a 1 . 2 cm. diameter bed packed with  the 1 cm. b a l l s to see i f a t u b e / p e l l e t diameter r a t i o < 2 could eliminate the w a l l e f f e c t . bed.  This l a t t e r case has been designated as a "single p e l l e t "  In a l l the foregoing runs, the t e s t system used was a methane pulse  i n a i r as a c a r r i e r  gas.  Following these t e s t s , runs 5 6 to 6 2 with porous p e l l e t s were carried out.  One of the porous p e l l e t s , an activated alumina, gave  - 50 abnormally low values for the e f f e c t i v e d i f f u s i v i t y .  Further i n v e s t i g a t i o n ,  which i s summarized i n Appendix IV, showed that methane was adsorbed to a s i g n i f i c a n t degree on the activated alumina.  The need for a non-adsorbing  system resulted i n further runs using the non porous p e l l e t s being carried out with a hydrogen pulse, as well as the methane pulse, i n a i r system. Runs 63 to 66 were carried out with O.568 cm. glass spheres, both gas systems and two column diameters, including one for a s i n g l e p e l l e t diameter.  Runs 69 to 72 were a s i m i l a r set of results with the 1 cm.  diameter spheres, two column diameters and two gas systems.  Runs 69 and 7 2  using methane are duplicates of runs 53 and 55* hut covered a wider range of Reynolds number.  Table l . I I summarizes the values o f the variables  pertaining to each run number.  TABLE l . I I SUMMARY OF THE PELLET AND TUBE TO PELLET DIAMETER RATIOS COVERED BY THE EXPERIMENTAL RUNS >  Pulse Gas Methane  N Tube/Pellet Ratio s  Pellet X. Diameter .208 cm. .568 cm. 1 . 0 cm.  Hydrogen  1  . 5 6 7 cm. 1 . 0 cm.  3 5  6k  k*  65  72 55 63  6  69 53 66  71  * p e l l e t diam. 0 . 2 9 cm.  70  12  25  51 51D  50 <52  - 51 0.  POROUS PELLETS IN PULSE APPARATUS Three samples of porous spherical p e l l e t s were acquired for  testing.  These included l/8" and l / U " diameter KL51 Alcoa activated  alumina p e l l e t s , and l/2" diameter Norton Alundum catalyst The physical c h a r a c t e r i s t i c s III.  supports.  of these p e l l e t s are summarized i n Appendix  One of the d i f f i c u l t i e s experienced i n setting experimental  conditions was that the activated alumina test p e l l e t s could not Le adequately dried i n the steady state apparatus, as the epoxy r e s i n holding the sample could not stand the necessary drying temperature.  In view of  t h i s problem, the pulse i n v e s t i g a t i o n was attempted on the "wet" p e l l e t s , because i t was found that the moisture content of the p e l l e t s which had been open to the atmosphere was quite stable even though the atmospheric humidity varied from 30$ RH to 100$ RH.  The only problem remaining  concerned the true porosity of the wet p e l l e t s , but the manufacturer's literature  (31) indicated that the water existed as l i q u i d water, and  hence could be assumed to have a density of 1.  Thus, the porosity could  be computed from the dry p e l l e t porosity and the moisture content.  The  d e t a i l s of these calculations and other confirming experiments with respect to the p o r o s i t i e s of the p e l l e t s are included i n Appendix I I I . The pulse technique was f i r s t applied using a methane pulse, i n run 56, to the l / V diameter H 1 5 1 activated alumna p e l l e t s i n a four foot long single p e l l e t diameter bed. with a i r at room temperature.  The p e l l e t s were i n equilibrium  Unexpectedly high d i s p e r s i o n of the pulse  (HETP) caused some doubt about the number of transfer u n i t s , so the bed was lengthened f o r run 57 hy adding two bends and two further four foot lengths to create a trombone configuration.  A 20$ change i n C was found  - 52 between t h e l o n g bed and t h e s h o r t .  As shown l a t e r i n th« " R e s u l t s " , the  s h o r t column was found to c o n t a i n i n s u f f i c i e n t t r a n s f e r units f o r a G a u s s i a n distribution.  In run ^Q, xhe same bed as t h a t used i n r u n 57 was employed  but the p e l l e t s were previously d r i e d .  At t h i s stage, the p o s s i b i l i t y of  surface adsorption of methane by the alumina was appreciated, and run 59 was conducted at higher flow rates i n the hope that the adsorption was a slow process and would not occur to a s i g n i f i c a n t extent under t h e s e conditions. In run 6 0 , a methane pulse was used i n a single p e l l e t  diameter  trombone bed, which was packed with l / 2 " diameter Norton c a t a l y s t c a r r i e r pellets.  In run 6 l ,  the use of a hydrogen pulse was tested on t h e same  dry l / 4 " diameter HI51 activated alumina p e l l e t s from runs 58 and 59, while i n run 6 2 the same bed was wetted back to the normal moisture content, and the hydrogen pulse applied again. Run 73 was carried out on a four foot long by 3/4" diameter bed packed with l / 8 " HL51 activated alumina p e l l e t s and using a hydrogen pulse. D.  INDEPENDENT EFFECTIVE DIFFUSIVITY MEASUREMENT A conventional steady state method was selected for a second  determination of e f f e c t i v e d i f f u s i v i t y , but the technique was adapted for use with spherical p e l l e t s .  This modification consisted of mount-  ing the p e l l e t s with epoxy r e s i n i n a hole i n a plate about O.75 p e l l e t diameter i n thickness.  The two spherical caps on each side of the p l a t e  were ground o f f when the r e s i n had d r i e d .  The s o l u t i o n for t h e d i f f e r e n t i a l  d i f f u s i o n equation with t h i s geometry i s included i n Appendix I, along w i t h the results and d e t a i l s of t h i s experiment.  Only the l / 4 " and l / 2 " p e l l e t s  - 53 were tested.  On the basis of the manufacturer's data Knudsen d i f f u s i o n  was expected i n the l/h" alumina p e l l e t s , while molecular d i f f u s i o n was expected i n the l / 2 " Norton p e l l e t s . The major problem with t h i s part of the i n v e s t i g a t i o n was the moisture content of the p e l l e t s .  The activated alumina could only be  dried i n s i t u , hut the epoxy r e s i n would not survive the drying temperature. Since the moisture content of the "wet" p e l l e t s remained r e l a t i v e l y constant, as mentioned previously, i t was decided to t e s t the p e l l e t s wet and correct the porosity accordingly. E.  PREPARATION OF THE TEST COLUMNS The packed beds (columns) were constructed  rubber bungs or tubing i n the ends.  from glass tubing with  The dimensions of the beds were  generally obtained with a metric rule except f o r small diameter where a c a l i p e r rule was used.  The bed porosities were obtained  tubes, either  by weighing the beds f u l l and empty i f the p e l l e t density was known, or by addition of water and weighing.  For the single p e l l e t diameter beds,  the porosity was calculated by counting the number of p e l l e t s i n a given length of bed, and c a l c u l a t i n g the p e l l e t volume from the mean p e l l e t diameter. The mean p e l l e t diameter was measured by placing a known number o f p e l l e t s i n l i n e and measuring the o v e r a l l length. For the porous p e l l e t beds, the porosity was calculated as for the single p e l l e t beds above, or from the weight of p e l l e t s i n the bed with the c h a r a c t e r i s t i c data of the p e l l e t .  A l l the columns were then  mounted i n a v e r t i c a l plane. •Joints i n trombone columns were made with rubber tubing.  - 5* F. 1.  OPERATION OF PULSE APPARATUS One of the four calibrated flow meter c a p i l l a r i e s was selected and fitted.  2.  The column was assembled, porosity c a l c u l a t i o n s ) ,  3.  (after taking the necessary data for the  and f i t t e d to the  apparatus.  The a i r supply was turned on with the flow c a p i l l a r y bypass open, and the column pressure was set at a convenient l e v e l (usually around 0.5" Hg), using the cartesian manostat.  h.  'The column was tested for leaks with soap s o l u t i o n .  5.  The appropriate detector was started up as described below.  6.  The appropriate pulse gas was set to flow at a low bleed rate using the cylinder regulator and valves.  The gas was bubbled i n water at  the e x i t to estimate the flow. 7.  A suitable a i r flow rate was passed through the column using the flow meter and c o n t r o l .  8.  The flow meter manometer reading was recorded.  The recorder chart was started at any speed (unless previous experiments suggested a s p e c i f i c chart speed), and a pulse i n j e c t e d . When the pulse was produced, the height of the pulse was adjusted on the attenuators (recorder attenuator for H2 flow or attenuator box for thermal conductivity detector) and the width noted.  Using  the i n i t i a l pulse, the equipment was adjusted to give a convenient peak height (e.g.  0.75 s c a l e ) , and a pulse width on the chart of  at l e a s t 1.5 cm. 9.  A series of pulses were i n j e c t e d , each at a d i f f e r e n t gas flow r a t e , to give about ten results covering the flow range d e s i r e d .  10. During the course of each run the room temperature, pressure and column pressure were recorded.  atmospheric  - 55 Hydrogen Flame Detector 1.  The detector was connected to the manifold with the correct c a p i l l a r y .  2.  Hydrogen flow was started at around 150 mls/min (using rotameter) and the flame was i g n i t e d .  J.  The power supply was turned on and- connected to a 6v battery for filament and event marker.  k.  The recorder was turned on and the zero of the recorder and detector adjusted.  The selector  switch on the detector a m p l i f i e r was always  set at the No. 4 p o s i t i o n f o r a l l runs. Thermal Conductivity Detector 1.  The manifold was connected with the correct c a p i l l a r y .  2.  The reference a i r bleed was turned on and adjusted to give a slow p o s i t i v e flow (e.g. by bubbling i n water).  3.  The filament current was adjusted to 100 ma. a f t e r connecting to 6v supply along with event marker leads.  h.  The recorder was set to zero and the detector to zero s i g n a l .  - 56 v  RESULTS A.  NON POROUS PELLETS  Treatment of Data f o r Non Porous Pellets For each pulse input the primary data consisted o f :  the Clow  rate of c a r r i e r gas, Q mis/sec, at 3'J?P, which i s a c t u a l l y recorded as a manometer reading and transformed using the c a l i b r a t i o n charts i n Appendix II to a f l o v r a t e , the width of the pulse at h a l f the height (WIDTH) taken from the recorder chart and also the "mean" or distance from the pulse i n j e c t i o n to the peak of the pulse (designated TOTAL).  These data  points are printed (in cm. units) i n columns 8 , 6 and 7 respectively of the tables of results i n Appendix I I . In addition to the above raw data, each table i n Appendix II i s headed with d e t a i l s of the columns pertinent to the i n d i v i d u a l run. These include a "Run number" which starts at 50 for the sophisticated apparatus, but a run (No. l ) from the preliminary r e s u l t s obtained on the i n i t i a l simple apparatus i s included.  The column length (L),  diameter (d^) and porosity (Eg) are included i n the heading along with p e l l e t porosity (Ep) and diameter (dp), and the c a r r i e r gas ( T ° K ) , molecular weight and pressure (P).  temperature  The pulse g a s - c a r r i e r  diffusivity  i s also printed f o r the run temperature and pressure. The values f o r the molecular d i f f u s i v i t y of the p u l s e - c a r r i e r gas systems are taken from the following sources: The d i f f u s i v i t y of hydrogen i n a i r was taken from the experimental r e s u l t s of Currie ( 6 ) .  Currie found a temperature dependence of  - 57 d i f f u s i v i t y t o t h e 1.715  power f o r this system, and t h i s was used t o  interpolate from the experimental results a d i f f u s i v i t y of 0.755 em /sec 2  a t 298°K and 1 atmosphere. The d i f f u s i v i t y of methane i n a i r was calculated from t h e Hirschfelder equation using the force constants tabulated i n B i r d , and  Lightfoot (52).  Stewart  The computation, which i s shown i n Appendix I I I ,  yielded a d i f f u s i v i t y f o r methane i n a i r of 0.212  cm /sec at 298°^ and 1 2  atmosphere. values corresponding to the table headings were fed d i r e c t l y to the computer except for the pulse gas-carrier  gas d i f f u s i v i t i e s which were  modified to the run temperature and pressure assuming an inverse pressure dependence and a temperature dependence to the 1.7 power. Provision was included to read i n the c a r r i e r gas v i s c o s i t y , but i n the computations shown i n Appendix II the v i s c o s i t y value read i n has been over-ruled i n the program by a v i s c o s i t y for a i r computed from the Sutherland equation (53)t  [  J\  = 0.01709 f 275 + 114) 114 1 ( / T  1-114  T \\ ^ 273 j 2  (l.6l)  3  The c a r r i e r gas density was calculated assuming the perfect gas law P  = M o l . Wt. 22400  2J3_ T  x  P  (1.62)  F i n a l l y , the hydraulic diameter was calculated from the following equation, h  D  = 4  Free Volume Wetted Area  =  T B / 3_ <*T ( l - € ) I 2 dp d  €  B  + 1 \  (I.63)  I  As mentioned e a r l i e r the primary data of flow rate at STP, WIDTH (= 2.360) and TOTAL (mean) are given i n columns 8, 6 and 7,  respectively,  i n Appendix I I .  - 58 l' -...' ' , ' I K . 1  -LU-T,  1  ane. .V >.  I.i  -  the  heading of each table, the C o l l o ^ i . i g calc i •.let.-o L E K are p;-_* • . In column 1 the intere ..j. ij.nl v e l o c i t y was calculeu. • -rom tube , .lameter d-jj, and the flow rate 4, correct, a f o r temperature T an. pressure :  ?.  1 f _„  u = Q J L  273  1  p [nv *J L  1_  E  (1.6*0  B  'The HETP was calculated as defined by equation (1.26) HETP = Lcr mean  =  L  I" WIDTH*] [2. ;6 J 7  2  2  f 1 ] [ TOTALJ  (I.65)  2  'Thi-ee Reynolds numbers were calculated f o r comparing the a x i a l dispersion data with data of other workers and are defined as follows: the p a r t i c l e Reynolds number shown i n column 4 of the table of results Appendix II i s given by u d p p  in  , the s u p e r i f i c i a l Reynolds number shown  y i n column 11, u€g d^p  , and the hydraulic Reynolds number based on the  r hydraulic diameter, u h ^ P / ^ , i n column 13. The dispersion c o e f f i c i e n t D^, was obtained from equation (1.34), which f o r non porous p e l l e t s reduces t o , HETP = 2 D u  (1.66)  L  so D = HETP u L  2  and t h i s value i s printed i n column 9 under the heading of "eddy diffusivity".  In f a c t , i t i s the sum of the molecular and eddy  d i f f u s i v i t i e s as given by equation (l.33). The number of transfer units (NTU), defined by uL , must be large 2D  L  for equation (1.30) to be s a t i s f i e d , however, the values calculated and  - 59 -  recorded i n column 5 are based on the molecular d i f f u s i v i t y rather than the dispersion c o e f f i c i e n t D^.  Inspection of the term shows that the KTU  i s smallest at low v e l o c i t i e s , and since low v e l o c i t i e s imply the existence of the molecular d i f f u s i v i t y regime, the NTU's based on these d i f f u s i v i t i e s are an adequate t e s t .  The use of "long" beds has generally eliminated the  NTU as a l i m i t i n g c r i t e r i o n i n this work. To make possible comparisons between the eddy d i f f u s i v i t y computed from t h i s work and the correlations and r e s u l t s of other workers, the Peclet and Schmidt numbers were also c a l c u l a t e d . The molecular and so-called "eddy" Peclet numbers are recorded i n columns 3 and 10, r e s p e c t i v e l y , and were computed from the following definitions, Molecular Peclet number  u dp  Eddy Peclet number  u dp  This eddy Peclet number should probably be c a l l e d the dispersion Peclet number, however, because the eddy d i f f u s i v i t y D x * has not been separated from the dispersion c o e f f i c i e n t D L i n this work the eddy Peclet or dispersion Peclet are  interchangeable.  The Schmidt number based on the dispersion c o e f f i c i e n t i s recorded as the inverse Schmidt number i n column 12, that  is,  Schmidt At the base of each table the l e a s t square error f i t of the HETP v s . u data to equation  (1.50) i s  computed and the best values of the  constants A, B and C are printed out.  The span of c e r t a i n runs was  r e s t r i c t e d to the eddy d i f f u s i o n regime, and the scatter of the data points could cause anomalous values of the B, or molecular d i f f u s i o n , term, which  - 60 was a r e l a t i v e l y small quantity i n this range.  To o f f s e t this problem a  second least squares computation i s carried out on the data to f i t the equation, HETP where H E T P ' =  =  HETP - B  AA + C'Cu  (1.67)  .  u Tne value of B i s set at 2 x .75 x Dj}, where 0.75 represents the inverse of the t o r t u o s i t y l/X  i n equation (1.49).  From the value of B derived from the three constant equation (1.50), the inverse of the t o r t u o s i t y has been calculated for each run. Since ^ varies from 1 to eo as discussed i n the i n t r o d u c t i o n , then the inverse ranges from 1 to 0. about O.67 to 0.8. is  The usual value expected i n a packed bed i s  The r e s u l t printed on the computer sheet (Appendix III)  i n the nomenclature originated by Van Deemter (l7)and so the inverse  t o r t u o s i t y computed from equation 1.49 as 1  X  =  2 D3  j . given under the s  3  heading GAMMA. S i m i l a r l y , the value of the constant c h a r a c t e r i s t i c d i f f u s i v i t y which has been designated  of the eddy  i s computed from the value of the  eddy d i f f u s i o n term A using equation (1.4-9), that i s Jf= A/2dp. Van Deemter et a l (17) suggest that  varies from about 8 f o r 200 mesh  p a r t i c l e s to about zero, f o r , say, l/8 inch p a r t i c l e s . printed the values of  The computer has  under the heading LAMDA (from Van Deemter et  al.)  i n Appendix I I I . Results for Beds of Non Porous Pellets Some t y p i c a l curves of the HETP v s . v e l o c i t y are shown i n Figure 1.10 and 1.11.  Figure 1.10 shows the results for run 52 which  covered both the molecular and eddy d i f f u s i o n regimes while Figure 1.11 shows the results f o r runs 51, 69 and 70.  The f i v e straight l i n e s shown  0  5 . 10 15 20 25 ' INTERSTITIAL VELOCITY (u) CM/SEC. Figure 1.11 HE P Vs. Veloci+y For Runs 51, 69 and 70. P  - 63 on the plots represent the equation HETP = A + Cu, using values of A and C determined from applying equation  (1.50) to the data.  Runs  69 and 70  were made i n a bed with a tube to p a r t i c l e diameter r a t i o of 6 containing the 1 cm. spheres, but a methane pulse was used i n run 69 and a hydrogen pulse i n run 70. intercept,  It may be noted that both sets of data have the same  i n d i c a t i n g that Van Deemter's d e f i n i t i o n of the eddy d i f f u s i v i t y  given by D i * = % u dp i s v a l i d , but a further mechanism which depends on the gas d i f f u s i v i t y and has a v e l o c i t y exponent of 2 must be added to account f o r the presence of a " C " term. Van Deemter et a l (17) suggested that the eddy d i f f u s i v i t y term i n equation  (1.49), A = 2 J*dp, decreased with increase i n p e l l e t diameter,  due to the decrease of the c o e f f i c i e n t # .  In Figure 1.12 i t may be noted  that with the larger p e l l e t s and generally higher flow rates i n t h i s work the trend has been reversed, and " A " increases with p e l l e t diameter.  If  a straight l i n e i s put through the points i n Figure 1.12 a slope around unity i s obtained, making }f = l/2 corresponding to the value obtained by McHenry and Wilhelm (15) with gases at Reynolds numbers greater than 10. Of the early runs, only run 1, which was c a r r i e d out on a 134 cm. bed with a cyclopropane pulse i n an a i r c a r r i e r gas stream i s included i n the data.  The results of t h i s run together with a d d i t i o n a l results from  i n i t i a l tests with methane pulses i n beds o f non porous p e l l e t s to 55) are summarized i n Table l . I I I .  (runs 50  The most s i g n i f i c a n t feature of  these results i s that over a range o f p e l l e t diameters from 0.2 cm. to 1 cm., with tube to p e l l e t diameter r a t i o s from 1 to 25, the wall  effect  or "C7 term, which might mask the dispersion e f f e c t due to p e l l e t p o r o s i t y , gave r e s u l t s which varied i n value only from 0.04 to 0.08.  - 6k -  In Runs 50 and 52 the C term calculated from tests i n the lower Reynolds number range (run 50) i s considerably higher than the value obtained i n the same column at higher Reynolds numbers (run 52).  This  suggests that either the wall effect term i s not constant or that the exponent of the v e l o c i t y i n the dispersion c o e f f i c i e n t i s less than 2. The plots of the dispersion c o e f f i c i e n t v s . u given i n Figure 1.13 would appear to substantiate the l a t t e r view. Comparisons of Runs 51 and 51D demonstrate that a r t i f i c i a l mixing devices or wall barriers do not reduce the wall dispersion e f f e c t . No data i n the regime i n which molecular d i f f u s i v i t y i s important were taken i n run 5XD, so that the comparison i s best made using the CC value from run 51D, which was calculated using equation (1.65) as described previously.  The value of B found from the results of run 51D represents  a molecular d i f f u s i v i t y more than double the normal gas d i f f u s i v i t y (GAMMA = 2.07), demonstrating the f a i l u r e of equation (I.50) when results i n the eddy regime only are used i n the l e a s t squares evaluation of the three constants A, B and C.  Values obtained i n run ^k also demonstrate  this point. The large diameter p e l l e t s i n runs 53 and 55 show a large intercept, or A term, compared to the other runs which show e s s e n t i a l l y zero intercept.  Inasmuch as A i s approximately proportional to d^, this  difference i s to be expected.  It i s rather interesting that a bed with  a single p e l l e t diameter (run 55) has e s s e n t i a l l y the same or less slope (i.e.  C value) at high Reynolds numbers as the bed s i x p a r t i c l e s i n  diameter of run 53.  T h i s , as well as other results given i n Table l . I I I ,  indicate that the dispersion due to the wall e f f e c t i s not a function of tube diameters.  TABLE l . I I I DISPERSION RESULTS WITH BEDS OF NON POROUS PELLETS Column to Pellet Diameter Ratio  Run  Pellet Diameter  Column Length  Column Diameter  1  0.22  134.6  2.6l  11.9  50  0.208  111.8  5.0  24  51  0.208  118.1  2.6  12.5  51 D  0.2C8  118.1  2.6  12.5  52  0.208  111.8  5.0  24  53  1.03  186.3  6.27  4  0.297  185.4  55  1.005  121.0  5  A  B  C  . AA  CC  Range of Reynolds Number  -0.150  C 5 - 2.4  0.13  0.18  O.07  -0.07  0.35  O.07  ^0.28  0.31  O.052  0.04  0.053  0.29 - 31.3  0.87  0.071  -O.O37  0.06h  2.6  - 32.6  0.001  0.37  0.041  0.032  O.O37  C8  - 33.0  .6.1  0.68  O.36  0.071  0.72  0.064  5.0  - u-.o  O.415  1.4  -0.22  3.88  0.079  0.177  O.069  16.0 - 79.0  1.15  11.1  0.16  0.060  0.260 0.3 22  3.0 - 48.0  0.050 -0.27  0.601  Continued...  TABLE l . I I I  (Continued)  Number of  Remarks  Run  Points  Gamma  1  10  50  9  0.76  51  15  0.73  51D  2.C7  52  8 30  53  13  0.85  54  10  s.2U  55  10  0.37  Doughnt rings i n column  0.87  Very small diameter (and hence plate volume) and high flow rates  - 67 The experimental pulse injector shovm i n Figure 1.8 was used i n run 53 on the s i x p a r t i c l e diameter bed containing 1 cm. spheres.  Methane  pulses are used and the effect of pulse size (as measured by peak height) a p a r t i c l e Reynolds number of 62.k i s shown i n Table l . I V .  at  Over a 13 fold  range d i f f e r e n t pulse sizes resulted i n e s s e n t i a l l y the same HETP values. It must be pointed out, however, that these values are not included In the data for Run 53«  At the time when the data were taken a maximum p a r t i c l e  Reynolds number o f 35  w a s  employed i n the hope that Van Deemter"s assumption  regarding the eddy d i f f u s i v i t y could be extended to a Reynolds number of 35 without serious e r r o r .  This l i m i t a t i o n was l a t e r discarded, and the four  points i n Table k were included with those of Run 53*  However, they were  found to change s e r i o u s l y the constants of the l e a s t square equation (1.50), i n d i c a t i n g an inconsistency.  Run 60 repeated the conditions of Run 53> but  employed the normal pulse i n j e c t i o n , and these data were consistent with the r e s u l t s at low flow rates i n Run 53, but not with the four points i n Table l . I V .  It i s concluded that the inconsistency was created by the  experimental i n j e c t o r at high flov; rates because of the f a i l u r e of the pop valve i n the i n j e c t o r to close c l e a n l y .  The requirements suggested by  Van Deemter to ensure that the feed pulse size does not influence the  exit  d i s t r i b u t i o n (equation 1.25) were e a s i l y s a t i s f i e d i n t h i s work, p a r t i c u l a r l y with a large diameter column such as that used i n Run 53. TABLE l . I V EFFECT OF PULSE SIZE (PEAK HEIGHT) ON HETP at P a r t i c l e Reynolds number of 62.k Run 53 HETP  PEAK HEIGHT  1.65 1.66 1.75 1.59  27.5 units 59 56 19  TABLE l . V FURTHER DISPERSION RESULTS WITH BEDS OF NON POROUS PELLETS  Pellet Diameter  Column Length  63  O.568  421.0  0.66  1.16  H  64  O.568  421.0  0.66  1.16  CH  65  O.568  119.5  2.175  3.83  66  O.568  119-5  2.175  69  1.03  186.3  70  1.03  71 72  Run  Column Diameter  Column to Pellet Diameter Ratio  Pulse  A  B  Inverse Tortuosity  C  AA  CC  0.11  1.88  1.25  0.019  0.51  -0.021  -0.06  0.79  1.88  0.081  0.24  0.048  CH4  C12  0.37  0.901  0.057  0.18  0.C49  3.83  H  0.12  O.87  0.59  0.027 -0.13  0.039  6.27  6.1  CH  0.7C -  O.32  O.76  O.O63 0.703  O.O63  186.3  6.27  6.1  H  2  0.68  0.86  0.57  O.C£3  0.54  0.030  1.005  122.0  1.15  1.1  H  2  0.31  1.14  0.79  0.028  0.34  0.C27  1.005  122.0  1.15  1.1  CH  0.64  0.17  0.41  0.06  0.59  0.062  2  4  2  4  4  A l l Dimensions , cms.  Continued  Table l . V (Continued)  Run  Range of Reynolds Numbers  Number of Points  63  6-35  10  6  - 33  Remarks  9  65  4.-28  14  66  O.6-125  20  69  5 -180  • 16  70  7 -130  13  71  4 -183  14  72  10 -181  12  OA vo  - 70 Table l . V shows t h e l a t e r results with non porous p e l l e t s  which  extend., the range o f the e a r l i e r d a t a , and a l l o w s comparison o f the hydrogen pulse technique with methane p u l s e r e s u l t s . dispersion effects  Once a g a i n t h e w a l l  (C value) f o r the methane vary only from 0.057 t o  0.08l with p e l l e t sizes from O.56 to 1.0 cms.  For the hydrogen p u l s e s ,  the value of the C term varied from 0.019 to 0.028 i n the same beds. These data confirm the previous conclusions regarding the effects of tube diameter and p e l l e t diameter. Runs 63 and 6h show h i g h B values (inverse  tortuosity),  i n d i c a t i n g that i n s u f f i c i e n t data has been obtained i n the molecular d i f f u s i v i t y region, and the AA and CC values are probably more  meaningful  than the A and C terms. The values of AA and A for a l l the data are plotted versus p e l l e t diameter i n Figure 1.12, which i n d i c a t e s , i n s p i t e of considerable s c a t t e r , the approximately l i n e a r dependence of the packed bed eddy d i f f u s i v i t y on p e l l e t diameter for a wide range of Reynolds numbers, as suggested by Van Deemter.  The deviation from l i n e a r i t y could be ascribed  to v a r i a t i o n of the constant  If In Van Deemter's eddy d i f f u s i o n expression.  However, the data from t h i s work aligns i t s e l f w e l l with the t y p i c a l values of If quoted by Van Deemter (17)> as shown i n Table 1.6, except that the trend i s reversed with larger p e l l e t s , and X increases with p e l l e t diameter. TABLE l . V T VALUES OF THE EDDY DIFFUSIVITY TERM CONSTANT,  Van Deemter (17  " This work  if = u d  P e l l e t diameters cms. .003 .0074  if 8  .015 .035  Z0  -  .025  .2  .C83  .6 1.0  3  0.06  0.13 0.37  >  AA  A  H. X CH O  A  1  V  4  •  van DeemterO A  X4  / A  -9  o  0-1 -O-ll  o : C_t .  O  0-3  0-5 0-7 0-9 10 PELLET DIAMETER CMS Figure 1.12  Eddy Diffusion Terra,  A,  (Equation  1.5°)  V s . Pellet Diameter  - 72 B.  LONGITUDINAL DISPERSION COEFFICIENT The d a t a o b t a i n e d  i n the beas o f non porous p e l l e t s were computed  as o v e r a l l d i s p e r s i o n c o e f f i c i e n t s  ( t h a t i s , eddy p l u s m o l e c u l a r  coeffic-  i e n t s ) and a r e compared w i t h t h e c o r r e l a t i o n s and t h e o r i e s o f o t h e r w o r k e r s i n F i g u r e s 1.15,  t o 1.17.  I n F i g u r e 1.13 a l l the' d a t a e x c e p t t h o s e  r u n 1 a r e p l o t t e d as d i s p e r s i o n c o e f f i c i e n t s v s . t h e i n t e r s t i t i a l (u).  from velocity  Tne d a t a p o i n t s form smooth c u r v e s b u t t h e s l o p e s i n t h e t u r b u l e n t  r e g i o n v a r y , showing a n e x p o n e n t i a l v e l o c i t y dependence o f 1.5 f o r t h e l a r g e r 1 cm p e l l e t s , i n c r e a s i n g t o a n exponent o f 2 f o r the packing  sizes.  smaller  A t low v e l o c i t i e s , t h e d i s p e r s i o n c o e f f i c i e n t s a p p r o a c h  the v a l u e o f the m o l e c u l a r d i f f u s i v i t y .  I n F i g u r e s 1.11+ _ 1.17, t h e  smoothed, d a t a from F i g u r e 1.13 has been u s e d , and i s shown as a  continuous  curve w i t h i d e n t i f y i n g symbols m a r k i n g t h e s t a r t and f i n i s h o f t h e  line.  111 F i g u r e 1.1-- t h e i n v e r s e d i s p e r s i o n P e c l e t number Dj^ i s ud_p p l o t t e d v s . the s u p e r f i c i a l Reynolds number  u€3d_f*/A/  t o 'compare t h e  r e s u l t s w i t h t h o s e o f McHenry and W i l h e l m (15) o b t a i n e d b y the r e s p o n s e method i n a bed o f 0.3 cm d i a m e t e r s p h e r e s .  frequency  The d a t a from t h i s  work a r e n o t e n t i r e l y c o n s i s t e n t w i t h McHenry and Wilhelm's, but the lack o f agreement i s p r o b a b l y due t o a d i f f e r e n c e i n the Schmidt number s i n c e McHenry and W i l h e l m used a  JQfo  hydrogen s t r e a m w h i l e the p u l s e s i n t h i s '  work used o n l y a t r a c e o f hydrogen.  The Reynolds number above i s t h u s not  a complete c r i t e r i o n , p a r t i c u l a r l y i n the t r a n s i t i o n flow regimes as p o i n t e d out b y H i b y (2l+).  The d a t a o f C a i r n s and P r a u s n i t z (3M f o r  l i q u i d s are a l s o i n c l u d e d i n F i g u r e 1.14. flow r a t e s (approaching  the molecular  H i b y (2k) suggested that a t low  regime) the i n v e r s e d i s p e r s i o n  P e c l e t number i s b e t t e r p l o t t e d a g a i n s t t h e m o l e c u l a r shown i n F i g u r e I.15.  U n f o r t u n a t e l y the m o l e c u l a r  Peclet numbers as ,•  P e c l e t numbers c o u l d n o t  be c a l c u l a t e d from McHenry and W i l h e l m s ' p u b l i c a t i o n w i t h o u t access to  •I  100  10  ~1—  T— PELLET TUBE/PELLET D I A . C M SDIAMETER RATIO 10 1 A 6v 0-6 IX 4+ 0-2 I2« 25 o 0-2 1 2D O U G H N U T 6 0-2 25 H I G H PRESS < D 03 2o  H Y D R O G E N J IN AIR M E T H A N E IN AIR  •  Ol  •  »  I 10 INTERSTITIAL VELOCITY CMS/SEC. Figure 1 . 1 3  Dispersion C o e f f i c i e n t , D Vs. Inl ers' i:-.ial V e i o c i y , u . L  H  100  t  t  PELLET TUBE / PELLET Hydrogen pulses in dark poDi In A.ts CMS DIAMETER RATIO 0-3 2X 10 1A 67 0-6 la 40 0-2 12 O 25® aims 8 Prausnitz.  Mc Henry SWilhelni. I 10 100 PARTICLE REYNOLDS N U M B E R (based on superficial veL) Figure  l.lk  I n v e r s e iuj.uy P e c l e t Lumber V s . S u p e r f i c i a l Reynolds Number  -a-  PELLET TUBE / PELLET DIA. CMS DIAMETER RATIO H, PULS E DARKENED PTS. 10 1A 6V 0-6 ID 40 0-2 120 25 0 0-3 2X  0-1  I 10 100 INTERSTITIAL VELOCITY,HYDRAULIC DIA. REYNOLDS NO. Figure 1.16  I n v e r s e lkldy Schmidt Huraiber Vs. H y d r a u l i c D i a u e t e r Reynolds 1,'ur.iber  - 77 -  M O M Eb &4  PELLET TUBE/ PELLET D I A M E T E RR A T I O DIA. C M S 10 IA 6V 0 6 ID 40 0 3 2X 2 l 2 o 25 rf 10• 0 Hydrogen pulse: Dark points e  «4  w o o  A  a  o M  i  cn  10  E-t  01 01  I  10  CALCULATED DISPERSION COEFFICIENT  Figure 1.17 Empirical Dispersion C o e f f i c i e n t Correlation  - 78 p r i m a r y d a t a and so a c o m p a r i s o n c o u l d n o t be made, b u t i t may be n o t i c e d t h a t i n F i g u r e 1.15 t h e r e s u l t s from t h i s work a r e n o t so s c a t t e r e d as i n the p r e v i o u s i l l u s t r a t i o n s .  Tne d a t a f o r l i q u i d s p u b l i s h e d  b y H i b y (2h)  a r e a l s o i n c l u d e d i n F i g u r e 1.15* b u t t h e v a l u e s a r e l o w e r t h a n t h e r e s u l t s f r o m t h i s work.  T h i s d e c r e a s e was t o be e x p e c t e d because H i b y  t o o k p a i n s t o e l i m i n a t e t h e h i g h p o r o s i t y w a l l s e c t i o n , and thus remove the d i s p e r s i o n due t c w a l l e f f e c t . considered  I t i s a l s o s i g n i f i c a n t t h a t Hiby  t h e , r e s u l t s o f McHenry and U i l h c l m t o show l o w e r v a l u e s o f t h e  eddy d i f f u s i v i t y o r d i s p e r s i o n c o e f f i c i e n t t h a n would be e x p e c t e d i n a bed w i t h w a l l e f f e c t s . I n F i g u r e 1.16, t h e c o r r e l a t i o n suggested t y B i s c h o f f and Levenspiel  (28) i s examined b y p l o t t i n g t h e d a t a as i n v e r s e  dispersion  Schmidt number v s . t h e Reynolds number based on h y d r a u l i c d i a m e t e r .  'The  covergence o f t h e d a t a i s no b e t t e r than i n t h e o t h e r p l o t s . The  Saffman model (29) c o u l d not be t e s t e d because the b o u n d a r i e  o f t h e d i s p e r s i o n regimes i n packed beds a r e n o t known as t h e y a r e i n the case o f d i s p e r s i o n i n p i p e s .  However, i t would appear t h a t t h e  Saffman model may have t h e g r e a t e s t p o t e n t i a l i n p r o v i d i n g a c o r r e l a t i o n f o r eddy d i f f u s i o n i n packed beds. I n t h e absence o f a more l o g i c a l c o r r e l a t i o n , an e m p i r i c a l c o r r e l a t i o n has been d e v e l o p e d below, w h i c h i s a n e x t e n s i o n  o f the  simpler  form proposed b y B i s c h o f f and L e v e n s p i e l (28). D  T  = 0.75 D  '  w  + 0.6 u h,, +  0.02 u  0.75 D  B  2  hn  0 , 6  + 0.022u hD  (1.68)  - 79 T h i s c o r r e l a t i o n i s p l o t t e d i n F i g u r e 1.17  as e x p e r i m e n t a l v s .  c a l c u l a t e d v a l u e s o f a x i a l d i s p e r s i o n c o e f f i c i e n t , anc. a].though  the  agreement i s not good, the method i s s u f f i c i e n t l y a c c u r a t e t o a l l o w a c o r r e c t i o n t o be c a l c u l a t e d f o r the "C" term i n e q u a t i o n (I.50), which w i l l c o r r e c t the v a l u e o f t h i s t e r n i n porous p e l l e t t e s t s where any e f f e c t s o f eddy d i s p e r s i o n a r e not a l l o w -:A f o r i n the eddy d i f f u s i o n , o r "A",  C.  term.  POROUS PELLETS  Porous P e l l e t Samples The p r o p e r t i e s o f the t h r e e porous p e l l e t samples t e s t e a a r e summarized i n T a b l e l . V I I .  However, t h e r e was  o r i g i n a l l y some q u e s t i o n  a b o u t the p e l l e t p r o p e r t i e s , the d e t a i l s o f w h i c h a r e d i s c u s s e d i n Appendix I I I .  A knowledge o f the p e l l e t p o r o s i t y i s e s s e n t i a l f o r t h i s  work, b u t t h e m a n u f a c t u r e r s '  d a t a s u p p l i e d w i t h t h e p e l l e t s seemed t o be  somewhat i n c o n s i s t e n t . The d a t a on the l / 2 " N o r t o n c a t a l y s t s u p p o r t p e l l e t were generally satisfactory. was  However, i n the t r a d e l i t e r a t u r e a 4l$  quoted f o r t h e s e p e l l e t s .  porosity  I n a p r i v a t e communication, a v a l u e o f  36-40$ was g i v e n , and a 'simple e x p e r i m e n t a l measurement d e s c r i b e d i n Appendix I I I found a 36$ p o r o s i t y . A v a l u e o f 38$ has thus been a c c e p t e d as a r e a s o n a b l e  average.  With the a c t i v a t e d alumina p e l l e t s , i n a d d i t i o n to the i n c o n s i s t e n c y o f the manufacturer's  and s u p p l i e r ' s d a t a , t h e amount o f m o i s t u r e  i n t h e p e l l e t p r e s e n t e d a problem.  contained  As d i s c u s s e d e a r l i e r , t h e epoxy r e s i n s  used t o mount t h e t e s t p e l l e t i n t h e s t e a d y s t a t e d i f f u s i o n  apparatus  'A-Jot.  l.VII  PROPERTIES OF P0R0U3 PELLE ' SAMPLES 1 1  Manufac :urers' Trade Description  Pellet Diameter  Pellet Porosity  Pellet Moisture Cont ent i n 6 0 $ RH Air  Porosity of Moist Sample  Catalysi suppor'" SA 203 mixiure  1.30 cm.  O.38  negligible  l/k" Alcoa ac:iva!.ed  O.597 cm.  0.30  12$  O.31 at 12$ wet 0.34 at 10$ wet  O.32 cm.  0.50  12$  0.31 a+ 12$ wet  l/2"  P J V , . _  Horton  1:  151  1/8" Alcoa activated r.lumina II 151  O.38  Pore Diame ter of por JS 2-IiC microns  Solid Densi iy gm/ml  90$  c 50 A o  50 A  3.5 3.2 3.2  Co  o  -  8 1  -  could not stand the drying temperature necessary, and so i t was decided to make the d i f f u s i o n tests on the w< t p e l l e t s .  The moisture content of the  wet p e l l e t s was found to he s t a b l e , and not sensitive to atmospheric humidity, remaining between 10-lU$ by weight. literature  In a d d i t i o n , the manufacturer's  (31) suggested that adsorbed water existed i n l i q u i d form, so  that i f the dry p e l l e t porosity could he found, the porosity of the wet p e l l e t s could be c a l c u l a t e d . • The suppliers quoted a dry p e l l e t porosity of manufacturer's l i t e r a t u r e stated with  60-65$, while the  50$« The moisture content i n equilibrium  60$ R . H . a i r was given as 20-24$, but at no time could more than 15$  water a c t u a l l y be found i n the p e l l e t s .  Examination of some of the  manufacturer's drying data indicated that a f t e r content was normal.  6 months a 12-15$ moisture  In order to obtain a better value of the dry p e l l e t  p o r o s i t y , s p e c i a l measurements were carried out.  One of the experiments  for t h i s purpose described i n Appendix III involved putting p e l l e t s under vacuum and then flooding them with water.  This t e s t suggested that the  50$ porosity was c o r r e c t , and t h i s value was l a t e r v e r i f i e d more exactly by placing dry p e l l e t s i n a chromatograph sample loop, and measuring the r e s u l t i n g reduction i n sample volume of the loop. at the sample gas i n the loop.  Hydrogen gas was used  This experiment gave a 50$ porosity f o r  the dry p e l l e t s and yielded 28$ and 33$ porosities f o r wet p e l l e t s having a 12$ moisture content.  A porosity of 31$ corresponds to a 50$ dry  porosity i n a p e l l e t containing 12$ by weight water i n the l i q u i d s t a t e . In a d d i t i o n , the alumina p e l l e t s are not homogeneous i n that they are apparently manufactured by seeding a c o l l o i d a l s o l u t i o n . Examination of a s l i c e of p e l l e t on a microscope s l i d e showed pores up to 0  150 microns i n the centre core, compared to a pore diameter of 50A i n the  outer s h e l l .  - 82 -  These p e l l e t s provide an excellent example of an instance i n  which the steady state method of measuring d i f f u s i v i t i e s would give a poor r e s u l t for use i n c a t a l y s i s work, while the unsteady state method would y i e l d an average d i f f u s i v i t y value which would he more l i k e l y to be suitable. Steady State Apparatus Results (Appendix I) The e f f e c t i v e d i f f u s i v i t y of hydrogen and nitrogen i n l/2" Norton SA2O5 spheres was found to be O.O667 cm /sec. at 23°C and 76O.7 ram. Hg. 2  The e f f e c t i v e d i f f u s i o n c o e f f i c i e n t of hydrogen in l/k" diameter Alcoa HI51 activated alumina p e l l e t s containing 12$ by weight of water was found to be O.OO67 cm /sec. at 26°C. 2  "reatment of Data for Pulse Apparatus For the porous p e l l e t s ,  the same measurements and computations  are recorded i n Appendix II as for the non porous p e l l e t s , except that the eddy d i f f u s i v i t y calculations in columns 8, and subsequent columns are omitted.  Column 3 contains the inverse v e l o c i t y rather than the molecular  Peclet number which was used with the non porous p e l l e t  results.  Equation I.50 was f i t t e d to the data, and the quantity C found thereby was corrected .using the d i f f e r e n t i a l of the l a s t term of the empirical c o r r e l a t i o n equation (1.68) to remove the eddy d i f f u s i v i t y contribution as follows, Correction = dHETP = 0.3 D B hp du (O.75 D B + 0.2 u* h )  (I.69)  , 6 6  D  where u* i s the mean v e l o c i t y from a l l the data points and allows the correction to be made i n the middle of the v e l o c i t y range of  interest.  The correction i s subtracted from the slope C and the  corrected  slope C applied i n the c a l c u l a t i o n of the e f f e c t i v e d i f f u s i v i t y from equation I.U9 and I.50 using the form,  Porous P e l l e t Results In Table l . V I I I , runs 56 to 62 were made with l / V Activated Alumina p e l l e t s , except run 60 which was made with the l/2" Norton Catalyst support, and run 75 i n which the l / 8 " Activated Alumina p e l l e t s were used. A l l the results were taken i n single p e l l e t diameter beds, except run 73 which used a bed having a 7 s i diameter  ratio.  Runs 56 and 57 d i f f e r only i n the length of column, while i n 58 the same bed was used as i n 57, except that the p e l l e t s were d r i e d . Run 59 & w  s  e s s e n t i a l l y unsatisfactory,  but i t shows the results of an  attempt to eliminate the adsorption e f f e c t with extremely high flow r a t e s . Run 6 l repeated 58, and 62 repeated 57> except that hydrogen pulses were used.  The hydrogen pulse was also used i n run 73•  Run 60 employed a  methane pulse i n the single p e l l e t diameter bed packed with the l/2" Norton Catalyst supports. The results for porous p e l l e t s are summarized i n Table l . V I I I . It may be noted that there appears to be an end e f f e c t i n comparing runs 56 and 57.  However, under Table l . V I I I the values of the c r i t e r i o n for  Gaussian d i s p e r s i o n , F i u  are given, and for run 56 these are greater than  oc the column length due to the large dispersion caused by the adsorption of the methane pulse.  Thus Van Deemter's s o l u t i o n (17) "to obtain equation  (1.31) would not h o l d . The effects of adsorption on a c a t a l y s t p e l l e t have been mentioned i n section C of the introduction under "Comparison of Methods". The adsorption of methane on dry Alcoa l/k" Activated Alumina p e l l e t s was  TABLE l . V I I I D I S P E R S I O N RESULTS FOR POROUS P E L L E T S  Run  56 57 58 59 60 61 62 75  Moisture Content  Pellet  vt., #  l / 4 " activated Alumina l/k" activated Alumina l/k activated Alumina l/k" activated Alumina 1/2" Norton Catalyst Support l/k" activated Alumina l/k" activated Alumina 1/8" activated Alumina 11  Run Number  56 60 61 73  Pellet Porosity  Column Length Cm.  Column Diamet er Cm.  Pulse Gas  Pellet Diameter  A  B  C  .12  O.Jl  129  0.66  CH4  0.597  -0.26  -0.64 1.32  12  0.31  k21  0.66  CH4  0.597  -0.22  O.58 1 . 6 l  0  0.50  421  0.66  CH4  0.597  -2.2  4.65 1.699  0  0.50  421  0.66  CH4  0.597  68  -  O.38  420  1.6  CH4  1.5  0.29  0.50 0.44  0  0.50  421  0.66  H  2  0.397  0.64  0.87  10  0.34  421  0.66  H  2  0.597  0.38  1.5  12  0.31  119.4  2.17  H  2  Height of Transfer Unit F J U < L at max. v e l o c i t y  l66 cm. 3.38 cm. 20.2 cm. 5.58 cm.  O.32  -0.015  -592 -C.33  0.22  0.237  1.43 O.O96  TABLE l . V I I I (Continued) Dispersion Results f o r Porous Pellets Diffusivity Assuming Equilibrium Adsorption  Bed Porosity  Run  Slope Correction  Corrected Slope  56  0.061  1.26  O.OOO85  0.471  3-48  57  0.062  1.5^  0.00069  0.471  2-42  58  0.06l  I.63  0.0012  0.471  2-42  59  0.019  0.31  0.471  47-318  60  0.12  0.32  0.0193  0.522  5-33  6l  0.019  0.20  0.0102  0.471  3-95  62  0.019  0.22  O.OO56  0.471  7-86  73  0.017  0.079  0.00^5  0.39  1-19  Diffusivity  —  0.0045  Reynolds Range  -  8 6  -  measured, and the procedure, which involved taking pressure and volume measurements of a gas p e l l e t sample trapped i n the l e g of a mercury manometer i s described i n Appendix I V . The results of t h i s experiment, which showed methane adsorbed to the extent of 1.J7 mis/ml of p e l l e t , are u t i l i z e d i n run 5 8 to calculate the e f f e c t i v e d i f f u s i v i t y assuming equilibrium of the adsorbed gas i n the pulse apparatus.  I f equilibrium  had been attained, then the d i f f u s i v i t y calculated with the increased capacity due to adsorption i n the p e l l e t taken into account, should be equivalent to the d i f f u s i v i t y found i n run 6 l using a hydrogen pulse with dry p e l l e t s (after correcting f o r the d i f f e r e n t gas  system).  In Table l . I X the d i f f u s i v i t i e s adjusted to those equivalent to hydrogen d i f f u s i o n are compared for a l l the runs.  The e f f e c t i v e d i f f u s i v i t y  with a hydrogen pulse i n run 6l l i e s between the two d i f f u s i v i t i e s calculated from run 5 8 with the methane pulse, (a) assuming no adsorption and (b) assuming equilibrium adsorption.  This r e s u l t would indicate that the methane  probably does not approach equilibrium adsorption c l o s e l y i n the pulse apparatus. The values of the constants A and B from equation 1.5° presented i n Table l . V I I I would appear to represent a breakdown o f the theory and/or an inconsistency with the results from the eddy d i f f u s i o n runs with nonporous p e l l e t s , but i f the fact that the C terms are extremely large due to the adsorption of methane i n runs 56 to 58 i s considered, tnen the A and B terms are n e g l i g i b l e , and correction of them has very l i t t l e influence on the slope, or C, term.  For the remaining runs, the C terms  are smaller but at the same time the values of the A and B terms are within the expected  range.  TABLE l.IX COMPARISON OF EXPERIMENTAL EFFECTIVE DIFFUSION COEFFICIENTS Diffusivities  Run  56  Puis e Gas  Pulse Method Experimental Result  CH*  2  T6  O.OOO85  CH4  0.00069  58.  CH  4  0.00127  60  CH  4  0.0193  6l  H  2  0.0102  62  H  2  O.OO56  73  H  2  0.0045  Assuming Adsorption a t A,from Equilibrium Column 5  Steady State O.OO67  0.0024  0  A  57  2  Result as a Hydrogen Diffusion  Factor to Convert to Hydrogen or H - N Diffusivity 2  cm /sec units  2  Jr  3.02  0.00195  fl6  O.OO36  f  0.017  O.OO67  2.64 O.O67  0.0694  4.14  0.0102  0-93  1  O.OO56  I.05  O.OO67  1  0.0045  1.31  O.OO67*  0.750  0.208 1  •For 1/4" pellets Interstitial Diffusivities Dg  Nitrogen and Hydrogen =  Dj£  Hydrogen i n 50 A pores at 296 °K  O.755 cm /sec. at 1 aim. and 296°K e  = 2 x 25_ x 1.84500 x 29_6 3 108 273  = 0.019 cm /sec. 2  Co  —J  - 88 Comparison of Steady S t a t e and F a l s e Apparatus  PL-suits  In o r d e r to compare' r e s u l t s , 'Cable l . I X p r e s e n t s t h e pulse d a t a converted to the equivalent hydrogen d i f f u s i v i t y bulk diffusion).  I f the r e s u l t s  (or hydrogen-nitrogen f o r  i t h t h e l / 2 " N o r t o n p e l l e t s i n r u n 60  a r e examined, i t i s seen t h a t t h e p u l s e method agrees v i l l i th.- •„. tency state value within 4$. Run 62 should give the same d i f f u s i v i t y f o r hydrogen i n t h e l/k" Alcoa activated alumina p e l l e t s as the s t e a d y s t a t e a p p a r a t u s , '.JUT. t h e l a t t e r r e s u l t i s 20$ higher than t h e p u l s e r e s u l t .  I f the  diffusivity  i n the l/8" p e l l e t s could he expected t o be t h e same as t h a t i n t h e  l/k u  s i z e , the steady state r e s u l t i s 52$ higher than the r e s u l t from r u n 73I n view of the lack of homogeneity of the alumina p e l l e t s these results are not s u r p r i s i n g . The  p e l l e t t o r t u o s i t y values calculated from the true i n t e r s t i t i a l  d i f f u s i v i t i e s and the p e l l e t p o r o s i t i e s shown below Table l . I X also indicate that the Alcoa activated alumina p e l l e t s are not homogeneous, as the tortuosities material.  are much lower than would be expected for t h i s type of  The steady state r e s u l t s should be even more influenced by the  macroporous p e l l e t centre or seed because of the removal of part of the microporous s h e l l , and i f the t o r t u o s i t y i s calculated from the steady state r e s u l t an impossible value of 0.88 i s obtained.  'The reason for t h i s  anomalous r e s u l t i s because the pore size has been assumed to be 5OA i n the c a l c u l a t i o n of the Knudsen d i f f u s i o n c o e f f i c i e n t , when i n fact the centre core has some pores up to 150 microns i n diameter as measured under a microscope.  - 89  -  VI  DISCUSblON A.  NON  POROUS PELLETS  HE UP v s . V e l o c i t y R e s u l t s The HETP v s . v e l o c i t y c u r v e s shown i n F i g u r e s 1.10 and appear t o f i t Van Deemter's e q u a t i o n ( l . 5 0 ) w e l l .  1.11  However, a v e l o c i t y  dependent, o r Cu t e r m , was n o t t o be e x p e c t e d w i t h non porous p e l l e t s t h e b a s i s o f Van Deemter's a n a l y s i s .  on  From F i g u r e s 1.10 and 1.11, as w e l l  as from t h e r e s u l t s i n T a b l e s l . I I I and l . V , t h e magnitude o f t h i s terra can be seen t o be independent o f p a r t i c l e d i a m e t e r , b u t i n v e r s e l y p r o p o r t i o n a l to t h e m o l e c u l a r d i f f u s i v i t y o f t h e gas system.  I f t h e Cu t e r m  i s a v e l o c i t y dependent a x i a l d i s p e r s i o n e f f e c t ) i n non porous  (vhich pellets  i s caused by t h e h i g h e r v e l o c i t y a n n u l u s v h i c h r e s u l t s from th/- h i g h p a c k i n g p o r o s i t y a t t h e w a l l , t h e n by a n a l o g y w i t h Van D e e m t e r ' o t r e a t m e n t , t h e r e l a t i v e v e l o c i t y between t h e f l o w i n t h e w a l l a n n u l u s and i n t h e p a c k i n g c o r e c o u l d c r e a t e a term w h i c h would be i n v e r s e l y p r o p o r t i o n a l t o the m o l e c u l a r d i f f u s i v i t y .  This reasoning implies that t h i s  d i s p e r s i v e e f f e c t f o r non porous p e l l e t s ' i s caused by a w a l l  additional effect.  On t h e o t h e r hand, t h e above model becomes l e s s s a t i s f a c t o r y i f s i n g l e p e l l e t d i a m e t e r beds a r e c o n s i d e r e d , so i t would appear t h a t a n o t h e r b u t s i m i l a r mechanism o c c u r s i n s i n g l e p e l l e t b e d s , o r t h a t t h e above physical explanation i s questionable. The i n t e r c e p t , o r A t e r m , (which i s a d i s p e r s i o n due t o t h e m i x i n g e f f e c t o f t h e p a c k i n g ) o f e q u a t i o n ( 1 . 5 0 ) depends on p e l l e t d i a m e t e r a t Reynolds numbers l e s s t h a n 1, a c c o r d i n g t o Van Deemter e t a l ( l ? ) , and a s i m i l a r r e l a t i o n s h i p f o r h i g h Reynolds numbers based on t h e m i x i n g s t a g e model has been o b t a i n e d by McHenry and W i l h e l m .  The r e s u l t s from t h i s  - 90 -  work as sho\m i n Tables l . I I I and l . V , and i n Figure 1.12, also show that the intercept A i s an approximately l i n e a r (t 50$) function of the packing diameter for diameters from about 0.2 to 1 cm., and for a wide range of tube:  p e l l e t diameter  ratios.  A x i a l Dispersion C o e f f i c i e n t I f the same data as above are considered i n terms of the dispersion c o e f f i c i e n t ( i . e . the data f o r non porous p e l l e t s are not f i t t e d to equation (I.50)) as defined by equation (l.66), then i t would appear that the wall e f f e c t i s not the major contribution to the mixing due to packing geometry. Figure 1.15 shows that the smaller p e l l e t s tend to y i e l d a dispersion c o e f f i c i e n t proportional to the square of the v e l o c i t y which jould correspond to the Cu term i n equation 1.50, but the larger p e l l e t s show a lower exponent of 1.5.Hiby (2h)  obtained t h e following empirical  c o r r e l a t i o n for l i q u i d s , D  L  = 0.67 D + O.65 (u d ) B  p  1 , 5  7jD^ + J u T  and at low flow rates where iJ^B exponent to the 1.5 power.  p  ^ J  U  d  t h i s expression has a v e l o c i t y  In t h e same work (2k),  workers with l i q u i d s are summarized.  the results of other  In general, t h e a x i a l dispersion  c o e f f i c i e n t found by other workers i s a l i t t l e larger than that obtained by Hiby, who eliminated the wall e f f e c t , but Hiby points out that the data of McHenry and Wilhelm, who worked with gas systems, gives the appearance of having the w a l l e f f e c t removed.  This e f f e c t may be due to the fact that  end corrections were applied to the bed data by McHenry and Wilhelm, because i n t h e i r work r e l a t i v e l y short beds were used ( l , 2* and 3' long), 1  with only the largest being comparable to the bed lengths i n the present work.  - 91 I n F i g u r e 1.14,  i t may  be seen t h a t t h e d a t a from t h i s work shows  h i g h e r d i s p e r s i o n c o e f f i c i e n t v a l u e s t h a n does t h a t o f McHenry and U i l h e l m (15),  so t h a t t h e d a t a from t h i s work would appear t o be c o n s i s t e n t  w i t h those o f H i b y ( 2 4 ) . The use o f the h y d r a u l i c d i a m e t e r t o d e s c r i b e the system as proposed i n B i s c h o f f and L e v e n s p i e l ' s work ( 2 8 ) , and as shown i n F i g u r e does n o t appear t o improve t h e c o r r e l a t i o n .  l.l6,  The h y d r a u l i c d i a m e t e r would  o n l y be e x p e c t e d t o a c c o u n t f o r t h e w a l l e f f e c t , and i f t h e w a l l e f f e c t i s n o t predominant, as suggested by H i b y , t h e n a major improvement i n c o r r e l a t i o n would not be l i k e l y t o r e s u l t . As mentioned i n the "Theory", Saffman's model (29)  of a series of  i n t e r c o n n e c t e d c y l i n d r i c a l c a p i l l a r i e s would appear t o show the most p o t e n t i a l f o r d e s c r i b i n g t h e l o n g i t u d i n a l d i s p e r s i o n i n a packed Since the r e s u l t s presented  bed.  h e r e were g e n e r a l l y o b t a i n e d between p a r t i c l e  Reynolds numbers o f 1 and 100  ( i . e . i n t h e i n t e r m e d i a t e r e g i o n between  l a m i n a r and t u r b u l e n t f l o w ) , t h e n i t i s q u i t e c o n c e i v a b l e t h a t a v e l o c i t y p r o f i l e mechanism e q u i v a l e n t t o t h a t d e s c r i b e d by T a y l o r (25)  occurs,  r e s u l t i n g i n r e g i o n s where t h e d i s p e r s i o n c o e f f i c i e n t i s p r o p o r t i o n a l t o t h e squares o f t h e v e l o c i t y and i n v e r s e l y p r o p o r t i o n a l t o t h e  molecular  diffusivity. The u p p e r l i m i t o f the r e g i o n was u «  10 L D / R  found t o be, from (i.57)  2  B  where u i s , t h e gas v e l o c i t y , L the tube l e n g t h , R t h e r a d i u s and D molecular  the  diffusivity. L e t t h e c a p i l l a r y l e n g t h be K i d  model, w h i c h s h o u l d be a r e a s o n a b l e sized  B  spheres.  p  and r a d i u s K2d  p  assumption f o r packings  i n the Saffman of uniformly  - 92 Then, u or  «  u <^  10 Kiclp D  K D£ 3  d  where K  3  - Kj./K  B  P  2  Thus, the smaller the p e l l e t diameter,  (dp), the larger the r i g h t hand side  of the above equation. This model would explain therefore, why the smallest  pellets  showed a v e l o c i t y exponent of 2 as compared to 1.5 or 1.7 f o r the pellets.  larger  A large molecular d i f f u s i v i t y would also increase the upper  l i m i t of the region, and may explain why a maximum i s seen i n McHenry and Wilhelm s results at a s u p e r f i c i a l Reynolds number of about 100 i n 1  Figure  1.1k. I t would appear that at l e a s t two mechanisms are operating here;  1.) the v e l o c i t y dependent dispersion described by equation (1.55) which i s caused by the difference i n flow paths between adjacent parts of the bed,  and which can also be described by the mixing stage theory, and 2.) the  effects of v e l o c i t y p r o f i l e (equation I.56) i n the i n d i v i d u a l channels, which y i e l d a v e l o c i t y exponent of 2 within the flow l i m i t s derived by Taylor, given i n equation (1.57).  Thus, the resultant dispersion c o e f f i c -  ient has a v e l o c i t y exponent between 1 and 2.  As pointed out above, a  high molecular d i f f u s i v i t y would r e s u l t i n a higher upper l i m i t of significance for the v e l o c i t y p r o f i l e range.  Nevertheless, McHenry and  Wilhelm s results f o r eddy d i f f u s i v i t y using hydrogen approach a v e l o c i t y 1  dependence of 1, possibly because although the molecular d i f f u s i v i t y i s h i g h , the magnitude of the contribution to the dispersion due to the v e l o c i t y p r o f i l e i n the c a p i l l a r i e s i s smaller with higher d i f f u s i v i t y  gases (equation I.56),  and so the mechanism o f equation (l.55) would  predominate. In pipes, when the flow becomes turbulent, the p r o f i l e contribution changes from the v e l o c i t y squared dependence of equation (I.56) to a function of v e l o c i t y and f r i c t i o n f a c t o r .  In t h i s turbulent region,  the d i s p e r s i o n c o e f f i c i e n t i s independent of the molecular d i f f u s i v i t y and  the same independence would be expected i n a packed bed.  Correlation of the A x i a l Dispersion C o e f f i c i e n t As may be seen from Figures 1.13 to l . l 6 ,  several attempts were  made to obtain a c o r r e l a t i o n for the dispersion c o e f f i c i e n t .  In addition  to these e f f o r t s , dimensional analysis and a l e a s t square c a l c u l a t i o n based on the r e s u l t i n g expression using a l l the non porous p e l l e t results yielded the following c o r r e l a t i o n , u hpl  ,DBJ  1  fu  2  IS h J D  ^  (1.70)  l D PJ h  u  where y andp are the c a r r i e r gas v i s c o s i t y and density.  The above  c o r r e l a t i o n shows an exponent f o r the hydraulic diameter of nearly u n i t y , and in  a v e l o c i t y exponent of I.67, which i s an average of the values shown Figure 1.13.  Equation (1.70) does not provide a p a r t i c u l a r l y good f i t  to the data, which i s not s u r p r i s i n g because the v e l o c i t y exponent i s obviously not constant, a f a c t c l e a r l y evident i n Figure 1.13.  Of the  correlations of the above type, that of Hiby recommended f o r the t r a n s i t i o n region and shown i n Figure 1.15 seems to be most s a t i s f a c t o r y ,  but due to  a dependence o n the packing diameter squared, the degree of c o r r e l a t i o n i s less satisfactory  than that given by equation (1.70).  Bischoff and Levenspiel (28) suggest the following expression, which does have the v i r t u e of allowing for the experimental f a c t that the  - 9k v e l o c i t y dependence i s 2 f o r small p e l l e t s and approaches 1.5 f o r l a r g e r ones.  The expression i s based on the Taylor t r a n s i t i o n regime i n which  v e l o c i t y p r o f i l e effects are s i g n i f i c a n t , but the molecular d i f f u s i v i t y i s replaced by a r a d i a l d i f f u s i v i t y which includes a v e l o c i t y dependent term. DL  = DB +  + K u dp  DB  a  Although better than equation (l.70), a further  considerable  improvement i n f i t was achieved by reducing the packing diameter from 2 to 1.  exponent  However, the equation then becomes dimensionally i n c o n s i s t e n t .  The c o r r e l a t i o n f i n a l l y u t i l i z e d e s s e n t i a l l y makes the l o n g i t u d i n a l d i s p e r s i o n c o e f f i c i e n t a summation of a molecular term, a mixing stage term as suggested from McHenry and Wilhelm's work (15) and a v e l o c i t y p r o f i l e term as suggested by Taylor (25) or by Saffman's model (29). DL = 0.75 D  B  + 0.6 u h  D  + 0.02 u (0.75 D  g  B  hn°'  6  + 0.0212 u hn)  The above expression i s plotted i n Figure 1.17 as experimental v s . calculated B.  results.  POROUS PELLETS The e f f e c t of gas adsorption on the measured d i f f u s i v i t y presents  i n t e r e s t i n g features of s i g n i f i c a n c e i n any type of unsteady state d i f f u s i o n measurement.  The method used to measure the degree of adsorption,  described i n Appendix I I I ,  has been developed since t h i s work was done and  reported as a technique f o r determining adsorption isotherms f o r gases on s o l i d s (40).  I f the amount of gas adsorbed from a methane pulse were close  to equilibrium, methods of estimating the d i f f u s i v i t y could s t i l l be worked out.  Unfortunately, the adsorption i s not indicated to be at equilibrium  on the alumina p e l l e t s i n t h i s work, but as the adsorption data were derived f o r large concentrations uses trace concentrations  ( l atm.)  of methane, while the pulse apparatus  i n the presence of a i r , the state of the  - 95 equilibrium cannot r e a l l y be claimed to be conclusively known. Inconsistency of Steady State and Pulse Results for Activated Alumina Pellets The results o f runs 6 2 and 7 3 with l/h" and l/8" activated alumina i l l u s t r a t e s the p o t e n t i a l l y serious errors possible with non homogeneous pelleted materials i n measuring the u n i d i r e c t i o n a l d i f f u s i o n through a part or a l l of a p e l l e t , as, for example, i n the steady state apparatus, when i n the actual reaction d i f f u s i o n occurs towards the centre and out again. There are, of course, other p o t e n t i a l reasons f o r differences i n the r e s u l t s from steady state and pulse methods, which have already been discussed. The pulse method i n t h i s work maintains either bulk equimolar counter d i f f u s i o n or Knudsen d i f f u s i o n i n the p e l l e t so that equation 1 . 1 3 i s v a l i d no matter what mechanism occurs.  In the case of the alumina  p e l l e t s , the outer s h e l l has a uniform structure with 50°A pores so that Knudsen d i f f u s i o n occurs, and s e t t l e s the choice of equation for the steady state apparatus.  Thus, the discrepancy between the steady state and pulse  apparatus must he caused l a r g e l y by the  macroporous seed which carries a  disproportionately large portion of the d i f f u s i o n f l u x i n the steady state apparatus. The l/k" alumina p e l l e t s were examined under a microscope and the seed i n the centre was seen to be approximately l/8" across with pores up to 150 microns, as compared to the 50°A pore size i n the deposited outer layer.  The seed i n the l/8" p e l l e t s was not v i s i b l e by eye and i t i s  possible that these p e l l e t s either had an extremely small seed or none at all.  This would account for the lower d i f f u s i v i t y of the l/8" p e l l e t s  compared to the l / U " ones.  -  96 -  I f the i n t e r s t i t i a l Knudsen d i f f u s i v i t y i s calculated f o r  50 A  c y l i n d r i c a l pores, the extremely large pores i n the seed would account f o r the t o r t u o s i t y value of less than unity obtained by the steady state method and given i n Table l . I X .  Another factor which could account f o r the  difference i n d i f f u s i v i t y values from the pulse and steady state apparatus i s that the alumina p e l l e t s were prone to break down i n annular l a y e r s . With caps ground o f f each side i n the steady state apparatus, the s t r a t a of these layers are exposed and may represent a low resistance  diffusion  path through the p e l l e t . Porosity One of the c r i t i c a l factors i n applying the pulse technique i s an accurate knowledge of the p e l l e t p o r o s i t y . r e s u l t i n a 4$ v a r i a t i o n i n d i f f u s i v i t y .  A lfo change i n porosity can  As a check on the manufacturer's  data, an experiment was carried out using a gas chromatograph and a 15' by l / 2 " diameter empty tube as a dispersing system.  Samples of the l / 8 "  "wet" alumina p e l l e t s were placed i n the sample loop of the chromatograph and a hydrogen pulse injected i n an a i r c a r r i e r gas.  The height of the  pulse output compared to the height obtained i n the same way from the empty sample loop gave a good measure of the s o l i d volume of the porous p e l l e t . 'The sample gas of hydrogen had to be d i l u t e d with a i r to keep the  detector  i n the l i n e a r range, but i t would appear reasonable that i f a pulse apparatus was to be u t i l i z e d , a porosity measuring device of t h i s type would be very u s e f u l , so that the porosity of the p e l l e t s as tested  is  measured. Non Spherical P e l l e t s There should be no reason why the e f f e c t i v e d i f f u s i o n c o e f f i c i e n t o f granular p e l l e t s of almost any form could not be measured by applying an  appropriate shape f a c t o r ,  - 97 and a surface-to-volume p e l l e t diameter as used  for effectiveness factor charts ( 3 ) .  A derivation was attempted to express  the mass transfer c o e f f i c i e n t for cylinders i n terms of an e f f e c t i v e diffusivity,  (as i n equation (1.46)) hut no s i m p l i f i e d approximation could  be made, due to the presence Of Bessel functions i n the s o l u t i o n .  Thus,  for shapes other than spheres, a constant based on experiment would seem to be required to r e l a t e the mass transfer c o e f f i c i e n t and e f f e c t i v e d i f f u s i v i t y i f the s i m p l i f i e d form o f equation (1.4-9) i s to be preserved. Methane Pulse The use of a methane pulse seems to be of l i t t l e value.  The  c o r r e c t i o n to the dispersion measured for a bed of porous p e l l e t s which i s due to eddy d i f f u s i o n effects i s no higher for hydrogen than the term for methane.  The desirable a m p l i f i c a t i o n of the p e l l e t  correction  capacity  dispersion term can be achieved by a high v e l o c i t y , rather than attempting to use a gas of lower molecular d i f f u s i v i t y .  The hydrogen flame detector  could conceivably have a lower response lag as compared to the thermal lag i n a hot wire detector (thermal conductivity), but this does not appear to be a problem i n t h i s work. Errors The errors i n the r e s u l t caused by the mathematical manipulations are not r e a d i l y estimated, however, the effects of inaccuracies i n the measured values are considered below.  _1  f  •1  I 1  +  6  B  s (1 -  The e f f e c t i v e d i f f u s i v i t y i s given  2 €33 ( d j  kit* c ( ! - « B T  where C i s the term from equation (I.50).  g  - 98 The o v e r a l l p o t e n t i a l error may be estimated, by adding the  effects  of individual errors for a given t y p i c a l set of values. In the following table t y p i c a l variable magnitudes are given along with the estimated error and the e f f e c t on the resultant e f f e c t i v e d i f f u s i v i t y .  Variable  c  Magnitude  TABLE l . A POTENTIAL ERRORS Degree of Uncertainty  Percentage e r r o r i n Effective Diffusivity  0.375  + 5$  5$  dp  1.0  + 2$  4$  €  0.33  - 10$  l6$  0.40  + 5$  _2$ 27?  €  cm/sec  B  It i s f a i r l y obvious that more accuracy i n the p e l l e t porosity values would r a d i c a l l y improve the r e s u l t s , but at the same time i t i s extremely improbable that a l l the errors would be i n the same sense and y i e l d the above o v e r a l l e r r o r .  It should be mentioned that the above  error estimates apply to inaccuracies i n mean values obtained from a reasonable sample.  For example, although the p e l l e t diameters could show  50$ v a r i a t i o n between i n d i v i d u a l p e l l e t s , the mean of 20 to 40 p e l l e t s was not found to vary when a grab sample was taken.  - 99 VII  CONCLUJIONri 1.  The e f f e c t i v e d i f f u s i v i t y o f gases i n porous p e l l e t s can b e  measured u s i n g a hydrogen p u l s e t e c h n i q u e .  A 27$  adequately  random e r r o r i s c o n c e i v -  a b l e due t o e r r o r s i n t h e measured v a r i a b l e s ; however, t h i s can b e w i t h b e t t e r methods o f m e a s u r i n g t h e p e l l e t and bed p o r o s i t i e s . a d d i t i o n , a p r o b a b l e e r r o r e x i s t s from t h e m a t h e m a t i c a l l a t t e r e r r o r s h o u l d be a r e l a t i v e l y c o n s t a n t p e r c e n t a g e , t o e l i m i n a t i o n by 2.  halved  In  derivations.  This  thus l e n d i n g i t s e l f  calibration.  An eddy d i f f u s i o n mechanism e x i s t s i n t h e t r a n s i t i o n r e g i o n between  l a m i n a r f l o w and t u r b u l e n t f l o w i n packed beds such t h a t t h e a x i a l d i s p e r s i o n c o e f f i c i e n t i s p r o p o r t i o n a l t o t h e square o f t h e v e l o c i t y .  VIII RECOIvlENDATIONS The method f o r t h e measurement o f t h e p o r o s i t y o f p e l l e t s  by  i n j e c t i n g a p u l s e o f hydrogen w h i c h has been purged f r o m t h e sample l o o p o f a chromatograph c o n t a i n i n g t h e t e s t p e l l e t s , s h o u l d be d e v e l o p e d f u r t h e r and i n c o r p o r a t e d i n t o t h e p u l s e a p p a r a t u s .  The main p r o b l e m t o overcome  i s t h a t o f m i n i m i z i n g t h e i n t e r p a r t i c l e volume by p a c k i n g i n as many p e l l e t s as p o s s i b l e . By e x t e n d i n g t h e f l o w r a n g e s c o v e r e d i n t h i s work, t h e range o f t h e r e g i o n where eddy d i f f u s i v i t y i s p r o p o r t i o n a l t o t h e square o f t h e v e l o c i t y may  be d e t e r m i n e d .  The r e s u l t s may  r e s u l t s o b t a i n e d i n empty p i p e s by T a y l o r  t h e n be compared w i t h t h e (25).  - 100 -  SECTION I I DEVELOPMENT OF AM UNSTEADY STA E FLOW METHOD FOR MEASURING BINARY GAS DIFFUSION COEFFICIENT'S rf1  I INTRODUC ?ION The  "bulk, or molecular  d i f f u s i o n c o e f f i c i e n t o f h i n a r y gas  mixtures i s not r e a d i l y measured experimentally.  One o f the  oldest  techniques i s the Loschmidt method which i s based on b r i n g i n g two cylinders containing the gases (lighter o n top) together and measuring v a r i a t i o n with time.  concentration  However, t h i s method i s sensitive to c o n v e c t i o n o r  thermal eddies. In  Stefan's method the rate of d i f f u s i o n - o f a vapour i n a  v e r t i c a l glass c a p i l l a r y tube i s measured by following the drop i n l e v e l o f a l i q u i d meniscus as evaporation occurs. tube i s flushed with the second component.  The open top end of the glass This method obviously cannot  be used f o r gases above the c r i t i c a l temperature,  and, i n p r a c t i c e ,  is  limited to narrow ranges of temperature and pressure. 'The  l o n g i t u d i n a l dispersion c o e f f i c i e n t i n a straight  tube,  within the l i m i t s described i n Section I on Taylor's work ( 2 5 ) , i s a function o f the molecular d i f f u s i v i t y .  Thus by measuring the dispersion  i n a straight tube by a method s i m i l a r to that described i n Section I, molecular d i f f u s i v i t y may be obtained. be used for this type of work.  Chromatography apparatus can also  Good results can be obtained, although t h e  apparatus i s not simple, and experimental conditions f e a s i b l y are limited  (55)(56).  the  - 101 'The molecular d i f f u s i v i t y at high temperatures has been measured by Walker and Westenberg (37) hy a point source technique i n vhich a trace pf one gas i s fed through a c a p i l l a r y which i s mounted i n the centre of a tube i n which the second gas i s flowing.  The p r o f i l e of the trace gas i n  the bulk stream i s measured downstream from the source, and the molecular d i f f u s i v i t y can be calculated using the appropriate s o l u t i o n of the d i f f u s i o n equation.  Very c a r e f u l experimental technique i s required to  obtain accurate values by t h i s method, although wide temperature  ranges  can be covered. Other methods, such as measurement of d i f f u s i o n rates through porous b a r r i e r s , have been employed by numerous workers, but these do not give absolute values, and require c a l i b r a t i o n , and a correct interpretation of r e s u l t s .  In p a r t i c u l a r , there appears to exist no absolute methods  which can be used to give acceptable values of the binary d i f f u s i o n c o e f f i c i e n t over wide ranges of both temperature and pressure, and which w i l l allow some i n v e s t i g a t i o n of concentration effects a l s o .  The present  work i s an attempt to develop a measurement technique which w i l l  satisfy  a l l these requirements. An unsteady state flow method s i m i l a r to the Stefan technique was selected, as o f f e r i n g the p o s s i b i l i t y of analysis o f an e f f l u e n t stream remote from the d i f f u s i o n c e l l by any convenient means and at any necessary conditions.  The c e l l i t s e l f could be maintained at any temperature and  pressure d e s i r e d .  By varying flowing and c e l l gas compositions, concentration  effects might be studied.  However, convection effects i n the c e l l must be  absent, and so some form of packing to produce c a p i l l a r y channels would also be a part of the construction.  - 102 II THEORY A.  SIMPLIFIED SOLUTION OF A DIFFUSION EQUATION I t has been sh6wn ( 6 ) that f o r equimolar d i f f u s i o n i n a porous  s o l i d F i c k ' s second law of d i f f u s i o n takes the following form, dCA  -  -  *t  2  & a CA  E  bx  «I  £  where D E i s the e f f e c t i v e d i f f u s i v i t y , 6 and t the time.  (2.1)  B  the porosity, C the  concentration  The absence of s i g n i f i c a n t surface adsorption i s also  implied by the above equation. The r e l a t i o n s h i p between the e f f e c t i v e d i f f u s i v i t y D and the E  true binary d i f f u s i o n c o e f f i c i e n t D-g of the free gas D i s given by, DE  = DB  E  (2.2)  B  X where X . i s the t o r t u o s i t y with values varying from 1.0 for straight pores to about 100 f o r a structure containing dead end pores.  parallel  I f Dg from  equation (2.2) i s substituted into ( 2 . 1 ) , then for a bed with t o r t u o s i t y 1.0 the s o l u t i o n of (2.1) would y i e l d the molecular d i f f u s i v i t y D  B  of  the gas. A simple s o l u t i o n of the d i f f u s i o n equation (2.1) for the model shown i n Figure 2.1 i s obtained i f the assumption i s made that the vessel i s i n i t i a l l y bathed i n a gas concentration Co and then at time zero the plane at x = L i s maintained at zero  concentration.  Mathematically, the boundary conditions  dc = 0  x = 0,  dx C  A  = C  C  A  = 0  0  for a l l x when x = L  when t * 0 for  t>0  are:  DISPLACING GAS  DISPLACED 8 DISPLACING GAS A TO ANALYSER  Y  END ZONE CONTAINING "WELL MIXED FLUID" PACKED SECTION  X «0  Figure  2.1  i b d e l oi" ?he Bed For Proposed D i f f u s i o n Kxporinen-t f  - 10k Crank (22)  ( p . 9 7 ) has given Viie j o i u t i c u c>C o q v c J i o n  (2.1) w i t h  boundary c o n d i t i o n s , except t h a n "ms s o l u t i o n a p p l i e d £-iov„ C  °°,  = L^Q.  A  Tf  .-  ~ L  J  hese t o +!•  n  >  L-lj  exp  (2n + l )  h=0  - D£ B  (gn + 3 ) ^ T r £ o 1| L ^  -0,  (2nn)7Tx ~£ L  (2.3)  In order to find the f l u x from the end of t h e v e s s e l ~ho above solution must be d i f f e r e n t i a t e d v i t h respect t o x, and t h e r e s u l t i n g expression solved to give t h e concentration gradient a t t h e end (x = L ) . This gradient may then be applied i n conjunction with Fick's f i r s t law of diffusion,  dx where  i s the f l u x of gas A i n moles/(sec)(cm ) under conditions of 2  equimolar counter d i f f u s i o n , which must e x i s t i n the model of Figure  2.1.  The series s o l u t i o n f o r the concentration gradient given by  (2.3)  when x = L i s ,  f&cA L  D  X  =  2 C  J x=L  L  n=0  L  (2nH) TT t  exp f - DE  y_  n  2  *^  B  (2.5)  2  T  This s o l u t i o n can be s i m p l i f i e d by taking i n t o account only times greater than the time when the second term o f the series i s l e s s than 1$ o f the f i r s t , or i n other words,  Ln 0.01 - Dg  e  B  7T t 2  =  - 9 DE  klF  k~e^~  ¥  2  t  (2.6)  ~Tr  I f a molecular d i f f u s i v i t y of 0.75 cm /sec ( e . g . hydrogen2  nitrogen) i s assumed, and a d i f f u s i o n path'of u n i t t o r t u o s i t y and length of 10 cm., then solving 2.6 gives t = 31 seconds.  S i m i l a r l y i f the gas  d i f f u s i v i t y i s taken as 0.1 cm /sec then the time before the second term 2  can be ignored becomes 232 sec.  - 105 I f now the boundary condition requiring that the end of the bed a t x = L should be at zero concentration i s achieved by sweeping the end r a p i d l y with a second gas, then the concentration of the displaced gas i n the exit stream w i l l be proportional to the f l u x at the end of the bed. I f , i n a d d i t i o n , s u f f i c i e n t time as calculated above i s allowed to elapse before concentrations are recorded so that the second and higher terms become n e g l i g i b l e , then the f l u x equation from (2.5) Ln  C  E  X  I  T  = Ln Z -  reduces to the form,  (2.7)  DTP,7T t 2  £B h L  2  where Z i s a constant including the unwanted terms from the material balance, and from equations S.k and Z = A  B  D  Q  E  .  2.5.  (2.8)  2_CQ L  Q i s the d i s p l a c i n g gas flow rate i n m i s / s e c , and A-g the area of bed at x = L . A semi-logarithmic p l o t of exit concentration v s . time f o r a constant flow rate should y i e l d a straight l i n e with slope D E IT .  If  2  1  U  2  the bed i s packed with p a r a l l e l tubes the t o r t u o s i t y should be 1, and the slope of the p l o t becomes D33 TT  2  , thus providing a means f o r measuring  k L the free gas molecular d i f f u s i v i t y without c a l i b r a t i o n of the apparatus. B. MORE RIGOROUS SOLUTION A problem with the above experimental model a r i s e s ,  i n that i f  a d i s p l a c i n g gas flow rate high enough to s a t i s f y the boundary condition that CA = 0 at x = L i s maintained, then by the time analysis i s  started  the concentration i s so small that an extremely s e n s i t i v e a n a l y t i c a l method i s required.  Possibly t h i s very high gas flow rate could be used  -  106  -  anyway, "but a second problem due to turbulence caused by the high v e l o c i t i e s entering above the packed section could a r i s e and cause eddies i n the d i f f u s i o n zone. In order to minimize the d i s p l a c i n g gas flow r a t e , the end zone through which the d i s p l a c i n g gas flows must be as small as p o s s i b l e , but too narrow an end zone would r e s u l t i n pressure drops which could cause bulk flow i n the d i f f u s i o n s e c t i o n . Experimentally, i t was not found possible to achieve the boundary conditions described above, but a solution f o r the d i f f u s i o n equation with a w e l l mixed f l u i d at the end of the d i f f u s i o n zone ( i . e .  a f i n i t e end zone)  has been obtained by Carslaw and Jaeger ( 3 8 ) f o r heat conduction from a solid.  E s s e n t i a l l y , the s o l u t i o n expressed i n terms of the mass d i f f u s i o n  case discussed above i s f o r the following boundary conditions, dC dx A  =  CA = C  at x =  0  for a l l  0  f o r a l l -t  0  2  at t  =  0  and a material balance around the w e l l mixed end zone y i e l d s , "  D  E B A  f OCA")  Ia  -  Q  = £A  x j X=L  B  at  ftCAc  (2.9)  where Agis the area of the end of the bed, & i s the height of the end zone, and C A O i s the well-mixed end zone concentration.  As before, Q i s the gas  flow r a t e , so that the loss of displaced gas from the system i s proportional to the concentration i n the end zone and also the gas flow r a t e . The s o l u t i o n obtained by Carslaw and Jaeger i n terms of heat gives the temperature v at time t i n the region 0 ^ x < L with i n i t i a l uniform temperature V, and no heat loss at the plane x = 0 . assumed with a mass of w e l l s t i r r e d f l u i d M*  At x = L , contact i s  per u n i t area of contact, and  , s p e c i f i c heat c temperature.  - 107 vhich i s cooling hy a r a d i a t i o n mechanism at H times  its  The i n i t i a l temperature of the f l u i d i s taken as zero.  The l a t t e r boundary condition i s not compatible with the apparatus proposed f o r this work but t h i s does not influence the s o l u t i o n at large times which i s the region o f i n t e r e s t i n t h i s work. 21 earn C K <*. t K h - k o t ) eo6 (»yx ) +c nZ n=l ( L ( h - k<* ) * ( ) + h) cos ) v  e  2  V  8  w  n  where h = H/K*  ,  2  L  +  (a.io)  3  k  k = M*c and of are the consecutive roots of n  octanoCL  = h - kod  (2.11)  2  In the above equations, ^ i s the density of the bed, with heat capacity c and thermal d i f f u s i v i t y K ' c m / s e c . 2  cm  2  The thermal conductivity i s K  cals/sec  (°K)/cm.  The above s o l u t i o n has been transposed to the equivalent d i f f u s i o n case, and the tabulation which follows may a s s i s t i n explaining the diffusion  parameters. Heat Transfer  vp c  cals/cm  (>c  cals/cm  Mass Transfer C  3  A  concentration moles/cm  1.0, unless i n a porous bed when  °K  equals porosity € cm /sec  K  3  B  D \ e f f e c t i v e d i f f u s i v i t y cm /sec i g " ) or D i f €jj= 1.0  2  2  E  B  K' =  K^c  H v  c a l s / s e c cm (°K/cm) 2  c a l s / s e c cm  2  Q, C / A moles/sec cm , where Q i s the gas flow r a t e , cm /sec, and A B the bed area, cm A o  2  B  3  2  Other quantities appearing i n equation  (2.10)  when written f o r the d i f f u s i o n  case are, v / V = C / C 0 , where C  Q  i s the i n i t i a l concentration i n the bed.  =  h  =  -  3  A  ( 2 . 1 2 )  B »  A  K  1 0 8  B  E  t  filj  6B  A  =  B  A  £  ( 2 . 1 3 )  B  where £> i s the molar density and i i s the length of the end zone. M  Rewriting equation ^ ( 2 . 1 0 ) and setting x = L y i e l d s , C  A O  =  2  k<)  (h -  CQ  L (h - k f r ^ ) 2  If the time, t ,  + et  2  Hf  exp [2 n  fr*f t ]  (2.U)  (L + k) + h  i s large then the second term i n the series becomes  n e g l i g i b l e compared to the f i r s t , and equation (2.1k) c  Ao =  2  C (h - k < * i ) e x p l L (h - k ^ ) * 'ac-f  becomes,  J  ( 2 . 1 5 )  t  ( 2 . 1 6 )  ~ (L + k) + h  2  +  or Ln  ( C ^ )  =  L  "(Z)  n  -  D E _ E  where Z = 2 C J h • k<*i )  2  B  _____  2  ITTfckSI2!  « I  (L + k) + h  Thus a p l o t of Ln ( C  A O  )  versus t f o r large times should y i e l d a  s t r a i g h t l i n e of slope - B _ o £ i / 6 . g . 2  It i s also of i n t e r e s t to note that  an absolute value of the concentration i s not needed.  For example, the  peak height of a chromatograph i s proportional to the concentration at low concentrations, and so the logarithm of peak heights rather than concentrations may be plotted versus time. Equation equation  ( 2 . 1 1 )  ( 2 . l 6 )  must be solved simultaneously with the a u x i l i a r y  i n order to obtain a d i f f u s i o n c o e f f i c i e n t from a set of  e x i t gas concentration versus time data.  Examination of the equations  shows that an a n a l y t i c a l s o l u t i o n i s not p o s s i b l e .  In order to obtain a  t r i a l and error s o l u t i o n the following i t e r a t i v e procedure was applied using the Newton-Raphson method (39).  - 109 The f i r s t root of equation 77*/ 2 L .  (2.11) must l i e between c< = 0 and  Selection of an i n i t i a l value approaching zero could r e s u l t i n a  break down of the i t e r a t i v e operation because the second approximation  toTT/2  f a l l s outside the zero  L range.  A further reason f o r selecting a '  root close to TT/2 L i s apparent, because on substitution of oL = TT/ 2 L back In  (2.16) the s i m p l i f i e d solution given by (2.7) i s obtained.  'Because  the apparatus was designed to approach the simpler boundary conditions, i t i s reasonable to assume that °C = 7T/2 L w i l l be close to the actual > root.  A l s o , due to the i t e r a t i v e nature o f the solutions to  (2.l6) and  (2.11) , the time when the second root can be ignored cannot e a s i l y be derived, but as the more rigorous s o l u t i o n approaches the s i m p l i f i e d s o l u t i o n i t i s reasonable to suppose that the time calculated from the simpler s o l u t i o n (2.6) and (2.7) i s an adequate c r i t e r i o n . From the assumed value of <*i = TT/2 L and the slope of the semilogarithmic concentration v s . time p l o t one gets, Slope =  DE  OC i  (2.17)  2  «B and  so an i n i t i a l value of the d i f f u s i v i t y Bjg/C-g i s obtained.  Equations  (2.12) and 2.13) may then be substituted i n equation (2.11), but since the value of  DE/^B  i s an i n i t i a l approximation, equation  by a term f o r the r e s u l t i n g e r r o r , A Differentiating  = h - koCi  2  +04  (2.11) i s corrected  , tan<*-iL  (2.l8)  (2.l8) with respect to  d A, dc<j  =  -  2 k<*i  - oC\ L „ (Cos<*iL)  - tan <*jL  (2.19)  The second approximation f o r the f i r s t r o o t ^ i can then be obtained from the f i r s t approximation, and equations 0^1  =  oC ( f i r s t approximation) x  -  A dA dot,  (2.l8) and (2.19). (2.20)  - 110 With the second approximation the process can.be repeated from equation (2.17) C  u n t i l a satisfactory  result i s obtained.  COMPUTATION OF SLOPE OF DECAY, CURyE_.WITH A RESIDUAL CONCENTRATION In cases where the gases are not pure, or where there are dead  zones i n the apparatus which are not e a s i l y purged, a plot of experimental data according to equation (2.7) may y i e l d a curve.  A problem was  experienced i n finding the value of the steady state (or i n f i n i t e time) concentration which the data should approach with time.  This value must  be subtracted from the results to y i e l d a straight l i n e .  It was found  that on a log p l o t a s l i g h t change i n the steady state value caused a large change i n the slope, and hence uncertainty i n the r e s u l t i n g value o f the d i f f u s i v i t y . To eliminate the need for judgment on the part of the  experimenter  i n deciding on a value of the steady state l e v e l a l e a s t squares s o l u t i o n —Bt was prepared for the equation Y-C = A . e " to be determined, Y represents  where A, B and C are the constants  the concentration (or peak height), and t  i s the time. There i s reason to question the use of an equation of the above form as i t tends to weight the s o l u t i o n i n favour of data at short times. However, as there i s evidence that the s o - c a l l e d steady state value i s dependent on the gas flow rate i n t h i s apparatus, weighting i n favour of shorter times where the steady state value i s n e g l i g i b l e would seem to be j u s t i f i a b l e .  The derivation of the expression for evaluating B i s  given below. E  2  =2[Y  :  - C - A " e  B t t  ]  2  (2.21)  - I l lv h e r e C r e p r e s e n t s t h e v a l u e o f Y approached a t i n f i n i t e time and A and . B a r e c o n s t a n t s o f t h e system when e q u a t i o n ( 2 . l 6 ) i s f i t t e d t o (2.21), w i t h E "being t h e e r r o r t o be m i n i m i z e d . D i f f e r e n t i a t i n g 2.21 b y A, B and C y i e l d s . =12  dE£ dA  (Y - C - Ae "  B t i  ;  d E f = £ 2 f Y. - C - A " dB  B t i  e  d E f = £ 2 [ Y, - C - A ~  B t i  e  )  (-e" <)  (2.22)  J  (-Afyf^)  (2.23)  Bt  ] (-1)  (2.24)  dC S e t t i n g ( 2 . 2 2 ) , (2.23) and (2.24) e q u a l t o z e r o and e l i m i n a t i n g A and C y i e l d s t h e f o l l o w i n g , where A  r e p r e s e n t t h e e r r o r r e s u l t i n g from assuming  a n i n c o r r e c t v a l u e o f B. . _ - B t , -Bt r : B t /X = - lie 2 te 2e u  r ^ + ZYe Zte ' B  T j  <r  -lie v  "  B  V-  t  ^.e  2  B  t  V  -  ~l_Y e v  t  2  B  v  B t  ^  - 2 B t v-  > te n  ^ ~ + 2_te t  B  t  r-vr " j Y l e n  -rBtAr - B t )  (2s 7  + 2.*te  2  2  B  t  (2.25) o  n  c  N  To a p p l y t h e Newton-Baphson method (39)> dA dB  =  - 5 V -Bt f /T+ - E t v - B t ^ , - B t l V , - B t - r -Bt-r _ - B t ZYe I (Z^e ) +Ze 2. te J +2 te 2. 2. Y t e n n 2  22Ye-BtIt2e"2Bt lYle"  2 E t  Vj  e  It e- ^ n 2  - ^ e ' ^ ^ Y t e ^ - 2I Y Xte"1* 2 t n 2 l Y t  B  +  2.t e  e  -  I t ' e -  B t  +2.Y2,te  (2e'  B t  )  2  l Y t  2 e  -  2  B  t  2.te n  n -  2 B t  e  B t  +  ^ e ^ I * te"2Bt - 22.Y t e n  Ze  2. t e (2.26)  n Hence B = B i 2  In  A dA  (2.27)  (2.27), B r e p r e s e n t s a b e t t e r v a l u e o f B t h a n t h e p r e v i o u s assumed 2  value, that i s , B i .  - 112 III APPARATUS  A c o n s t a n t t e m p e r a t u r e a i r b a t h was f i t t e d w i t h t h e hardware f o r a gas chromatograph, and a v e s s e l c o n t a i n i n g t h e bed f o r t h e d i f f u s i o n measurement.  The t e s t v e s s e l s were s o l d e r e d from p i e c e s o f b r a s s o r copper  p i p e and were f i l l e d t o t h e b r i m w i t h t h e p a c k i n g m a t e r i a l .  A rubber  g a s k e t was used t o p r o v i d e t h e s p a c e r f o r t h e " w e l l mixed end zone" a s shown i n t h e s k e t c h e s i n F i g u r e 2 . 2 .  Two e n t r a n c e f l o w p a t t e r n s were  used i n t h e beds, a t a n g e n t i a l e n t r y i n t h e 5 cm. d i a . v e s s e l , and a d i r e c t sweep a c r o s s t h e bed i n one d i r e c t i o n i n t h e 2 . 5 cm. c e l l . A schematic  diagram o f t h e apparatus  i s shown i n F i g u r e 2 . 2 .  Moore c o n s t a n t d i f f e r e n t i a l f l o w c o n t r o l l e r s were used t o m a i n t a i n  constant  gas f l o w r a t e s , w h i l e a soap b u b b l e meter was used f o r measurement o f t h e e f f l u e n t stream f l o w s .  I n o r d e r t o reduce t h e h o l d up o f t h e apparatus  due t o v a l v e s and f i t t i n g s , t h e two gas f e e d systems were connected t o t h e d i f f u s i o n c e l l w i t h l/k"  p o l y e t h y l e n e t u b i n g , and s w i t c h i n g from one  gas t o t h e o t h e r was done b y d i s c o n n e c t i n g one tube a t t h e e n t r a n c e t o t h e c o n s t a n t t e m p e r a t u r e zone and c o n n e c t i n g t h e second.  A bypass v a l v e  a t t h e e n t r a n c e t o t h e c o n s t a n t t e m p e r a t u r e b a t h a l l o w e d gas t o f l o w d i r e c t l y t o t h e f l o w meter.  The u s e o f l / 8 " t u b i n g t o connect t h e t e s t  v e s s e l t o t h e chromatograph sample v a l v e p r o v i d e d s u f f i c i e n t r e s i s t a n c e to f l o w t o make t h e bypass v a l v e e f f e c t i v e w i t h o u t s h u t - o f f v a l v e s . T e s t gases used i n d i f f u s i o n runs were: Nitrogen  P r e p u r i f i e d Matheson Co.  Ethane Hydrogen  CP Prepurified  " "  CP  "  Butane  99.9$ ,99-0$ 99-9$ 99$  BY-PASS VALVE SOAP BUBBLE 'FLOW METER  CONSTANT TEMPERATURE AIR BATH  3  CHROMATOGRAPH SAMPLE VALVE  TOLYETHYLENE TUBING  N  MOORE FLOW CONTROLS  V RIGHT ANGLE  NOZZLE ON C ' FITTING  -GASKET  /  / PACKED BED  /  GAS I  GAS 2  GASKET FOR CROSSFLOW PATTERN AS IN ABOVE BED  20 f—  DIMENSIONS! MILLIMETERS  Figure 2 . 2 Diffusion Apparatus  - J 20 GASKET AND POSITION OF PORTS FOR TANGENTIAL FLOW PATTERN  The d e t a i l s o f the packed beds t e s t e d a r e shovn i n Table The chromasorb. a 9'  x l/k"  2.1.  chromatograph columns were packed w i t h 25$ N u j o l on The  s e p a r a t i o n o f n i t r o g e n and  ethane was  diam. column u s i n g a h e l i u m c a r r i e r .  accomplished w i t h  Hydrogen and  nitrogen  were a n a l y z e d on t h e same column b u t w i t h a hydrogen c a r r i e r so t h a t o n l y n i t r o g e n showed as a peak.  Butane and n i t r o g e n were a n a l y z e d b y an  column w i t h h e l i u m c a r r i e r gas.  Ten p s i g c a r r i e r gas p r e s s u r e was  18" used i n  t h e l o n g columns b u t t h e s h o r t column needed o n l y 2 p s i g .  IV PROCEDURE A.  SELECTION OF THE  DISPLACED AND  DISPLACING  GAS  I n t h e s e l e c t i o n o f d i s p l a c e d and d i s p l a c i n g gas two  f a c t o r s must be c o n s i d e r e d .  The  from a gas p a i r  t a i l n o r m a l l y encountered i n gas  chromatography peaks tends t o mask a f o l l o w i n g peak, and t h i s e f f e c t  may  be p a r t i c u l a r l y s e r i o u s when the columns a r e made as s h o r t as p o s s i b l e to  reduce a n a l y s i s t i m e .  'Thus, i t was  necessary  the f i r s t peak t o appear on the chromatograph. c o n s i d e r e d i s t h a t t h e l i g h t e r gas  t o make the d i s p l a c e d The  second e f f e c t t o be  s h o u l d be p l a c e d on t o p , and i f t h e  end zone i s a l s o a t t h e t o p o f t h e bed t h e n t h i s l a t t e r r e q u i r e m e n t i s c o n t r a d i c t o r y t o t h e f i r s t , as the l i g h t e r gases u s u a l l y tend t o appear f i r s t i n the c h r o m a t o g r a p h i c t r a c e .  gas  TABLE 2.1 DIFFUSION CELL PROPERHES  P a r a l l e l 'Tube P a c k i n g Bed L e n g t h , cms. Bed D i a m e t e r , cms. Length o f "End Zone", cms,  Porous S o l i d  Packing  S p h e r i c a l P a c k i n g Spheres  10.0  7.0  7.0  5.0  2.6l  2.6l  0.27  0.27  0.27  0.52  0.59  0.39 H H  Porosity-  vn  Properties o f Packing m a t e r i a l  "Kimax" m e l t i n g p o i n t tubes 10 cm. l o n g x 1.2 mm O.D. x 0.8 mm I.D.  S o l a s 01 M i c r o p o r o u s s y n t h e t i c ceramic average pore s i z e h.5 Specific surface area 0.577 m /cm o r 1.10 m /cm by B.E.T. Ref. (5) 2  2  3  3  1  B o r o s i l i c a t e Glass  -  116  -  Three gas systems were tested on each bed, hydrogen-nitrogen, ethane-nitrogen and butane-nitrogen.  The problems described above were  overcome f o r the f i r s t p a i r by using a hydrogen c a r r i e r gas so that the hydrogen peak was l o s t completely.  For ethane-nitrogen, i t was hoped  that because of the i d e n t i c a l molecular weights density effects would not be s i g n i f i c a n t , however, t h i s system does represent a more d i f f i c u l t separation i f chromatography i s used f o r a n a l y s i s .  I f the bed packing i s  f i r m l y held then obviously an inverted bed can be r e a d i l y used also with gas chromatography f o r the a n a l y s i s .  Butane and nitrogen were r e a d i l y  separated i n the a n a l y s i s , providing butane was used as the d i s p l a c i n g gas. B.  OPERATION OF EQUIPMENT To s t a r t a run the constant temperature a i r bath was brought  up to i t s control temperature,  (95°E)>  the c a r r i e r gas was put on stream,  and a purge of about one ml/sec. of the displaced gas was passed across the bed (by-pass closed).  When the bed had been thoroughly purged, a  sample of the purge gas was taken. A f t e r purging, the bypass was opened, and the d i s p l a c i n g gas l i n e connected and put on stream.  The d i s p l a c i n g gas was allowed to  purge f o r about 1 0 minutes while the flow rate was measured on the soap bubble meter and adjusted to the desired range. at the same time as the bypass valve was closed.  The stop watch was started Samples were taken and  i n j e c t e d into the chromatograph at convenient times, u n t i l the displaced gas peak had become too small to give a s a t i s f a c t o r y a n a l y s i s , or u n t i l s u f f i c i e n t r e s u l t s had been obtained.  In general, the highest concentration  - 117 -  included i n a run was about 25$ b y volume of the displaced gas and c a l i b r a t i o n s of the chromatograph indicated a l i n e a r response up to about 40$.  Therefore, absolute values of concentrations were not usually used,  but rather peak height readings. At the end of the run the flow rate was checked.  I f any  discrepancy from the i n i t i a l value was found, the l a t e r measurement was u t i l i z e d because the Moore flow controls were found to d r i f t f o r the f i r s t few minutes a f t e r a setting change.  -No flow measurements were taken  during a run as the soap bubbles caused a v i s i b l e increase i n pressure i n the system.  The room temperature and atmospheric pressure were recorded  f o r each run, and the temperature of the a i r bath was checked.  V RE5ULT5 A.  '  TREATMENT OF DATA The raw data, computer program and computed r e s u l t s are recorded  i n Appendix V f o r each r u n .  The value of the d i f f u s i v i t y recorded i s  a c t u a l l y the Dg/^_ value which i s obtained by t h i s experiment. d i f f u s i v i t y value i s f o r the temperature of the bed, but i s  The  corrected  to one atmosphere assuming no pressure drop i n the vent l i n e s .  The  e f f e c t i v e d i f f u s i v i t y i s computed f o r the same conditions. The data f o r each bed are printed along with the constants and sums f o r the l e a s t mean square l i n e computed from the data.  Ten  i t e r a t i o n s were used f o r t h i s l e a s t square c a l c u l a t i o n , but k or 5 were generally s u f f i c i e n t to obtain four figure accuracy.  The number of  i t e r a t i o n s f o r the d i f f u s i v i t y c a l c u l a t i o n was set by a t e s t of the  -  118  -  magnitude of the e r r o r , and t h i s number i s recorded. were rejected as described i n the following.  Certain data points  These points are recorded,  but they were not used by the computer. The r e s u l t s were calculated by a two-part computer program.  A  subroutine used the Wewton-Raphson (39) i t e r a t i o n described i n the "Theory" to compute the l e a s t mean square f i t of the equation Y - C = A exp(-Bt) to the data of peak heights (Y) v s . time, ( t ) .  Then using the  s o l u t i o n of the d i f f u s i o n equation described i n "Theory" (equation 2.l6), the main program calculated the d i f f u s i v i t y from the slope of the l e a s t squares l i n e with a second Newton-Raphson i t e r a t i o n . The l e a s t squares f i t of the equation i n the form Y-C = A exp (-Bt)  weighs the l i n e i n favour of the small time (large Y) p o i n t s .  Thus, i f the f i r s t or second point was inconsistent with the rest of the results,  the computed slope showed t h i s inconsistency i n spite of a l l the  other p o i n t s .  From the p l o t of Log Y v s . t ,  points which appeared to be  inconsistent when plotted have been discarded before a r r i v i n g at the values i n the following t a b l e s . The residence time of analysis gases i n the chromatograph was extremely short f o r the butane-nitrogen system, with the r e s u l t that the recorder was not able to follow the sharp narrow peaks.  The lag of the  recorder caused the peak heights to be non-linear with composition unless small peak heights were used.  Thus, computations for the butane-nitrogen  system are based on considerably longer times than the minimum for acceptable data indicated i n the discussion of theory.  Other reasons  f o r r e j e c t i n g data points are discussed where a p p l i c a b l e . In order to compare the data, the t o r t u o s i t y of the beds as calculated from each data point o f f e r s a convenient parameter.  This  c a l c u l a t i o n requires a knowledge of the value of the molecular d i f f u s i v i t y  - 119 -  for each gas p a i r used, and the values i n 'Cable 2.V show that available published results are not r e l i a b l e beyond t 5$.  Because of this  discrepancy the t o r t u o s i t y only gives a good i n d i c a t i o n of the consistency of the method, but i t s absolute value depends upon the value of molecular d i f f u s i v i t y selected.'  The t o r t u o s i t y i s shown i n the following tables,  but i n Table 2 . V the computed value D Q / X . for each set of gas systems and beds are averaged, and then ratioed with the results f o r the ethanenitrogen system.  These r a t i o s may then be compared with the same ratios of  the published experiments and calculated values, and give a comparison less dependent upon experimental e r r o r . 3.  1  PARALLEL TUBE PACKING The f i r s t experiments were carried out on a bed packed with 1.2 mm  diameter melting point tubes, thus providing a bed with unit t o r t u o s i t y p a r a l l e l to the tube bundle.  The d e t a i l s of t h i s bed are given i n  Table 2.1, while the d i f f u s i o n results are summarized i n 'Table 2.II and shown g r a p h i c a l l y i n Figures 2.3, 2.4 and 2.5 as plots of the log peak heights v s . time.  In some of the runs shown a M i l l i p o r e Type HA f i l t e r  (80$ porosity) was placed over the bed of tubes to prevent eddy currents i n the d i f f u s i o n channels due to the flowing displacing gas.  The results  shown suggest that such currents are not s i g n i f i c a n t . An inspection of the t o r t u o s i t i e s results scatter over a t 9$> range.  i n Table 2.II shows that the  Turning the bed on i t s side so that  g r a v i t y effects became i n f l u e n t i a l increased the d i f f u s i v i t y by  5O70.  The  reason f o r the scatter can be seen i n the run with the hydrogen-nitrogen system at a flow rate of O.563 ml/sec.  Three data points had to be  discarded because the recorder automatic standardization operated and thus  TABLE 2.II R E S U L T FOR PARALLEL TUBE BED Bed Temp. 306°K  Displacing Gas  Displaced Gas  Nitrogen  Hydrogen  Hydrogen  Nitrogen  Slope sec"  D /Xcm /sec.  0.510  0.00364  O.55I  0.82  1.49  0.544 O.565 2.81  o.oo4o6 0.00434 0.0115  0.715 0.873  0.82 0.82 0.82  1.15 0.940 I.05  0.151 0.151 0.151 0.151 0.151  1.11 1.02 1.08 0.915 1.015  0.095 O.O95 O.O95  1.16 1.24 1.016  Flovr Rate cm /sec. 3  Average Ethane  Nitrogen  0.485 1.461 2.27 2.94 5.08  0.00200 0.00295 0.00300 O.OO357 O.OO329 Average  Nitrogen  Butane  Molecular Diffusivity Used f o r X Computation cm /sec.  0.460 O.903 2.05  6.00143 O.OOI58 0.00202 Average  B  2  2  Tortuosity X* -  Remarks  '  Bed on S i d  Millipore  0.788 0.135 0.148 0.140 O.165 0.149  Millipore Millipore Millipore  0.1505 O.0817 0.0766 0.0905 0.084  Millipore Millipore Millipore  TIME  (SECONDS) Figure 2.J>  Results With. P a r a l l e l  ,n  ube Bed. Ilydrog,-n-!Iitrog-'n  TIME  (SECONDS) Figure 2.k  Results With P a r a l l e l Tube Bed. Ethane-Nitrogen  0  1000 TI^E  2000 (SECONDS )  Figure 2.5 Results With P a r a l l e l  fubo Bed. Butane-Nitrogen  - 124 caused a s h i f t i n t h e peak h e i g h t p r o p o r t i o n a l i t y w i t h c o n c e n t r a t i o n . Removal o f t h e s e p o i n t s caused t h e l e a s t square l i n e s l o p e t o change from 0.00446 t o 0.00434 w i t h a r e s u l t i n g change i n the D-Q/\ to  0.873 c m / s e c . 2  t h a n 4$,  yet  i n a t o t a l o f 15 p o i n t s t h e s e t h r e e cause a 3$ v a r i a t i o n i n eauees  a 15$ d i f f e r e n c e m the a i f f u s i v i t y .  POROUS SOLID PACKING The r e s u l t s o f t h e runs u s i n g p a r a l l e l tubes were  c a l c u l a t e d by hand from s l o p e s o b t a i n e d by g r a p h i c a l means. of  1.008  The t h r e e d i s c a r d e d p o i n t s a r e n o t i n e r r o r by more  t h e s l o p e , whieh i n turn C.  v a l u e from  initially The  sensitivity  t h e method t o s l i g h t e r r o r s was n o t a p p r e c i a t e d a t t h e t i m e , and  errors  were t o l e r a t e d i n t h e i t e r a t i v e c a l c u l a t i o n as w e l l as t h o s e caused b y t h e u n c e r t a i n t y o f p l a c i n g a s t r a i g h t l i n e through a s l i g h t l y curved s e t o f p o i n t s t o o b t a i n the s l o p e . Because i t was o r i g i n a l l y f e l t t h a t t h e s e e r r o r s c o u l d a l s o be due i n some measure t o eddy d i f f u s i o n w i t h i n t h e r e l a t i v e l y c o a r s e - p o r e d t u b u l a r p a c k i n g , a d d i t i o n a l experiments were c a r r i e d o u t u s i n g f i n e  porous  s o l i d s as a d i f f u s i o n medium. A S e l a s 01 ceramic f i l t e r medium s o l i d r o d was  fitted tightly into  a 2 . 6 l cm. d i a m e t e r v e s s e l t h e r e b y h a l v i n g t h e former bed d i a m e t e r , b u t t h e pore d i a m e t e r was a l s o reduced from 0.8  mm  (800 m i c r o n s ) t o 4.5  microns.  The d e t a i l s o f t h i s bed a r e g i v e n i n T a b l e 2.1, and t h e r e s u l t s a r e summarized i n T a b l e 2 . I I I . Because o f t h e s m a l l e r d i a m e t e r end zone, t h e f l o w p a t t e r n changed from  was  the former tangential i n l e t arrangement t o one h a v i n g f l o w i n  one d i r e c t i o n a c r o s s t h e chamber.  - 125 TABLE 2 . I l l RESULTS FOR POROUS SOLID PACKING Bea Temp. 306°K D i s p l a c i n g Gas  Displaced Gas  Flow Rate cm /sec  Slope - sees  0.562 0.795 0.928 1.25 1.85  0.01J9 0.0159 0.0171 0.0191 0.0245  3  Hydrogen  Nitrogen  DB _ DE 1  Average Ethane  "'0.39 0.82 1.32 1.90  0.00450 0.00490 O.OO525 0.00523  Average  0.565 0.596 0.533 0.538 0.651  0.82 0.82 0.82 D o  0.82  1.115 1.37 1.5U 1.52 1.26 1.4.3  0.114 0.108 0.112 0.108  0.151 0.151 0.151 0.151  0.1105  Nitrogen  Average  2  x  2  B  0.577  Nitrogen  n Butane  A _ ,« cm /sec  Molecular Diffusivity cm /sec  0.594 1.14 2.06  O.OO337 O.OO38O  0.00400  0.0747 0.0802 0.0823  1.3^ 1.40 1.35 1.39 1.37  O.C99 0.099 0.099  1.32 I.23 1.20 1.25  .0791  An e x a m i n a t i o n o f t h e r e s u l t s i n Table 2 . I I I . shows t h a t t h e ' t o r t u o s i t i e s a r e f a i r l y c o n s i s t e n t , w i t h each gas system showing about a t 5$ s c a t t e r from t h e mean.  However, t h e b u t a n e - n i t r o g e n system t o r t u o s i t i e s  a r e l o w e r t h a n t h o s e o b t a i n e d from t h e o t h e r g a s e s , i n d i c a t i n g t h a t a t r u e d i f f u s i v i t y v a l u e h i g h e r t h a n t h a t used would be a p p r o p r i a t e .  I n order t o  a v o i d t h e " t a i l e f f e c t " mentioned e a r l i e r , n i t r o g e n was made t h e d i s p l a c e d gas.  The f a c t t h a t butane i s a l m o s t d o u b l e t h e d e n s i t y o f n i t r o g e n would  p r o b a b l y l e a d t o g r a v i t y e f f e c t s and c o u l d cause a n a p p a r e n t i n c r e a s e i n the d i f f u s i v i t y .  The d i f f e r e n c e between t h e average t o r t u o s i t y o f t h e  h y d r o g e n - n i t r o g e n and e t h a n e - n i t r o g e n systems i s n o t s i g n i f i c a n t a s i t  - 126 depends upon t h e assumed v a l u e o f t h e d i f f u s i v i t y . o f 0.80 0.82  For example, i f a v a l u e  em /sec i s assumed f o r t h e h y d r o g e n - n i t r o g e n d i f f u s i v i t y r a t h e r t h a n 2  c m / s e c , b o t h systems g i v e an average t o r t u o s i t y o f 1.37 2  "to I.38.  The r e s u l t a t h i g h f l o w r a t e f o r t h e hydrogen c o n t a i n i n g system i s included i n the averages.  I f t h i s r e s u l t i s i g n o r e d i t would appear  t h a t t h i s system i s showing about 5$ l o w e r d i f f u s i v i t y r e l a t i v e t o t h e ethane system.  The.average pore d i a m e t e r o f t h e S e l a s Bed i s 4.5 m i c r o n s ,  w h i l e t h e mean f r e e p a t h o f hydrogen a t NTP i s 0.18  microns.  I t i s unlikely  t h a t t h e p o r e s i z e d i s t r i b u t i o n i s so narrow t h a t some p e r c e n t a g e o f t h e pores a r e n o t s m a l l e r t h a n , s a y , 1.8  , a t w h i c h pore s i z e t h e r e s u l t a n t  o f t h e mixed Knudsen and b u l k d i f f u s i o n r a t e s c o u l d be 5$ l e s s t h a n t h e bulk d i f f u s i o n alone. 'Thus, i n s p i t e o f t h e f a c t t h a t t h e r e s u l t s l o o k f a i r l y good, use "o f t h e S e l a s 01 bed i s q u e s t i o n a b l e w i t h h i g h d i f f u s i v i t y gases a t room temperature.  Such a p a c k i n g a l s o s u f f e r s from t h e need t o c a l i b r a t e t h e  bed t o f i n d t h e t o r t u o s i t y b e f o r e i t can be used on gases o f unknown diffusivity. D.  SPHERICAL PACKING The r e l a t i o n s h i p o f p o r o s i t y t o t o r t u o s i t y has been p u b l i s h e d  (6)  f o r beds o f s p h e r i c a l p a r t i c l e s , and t h i s p r o v i d e s an o b v i o u s means o f overcoming t h e need t o  c a l i b r a t e a porous s o l i d t y p e o f p a c k i n g t o f i r s t  determine i t s t o r t u o s i t y . the  The bed v e s s e l was t h e same as t h a t w h i c h h e l d  S e l a s 01, b u t i t was packed w i t h hi y d i a m e t e r g l a s s s p h e r e s .  However,  the p o r o s i t y o b t a i n e d w i t h t h e s p h e r i c a l p a c k i n g was c o n s i d e r a b l y l e s s t h a n f o r t h e porous s o l i d , and t h e r e s u l t i n g r e d u c e d bed c a p a c i t y l e d t o a decay c u r v e t h a t r a p i d l y d e c r e a s e d below t h e range o f a n a l y s i s b y chromatography. The h y d r o g e n - n i t r o g e n r e s u l t s were most i n f l u e n c e d b y t h i s  effect.  - 127 TABLE 2.IV RESULTS FOR SPHERICAL PACKING Bed Temp. 306°K Displacing Gas  Displaced Gas  Flow Rate cm /sec 3  Hydrogen  Nitrogen  0.1)56 0.832 1.25  - 1  0.0164 0.0218 0.0242  Average Ethane  Nitrogen  0.604 0.919 I.36  Nitrogen  O.596 0.979  1.24  Molecular Diffusivity cm /sec  0.00524 0.00521 0.00520  2  0.682 O.687 0.66l  0.82 0.82 0.82  1.20 1.19  0.151 0.151 0.151  1.29 1.35 1.39  0.099 0.099 O.O99  1.25 1.26 1.28  1.24  0.117 0.112 O.IO9 0.113  0.00364 0.00370 O.OO369  Average  X  2  0.677  Average Butane  D_=D A "B~ cm /sec. E  Slope sec  0.0795 O.0784 0.0775 O.0785  The graphical plots of the data i n Figures 2.6, 2.7, and 2.8 shows a sharp change of slope at longer times.  It i s possible that the d i f f u s i o n  f l u x measured i s the resultant of two decay processes,  one due to the  d i f f u s i o n from the bed and the other due to d i f f u s i o n from stagnant portions o f the p i p i n g .  This l a t t e r contribution would normally be n e g l i g i b l e for a  d i f f u s i o n c e i l with a s u f f i c i e n t l y large capacity.  It may be noticed that  for the hydrogen data with t h i s bed (Appendix V ) , the l e a s t square computation has shown the decay curve to approach a value higher than the data for larger times.  For t h i s reason, the slope and hence the d i f f u s i v i t y  Table 2.Ill) i s higher for these runs, giving a lower t o r t u o s i t y .  (see  - 128 -  Figure 2.6 Results With Spherical Packing Bed.  Hyeiogor.-Litrog,.n  - 129 -  0  200  400 600 800 TIME (SECONDS)  1000  Figure 2.7 Results With Spherical Packing Bed. Ethane-Nitrogen The lover graph shows the above points a f t e r the steady state constant has "been subtracted  - 150 -  100  1-23 PARAMETERS :  I  FLOW RATE  I • • • • . •  0  200  400  Tl^E Figure  600  . .  800  (SECONDS)  2.8  Results With Spherical Packing Bed. Butane-Nitrogen  mis/sec. n  1000  .  TABLE 2 . V COMPARISON OF RESULTS PUBLISHED DIFFUSIVITIES REF 40 Calculated Diff. Temp. H -N 273.2 288.2 293.2 306* 2  2  Ratio  O.656 0.718 0.739 0.790  C H -N 298.2 0.144 306* 0.1^98  1.0  nC Hj -N 298.2 O.O986 306* 0.1025  O.685  2  6  4  Experimental Temp. Diff. °K  Ratio  EXPERIMENTAL RESULTS FROM THIS WORK Selas 01 42 Micron Spheres Melting Point DB Ratio D Ratio D Ratio B  3  273.2 288.2 293.2  0.674 0.743 0.76 0.814  5.28  O.788  298.2  0.148 0.154  1.0  0.615  O.577  5.22  0.677  5.99  0.1505 l i P _  0.1105  1^0  0.113  1.0  o.o84  0.0791  o.7i6  0.0785  0.695  ^2k  2  0  2  •Extrapolated Values  298.2  O.O908  0.0944  0.567  - 132 Again the use of the heavier gas as the displacing gas i n the butane-nitrogen experiments may have caused the d i f f u s i v i t y to be r e l a t i v e l y somewhat higher than that of the ethane-nitrogen system, s i m i l a r to the e f f e c t apparent i n Table 2 . I l l a l s o .  IV DISCUSSION The o v e r a l l p o t e n t i a l error of the method cannot be estimated by the conventional methods due to the i t e r a t i v e nature of the s o l u t i o n . Nevertheless, the extreme s e n s i t i v i t y of the procedure to errors  is  indicated by the example i n the "Results" section (for the hydrogen-nitrogen system i n a bed of p a r a l l e l tubes) where a 3$ change i n the slope causes a 15$ change i n the d i f f u s i v i t y .  An understanding of the p o t e n t i a l accuracy  of the method may be aided by examining the f i r s t root of the a u x i l i a r y equation, tanocL  =  h £  -  k = at  Q, A D X B  E  - _ _ € B  =  Q. PL A D € oc B  B  B  -  _ _ e B  I f the r i g h t hand side (RHS) of the equation i s large, then «- L approaches 77/2 and oc becomes independent of the flow rate (Q), bed porosity (6 ),  bed area ( A ) , end zone length ( t ) and gas d i f f u s i v i t y ( D ) .  B  B  Thus i n order to reduce the present 10$ scatter of the experiments,  B  it  would appear to be necessary to achieve a large value o f X L , that i s to increase h/c*, and minimize k oc. Both h and k are i n v e r s e l y proportional to the p o r o s i t y , so i f  h/tx,>?  \LOC  }  a porosity decrease w i l l increase the RHS, however, the  reduction of bed capacity which r e s u l t s , decreases the time a v a i l a b l e f o r  -  analysis of e f f l u e n t concentrations.  1 3 3  -  I t i s noticeable that the results from  the low porosity h2 micron bed are less  scattered.  The term h increases as the bed area decreases, but experimentally, reduction of the area has the same l i m i t a t i o n s as a decrease of the p o r o s i t y , except that the benefits do not depend upon the r e l a t i v e magnitude of h and k. The gas d i f f u s i v i t y has the same influence as the bed area, and so high d i f f u s i v i t y gases are most susceptible to e r r o r .  The p a r a l l e l tube  bed with the hydrogen-nitrogen system would be expected to have most scatter of the experimental r e s u l t s .  Unfortunately, there are not enough data  points to carry out any form of s t a t i s t i c a l comparison. High flow rates of the displacing gas increase the term Q and therefore h , but once again experimental factors w i l l r e s t r i c t the maximum flow rate because of the turbulence, which can enter the bed packing to some extent, thereby increasing the e f f e c t i v e d i f f u s i v i t y and making the r e s u l t flow dependent.  Coupled to t h i s i s the e f f e c t of pressure gradients  from f r i c t i o n l o s s e s , or changes i n k i n e t i c energy at the entry p o r t , which could cause bulk flow i n the bed.  Even the p a r a l l e l tube bed i s  susceptible to bulk flows as the tubes are not sealed at the blank end. An increase i n the end zone length w i l l have the deleterious e f f e c t of increasing k and hence decreasing the R H 3 .  In the experimental  apparatus used i n t h i s work, k was n e g l i g i b l e , so that the end zone depth could probably be doubled without too much influence on the magnitude of " C . This depth increase might a s s i s t i n minimizing another p o t e n t i a l source of e r r o r , i n that the solution to the d i f f e r e n t i a l equation assumes perfect mixing i n the end zone.  The use of a deeper end zone would allow larger  - 13k scale eddies to increase the mixing, but at the same time the larger eddies should not be able to penetrate too far into the bed.  The m i l l i p o r e f i l t e r  used to discourage eddy penetration does not show any influence on the r e s u l t s , but t h i s i s probably to be expected because the added resistance would not amount to more than 0.3$ of the t o t a l while the results scatter to t lOJa.  The m i l l i p o r e f i l t e r may help to reduce the penetration of  eddies into the bed but i t would not be expected to stop the bulk flow effects discussed e a r l i e r . F i n a l l y , the length of the bed, L , may be increased to make oCsmall and hence h/d l a r g e .  On f i r s t inspection t h i s i s an obvious improvement,  however, there are l i m i t a t i o n s .  The dead time, before the second term of  the series may be dropped, i s increased four f o l d by doubling the bed length.  In the case of the 0.1 cm /sec d i f f u s i v i t y gas the dead time was 2  found to be 232 sees f o r a 10 cm bed (see i n t r o d u c t i o n ) .  In a 20 cm bed a  15 min dead time would be required. At the same time the e f f l u e n t gas concentration must be considered. From equation 2.7, the e f f l u e n t concentration would change only l i n e a r l y , with bed length.  Thus, at the time when the second term represents l°jo of  the f i r s t , the 10 cm bed a f t e r 232 sees would have doubled the concentration of the 20 cm bed a f t e r 928 sees.  The longer bed thus has the effect of  lengthening the time s c a l e , and would allow more gas chromatograph analysis to be carried out before the samples are too d i l u t e , but at the same time would s t a r t from a lower concentration. The use of the three constant equation to f i t the curved data would not appear to be responsible for the variations i n the results because the data f o r the butane-nitrogen system as shown i n Figure 2.5 f o r  - 135 the tubular bed are not curved, yet the d i f f u s i v i t i e s calculated are badly scattered.  It may be noticed, however, that at a flow rate of O.903 mis/  sec the points at 200 and 300 seconds i n Figure 2.5 deviate s l i g h t l y from the other p o i n t s , and since the l e a s t square equation favours the lower times the scatter may be caused by the small deviations of the f i r s t two points. Nitrogen decay was followed i n most of the runs, and so the trace o f a i r i n the gas systems appeared as nitrogen, r e s u l t i n g i n a curve -Bt _Bt / Y = A.e + C instead of Y = A,e (where Y and t are the v a r i a b l e s ) , which can be plotted as a straight l i n e .  The use of high p u r i t y gases might  simplify the i n t e r p r e t a t i o n of r e s u l t s . In summary, there are at least three major sources of error which may influence the d i f f u s i v i t y obtained by this method:  (a) eddies from the  end zone penetrating the bed to increase the d i f f u s i v i t y (b) bulk flow i n the bed caused by pressure gradients i n the end zone, which also act  to  increase the d i f f u s i v i t y and (c) poor mixing i n the end zone causing a lowered d i f f u s i v i t y . There i s some i n d i c a t i o n of the presence of the l a s t of these errors i n the large diameter p a r a l l e l tube bed (see Table 2.II). of the data i n Tables 2.II, bed (Table 2.II)  Examination  2 . I l l and 2.IV shows that f o r the p a r a l l e l tube  there was no s i g n i f i c a n t increase i n d i f f u s i v i t y with  increasing gas flow r a t e .  The other beds used, which were more i s o t r o p i c  i n structure, do tend to show such an increase with flow r a t e .  As the  range of flow rates used i n a l l beds was comparable, there appears to be a s l i g h t e f f e c t of the f i r s t two sources of error mentioned i n a l l but the p a r a l l e l tube bed.  The results f o r t h i s l a t t e r bed (see Table 2.II)  also  indicate that there may be some evidence for poor mixing i n the end zone.  - 136 VII CONCLUSION The method as used i n the present apparatus i s s a t i s f a c t o r y for measuring gas molecular d i f f u s i v i t i e s for binary systems within plus or minus 10$.  Analysis of sources of error suggest that by redesigning the  apparatus a probable accuracy of 2 l/2$ could be r e a d i l y achieved.  VIII RECOMMENDATIONS On the basis of t h i s work, i t i s apparent that a bed of the following dimensions could minimize the p o t e n t i a l sources of error i n the present  encountered  experiments. P a r a l l e l tube packing 1 mm or less OD. Length 20 to 30 cms. Diameter  5 cms.  End Zone length 0.25 -0.5  cms.  The sealing o f f of the tubes and prevention of bulk flows through the bed i s also advisable.  It would be advantageous to be able to invert  the bed and also much time could be saved i f the displaced gas could be purged through the bed, p a r t i c u l a r l y i f the dead time i s increased  to  15 mins. or h a l f an hour by the larger bed. Further Study The advantages of the larger bed should be experimentally v e r i f i e d and the magnitude of the flow e f f e c t s , l i k e bulk flow and turbulence, should be investigated i f the larger beds are used to reduce the s c a t t e r . The e f f e c t of the end zone length on the mixing should also be investigated.  - 137 HOI gSKrjLA'i'TJIHil  A , B and C,Constants i n l e a s t square equation. AQ  Area of bed, cm  Ap  S p e c i f i c surface area/unit volume of bed, c m "  A C  g  n  Concentration i n stage n , moles/cm  molcs/cm  Concentration i n mobile phase, moles/cm  C  Concentration i n stationary phase, moles/cm  C^  Concentration of component A, moles/cm  C^Q  End zone concentration,  C  Concentration at p e l l e t surface,  s  molcs/cm  3  3  moles/cm  C g  Average concentration,  C  I n i t i a l concentration, molep/cm.  D  Diffusion coefficient,  D-Q  Molecular d i f f u s i o n c o e f f i c i e n t ,  Dj^  Knudsen c o e f f i c i e n t ,  Dg  Effective diffusion coefficient,  D-^  Longitudinal dispersion c o e f f i c i e n t term c o n t r i b u t i o n ) , cm /scc  aV  3  3  Ci 2  1  . Sample or pulse volume, mis. I n i t i a l gas concentration i n Section I I ,  Q  C  2  3  moles/crru  cm /sec 2  cm /sec 2  cm /sec 2  cm /sec 2  (Overall including molecular  2  D *  Eddy d i f f u s i o n c o e f f i c i e n t  E  Effectiveness  Fa  Area f r a c t i o n of mobile phase = 6-g  F2  Area f r a c t i o n of stationary phase = ( l - €3 )  H  HETP, cms.  HETP  Height equivalent to a t h e o r e t i c a l p l a n t , cms.  K  Constants  L  Length of bed, cms.  L  (excluding molecular d i f f u s i o n ) cm /sec  factor  2  - 138 lip 11^  Molar f l u x , moles/sec of component A per 1  cm  2  Molar f l u x per unit geometrical area, moles/(cm)  2  sec.  P  Pressure, atm.  Q  Total f l u x , moles/sec, or gas flow rate, mis./sec. or p e l l e t volume i n Appendix IV  K  Gas constant or radius dimension  T  Temperature, °K  U-  Volume of gas, mis.  V  Volume of gas phase i n t h e o r e t i c a l plate, cm  V  Volume of t h e o r e t i c a l plate cms  3  3  c  W  Adsorp-ion or p a r t i t i o n c o e f f i c i e n t s .  Z  C o e f f i c i e n t c f exponential.  d  P  P e l l e t diameter, cms.  d'n  Column diameter, cms.  f  Fanning f r i c t i o n f a c t o r .  h  D  Hydraulic diameter, cms.  h  Thiele modulus or i n section I I h = Q/A^  D^,  j  1 +  k  F i r s t order rate constant, sec.  ki  Mass transfer c o e f f i c i e n t i n mobile 'phase, cm/sec.  k_  Mass transfer c o e f f i c i e n t i n stationary phase, cm/sec.  JL  End zone length, cms.  n  Number of t h e o r e t i c a l plates, or number of term i n series solution,  r  Pore radius, or radius variable i n d i f f e r e n t i a l equation, cms.  t  Time, seconds  \i  I n t e r s t i t i a l v e l o c i t y , cm/sec.  v  Volume of l i q u i d sido of t h e o r e t i c a l plate,  v  Average v e l o c i t y of a gas molecule, cms/sec.  % / H B  r  cm  3  - 139 Distance i n d i r e c t i o n of f l u x or flow, eras. Mole f r a c t i o n or peak height. Mass transfer c o e f f i c i e n t ,  sec.  - 1  (Section  l)  Consecutive roots of equation (2.11) (Section Error i n equality of equation. P e l l e t porosity Bed p o r o s i t y . Molar density, moles/ml. Density, grams/ml. Viscosity,  cps  Eddy d i f f u s i v i t y c o e f f i c i e n t . Tortuosity Standard d e v i a t i o n .  - 140 LITERATURE  CITED  1.  T h i e l e , E.W.,.Ind. Eng. Chem., _1, 9l6, (1939).  2.  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Eng. S c i . , _, 285, (1956).  - 141 23.  C a r b e r r y , J . J . and Bret/con, R.H., A.I.Ch.E. J o u r n a l , 4, 367, (1958).  24.  H i b y , J.W., " I n t e r a c t i o n between F l u i d s and P a r t i c l e s " , I n s t . Chem. Eng. Symposium, C 71, London, June (1962).  25.  T a y l o r , G.I., P r o c . Roy. S o c , A 225,  26.  A r i s , R.,  27.  T u r n e r , J.C.R., B r i t . Chem. Eng., 9_, 376, (196k).  28.  B i s c h o f f , K.B. and L e v e n s p i e l , 0., Chem.. Eng. S c i . , ±7_, 257, (1962), a l s o "Advances i n C h e m i c a l E n g i n e e r i n g " , V o l . I V , Academic P r e s s , New Y o r k , (1964).  29.  Saffman, P . C , Chem. Eng. S c i . , 11, 125, (1959), J . F l u i d Mech., 6, 321, (1959); L 19^, (I960); 8, 273, (I960).  30.  H a r l e y , J . , N e l W., and P r e t o r i u s , V., N a t u r e , l 8 l , 177, (1958).  31.  Newsorae, J.W., H e i s e r , H.W., R u s s e l l , A.S. and Stumpf, H.C., "Alumina P r o p e r t i e s " , Tech. Paper No. 10, A l c o a R e s e a r c h L a b o r a t o r i e s , Aluminum Co. o f A m e r i c a , P i t s b u r g h , (i960).  32.  B i r d , R.B., S t e w a r t , W.E. and L i g h t f o o t , E.N., " T r a n s p o r t Phenomena", John W i l e y and Sons, New Y o r k , (i960).  33.  P e r r y , J.H., "Chemical E n g i n e e r ' s Handbook", M c G r a w - H i l l Book Co., New Y o r k , (1950).  34.  C a i r n s , E . J . and P r a u s n i t z , J.M., Chem. Eng. S c i . , 12, 20,  35-  G i d d i n g s , J.C. and Seager, S.L., I n d . Eng. Chem. Fundamentals, 1,  4-73, (195 0. [  P r o c . Roy. S o c , A 235, 473, (1956).  3rd Ed., P. 370, (i960).  277, (1962). 36.  Kenney, N., Dept. o f C h e m i c a l E n g i n e e r i n g , U n i v e r s i t y o f Cambridge, P r i v a t e communication.  37.  W a l k e r , R.E. and Westenberg, A.A., J.. Chem. Phys., 29_, 1159, (1958).  38.  C a r s l a w , H.S. and J a e g e r , J . C , " C o n d u c t i o n o f Heat i n S o l i d s " , 2nd Ed., P. 129, U n i v e r s i t y P r e s s , O x f o r d , (1959).  39.  R a l s t o n , A. and W i l f , U.S., X'lathematical Methods f o r D i g i t a l Computers", P. 33, John W i l e y and Sons I n c . , New Y o r k , (1964).  40. ' H i r s c h f e l d e r , J.O., C u r t i s s , C F . and B i r d , R.B., " M o l e c u l a r Theory o f Gases and L i q u i d s " , P. 579, John W i l e y and Sons I n c . , New York(l954). 41.  Cox, K.E., MASc. Thesis, Columbia, (1959).  " D i f f u s i o n o f Gases", U n i v e r s i t y o f B r i t i s h  42.  B l u h , 0. and E l d e r , J.D., " P r i n c i p l e s and A p p l i c a t i o n s o f P h y s i c s " , P. 492, O l i v e r and Boyd L t d . , E d i n b u r g h , (1955).  - 14£ 4-3.  Kipping, P . J . , J e f f t r y , P . C . and Savage, C.A., Keccarch. and 0-y l-jpment for Industry, Wo. 39, P. l 8 , Feb. - March, (iS'^y).  - lUp APPENDIX I DETERMINATION OF THE EFFECTIVE GAS DIFFUSIVITY I N A POROUS SPHERICAL PELLET BY A STEADY STATE METHOD INTRODUCTION In order t o evaluate the r e s u l t s obtained by the pulse an a p p a r a t u s  technique  was c o n s t r u c t e d t o measure t h e e f f e c t i v e gas d i f f u s i v i t y i n t h e  t e s t m a t e r i a l s by a w e l l e s t a b l i s h e d procedure.  The s t e a d y s t a t e method  d e s c r i b e d b y Weisz (13) was s e l e c t e d . THEORY D i f f e r e n t d i f f u s i o n r e g i m e s , Knudsen and b u l k , were a n t i c i p a t e d i n t h e two samples w h i c h were examined and so two s o l u t i o n s a r e needed for the d i f f u s i o n  equation.  Knudsen D i f f u s i o n The m o l a r f l u x %  i s g i v e n i n terms o f t h e E f f e c t i v e D i f f u s i v i t y  Dg and t h e c o n c e n t r a t i o n g r a d i e n t b y F i c k ' s f i r s t l a w , N  A  = - Djj d C dx  a t any p l a n e x, moles/sec c m  A  2  R e f e r r i n g t o F i g u r e A 1.1, 'The t o t a l f l u x Q i s g i v e n b y Q dx Rix  =  f D  E  = N  A  A  y  2  =  TT ( R - x ) moles/sec 2  A  2  2  P^TLdCA 2  = QA_ 2 7TR  A  (Area o f p l a n e ) = N  7T ( R - x ) dCA dx  E  QA +t  Ln R - x  DE  A  = D  R + x  _1 2R  Since C  A  Q  C A  QA  2  -  C  Ln A  L  R + t U  -  •  where y i s t h e mole f r a c t i o n and  _QA 2 TTR  Pm  Ln  fR + t f R - t j L  t h e molar d e n s i t y cms/sec  yAi - y A  2  - Ikh _  Figure A 1 . 1 Sample Mounting In Steady State Apparatus  - 145 Bulk D i f f u s i o n F i c k ' s law i s again a p p l i e d b u t w i t h a c o r r e c t i o n f o r the b u l k f l o w caused b y non-equimolar c o u n t e r %  = " E D  QA = %  °A + ( A dx  D  IJ  +  N  diffusion.  B)  m  °l  e s  /  (Area) = - D_TT(R - x ) d £ 2  s e c  c  »  2  + (Q + % ) y  2  A  A  A  dx D E TT ( R  2  djy_  - x ) Pro. 2  =  - Q  A  + Q y A  A  + Q_y  A  dx Q  _£A.  l(  i + Sa\ y  dx 2  2  E  QAJ  1 l +  D  A  A  D TrV>m ( R - x )  -i  B  QB\  Ln JAf  ^A)  -  2  TT R  + 0B\ ^A)  1  \  i'+ p  m  1  D_TTA  m  2R "  R + t R - t  Ln'  9_a\  +t  7A,  R^T  n  cm /sec 2  ^  'yA A + Q B \ - l '  QA;  2  Ln y  A l  \  / i + _B\ - l \  ^A)  APPARATUS The a p p a r a t u s shown i n F i g u r e A 1.2 was assembled around a" p a i r o f b r a s s b l o c k s between w h i c h a t h i r d b l o c k (shown i n F i g u r e A I . l ) c o n t a i n i n g t h e sample p e l l e t was b o l t e d .  The b r a s s b l o c k s were c o n s t r u c t e d  i n such a way t h a t two d i f f e r e n t gases c o u l d be f l u s h e d t h r o u g h image passages a c r o s s t h e two f a c e s o f t h e sample.  mirror  Streams o f hydrogen  and n i t r o g e n from t h e i r r e s p e c t i v e c y l i n d e r s f l o w e d t h r o u g h r e s p e c t i v e c y l i n d e r p r e s s u r e r e g u l a t o r s and "Moore" c o n s t a n t  their differential  f l o w c o n t r o l s t o t h e r e f e r e n c e s i d e s o f a p a i r o f "Gow Mac" NI3 model 9220 thermal c o n d u c t i v i t y c e l l s .  From the r e f e r e n c e c e l l s t h e gases c o u l d be  - 146 -  POROUS PELLET a COUNTING  CONNECTORS FOR CALIBRATION  INCLINED MANOMETER  BYEPASS VALVE GO W-MAC  MODEL h 9220 "I THERMAL CONDUCTIVITY DETECTORS  f  MOORE FLOW CONTROL  NITROGEN' CYLINDER  r- NEEDLE -su VALVES FOR PRESSURE ADJUSTMENT  \ f  SOAP BUBBLE FLOW METERS  Figure A 1.2 Steady State Apparatus  MOORE FLOW CONTROL  HYDROGEN CYLINDER  -  Ikl  -  diverted either across the sample faces,  or v i a a by-pass to the measuring  side of the c e l l s , and from here the gases were vented to atmosphere through a needle valve and soap bubble flow meter. Manometer taps i n the sample blocks were located opposite the centre of the sample faces and were connected to an i n c l i n e d o i l f i l l e d manometer.  Polyethylene l / V diameter tubing was used f o r connecting the  apparatus, and t h i s allowed flushing of dead end l i n e s by loosening of the fittings. The t e s t samples were mounted by bathing them i n epoxy r e s i n , and then f i t t i n g them into the brass block which was d r i l l e d with a clearance hole.  A f t e r the r e s i n had set the faces of the p e l l e t were ground by  means of sand paper oh a glass  plate.  Two "DORION" potentiometers were used to measure the output of the c e l l s which formed part of a conventional bridge c i r c u i t . PROCEDURE C a l i b r a t i o n of Thermal Conductivity Cells These " d i f f u s i o n - t y p e " c e l l s have the property of being r e l a t i v e l y independent of flow r a t e , and at low concentrations a l i n e a r output with concentration can be assumed.  In order to calibrate the nitrogen c e l l a  f a i r l y high flow was set through the c e l l and sample block by-pass.  The  nitrogen flow rate was measured with the bubble meter and the c e l l zero adjusted e l e c t r i c a l l y .  A flow of hydrogen was set through i t s system and  measured on the appropriate bubble meter.  The polythene tube from the  hydrogen was then disconnected and reconnected into a point on the by-pass of the nitrogen system so that the hydrogen now appeared i n the measuring  -  side of the nitrogen c e l l .  148 -  The system was allowed to come to equilibrium  and the output measured on the potentiometer.  The concentration was  calculated from the flow rates of the two gases. Operation The two gas flows were set to convenient l e v e l s and measured while passing through the sample by-pass system.  The outputs of the two  detectors were set to zero, and then the flows were diverted to pass across the sample faces.  The manometer legs were bled and the o u t l e t measuring  valves were adjusted to be at maximum opening but maintaining zero pressure difference across the p e l l e t .  The system was allowed to come to equilibrium,  and then detector outputs were taken at convenient i n t e r v a l s over a period of twenty minutes.  The gas streams were set back on the by-pass and the  zero d r i f t of the detectors i n the course of the experiment recorded along with the flow rates of the gases. RESULTS C a l i b r a t i o n of thermal conductivity detectors Nitrogen content i n hydrogen c e l l Nitrogen flow:  25 mis i n 72.0,  Hydrogen flow:  50 mis i n 8.2,  Mole $ nitrogen =  72.2 seconds = 0.546 mis/sec. 8.2,  8.6 seconds = 6.1 m i s / s e c .  0.546 6.1  x 100  = 5*37$  + 0.346  Output o f detector 9.56 m i l l i v o l t s  or  1 . 7 8 mv/lfa nitrogen  Hydrogen content i n nitrogen c e l l Nitrogen flow:  50 mis i n 7-5,  Hydrogen flow:  25 mis i n 6 8 . 0 , 68.5 seconds  Mole $ hydrogen =  7.5 seconds  =  6.66  mis/sec.  -  O.367 mis/sec.  0.567 x 100 = 5.22$ 6.66 + O.367  Output o f detector 11.205 x 5 m i l l i v o l t s or 10.72 mv/l$ hydrogen  ,  - Ik9 I t i s of interest to compare the above r e s u l t with the c a l i b r a t i o n  of Cox (41)who obtained several points with a s i m i l a r apparatus and v e r i f i e d the l i n e a r i t y of the response.  He obtained a slope of 10.85 mv/l%  hydrogen. Activated Alumina P e l l e t l A " Diameter Pellet  Characteristics  "Alcoa HI51 Activated alumina sphere" having h2 A mean pore diameter. Diameter of p e l l e t used i n t e s t Average D i a .  =  0.255", 0.262", 0.262"  =  0.66 cms  Thickness of mounting p l a t e , i . e . across f l a t s of p e l l e t = 3/l6" =  O.476 cms. The mean free path of hydrogen at 0°C and 1 atmosphere = 180 x 10~  7  cms. (Ref.42) = 1800 A versus 42 A pore size hence Knudsen d i f f u s i o n w i l l be the predominant mechanism. The amount of nitrogen which diffused into the hydrogen stream i n t h i s experiment was so small that with the lower s e n s i t i v i t y of t h i s detector the output was of the same order as the zero d r i f t during the course o f the experiment.  For t h i s reason the d i f f u s i v i t y i s calculated from  the hydrogen f l u x , Hydrogen flow r a t e :  50 mis i n 20.5, 20.5 sec. before t e s t 22.0, 22.2 sec. a f t e r test  Nitrogen flow r a t e :  50 mis i n 18.8, 18.8 sec. before test 19.0, 19.0 sec. a f t e r t e s t  Room temperature 26 °C Atmospheric Pressure 755.6 mm Hg.  - 150 Analysis of Streams Hydrogen i n nitrogen mv:  1.47, 1.44, 1.42, 1.4l, 1.405, 1.405, 1.405 zero d r i f t add 0.17 mv y i e l d i n g 1.575 mv.  Nitrogen i n hydrogen mv:  0.07, 0.07, 0.07, O.O95, 0.07, 0.08, 0.08 zero d r i f t add 0.045 mv y i e l d i n g 0.12 mv  Subscript A refers to hydrogen °-A = 50 x 1.575 <°m 19-0 IO.85 x 100  =  yA, = 1.0 - .125 x .01 = 1.78 V  A  = 1.575 10.72  2  27TR p =  moles/sec  O.9995 mole f r a c t i o n  0.147$ = .00147 mole f r a c t i o n 2R + 2t 2R - 2t  m  .00582P_ 2  =  =  .00382 p  Ln  ( # *  .2  .66 + .476' 2  .66 -  \  y i - y2  .476J  \  0.9993 - .00147  O.OO67 cm /sec 2  Knudsen d i f f u s i v i t y of hydrogen i n p e l l e t = O.OO67 cm /sec at 2  26°C  "Norton" Catalyst Support l/2" Diameter (Alundum) Pellet  Characteristics  Maximum diameter of p e l l e t = O.55" Minimum diameter of p e l l e t = O.525" Mean diameter = O.538" or I.365 cms. Thickness of samples plate 0.90 cms. Pore diameter 90$ i n range 2 to 40 microns.  Hydrogen has a mean  free path around 0.18 microns (Ref. 42) so that bulk d i f f u s i o n w i l l be the predominant mechanism.  - 151 Nitrogen flow rate 50 mis i n 28.0, 27.0, 27.5 seconds before t e s t 28.1, 28.0 seconds a f t e r test Hydrogen flow rate 50 imLs in. 27.2, 27.1 seconds before test 29.8, 50.0 seconds a f t e r t e s t Room temperature 25°C  -  ,  •  Atmospheric pressure 160.1 mm Hg Analysis o f streams Hydrogen i n nitrogen millivolts  Nitrogen i n hydrogen millivolts  17.44 x 5 17.46 17.^5 17.44 17.37 17.35 17.39 U.h6  17.4-95 x 5 17.485 17.^91 17.1+6 17.^7 17.U5 17.^5 17.39  Average = 17.421 mv Zero d r i f t :  4.09 4.10 4.10 4.10 ^.13 4.13 4.15  3-91 3.95 3-97 4.00 4.02 4.04 4.07 4.10 4.10 - 4.103  add 0.0  mv  add 0.22  17.421 x 5/ 10.72 = 8.11$  4.32/ 1.78 = 2.425$  Subscript A refers to hydrogen y  A a  = 0.081  QA =  y  50 . x .0811 p m 28.05  QB = _5_9_ x .02425 C 29.9 D  Effective =  m  == (1  2TT R ?  m  = 0.1445 p m 2TT l^pm 2 . }  =  0.0667 cm /sec 2  0.1445 P  =  0.0405  +  A j  = 0.97575  moles/sec  m  P moles/sec m  f 2R + 2 t 12R - 2t J  QB\ Ln  0.0405_ 0.1445  a v Ln  1 + QB\  ••  2  1  QA/  /I.565 +  0.?0\ ,565 _ I 1.565 - 0.90 0.90 / Ln /0.0811 (.7195) - 1 l o . ^97575 (.7191) ~ 2  )  - 152 E f f e c t i v e bulk d i f f u s i v i t y of hydrogen and nitrogen i n the l/2" Norton catalyst supports was found to be O.O667 cm /sec at 23°C and 76O.7 E ™ 1 2  Hg pressure. Scott and D u l l i e n (5) pointed out that the r a t i o of fluxes of two gases d i f f u s i n g at constant pressure i n c a p i l l a r i e s should be inversely proportional to the r a t i o of the square root of t h e i r molecular weights. In t h i s experiment a r a t i o of 3«57 was obtained as compared with a value of 3.74 for the square root of the molecular weights.  The difference i s  probably caused by the d i f f i c u l t y i n keeping the pressures i d e n t i c a l across the p e l l e t . No absolute pressure measurements were taken i n the t e s t c e l l and so the actual pressure of the measurement may be expected to be s l i g h t l y higher than the ambient atmospheric pressure.  However, care was taken to  operate with the valves wide open except to balance the d i f f e r e n t pressure drops caused by the difference i n v i s c o s i t y of the gases. equipment at s i m i l a r flow r a t e s ' i n d i c a t e  Results on other  that a l/k" tube at flow rates  such as used here, the pressure drop i s not measurable on a mercury manometer. CONCLUSION The d i f f u s i o n c o e f f i c i e n t for the Knudsen d i f f u s i o n o f hydrogen i n a l/k" d i a . Alcoa H 151 activated alumina spheres was found to be O.OO67 cm /sec at 26°C. 2  The moisture content of the p e l l e t i s taken to be  12$ by wt from analysis of s i m i l a r p e l l e t s , however, the actual moisture o f the t e s t p e l l e t during the t e s t was not obtainable. The d i f f u s i o n c o e f f i c i e n t for the bulk d i f f u s i o n of hydrogen and nitrogen i n l/2" d i a . Norton SA 203 Alundum catalyst c a r r i e r spheres was found to be O.O667 cm /sec at 23°C and 760.7 mm Hg. 2  was found i n these p e l l e t s .  No moisture adsorption  I  1 f  •111-  e  S  z  i :  *  9  S  & i ii  Hi  iSlIBWHi!  I  » a i » ^  liiii§ililiiiiilihlitii t^i^aia^iiiiiiiai^iiini s  oSS  A  3 *  »  S  ? s  !  l  [  -IJ4-  s  s  « M 1 • Z _ » » X •  or  s CL  o  —  M  3  S  • t u • ± o  O  * 3  • •  O  _ » X X X • • 3 I 3 1 3 M M -  X ft.  •  s at M i l  M X 09  ar a • •  *• •«  U  • O * 11 l U U I L * • • * • * •  _  •  *  *  m  m  « • • • »SXXX**N*** 0  3 M M M M  •3  •  •  •  • * tfl tfl B X - f  « • t  • • • • • •  «O O  t aa • *t x 4 * o  I H W W W U I M I K U  VI3  -«  •  • H.  I t I I I f 3 3 Z X  «t I _ 19 • K « • * Klf X M U » 9 4 - W H U O W I u « > v i ( a«4  <A*V»A*t«*  - -  3a  4  O  t a t • *X « •• — O K  - - --  v» %* _ . . * ! » < » • • • « • • _. * J f »* » _. O I I I l - - f f I » C I f M W M r t M M t r l A M i i A M M i M M 3 l l l « i U 0 I H I b U « ) 4 U 4 a - M « - »  xo xx  o —*  • e  «- O fl O f f  3MM  * •* « + + +* * +>  =i mmmmm » SSSssSsSSissSsBSsSs  „°  c  cSooc^ooo-oooooo'oooco  sl»i!lfilESilif!iil •*  11 * Aiiiiiii i! jj •! -1=1111111111111111111 i i J  '! ! I! j ilillillllililllgll . 1 1 5  .hiHlHliiiliiiiiil!- s 1 A Il ' fllllllfflBHHII! i\ \l ss s S|i].i " S K » " « » " ' « ' " n » T.Y  l  RUN NO 10  COLUMN LENGTH CMS 11I.8000  CARRIER MW 29.00000  COLUMN OIAHETER CMS 9.0000  PELLET BED PELLET OIFFUSIVITV OIANETER POROSITY POROSITY CMS 0.162)81388 0.2080 0.3660000 0.  TEMP KELVIN PRESS ATM 294.10000 1.26900  I 2 1 VELOCITY HETP MOLECULAR CM/SEC CMS PECLET I.3TT19 0.25976 1.76410 1.32060 0.26547 1.70169 1.26873 0.28468 1.62319 1.16031 0.34131 1.48629 1.06645 0.33333 1.34066 0.92174 0.38881 1.18069 0.80266 0.44057 1.02790 0.49003 0.66467 0.97646 0.29279 1.16811 0.37504 -0.06964026 AA« 0.2799876 GAMMA.  1.06)680  VISCOSITY 0.01815*6  • PELLET RE 2.60161 2.31632 2.21232 2.02323 1.82669 1.60726 1.39923 0.78471 0.31054  0.369)7887  *  NTU 6T4 637 636 399 360 317 276 154 100  HVO D M 0.0766935 6 WIDTH CN 2.1900 0.6600 0.3000 0.6000 0.6700 0.8100 1.0000 2.2200 4.9000  0.07176007  CC—0.1302027 LANDA. -0.167404  DENSITY 0.0011219  1 S 9 10 11 TOTAL 0 EOOY OIFF PECLET EMPTY RE CN MLS/SEC 11.9000 12.7000 0.1T9 0.626 0.879 6.0000 12.2300 0.176 0.63B 0.8*8 6.2000 11.7000 O.180 0.684 0.810 6.6000 10.7000 0.198 0.821 0.761 3.0300 9.6900 0.185 0.869 0.668 9.8200 8.9000 0.179 0.935 0.388 6.7500 7.6000 0.177 1.059 3.512 12.2000 4.1500 0.190 1.998 0.287 20.4000 2.7000 0.170 2.786 0.187 SIGMA 0.02973078  12 13 1/SCH «E M O 1.699 1.678 1.313 1.661 1.390 1.302 1.682 1.254 1.421  0. 3. 3. 3. 9. 3. 3. 3. . 3.  RIM NO SIP  COLUMN LENGTH CMS 118.1000  CARRIER MM 24.00000  COLUMN OIAMETER CMS 2.6000  PELLET BEO PELLET DIFFUSIVITY DIAMETER PORO$ITV POROSITY CMS 0.2080 0.1830000 0. 0.20921196*  TEMP KELVIN PRESS ATM 296.00000 1.02000  1 2 1 VELOCITY HETP MOLECULAR CMS PECLET CN/SEC 11.64811 0.69431 11.56801 17.096*% 1.17195 16.99572 22.02917 1.16291 21.89925 21.4658* 1.64875 23.32746 S.8S9S6 0.47216 8.80711 9.26784 0.34159 5.23678 1.91558 0.11749 1.90428 10.91881 0.58901 10.85441 A -0.26517320 AA>-0.0705450 GAMMA*  2.077865  VISCOSITY 0.0182169  4 PELLET RE 18.97991 21.77486 10.61427 12.61216 12.12011 7.32559 2.66185 15.18194  8 0.86951987  S  HVO DIA 0.0T92282  NTU  6 K10TM  1851 4824 6217 6622 2500 I486 340 1081  0.3800 0.4000 1.3700 1.4500 0.5000 0.6600 1.4500 0.4500  CN  C  0.07300184  CC" 0.0640317 IANOA* •0.637416  OENSITV 0.00121T9 9 10 11 12 13 a TOTAL 0 EDOV OIFF PECLET EMPTY RE 1/SCH IE Hf 3 CM MLS/SEC 2.1000 1.659 7.269 31.678 28.5000 4.718 7.233 1.7000 35.7000 10.035 2.822 9.106 67.092 9.355 46.0000 12.899 5.8590 2.796 11.731 85.619 11.66* 5.2000 49.0000 19.145 3.963 12.498 129.133 12.433 3.3500 18.5000 2.092 4.719 1.HS 11.990 4.693 5.2000 11.0000 0.900 0.821 6.315 2.806 2.713 11.8500 4.0000 0.304 0. 763 1.329 2.311 1.315 2.7000 22.8000 1.216 1.416 5.784 5.815 21.499 T  SIGMA 0.11659287  N 8  R U N  C O L U M N  C O L U M N  P E L L E T  N O  L E N G T H  D I A M E T E R  D I A M E T E R  C M S 9 2  C M S  1 1 1 . 6 0 0 0  C A R R I E R  M M  2 9 . 0 0 0 0 0  0 . 2 0 8 0  K E L V I N  P R E S S  2 9 6 . 0 0 0 0 0  2 V E L O C I T Y C M / S E C  0 . 3 7 2 0 0 0 0  M O L E C U L A R  C M S  O I F F U S I V I T V  0 .  A T M  V I S C O S I T Y  1 . 0 2 0 0 0  0 . 0 1 8 2 1 6 9  4  H E T P  P E L L E T P O R O S I T Y  C M S  5 . 0 0 0 0  T E M P  B E O P O R O S I T Y  P E L L E T  9 R E  0 . 2 0 9 2 3 1 9 6 *  H Y D  D E N S I T Y  O . 0 T B 6 6 6 1  6 N T U  D I A  O . O S 1 2 1 7 9  a  T  W I D T H  P E C L E T  C M  9  0  T O T A L  E D D Y  C N  M L S / S E C  1 0 O I F F  P E C L E T  1 1 E M P T Y  1 2 R E  1 / S : H  1 3 I E  1 Y 3  2 3 . 9 9 8 0 2  1 . 0 4 8 7 3  2 3 . 8 5 6 4 9  1 3 . 3 7 2 1 9  6 4 1 1  1 . 2 0 0 0  5 . 2 5 0 0  1 8 0 . 0 0 0 0  1 2 . 9 1 4  2 . 9 2 1  1 2 . 4 1 4  8 4 . 1 3 0  2 1 . 3 3 1 5 7  0 . 8 8 5 9 4  2 1 . 2 0 5 7 7  2 9 . 6 6 4 1 7  5 6 9 9  1 . 2 5 0 0  5 . 9 5 0 0  1 6 0 . 0 0 0 0  9 . 4 4 9  2 . 1 3 3  1 1 . 0 3 5  6 3 . 1 7 9  1 1 . 2 1 9  2 0 . 3 1 1 6 5  0 . 7 6 5 7 3  2 0 . 2 1 1 7 5  2 8 . 2 7 3 6 7  5 4 1 1  1 . 2 5 0 0  6 .  1 5 2 . 5 0 0 0  7 . 7 8 4  1 . 8 4 1  1 0 . 5 1 8  5 2 . 3 4 4  1 3 . 5 9 3  1 8 . 6 6 5 1 2  0 . 8 4 0 7 1  1 8 . 5 5 5 0 4  2 5 . 9 5 6 1 5  4 9 8 6  1 . 3 2 0 0  6 . 4 S O 0  1 4 0 . 0 0 0 0  7 . 8 4 6  2 . 0 2 1  9 . 6 5 6  9 2 . 4 5 6  9 .  1 7 . 3 3 1 9 0  0 . 7 5 7 3 8  1 7 . 2 2 9 6 8  2 4 . 1 0 2 1 4  4 6 3 0  1 . 3 5 0 0  6 . 9 S 8 B  1 3 0 . 0 0 0 0  6 . 9 6 3  1 . 8 2 1  8 . 9 6 6  4 3 . 8 8 1  9 . 1 1 5  1 6 . 3 9 8 6 4  0 . 6 3 8 7 1  1 6 . 3 0 1 9 3  2 2 . 8 0 4 3 3  8 . 6 2 9  1 4 . 6 6 5 4 5  0 . 5 4 6 8 9  1 4 . 5 7 8 9 6  1 0 . 6 6 5 7 8  0 . 4  1 0 . 6 0 2 8 8  1 4 . 3 3 2 1 5  0 . 6 5 1 2 7  1 4 . 2 4 7 6 2  1 1 . 0 6 5 7 5  0 . 5 0 5 2 6  1 1 . 0 0 0 4 9  8 . 8 6 5 9 3  0 . 4 3 2 7 3  8 . 8 1 3 6 5  7 . 1 ) 6 6 0 2  0 . 1 3 1 3 1  7 . 8 1 9 6 3  7 . 8 6 6 0 2  0 . 1 7 9 7 7  7 . 8 1 9 6 3  7 . 7 9 9 3 9  0 . 3 6 0 7 9  7 . 7 5 3 3 6  7 . 2 6 6 0 7  0 . 3 5 4 1 7  7 . 2 2 3 2 1  7 . 2 6 6 0 7  0 . 1 4 9 4 6  7 . 2 2 3 2 1  6 . 5 1 2 7 9  0 . 1 1 5 7 9  6 . 4 9 4 2 7  6 . 1 1 2 8 1  0 . 3 2 2 1 6  6 . 0 9 6 6 6  9 . 8 2 6 1 8  0 . 2 9 1 3 1  5 . 7 9 1 8 2  9 . 0 1 2 9 2  0 . 2 6 2 1 8  4 . 9 8 3 3 5  9 . 0 1 2 9 2  0 . 2 6 2 1 8  4 . 9 8 3 3 5  4 . 2 3 2 9 8  0 . 2 6 7 5 4  4 . 2 0 8 0 2  7 0 5 7  1 . 6 2 6 1 7  0 . 2 4 6 4 5  3 . 6 0 4 9 8  2 . 7 1 3 1 1  0 . 2 6 0 8 2  2 . 7 1 6 9 9  1 . 6 6 6 5 3  0 . 2 9 2 7 1  1 . 6 5 6 7 0  1 . 6 7 9 8 6  0 . 2 8 8 4 5  1 . 6 6 9 9 5  1 . 4 3 9 8 8  0 . 3 0 3 4 5  1 . 4 3 1 3 9  1 . 1 9 9 9 0  0 . 3 6 0 9 2  1 . 1 9 2 8 2  0 . 8 7 9 9 3  0 . 9 0 3 2 3  0 . 8 7 4 7 4  0 . 6 1 9 9 5  0 . 5 8 4 1 4  0 . 6 1 6 2 9  4 3 8 1  1 . 3 2 0 0  7 . 4 0 8 0  1 2 3 . 0 0 0 0  9 . 2 3 7  1 . 9 3 5  8 . 4 8 3  3 5 . 3 1 3  3 9 1 8  1 . 3 7 0 0  8 . 3 0 0 8  1 1 0 . 0 0 0 0  4 . 0 1 3  1 . 3 1 5  7 . 5 8 7  2 6 . 8 1 1  7 . 7 1 3  1 4 . 8 3 2 0 9  2 8 4 9  1 . 6 0 0 0  9 . 6 1 3  1 9 . 9 3 0 6 2  3 8 2 9  1 . 4 5 0 0  1 5 . 3 8 8 2 9  2 9 5 6  1 . 6 9 0 0  8 0 . 0 0 0 0  8 . 0 9 O O  1 0 7 . 5 0 0 0  1 0 . 4 0 0 0  8 1 . 0 0 0 0  2 . 5 1 9  1 . 1 3 1  9 . 5 1 8  4 . 6 6 7  1 . 5 5 6  7 . 4 1 4  3 1 . 2 0 3  7 . 5 3 8  2 . 7 9 6  1 . 2 1 5  5 . 7 2 4  1 8 . 6 9 0  9 . B 2 0  1 2 . 8 2 5  4 . 6 6 3  2 3 6 8  1 . 8 9 0 0  1 2 . 6 0 8 0  6 6 . 5 0 0 0  1 . 9 1 8  1 . 0 4 3  1 0 . 9 3 8 6 6  2 1 0 1  1 . 8 9 0 0  1 4 . 4 0 0 0  9 9 . 0 0 0 0  1 . 3 0 1  0 . 7 9 5  4 . 9 6 9  8 . 7 1 2  4 . 1 3 7  1 0 . 9 3 8 6 6  2 1 0 1  1 . 8 9 0 0  1 3 . 4 5 0 0  9 9 . 0 0 0 0  1 . 4 9 4  0 . 9 1 3  4 . 3 6 9  9 . 9 8 6  4 . 1 3 7  1 0 . 8 4 5 9 6  2 0 8 3  1 . 8 3 0 0  1 3 . 6 5 0 8  9 8 . 9 0 0 0  1 . 4 0 7  0 . 8 6 7  4 . 9 3 9  9 . 4 0 7  4 . 1 3 2  8 . 6 0 7  3 . 3 2 1  8 . 4 8 8  3 . 8 2 1  1 0 . 1 0 4 3 6  1 9 4 1  1 . 9 0 0 0  1 4 . 3 0 0 9  9 4 . 5 0 0 0  1 . 2 8 7  0 . 8 5 2  4 . 9 8 6  3 . 7 9 9  1 0 . 1 0 4 3 6  1 9 4 1  1 . 9 0 0 0  1 4 . 4 0 0 0  9 4 . 9 0 0 0  1 . 2 7 0  0 . 8 4 3  3 . 7 3 9  9 . 0 8 4 6 5  1 7 4 5  2 . 0 9 0 0  1 5 . 8 5 0 0  4 9 . 0 0 0 0  1 . 0 9 7  0 . 8 0 7  3 . 3 7 9  8 . 5 2 8 4 5  1 6 3 8  2 . 3 9 0 0  1 8 . 9 5 0 8  4 6 . 0 0 0 0  0 . 9 8 8  0 . 7 7 4  3 . 1 7 3  8 . 1 0 2 0 3  1 5 5 6  0 . 9 9 0 0  4 . 5 9 0 8  4 1 . 7 0 0 0  0 . 8 9 4  0 . 7 9 5  3 . 9 1 4  6 . 9 7 1 0 8  1 3 3 9  0 . 6 0 0 0  5 . 2 9 0 0  3 7 . 6 0 0 0  0 . 6 9 7  0 . 6 3 3  2 . 3 9 3  6 . 9 7 1 0 8  1 3 3 9  0 . 6 0 0 0  9 . 2 9 0 8  3 7 . 6 0 0 0  0 . 6 9 7  0 . 6 3 3  2 . 5 9 3  5 . 8 8 6 4 8  1 1 3 0  O . 7 I 0 O  6 . 1 9 0 8  3 1 . 7 9 0 0  0 . 9 6 6  0 . 6 4 3  2 . 1 9 0  9 . 0 4 2 9 1 3 . 8 0 0 7 2  9 6 8  0 . 8 0 0 0  7 . 2 2 0 8  2 7 . 2 0 0 0  0 . 4 4 7  9 . 6 5 0 8  2 0 . 9 0 0 0  0 . 3 9 6  0 . 6 2 7  0 . 5 9 2  1 . 8 7 6  7 3 0  1 . 1 0 0 0  2 . 3 1 7 5 1  4 4 5  1 . 8 9 0 0  1 5 . 3 2 0 8  1 2 . 9 0 0 0  0 . 2 4 4  0 . 7 0 4  0 . 8 6 2  2 . 3 3 6 0 5  4 4 8  1 . 9 0 0 0  1 5 . 8 5 0 8  1 2 . 6 0 0 0  0 . 2 4 2  0 . 6 9 3  0 . 8 6 9  1 8 . 3 0 0 8  1 0 . 8 0 0 0  0 . 2 1 8  0 . 7 2 9  0 . 7 4 5  0 . 2 1 7  0 . 8 6 8  0 . 5 2 1  2 . 0 0 2 3 3  3 8 4  2 . 2 5 0 0  1 . 4 1 4  I . 6 6 8 6 1  3 2 0  2 . 9 5 0 0  2 2 . 0 0 S O  9 . 0 0 0 0  1 . 2 2 3 6 5  2 3 5  0 . 9 9 0 0  6 . 0 0 O 8  6 . 6 0 0 0  0 . 2 2 1  1 . 2 1 3  0 . 4 9 9  0 . 8 6 2 1 2  1 6 5  1 . 4 9 0 0  8 . 5 0 0 8  4 . 6 5 0 0  0 . 1 8 1  1 . 4 3 4  0 . 3 2 1  C  A A *  C C "  0 . 8 7 5 8 7 9  1 0 . 4 5 O O  1 6 . 7 7 8  1 2 . 3 2 9 1 7  0 . 3 6 6 5 2 5 5 8  G A M M A "  B I T  2 0 . 3 9 4 1 2  0 . 0 0 1 9 3 6 8 6  0 . 0 3 2 7 5 4 2  4 3 1 * 0  1 2 . 6 2 1  S I G M A  0 . 0 4 0 7 5 3 3 6  0 . 0 3 8 9 1 8 1  L A M U A "  0 . 0 0 3 6 9 4  0 . 0 3 2 7 1 1 6 9  N 1 0  7 . 3 3 3  3 . 6 3 6  6 . 6 0 9  3 . 2 2 3  9 . 7 1 2  3 . 3 6 4  4 . 3 9 3  2 . 6 3 6  4 . 9 9 3  2 . 5 3 6  3 . 7 8 6  2 . 2 2 6  2 . 9 8 8  1 . 3 0 7  2 . 3 8 3  1 . 4 3 7  1 . 6 3 1  3 . 1 7 5  1 . 6 2 0  3 . 8 8 4  1 . 4 6 1  3 . 7 9 7  1 . 4 4 8  3 . 5 1 1  1 . 4 8 0  9 . 4 5 1  1 . 2 1 1  0 . 1 2 5  RUN NO 11  COLUMN LENGTH CMS 186.1000  CARRIER MM 29.00000  COLUMN OIANETER CMS 6.2700  PELLET BED PELLET OIFFUSIVITV DIAMETER POROSITY POROSITY CMS 1.0300 0.4010000 0. 0.209233164  TEMP KELVIN PRESS ATM 296.00000 1.02000  I 2 3 VELOCITY HETP MOLECULAR CM/SEC CMS PECLET 0.21928 1.24171 3.73771 1.38227 0.92379 6.80414 2.06168 1.01114 10.13890 2.14610 1.01431 12.13170 1.01268 0.998)0 11.02710 1.14329 1.12271 17.44264 1.91320 1.21201 19.26117 4.36098 1.03526 21.46786 4.69194 1.08964 23.09712 S.81181 1.00200 18.78438 1.02629 22.61792 4.19460 1.41122 1.10811 26.83482 6.10784 1.18432 31.01172 A 0.68256106 . AA» 0.7240111 GAMMA.  0.810906  VISCOSITY 0.0182169  4 PELLET RE 1.2281B 9.11869 14.21101 17.13181 21.02117 24.40001 26.94729 30.03081 32.30993 26.27696 31.63960 37.13811 43.43742  B 0.31607676  1  NTU 338 611 918 113) 1319 1377 1742 1941 2088 1698 2043 2426 2808  HVD OIA 0.3947390  *WIDTH  CN 2.2000 1.7100 1.8100 3.1000 2.6000 2.4000 2.2100 1.9000 1.8100 2.2100 1.8900 1.7200 1.1900  C 0.07115107  CC- 0.0641448 LAHDA-  0.111162  OENSITY 0.0012179 T  T3TAL  CN 11.4000 14.6000 22.1300 17.8000 11.0300 11.1000 11.8200 10.8000 10.2600 13.0000 10.7900 9.4100 8.4100  9 10 11 S 0 EOOV DIFF PECLET EMPTY RE MLS/SEC 9.7100 0.471 2.118 0.631 17.7100 3.811 0.638 0.448 1.067 1.711 26.1000 0.493 1.292 0.412 7.102 12.7000 8.114 1.124 0.48> 39.2000 1.989 0.16* 9.882 61.1000 10.914 2.171 10.2100 0.188 2.217 12.162 16.0000 0.131 60.2100 11.386 2.116 0.129 10.642 49.0000 1.912 0.486 19.0000 2.118 0.498 T 2 . S U 11.203 70.0000 1.020 0.118 0.171 17.192 81.0000 1.711  SIGMA 0.06606999  H 11  IS 12 1/SCH »fc HV3 1.162 4.269 7.003 8.636 10.187 13.298 11.811 11.092 17.090 12.781 11.763 23.193 24.973  2.334 1.648 1.446 6. 723 9.116 9.311 13.327 11.139 12.38) 13.173 12.126 14.386 16.647  RIM NO I*  COLUMN LENGTH CMS IBS.4000  PELLET BED PELLET DIFFUSIVITY DIAMETER POROSITY POROSITY CMS 0.209233964 0.2975 0.6300000 0 .  3 2 HETP MOLECULAR PECLET CMS 0.87)73 11.86113 1.12457 19.66026 1.44479 25.02214 1.33151 32.33381 1.86726 35.74592 2.19)78 42.40766 2.82904 54.43129 2.3)873 40.62036 1.69844 29.89659 1.15076 18.19792  -0.22863926 AA'  VISCOSITY  TEMP K E L V I N P R E S S ATM 296.00000 1.02000  C A R R I E R MW 29.00000  1 VELOCITY CM/SEC 8.34203 13.82720 17.59826 22.74061 25.14037 29.82562 38.2819) 28.56860 21.02649 12.79874  COLUMN OIAMtTER CMS 0.4150  0.1867108  GAMMA'  9.271126  0.0182169  6 PELLET  16.59226 27.50220 35.00280 45.2)089 50.00400 59.32293 76.14246 56.82273 41.82153 25.45638  3.8813*199 C C  6  3 RE  NTU 3695 6126 7796 10075 11138 13214 16960 12657 9315 5670  DENSITY  HVD OIA  0.0012179  0.1673621  WIDTH CN 1.4500 1.0500 1.0000 0.4200 0.9000 0.8600 0.8600 0.9600 0.9600 1.1100  0.07882721  0.0688371  LANDA" -0.38*100  a  7 TOTAL CM 8.9500 5.7000 6.8000 4.6000 3.8000 3.3500 2.9500 3.4400 4.2500 5.9700  9 0 EDDY D I F F MLS/SEC 0.7300 3.666 7.809 1.2100 1.5400 12.713 1.9900 16.160 23.672 2.2000 2.6100 32.715 3.3500 54.150 2.5000 36.266 1.8400 17.856 7.166 1.1200  SIGMA 0.18221121  N 10  10 PE : L E T 1.45B 1.898 t.428 (.238 1.138 1.687 k.7S5 <k . 2 5 7 1.855 1.934  11 EMPTY  3  12 RE  10.433 17.126 22.952 28.495 31.503 37.373 67.370 35.798 26.36B 16.938  I/SCH 24.366 62.211 84.994 101.219 156.925 218.725 362.033 242.449 119.383 49.236  <t  • YD  li 11 21 I< 21 31 21 23 11  RUN NO 55  COLUMN LENGTH CMS 121.0000  CARRIER MW 29.00000  COLUMN OlAHETER CMS 1.1500  PELLET BED PELLET OlffUSIVlTY OIAMETER POROSITY POROSITY CMS 0.19&6992U 1.0050 0.4540000 0.  TEMP KELVIN PRESS ATM 296.00000 1.0S500  2 3 I VELOCITY HETP MOLECULAR CMS PECLET CM/SEC 1.10204 34.12138 6.67826 0.81204 24.00399 4.69808 4.6980S 0.92790 24.00399 0.86016 18.25097 3.57209 0.79122 2.64024 13.48985 2.09666 0.75765 10.71253 1.70839 0.79877 8.72872 0.88864 1.04833 5.35626 0.91842 3.86841 0.75713 0.43680 0.93106 2.23178  4 PELLET RE 47.73147 33.57854 33.57854 25.53079 18.87058 14.98546 12.21038 7.49273 5.41142 3.12197  A 0.60157888  8 0.15542669  AA* 0.4058721  CC* 0.0954678  GAMMA*  0.395087  VISCOSITY 0.0132169 5  NTU  2054 1445 1445 1098 812 644 525 322 232 134  HYD OIA 0.2695177 6 WIDTH CM 1.2500 1.4500 1.5500 1.9500 2.5000 3.1000 3.9500 7.2000 2.2000 5.9000  C 0.06020300  LAMDA*  0.299293  DENSITY 0.0012955  10 7 9 8 Q EDDY OIFF PECLET TOTAL CM ' MLS/SEC 3.680 5.5500 0.548 3.4400 7.5000 2.4200 1.908 0.404 0.452 2.180 7.5000 2.4200 9.8000 1.536 0.428 1.8400 1.045 0.394 13.1000 1.3600 0.794 16.6000 0.377 1.0800 20.6000 0.682 0.397 0.8800 35.6000 0.5400 0.466 0.442 0.348 0.457 10.7000 0.3900 0.233 28.5000 0.2250 0.453 SIGMA 0.05700413  N 10  11 21.670 15.245 15.245 11.591 8.567 6.803 5.544 3.432 2.457 1.417  12 13 1/SCH »E HYD 26.170 13.566 15.501 13.926 7.428 5.649 4.852 3.313 2.473 1.446  12.80} 9.005 9.005 6.347 5.361 4.319 3.275 2.309 1.451 0.837  RUN NO 6)  COLUMN LENGTH CMS  COLUMN DIAMETER CMS  PELLET DIAMETER CMS  BED POROSITY  0.6600  0.5680  0.4470000  421.0000  CARRIER MM 29.00000  0.  DlfFUSIVITV 0.7614962)8  TEMP KELVIN PRESS ATM  VISCOSITY  HVD OIA  296.50000  0.0182409  0.1747848  0.98)00  t 2 3 VELOCITY HETP MOLECULAR CM/SEC CMS PECLET 9.46495 0.49963 7.05990 8.6)156 0.49816 6.43828 7.91722 0.49359 5.90546 7.20789 5.37263 0.49283 5.01742 6.72666 '0.49366 5.95280 0.53257 4.44019 6.29799 3.93177 0.34366 3.92886 0.64628 2.93053 2.67876 1.99809 0.91661 1.66678 1.24344 1.24325  •  4 PELLET  S RE  34.53528 31.44443 28.88800 26.28156 24.54344 21.72030 19.33107 14.33540 9.77413 6.08168  0.10611326  B 1.87872801  AA.  CC—0.02096S4  0.6070601  GAMMA.  PELLET POROSITY  1.21137T  NTU 2616 2386 2188 1491 1859 1645 1464 1086 740 460  6 WIDTH CN 1.0000 1.1000 1.2000 1.3000 1.3900 1.6200 1.8700 2.7000 , 0.9300 9.3300  C 0.01889421  DENSITY 0.0011718  T TOTAL CN 12.3300 13.5600 14.8300 16.1000 17.2900 19.3000 22.0600 29.2000 8.4500 72.9000  B 9 10 11 0 EDOV 0 1 FF P E C L E T E M P T Y RE MLS/SEC 2.366 0.4621 17.164 1.5900 0.41« 1.4500 2.130 t 15.653 0.6)4 > 1.956 14.3S7 1.3300 0 . 4 3 «> 13.062 1.2100 1.773 0.43! 1.1300 1.660 12.198 0.4641 1.0000 1.585 10.795 0.8900 0.471» 1.440 9.698 7.125 0.6600 0.569 1.270 0.4500 1.226 4.868 0.2800 1.093 1.036 1.323  SIGMA 0.02249600  o.eje  N 10  12 1/SCH 11.189 13.811 12.552 11.402 13.666 13.181 9.211 8.116 7.878 6.617  13 « E NVO 10.627 9.691 B.B89 6.367 7.111 6.686 1.949 4.411 1.008 1.871  RUN NO 6*  COLUMN LENGTH CMS 421.0000  CARRIER MM 29.00000  COLUMN DIAMETER CMS 0.6600  PELLET BEO PELLET OlFFUSIVITV OIAMETER POROSITY POROSITY CMS 0.5680 0.4970000 0. 0.200754301  TEMP KELVIN PRESS ATM 297.00000 1.05000  HETP VELOCITY MOLECULAR CMS CM/SEC PECLET 0.71677 -23.69146 8.57)53 0.67842 7.81529 22.11203 0.64067 20.21671 7.14541 0.59639 6.36388 18.00651 0.52179 5.58235 15.79431 0.46804 4.63335 13.10928 0.47013 3.62853 10.26630 0.47348 2.45623 6.94950 0.58603 1.50723 4.26446 -0.05827764 AA> 0.2393774 GAMMA.  1.975639  VISCOSITY 0.0182648  PELLET RE 32.53779 30.36861 27.76558 24.72872 21.69186 18.00424 14.09971 9.54442 5.85680  0.79323625  NTU 8780 8194 7492 6672 5853 4858 3804 2575 1580  HVO OIA 0.1747848  6  MIOTM CN 1.3000 1.3500 1.4500 1.5500 1.6700 1.9200 2.5000 3.7000 1.4000  0.0813202*  CC« 0.0486036 LAMOA. -0.051301  OENSITV 0.001249S 7 B 9 10 11 T3TAL 0 EOOV OIFF PECLET EMPTY RE CM MLS/SEC 13.3500 1.5000 3.001 0.631 16.171 14.2500 1.4000 2.631 0.597 15.093 15.7500 1.2800 2.289 0.564 13.799 17.4600 1.1400 1.898 0.525 12.290 20.1000 1.0000 1.456 0.459 10.781 24.4000 0.8300 1.084 0.412 8.948 31.7000 0.6500 0.853 0.414 7.008 46.7500 0.4400 0.581 0.417 4.744 15.9000 0.2700 0.442 0.516 2.911 SIGMA 0.01178047  N 9  12  H IE  i/s: 20.530 18.131 15.65' 12.98; 9.96' 7.41 5.83! 3.37 3.021  RUN NO 65  COLUMN LENGTH CMS 119.5000  CARRIER MM 29.00000  COLUMN OIAMETER CMS 2.1750  PELLET BED PELLET DIAMETER POROSITY POROSITY CMS 0.5680 0.4630000-0.  TEMP KELVIN PRESS ATM 291.50000 1.05000  1 2 3 VELOCITY HETP MOLECULAR CM/SEC CMS . PECLET 7.08854 0.60142 20.46409 6. 78864 0.55490 19.59830 6.70685 0.54847 19.16218 6.18864 0.51881 17.86671 5.61611 0.51662 16.21386 5.34367 0.48800 15.42678 4.77111 0.47337 13.773V1 4.49850 0.47137 12.98663 1.57151 0.42841 10.31075 2.90085 0.41559 6.37454 2.13747 0.40664 6.17071 2.26288 0.41787 6.53277 1.55401 0.44896 4.48616 0.98149 0.55904 2.83349  4 PELLET RE 28.11152 26.94114 26.61675 24.56098 22.28882 21.20684 18.93468 17.85270 14.17396 11.51228 6.48274 8.98045 6.16729 1.89511  A 0.12397121  B 0.17161307  AA* 0.1811939  CC- 0.0491846  GAMMA*  0.944181  VISCOSITY 0.0180970 9  MTU 2152 2061 2016 1879 1705 1622 144B 1166 1084 880 649 687 471 298  0.1967*9071  MVD OIA 0.2465515 6 MIOTH CN 2.7000 2.7500 2.7500 2.9000 1.1500 1.2500 1.5500 3.7500 4.5500 9.4000 7.2000 1.6000 2.1000 4.1000  C  0.09709327  LAHOA*  DIFFUSIVITY  0.109131  OEMSITV 0.0012666  T  TOTAL  CN 16.1000 17.1000 17.2000 18.1000 20.1000 21.9500 21.9000 25.3000 12.2000 38.8000 52.3000 11.2000 19.9000 25.4000  8 9 10 11 0 EOOY OIFF PECLET EMPTY RE MLS/SEC 13.0000 2.119 0.911 11.825 12.4500 1.864 0.488 12.474 12.3000 1.839 0.481 12.324 11.3500 1.66T 0.474 11.372 10.3000 0.455 1.451 10.320 9.8000 1.304 0.410 9.619 8.7500 1.129 0.417 8. 767 8.2500 0.419 1.060 8.266 6.5500 0.765 0.177 6.563 5.3200 0.366 0.691 5.330 3.9200 0.439 0.158 3.928 0.499 4.1500 0.185 4.158 2.8500 0.149 0.193 2.899 1.8000 0.274 0.492 1.801  SIGMA 0.01086196  N 16  12 l/SCH « 14.943 13.160 12.691 11.644 10.116 9.110 T.690 7.408 9.349 4.212 3.016 1.462 2.417 1.917  13 HYD 12.211 11.696 11.954 10.661 4.979 9.209 8.219 7.769 6.152 4.997 1.6B2 1.698 2.677 1.691  RUN NO  COLUMN LENGTH CMS 119.5000  CARRIER MM 29.00000  COLUMN OlAMETER CMS 2.1750  PELLET SEO PELLET DIFFUSIVITY DIAMETER POROSITY POROSITY CMS 0.5680 0 *630000-0. 0.732060581  TEMP KELVIN PRESS ATM 293.50000 1.00500  1 2 VELOCITY HETP CM/SEC CMS 33.0*188 1.08620 26.*90*7 0.9562* 21.79055 0.72807 16.09367 0.59599 8.260*7 0.37585 7.*059* 0.37903 7.1*958 0.*0062 6.57989 0.37821 5.78233 0.39897 5.32658 0.39289 5.01325 0.1001*. 3.81691 0.*2907 3.30*19 0.*51S2 2.53511 0.52718 0.71956 1.70906 2.07936 0.5*927 1.39573 0.82132 1.36681 0.82605 0.*8*23 2.11930 0.17091 5.09239  VISCOSITY 0.0180970  4 3 MOLECULAR PELLET RE PECLET 25.63693 125.50986 20.55375 100.62*29 16.90711 82.77159 12.*869S 61.13196 6.*0923 31.377*7 5.7*621 28.13152 5.5*730 27.1577* 5.10529 2*.99377 4.*86*6 21.96*23 4.13285 20.23306 3.8897* 19.0*288 2.96151 l*.*9855 2.56369 12.55099 1.96697 9.6296* 1.32605 6.*9189 1.61336 7.898*7 1.0829* 5.30171 0.6*092 3.13775 0.37571 1.83937 0.13260 0.6*919  0.12011538  0 0.86630061  AA—0.129712*  CC- 0.0396874  A  GAMMA-  0.591686  5  NTU  2696 2162 1778 1313 67* 60* 583 537 *7l *3* *09 311 269 206 139 169 113 67 39 13  HYO DIA 0.2*65515 6 WIDTH CM 0.9000 0.9500 1.0500 1.2000 1.8000 2.1000 2.2000 2.3500 2.7000 2.9500 3.1000 *.2000 5.0500 6.8500 12.0500 2.0000 3.6000 7.9000 3.3000 13.3000  C 0.027*5608  LAMOA*  0.105735  DENSITY 0.0012102 7 8 9 10 11 12 0 EDDY OIFF PECLET EMPTY RE TOTAL MLS/SEC CM I/S:H « 58.0000 17.9*5 0.956 58.111 120.008 *.0000 *6.5000 12.666 0.8*2 *6.589 8*.701 *.5000 38.2500 7.933 0.6*1 38.323 53.3*9 S.7000 28.2500 *.796 0.525 28.30* 32.072 7.2000 1*.5000 1.552 0.331 1*.628 10.381 13.6000 9.386 13.0000 0.33* 13.325 15.8000 i.*o* 3.578 12.5500 0.353 12.57* 16.1000 l.*32 8.321 11.5500 0.333 11.572 17.7000 1.2** 7.71* 10.1500 0.351 10.167 19.3000 1.153 6.998 0.3*6 9.363 21.8000 9.3500 1.0*6 6.708 0.352 8.817 22.7000 8.8000 1.003 5.*76 0.378 6.713 29. 7000 6.7000 0.819 *.992 0.398 5.811 3*.8000 5.8000 0.7*6 *.*69 0.*6* *.*S9 43.7000 *.*500 0.668 *.112 0.633 3.006 65.8000 3.0000 0.615 3.819 0.*8* 3.6S7 12.5000 3.6500 0.571 3.833 0.723 2.*5S 18.*000 2.4500 0.573 3.775 1.233 l.*53 31.3000 l.*500 0.565 3.431 1.866 0.852 10.5000 0.8500 0.513 2.910 *.*83 0.301 27.3000 0.3000 0.*35  SIGMA 0.07*73090  N  20  13  *»o  5*.*83 *3.678 35.929 26.536 13.520 12.211 11.783 13.3*9 9.53* 8.783 8.266 6.293 S.**8 *.181 2.818 3.*28 2.301 1.362 3.798 0.282  RUN NO 69  COLUMN LENGTH CMS 186.1000  CARRIER MM 29.00000  COLUMN OIAMETER CMS 6.2700  PELLET BED PELLET DIFFUSIVITY DIAMETER POROSITY POROSITY CMS 1.0100 0.4040000 0 . 0.209211964  TEMP KELVIN PRESS ATM 296.00000 1.02000  1 2 VELOCITY HETP CM/SEC CMS 26.08798 2.35665 21.90749 2.09059 21.41551 2.08198 I.91579 18.61202 15.57491 1.81694 15.57491 1.65182 12.45991 1.66300 8.72195 1.27502 1.58221 1.00956 4.61154 1.03061 1.02574 1.95211 1.12914 0.92915 0.92915 2.60880 2.12208 0.95168 1.11456 1.46015 1.14091 0.79821  1 MOLECULAR PECLET 128.42380 117.68987 105.4225? 91.62175 76.6709? 76.6709? 61.33674 42.93572 17.63431 22.80960 19.45525 16.38841 12.84218 10.44641 7.18790 1.92918  VISCOSITY 0.0182169  4 PELLET RE 179.64859 164.61318 147.47272 128.16720 107.25289 107.25289 85.80211 60.06162 24.66816 11.90771 27.21542 22.92510 17.96486 14.61321 10.05496 5.49671  A 0.70091049  B 0.11824411  AA* 0.7012348  CC* 0.0610176  GAMMA*  0.760496  5 NTU 11614 10641 9514 8285 6913 6931 5547 1882 1594 2062 1759 1462 1161 944 650 155  HYD OIA 0.1947190 6 MIOTH CM 2.1500 2.2500 2.4200 2.5600 2.6900 2.8000 3.3000 4.1000 2.0500 1.6500 1.9000 2.2000 2.8000 1.5000 1.1500 2.0500  C 0.06312746  LANOA*  0.34024S  OENSITV 0.0012179  7 TOTAL CM 8.1000 9.0000 9.7000 10.6000 12.4000 12.6000 14.8000 21.0000 11.8300 9.4000 10.6500 11.2000 16.8000 70.7500 6.1000 11.1000  8 9 10 11 0 EDDY OIFF PECLET EMPTY RE MLS/SEC 72. 7!18 1.144 115.0000 30.740 66. 61r6 1.015 107.0000 24.990 275.0000 22.293 1.011 59. 7,16 51. 908 219.0000 18.014 0.940 41, 43I f 14.149 200.0000 0.862 41. 431? 12.864 200.0000 3.832 0.607 34. 7!10 160.0000 10.160 24. 3i E5 112.0000 0.619 5.560 9, 9111 46.0000 1.808 0.493 59.5000 2.188 0.500 12. 91t l 50.7500 2.027 0.498 11. Si•2 9. 2115 42.7500 1.547 0.461 13.5000 7. 21r6 0.451 1.212 S. 9 18 27.2500 1.010 0.462 0.341 4. 3 r2 18.7500 0.814 10.2500 0.554 0.455 2..21E6  SIGMA 0.07076120  N 16  12 13 1/SCM »E « V » 205.514 167.378 149.046 120.439 94.598 86.301 69.26? 17.175 12.089 15.964 19.551 10.340 8.109 6.751 5.440 9.944  68.849 63.994 56.918 49.119 41.104 41.194 32.889 23.918 9.494 12.228 10.499 8.789 6.889 9.609 9.B99 2.10?  RUM NO TO  COLUMN LENGTH CMS 186.3000  CARRIER MK 29.00000  COLUMN DIAMETER CMS 6.2700  PELLET BED PELLET DIFFUSIVIfV DIAMETER POROSITV POROSITY CMS 1.0300 0.6030000 0 . 0.768330810  TEMP KELVIN PRESS ATM 296.30000 1.00000  VISCOSITY 0.0182409  4 1 1 2 VELOCITY HETP MOLECULAR PELLET RE PECLET CMS CM/SEC 0.92911 6.37736 31.19642 4.61473 4.62164 22.62741 3.36167 1.00033 3.96874 19.41408 1.00200 2.88428 1.10177 3.31181 16.20072 2.40688 1.19846 2.32650 11.38067 1.69078 6.69411 1.18211 1.16811 0.99418 4.19827 22.49316 1.01170 1.34178 11.10819 73.90740 10.98014 0.98300 1.11381 19.26892 94.21871 14.00366 17.82284 1.20311 24.32408 119.96563 I.12841 29.34111 143.51031 21.37371 1.16736 33.88484 165.71609 24.62171 1.33798 26.71373 130.67681 19.41416  B  A 0.67149119  0.81181611  AA* 0.1)90088  CC- 0.0297571  GAMMA.  0.171649  1  NTU 176 418 318 299 210 121 411 1166 1742 2217 2613 1064 2411  •  HVD DIA 0.3967)90  *KlDTH CN 1.6000 2.3000 2.TO0O 3.4000 1.1000 10.2000 2.4000 1.0000 2.1000 2.2000 1.8000 1.7000 2.1000  C 0.0232)081  LAMDA*  0.127910  DENSITY 0.0011920 T  TOTAL  CN 9.6000 11.3000 11.6000 18.TO0O 28.0300 46.9000 11.8000 17.1000 11.7000 11.6000 9.8000 9.1300 10.1000  B 9 10 11 0 EOOV DIFF PECLET EMPTY RE MLS/SEC 12.611 0.411 18.2100 2.111 9.164 0.486 42.2100 1.6SI 0.486 1.441 7.461 16.2100 0.117 6.161 10.2100 1.111 1.013 3.182 4.60* 21.2100 0.787 0.768 12.1000 2.711 0.491 4.110 42.0000 1.690 0.477 29.9)2 118.0000 3.1*7 7.799 0.141 18.171 176.0000 0.184 48.186 224.0000 10.722 0.148 58.1)6 268.0000 12.0)1 0.167 67.111 309.3000 14.374 244.0000 12.988 0.610 12.924  SIGMA 0.06791941  N IS  If 11 1SSCM *E HYO 14.371 10.988 9.441 8.696 6.621 1.142 11.042 11.268 13.966 TO.066 78.621 91.910 84.871  11. * 8. 6 T. 4 6. 2 4. 1 2. 1 a. 6 20. 1 16. 1 41. 9 11. 0 61..1 13..3  RUN NO TI  COLUMN LENGTH CMS 122.0000  CARRIER MM 29.00000  COLUMN DIAMETER CMS I.1100  PELLET BEO PELLET DIFFUSIVITY DIAMETER POROSITV POROSITY CMS 1.0050 0.4540000 0. 0.725219170  TEMP KELVIN PRESS ATM 290.00000 0.99400  VISCOSITY 0.0179282  1 2 3 4 VELOCITY HETP MOLECULAR PELLET RE CM/SEC CMS PECLET 26.98967 1.10044 37.40195 183.28539 26.47064 1.05343 36.68268 1T9.7606T 24.39451 1.04817 13.80560 165.66179 20.76129 1.01516 28.77073 140.9B8T6 18.16613 0.88690 25.17439 123.36516 14.13290 O.B0681 20.13951 98.69213 11.93774 0.74504 16.54317 81.06854 0.73362 12.22756 59.92022 B.82155 0.69510 7.91195 38.77191 S.70935 1.55710 1.01222 2.15780 10.57416 5.50174 0.62926 7.62424 37.36202 1.J2I81 0.77561 4.60332 22.55820 1.45329 1.11689 2.01395 9.86921 0.5T094 2.34615 0.79119 3.87719  a 0.30983961  B 1.13807026  AA- 0.3436004  CC" 0.0267854  GAMMA*  0.784638  5  NTU  22TO 2226 2051 1746 1527 1222 1006 742 480 130 462 279 122 48  HVO DIA 0.2695177 6 WIDTH CN 1.3000 1.2500 1.4000 1.5500 1.6500 1.9000 2.2500 2.8000 4.4000 15.8000 1.0000 I.T500 4.9000 3.6000  c  0.02831543  LAHOA*  0.154149  DENSITY 0.0012114 7 TOTAL CN  5.8000 5.TO00 6.4000 7.2000 8.2000 9.9000 12.2000 15.3000 24.7000 73.5000 5.9000 4.3000 21.7000 11.0000  8 9 10 11 12 IS 0 EOOV OIFF PECLET EMPTY RE l / S C H «E HYO MLS/SEC 13.0000 16.850 0.547 83.212 100.346 49.15) 12.7500 13.942 0.524 81.611 94.212 48.208 11.7500 12.785 0.521 75.210 86.389 44.427 10.0000 10.538 0.535 64.009 71.207 37.113 0.441 56.308 54.4)4 S3.384 8.7500 8.056 0.401 44.806 39.615 26.467 7.0000 5.863 0.371 36.835 33.350 21.741 5.7500 4.467 0.365 27.204 21.870 16.369 4.2500 3.2)7 0.346 17.602 13.408 13.398 2.7500 1.9B4 0.536 0.7500 0.788 4.801 5.325 2.8)6 0.313 16.362 11.697 10.323 2.6500 1.731 0.386 10.241 1.6000 1.288 8.705 6. 353 0.556 0.7000 0.812 4.481 5.484 2.647 1.167 0.2750 0.670 1.760 4.526 1.040  SIGMA 0.03608801  N 14  RUN NO 72  COLUMN LENGTH CMS 122.0000  CARRIER MW 29.00000  COLUMN OIAMETER CMS 1.1500  PHIET BED PEllET OIAMETER POROSITY POROSITY CMS 1.0050 0.4540000 0 .  TEMP KELVIN PRESS ATN 294.00000 1.00000  2 VELOCITY HETP CM/SEC CMS 27.19777 2.35064 26.67474 2.13912 25.62867 2.17880 24.05957 2.21762 21.96743 1.93606 19.87530 1.87034 17.78116 1.73712 15.69102 1.57176 12.02978 1.39214 8.36855 1.15284 5.23034 0.99599 1.56910 0.84553  3 MOLECULAR PECLET 129.56047 127.06892 122.08583 114.61119 104.64500 94.67881 84.71262 74.74641 57.30559 39.86476 24.91548 7.47464  4 PELLET  •  0.17188688  AA-  CC«  GAMMA*  0.407367  TINE 16HRS 41MIN I2.6SEC  RE  181.33542 177.84820 1T0.87376 160.41210 146.46322 132.51434 118.56546 104.61659 80.2060$ 55.79551 34.87220 10.46166  0.64111141 0.9882264  VISCOSITY 0.0181210  S NTU 7863 7712 7410 6956 6351 5746 5141 4936 3478 2419 1912 493  0.210971311  HYD 0 1 * 0.2695177  6 U10TH CM 1.9000 2.0000 2.090O 2.1000 2.2000 2.2500 2.4900 2.6000 3.0000 3.9000 S.8000 16.7000  C 0.06044230  0.0629386  LAMOA*  DIFFUSIVITY  0.318961  DENSITY 0.0012022  7 TOTAL  CN 9.8000 6.4000 6.5000 6.6000 7.4000 7.7000 8.7000 9.7000 11.9000 17.0000 27.2000 83.0000  8 9 10 11 12 0 EOOV OIFF PECLET EMPIY RE 1/SCH « MLS/SEC 13.0000 1.16 31.966 82.326 212.967 12.7500 1.06' 28.930 80.743 189.279 1.08 12.2900 T T . 5 T 7 189.224 27.920 1.13 11.9000 7 2 . 8 2 7 176.981 26.67? 0.96i 66.696 141.375 10.5000 21.269 0.99 60.162 123.307 9.9000 18.987 0.86 93.829 102.469 8.9000 19.446 0.79 4T.496 7.3000 12.347 81.911 0 . 6 9 16.414 5.7900 8.374 99.951 0.57 2 5 . 3 3 1 4.0000 4.824 32.002 0.491 15.832 2.5000 2.609 17.280 0 . 4 2 4 . 7 9 ) 0.7900 0.663 6.401  SIGH A 0.09391870  N  12  19 HYJ 48.633 47.699 45.826 49.319 39.2TB 39.937 91.79? 28.099 21.90* 16.969 9.392 2.9)9  • OAVIS  IM  FORTRAN SOURCE LIST GAAA  SOURCE STATEMENT  C I 61 1*1 * 1 64 A T 16  01/27/69  PROGRAM FOR COMPUTATION OF RESULTS FROM POROUS PELLET RUNS REACH,NO,CL,CO,OP ,EB jEP*D08S_ FORMAT lllr)Fi0.4,2FiO.T FIZ.9 I IF <NOI 64,61.84 * CALL SKIP TO III PRINT 16 OFORKATIRH COLUMN. 2X,6HC0LU*N,)X.6HC0LURN,4K,6HPELLET.1S,1MBED,68« liHPeilET.^X.llHDIfFUSIVITT I _ 10 PRINT IT 11 IT 0F0RMATt4X,2KN0 .4I6HLENGTH,2X,6H0IAMETER,2X,8H0IARETCRt»fBNPOROS 1ITV,2X,8HP0R0SITV /11X,21HCMS CMS CMS I 12 READ 61.OH.T,TOSS.P.VISC .ADS 11 81 F0RNAT(4FI2.1.2F12.T I 14 P»D0BS»IT/T08SI»»I.T tP 11 PRINT S30.NO.Cl . CD'. OP . EB~7~t>. 6" 16 1*0 F0RNATIIX,l9,SF10.4,2Ft0.T,FI2.9> IT PRINT 61 20 61 FORMATI/11M CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY , I I2H MVO DIA ,10H OENSITV • 21 HO .E6«CD/l3.«CO>tl.-EBI/t2..0PI*l.l 22 VISC* 0.01709'fl27).l»ll4.l/imi4.ll><Tmi.l««l.S 21 RHO.29..2T3.*P/(22400..TI 24 PRINT 64.GM , T .P . VISC .MD.RHO 21 64 F0RHATI1X.1F12.1.SF12.7 // I 26 M-0 27 SMS* 0.0 M IH • 0.0 "~ 11 SU • 0.0 12 SUS" 0.0 11 SM • 0.0 14 IMS' 0.0 18 SHU- 0.0 _ 16 SMM. 0.0 " " 17 SHMS « 0.0 40 SHM2 • 0.0 41 SHM1 • 0.0 42 SMI • 0.0 41 SM4 "JI.O__ _ 44 S6-1.0 41 SGU'0.0 46 PRINT60 47 60 F0RHATI74H VELOCITY HETP RICIP VEL Ri NTU MID 1TH TOTAL 0 / 228H CM/SEC CMS SECSCN.tTl.tOHCH CM .7X.6HCC/SCC I SO CRE • OP'GH.160./TZ24OSYWISC .7). 2TI..F 11 cm • CL/12. • oi 12 10 READ 19.0.MOTH, TOTAL 11 19 FORMAT! 3F12.1 I 14 IFIB* 1.6.1 11 1 HN '12.36* TOTAL/ NIDTN|M|__ _ 16 H • CL/HN . IT V* 0*4./(1.14119*fB*P*CO»2l »T/296. 60 M • l./U 61 RE«CRE*U 62 NTU* CUL'U (  t  PAGE 1  -17J-  COLUMN NO 96  COLUMN LENGTH CMS 129.6000  C A R R I E R MW 29.00000  COLUMN OIAMETER CMS 0.6600  PELLET BEO PELLET OIAMETER POROSITY POROSITY CMS 0 . 5 9 T 0 0.•.710000 0 . 3 1 0 0 0 0 0  TEMP K E L V I N P R E S S ATM 296.00000 1.02800  VELOCITY HETP R E C I P VEL CM/SEC CMS SEC/CM 0.95941 0.58797 1.04231 1.31919 1.00155 0.75804 2.27860 2.14265 0.43887 2.87823 2.85963 0.34744 3.05812 3.46638 0.32700 4.79705 5.35024 0.20846 5.75646 7.57958 0.17372 6.17620 7.30867 0.16191 7.37546 9.87107 0.13558 7.97509 10.41804 0.12539 8.63469 12.80894 0.11581 8.75461 11.75458 0.11423 9.59410 12.69080 0.10423 10.31365 12.82180 0.09696 11.99262 14.12322 0.08338  -0.26101748  RE 3.85938 5.3066S 9.16602 11.57814 12.30177 19.29689 23.15627 24.84475 29.66898 32.08109 34.73441 35.21683 38.59379 41.48832 48.24224  B -0.63914265  VISCOSITY 0.0182169  NTU 299 411 711 898 954 1497 1796 1927 2302 2489 2695 2732 2994 3219 3 743  OIFFUSIVITY  0.207605682  HYD O I A 0.16559*5  DENSITY 0.0012275  WIDTH TOTAL CM CM 17. 3 0 0 0 2. 7500 2. 3900 11. 5200 1. 7 6 0 0 5. 8000 1. 5 6 0 0 4. 4 5 0 0 7. 7 0 0 0 19 9500 5. 8500 12.2000 1. 1 7 0 0 2 0500 5. 1000 9. 1 0 0 0 4. 9 5 0 0 7. 6 0 0 0 4. 5 5 0 0 6. 8 0 0 0 4. 6 0 0 0 6. 2 0 0 0 4. 3 0 0 0 6. 0 5 0 0 4. 1000 5. 5500 3. 8 6 0 0 5. 2 0 0 0 3. 3 5 0 0 4. 3 0 0 0  SIOMA 0.63283886  1.32167980  EFFECTIVE OIFFUSIVITY * 0.00085020 C O R R E C T I O N OF S L O P E FOR O I F F U S I V I T Y TERM « 0 . 0 6 0 5 3 1 8  AND  Q CC/SEC 1600 2200 3800 4800 5100 8000 9600 0300 2300 3300 4400 4600 6000 7200 0000  NEW  SLOPE C -  H  15  1.26115  COLUMN NO 57  COLUMN LENGTH CMS 421.0000  CARRIER Mil 29.00000  COLUMN OIAMETER CMS 0.6600  PELLET BEO PELLET OIFFUSIVITY OIAMETER POROSITY POROSITY CMS 0.5970 0.4.710000 0.3100000 0.210264673  TEMP KELVIN PRESS ATM 296.00000 1.01500  VELOCITY HETP RECIP VEL CN/SEC CMS SEC/CM 10.62794 0.09409 16.53753 0.10488 9.53478 15.40599 0.12019 8.32016 13.49911 0.13722 7.28773 11.65843 0.16466 6.07311 9.47155 0.19147 5.22287 8.50542 0.24215 4.12971 6.47520 0.33604 2.97582 4.53345 0.49897 2.00413 3.37384 0.82330 1.21462 2.20735 2.05825 0.48585 1.75714 -0.22380747  VISCOSITY 0.0182169  RE  NTU  42.21196 10639 37.87016 9545 33.04593 8329 28.94534 7295 24.12112 6079 20.74416 5228 16.40236 4134 11.81935 2979 7.95997 2006 4.82422 1215 1.92969 486  B 0.57967149  EFFECTIVE OIFFUSIVITY » 0.00069313 CORRECTION OF SLOPE FOR OIFFUSIVITY TERM  HYD OIA 0.1655945  DENSITY 0.0012120  WIDTH TOTAL CN CM 18. 6000 8. 7000 9 ,3000 20. ,6000 10. 1000 23. ,9000 10 8000 27, ,5000 12 ,0000 33. ,9000 2 ,6500 7,,9000 3, 0000 10.,2500 3 6000 14,,7000 4, ,5000 21, ,3000 I ,3500 7.,9000 3, ,4000 22, ,3000  1.60895641  0 CC/SEC ,7500 ,5700 ,3700 ,2000 ,0000 ,8600 ,6800 ,4900 ,3300 ,2000 ,0800  SIGMA 0.19640320  0.0620292 ANO NEW SLOPE C >  N 11 1.54693  COLUMN NO SB  COLUMN LENGTH CMS 421.0000  CARRIER MM 29.00000  COLUMN DIAMETER CMS 0.6600  PELLET BED PELLET OIFFUSIVITY OIAMETER POROSITY POROSITY CMS 0.9970 0.4710000 0.5000000 0.204915337  TEMP KELVIN PRESS ATM 296.00000 1.00790  VELOCITY HETP RECIP VEL CM/SEC CMS SEC/CM 0.73391 5.41801 1.36257 2.20172 0.45419 3.35020 0.31444 3.18026 4.786*6 0.18794 S.32082 8.01792 0.16351 6.11589 8.98325 0.14864 6.72748 10.17581 0.13626 7.33907 11.14604 0.11515 8.68456 12.20968 0.10901 9.17384 13.47536 0.09290 10.76397 17.10708 -2.20553330  VISCOSITY 0.0182169  RE  DENSITY 0.0012035  WIDTH TOTAL CM CN 81. 8000 21c .9000 4. 8000 22. 8000 15. 3000 3, 8500 2, 8400 8. 7200 2, 6200 7. 6000 2, 5500 6. 9500 2, 4000 6. 2500 2, 1100 5. 2500 4. 9500 2. 0900 20. 6000 9, 8000  NTU  2.89453 753 8.68360 2261 12.54298 3266 20.98537 5465 24.12112 6282 26.53323 6910 28.94534 7539 34.25199 8921 36.18168 9423 42.4531711057  B 4.65074056  HYO DIA 0.1655945  1.69947962  SIGMA 0.40248025  EFFECTIVE OIFFUSIVITY « 0.00126920 CORRECTION OF SLOPE FOR OIFFUSIVITY TERM - 0.0613776 AND NEW SLOPE C * OIFFUSIVITY WITH K » l / ( b P » A D S > - 0.00450397 TORTUOSITY* 22.74828 AOSORPTION 1.37000MLS GAS/ML OF PELLET  Q CC/SEC 1200 3600 5200 8700 0000 1000 2000 4200 5000 7600 N 10 1.63810  COLUMN NO 39  COLUMN LENGTH CMS 421.0000  CARRIER HM 29.00000  COLUMN OIAMETER CMS 0.6600  PELLET BED PELLET OIFFUSIVITY DIAMETER POROSITY POROSITY CMS 0.5970 0.4710000 0.5000000 0.211745851  TEMP KfcLVIN PRESS ATM 296.00000 " 1.00790  VELOCITY HETP RECIP VEL CM/SEC CMS SEC/CM 11.98715 0.08342 17.32317 17.43029 25.13199 0.05737 0.04360 22.93459 30.70657 0.03586 27.88846 36.22900 0.03114 32.1084J 40.22707 0.02535 39.44749 42.23292 0.02096 47.70395 42.51876 0.01827 54.73722 42.09674 0.01565 63.91106 35.33998 0.01386 72.16751 35.09355 0.01287 77.67181 31.52819 0.01239 80.729 76 35.13947 68.01311398  VISCOSITY 0.0182169  RE  NTU  47.2773911916 68.7451917327 90.4541922799 109.9923027724 126.6358731919 155.5812139215 188.1447247423 215.8840154415 252.0656963535 284.6291971742 306.3382077214 318.3987680254  B -592.80078888  HYD DIA 0.1655945  DENSITY 0.0012095  MIDTH TOTAL CM CM 9, ,0000 18. 8000 7 1500 12. 4000 5, 8000 9. 1000 4, 9500 7. 1500 4 4500 6. 1000 3 7000 4. 9500 4. 0000 3 0000 2, 5000 3. 3500 8, 0000 11. 7000 10. 2000 6 .9500 9. 6000 6, 2000 6, 0000 8. 8000  -0.33316236  0 CC/SEC 9600 ,8500 ,7500 ,5600 ,2500 ,4500 ,8000 ,9500 ,4500 ,8000 ,7000 ,2000  SIGMA 2.48643523  INVALIO OUTPUT FORMAT. -0.59035632E-02 EFFECTIVE OIFFUSIVITY » XXXXXXXXXX CORRECTION OF SLOPE FOR OIFFUSIVITY TERM = 0.0190111 AND NEM SLOPE C -  N 12  -0.35217  COLUMN NO 60  COLUMN LENGTH CHS 420.0000  CARRIER MU 29.00000  COLUMN DIAMETER CMS 1.6000  PELLET BED PELLET DIFFUSIVITY DIAMETER POROSITY POROSITY CMS 1.3000 0.5220000 0.3800000 0.205018722  TEMP KELVIN PRESS ATM 295.00000 1.03500  VELOCITY HETP RECIP VEL CM/SEC CMS SEC/CM 0.58324 1.43971 1.71457 0.95687 1.04507 1.17827 0.85728 1.16647 1.20172 0.65708 1.52189 1.25354 0.52253 1.913TS 1.45079 0.41723 2.39674 1.59910 0.36577 2.73393 1.72312 0.30911 3.23515 1.88001 0.26895 3.71814 2.01567 0.26895 3.71814 2.04571 0.29405186  VISCOSITY 0.0181690  RE 5.17474 8.48981 10.34948 13.50284 16.97962 21.26496 24.25660 28.70365 32.98898 32.98898  0.49/73393  NTU 597 980 1194 1558 1960 2454 2800 3313 3808 3808  HYO OIA 0.4436744  OENSITV 0.0012400  UIOTH TOTAL CN CN 5.9000 42, ,7000 3.1500 25, ,2000 2.5500 20. ,2000 2.0500 15.,9000 8.3500 60. ,2000 7.1500 49. ,1000 6.5000 43. ,0000 5.7000 36. ,1000 5.1500 31.,5000 5.1800 31.,4500  0.43967650  Q CC/SEC ,6400 .0500 ,2800 ,6700 ,1000 ,6300 .0000 ,5500 ,0800 ,0800  SIGMA 0.04052734  EFFECTIVE DIFFUSIVITY * 0.01938486 CORRECTION OF SLOPE FOR DIFFUSIVITY TERM - 0.1182664 ANO NEW SLOPE C -  N 10 0.32141  COLUMN NO 61  COLUMN LENGTH CMS 621.0000  CARRIER MM 29.00000  COLUMN OIAMETER CMS 0.6600  PELLET BEO PELLET OIFFUSIVITY OIAMETER POROSITY POROSITY CMS 0.S970 0.6710000 0.5000000 0.773230260  TEMP KELVIN PRESS ATM 299.00000 0.98200  VELOCITY HETP REC1P VEL CM/SEC CMS SEC/CM 0.95112 1.80907 1.05139 1.71202 0.58411 1.48969 0.43808 2.28269 1.55270 0.29205 3.42604 1.58861 0.19714 3.07265 1.87352 0.16259 6.15059 2.07753 0.139S6 7.16512 2.33134 0.12822 T.79920 2.57822 0.11185 8.94055 2.79545 0.09501 10.52575 3.05998 0.08860 11.28665 3.42417 0.06334 15.78863 4.24708 0.03963 25.23645 6.23979 0.64075078  VISCOSITY 0.0183602  RE 3.58993 6.46187 8.61583 12.92374 19.14628 23.21486 27.04412 29.43741 33.74532 39.72853 42.60047 59.59280 95.25274  0.87210897  NTU 258 466 621 932 1380 1674 1950 2123 2433 2865 3072 4298 6870  HYO OIA 0.1635965  DENSITY 0.0011608  WIDTH TOTAL CN CM .2500 40. 4000 9200 20. 8000 2000 15. 3500 3500 SO. 7000 4000 34. 3000 7000 28. 3500 2500 24. 2000 1000 22. 2000 7500 19. 5000 3500 16. 6500 1500 14. 8000 5600 10. 8000 7000 26. 8000  0.22362800  Q CC/SEC ,1500 ,2700 ,3600 ,5400 ,8000 ,9700 ,1300 ,2300 ,4100 ,6600 ,7800 ,4900 ,9800  SIGMA 0.074S0189  EFFECTIVE OIFFUSIVITY » 0.01016580 CORRECTION OF SLOPE FOR OIFFUSIVITY TERM » 0.0191109 AND NEW SLOPE C >  N 13 0.20452  COLUMN NO 62  COLUMN LENGTH CMS 421.0000  CARRIER MM 29.00000  COLUMN OIAMETER CHS 0.6600  PELLET BEO PELLET DIFFUSIVITY DIAMETER POROSITY POROSITY CMS 0.5970 0.4710000 0.3400000 0.767428882  TEMP K E L V I N PRESS ATM 297.50000 0.98100  VELOCITY HETP RECIP VEL CN/SEC CMS SEC/CM 1.89463 1.61910 0.52781 3.41034 1.64960 0.29323 4.42081 1.76095 0.22620 5.36812 1.89269 0.18628 6.31544 2.13187 0.15834 7.01014 2.208 75 0.142A5 7.83114 2.39186 0.12770 7.83114 2.49881 0.12770 8.58899 2.56615 0.11643 9.34685 2.73856 0.10699 10.23101 3.06171 0.09774 22.79873 5.85278 0.04386 0.37897281  VISCOSITY 0.0182887  RE 7.20795 12.97431 16.81855 20.42253 24.02650 26.66942 29.79286 29.79286 32.67604 35.55922 38.92293 86.73567  1.49184255  NTU 519 935 1212 1472 1732 1922 2148 2148 2355 2563 2806 6253  HYO O I A 0.1655945  DENSITY 0.0011655  MIOTH TUTAL CM CM 13, ,1500 89. 8500 7, 1500 48, 4000 38. 0000 5, 8000 30. 6500 4, 8500 26. 2000 4, 4000 23, 4000 4, ,0000 20. 8000 3, 7000 20. 9000 3, 8000 1». 0500 3, 5100 17. 6000 3, 3500 15. 9000 3, 2000 6. 9000 1, 9200  0.23788515  0 CC/SEC ,3000 ,5400 ,7000 ,8500 ,0000 ,1100 ,2400 ,2400 ,3600 ,4800 ,6200 ,6100  SIGMA 0.04411453  EFFECTIVE D I F F U S I V I T Y » 0.00561640 CORRECTION OF SLOPE FOR D I F F U S I V I T Y TERM « 0.0192997 ANO NEH SLOPE C -  N 12 0.21859  COLUMN NO 73  COLUMN LENGTH CMS 119.4000  CARRIER MW 29.00000  COLUMN DIAMETER CMS 2.1700  PELLET BED PELLET DIFFUSIVITY OIAMETER POROSITY POROSITY CMS 0.3200 0.3900000 0.3100000 0.742124423  TEMP KELVIN PRESS ATM 295.00000 1.00000  VELOCITY HETP RECIP VEL CM/SEC CMS SEC/CM 1.01778 9.09386 0.10996 0.98153 0.11208 8.92227 0.96925 0.11894 8.40753 0.89770 0.12951 7.72120 0.88500 0.14215 T.03487 0.81740 0.16652 6.00538 0.72322 0.19427 *.14747 0.72916 0.24284 4.11797 0.76273 0.34283 2.91690 0.92221 0.52983 1.88740 2.14016 1.45703 0.68633 -0.01542345  VISCOSITY 0.0181690  RE 19.18923 18.82717 17.74099 16.29275 14.84450 12.67214 10.86183 8.68946 6.15504 3.98267 1.44824  1.43413471  NTU  HYO OIA 0.1174626  OENSITV 0.0011981  WIDTH TOTAL CN CN 4.7500 21. 8000 4.9000 22. 9000 5.0500 23. 7500 5.3000 25. 9000 5.7500 28. 3000 1.6500 8. 4500 1.8000 9. 8000 2.2500 12 2000 3.1500 16. 7000 -.6000 27. 0000 4.8500 15. 3500  0.09974282  Q CC/SEC ,2500 ,0000 ,2500 ,2500 ,2500 ,7500 .5000 ,0000 ,2500 ,7500 ,0000  SIGNA 0.01539098  EFFECTIVE DIFFUSIVITY - 0.00447061 CORRECTION OF SLOPE FOR DIFFUSIVITY TERM « 0.0166358 AND NEW SLOPE C » TIME 17HRS 21MIN 55.4SEC  N 11 0.07911  - 181 -  TABLE A I I . 1  Manometers Inches O i l Low Flow (LF) •1.8 3.65 7.9 10.7 17.5 23.7 26.1  Flow Rate cm /sec. at 298°K 760 mm Kg 3  O.338 0.674 1.416 1.87 2.82 3.65 3.96  High Flow (HF) .91 1.5 2.8 4.1  5.1 7.7 7.0  10.2  15.1 8.1 14.4  18.1 21.1  2k.9 24.0  Very High Flow (VHF)  I.65 2.35 5.3 8.7 14.4  20.05 26.1  .811 1.31 2.37 3.21 3.91 5.88 5.38 6.90 9.46 5.85 9.00 10.50 11.70 12.68 13.45 12.9 15.9 26.3 33.9 44.0 52.0 59-5 Continued....  - 182 -  TABLE A I I . 1 (Continued)  Manometers Inches O i l  Flow Rate c m / s e c . a t 298°F 760 mm Hg 3  U l t r a High Flow (UHF) 26.7 23.3 19.6 17.6 13.9 10.0 6.8 ^.35 1.85 1.1 E x t r a H i g h Flow  287.5 271 246 233 204 171 139 106 6  4  4  8  4  8  .  4  (EHF) 26.75 23.3 17.6 II.85 8.25 6.35 5.77 1.88 .95  3  323 278 221 180 152 112.5 73.6 ^5.3  J  - 183  -  1 0 0 0 .  CL*  5 too  V H F  J  O LU  10 UJ  cr o 0-1  i  i i I m i  I  10  M A N O M E T E R INCHES  R E A D I N G O I L  Figure A II.1 Flow Meter C a l i b r a t i o n  Too  - 184 APPENDIX III TIME OF DIFFUSION OF A GAS FROM A SPHERICAL PELLET WITH A STEP CHANGE IN SURFACE CONCENTRATION RATE OF DIFFUSION FROM A SPHERICAL PELLET The d i f f u s i o n of a gas from a spherical p e l l e t with a step change i n the surface concentration i s given by Crank (Ref. 2, page 86). The amount of gas diffused at time t.(M-t) as compared to the amount of gas diffused at i n f i n i t e time (M ) i s given by, M  M  OO  = 1 = 6_  t  1 exp / - D E n 2 T T t \ 2  n  {  a*  J  where a i s the p e l l e t r a d i u s . For the second term i n the series to be less than 1$ of the f i r s . t , { (  ^P  D  E TT  2  t  Ln ( 5 ) = ^ T T D t a 2  g  \ = 100 exp / - k DE TT t \ ) — ( — £ 2 — ) 2  - 7T D t a 2  E  2  E  =  2  5TT DEt a 2  2  I f D£ = 0.01 cm /sec and a = 0.5 cms 2  t = .25 Ln(25) 3 TT x .01  '=-  2.82 seconds  2  Thus a f t e r ten seconds M+ Moo  =  1 - 6_ e TT  2  (\  .01 .25  2  x ION  J  = .9881  = 93.81$ Thus i n ten seconds 98.8$ of the gas w i l l d i f f u s e out of a 1 cm diameter p e l l e t i f the e f f e c t i v e gas d i f f u s i v i t y i s 0.01 cm /sec 2  MANUFACTURERS DATA ON POROUS PELLETS (INCLUDING CONTRADICTIONS AND VERIFICATIONS) Norton Catalyst Supports, l / 2 " diameter SA 205 mixture Information from two sources i s summarized below, the f i r s t from the manufacturer's general information sheet, and the second from data supplied by the manufacturer i n a private communication.  - 105 -  Manufacturer's Data Apparent P o r o s i t y Water absorption Bulk d e n s i t y Apparent S p e c i f i c G r a v i t y Packing Density Surface Area Pore Diameter Range  P r i v a t e Communication  0.4l 20$ 2.1 grams/ml 75 - 73 l b / f t l e s s than 1 meter /gram  .36 - AO 15 - 19$ 2.1 - 2.3 grams/ml 3.4 - 3.6  3  2  9°$ i n 2-40 microns  Using the bulk d e n s i t y value o f 2.1 and the s p e c i f i c g r a v i t y of 3-5>  a  p o r o s i t y o f O.382 can be c a l c u l a t e d .  As a f u r t h e r check on the  consistency o f the data, i f the water adsorbed i s assumed t o e x i s t as l i q u i d water occupying the pores, then 17$ water i n d i c a t e s a p o r o s i t y o f 0.37$. I n an experimental check, a p e l l e t was d r i e d by heating t o 500°F, and then put i n a vacuum which was released by water so t h a t the p e l l e t absorbed as much water as p o s s i b l e .  Weighing the p e l l e t between each  operation showed that the i n i t i a l water content was n e g l i g i b l e , but the evacuation and s a t u r a t i o n procedure y i e l d e d a water content o f 16.8$ which again i s an i n d i c a t i o n o f a 37$ p o r o s i t y i f the p e l l e t s p e c i f i c g r a v i t y o f 3.5 i s accepted. In c o n c l u s i o n , a value o f 38$ p o r o s i t y has been taken f o r the pulse apparatus experiments and the information a v a i l a b l e suggests an  error  l i m i t o f ± 1$.  *  A c t i v a t e d Alumina P e l l e t s Alcoa H 151 l / 4 " and l / 8 " diameter Two sources o f information were again used to determine the p r o p e r t i e s o f these p e l l e t s , but some of these data were c o n t r a d i c t o r y . Because one o f these sources was p r i v a t e communication i n the form o f a l e t t e r from the s u p p l i e d , which d i f f e r e d from the manufacturer's data i t was concluded t h a t the s u p p l i e r had not furnished the c o r r e c t data. ' This conclusion i s j u s t i f i e d i n the f o l l o w i n g paragraphs.  - 186 I ianufacturer' s Data v  Packing Density  52-55 l b / f t 51-53 I b / T t s 5.1-3.3 0.3 mls/gm 50 A° 350 meter /gm  Private Communication  3  S p e c i f i c Gravity Pore .Volume Pore Diameter Surface Area (BET) Pore Volume greater than 3OA Average Pore Diameter i n 90 to 30A° region Static Adsorption at 60% RH 22-25$ 2  0  O.5-G.55 ials/gm 40 A " 390 meter /gm 0.28 mls/gm 2  k2A°  A pore volume of 0.3 mis per gram represents a porosi'cy o f 50$ i f the s p e c i f i c gravity of the p e l l e t s i s taken as 3.2, while a pore volume o f 0.5 indicates a porosity of 63$.  The problem amounts to deciding  which set of data above are consistent. Placing the p e l l e t s i n a vacuum and releasing with water to measure the water absorbed showed a porosity of around 50 "to 55$ which i s somewhat indeterminate, l y i n g as i t does between the data from the two sources.  However, f o r the l/8 inch activated alumina p e l l e t s the t e s t  described below was a p p l i e d . The t e s t p e l l e t s were placed i n the sample loop of a gas chromatograph and a hydrogen purge put on the loop.  A i r c a r r i e r gas was  put on the column, which consisted of a 20 f t . length o f l / 2 " p l a s t i c hose to cause dispersion and create a Gaussian pulse d i s t r i b u t i o n .  The height  of the pulse was proportional to the gas i n the pulse, and so by noting the difference i n height between the pulse with the p e l l e t s i n the sample loop, and without, the volume of the s o l i d s i n the p e l l e t s could be determined.  The peak heights were calibrated i n terms of gas volume by  i n j e c t i n g known volumes o f hydrogen with a syringe.  The volume of the sample loop was found to be 4.80 mis while the volume of gas when kj dried p e l l e t s occupied the tube was 4.40 mis.  From the  mean diameter the o v e r a l l volume of the p e l l e t s was calculated to he 0.8l mis and so the porosity i s given by (0.8l-0.40)/0.8l = 50.5% S i m i l a r l y for 47 wet (12$) p e l l e t s , porosities of 28.4 and 33$ were obtained. I f the water i s assumed to exist as l i q u i d water (3l)> then the porosity of the wet p e l l e t s can be calculated from the moisture content i f the dry p e l l e t porosity i s known.  I f a 50$ dry p e l l e t porosity i s  assumed, then f o r a 12$ wet p e l l e t the porosity comes out to be 30.8$. Thus, the manufacturer's data appears to give the best agreement with the observations. F i n a l l y , two i n d i v i d u a l p e l l e t s were weighed and the dimensions of three diameters measured on c a l l i p e r s .  'This allowed the apparent  density of the p e l l e t s to be calculated, and i f the water content i s taken into account, the porosity of the dry p e l l e t s can be obtained from a knowledge of the true s p e c i f i c g r a v i t y . 0.699 cm p e l l e t weighed 0.3140 grams  so density = 1.76" gms/ml  0.617 cm p e l l e t weighed O.23OO grams  so density = 1.87 gms/ml  I f 12$ water i n the p e l l e t i s assumed then the densities become 1.57 and I.67 r e s p e c t i v e l y .  In conjunction with the s p e c i f i c gravity the  porosities of these two p e l l e t s when dried are;  1.57/3.2  = 49$  1.67/3.2  = 52$  The only anomaly l e f t i n the manufacturer's data i s the claim of 22 to 25$ s t a t i c adsorption of moisture i n a i r of 60$ RH. i n water only i4$ water was adsorbable.  Even with soaking  However, i n an Alcoa product data  - 188 ("Activated and Catalytic Aluminas", Feb. 1, 1963, Section GB2A,  bulletin,  Figure 2, page 8), i t i s shown that a f t e r about s i x months operation the adsorptive capacity of t h i s material drops to 13 or  lh /a. c  The samples used  i n t h i s work were stored for s i x months before work was started,  and so i t  i s possible that the low moisture contents are to be expected. The dry p e l l e t s were assumed to have a 50$ porosity i n the pulse apparatus determinations and the porosity of the wet p e l l e t s was taken as 31$ corresponding to a 12$ wet p e l l e t .  The l/k"  and l/8" p e l l e t s were  assumed to have the same properties. POROSITY OF PACKED BEDS Two general methods were used to obtain the porosity of the non porous p e l l e t beds A.)  I f the density of the p e l l e t packing was known, the bed was weighed  before and a f t e r f i l l i n g and the p e l l e t density used to convert the packing weight to a packing volume.  The o v e r a l l volume of the vessel was calculated  from the i n t e r n a l dimensions of the bed. Example Run 50:  5 cm. diameter by 111.8 cm. length bed packed with No. 9  lead shot having a density of 10.808 gm/ml. Weight of column + bungs + screens + packing  Weight of packing Volume of packing Volume of bed  15,051/10.808  T(5.0) 111.8 4 Bed Porosity = 2195 - 1592 2  2195  J  =  1051 grams  =  35.5 l h .  =  161028 grams  =  15,°51 grams  =  1392 m l .  =  2195 m l .  =  36.6$  - 189 B.)  The alternate method of porosity measurement was to weigh the bed,  (a)  empty, (b) packed, (c) packed and f i l l e d with water, (d) unpacked and  f i l l e d with water.  I f the density of water i s taken as unity the bed  porosity i s given by ( c - b ) / ( d - a ) . Example Run 54:  1/4" polyethylene tube packed with 2.975 mm glass beads.  Column length 184.5 cm and diameter 0.415 cm. Weight of tube " " " + packing " " " + water " " " + water  = 4J.0 grams =72.0 grams = 89.0 grams = 70.0 grams  11  Porosity of bed = (89.0 - 72.0)/(70.0 - H 3 . 0 ) =  63$  The p o r o s i t i e s of the beds of porous p e l l e t s are treated i n d i v i d u a l l y depending on the r e l i a b i l i t y of the available manufacturer's data. Norton Catalyst Support l/2" diameter SA 203 Mixture, RUN 60 The moisture content of these p e l l e t s was found to be n e g l i g i b l e and so the manufacturer's p e l l e t density was accepted as a value of 2.05 grams/ ml.  With the weight of p e l l e t s i n the bed measured, the porosity of the  bed (not including p e l l e t pores) was calculated by method A) above. Activated Alumina P e l l e t s Alcoa  iii51  l/4" diameter, RUNS 56,57,58,59,61,62  The bed used f o r these runs was a single p e l l e t diameter and the porosity was measured as f o l l o w s .  The average diameter was used to  calculate  the mean p e l l e t volume and then the number of p e l l e t s i n a measured length of tube was measured.  The volume of the p e l l e t s i s thus known and the volume  of the bed over the measured length can be calculated from the i n t e r n a l diameter of the v e s s e l .  -  1 9 0 -  Example Forty-four p e l l e t s i n l i n e occupy 2 5 . 0 cm. making a mean p e l l e t diameter of  O.568  cm.  (Note the p e l l e t s were graded so that a small p e l l e t  was not adjacent to a large p e l l e t , which would introduce an error into the result.) In the paeH©d "baa 18 p e l l e t s occupied 10 cm. p e l l e t s i s thus i s given "by  1 8 T T ( 0 . 5 6 8 )  1 0 1 T ( 0 . 6 6 )  2  / 4  =  3  / 6  =  3.I1-2  1 . 7 2 7  The voluma of the  mis. and the volume of the vessel  mis.  The bed porosity i s ( 3 . 4 2 -  1.727)/3.42  =  49.7$  Activated Alumina P e l l e t s , Alcoa H 1 5 1 1 - / 8 " Diameter, RUN 7 3 The above method could not be applied to a bed of several p a r t i c l e diameters t h i c k . 12$,  The moisture content of the p e l l e t s was determined to be  and weighing the bed before and a f t e r f i l l i n g showed that 4 6 7 grams  of the wet p e l l e t s packed the bed. 4ll  Thus the weight of dry p e l l e t s was  grams and since the density of the dry p e l l e t s was given as 3 . 1 "to  3 . 3 gm/cm. by the manufacturer, the volume of s o l i d can be calculated to be 1 2 9 m l .  'The 1 1 9 . 4 cm. long by 2 . 1 7 5 cm. diameter bed contains a volume  of 4 4 l m l . and so the porosity of the bed can be evaluated by method A i f the dry p e l l e t s are assumed to be 5 0 $ porous. The porosity i s thus  ( 4 4 l  -  1 2 9 / 0 . 5 ) / 4 4 l  = 4 l $  As a c o r o l l a r y to the above c a l c u l a t i o n , the porosity of the dry activated alumina p e l l e t s i s u n l i k e l y to be 6 5 $ as claimed i n the supplier's literature.  The assumption of a 6 5 $ p e l l e t porosity y i e l d s  impossible bed p o r o s i t i e s , f o r example, a value of 5 ^ $  i  s  obtained and a  5 4 $ bed porosity i s extremely u n l i k e l y i n a random packed bed of uniform spheres.  - 191 -  E s t i r a a t i o n o f t h e M o l e c u l a r D i f f u s i v i t y o f t h e Methane A i r System Ref vj>2) Mol Wt.  or  Air  28.97  3.617 A  Methane  16.04  3.822  °AB  -  AB  J  =  K  KT_ 6  From Table B.2  =  97.0 137.0  3.7195 A  €  At 298°K  K  =  AB Jl  0.212 cm /sec 2  91 x 137  298 115.4 O.990  =  115.4°K  2.54  Tc  Pc  132  36.4  86.6  190.7  45.8  99-7  - 192 APPENDIX IV ADSORPTION OP GASES BY ACTIVATED ALUMINA PELLETS THEORY AND APPARATUS This experiment was carried out i n order to obtain the degree of methane adsorption i n dry alumina p e l l e t s , adsorption i s known to influence the e f f e c t i v e d i f f u s i v i t y i n a porous p e l l e t (2). The following two assumptions were made, the adsorption isotherm i s l i n e a r , i . e . moles adsorbed/gm. s o l i d = W(partial press.) and the presence of other gases does not a f f e c t the adsorption isotherm. The apparatus i s shown i n Figure A I V . 1 .  The test chamber BC  could be evacuated while the burette zone AB was purged with the t e s t gas. The stop cock A.was then turned to shut o f f the purge gas and open the mercury tube to the burette.  The amount o f gas used i n the t e s t could be  adjusted by regulating the burette zone vent at B and adjusting the manometer level. The vacuum i n the t e s t chamber could be shut o f f at C, and the "burette and t e s t chamber connected at B.  Thus by knowing the volume o f  the test chamber and the tube connecting to the burette zero, a series of measurements o f the volume and pressure o f trapped gas could be made by a l t e r i n g the manometer p o s i t i o n .  A series of burette readings (volume) and  manometer readings (pressure) were taken at corresponding points as w e l l as the atmospheric pressure.  The volume of sample s o l i d s was also obtained.  Total gas i n the system =  PQVQ  Moles  RT A material balance of the trapped gas y i e l d s , PQVQ  R T  =  P_V R  T  + WP p Q p  - 195 -  VACUUM CONNECTION TEST CHAMBER  METER RULE  VENT  BURETTE  TEST GAS  >!  MERCURY  1  RESERVOIR  v  y  Figure A VI.1 Adsorption Measurement Apparatus  or V - FQVQ  P  Where p  - WRT (QLQ  - 19k -  i s the p e l l e t density, and Q the volume of p e l l e t s .  1?  Hence a p l o t of volume V against 1  should y i e l d a straight l i n e  P having an intercept WRT eppQ which i s the volume o f gas adsorbed.  I f the  o v e r a l l volume o f the p e l l e t s (including pores) i s known then the volume of adsorbed gas per u n i t volume of p e l l e t s i s e a s i l y obtained. RESULTS Table A IV. I shows the manometer and burette readings along with data i n terms of volume and inverse pressure f o r the methane, hydrogen and nitrogen.  Two sets of data are recorded f o r methane, one assumed atmospheric  pressure and the other at about l / 2 atmospheres i n order to t r y and approach the concentration i n the pulse apparatus.  These points are plotted i n  Figure A IV.2. Following are the c h a r a c t e r i s t i c s  o f the apparatus which had to  be known to prepare Table A IV. 1, Volume from zero o f burette to stop cofek B  =  6.7 m l .  Volume of empty t e s t chamber  =  25.29 m l .  P e l l e t s ALCOA H 151 activated alumina spheres 1 / V ' diameter. Weight o f dry p e l l e t s i n sample  =  Volume o f s o l i d excluding pores  =  Overall volume of p e l l e t (50$ porosity)  =  7.68 grams 7.65/3.2 = 2.k m l . 4.8 m l .  The o v e r a l l volume of the p e l l e t s was also computed from the dimensions of the p e l l e t s and the same r e s u l t was obtained. Hence volume to be added to the burette reading = 25.29 .+ 6.7 - 2.4  = 29.59 m l . Volume of gas = 29.59 + Burette reading Pressure i n chamber = Atmospheric ±  manometer pressure d i f f e r e n c e  - 195 -  RESULTS AND CONCLUSIONS  The intercept of the hydrogen data i s of the order of the accuracy of the experiment, and so i t may be concluded that the hydrogen intercept represents zero adsorption.  The intercepts were computed by the l e a s t  squares technique and methane showed 5.22 m l . adsorbed at a h a l f atmosphere and 5«387 ml, at one atmosphere.  Since the hydrogen intercept of + O.316  i s taken as zero this must be added to the methane r e s u l t giving 5«22 + .31 = 5.53 m l . i n k.8 ml. of p e l l e t at a h a l f atmosphere and 6.60 ml,, i n k.Q m l . p e l l e t at one atmosphere. Hence the methane adsorbed per u n i t volume of p e l l e t material i s 1.15 m l . at a h a l f atmosphere and 1.375 m l . / m l . p e l l e t at one atmosphere. The results for nitrogen are not p a r t i c u l a r l y of i n t e r e s t but i t can be seen from Figure A IV.2 that nitrogen adsorption i s s l i g h t l y higher than that of hydrogen as may be expected. nitrogen data was not carried out.  A l e a s t squares computation for the  Figure A IV.2 Gas Volume v s . Inverse Pressure For Adsorption Measurement  TABLE A I V . 1 RESULTS FOR ADSORPTION APPARATUS METHANE Low Pressure Atm. Pressure 759.0 mm Hg Boom Temp. 22°C  HYDROGEN  765.2 mm Hg 22 °C atm  Vol. mis.  62.5  1.215  70.89  -15.4  61.1  1.242  72.59  36.7  - 9.7  66.8  1.138  66.29  30.7  - 3.2  73.3  I.O38  60.29  41.49  25.3  + 4.3  80.8  .940  59.89  1.55  48.69  23.9  + 6.4  82.9  .917  55.99  44.1  1.72  5^.99  21.6  +10.7  87.2  .872  51.19  -35.0  40.9  I.85  58.99  28.4  0  76.5  .994  57.99  34.01  -57.5  38.4  1.98  63.60  49.0  -45  30.9  2.45  78.59  Burette mis.  Man. cm.  Press. mm Hg  29.59  41.3  -14.0  1.09  51.89  43.0  67.7  1.12  35.79  63.I  1.20  36.09  Man. cm.  Press. mm Hg  0  - 1.5  74.4  .1.02  2.5  - 6.2  69.7  4.2  - 8.2  6.5  -12.8  11.9  -19.7  56.2  1.35  19.1  -26.8  49.1  25.4  -31.8  29.4  Burette mis.  <— p  atm."  1  Vol. mis.  25.4517  8.86418  15.33  8.356  477.71  487.62  799.7635  Intercept= -5.2208 mis.  = Intercepts  517.23384 O.3167 mis,  H VO  -4  TABLE A IV. 1 (Continued) NITROGEN Atm. Pressure Room Temp.  METHANE High Pressure 7 6 5 . 2 22  Man. cm.  Burette mis.  °C  mm  Hg  7 ^ 9 . 5  22.5°C  Press. mm Hg  atm.  1  Vol. mis.  Burette mis.  Man. cm.  Press. mm Hg.  atm.  Vol. mis.  42.3  -17.0  5 9 - 5  1 . 2 7 9  7 1 . 8 9  31.0  -10.6  64.3  I.I85  60.6  39.0  -14.0  6 2 . 5  1 . 2 1 5  6 8 . 5 9  3 3 . 3  -12.7  62.2  1.225  62.9  35-2  -10.5  6 6 . 0  1 . 1 5 2  64.79  37.55  -16.4  5 8 . 5  1.30  67.2  31.0  -  6.1  70.4  1 . 0 8  6 0 . 5 9  40.8  -18.9  56.O  1.354  70. ii  0  7 6 . 5  .99^  55.54  26.8  -  6 . 5  68.4  1.11  56.4  -  2.3  72.6  1.045  52.9  7 ^ . 9  1.018  50.8  78.2  O.965  48.6  2 5 - 9 5  23.4  +  3 . 3  7 9 . 8  . 9 5 3  5 2 . 9 9  23.3  20.6  +  7.7  84.2  .901  50.19  21.2  0  19.0  7  +  1  3 . 3  =  -— p 2  n In"l ex c cpt  10.71984 9.202  =  548.0431  =  1:69.8  =  =  8  -6.387 moles  APPENDIX V  PROGRAM B DAVIS  IStt 1 2 3 <V S 7 10 11 12 13 14 15 16 17 20 21 22 23 24 25 26  27 30 31 32 33 34 35 36 37 40 41 42 43 44 46 46 47 50 51  SOURCE STATEMENT 6 1 5 2 9  101 61 102 62 103 63 105 3  Jl 20  50 71 51  52  52 53 70 54  TIME  FORTRAN SUURCE LIST GAAA  04/29/65  PI*3.14159 READ l.DB.EL.ELE.E FORMAT (4F10.5I IFIDB>70.70,5 READ3,S,q,P,T,TB,00.J FaRNATIFlO.a.SFlO.5 .12) 1F1S16.6.9 0«.0*TB/T CALL SKIP TO 111 1-0 IFIJ-621101,102,103 PRINT61 F0RMATI20H HYDROGEN-NITROGEN /I G0T0105 PRINT62 FORMATI 20H NITROGEN-ETHANE / ) G0T0105 PRINT63 FORMAT! 20H NITROGEN-BUTANE /) PRINT3.DB.0.EL,T,ELEtP F0RMATI9X.3HBED DATA.50X.8HRUN DATA// U8H BEO OIAMETER CMS ,F10.5,20X,18HFLOK RATE CC/SEC- .F10.5/ 218H BEO LENGTH CMS .F10.5.20X,IUHRUOM TEMPERATURE* .F10.4/ 318H ENO ZONE HEIGHT .F10.S.20X.18HATM.PRESSURE MM HG .F10.11 PRINT31.E.TB FORMAT(9H POROSITY .F10.5.20X,18HBED TFMPERATURE K> .F10.4//1 CALLABCOIS) A.1.S6/EL D-S/A..2 H*.4.*0/(D<>E*PI*0B**2> tK-ELE/E 0EL-H-EK*A*.2-A*SINIA.ELI/COS!4*EL1 1*I»1 IFUBSIDELI-.00001) 31.50.50 DDEL«-2.«fcK«A-A«EL/lC0SIA*El»>**2-SlN(A«EL)/C0S(A»Elt A.A-OEL/DOEL IFII-20) 71.71,51 G0TO20 0-<D»760./P AMDA*D0/D OEF'OoE PRINTS2.0.AMDA,DEF,00.A.I FORMAT(10X*15H0IFFUSIVITY. .F16.8/10X.7HLAM0A • ,F16.8/10X.25HEF 1FECTIVE OIFFUSIVITY. .F16.8/10X.23HPUBLISHED DIFFUSIVITY* ,M6. 2BM0X.6HALPHA* ,F16.S.20X,20H NUMBER ITERATIONS* .12) GOT06 STOP ENO  NO MESSAGES FOR ABOVE 17HRS OOMIN 0 6 . 0 S E C  ASSEMBLY  6  OAVIS  HIRAM ISM  SOURCE 1 2 i *  2  5 6 7 10 11 12 13 14 15 16  SO 26 2S 41 27  17 20 21 22 23 24 25 26 27  LIST  JAAA  04/i">/t»  SUBROUTINE ABCO(B) DIMENSION V(1001 OlMCNSION T1100) SY-0.0  U'l  P R I NT 5 0 FORMAT I ? O X i 3 0 H  T I ME  StC.  PEAK  HEIGHTS  )  CONTINUE REAI)25,T(N>,Y(NI FCRHATIF10.1.F10.21 IFITINM -.0,40,41 PRINT27.TIN),V(N) FORMATI20X.F10.1.F10.2 SY-SY»Y(NI  )  N-N»l 40 tl  C0T026 NU»N-1 00711-1,10 SEBT-0.0 STtBT-0.0 SYE6T-H.0 SYTEBT-0.0 STE2BT-0.0  30  SE2dT»0.0  31  ST2E8T*0.0  32 33 34 34 36 J7 40 41 42 43 44 45 46 47 50  ST2E2B-0.0 SYT2EB -0.0 0012N-I.NU EBT"£XPI-8«T(N>) E2BT»EXPt-B»T<N>«2.) SE8T«SCBT»EBT SYTEBT«S»TE8T»Y(NI«TINI«EBr SYEBT"SYtBT»Y(N)«EBT STEHT"STfcBT»T(N>»£BT SYT2EB •SYT2E8 •Y(N)•!INI••2«E8T ST2E2B -ST2E2B »TIN)••?«E2BT S T 2 E B T .ST<:£BT •T(N)««2«EBT STE2BT •ST£2BT«TCN)«E2BT SE23T* SE2BT »E2BT CONTINUE  12  52 53 54  16  55 67 60 61 62  7 15  63  SOURCE  STATEMENT  il  EN-NU EROR-SYEBT»STEBT«SE8T/EN»SYeBT«SrE2Br»irtaT»SY»iE2lJT/E%-5YTF(ir« 1SE2BT -SY«SEBT«STE2BT/EN •SYTEBT«SEBT»«2/EN DEROR • S Y E B T o ( S T E B T « 2 • S E B T » S T 2 E B T I / E N • ST£ST«SFrlI»SYTEl!T/t"« l-2.»SY>:BT«ST2t2D -5TE2BT»SYTtBT-(2.»SY«STtBT»STE2bI»SY« SE2HT«SI2F 2BTI/EN •2.«SYTE6T«STE2oT • SE2(iT»SYT2t8 «2.'SY<iSFBT«ST2E2B/LN • 3SY»STfc2BT»STEBT/EN -2.«SVTE8T«SE»T»iTEdT /EN-SEBT«»<«SYT2f.6/fcN B-B-ERUR/IJEROR A" (STEBT.SY/tN-SYTEBTI/(ST£BT«SEBT/EN-STE2Bll C-(SY-A»SbBTI/EN PRINT37,A,B.C F O R M A T ( / / 6 0 H C O N S T A N T S FOR L E A S T S A U A R E S F I T OF O A T A I N Y - C " A»tXP K-BTI /20X.3HA < i H t . t /20X.3HB • IF16.S/20X.3HC •,F|6.B /// 250H SUMMATIONS FROM L E A S T SQUARES C A L C U L A T I O N I PRINT D t S E B T i S Y E B T . SV»STkBTfSTE2BTiSfc2aTfST2EBT»ST2E?b .  1  -2008  FORTRAN SOURCfc LIM  DAVIS  SOURCE  ISN  6*,  65 66  13  JAAA  STATEMENT  ISYT2EB .SYTEBT F0«MATU6X,7HSEBT » ,£12.5/16X,7HSYEBT » ,F12.b/16X.7HSY Ul2.5/l6X,7HSrE8T » , E 1 2 . 5 / 16X , 7HSTE26T" ,fc12.5/16X,7HSfc?8T » 2 E 1 2 . 5 , / 1 6 X , 7 H S T 2 E B T » , E 1 2 . 5 / 1 6 X , 7 H S T 2 E 2 . * , £ 1 2 . •»/16X, 7hSVT?£B« . 3 E 1 2 . 5 /16Xi7MSYTEBT* ,E12.5 I RETURN  END  NO MESSAGES FOR ABOVE ASSEMBLY TIME 17HRS OONIN 39.JSEC  -201  PARALLEL TUBE BED HYDKOGEN-NlTROGtU hfcO  DATA  BfcO  DIAMETER  HEO  LENGTH  ENO  ZONfc  KU'4  CMS  10.01000  Mfcl&HT  PORUSITY  FLOW SATE  5.03000  CMS  MUtlH  0.27000  ATM.PRcSSUKE  0.52000  bEO  UMfc  S E C .  PEAK  200.0  CONSTANTS  FOR LEAST  *  11 » C  SUMMATIONS  FRO  M  »  LtAST  ^050.00  8EJB0TB) K U B T S  1400.00  MO  1000.OC  600.0  700.00  700.0  4H0.0O  H I  DATA  IN  V-C«  *°«>  A»EXP|-BT>  0.00384182 13.96370361  SQUARES  CALCULATION 01  »  0.27037E  04  SV  »  0.85550E  04  STEBT  •  ft.4<l346t  0 3  O.12707E  0 3  (I.44471E-U0  ST^£OT>  0.20602t  06  ST2f2H«  .I.42225E  05  SVT^EB.  (>.25»73F  09  SYTEST"  «.77353C  06 0.55121716  1.48761696  EFFECTIVC  OIFFUSIVITY.  PUBLISPEO  0|FFl'SIVITY«  ALPHA*  OF  SVLUT  •  7 5 7 .  6035.51483  UIFFUSIV|TV» LAMOA  HI}  300.0  0.14035E  «  MM  104.000U  HEIGHTS  500.0  »  St2BT  K»  O.SIO?') 2">5.5C00  2925.00  SEbT  STE2BT'  TEMPERATURE  4C0.0  SAUARES A  CC/SEC«  TfMPtRATURC*  OATA  U.I»12?"»T7  0 . ? H  6  6 3 . " ) 3  O.H7000000 NuMufcR  ITFRA TIUNS*  21  h  -202HYQRUGtN-MTRQGEN BED  DATA  HEO D I A M E T E R C C S 060 LENGTH CMS b N O IQHt H E I G H T POROSITY 0.52000  RU b.JJOOO lU.uSOOO 0.27000 •tED  l i f t SfcC. 30'). 0 400.0 500.0 600.0 700.0 BCO.O 1010.0 1100.0 1200.0 13U0.0  1400.0 13U0.0  CONSTANTS  FOR LEAST  A B C  FLOW K A T E CC/ScC* ROOM T E M P E R A T U R E * A T M . P R C S S U R t M " HG TfcMPfcRATURe K * 109.0000  PEAK MHOMTS 3 6 0 . OQ 245.CO 167.60 115.00 R2.00 6S.00 36.50 30.00 24.iO 20.70  i  DAT  £21X0233) 800.  490.  IB.20  16.50  S A U A 4 E S F I T OF D A T A • 1164.03903 • 0.00406218 • IS.63100403  IN Y - C - A*E*P(-BTI  FROM L E A S T SUUARES C A L C U L A T I O N SEUT > C.8S436E 00 SVEBT • 0 . 1 9 S 3 0 E 0 3 SY « O . U B O t - 04 STCBT • C . 4 1 0 4 0 L 0 3 STE2BT* 0.5'(025t 02 SE2BI • 0.lb6»HE-00 ST2EBI* 0.23771F 06 S T 2 E 2 H - G . 2 4 3 3 8 E Ob SYT2EU* 0.32040t 08 S Y I E B I * 0 . 7 i O « F OS DIFFUSIVITY* 0.71461010 LAMOA • 1.14747888 EFFL-CTIVE O I F F U S I V I T Y 0.3?li972S PUBLISHED O l F F U i l V I T Y * 0.B2QO0000 ALPHA* 0 . 0 7522820  SUMMATIONS  NUMBER  4  0.54i7b 29S.5O00 767.6  ITERATIONS-  21  -203HYDRUGCN-NITRUGEN BEO BEO 060 ENO  OlAMEtEH CMS LENGTH CMS ZONE HEIGHT  POHUSm  RUN  OATA  0.52000  5.03000 10.05000 0.270CO  TIME  bEO  SEC.  150.0 200.0  250.0 300.0 400.0 700.0 800. 0 900.0 IOOO.O iloo.o 12U0.O 1300.1 CONSTANTS  FUR LtAST  A b C  SVJARES * » •  PEAK  FLOW RATE CC/SEC' R O O M TfcMHC*ATU*E» A T M . P R E S S U R E MM H G T E M P E R A T U R E K» 309.0000  F I T fJF O A T A 12V2.H0475 0 . 0 0 4 34213 15.919S9U9  «M> TOO. £00.  I N Y-C«  0.56248 795.0000 749.8  HUSCTEU P0JHX3  HEIGHTS  655.00 530.00 430.00 350.00 227.50 HO. 00 " "58.1)0 43.00 3 3.00 26.00 21.50 18.50  DA I 4  173. ISO.  lot.  A»E»(M-HTI  SUMMATIONS  FROM LcAST SUUARbS CALCULATION SE8T • 0.185.S0E 01 S Y t b T • C1.85162E 0 1 SY • P.24725E 04 STEflT • C . 5 0 8 3 3 C 0 3 SIE2BT" 0.14210E 03 SE26T • 0 . 6 7 0 / 6 6 0 0 S T 2 F B T - C 19886F. 0 6 5T2fc2tt« 0 . 3 4 2 H 0 E 0 5 SYT2Eb» C . 4 5 3 ? 7 t 0 8 SYTFBT» 0 . 1 8 2 3 6 E 0 6 OIFFUSIVITY* 0.87263739 LAMOA • 0.93968011 EFFECTIVE O I F r U S I V I Y . 0.45377145 PUBLISHED DIFFL'SIVHY0.82000000 ALPHA' 0.07060957 T  NUM8ER  ITERATIUNS*  21  -204H Y D R U G E N - N I T R O G k N B E D  RUN  O A T A  ttEO  DIAMETfcR  H8D  L E N G T H  kHO  ZCNfc  C M S  5 . 0 3 0 0 0  C M S  1 0 . 0 5 0 0 0  H E I G H T  P O R O S I T Y  0 . 7 7 0 0 0  iiCO  S b C .  P E A K  1 C U . 0  C O N S T A N T S  FOR  S U M M A T I O N S  L E A S T  FROM  I " . 5 0 l<r.OU  2 5 0 . 0  7 . 5 0  3 C 0 . 0  5 . 0 0  3 5 0 . . 1  i . 5 0  4 C G . 0  3 . 0 0  H I T  0 . 0 1 1 5 4 3 6 4  C  •  1 . 8 9 3 5 1 0 1 3  S C U A R t S  •  C . 7 0 6 2 6 E  0 0  SYErtT  •  0 . 1 5 8 7 7 E  0 2  SV  »  0 . 6 4 0 " 0 C  0 2  STfcBl  «  0 . l l l 4 l t  0 3  0 . 1 7 8 3 5 E  0 7  •  EJECTS' SO.  Ii4  Y - C "  MM  HG  755.3  309.0000  rvJHTS i»5.  A ' E X P I - b T )  O . i 4 M 0 t - O ( >  S T 2 E B T »  G . 2 U u 4 t  S I 7 E 2 0 '  0 . 2 4 3 " 5 t  0 4  S Y T 2 E B '  0 . 2 8 3 - > ? E  0 6  S Y T E B T '  0 . 1 9 9 7 4 F  0 4  D I F F U S I V I T Y .  0'.  0 . 7 7 6 0 2 > 9 l l . 0 5 6 6 o 0 7 9  E F F E C T I V E  D I F U ' S l V l W '  P U B L I S H E D  O I F F U S I V f T Y .  A L P H A '  K*  z.nan-t 295.0GUO  C A L C U L A T I O N  S E B T  «  D A T A  1 0 0 . 1 6 8 7 8  »  S E 2 H T  LAMOA  b f  •  R  S T E 2 B T '  T F V PFRATURC  W I G H T S  1 5 0 . 0  A  L E A S T  TtNPt«ATURfc=  3 3 . S O  2 0 0 . 0  S A U A R E S  R A T E  RUUM  ATM.fMt.SSURE  0 . 5 2 U 0 0  T I K E  cc/sfc«  Kov.  DATA  0 . 1 2 7 . 3 4 J u l  0 . 4 0 3 5 3 5 5 6 O . f l Z O U O O O O DUMBER  t T F R A T I O N S »  18  NHHOGEN-ETHANi BED  DATA  RUN  BfcU O I A M E T E R C H S BEO L E N G T H C M S ENO ZONE H E I G H T POROSITY O.H2000  S.03COO 10.05000 0.27000 HEO  Tl*fc  SEC•  too.o  5 0 0 . 0  700.0 100.1 I050.il 1200.0 1150.a ISOO.O 1630.0 1800.0 1930.0 2100.0 2300.0  CONSTANTS  EQ* LEAST  SUIA4ES  A * B  •  C • SUMMATIONS  FROM  LFASl  star  F L O W «ATE CC/SEC* * 0 0 « r£.*PtrRATURE = A T H . P R E S S U R E MM H G TEMPtKATURE K* 304.0000  DATA  0.48497 29S.0n.10 7*9.8  PEAK HEIGHTS 470.00 118.00 217.00 15,?.00 116.JO 90.00 72. 5 7.00 45.50 37.00 31.00 26.00 2 1 . 50  SO  F I T tit DATA  R27.H814')  IH  Y - C * A'EXPI-BT)  0.00l')9940  14.535 )2410 SQUARES  * o.i7t'*ot-  CALCU11TI0N  oi  SYEBT - 0 . 4 8 / 7 U 01 SY • O . l G S l S t 04 STtBT • 0 . 1 2 / 7 7 t 04 STE2BT* 0 . 2 6 S 9 U 03 SE.Hr • 0.55P05E 00 S T 2 E B T - 0.12«24t 07 ST2C2B* 0.16443<" 0 6 SYT2bb' 0.16489t 09 SYTEHT- C.23872t 06 DIFFUSIVITY* 0.13564177 LAMOA • 1.113/2641 EFFtCTIVE O U F u S l V I l Y 0.07051372 PUBLISHED DIFFUSIVITY' 0.16100000 ALPHA* 0.122232S2  NOMHFR  ITERATIONS*  Id  -206Nr-TRUGEN-fc THANE BED  DATA  B E D C1AHETEH C H S HBO L E N G T H C M S END ZONE HEIGHT PORUSITY 0.52000  ».03000 lO.OiOOO 0.27000  FOR. L c A S T  SUPINATIONS  FROM  SAUARES  ALPHA*  K *  M M tr»  7 5 5 . 1  t O . 0 0 0 0  HEIGHTS  76.00 46.fn! 2B.0I) I'). 0 0  Of  O A TA  *  '). 0 0 ^ 9 4 5 9 4  C  -  /.47580498  ST2t2B* SYT2EB" SYTtBT* OIFFUSIVITY* LAHOA •  rt«°e«ATL;i«C  l . 4 h l ? 0 295.00i/ii  59.00  B  * * *  C C / S E C *  130.00  400.>>7'>2l  STEBT * STE2BT* SE20T * ST2EBT*  EFFECriVE PUBLISHED  'IT  PEAK  *  Ll-AST  IE>P£«ATUf<E»  230.00  A  SLOT SYEBT SY  KATfc  «(J0«  ATM.PRESSIME tttO  TIKE S E C . 200.0 400.0 6C0.0 '00.0 1)00.0 1000.> 1200.0  CONSTANTS  FLCw  I N Y - C * A»FXP(-8T»  SOUAKLS CALCULAtlON 0 . 1 135'JF 0 1 0 .1 m Ic 0 3 C.5il«)0£ 03 O.SBBrlhu 0 . i i 9 ? 2 E 0.46042F 0.3503KE  0 3 0 3 -00 0 6  O . V > 6 l / t 0.24912L" 0.601H5F  0 5 08 0 5 P.147S240* 1.0214B467  OIFFUSIVITY* DIFFUSIVITY* 0.14160762  0.07686850 0.15100000 NUMnfcK  ITERATIONS*  12  -207N U R O G E N - E THANE BEO  OA TA  BEO  DIAMETER  H60  LENGTH  ENO  ZONE  POROSITY  RUN  CMS  5.03000  ELOw  RATE  CMS  10.05000  RUO"  TEMPERATURE-'  HEIGHT  0.27000  ATM.PRESSURE  0.S2C00  KEO  TIKE S E C . 200.0 300.0  CONSTANIS  FUR LEAST  PEAK  TCMPERATURE  K*  DATA  2.26609 296.5000  MM H G  755.3  309.0000  HEK.HTS  150.00 111.00  400.0  84.00  500.0  62.50  600.0  46.50  700.0  35.50  SAUARES  CC/SEC«  F I T OF DATA  A  *  268.37303  b C  « •  0.00799714 2.44440460  IN Y - C * A*EXP(-BT)  SUMMATIONS FROM LEAST SQUARES CALCULATION SEfiT SYEBT SY STErtT  * 0 . 1 7693E 01 « 0.17689E 03 * C.4B9*0E 03 * C . 6 4 0 4 8 F 0 3  ST£2ei»  0.1983OL 0 3  5E2BI  «  C.65044E  0 0  5 T2E8T*  0.28243E  06  ST2E28* SY T 2 t b "  0.71241E C.19810E  0 5 0C  SYTEBT* 0.54807E 05 DIFFUSIVITY* 0.13953946 LAMOA « 1.08213115 EFFECTIVE PUbLIShkU ALPHA*  UIFFUSIVITY* OIFFUSIVITY* 0.14701164  0.07256052 0.15103000 '1UMBEK  ITERATIONS"  II  •208'  N I T R O G E N - E T H A N E  BED  RUN  D A T A  BED  C F A M E T E H  rtEO  L E N G T H  E N O  ZONE  5 . 0 3 0 0 0  FLO«  HATE  1 0 . 0 5 0 0 0  ROOM  TEMPERATURE*  C M S  C M S  HE I J H T  P O R O S I T Y  ATM.PRESSURE  G . 2 7 0 0 0  BED  0 . 5 2 0 0 0  I l e t  PEAK  S E C .  FOR LEAST  SUMMATIONS  FROM  4C0.O  65.  500.0  4 7 . 5 0  oOO.O  35.50  «  700.0  2 7.00 2 1 . 5 0  900.0  17.50  too.o  12.00  F I T OF  0.00356950 7.68526965  LEAST  SOUARtS  «  0.15573E  01  SVEHT  «  0.12301L  0 3  SY  *  0.43800E  0 3  S TE8T  •  0.612561  0 3  STE28T*  0.13784E  0 3  •  A*EXP(-aTI  0 . 4 6 8 S 5 E - 0 0  ST2EBT'  0.30660E  0 6  ST2C7U'  0.48517E  0 5  SYT2ctt»  0.13858E  08  SYTE8T•  0.37354E  0 5  O I F F U S I V I T Y '  0.16494461 0.91545884  EFFECTIVE  OIFTUSIVITY'  PUIJL I S H E U  DIFFUSIVITY'  ALPHA*  I N Y-C»  CALCULATION  SEdT  •  OATA  2 36.84694  tt •  SE78T  LAMUA  00  C  »  757.6  »09.0000  H F I G H T S  8C0.0  SAUARfcS A  K*  295.0000  MM H G  8 8 . 0 0  300.0  CONSTANTS  TfMPMATURt  OATA  7.94336  124.00  200.0  1  C C / i E C '  0.14782880  0.08577120 0.15100000 NUMBER  ITERATIONS'  l u  209NHROGEN-ETHANb BED  OATA  BED OIAMETSK CMS atC L E N G T H C M S E N D ZO'it H E I G H T POKOSITY 0.52000  RUN 5.03000 10,05000 J . 27000 OEO  T 1 M b ScC. 200.0 300.0 4C0.O 500.0 600.') 700.0 800. 0 9C0.0 1000. I 1100.J  CONSTANTS  FOR L E A S T  SAUARbS A « b C «  FLOW R A T E CC/SECROHM T I M P E R A T U R t * A T H . P R t S S U R b M.M H G T t M P E R A T U R E K» 309.0000  DATA  3.0 7 9 J ) 295.0000 75c-.O  PtA« H E I G H T S 115.00 d9.0u 67.00 52.00 42.00 34.00 29.00 25.00 22.50 20.50  F I T OF DATA 194.37181 0.00328754 14.997063*0  I t Y - C - A*EXt><-fcT»  SUMMATIONS  FRCM L E A S T SQUARES C A L C U L A T I O N SEBT - 0.l7B')2t 01 SYkBT = 0.13*'<*E 0 3 SY • 0.44600E 0 3 STEBT • 0.7443'>E 0 3 S T E 2 8 T - 0 . 1 7032E 0 3 SE2BT » 0 . 5 5 6 3 7 E 0 0 ST2EBT- 0.40273E 06 ST2b2b- 0.63779E 0 5 SYT2EB- 0.18*281 0 8 SrTcttT- 0.4*2691- 0 5 DIFFUSIVITY* 0.14867555 LAMOA 1.01563*47 tFFtCTlVE DIFFUSIVITY0.07731129 PUBLISHEO DIFFUSIVITY0.16100000 ALPHA* 0.1*88^776  NUMBER  ITERATIUNS* 10  •210M'TRUGEN-ETHANE ato  DATA'  RU*  d E U DIAMETER CMS UCD  LENGTH CMS  END  ZUNt  S.OJGOO  lO.COOOG  Ht1GHT  POROSITY  BEO  S E C .  PEAK  2 0 0 . 0  FOR LEAST  SUMMATIONS  FRUM  5 2 . 0 0 39.00  5 0 0 . 0  2 6 . 0 0  6 0 0 . 0  20.5C  700.0  1 5 . 5 0  8 0 0 . 0  1 2 . C C  FIT  H  »  0 . 0 0 399948  C  •  5.92013363  SCUARtS  *  O.I7803E  01  SYEbT  «  C.656746  02  iY  •  C.24500F  03  STcbT  *  P.45836E  03  STC2iiT»  0.1O195L-  03  Sh2tlT  0 .  =  0 . 1 9 9 R U  0 6  S T2E2H*  C . J H 9 3 F  05  SYT2S8*  0.64644E  0 7  SYTE8T *  0.189? It  PUaLISHED ALPHA*  IN  K*  >,  l0C  4.55o44 205.0000  MM HG  755.6  309.0O0C  ' i'  18  Y - C ' A ' E X P I - U T )  16539E-00  ST2E8T"  *  CC/SEC=  CALCULATION  SECT  EFFECTIVE  DATA  158.99557  DIFFUSIVITY" LAMDA  uf  =  LEAST  TEMPERATURE  HEIGHTS  4 0 0 . 0  SAUARES A  TF M P l . R A T U R E =  78.00  100.0  CONSTANTS  RATE  ROOM  ATM.PRESSURE  0 . 2 / 0 0 0  0.52000  TIME  FLOW  DATA  05 D.17814471  0 . 0 4 7 6 ? 5 4 9 OIFFuSIVITY= OIFFUSIVITY" 0.15027141  0.09261525 0.15100000 NtUtoBER  ITERATIONS*  9  -211-  N1TR0GEN-BUTANE BED  OATA  BED DIAMETER CMS BEO L E N G T H CMS ENO ZONE H E I G H T POROSITV 0.S2000  RUN  5.03000 10.05000 0.27000 BED  TIME S E C . 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 1200.0 1400.0 1600.0 1800.0 2000.0 2200.0 2400.0 2600.0  CONSTANTS  FOR  LEAST  A B  CATA  0.461M FLOW R A T E CC/SEC246.0000 RUOM T E M P E R A T U R E 761.0 A T M . P R E S S U R E MM HG TEMPERATURE K 309.0000  PEAK HEIGHTS 510.00 440.00 380.00 325.00 285.00 250.00 215.00 190.00 142.00 108.00 82.00 61.00 45.00 34.00 26.00 19.00  S A O A R E S F I T OF D A T A 776.69521 0.00143133 1.50567818  IN  Y-C-  A-EXP1-BT)  c  SUMMATIONS FROM L E A S T SEBT » SYEBT ° SY STEBT STE2BTSE2BT • ST2EBTST2E2B" SYT2EBSYTE8TDIFFUSIVITYLAMOA • EFFECTIVE PUBLISHED ALPHA-  SQUARES CALCULATION 0.39757E 01 0 . 1 2 S 0 B E 04 O.31120E 04 0.29702E 04 0.88801E 03 0.160276 01 0.30831E 07 0.62188F. 06 0 . 4 8 8 0 2 b 09 0.69419E 06 0.08172526 1.16243123  DIFFUSIVITYDIFFUSIVITY0.13225333  0.04249714 0.09500000  NUMBER  ITERATIONS-  15  -212-  NITROGEN—BUTANE BED  OATA  BED OIANETER CNS BED LENGTH CHS END ZONE H E I G H T POROSITY 0.52000  RUN 5.03000 10.05000 0.27000 BED  TIME S E C . 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 1100.0 1200.0 1400.0 1600.0 1800.0  CONSTANTS  FOR  LEAST A B C  FLOW R A T E CC/SEC« ROOM T E M P E R A T U R E * A T M . P R E S S U R E MM H G TEMPERATURE K* 309.0000  OATA  0.90294 296.0000 761.0  PEAK HEIGHTS 285.00 240.00 205.00 174.00 148.00 126.00 104.00 90.00 76.00 63.00 46.00 33.00 23.00  S A U A R E S F I T OF DATA • 466.99112 • 0.00163191 » -1.74857622  IN  Y - C *  A.EXPI-BT)  SUMMATIONS  FROM L E A S T SQUARES C A L C U L A T I O N SEBT « 0.35027E 01 SYEBT * 0.60839E 03 SY « 0 . 1 6 1 JOE 04 STEBT = 0 . 2 3 8 9 0 E 04 STE2BT* 0.70196E 03 SE2BT • 0 . 1 3 1 5 9 E 01 ST2EBT* O.20596E 07 ST2E2B" 0.46022E 06 SYT2EB" 0.21130E 09 SYTEBT* 0.32363t 06 DIFFUSIVITY" 0.07939063 LAMOA * 1.1966146B EFFECTIVE DIFFUSIVITY* 0.04128313 PUBLISHED DIFFUSIVITY* 0.09500000 ALPHA* 0.14327731  NUMBER  ITERATIONS*  12  -213N1TR0GEN-BUTANE BEO  OATA  RUN  BED D I A M E T E R CMS BED L E N G T H CMS t N O ZONE H E I G H T POROSITY 0.52000  5.03000 10.05000 0.27000 BEO  TIME S E C . 300.0 400.0 500. 0 600.0 700.0 800.0 900.0 1000.0 1150.0  CONSTANTS  FOR  LEAST  FLOW R A T E CC/StC' ROOM T E M P E R A T U R E A T M . P R E S S U R E MM HG TEMPERATURE K" 309.0000  OATA  2.0*60') 296.00U0 761.0  PEAK HEIGHTS 138.00 110.00 92.00 76.00 61.00 51.00 42.00 34.00 25.00  S A U A R E S F I T OF D A T A A • 249.21826 B • 0.00202606 C 1.23396216  IN  Y-C'  A'EXPI-BTI  SUMMATIONS  FROM L E A S T SQUARES C A L C U L A T I O N SE6T • 0.74793E 01 SYEBT • 0 . 2 1 8 5 5 E 03 SY - 0.62900E 03 STEBT « 0 . 1 4 1 7 4 E 04 STE2BT' 0.41079E 03 SE2BT • 0 . B 6 4 6 8 E 00 ST2EBT' 0.95420E 06 ST2E2B' 0.22771E 06 S Y T 2 E 6 ' 0 . S 7 9 3 B E 08 SVTEBT' 0.10412E 06 DIFFUSIVITY' 0.09057913 LAMOA ' 1.01568647 EFFECTIVE OIFFUSIVITY' 0.04710115 PUBLISHEO O I F F U S I V I T Y ' 0.09200000 ALPHA'  ENO-OF-OATA TIME  17HRS  0.14946068  E N C O U N T E R E D ON 01MIN  2S.3SEC  SYSTEM INPUT  '  NUMBER FILE.  ITERATIONS'  10  POROUS S O L I D  -214-  BED  HYOROGCN-NlTROCtfJ BEO  OATA  6b0  0SAMET6R  BOO  LENGTH  END  ZONE  RUN  CMS  CMS  HEIGHT  POROSITY  2.610C0  FLOn  ,UTE  7.00000  ROOM  TEMPERATURE*  0.27000  ATM.PRESSURE  O.59C00  BEO  TIME  SEC.  PEAK  150.0  123.00  200.0  6 1 . 0 0  250  34.00  .0  300.0  18.00  350.0  10.00  4 0 0 . 0 4 5 0 . 0  FOR  LEAST A  SAUARES *  SUPMATIQNS  FROM  LEAST  SEBT  *  SQUARES  V-C=  AoEXP(-BT)  CALCULATION  *  0.259906  0.20592E""0"2  STErtT  *  0.49247C  02  STC2BT*  0.34536C  Ol  0.2072  03  7E-01  ST2EBT*  0.10B52C  ST2E2B*  0.59838b  0*  SYT2EB*  0.60443E  06  SYTEBT*  0 . 3 4 5 2 0 t  05  04 0.56476959  1.45191953  EFFECTIVE  DIFFUSIVITY*  PUBLISHED  DIFFUSIVITY*  ALPHA*  IN  0 . 2 4 8 2 7 E - 0 0  «  «  765.1  0.01387832 2 . 6 J 4 5 8 3 9 5  DIFFUSIVITY* LAMOA  HG  3.20  SY  *  MM  306.0000  4 . 5 0  SYEBT  SE2BT  K*  H H G H T S  FIT OF OAT A 961.06734  B"» C *  TEMPERATURE.  0.56238 296.0000  _t>j-JO  "  500.0  CONSTANTS  C C / S E C *  OATA  0.15623251  0.31321406 0.82000000 DUMBER  ITERATIONS*  21  £1> HYDROGEN—NITROGtN BEO  DATA  OED  DIAMETER  BED  LENGTH  END  2 . 6 1 0 0 0  FLOW  RATE  CMS  7.C0O00  ROOM  TEMPERATURE*  ZONE  HEIGHT  0.27000  POROSITY  CMS  BED  FOR LEAST  SUMMATIONS  FROM  S E C .  6 6 . 5 0  2 0 0 . 0  2 9 . 0 0  2 5 0 . 0  13.50  3 C 0 . 0  6 . 5 0  3 5 0 . 0  3.50  4 0 0 . 0  2 . 2 0  F I T  b  *  0 . 0 1 6 9 8 6 1 3  C  *  1.35669661  LEAST  SEBT  '  SC-UARFS  MM  HG  7 6 5 . 1  3 0 6 . 0 0 0 0  IN  Y - C -  A»EXPI-bT)  CALCULATION  0 . 1 3 5 8 8 F - 0 0  SYE8T  *  0 . 6 4 1 B 4 E  0 !  *  0 . 1 2 1 2 0 F  C 3  STEOT  =  0 . 2 5 2 0 9 E 0 2 O. 12077E  01  C . 7 4 9 2 6 E - 0 2  ST2EttT»  0 . 5 0 4 4 7 E  04  ST2E2B*  0 . 1 9 9 7 7 E  0 3  SYT2EH*  0 . 1 7 3 0 7 E  06  SYTEBT*  U . 1 0 J 9 0 E  04  O I F F U S I V I T Y " «  K»  8 3 2 . 0 2 7 1 6  SY  *  TEMPERATURE  0 . 7 9 2 9 1 2 9 6 . 0 0 0 0  HEIGHTS  DATA  *  SE2BT  0 . 5 9 6 2 0 0 1 5 1 . 3 7 5 i 7 704  EFFECTIVE  DI F F U S I V H Y »  PUBLISHED  O I F F U S I V I T Y -  ALPHA*  OF  A  STE2BT*  LAMUA  PEAK  1 5 0 . 0  SAUARES  C C / S t C -  ATM.PRESSURE  O.59C00  TIME  CONSTANTS  O A T A  RUN  0 . 1 6 8 2 2 8 2 2  0 . 3 5 1 75809 0 . 8 2 0 0 0 0 0 0 NUMBER  ITERATIONS*  20  -216H^DRUGEN-MTROGfcN «tO  DATA  BED OIAMETER CHS H8U LENGTH CMS END ZONE HEIGHT PUROSHV 0.59C00  RUN DATA 2.610C0 7.00000 0.27000  TIME S E C . 100.0 150.0 200.0 250.C 300.0 350.0 400.0 CONSTANTS FOR LEAST  FLOW RATE CC/SEC* 0.92B34 RllOM TEMPERATURE* 246.0000 ATM.PRESSURE MM HG 76b.I BED TFNPFHATURE K« 106.0000 PEAK HEIGHTS 124.00 54.00 23.00 10.00 5.00 i.00 2.00 " ~  SAUARES F I T uF OATA IN V - C " A * E X P ( - B T > A * 673.31601 B » 0.01706416 C * I.O01777O9  SUMMATIONS FROM L t A S T i 0 U A f . C S CALCULATION SFBT * C.31545F-00 S V E B l » 0 . 2 7 6 2 2 F 0? SY * C . 2 2 1 0 0 E 03 STtBT « 0 . 4 2 9 / l E 02 STfc2ttT« 0 . 4 4 7 1 7 E U l SC2BT • 0 . 4 0 2 5 4 t - 0 l S T 2 E B T " 0 . 6 7 7 4 5 E 04 ST2E2H* 0.5239HC 03 S Y T 2 E B * 0 . 3 6 2 1 0 C Ob S Y T E I T " 0 . 3 0 7 6 4 b 04 OIFFUSIVIIY0.5J3096B5 LAMOA * 1.53811)204 EFFECTIVE OIFFUSIVITY* 0.31452714 PUBLISHED D I F F U S I V I T Y * 0.82000000 ALt»HA« 0.17331466  NUMBER I T E R A T I O N S *  18  217HYUKUGbN-NITKOGEN BED DATA BbU DIAMETER CMS i)60 LEN6TH CMS ENO ZONE HEIGHT POROSITY 0.59000  RUN OATA 2.610C0 7.00000 0.27000  IIMt SEC. 100.0 150.0 200.0 250.0 1C0.0 350.0 CONSTANTS FUR LEAST  FLOw RATE CC/SEC" 1.24571 ROOM TEMPtftATURfc* 296.0000 ATM.t»ReSSUKE MM HG 765.1 BCD TEMPERATURE K» 306.0000 PEAK M U G H T S 71.00 2 b . 00 10.50 '..50 2.20 1.10  SAUARES F I T OF OAIA li Y - C » A » E X P | - H I ) A » 474.73421 B « 0.0190SUU6 C • 0.4624368)  SUMMATIONS FROM LEAST SQUARES C A L C U L A T I O N StBT » 0.241246-00 SVEBT « '>.1244 3E 02 SY « ' I . U T 3 0 F 03 STCJT • 0 . J 1 4 S 6 E 02 S T E 2 B T * U . 2 8 2 4 3 E 01 SE2BT • O . 2 5 9 7 5 C - 0 1 S T 2 E B T " ' . 4 6 4 5 9 t 04 ST2E2B* 0.32038E O i SYT2EB* U.15421E 06 SYTEBT* 0 . 1 3 5 5 3 f 04 DIFFUSIVITY" 0.5377I2G2 LAMOA « 1.52497758 EFFECTIVE Olf F U S I V H Y * 0.31725056 PUBLISHED D I F F U S I V I T Y * 0.82000000 ALPHA" 0.18763416 ,  .4UMBER  ITERATIONS"  lo  218HYDROGEN-NlTROGfcN BEO OAT A BEO OIAMETER CMS BEO L6NSTH CMS ENO ZONE HEIGHT POROSITY 0.59C00  MUN DATA 2.61000 7.00000 0.27000  TIME  SLC. 50.0 " 100.0 150.0 200.0 250.0  FLCW RATE CC/SEC* 1.83497 ROOM TEMPERATURE" 296.0000 A T K . P R E S S U R t MM HG 765.1 8ED TEMPERATURE K« 306.0000 PEAK HEIGHTS 124.00 37.00 14.00 5.00 2.00  CONSTANTS FOR LEAST SAUARES F I I UF DATA IN Y - C " A » £ K P ( - B T t A « 414.35182 B • 0.02445598 C • 1.89934120 SUMMATIONS FROM LEAST SQUARtS C A L C U L A T I O N SE8T » 0.41L32C-00 SYtBT • 0 . 4 0 1 1 2 E 02 SY « 0 . 1 8 7 0 0 E 03 STEbT • 0 . 2 9 2 7 1 E 02 S T E 2 B T * O.Sr?51f.~CT SE2UT • 0 . 9 4 8 9 9 E - 0 1 S T 2 E B T * 0 . 2 6 1 5 6 L 04 S T 2 E 2 8 - 0 . J 0 9 0 2 C 03 S Y T 2 E H - 0 . 1 3 3 1 5 C 06 SYTE8T« 0 . 2 2 C 8 2 E 04 DIFFUSIVITY' 0.65093117 LAMDA • 1.25973380 EFFECTIVE OIFFUSIVITY* 0.38404939 PUBLISHED D I F F U S I V I T Y * O.B2OU0O0O ALPHA* 0.1931C470  NUMBER I T E R A T I O N S *  15  -21* MiTRUGEN-ETHANfc BEO  OATA  BEO DIAMETER CMS BEO LENGTH CMS ENO ZONE HEIGHT POROSITV 0.59COO  RUN 2.61000 7.00000 0.27000 BEO T IMF. S t C . 250.0 " 400.0 550.0 700.0  aso.o  10C0.0  05070  OATA  FLOW RATE CC/SEC* 0.39180 ROOM TEMPERATURE* 296.0000 ATM.PRESSURE MM H i 766.9 TEMPERATURE K> 306.0000  PtAK HEIGHTS 212.50" 117.00 68.00 43.50 30.50 24.50  ?0T5fT  CONSTANTS FOR LEAST SAUARES F I T OF OATA IN Y-C- A ' E X P ( - B T ) A » 598.64044 B * 0.00448394 C » " "IT. 3 6 4 9 1 0 6 0 SUMMATIONS FROM LEAST SUUARFS CALCULATION SEBT * 0.65974C 0 0 SVEBT » 0 . 974 5 9 E 02 SY - C5T6"50E~0~3 STEBT « 0 . 2 6 1 7 9 f O i STE2BT* 0 . 4 J 4 9 4 E 0 ? SC2BT * 0 . 1 4 3 6 6 E - 0 0 ST2EBT* C.12BB0L 0 6 ST2E2B» 0.1469*C 0 5 SYT2EB* 0 . 1 l 0 3 4 t 08 SYTEBT* 0 . 3 0 5 8 3 E 0 5 DIFFUSIVITY0.11308980 LAMOA • 1.33522204 EFFECTIVE DIFFUSIVITY* 0.06672298 PUBLISHED O I F F U S I V I T Y * 0.15100000 ALPHA* 0.19822170  NUMBER ITERATIONS-  U  -220' N H R U G t N - E THANE BEO OATA BEO OtAMETER CMS BEO LENGTH CMS END ZONE HEIGHT PURQSITY 0.59000  RUN DATA 2.61000 7.00000 0.27000  TIME S E C . 2C0.0 300.0 400.0 500.0 600.0 700.0 120O.')  FLOW RATE CC/SEC0.82082 ROOM I f M P F R A T U R t 296.0000 766.9 ATM.PRESSURE MM HG BEO TfcMPtRATURC K« 306.0000 PEAK HEIGHTS 116.00 77.50 53.00 38.50 29.50 24.00 "16.Off  CONSTANTS FOR LEAST SAUARES F I T OF OATA I N Y - C - A»EXK(-tJT( A « 2h7. 69002 B 0.C0487665 C 15.17382121 SUMMATIONS FROM LEAST SOUARtS C A L C U L A T I O N SEOT » 0 . 9 2 7 5 0 r 00 SYEBT = 0 . 7 4 9 9 9 F 0 2 SY - 0 ; i ? 4 5 0 t 03 -STE8T » C . 3 0 4 0 6 E 03 S T E 2 B T - 0 . 5 8 9 1 0 c 02 SE2i3T » O . 2 2 7 A 0 E - 0 O S T 2 E B T - 0 . 1 2 0 0 7 L 06 S T 2 E 2 B * 0 . 1 7 2 3 0 F . 05 SYT2EB- 0 . 6 4 * 3 3 F 07 S Y T E B T - 0 . 2 0 3 8 3 t OS DIFFUSIVITY0.107*6807 LAMOA • 1.399S5816 EFFECTIVE D I F F U S I V I T Y ' 0.06364216 PUBLISHED D I F F U S I V I T Y 0.15100000 ALPHA0.21I6&63'~> NUMBER I T E R A T I O N S -  11  22b NHRUGEN-ETHAME bEO DAT« BED DIAMETER CMS BCD LENGTH CMS END ZONE HEIGHT POROSITY 0.59COU  RUN DATA 2.61000  '.ooooo  0.27000  TIME S E C . 100.0 2C0.0 300.0 400.0 500.0 uOO.O CONSTANTS FOR LEAST  F L O * HATE C C / S E C " ROOM TEMPERATURE" ATM.PRESSURE MM HG BEO TEMPERATURE K» 306.0000  1.32324 296.0000 766.9  PEAK HEIGHTS 122.00 78.50 53.00 37.50 28.50 23.50  SAUARES F I T OF DATA I N Y - C « A»fcXP(-BT) A » 179.89097 8 « 0.00525755 C » 15.67274153  SUPMATIONS FRUM LEAST SQUARES C A L C U L A T I O N SCBT » 0 . 1 3 8 4 0 E 01 SYEBT * 0 . 1 l 8 1 3 t 0 3 SY • 0 . 3 4 3 0 0 E 03 STEBT * 0 . 3 0 « 7 E ~ 0 3 S T E 2 B T " 0 . 8 1 8 1 4 E 02 SC26T 0.53609E 00 S T 2 E K T " 0 . 9 1 4 0 9 k 05 ST2E2B" 0 . 1 6 5 5 9 E 05 S Y T 2 E B " 0 . 4 4 H 1 E 07 SYTEBT" 0 . 1 9 4 4 2 F 05 OIFFUSIVITY" 0.11160270 LAMOA « 1.35301381 EFFECTIVE OIFFUSIVITY" 0.06184559 PUBLISHED O I F F U S I V I T Y " 0.15100000 ALPHA" 0.21606873  NUMBER I T E R A T I O N S "  10  -222NHRLiGEN-ETHANfc BED BED  UIAMETEK  B60  LENGTH  END  ZONE  C M S  CMS  HEIGHT  POROSITY  2 . 6 1 0 0 0  FlCk»  RATE  /.COOOO  ROHM  T tMPfcRATURfc"  0 . 2 7 0 0 0 HEO  PEAK  1 0 0 . 0 -  5 8 . 0 0  3 0 0 . 0  4 0 . 5 0  4 0 0 . 0  SUMMATIONS  FROM  F I T OF OATA  8  •  0.00522691  C  *  1 5 . 0 4 5 6 7 6 3 5  SYEBT  =  SQUARES  0 1  0 . 8 7 S 4 9 E  0 2  0.27370CO3  STtBT STF28T*  0 . 3 2 2 J 2 6  0 3  0 . 8 3 3 0 2 E  0 2  SC2bT  0 . 5 4 1 8 0 F  0 0  ST2E81"  0 . 1 0 5 1 1 E  0 6  ST2E28-  0 . 1 7 1 6 1 E  0 5  SYT2£B=  0.3O787E  0 7  SYTfcBT"  0 . 1 5 0 2 8 E  0 5  •  ~  0 . 1 0 8 2 8 4 3 3 1 . 3 9 4 4 7 6 9 1  EFFECTIVE  D I F r U S l V I T Y *  PUBLISHEC  D I F F U S I V I T Y "  ALPHA"  A»fXP(-BT)  CALCULATION  0 . 1 4 1 9 0 E  D I F F U S I V I T Y " LAMCA  I N Y-C»  1 2 2 . 1 8 3 7 9  SY  *  HEIGHTS  1 8 . 0 0  *  LEAST  7 6 6 . 9  2 0 . 5 0  A  sear  MM H G 3 0 6 . 0 0 0 0  2 4 . 2 0 J  70ovor~~ SAUAHES  K •  30.00  5 0 0 . J  FOR LEAST  TEMPtRATURt  1.90216 2 9 6 . 0 0 0 0  8 7 . 50  2 0 0 . 0  6 0 0 .  C C / S E C *  ATM.CRfSSURE  0.59COO  TIME S F t .  CONSTANTS  OAIA  RUN  CAIA  0 . 2 1 6 7 1 4 3 2  0 . 0 6 3 8 8 7 7 6 0 . 1 5 1 0 0 0 0 0 "  NUMBER  ITERATIONS"  8  -223-  N I F R U G E N - B U T A N E  B E O  L A T A  B E D  D I A M E T E R  HED  L E N G T H  LNL)  Z O N E  R U N  C M S  C M S  H E I G H T  P U R O S I T Y  2 . 6 1 0 0 0  FLOW  R A T E  7 . 0 C O O O  ROOM  T E M P E R A T U R E *  O . 2 7 0 C O  C U N S I A N T S  A T M . P R E S S U R E  O . 5 9 C 0 O  B E O  T I M E  FOR  L E A S T  S U M M A T I O N S  F R O M  S E C .  5 8 . 2 0  7 0 0 . 0  4 2 . 0 0  B C O . O  3 0 . 5 0  9 C 0 . 0  2 2 . 0 0  1 0 0 0 . 0  1 6 . 0 0  1 1 0 0 . 0  1 2 . 0 0  1 2 0 0 . 0  9 . 0 0  F I T  B  '  0 . 0 0 3 3 7 3 2 b  C  '  1 . 4 3 0 0 4 9 b !  L E A S T «  S Q U A R E S  »  0 . 1 5 7 T 9 E  »  G . 1 8 9 / 0 E  0 3  STErtT  »  0 .  3 2 4 - . 9 E  0 3  0 . 2 4 S 9 8 F  0 2  HG  7 6 1 . 4  IN  Y - C >  A » E X P ( - B T )  0 2  0 . 3 5 2 u 5 C - O l  S T 2 E B T *  0 . 2 6 4 7 7 E  0 6  S T 2 E 2 B «  0 . 1 7 7 ' . 8 E  0 5  S Y T 2 E 8 "  0 . 8 0 0 3 6 E  0 7  S Y T E B T *  ( . . 1 1 0 3 9 E  1)5  D I F F U S I V I T Y * *  MM  3 0 6 . 0 0 0 0  C A L C U L A T I O N  SY  «  K *  0 . 4 1 7 O 6 E - 0 0  S Y E 8 T  S E 2 8 T  T F K P E R A T U R E  O . S 9 4 4 3 2 9 6 . 0 0 0 0  H E I G H T S  D A T A  4 2 9 . 9 2 2 0 7  S T E 2 B T »  0 . 0 7 4 1 . 3 2 4 8  7 2 3  75  7 9 9 9  E F F E C T I V E  D l F r u S I V I ' T Y *  P U U L I S H E O  O I F f U S I V i  A L P H A *  (IF  *  S E B T  L A M U A  P E A K  6 0 0 . 0  S A U A R E S A  C C / S E C »  D A T A  T  Y *  0 . 2 1 2 2 7 3 6 6  0 . 0 4 4 0 8 7 0 1 0 . 0 9 9 0 0 0 0 0 N U M B E R  I T E R A T I O N S *  11  - 2 2 4 -  NHROGLN-BUTANfc BEO BEC  DIAMETER  DdTA  HUN  CMS  2.11000  FLOW  KATE  ROOM  TEMPERATURE*  1360  LENGTH CMS  7.CO00O  END  2.CNE  0 . 2 / 0 0 0  HEIGHT  POROSITY  0.59C00  BEO  TIME  S E C .  PEAK  500.0  SUMMAIIUNS  FKL-M  7.50  1000.)  5.50  F I T  »  »  0.00380157  C  •  1.12496  •  SQUARES  SYEBT  •  U.H621BF  01  *  O.VOOJOt  U2  •  0.274906  0 )  STt2BT»  0.241JbE  02  320  0 . 4 I 5 0 7 E - 0 1  ST2E1T*  0.187  ST2E2B*  C.l45->9F  ?6t  SYT2E0*  0.30610F  0 /  SYTCBl*  0.50450F  04  DIFFUSIVITY*  06 0 5  0.08021061 1.2342S071  EFFECTIVE  O l F F l l S I V l TV*  PUBLISHED  DIFFUSIVIIY*  ALPHA*  Y - C * A»EXPC-HTI  CALCULATION  STtBT  *,  IN  U.424.'»9C-U0  SY  •  OATA  196.2112a  SE2hT  LAMCA  OF  •  LEAST  HEIGHTS  10.50  9 0 0 . 0  A  SEBT  761.4  15.00  800.0  SAUARES  1.14750 296.0000  ATM.PRESSURE MM H G TLMPLRATURE K« 306.0000  21.00  700.0  FOR LEAST  OATA  30.50  600.0  CONSTANTS  CC/SEC*  0.21750321  0.04732426 0.09900000 NUMBER  ITERATIONS*  9  •225NHRCGEN-BUTANk BED  OAT A  RON  etC D I A M E T E R CMS B60 LENGTH CMS t N D ZONE H E I G H T POROSITY 0.59000  2.61CC0 7.CO0CO  0.27000  BEO  TIME S C C . 100.0 i50.0 400.0 45U.0 500.0  CONSTANTS  FUR  LEAST A B C  SAUARES * * *  FLOU RATt CC/SEC* ROOM T£MPtRATURc» A T M . P R t S S U R E MM H G T E M P E R A T U R E K* 306.0000  OAT A  2.06757 296.0000 761.4  P E A K HE1GHTS 13.00 2 7.CO 23.00 19.50 16.00  F I T OF DATA 100.537S7 0.0040O64J 2.6322294?  I N Y-C* A - E X P I - B T I  SUMMATIONS  FROM L E A S T SQUARE S C A L C U L A T I O N SEBT « 0.104/HE 01 SYERT * 0.26568E 0 2 SY « 0.U850E 03 STEBT = 0.39847E 0 3 STE28T« 0.85844E 0 ? SE28T » 0.23682E-O0 ST2EBT* 0.15657E 06 ST2E28« 0.32018E 05 •SYT2EB* U.36381E 0 7 SYTEBT* 9.96794E 04 DIFFUSIVITY* 0.08230689 LAMOA • 1.20281547 EFFECTIVE OIFFUSIVITY* 0.04856106 PUBLISHED OIFFUSIVITY* 11.09900000 ALPHA* 0.22042512  NUMBER  ITERATIONS*  8  SPHERICAL PACKING BEO HYDROGEN-NITROGEN BEO  OATA  BEO  OIAMETER  BEO  LENGTH  ENC  ZONE  RUN  CMS  2.61C0O  CMS  7.00000  HEIGHT  0.27000  POROSITY  BEO  TIN£ S t C . 100.0 150.0  TEMPERATURE  MM H G  742.5  309.0U00  28.00 18.OC ~TJVO~0  300.0 -  400.0  11.00 10.OC 9.O0  450.0 500.0  L E A S T " STACl«R£S F I T o r O A T A " I N V - C » A « 1023.41085 8 « 0.01643952 C > 9.95351410  FROM  K«  0.45564 295.0000  50.50  250.0  SUMMATIONS  CC/SCC"  95.00  200.0  FOR  TEMPERATURE*  PEAK HEIGHTS 460.00 207.50  50.0  CONSTANTS  SATE  RuOM  ATM.PRESSURE  0.39300  J 5 0 T O  FLOW  OATA  LEAST"SOUAKES  A*E"5SPT-BT>  CALCULATION  SE6T  •  0.7841 IE  00  SYEBT  •  O.25290t  03  SY • STtBT " STE2BT" SE2BT •  0 0 0 0  03 02 02 00  ST2EBT"  0.B912BE  .90200t .69B49E .143426 .23949E-  - - - - -  04  ST2E2B"  0.10975E  04  SYT2EB*  0.12131E  07  SYT6BT" G.158856 05 DIFFUSIVITY" 0.68188812 LAMOA • l.2025«Y3o ~ EFFECTIVE PUBLISHED ALPHA"  DIFFUSIVITY. DIFFUSIVITY" 0.15708618  0.2679B203 0.82000000 NUMBER  ITERATIONS"  21  •22T HY0RU6EN-NITRUGlN BEO OATA BED DIAMETER CMS HBO LENGTH CHS fcNO /ONE HEIGHT POROSITY 0.39300  RUN DATA 2.61000 7.00000 0.27000  TIME  '  SEC. 50.0 100.0 150.0 '200.0 250.0 100.3 350.0 400.9  CC/SEC* 0.83168 TEMPERATURE* 295.0000 ATM.PRESSURE MM HG 7*2.5 BED TEMPERATURE M 309.0000 FLO*  RATE  ROOM  PEAK HEIGHTS 229.00 80.00 31.40 16.00 9.50 6.80 5.80 S.00  CUN3TANTS FOR LEAST SAUARES F I T OF OATA I N Y - C " A « E X P I - B T I A • 664.12617 B • 0.02184377 C * 6.033H1014 SUMMATIONS FROM LEAST SQUARES CALCULATION StOT * U . 5 0 4 77E 00 SYEBT - H . 8 7 2 7 1 C 02 " SY « U.33350E Ot STErtT * 0 . 3 7 9 4 8 E 0 ? S T E 2 B T ' 0 . 7 1 4 5 2 k Ot SE2BT * P . 1 2 6 8 2 E - 0 0 ST2EHT* 0 . 3 7 9 8 6 E 04 S T 2 E 2 8 - 0 . 4 4 7 3 8 E 03 S Y T 2 E B ' O . 3 2 0 7 5 E 06 S Y T E B T * 0 . 4 9 7 4 3 E 04 OIFFUSIVITY* 0.6871158* LAMDA « 1.193)9411 EFFECTIVE OIFFUSIVITY* 0.27003652 PUBLISHED O I F F U S I V I T Y * 0.82000000 ALPHA* 0.I803H79I  NUMBER I T E R A T I O N S *  17  228' H¥DROG£N-NITRUGEN BED OATA BEO DIAMETER CMS REO LENGTH CMS tNO ZONE HEIGHT POROSITV 0.39300  RUN DATA 2.61000 7.00000 0.27000  FLOW RATE CC/SEC' 1.25171 ROOM TEMPERATURE" 295.0000 ATM.PRESSURE MM HO T*2.5 BCD TEMPERATURE K> 309.0000  Tine s e c 50.0 100.0 150.0 200.0 250.0 300.0 CONSTANTS FOR LEAST SAUARES * * B • C •  PEAK HEIGHTS 122.25 38.00 15.00 6.80 *.10 3.20  FIT OF DATA IN 399.9892.* . 0.02*29557 3.4*35383*  Y-C*  A»EXPI-BTI  SUPMAIIONS FROM LEAST SQUARES CALCULATION SEOT « 0.*2173O00 SYEBT • O.*00d*b 02 SY • 0.18935E 03 STEHT « 0 . ? 9 8 9 9 t 0?" STE2BT* 0.52955fc 01 St2BT * 0 . 9 6 5 8 2 F - 0 1 ST2EBT* 0.27265E 0* ST 2fc?B* 0.31592k 03 SVI2Ert* 0.13589E 06 SYTEBT* 0.22211E 0* DIFFUSIVITY* 0.66116180 LAMDA * 1.2*02*105 EFFECTIVE OIFFUSIVITY* 0.25983658 PUBLISHED OlFFiiSIVITV* 0.82000000 ALPH4* 0.1939*036  NUMBER ITERATIONS* 15  -229-  NIT ROGE N-E THANE HEP DATA BEO DIAMETER CMS ilBO L E N t T H CMS END ZONE HEIGHT POROSITY 0.39400  RUN OATA 2.61000 T.00000 0.27000  TIME S E C . 150.0 300.0 450.0 600.0 700.0 800.0 900.0 1050.0  FLOW RATE C C / S E C * 0.604)8 ROOM TEMPERATURE" 295.0D00 ATM.PRFSSURf MM HG 754.5 BED TEMPERATURE K» 309.0000 PEAK HEIGHTS 131.00 68.00 40.50 27.00 22.50 20.00 18.50 17.00  CONSTANTS TOR LEAST SAUARES F I T CF DATA IN Y - C - A * t X P I - B T I A « 251.88520 b « 0.00574220 C ' 16.16684971 SUMMATIONS FROM LEAST SOUARES CALCULATION SCBT • 0 . 8 5 4 I 6 E 00 SYEBT * 0 . 7 9 8 S 2 E 0? SY « 0 . 3 4 4 5 0 E 03 STEBT * 0 . 2 4 I I 6 E 03 STE2BT* 0.49.->98E 07 SE2BT • 0 . 2 6 7 3 1 C - 0 0 ST2EBT* 0 . 9 7 4 3 1 E 05 ST2E2B" 0 . 1 1 5 6 7 E US SVT2EB" 0 . 4 4 9 0 4 L 07 SYTEBT" 0 . 1 6 4 S 8 E 05 DIFFUSIVITY" 0.11735015 LAMOA > 1.28674738 EFFECTIVE O I F F U S I V I T Y " 0.04611861 PUBLI SHEU D I F F U S I V I T Y " 0.15100000 ALPHA* 0.21198461  .UMBER I T E R A I I U N S *  11  -230NITROOEN-ETHANE BEO OATA BEO OIAMETER CMS BEO L6.U6TH CMS END ZONE HEIGHT POROSITY 0.39300  RUN DATA 2.61000 T.00000 0.27000  TIME SEC. 200.0 300.0 400.0 5C0.0 600.0 700.0 BOO.U  900.0 looo.r  F L O W RATE CC/SEC* 0.91862 ROOM TEMPERATURE* 295.0000 ATM.PRESSURE MM HG '55.5 BEO TEMPERATURE «.* 309.0000  PEAK HEIGHTS 66.00 45.50 32.50 25.00 20.50 18.00 16.50 15.80 15.20  CONSTANTS FOR LEAST SAUARES FIT OF OATA IN Y-C* A*EXP(-BTI A • 147.09993 b • 0.00520905 C • 14.26128089 SUMMATIONS FROM LEAST SUUARES CALCULATION SEBT • 0.86097E 00 SYEBI • 0.40569E 02 SV • 0.25500E 03 STEBT • 0.29095b 03 STE2BT* 0.48934b 02 SE2BT • 0.192 42E-00 ST2t.lT* 0 . 1 2 2 H F 06 ST2E2B* U.14058E 05 SYT2EB* 0.38177E 07 SYTEBI* 0.11348E 05 01FFUSIVITV* 0.11182137 LAMOA • 1.35036799 EFFECTIVE OIFFUSIVHV* 0.04394580 PUBLISH60 DIFFUSIVITY* 0.15100000 ALPHA* 0.21647434  NUMBER ITERATIONS*  9  -231  NHROQEN-E THANE BED DATA OEC DIAMETER CMS BEO LENSTH CMS ENO ZONE HEIGHT POROSITY 0.39300  RUN D M A  2.610C0 7.00000 0.27000  TIME SEC. 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0  FLO* HATE CC/SEC* 1.36169 ROOM TEMPERATURE* 295.0000 ATM.PRFSSURE MM HG 755.5 BED TEMPERATURE K> 309.0000 PEAK  HEIGHTS  4 7 . OC  34.00 25.50 21.00 18.CO  16.50 15.50 IS.00  CONSTANTS FOR LEAST SAUARES FIT OF DATA IN Y-C A»6»P(-8T1 4 • 93.58899 ft • 0.00520108 C • 14.02765203 SUMMATIONS FROM LEAST SQUARES CALCULATION SEBT • 0.35778E 00 SVfcBT * 0.30102E 02 ~ SY • 0.19250E 03 STEdT * 0.286426 03 STE2BT* 0.49129F 02 SE2BT « 0.19307E-00 ST2EBT* 0.11731E 06 ST2C28' 0.14103E 05 SVT2E8' 0.2971rt 0 I SYTEBT* 0.86157E 04 OIFFUSIVITY' 0.10895333 LAMOA • 1.38591449 EFFECTIVE OIFFUSIVITY' 0.04281866 PUBLISHED OIFFUSIVITY' 0.15100000 ALPHA* 0.21913726  NUMBER ITERATIONS* 8  -232NMRUGEN-BUJANE BED OATA  RUN  BEO DIAMETER CMS BSD LkNGTH CMS ENO ZONE HEIGHT POROSITY 0.39300  2.6100U f.OOOCO 0.2/000  FLOW R A T E CC/SEC* ROOM T E M P E R A T U R E * A T M . P R E S S U R E M M H(, DEO  TIME SEC. 400.0 500.0 600.0 700.0 800..3 700.0 1000.0 1100.0  CONSTANTS  FOR LEAST A B C  SUMMATIONS  SAUARES * • •  FRUM LEAST Sfc-BT * SYE8T SY  *  S I E 6 T =• STE2BT* SE281 * ST2EBT* ST2E2B* SYT2EB* SYTEBT* OIFFUSIVITY* LAM'IA * EFFECTIVE PU8LISHEO ALPHA*  TEMPfcRATURC  K*  309.0000  OATA  0.59619 297.5000 7*6.B  PEAK HEIGHTS 63.00 45.00 32.00 23.50 17.00 13.00 10.00 8.00  F I T U F OATA 256.29736 0.00163822 3.26159501  IN Y - C *  AoFXPI-bTI  SQUARES CALCULATION 0 .72341E 0 0 0.29272E 0.21150E  02 0 3  0.420HJE 0 3 C . 5 l 5 o 4 E O? U.10SnOL-00 0.27074E 06 0.2'OIOE 05 C.78076E 07 0.14588E 05 0.0794745) 1.245682U6  DIFFUSIVITY* OIFFUSIVITY* (I.21584U0  0.03123349 0.09900000 NUMBER  ITERATIONS*  10  -233-  NlTRCGFN-BUTANfc UEO  OATA  bEO  DIAMETER  BEO  LENGTH  t-NC  ZONE  RUN 2.61000  FLOW  4ATE  CMS  7.00000  ROOM  Tf-MPPKATURE"  HEIGHT  0.27000  POROSITY  CMS  ATM.PRtSSURt BET)  0.39300  FOR LEAST A B C  SUMMATIONS  SAUARES « • •  FROM LEAST SEBT «  TEMPERATURE  K«  0.-77945 297.5000  MM H I .  746.8  309.0000  PEAK HEIGHTS  TIME S E C . 300.0 400.0 500.0 600.0 700.0 800.0 9C0.0 1000.0  CONSTANTS  C C / S E C *  OATA  55.00  38.00 27.50 20.00 14.20 10.70 8.00 6.20  F I T OF O A T A 159.15279 0.00370354 2.38304090  IN Y - C A.FX«<-BTI  SQUARE S C A L C U L A T I O N 0 . 1 0 0 8 f t 01  SYEbT » 0 . 3 5 2 3 2 E SY « 0.179606 STtBT > 0.4H367E S1E2BT" 0.B0356E SE2BT • 0.2065BE  02 0 3 0 3 02 -00  ST2EBT"  0.26753E  0 6  ST2C2B'  0.34501E  0 5  SYT2EB" 0.61296t 07 SYTCBT« 0 . 1 3 9 4 2 E 0 5 DIFFUSIVITY' 0.078447)4 LAMOA « 1.26199301 E F F E C T I Vt PUBLISHED  ALPHA"  O i r r u S I V K Y DIFFUSIVITY" 0.21919160  0.03082981 0.09900000 NUMBER  ITERATIONS"  8  •libNil TRUGEN-BUTANE  BEO DATA BED DIAMETER CMS BEO LENGTH CMS END ZONE HEIGHT POROSITY 0.39300  RUN DATA 2.O1000 T.00000 0.2T0OO  TIME SEC. 350.0 450.0 550.0 650.0 750.0 850.0 950.0 1050.0  F L O N RATE CC/SEC* 1.23600 RuOM TEMPERATURE' 297.5000 ATM.PRfcSSUKt MM HO 746.8 BED TEMPERATURE K» 309.0000 PEAK HEIGHTS 31.00 22.00 15.50 11.00 8.00 6.00 4.60 3.50  CONSTANTS FOR ItAST SAUARES FIT OF OATA IN Y - C A > 108.61678 B • 0.00369706 C • 1.28090966  A*EXP(-BT)  SUMMATIONS FROM LEAST SCUARES CALCULATION SbbT * 0.84106E 00 SYEBT » 0.16659E 02 SY * 0.10160E 03 STEBT • 0.44553E 03 STE2BT" 0.63004E 02 SE2BT * 0.14346E-00 ST2EBT* 0.26574E 06 ST2E2B- 0.29929E 05 SYT2EB» 0.35909F 07 SYTEBT* 0.74140E 04 DIFFUSIVITY* O.O7752087 LAMOA • 1.2 7707544 EFFECTIVE 01FFUSIVITY* 0.03046570 PUBLISHEO DIFFUSIVITY* 0.09900000 ALPHA*  U.22030460  TIME 20HRS 98MIN 39.5SEC  NUMBER ITERATIONS*  6  

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